Mechanics of 21st Century - ICTAM04 Proceedings
NANO MECHANICAL ANALYSIS OF IFM
FORCE PROFILES ON SELF-ASSEMBLED
MONOLAYERS
Mingji Wang, Kenneth M. Liechti
Center for Mechanics of Solids, Structures and Materials,
Department of Aerospace Engineering and Engineering Mechanics,
The University of Texas, Austin, TX 78712
Vibha Srinivasan
Institute for Theoretical Chemistry,
The University of Texas, Austin, TX 78712
John M. White
Center for Nanomolecular Science and Technology,
The University of Texas, Austin, TX 78712
Peter J. Rossky
Institute for Theoretical Chemistry,
The University of Texas, Austin, TX 78712
Abstract
In this study, a defect-free, self-assembled monolayer of octadecyltrichlorosilane (OTS) was deposited on a silicon substrate. Nanoindentation
experiments were performed with an interfacial force microscope (IFM)
on these 2.5 nm monolayers. As a first step in continuum finite element
analyses, the OTS was assumed to be linearly elastic and isotropic. Adhesive interactions were also accounted for via a cohesive zone model.
However, the assumption of linearity gave rise to force profiles that did
not match the measurements. Molecular dynamics simulations were
therefore employed in order to provide further insight into the behavior of OTS. These simulations indicated that the OTS had a highly
non-linear and nearly incompressible response. Based on these results,
a hypo-elastic material model was developed as a convenient continuum
representation of the mechanical behavior of OTS. This was then used
in finite element analyses, which were able to fully reproduce the IFM
force profiles. As a result, molecular and microscopic scales were linked
in a relatively simple but very effective manner. This suggests that
there is a class of problems where the continuum representation of the
material behavior may be directly obtained from molecular analyses.
Keywords: Interfacial force microscope, self-assembled monolayer, nanoindentation.
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1.
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Introduction
Self-assembled monolayers are used in MEMS devices for the reduction of stiction, friction and wear [1–4]. The frictional behavior between
scanning probe microscope tips (functionalized [5] and bare [5] and surfaces coated with self-assembled monolayers has been examined. Selfassembled monlayers have also been used to control interfacial toughness [7–9]. In the latter experiments, the locus of fracture was close the
self-assembled monolayer. In both the friction and fracture studies, the
mechanical behavior of the self-assembled monolayers is needed for the
determination of fracture and frictional properties.
The objective of the present study was to provide a nanomechanical
analysis of interfacial force (IFM) force profiles produced by a tungsten tip probing a self-assembled monolayer. An octadecyltrichlorosilane (OTS) self-assembled monolayer was deposited on a silicon (100)
surface covered with an oxide layer. This configuration was first numerically analyzed based on the assumption that the OTS was isotropic and
linearly elastic. Surface interactions were also included in the model.
An isotropic, non-linear elastic constitutive model for the OTS was then
developed that was based on molecular dynamics simulations. This was
used in a continuum analysis of the IFM experiment.
2.
Experimental
Self-assembled monolayers of OTS were deposited on Si (100) surfaces
whose native oxide layer had been removed by a HF etch and replaced
with a well hydroxylated oxide layer using a piranha solution. The OTS
deposition was conducted in a dry box using an anhydrous solution of
OTS in dicyclohexyl. Clusters were removed from the monolayer using
sonication in various solvents. This process [10] resulted in stable monolayers with an rms roughness less than 0.07 nm. Ellipsometry was used to
determine the thickness of the OTS monolayers was 2.5±0.1 nm, which
means that they were tilted at 16◦ from the normal.
The coated specimens were probed with an IFM [11], which has a self
balancing force sensor (Fig. 1) that allows nanoindentation experiments
to be conducted in displacement control and avoids the instabilities associated with AFM experiments. Electrochemically etched tungsten wires
with tip radii ranging from 100 to 200 nm were used as the probes. The
tips were examined in a scanning electron microscope after each experiment in order to ensure that no blunting had occurred.
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Nano Mechanical Analysis of IFM Force Profiles . . .
3
Figure 1.
A schematic of the IFM. The orientation of the common plate is maintained parallel via a feedback loop with the force being proportional to the feedback
signal. This provides a force sensor which is capable of being used in displacement
controlled experiments.
