Supporting Information - Ordering Poly(trimethylsilyl styrene-block-D,L-lactide) Block
Copolymers in Thin Films by Solvent Annealing Using a Mixture of Domain-Selective Solvents
Calculations for estimation of the solubility parameter of PTMSS by the group contribution method
The solubility parameter of a polymer can be estimated by using group contribution methods as
summarized in the Polymer Handbook.[1] The groups used to estimate the solubility parameter of
PTMSS, whose structure is shown in Figure S1, are summarized in Table S1.
Figure S1. Structure of PTMSS
Table S1. Molar attraction constants and number of groups used in estimating the solubility parameter
of PTMSS
-Si-CH3
-CH2=CHp-phenylene
Fj (MPa1/2*cm3/mol)
-77
437
57
272
1350
Total F
Number of groups
1
3
1
1
1
1/2
2913 MPa *cm3/mol
Equation S1 was then used to estimate the solubility parameter of PTMSS, where p represents the
density of the polymer, Fj represents the molar attraction constant of each group, and Mi is the
molecular weight of the repeat unit. p = 0.963g/cm3 for PTMSS and Mi = 176.35 for TMSS. It should be
noted that this equation is simplified for a polymer by canceling out the number of repeat units and end
groups are neglected. It should also be noted that in the absence of data for the molar attraction
constant of a silicon group with four bonds, the molar attraction constant for a silicon group with two
bonds was used. The solubility parameters of the solvents and polymers used in this study are
summarized in Table S2.[1]
(S1)
Table S2. Solubility parameters of solvents and polymers used in this study
Polymer
PTMSS
PLA
Solubility parameter, P
(MPa1/2)
17.2
20.5
Solvent
Cyclohexane
Acetone
Solubility parameter, S
(MPa1/2)
16.8
20.3
Vs
(cm3/mol)
108.0
73.4
Estimation of P-S using solubility parameters
The polymer-solvent interaction parameter can be estimated based on solubility parameters as
summarized in the Polymer Handbook in Equation S2 where Vs represents the molar volume of the
solvent, R is the ideal gas constant, and S and P are the solvent and polymer solubility parameters,
respectively. The values evaluated for this study are summarized in Table S3.
𝜒𝑃−𝑆 =
𝑉𝑠
2
(𝛿𝑠 − 𝛿𝑝 )
𝑅𝑇
+ 0.34
(S2)
Table S3. Polymer-solvent chi-parameters in this study
PTMSS-cyclohexane
PTMSS-acetone
PLA-cyclohexane
PLA-acetone
0.347
0.937
0.625
0.341
Terracing in lamellae sample after thermally annealing
Figure S2. AFM image of terracing in lamellae sample after thermally annealing at 160°C for 15 minutes.
Terrace height is measured to be 14-15 nm and agrees well with bulk spacing after thermal annealing.
Extracting Lo from image analysis
The periodicities of the domains in the SEM images were determined by performing a standard fast
Fourier transform on the image as detailed elsewhere.[2] The power spectral density was circularly
averaged and the averaged PSDs for PTMSS5.5-b-PLA6.6 and PTMSS2.3-b-PLA1.7 are shown in Figures S3a
and b, respectively. Lo was determined from the relationship Lo = 2/ko, where ko is the location of the
primary peak.
Figure S3. Circularly averaged PSDs and resulting Lo values for a) PTMSS5.5-b-PLA6.6 and b) PTMSS2.3-bPLA1.7
Identifying neutral surface treatments, optimal film thickness, and solvent annealing conditions
Water contact angles were measured for homopolymer brushes (hydroxyl-functionalized constituent
homopolymers grafted to the substrate) and the island/hole behavior was observed for identical
thicknesses of PTMSS5.5-b-PLA6.6 on the brushes after solvent annealing. The presence of neither islands
nor holes on the PS-r-PMMA surface treatment was used to identify it as a neutral surface treatment. A
cross-linked polystyrene surface treatment was also identified as a neutral surface treatment for
PTMSS5.5-b-PLA6.6 during solvent annealing using this method.
Figure S4. Water contact angles of surface treatments and island/hole behavior on the surface
treatments. a) water contact angle on PLA brush b) holes form on PLA brush c) water contact angle on
PS-r-PMMA d) no islands or holes form e) water contact angle on PTMSS brush f) islands form on PTMSS
brush
Optimal film thickness and solvent annealing conditions were identified on the PS mat surface
treatment. These conditions corresponded to optimal conditions on the PS-r-PMMA surface treatment
as well. Film thickness was optimized by solvent annealing the block copolymer and observing the selfassembly by top-down SEM. As observed in Figure S5a, a film thickness of 15 nm was too thick and
