SURFACE DYNAMICS OF PARTIALLY TETHERED POLYMER FILMS
A Dissertation
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Jin Kuk Lee
May, 2013
SURFACE DYNAMICS OF PARTIALLY TETHERED POLYMER FILMS
Jin Kuk Lee
Dissertation
Approved:
Accepted:
Advisor
Dr. Mark D. Foster
Department Chair
Dr. Coleen Pugh
Committee Chair
Dr. Ali Dhinojwala
Dean of the College
Dr. Stephen Z. D. Cheng
Committee Member
Dr. Gustavo Carri
Dean of the Graduate School
Dr. George R. Newkome
Committee Member
Dr. Matthew Becker
Date
Committee Member
Dr. Alamgir Karim
ii
ABSTRACT
The surface dynamics of thin polymer films were tailored via tethering a fraction of
the chains to the supporting substrate. This fractionally tethered state was called a
“partially tethered film”. A partially tethered film was prepared by coating an untethered
chain layer on top of a tethered chain layer, followed by annealing. The tethered chain
layers for this study had values of grafting density smaller than 0.15 chains/nm2. This low
grafting density provided better miscibility between untethered and tethered chains, as
compared to previous work with densely grafted brushes, therefore allowing the
untethered chains to penetrate to the substrate region.
Partially tethered films had slower surface fluctuation dynamics than did
corresponding untethered reference films. These tailored surface dynamics after partial
tethering were indicated by variations in surface relaxation time measured with X-ray
photon correlation spectroscopy. Using the hydrodynamic continuum theory, the surface
relaxation behavior of each sample was rationalized. However, since the hydrodynamic
continuum theory hypothesizes a homogeneous viscous layer, it is not perfectly
applicable to partially tethered films.
To address this problem, the surface dynamics of the partially tethered film were
described with models designed to account for various effects.
A confinement effect
due to the covalent bonds between the polymer layer and the substrate was described by
modeling the hydrodynamics of a layer adjacent to the substrate and containing a
substantial fraction of tethered chains as being extremely slow. The surface dynamics of a
iii
partially tethered film for which surface relaxation could still be observed was explained
using a two-layer model composed of a top layer of untethered chains moving essentially
like bulk untethered chains on top of the bottom layer with very slow dynamics designed
to capture the effect of tethering. For sufficiently thick films this model looks very
much like that already described by others for the flow of solvent past a solvent swollen
layer of adsorbed polymer chains.
However, as the partially tethered film becomes
thinner, a more complicated model is required.
At the least, the thickness of the layer
with very slow hydrodynamics has to be varied as a fitting parameter and can become as
large as the entire film thickness in some cases.
"neighboring layer" effect.
We attribute this behavior to a
That is, the dynamics of the nominally untethered top layer
are slowed by the influence of the practically immobile lower layer. The width of the
interface between the top region of nearly pure untethered chains and the bottom region
rich in tethered chains apparently plays an important role.
When the degree of
interpenetration between the two regions is altered by varying miscibility between the
tethered and untethered chains, through the control of molecular weights or the segmentsegment exchange interaction parameter, χ, the surface dynamics can be strongly
impacted. The relationship of the glass transition temperatures (Tg) of the tethered and
untethered chains to one another was yet another parameter found to influence the
tethering effect.
When an oligiomeric untethered chain was used, the plasticization of
the overall film by the low Tg untethered material dominated in determining the surface
dynamics in the range of temperatures studied.
More realistically accounting for all
these effects requires a gradient model envisioning a gradual change in dynamical
properties with depth into the film.
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ACKNOWLDGEMENTS
I would like to appreciate my advisor Dr. Foster for his kind and valuable help,
instructions and discussions. Also, my committee members Dr. Dhinojwala, Dr. Carri, Dr.
Becker and Dr. Karim gave me lots of help.
Also, the staffs and students in the polymer science department gave me kind help
and useful discussions. Mr. Page helped me so much for DSC and GPC measurement.
Without him, I could not characterize my polymers. I appreciate my group members for
AFM and XR experiment. Also, Dr. Wang gave me the instructions for optical
microscope use.
For scientists outside of the University of Akron, Dr. Akgun gave me great
discussions and helped a lot for NR measurements. Also, Dr. Jiang and Dr. Narayanan did
a lot of job for me to do XPCS measurement. Dr. Jennings in Case Western Reserve
University helped me for XPS experiment.
While applying for Ph.D. courses, many people helped me a lot for
encouragements and advices. I would like to thank to Dr. Jin Chul Jung who was my
professor for my M.S. course, Dr. Byeong-hyeok Sohn, Dr. Hwan-seung Cheon and Mr.
Chang-il Oh.
v
My wife Barbara Kim, my son Jason Lee, my parents and my parents in law
supported and encouraged me very much. I appreciate their efforts for me. Especially, my
wife had lots of wise discussions and advices for my life in the US. My friends, Sung-suk
Oh and Jea-ho Hwang gave me a lot of help during the 4.5 years of this hard course.
Finally, my friend Ju Hyung Lee (deceased) gave so much mental supports to me.
Also, he gave me the previous chance to get married with Barbara. Even though he is not
here any more, our friendship for 15 years will not be forgotten.
DURIP grants W911NF-09-1-0122 and W911NF-10-1-0167.
I acknowledge the support of the National Institute of Standards and Technology,
U.S. Department of Commerce, in providing the neutron research facilities used in this
work.
Use of the Advanced Photon Source, an Office of Science User Facility operated
for the U.S. Department of Energy (DOE) Office of Science by Argonne National
Laboratory, was supported by the U.S. DOE under Contract No. DE-AC02-06CH11357.
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TABLE OF CONTENTS
Page
LIST OF FIGURES.........................................................................................................xii
LIST OF TABLES..........................................................................................................xxvi
CHAPTER
I. INTRODUCTION………………...……………………………………………………1
1.1. Surface Dynamics…………………………………………………………....1
1.2. Surface Dynamics of Polymer Melts………………………………….......…5
1.3. Tethered Chains.............................................................................................21
1.3.1. Concepts of Tethered Chains…………………………….………21
1.3.2. Preparation of Tethered Chains………………………………….25
1.3.3. Surface Dynamics of Tethered Chains………………………..…28
1.4. Hypothesis of This Study……………………………………………..……30
II. BACKGROUND………………………………………………………………...…..33
2.1. Diffusion .................................................…………………………………..33
2.1.1. Rouse Model.....…………………………………………..……..33
2.1.2. Reptation Model………………………………..................……..34
2.1.3.Diffusion Measurement..................................................................35
2.2. Friction of Tethered Chains……………………………………...…............38
vii
2.3. Conformation of Tethered Chains.................................................................41
2.4. Miscibility between Tethered and Untethered Chains…………………..….47
2.4.1. Theory for Chemically Identical System……….....…………….47
2.4.2. Miscibility and De-wetting...........................................................51
2.4.3. Flory-Huggins Theory for Chemically Different System.............57
2.4.4. Theories for Partially Tethered Layers..........................................61
2.4.5. Experimental Results.....................................................................62
2.5. State of the Problem......................................................................................70
III. X-RAY AND NEUTRON SCATTERING…………………………………………..71
3.1. Introduction………………………………………………………………...71
3.2. Basic Principles………………………………………………………….…72
3.3. Specular Reflectivity…………………………………………………….…74
3.4. Off-Specular Scattering…………………………………………………….83
3.5. XPCS (X-Ray Photon Correlation Spectroscopy)………………………….87
3.5.1. Coherence…………………………………………………..........87
3.5.2. Speckle Pattern..............................................................................88
3.5.3. Principles of XPCS……………………………………………...90
IV. EXPERIMENTAL…………………………………………………………………...94
4.1. Tethered Chain Preparation Method……………………………………..…94
4.1.1. Materials………………………………………………………....94
4.1.2. Substrate Preparation…………………………………………….95
4.1.3. Deposition of Epoxide Funtionalized Compound on the
Substrates............................................................................................…95
4.1.4. Grafting hPS-COOH……………………………....…………….95
viii
4.2. Characterization of Tethered Chains………………………....………….…96
4.2.1. The Reaction of Silane Coupling Agent…………………..……..96
4.2.2. Tethering Process……………………………………………..…98
4.2.3. Characterization………………………………………………..100
4.3. Polymer Layer Preparation………………………………………………..112
4.3.1. Substrate Preparation…………………………………………...112
4.3.2. Spin Coating……………………………………………………112
4.3.3. Annealing and Surface…………………………………………113
4.3.4. Sample Naming Rules………………………………………….116
4.4. Radius of Gyration………………………………………………………...116
4.5. NR (Neutron Reflectivity)………………………………………………...118
4.6. XPCS (X-Ray Photon Correlation Spectroscopy)………………………...123
4.7. Other Measurements………………………………………………………130
V. SURFACE DYNAMICS AFFECTED BY CONFINEMENT AND COVALENT
BONDS...........................................................................................................................132
5.1. Confinement Effect of Thin Film of Untethered Chain…..………...……..132
5.1.1. Viscosity Calculation…………………………………………...132
5.1.2. Comparison with Bulk Viscosity………………………………139
5.2. Confinement Effect from Covalent Bond…………………………………143
5.2.1. Comparison with Molecular Weights Blends………………..…143
5.2.2. Surface Dynamics for Higher Covalent Bond Density……...…148
5.2.3. Other Results Supporting the XPCS Data……………………..157
5.3. Summary…………………………………………………………………..162
ix
VI. SURFACE DYNAMICS AFFECTED BY THICKNESS AND TETHERED CHAIN
EXTENSION…………………………………………………………………..………164
6.1. Objective………………………………………………………………….164
6.2. Sample Description for Thickness Varied Samples.....................................164
6.3. Models and Parameters to Explain the Thickness Dependence..................167
6.4. Tethered Chain Conformation and Surface Dynamics................................190
6.4.1. Conformation Observed with NR...............................................192
6.4.2. XPCS Result...............................................................................194
6.5. Summary.....................................................................................................203
VII. SURFACE DYNAMICS AFFECTED BY MISCIBILITY BETWEEN TETHERED
AND UNTETHERED CHAINS.....................................................................................205
7.1. Objective......................................................................................................205
7.2. Miscibility Determined by Molecular Weight.............................................205
7.2.1. Sample Description.....................................................................206
7.2.2. NR Results..................................................................................208
7.2.3. XPCS Results..............................................................................214
7.3. Plasticization Effect.....................................................................................223
7.3.1. Sample Description.....................................................................223
7.3.2. XPCS Result...............................................................................223
7.3.3. Discussion...................................................................................229
7.3.4. Sample Damage..........................................................................230
7.4. Effect of Miscibility Changes Due to Interaction Parameter......................233
7.4.1. Sample Description....................................................................233
7.4.2. Results and Discussions..............................................................236
x
7.4.3. Thermal Damage.........................................................................250
7.5. Summary......................................................................................................259
VIII. CONCLUSION......................................................................................................261
REFERENCES AND ENDNOTES................................................................................264
APPENDICES.................................................................................................................277
APPENDIX A. THICKNESS VARIATION WITH TEMPERATURE...........................278
APPENDIX B. REFLECTIVITY CURVE FITTING QUALITY...................................280
APPENDIX C. XPCS MEASUREMENT CONDITION...............................................281
APPENDIX D. FITTING RESULTS FOR 2748 37........................................................285
APPENDIX E. g2 FUNCTIONS.....................................................................................287
APPENDIX F. VISCOSITY OF POLY(4-BROMO STYRENE) (PSBR)......................315
xi
LIST OF FIGURES
Figure
Page
1.1: The relationships among the wave vectors for incident and scattering radiation,
scattering angle, 2θ, and scattering vector. ………………………………………………2
1.2: The surface mode diagram for polymer gel regime is plotted with logarithm scale of
wave number (k) and shear modulus(E0). Modes are distinguished with characteristic
wave number, k   02 and modulus E    0  where ρ is the density of the gel.
Reprinted with permission from J. L. Harden, H. Pleiner and P. A. Pincus,
“Hydrodynamic Surface Modes on Concentrated Polymer Solutions and Gels”, J. Chem
Phys. 94, 5208 (1991). Copyright 1991, American Institute of Physics.
……………………………………………....................................………………………3
2
1.3: The surface mode diagram for polymer solution regime is plotted with logarithm
scale of wave number (k) and shear modulus(E0). Modes are distinguished with

characteristic wave number, k   02 , k   2 



1 3
and modulus, E    0  ,
2
13
E   2  2
where ρ is the density of the solutoin. Reprinted with permission from J.
L. Harden, H. Pleiner and P. A. Pincus, “Hydrodynamic Surface Modes on Concentrated
Polymer Solutions and Gels”, J. Chem Phys. 94, 5208 (1991). Copyright 1991, American
Institute of Physics. …………...................................................................................….…4
1.4: A schematic illustrating the significance of the surface relaxation time, τ. A film of
low molecular weight liquid or polymer at a temperature above Tg has surface
fluctuations. After a time, τ, has passed, the surface structure has become completely
different from that of the initial state. No structural correlation exists between the two
surface states. ………....................................................................................…………….5
1.5: The τ vs. q|| for PS melt films measured using XPCS. a) τ vs. q|| at different
temperatures with fixed molecular weight and film h, 123 kg/mol. b) τ vs. q|| at different
hs at the same temperature and molecular weight. c) τ vs. q|| for different molecular
weights at the same temperature and h. Figure reprinted with permission from H. Kim, A.
Rühm, L. B. Lurio, J. K. Basu, J. Lal, D. Lumma, S. G. J. Mochrie and S. K. Sinha,
“Surface Dynamics of Polymer Films”, Phys. Rev. Lett. 90, 068302 (2003). Copyright
2003 by the American Physical Society(a and b). Reprinted from H. Kim, A. Rühm, L. B.
xii
Lurio, J. K. Basu, J. Lal, S. G. J. Mochrie and S. K. Sinha, “X-ray Photon Correlation
Spectroscopy on Polymer Films with Molecular Weight Dependence”, Physica B 336,
211 (2003). Copyright 2003, with permission from Elsevier(c). ......................................8
1.6: g2 of poly(octadecyl acrylate) in a thin film at q|| = 1.65×10-4 Å -1. The h for (a)-(c)
was 220 Å and for (d)-(f) was 660 Å , respectively. The temperature was at T < Tm [(a),
(d)], Tm < T < Ts [(b), (e)] and T > Ts [(c), (f)], where Tm is the melting temperature and Ts
is the order-disorder transition temperature of the side chains. The 660 Å film had
measurable relaxation at T < Tm, but for the 220 Å film the relaxation behavior was seen
only at T > Ts. Figure reprinted with permission from S. Prasad, Z. Jiang, M. Sprung, S. K.
Sinha and A. Dhinojwala, “Effect of Surface Freezing on Meniscus Relaxation in Side
Chain Comb Polymers”, Phys. Rev. Lett. 104, 137801 (2010). Copyright 2010 by the
American Physical Society. ……………………..…..................................................…..10
1.7: The entanglement effect on surface fluctuation dynamics. a) τ was measured with
XPCS at 111 °C. The value of τ shows universal behavior when the molecular weight is
more than 30 kg/mol. The solid line connecting data points is the result from fitting using
the hydrodynamic continuum theory Equation 1.2. b) The plot of η with respect to
molecular weight for the polymer thin film (○) and bulk (◇). The dash-dotted line
denotes the average η of the polymer film for Mw > 30 kg/mol. The solid red line is the
result from fitting a power law to the bulk behavior in the entanglement regime. Figures
reprinted with permission from Z. Jiang, M. K. Mukhopadhyay, S. Song, S. Narayanan, L.
B. Lurio, H. Kim and S. K. Sinha, “Entanglement Effects in Capillary Waves on Liquid
Polymer Films”, Phys. Rev. Lett. 101, 246104 (2008). Copyright 2008 by the American
Physical Society. ………………......................................................................................12
1.8: The off specular scattering curve of PS thin film at various h. The thin solid lines on
the data point are fitting to find ql,c. Figure reprinted with permission from J. Wang, M.
Tolan, O.H. Seeck, S. K. Sinha, O. Bahr, M. H. Rafailovich and J. Sokolov, “Surfaces of
Strongly Confined Polymer Thin Films Studied by X-Ray Scattering”, Phys. Rev. Lett.
83, 564 (1999). Copyright 1999 by the American Physical Society. ……......................16
1.9: The τ of a PS thin film with Mn = 123 kg/mol and Mn = 400 kg/mol at h = 2Rg
measured with XPCS. The dashed lines present the fitting using the viscous liquid model,
Equation 1.2, and the solid lines present fits from the viscoelastic models using Equation
1.8. Figure reprinted with permission from J. Wang, M. Tolan, O.H. Seeck, S. K. Sinha,
O. Bahr, M. H. Rafailovich and J. Sokolov, Z. Jiang, H. Kim, X. Jiao, H. Lee, Y. J. Lee, Y.
Byun, S. Song, D. Eom, C. Li, M. H. Rafailovich, L. B. Lurio and S. K. Sinha, “Evidence
for Viscoelastic Effects in Surface Capillary Waves of Molten Polymer Films”, Phys. Rev.
Lett. 98, 227801 (2007). Copyright 2007 by the American Physical Society……….......18
1.10: The Tg of a PS layer as measured by a fluorescent dye method varies with the type
of supporting polymer layer. Reprinted with permission from C. B. Roth, K. L. McNerny,
W. F. Jager and J. M. Torkelson, “Eliminating the Enhanced Mobility at the Free Surface
of Polystyrene: Fluorescence Studies of the Glass Transition Temperature in Thin Bilayer
Films of Immiscible Polymers”, Macromolecules 40, 2568 (2007). Copyright 2007
xiii
American Chemical Society. ………...……………...........................………….……….20
1.11: Tethered chain systems can have various structures depending on the type of
substrate and the type of chain being tethered. Reprinted from M. Zhang and A. H. E.
Müller, “Cylindrical Polymer Brushes”, J. Polym. Sci. Part A : Polym. Chem. 43, 3461
(2005). Copyright 2005, with permission from John Wiley and Sons. …………………23
1.12: The conformation of tethered chains varies with σ……..........…....……………...24
1.13: Schematic of tethered chain surface dynamics used by Frederickson et al. Reprinted
with permission from G. H. Fredrickson, A. Ajdari, L. Leibler and J. P. Carton, “Surface
Modes and Deformation Energy of a Molten Polymer Brush”, Macromolecules 25, 2882
(1992). Copyright 1992 American Chemical Society. …………................……......……28
1.14: g2 vs. delay time, t, at q|| = 5.3×10-3 nm-1 for PS tethered chains (PS Brush) and
untethered thin film. The untethered chain layer (PS, Mn = 65 kg/mol) showed relaxation
at 170 °C, but the tethered PS layer did not show a relaxation time until 225 °C.
Reprinted with permission from B. Akgun, G. Uğur, Z. Jiang, S. Narayanan, S. Song, H.
Lee, W. J. Brittain, H. Kim, S. K. Sinha and M. D. Foster, “Surface Dynamics of “Dry”
Homopolymer Brushes”, Macromolecules 42, 737 (2009). Copyright 2009 American
Chemical Society. ………….....................................................................................……29
1.15: Schematic illustrating how the surface relaxation time can be tailored by changing
the degree of tethering in the layer. …………………………….....……….................…30
1.16: Factors determining the surface dynamics of partially tethered layers. ..................32
2.1: Plot of the self diffusion coefficient of a labeled linear PS in a cross-linked matrix vs.
Pn at various T. Open circles are for the PS network with pc = 0.02 and ΦPS = 0.2 where pc
is the number of crosslinks per monomer and ΦPS is the volume fraction of PS. Full
circles are for the PS network with pc = 0.03 and ΦPS ≤ 0.01 at 194 °C. Diffusion in uncrosslinked PS matrix with Mw = 110 kg/mol at 194 °C is displayed as triangles. Higher T
and lower crosslink density allows faster diffusion. Reprinted with permission from M.
Antonietti and H. Sillescu, “Self-Diffusion of Polystyrene Chains in Networks”,
Macromolecules 18, 1162 (1985). Copyright 1985 American Chemical Society. ……...37
2.2: The schematic of shear flow of polymer melt on the surface. The slippage length, b is
a means of quantifying the degree of slip at the wall. The relation between b and the
dV
degree of slip is given by b  V ( z )
. The shear stress is the same at all z > 0.
dz
…………………......................................................................................................…….39
2.3: The entanglement behavior of tethered chains with respect to V. (a) The chains are
entangled with an untethered melt at low V, but (b) disentanglement and stretching will
happen when the shear stress exceeds a threshold value. ........................................……40
xiv
2.4: The schematic of lateral force microscopy (LFM) set up to measure tethered chain
friction. Reprinted with permission from L.J.T. Landherr, C. Cohen, P. Agarwal and L. A.
Archer, “Interfacial Friction and Adhesion of Polymer Brushes”, Langmuir 27, 9387
(2011). Copyright 2011 American Chemical Society. …………......................…………40
2.5: The conformation of tethered chains with high σ in good solvent. The strongly
stretched chains can be described using the blob model as shown in (a). The theoretic
profile of the tethered chain concentration is displayed in (b). Reprinted with permission
from P. G. de Gennes, “Conformations of Polymers Attached to an Interface”,
Macromolecules 13, 1069 (1980). Copyright 1980 American Chemical Society. ……...42
2.6: The system volume considered for this study (a) and the dependence of the
concentration of tethered chain segments in a brush, ϕN, on σ (b). The conformation of the
tethered chains can be separated into three regimes i), ii) and iii). Reprinted with
permission from P. G. de Gennes, “Conformations of Polymers Attached to an Interface”,
Macromolecules 13, 1069 (1980). Copyright 1980 American Chemical Society. …..…45
2.7: ϕN profiles of tethered chains (solid line on left of each plot) adjacent to an
untethered chain layer together with the depth profiles of volume fraction of untethered
chain (right solid line on right of each plot). Here, σ varies from 0.004 to 0.1. P has two
values: 30 and 300 and N remains constant at 600 in all cases. Reprinted with permission
from C. M. Wijmans, E. B. Zhulina and G. J. Fleer, “Effect of Free Polymer on the
Structure of a Polymer Brush and Interaction between Two Polymer Brushes”,
Macromolecules 27, 3238 (1994). Copyright 1994 American Chemical Society. ……..50
2.8: The definition of penetration depth (λ) for a simplified ϕN profile. In contrast to the
picture from Alexander’s theory, the composition of the tethered chains gradually
decreases at the interface with the untethered chain layer. Reprinted with permission from
C. Gay, “Wetting of a Polymer Brush by a Chemically Identical Polymer Melt”,
Macromolecules 30, 5939 (1997). Copyright 1997 American Chemical Society. ……..51
2.9: Simulation of Helmholtz free energy, H, vs. h of the tethered chain layers. Rg is the
radius of gyration of the tethered chains. The way in which H changes with h is
determined by the chain length ratio between the untethered and tethered chains, P/N.
When P/N > 1 (red line) the curve shows an absolute minimum, while for P/N = 1 (blue
line) the curve has a monotonic decrease without any absolute minimum. This figure was
adapted from [84]. …………….................................................................................…...52
2.10: Wetting phase diagram of a PS untehered layer on a PS tethered chain layer with
various N and σ. The annealing condition was 12 days at 145 °C under reduced pressure.
The initial h of the total PS layer was 5 ± 0.1 nm. Reprinted with permission from J. H.
Maas, G. J. Fleer, F. A. M. Leermakers, and M. A. Cohen Stuart, “Wetting of a Polymer
Brush by a Chemically Identical Polymer Melt: Phase Diagram and Film Stability”,
Langmuir 18, 8871 (2002). Copyright 2002 American Chemical Society. ................….56
xv
2.11: Schematic of thermoresponsive sensor prepared with polymer tethered chains.
When layers of tethered chain are mixed with a solvent, the value of χ depends on T. For
LCST behavior, phase separation will happen above the transition temperature, Ttr,
resulting in a sudden change in tethered chain conformation. Reprinted with permission
from X. Laloyaux, B. Mathy, B. Nysten and A. M. Jonas, “Surface and Bulk Collapse
Transition s of Thermoresponsive Polymer Brushes”, Langmuir 26, 838 (2010).
Copyright 2010 American Chemical Society. …..............................................................63
2.12: The morphology and conformation changes of polystyrene-b-poly(methyl
methacrylate) (PS-b-PMMA) tethered chains under different solvents. (a) CH2Cl2 is a
good solvent for both blocks and the AFM image shows a smooth surface after treating
with CH2Cl2. (b) When treated with cyclohexane, the PMMA block will aggregate to
reduce contact with the poor solvent. And then the surface will become rougher. (c) A
cartoon of a possible conformation change process due to different solvent qualities.
Reprinted with permission from B. Zhao, W. J. Brittain, W. Zhou and S. Z. D. Cheng,
“AFM Study of Tethered Polystyrene-b-poly(methyl methacrylate) and Polystyrene-bpoly(methyl acrylate) Brushes on Flat Silicate Substrates”, Macromolecules 33, 8821
(2000). Copyright 2000 American Chemical Society. ……..............................………...65
2.13: The g2 function with respect to measurement time, Δt, measured with XPCS at
176 °C. The polymer films were (a) a bilayer of PS and PSBr and (b) a single layer of PS.
For the bilayer, the interface relaxation behaviors of the two interfaces were separately
measured. Figure reprinted with permission from X. Hu, Z. Jiang, S. Narayanan, X. Jiao,
A. R. Sandy, S. K. Sinha, L. B. Lurio and J. Lal, “Observation of a Low-Viscosity
Interface between Immiscible Polymer Layers”, Phys. Rev. E 74, 010602(R) (2006).
Copyright 2006 by the American Physical Society. ………….............................…..…68
2.14: The segment volume fraction profiles of tethered PS (Mw = 79.8 kg/mol) from the
substrate for various chemical structures of the untethered chain layers were measured
with NR. Further stretching from the substrate means more swelling of the tethered
chains due to better miscibility with the untethered chains. For favorable mixing, PVME
was used. For unfavorable mixing, PBD and PMMA were used. Reprinted with
permission from C. J. Clarke, R. A. L. Jones, J. L. Edwards, K. R. Shull and J. Penfold
“The Structure of Grafted Polystyrene Layers in a Range of Matrix Polymers”,
Macromolecules 28, 2042 (1995). Copyright 1995 American Chemical Society. .…….69
3.1: The geometry of reflection and refraction of incident radiation. In many reflectivity
measurements, medium 0 is air or vacuum. …………………..................…………..…75
3.2: The geometry of reflection and refraction in a thin film supported by a substrate. The
refracted radiation from the 0-1 interface experiences another refraction and reflection at
the 1-2 interface. The reflected radiation from the 1-2 interface goes toward the 0-1
interface and experiences another reflection and refraction. As a result, at the top of the
sample the reflected beam consists of radiation reflected not only from medium 1, but
also from medium 2. ……...........................................................................................…77
xvi
3.3: Schematic of a rough surface. The local surface or interface shows height variations
with respect to the nominal interface height. ……………………......................………78
3.4: An example of analysis of the reflectivity curve shown at left. The curve is fitted
with the simulated reflectivity from a mathematical model of the sample structure until
the best fit is obtained. The best fit corresponds to the SLD profile on the right. When
there are sufficient constraints on possible physical models the resulting SLD profile may
be assumed to be close to the actual profile. ……………..........................……………..80
3.5: The SLD profile can be obtained by approximation as a series of discrete sections.
…………………………………………………………………………..............…....…81
3.6: Schematic of one type of off-specular scattering experiment………..........…….....84
3.7: The cut off scattering vector, ql,c, calculated from Figure 1.7 in Chapter 1, was fitted
with respect to h. The solid line in is fitting using power law, ql,c = b/dm where m is 1 for
PS. The broken line is the calculation from capillary wave theory. Reprinted figures with
permission from J. Wang, M. Tolan, O.H. Seeck, S. K. Sinha, O. Bahr, M. H. Rafailovich
and J. Sokolov, “Surfaces of Strongly Confined Polymer Thin Films Studied by X-Ray
Scattering”, PRL 83, 564 (1999). Copyright 1999 by the American Physical Society.
…….........................................................................................................................……86
3.8: The schematics of coherent (a) and incoherent (b) waves. The criterion of coherence
is that waves should have the constant relative phase. …………..............………........88
3.9: Frequency, energy, q and length scale of XPCS and complementary techniques : PCS,
Raman and Brillouin scattering, inelastic neutron(INS) and inelastic X-ray(IXS)
scattering, neutron spin-echo and nuclear forward scattering(NFS). Reprinted from G.
Grübel and F. Zontone, “Correlation Spectroscopy with Coherent X-rays”, Journal of Alloys and
Compounds 362, 3 (2004). Copyright 2004, with permission from Elsevier. ….........................91
4.1: Schematic of deposition to prepare an ultrathin layer of a silane compound on a polar
inorganic surface. ………………………………………………………..........……......97
4.2: Reaction schemes to prepare covalently tethered PS chains on a silica surface. First
of all, epoxide silane coupling agents form a SAM on the substrate surface. And then
polymers with –COOH end group are coated and react on the substrate. Finally, after
removing un-reacted chains, only tethered chains remain. Reprinted with permission from
S. Minko, S. Patil, V. Datsyuk, F. Simon, K. J. Eichhorn, M. Motornov, D. Usov, I.
Tokarev and M. Stamm, “Synthesis of Adaptive Polymer Brushes via “Grafting To”
Approach from Melt”, Langmuir 18, 289(2002). Copyright 2002 American Chemical
Society. ………………….................................................................……………………99
4.3: Water contact image on a piranha solution treated silicon oxide (a) and on a (3glycidoxypropyl)trimethoxy silane deposited surface (b). The contact angle was increased
due to the ultrathin epoxide functionalized layer. ………………..............….....……..101
xvii
4.4: AFM image of a (3-glycidoxypropyl)trimethoxy silane ultra thin layer on a silica
surface. a is the top view of the surface and b is the cross section view. ….........……102
4.5: XPS survey scan data for the epoxide functionalized SAM and the 28 kg/mol hPS
tethered chain layer. .................................…………………………….......……………104
4.6: XPS high resolution C1s scan for a PS tethered layer (a) and an epoxide
functionalized layer (b). ………………………………………………………………..106
4.7: (a) the XR curve of a 28 kg/mol hPS tethered chain layer and the fit to a model curve
and (b) the SLD depth profile calculated with the model fit. ……........................…….108
4.8: AFM result of 28k tethered PS surface. (a) the surface image from the top and (b)
cross section images. ………………………………..…………………….....………...111
4.9: The surface image of an annealed, partially tethered layer having a Mn = 200 kg/mol
hPS tethered chain layer mixed with Mn = 2.6 kg/mol dPS untethered chains. (a) the
image from the top and (b) cross section images. ………………..………......……..…114
4.10: AFM image of an annealed, partially tethered layer having a Mn = 200 kg/mol hPS
tethered chain layer mixed with Mn = 120 kg/mol dPS untethered chains. (a) the image
from the top and (b) cross section images. ………………………..…….................…..115
4.11: The intrapolation of calculated Rg using Eq (4.1) with experimentally measured
value from [129]. ………………………….......……………………................……….117
4.12: Reflectivity from various layers in a sample film gives information about h,
composition and roughness. …………………………………………………….…..…120
4.13: Providing higher contrast makes it easier to distinguish different components in a
layer. ……………………………………………...………………………........………121
4.14: Schematic of the horizontal reflectometer installed on NG7 beam line in NCNR.
…………....………………………………………...……………………............…….121
4.15: (a) The NR curve (○), model fit (
) and the SLD profile calculated from the
model fit of 2802 12. A model assuming partially tethered layer was used for the fit. (b)
The NR curve (○), model fit (
) and the SLD profile calculated from the model fit of
2802 12. A model assuming untethered blend was used for the fit. .........…………….122
4.16: Geometry of scattering and detection in XPCS. …………………….........….....124
4.17: XPCS set up on 8-IDI beam line at APS. …………………………..........….......124
4.18: An example of XPCS data. ◇ is g2 measured with XPCS. The solid line is the
fitting of g2 using Equation 4.4 to calculate τ. ………………………………......……126
xviii
4.19: Speckle patterns on CCD camera with full frame mode (a) and kinetics mode (b).
………………………………………………………………………................………127
4.20: The speckle pattern from XPCS is partitioned. The value of q|| for each partition is
related to the scale of surface fluctuations.……………………...................………..….128
4.21: The degree of speckle pattern partitioning results in various data quality. A sparse
partitioning (a) provides more readily interpreted data, but poor q|| resolution. A dense
partitioning (b) has values of τ for more q|| points, but the τ vs. q|| curve becomes noisy.
……………….................................................................................................................128
5.1: The τ vs. q|| curves for 2k Ref 43 at 100 °C (□), 110 °C (○) and 120 °C (◇).
.........................................................................................................................................135
5.2: The τ vs. q|| curves of 48k Ref 35 at 150 °C (□) and 160 °C (○)...........................136
5.3: τ/h vs. q||h curves of 2k Ref 43 at 100 °C (□), 110 °C (○) and 120 °C (◇). The solid
lines are fits with the HCT to calculate η. .……………….……...………….................……….…137
5.4: τ/h vs. q||h curves of 48k Ref 35 at 150 °C (□) and 160 °C (○). The solid lines are
fits with the HCT to calculate η. ……………………………………................……….138
5.5: (a) Comparison of PS viscosities of 2k Ref 43 (▲) and of bulk samples (◆) at
110 °C. The solid line is the fitting using the equation η = kMa. The bulk η values are
form [133]. (b) 2k Ref 43 still has lower η than the interpolation of 1.1k and 3.4k bulk
PS.…........……………………………..……..................................................................141
5.6: The η values of 48k Ref 35 (▲) compared with those of 15.7 kg/mol (◆) and 47.0
kg/mol (■) bulk PS. The bulk η values are form [133]. .........................................…….142
5.7: τ vs. q|| curves of 200k 10Prc 38 (□) and 2h02 45 (○) at 110 °C. .....................…146
5.8: τ/h vs. q||h curves of 200k 10Prc 38 (□) and 2h02 45 (○) at 110 °C. The solid lines
are the fits with the HCT to calculate η values. ………………….......................….......147
5.9: g2 functions of 2802 45 for several values of q|| at 120 °C. The red line is the attempt
to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too large to
be calculated. ………………………………………...…….........................................150
5.10: g2 functions of 2802 45 for several values of q|| at 130 °C. The red line is the
attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too
large to be calculated. ……..……............................................................................…151
xix
5.11: g2 functions of 2802 45 for several values of q|| at 180 °C. The red line is the
attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too
large to be calculated. ................................................................……….......………..152
5.12: The τ/h vs. q||h curves of 48k Ref 35 (□) and 2k Ref 43 (○) at T - Tg ≈ 50 °C. The
dotted curve is to illustrate the surface relaxation time of 2802 45 which is beyond the
measurement limit of the XPCS up to 180 °C (i.e. > 10,000 s in the q|| range accessed).
………….……………...….………….......................................................................….154
5.13: Tg from the DSC data of hPS with Mn = 28 kg/mol. ..……...................................155
5.14: The thermal resistance of a bulk state of Mn = 2.6 kg/mol dPS was measured with
TGA under a nitrogen flow. ................................................................................……...155
5.15: Optical microscope images of samples before and after annealing in a high vacuum.
(a) 2k Ref 43 after annealing at 180 °C for 17hrs, (b) 2802 45 before any annealing, (c)
2802 45 after annealing at 180 °C for 19hrs and (d) 2082 40 after annealing at 180 °C for
43hrs. ………………………….................................................................................…159
5.16: The surface image of 2802 45 via AFM after 43hrs annealing at 180 °C in a high
vacuum. (a) the image from the top view and (b) a cross section image. ...................157
5.17: Off specular scattering from a sample of pure untethered 2.6k PS (2k Ref 43) (○)
and from a partially tethered layer (2802 45) (□) corresponding to that for which XPCS
relaxation times are shown in Figure 5.19. The value of qz is fixed at 0.35 Å -1 and the
temperature is 110 °C. …………………………………….....……………................161
6.1: τ vs. q|| curves of 2802 70 (○) and 2802 90 (◇) at 120 °C. The red and black solid
lines are the simulated τ values of 2.6k dPS layers assuming h = 42 nm and 66 nm at
120 °C, respectively. The η and γ values for the simulation were 250 Pa·s and 32 mN/m
which were used to fit 2k Ref 43 at 120 °C. ……..........................................................168
6.2: τ vs. q|| curves of 2802 70 (□) and 2802 90 (△) at 110 °C. The red and black solid
lines are the simulated τ values of 2.6k dPS layers assuming h = 41 nm and 65 nm at
110 °C, respectively. The η and γ values for the simulation were 1100 Pa·s and 33 mN/m
which were used to fit 2k Ref 43 at 110 °C. …........................................................…..169
6.3: Schematic of the structure of the partially tethered layer suggesting ways of thinking
about the variation of viscosity with depth and its influence on the surface fluctuation
dynamics. …………...................................................................................................…170
6.4: τ/h vs. q||h curves of 2802 70 at 110 °C (□) and 120 °C (○), of 2802 80 at 110 °C
(△) and 120 °C (◇) and of 2k Ref 43 at 110 °C (□) and 120 °C (○). The solid lines are
fits with the HCT fits to calculate η. …………………..............................................…170
xx
6.5: Using Model I in Figure 6.3, 2802 70 was divided by two layers : tethered and
untethered layer. The h of untethered layer was 41 nm and 42 nm at 110 °C and 120 °C,
respectively. ………………………......................……..................................................173
6.6: (a) NR curve (○) and its model fit (
) of 2802 85 and (b) the SLD profile
calculated from the model fit. …………….................................…….......................…175
6.7: Cartoon of the tethered chain conformation in the partially tethered layer sample
2802 85.………........................…………….....................…….............……….............176
6.8: τ vs. q|| curve of simulated τ values for 2.6k dPS layers assuming h = 17 nm at
110 °C (red line) and 120 °C (blue line). The τ values are within the measurement
window of the XPCS (τ < 4000 s) for most or all of the q range studied in this work. The
η and γ values for the simulation were η = 1100 Pa·s and 250 Pa·s and γ = 33 mN/m and
32 mN/m which were used to fit 2k Ref 43 at 110 °C and 120 °C, respectively. …..…178
6.9: Plot of slowest simulated dynamics readily observable with the XPCS setup used,
(i.e. τ > 3000 s over appreciable range of q||) corresponds to 5 nm of untethered 2.6k
chains at 110 °C and 3 nm at 120 °C.………..................................................................179
6.10: The calculated htethered and huntethered for 2802 45, 2802 70 and 2802 90 at 110 °C and
120 °C. ............................................................................................................................181
6.11: (a) Schematic illustrating the definition of RH and D for flow of solvent between
opposing brush covered surfaces as in Dhinojwala et al. [156, 157]. The red broken line
corresponds to the plane at which solvent velocity is zero. Solvent does not flow between
this plane and the hard surface. (b) RH decreases as D decreases. ................................181
6.12: Since the tethered chain segment density has a gradient through the film, the surface
dynamics of 2802 45 could also be viewed as resulting from a gradual change (gradient)
in dynamics (viscosity) through the film depth. Also there is a gradient in elastic character,
potentially. The surface region with no tethered chains is just viscous. Deep in the
tethered chain region there may be a viscoelastic character. ….....................................184
6.13: τ vs. q|| curves of 2h48 42 (○), 48k Ref 35 (□), 2h48 99 (△) and 48k Ref 95 (◇)
at 160 °C. ……………….....................................…...............................................…185
6.14: τ/h vs. q||h curves of 2h48 42 (○), 48k Ref 35 (□), 2h48 99 (△) and 48k Ref 95
(◇) at 160 °C. The solid lines are fits with the HCT to calculate η. …...........…….….186
6.15: (a) NR curve (○) and its model fit (
) of 4848 20 and (b) SLD profile of 4848 20
calculated from the fitting.………………………..........................................................189
6.16: (a) NR curve (○) and its model fit (
) of 2h02 90 and (b) the SLD profile
calculated from the model fit . …………........................................................…...…..193
xxi
6.17: The expected conformations of tethered chains (black lines) mixed with untethered
chains (blue lines) were referred from the SLD profiles of (a) 2h02 90 and (b) 2802 85..
………………….............................................................................…….................……194
6.18: τ vs. q|| curves of 2809 92 at 110 °C (◇), 120 °C (□), 130 °C (○) and 9k Ref 80 at
110 °C (◇), 120 °C (□), 130 °C (○).
….........….....................................................196
6.19: τ/h vs. q||h curves of 2809 92 at 110 °C (◇), 120 °C (□), 130 °C (○) and 9k Ref 80
at 110 °C (◇), 120 °C (□), 130 °C (○). The solid lines are fits with the HCT to calculate
the η values of the polymer films. …...........................................................................197
6.20: τ vs. q|| curves of 2h09 100 (○) and 9k Ref 80 (○) measured with XPCS at 130 °C.
……………..............................................................................................................……198
6.21: τ/h vs. q||h curves of 2h09 100 (○) and 9k Ref 80 (○) measured with XPCS at
130 °C. The solid lines are fits with the HCT model to calculate the η values of the
polymer films. …..........................................................................................................199
6.22: τ vs. q|| curves of 2h09 100 (○) and 2809 92 (○) at 130 °C. The blue and yellow
solid lines are the simulated τ values of 9.4k dPS layers assuming h = 55 nm and 76 nm at
130 °C, respectively. The η and γ values for the simulation are 9000 Pa·s and 34 mN/m,
which are the values that were used to fit 9k Ref 80 at 130 °C.…………......................201
7.1: (a) NR curve (○) and its model fit (
) of 2h02 13 and (b) the SLD profile
calculated from the model fit. .........................................................................................209
7.2: (a) NR curve (○) and its model fit (
) of 2h1h 15 and (b) the SLD profile
calculated from the model fit. .........................................................................................210
7.3: (a) NR curve (○) and its model fit (
) of 2h02 45 and (b) the SLD profile
calculated from the model fit. .........................................................................................212
7.4: (a) NR curve (○) and its model fit (
) of 2h09 100 and (b) the SLD profile
calculated from the model fit. .........................................................................................213
7.5: τ vs. q|| curves of 9k Ref 80 (□), 2h09 100 (◇) at 130 °C and of 48k Ref 95 (○),
2h48 99 (△) at 140 °C. ...................................................................................................215
7.6: τ vs. q|| curves of 2h09 42 (□), 9k Ref 80 (○) at 130 °C and of 2h02 45 (◇), 2k Ref
43 (△) at 110 °C. ............................................................................................................216
7.7: τ vs. q|| curves of 2809 92 (□), 9k Ref 80 (○) at 130 °C and of 2802 90 (◇), 2k Ref
43 (△) at 110 °C. ............................................................................................................217
xxii
7.8: τ/h vs. q||h curves of 9k Ref 80 (□), 2h09 100 (◇) at 130 °C and of 48k Ref 95 (○),
2h48 99 (△) at 140 °C. The solid lines are fits with the HCT to calculated values of
η. ................................................................................................................................218
7.9: τ/h vs. q||h curves of 2h09 42 (□), 9k Ref 80 (○) at 130 °C and of 2h02 45 (◇), 2k
Ref 43 (△) at 110 °C. The solid lines are fits with the HCT to calculate η
values. ...........................................................................................................................220
7.10: τ/h vs. q||h curves of 2809 92 (□), 9k Ref 80 (○) at 130 °C and of 2802 90 (◇), 2k
Ref 43 (△) at 110 °C. The solid lines are fits with the HCT to calculate η
values. .............................................................................................................................222
7.11: g2 functions of 28Oli for several values of q|| at 20 °C. The red line is the attempt to
fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too large to
be calculated. ...............................................................................................................224
7.12: g2 functions of 28Oli for several values of q|| at 30 °C. The red line is the attempt to
fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too large to
be calculated. .................................................................................................................225
7.13: g2 functions of 28Oli for several values of q|| at 40 °C. The red line is the attempt to
fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too large to
be calculated. .................................................................................................................226
7.14: g2 functions of 28Oli for several values of q|| at 60 °C. The red line is the fit using
Equation 4.4 to calculate τ..............................................................................................227
7.15: (a) τ vs. q|| and (b) τ/h vs. q||h curves of 28Oli at 60 °C. The solid line on b is the
HCT fit to calculate η value. ..........................................................................................228
7.16: The TGA result of the oligomer used to prepare 28Oli. ........................................231
7.17: The MALDI result of the oligomer used to prepare 28Oli. ...................................232
7.18: The XR curves of 28Oli at 20 °C (blue curve) and at 60 °C (red curve). These
curves were measured before any exposure for XPCS. .................................................232
7.19: (a) The XR curve of 2h03 PSBr 38 obtained from APS 8-IDI (○) and its model fit
(
). (b) The SLD profile calculated from the model fit. Air interface is on the
right. ...............................................................................................................................237
7.20: τ vs. q|| curves of 2h03 PSBr 38 (◇), PSBr Ref 32 (□) at 150 °C and of 2h02 45
(△), 2k Ref 43 (○) at 120 °C. ......................................................................................240
xxiii
7.21: τ/h vs. q||h curves of 2h03 PSBr 38 (◇) and PSBr Ref 32 (□) at 150 °C and of 2h02
45 (△) and 2k Ref 43 (○) at 120 °C. The solid lines are fits with the HCT to calculate η
values. ...........................................................................................................................241
7.22: Schematics of partially tethered layer structure at different h. In a thinner layer (a),
tethered chains reach to the mobile near surface region, but a thicker layer (b) does not
have any tethered chain segments on or near the surface. .............................................242
7.23: The number of segments between entanglements as a function of χ was calculated
with mean field technique. Reprinted with permission from R. Oslanec and H. R. Brown,
“Entanglement Density at the Interface between Two Immiscible Polymers”,
Macromolecules 36, 5839 (2003). Copyright 2003 American Chemical Society. .........242
7.24: g2 functions of PCHMA Ref 38 for several values of q|| at 140 °C. The red line is
the fit using Equation 4.4 to calculate τ. .........................................................................244
7.25: τ vs. q|| curve of PCHMA Ref 38 at 140 °C. .........................................................245
7.26: g2 functions of 2h03 PCHMA 40 for several values of q|| at 150 °C. The red line is
the attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were
too large to be calculated. ...............................................................................................246
7.27: g2 functions of 2h03 PCHMA 40 for several values of q|| at 180 °C. The red line is
the attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were
too large to be calculated. ...............................................................................................247
7.28: g2 functions of 2h03 PCHMA 40 for several values of q|| at 200 °C. The red line is
the attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were
too large to be calculated. ..............................................................................................248
7.29: τ/h vs. q||h curve of PCHMA Ref 38 at 140 °C. The broken line indicates that 2h03
PCHMA 40 did not have any measurable value of τ until 200 °C. ................................249
7.30: XR curves of 2h03 PSBr 38 measured at APS 8-IDI. The blue and red curves were
measured before and after the XPCS measurement, respectively at 150 °C. ................251
7.31: XR curve of 2h03 PSBr 38 at 160 °C. ..................................................................252
7.32: Surface images of 2h03 PSBr 38 after cooling from 160 °C obtained from an
optical microscopy (a) and an AFM (b). ........................................................................253
7.33: The surface image of PSBr Ref 41 after annealing at 197 °C for 16 hrs in a high
vacuum obtained from optical microscopy. The polymer layer was on the top of a self
assembled monolayer of (3-glycidoxypropyl)trimethoxy silane. ..................................254
xxiv
7.34: MALDI result of the PCHMA used to prepare samples. ......................................255
7.35: TGA result of bulk PCHMA under nitrogen flow. ................................................256
7.36: XR curves of 2h03 PCHMA 40 measured at APS 8-IDI. The blue curve is measured
at 140 °C before XPCS and the red curve was measured at 180 °C after XPCS.
...................................................................................................................................... 256
7.37: Optical microcopy image of 2h03 PCHMA 40 surface after XPCS measurement at
200 °C. ............................................................................................................................257
7.38: AFM images of 2h03 PCHMA 40 surface after XPCS measurement at 200 °C. (a) is
the topography image and (b) is cross-section plots. .....................................................258
xxv
LIST OF TABLES
Table
Page
4.1: Element analysis from the XPS survey scan. ..........................................................104
4.2: Binding energy of various carbon bonds. ................................................................107
4.3: Comparison of 28k tethered PS layer thicknesses with two measurement
methods. ..........................................................................................................................110
4.4: Untethered chains used for partially tethered layers and for reference
layers. ..............................................................................................................................113
5.1: The information of two reference samples : 2k Ref 43 and 48k Ref 35. .................134
5.2: Viscosities calculated from the HCT fits. ................................................................138
5.3: η of 47 kg/mol – 48 kg/mol linear PS in a thin layer and bulk state. .......................142
5.4: Samples to be studied. .............................................................................................145
5.5: Viscosities calculated with the HCT fits. .................................................................147
5.6: Samples to be studied in this section. ......................................................................149
5.7: h values of 2802 45 at various XPCS measurement T. ............................................156
6.1. Samples to be studied. ..............................................................................................166
6.2: η values of partially tethered layers and reference samples calculated from the HCT
fits. ..................................................................................................................................171
6.3: Samples to be studied. .............................................................................................191
xxvi
6.4: η values of partially tethered layers and reference samples calculated from the HCT
fitting. .............................................................................................................................197
7.1: Samples to be studied. .............................................................................................207
7.2: η values calculated from the HCT fits. ...................................................................218
7.3: η values calculated from the HCT fits. ...................................................................220
7.4: η values calculated from the HCT fits. ...................................................................222
7.5: h of 28Oli at XPCS measurement T. ......................................................................233
7.6 : Samples to be studied. ............................................................................................235
7.7: η values calculated from the HCT fits. ....................................................................241
7.8: h values of 2h03 PCHMA 40 with T before any XPCS exposure. ..........................257
xxvii
CHAPTER I
INTRODUCTION
1.1. Surface Dynamics of Liquid
One of the most important objectives of this study is to observe the surface
dynamics of liquids, in particular, polymer melts. A liquid surface is always stimulated by
thermal energy and therefore exhibits height fluctuations. The momentum of the wave is
related to the system energy. The surface fluctuation can be described as a wave.
Therefore, a dispersion relation,  ~ k  , where ω is angular frequency and k is wave
number, can be established. The definition of the scattering vector, q, shown in Figure 1.1
for the special case in which the incident and the scattered radiation both propagate in the
plane of the paper, is given by
ki : Wave vector of incoming radiation, |k| = 2π/λ
k : Wave vector of scattered radiation
q : Scattering vector, q≡k- ki, |q| = 4πsinθ/λ
where θ is the incident and reflection angle of the radiation with respect to the surface.
Therefore,  ~ k  is equivalent to  ~ q where α depends on the properties of liquids.
The scale of the surface fluctuation is ~10Å in amplitude and covers a wide range in
wave length. These surface waves are determined by two factors: surface tension,γ, and
viscosity, η, of the liquid (in the absence of elasticity).
1
Figure 1.1: The relationships among the wave vectors for incident and scattering radiation,
scattering angle, 2θ, and scattering vector.
The surface modes of low molecular weight liquids have been studied for a long
time using light scattering [1-3]. When a low molecular weight liquid, a solvent, is mixed
with a high molecular weight polymer, the solution will have multi-component properties.
Harden et al. [4] presented a theory for the dispersion relation of polymer solution and
gel. As demonstrated by Figures 1.2 and 1.3, different trends of dispersion relation are
shown at different wave number, k and at different shear modulus of the solution, E0. At
 ~ k  , α=1 indicates elastic modes and α=3/2 is for capillary liquid modes. For
overdamped liquid, the relation becomes  ~ ik 2 . In the gel regime, the amount of
polymer is sufficiently high to make permanent physical crosslinks via entanglement. On
the other hand, in the polymer solution regime, since the concentration is not so high, the
chains will disentangle and diffuse out after some relaxation time, τ. The relationship
between τ and degree of polymerization, N is given by  ~ N 3.4 .
Cao et al. [5] has experimentally shown the relationship between surface mode and
polymer concentration. The surface of a solution of polyisobutylene in decane was
observed using the surface heterodyne light scattering technique. The mode of polymer
solution was continuously changed with polyisobutylene concentration. At 0.5g/dl,
2
capillary modes, for which  ~ k 3 / 2 , were seen, but α decreased as the concentration
increased. Elastic modes, for which  ~ k , were observed at 8g/dl. This result agreed
well with the theory [4].
Figure 1.2: The surface mode diagram for polymer gel regime is plotted with logarithm
scale of wave number (k) and shear modulus(E0). Modes are distinguished with
characteristic wave number, k   02 and modulus E    0  where ρ is the
2
density of the gel. Reprinted with permission from J. L. Harden, H. Pleiner and P. A.
Pincus, “Hydrodynamic Surface Modes on Concentrated Polymer Solutions and Gels”, J.
Chem Phys. 94, 5208 (1991). Copyright 1991, American Institute of Physics.
3
Figure 1.3: The surface mode diagram for polymer solution regime is plotted with
logarithm scale of wave number (k) and shear modulus(E0). Modes are distinguished with