3.
Analysis
The specimens consisted of an OTS monolayer covering the SiO2 layer
(2.0±0.1 nm) that had been grown in the piranha solution. These two
thin layers on top of the Si meant that substrate effects would be present,
thereby disallowing [12] the use of classical contact mechanics theories
for extracting the mechanical behavior of the OTS layer. Accordingly,
a finite element analysis of the IFM experiment was conducted that
incorporated normal surface interactions via a traction-separation law
of the form shown in Fig. 2. The area underneath the curve gives the
thermodynamic work of adhesion ω = 0.5σ0 δt between the tungsten tip
and the OTS. In all analyses, the tungsten tip, silicon and silica were
assumed to be linearly elastic and isotropic following the properties given
in Table 1. The OTS was first assumed to be linearly elastic and isotropic
and its modulus was extracted by matching finite element solutions for
the force profiles with measured ones. The OTS was also taken to be
an isotropic hypo-elastic material based on the results of a molecular
dynamics analysis of uniaxial compression straining of the OTS.
The molecular analysis was conducted using the code DL POLY 2.0.
A simulation cell of 30 OTS molecules was compressed between rigid
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σ
σ(δ)
δ
a
δ0
δτ
δ
σ0
c
Figure 2.
Contact geometry and normal traction-separation law.
tungsten and tridymite planes. Periodic boundary conditions were used
to simulate the infinite in-plane extent of the OTS layer relative to its
thickness. The CH3 , CH2 and OH groups in the OTS molecules were
treated as united atoms in a room temperature calculation of the stresses
that resulted from moving the rigid planes together at a strain rate of
107 /s. The harmonic, valence angle, cosine dihedral angle and van der
Waals potentials that were used in the analysis are given in [13].
Table 1.
Properties of tungsten, silica and silicon that were used in all analyses.
Material
Tungsten
Silica
Silicon
4.
Young’s Modulus
E [GPa]
392
73.6
168
Poisson’s Ratio
ν
0.28
0.17
0.22
Results
Detailed IFM force profiles resulting from the probing of hydroxylated
silicon and OTS-coated silicon are shown in Fig. 3. The IFM clearly
distinguished the presence of the OTS monolayer in the compressive
regime. There were some interesting differences in the tensile regime
shown in the magnified insert. For the hydrolyzed silicon, there was
an adhesive step in the response of about 0.2 mN as a water bridge
presumably formed in the ambient humidity in which the experiment
was conducted. There was a slightly smaller initial step when the OTScoated specimen was probed. We think that this was due to a reduction
in the 16◦ tilt as the tungsten tip approached. The water explanation is
also possible, but less likely, due to the hydrophobic nature of OTS.
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Nano Mechanical Analysis of IFM Force Profiles . . .
2.5
2.5
P
(µN)
2
P
(µN)
2
Hydroxylated silicon OTS coated silicon
Loading
Loading
Unloading
Unloading
1.5
1.5
1
0.5
1
0
0.5
(a)
-0.5
28
28.5
29
29.5
30
δ (nm)
0
(a)
-0.5
18
20
22
24
26
28
30
δ (nm)
0.4
P
(µN) 0.2
0
(b)
Hydroxylated silicon
Loading
Unloading
-0.2
0.4
P
(µN) 0.2
0
-0.2
16
(c) OTS coated silicon
Loading
Unloading
18
20
22
24
26
28
30
δ (nm)
Figure 3. IFM force profiles from bare and OTS-coated silicon. (a) overall response,
(b) detail of the adhesive response for bare silicon and (c) detail of the adhesive
response for OTS-coated silicon.