resulted in regions of parallel orientation of the block copolymer. When the film thickness was reduced
to 12 nm in Figure S5b, the regions of parallel orientation were reduced. When the film thickness was
further reduced to 10 nm as shown in Figure S5c, perpendicular orientation was observed everywhere.
When the thickness was further reduced to 9 nm and became too thin in Figure S5d, the self-assembly
appeared to contain significant defects.
Figure S5. Optimization of film thickness for solvent annealing. a) 15 nm thick film b) 12 nm thick film c)
10 nm thick film d) 9 nm thick film
The optimal solvent annealing condition for self-assembly was also investigated. When a 10 nm thick
film was annealed with a 50 sccm N2 flow rate through the cyclohexane reservoir and a 50 sccm rate
through the acetone reservoir, the pattern shown in Figure S6a was observed. Defectivity appeared to
be reduced when changing the annealing condition to 60 sccm through the cyclohexane reservoir and
40 sccm through the acetone reservoir, shown in Figure S6b. The same trend was observed for a film
thickness double this, shown in Figures S6c and S6d. Thus, this condition was identified as the best for
the annealing experiments.
Figure S6. Optimization of solvent annealing conditions for. a) 10 nm thick film of after annealing with
50 sccm N2 through the cyclohexane reservoir and 50 sccm through the acetone reservoir, b) 10 nm
thick film of after annealing with 60 sccm N2 through the cyclohexane reservoir and 40 sccm through
the acetone reservoir, c) 20 nm thick film of after annealing with 50 sccm N2 through the cyclohexane
reservoir and 50 sccm through the acetone reservoir, d) 20 nm thick film of after annealing with 60 sccm
N2 through the cyclohexane reservoir and 40 sccm through the acetone reservoir,
It has been previously established that  varies inversely with temperature for PTMSS-b-PLA.[3]
To rule out the possibility that the increase in  observed when solvent annealing with a
mixture of selective solvents is due to the difference in temperature between thermal
annealing (performed at 160°C) and solvent annealing, which is done at room temperature,
SAXS and rheological data were analyzed for PTMSS-b-PLA samples over a range of
temperatures. SAXS data was collected for PTMSS5.5-b-PLA6.6 at temperatures between 120°C and
180°C. As seen in Figures S7a and S7b, it was observed that domain spacing of this polymer
does not change significantly with temperature and when extrapolated to room temperature
and assuming a linear relationship between domain size and annealing temperature, could only
account for a 1 nm increase in the domain spacing. Likewise, SAXS data was collected for
PTMSS2.3-b-PLA1.7 over the same temperature range and it was observed that the sample remains
in the disordered state the entire time, seen in Figure S7c. Rheology data in Figure S7d for
PTMSS2.3-b-PLA1.7 also indicates that it remains disordered for all temperatures above 105°C.
Rheology data were collected on a TA Instruments AR-2000ex in strain controlled mode at 1
rad/s frequency in the linear viscoelastic regime. This data suggests that the domain selectivity
of the solvent annealing process likely contributes more significantly to the increase in  than
the lower temperature at which the solvent annealing process takes place.
Figure S7. A) SAXS patterns of PTMSS5.5-b-PLA6.6 at temperatures ranging from 120-180°C b) Plot
of domain spacing vs. temperature for PTMSS5.5-b-PLA6.6 c) SAXS patterns of PTMSS2.3-b-PLA1.7 at
temperatures ranging from 120-180°C d) Rheology data for PTMSS2.3-b-PLA1.7 at temperatures
ranging from 100-150°C
The self-consistent field theory formalism:
We used polymer self-consistent field theory (SCFT) to predict the domain spacing and spinodal  for
AB diblock copolymer having volume fraction, f  0.5 in two selective solvents
 S1 , S2  .
The
copolymer chains are modeled as Gaussian chains having N segments whose conformations are
described by a continuous functions Ri ( s ) where s is a continuous chain index coordinate running from
0 to 1 for the i th copolymer chain. The solvent conformations for j th solvent are denoted by Rs1 j and
Rs2 j for solvent S1 and S2 respectively. We assume that the segment densities of monomers and solvents
are identical and denoted as  0 .
Using above notation, the partition function of the system in canonical ensemble is given by,
  U 0  Ri ( s )  U 0  Rs j   U 0  Rs j  
ns
ns
n
 1 
 2 
Z    DRi ( s ) DRs1 j DRs2 j exp 
  U  R ( s), R , R 