characteristic wave number, k   02 , k   2 

E   2  2

13

1 3
and modulus, E    0  ,
2
where ρ is the density of the solutoin. Reprinted with permission from J.
L. Harden, H. Pleiner and P. A. Pincus, “Hydrodynamic Surface Modes on Concentrated
Polymer Solutions and Gels”, J. Chem Phys. 94, 5208 (1991). Copyright 1991, American
Institute of Physics.
4
1.2. Surface Dynamics of Polymer Melts
A melt of pure untethered polymer behaves like a liquid at temperatures above the
glass transition temperature, Tg, showing surface fluctuations. Mochrie and Kim et al. [6,
9] observed surface fluctuation of a polymer melt layer supported on a silicon wafer at
50 °C above its Tg. The thickness of the polymer layer, h, was thicker than four times the
radius of gyration, Rg (h ≥ 4Rg). The surface modes agreed well with the overdamped
thermal capillary wave theory [7]. The parameter describing the rate at which surface
fluctuations relax is the surface relaxation time, τ. The surface fluctuations continuously
change the surface structure. After some amount of time, τ, the surface is so completely
changed that there is no correlation with the initial state as shown by Figure 1.4. A
smaller value for τ means that the surface moves more quickly.
Figure 1.4: A schematic illustrating the significance of the surface relaxation time, τ. A
film of low molecular weight liquid or polymer at a temperature above Tg has surface
fluctuations. After a time, τ, has passed, the surface structure has become completely
different from that of the initial state. No structural correlation exists between the two
surface states.
5
Assuming that the h of the layer is sufficiently large to ignore any confinement
effects, the polymer melt layer can be considered as a homogeneous liquid continuum.
Then in the special case of non-slip boundary condition at the substrate, the relaxation
time, τnonslip, is given by [7]
 n o n sql ||i p 
2H
,
q|| F
(1.1)
where H and F are
H  coshq|| h  q|| h and
2
2
F  sinhq|| hcoshq|| h  q|| h .
If there is slip at the substrate interface, the surface relaxation time, τslip, is given by
 slip q||  
2 cosh q|| h sinh q|| h   q|| h 
q|| sinh q|| h 2
.
(1.2)
When τ is divided by h and then  nonslipq||  h and  slip q||  h become functions of the
dimensionless scattering vector q||h . In addition, since there is a relation  ~   , the
τ/h vs. q||h plot measured at the same η and γ is independent of h. In other words, if the τ
is measured at the same temperature and molecular weight, the τ/h values converges to
the the same value at different hs. Furthermore, if γ is known, fitting the τ/h vs. q||h curve
with the equation,

h

2 cosh q|| h sinh q|| h   q|| h 
q|| h sinh q|| h 2
yields the η of the polymer melt layer, at the temperature and molecular weight.
6
(1.3)
X-ray photon correlation spectroscopy (XPCS) is one of the best instruments to
observe surface dynamics [8]. The raw data from XPCS is scattered x-ray intensity, I, at
each in-plane wave vector, q||. The intensity-intensity correlation function, g2,
characterizes the rate at which the surface height fluctuations relax.
g 2 (q|| , t ) 
I q|| , t 'I q|| , t 't 
I q|| , t '
2
.
(1.4)
Since value of η and γ depends on molecular weight and measurement temperature,
the τ is affected by molecular weight and temerature as well. Kim et al. [9, 10] measured
PS thin films on silicon wafers with various molecular weights and hs. As shown by the
XPCS result (Figure 1.5), τ decreases as q|| increases, meaning that the smaller scale
structure of the surface relaxes faster than the larger scales. As Figure 1.5a displays, at the
same molecular weight (123 kg/mol) and h, τ decreases as temperature, T, increases. In
particular, as T - Tg gets higher, the surface moves faster due to thermal fluctuation.
Figure 1.5b demonstrates that τ decreases when the h increases at the same molecular
weight and T. The surface fluctuation is not only sensitive to the surface, but depends on
the flow throughout the film. Finally, in Figure 1.5c, τ increases with molecular weights.
This is due to larger η at high molecular weight. For the molecular weights shown here,
the η is influenced by chain entanglement. The solid lines on Figure 1.5 represent fits to
the data with Equation 1.2. The quality of the fits suggests that the surface dynamics of
these films of untethered chains at 65k and 123k molecular weights are presumably
entangled. Sufficiently large h can be described by the purely viscous liquid model.
7
a)
b)
c)
10-3
q||(nm-1)
10-2
Figure 1.5: The τ vs. q|| for PS melt films measured using XPCS. a) τ vs. q|| at different
temperatures with fixed molecular weight and film h, 123 kg/mol. b) τ vs. q|| at different
hs at the same temperature and molecular weight. c) τ vs. q|| for different molecular
weights at the same temperature and h. Figure reprinted with permission from H. Kim, A.
Rühm, L. B. Lurio, J. K. Basu, J. Lal, D. Lumma, S. G. J. Mochrie and S. K. Sinha,
“Surface Dynamics of Polymer Films”, Phys. Rev. Lett. 90, 068302 (2003). Copyright
2003 by the American Physical Society(a and b). Reprinted from H. Kim, A. Rühm, L. B.
Lurio, J. K. Basu, J. Lal, S. G. J. Mochrie and S. K. Sinha,
“X-ray Photon Correlation
Spectroscopy on Polymer Films with Molecular Weight Dependence”, Physica B 336,
211 (2003). Copyright 2003, with permission from Elsevier(c).
8
Compared to low molecular weight liquids, the melting transition of a polymer is
not sharp and other transitions exist due to the architecture of the polymer. Long sidechains attached to the main backbone may exist in ordered or disordered structures
depending on temperature [62 - 65]. The transition that occurs between the ordered and
disordered structures also affects the surface dynamics of the whole polymer film. Using
XPCS, Dhinojwala et al. [66] observed the surface capillary fluctuations of poly(n-alkyl
acrylate) films supported on silicon wafers (Figure 1.6). Around the order-disorder
temperature of the n-alkyl side chains, Ts, the surface dynamics of the polymer film were
determined by the ratio between the total h and the h of the ordered structure. In a thinner
film, in which the ordered structure constitutes a large fraction of the entire film volume,
the surface dynamics slowed down due to the ordered structure.
9
Figure 1.6: g2 of poly(octadecyl acrylate) in a thin film at q|| = 1.65×10-4 Å -1. The h for
(a)-(c) was 220 Å and for (d)-(f) was 660 Å , respectively. The temperature was at T < Tm
[(a), (d)], Tm < T < Ts [(b), (e)] and T > Ts [(c), (f)], where Tm is the melting temperature
and Ts is the order-disorder transition temperature of the side chains. The 660 Å film had
measurable relaxation at T < Tm, but for the 220 Å film the relaxation behavior was seen
only at T > Ts. Figure reprinted with permission from S. Prasad, Z. Jiang, M. Sprung, S. K.
Sinha and A. Dhinojwala, “Effect of Surface Freezing on Meniscus Relaxation in Side
Chain Comb Polymers”, Phys. Rev. Lett. 104, 137801 (2010). Copyright 2010 by the
American Physical Society.
10
The other property of polymer chains that distinguishes them from low molecular
weight species is that chains can form entanglements. At higher molecular weights, the
chains in bulk are in an entanglement regime showing a very strong dependence of η on
molecular weight [11]
 ~ M 3.4 ,
(1.5)
where M is the molecular weight. However, this entanglement effect could possibly lead
to a different trend of polymer η with molecular weight in a thin film. Using XPCS, Jiang
et al. [12] measured τ of 160 nm thick polystyrene films supported on silicon wafers. For
films of low molecular weight chains, where entanglement did not occur, the value of τ
increased with molecular weight. Entanglement of PS chains takes place for weight
average molecular weight, Mw, greater than 30 kg/mol. And as shown by Figure 1.7a, the
value of τ became independent of Mw for Mw > 30 kg/mol. The τ vs. q|| data points were
fitted using Equation 1.2 to calculate η. The η of the thin film calculated in this way did
not increase for molecular weights higher than ~20 kg/mol (Figure 1.7 b).
11
τ (sec)
a)
10-3
10-2
q(nm-1)
10-1
b)
Figure 1.7: The entanglement effect on surface fluctuation dynamics. a) τ was measured
with XPCS at 111 °C. The value of τ shows universal behavior when the molecular
weight is more than 30 kg/mol. The solid line connecting data points is the result from
fitting using the hydrodynamic continuum theory Equation 1.2. b) The plot of η with
respect to molecular weight for the polymer thin film (○) and bulk (◇). The dash-dotted
line denotes the average η of the polymer film for Mw > 30 kg/mol. The solid red line is
the result from fitting a power law to the bulk behavior in the entanglement regime.
Figures reprinted with permission from Z. Jiang, M. K. Mukhopadhyay, S. Song, S.
Narayanan, L. B. Lurio, H. Kim and S. K. Sinha, “Entanglement Effects in Capillary
Waves on Liquid Polymer Films”, Phys. Rev. Lett. 101, 246104 (2008). Copyright 2008
by the American Physical Society.
12
The main topic of this dissertation research is the effect of confinement on the
dynamics of thin (< 100 nm) polymer film on the silicon wafer. The confinement effects
originate from the interfaces. First, we consider the effect of the air/vacuum interface.
Using ellipsometer, Cory et al. [13] measured Tg of supported polystyrene thin films with
various hs. To fit the results, they suggested an equation describing the variation of Tg
with h as follows:
  A  
Tg d   Tg  1     ,
  h  
(1.6)
where Tg   is the Tg of the bulk polymer (374K for PS), A is a characteristic length,
3.2 ± 0.6 nm, and δ is 1.8 ± 0.2. The films with lower h had lower Tg. Polymer films of
chains with various molecular weights from 120 kg/mol to 900 kg/mol showed the same
trend. They argued that this Tg trend is due to the more liquid like properties of film
surfaces.
Torkelson et al. [14, 15] showed that when a fluorophore was mixed with polymers,
the fluorescence emission spectra intensity decreased at higher temperature. The slope of
the emission intensity vs. temperature changed at Tg. Using this change as a signature of
Tg, the Tg's of thin films of PS, poly(4-methyl styrene) (P4MS) and poly(4-tertbutylstyrene) (PTBS) were measured using the fluorescence emission spectra method. As
the hs of the thin films became smaller, Tg - Tg,bulk became more negative. And these
phenomena are consistent with the data from Cory et al. [13]. Also, Torkelson et al. [16]
argued that if a polymer surface faces air or vacuum, the density of the polymer near the
surface would be lower than that of the polymer in the bulk, providing more free volume.
Due to this extra free volume, the segment mobility would be enhanced at the interface
13
facing air or vacuum. After all, these interfaces would contribute to the whole polymer
layer to reduce Tg.
Next we consider the confinement effect due to the interface between the substrate
and the polymer layer. Depending on the surface treatmentused, a silicon wafer's surface
is polar (piranha solution cleaning) or moderately hydrophobic (hydrofluoric acid (HF)
etching). Vignaud et al. [17] showed that the interaction determined by the polarity of the
substrates could affect the glass transition behavior of the polymer film on the top. PS
layers of 5-100 nm were deposited on silicon wafers with either polar or hydrophobic
(less polar) surface and multilayer wavelength ellipsometer used to observe the trend of
Tg with h. It was observed that the ellipsometric angle, Δ, varied with temperature and the
temperature at which the slope changed is Tg. The PS film had only one Tg on the less
polar surface, showing a reduced value of Tg at lower h. On the other hand, the PS films
showed two or three Tgs on polar substrates. Also, on polar surfaces, the variation of Tg
with h was not as clear as that for films on non-polar surfaces.
The van der Waals interaction between the substrate and the polymer layer is able
to provide a confinement effect. As a result, some surface fluctuations at some
lengthscales are suppressed at T above Tg. One manifestation of the suppression of these
surface fluctuations is the appearance of a cutoff value of in-plane wave vector for an Xray diffuse scattering curve. The fluctuations with wavelengths larger than the inverse of
the cutoff wave vector are suppressed. Therefore, larger values of the cutoff wave vector
indicate that wider range of surface fluctuations is suppressed [18, 19]. The value of the
cutoff wave vector resulting from the van der Waals interaction, qvdW, is given by
Aeff 2
qvdW d  
h2
14
,
(1.7)
where Aeff is the Hamaker constant. Wang et al. [20] observed changes in the cutoff wave
vector value with film thickness for the surface fluctuations of PS film with Mw = 90
kg/mol using a static diffuse X-ray scattering measurement. That is, they measured
scattering intensities averaged over times that could be long relative to the characteristic
time of the surface relaxation. As displayed in Figure 1.8, the lower wave number cut off,
ql,c, could be seen until the film was thinner than 6Rg. The authors interpreted these
results as being were due to van der Waals force at a polymer layer/substrate interface.
15
Figure 1.8: The off specular scattering curve of PS thin film at various h. The thin solid
lines on the data point are fitting to find ql,c. Figure reprinted with permission from J.
Wang, M. Tolan, O.H. Seeck, S. K. Sinha, O. Bahr, M. H. Rafailovich and J. Sokolov,
“Surfaces of Strongly Confined Polymer Thin Films Studied by X-Ray Scattering”, Phys.
Rev. Lett. 83, 564 (1999). Copyright 1999 by the American Physical Society.
16
The capillary wave model needs to be modified for use with thinner supported
polymer films due to the confinement effect with a substrate. Jiang et al. [21] measured
the τ value of PS films on silicon wafers using XPCS. For h ≥ 4Rg , the variation of τ with
wave vector showed a behavior characteristic of a simple viscous liquid. However, for a
thinner film, (h = 2Rg), the conventional capillary wave model (Equation 1.2) could not
represent the trend of τ vs. q|| well, but a surface dynamics model (Equation 1.8) into
which elasticity had been added fitted the τ vs. q|| curve better. Figure 1.9 shows the
difference in goodness of fit provided by the two models. The shear modulus, μ, of the
polymer film becomes a factor in the calculation of τ as
 q||  
 0 q|| 
,
1   0 q||   
(1.8)
where τ0(q||) is the surface relaxation time from the purely viscous model (Equation 1.2).
17
Figure 1.9: The τ of a PS thin film with Mn = 123 kg/mol and Mn = 400 kg/mol at h = 2Rg
measured with XPCS. The dashed lines present the fitting using the viscous liquid model,
Equation 1.2, and the solid lines present fits from the viscoelastic models using Equation
1.8. Figure reprinted with permission from J. Wang, M. Tolan, O.H. Seeck, S. K. Sinha,
O. Bahr, M. H. Rafailovich and J. Sokolov, Z. Jiang, H. Kim, X. Jiao, H. Lee, Y. J. Lee, Y.
Byun, S. Song, D. Eom, C. Li, M. H. Rafailovich, L. B. Lurio and S. K. Sinha, “Evidence
for Viscoelastic Effects in Surface Capillary Waves of Molten Polymer Films”, Phys. Rev.
Lett. 98, 227801 (2007). Copyright 2007 by the American Physical Society.
18
Another large body of literature has addressed the experimental determination of
values of effective glass transition temperature for thin polystyrene films [14] and the
variation of Tg,film with film thickness and the manner in which the film is supported or
not.
These studies have not explicitly considered surface fluctuations, but the Tg
phenomena and the surface fluctuation phenomena are interrelated. Torkelson and
coworkers have considered particularly the way in which the apparent Tg,film is influenced
by the character of the substrate upon which the film sits. The Tg of a 12 nm - 14 nm
thick PS layer supported by layers of various polymer species such as PS, poly(methyl
methacrylate) (PMMA) and poly(2-vinylpyridine) (P2VP). Their results are summarized
by Figure 1.10. When the PS thin film was supported by a layer of PS, the Tg of the top
layer was reduced by 10 K - 33 K, depending on the thickness of the underlying layer.
However, when supported by a PMMA layer or a P2VP layer, the Tg of top PS layer was
decreased less. They argue that the presence of a narrow interface between immiscible
polymers close to the free surface disrupts the propagation of enhanced mobility into the
PS film which is important for achieving the large drop in Tg at the surface seen for the
standard PS film.
19
Figure 1.10: The Tg of a PS layer as measured by a fluorescent dye method varies with
the type of supporting polymer layer. Reprinted with permission from C. B. Roth, K. L.
McNerny, W. F. Jager and J. M. Torkelson, “Eliminating the Enhanced Mobility at the
Free Surface of Polystyrene: Fluorescence Studies of the Glass Transition Temperature in
Thin Bilayer Films of Immiscible Polymers”, Macromolecules 40, 2568 (2007).
Copyright 2007 American Chemical Society.
20
1.3. Tethered Chains
So far, only polymer films composed of linear untethered chains have been
discussed. In other words, the effect of chemical bonds between the substrate and
polymer layer has not been considered yet. Dipole interactions and other van der Waals
interactions can result in a confinement effect on untethered chains, but these non-bonded
interactions are not as strong as the interactions represented by covalent bonds. Methods
to prepared covalently tethered polymer chains and the dynamics of tethered chains will
be discussed in this section.
1.3.1. Concepts of Tethered Chains
The term "tethered chain" indicates in this work a state in which one end of a
polymer chain is covalently attached to a substrate. Tethered chains can be placed on a
substrate to provide a sample surface that is modified or improved relative to the surface
of the original substrate and this means that tethered chains can be applied to modifying a
wide array of properties such as adhesion [22 - 28], lubrication [29, 30] and wetting [31 33]. The "substrates" can take on various shapes. For example, a linear polymer backbone
is a one-dimensional "substrate", a planar substrate is two-dimensional and particles can
be three-dimensional (Figure 1.11). The architecture of the tethered chains can also vary.
They can be linear polymers or branched polymers. Also, either homopolymers and
copolymers can be attached to the substrates. The type of tethered chain system of
interest for this dissertation is that of linear homopolymers [34] tethered on a twodimensional substrate.
21
One factor determining the conformations of tethered chains is the grafting density
(σ), or, expressed another way, the distance between grafting points (d). The parameter σ
describes how many chains are tethered in a unit area as given by [35]

hN A
,
Mn
(1.9)
where ρ is the bulk mass density of the tethered chain component and NA is Avogadro’s
number. Alternatively, one polymer chain can be considered to occupy an area, d2.
Therefore, the relationship between σ and d is :   d 2 .
Wu et al. [36] measured the h of tethered chain layers depending on σ using poly
acrylamide (PAAm) tethered chains in water, a good solvent of PAAm, using variable
angle spectroscopic ellipsometer. For lower grafting densities, h was independent of σ
( h ~  0 ), but a cube root dependence on σ was observed at higher grafting densities (σ >
0.065 chains/nm2) ( h ~  1/ 3 ). As d gets smaller, overlap between tethered chains takes
place. Once d is smaller than Rg, the tethered chain conformation becomes stretched due
to overlapping of the projections of the chain pervaded volumes on the substrate surface.
It has been suggested [37, 38] that the conformations of tethered chains can be
categorized as belonging to one of several regimes. Since the conformation changes due
to chain volume overlap, the conformation depends on Rg and σ. Therefore, a reduced
tethering density, Σ, is a useful parameter to distinguish each regime,
   Rg2
.
(1.10)
Σ represents the number of chains that can be found in the volume occupied by one
untethered, unperturbed chain having the same molecular weight.
22
Figure 1.11: Tethered chain systems can have various structures depending on the type of
substrate and the type of chain being tethered. Reprinted from M. Zhang and A. H. E.
Müller, “Cylindrical Polymer Brushes”, J. Polym. Sci. Part A : Polym. Chem. 43, 3461
(2005). Copyright 2005, with permission from John Wiley and Sons.
Figure 1.12 shows the tethered chain conformations that can be realized by varying
σ. The “mushroom” regime is found for   1 , where d  Rg . Since there is no overlap
between chain volumes, the chains do not stretch, but have a typical random coil
configuration. At   1 or d  Rg , the conformation enters the “pancake-like” regime.
The projections of chain volumes on the substrate start to overlap, but a chain still
experiences attractive interactions with other chains and with the substrate. Therefore, the
chains will flatten out like pancakes on the substrate. The "brush" regime begins at   1
or d  Rg . Chains start to interpenetrate the unperturbed volume of neighboring chains
and they stretch away from the substrate.
23
Figure 1.12: The conformation of tethered chains varies with σ.
All chains significantly stretch in the true brush regime, usually understood as
corresponding to values   5 . However, the value at which this transition occurs is
affected by the structure of the tethered chain. Cheng et al. [39, 40] prepared polymer
crystals using poly(ethylene oxide)-block-PS (PEO-b-PS) and poly(L-lactic acid)-blockPS (PLLA-b-PS). The crystals were used as basal templates to obtain tethered PS layers.
The thickness of crystal layers of the PEO or PLLA blocks and other polymer blocks
were measured with atomic force microscopy (AFM) to calculate the surface free energy
of the PS tethered chains (ΓPS). ΓPS was observed to increase with Σ, with the increase in
ΓPS entering a new regime for Σ > 14. This transition to a new regime was considered to
show that the true brush regime for tethered PS chains on crystals of PEO or PLLA starts
from Σ = 14.
Not only σ and d, but also the medium in which the tethered chains are located, can
determine the tethered chains conformation. For example, this medium could be vacuum,
ambient, a polymer or some solvent. In poor solvent, the tethered polymer chains will be
collapsed because the polymer-solvent interaction is less favorable than the polymer24
polymer segment interaction. For a theta (θ) solvent, the polymer-polymer interaction is
as favorable as the polymer-solvent interaction, resulting in zero excluded volume. For
good solvent, the tethered chains will swell to increase the contact between polymer and
solvent. If untethered chains are surrounding the tethered chains, the conformation of
tethered chains will depend on the molecular weight of the untethered chains and the
Flory-Huggins segment-segment exchange interaction parameter. This will be described
in more detail in the results and discussion chapters.
1.3.2. Preparation of Tethered Chains
Both "grafting from" and "grafting onto" methods are used to prepare tethered
chain layers. "Grafting from" involves polymerization on the substrate using initiators
attached on the substrate surface. "grafting onto" [41] is attaching already polymerized
chains onto substrates via chemical reaction or physical adsorption.
Free radical polymerization was first used for the grafting from method. Rühe et al.
[42 - 44] prepared PS tethered chains via free radical polymerization initiated by azo
compounds attached on the substrate. The h of tethered chain layer could be more than 60
nm. However, free radical polymerization was not suitable to control % conversion and
molecular weight. In addition, due to termination and chain transfer steps, the molecular
weight distribution was quite broad. The polydispersity index, PDI, which is Mw is
divided by number average molecular weight (Mn), was 2.0. To overcome this weakness,
living polymerization was used for the grafting from method. Fukuda et al. [42, 45] used
atom transfer radical polymerization (ATRP) to prepare tethered PMMA chains. A linear
increase of molecular weight with monomer % conversion was observed during the
25
ATRP. Also, molecular weights higher than 100 kg/mol could be acquired with PDI
between 1.0 and 2.0. Hawker et al. [46] have prepared PS tethered chains using nitroxide
mediated radical polymerization (NMRP). Molecular weights as high as Mn = 51 kg/mol
could be achieved while maintaining narrow molecular weight distribution (PDI < 1.2).
Since low molecular weight monomers are attached to the reactive tethered chain
ends in the grafting from approach, the steric hindrance during tethering is far smaller
than during the grafting onto method. Therefore, the advantage of the grafting from
method is that it can provide high σ. However, the weakness is that the tethered chains
can not be characterized without cleaving them from the substrate.
Chains tethered via the grafting onto method are acquired usually by chemical
reaction between the substrate surface and functional groups on the polymer chain ends.
Depending on the type of functional groups on the polymer, the substrate can be used "as
is" or it needs to be chemically modified. Polymers having certain functional groups can
react with silicon oxide surfaces to form covalent bonds. In this case, tethering can be
accomplished by spin casting a polymer solution on a substrate and annealing the coated
substrate at high temperature. In one case, carboxylic acid end groups on polymer chains
were converted into triethoxysilane groups and these groups could make covalent bonds
with a silica surface [47, 48]. Polymers having trichlorosilane end groups could be
reacted with a piranha solution cleaned silicon wafer to prepare tethered chains [49].
Using atomic force microscopy (AFM), Zhao et al. demonstrated that a PS chain with a –
Si(OH)3 end group can be grafted onto a silica surface [50].
26
In the second method chains are grafted onto a chemically modified substrate. This
modification usually involves deposition of a functionalized self-assembled monolayer
(SAM) on the substrate via reaction. Functionality to react with the polymer is an
essential property of the SAM reagent. For example, SAMs with epoxide, carboxylic acid,
or amine functionalities on their surfaces can make covalent bonds [51 - 53] with chains.
Chains with functional groups are coated onto these modified surfaces and then covalent
bonds can be formed via chemical reaction [54, 55] after heating.
Not only forming covalent bonds, but also physical adsorption can be used in the
grafting onto method. A substrate-attractive block in a diblock copolymer can play a role
as an “anchor” to the substrate, while the rest of chain is a “buoy” block. Or telechelic
polymers having end-groups that adsorp readily to the substrate can be used for the
tethering [67 - 72].
The advantage of the grafting onto method is that since polymer chains are attached
to a substrate, characterization of the polymer is easier than in the case of grafting from.
While tethered chains prepared with the grafting from method must be cleaved from the
substrate for characterization, polymer chains can be characterized before attaching with
the grafting onto method. However, since the preformed polymers have larger volumes
than do monomers, in the grafting onto technique steric hindrance between polymer
chains already on the surface and those approaching the surface restricts σ to low values.
Furthermore, since Rg is determined by the molecular weight, the value of σ is not
independent of molecular weight. The worst of these drawbacks is that the hydrodynamic
size of the polymer chains approaching the surface to be tethered is not optimized.
However, when the tethering occurs in a polymer solution, the chain size can be
27
controlled with solvent quality and concentration. Making the chain size in the solution
smaller can result in higher σ [59 - 61].
1.3.3. Surface Dynamics of Tethered Chains
Frederickson et al. [56] predicted the surface dynamics of a melt of only tethered
chains. According to their work, the fluctuation in surface height of the tethered chain
layer, h(x) can be described as a sinusoidal fluctuation given by
hx   h0   cosq|| x  ,
(1.11)
where the wavelength of the surface fluctuation is defined as 2 q|| , h0 is the
equilibrium uniform h and ε is the fluctuation amplitude. Since lateral chain stretching is
expected to maintain the polymer layer density constant, in contrast to the case of a liquid
surface, the fluctuation modes with 2 q||  h0 will be suppressed.
Figure 1.13: Schematic of tethered chain surface dynamics used by Frederickson et al.
Reprinted with permission from G. H. Fredrickson, A. Ajdari, L. Leibler and J. P. Carton,
“Surface Modes and Deformation Energy of a Molten Polymer Brush”, Macromolecules
25, 2882 (1992). Copyright 1992 American Chemical Society.
28
This theory was experimentally tested by Foster et al. [57]. The surface dynamics
of PS tethered chain layers having various values of h between 9 nm and 107 nm and
with σ in the range of 0.12 - 0.80 chains/nm2 were measured with XPCS. Interestingly,
even at T - Tg = 90 °C, where Tg is of the bulk PS, τ could not be measured. This means
that the surface dynamics were slowed so much that the relaxation time became longer
than 10000 seconds which is the limit of the XPCS measurement window (Figure 1.14).
Figure 1.14: g2 vs. delay time, t, at q|| = 5.3×10-3 nm-1 for PS tethered chains (PS Brush)
and untethered thin film. The untethered chain layer (PS, Mn = 65 kg/mol) showed
relaxation at 170 °C, but the tethered PS layer did not show a relaxation time until 225 °C.
Reprinted with permission from B. Akgun, G. Uğur, Z. Jiang, S. Narayanan, S. Song, H.
Lee, W. J. Brittain, H. Kim, S. K. Sinha and M. D. Foster, “Surface Dynamics of “Dry”
Homopolymer Brushes”, Macromolecules 42, 737 (2009). Copyright 2009 American
Chemical Society.
29
1.4. Hypothesis of This Study
The surface dynamics of tethered and untethered chains have been reviewed so far.
The summary of this review is that for layers of untethered chains, the surface dynamics
are fast enough to be measured, but the surface dynamics of tethered chain layers are too
slow to be measured with XPCS. A hypothesis can be derived from these observations for
two extreme cases. If the degree of tethering can be controlled, the surface dynamics will
be tailored as well (Figure 1.15).
Figure 1.15: Schematic illustrating how the surface relaxation time can be tailored by
changing the degree of tethering in the layer.
Uğur [58] spun cast a layer of untethered deuterated PS (dPS) chains on a layer of
tethered hydrogenous PS (hPS) having a large value of σ (0.6 chains/nm2). Neutron
reflectivity (NR) was used to observe the composition depth profile of dPS. According to
the composition profiles from the best fits to the NR curves, clear separation of the
untethered chains from the tethered chains was seen with a narrow interface between the
two regions. Due to the large value of σ, the tethered chains did not mix favorably with
the untethered chains. Therefore, the sample could be considered to be a “bilayer”.
30
To investigate our hypothesis about tailoring surface fluctuations with tethering, we
will use “partially tethered layers”, i.e. mixtures of tethered and untethered chains. The
properties of partially tethered layers may be different than those of bilayers or of layers
of pure untethered chains. Tsui et al. [73] measured the Tg of untethered PS layers on the
tops of layers of tethered PS chains having various values of σ. For large σ over 0.3
chains/nm2, as the layer of untethered PS became thinner, it showed a lower value of Tg.
For the case of 0.1 chains/nm2, the Tg variation trend was not distinct when the h of the
untethered PS layer decreased, displaying the Tg value similar to that of the bulk state. In
this study, to obtain better miscibility with the untethered chains, the tethered chain layer
will be prepared via the grafting onto method to obtain values of σ smaller than 0.15
chains/nm2. Untethered chains will be spun cast on the top of the tethered chains and
annealed to make the untethered chains diffuse into the layer of tethered chains. Thanks
to the favorable mixing with lower σ, at least for the case in which the untethered and
tethered chains are of the same type (e.g. PS) the whole sample could be seen as a single
polymer layer having a fraction of the chains covalently bonded to the substrate.
The factors determining the surface dynamics of partially tethered layers would be
i) the covalent bonds with the substrate, ii) the conformation of tethered chains (resulting
from the penetration of the tethered chains into the untethered chains) and iii) the
miscibility between the tethered and untethered chains determined by the Flory-Huggins
interaction parameter and the molecular weight of the untethered chains. In this study, our
existing understanding of the factors mentioned above will be discussed. And then their
actual effects on the surface dynamics observed using XPCS and NR, will be discussed.
31
Figure 1.16: Factors determining the surface dynamics of partially tethered layers.
32
CHAPTER II
BACKGROUND
2.1. Diffusion
Key for understanding the role of tethered chains in determining the surface
dynamics of partially tethered layers is an understanding of how chains pass by one
another in a melt. We consider first the extensively studied case in which all the chains
are untethered.
2.1.1. Rouse Model
The Rouse model has been used to describe the chain dynamics in polymer melts
and solutions. Each chain is considered as a string of N beads, with each pair of beads
connected with a spring type bond with root-mean-square size, b. When the chain moves,
each bead experiences a friction coefficient, ζ. If one assumes that the friction for the
entire chain is simply the sum of frictions for all beads, then the friction coefficient of a
chain, ζR is given by
 R  N
.
(2.1)
Using the Einstein relation, the diffusion coefficient of a chain moving due to thermal
energy is given by
DR 
kT
R
33

kT
.
N
(2.2)
The Rouse time, τR , is the time the chain takes to move a distance on the order of the size
of the chain,
R 2 R 2 N
R 

,
DR
kT
(2.3)
where R is the radius of a polymer coil. As Equation 2.3 shows, a larger friction
coefficient would make the diffusion slower.
The viscosity, η, of the polymer melt or solution depends on the chain friction
coefficient and the size of the polymer coil. The friction coefficient has the relationship
given by
 ~ R .
(2.4)
For a larger size of polymer or at higher η, the chain would have less mobility. And If we
assume that the hydrodynamic continuum theory (HCT) of Jäckle [7] applies (Equation
1.3) then it follows that when the friction is higher for unentangled chains the relaxation
time for surface height fluctuations will be larger.
2.1.2. Reptation Model
If the polymer chains are long enough to entangle with each other, the chain diffusion
will moved to the entanglement regime. In the entanglement regime, one chain is
entangled with multiple chains. Therefore, a chain is conceptually considered to be
composed of strands with an entanglement point at each strand end. Since the motion of a
chain is restricted by the surrounding chains, the chain motion can be thought of as a
movement through the contour of a confining tube. The size of entanglement strand, e, is
assumed to be the diameter of the confining tube, because there is no effect of
entanglement within a strand. The value of e is related with b and the number of repeating
34
units per strand, Ne, given by
e  b Ne .
(2.5)
The contour length of the tube, L , is the sum of the length of strands.
L e
N b2 N
bN


.
Ne
e
Ne
(2.6)
The reptation time, τrep, indicates the time needed for the chain to move distance
L
and it is given by
 rep
 bN 



 N 
D
e


L
2
2
 kT

 N
3
 b 2 2  N 
 .
 