The results from the analyses that assumed that the OTS was linearly elastic are shown in Fig. 4. The first result (Fig. 4a) was for bare
silicon. In this case, all the properties were known and the finite element solution agreed well with the measured force profile. Solutions for
OTS coated specimens are shown in Figs. 4b,c for Poisson’s ratios of 0
and 0.44, respectively. As will be seen shortly, the latter was obtained
from the molecular analysis. The match between solutions and measurements was not as good as was just noted for bare silicon, particularly
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2.5
2.5
2
P
(µN)
P
(µN)
1.5
2
Data
1.5
FEM
1
0.5
1
0
-0.5
27
0.5
28
29
30
δ (nm)
0
-0.5
22
24
26
28
30
δ (nm)
(a)
2.5
P
(µN)
ν
2
OTS
=0.0
0. 2
P
(µN) 0. 1
0
1.5
Data
-0. 1
E
1
OTS
(GPa)
-0. 2
24.5
25
2 5.5
0.5
26
2 6.5
δ (nm)
10
15
20
0
-0.5
20
22
24
26
28
δ (nm)
(b)
2.5
P
(µN)
ν
2
OTS
=0.44
0.2
P
(µN) 0.1
0
1.5
Data
1
E OTS (GPa)
0.5
-0.1
-0.2
2 4.5
25
2 5.5
26
2 6.5
δ (nm)
4.5
7.5
0
-0.5
20
(c)
22
24
26
28
δ (nm)
Figure 4. Comparison of linear elastic solutions and measured force profiles for (a)
bare and OTS coated silicon with (b) ν = 0 and (c) ν = 0.44 for the OTS.
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Nano Mechanical Analysis of IFM Force Profiles . . .
at low and high force levels. The reduced moduli extracted from the
best fit between solutions and measurements are shown in Table 2. The
extracted values are higher than those commonly associated with amorphous polymers and are consistent with the higher degree of order in
self-assembled monolayers. The parameters for the traction-separation
laws for bare and coated silicon are also listed in Table 2. The work of
adhesion between the tungsten tip and the bare silicon was higher than
the value obtained for tungsten probing an OTS monolayer. Such values are consistent with the formation of water bridges and the ordering
reflects the hydrophobic nature of OTS.
Table 2.
Extracted modulus and adhesion parameters for for bare and coated silicon.
Bare silicon
OTS on silicon
OTS on silicon
Poisson’s
Ratio
ν
0.22
0.0
0.44
Reduced
Modulus
E ∗ [GPa]
168
15±5
6±1.5
σ0
[MPa]
60
35
35
δ0
[nm]
3
5
5
δt
[nm]
7
7
7
ω
[mJ/m2 ]
210
122.5
122.5
The stress-strain behavior of the OTS under uniaxial compressive
strain as obtained from the molecular analysis is shown in Fig. 5. The response was quite nonlinear and reminiscent of rubbery materials. There
were several discontinuities in the response, which are probably due
to configurational changes. At higher stress levels, the three normal
stresses were nearly the same, indicating an approach to incompressibility. The tangent modulus and Poisson’s ratio of the hypo-elastic
model were extracted from the molecular analysis results. The values
are shown in Fig. 6 and reflect the nonlinear behavior with an initial
modulus of 1 GPa, rising to 30 GPa at high compressive strains. The
early, relatively soft response suggests some partial contact and localized rearrangements of the end groups as the rigid tungsten plate first
approaches. The increasing stiffness reflects the interaction between the
lateral expansion of each chain that is associated with the zig-zag assembly of the individual molecules and the confinement provided by the
infinite extent of the monolayer in the analysis. The Poisson’s ratio remained constant at 0.44. These values of tangent modulus and Poisson’s
ratio were used in a hypo-elastic model of uniaxial compressive strain
and compared (Fig. 5) with the results of the molecular analysis. There
was very reasonable agreement up to about 40% nominal strain, which
was sufficient for modeling the IFM experiment.
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40
σ
(GPa)
(a)
30
σ
z
σ
x
Hypoelastic σ
20
z
Hypoelastic σ
x
10
0
0
0.1
0.2
0.3
0.4
0.5
ε (Nominal)
z
200
(b)
σ
(GPa) 150
σ
σ
100
z
x
Hypoelastic σ
Hypoelastic σ
z
x
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ε (Nominal)
z
Figure 5. Stress-strain behavior of OTS under uniaxial strain. Data obtained from
molecular analysis is compared with a hypo-elastic fit.
This continuum representation of the mechanical behavior of an OTS
monolayer was then used in a finite element analysis of the IFM experiment. The same traction-separation law that had been derived from the
linear analysis was used because it mainly affects the tensile response
while the mechanical behavior is reflected in the compressive response
with very little interaction between the two regimes. The finite element
solution compared well with the measurements (Fig. 7) even in the low
compressive force regime without any further adjustments to the hypoelastic model.