i 1 j 1 j 1
1 i
s1 j
s2 j 


(1)
   A   B   S   S  0 
1
2
where  A ,  B ,  s1 and  s2 denote the microscopic densities of A, B, S1 and S2 respectively. The total
number of polymers and solvent molecules are denoted by n and ns respectively. The incompressibility
of the overall system is enforced by the delta function by requiring the sum of local densities of all the
species in the system to equal the average system density  0 . U 0 corresponds to the bonded
interactions represented by Gaussian chain model given by,
3
U 0  Ri ( s )   2
2b
R ( s )
ds i


s
i 1 0
n
1
2
(2)
where b denotes the statistical segment length. In the partition function (Eq. (1)), U1 describes the nonbonded interactions represented by Flory model:
 U1 
1
0
 dr  
AB
 A  B   AS  A  S   AS  A  S   BS  B  S   BS  A  B  .
1
1
2
2
1
1
2
2
(3)
The Flory parameter  ij characterizes the energetic interactions between i and j species.
The partition function given by equation (1) is transformed to a field theory by the introduction of
potential fields WA (r ) , WB (r ) , WS1 (r ) and WS2 ( r ) conjugate to the local volume fractions
 A   A / 0 , B   B / 0 ,  S   S / 0 and  S   S / 0 of components.
1
1
2
2
In the mean-field approximation adopted in this study, the free energy of the system, f  fb ( fb is the
free energy of the homogenous phase) can be expressed as:

 N  f  fb 
0VRg 3





  AB  A (r )   A  B (r )   B   AS  A (r )   A  S (r )   S
1
1
1

 
 S2 (r )   S2   BS1  B (r )   B  S1 (r )   S1
AS2  A ( r )   A
1
  dr 

V
   BS2  B (r )   B  S2 (r )   S2  WA (r ) A ( r )  WB ( r ) B ( r ) 

 WS (r ) S (r )  WS (r ) S (r )   (r ) 1   A ( r )   B ( r )   S ( r )   S (r )
1
2
2
1
2
 1
 AB ln QAB WA ,WB   N  S1 ln QS1 WS1   N  S2 ln QS2 WS2 




















(4)
where i represents the average volume fraction of species i . In above equation all the lengthscales
are normalized by the unperturbed radius of gyration ( Rg ). QAB WA ,WB  represents the single chain
partition function of copolymer chains under external fields WA , WB given by,
QAB WA , WB  
1
drq AB (r , s  1)
V
(5)
Where V represents the volume of the system and q AB (r , s ) is the end segment distribution function
which satisfies the modified diffusion equation:
q AB
  2 q AB  WA (r )q AB
s
q AB
  2 q AB  WB (r )q AB
s
0s f
(6)
f  s 1
with an initial condition qAB (r ,s = 0)= 1 . The evaluation of the component volume fraction fields
requires solution of a “complementary” propagator, q †AB ( r , s ) which satisfies the following diffusion
equation:
q †AB
  2 q †AB  WA (r )q †AB
s
q †AB
  2 q †AB  WB (r )q †AB
s
0  s  1 f
(7)
1 f  s  1
with an initial condition q †AB (r ,s = 0)= 1 . QS1 WS1  and QS2 WS2  represent the partition functions
for solvent species given by:
Qs1 Ws1  
 Ws1 (r ) 
1
d
r
exp


V
N 

 Ws (r ) 
1
Qs2 Ws2    dr exp   2

V
N 

The component volume fraction fields are then calculated using:
(8)
 A (r ) 
 B (r ) 
 AB
f
 dsq
QAB
AB
(r ,s)q AB † (r ,1- s)
AB
(r ,s)q AB † (r ,1- s)
0
 AB
1
 dsq
QAB
f
(9)
S
 W (r ) 
 S1 (r )  1 exp   s1 
QS1
N 

 S (r ) 
2
S
 Ws (r ) 
exp   2

QS2
N 

2
In the mean-field approximation, the potential fields are calculated using:




N   ( r )      N  ( r )      ( r )
N  (r )      N  (r )      (r )
N   ( r )      N  ( r )      ( r )
WA (r )   AB N  B (r )   B   AS1 N  S1 (r )   S1   (r )
WB (r )   AB
A
A
BS1
WS1 (r )   AS1
A
A
BS1
WS2 (r )   AS2
A
A
BS2
S1
S1
B
B
B
(10)
B
 (r )    A (r )   B (r )   S (r )   S (r )  1
1
2
 is incremented until the maximum error in incompressibility field reaches below 0.015 . Above
equations form a closed set of self-consistent equations solved in one-dimension using periodic
boundary conditions. We implement the pseudospectral method to solve the diffusion equation for
qAB (r ,s) and q AB † (r ,s) with a discretization s  2.5  103 . The density fields are evaluated using
the standard Trapezoidal rule. The condition of self-consistency is enforced by restricting the maximum
error in consecutive field updates to 103 .
1. J. Brandrup; E. H. Immergut; E. A. Grulke; A. Abe; D. R. Bloch, John Wiley & Sons.
2. R. Ruiz; R. L. Sandstrom; C. T. Black, Advanced Materials 2007, 19, 587.
3. J. D. Cushen; C. M. Bates; E. L. Rausch; L. M. Dean; S. X. Zhou; C. G. Willson; C. J. Ellison,
Macromolecules 2012, 45, 8722.