N e 
 kT
 Ne 
(2.7)
Since τrep is strongly depended (to the 3rd power) on the number of segments, longer
chains will have longer reptation times. Therefore, the rate of chain diffusion scales with
molecular weight, M, as follows:
 rep ~ N 3 ~ M 3 .
(2.8)
However, the experiment results suggest that the relationship between τrep and M is given
by [11]
 rep ~ M 3.4 .
(2.9)
2.1.3. Diffusion Measurement
In the dissertation research we focus on the surface fluctuations, but these may be
supposed to depend on the mobility of the chains in the melt layer. This mobility, should,
in turn be related to the diffusion of the chains. The untethered chains in a partially
tethered layer will experience contacts with the tethered chains. The diffusion of
35
untethered chains might be hindered due to the friction with the tethered chains.
Therefore, reviewing the friction of linear chains in the various polymer matrices will be
helpful to understand the results discussed in this study. The diffusion coefficient of a
polymer chain in a matrix of chemically identical untethered polymers has been measured
using a fluorecein labeling technique by Sillescu et al. [74]. The diffusion coefficient of a
linear polystyrene (PS) with Mn = 4 kg/mol, was in the range between 10-11 and 10-10
cm2/s at 138 °C - 153 °C in a matrix of linear PS of the same Mn. Slower diffusion was
observed for higher values of M of the polymer matrix.
In addition, the diffusion in a cross-linked matrix has been studied [75]. A crosslinked PS matrix was prepared via cross-linking with the Friedel-Crafts reaction using pdichloroxylene. Then photo-labeled linear PS was mixed into the matrix. Factors
determining the linear PS diffusion in the cross-linked matrix were the degree of
polymerization of the linear PS (Pn), temperature (T), and the crosslink density of the
matrix, (ρc). The self diffusion coefficient of linear PS, D, that is, the coefficient for
diffusion of a single chain, followed a power law relationship with Pn, D ~ Pn with α ~
2 (Figure 2.1).
36
Figure 2.1: Plot of the self diffusion coefficient of a labeled linear PS in a cross-linked
matrix vs. Pn at various T. Open circles are for the PS network with pc = 0.02 and ΦPS =
0.2 where pc is the number of crosslinks per monomer and ΦPS is the volume fraction of
PS. Full circles are for the PS network with pc = 0.03 and ΦPS ≤ 0.01 at 194 °C. Diffusion
in un-crosslinked PS matrix with Mw = 110 kg/mol at 194 °C is displayed as triangles.
Higher T and lower crosslink density allows faster diffusion. Reprinted with permission
from M. Antonietti and H. Sillescu, “Self-Diffusion of Polystyrene Chains in Networks”,
Macromolecules 18, 1162 (1985). Copyright 1985 American Chemical Society.
37
2.2. Friction of Tethered Chains
DeGennes et al. [76] calculated the force imposed on a tethered chain surrounded
with untethered chains due to a shear stress. As shown by Figure 2.2, the polymer melt is
under a constant shear stress regardless of the distance from the surface, z. Imagine
tethered chains are attached on the interface between the polymer melt and the substrate
(z = 0). When the shear flow velocity, V(z), is low, the tethered chains would be a little
stretched and they would be still entangled with untethered chains. However, the tethered
chains will be stretch much more and will be disentangled from the untethered chains
larger V(z). The friction coefficient of a repeating unit of the tethered chains is given by
[76] :
 N3 
,
2 
 Ne 
 1  a 
(2.10)
where a is the monomer size. The force (f) imposed on an untethered chain entangled
with the tethered chain is related to η and V. f is given by
f  aV .
(2.11)
Assuming that X untethered chains are entangled with a tetethered chain, the force
imposed on the tethered chain will be
F  Xf .
(2.12)
However, entanglements with untethered chains will be lost at higher V (Figure 2.3).
Therefore, F starts to follow behavior characteristic of the Rouse regime :
F  NV .
38
(2.13)
The interfacial friction behavior of tethered chains with respect to grafting density,
σ, has been observed using Lateral Force Microscopy (LFM) (Figure 2.4) [77]. The
friction coefficient per tethered chain was measured using scanning probe microscope
equipped with a bead probe which provides constant shear stress on the layers of tethered
chains at fixed V. The σ range was between 0.04 chains/nm2 and 4 chains/nm2. The
transition of the tethered chain conformation from mushroom to true brush regime caused
a sharp decrease in the friction coefficient. This reason for this phenomenon was
understood to be the formation of an uniform morphology at the larger σ. The measured
friction coefficient was in the range between 0.001 and 0.01.
z
V(z)
b
Figure 2.2: The schematic of shear flow of polymer melt on the surface. The slippage
length, b is a means of quantifying the degree of slip at the wall. The relation between b
and the degree of slip is given by b  V ( z )
dV
. The shear stress is the same at all z > 0.
dz
39
(a)
(b)
Figure 2.3: The entanglement behavior of tethered chains with respect to V. (a) The
chains are entangled with an untethered melt at low V, but (b) disentanglement and
stretching will happen when the shear stress exceeds a threshold value.
Figure 2.4: The schematic of lateral force microscopy (LFM) set up to measure tethered
chain friction. Reprinted with permission from L.J.T. Landherr, C. Cohen, P. Agarwal and
L. A. Archer, “Interfacial Friction and Adhesion of Polymer Brushes”, Langmuir 27, 9387
(2011). Copyright 2011 American Chemical Society.
40
2.3. Conformation of Tethered Chains
De Gennes has provided theories for tethered chain conformations depending on σ
when the tethered chains are mixed with a good solvent or a polymer melt [38]. First of
all, let us discuss tethered chains in good solvent. The hydrodynamic size of a linear
untethered chain, R, follows a power law given by
R  aN m ,
(2.14)
where m is 3/5 in a good solvent and is 1/2 in a theta solvent [78]. Assuming that the
tethered chains are in a solvent, the quality of solvent will solely determine the
conformation of the tethered chains at low σ, but the influence from neighboring chains
would become significant due to the overlap of chain volume at higher σ. The stretching
of tethered chains in a good solvent follows a trend similar to that in layers of untethered
chains, but the effect of σ also needs to be considered. The tethered chain volume fraction
depth profile,  N z  , will follow a power law given by
N z    z a m .
(2.15)
For z < R, m is equal to m = 3/2, but for longer distance as z > R, ϕN(z) drops off very fast.
Figure 2.5 presents the tethered chain segment volume fraction profile with respect
to z. When σ is high enough so that the projections of the chain pervaded volumes overlap
on the substrate surface, the tethered chains will stretch out. Therefore, both the chain
volume overlap and the solvent quality need to be considered. Also, a stretched chain can
be divided into several blobs (Figure 2.5a). Each blob has σ-5/6 repeating units. So long as
z < d where d is the distance between tethering points, the tethered chain segment volume
fraction profile follows Equation 2.15. However, the ϕN(z) will become a constant value
equal to σ2/3 for d <z < L where L is a nominal layer thickness. However, at the longer
41
distances, i.e. for z > L, ϕN(z) will decrease.
(a)
(b)
Figure 2.5: The conformation of tethered chains with high σ in good solvent. The strongly
stretched chains can be described using the blob model as shown in (a). The theoretic
profile of the tethered chain concentration is displayed in (b). Reprinted with permission
from P. G. de Gennes, “Conformations of Polymers Attached to an Interface”,
Macromolecules 13, 1069 (1980). Copyright 1980 American Chemical Society.
42
A theory has also been developed for the case when tethered chains of length N are
mixed with a melt of untethered chains of length P of the same sort of polymer (N and P
monomers per chain, respectively). We consider as our system volume, just the volume of
the tethered chains plus the untethered "inside" the tethered chain layer as shown in
Figure 2.6a.
We ignore for the moment the variations in tethered chain segment density
with depth and consider just the overall tethered chain segment density in the film as a
function of the tethering density. The conformation is determined by two factors, the
elastic energy of a tethered chain, Fel, and the mixing free energy, Fmix. They are given by
Fmix LD 2 1
 3
 p ln  p ,
kT
a P
(2.16)
Fel 3  L2 R02 
,
 

kT 2  R02 L2 
(2.17)
and
where ϕP is the volume fraction of untethered polymer melt. Since Equation 2.17 assumes
mixing between chemically identical chains, R0, which is the size of a linear untethered
polymer in the mixture, is the same as that of a chain in the theta state. Since the sum of
the free energies, Fmix + Fel, is minimized at equilibrium, Fmix is best described as the
contribution to the overall free energy due to the entropy of mixing at equilibrium. The
relationship among ϕN, P and σ is given by
,
(2.18)
where k is the Boltzmann constant. The variation in concentration of tethered chain
segments in the brush with σ is presented in Figure 2.6b. DeGennes argues that there are
three regimes of behavior for the conformations of the tethered chains. These regimes are
marked on the Figure 2.6b. Since they are not apparent from that figure alone, for a low
43
value of σ such that σ < PN-3/2(regime i), the tethered chains have good miscibility with
the untethered chains without any stretching. For a moderate value of σ where σ is in the
range PN-3/2 < σ < P-1/2 (regime ii), the overlap of the projections of chain volumes on the
substrate starts and the tethered chains will be stretched. In spite of stretching, the
untethered chains are still mixed with tethered chains. Expulsion of untethered chains
happens for a high value of σ such that σ > P-1/2, so that tethered chains significantly
stretch. Ultimately, for σ→1, the tethered chains become completely stretched and the
untethered chains are completely separated from the tethered chains.
44
a)
b)
Figure 2.6: The system volume considered for this study (a) and the dependence of the
concentration of tethered chain segments in a brush, ϕN, on σ (b). The conformation of the
tethered chains can be separated into three regimes i), ii) and iii). Reprinted with
permission from P. G. de Gennes, “Conformations of Polymers Attached to an Interface”,
Macromolecules 13, 1069 (1980). Copyright 1980 American Chemical Society.
45
A model of the tethered chain layer thickness determined by N was provided by
Alexander [79]. The assumption of this model is that a tethered chain layer has a step-like
segment density profile with a constant density throughout the whole brush thickness.
The tethered chain conformation is determined by two free energy terms: Fint, the free
energy from monomer-monomer or monomer-solvent interactions and Fel. The total free
energy is the sum of Fint and Fel. For the elastic energy term, the polymer chains can be
2
considered to be entropic springs with the spring constant, kT Rg . Therefore, the free
energy per chain is given by
,
(2.19)
where ν is the excluded volume parameter. The thickness, h, of the layer of tethered
chains has the relationship with N at the minimum free energy :

h ~ N a 2

13
.
(2.20)
The scaling behavior of tethered chains from this equation is that tethered chains would
be stretched more at higher σ and higher N. Due to chain volume overlap among tethered
chains, stronger dependence of chain size on N is shown in Equation 2.20 than is seen in
the well-known case of chains in good solvent, R ~ aN3/5.
46
2.4. Miscibility between Tethered and Untethered Chains
The theory and experimental findings about the miscibility between tethered and
untethered chains will be dealt with in this section.
2.4.1. Theory for Chemically Identical System
When chemically identical layers of tethered and untethered chains are mixed, the
enthalpic interactions can be assumed to be 0. Therefore, the miscibility between the
layers of tethered and untethered chains is determined by other factors including the sizes
of N, P and σ. For an extreme case, if σ is very large, untethered chains would be
excluded from tethered chains. This critical value of σ for the exclusion is specified by
the inequality [38]
  P 1 2
P  N 
(2.21)
  N 1 2
P  N  .
(2.22)
or
However, untethered chains start to infiltrate into the layer of tethered chains at
lower values of σ. Fleer et al. [80] calculated the dependence of theoretical tethered chain
segment volume fraction depth profiles on P and σ at a fixed N. Figure 2.7A shows that
the untethered chains penetrate all the way to the substrate surface (z = 0) at low P and σ.
However, untethered chains do not reach the substrate at larger σ and P (Figure 2.7B-F).
The case corresponding most closely to the samples of primary interest in this
dissertation research, is that of Figure 2.7 A (low P and low σ).
47
Mixing with untethered chains happens at lower values of σ with the tethered chain
layer conformation in the “mushroom” regime. The elastic free energy of a single chain
in the mushroom conformation is [81]
2
Fel  Rmush
Na 2 .
(2.23)
The total free energy per chain including the entropic excluded volume contribution is
given by
F kT  Rm2 u s hNa 2  a 3 P 1 N 2 Rm3 u s .h
(2.24)
Minimizing the free energy leads to the relationship :
2
Rmush
a 2  N 3 / 5 P 1/ 5 .
(2.25)
As shown by Equation 2.25, Rmush decreases as P increases for a given N. As a result,
for higher M of the untethered melt, the conformations of the tethered chains are
predicted to be collapsed, reflecting the poorer miscibility between tethered and
untethered chains.
Penetration depth, λ, is used as a parameter indicating the miscibility between the
layers of tethered and untethered chains [82, 83]. Penetration means there must be extra
stretching of the tethered chains into the untethered chain layer with an accompanying
sacrifice of elastic free energy. Figure 2.8 displays how λ can be defined for the case of a
simplified ϕN profile in the work of Gay [82]. The height of the tethered chain layer, h,
without any untethered chain is given by
h  aN~ ,
(2.26)
where ~ is a dimensionless number indicating the number of tethered chains per
substrate surface area a2. The ratio,
shows the degree of penetration. The
48
penetration into the untethered melt with high M in the case that P is sufficiently large (P
> P*) is given by:
  1  a~ 1 3 N 1 3
P  P
*

 N 2 3~ 2 3 .
(2.27)
Partial wetting, in which only a fraction of the tethered chains is wet with untethered
chains, will happen at higher value of P. At a lower value of P, the behavior of λ displays
another regime in which λ depends on both N and P. The relationship is given by
  2  a~ 1 NP 1
49
P  P
*

 N 2 3~ 2 3 .
(2.28)
Figure 2.7: ϕN profiles of tethered chains (solid line on left of each plot) adjacent to an
untethered chain layer together with the depth profiles of volume fraction of untethered
chain (right solid line on right of each plot). Here, σ varies from 0.004 to 0.1. P has two
values: 30 and 300 and N remains constant at 600 in all cases. Reprinted with permission
from C. M. Wijmans, E. B. Zhulina and G. J. Fleer, “Effect of Free Polymer on the
Structure of a Polymer Brush and Interaction between Two Polymer Brushes”,
Macromolecules 27, 3238 (1994). Copyright 1994 American Chemical Society.
50
Figure 2.8: The definition of penetration depth (λ) for a simplified ϕN profile. In contrast
to the picture from Alexander’s theory, the composition of the tethered chains gradually
decreases at the interface with the untethered chain layer. Reprinted with permission from
C. Gay, “Wetting of a Polymer Brush by a Chemically Identical Polymer Melt”,
Macromolecules 30, 5939 (1997). Copyright 1997 American Chemical Society.
2.4.2. Miscibility and De-wetting
When a layer of tethered chains are covered with chemically identical untethered
chain melt on the top, the wetting/de-wetting behavior of the untethered melt manifests
the miscibility between the layers of tethered and untethered chains. If the untethered
chains cannot wet the brush, a melt layer of untethered chains will decompose into
droplets. This is"de-wetting". The transition between wetting and de-wetting has been
studied a great deal using both theory and simulation. The theoretical considerations can
be framed using the free energy of the system consisting of tethered and untethered
chains. Shull argued that the driving force for de-wetting is to keep a certain h which is
equivalent to the absolute minimum of Helmholtz free energy [84]. Considering this
assumption, the free energy was simulated as a function of N and P. Two regimes of
51
behavior are exemplified by the generic results plotted in Figure 2.9 For P/N < 1, the
Helmholtz free energy (H) decreases monotonically without becoming less than 0.
Therefore, miscibility and wetting can be maintained. However, for P/N > 1 the free
energy curve has a minimum for a specific value of brush height. To maintain the height
at this value requires that the untethered chains de-wet from the surface.
H(h)/Rg
+
0
h/Rg
Figure 2.9: Simulation of Helmholtz free energy, H, vs. h of the tethered chain layers. Rg
is the radius of gyration of the tethered chains. The way in which H changes with h is
determined by the chain length ratio between the untethered and tethered chains, P/N.
When P/N > 1 (red line) the curve shows an absolute minimum, while for P/N = 1 (blue
line) the curve has a monotonic decrease without any absolute minimum. This figure was
adapted from [84].
52
At a particular value of M, varying σ can cause a wetting/de-wetting transition.
MacDowell et al. [85] predicted trends wetting with respect to σ. At low σ, de-wetting
was expected, but for intermediate σ, the untethered melt would wet the tethered chain
layer. However, for higher σ, an autophobic de-wetting was expected.
The wetting/de-wetting transition has been investigated experimentally as well.
Experiments showed the transition is also determined by h of the tethered chain layer and
σ. Voronov et al. [31, 86] prepared PS tethered chains on planar silica surfaces using free
radical polymerization with σ < 0.2 chains/nm2. An untethered chain melt was coated on
the top of the tethered chain layer and the sample was annealed at 155 °C. When h of the
tethered chain layer was less than 2 nm, de-wetting was seen, but for h of 20 nm – 30 nm,
the surface showed wetting. However, for h over 60 nm, de-wetting happened again. The
de-wetting at the low value of h was thought to originate from the unfavorable interaction
between the polymer melt and the substrate. In addition, since a thicker tethered chain
layer could have higher σ, the thicker tethered chains also provided unfavorable mixing
with untethered chains on top due to smaller conformational entropy.
The problem of the wetting/de-wetting transition of a polymer liquid on a tethered
chain layer can alternatively be considered from the viewpoint of the interfacial tension.
Generally, interfacial tension, γ, can be decomposed into two terms
  H  TS ,
(2.29)
where ΔH and ΔS are the differences in the system enthalpy and entropy between a state
in which two phases are infinitely far apart and a state in which they are in intimate
contact. The surface tension of a pure organic liquid on an inorganic substrate has been
found to decrease with increasing T [87 - 89]. This means that ΔS is positive. Using a self
53
consistent field theory, Matsen provided a description for the interfacial tension between
tethered chains and an untethered homopolymer liquid at high σ [90]. This theoretical
prediction was experimentally tested by Khanna et al.
 TU 
3kT
 TSTU ,
8a
(2.30)
where ΔSTU is the excess interfacial entropy between the tethered and untetehred chains.
Interestingly, according to Khanna et al.’s work, ΔSTU was negative [91], resulting in
increased interfacial tension, γTU, at higher T. In addition, γTU depended on λ. Lower M
and lower σ results in deeper penetration, corresponding to smaller interfacial tension.
The variation in miscibility between tethered and untethered chains with M has
been studied, as well. Sharma et al. [92] prepared PS tethered chain layers with σ of 0.22
chains/nm2 and observed de-wetting when chemically identical layers of untethered
chains were coated on the top. De-wetting was observed when P was larger than the
critical value P = 0.5N. The significant feature of this work was that the wetting/dewetting transition was observed by varying only M while σ was almost constant.
The miscibility for a wide range of M and σ was researched by Mass et al. [33].
The σ of the PS tethered chain was varied between 0 chains/nm2 and 1.2 chains/nm2, and
the M range of the PS untethered melt was from 0.1 kg/mol to 100000 kg/mol. The
annealing condition was 12 days at 145 °C and the topology after the annealing was
measured with an atomic force microscopy (AFM). Figure 2.10 shows the wetting
diagram. Complete wetting appeared only at intermediate M and σ. However, for σ
smaller than 0.2 chains/nm2, complete wetting did not occur, indicating the allophobic
behavior between the untethered melt and substrate. This trend is consistent with the
results from Voronov and Shafranska [31, 86]. Furthermore, at higher σ, wetting
54
depended on M of the untethered chains with autophobic dewetting appearing for σ > 0.5
chains/nm2. This agrees well with the theory suggesting that the miscibility is lower for
higher P/N .
The results discussed so far have dealt with the relationship between miscibility
and the wetting/de-wetting transition only by observing sample topographies. However,
Uğur [58] probed interface width as it varies with the miscibility. The interfaces between
the layers of tethered and untethered chains were observed using neutron reflectivity
(NR). Layers of hydrogenous tethered PS (hPS) were prepared by Atom Transfer Radical
Polymerization (ATRP). The untethered chain layers were dueterated PS (dPS) and were
spun cast on the top of the tethered chain layers and then the "bilayer" annealed. Two
cases of σ were considered: 0.5-0.6 chains/nm2 and 0.2 chains/nm2. It was experimentally
shown that the interfaces at the lower σ were 2 - 3 times wider than those at the higher σ.
Furthermore, the M dependence of the interface was observed. For 2.2 kg/mol untethered
chains, the interface was 2.5 times wider than for 40 kg/mol. However, the M effect on
the width became smaller at the higher σ.
Preparation of polymer nano-composites is one application for which the
miscibility of tethered chains with untethered polymer matrix has been extensively
studied. The performance of the nano-composite is determined by the dispersion of
inorganic filler in the polymer matrix. The interfacial energy between the particle and
polymer should be minimized to achieve a fine dispersion. Therefore, the particle surface
needs to be modified with tethered chains [93]. The contribution of the tethered chain to
the particle dispersion has been judged by observing wetting/de-wetting on the surface of
the particle or by studying dispersion/aggregation of the particles using Transmission
55
Electron Microscopy (TEM). If σ is fixed, the value of P/N is an important parameter for
dispersion. Generally, as P increased the mixing entropy decreased. P/N ~ 1 was reported
as the critical ratio for aggregation [94, 95]. Aggregation was seen for values of P/N > 1.
Figure 2.10: Wetting phase diagram of a PS untehered layer on a PS tethered chain layer
with various N and σ. The annealing condition was 12 days at 145 °C under reduced
pressure. The initial h of the total PS layer was 5 ± 0.1 nm. Reprinted with permission
from J. H. Maas, G. J. Fleer, F. A. M. Leermakers, and M. A. Cohen Stuart, “Wetting of
a Polymer Brush by a Chemically Identical Polymer Melt: Phase Diagram and Film
Stability”, Langmuir 18, 8871 (2002). Copyright 2002 American Chemical Society.
56
2.4.3. Flory-Huggins Theory for Chemically Different System
A binary mixing means mixing of two polymer species, let us say A and B. The
change in the entropy and energy due to the binary mixing will be dealt with in this
section. The Flory-Huggins theory assumes that each of the two polymers in the blend is
in a single amorphous state of aggregation and the mixing can be described by
envisioning the segments sitting on a lattice. The volume fraction of each component, i ,
is given by
A 
VA
VB
and  B 
 1  A ,
VA  VB
VA  VB
(2.31)
where VA and VB are the volumes occupied on the lattice by chains A and B. Therefore,
the relation between the number of total lattice sites, n, and the volume of a repeating unit,
v0, is as follows if the volumes of the repeating units of the two chains are the same:
n
V A  VB
.
v0
(2.32)
VA and VB can by calculated by
VA  N Av0 and VB  N B v0 ,
(2.33)
where NA and NB are the numbers of repeating units in the two polymers. The entropy, S,
of the system is the product of the Boltzmann constant, k, and the logarithm of the
number of ways to arrange the molecules on the total lattice sites, Ω.
S  k ln  .
(2.34)
When mixing one A chain in a melt of pure B chains, the entropy change with mixing, Δ
SA, is
57

S A  k ln  AB  k ln  A  k ln  AB
 A

 n 
  k ln 
  k ln  A .

 n A 
(2.35)
Since  A is the same or less than 1, ΔSA is always positive or zero. The total change in
entropy upon mixing is the sum of the entropy changes of all of the molecules. Thus, the
total entropy change, S mix , can be expressed as follows
Smix  nAS A  nB S B  k (nA ln  A  nB ln B ) .
(2.36)
Since n A  n A N A and nB  nB N B , the mixing entropy per lattice site, S mix is
S mix 


 

S mix

1
 k  A ln  A  B ln  B   k 
ln  
ln 1    . (2.37)
n
NB
NB
 NA

 NA

Also, if ϕA is equal to ϕ then, ϕB becomes 1 - ϕ. If component B is a low M liquid such as
a solvent, NB is 1. Therefore, S mix becomes


S mix  k  ln   1    ln 1    ,
N

(2.38)
where N=NA.
The change in enthalpy upon mixing is determined by the interactions between
pairs of monomers. Therefore, it is composed of the change in interaction energy between
identical monomers (uAA and uBB) and between different monomers (uAB). The average
interaction energy of monomer A with its one neighbor, UA, is composed of the
contributions from the volume fraction of each species as shown by
U A  u AA A  u ABB
and U B  u AB A  uBBB .
(2.39)
To calculate the interaction energy at lattice sites, the coordination number, z, which
indicates the number of neighboring sites contacting with a lattice site, must be
considered. In a 2- dimensional square lattice, z is 4 and in a 3-dimensional cubic lattice,
58
z is 6. The total interaction energy of all lattice sites in the system is
U
zn
U A A  U BB  .
2
(2.40)
Since it is assumed that two monomers are involved in each interaction, the interaction
energy has to be divided by two. With  A   and B  1   , the energy of the binary
mixture is given by
U


zn
u AA  u AB 1     u AB  u BB 1   1     zn u AA 2  2u AB 1     u BB 1   2 .(2.41)
2
2
No interaction between A and B needs to be considered before mixing. That is the initial
state for our calculation of the change upon mixing is one in which the two pure polymers
are well separated from one another. Therefore, the internal energy before mixing, U0, is
determined by the sum of energy for each pure component. Therefore, U0 is
U0 
zn
u AA  u BB 1    .
2
(2.42)
The energy change with mixing, U mix , is
U mix  U  U 0 
zn
 1   2u AB  u AA  u BB  .
2
(2.43)
The energy change per lattice site is therefore
U mix 
U mix z
  1   2u AB  u AA  u BB  .
n
2
(2.44)
The Flory-Huggins exchange interaction parameter, χ, is defined as the change in the
interaction energy for a single repeat unit when a contact between like repeating units is
replaced with a contact between different repeating

z 2u AB  u AA  u BB 
.
2kT
59
(2.45)
  0 indicates that the internal energy increases after mixing the two components.
Therefore,   0 means the interaction between repeating units of two different types is
less favorable than the interaction between repeating units of the same type.   0
means that mixing is energetically favorable. In addition, χ shows a T dependence. It can
be written as
 T   A 
B
.
T
(2.46)
The T independent part, A, is the entropic part and T dependent part, B/T, is the enthalpic
part.
The Helmholtz free energy of mixing per lattice site,  H mix , has two terms :
interaction energy and entropy per lattice site, U mix and S mix , respectively and then
 H mix is given by
 

1
 H mix  TS mix  U mix  kT 
ln  
ln 1      1    . (2.47)
NB
NA

Positive free energy of mixing indicates that mixing at that composition does not occur
spontaneously and negative free energy of mixing indicates a spontaneous mixing. Since
  1, the entropic part is always negative, indicating that the entropy always increases
with mixing. To obtain negative free energy, the degree of polymerization needs to be
small and χ needs to be sufficiently small.
60
2.4.4. Theories for Partially Tethered Layers
Bilayers composed of immiscible untethered polymers still form an interface of
finite width. Helfand et al. [96] developed a theory that predicted that in the limit of
infinite chain size (N → ∞) the characteristic width of an immiscible bilayer interface, aI ,
depends on χ as given by
aI 
2a
.
6
(2.48)
However, for chains of finite length, not only χ, but also the length of each polymer is an
important factor determining the interface width. Boresta et al. theoretically predicted aI
to be given in this case by [97]
2
aI 
6
 a A2  a B2

 2
12
 
 1
1
 1  ln 2

 
 N A  AB N B  AB

 ,

(2.49)
where aA and aB are the repeating unit size of each polymer.
For mixing between chemically different tethered and untethered chains, the
contribution from χ needs to be added to deGennes’ theory. The theoretical expectation
for the variation of the free energy of mixing, Fmix, with χ was given by [98]
Fmix hD 2 P
 3 ln  P  N P ,
kT
a P
(2.50)
where h is the height of the tethered chain layer. In the context of the Flory-Huggins
picture χ is necessarily always positive, but we may allow for a generalized
understanding of χ in which the case of specific favorable interactions between repeating
units of two kinds result in χ less than 0. In this case, to increase contact with the
untethered chains, the tethered chains will be stretched. In other words, smaller χ would
give larger h. Also, as discussed above, σ determines the conformation and h of the
61
tethered chain layer. Therefore, for a negative χ, the relationship among χ, h and σ is
given by [98]
13
 
13
h~
 Na .
2


(2.51)
2.4.5. Experimental Results
For a polymer solution, χ is determined by the quality of interaction between a
polymer segment and a solvent molecule and by T. When a tethered chain is wet with a
solvent, the favorability of solvent with the tethered chain determines the stretching and
swelling of the tethered chain. The collapse/swelling behavior of the tethered chains due
to the solvent quality was observed by Gutmann et al. [99]. Solvent mixtures containing
different ratios between the volume of good solvent (THF) and the volume of poor
solvent (methanol, MeOH) were spread on the top of PMMA tethered chain layers to see
the variation in the chain conformation with solvent quality. The χ calculated for ternary
system was as follows
  THF PMMA bulk THF    MeOH PMMA 1   bulk THF    MeOH THF  bulk THF 1   bulk THF  ,(2.52)
where  bulk THF  was the volume fraction of THF in the solvent, χTHF/PMMA was the χ
between THF and PMMA and χMeOH/PMMA was the χ between methanol and PMMA and
χMeOH/THF was the χ between methanol and THF. Using NR, the h of the tethered chain
layer was observed with different mixtures of solvents. Mixtures with a higher fraction of
good solvent [larger  bulk THF  ] resulted in higher values of h. Also, since the upper
critical solution temperature (UCST) includes phase separation at lower T, the pair of
solvent and tethered chain having UCST behavior will have a sudden collapse of the
62
tethered chains at T < Ttr where Ttr is the critical temperature. Also, tethered chains will
experience a sudden conformation change in a lower critical solution temperature (LCST)
system when T increases above Ttr. This “thermoresponsive” behavior of tethered chains
can be the basis for expanding their application to thermal sensors (Figure 2.11) [99 101].
Figure 2.11: Schematic of thermoresponsive sensor prepared with polymer tethered
chains. When layers of tethered chain are mixed with a solvent, the value of χ depends on
T. For LCST behavior, phase separation will happen above the transition temperature, Ttr,
resulting in a sudden change in tethered chain conformation. Reprinted with permission
from X. Laloyaux, B. Mathy, B. Nysten and A. M. Jonas, “Surface and Bulk Collapse
Transition s of Thermoresponsive Polymer Brushes”, Langmuir 26, 838 (2010).
Copyright 2010 American Chemical Society.
63
A block copolymer tethered chain can be prepared via living polymerization. The
favorability and unfavorability of the solvent for each block can determine the
conformation and topology of the tethered chains. Zhao et al. [102 - 105] prepared a layer
of polystyrene-b-poly(methyl methacylate) (PS-b-PMMA) chains with the PS blocks
tethered to a silicon wafer and observed the variation of morphology with AFM under
various solvents. According to the AFM data, the surface roughness increased and chains
were aggregated after treatment with cyclohexane. The block for which cyclohexane was
an unfavorable solvent, PMMA, was collapsed to minimize contact with the cyclohexane,
while the PS chains extended due to better miscibility with it (Figure 2.12).
64
(a)
(b)
(c)
Figure 2.12: The morphology and conformation changes of polystyrene-b-poly(methyl
methacrylate) (PS-b-PMMA) tethered chains under different solvents. (a) CH2Cl2 is a
good solvent for both blocks and the AFM image shows a smooth surface after treating
with CH2Cl2. (b) When treated with cyclohexane, the PMMA block will aggregate to
reduce contact with the poor solvent. And then the surface will become rougher. (c) A
cartoon of a possible conformation change process due to different solvent qualities.
Reprinted with permission from B. Zhao, W. J. Brittain, W. Zhou and S. Z. D. Cheng,
“AFM Study of Tethered Polystyrene-b-poly(methyl methacrylate) and Polystyrene-bpoly(methyl acrylate) Brushes on Flat Silicate Substrates”, Macromolecules 33, 8821
(2000). Copyright 2000 American Chemical Society.
65
The interfacial width between immiscible homopolymers or between the domains
in an ordered block copolymer depends on χ, which describes the miscibility of the two
homopolymers or two blocks. Higgins et al. [106] prepared a homopolymer bilayer of PS
and PMMA having positive χ. The interfacial width between PS and PMMA measured
with NR after annealing above Tg was around 2 nm. Block copolymers having blocks
which are immiscible can form various ordered structures in the melt. PS-b-PMMA block
copolymers of nearly symmetric composition can form lamellar structures and if the
lamellae in a film are parallel to the substrate the interfacial width can be measured with
NR. Anastasiadis et al. found that the interfacial width between the layers composing the
lamellar structure decreased with increasing χN where N is degree of polymerization. A
similar trend was reported by Foster et al. [107] for another diblock copolymer system.
The chains of one block can be thought of as being tethered in the domain rich in the
other block. This concept was further tested by Bucknall et al. [109] using blends of
PMMA homopolymer with PS-b-PMMA block compolymer. The PMMA blocks were
extended into regions rich in PMMA homopolymer while the PS blocks were collapsed in
PS domains. Since this structure can be viewed as extended chains grafted to aggregated
blocks, the PMMA blocks showed the properties of end tethered chains [108, 109].
For positive χ, the interface between two polymers of a bilayer is narrow. As a
result, the surface dynamics of an immiscible bilayer do not display an average behavior
of the total bilayer, but rather the dynamics of each interface. Using X-ray photon
correlation spectroscopy (XPCS), Hu et al. [110] measured the surface and interface
fluctuations of a bilayer containing an untethered linear PS layer on top of a layer of
poly(4-bromo styrene) (PSBr). As Figure 2.13 presents, the relaxation behaviors for the
66
surface region and for the interface between the two polymer layers could be obtained
separately using two different incident angles. The top surface showed both fast and slow
modes of surface relaxation with two relaxation times, while the buried interface showed
a slow mode only. Considering that the relaxation time of the slow mode of the surface is
similar to that of the buried interface, there might be a capillary wave fluctuation
coupling between the PS layer surface and the buried interface.
Just as low M solvent can cause swelling of tethered chains, sufficiently short
untethered chains can cause a swelling of tethered chains and the swelling will depend on
the value of χ between the tethered and untethered chains. Clarke et al. [111] prepared
layers of PS tethered chains mixed with various untethered chain layers on the top. The
interfacial width between the layers of tethered and untethered chains varied with
miscibility and was studied using NR. As the PS miscible untethered chains, poly (vinyl
methyl ether) (PVME) chains were used. PMMA and polybutadiene (PBD) chains were
used as immiscible untethered chains. As shown by the volume fraction profiles of
tethered chain segment in Figure 2.14, the tethered chains were stretched by more than 40
nm when mixed with PVME. However, for immiscible untethered chains, PMMA and
PBD, the tethered chains stretched no more than 20 nm from the substrate. Further
stretching from the substrate means that tethered chains were more swelled due to better
miscibility with the untethered chains.
67
PS
PSBr
Si Wafer
(a)
PS
Si Wafer
(b)
Figure 2.13: The g2 function with respect to measurement time, Δt, measured with XPCS
at 176 °C. The polymer films were (a) a bilayer of PS and PSBr and (b) a single layer of
PS. For the bilayer, the interface relaxation behaviors of the two interfaces were
separately measured. Figure reprinted with permission from X. Hu, Z. Jiang, S.
Narayanan, X. Jiao, A. R. Sandy, S. K. Sinha, L. B. Lurio and J. Lal, “Observation of a
Low-Viscosity Interface between Immiscible Polymer Layers”, Phys. Rev. E 74,
010602(R) (2006). Copyright 2006 by the American Physical Society.
68
Figure 2.14: The segment volume fraction profiles of tethered PS (Mw = 79.8 kg/mol)
from the substrate for various chemical structures of the untethered chain layers were
measured with NR. Further stretching from the substrate means more swelling of the
tethered chains due to better miscibility with the untethered chains. For favorable mixing,
PVME was used. For unfavorable mixing, PBD and PMMA were used. Reprinted with
permission from C. J. Clarke, R. A. L. Jones, J. L. Edwards, K. R. Shull and J. Penfold
“The Structure of Grafted Polystyrene Layers in a Range of Matrix Polymers”,
Macromolecules 28, 2042 (1995). Copyright 1995 American Chemical Society.
69
2.5. State of the Problem
The factors determining the surface dynamics of the partially tethered layers were
hypothesized as the conformation of tethered chains and the miscibility between tethered
and untethered chains. In this chapter, the trends of these factors depending on molecular
weight, σ, and χ were discussed. In Chapters 6 and 7, the tethered chain segment
composition profiles calculated from the reflectivity curves of each sample will give the
comparison between the theoretical pictures and experimental results. Also, the variation
in the surface dynamics determined by the characteristics of the tethered chains will be
discussed.
70
CHAPTER III
X-RAY AND NEUTRON SCATTERING
3.1. Introduction
X-ray and neutron scattering can be used to analyze a variety of materials including
not only crystalline inorganic materials, but also soft matter with amorphous structures
such as low molecular weight liquids, polymer films and organic multilayers. Analysis of
the reflectivity vs. scattering vector curve can provide, in the case of X-ray scattering, an
electron density profile, or, in the case of neutron scattering, information on the
distributions of different atomic nuclei. This data can be used to learn the thickness of the
organic layer, and width and the roughness of interfaces down to atomic scale. Since
microscopy methods give direct image of a sample, data from microscopy are easy to
understand. However, the raw data from scattering are difficult to understand because
they are presented as intensity in reciprocal space. In spite of this weakness, scattering
methods are still widely being used due to several advantages. First of all, altering the
size of the scattering vector allows one to observe various length scales of the sample
from ~1 nm to ~ 100 nm. In addition, while microscopic methods give only local images
of the sample, scattering methods show the statistical average result for the illuminated
volume of the sample, providing a more global picture of the sample. Furthermore, if a
sufficiently low beam dose is used, radiation damage of the sample during the
measurement can usually be avoided. Therefore, non-destructive analysis is available. In
71
this section, basic concepts of specular X-ray and neutron reflectivity (XR and NR), offspecular X-ray scattering and X-ray photon correlation spectroscopy (XPCS) will be
discussed in detail with theoretical models.
3.2. Basic Principles.
When radiation is incident on an interface, part of the radiation energy is
transmitted or absorbed via refraction while another part of the energy is reflected from
the interface. The refractive index, n, characterizes the interaction of the incident
radiation with a material. In fact, the values of n on the two sides of an interface
determine the ratio of the reflected intensity to the refracted intensity at a given incident
angle. For X-rays and neutron beams, n is given by
n 1    i .
(3.1)
2  b 
 ,
2  V 
(3.2)
The real part of n, δ, is as follows

where b is the scattering length of a reference of the material and V is that reference
volume. Therefore, b/V indicates the scattering length per unit volume. In other words, it
is the scattering length density (SLD). The real part is composed of the product of
incident wavelength, λ, and SLD. Mostly, the value of δ is on the order of 10-6 Å -2 for
both X-rays and neutrons. The imaginary part, β, represents the absorption component
and contains the product of the mass absorption coefficient, μ, and λ as given by


.
4
(3.3)
Since a neutron beam is highly penetrating, β is usually negligible, except for a few
72
materials containing neutron absorbing elements such as Li, B, Cd and Gd. In the case of
X-rays, β is only on the order of 10-7 or 10-8 for common organic polymers, but the
absorption increases for elements with high atomic numbers. With CuKα X-ray, β is a
factor of 10-2 or 10-3 smaller than δ.
For XR, the contrast between layers or components is determined by the difference
in SLD [(b/V)X] . A SLD value can be given by the following formula :
b
   r0  e ,
V X
(3.4)
where r0 is the classical radius of an electron (2.82×10-13 cm) and ρe is the electron
density of a materials. XR is sensitive to differences in the values of ρe of the components
and a larger difference in ρe gives more contrast among the components. ρe is
proportional to atomic number, Zi, of element i in the material, and mass density, ρ. The
value of ρe is given by
e 
N A   bi
M
,
(3.5)
where NA is Avogadro’s number, M is the molecular weight of a repeating unit or a
structural unit and
is the total scattering length of the structural unit. This sum is the
same as the summation of all the atomic numbers of elements in the repeating unit (bi =
Zi ).
Zi and ρ are the most important contrast sources for XR, but for NR, variation in
the number of neutrons in an atom's nucleus can give contrast in the SLD, (b/V)n. In fact,
elements which differ only slightly in atomic number and isotopes of a single element
may have large differences in their neutron scattering lengths. In fact, the scattering
length of a hydrogen atom, bH (-3.74×10-15m) has a large difference with that of a
73
deuterium atom, bD (6.67×10-15m). The (b/V)n of a substance can be calculated as follows :
b N A
b
   i 
 bi .
Vi
M
 V n
(3.6)
Just as with XR, (b/V)n is a determined as a summation over a structural unit or a
repeating unit. Here, (b/V)n also shows a dependence on the density. However, in this
dissertation, since all the polymers considered have similar densities of around 1 g/ml,
the density is not an effective factor to distinguishing different components. Therefore,
the difference in bi through deuteration is used to obtain contrast.
3.3. Specular Reflectivity
As shown by Figure 3.1, when radiation is incident on a boundary between two
media having different refractive indices, n0 and n1, both reflection and refraction is takes
place. The two media might be optically homogenous with high densities. However, in
many studies including this research, the radiation is incident upon the interface between
an gas (n0 ~ 1) or a vacuum having n0 = 1 and condensed matter. According to the law of
reflection, the reflection angle, θ, has a relationship with the incident angle, θ0, given by
cos 0  cos .
(3.7)
The refraction angle, θ1, can be calculated using Snell’s law :
n0 c o s 0  n1 c o s 1 .
(3.8)
The wavelength of the radiation in the medium with n is λ/n where λ is the wavelength in
vacuum. As a result, the magnitude of the scattered wave vector, k, becomes 2πn/λ. When
the medium 0 is a vacuum, the refraction angle, θ1, is smaller than θ0 because n1 is less
than 1. As a result, “total reflection” can occur. If θ0 is less than a certain critical value,
74
θc=
(in units of radians), θ1 becomes 0.
z
incident
radiation
k0
reflected
radiation
x
k
y
medium 0
θ0
θ
θ1
medium 1
refracted
radiation
Figure 3.1: The geometry of reflection and refraction of incident radiation. In many
reflectivity measurements, medium 0 is air or vacuum.
As Figure 3.1 presents, the intensity of the incident radiation is split between the
refracted and reflected radiation. The reflectivity, R, by definition, is the ratio of the
reflected radiation intensity to the incident intensity,
.
(3.9)
The wave vector k is composed of two components, kx and kz, where x is parallel
to the interface and z is perpendicular to it. The magnitude of the z component of k as
defined in the vacuum is given by
k z ,0 
2 sin  0

.
(3.10)
The magnitude of the z component of the wave vector describing the refracted radiation
in medium 1, kz,1, can be written as
75
k z ,1  k z2,0  k z2,c ,
(3.11)
where kz,c is defined using the refractive index of medium 1. R is the absolute square of
reflectance, r, and it is given by
Rr r 
*
0 ,1 0 ,1
2
k z ,0  k z ,1
k z ,0  k z ,1
 RF q z ,0 ,
(3.12)
where r* is the complex conjugate of r. This reflectivity from a single ideal interface is
often referred to as the Fresnel reflectivity. Here, q is the scattering vector of the radiation
it is defined as q = k – k0. qz is the z component of the scattering vector. The calculation
of q from k is presented in Figure 1.1.
At high θ0, kz,0 is much larger than kz,c (kz,0 >> kz,c). In this regime R can be
approximated as
1  1  k z ,c k z , 0 
2
R
1  1  k z ,c k z ,0 
2
2
4
1 k 
  z ,c  .
16  k z ,0 
(3.13)
As Equation 3.13 presents, a power law is found between R and kz,0
R ~ k z,40 for kz,0 >> kz,c.
(3.14)
The samples for reflectivity of interest here are thin films with a thickness, h,
supported on a substrate. If the film is sufficiently thin and the absorption coefficient is
not extremely high, the refracted radiation will not be completely absorbed in the sample
layer, but will experience another reflection and refraction at the interface between the
substrate and the film. Figure 3.2 demonstrates the reflection and refraction geometry of
such a sample. Since reflection and refraction still take place aton the buried 1-2 interface,
the radiation striking the 0-1 interface is not only approaching from medium 0, but also
from medium 1.
76
medium 0
medium 1
θ0
θ0
θ1
h
θ2
medium 2 (Usually Substrate)
Figure 3.2: The geometry of reflection and refraction in a thin film supported by a
substrate. The refracted radiation from the 0-1 interface experiences another refraction
and reflection at the 1-2 interface. The reflected radiation from the 1-2 interface goes
toward the 0-1 interface and experiences another reflection and refraction. As a result, at
the top of the sample the reflected beam consists of radiation reflected not only from
medium 1, but also from medium 2.
For a substrate supported film of uniform refractive index, R can be calculated as
R
r02,1  r12, 2  2r0,1r1, 2 cos 2k z ,1h 
1  r02,1r12, 2  2r0,1r1, 2 cos 2k z ,1h 
.
(3.15)
Radiation reflected from various interfaces interferes constructively or destructively at a
detector placed far from the sample (far field approximation). As a result, the overall R
profile has a series of maxima and minima. These maxima and minima are called as
Kiessig fringes [112]. As the film thickness, h, grows, the separation of two successive
minima, Δqz decreases.
77
R from an extremely sharp interface has been dealt with so far. However, interfaces
with flatness at the atomic level do not exist in the real world. As shown by Figure 3.3,
height fluctuations, or "roughness", always exist on all surfaces and interfaces. The root
mean squared (rms) roughness, Rq, is one of the most popularly used parameters to
indicate the roughness and it is defined by
L
Rq 
1 2
z x dx .
L 0
(3.16)
Here, L is the evaluation length, z is the height of the surface at position x and the x
direction is parallel to the nominal surface. Reflectivity data manifests the statistical
average of roughness over the footprint of the beam on that sample.
z
x
L
z(x)
z(x’)
Figure 3.3: Schematic of a rough surface. The local surface or interface shows height
variations with respect to the nominal interface height.
Taking the roughness, Rq, into account leads to a different behavior of R with
respect to the scattering vector, q. In general, the reflectivity of a rough interface is lower
than that of a smooth surface.
The exact expression for R in the presence of roughness
78
depends upon details of the kind of roughness present. However, for a common case of
random roughness of typical frequencies for samples of interest here, the reflectivity for
values of q far above qc is given to a good approximation by:
Rq z ,0  


