A fast Ising-type model for binary polymer blends containing their block copolymer

Computational

and Theoretical

Polymer

PII: s1089-3156(9@oooo4-x

A fast Ising-type blends containing Senthil

Kandasamy

model their

and E. Bruce

Sciencr Vol. I, No. 314, pp. 183-189, 1997 ICI 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0189-3156/98/S19.00 + 0.00

for binary polymer block copolymer

Nauman”

The lsermann Depattment of Chemical Engineering, Rensselaer 12180-3590, USA (Received 27 October 1997; revised 12 December 1997)

Polytechnic

Institute.

Troy, NY,

Monte Carlo simulationswere done to predict the rate of domain growth in an amphiphilic system.An Ising-type modelwasusedfor a systemcomprisedof two homopolymersA and B and the corresponding copolymer A-B. The power-law exponent for phasegrowth decreasedfrom its normal value of 0.33 as block copolymerwasadded.The pure block copolymerexhibited lamellaeand had a growth exponent of zero. There is a fair agreementbetweenthe simulatedscalingexponentsand experimentalvalues.The model is fast from a computationalviewpoint. It is a good representationof a systemcomprisedof two homopolymerswith equal chainlengthsand the correspondingblock copolymer with a chainlengthtwice aslong. Extensionsof the model may be usefulin representingother systems,but this hasnot beentested yet. 0 1998ElsevierScienceLtd. All rights reserved (Keywords:

block copolymer;

spinodal

decomposition;

Ising model)

INTRODUCTION There has been considerable theoretical interest in the kinetics of phase transition in quenched systems. Transitions from disorder to order are common in metal alloys and polymer blends. Often, the quenched system will evolve from its initial unstable state toward a final equilibrium state by a dynamic process known as spinodal decomposition. Monte Carlo simulations have been used to study such phase separation processes in binary systemslP5.These computer simulations show that for late times, the growth rate of ordered domains satisfies the power-law of growth, R(r) cx tn

where R is a characteristic length scale. Some studies have shown that the scaling exponent, n ranges from l/4 to l/3 but the commonly accepted value for a simple binary phase separation process is 1/32P5. The presence of amphiphiles, such as surfactants or block copolymers influences the properties of binary systems. Block copolymers act as compatibilising agents at the interface by reducing the interfacial tension. There is ample experimental

*To whom

correspondence

should

be addressed

evidence for the compatibilizing properties of block copolymers 6-8. The other relevant observation is that block copolymers decrease the rate of phase coarsening9,10. Theoretical models for block copolymers include mean field treatments”-13, quaternary models14 and lattice models 15-19. Mean field models and other continuum models fail to capture the essence of the dual nature of the surfactants and are fairly inept in representing the properties of the copolymer molecule. Lattice models for block copolymers range from simple Ising-type models, in which the block copolymer occupies only one or two sites in the lattice, to Flory-type models in which the block copolymer occupies more than two sites and diffuses by reptation. The Flory-type models are better representations of systems containing block copolymers, but require enormous computational effort to simulate even very short physical times. In polymer systems, phase growth may occur over time scales from milliseconds to hours, and even the low end of this range is inaccessible with available computers. Extensive literature can be found on the static properties and behaviour of systems containing block copolymers. However, the dynamic properties are still not clearly understood. The fact that block copolymers decrease the rate of coarsening is well agreed upon, but the exact exponent to the power law is under dispute. Some studies show the growth exponent to be l/5 while other studies have

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A fast king-type

model

for binary polymer

blends:

S. Kandasamy

shown it to be closer to l/4. It is quite understood that the coarsening rate depends on various factors such as the relative concentration of the copolymer in the system and nature of the copolymer. In this paper, we investigate the effect of the concentration of block copolymers on the rate of phase coarsening in ternary systems consisting of homopolymers A and B and the AB block copolymer. An Ising-type Monte Carlo simulation is used. The model is briefly described and the results are presented and compared with the literature. MODEL

SYSTEM

The model presented in this paper is an extension of the spin-Ising model with spin-exchange dynamics for binary systems. This is similar to the model as suggested by Kawakatsu and Kawasaki]‘. Both the approaches use nearest neighbour interactions to calculate the Hamiltonian of the system. But this model considers nearest neighbour spin-exchange as the mode of phase separation, whereas Kawasaki’s model suggests diagonal spinexchanges. This model, unlike theirs, has no forbidden exchanges due to topological constraints, which is a direct consequence of the elaborate spinexchange scheme. In this model, a three component mixture of polymer A, polymer B and block copolymer A-B is considered. The simulations are performed on a two dimensional N x N lattice with periodic boundary conditions. Homopolymer molecules A and B occupy a single square lattice site each, while the copolymer molecule A-B occupies two adjacent lattice sites. Molecules of type A are assigned a spin of + 1 and molecules of type B have a spin of -1. The copolymer molecule occupies two nearest neighbour sites with opposite spins. Spins are exchanged by following the simple algorithm described below. A schematic representation is given in Figure 1. On the N x N lattice, two nearest neighbours are chosen at random. If the points contain two homopolymers, then a simple swap is made. If only one of the adjacent neighbours contains a block copolymer, then a swap is made in such a way that the block copolymer occupies the two chosen points. If both the points chosen contain blocks and they do not belong to the same block copolymer, then an end to end swap is made for one of the block copolymer molecules. When both the chosen points belong to the same copolymer, it is determined whether there is a nearby parallel copolymer molecule. When there is. a swap is made in such a way that the original copolymer molecule and the nearby parallel molecule change orientations. When there is no nearby parallel molecule, 184