The agreement between analysis and experiment in Fig. 7 is remarkable, given that the constitutive law that was used in the continuum
model was based directly on molecular dynamics simulations without
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Nano Mechanical Analysis of IFM Force Profiles . . .
30
E
Et
t
(GPa)
MD data
20
Fit
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
ε (Log)
z
0.5
ν
t
0.4
νt
MD data
0.3
0.44
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
ε (Log)
z
Figure 6.
behavior.
Tangent modulus and Poisson’s ratio for a hypo-elastic model of OTS
2.5
P
(µN)
Data
2
Nonlinear analysis
1.5
0.2
P
(µN) 0.1
1
0
-0.1
0.5
-0.2
24.5
25
0
25.5
26
δ (nm)
26.5
-0.5
20
22
24
26
28
δ (nm)
Figure 7.
Comparison of the hypo-elastic solution for the IFM experiment with
measurements.
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any adjustment. This result suggests that the behavior of the OTS is
indeed simple enough that differences in the time scales of the molecular dynamics analyses and the actual experiments are not important.
It appears that a class of problems has been opened up where spatial
and temporal scales can be crossed in a relatively simple manner, and
molecular dynamics analyses may be used to motivate continuum representations of self-assembled monolayers in a simple but direct manner.
This result has a direct bearing on analyses of the frictional behavior
of self-assembled monolayers where the contact area is required for the
determination of the frictional strength. The usual approach is to use the
classical JKR contact mechanics analysis developed for contact of monolithic bodies or layered media with sufficiently thick layers that substrate
effects are not important. The load versus contact radius response of the
OTS monolayer was obtained from Hertz, DMT and JKR analyses of
the contact using a reduced modulus of 6 GPa and a Poisson’s ratio of
0.44 for the OTS and 122.5 J/m2 for the work of adhesion between tungsten and OTS. These were the same parameters were used in the linear
elastic finite element analysis of the IFM experiment. The results of the
classical contact mechanics analyses are compared (Fig. 8) with the con20
a (n m )
15
10
H y p o - e l a s tic
L i n e a r e l a s ti c
JK R
DMT
H e rt z
5
0
-0.5
0
0 .5
1
1.5
2
2 .5
3
P ( µN )
Figure 8. Contact radii from classical contact mechanics and finite element analyses
of IFM probing of an OTS monolayer.
Mechanics of 21st Century - ICTAM04 Proceedings
Nano Mechanical Analysis of IFM Force Profiles . . .
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tact radii that were extracted from the analyses that produced the force
profiles in Fig. 7. The JKR contact mechanics analyses underestimated
the contact radius by as much as 50%. The DMT and Hertz solutions
differed even more from the finite element analyses. This means that
frictional strengths derived from JKR analyses are about 50% higher
than the actual values.
5.
Conclusions
The experiments conducted in this study demonstrated that the IFM
has sufficiently high resolution in both force and displacement for conducting nanoindentation experiments on ultra thin films whose thickness
is on the order of nanometers. The interpretation of the force profiles
from such experiments is complicated by substrate effects and interface
mismatch, requiring a numerical approach rather than classical contact
mechanics analyses.
The IFM experiments showed that the OTS monolayers were elastic,
even when the indentation depth was relatively large. Linear elastic
analyses of the IFM experiments suggested that the Young’s modulus of
OTS monolayers is relatively high. These high values are not surprising
given the highly ordered structure of the OTS monolayers. However,
the linear elastic analyses were unable to fully match the measured force
profiles. Molecular dynamics simulations were used to develop a nonlinear constitutive model for OTS, which could be represented as a hypoelastic material. The measured response was fully reproduced when the
hypo-elastic behavior was incorporated in the finite element analyses of
the IFM experiments. This result suggests that representations of the
behavior of self-assembled monolayers that are obtained from molecular
simulations can be readily incorporated in continuum analyses.
The analysis also showed how important it is to account for the nonlinear behavior of the OTS monolayer in determining the true contact
radius. This has a large impact on estimates of frictional strength.
Acknowledgements
This work was supported by National Science Foundation (Grant
number CMS9978678). PJR acknowledges the support of the R.A. Welch
Foundation. The authors would like to thank Dr. Jack Houston at Sandia National Laboratory for his generous assistance in setting up and
operating the IFM facility at UT Austin.
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