16 2 r02  e2
2
exp  q z2,0 Rq for q z ,0  q z ,c ,
4
q z ,0
(3.17)
where Δρe, is the difference between the electon density of the film and that of the
substrate. Here, the relationship between R and Fresnel reflectivity, RF, is given by


Rqz ,0   RF qz ,0 exp  qz2,0 Rq .
2
(3.18)
As made clear by Equations 3.17 and 3.18, R depends sensitively on Rq. In particular,
reflectivity drops rapidly at high qz,0 and for high Rq. The electron density gradient is a
delta function for an ideal interface with Rq = 0. On the other hand, for a real interface
having roughness Rq, the specular reflectivity can calculated by assuming that the rough,
but locally sharp interface is equivalent to an interface with a ρe profile that has the shape
of an error function. Therefore, ρe(z) across the air and film interface is given by
e z   0  1  0 erf z, Rq  ,
(3.19)
where ρ0 is electron density of the air and ρ1 is electron density of the film. The error
function is defined as
erf z, Rq  
1
Rq
  2 

d .
exp
2 



2
 2 Rq 
z
(3.20)
The geometry of any periodic structure in the direction normal to the surface and
the approximate thickness of any key component layer in the film can be learned from
observation of R(q) without detailed analysis. However, information about an interface,
and in particular, the interface profile including width, roughness and the full SLD profile,
79
only can be obtained from a detailed analysis. The reflectivity is defined in reciprocal
space. Due to loss of phase information in the measurement, directly inverting R to give
the structure of the thin film [113 - 115] is not possible, in general. Therefore, for the best
analysis, a candidate model which can explain R(q) best needs to be suggested first. Then
the model parameter values are varied through non-linear regression until the model
reflectivity agrees with the data very well. Through this “model fit” process, the SLD of
the whole sample can be learned. Figure 3.4 presents an example of fitting. There are
many methods to fit R curves. Here we review just three common approaches. Among
them, two methods calculate the SLD profile using an approximation of the continuous
SLD profile consisting of many discretized layers, where each layer i has given values of
SLDi, ρi and hi as shown in Figure 3.5.
0
Scattering length density
○ : Reflectivity
-1
: Simulation curve from model
Log (R )
-2
-3
-4
-5
Distance from substrate
-6
0.00
0.05
0.10
0.15
0.20
0.25
qz (Å-1)
Raw Data
Fitting data
Figure 3.4: An example of analysis of the reflectivity curve shown at left. The curve is
fitted with the simulated reflectivity from a mathematical model of the sample structure
until the best fit is obtained. The best fit corresponds to the SLD profile on the right.
When there are sufficient constraints on possible physical models the resulting SLD
profile may be assumed to be close to the actual profile.
80
The first calculation formalism is called the Parratt formalism [116]. The Parratt
formalism considers dynamic scattering effects which become more important when qz
gets close to qz,c. In the Parratt formalism, the reflection and transmission coefficient for
each interface in the discretized structure are calculated recursively from the bottom,
film/substrate interface. The reflectivity coefficient can be acquired through Equation
3.12. The general recursive equation is given by
ri 1,i 
r 'i 1,i  r 'i ,i 1 exp 2ihi k z ,i 
1  r 'i 1,i r 'i ,i 1 exp 2ihi k z ,i 
,
(3.21)
where hi is the thickness of ith layer and r′i,i+1 is the reflection coefficient between media i
and i+1. The recursive equation is sequentially solved from the interface layer with the
substrate to the top layer having air/sample interface. The overall R is obtained as the
Scattering length density
modulus of the air/top layer reflection coefficient.
Distance from substrate (z)
Figure 3.5: The SLD profile can be obtained by approximation as a series of discrete
sections.
81
The second method is the optical transfer matrix formalism [117]. This method also
considers the dynamic scattering effect. Similar to the Parratt formalism, this method also
assumes many discretized layers having fixed properties for each layer. The optical
properties of the ith layer are represented using a matrix, Mi. Mi is defined as follows
 m11,i
Mi  
m21,i
m12,i   cosk z ,i hi 
sin k z ,i hi  k z ,i 

.

m22,i   k z ,i sin k z ,i hi 
cosk z ,i hi  
(3.22)
Mi shows how the amplitude and its derivative of the propagating wave (beam) from the
preceding boundary (i-1, i), vary when the wave is reaching the other boundary (i, i+1).
The total reflectance of the sample can be calculated by multiplying together the matrices
of all the discretized layers. If it is assumed that the film can be divided into N discrete
layers, the optical property of the total film, Mtotal, is given by
M total  M1M 2 M N 1M N .
(3.23)
Using elements in Mtotal, the reflectance, r, of the top surface can be calculated as follows:
r
k z ,0 k z , N 1m12  m21  ik z , N 1m11  ik z ,i m22
.
k z ,0 k z , N 1m12  m21  ik z , N 1m11  ik z ,i m22
(3.24)
The third formulation makes use of the Born Approximation. The differences from
other two methods are that, first, the dynamic scattering effect is not considered, and,
secondly, that no discretization is used. In this method, it is assumed that the intensity of
the scattered radiation is negligibly small compared to that of an incident radiation.
Therefore, this formalism can only be used for q > 4qc, where the value of R is small.
Also, the approach works best when there are only small ρe gradients in the sample. The
R averaged over the coherence of the beam is
Rq z   RF q z 
1
s


0
2

dz exp iq z z  z ,
z
82
(3.25)
where ρs is a reference electron density, which is usually ρe of the substrate.
3.4. Off-Specular Scattering
The geometry of one specific sort of "off-specular" scattering experiment is
depicted in Figure 3.6. Here the incident angle defined in the plane of incidence is θi and
the angle of the scattered beam relative to the surface plane is θf. If the incident angle, θi,
is not equal to the scattering angle, θf, q will be composed of not only a perpendicular
component qz, but also a surface parallel (lateral) component, qx. Since the magnitude of
the parallel component is not 0, the lateral structure of the interface can be observed. In
this study, the laterally varying structure of the surface resulting from fluctuation at
temperatures substantially above Tg was studied using off-specular method. For a
perfectly smooth surface, the value of R measured with in the specular mode for θi
under θc is 1. However, in reality, due to roughness on the surface, the R value becomes
less than 1 even at very low angles because the detector is set for θi = θf, but the missing
intensity is due to radiation scattered diffusely at θi ≠ θf. Specular reflectivity always only
provides information on the structure in the direction normal to the surface. Alone it
cannot provide unambiguous information about composition or morphological variation
in the lateral direction. In other words, in the specular method, only information about the
lateral structure averaged over the whole measured area is available. On the other hand,
the off-specularly scattered intensity gives information about the lateral correlations and
in-plane surface fluctuation.
83
qx
ki
qz
q
α
θi i
αθff
kf
Polymer
Layer
Sample Layer
Figure 3.6: Schematic of one type of off-specular scattering experiment.
The electron density of a sample can be 3-dimensionally expressed as ρ(x, y, z). In
addition, the density can be expressed as the sum of two components. The first
component is the nominal density variation with depth, ρ(z), averaged in the x and y
directions and the second one is the fluctuation from this laterally averaged value, δρ(x, y,
z). Therefore, ρ(x, y, z) can be given by
 x, y, z    z    x, y, z  ,
(3.26)
where ρ(z) and δρ(x, y, z) are obtained from the specular and off-specular data,
respectively. The Born approximation is popularly used to calculate δρ(x,y,z).
The Born approximation has been used to describe the scattering from a single
rough interface [118]. This approximation is valid only in a “weak scattering regime”.
For a weak scattering regime, the cross section for the scattered radiation is small.
Therefore, multiple scattering effects can be neglected and only single scattering events
are taken into account in Born approximation. The scattering function dependence on q in
the plane of the surface is S(q) is given by
84
S q  
e 2 exp  q 2 R 2 
z
q z2
  exp[q C X , Y ] exp[iq
q
2
z
x
X  q yY ]dXdY ,
(3.27)
where C(X, Y) is the height-height correlation function with X  x  x' , Y  y  y' . S(q)
is proportional to the scattered radiation intensity at q. The extreme value of C(X, Y) for
an infinite value of X and Y is 0.
lim C  X ,Y   0
(3.28)
XandY
Equation 3.27 has a delta function component corresponding to the specular scattering as
well as the off-specular scattering. For
S q   S spec q   Soffspecq  ,
(3.29)
the specular contribution is
2

 e 
S spec q  
exp  q z2 2  q|| 
2
(3.30)
qz
and the off-specular contribution is
S offspecq  
 e 2 exp  q 2 2 
q
2
z
z
 exp q C L 1exp  iq  LdL .
2
z
l
Here, ql  ql qx , q y  is the lateral scattering vector and L 
(3.31)
X 2  Y 2 . Wang et al. [20],
used Equation 3.31 to fit their off-specular scattering curves from PS thin films. While
other values were fixed, C(L) was varied for fitting to find the cut off scattering vector,
ql,c, of the diffuse scattering curve. C(L) was given by


C L   B 2K 0 ql ,c L2  r02 ,
(3.32)
where K0 is modified Bessel function with adjustable parameters: B and ql,c and r0 is size
of a molecule or a bond. In Figure 1.7, the thin solid lines on each data point were the
fitting using Equation 3.32 to calculate ql,c at each h of the PS layers on etched silicon
85
wafers. Figure 3.7 is the plot of the variation of ql,c with h. As the Figure presents, ql,c
decreases as thickness increases, indicating that the confinement became less severe for
thicker layers.
Figure 3.7: The cut off scattering vector, ql,c, calculated from Figure 1.7 in Chapter 1, was
fitted with respect to h. The solid line is a fit using the power law, ql,c = b/dm, where m is 1
for PS. The broken line is the calculation from capillary wave theory. Reprinted figures
with permission from J. Wang, M. Tolan, O.H. Seeck, S. K. Sinha, O. Bahr, M. H.
Rafailovich and J. Sokolov, “Surfaces of Strongly Confined Polymer Thin Films Studied
by X-Ray Scattering”, PRL 83, 564 (1999). Copyright 1999 by the American Physical
Society.
86
3.5. XPCS (X-Ray Photon Correlation Spectroscopy)
Synchrotron sources have only existed about 50 years, but the strengths of the
synchrotron, such as high intensity, good coherence and variable λ, have resulted in a
dramatic increase of its use in various fields. Also, the design of the synchrotron has been
improved as well. Third generation light sources provide X-ray beams with higher
coherence, expanding the applications. In particular, XPCS is one technique using a
coherent beam and it is a powerful tool to observe the surface dynamics. In this section,
the introduction and basic theories of XPCS will be dealt with.
3.5.1. Coherence
When there is coherence between waves it means that the difference in the phase of
the multiple waves is time constant. To achieve this condition, the frequency of the waves
must be identical. Figure 3.8 shows schematics of coherent and incoherent waves. When
the waves have different λ values, as presented by Figure 3.8b, they must be “incoherent”.
Laser and synchrotron X-ray beams are common coherent light sources.
87
amplitude
time
amplitude
a)
time
b)
Figure 3.8: Schematics of coherent (a) and incoherent (b) waves. The criterion of
coherence is that waves must have the constant relative phase.
3.5.2. Speckle Pattern
A speckle pattern is a random intensity pattern originating from the interference of
multiple wave fronts. Waves with identical λ, but various phases and amplitude can
provide random intensity patterns upon interference. Sutton et al. [119] first observed a
speckle pattern generated from a coherent X-ray beam. The experiment was conducted
with a synchrotron beam irradiating a Cu3Au alloy crystal. The speckle patterns of
organic materials such as polymer films [120] and disordered aerogels [8] were observed
after Sutton’s work.
Since a speckle pattern represents an ensemble average containing information on
the average correlations in the sample. It can provide the information on the sample
which cannot be learned with an incoherent light experiment. If the disordered
arrangement of a sample varies with time, the speckle pattern also varies with time.
Therefore, the speckle intensity variation with time can give information on the sample
88
dynamics.
Several conditions must be met to obtain speckle patterns. First of all, the
transverse coherence length of the beam, ξt, should be equal to or greater than i) the beam
size and ii) the length scales of the inhomogeneities of the sample to be studied. The
value of  t is given by
t 

2 

D
2s
,
(3.33)
where λ is of the beam and  is the angular source size with   s D . s is the
monochromatic disk source size and D is the distance from the source. The second
condition is that the sample should be illuminated coherently. In order to obtain coherent
illumination, the maximum path length difference (PLD) needs to be equal to or less than
the longitudinal coherence length, ξl. ξl is given by
  
.
  
l   
(3.34)
Monochromaticity of the beam source determines  . PLD is given by
PLD  2 sin 2  or PLD  2h sin 2   d sin 2 ,
(1.35)
where μ is the absorption length, d is the beam size and Θ is the scattering angle [120, 8].
Scattered coherent light from a disordered system can form a speckle pattern. The total
intensity of the speckle pattern is either the square of total field, E, or the sum of intensity
from each speckle point given by
I q, t   E q, t  
2
 exp iq  r t  f q 
n
n
2
,
(3.36)
n
where fn(q) is the amplitude from the n-th scattering point and the location of the point is
89
described by rn(t). In reality, the measured intensity, I q, t  T , is the time average during
the acquisition time, T. If the sample is non-ergodic, in other words, if the system has
static random disorder, I q, t  T can be displayed as a function of q and the distinct and a
sharp change of intensity for each position, that is a speckle pattern, will be present. On
the other hand, in an ergodic system, where the fluctuation time is far shorter than the
counting time, I q, t  T can be considered to be the same as the ensemble average,
I q, t  .
3.5.3. Principles of XPCS
Figure 3.9 illustrates the light energy and measurable scale for each of a variety of
techniques for measuring dynamics. Considering that photon correlation spectroscopy
(PCS) and XPCS use coherent optical sources to see the surface and interior structure of
the sample, these two methods are very similar. However, PCS uses a visible light laser,
while XPCS makes use of X-rays. Furthermore, PCS is not able to measure many types
of samples due to poor transparency and the large λ of visible light does not allow the
study of nano-scale dynamics. However, since XPCS uses λ ~ 1 Å , the nano-scale
movement of the surface can be observed. Also, the use of X-rays enables measurement
of dynamics within an opaque polymer sample.
The conditions to meet in order to obtain high signal to noise ratio for an XPCS
measurement are as follows [121 - 123]:
1. The scattering volume should be comparable size to the coherent volume.
2. The sample surface should contain some amount of disorder to make the scattering
broad.
90
3. At least one count should exist per pixel in the speckle pattern. This means that a
sufficient count for each correlation time is necessary.
4. To obtain the necessary statistics, the measurement time should be sufficiently long.
Figure 3.9: Frequency, energy, q and length scale of XPCS and complementary
techniques: PCS, Raman and Brillouin scattering, inelastic neutron (INS) and inelastic Xray (IXS) scattering, neutron spin-echo and nuclear forward scattering (NFS). Reprinted
from G. Grübel and F. Zontone, “Correlation Spectroscopy with Coherent X-rays”,
Journal of Alloys and Compounds 362, 3 (2004). Copyright 2004, with permission from
Elsevier.
91
When the arrangement of scattering points varies with time, the speckle pattern
changes correspondingly and the intensity fluctuations of the pixels in the speckle pattern
indicate the dynamics of the sample. If an ergodic system is measured, the time averaged
correlation function can be written as an ensemble averaged time correlation function.
The time correlation function, g2(q,t) of intensity and scattered field is then given by
g 2 q, t  
I q,0I q, t 
I q 
2
 1
 q  E q,0E q, t 
I q 
2
2
,
(3.37)
where β(q) is the setup constant of the instrument. g2(q,t) can be rewritten with the
normalized intermediate scattering function f(q,t) as
g q, t   1   q  f q, t  ,
2
(3.38)
where f(q,t) is
f q, t  
F q, t 
.
F q,0
(3.39)
F(q,t) is as follows :


F q, t   1 Nf 2 q   f n q  f m q  exp iqrn 0  rm t  .
n
(3.40)
m
N indicates the total number of scattering points. Values inside
is ensemble average
and the F(q,0) at t = 0 can be taken as the static structure factor. Assuming that there is
no interaction between particles and molecules within the sample, each particle position
can be considered to be uncorrelated and the sum of cross term m  n in Equation 3.40
is zero, resulting in F(q,0)=1. Displacement happens for free Brownian particles within
the sample with time and the ensemble average of the displacement is given by
r 0  r t 2
92
 6 D0t ,
(3.41)
where D0 is the diffusion coefficient of a free particle. D0 depends on the particle radius,
d, viscosity of surrounding medium, η, and temperature, T. Therefore, D0 is given by
D0 
k BT
.
6 d
(3.42)
Thus, f(q,t) reduces to


f q, t   exp  D0 q 2t .
(3.43)
If any interaction between particles exists, equation 3.42 for D0 is not valid anymore and
the time- and wavevector- dependent diffusion coefficient needs to be considered [8].
93
CHAPTER IV
EXPERIMENTAL
4.1. Tethered Chain Preparation Method
The preparation of tethered chain will be discussed in detailed manner in this
section. The reaction conditions and parameters will be presented.
4.1.1. Materials
98% (3-glycidoxypropyl)trimethoxy silane was purchased from Sigma-Aldrich Co.
and was used without further purification. Hydrogeous polystyrene with one carboxyl end
group(hPS-COOH) with Mn = 28 kg/mol, Mw/Mn = 1.03 was synthesized via anionic
polymerization by Camila Garces and Roderic Quirk and hPS-COOH with Mn = 200 kg/mol,
Mw/Mn = 1.08 was purchased from Sigma-Aldrich Co. 99.8% anhydrous toluene was
purchased from Sigma-Aldrich Co. or from Alfar Aesar and was used without further
purification. Silicon wafers (100), with a thickness range of 0.6
0.05 mm, were
purchased from EL-CAT Inc. Since neutron reflectivity (NR) instruments have a large slit
opening, the size of silicon wafer needed to be at least 60 mm × 40 mm. X-ray photon
correlation spectroscopy (XPCS) and off-specular scattering required smaller samples
approximately 20 mm × 30 mm.
94
4.1.2. Substrate Preparation
Each wafer was tested for planarity and microroughness by measuring an X-ray
scattering "rocking curve" (described below) before being selected for sample preparation.
Selected wafers were then cleaned using piranha solution [124] (a 7v/3v mixture of
sulfuric acid and 30% aqueous hydrogen peroxide) to remove organic impurities from the
surface.
4.1.3. Deposition of Epoxide Functionalized Compound on the Substrates
The reaction mixture was composed of 0.4ml of (3-glycidoxypropyl)trimethoxy
silane and 40ml of anhydrous toluene, and was poured into a 200ml crystallization dish
inside a nitrogen filled dry box. A freshly cleaned silicon wafer was submerged into the
reaction mixture. The reaction proceeded at 55 °C for 16 hr. After the deposition reaction,
the wafer was rinsed with toluene and ethanol followed by sonication in ethanol for 15
min. The thickness (h) of the deposited layer was measured with an ellipsometer using a
single value of n = 1.429 for refractive indices of both the epoxide and the silicon oxide
layers. The average h was 0.8 nm, which was consistent with the self-assembled
monolayer (SAM) h of (3-glycidoxypropyl)trimethoxy silane reported in the
literature[53].
4.1.4. Grafting hPS-COOH
hPS-COOH with Mn = 28 kg/mol or 200 kg/mol was dissolved in toluene to obtain
1.0-1.2 wt% solutions, which were further agitated for 90 min to obtain complete mixing.
The hPS-COOH solution was dropped on an epoxide covered substrate and was spun at
95
500 rpm to obtain a layer roughly 100 nm thick. To promote the grafting reaction
between the epoxide and carboxylic groups, the spun cast samples were annealed in a
vacuum for 24 hr at 175 °C. The un-grafted hPS-COOH chains were washed away with a
toluene Soxhlet extraction until no further decrease in h was observed (24 hr). The h of
the grafted PS was measured with a 632 nm laer ellipsometer (n = 1.589). Grafting
density (σ) of the PS tethered layer was calculated using the equation   N A h M n
where NA is Avogadro’s number and ρ is the bulk density of polystyrene (PS). For 28
kg/mol tethered chains the film h values ranged between 3.6 and 5.3 nm. This
corresponded to σs of 0.08 - 0.12 chains/nm2. h of the layer of 200 kg/mol tethered chains,
was between 5.0 and 7.5 nm, resulting in σs of 0.016 to 0.024 chains/nm2.
4.2. Characterization of Tethered Chains
In this section, the reaction scheme used to prepare the tethered PS layer will be
discussed. Also, using a variety of analysis methods, the characteristics of epoxide layers
and tethered PS layers will be dealt with.
4.2.1. The Reaction of Silane Coupling Agent
Highly ordered, ultrathin coatings with molecular level h, in spite of the very small
h, can modify or improve the properties of the substrate including lubrication,
biocompatibility and adhesion. Spin coating, Langmuir Blodgett deposition and SAM
deposition are the most popular methods for preparing ultrathin coatings [125, 126]. A
silane group is a functional group containing a silicon atom bonded with methoxy, ethoxy
or halide species, which is readily attached to the polar inorganic substrate via hydrolysis
96
(Figure 4.1).
A silane coupling agent contains a silane group at one end and organic friendly
alkyl chains or functional groups at the other end. Therefore, it has been widely used to
increase the stability and the integrity of a polymer/inorganic interface. In the presence of
a functional group on a silane coupling agent, a polymer chain with a functional end
group can be covalently bonded to the substrate [127, 128]. In this study, preparation of
the epoxide SAM using a silane coupling agent was performed to prepare “glue layers” to
make the polymer chains tether onto the substrates.
R'
R'
RO
Si
H+ or OH-
OR
HO
H2O
Si
OH
OR
R'
R'
OH
RO
OH
OH
Si
OR
+
OHH or OH
H2O
+
HO
Si
OH
OH
OR
R'
O
Si
O
O
+
3H2O
Figure 4.1: Schematic of deposition to prepare an ultrathin layer of a silane compound on
a polar inorganic surface.
97
4.2.2. Tethering Process
The ultrathin layer of epoxide has excellent reactivity with other substance like a
molecular glue [53]. As shown in Figure 4.2, due to the thin epoxide layer, a PS-COOH
chain can form a covalent bond on the substrate. The epoxide layer was obtained via
SAM preparation and then PS-COOH chains were spun cast on the SAM. The reaction
between functional groups proceeded at a temperature above the glass transition
temperature (Tg) of the hPS-COOH in a vacuum. Even though an epoxide group is
oxidized to yield two hydroxyl groups, it still reacts with a carboxylic group to form a
covalent bond [54].
98
Figure 4.2: Reaction schemes to prepare covalently tethered PS chains on a silica surface.
First of all, epoxide silane coupling agents form a SAM on the substrate surface. And
then polymers with –COOH end group are coated and react on the substrate. Finally, after
removing un-reacted chains, only tethered chains remain. Reprinted with permission from
S. Minko, S. Patil, V. Datsyuk, F. Simon, K. J. Eichhorn, M. Motornov, D. Usov, I.
Tokarev and M. Stamm, “Synthesis of Adaptive Polymer Brushes via “Grafting To”
Approach from Melt”, Langmuir 18, 289(2002). Copyright 2002 American Chemical
Society.
99
4.2.3. Characterization
The ultrathin layer of (3-glycidoxypropyl)trimethoxy silane on the silicon oxide
was prepared as described in the previous section. The h of the layer, measured with an
ellipsometer, was between 8 Å and 10 Å . However, in spite of such small h, as displayed
in Figure 4.3, the variation in water contact angle before and after the deposition of the
layer was large. Due to hydroxyl groups, the silica surface after piranha solution cleaning
becomes very hydrophilic. Therefore, in Figure 4.3a good wetting with water is seen, but
after the layer was deposited, the surface became hydrophobic. In Figure 4.3b, the
average water contact angle was 50˚.
The surface morphology of the epoxide functionalized layer was observed with
AFM tapping mode as presented by Figure 4.4. The root mean squared roughness, Rq, of
the surface was calculated as 1.12 nm. The silane coupling agent was evenly spread on
most of the substrate area, but showed aggregation in some locations. The maximum
height of the aggregation was 15 nm. These aggregates are thought to be due to the high
reactivity of the silane groups. Polymerization among the silane groups could happen in
the presence of a trace amount of water.
100
a)
b)
Figure 4.3: Water contact image on a piranha solution treated silicon oxide (a) and on a
(3-glycidoxypropyl)trimethoxy silane deposited surface (b). The contact angle was
increased due to the ultrathin epoxide functionalized layer.
101
8.0nm
10μm
0μm
-2.6nm
a)
15
10
5
nm 0
0
2
4
6
8
10 μm
0
2
4
6
8
10 μm
15
10
5
nm 0
b)
Figure 4.4: AFM image of a (3-glycidoxypropyl)trimethoxy silane ultra thin layer on a
silica surface. a is the top view of the surface and b is the cross section view.
102
The surfaces of both the epoxide functionalized layer and the tethered chain layer
were characterized with a PHI 5000 VersaProbeTM X-ray Photoelectron Spectrometer
manufactured by Ulvac-PhI Inc. using an AlKα (1486.7 eV) source. Multipak ver. 9.0.0
was used as the analysis software. Pass energy for the survey scan was 93.9 eV. For the
C1s scan, it was 23.5 eV. The takeoff angle was 45o. Figure 4.5 presents the X-ray
photoelectron spectroscopy (XPS) survey scan results and the corresponding elemental
analyses from the peak analysis are given in Table 4.1. In the scan for the epoxide
functionalized layer the silicon and oxygen peaks are prominent because the sampling
depth (ca. 50 Å ) exceeds the h of the functionalized layer. In the literature [53], the
elemental composition (Si/O/C) of epoxide layer was between 1/0.99/0.39 and 1/1.2/0.68.
However, in this study, the composition was 1/0.93/0.13. Possible reason for such
difference in atomic ratio would be electron penetration depth of the XPS. The presence
of the 55 Å thick PS tethered layer is evidenced in the spectrum for that sample by
several differences from the spectrum of the epoxide functionalized layer. The C1s peak
has become very prominent, while the only one oxygen peak can be seen and that is quite
weak.
103
PS Tethered Layer
Epoxide Fuctionalized Layer
Figure 4.5: XPS survey scan data for the epoxide functionalized SAM and the 28 kg/mol
hPS tethered chain layer.
Table 4.1: Element analysis from the XPS survey scan.
Sample
Epoxide functionalized
Layer
PS Tethered Layer
Si2s
Atomic %
C1s
O1s
48.5
6.3
45.2
5.7
89.3
5.0
104
A high resolution XPS scan for C1s peaks is presented in Figure 4.6. Since oxygen
has higher electronegativity than carbon, the -C-O- bond has higher bonding energy than
the –C-C- bond. (3-glycidoxypropyl)trimethoxy silane contains –C-O- bonds and more –
C-O- bonds could be made when the epoxy groups are oxidized. Therefore, peaks or
shoulders for higher energy on the C1s scan indicate that the sample had more epoxy
compound, but less PS. Table 4.2 presents the binding energy of various carbon bonds.
The PS tethered layer showed a slight shoulder for higher bonding energy, but most of the
peak was located between 284 and 286 eV in which the range is equivalent to CxHy and –
C=C- bonding energy. The epoxide layer showed distinct shoulders at 286 – 288 eV
indicating that the amount of alkyl group is far less than that of the PS tethered layer.
105
Binding Energy (eV)
a)
Binding Energy (eV)
b)
Figure 4.6: XPS high resolution C1s scan for a PS tethered layer (a) and an epoxide
functionalized layer (b).
106
Table 4.2: Binding energy of various carbon bonds.
Bonding
Energy
(eV)
284.8
285.0
286.3
286.7
288.2
Component
CxHy
-C=C-
-C-O-Cand
-C-OH
-C-O
-C-OH
C in epoxy
group
The 28 kg/mol hPS tethered chain layer was measured with X-ray reflectivity (XR),
(λ = 1.54 Å ) to characterize the interface with the substrate and the general morphology
of the layer. The X-ray generator for the XR was produced at Rigaku Inc. An example
XR curve is shown in Figure 4.7a together with a model fit. The scattering length density
(SLD) of the sample was inferred from the XR data by fitting the data to a model
reflectivity curve using Igor Pro version 6.2.2.2 with the data analysis module Motofit
(ANSO). The SLD of the silicon substrate was taken to be 20.1×10 -6 Å -2 and the values
assumed for the silicon oxide and PS were 18.9×10-6 Å -2 and 9.9×10-6 Å -2, respectively.
The SLD profile is shown in Figure 4.7b. On the far right is the silicon.
107
0
Experimental
-1
Reflectivity
Model
Fitting
Log(Reflectivity)
-2
Fit
-3
-4
-5
-6
-7
-8
0
0.05
0.1
0.15
0.2
0.25
0.3
qz(Å-1)
a)
SLD profile
25
PS
SAM
SiOx
Substrate
SLD (10-6Å-2)
20
15
10
5
 Air/Substrate 
0
0
10 20 30 40 50 60 70 80 90 100 110
z (Å)
z(Å, ←Air,
Substrate→)
b)
Figure 4.7: (a) the XR curve of a 28 kg/mol hPS tethered chain layer and the fit to a
model curve and (b) the SLD depth profile calculated with the model fit.
108
The roughness between the SiOx and Si found to fit the data is 17 Å and then the
roughness used between the oxide and the SAM is 5 Å . Since the oxide is quite thin, the
variations in SLD associated with these two interfaces overlap when laterally averaged
and those two interfaces cannot be clearly distinguished. The epoxide functionalized
SAM has a h of about 12 Å and no interface between this layer and the tethered hPS
chains is resolved since these two components have SLDs that are not too strongly
differentiated. The PS layer extends approximately from the air surface to a depth of 55 Å .
The thickness of the tethered chain layer measured with XR is compared with thickness
values measured with an ellipsometer, using a 632 nm laser beam, in Table 4.3. The
thickness values were obtained using a one layer model with n = 1.59. The ellipsometer
data include thicknesses from five separate points on a tethered PS layer. The average
thickness from the ellipsometer data is 51 Å . At the interface with the substrate, the
density is somewhat low, probably due to mixing with the epoxide functionalized SAM,
which has a lower SLD. The form of the SLD profile at the air interface is unusual for a
PS layer. Comparison with AFM images from such a film, shown in Figure 4.8 and AFM
images from a SAM itself (Figure 4.4), suggests that this shape of the polymer film SLD
is due to bumps on the PS surface. It appears that the low density of the top layer in
Figure 4.7b is due to bumps on the PS surface.
The grafting densities of PS tethered layers are calculated using Equation 1.9 with
a fixed PS density value of 1.05 g/mL. However, since the thickness and molecular
weight measurements include uncertainties, the calculated grafting density values also
have uncertainties. The thicknesses from ellipsometry (632 nm) are always determiuned
using measurements of the thickness at five separate points on the tethered PS layer. The
109
deviation in the thickness value at each point from the average value is less than 1 nm.
Therefore, the uncertainty of the average thickness value is ± 1 nm. The uncertainty in the
molecular weight is assumed as ± 10%. Considering these uncertainties in thickness and
molecular weight, the uncertainty of grafting density is estimated as ± 0.012 chains/nm2
for 28k tethered chains and as ± 0.002 chains/nm2 for 200k tethered chains.
Table 4.3: Comparison of 28k tethered PS layer thicknesses with two measurement
methods.
Measurement Method
Thickness (Å )
X-ray Reflectometry
55
Ellipsometry
50 53 50 49 51
110
5.6nm
10μm
0μm
-2.4nm
a)
b)
Figure 4.8: AFM result of 28k tethered PS surface. (a) the surface image from the top and
(b) cross section images.
111
4.3. Polymer Layer Preparation
Once a tethered layer is parepared, to obtain a partially tethered film, untethered
chains need to be coated on the top of the tethered layer followed by annealing for mixing.
The detailed preparation method and surface anaylsis of partially tethered films will be
discussed.
4.3.1. Substrate Preparation
To prepare partially tethered layers, PS chains needs to be covalently attached to
the substrate surface. The procedure for tethering was described in the previous section.
To prepare reference samples, untethered layers were directly spun cast onto silicon wafer
from which the native oxide had been removed by etching in 1% aqueous hydrofluoric
acid solution for 60 seconds and rinsing with water.
4.3.2. Spin Coating
Untethered polymer chains were purchased from Polymer Source Inc. and were
used without further purification. GPC was used to observe molecular weight and
molecular weight distributions of all polymers. Table 4.4 displays the list of polymers
used for untethered chains. Since all polymers were toluene soluble, they were spun cast
as toluene solutions. To obtain the solution concentration for desired h, solutions with
various concentrations were spun cast on etched silicon wafers to plot concentration vs. h
curves. The rpm was 2000 for all coatings. To prepare partially tethered layers, untethered
chain solutions were spun cast on the top of tethered chain layers and then the polymer
layer was annealed in a high vacuum at the temperature over Tg of the untethered chains.
112
Detailed annealing conditions are displayed in sample description tables presented in
each chapter.
Table 4.4: Untethered chains used for partially tethered layers and for reference layers.
Mn
(kg/mol)
0.9
2.6
9.4
48
120
3.8
3.8
Mw/Mn
Polymer
1.10
1.02
1.01
1.01
1.11
1.10
1.04
Dueterated Polystyrene (dPS)
dPS
dPS
dPS
dPS
Poly (4-bromo styrene) (PSBr)
Poly (cyclohexyl methacrylate) (PCHMA)
4.3.3. Annealing and Surface.
Partially tethered layer having Mn = 200 kg/mol tethered PS mixed with 2.6 kg/mol
or 120 kg/mol untethered PS were prepared by via spin coating a solution of untethered
chain solution on the top of a tethered chain layer. The h of the whole layer measured
with ellipsometer was ca. 40 nm. To check any de-wetting due to the annealing, samples
for AFM were annealed for 8 hrs at 120 °C for 2.6kg/mol layer. For 120 kg/mol layer, the
sample was annealed by two steps: 8hr at 110 °C followed by 8hr at 150 °C. All
annealing was done under high vacuum.
Figure 4.9 is the surface image of a 2.6 kg/mol partially tethered layer measured
with AFM. Peaks with 5 – 10 nm height were observed (Figure 4.9b). However, the Rq
calculated from the topography was 2.2 nm. Therefore, the surface height variation was
still far from complete de-wetting. The partially tethered layer with 120 kg/mol
untethered chain showed Rq = 3.6 nm (Figure 4.10).
113
0μm
20μm
a)
nm 44
33
22
11
0
0
4
8
12
16
4
8
12
16
20
μm
nm 44
33
22
11
0
0
20
μm
b)
Figure 4.9: The surface image of an annealed, partially tethered layer having a Mn = 200
kg/mol hPS tethered chain layer mixed with Mn = 2.6 kg/mol dPS untethered chains. (a)
the image from the top and (b) cross section images.
114
0μm
20μm
a)
nm 94
70.5
47
23.5
0
0
4
8
12
16
20
μm
0
4
8
12
16
20
μm
nm 94
70.5
47
23.5
0
b)
Figure 4.10: AFM image of an annealed, partially tethered layer having a Mn = 200
kg/mol hPS tethered chain layer mixed with Mn = 120 kg/mol dPS untethered chains. (a)
the image from the top and (b) cross section images.
115
4.3.4. Sample Naming Rules
A sample composed of PS only has a 6-digit number as its sample name. The first
2- digit indicates the molecular weight of the used tethered chain as kg/mol. The middle
2-digit indicates the molecular weight of the used untetherd chain as kg/mol. If the
molecular weight is more than 100 kg/mol, h which means “hundred” was used. The last
2-digit indicates the h value of the sample at the XPCS or NR measurement temperature.
For example 2h02 45 means the molecular weight of the tethered chain is 200 kg/mol and
the molecular weight of the untethered chain is near 2 kg/mol with h ≈ 45 nm at the
XPCS or NR measurement temperature. If the untethered chain is not PS, the name of
polymer is inserted between the 4th and 5th digit. For example, 2h03 PCHMA 40 means
that the molecular weight of the PS tethered chain is 200 kg/mol and the molecular
weight of the PCHMA [poly(cyclohexyl methacrylate)] untethered chain is near 3 kg/mol
with h ≈ 40 nm. The h value of a sample with temperature is not constant. The h values
variation with temperature is presented in the Appendix (Figure A.1).
4.4. Radius of Gyration
From the literature, the radii of gyration (Rg) of linear ideal polymers can be
calculated as follows
Rg 
Na2
,
6
(4.1)
where N the number Kuhn segments unit and a is the size of Kuhn segment as 0.68 nm.
For 2.6 kg/mol of dPS the N is
2600 g / mol
 23 .
112 g / mol
116
Therefore, the Rg of 2.6 kg/mol of dPS is 13 Å . Also, for 48kg/mol of dPS the Rg is
calculated as 57 Å . In case of 28 kg/mol of hPS, the N is
28100 g / mol
 270 .
104 g / mol
Therefore, the Rg of 28k of hPS is 46 Å . Using small angle X-ray scattering (SAXS),
Konish et al. [129] measured the Rg of linear, atactic PS in cyclohexane at 34.5 °C (theta
temperature). The Rgs of PS with Mw = 2270 g/mol, 20500 g/mol and 40000 g/mol were
11 Å , 39 Å and 56 Å , respectively. As shown by Figure 4.11, the calculated Rg shows
good match with experimentally determined values.
70
60
Calculated
Measured
R g (Å)
50
40
30
20
10
0
10
100
1000
N
Figure 4.11: The intrapolation of calculated Rg using Eq (4.1) with experimentally
measured value from [129].
117
4.5. NR (Neutron Reflectivity)
Specular NR provides the composition profile of a sample layer in the direction, z,
perpendicular to the substrate surface. Reflectivity is the ratio of the reflected beam to the
incident beam and it depends on the roughness and composition of the sample layer. The
NR data is obtained as the curve of reflectivity vs. scattering vector. Scattering vector, qz,
is given by
qz 
4

sin ,
(4.2)
where λ is the beam wavelength and θ is incident angle.
As Figure 4.12 presents, NR gives data about total film h, the h and composition of
each layer in the sample, the h of interface between layers and the surface and interface
roughness. This information is displayed as a scattering length density (SLD) profile. The
SLD profile was generated by the best model fit of the reflectivity curve using Reflfit
verson 2006.06.22 developed in National Institute of Standards and Technology (NIST).
Since most polymers are composed of carbon and hydrogen, they are hard to be
distinguished in spite of different chemical structures. For NR, selective deuteration is
one of the most popular methods to provide contrast among different polymers in a
sample. In this study, hPS was used for tethered chains and dPS was used for untethered
chains (Figure 4.13). The substrate was a silicon wafer larger than 4 cm × 6 cm and was
large enough to cover the beam’s footprint. If the substrate is too thin, micro bending
could be generated, resulting in poor beam alignment. Therefore, silicon wafers thicker
than 0.6mm were used to avoid bending.
118
The horizontal reflectometer installed in the NG7 cold neutron beam line at the
NIST Center for Neutron Research (NCNR) was used for NR. The schematic of the
reflectometer is in Figure 4.14. The neutron beam wavelength was 0.475nm with 1nm of
depth resolution. To maintain maximum reflectivity, the collimating slit opening and
detector slit opening increased at larger incident angles keeping relative qz resolution
constant as qz qz  0.04.
One might be concerned that some tethered chains are not well grafted to the
substrate, making a blend layer upon mixing with an untethered layer. We offer one piece
of indirect evidence that the polymer layer was tethered. A NR curve of a partially
tethered layer, 2802 12, in which a 28k tethered chain layer was mixed with a 2.6k
untethered chain layer, was fitted with two models : a partially tethered layer model and a
fully untethered blend model. First of all, 2802 12 was fitted with a model assuming a
partially tethered layer. Figure 4.15a presents the fitting result. The SLD value for pure
dPS untethered chains known from other measurements was 5.8 × 10-6 Å -2 and for pure
hPS tethered chains was 1.4 × 10-6 Å -2. Due to tethering, the hPS was not evenly
distributed in the layer, but rather more hPS segments were observed near the substate,
resulting in lower SLD values near the substrate. The value of χ2 for this model was 3 and
there was little discrepancy with the real data. Figure 4.15b presents a comparison
between the reflectivity calculated for a model assuming an untethered blend of uniform
composition and the experimental data. Comparison was made for various SLD values
for the blend layer. The value providing the best agreement between the model curve and
the experimental data was 4.3 × 10-6 Å -2, which was the average SLD value of partially
tethered model. However, even though this SLD value provides the minimum χ2, the
119
model curve in Figure 4.15b does not come close to providing a good match with the real
data. The χ2 value was 93. Clearly the film can not be modeled as a single uniform blend
layer.
Figure 4.12: Reflectivity from various layers in a sample film gives information about h,
composition and roughness.
120
Figure 4.13: Providing higher contrast makes it easier to distinguish different components
in a layer.
Figure 4.14: Schematic of the horizontal reflectometer installed on NG7 beam line in
NCNR.
121
Scattering Length Density (Å-2)
1.E+00
Reflectivity
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
1.E-06
0
0.05
0.1
0.15
6.E-06
5.E-06
4.E-06
3.E-06
2.E-06
1.E-06
0.E+00
0.2
0
qz (Å-1)
20
40
60
80 100 120 140
Distance from Air (Å )
a)
Data
1.E+00
Scattering Length Density (Å-2)
Fitting
1.E-01
Reflectivity
1.E-02
1.E-03
1.E-04
1.E-05
1.E-06
1.E-07
0
0.05
0.1
0.15
6.E-06
5.E-06
4.E-06
3.E-06
2.E-06
Data
Fitting
1.E-06
0.E+00
0.2
0
qz (Å-1)
50
100
150
Distance from Air (Å)
b)
Figure 4.15: (a) The NR curve (○), model fit (
) and the SLD profile calculated from
the model fit of 2802 12. A model assuming partially tethered layer was used for the fit.
(b) The NR curve (○), model fit (
) and the SLD profile calculated from the model fit
of 2802 12. A model assuming untethered blend was used for the fit.
122
4.6. XPCS (X-Ray Photon Correlation Spectroscopy)
Figure 4.16 shows the schematic presenting the scattering on a fluctuating surface
in the XPCS. To make a polymer surface fluctuate, the polymer film supported on a
substrate was put into a sample chamber and heated over Tg under high vacuum. A
coherent X-ray beam to the fluctuating surface scattered yielding a speckle pattern. A 2dimensional charged couple device (CCD) detected this speckle pattern. XPCS was
available in the synchrotron source of Argonne National Laboratory, Advanced Photon
Source (APS) and was installed on the 8-IDI beam line. The scale resolution of APS
XPCS was ~10 Å for z direction and few Å – 100 μm for the lateral (x) direction.
Figure 4.17 presents the XPCS configuration in 8-IDI. The monochromator is
made up of two pieces of germanium (111) crystal and the beam is monochromatized by
double bouncing on the crystals. This set-up provides coherent monochromatic X-ray
with 7.5keV energy. The beam size is determined by guard slits and defining slits. The
beam size incident on the sample was 20 μm × 20 μm with typical beam flux ≈ 3×109
photons/s. The incident angle was 0.14˚ and it is smaller than the critical angle of PS film,
0.17˚. The penetration depth in this angle is around 9 nm. Considering that the sample h
for this study was at least 30 nm, scattering from the film/substrate interface was
negligible and only surface height fluctuations at the interface with the vacuum were
probed. The wave vector spread was 1%, the detector opening was kept small, and data
were collected at sufficiently large values of q|| so that the shape of the correlation
function was not perturbed by partial coherence [131]. CCD was composed of 1340 ×
1300 pixel array with each pixel size of 22.5 × 22.5 μm2. The read-out time at full frame
CCD mode was about 1.8 seconds. During read-out time, the front shutter, which is
123
synchronized with the CCD, blocks the beam to protect the sample. The maximum time
span for XPCS at 8-IDI was ~1000 s in the reflection geometry and was longer than 5000
s in the transmission geometry.
Figure 4.16: Geometry of scattering and detection in XPCS
Figure 4.17: XPCS set up on 8-IDI beam line at APS.
Since the measurement temperature was higher than Tg and the samples have been
exposed to high intensity radiation, the sample surface might have suffered from beam
damage. XR was measured on the same point before and after measurement to observe
124
any reflectivity change due to irradiation. Any significant change in the reflectivity curve
means beam damage. If one measurement batch is composed of multiple points, the beam
irradiation is shared by these points. Therefore, to avoid any serious beam damage,
usually, several points were taken for one measurement. Also, in higher temperature, the
exposure time was reduced. The detailed measurement conditions for each sample
studied are presented in the Appendix (Table A.3). In this study, a point is allowed to be
exposed no longer than 5 min. Since all frames are measured time sequentially,
comparing the first-half frame with the later-half frame of a point can indicate the amount
of the beam damage. Normalized intensity-intensity time autocorrelation function, g2, of
each half was calculated to yield surface relaxation time, τ, to be compared each other.
Any serious deviation between two halves means beam damage and then the later half is
discarded.
The measured XPCS data is the scattering intensity variation from measuring
points. This variation can be expressed as The intensity-intensity correlation function, g2,
given by
g 2 q|| , t  
I q|| , t 'I q|| , t 't 
I q|| , t '
2
,
(4.3)
where I q|| , t ' indicates the scattering intensity at a certain time and at a wave vector
and t  t ' denotes time delay. A g2 decreases as shown by Figure 4.18 and approaches a
value near 1 after a long measurement time. If a g2 decreases and converges to a value in
10000 s, it means that the τ can be measured with a XPCS configuration of the 8-IDI APS.
However, if decrease in the g2 is not clear in 10000 s, the τ is too slow to be measured. To
calculate τ values, the g2 data was fitted with the equation below :
125
g2  1  exp 2t   ,
(4.4)
where β is the speckle constant.
g2
Time delay (s)
Figure 4.18: An example of XPCS data. ◇ is g2 measured with XPCS. The solid line is
the fitting of g2 using Equation 4.4 to calculate τ.
There are two modes for a CCD camera to collect dynamics data: full frame mode
and kinetics mode [132]. Figure 4.19 shows speckle patterns on a CCD camera in each
mode. Figure 4.19a indicates that full frame mode makes use of the whole area of the
CCD for a single exposure. This method is advantageous to measure relatively slower
dynamics and it allows seeing smaller scale of dynamics with larger in-plane wave vector.
Figure 4.19b is a CCD image of kinetics mode. Only a fraction of the CCD is used for
one frame and the whole area is used for multiple exposures. Though the read out time is
1.8s, multiple frames in a CCD provide a 30 ms time scale. However, using a fraction of
the CCD could not measure the wide range of an in-plane wave vector. Therefore, the
length scale of the kinetics mode becomes limited. Furthermore, since the exposure is
carried out in a short time with a pulse-like way, sometimes it fails to provide sufficient
signal to noise ratio. In this study, kinetics mode and full frame mode were applied
selectively depending on the rate of surface dynamics of a sample.
126
a)
b)
Figure 4.19: Speckle patterns on CCD camera with full frame mode (a) and kinetics
mode (b).
As shown by Figure 4.16, when an X-ray beam is scattered from a surface
fluctuation, it will be off-specularly scattered. Therefore, the scattering vector includes a
component parallel to the sample surface. Since we assume the sample surface is laterally
isotropic, we ignore any distinction between x and y components and simply use the inplane wave vector, q||. Each position in the speckle pattern on the detector corresponds to
a particular value of q|| and qz. However, in the data analysis, to obtain useful statistics the
intensities from multiple detector pixels are binned together. The pixels that are binned
together are those corresponding to a particular range of values of q|| (Δq||). Figure 4.20
shows an example of this partitioning. Sparse partitioning, i.e. larger Δq||, makes the τ vs.
q|| curve easier to understand, because the plot of τ vs. q|| is more smooth, but sacrifices
the q|| resolution (Figure 4.21a). On the other hand, dense partitioning, i.e. smaller Δq||,
allows better q|| resolution. The data becomes noisier for denser partitioning. In this study,
Δq|| is between 0.0001 Å -1 and 0.0005 Å -1. In reality, due to poorer counting statistics at
large q||, the τ does not always decrease at the highest q|| (Figure 4.21b).
127
Figure 4.20: The speckle pattern from XPCS is partitioned. The value of q|| for each
τ (sec)
τ (sec)
partition is related to the scale of surface fluctuations.
q|| (Å-1)
q|| (Å-1)
a)
b)
Figure 4.21: The degree of speckle pattern partitioning results in various data quality. A
sparse partitioning (a) provides more readily interpreted data, but poor q|| resolution. A
dense partitioning (b) has values of τ for more q|| points, but the τ vs. q|| curve becomes
noisy.
128
The surface dynamics of a polymer layer were studied by observing the surface
fluctuations using XPCS. At T > Tg, the thermally stimulated polymer surface manifests
capillary waves. The capillary wave can be identified with a complex frequency, f, given
by [151, 152]
f  P  i ,
(4.5)
where the real part, ɷp, corresponds to the propagation frequency and the imaginary part,
Γ, corresponds to damping. For overdamped waves, ɷp is equal to 0. Depending on the
surface tension, viscosity and density of the layer, the surface fluctuation can be either
propagating or overdamped. If the surface fluctuation is propagating, the g2 function has
an oscillating character. However, if the surface fluctuation is overdamped, the g2
function has a single exponent decay [151]. In this study, all polymer samples had
overdamped waves and this was consistent with the previously reported data [9].
An untethered chain layer can be either viscous or viscoelastic. A favorable
interaction with the substrate, such as chain adsorption, can lead to viscoelastic properties
in the polymer layer for very small thickness [21]. Also, the XPCS data (τ vs. q|| curves)
can be fitted using a viscoelastic model (Equation 1.8) in that case. For a thicker layer of
untethered chains of molecular weight below the entanglement molecular weight, the
viscous character becomes dominant, and the layer can be treated as a homogenous
viscous layer for h/Rg > 4 [9]. In this study, a purely viscous model (Equation 1.1) was
used to fit τ with respect to q||. Since a partially tethered layer is composed of tethered
and untethered chains, a purely viscous model is not perfectly suitable to explain its
surface dynamics. In the later chapters, models suggested by presence of various effects
will be discussed to address this problem.
129
4.7. Other Measurements
Off-specular scattering was done at APS, synchrotron line 33BM where the beam
width was 200 μm. The beam energy was 11 kev with the intensity, 9.9 1010 photons/sec.
Samples were measured for qz values of 0.15, 0.2, 0.3 and 0.35 Å -1 and qx was varied at a
fixed qz. The temperature of the sample was 110 °C during the measurement.
AFM(Bruker Dimension® Icon® Atomic Force Microscope) was used to observe
sample surface morphology change due to annealing and beam damage. MikroMasch tips
were used with tapping mode. The scan size varied from 1 × 1 μm2 to 10 × 10 μm2.
For thermal analysis differential scanning calorimetry (DSC) and thermal
gravimetric analysis (TGA) were used. DSC was used to measure the bulk Tg, of each
polymer using a DSC Q 2000 manufactured by TA Instruments. The heating and cooling
rates were 10 oC/min and the sample was heated to 150 °C in the first heating scan. The
Tg was determined using the data from the second heating. TGA was measured under
nitrogen flow from room temperature to 500 °C at 10 °C /min heating rate. TGA with
model number Q 500 produced by TA instruments was used. Both TGA and DSC raw
data were plotted and analyzed using the Universal Analysis 2000 version 4.4A software
obtained from TA Instruments.
Matrix-assisted laser desorption/ionization, MALDI, mass spectra were obtained
using Bruker Ultraflex-III TOF/TOF mass spectrometer manufactured at Bruker
Daltopnics, Inc. All spectra were measured via positive reflection linear mode with a
Nd:YAG laser source. Standard samples were measured prior to sample measurement and
Poly(methyl methacrylate), PMMA, within desirable range of molecular weight was used
as the standard. All samples were measured as a solution in CHCl3. The cationizing agent
130
was sodium trifluoroacetate or silver trifluoroacetate in MeOH/CHCl3 (v/v = 1/3).
Gel Permeation Chromatography (GPC) was used to measure the molecular weight
of PS chains before tethering. The GPC setting contained WatersTM 2412 differential
reflractometer concentration detector and Wyatt DAWNTM EOS multi angle later light
scattering detector. Tetrahydrofuran (THF) was used as a solvent and an eluent with 1
mL/min flow rate.
131
CHAPTER V
SURFACE DYNAMICS AFFECTED BY CONFINEMENT AND COVALENT BONDS
5.1. Confinement Effect of Thin Film of Untethered Chains.
The confinement effect on the surface dynamics of a polymer layer supported on a
substrate depend on bonds and interactions between the polymer layer and the substrate.
The main topic of this chapter is the role of the confinement effect originating from the
covalent bonds. However, before considering covalent bonds, other weak effects such as
the van der Waals interaction [19, 20] need to be screened out to see the effect of covalent
bonds. Therefore, to learn how large the role of van der Waals is, the untethered polymer
layers on etched silicon wafers will be dealt with in this section. The surface relaxation
time (τ) and the viscosity (η) of thin untethered layers will be measured and compared
with that of the bulk state.
5.1.1. Viscosity Calculation
Before dealing with partially tethered layers, linear dPS chains with different
molecular weight were coated on etched silicon wafers to prepare reference samples. The
surface dynamics of two references, 2k Ref 43 and 48k Ref 35 were measured with X-ray
photon correlation spectroscopy (XPCS). The information for the two reference samples
is presented in Table 5.1. The measurement temperature for 2k Ref 43 was 100 °C,
110 °C and 120 °C. For 48k Ref 35, the measurement temperature was 150 °C and
132
160 °C. Figures A.2, A.3, A.4, A.5 and A.6 present the g2 functions. The XPCS raw data
was expressed as intensity-intensity time autocorrelation function, g2, given by Equation
1.4. Then the τ value for each in-plane wave vector (q||) was determined by fitting the
correlation the functional form that assumed there was a single predominant
characteristic τ. The equation to be used is given by Equation 4.4.
Once the values of τ were determined over a range of q||, those data were fit using
the hydrodynamic continuum theory (HCT) model. From the HCT, τ for a viscous film
with a no-slip boundary condition at the substrate should be given by
 q||  

2 cosh 2 q|| h   q||2 h 2

q|| sinh q|| h cosh q|| h   q|| h 
,
(5.1)
where h is the thickness and γ is the surface tension of the polymer layer. If the surface
tension is known, the viscosity may be treated as fitting parameter if this expression is
forced to fit experimental data of τ vs. q||. In this study, γ values were adapted from
elsewhere [3]. The h of a sample was measured using two instruments: an ellipsometer (λ
= 632 nm) at room temperature and a X-ray reflectometer at the APS 8-IDI line at the
XPCS measurement temperature. The h value use for the HCT fit is from the X-ray
reflectometer. The h values obtained from the X-ray reflectivity (XR) data are presented
in the Appendix (Table A.1).
For both reference samples, the τ calculated from Equation 4.4 falls within the
experimentally measurable window for some range of in-plane wave vector within our
measurement window (τ < 10000 s). The dynamics got faster at higher temperatures as
shown by Figures 5.1 and 5.2. This trend was consistent with the previously reported
results [9, 10]. The glass transition temperatures (Tg) of the polymers used for sample
preparation, were measured using differential scanning calorimetry (DSC). For 2.6kg/mol
133
dPS, the Tg was 68 °C, and for 48 kg/mol dPS the Tg was 99 °C. Therefore, at T = 120 °C,
where T is the XPCS measurement temperature, T - Tg for 2k Ref 43 was 53 °C and at T
= 150 °C, T - Tg for 48k Ref 35 was 52 °C. Due to the similar values of T - Tg, comparing
τ values of 2k Ref 43 at 120 °C with 48k Ref 35 at 150 °C is possible in spite of different
measurement temperatures.
However, τ also depends on h of the polymer layer. A thinner layer had a larger
value of τ at the same molecular weight and T [9]. Assuming that the polymer layer is a
homogenous and pure viscous liquid, the h effect could be removed after the data was
normalized with h. After the normalization, the values of τ/h depended only on η and γ of
the samples. In other words, if we can ignore any confinement effect of the polymer film,
the values of τ/h of reference samples in this study will only depend on T and the
molecular weight.
Table 5.1: The information of two reference samples : 2k Ref 43 and 48k Ref 35.
Sample
2k Ref 43
48k Ref 35
Mn (kg/mol) of
untethered chains
2.6
48
Mw/Mn
h (nm)a
1.02
1.01
43±1
39±1
a
Annealingb
(temp(°C)/time(hr))
Annealed in XPCS
140/15
The thickness of total PS layer was measured by ellipsometer with 632nm laser source
and n=1.589 at room temperature. For HCT fit, the h was obtained from the XR curve
analysis which is measured at APS, 8-IDI at the XPCS measurement temperatures. The h
values from the XR curves are presented in the appendix (Table A.1).
b
The annealing was done under high vacuum condition (10-5-10-6 Pa).
134
1000
100 Cel-deg
110 Cel-deg
120 Cel-deg
τ(s)
100
10
1
0.1
0.0001
0.0002
0.0003
0.0004
q||(Å
0.0005
0.0006
0.0007
0.0008
-1)
Figure 5.1: The τ vs. q|| curves for 2k Ref 43 at 100 °C (□), 110 °C (○) and 120 °C (◇).
135
10000
1000
150 Cel-deg
τ (s)
160 Cel-deg
100
10
1
0.0001
0.0002
0.0003
0.0004
q||(Å
0.0005
0.0006
0.0007
0.0008
-1)
Figure 5.2: The τ vs. q|| curves of 48k Ref 35 at 150 °C (□) and 160 °C (○).
As presented by the solid lines in Figures 5.3 and 5.4, using the HCT, [7], the
normalized data were fitted with
 q|| 
h


2 cosh 2 q|| h   q||2 h 2

q|| hsinh q|| h cosh q|| h   q|| h
.
(5.2)
Since values of h, q|| and γ were already known, the HCT fit provide an estimate of the
value of η for each T and molecular weight if the HCT were applicable. In this study, the
γ values were from elsewhere [140]. The calculated η values are shown in Table 5.2. A
higher T and a lower molecular weight resulted in a smaller value of η.
136
1.E+00
100 Cel-deg
110 Cel-deg
120 Cel-deg
1.E-01
100 Cel-deg fitting
τ/h
(s/Å )
τ/h(sec/Å)
110 Cel-deg fitting
120 Cel-deg fitting
1.E-02
1.E-03
1.E-04
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
q||h
Figure 5.3: τ/h vs. q||h curves of 2k Ref 43 at 100 °C (□), 110 °C (○) and 120 °C (◇).
The solid lines are fits with the HCT to calculate η.
137
1.E+01
150 Cel-deg
160 Cel-deg
150 Cel-deg fitting
160 Cel-deg fitting
τ/hτ/h(sec/Å)
(s/Å )
1.E+00
1.E-01
1.E-02
0.05
0.10
0.15
0.20
0.25
0.30
0.35
qq||||hh
Figure 5.4: τ/h vs. q||h curves of 48k Ref 35 at 150 °C (□) and 160 °C (○). The solid
lines are fits with the HCT to calculate η.
Table 5.2: Viscosities calculated from the HCT fits.
Sample
2k Ref 43
48k Ref 35
T (°C)
100
110
120
150
160
γ (mN/m)
34
33
32
32
31
138
η (Pa·s)
5800 ± 290
1100 ± 55
250 ± 13
55000 ± 2750
10000 ± 500
5.1.2. Comparison with Bulk Viscosity
Values of bulk η from elsewhere [133] were compared with the values η of thin
polymer layers calculated from the HCT fits. h is a key factor determining both τ and η.
However, the dynamics not only depend on the absolute h, but also depend on the relative
h [21]. The ratio of h to the unperturbed radius of gyration, Rg, (h/Rg) is the parameter
manifesting the relative h. For 2k Ref 43 and 4k Ref 35, the values of h/Rg were ~30 and
~7, respectively.
In Figure 5.5, the values of η for the nominally 43 nm thick 2.6 kg/mol dPS layer,
2k Ref 43, inferred from fitting the HCT to the data are compared with bulk η values for
1.1k, 3.4k, 15.7k, 16.4k and 47.0k hPS from the literature [133] at 100 °C and 110 °C.
Except the lowest molecular weight, 1.1k, the data points followed the trend: η = kMa
where M is the molecular weight and the k was fitted as the value, 300 Pa·s·(10-3g/mol)3.4
and a had the value as 3.4. Values of η from the literature are plotted as a function of
molecular weight for hPS for the purpose of interpolating to determine an expected value
for the 2.6k dPS chains, assuming that the actually measured η of 2k Ref 43 was 1100
Pa·s at 110 °C. Being compared with the fit using η = kM3.4 the η value of 2k Ref 43 is
lower than the η value from the fit.
Since h of the 2.X kg/mol reference sample is 30Rg, it is anticipated that it is thick
enough to avoid confinement effects due to van der Waals interactions with the substrate.
However, for the 48 kg/mol chains, for which h is only 38 nm or ~6Rg, it is possible that
there may be some sort of confinement effect present. The η is 2 - 3 times higher than
47.0 kg/mol bulk PS (Figure 5.6 and Table 5.3) [133]
(160°C) will be discussed in the next chapter.
139
Data for one addition temperature
Since we do not have data for other
thicknesses or for several temperatures covering a broader range it is not possible to
further define why this viscosity is somewhat above the bulk value.
For the thicker layer with h/Rg ~ 30, the value of η was comparable to that for the
bulk state (Figure 5.5). This observation was consistent with the result from the off
specular scattering experiments conducted by Wang et al., but seems to follow an
opposite trend from that in the results obtained by Tsui et al. and by Cory et al. with
AFM and ellipsometer[134, 13]. In this study, the silicon wafers supporting the reference
(untethered) PS layers were etched with hydrofluoric acid and became hydrophobic just
before depositing the film. Since PS is somewhat hydrophobic its interaction with the
etched silicon wafer is more favorable than that with a wafer having its native oxide.
Perhaps this difference plays some role in the value of η that is seen. The other important
point is that even though the thinner film had higher value of η, the surface dynamics of
48k Ref 35 were still fast enough to be measured by XPCS. This is surprising, since this
sample has the highest effective molecular weight considered in this study.
important result will be dealt with in the next section.
140
This
1.E+10
1.E+09
1.E+08
Viscosity (Pa·s)
1.E+07
1.E+06
1.E+05
1.E+04
1.E+03
1.E+02
1.E+01
1.E+00
1
10
Molecular Weight (kg/mol)
100
a)
Bulk viscosity at 110Cel-deg
1.E+05
2k ref 40 from XPCS at 110Cel-deg
Fitting
Viscosity (Pa·s)
1.E+04
1.E+03
1.E+02
1.E+01
1.E+00
1
10
Molecular Weight (kg/mol)
b)
BulkFigure
viscosity
at 110Cel-deg
5.5:
(a) Comparison of PS viscosities of 2k Ref 43 (▲) and of bulk samples (◆)
2k ref 40 from XPCS at 110Cel-deg
at 110 °C. The solid line is the fitting using the equation η = kMa. The bulk η values are
form [133]. (b) 2k Ref 43 still has lower η than the interpolation of 1.1k and 3.4k bulk PS.
141
10000000
15.7k Literature
1000000
47.0k Literature
Viscosity (Pa*s)
100000
47k ref HCT fitting
10000
1000
100
10
1
110
120
130
140
150
160
170
180
190
Temperature (℃)
Figure 5.6: The η values of 48k Ref 35 (▲) compared with those of 15.7 kg/mol (◆) and
47.0 kg/mol (■) bulk PS. The bulk η values are form [133].
Table 5.3: η of 47 kg/mol - 48 kg/mol linear PS in a thin layer and bulk state.
Sample
Temperature (°C)
Viscosity (Pa·s)
150
55000 ± 2750
160
10000 ± 500
139
255000
145
30000
160
3000
48k Ref 35
47k Bulka
a
The bulk η values are form [133].
142
5.2. Confinement Effect from Covalent Bond
As discussed in Chapter 1, van der Waals interaction could suppress the surface
fluctuation of a thin polymer film. However, the interaction energy from the van der
Waals force converges to 0 even in a short distance (~ 1 nm) and repulsion occurs for
closer contact than 0.2 nm. In this study, to provide a means of altering surface
fluctuation dynamics that can be tailored, partially tethered layers have been prepared. If
tethered chains are mixed with untethered chains on a substrate, the whole sample could
be seen as a single polymer layer having a fraction of the chains covalently bonded to the
substrate. Since the covalent bond energy is far higher than the van der Waals interaction
energy, the polymer layer will be more bound to the substrate, leading to much more
slowed or suppressed surface fluctuation. In addition, increasing the grafting density (σ)
of the tethered chains also will increase the number of covalent bonds to the substrate per
unit area. In this section, we will mainly focus on the effect of strong confinement caused
by covalent bonds. Also, how the the surface dynamics are changed for two values of
covalent bond density (0.10/nm2 and 0.02/nm2), will be compared.
5.2.1. Comparison with Molecular Weights Blends
The tethered chains in this study, have larger molecular weight than the untethered
chains do. Once the tethered chain layer is mixed with the untethered chain layer, the
average molecular weight of the partially tethered layer will be higher than that of the
pure untethered chain. Therefore, compared with untethered reference samples, the
variation in the surface dynamics is also contributed by increased average molecular
weight.
143
Therefore, to see the increase of τ excluding this larger molecular weight effect, the
same polymers used for tethered and untetherd chains were mixed as a solution and then
were spun cast on an etched silicon wafer without any process for tethering. The τ values
of this untethered blending sample, 200k 10Prc 38, was compared with those of a
partially tethered layer, 2h02 45. Table 5.4 presents the sample list. Considering the
neutron reflectivity study (Figure 7.3 in Chapter 7) of 2h02 45, the volume % of 200k
hPS (1.05g/mol) is 9.8 %. Considering the lower density of dPS at a lower molecular
weight (0.95g/ml, 2.6kg/mol), The weight % of 2.6kg/mol untethered chains in 2h02 45
was 10.7%.
The XPCS data (τ vs. q|| curves) of both samples were presented by Figures A.7, A.8
and 5.7. At 110 °C for both samples showed measurable surface dynamics (τ < 10000
sec). However, as demonstrated by Figure 5.8, the τ/h of 2h02 45 was 2 times higher than
that of 200k 10Prc 38. Considering that 200k 10Prc 38 is thinner than 2h02 45, the larger
value of τ in 2h02 45 was not originated from the h contribution. Using the HCT, the data
were fitted and to obtain η values (Table 5.5). The η of 2h02 45 was 2 times higher than
that of 200k 10Prc 38. The only distinctive difference between the two samples is that
2h02 45 has the confinement effect due to the covalent bonds, while 200k 10Prc 38 does
not.
144
Table 5.4: Samples to be studied.
Sample
2h02 45
200k 10Prc
38
Mn(kg/mol)
Tethered
Untethered
200
2.6
2.6 (90wt%)
200 (10wt%)
a
σ
(chains/nm2)a
h (nm)b
0.02±0.002
43±1
Annealing
(temp(°C)
/time(hr))
119/24
39±1
115/15
Calculated from σ = hρNA/Mn , where σ is grafting density, h is thickness of tethered
chains, ρ is bulk density of the tethered chain component and NA is Avogadro’s number.
b
The thickness of total PS layer was measured by ellipsometer with 632nm laser source
and n=1.589 at room temperature. For the HCT fit, the h was obtained from the XR curve
analysis which is measured at APS, 8-IDI at the XPCS measurement temperatures. The h
values from the XR curves are presented in the appendix (Table A.1).
145
1000
τ (sec)
100
10
2h02
200k 10Prc
1
0.0002
0.0004
0.0006
0.0008
q|| (Å-1)
Figure 5.7: τ vs. q|| curves of 200k 10Prc 38 (□) and 2h02 45 (○) at 110 °C.
146
0.0010
10.00
τ/h (sec/Å)
1.00
0.10
0.01
0.10
0.20
0.30
0.40
q||h
200k 10% 110C 39nm
2h02 110C 44nm
Figure 5.8: τ/h vs. q||h curves of 200k 10Prc 38 (□) and 2h02 45 (○) at 110 °C. The
2h02 110C fitting
200k 10Prc fitting
solid lines are the fits with the HCT to calculate η values.
Table 5.5: Viscosities calculated with the HCT fits.
Sample
2h02 45
200k 10Prc 38
T (°C)
γ (mN/m)
110
33
147
η (Pa·s)
90000 ± 4500
45000 ± 2250
5.2.2. Surface Dynamics for Higher Covalent Bond Density
Using a PS-COOH with Mn = 28kg/mol, the σ value of the tethered PS was
obtained as 0.1 chain/nm2. This tethered chain of 2802 45 could provide 5 times higher
covalent bond density to the substrate than 2h02 45 could do. The h of both 2802 45 and
2h02 45 was maintained around 40nm. Table 5.6 presents the samples to be studied in
this section.
2802 45 is a 43 nm thick film (at room temperature) containing 9 vol% of sparsely
grafted 28k chains (0.1 chains/nm2). However, the τ value was larger than the maximum
measureable value even at a temperature as high as 180 °C. XPCS data were collected at
120, 130 and 180 °C for sample 2802 45. No surface relaxation was seen at any of these
temperatures and the value of g2 was always above the baseline (Figures 5.9, 5.10 and
5.11), excluding the possibility that the relaxation time was too short to be observed.
Even though the highest temperature was about 110 °C above the Tg of the untethered
chains (68 °C) [58], the value of τ was outside the measurement window (i.e. > 10,000 s
in the q|| range accessed).
148
Table 5.6: Samples to be studied in this section
Sample
2k Ref 43
48k Ref 35
2802 45
Mn(kg/mol)
Tethered
Untethered
28
2.6
48
2.6
σ
(chains/nm2)a
0.097±0.012
a
h (nm)b
43±1
39±1
43±1
Annealing
(temp(°C)
/time(hr))
Annealed in XPCS
140/15
110/15
Calculated from σ = hρNA/Mn , where σ is grafting density, h is thickness of tethered
chains, ρ is bulk density of the tethered chain component and NA is Avogadro’s number.
b
The thickness of total PS layer was measured by ellipsometer with 632 nm laser source
and n=1.589 at room temperature. For the HCT fits, the h was obtained from the XR
curve analysis which is measured at APS, 8-IDI at the XPCS measurement temperatures.
The h values from the XR curves are presented in the appendix (Table A.1).
149
1.10
1.10
1.08
1.08
1.06
g2
g2
1.12
1.04
1.06
1.04
1.02
q||=0.000317Å -1
q||=0.000408Å -1
1.00
1.02
1
10
100
1000
Time delay (s)
1
10000
1.06
1.04
1.04
1.02
1.02
100
1000
Time delay (s)
10000
g2
g2
1.06
10
1.00
1.00
0.98
0.98
q||=0.000506Å -1
0.96
q||=0.000604Å -1
0.96
1
10
100
1000
Time delay (s)
10000
1
1.06
1.04
1.04
1.02
1.02
100
1000
Time delay (s)
10000
g2
g2
1.06
10
1.00
1.00
0.98
0.98
q||=0.000700Å -1
0.96
q||=0.000800Å -1
0.96
1
10
100
1000
Time delay (s)
10000
1
10
100
1000
Time delay (s)
10000
Figure 5.9: g2 functions of 2802 45 for several values of q|| at 120 °C. The red line is the
attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too
large to be calculated.
150
1.10
1.09
1.08
1.07
1.06
q||=0.000411Å -1
g2
g2
1.11
1.05
1.04
1.03
1.02
q||=0.000317Å -1
1.01
1.00
1
10
100
1000
Time delay (s)
10000
1
10
100
1000
Time delay (s)
10000
1.10
1.10
1.08
1.08
q||=0.000510Å -1
q||=0.000610Å -1
1.06
g2
g2
1.06
1.04
1.04
1.02
1.02
1.00
1.00
1
10
100
1000
Time delay (s)
1
10000
1.05
1.03
1.03
1.01
1.01
100
1000
Time delay (s)
10000
g2
g2
1.05
10
0.99
0.99
q||=0.000707Å -1
0.97
0.97
q||=0.000809Å -1
0.95
0.95
1
10
100
1000
Time delay (s)
1
10000
10
100
1000
Time delay (s)
10000
Figure 5.10: g2 functions of 2802 45 for several values of q|| at 130 °C. The red line is the
attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too
large to be calculated.
151
1.11
1.13
1.09
1.11
1.07
g2
g2
1.15
1.09
1.07
1.03
q||=0.000316Å -1
q||=0.000408Å -1
1.01
1.05
1
10
100
Time delay (s)
1
1000
1.10
1.10
1.08
1.08
1.06
1.06
g2
g2
1.05
1.04
1.02
10
100
Time delay (s)
1000
q||=0.000605Å -1
1.04
1.02
q||=0.000507Å -1
1.00
1.00
1
10
100
Time delay (s)
1000
1
10
100
Time delay (s)
1000
1.08
1.06
g2
1.04
1.02
1.00
q||=0.000701Å -1
0.98
1
10
100
Time delay (s)
1000
Figure 5.11: g2 functions of 2802 45 for several values of q|| at 180 °C. The red line is the
attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too
large to be calculated.
152
This result is represented schematically in Figure 5.12 for comparison with results
from other samples for which the τs are measurable. The remarkable slowing of surface
fluctuations seen in Figure 5.12 is not due to an increase in Tg of the film caused by
tethering. Nor is the strong slowing of the film surface due simply to an increase in film
Tg resulting from mixing of higher Mn chains with the 2.6kg/mol chains. The Tg of the 28
kg/mol chains is 95 °C in the bulk (Figure 5.13). The effective Tg of a bulk blend of 9 vol%
28 kg/mol chains can be determined with the equation given by [135]
Tg ,PS  1  wPS Tg ,1  wPS Tg ,2 ,
(5.3)
where Tg,1 and Tg,2 represents the Tgs of the low and high molecular weight PS,
respectively and wPS is the weight fraction of the high molecular weight PS. The density
of 28k hPS was 1.05 g/ml and that of 2.6k dPS was 0.95 g/ml. Considering the vol% and
density, the Tg of the 2802 45 calculated using Equation 5.3 was (ϕPS = 90.1 wt%) 71 °C.
The effective Mn of chains in the 2802 45 was 4.9 kg/mol. This is much less than
48 kg/mol, and yet a 35 nm thick film of pure 48 kg/mol untethered chains still has
observable surface fluctuations for a temperature such that T - Tg ≈ 50 °C, while the 45
nm partially tethered layer of Figure 5.12 has unmeasurably slow surface fluctuations
even for T - Tg as high as 110 °C.
Since the T was 180 °C, one might think of that the high value in τ could be due to
the thermal degradation of 2802 45 making the h smaller. Therefore, checking the thermal
resistance of the 2.6 kg/mol dPS in a bulk and a film state is essential. First of all, thermal
gravimetric analysis (TGA) study to see the thermal resistance as a bulk state, was
performed. Figure 5.14 presents only a 0.5% weight loss at 200 °C. Second, the XR
curves collected when running the XPCS measurements provides some secondary
153
evidence on the thermal stability of the films in vacuum. The thickness was measured at
one spot for each temperature. For sample 2802 45, h measured at 120 °C - 180 °C are
summarized in Table 5.7. The values reflect the thermal expansion with increasing T. The
apparent expansion is not exactly uniform because the spots measured for adjacent T
were about 0.4 mm apart and there are some small lateral variations in film h. No
reduction in h due to thermal degradation is evident even at 180 °C.
1.E+01
τ/h(sec/Å)
1.E+00
1.E-01
1.E-02
1.E-03
1.E-04
0
0.1
0.2
q||h
0.3
0.4
Figure 5.12: The τ/h vs. q||h curves of 48k Ref 35 (□) and 2k Ref 43 (○) at T - Tg ≈
50 °C. The dotted curve is to illustrate the surface relaxation time of 2802 45 which is
beyond the measurement limit of the XPCS up to 180 °C (i.e. > 10,000 s in the q|| range
accessed).
154
-2.5
Heat Flow (W/g)
-2.7
-2.9
Tg = 94.49℃
-3.1
-3.3
-3.5
-3.7
-3.9
50
60
70
80
90
100
110
120
130
140
400
450
150
Temperature ℃
Figure 5.13: Tg from the DSC data of hPS with Mn = 28 kg/mol.
100%
90%
Remaining Weight %
80%
70%
60%
50%
40%
30%
20%
10%
0%
0
50
100
150
200
250
300
350
500
Temperature (℃)
Figure 5.14: The thermal resistance of a bulk state of Mn = 2.6 kg/mol dPS was measured
with TGA under a nitrogen flow.
155
Table 5.7: h values of 2802 45 at various XPCS measurement T.
T (°C)
120
130
160
180
h (Å )
442
439
450
453
Another possibility of such high value of τ in 2802 45 might be due to the large
value of Tg of the tethered chain resulting in a high value of the overall Tg. Uğur [58]
measured the h variation of a PS tethered chain layer having σ ≥ 0.6chain/nm2 with
respect to T. The h vs. T curve obtained from XR, could indicate the gradient changing T
which is equivalent to Tg. However, the measured Tg (95 °C) value was very close to the
value of a bulk PS. Since a lower σ gives more space for better mobility, the tethered
layers for this study (σ < 0.15 chain/nm2) cannot have higher Tg than σ = 0.60 chain/nm2.
The average molecular weight of 2h02 45 was calculated as 22.3 kg/mol by fitting
its NR curve (Figure 7.3). Therefore, the average molecular weight of 2h02 45 is 4 - 5
times higher than that of 2802 45. However, in spite of smaller average molecular weight,
2802 45 had far slower surface dynamics than 2h02 45 did. For σ, 2802 45 had the value
factor of 5 larger than 2h02 did 45 indicating that the covalent bond density with the
substrate is larger.
After all, a conclusion that one of the factors resulting in slower surface dynamics
of 2802 45 was contributed by the confinement effect with the substrate via covalent
bonds, can be derived. In addition, considering the difference in surface dynamics
between 2802 45 and 2h02 45, it can be learned that varying the density of covalent bond
to the substrate tailors the surface dynamics.
156
5.2.3. Other Results Supporting the XPCS Data
Slowing of the surface fluctuations should be associated with greater stability with
respect to film de-wetting. That this was actually the case was tested by heating sample
films at 180 °C under high vacuum (10-6 Pa) and then imaging with optical microscopy
before annealing and after annealing for 19 and 43 hours (Figure 5.15) and with an
atomic force microscope (AFM) after annealing (Figure 5.16) for 43 hours. For AFM,
MikroMasch tips were used with tapping mode. The images show that for the reference
film of untethered chains, 2k Ref 43, significant de-wetting was seen after 17 hr of
annealing (Figure 5.15a) while the surface of 2802 45 was almost flat (root mean squared
roughness of 0.56 nm) after 43 hr. Although a few more holes appear in the optical
images of the 2802 45 film after 43 hr annealing, most of the surface still looks very
smooth, as shown in Figures 5.15d and 5.16. At T > Tg, the large fluctuation amplitude
and the aggregation of polymer liquid will cause polymer layer de-wetting via spinodal
rupture [136]. The de-wetting shape by the spinodal rupture has slightly periodic “hills
and valleys” with a wavelength as λs. However, the τ value of 2802 45 was beyond the
XPCS measurement window. It could mean that the slow surface movement 2802 45
resulted in a slow height fluctuation. In addition, the aggregation of untethered chains
might have been slowed by substrate-tethered chains. It could not only be related to the
covalent bond-confinement effect, but also be related to the friction with the tethered
chain. More detailed discussion about the friction will be dealt with at the next chapters.
Placing tethered chains in the film also changes the spectrum of the surface
fluctuations, which can be studied using off-specular scattering [114, 137, 138]. Figure
5.17 shows a plot of off-specular scattering that reflects the power spectral density of the
157
surface fluctuations. For a polymer melt surface with suppression of surface modes one
expects that in the power law region between the resolution limited scattering from the
specular beam (at low qx) and the Yoneda peak [139] (at qx about 0.01 Å -1) there will be a
change of slope corresponding to a cutoff value of qx. For qx < qx,cutoff, that is, for length
scales larger than those corresponding to the cutoff wavelength, surface fluctuations are
suppressed and the slope is less negative. While assignment of the exact location of the
cutoff is subject to some uncertainty, it is clear that the value of qx,cutoff is larger for the
film containing the tethered chains. That is, the surface fluctuations are suppressed for a
larger range of wavelengths when some chains are tethered. This is consistent with
slowing of surface modes in the window for XPCS.
158
50um
50um
a)
b)
50um
50um
c)
d)
Figure 5.15: Optical microscope images of samples before and after annealing in a high
vacuum. (a) 2k Ref 43 after annealing at 180 °C for 17hrs, (b) 2802 45 before any
annealing, (c) 2802 45 after annealing at 180 °C for 19hrs and (d) 2082 40 after annealing
at 180 °C for 43hrs.
159
a)
nm
1.5
1.0
0.5
0
- 0.5
0
2
4
6
8
10
μm
b)
Figure 5.16: The surface image of 2802 45 via AFM after 43hrs annealing at 180 °C in a
high vacuum. (a) the image from the top view and (b) a cross section image.
160
106
1.00E+06
105
Reflectivity
1.00E+05
104
1.00E+04
103
1.00E+03
102
1.00E+02
2k ref 40
101
1.00E+01
100
10-1
1.00E+00
1.00E-06
2802 40
10-2
1.00E-05
10-3
1.00E-04
10-4
1.00E-03
10-5
1.00E-02
10-6
1.00E-01
qx(Å-1)
Figure 5.17: Off specular scattering from a sample of pure untethered 2.6k PS (2k Ref 43)
(○) and from a partially tethered layer (2802 45) (□) corresponding to that for which
XPCS relaxation times are shown in Figure 5.19. The value of qz is fixed at 0.35 Å -1 and
the temperature is 110 °C.
161
5.3. Summary
Two major results were discussed in this chapter. The first result was that
introducing covalent bonds between a polymer layer and a substrate was able to slow the
surface dynamics of a polymer film. To account for the effect of molecular weight and Tg,
the relaxation time of 2h02 45 (σ = 0.02 chains/nm2) was compared with the relaxation
time of a blend of 200k and 2.6k PS chains of equivalent composition. 2h02 45 showed
surface dynamics that were a factor of two slower than the dynamics of the blend sample.
For 2802 45, in which 28k tethered PS was mixed with 2.6k untethered PS with σ = 0.1
chains/nm2, the surface relaxation behavior was too slow to be measured with XPCS,
even at T – Tg = 110 °C (T = 180 °C). These slowed surface dynamics were not due to
thermal damage of the sample. While the average Mn of 2802 45 was smaller than 5k, a
reference sample composed of only 48k untethered chains, 48k Ref 35, showed a
measurable relaxation time at 180 °C. This result indicated that the slow dynamics of
2802 45 was not due simply to the larger molecular weight of the tethered chains.
Considering that the Tg of the PS brush is similar to the bulk PS Tg [58], the surface
dynamics of 2802 45 were not dictated by the Tg of the tethered chain layer. This suggests
that designing the partial tethering with covalent bonds can be used to tailor the surface
dynamics of a polymer layer.
The second result was that increasing the density of covalent bonds connecting the
polymer layer and the substrate slows the surface dynamics more. 2h02 45 showed a
measurable surface relaxation, but 2802 45 did not. The grafting density for 2802 45 was
a factor of five higher than that of 2h02 45. The surface dynamics that were observed for
the layer with higher grafting density were at least 12 times slower than the surface
dynamics of the less highly grafted layer (η2802 45/η2h02 45 > 1100000 Pa·s/90000 Pa·s).
162
The role of the tethered chain molecular weight, which was also varied here, remains to
be clarified in subsequent chapters.
163
CHAPTER VI
SURFACE DYNAMICS AFFECTED BY THICKNESS AND
TETHERED CHAIN EXTENSION
6.1. Objective
The effect of covalent bonding of chain ends to a substrate on surface fluctuation
dynamics of a partially tethered layer was focused on in Chapter 5. In this chapter, we
focus on the models to explain the surface dynamics depending on total film thickness,
htotal. Also, changes in the conformation of the tethered chain and their effect on surface
dynamics will be discussed.
6.2. Sample Descriptions for Thickness Varied Samples
Table 6.1 presents characteristic properties of samples discussed in this section.
These fall roughly into three series. The first series contains three samples with the same
sorts of tethered layers and untethered chains, but with three different thicknesses (h ≈ 40,
70 and 90 nm). These three films show vastly different behaviors. The purpose of these
samples was to see specifically how varying the fraction of the film penetrated by the
tethered chains altered the surface fluctuation dynamics. A ca. 40 nm thick film with 28k
tethered chains and 2k untethered chains (2802 45) was investigated in the previous
chapter. That sample had a surface relaxation time, τ > 10000 s even for a measurement
temperature, T, as high as 180 °C. Samples 2802 70 and 2802 90 have similar grafting
164
densities (σ) (0.08 – 0.11 chains/nm2), but they are thicker than 2802 45. A second series
of samples involves untethered chains of higher molecular weight, Mn = 48 kg/mol
specifically. Partially tethered films of h values ca. 40 nm and 90 nm are considered. The
effect upon τ of tethering is elucidated by comparing the behaviors of each partially
tethered layer with a corresponding sample of only untethered chains ("reference layers"
48k Ref 35 and 48k Ref 95). Comparisons of the behaviors of these various samples are
facilitated by normalizing the relaxation times (τ) by film h. This also allows us to readily
calculate effective film viscosities using the hydrodynamic continuum theory (HCT) for
purposes of comparing in a systematic way how much various tethering schemes alter the
surface fluctuations.
165
Table 6.1: Samples to be studied.
σ
(chains/nm2)a
h (nm)b
0.082±0.012
61±1
Annealingc
(temp(°C)
/time(hr))
110/15
2.6
0.091±0.012
82±1
110/15
28
2.6
0.105±0.012
85±1
118/16
4848 20
48
48
0.039±0.007
21±1
139/15
2h48 42
200
48
0.028±0.002
42±1
140/17
2h48 99
48k Ref
35d
48k Ref
95d
200
48
0.023±0.002
97±1
140/17
48
38±1
140/17
48
91±1
135/8
2802 70
Mn(kg/mol)
of tethered
chains
28
Mn(kg/mol) of
untethered
chains
2.6
2802 90
28
2802 85
Sample
a
Calculated from σ = hρNA/Mn , where σ is grafting density, h is thickness of tethered
chains, ρ is bulk density of the tethered chain component and NA is Avogadro’s number.
b
The thickness of total PS layer was measured by ellipsometer with 632nm laser source
and n=1.589 at room temperature. For HCT fitting, the h was obtained from the XR curve
analysis which is measured at APS, 8-IDI at the XPCS measurement temperatures. The h
from the XR curves were presented in the appendix (Table A.1).
c
The annealing was done under high vacuum condition (10-5-10-6Pa).
d
Reference samples were spin coated on etched silicon wafers.
166
6.3. Models and Parameters to Explain the Thickness Dependence.
Both samples 2802 70 and 2802 90 had τ values within the experimental window
of the XPCS experiment at 110 °C and 120 °C and their g2 functions, shown in Figures
A.9, A.10, A.11, and A.12 were well represented by a single exponential decay. Figures
6.1 and 6.2 present the τ values for 2802 70 and 2802 90, respectively, for q|| between
0.0001 Å -1 and 0.0005 Å -1 at 110 °C and 120 °C. Considering that 2802 45 did not show
any relaxation within the XPCS measurement window, the change in dynamics was
dramatic when h was increased by factors of 1.5 and 2.0. The first question we ask is
whether the data may be understood with the help of the HCT. If the viscosity of each
film is uniform through its depth the presence of tethered chains could result in an
enhanced viscosity through that whole depth and the HCT model would still apply. This
would be the zeroth order model illustrated in Figure 6.3 as "Model 0".
In the spirit of
this model the data can be plotted in the form of τ/h vs. q||h, as in Figure 6.4 and effective
viscosities for the whole film extracted if a value for surface tension is assumed. For the
purposes of argument we assume here a surface tension equal to that of the pure
untethered chains.
We see immediately from Figure 6.4 that such an analysis finds that
the shape of the τ/h vs. q||h curve anticipated by the HCT agrees quite well, in fact, with
the observed shape. Therefore it is possible to fit the τ/h vs. q||h data with an effective
viscosity. The value of ηeff, must be different, however, for each film. For 2802 90 ηeff is
a factor of 2.4 larger than ηref for pure untethered 2.6k chains and for 2802 70 ηeff is a
factor of 4.5 larger than the reference value at a temperature of 110 °C, as shown in Table
6.2. The results are similar for 120 °C. For 2802 45 the increase in effective viscosity is
at least a factor of 2000.
167
1.E+03
τ (sec)
1.E+02
1.E+01
1.E+00
1.E-01
0.0001
0.0002
0.0003
q||
2802 60 120C
2802 80 120C
0.0004
0.0005
(Å-1)
Figure 6.1: τ vs. q|| curves of 2802 70 (○) and 2802 90 (◇) at 120 °C. The red and black
2k Ref simulated 66nm
solid
2k Ref simulated 42nm
lines are the simulated τ values of 2.6k dPS layers assuming h = 42 nm and 66 nm at
120 °C, respectively. The η and γ values for the simulation were 250 Pa·s and 32 mN/m
which were used to fit 2k Ref 43 at 120 °C.
168
1.E+03
τ (sec)
1.E+02
2802 60 110C
1.E+01
1.E+00
2802 80 110C
2k Ref simulated at
41nm at 110C
2k Ref simulated as
65nm at 110C
1.E-01
0.0001
0.0002
0.0003
0.0004
0.0005
q||(Å-1)
Figure 6.2: τ vs. q|| curves of 2802 70 (□) and 2802 90 (△) at 110 °C. The red and black
solid lines are the simulated τ values of 2.6k dPS layers assuming h = 41 nm and 65 nm at
110 °C, respectively. The η and γ values for the simulation were 1100 Pa·s and 33 mN/m
which were used to fit 2k Ref 43 at 110 °C.
169
Figure 6.3: Schematic of the structure of the partially tethered layer suggesting ways of
thinking about the variation of viscosity with depth and its influence on the surface
fluctuation dynamics.
1.E+01
1.E+00
2802 60 110C
2802 60 120C
2802 60 120C fitting
2802 80 110C
2802 80 110C fitting
2802 80 120C
τ/h (sec/Å)
2802 60 110C fitting
1.E-01
1.E-02
2802 80 120C fitting
2k Ref 110C
2k Ref 120C
1.E-03
2k Ref 110 C fitting
2k Ref 120C fitting
1.E-04
0.0
0.1
0.2
0.3
0.4
0.5
q||h
Figure 6.4: τ/h vs. q||h curves of 2802 70 at 110 °C (□) and 120 °C (○), of 2802 80 at
110 °C (△) and 120 °C (◇) and of 2k Ref 43 at 110 °C (□) and 120 °C (○). The solid
lines are fits with the HCT fits to calculate η.
170
Table 6.2: η values of partially tethered layers and reference samples calculated from the
HCT fits.
Sample
2k Ref 43
2802 70
2802 90
Temperature
(°C)
γ (Surface Tension)
(mN/m)
η
(Pa·s)
110
120
110
120
110
33
32
33
32
33
1100b ± 55
250b ± 13
5000a ± 250
1000a ± 50
2600a ± 130
120
32
550a ± 28
70000a ± 3500
2h48 42
2h48 99
48k Ref 35
160
31
b
10000b ± 500
10000b ± 500
48k Ref 95
a
10000a ± 500
ηeff
ηref
The only way for these effective viscosities to be equal to the actual viscosities
would be for the impact of the tethered chains on the mobility of the material in the film
to somehow be uniform with depth and this seems unlikely.
At the very least, for the
impact of the tethered chain to be uniform with depth the tethered chain would need to
stretch through the entire film, no matter what the h, and it is clear this cannot be the case.
It is already known [111] that such tethered chains, when in contact with untethered
chains, can stretch only to a finite extend, and that the tethered chain segment density
profile is never uniform with depth. The tethered chain segments are preferentially
located near the substrate in a manner similar that suggested in the schematic of Figure
6.3. The most likely situation, then, is that there is a gradient in viscosity with depth as
suggested by the notations to the left of the film schematic. An effective viscosity
171
calculated from the HCT would then represent a lower bound for the value of the
viscosity in the bottom of the layer and an upper bound for the viscosity near the surface.
We do not have available to us a model capable of describing the surface
fluctuations of film with an arbitrary profile of viscosity with depth.
Therefore, to
capture some basic features of the tethering effect for purposes of providing some ability
to predict the behavior of a partially tethered system we turn a first order model in which
we imagine the film to consist of two layers, one at the substrate with substantially
enhanced viscosity, ηtethered, due to tethering, and another near the surface containing only
untethered chains or only a small density of segments from tethered chains of much
smaller viscosity, ηuntethered. The viscosity of this upper layer might still be larger than
that of pure untethered chains in bulk, due to the influence of the underlying high
viscosity layer. This model is suggested in the Figure 6.3 by the designation "Model I". In
the simplest version of Model I we might take ηtethered to be effectively infinite so that it
plays no role in the surface dynamics and then estimate a value of ηuntethered that would
yield the observed behavior. The simulated τ vs. q|| data for such a model to match the
experimental data for 2802 70 and 2802 90 are shown in Figures 6.1 and 6.2. We find
then that the surface behavior of 2802 70 is equivalent to that of a 41nm film of pure 2.6k
untethered chains and the behavior of 2802 90 is equivalent to that of a 65nm thick film
of pure untethered chains at 110 °C, as illustrated in Figure 6.5.
For 120 °C the values
are 42 and 66 nm. If this model were strictly applicable we could expect that if h
increases by 20 nm from 70 to 90nm, then the h of the effective layer should also increase
by 20 nm, from 41 nm to 61 nm.
Since the effective layer increases a bit more than this
one can see that this approximation is still missing some aspect of the phenomenon,
172
which is probably a broader gradient in viscosity rather than a step change. To refine the
model further we consider the structural information on the tethered chain conformation
that can be obtained from the NR data.
Figure 6.5: Using Model I in Figure 6.3, 2802 70 was divided by two layers : tethered and
untethered layer. The h of untethered layer was 41 nm and 42 nm at 110 °C and 120 °C,
respectively.
The conformations of the 28k tethered chains were inferred from a measurement of
85 nm partially tethered layer with 2.6k untethered chains, 2802 85.
Figure 6.6 shows
the NR curve and SLD profile for 2802 85. The SLD of the pure tethered chains should
be 1.4 × 10-6 Å -2 and the SLD value of pure untethered 2.6k chains was determined from
the SLD value in the plateau region on the left of the plot to be 5.8 × 10-6 Å -2. The SLD
value increased from a point at the top of the epoxide layer, which we refer to here for
convenience as the "border point", to the point where the plateau corresponding to melt
composed of only untethered chains begins. This gradient exists because the sample is
not a blend of untethered chains, but rather a complex single phase blend of tethered and
untethered chains. The fraction of untethered chains at any depth can be estimated, in the
173
limit of incompressibility, by assuming that the SLD of the blend is a linear combination
of the SLDs of the components, i.e.
SLD  tethered SLDtethered  1  tethered SLDuntethered ,
(6.1)
where ϕtethered is the volume fraction of tethered chain segments and SLDtethered their SLD.
The degree to which the SLD at the border point is determined by untethered chains is a
measure of the ease with which the untethered chains penetrate the brush. The border
point SLD values was 3.3 × 10-6 Å -2 for 2802 85, which corresponds to a composition of
57 vol% of tethered chain.
The penetration depth (λ) of each tethered chains in the untethered layer could be
learned by measuring the depth between the region of the film with the highest tethered
chain segment density and the region corresponding to a pure untethered chain layer.
Before calculating the experimental values let us deal with the theoretical values. Since
the molecular weight of untethered chains is low for 2802 85, Equation 2.28 (Chapter 2)
can be used to calculate λ. For 2802 85 and the λ value was calculated to be 173 nm. This
clearly dramatically overestimates the experimental value. For 2802 85 the region over
which the tethered chains extend reaches from the border point at a depth of 86 nm to the
start of the plateau at 60 - 64 nm, depending how an approach to the plateau value is
demanded. This corresponds to a 22 – 26 nm h of the region containing the tethered chain
segments (Figure 6.7). It may be supposed that in any partially tethered layer of this type
the tethered chains are essentially stretched out as far as possible for film h values of ca.
30 nm or greater.
174
0
Log (Reflectivity)
-1
Data
-2
Fitting
-3
-4
-5
-6
0.00
0.05
0.10
0.15
0.20
qz (Å-1)
Scattering Length Density (Å-2)
a)
7.E-06
6.E-06
5.E-06
4.E-06
3.E-06
2.E-06
1.E-06
0.E+00
00
20
200
40
400
60
600
80
800
z (<--Surface/Substrate-->)
Distance from Air (nm) (Å)
b)
Figure 6.6: (a) NR curve (○) and its model fit (
calculated from the model fit.
175
) of 2802 85 and (b) the SLD profile
22 – 26 nm
Figure 6.7: Cartoon of the tethered chain conformation in the partially tethered layer
sample 2802 85.
Returning to trying to create put together a picture of the dynamic behavior of the
partially tethered layer, we make the following observation. Although the 28k tethered
chains only extend about 26 nm into the partially tethered film they are so effective at
slowing flow in the film that even at the h value of 40 nm the surface fluctuations cannot
be seen.
On the time scales observable in our experiment a region from 26 to 40 nm at
the bottom of the partially tethered film can be viewed as immobile. The "2802 70"
sample is actually 69 nm thick at the 110 °C.
layer of pure untethered chains.
modeled in Model I as immobile.
It behaves as though it were a 41 nm thick
That leaves a bottom region of h 28nm that would be
If we assume that this same 28 nm bottom layer is
immobile in 2802 90 (which is really 87 nm at 110 °C), can we reconcile the 2802 90
data?
That sample behaves in a manner equivalent to a 66 nm thick film of pure
untethered chains.
The sum of 28 nm and 66 nm gives 94 nm, somewhat overestimating
the actual film h. That is, in the thicker partially tethered layer, in the context of Model
I the immobile layer at the bottom must be taken to be a bit thinner, about 22 nm.
176
We may now return to the 2802 45 sample. If the bottom 28 nm section of that
sample were essentially immobile on the times scales to which we are sensitive, should
we expect to see the fluctuations due to a 17 nm layer atop that bottom section?
If no
confinement effects were present the predicted τs are shown in Figure 6.8. We find that
if the sample had had a top layer of 17 nm PS unperturbed by confinement effects, the
surface dynamics would easily have been observable at 110 °C and 120 °C. Since they
were not observable even at 180 °C there clearly was a strong effect of the bottom layer
containing tethered chains on the top region that was free of segments from tethered
chains. In fact, simulations (Figure 6.9) show that if no confinement effects were present
it should have been possible at 110 °C and 120 °C to observe dynamics even from
untethered chain films as thin as 3 and 5 nm, respectively, if they were sitting atop rigid
substrates not interacting with the untethered chain layers, free of confinement effects.
For a 2.6k linear PS film as thin as 17 nm we have no XPCS measurements to indicate
how strong confinement effects might be. For example, it is possible that there could be
confinement effects due to van der Waals interactions for such a thin film [158]. The
most pertinent information we have for thin polymer films is for films of cyclic PS chains.
Wang et al. [159] see that for 14k cyclic PS chains at a film thickness of 40 nm on silicon
there are no confinement effects.
For films of thickness 22 nm, confinement slows the
surface relaxation by a factor of two. A factor of two slowing due to confinement on a
rigid substrate is not sufficient to explain the fact that we are unable to see any surface
fluctuations for the 2802 45 sample.
177
2k Ref simulated at 41nm at 110C
120C
1.E+04
τ (sec)
1.E+03
1.E+02
1.E+01
1.E+00
0.0001
0.0002
0.0003
0.0004
0.0005
q||(Å )
-1
Figure 6.8: τ vs. q|| curve of simulated τ values for 2.6k dPS layers assuming h = 17 nm at
110 °C (red line) and 120 °C (blue line). The τ values are within the measurement
window of the XPCS (τ < 4000 s) for most or all of the q range studied in this work. The
η and γ values for the simulation were η = 1100 Pa·s and 250 Pa·s and γ = 33 mN/m and
32 mN/m which were used to fit 2k Ref 43 at 110 °C and 120 °C, respectively.
178
1.E+06
τ (sec)
1.E+05
1.E+04
1.E+03
0.0001
0.0002
0.0003
q||
110C
120C
0.0004
0.0005
(Å-1)
Figure 6.9: Plot of slowest simulated dynamics readily observable with the XPCS setup
used, (i.e. τ > 3000 s over appreciable range of q||) corresponds to 5 nm of untethered
2.6k chains at 110 °C and 3 nm at 120 °C.
Such slow surface dynamics of 2802 45 can only be explained if the lower part of
the film containing tethered chains is imagined to have a strong effect on the upper part
of the film, for example, as in a two-layer model with having strong confinement effect.
As others [20, 21, 65] have reported, for a thin polymer layer, the favorable interaction
between a substrate and chains slows the surface dynamics. If any interaction such as van
der Waals force or physical pinning between the two layers exists, the dynamics of the
thin untethered layer of 2802 45 with huntethered = 17 nm can be slowed down to be out of
the XPCS measurement window. However, for thicker layers such as 2802 70 and 2802
90, the effect of the interaction with the substrate is less important. It has less effect on
the surface fluctuations. Figure 6.10 shows values calculated for htethered and huntethered for
179
2802 45, 2802 70 and 2802 90 for the two layer model to agree with the experimental
data. The htethered value increases as htotal decreases. This suggests that the interaction with
the tethered layer, whatever it may be, leads to larger htethered for 2802 45 and becomes
diluted for larger htotal.
To some degree it may be possible to explain the results for 2802 70 and 2802 90
by appealing to an analogy to the flow of solvent between opposing hard surfaces
covered with adsorbed polymer layers, a phenomenon studied by others [153 - 157]. Our
short chain polymers are like a solvent.
In this analogy we further imagine our surface
with tethered chains to correspond to the lower of the two opposing surfaces. Wu and
Cates [153 - 155] defined a hydrodynamic thickness, RH, as illustrated in Figure 6.11a.
When tethered chains are mixed with a solvent, they hinder the flow of solvent. Below a
certain depth within the tethered chain layer, the solvent becomes immobile. However,
above the immobile layer, the solvent is mobile. The thickness of this immobile layer is
RH [153 - 155] and we liken RH to htethered. Dhinojwala et al. [156, 157] argued that the RH
value gets smaller as the distance between two tethered layers, D, becomes smaller
(Figure 6.11b).
The calculated values shown in Figure 6.10 suggest that this
correspondence between RH and htethered works reasonably well for 2802 90. That is, the
apparent thickness of the layer in which there is normal hydrodynamic flow of the 2.6k
chains, huntethered, is the same as what would be expected for (D-RH)/2 for two surfaces far
apart.
180
Figure 6.10: The calculated htethered and huntethered for 2802 45, 2802 70 and 2802 90 at
110 °C and 120 °C.
a)
b)
Figure 6.11: (a) Schematic illustrating the definition of RH and D for flow of solvent
between opposing brush covered surfaces as in Dhinojwala et al. [156, 157]. The red
broken line corresponds to the plane at which solvent velocity is zero. Solvent does not
flow between this plane and the hard surface. (b) RH decreases as D decreases.
181
However, the analogy is not able to explain why htethered for 2802 70 is larger than
that of 2802 90. If one assumes that the surface fluctuations of untethered chains can be
viewed as a manifestation of hydrodynamic flow, htotal is that analog of D/2. So, we
would expect htethered to show the same trend with htotal as RH with D/2. In contrast to this
expectation, htethered increases slightly as htotal drops from 90 to 70 nm, while the analog
system shows a decrease in RH as D/2 drops.
For the analog system studied by
Dhinojwala, et al. [156, 157], the solvent above the tethered chain layer is always mobile.
This is certainly different from the surface dynamics of 2802 45 for which the whole
layer appeared to be immobile. However, already for htotal of 70 nm the analogy breaks
down, meaning that there is a phenomenon operating in our case that does not play a role
in the system studied by Dhinojwala et al.[156, 157].
The fact that the tethered layer influences chains that are adjacent to, but outside of,
the tethered layer may result from the "neighboring layer" effect seen by Torkelson et al.
[14, 160, 161] for films of untethered chains. They report that the Tg of a 14 nm thick
layer of PS untethered chains is increased by the presence of a several nanometers wide
interface with an adjacent layer of less mobile (higher Tg), immiscible polymer such as
poly(methyl methacrylate) or poly(2-vinylpyridine).
This occurs even though the
distance over which intermixing of chains from adjacent layers occurs is smaller than for
our samples. Furthermore, in their most recent work Torkelson et al. [160] have shown
that the surface fluctuations on a layer of untethered, but entangled, PS chains are slower
when the underlayer's modulus is increased.
Even though both molecular weights in our
system are below the critical molecular weight for entanglement, and therefore there are
no entanglements, the bottom layer containing tethered chains does have some elastic
182
character. However, no modulus is evident in the shape of τ/h vs. q||, so it appears that the
much larger viscosity of the lower layer is the more important characteristic.
We have no data from the XPCS or NR which demands that there be only two
layers in the film of two strongly different viscosities. An alternative model, albeit more
complex, is one in which the effect of tethering effect varies with depth within the film.
Figure 6.12 shows a schematic of such a model to explain the dramatically slowed
surface dynamics of 2802 45. Where the composition of tethered chain segments is larger,
down at the bottom of the film, the polymer layer will have more slowed dynamics.
These could involve both higher viscosity as well as some elastic character, because
tethered chain layers have elastic character even if the molecular weight of the tethered
chains is below the entanglement molecular weight [56]. As the composition of tethered
chain segments decreases with approach to the surface, the chains will have better
mobility and viewed as a continuum the layer material has properties closer to those of a
viscous liquid. Therefore, the dynamics of the 2802 45 probably were not homogenous
with depth, but varied gradually with depth. Though the surface region and untethered
layer would have had enough mobility for surface dynamics measurable with XPCS if
that layer had been on a silicon wafer, the surface dynamics of 2802 45 were not
measurable because they depended on the summation of all dynamics effects through the
whole layer.
183
Figure 6.12: Since the tethered chain segment density has a gradient through the film, the
surface dynamics of 2802 45 could also be viewed as resulting from a gradual change
(gradient) in dynamics (viscosity) through the film depth. Also there is a gradient in
elastic character, potentially. The surface region with no tethered chains is just viscous.
Deep in the tethered chain region there may be a viscoelastic character.
The surface dynamics dependence on h of a partially tethered film was further
studied with a series of samples in which the untethered chains had a higher molecular
weight of 48k, but also the tethered chains were of higher molecular weight (200k) and
lower σ. For 2h48 42, 2h48 99, 48k Ref 35 and 48k Ref 95 (Figures A.6, A.13, A.14 and
A.15), even with Mn = 48 kg/mol, the τ values were within the measurement window of
XPCS at 160 °C. The data of τ vs. q|| are shown in Figure 6.13 and plots of τ/h vs. q||h are
presented in Figure 6.14. Three important conclusions are immediately apparent without
more detailed analysis.
First, for the h of 40 and 90 nm and chain molecular weight of
48k, there are no confinement effects apparent for the films of just untethered chains. The
scaling with h expected from the HCT for films of pure untethered chains is observed for
the 40 and 90 nm thicknesses. Secondly, there is still a strong tethering effect in the 40
nm partially tethered film, even though the Tg of the untethered chain (Tg = 99 °C) is now
very close to that of the tethered chain (Tg ≈ 100 °C) and the σ is low (ca. 0.02
184
chains/nm2). Thirdly, and surprisingly, when the h of 90 nm is reached for the partially
tethered film, the effect of tethering can no longer be resolved.
1.E+04
τ (sec)
1.E+03
2h47 40
1.E+02
1.E+01
1.E+00
47k Ref 40
2h47 90
47k Ref 90
1.E-01
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
q∥(Å-1)
Figure 6.13: τ vs. q|| curves of 2h48 42 (○), 48k Ref 35 (□), 2h48 99 (△) and 48k Ref
95 (◇) at 160 °C.
185
1.E+02
τ/h (sec/Å)
1.E+01
1.E+00
1.E-01
1.E-02
1.E-03
1.E-04
0.0
0.2
0.4
0.6
0.8
q∥ h
2h47 40
47k Ref 40
Figure 6.14: τ/h vs. q||h curves of 2h48 42 (○), 48k Ref 35 (□), 2h48 99 (△) and 48k
2h47 90
47k Ref 90
fitting
47k Ref
40 fitting η.
Ref 95 (◇) at 160 °C. The2h42
solid40
lines
are fits with the HCT
to calculate
2h47 90 fitting
47k Ref 90 fitting
Again, as a zeroth order analysis the τ vs. q|| curve of each sample was normalized
with h and the resulting curve fitted with the HCT to yield ηeff values. As Table 6.2
presents, the thinner partially tethered layer, 2h48 42 had surface dynamics consistent
with a film η value of 70000 Pa·s, which is factor of 7 larger than the viscosity of its
reference, 48k Ref 35. For the other samples the ηeff values were identical, 10000 Pa·s,
even for the partially tethered layer, 2h48 99. This result suggests that for the 90 nm h the
effect of tethering is no longer resolvable.
This result does not seem physically
reasonable, and as yet we have identified no way of explaining it.
186
Obtaining the conformation of the tethered layer for the 2h48 series samples was
more difficult that for the 2802 series.
NR data from all samples with 200k tethered
layers were difficult to fit using a models constructed from step changes in SLD
convoluted with error function broadening of the interface. This appeared to be due to
the fact that there were shallow gradients through a large fraction of each film.
An
example of an attempt to fit the NR data from a 2h48 37 sample is shown in Appendix
(Figure A.1).
The quality of fit is poor, but the SLD of the best attempt nonetheless
suggests some general features of the film structure. Most of the 200k tethered chain
segments are found in the bottom 19 nm or so of the film.
However, above that bottom
layer there is a shallow gradient in composition of 200k chain segments, indicating some
tethered chain stretch all the way to the surface.
A much better fit was obtained with a
sample in which the tethered chains are 48k and untethered chains 48k (4848 20), for
which the NR curve and SLD profile are shown in Figure 6.15. For that sample, which
has essentially the same untethered chain molecular weight as the 2h48 samples, the 48k
tethered chain, with σ of 0.039 chains/nm2 stretch roughly 15 - 16 nm.
However, this
film is very thin, so the 48k chains may have stretched farther in a thicker film.
Returning to the analysis of the 2h48 42 sample, for which strong slowing of the
surface is seen, we may once again attempt to model the partially tethered film behavior
using a model of rigid bottom layer and top layer of unconfined untethered chains.
this case we find the top layer of such a model to have a h of 22 nm.
In
This is roughly
consistent with the notion that there is a bottom layer of about 20 nm h that is
tremendously slowed by tethering.
187
An important question that arises in the case of the 2h48 42 sample is whether the
slowing of the surface fluctuations is qualitatively different from that seen in the case of
the 2802 series of samples, for which molecular weights of both chains are below the
critical molecular weight for entanglement, Mc (38k) [12]. For the 2h48 series the
molecular weights of both chains are above Mc.
188
1.E+00
1.E-01
data
Reflectivity
1.E-02
Fitting
1.E-03
1.E-04
1.E-05
1.E-06
1.E-07
0
0.05
0.1
a)
0.15
0.2
0.25
qz(Å-1)
χ = 1.97
2
Scattering Length Density (Å-2)
7.E-06
6.E-06
5.E-06
4.E-06
3.E-06
2.E-06
1.E-06
계열1
0.E+00
0
0
25
50
5
75
100
10
125
150
15
175
200
20
225
Distance From Air (nm)(Å)
z (<--Surface/Substrate-->)
b)
Figure 6.15: (a) NR curve (○) and its model fit (
4848 20 calculated from the fitting.
189
) of 4848 20 and (b) SLD profile of
6.4. Tethered Chain Conformation and Surface Dynamics
The hypothesis of this section is that the surface dynamics of a partially tethered
layer depends on the conformation of the tethered chains. To see the effect of tethered
chain conformation, two partially tethered films having different tethered chain layers,
but the same molecular weight of the untethered melt layers were prepared. NR sample
2h02 90 had the same 2.6 kg/mol deuterated polystyrene (dPS) untethered chains as 2802
85, but the molecular weight and σ are different. The dependence of tethered chain
conformations on molecular weight and σ will be observed by comparing 2h02 90 and
2802 85. After considering the difference in tethered chain conformation, the dependence
of surface dynamics on this conformation will be considered using samples 2809 92 and
2h09 100 (Table 6.3).
190
Table 6.3: Samples to be studied.
Mn (kg/mol)
of tethered
chains
200
Sample
2h02 90
d
9k Ref 80
Mn (kg/mol)
of untethered
chains
2.6
σ
(chains/nm2)a
h (nm)b
0.027±0.002
91±1
Annealingc
(temp(°C)
/time(hr))
118/16
85±1
137/9
9.4
2809 92
28
9.4
0.074±0.012
88±1
128/8
2h09 100
200
9.4
0.023±0.002
94±1
128/8
a
Calculated from σ = hρNA/Mn , where σ is grafting density, h is thickness of tethered
chains, ρ is bulk density of the tethered chain component and NA is Avogadro’s number.
b
The h of total PS layer was measured by ellipsometer with 632nm laser source and
n=1.589 at room temperature. For HCT fitting, h was obtained from analysis of the XR
curve measured at APS, 8-IDI at the XPCS measurement temperatures. The h from the
XR curves were presented in the Appendix (Table A.1).
c
The annealing was done under high vacuum (10-5-10-6Pa).
d
Reference samples were spin coated onto etched silicon wafers.
191
6.4.1. Conformation Observed with NR
Figures 6.6 and 6.16 show the NR curves and SLD profiles of 2802 85 and 2h02 90,
respectively. The distance over which the tethered chains stretched into the film was
about 45 nm for the 200k tethered chains at 0.027 chains/nm2 (Figure 6.16) and 25 nm for
the 28k tethered chains at 0.105 chains/nm2, so the 200k chains penetrate far more into
the film of 2.6k untethered chains.
Using the NR results, highly schematized illustrations of the relative conformations
of the two types of tethered chains are drawn in Figure 6.17. The segment density of the
tethered chain is higher close to the substrate and drops off as one moves up into the
sample as seen from the SLD profile in Figure 6.16b.
In Figure 6.17 we illustrate this
simplistically with a kind of triangular shape to the silhouette of the tethered chain,
though the segment density does not vary linearly with depth. As discussed in Chapter 2,
when the tethered chains are tethered more sparsely, untethered chains can penetrate
deeper into the tethered chains, resulting in more extension of tethered chains, or
equivalently, less concentration of the tethered chains down near the substrate. This
corresponds to a larger value of λ.
192
1.E+00
Data
Fitting
Reflectivity
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
1.E-06
0
0.05
0.1
0.15
0.2
qz (Å-1)
a)
2h02 90
Scattering Length Density (Å-2)
7.E-06
6.E-06
5.E-06
4.E-06
3.E-06
2.E-06
1.E-06
0.E+00
0
0
200
20
400
40
600
60
800
80
1000
100
z Distance
(<-- Surface/Substrate
from Air (nm)-->) (Å)
b)
Figure 6.16: (a) NR curve (○) and its model fit (
calculated from the model fit .
193
) of 2h02 90 and (b) the SLD profile
a)
b)
Figure 6.17: The expected conformations of tethered chains (black lines) mixed with
untethered chains (blue lines) were referred from the SLD profiles of (a) 2h02 90 and (b)
2802 85.
6.4.2. XPCS Result
The variation in τ values with change in the conformation of the tethered chains, is
seen via comparison between the results for 2809 92 and those for 2h09 100. The τ values
of the partially tethered layers are compared with those of the reference layer, 9k Ref 80.
The greater the increase in τ value of the partially tethered layer over that of the reference,
the greater is the effect of the tethered chains. For 9k Ref 80 and 2809 92 the values of τ
are less than 10000 s in the measured q|| range at 110 °C, 120 °C and 130 °C (Figures
A.16, A.17, A.18, A.19, A.20 and A.21). The τ vs. q|| curves of 2809 92 and 9k Ref 80 are
presented in Figure 6.18 and the most remarkable thing is that the difference in the τ
values between those two layers is very small. The same trend is seen for the h
normalized results in Figure 6.19. The slowing of the partially tethered film is seen at
194
130 °C, but hardly at the lower temperatures. The sample was annealed at 128 °C for 8
hours before the first XPCS measurement was done at 110 °C. The measurements were
done in ascending order of temperature. Table 6.4 presents the calculated η values of 9k
Ref 80 and 2809 92. At any temperature among 110 °C, 120 °C and 130 °C, the η
variation was less than factor of 1.5.
However, when it comes to the longer and more sparsely grafted 200k chains the
effect of the tethered chains becomes more distinctive. 2h09 100 still shows measurable
surface dynamics (Figures A.22, 6.20), but the difference in τ values with 9k Ref 80 is far
more distinctive than for 2809 92. The calculated η is a factor of five larger than the
reference value (Figure 6.21 and Table 6.4). The much stronger effect of the longer, more
sparsely grafted chain on surface dynamics may be explained by considering the friction
between the untethered chains and the tethered chains during hydrodynamic flow in the
film.
195
1.E+04
τ (sec)
1.E+03
1.E+02
1.E+01
1.E+00
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
q|| (Å-1)
Figure 6.18: τ vs. q|| curves of 2809 92 at 110 °C (◇), 120 °C (□), 130 °C (○) and 9k
Ref 80 at 110 °C (◇), 120 °C (□), 130 °C (○).
2809
2809
2809
9k R
9k R
9k R
196
9 80 110C
9 80 130C
9 80 120C fitting
ef 80 110C
1.E+02
τ/h(sec/Å)
1.E+01
1.E+00
1.E-01
1.E-02
1.E-03
0.0
0.2
0.4
0.6
0.8
1.0
1.2
q||h
2809
806.19:
120Cτ/h vs. q||h curves of 2809 92 at 110 °C (◇), 120 °C (□), 130 °C (○) and 9k
Figure
2809 80 110C fitting
Ref 80
110fitting
°C (◇), 120 °C (□), 130 °C (○). The solid lines are fits with the HCT to
2809
80at
130C
ef 80 130C
9k
Ref 80 120C
calculate
the η values of the polymer films.
9k Ref 80 110C fitting
ef 80 120C fitting
9k Ref 80 130C fitting
Table 6.4: η values of partially tethered layers and reference samples calculated from the
HCT fitting.
Sample
9k Ref 80
2809 92
2h09 100
Temperature
(°C)
110
120
130
110
120
130
130
γ (Surface Tension)
(mN/m)
32
33
34
32
33
34
34
197
η
(Pa·s)
1100000 ± 55000
80000 ± 4000
9000 ± 450
1200000 ± 60000
90000 ± 4500
13000 ± 650
45000 ± 2250
1.E+03
2h09 90 130Cel-deg
9k Ref 80 130Cel-deg
τ (sec)
1.E+02
1.E+01
1.E+00
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
q|| (Å-1)
Figure 6.20: τ vs. q|| curves of 2h09 100 (○) and 9k Ref 80 (○) measured with XPCS at
130 °C.
198
130C
τ/h (sec/Å)
1.E+00
1.E-01
1.E-02
1.E-03
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
q||h
0C fitting
Figure 6.21: τ/h vs. q||h curves of 2h09 100 (○) and 9k Ref 80 (○) measured with XPCS
95nm 130C
130C fitting
at 130 °C. The solid lines are fits with the HCT model to calculate the η values of the
polymer films.
One possible means of comparing the flow in the partially tethered layers to
molecular movement already studied by others is to consider diffusion of chains in a
cross-linked matrix. The cross-linked chains may also be thought of as "tethered."
As
studied by Silliescu et al. [75], linear chains in a cross-linked matrix have smaller
diffusivity. This smaller diffusivity is caused by the contacts and friction between the
chains and the matrix. The mobility of the tethered chains is far less than that of
untethered chains [57]. Therefore, just as with linear chains in a cross-linked matrix, the
tethered chains will provide contacts and friction with the untethered chains, resulting in
slower chain motion. Somehow this leads to slower surface fluctuation dynamics. A
larger penetration depth of untethered chains into the tethered chain layer provides more
199
contacts with the untethered chains. There are two factors affecting the amount of friction
that are tied up in the penetration length.
First is the fraction of the film thickness in
which this friction between tethered and untethered chains can take place. The second
is the areal density of tethered chains.
The 200k chains are less densely grafted, so there
are fewer chains per nm2 with which the untethered chains can interact.
However, this
lower grafting density also allows the untethered chains to access more readily the
bottom of the tethered chain near the substrate.
Apparently, the greater contact between
each tethered chain and the untethered chains dominates over the loss of tethered chains
when going to the lower grafting density.
This friction effect is able to explain the
difference in the surface fluctuation dynamics of 2809 92 and 2h09 100.
We consider again whether the behaviors of these samples may be modeled using
simply two layer models of the film dynamics. Figure 6.22 presents comparison of the
experimental data for 2809 92 and 2h09 100 at 130 °C with simulated curves of
normalized relaxation times for two layer models.
The bottom layer of the model is
imagined to consist of primarily tethered chains and be immovable.
This bottom layer
has thickness htethered. The top layer is imagined to consist of only 9.4k untethered dPS
chains and have thickness huntethered.
For 2h09 100 huntethered is 55nm and for 2809 92 it is
76 nm, corresponding to the high mobility of the surface of 2809 92. For 2h09 100 htethered
is 45nm. For 2809 92, htethered is 17nm. These simulated values are consistent with the
illustration in Figure 6.17. We recall that for 2802 70 htethered in the two layer model was
28 nm, more than 50% higher that htethered for the 2809 92 sample. The tethered chain
is more swollen or extended with the 2.6k untethered chains than with the 9.4k untethered
chains. The greater miscibility of untethered with tethered chains determined by the
200
untethered chain molecular weight influences the surface fluctuation dynamics.
1000
τ (sec)
100
2809 92 130C
2h09 100 130C
2h09 100 simulation
10
1
0.0000
2809 92 simulation
0.0002
0.0004
0.0006
q|| (Å-1)
Figure 6.22: τ vs. q|| curves of 2h09 100 (○) and 2809 92 (○) at 130 °C. The blue and
yellow solid lines are the simulated τ values of 9.4k dPS layers assuming h = 55 nm and
76 nm at 130 °C, respectively. The η and γ values for the simulation are 9000 Pa·s and 34
mN/m, which are the values that were used to fit 9k Ref 80 at 130 °C.
As discussed in Chapter 5, one interesting point was that, for the films that had
2.6k untethered chains and were ca. 45 nm thick the surface dynamics for the film with
more densely grafted 28 k chains (2802 45) were far slower than those for the less
densely grafted 200k chains (2h02 45). On the other hand, in this chapter, we have seen
that for thicker films containing 9.4k untethered chains the surface dynamics of the film
with less densely grafted 200k chains (2h09 100) were slower than those of the film with
more densely grafted 28k chains (2809 92). Altering the molecular weight of the
201
untethered chains resulted in a reversal of the behavior. A possible reason for this
phenomenon is that the effect of molecular weight on the miscibility between the tethered
and untethered chains was dominant in determining the surface fluctuation dynamics. For
the 2.6k untethered chains, the 28k tethered chains will be well mixed with untethered
chains, forming a tethered layer having a very large value of ηtethered and this value will be
mainly affected by the confinement due to the covalent bonds to the substrate. However,
the 9.4k untethered chains will be much less well mixed with the untethered chains than
will be the 2.6k chains. Therefore, the movement of 9.4k chains will be less affected by
the confinement effect of covalent bonds. Rather than the effect from the substrate, the
effect from the penetrating chains will be more dominant. How controlling miscibility
between the tethered and untethered chains alters the surface fluctuation dynamics will be
discussed in more detail in Chapter 7.
202
6.5. Summary
Models to explain the dynamics behavior of partially tethered films were proposed.
For the first model, it was assumed that the whole film had a single ηeff and this value was
determined by fitting to the HCT. Though the fits of eachτ/h points looked successful, the
HCT hypothesizes a homogeneous viscous layer, not a mixture of tethered chains with
untethered chains. To overcome this discrepancy, a two-layer model composed of a
tethered layer and an untethered layer was proposed. This model explained the dynamic
behavior of the 2802 series sample with htotal = 90 nm well. The chains in the tethered
layer are immobile and the chains in the untethered layer are assumed to show normal
hydrodynamic flow. However, as htotal decreased, htethered increased and this trend was the
opposite to the trend seen for hydrodynamic flow between two opposing brush covered
surfaces, for which the immobile layer thickness decreased as the distance between the
surfaces decreased [156, 157].
The surface dynamics for htotal = 45 nm were too slow to be measured. For the twolayer model, to describe this behavior the bottom, immobile layer has to be as thick as the
entire film. The reason for this slow dynamics was suggested as that huntethered of 2802 45
(17 nm) was thin enough to be affected by confinement effect from the tethered layer
giving slower dynamics to the untethered layer. However, this effect will be less
significant for thicker layers. The fact that htethered of 2802 90 was at least 4 nm smaller
than that of 2802 70, supported the argument that the confinement effect is more
important for thinner films. Based on the arguments from Torkelson et al., [14, 160, 161]
the neighboring layer effect is reponsible for the slow dynamics of the untethered layer of
2802 45 as well. The modulus and high viscosity of the tethered layer, would have caused
203
the slow dynamics of the untethered layer.
The NR data supported the third model presenting that the tethered chain segment
composition varied as a gradient within the layer. Since the friction between tethered and
untethered chains slowed the dynamics, the hydrodynamics of a partially tethered layer
changed gradually with depth.
The surface dynamics of a partially tethered layer were affected by both the
confinement and penetration effects. The molecular weights of the untethered chains
determined which effect was dominant. For 2.6k untethered PS chains, 200k tethered
chains with σ = 0.02 chains/nm2 had a factor of two larger penetration depth than did 28k
tethered chains with σ = 0.1 chains/nm2. For 2.6k untethered chains, the sample with
greater penetration depth had faster surface dynamics. Due to better miscibility with
tethered chains, the 2.6k untethered chains penetrated to the substrate where a favorable
confinement effect is present. For such a structure with higher untethered chain segment
composition near the substrate, the surface dynamics varied primarily with the density of
the covalent bonds to the substrate (indicated with grafting density). On the other hand,
for 9.4k untethered PS chains, the sample with greater penetration depth had slower
surface dynamics. Due to poorer miscibility, fewer untethered chains penetrated to the
substrate region. Most of the untethered chains remained in the middle and top region of
the layer. As a result, the larger penetration of the 200k chains as compared to that of the
28k chains, was the dominant effect.
204
CHAPTER VII
SURFACE DYNAMICS AFFECTED BY MISCIBILITY BETWEEN TETHERED AND
UNTETHERED CHAINS
7.1. Objective
In the previous chapter, it was shown that when the conformations of the tethered
chains are altered by changing the grafting density (σ) and chain molecular weight, the
surface fluctuation dynamics of the partially tethered layer can be strongly affected. In
this chapter, we determine in greater detail how the interactions between tethered and
untethered chains impact the surface dynamics through changes in conformation and
through plasticization. In section 7.1 we look at how the variation of the conformation
of the tethered chains with mixing between the tethered and untethered chains affects the
surface dynamics. Then in section 7.2 we consider the relative importance of the effects
of chain plasticization and chain swelling when the untethered chains are very short.
Finally, in section 7.3 we delve further into the effect on chain conformations of χ
between the tethered and untethered chains.
7.2. Miscibility Determined by Molecular Weight
In this portion of the dissertation the objective is to study the effect of chain
molecular weight on tethered chain conformation and then on the surface dynamics. To
do this layers of tethered chains with a given molecular weight and similar σ were mixed
205
with melts of untethered chains of various molecular weights. The variation of
conformations of the tethered chains with molecular weight of the untethered chain was
observed using neutron reflectivity (NR) and comparisons made between partially
tethered layers and reference layers containing no tethering.
7.2.1. Sample Description
Therefore, for a given molecular weight of untetehred chain, a partially tethered
layer and a reference sample having only untethered chains were prepared for X-ray
photon correlation spectroscopy (XPCS) measurements. Table 7.1 presents the
characteristic properties of the samples used in this section. The number average
molecular weight of untethered chains, Mn, was varied among four values: 2.6, 9.4, 48
and 120 kg/mol. Bulk Tgs values measured with differential scanning calorimetry (DSC)
for the 2.6k, 9.4k and 48k materials were 68 °C, 88 °C and 99 °C, respectively. To
provide NR contrast, the tethered chains were hydrogenous PS (hPS) and the untethered
chains were deuterated PS (dPS). Also, to provide better consistency between NR and
XPCS data, the untethered chains for XPCS samples were also dPS. The thicknesses of
the total samples (h) were around 10 nm, 40 nm and 90 nm. The tethered chains were
prepared using hPS-COOH having Mn = 200 kg/mol and Mn = 28 kg/mol providing σ
values of 0.02 ± 0.004 chains/nm2 and 0.10 ± 0.03 chains/nm2, respectively.
206
Table 7.1: Samples to be studied
2k Ref 43
2.6
43±1
9k Ref 80
9.4
85±1
Annealingc
(temp(°C)/
time(hr))
Annealed
in XPCS
137/9
48k Ref 95
48
91±1
135/8
Sample
Mn(kg/mol)
of tethered
chains
Mn(kg/mol)
of untethered
chains
σ
(chains/nm2)a
h (nm)b
2802 90
28
2.6
0.105±0.012
83±1
110/15
2809 92
28
9.4
0.074±0.012
88±1
128/8
2h02 13
200
2.6
0.023±0.002
13±1
115/17
2h02 45
200
2.6
0.020±0.002
43±1
119/14
2h09 42
200
9.4
0.016±0.002
40±1
128/14
2h09 100
200
9.4
0.023±0.002
94±1
128/8
2h48 99
200
48
0.019±0.002
96±1
135/8
2h1h 15
200
120
0.021±0.002
15±1
145/16
a
Calculated from σ = hρNA/Mn , where σ is grafting density, h is thickness of tethered
chains, ρ is bulk density of the tethered chain component and NA is Avogadro’s number.
b
The h of total PS layer was measured by ellipsometer with 632nm laser source and
n=1.589 at room temperature. For HCT fitting, the h was obtained analysis of the XR
curve measured at APS, 8-IDI at the XPCS measurement temperature. The h from the XR
curves are presented in the Appendix (Table A.1).
c
The annealing was done under high vacuum (10-5-10-6Pa).
207
7.2.2. NR Results
Figures 7.1 and 7.2 present the NR curves, model fits and SLD profiles of 2h02 13
and 2h1h 15. The SLD profiles indicate the conformations of the tethered chains for these
two values of molecular weight of surrounding untethered chains. The SLD of pure
tethered chains, hPS, was 1.4 × 10-6 Å -2. For untethered chains, because of the different
densities due to molecular weight, the pure 2.6k and pure 120k dPS had somewhat
different values of SLD, 5.8 × 10-6 Å -2 and 6.6 × 10-6 Å -2, respectively. Comparing the
two SLD profiles, the tethered and untethered chains 2h02 13 seem to have far better
miscibility (more stretched tethered chain) in sample 2h02 13 than in 2h1h 15. In fact, the
vol% of hPS on the bottom (border point with the epoxide layer) was 67 vol% and 91 vol%
for 2h02 13 and 2h1h 15, respectively. In addition, on the surface, the 2h02 13 had 56 vol%
hPS, while 2h1h 15 had only 13 vol%. As the profile shows, the hPS concentration
gradient with respect to the depth, z, was more dramatic in 2h1h 15 (4.3 vol%/nm) than in
2h02 13 (0.8 vol%/nm).
208
0
Log(Reflectivity)
-1
data
fitting
-2
-3
-4
-5
-6
-7
0.00
0.05
0.10
0.15
0.20
qz(Å-1)
a)
Scattering Length Density (Å-2)
4.E-06
4.5E-6
4.E-06
4.0E-6
3.5E-6
3.E-06
계열1
3.0E-6
3.E-06
2.5E-6
2.E-06
0
0
100
10
50
5
150
15
200
20
z (<-- Surface/Substrate-->) (Å)
Distance from Air (nm)
b)
Figure 7.1: (a) NR curve (○) and its model fit (
calculated from the model fit.
209
) of 2h02 13 and (b) the SLD profile
0
Log (Reflectivity)
-1
data
fitting
-2
-3
-4
-5
-6
0.00
0.05
0.10
0.15
0.20
0.25
qz (Å )
-1
a)
Scattering Length Density (Å-2)
7.E-06
6.E-06
5.E-06
4.E-06
3.E-06
계열1
2.E-06
1.E-06
0
50
5
100
10
150
15
200
20
250
25
z (<-Surface/Substrate
-->) (Å)
Distance
from Air (nm)
b)
Figure 7.2: (a) NR curve (○) and its model fit (
calculated from the model fit.
210
) of 2h1h 15 and (b) the SLD profile
To see the length to which a tethered chain could stretch, partially tethered layers
with h over 40 nm were used. The NR curve fittings and SLD profiles of 2h02 45 and
2h09 100 are shown by Figures 7.3 and 7.4. The proportion of tethered chains at the top
of 2h02 45 is 3 vol% indicating that the 200 kg/mol tethered PS can extend longer than
40 nm into the 2.6 kg/mol of untethered PS. However, for 2h09 100, the point equivalent
to 3 vol% of tethered chain is 58 nm from the air surface. Considering the polymer layer
starts from 89 nm on the depth scale, the maximum length of the stretched tethered chain
is 31 nm. The extension length of 200 kg/mol tethered chains into the 9.4 kg/mol
untethered chains was a factor of 3/4 of that into the 2.6 kg/mol untethered chains. In
addition, the composition of hPS on the bottom for 2h02 45 was 16 vol%, but, with
poorer miscibility, for 2h09 100 the value was 42 vol%. This means that the tethered
chains showed more compact conformations with higher molecular weight of untethered
chains. The literature provides one value of extension length [111] for comparison. When
29k dPS untethered chains were mixed with 80k hPS tethered chains with unknown σ, the
tethered chains stretched to 30 nm. Since the σ values of PS tethered chain layers were
not presented in the literature, the literature did not show the relationship between σ and
tethered chain extension. Simply considering the value of stretching length itself, samples
in this study show comparable extension behavior with the data in the literature [111].
211
1.E+00
Reflectivity
1.E-01
1.E-02
Data
Fitting
1.E-03
1.E-04
1.E-05
1.E-06
0
0.05
0.1
a)
0.15
0.2
0.25
qz(Å-1)
Scattering Length Density (Å-2)
7.E-06
6.E-06
5.E-06
4.E-06
3.E-06
2.E-06
1.E-06
0.E+00
0
0
100
10
200
20
300
30
400
40
500
50
z (<-Surface/Substrate-->)
Distance
from Air (nm)(Å)
b)
Figure 7.3: (a) NR curve (○) and its model fit (
calculated from the model fit (b).
212
) of 2h02 45 and (b) the SLD profile
1.E+00
data
Reflectivity
1.E-01
fitting
1.E-02
1.E-03
1.E-04
1.E-05
0
0.05
0.1
0.15
qz(Å )
-1
a)
Scattering Length Density (Å-2)
7.E-06
6.E-06
5.E-06
4.E-06
3.E-06
2.E-06
2h099 SLD
1.E-06
0.E+00
0
0
20
200
40
400
60
600
80
800
100
1000
Distance from Air (nm) (Å)
z (<--Surface/Substrate-->)
b)
Figure 7.4: (a) NR curve (○) and its model fit (
calculated from the model fit.
213
) of 2h09 100 and (b) the SLD profile
7.2.3. XPCS Results
Results from studies of the surface fluctuations of layers of pure untethered chains
of different molecular weights have shown that the value of bulk Tg itself has a very large
effect on the dynamics. To ensure that we study differences due to tethering and not
simply due to variations in Tg with molecular weight, we compare surface fluctuation
dynamics at comparable values of T - Tg rather than at constant T. To make the XPCS
comparison at T - Tg ≈ 40 °C, the value of T for the reference sample and partially
tethered layer sample with 48 kg/mol untethered chain was 140 °C, with 9.4 kg/mol
untethered chain was 130 °C and with 2.6 kg/mol untethered chain was 110 °C. All
samples showed fast enough surface dynamics to be measured with the XPCS (Figures
A.3, A.12, A.18, A.21, A.23, A.24 and A.25). The g2 functions were fitted with Equation
4.4 to yield τ vs. q|| curves for each sample (Figures 7.5, 7.6 and 7.7). Figure 7.8 presents
τ/h vs. q||h curves of partially tethered layers with 48 kg/mol and 9.4 kg/mol untethered
chains and thicknesses, h, ca. 90 nm. Table 7.2 presents the η values of each sample
calculated from the hydrodynamic continuum theory (HCT) fitting. The η value of 2h48
99 was only a factor of 1.5 - 2 higher than the value for its reference (48k Ref 95).
However, for the samples with 9.4 kg/mol chains, the η value increased by nearly a factor
of four due to partial tethering.
214
1000
τ (sec)
100
10
1
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
q|| (Å-1)
9k Ref 80Figure 7.5: τ vs. q|| curves of 9k Ref 80 (□), 2h09 100 (◇) at 130 °C and of 48k Ref 95
2h09 90
47k Ref 90
(○), 2h48 99 (△) at 140 °C.
2h47 90
215
10000.0
2h09 40
9k Ref 80
2h02 40
2k Ref 40
τ (sec)
1000.0
100.0
10.0
1.0
0.1
0.0000
0.0003
0.0006
0.0009
0.0012
0.0015
q||(Å-1)
Figure 7.6: τ vs. q|| curves of 2h09 42 (□), 9k Ref 80 (○) at 130 °C and of 2h02 45 (◇),
2k Ref 43 (△) at 110 °C.
216
1000.0
2809 80
9k Ref 80
τ (sec)
100.0
2802 80
2k Ref 40
10.0
1.0
0.1
0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008
q||(Å-1)
Figure 7.7: τ vs. q|| curves of 2809 92 (□), 9k Ref 80 (○) at 130 °C and of 2802 90 (◇),
2k Ref 43 (△) at 110 °C.
217
τ/h(sec/Å)
1.000
0.100
0.010
0.001
0.1
0.3
0.5
0.7
q||h
Figure 7.8: τ/h vs. q||h curves of 9k 9k
RefRef
80 (□),
and80
offitting
48k Ref
80 2h09 100 (◇) at 1309k°CRef
90 fitting
95 (○), 2h48 99 (△) at 140 °C. The2h09
solid 90
lines are fits with the HCT2h09
to calculated
values
of η.
47k Ref 90
47k Ref 90 fitting
2h47 90
2h47 90 fitting
Table 7.2: η values calculated from the HCT fits.
Sample
Temperature (°C)
9k Ref 80
2h09 100
48k Ref 95
2h48 99
130
130
140
140
Surface Tension (γ)
(mN/m)
32
32
33
33
218
η (Pa·s)
10000 ± 500
45000 ± 2250
140000 ± 7000
240000 ± 12000
However, since the molecular weight of 48 kg/mol is high enough to be in the
entanglement regime [12], and the h was rather large compared to the length to which the
tethered chain can stretch, one might be afraid that the variation in surface dynamics
could not be observed distinctively. Therefore, to see the variation in τ values more
clearly, partially tethered layers with h ≈ 40 nm and having lower molecular weights of
untethered chains of 2.6 kg/mol and 9.4 kg/mol were prepared. Figure 7.9 shows the τ/h
vs. q||h curves of 2h09 42 and 2h02 45 with those of their reference samples. The τ value
was increased more by partial tethering when the untethered chain molecular weight was
lower. In fact, the η value was increased by factor of 80 due to partial tethering for 2.6
kg/mol, but by only factor of 15 for 9.4 kg/mol (Table 7.3).
In Figure 7.9, while other samples have h around 40 nm, the thickness of 9k Ref 80
is 85 nm. The τ/h value of the 40 nm reference for 2k chains sample could potentially be
larger than the value would be if a 85 nm thick reference had been used for the 2k chains
as well, due to pinning of chains to the substrate (modulus effect) or interaction with the
substrate. According to Jiang et al. [21], the modulus effect plays a role at h < 4Rg where
Rg is the unperturbed radius of gyration of the untethered chain in a melt film of only
untethered chains. The calculated Rgs of 9.4k and 2.6k dPS are 2.5 nm and 1.3 nm,
respectively. Since the h values of 9k Ref 80 and 2k Ref 43 were 85 nm and 43 nm, the
values of h/Rg for both samples are over 30. Therefore, we assume that h ≈ 40 nm is large
enough to ignore the influence of any modulus effect or weak interaction (van der Waals
force) with the substrate on the surface relaxation time.
219
1.E+01
τ/h (sec/Å)
1.E+00
1.E-01
1.E-02
1.E-03
0.05
0.15
0.25
0.35
0.45
q||h
Figure 7.9: τ/h vs. q||h curves of 2h09 422h09
(□),409k Ref 80 (○) at2h09
13040
°Cfitting
and of 2h02 45
9k Ref 80
9k Ref 80 fitting
(◇), 2k Ref 43 (△) at 110 °C. The solid lines are fits with the HCT to calculate η values.
2h02 40
2h02 40 fitting
2k Ref 40
2k Ref 40 fitting
Table 7.3: η values calculated from the HCT fits.
Sample
Temperature (°C)
γ (mN/m)
η (Pa·s)
9k Ref 80
2h09 42
2k Ref 43
130
130
110
32
32
33
10000 ± 500
150000 ± 7500
1100 ± 55
2h02 45
110
33
90000 ± 4500
220
Finally, the trend in τ values discussed above was also tested using a tethered chain
with a smaller length, but with a larger value of σ. For 28 kg/mol tethered PS, the
dependence of the surface dynamics variation on molecular weight of the untethered
chain followed the same trends as presented by Figure 7.10. However, due to smaller
value of penetration depth, λ, into the untethered layer, the variation in surface dynamics
with tethering is not as large as that shown in Tables 7.2 and 7.3 for the samples with
larger penetration. Here the η value only increased by a factor of two for 2.6 kg/mol
untethered chain and for 9.4 kg/mol the η value of the partially tethered layer was barely
above that of its reference (Table 7.4).
A contributing factor for such differences in surface dynamics will be the
conformation of the tethered chain as determined by the entropy of mixing. As shown by
NR data, more penetration of tethered chains into the untethered chains was observed at
lower molecular weight of untethered chains. If the tethered chains have more penetration
into untethered chains, the contact between these two chains will increase, causing more
friction between the tethered and untethered chains.
221
1.E+00
τ/h (sec/Å)
1.E-01
1.E-02
1.E-03
0.05
0.15
0.25
0.35
0.45
0.55
q||h
2809 80
Figure
7.10: τ/h vs. q||h curves of 2809 92 (□), 9k Ref 80 (○) at 130 °C and of 2802 90
2809
80 fitting
9k Ref 80
9k Ref 80 fitting
(◇), 2k Ref 43 (△) at 110 °C. The solid lines are fits with the HCT to calculate η values.
Table 7.4: η values calculated from the HCT fits.
Sample
Temperature (°C)
γ (mN/m)
η (Pa·s)
9k Ref 80
2809 92
2k Ref 43
130
130
110
32
32
33
10000 ± 500
12000 ± 600
1100 ± 55
2802 90
110
33
2300 ± 115
222
7.3. Plasticization Effect
An untethered chain layer with an extremely low molecular weight was used to
maximize the miscibility with the tethered chains. The surface dynamics were expected to
be the most profoundly affected for this case, but the Tg of the polymer film also needed
to be considered. In this section, not only the miscibility effect, but also the plasticization
effect which can be expected in an oligomer untethered layer will be discussed.
7.3.1. Sample Description
To see the surface dynamics at extreme low molecular weight of untethered chains,
28Oli was prepared by spun casting hPS oligomer on the top of the tethered chain layer.
The molecular weight of the untethered chain was 880 g/mol with Mw/Mn = 1.10. The Tg
of the oligomer measured with DSC was 9 °C. hPS-COOH with 28 kg/mol was selected
for tethered chain with a σ value of 0.125 ± 0.012 chains/nm2. The total h of 28Oli
measured with an ellipsometer at room temperature was 45 nm. The annealing condition
was at room temperature for 5 hrs under high vacuum.
7.3.2. XPCS Result
Figures 7.11, 7.12 and 7.13 present g2 functions of 28Oli at 20 °C, 30 °C and 40 °C,
respectively. No clear relaxation behavior was observed in the range of q|| available,
indicating that the τ value was out of the XPCS measurement window. However, as
Figure 7.14 shows, at 60 °C where T - Tg ≈ 50 °C, the surface relaxation could be
observed and the g2 function could be fit with the HCT as well (Figure 7.15). The η value
of 28Oli was 12000 Pa·s for the γ value of 34 mN/m at 60 °C.
223
1.20
1.24
1.18
1.22
1.16
g2
g2
1.26
1.20
1.12
1.18
q||=0.000138Å -1
1.16
10
100
Time delay (s)
1
1000
1.17
1.17
1.15
1.15
1.13
1.13
1.11
1.09
1
10
100
Time delay (s)
1000
1
1.15
1.15
1.13
1.13
1.11
g2
1.17
1.09
1
10
100
Time delay (s)
10
100
Time delay (s)
1000
1.09
1.07
q||=0.000373Å -1
1.07
1000
q||=0.000315Å -1
1.07
1.11
10
100
Time delay (s)
1.11
1.09
q||=0.000254Å -1
1.07
q||=0.000194Å -1
1.10
g2
g2
1
g2
1.14
q||=0.000618Å -1
1.05
1
1000
10
100
Time delay (s)
1000
Figure 7.11: g2 functions of 28Oli for several values of q|| at 20 °C. The red line is the
attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too
large to be calculated.
224
1.16
1.14
g2
g2
1.18
1.22
1.20
1.18
1.16
1.14
1.12
1.10
1.10
q||=0.000137Å -1
q||=0.000192Å -1
1.08
1
10
100
Time delay (s)
1
1000
1.16
1.17
1.14
1.15
1.12
1.13
g2
g2
1.12
1.10
1.08
1000
1.11
1.09
q||=0.000250Å -1
10
100
Time delay (s)
q||=0.000426Å -1
1.07
1.06
1
10
100
Time delay (s)
1
1000
10
100
Time delay (s)
1000
1.16
1.14
g2
1.12
1.10
1.08
q||=0.000604Å -1
1.06
1
10
100
Time delay (s)
1000
Figure 7.12: g2 functions of 28Oli for several values of q|| at 30 °C. The red line is the
attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too
large to be calculated.
225
1.18
1.22
1.16
1.20
1.14
g2
g2
1.24
1.18
1.16
1.10
q||=0.000138Å -1
1.14
q||=0.000254Å -1
1.08
1
10
100
Time delay (s)
1000
1
1.18
1.17
1.16
1.15
1.14
1.13
g2
g2
1.12
1.12
1.10
1.08
1000
1.11
1.09
q||=0.000373Å -1
10
100
Time delay (s)
q||=0.000495Å -1
1.07
1
10
100
Time delay (s)
1000
1
10
100
Time delay (s)
1000
1.17
1.15
g2
1.13
1.11
1.09
q||=0.000617Å -1
1.07
1
10
100
Time delay (s)
1000
Figure 7.13: g2 functions of 28Oli for several values of q|| at 40 °C. The red line is the
attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of τ were too
large to be calculated.
226
1.15
1.20
1.13
1.18
1.11
g2
g2
1.22
1.09
1.16
1.14
1.07
q||=0.000191Å -1
1.05
1.12
10
100
Time delay (s)
1
1000
1.16
1.13
1.14
1.11
1.12
1.09
g2
g2
1
1.08
1.05
q||=0.000309Å -1
1000
q||=0.000367Å -1
1.03
1.06
1
10
100
Time delay (s)
1
1000
1.12
10
100
Time delay (s)
1000
1.13
1.10
1.11
1.08
g2
g2
10
100
Time delay (s)
1.07
1.10
1.06
1.09
1.07
1.04
q||=0.000426Å -1
q||=0.000485Å -1
1.05
1.03
1.02
1
10
100
Time delay (s)
1
1000
10
100
Time delay (s)
1000
1.12
1.12
1.10
1.10
q||=0.000545Å -1
q||=0.000604Å -1
1.08
1.08
g2
g2
q||=0.000250Å -1
1.06
1.06
1.04
1.04
1.02
1.02
1.00
1
10
100
Time delay (s)
1000
1
10
100
Time delay (s)
1000
Figure 7.14: g2 functions of 28Oli for several values of q|| at 60 °C. The red line is the fit
using Equation 4.4 to calculate τ.
227
1.E+04
28Oli
τ (sec)
1.E+03
1.E+02
1.E+01
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
q||(Å-1)
a)
1.E+01
τ/h(sec/Å)
28Oli
Fitting
1.E+00
1.E-01
1.E-02
0.0
0.1
0.2
0.3
q||h
b)
Figure 7.15: (a) τ vs. q|| and (b) τ/h vs. q||h curves of 28Oli at 60 °C. The solid line on b is
the HCT fit to calculate η value.
228
7.3.3. Discussion
As shown in Chapter 5, the τ value of 2802 45 was out of the measurement window,
even at T - Tg ≈ 110 °C. As data from the previous section showed, when the molecular
weight of the untethered chains was lower the surface dynamics were slowed more
relative to a reference of pure untethered chains. Therefore, if that were the whole story,
the surface dynamics of 28Oli should have been much slower than those of 2802 45 at
the same T - Tg. However, even at T - Tg ≈ 50 °C, 28Oli showed measurable values of τ.
Two factors play roles in determining the phenomenon in the film with very short
untethered chains.
First of all, thanks to the extremely low molecular weight of the untethered chains,
the partially tethered layer has the best miscibility of all the combinations considered
here and the effect of the covalent bond and the friction between tethered and untethered
chain should have been maximized. As a result, the surface dynamics of this sample
should had been slower than those of 2802 45. However, effects on Tg of the film also
play a role. Using Equation 5.3, the effective Tg of the 28 kg/mol and 0.9 kg/mol blend
can be estimated. Referring to the NR data of 2802 85 (Figure 6.6), the volume fraction
of 28 kg/mol tethered chains in 28Oli (h ≈ 40 nm) can be expected to be 10 vol%. As
discussed in Chapter 5, the Tg of the tethered chains is similar to that of bulk chains.
Therefore, the Tg value of the 28 kg/mol tethered chains was assumed to be the same as
the bulk value. The density of 28 kg/mol hPS was assumed to be 1.05 g/ml and that of
0.9 kg/mol dPS was assumed to be 0.95 g/ml . Considering the vol% of tethered chain in
2802 85 and chain densities, the estimated Tg of 28Oli using Equation 5.3 [141] was
18 °C. The calculated Tg of 2802 45 was 71 °C. Such a lowering of Tg of the layer due to
229
plasticization by the oligomer chains can result in faster surface dynamics than would
otherwise be expected just on the basis of degree of interpenetration. The surface
dynamics of 28Oli were not determined by a single factor. There were contributions to
increase the τ values. However, other “faster dynamics” factors were dominant. As a
result, the surface dynamics of 28Oli became faster than those of 2802 45.
7.3.4. Sample Damage
Since the very short chains may be more susceptible to damage or problems with
the vacuum environment those might be complicating factors in comparing with the films
with higher molecular weight untethered chains. Therefore we consider those factors
briefly. Figure 7.16 presents the thermal gravimetric analysis (TGA) result for the
oligomer under nitrogen flow. The residual weight fraction at 60 °C was 99.8 wt% and so
the thermal degradation at the measurement T was negligible. We should also consider
how the low molecular weight material may behave in vacuum.
The mean molecular
weight is 880 g/mole. MALDI-ToF analysis of the molecular weight distribution can
augment our information from a size exclusaion chromatography (SEC) on what
molecular weights are present.
However, we have to be aware that MALDI-ToF MS
does not give distributions that are centered at the same molecular weight as seen with
SEC. The MALDI result presented by Figure 7.17 suggests that indeed the molecular
weight distribution of the oligomer covers molecular weights substantially lower than the
mean value. Since the XPCS measurements were run with the sample in high vacuum
condition, some of the chains of lowest molecular weight might have evaporated during
the experiment. To check for loss from the oligomer layer, X-ray reflectivity (XR) curves
230
for 30 °C, 40 °C and 60 °C were obtained during the XPCS measurements and fitted to
learn the h values at each T (Figure 7.18). Fitting the data to a model reflectivity curve
was done using Igor Pro version 6.2.2.2 with the data analysis module Motofit (ANSO).
Table 7.5 presents the values of h determined in this way and one sees a reduction in
thickness of 28Oli with increasing T. Considering that the surface fluctuation dynamics
are slower with thinner films, the fact that the surface dynamics of 28Oli are faster than
those of 2802 45 did not originate from thermal damage. We note that the values of τ/h
plotted in the figures are all calculated using the actual film thickness measured by XR at
temperature before the XPCS measurement was done and so loss of material with time
has been accounted for to at least a first order approximation.
100
90
Residual Weight %
80
70
60
50
40
30
20
10
0
0
50
100
150
200
250
300
350
400
Temperature (℃)
Figure 7.16: The TGA result of the oligomer used to prepare 28Oli.
231
450
500
Intens. [a.u.]
Intensity (A.U.)
x104
15000
1.5
1282.111
1282
1394
10000
1394.266
1.0
1506.428
1506
5000
0.5
0
0.0
600
600
800
800
1000
1000
1200
1200
1400
1400
1600
1600
1800
2000
1800
m/z
2200
m/z
Figure 7.17: The MALDI result of the oligomer used to prepare 28Oli.
1.E+08
Counts
Detector
Reflectivity
1.E+07
20 Cel-deg
60 Cel-deg
1.E+06
1.E+05
1.E+04
1.E+03
1.E+02
1.E+01
0.00
0.05
0.10
0.15
0.20
0.25
qz (Å-1)
Figure 7.18: The XR curves of 28Oli at 20 °C (blue curve) and at 60 °C (red curve).
These curves were measured before any exposure for XPCS.
232
Table 7.5: h of 28Oli at XPCS measurement T.
Temperature (°C)
Thickness (nm)
30
43 ± 1
40
40 ± 1
60
39 ± 1
7.4. Effect of Miscibility Changes Due to Interaction Parameter
The effect on surface fluctuation dynamics of changes in miscibility determined by
the Flory-Huggins interaction parameter (χ) between the tethered and untethered chains is
the main topic of this section. The variation of surface dynamics due to partial tethering
will be observed with different combinations of tethered and untethered chains.
7.4.1. Sample Description
Table 7.6 presents the samples studied in this section. PS tethered chains were used
for partially tethered layers, but to provide various χ values, the species of untethered
chains were altered. Poly(4-bromo styrene) (PSBr) has χ = 0.064 at 180 °C [110] with PS
and it was selected as untethered chain to see the effect of poor miscibility on surface
dynamics. To provide good consistency between NR and XPCS data, after the NR
measurement (Figure 7.3) sample 2h02 45 was cut into small pieces and was measured
with XPCS. The χ value between hPS and dPS is on the order of 10-4 at 160 °C [142] and
thus this pair was used to represent the case in which the exchange interaction energy is
extremely small. On the other hand, poly(cyclohexyl methacrylate), PCHMA, has two
reported values of χ with PS at 150 °C of -0.034 or -0.015 [143-145]. Therefore, PCHMA
was selected to study the surface dynamics in the case that the miscibility is very good
233
and interactions between the tethered and untethered chains are particularly strong.
To see the effect of tethered chains clearly, the molecular weight of the untethered
chains needed to be smaller than those in the entanglement regime [21]. Also, to avoid
undesirable effects due to variations in molecular weights among samples, the molecular
weights of the untethered chains needed to be similar to each other. Therefore, the Mn of
the untethered PSBr, PCHMA and PS were selected to be 3.8 kg/mol, 3.8 kg/mol and 2.6
kg/mol, respectively, with Mw/Mn < 1.10. The Tg values of untethered chains measured
with DSC for PCHMA, PSBr and PS were 81 °C, 89 °C and 68 °C, respectively. The Mn
of hPS-COOH used as tethered chain was 200 kg/mol, yielding σ = 0.020 chains/nm2.
2h03 PCHMA 40 and 2h03 PSBr 38 were partially tethered layers composed of PS
tethered chains with PCHMA and PSBr untethered chains, respectively, and 2h02 45 was
only composed of PS. Since de-wetting was seen for 2h03 PSBr 38 at 160 °C, more
annealing and surface observation was done for PSBr. To see the compatibility with the
epoxide layer, a thin layer of PSBr was coated on a self-assembled monolayer of (3glycidoxypropyl)trimethoxy silane which was used to form covalent bonds with the hPSCOOH chains (PSBr Ref 41).
234
Table 7.6 : Samples to be studied
Sample
2h03
PCHMA 40
2h03
PSBr 38
2h02 45
PCHMA
Ref 38
PSBr
Ref 32
Species a
Mn (kg/mol) b
T
U
T
U
PS
PCHMA
200
3.8
PS
PSBr
200
3.8
PS
PS
200
2.6
σ
(chains
/nm2) c
0.022
±0.002
0.023
±0.002
0.020
±0.002
h
(nm) d
Annealing
(temp(°C)
/time(hr)) e
41±1
110/15
44±1
110/15
44±1
119/14
PCHMA
3.8
40±1
110/15
PSBr
3.8
41±1
110/15
2k Ref 43
PS
2.6
43±1
Annealed
in XPCS
PSBr
Ref 41f
PSBr
3.8
41±1
197/16
a
T and U means the species of the tethered and untethered chain, respectively.
T and U means the Mn of the tethered and untethered chain, respectively.
c
Calculated from σ = hρNA/Mn , where σ is grafting density, h is thickness of tethered
chains, ρ is bulk density of the tethered chain component and NA is Avogadro’s number.
d
The h of total PS layer was measured by ellipsometer with 632nm laser source and
n=1.589 at room temperature. For HCT fitting, the h was obtained analysis of the X-ray
XR curve measured at APS, 8-IDI at the XPCS measurement temperature. The h from the
XR curves are presented in the Appendix (Table A.1).
e
The annealing was done under high vacuum (10-5-10-6 Pa).
f
The sample is composed of a polymer layer on the self assembly monolayer of (3glycidoxypropyl)trimethoxy silane.
b
235
7.4.2. Results and Discussions
2h03 PSBr 38 was measured with XR at the APS 8-IDI beam line at room
temperature before the XPCS measurement. Figure 7.19b shows the SLD profile
calculated from the XR curve model fit. The SLD value of the hPS tethered chain layer
with PSBr mixed in with it was 9.0 × 10-6 Å -2 and that of the top layer containing
essentially pure PSBr was 13.7 × 10-6 Å -2. On the SLD profile, the interface between the
PS rich layer and the pure PSBr is from 42 nm to 38 nm. The width of interface between
tethered and untethered chains was at most 4 nm. In spite of similar σ and identical
molecular weights, the tethered chain penetration in 2h03 PSBr 38 was smaller than that
in 2h02 45 ( > 40 nm). As the reflectivity data for 2h02 45 and 2h03 PSBr 38 show
(Figures 7.3 and 7.19), the composition of the tethered chain (hPS) on the border point
with the epoxide layer was 16 vol% and 99 vol%, respectively. This trend was consistent
with the miscibility of the two chains. Springer et al. [146] studied bilayers of dPS and
brominated PS with NR and the width of the interface between the layer of dPS and layer
of PS depended on the degree of bromination. The interface width was 4.7 nm when the
degree of bromination was 0.16, but was decreased to 1.8 nm for degree of bromination
of 1.0.
236
1.E+00
Reflectivity
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
1.E-06
0
0.05
0.1
0.15
0.2
qz (Å-1)
Scattering Length Density (Å-2)
a)
25
20
15
10
5
0
0
100
10
200
20
300
30
400
40
500
50
Distance
from Air
(nm)
Distance
from
Air
(nm)
b)
Figure 7.19: (a) The XR curve of 2h03 PSBr 38 obtained from APS 8-IDI (○) and its
model fit (
). (b) The SLD profile calculated from the model fit. Air interface is on the
right.
237
The XPCS results show how the surface fluctuations vary with different species of
untethered chains. The τ values of 2h02 45 and 2h03 PSBr 38 were calculated by fitting
each g2 function with Equation 4.4 and are plotted as τ vs. q|| curves (Figures A.4, A.26,
A.27, A.28 and 7.20). Figure 7.21 shows τ/h vs. q||h curves. The difference in the impact
on τ/h of tethering in the case of the PS/PS interaction in 2h02 45 and 2k Ref 43 and in
the case of the PS/PSBr interaction for 2h03 PSBr 38 and PSBr Ref 32 is the key result
manifesting the effect of the χ value. The PS/PS pair was measured at 120 °C, for which
T - Tg = 52 °C and then η values calculated from the HCT fit of each curve may be taken
as one indication of the difference in the surface dynamics (Table 7.7). The value of η
increased by a factor of 64 from that of the reference to that of the partially tethered layer.
For the PS/PSBr pair, the measurement temperature was 150 °C, or T - Tg = 61 °C. For
this case in which χ is substantially greater than 0, the increase in the η value was only a
factor of 2.5. The Mn of the PSBr untethered chains was 3.8 kg/mol, somewhat higher
than that of PS untethered chains, 2.6 kg/mol. This difference in molecular weight may be
part of the reason for the smaller effect in the case of the PS/PSBr pair. However, as
shown by Table 7.3, for two other pairs of PS/PS, increasing the molecular weight of the
untethered chain by four times, from 2.6k to 9.4k, only decreased the difference in η due
to tethering by a factor of six. So the roughly factor 1.5 mismatch of molecular weights
between PS and PSBr cannot explain more than a small part of the effect seen in Figure
7.21.
If the surface of the partially tethered layer is a mixture containing tethered chains,
as pictured in Figure 7.22, the surface dynamics are much more strongly impacted than if
the tethered chain does not reach the surface. It is widely thought that the surface region
238
adjacent to air or vacuum has more free volume for enhanced chain mobility, causing
lower Tg and η values for thin films [134, 147]. Since the near surface region has more
free volume, faster chain diffusion due to enhanced mobility can be expected [148]
contributing to the dynamics of the whole layer. However, for a thin partially tethered
film, tethered chains can reach to the surface and they will provide friction in the most
mobile volume. As the NR data show, the sample 2h03 PSBr 38 does not have any parts
of hPS tethered chains on the surface of the film, while the 2h02 45 sample has about 3
vol% of tethered chain (segments) on the surface. The existence of tethered chain
segments on the top layer is one of the factors resulting in such slower surface dynamics.
Brown et al. [149] calculated the number of segments between entanglements for
interfaces between layers of two different polymers as a function of the interaction
parameter (
) using a self-consistent field technique. The results are summarized in
Figure 7.23. For larger χ values the more number of segments between entanglements, Ne
was larger, meaning that the mechanical interaction of chains of the two types is less
strong when χ is large. The Mn of PSBr used for this study was 3.8kg/mol and it was too
small for entanglement. However, as shown by the XR result, the contacts between two
types of chains were far fewer for
than for
. As a result, it was expected
that less contact caused smaller friction between the tethered and untethered chains,
resulting in a smaller effect of the tethered chains on the surface dynamics.
239
1.E+04
τ (sec)
1.E+03
1.E+02
2h03 PS
PSBr Ref
2h02 40
1.E+01
2k Ref 4
1.E+00
1.E-01
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
q|| (Å-1)
Figure 7.20: τ vs. q|| curves of 2h03 PSBr 38 (◇), PSBr Ref 32 (□) at 150 °C and of
2h02 45 (△), 2k Ref 43 (○) at 120 °C.
240
1.E+01
τ/h (sec/Å)
1.E+00
1.E-01
1.E-02
1.E-03
0.00
0.10
0.20
0.30
0.40
q||h
2h03 PSBr 40 150Cel-deg
PSBr Ref 40 15
Figure 7.21: τ/h vs. q||h curves of 2h03 PSBr 38 (◇) and PSBr
32 (□) at 150 °C and 2k Ref 40 120C
2h02Ref
40 120Cel-deg
2h03 PSBr
40 fitting
of 2h02 45 (△) and 2k Ref 43 (○) at 120 °C. The solid lines
are fits
with the HCT to PSBr Ref 40 fitt
2h02 40 fitting
calculate η values.
Table 7.7: η values calculated from the HCT fits.
Sample
PSBr Ref 32
2h03 PSBr 38
2k Ref 43
2h02 45
PCHMA Ref 38
T (°C)
γ (mN/m)
150
33
120
32
140
25
241
η (Pa·s)
200000 ± 10000
500000 ± 25000
250 ± 13
16000 ± 800
35000 ± 1750
2k Ref 40 fittin
a)
b)
Figure 7.22: Schematics of partially tethered layer structure at different h. In a thinner
layer (a), tethered chains reach to the mobile near surface region, but a thicker layer (b)
does not have any tethered chain segments on or near the surface.
Figure 7.23: The number of segments between entanglements as a function of χ was
calculated with mean field technique. Reprinted with permission from R. Oslanec and H.
R. Brown, “Entanglement Density at the Interface between Two Immiscible Polymers”,
Macromolecules 36, 5839 (2003). Copyright 2003 American Chemical Society.
242
As shown by Figures 7.24 and 7.25, PCHMA Ref 38 showed a measurable
relaxation behavior at 140 °C. Using reported values of surface tension for series of
poly(alkyl methacrylate) polymers [150], the η value of PCHMA Ref 38 was calculated
and reported in Table 7.7. For χ < 0, Figures 7.26, 7.27 and 7.28 present the g2 functions
of 2h03 PCHMA 40 at 150 °C, 180 °C and 200 °C, respectively. The τ values were too
large to be measured with XPCS (Figure 7.29). In order for the τ value for 2h03 PCHMA
40 to have been greater than 10000s at q|| = 0.0007 Å -1 η would have to have been at least
8000000Pa·s. Therefore, 2h03 PCHMA 40 has surface dynamics at least 228 times
(8000000/35000) slower than those of its reference at 140 °C.
The 2h03 PCHMA 40 sample had an h value of ca. 40 nm with σ ≈ 0.02 chain/nm2.
In other words, this sample had similar thickness, but had fewer covalent bonds to the
substrate than did 2802 45. As shown by 2h02 45, considering covalent bond density only,
2h03 PCHMA 40 should have shown a τ value less than 10000 s. Therefore, not only the
confinement effect by covalent bonds, but also another factor, miscibility between the
tethered and untethered chains, played a significant role in the surface dynamics of 2h03
PCHMA 40. In addition, the partially tethered layer h ≈ 40 nm, was thin enough to have
portions of the 200 kg/mol tethered chains reach to the surface at χ ≈ 0. Since for χ < 0
the tethered chains should stretch even farther into the film, 2h03 PCHMA 40 should
have had portions of even more tethered chains in its near surface region than did 2h02
45. The mobility of the near surface region of the partially tethered layer with χ < 0
would be suppressed even more than was that of 2h02 45.
243
1.15
1.17
1.13
1.15
1.11
g2
g2
1.19
1.13
q||=0.000310Å -1
1.11
1.07
1.09
10
100
Time delay (s)
1000
1
1.14
1.13
1.12
1.11
1.10
1.09
g2
g2
q||=0.000393Å -1
1.05
1
1.08
1.06
10
100
Time delay (s)
1000
1.07
1.05
q||=0.000482Å -1
q||=0.000571Å -1
1.03
1.04
1
10
100
Time delay (s)
1
1000
1.13
1.12
1.11
1.10
1.09
1.08
g2
g2
1.09
1.07
1.05
1000
1.06
1.04
q||=0.000659Å -1
10
100
Time delay (s)
q||=0.000747Å -1
1.02
1.03
1
10
100
Time delay (s)
1
1000
10
100
Time delay (s)
1000
Figure 7.24: g2 functions of PCHMA Ref 38 for several values of q|| at 140 °C. The red
line is the fit using Equation 4.4 to calculate τ.
244
PCHMA 40nm 140C
τ (sec)
1000
100
10
0.0002
0.0004
0.0006
q|| (Å -1)
Figure 7.25: τ vs. q|| curve of PCHMA Ref 38 at 140 °C.
245
0.0008
1.10
1.11
1.09
1.10
1.08
g2
g2
1.12
1.09
1.07
q||=0.000314Å -1
1.06
1.08
10
100
Time delay (s)
1
1000
1.09
1.10
1.08
1.09
1.07
1.08
g2
g2
1
1.06
1.07
q||=0.000496Å -1
10
100
Time delay (s)
1000
q||=0.000590Å -1
1.06
1.05
1
10
100
Time delay (s)
1
1000
1.08
1.08
1.07
1.07
1.06
1.06
g2
g2
q||=0.000402Å -1
1.05
1.05
q||=0.000683Å -1
10
100
Time delay (s)
1000
q||=0.000778Å -1
1.04
1.04
1
10
100
Time delay (s)
1
1000
10
100
Time delay (s)
1000
Figure 7.26: g2 functions of 2h03 PCHMA 40 for several values of q|| at 150 °C. The red
line is the attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of
τ were too large to be calculated.
246
1.11
1.12
1.10
1.11
1.09
g2
g2
1.13
1.10
1.08
q||=0.000313Å -1
1.09
1.07
10
100
Time delay (s)
1000
1
1.09
1.09
1.08
1.08
1.07
1.07
g2
g2
1
1.06
10
100
Time delay (s)
1.06
q||=0.000495Å -1
1000
q||=0.000589Å -1
1.05
1.05
1
10
100
Time delay (s)
1
1000
1.08
1.07
1.07
1.06
1.06
1.05
g2
g2
q||=0.000401Å -1
1.05
1.04
q||=0.000682Å -1
1.04
10
100
Time delay (s)
1000
q||=0.000777Å -1
1.03
1
10
100
Time delay (s)
1000
1
10
100
Time delay (s)
1000
Figure 7.27: g2 functions of 2h03 PCHMA 40 for several values of q|| at 180 °C. The red
line is the attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of
τ were too large to be calculated.
247
1.10
1.12
1.09
1.11
1.08
g2
g2
1.13
1.10
1.07
q||=0.000314Å -1
1.06
1.09
10
100
Time delay (s)
1
1000
1.08
1.09
1.07
1.08
1.06
1.07
g2
g2
1
1.05
10
100
Time delay (s)
1000
q||=0.000592Å -1
1.06
q||=0.000497Å -1
1.05
1.04
1
10
100
Time delay (s)
1
1000
1.07
10
100
Time delay (s)
1000
1.06
1.05
q||=0.000684Å -1
1.06
1.04
1.05
g2
g2
q||=0.000402Å -1
1.04
1.03
1.02
1.03
q||=0.000780Å -1
1.01
1
10
100
Time delay (s)
1000
1
10
100
Time delay (s)
1000
Figure 7.28: g2 functions of 2h03 PCHMA 40 for several values of q|| at 200 °C. The red
line is the attempt to fit the g2 functions with Equation 4.4 to calculate τ, but the values of
τ were too large to be calculated.
248
τ/h (sec/Å)
1.E+01
1.E+00
1.E-01
1.E-02
0.0
0.1
0.2
0.3
0.4
q||h
Figure 7.29: τ/h vs. q||h curve of PCHMA Ref 38 at 140 °C. The broken line indicates that
2h03 PCHMA 40 did not have any measurable value of τ until 200 °C.
249
7.4.3. Thermal Damage
It is known that materials with carbon-Br bonds are more susceptible to X-ray
damage than is PS. Therefore, radiation damage could have been an issue with the
samples containing PSBr. Figure 7.30 presents the reflectivity curve (presented only as
detector counts) of 2h03 PSBr 38 before and after the XPCS measurement at 150 °C.
Though the amplitude of the fringes in the curve decreased some after the measurement,
the roughness did not seem to increase dramatically. The time the sample spent at 150 °C
was about 1 hr in XPCS. Another XPCS measurement at 160 °C was planned, but as
shown by Figure 7.31, which was measured after 20 min. at 160 °C, the fringe amplitude
had already almost disappeared after 20 min. due to loss of definition in the thickness of
the film, perhaps due to high roughness. However, the change in shape is not what is
generally seen due simply to an increase in roughness.
After cooling back down to
room temperature, optical microscopy and AFM were used to study the surface of 2h03
PSBr 38 and advanced de-wetting on the surface was observed (Figure 7.32). Assuming
that the de-wetting might not be due to immiscibility of the PSBr with the PS tethered
chain, but rather might be due to the immiscibility with the epoxide layer, PSBr Ref 41
with h ≈ 40 nm on a self assembled monolayer of (3-glycidoxypropyl)trimethoxy silane,
was prepared. PSBr Ref 41 was annealed for 16 hr at 197 °C. Optical microscopy
indicated no serious de-wetting happened upon annealing (Figure 7.33). Therefore, the
de-wetting of the PSBr layer at 160 °C can be considered to be due to the immiscibility
with the PS tethered layer.
250
1.E+09
Before Measurement at 150Cel-deg
1.E+08
After Measurement at 150Cel-deg
Detector Counts
Reflectivity
1.E+07
1.E+06
1.E+05
1.E+04
1.E+03
1.E+02
0.00
0.05
0.10
0.15
0.20
0.25
qz (Å-1)
Figure 7.30: XR curves of 2h03 PSBr 38 measured at APS 8-IDI. The blue and red curves
were measured before and after the XPCS measurement, respectively at 150 °C.
251
1.E+09
Detector Counts
Reflectivity
1.E+08
1.E+07
1.E+06
1.E+05
계
1.E+04
1.E+03
1.E+02
0.00
0.05
0.10
0.15
qz (Å-1)
Figure 7.31: XR curve of 2h03 PSBr 38 at 160 °C.
252
0.20
0.25
100μm
a)
437nm
-256nm
10 μm
0
b)
Figure 7.32: Surface images of 2h03 PSBr 38 after cooling from 160 °C obtained from an
optical microscopy (a) and an AFM (b).
253
100μm
Figure 7.33: The surface image of PSBr Ref 41 after annealing at 197 °C for 16 hrs in a
high vacuum obtained from optical microscopy. The polymer layer was on the top of a
self assembled monolayer of (3-glycidoxypropyl)trimethoxy silane.
As discussed in section 7.4.2, 2h03 PCHMA 40 showed surface relaxation behavior
too slow to be measured at temperatures up to 200 °C. However, this phenomenon might
have resulted from a seriously damaged surface leaving little PCHMA on the top.
Therefore, MALDI and TGA under nitrogen flow were measured for bulk PCHMA. The
minimum m/z was 1000 and most of the polymer is composed of m/z > 2000 as shown by
Figure 7.34, so evaporation of chains from the sample was not a problem. In addition,
TGA data in Figure 7.35 showed a weight loss of only 1.6% at 200 °C. To see how h
varied with respect to T, the XR curve of 2h03 PCHMA 40 was measured on the 8-IDI
beamline at 140 °C, 160 °C and 180 °C.
Two example curves are shown in Figure 7.36.
As Table 7.8 shows, the h value increased with T as expected from thermal expansion.
After XPCS measurement at 200 °C, the surface of 2h03 PCHMA 40 was observed using
an optical microscopy at room temperature, but no significant de-wetting was observed as
254
shown by Figure 7.37. Also, the root mean squared roughness calculated from the AFM
Intens. [a.u.]
image in Figure 7.38 was only 0.6 nm.
2783.010
2614.919
2500
2951
2500
2951.116
2614
2783
Intensity (A.U.)
2000
2000
1500
1500
1000
1000
500
500
0
0
1500
1500
2000
2000
2500
2500
3000
3000
3500
3500
4000
4000
Figure 7.34: MALDI result of the PCHMA used to prepare samples.
255
4500
4500
5000
5000
m/z
m/z
100
90
Residual Weght %
80
70
60
50
계
40
30
20
10
0
0
50
100
150
200
250
300
350
400
450
Temperature (℃)
Figure 7.35: TGA result of bulk PCHMA under nitrogen flow.
1.E+07
Before measurement at 140Cel-deg
Reflectivity
Counts
Detector
1.E+06
After measurement at 180Cel-deg
1.E+05
1.E+04
1.E+03
1.E+02
1.E+01
0.00
0.05
0.10
0.15
0.20
0.25
qz(Å-1)
Figure 7.36: XR curves of 2h03 PCHMA 40 measured at APS 8-IDI. The blue curve is
measured at 140 °C before XPCS and the red curve was measured at 180 °C after XPCS.
256
Table 7.8: h values of 2h03 PCHMA 40 with T before any XPCS exposure.
T (°C)
140
160
180
h (nm)
40 ± 1
40 ± 1
42 ± 1
100μm
Figure 7.37: Optical microcopy image of 2h03 PCHMA 40 surface after XPCS
measurement at 200 °C.
257
2.4nm
-2.1nm
10 μm
0
a)
4
2
0
nm -2
2
4
6
8
10 μm
2
4
6
8
10 μm
4
2
0
nm -2
b)
Figure 7.38: AFM images of 2h03 PCHMA 40 surface after XPCS measurement at
200 °C. (a) is the topography image and (b) is cross-section plots.
258
7.5. Summary
The penetration length of tethered chains into untethered chain layers depended on
the miscibility between tethered and untethered chains.
For films with 200k tethered
chain with σ = 0.02 chains/nm2, tethered chains showed a more expanded conformation
for smaller molecular weight of untethered chains and the composition of the tethered
chain segments in the surface region was higher. For samples with different χ values, a
tethered showed more expanded conformations when χ was smaller. For 2h02 45, for
which χ ≈ 0, the 200k PS penetrated over 40 nm, while for 2h03 PSBr 38, for which χ > 0,
the 200k PS penetrated only 10 nm. Better miscibility caused more penetration of
tethered chains into untethered chains.
More penetration resulted in slower surface dynamics, due to segmental friction. In
particular, the presence of tethered chain segments in the surface region of the partially
tethered layer hindered the enhanced mobility at the air/polymer interface. For PS
untethered chains mixed with PS tethered chains, as the molecular weight of the
untethered chains became larger, the tethering effect on the surface dynamics became
smaller. In addition, the value of χ impacts the dynamics. For χ > 0, for which tethered PS
was mixed with PSBr, the η value of the partially tethered layer increased only by a factor
of 2.5 from the value for the reference layer. For χ ≈ 0, for which the partially tethered
layer was composed of PS only, the τ values increased by 64 times. In the case of χ < 0,
for which PS tethered chains were mixed with PCHMA, the τ values of the partially
tethered layer became too large to be measured with XPCS, while the reference layer
presented a measurable relaxation behavior.
259
However, not only the penetration determined by miscibility, but also the
plasticization effect played a role in determining the surface dynamics. 28Oli was
composed of PS oligomer spun cast on the top of a 28k tethered chain layer and showed
measurable surface relaxation behavior at T – Tg = 50 °C. However, for 2802 45, for
which the molecular weight of the untethered chain was a factor of 2.5 higher, the surface
relaxation was not measurable even at T – Tg = 110 °C. This result can not be explained
just by considering the miscibility effect. For these short chains, the plasticization effect,
which resulted in a lower effective Tg of the partially tethered layer (Tg = 18 °C for 28Oli
and Tg = 71 °C for 2802 45), determined the surface dynamics.
260
CHAPTER VIII
CONCLUSION
A number of conclusions about the factors controlling surface fluctuation dynamics
in partially tethered layers can be drawn from the research results presented in the
preceeding chapters. These conclusions are pertinent to films containing grafted chains
for which the σ of the tethered layer is smaller than 0.15 chains/nm2. This low value of σ
is the property of these films that most distinguishes the partially tethered layers from the
bilayers studied by Uğur [58].
Comparison of the results from a partially tethered layer composed of a 28k
tethered PS layer with σ ≈ 0.10 chains/nm2 and 2.6k PS untethered chain and those from a
film of pure 48 kg/mol untethered chains provide the first insight into the phenomenon.
The slowed surface dynamics of partially tethered layers not only originate from an
increase in Tg of the film caused by adding some chains of higher Tg, but also arise due to
the covalent bonds to the substrate. The slowing of surface fluctuations achieved in the
case of a 45 nm film of 2.6k chains partially tethered by 28k chains is remarkable. In
addition, the confinement effect determined by the density of covalent bonds to the
substrate (indicated by grafting density) tailored the surface dynamics. A partially
tethered layer with smaller σ value (0.02 chains/nm2) showed at least 12 times faster
dynamics than did more densely tethered film (0.1 chains/nm2).
261
Comparison of the surface dynamics of partially tethered layers of PS with PS, but
with different thicknesses, provided a means of evaluating models to explain the surface
dynamics. A two layer-model for a partially tethered layer has an upper layer (huntethered)
with faster dynamics and a lower layer (htethered) with very slow dynamics. It can readily
explain the surface dynamics of partially tethered layers with htotal = 70 and 90 nm where
2.6k untethered chains are mixed with 28k (2802) tethered chains having 22 – 26 nm of
penetration into the untethered chains. For the 2802 sample with htotal = 90 nm the two
layer-model coincides with a model already presented for hydrodynamic flow of solvents
adjacent to layers of adsorbed polymer chains [153 – 157]. However, the dynamics of
htotal = 45 nm with huntethered = 19 nm is unmeasurably slow. There are possible reasons for
such slow surface dynamics. First, the top untethered chains of htotal = 45 nm sample are
affected by the “neighboring layer” effect. The bottom has high viscosity and some
elasticity. This slowed the top layer. Second, there is distribution of tethered chain
conformations due to thermal fluctuations of the tethered chains. A very small fraction of
the tethered chain segments penetrate farther into the the top layer than the penetration
length observed with the NR data.
The miscibility between tethered and untethered chains determines the surface
dynamics of a partially tethered layer. For tethered PS chains having the same molecular
weight and σ, smaller χ or lower molecular weight of the untethered chains results in
slower surface dynamics. On the other hand, not only the miscibility, but also the
effective Tg of the partially tethered layer determines the surface dynamics. The
plasticization effect with an untethered oligomer layer results in lower effective Tg of a
262
partially tethered layer and makes the surface dynamics faster. For 28k tethered PS, in
spite of lower molecular weight, the partially tethered layer having 0.9k PS has faster
surface dynamics than does the partially tethered layer having 2.6k PS. The calculated
effective Tg of the oligomer sample was 53 °C lower than that of the 2.6k sample.
In this study, a variety of models and effects were used to explain the surface
dynamics of partially tethered layers. However, there has been no clear theoretical picture
of the role of molecular motion in polymer film surface fluctuations. The only theory
available is a hydrodynamic one that views the film as a continuum.
The HCT was used
here to rationalize the XPCS results, but the samples used in this study are mixtures of
two different types of chains. This engineering of surface fluctuation dynamics could be
done more effectively if a theory illuminating the mechanism of surface fluctuations at
the molecular scale were provided. This study provides a substantial body of
experimental results against which potential theories can be tested or validated.
263
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276
APPENDICES
277
APPENDIX A
THICKNESS VARIATION WITH TEMPERATURE
The surface relaxation time (τ) vs. in-plane wave vector (q||) curves were
normalized with thickness (h) and fitted using the hydrodynamic continuum theory
(HCT). The value of h used for fitting with the HCT was not the value at room
temperature, but rather the value at the measurement temperature of XPCS, shown in
Table A.1. The X-ray reflectivity (XR) curves were obtained from the 8-IDI APS beam
line and were fitted with a model curve which showed the best match with the
experimental data. The fit program was Igor Pro version 6.2.2.2 with the data analysis
module Motofit (ANSO).
278
Table A.1: The h values of XPCS samples at the measurement temperature.
Sample
9k ref 80
48k Ref 35
48k Ref 95
PSBr Ref 32
PCHMA Ref
38
2802 45
2802 70
2802 90
T (°C)
110
120
130
140
150
160
120
140
150
160
150
h (nm)
80 ± 1
82 ± 1
83 ± 1
35 ± 1
35 ± 1
35 ± 1
92 ± 1
96 ± 1
96 ± 1
95 ± 1
32 ± 1
140
38 ± 1
120
130
160
180
110
120
110
120
44 ±
44 ±
45 ±
45 ±
69 ±
70 ±
87 ±
90 ±
Sample
2809 92
2h02 45
2h09 42
2h09 100
2h48 42
1
1
1
1
1
1
1
1
2h48 99
200k 10Prc 38
2h03 PSBr 38
2h03 PCHMA
40
279
T (°C)
110
120
130
110
120
120
130
110
120
130
150
160
120
140
150
160
100
110
120
150
140
160
180
h (nm)
92 ± 1
92 ± 1
93 ± 1
44 ± 1
46 ± 1
42 ± 1
42 ± 1
99 ± 1
99 ± 1
100 ± 1
42 ± 1
42 ± 1
96 ± 1
100 ± 1
100 ± 1
100 ± 1
37 ± 1
38 ± 1
39 ± 1
38 ± 1
40 ± 1
40 ± 1
42 ± 1
APPENDIX B
REFLECTIVITY CURVE FITTING QUALITY
The χ2 value indicates how far the model fit is away from the data. Table A.2
presents the χ2 value and fitting software used for each reflectivity curve presented in the
dissertation.
Table B.1: Fitting quality of reflectivity curves in this dissertation.
Sample
28k PS Tethered
Layer
2h02 90
2802 85
4848 20
2h02 13
2h1h 15
2h02 45
2h09 100
2h03 PSBr 38
Software Used
χ2
Igor
2.8
Reflpak
Watcom
Reflpak
Watcom
Watcom
Reflpak
Reflpak
Igor
3.6
0.5
2.0
0.5
0.1
5.7
5.6
3.5
280
APPENDIX C
XPCS MEASUREMENT CONDITION
The experimental details for collection of XPCS data for each sample are shown in
Table A.3. The mode designations "F" and "K" indicate "full frame mode" and "kinetics
mode", respectively.
Table C.1: XPCS Measurement Conditions for Each Sample.
2k Ref 43
T
Mode
(C)
100
K
110
K
120
K
Frame
Number
50
50
40
Exposure Time
(sec)
0.1
0.1
0.1
Number of
measured points
5
5
5
Sleep Time
(sec)
0.0
0.0
0.0
9k Ref 80
T
Mode
(C)
110
F
120
F
130
K
Frame
Number
260
260
70
Exposure Time
(sec)
0.4
0.4
0.1
Number of
measured points
3
3
6
Sleep Time
(sec)
1.0
1.0
0.0
48k Ref 35
T
Mode
(C)
140
F
150
F
160
F
Frame
Number
256
256
224
Exposure Time
(sec)
0.18
0.15
0.12
Number of
measured points
2
3
3
Sleep Time
(sec)
1.0
1.0
1.0
281
48k Ref 95
T
Mode
(C)
140
K
160
K
Frame
Number
70
32
Exposure Time
(sec)
0.05
0.05
Number of
measured points
7
8
Sleep Time
(sec)
0.0
0.0
PSBr Ref 32
T
Mode
(C)
150
F
Frame
Number
256
Exposure Time
(sec)
0.1
Number of
measured points
4
Sleep Time
(sec)
1.1
PCHMA Ref 38
T
Mode
(C)
140
F
Frame
Number
192
Exposure Time
(sec)
0.1
Number of
measured points
4
Sleep Time
(sec)
2.0
28Oli
T
(C)
30
40
60
Frame
Number
128
128
128
Exposure Time
(sec)
0.2
0.3
0.1
Number of
measured points
1
3
4
Sleep Time
(sec)
0.0
0.0
0.0
2802 45
T
Mode
(C)
120
F
130
F
160
F
180
F
Frame
Number
224
224
224
224
Exposure Time
(sec)
0.4
0.3
0.1
0.03
Number of
measured points
3
3
1
3
Sleep Time
(sec)
2.0
2.0
2.0
2.0
2802 70
T
Mode
(C)
110
K
120
K
Frame
Number
40
32
Exposure Time
(sec)
0.03
0.025
Number of
measured points
8
10
Sleep Time
(sec)
0.0
0.0
2802 90
T
Mode
(C)
110
K
120
K
Frame
Number
80
40
Exposure Time
(sec)
0.04
0.02
Number of
measured points
8
8
Sleep Time
(sec)
0.0
0.0
Mode
F
F
F
282
2809 92
T
Mode
(C)
110
F
120
K
130
K
Frame
Number
260
112
112
Exposure Time
(sec)
0.2
0.1
0.1
Number of
measured points
4
4
4
Sleep Time
(sec)
1.0
0.0
0.0
2h02 45
T
Mode
(C)
110
K
Frame
Number
258
Exposure Time
(sec)
0.1
Number of
measured points
4
Sleep Time
(sec)
0.0
2h09 42
T
Mode
(C)
130
F
Frame
Number
260
Exposure Time
(sec)
0.2
Number of
measured points
4
Sleep Time
(sec)
1.0
2h09 100
T
Mode
(C)
130
K
Frame
Number
112
Exposure Time
(sec)
0.1
Number of
measured points
4
Sleep Time
(sec)
0.0
2h48 42
T
Mode
(C)
160
F
Frame
Number
224
Exposure Time
(sec)
0.12
Number of
measured points
3
Sleep Time
(sec)
1.0
2h48 99
T
Mode
(C)
140
K
160
K
Frame
Number
70
256
Exposure Time
(sec)
0.1
0.05
Number of
measured points
5
8
Sleep Time
(sec)
0.0
0.0
200k 10Prc 38
T
Mode
(C)
110
F
Frame
Number
256
Exposure Time
(sec)
0.3
Number of
measured points
4
Sleep Time
(sec)
0.0
2h03 PSBr 38
T
Mode
(C)
150
F
Frame
Number
256
Exposure Time
(sec)
0.1
Number of
measured points
4
Sleep Time
(sec)
1.1
283
2h03 PCHMA 40
T
Frame
Mode
Number
(C)
140
F
192
160
F
192
180
F
192
200
F
192
Exposure Time
(sec)
0.1
0.1
0.06
0.06
284
Number of
measured points
1
1
6
3
Sleep Time
(sec)
2.0
2.0
2.0
2.0
APPENDIX D
FITTING RESULTS FOR 2748 37
The sample had a grafting density of 0.02 chains/nm2. The molecular weight of the
tethered chain was 200 kg/mol and of the untethered chain was 48 kg/mol. Annealing was
done at 140 C for 15 hr before the NR measurement. The model fit to the NR curve was
done with Reflpak. The χ2 value for this model fit was 10.7.
285
1.E+00
Reflectivity
1.E-01
1.E-02
1.E-03
1.E-04
Data
Model Fit
1.E-05
1.E-06
0
0.05
0.1
a)
0.15
0.2
0.25
qz (Å-1)
Scattering Length Density (Å-2)
7.E-06
6.E-06
5.E-06
4.E-06
3.E-06
계열1
2.E-06
1.E-06
0.E+00
0
10
20
30
40
Distance from Air (nm)
b)
Figure D.1: (a) NR curve (○) and its model fit (
calculated with the model fit.
286
) of 2h48 37 and (b) the SLD profile
APPENDIX E
g2 FUNCTIONS
Plots of the g2 functions for various samples are provided in the figures A.2
through A.28.
287
1.11
1.09
1.09
1.07
1.07
g2
g2
1.11
1.05
1.05
1.03
1.03
q||=0.000225Å -1
1.01
1.01
1
10
100
Time delay (s)
0.1
1000
1.11
1.11
1.09
1.09
1.07
1.07
g2
g2
0.1
1
10
100
Time delay (s)
1000
1.05
1.05
1.03
1.03
q||=0.000375Å -1
q||=0.000475Å -1
1.01
1.01
0.1
1
10
100
Time delay (s)
0.1
1000
1.11
1.11
1.09
1.09
1.07
q||=0.000575Å -1
g2
g2
q||=0.000275Å -1
1
1.07
1.05
1.05
1.03
1.03
10
100
Time delay (s)
1000
q||=0.000626Å -1
1.01
1.01
0.1
1
10
100
Time delay (s)
0.1
1000
1
10
100
Time delay (s)
1000
Figure E.1: The g2 functions of 2k Ref 43 for several values of q|| at 100 °C. This
measurement was done by Uğur et al. [58].
288
1.12
1.1
1.1
1.08
1.08
g2
g2
1.12
1.06
1.06
1.04
1.04
q||=0.000285Å -1
1.02
1.02
0.1
1
1.12
10
100
Time delay (s)
1000
0.1
1.1
1.1
1.08
1.08
1.06
1000
q||=0.000453Å -1
1.04
q||=0.000397Å -1
1.02
1.02
0.1
1
1.12
10
100
Time delay (s)
1000
0.1
1
1.12
10
100
Time delay (s)
1000
1.1
1.1
q||=0.000508Å -1
1.08
g2
g2
10
100
Time delay (s)
1.06
1.04
q||=0.000564Å -1
1.08
1.06
1.06
1.04
1.04
1.02
1.02
0.1
1
1.12
10
100
Time delay (s)
0.1
1000
1
1.12
1.1
10
100
Time delay (s)
1000
1.1
q||=0.000620Å -1
1.08
g2
g2
1
1.12
g2
g2
q||=0.000341Å -1
1.06
q||=0.000675Å -1
1.08
1.06
1.04
1.04
1.02
1.02
0.1
1
1.12
10
100
Time delay (s)
1000
0.1
1
10
100
Time delay (s)
1000
g2
1.1
q||=0.000731Å -1
1.08
1.06
1.04
1.02
0.1
1
10
100
Time delay (s)
1000
Figure E.2: The g2 functions for 2k Ref 43 for several values of q|| at 110 °C. This
measurement was done by Uğur et al. [58].
289
1.15
1.13
1.13
1.11
1.11
g2
g2
1.15
1.09
1.09
1.07
1.07
q||=0.000175Å -1
1.05
1.05
0.1
1
10
100
Time delay (s)
1000
0.1
1
1.15
1.13
1.13
1.11
1.11
g2
g2
1.15
1.09
10
100
Time delay (s)
1000
q||=0.000337Å -1
1.09
1.07
1.07
q||=0.000283Å -1
1.05
1.05
0.1
1
1.15
10
100
Time delay (s)
1000
0.1
1
1.15
1.13
10
100
Time delay (s)
1000
1.13
q||=0.000391Å -1
1.11
g2
g2
q||=0.000229Å -1
1.09
1.09
1.07
1.07
1.05
q||=0.000445Å -1
1.11
1.05
0.1
1
1.15
10
100
Time delay (s)
1000
0.1
1
10
100
Time delay (s)
1000
g2
1.13
q||=0.000500Å -1
1.11
1.09
1.07
1.05
0.1
1
10
100
Time delay (s)
1000
Figure E.3: The g2 functions of 2k Ref 43 for several values of q|| at 120 °C. This
measurement was done by Uğur et al. [58].
290
1.10
1.10
1.08
1.08
1.06
g2
g2
1.12
1.04
1.06
1.04
1.02
q||=0.000312Å -1
1.00
1.02
1
10
100
1000
Time delay (s)
1
10000
1.10
10
100
1000
Time delay (s)
10000
1.09
1.08
1.07
1.06
g2
g2
q||=0.000417Å -1
1.04
q||=0.000628Å -1
1.05
1.03
1.02
1.01
q||=0.000519Å -1
1.00
0.99
1
10
100
1000
Time delay (s)
10000
1
10
100
1000
Time delay (s)
10000
1.08
g2
1.06
q||=0.000735Å -1
1.04
1.02
1.00
0.98
1
10
100
1000
Time delay (s)
10000
Figure E.4: The g2 functions of 48k Ref 35 for several values of q|| at 150 °C. The red line
is the fit using Equation 4.4 to calculate τ.
291
1.19
1.12
1.10
1.15
g2
g2
1.17
1.13
1.06
1.11
1.04
q||=0.000213Å -1
1.09
q||=0.000312Å -1
1.02
1
10
100
Time delay (s)
1000
1
1.09
1.10
1.07
1.08
1.05
1.06
g2
g2
1.08
1.03
10
100
Time delay (s)
1000
1.04
1.01
1.02
q||=0.000418Å -1
0.99
q||=0.000520Å -1
1.00
1
10
100
Time delay (s)
1000
1
10
100
Time delay (s)
1000
1.09
g2
1.07
q||=0.000630Å -1
1.05
1.03
1.01
0.99
1
10
100
Time delay (s)
1000
Figure E.5: The g2 functions of 48k Ref 35 for several values of q|| at 160 °C. The red line
is the fit using Equation 4.4 to calculate τ.
292
1.14
1.24
1.12
1.22
1.10
g2
g2
1.26
1.08
1.20
1.18
1.06
q||=0.000315Å -1
1.04
1.16
10
100
Time delay (s)
1
1000
1.11
1.11
1.09
1.09
1.07
1.07
g2
g2
1
1.05
1.03
10
100
Time delay (s)
1000
1.05
1.03
q||=0.000503Å -1
1.01
q||=0.000600Å -1
1.01
1
10
100
Time delay (s)
1000
1
1.10
1.10
1.08
1.08
1.06
1.06
g2
g2
q||=0.000406Å -1
1.04
1.02
10
100
Time delay (s)
1000
q||=0.000892Å -1
1.04
1.02
q||=0.000695Å -1
1.00
1.00
1
10
100
Time delay (s)
1
1000
10
100
Time delay (s)
1000
Figure E.6: g2 functions of 200k 10Prc 38 for several values of q|| at 110 °C. The red line
is the fit using Equation 4.4 to calculate τ.
293
1.12
1.10
1.10
1.08
1.08
g2
g2
1.12
1.06
1.04
1.04
q||=0.000423Å -1
0
1
10
100
Time delay (s)
0
1000
1.12
10
100
Time delay (s)
1000
1.08
g2
1.08
g2
1
1.10
1.10
1.06
q||=0.000673Å -1
1.06
1.04
1.04
q||=0.000611Å -1
1.02
1.02
0
1
10
100
Time delay (s)
1.00 0
1000
1
10
100
Time delay (s)
1000
1.10
1.10
1.08
1.08
q||=0.000737Å -1
q||=0.000800Å -1
1.06
g2
1.06
g2
q||=0.000548Å -1
1.02
1.02
1.04
1.04
1.02
1.02
1.00
1.00
0
1
10
100
Time delay (s)
0
1000
1
10
100
Time delay (s)
1000
1.10
1.10
1.08
1.08
q||=0.000862Å -1
q||=0.000925Å -1
1.06
g2
1.06
g2
1.06
1.04
1.04
1.02
1.02
1.00
1.00
0
1
10
100
Time delay (s)
0
1000
1
10
100
Time delay (s)
1000
Figure E.7: g2 functions of 2h02 45 for several values of q|| at 110 °C. The red line is the
fit using Equation 4.4 to calculate τ.
294
1.20
1.23
1.18
1.21
1.16
g2
g2
1.25
1.19
1.17
1.14
1.12
q||=0.000135Å -1
1.15
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
q||=0.000184Å -1
1.10
1.E-02
1.E+02
1.17
1.15
1.15
1.13
1.13
1.E+00
1.E+01
Time delay (s)
1.E+02
g2
g2
1.17
1.E-01
1.11
1.11
1.09
1.09
q||=0.000236Å -1
1.07
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.15
1.13
1.13
1.11
1.11
1.09
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
g2
g2
q||=0.000289Å -1
1.07
1.E-02
1.09
1.07
1.07
1.05
q||=0.000342Å -1
1.05
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.03
1.E-02
q||=0.000393Å -1
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.14
1.12
g2
1.10
1.08
1.06
1.04
1.E-02
q||=0.000447Å -1
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
Figure E.8: g2 functions of 2802 70 for several values of q|| at 110 °C. The red line is the
fit using Equation 4.4 to calculate τ.
295
1.20
1.23
1.18
1.21
1.16
g2
g2
1.25
1.19
1.14
q||=0.000135Å -1
1.17
1.15
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.10
1.E-02
1.14
1.14
1.12
1.12
1.10
1.10
q||=0.000236Å -1
1.08
1.06
1.E-02
1.E+00
1.E+01
Time delay (s)
1.E+02
1.E-01
q||=0.000289Å -1
1.08
1.E+00
1.E+01
Time delay (s)
1.06
1.E-02
1.E+02
1.12
1.13
1.10
1.11
1.08
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
g2
g2
1.15
1.06
1.09
1.04
q||=0.000341Å -1
1.05
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
q||=0.000393Å -1
1.02
1.E-02
1.10
1.08
1.08
1.06
1.06
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
g2
g2
1.10
1.04
1.04
1.02
1.E-01
g2
1.16
g2
1.16
1.07
q||=0.000184Å -1
1.12
1.02
q||=0.000446Å -1
1.00
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
q||=0.000499Å -1
1.00
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
Figure E.9: g2 functions of 2802 70 for several values of q|| at 120 °C. The red line is the
fit using Equation 4.4 to calculate τ.
296
1.20
1.23
1.18
1.21
1.16
g2
g2
1.25
1.19
1.17
1.14
1.12
q||=0.000133Å -1
q||=0.000180Å -1
1.15
1.10
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
Time delay (s)
Time delay (s)
1.17
1.15
1.15
1.13
1.13
g2
g2
1.17
1.11
1.09
1.11
1.09
q||=0.000229Å -1
q||=0.000279Å -1
1.07
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
Time delay (s)
1.14
1.12
1.12
1.10
1.10
1.08
g2
g2
1.07
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
Time delay (s)
1.08
1.06
1.06
q||=0.000328Å -1
1.04
q||=0.000377Å -1
1.04
1.02
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
Time delay (s)
Time delay (s)
1.12
1.10
g2
1.08
1.06
1.04
q||=0.000427Å -1
1.02
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03
Time delay (s)
Figure E.10: g2 functions of 2802 90 for several values of q|| at 110 °C. The red line is the
fit using Equation 4.4 to calculate τ.
297
1.25
1.28
1.23
1.26
1.21
g2
g2
1.30
1.24
1.22
1.19
1.17
q||=0.000134Å -1
1.20
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.22
1.18
1.20
1.16
1.18
1.E+00
1.E+01
Time delay (s)
1.E+02
g2
1.16
1.14
1.12
1.14
q||=0.000231Å -1
1.10
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
q||=0.000282Å -1
1.12
1.E-02
1.E+02
1.12
1.18
1.10
1.16
1.08
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
g2
g2
1.20
1.06
1.14
1.10
1.E-02
1.E-01
g2
1.20
1.12
q||=0.000181Å -1
1.15
1.E-02
1.04
q||=0.000333Å -1
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.02
1.E-02
q||=0.000382Å -1
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.12
q||=0.000434Å -1
1.10
g2
1.08
1.06
1.04
1.02
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
Figure E.11: g2 functions of 2802 90 for several values of q|| at 120 °C. The red line is the
fit using Equation 4.4 to calculate τ.
298
1.13
1.18
1.11
1.16
1.09
g2
g2
1.20
1.07
1.14
1.12
1.05
q||=0.000214Å -1
q||=0.000316Å -1
1.03
1.10
1
10
100
Time delay (s)
1
1000
1.08
1.08
1.06
1.06
1000
g2
1.10
g2
1.10
10
100
Time delay (s)
1.04
1.04
1.02
1.02
q||=0.000316Å -1
1.00
q||=0.000528Å -1
1.00
1
10
100
Time delay (s)
1000
1
10
100
Time delay (s)
1000
1.10
1.08
g2
1.06
1.04
1.02
q||=0.000640Å -1
1.00
1
10
100
Time delay (s)
1000
Figure E.12: g2 functions of 2h48 35 for several values of q|| at 160 °C. The red line is the
fit using Equation 4.4 to calculate τ.
299
1.16
1.22
1.14
1.20
1.12
g2
g2
1.24
1.18
1.16
1.14
1.E-02
1.10
1.08
q||=0.000203Å -1
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.06
1.E-02
1.16
1.11
1.14
1.09
1.12
g2
1.03
1.E-02
1.E+00
1.E+01
Time delay (s)
1.E+02
1.10
1.07
1.05
1.E-01
g2
1.13
q||=0.000280Å -1
1.08
q||=0.000363Å -1
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.06
1.E-02
q||=0.000446Å -1
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.10
1.08
g2
1.06
1.04
1.02
1.00
1.E-02
q||=0.000523Å -1
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
Figure E.13: g2 functions of 2h48 99 for several values q|| at 160 °C. The red line is the fit
using Equation 4.4 to calculate τ.
300
1.15
1.14
1.13
1.12
1.10
1.11
g2
g2
1.08
1.09
1.07
1.06
q||=0.000198Å -1
1.05
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.04
1.02
1.E-02
1.E+02
1.13
1.10
1.11
1.08
1.09
1.E+00
1.E+01
Time delay (s)
1.E+02
g2
1.07
1.06
1.05
q||=0.000270Å -1
1.02
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.10
q||=0.000424Å -1
1.03
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
1.08
1.08
1.05
g2
g2
1.06
1.02
1.04
1.02
1.E-01
g2
1.12
1.04
q||=0.000270Å -1
0.99
q||=0.000499Å -1
1.00
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
q||=0.000576Å -1
0.96
1.E-02
1.E-01
1.E+00
1.E+01
Time delay (s)
1.E+02
Figure E.14: g2 functions of 48k Ref 95 for several values of q|| at 160 °C. The red line is
the fit using Equation 4.4 to calculate τ.
301
1.10
1.11
1.08
1.09
1.06
g2
g2
1.13
1.04
1.07
1.05
1.02
q||=0.000333Å -1
1.00
1.03
1
10
100
Time delay (s)
1
1000
1.14
1.13
1.12
1.11
1.10
10
100
Time delay (s)
1000
g2
g2
1.15
1.08
1.09
1.07
1.06
q||=0.000582Å -1
q||=0.000711Å -1
1.04
1.05
1
10
100
Time delay (s)
1
1000
10
100
Time delay (s)
1000
1.10
1.12
1.08
1.10
1.06
1.08
g2
g2
q||=0.000457Å -1
1.04
1.06
1.04
1.02
q||=0.000838Å -1
1.02
q||=0.000966Å -1
1.00
1
10
100
Time delay (s)
1000
1
10
100
Time delay (s)
1000
1.09
q||=0.00109Å -1
g2
1.07
1.05
1.03
1.01
0.99
1
10
100
Time delay (s)
1000
Figure E.15: g2 functions of 2809 92 for several values of q|| at 110 °C. The red line is the
fit using Equation 4.4 to calculate τ.
302
1.12
1.12
1.10
1.10
g2
1.14
g2
1.14
1.08
1.08
1.06
1.06
q||=0.000227Å -1
1.04
q||=0.000285Å -1
1.04
0
1
10
100
Time delay (s)
1000
0
1.15
1.13
1.13
1.11
1.11
10
100
Time delay (s)
1000
g2
g2
1.15
1
1.09
1.09
1.07
1.07
q||=0.000343Å -1
q||=0.000401Å -1
1.05
1.05
0
1
10
100
Time delay (s)
0
1000
1.14
1.13
1.12
1.11
1.10
10
100
Time delay (s)
1000
g2
g2
1.15
1
1.09
1.08
1.07
1.06
q||=0.000459Å -1
1.05
q||=0.000516Å -1
1.04
0
1
10
100
Time delay (s)
1000
0
1
10
100
Time delay (s)
1000
1.13
1.11
g2
1.09
1.07
1.05
q||=0.000575Å -1
1.03
0
1
10
100
Time delay (s)
1000
Figure E.16: g2 functions of 2809 92 for several values of q|| at 120 °C. The red line is the
fit using Equation 4.4 to calculate τ.
303
1.13
1.15
1.11
1.13
1.09
g2
g2
1.17
1.11
q||=0.00177Å -1
1.09
1.03
0
1
10
100
Time delay (s)
1000
0
1.10
1.10
1.08
1.08
1.06
1.06
g2
g2
q||=0.000236Å -1
1.05
1.07
1.04
q||=0.000300Å -1
1.02
1
10
100
Time delay (s)
1000
1.04
q||=0.000363Å -1
1.02
1.00
1.00
0
1
10
100
Time delay (s)
1000
0
1
10
100
Time delay (s)
1000
1.10
1.10
1.08
1.08
q||=0.000427Å -1
q||=0.000490Å -1
1.06
g2
1.06
g2
1.07
1.04
1.04
1.02
1.02
1.00
1.00
0
1
10
100
Time delay (s)
0
1000
1
10
100
Time delay (s)
1000
1.10
1.08
q||=0.000555Å -1
g2
1.06
1.04
1.02
1.00
0
1
10
100
Time delay (s)
1000
Figure E.17: g2 functions of 2809 92 for several values of q|| at 130 °C. The red line is the
fit using Equation 4.4 to calculate τ.
304
1.18
1.23
1.16
1.21
1.14
g2
g2
1.25
1.12
1.19
1.17
1.10
q||=0.000213Å -1
1.08
1.15
1
10
100
Time delay (s)
1
1000
1000
1.10
1.11
g2
g2
1.13
1.09
1.05
1.07
q||=0.000449Å -1
q||=0.000571Å -1
1.05
1.00
1
1.11
1.09
1.07
1.05
1.03
1.01
0.99
10
100
Time delay (s)
1000
1
10
100
Time delay (s)
1000
1.10
1.07
g2
g2
10
100
Time delay (s)
1.15
1.15
1.04
1.01
0.98
q||=0.000696Å -1
q||=0.000821Å -1
0.95
1
g2
q||=0.000328Å -1
1.13
1.11
1.09
1.07
1.05
1.03
1.01
0.99
10
100
Time delay (s)
1000
1
10
100
Time delay (s)
1000
q||=0.000945Å -1
1
10
100
Time delay (s)
1000
Figure E.18: g2 functions of 9k Ref 80 for several values of q|| at 110 °C. The red line is
the fit using Equation 4.4 to calculate τ.
305
1.16
1.20
1.14
1.18
1.12
g2
g2
1.22
1.16
1.14
1.10
1.08
q||=0.000215Å -1
1.12
1.10
1.04
1
10
100
Time delay (s)
1000
1
10
100
Time delay (s)
1000
1.10
1.11
1.08
1.09
1.07
g2
g2
q||=0.000331Å -1
1.06
q||=0.000578Å -1
1.06
1.04
1.05
1.03
1.02
q||=0.000454Å -1
1.00
1.01
1
10
100
Time delay (s)
1
1000
10
100
Time delay (s)
1000
1.08
g2
1.06
q||=0.000705Å -1
1.04
1.02
1.00
0.98
1
10
100
Time delay (s)
1000
Figure E.19: g2 functions of 9k Ref 80 for several values of q|| at 120 °C. The red line is
the fit using Equation 4.4 to calculate τ.
306
1.14
1.16
1.12
1.14
1.10
g2
g2
1.18
1.12
1.08
q||=0.000172Å -1
1.10
1.08
1.04
1
10
100
Time delay (s)
1000
0
1.14
1.16
1.12
1.14
1.10
1.12
1.08
1.10
g2
g2
0
1.06
1
10
100
Time delay (s)
1000
q||=0.000333Å -1
1.08
q||=0.000278Å -1
1.06
1.04
1.04
1.02
0
1
10
100
Time delay (s)
0
1000
1.15
1.13
1
1.12
q||=0.000388Å -1
g2
1.09
1000
1.08
1.07
1.06
1.05
1.04
1.03
10
100
Time delay (s)
q||=0.000443Å -1
1.10
1.11
g2
q||=0.000223Å -1
1.06
1.02
0
1
10
100
Time delay (s)
1000
0
1
10
100
Time delay (s)
1000
1.14
1.12
q||=0.000497Å -1
g2
1.10
1.08
1.06
1.04
1.02
0
1
10
100
Time delay (s)
1000
Figure E.20: g2 functions of 9k Ref 80 for several values of q|| at 130 °C. The red line is
the fit using Equation 4.4 to calculate τ.
307
1.21
1.21
1.19
1.19
q||=0.000225Å -1
g2
1.23
g2
1.23
1.17
1.17
q||=0.000172Å -1
1.15
1.15
1.13
1.13
0
1
10
100
Time delay (s)
1000
0
1.18
1.16
1.16
1.14
1.14
10
100
Time delay (s)
1000
g2
g2
1.18
1
1.12
1.12
q||=0.000281Å -1
1.10
q||=0.000338Å -1
1.10
1.08
1.08
0
1
10
100
Time delay (s)
1000
0
1.15
1.13
1.13
1.11
1.11
10
100
Time delay (s)
1000
g2
g2
1.15
1
1.09
1.09
q||=0.000394Å -1
1.07
1.07
1.05
q||=0.000451Å -1
1.05
0
1
10
100
Time delay (s)
1000
0
1.10
1.10
1.08
1.08
1.06
10
100
Time delay (s)
1000
g2
g2
1.12
1
1.06
1.04
1.04
1.02
q||=0.000506Å -1
1.02
q||=0.000564Å -1
1.00
0
1
10
100
Time delay (s)
1000
0
1
10
100
Time delay (s)
1000
Figure E.21: g2 functions of 2h09 100 for several values of q|| at 130 °C. The red line is
the fit using Equation 4.4 to calculate τ.
308
1.15
1.15
1.13
1.13
q||=0.000298Å -1
1.11
g2
g2
1.11
1.09
1.07
1.05
0
1
10
100
Time delay (s)
1000
0
1.13
1.10
1.11
1.08
1.09
1.06
g2
g2
1.09
1.07
1.05
1.07
1.05
1
10
100
Time delay (s)
1000
1.04
q||=0.000486Å -1
1.02
q||=0.000424Å -1
1.00
1.03
0
1
10
100
Time delay (s)
0
1000
1.12
1.11
1.10
1.09
1.08
1.07
g2
g2
q||=0.000361Å -1
1.06
1
10
100
Time delay (s)
1000
1.05
q||=0.000550Å -1
1.04
q||=0.000613Å -1
1.03
1.02
1.01
0
1
10
100
Time delay (s)
1000
0
1
10
100
Time delay (s)
1000
Figure E.22: g2 functions of 2h48 99 for several values of q|| at 140 °C. The red line is the
fit using Equation 4.4 to calculate τ.
309
1.16
1.16
1.14
1.14
g2
1.18
g2
1.18
1.12
1.12
1.10
1.10
q||=0.000356Å -1
1.08
q||=0.000418Å -1
1.08
0.0
0.1
1.0
10.0
Time delay (s)
100.0 1000.0
0.0
1.12
1.10
1.10
1.08
1.08
1.0
10.0
Time delay (s)
100.0 1000.0
g2
g2
1.12
0.1
1.06
1.06
1.04
1.04
q||=0.000479Å -1
q||=0.000541Å -1
1.02
1.02
0.0
0.1
1.0
10.0
Time delay (s)
0.0
100.0 1000.0
1.10
1.09
1.08
1.07
1.06
1.0
10.0
Time delay (s)
100.0 1000.0
q||=0.000665Å -1
g2
g2
1.11
0.1
1.05
1.04
1.03
1.02
q||=0.000604Å -1
1.01
1.00
0.0
0.1
1.0
10.0
Time delay (s)
100.0 1000.0
0.0
0.1
1.0
10.0
Time delay (s)
100.0 1000.0
Figure E.23: g2 functions of 48k Ref 95 for several values of q|| at 140 °C. The red line is
the fit using Equation 4.4 to calculate τ.
310
1.11
1.13
1.09
1.11
1.07
g2
g2
1.15
1.09
1.07
1.03
q||=0.000346Å -1
1.05
10
100
Time delay (s)
1000
1
1.13
1.10
1.11
1.08
1.09
1.06
g2
g2
q||=0.000475Å -1
1.01
1
1.07
1.05
10
100
Time delay (s)
1000
1.04
1.02
q||=0.000611Å -1
1.03
q||=0.000747Å -1
1.00
1
10
100
Time delay (s)
1000
1
1.10
1.10
1.08
1.08
1.06
1.06
g2
g2
1.05
1.04
1.02
10
100
Time delay (s)
1000
q||=0.00102Å -1
1.04
1.02
q||=0.000883Å -1
1.00
1.00
1
10
100
Time delay (s)
1
1000
10
100
Time delay (s)
1000
1.08
1.06
q||=0.00115Å -1
g2
1.04
1.02
1.00
0.98
1
10
100
Time delay (s)
1000
Figure E.24: g2 functions of 2h09 42 for several value of q|| at 130 °C. The red line is the
fit using Equation 4.4 to calculate τ.
311
1.11
1.08
1.09
1.06
1.07
g2
g2
1.10
1.04
1.02
1.03
q||=0.000477Å -1
1.00
q||=0.000565Å -1
1.01
1
10
100
Time delay (s)
1000
1
1.10
1.08
1.08
1.06
1.06
1.04
g2
g2
1.05
1.04
1.02
1000
1.02
1.00
q||=0.000653Å -1
10
100
Time delay (s)
q||=0.000739Å -1
0.98
1.00
1
10
100
Time delay (s)
1
1000
10
100
Time delay (s)
1000
1.10
1.08
g2
1.06
1.04
1.02
q||=0.000828Å -1
1.00
1
10
100
Time delay (s)
1000
Figure E.25: g2 functions of PSBr Ref 32 for several values of q|| at 150 °C. The red line
is the fit using Equation 4.4 to calculate τ.
312
1.10
1.11
1.08
1.09
1.06
g2
g2
1.13
1.04
1.07
1.05
1.02
q||=0.000408Å -1
q||=0.000506Å -1
1.00
1.03
1
10
100
Time delay (s)
1
1000
1.10
1.08
1.08
1.06
1.06
1000
g2
g2
1.10
10
100
Time delay (s)
1.04
1.04
1.02
1.02
q||=0.000605Å -1
q||=0.000700Å -1
1.00
1.00
1
10
100
Time delay (s)
1
1000
1.10
1.06
1.08
1.04
1.06
1000
g2
g2
1.08
10
100
Time delay (s)
1.04
1.02
1.00
1.02
q||=0.000801Å -1
q||=0.000901Å -1
1.00
0.98
1
10
100
Time delay (s)
1
1000
10
100
Time delay (s)
1000
Figure E.26: g2 functions of 2h03 PSBr 38 for several values of q|| at 150 °C. The red line
is the fit using Equation 4.4 to calculate τ.
313
1.16
1.14
1.14
1.12
g2
g2
1.16
q||=0.000348Å -1
1.10
1.10
1.08
1.06
1
10
100
Time delay (s)
1000
0
1.15
1.15
1.13
1.13
1.11
1.11
g2
g2
0
1
10
100
Time delay (s)
1000
1.09
1.09
1.07
1.07
q||=0.000467Å -1
q||=0.000526Å -1
1.05
1.05
0
1
1.15
1.13
10
100
Time delay (s)
0
1000
1
1.13
1.11
q||=0.000586Å -1
1.11
g2
g2
q||=0.000408Å -1
1.08
1.06
10
100
Time delay (s)
1000
q||=0.000646Å -1
1.09
1.09
1.07
1.07
1.05
1.03
1.05
0
1
1.13
1.11
g2
1.12
10
100
Time delay (s)
0
1000
1
10
100
Time delay (s)
1000
q||=0.000705Å -1
1.09
1.07
1.05
1.03
0
1
10
100
Time delay (s)
1000
Figure E.27: g2 functions of 2h02 45 for several values of q|| at 120 °C. The red line is the
fit using Equation 4.4 to calculate τ.
314
APPENDIX F
VISCOSITY OF POLY(4-BROMO STYRENE) (PSBR)
The viscosity (Pa·s) of PSBr in the bulk state was measured for a range of angular
frequency at 150 °C, 160 °C and 170 °C under air. A rheometer manufactured by Anton
Paar Inc. with a parallel plate was used.
Rheoplus
10
4
10
6
Jin Kuk PSBR 150C 5
Pa
Pa·s
10
10
3
10
CP-8--SNCust; d=0.052 mm
5
4
10
3
G''
10
2
10
Complex Viscosity
G'
Storage Modulus
G''
Loss Modulus
Jin Kuk PSBR 160C 1
G'
| *|
| *|
2
CP-8--SNCust; d=0.052 mm
| *|
Complex Viscosity
G'
Storage Modulus
G''
Loss Modulus
Jin Kuk PSBR 170C 1
CP-8--SNCust; d=0.052 mm
10
1
10
0.001
0.01
0.1
1
Angular Frequency
10
10 1/s 100
1
| *|
Complex Viscosity
G'
Storage Modulus
G''
Loss Modulus
0
Anton Paar GmbH
Figure F.1: Rheological data for the PSBr material used to prepare 2h03 PSBr 38.
315