and E. B. Nauman

the original copolymer molecule undergoes an endto end swap. In Figure I, the two types of molecules A and B are represented by two different types of lattice shadings. The block copolymer is represented by a bond between the two lattice sites occupied by it. The representation on the left in each of the pictures shows a possible configuration before or after exchange. The picture on the right shows its counterpart, after or before the exchange. Figure lu represents the simple binary case. Figure lb and c represents the case where a homopolymer and a copolymer molecule are chosen. Figure Id-h represents the case where the two chosen sites both have copolymer molecules. Such an elaborate spin-exchange scheme is required to make sure that all the random pairs chosen have a way to diffuse and that no exchange is forbidden. The Hamiltonian of the system is calculated by considering the nearest neighbour interactions for a cluster of 25 points around the first random site. This involves 36 nearest neighbour interactions. The Hamiltonian of the system is given by H = C SiSj (i. j)

where si and sj can take values of 1 or - 1. The sites under consideration are represented schematically in Figure 2. The central site represents the first random site chosen (x) and the other 24 sites surrounding it (0) could possibly contribute to the Hamiltonian depending on the type of molecule chosen. The probability of spinexchange is given by ,-(AHiRT,) Pexchange

=

1 + e-(AWRT,)

where Tg is the final temperature to which the system is quenched. RESULTS AND DISCUSSION Simulations were run for varying block concentrations for long times and the phase coarsening behaviour was studied. A simple 20/80 homopolymer binary system was taken to be the base case, instead of the critical quench normally used in this type of a study. The reason for this is that polymer blends typically have a particulate minor phase. The block concentrations were varied in such a way that the sum of the homopolymer

COMP. AND THEOR. POLYMER SCIENCE Volume 7 Number 314 1997

A fast king-type

Figure

1

Schematic

representation

model for binary polymer blends: S. Kandasamy

of the spin exchange

concentration and half the copolymer concentration was maintained at 20/80. This enabled the comparison of the system under different block concentrations. Simulations were run on a 100x 100, 2-D lattice for times up to 100000 MCS (Monte Carlo seconds) where one MCS is defined as 100x 100 attempted spin exchanges. Properties were averaged over 20 runs in each case. The temperature was chosen such that kT, = 1. The lattice of spins I

I

I

I

I

I

I

Cl

00 0 0

I

I

I

I

I

I

i

sites under

con-

0

0

0

0

0

0

0

0

0

0

0

0

00x000 0

0 Figure 2 sideration

and E. B. Nauman

Schematic representation of the 25 lattice for the free energy calculations

COMP. AND THEOR.

was initialized at a random state, corresponding to infinite temperature. Then, the system was quenched to a temperature Tq and the simulation was performed till the specified time. The spins were stored at periodic intervals, and these data files were used for analysis. Once the simulations were done, the domain sizes at different times were calculated. For the particulate domains examined here, it was satisfactory merely to count the number of particles in the lattice. The equivalent radius was then averaged over the 20 runs performed in each case. Two different measures of domain sizeswere calculated, using the method specified above: one by including the block-part of the corresponding copolymer in the domain and the other by ignoring it. These methods yielded different but consistent measures of the domain size. The different cases that were run included 20/80/ 0 (20% A, 80% B and 0% block), 18/78/4, 17/77/6, 15/75/10, 10/70/20 and O/60/40. Thus the system consisted of 20% type A molecules and 80% type B molecules in all the different cases. The morphologies obtained for the 17/77/6 case at different times are shown in Figure 3. We can observe that the block copolymers (represented in black) which are initially random, gradually migrate to the interface between the two homopolymers (represented by grey and white) at later times. At still later times, the domains grow at a rate POLYMER

SCIENCE

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S. Kandasamy

Figure 3 Morphologies at different times for the case with low block concentration (6(X,). White and grey represent the homopolymers while black represents the copolymer

governed by the power law. but with a different exponent than the simple binary case. Figure 4 shows the morphologies obtained for different block concentrations at a low quench time of 100 MCS. It can be observed that the domains are fairly small and the block copolymers are still randomly placed and have not yet completely covered the interface. Figure 5 shows the morphologies for the different block concentrations at a long time of 100 000 MCS. The domains are fairly large, and it can be observed that the block copolymer molecules engulf the minor component almost totally, especially at high block concentrations. We can also observe the formation of micelles in the 10 and ZOO/6block cases. suggesting that the extra block that is not used to saturate the interface forms miceffes. It can be observed that the domain sizes are smaller for high block concentrations at the same ripening time. This suggests different 186

and E. 8. Nauman

(a) 0% block

(b) 4% block

(c) 6 /c block

Cd) 10% block

(e) 20% block

(f) 40% block

Figure 4 Morphologies at the same early time (100 MCS) for different block concentrations. White and grey represenl the homopolymers while black represents the copolymer

scaling exponents at long times for different block concentrations. To estimate the scaling exponents, the domain sizes were calculated for all the different cases by the method described previously. Single molecules of homopolymers or block copolymers, not adhered to a larger domain were not considered in calculating the domain size, since they just act as diffusing entities responsible for the domain growth. The exponent was estimated by finding the slope of the log-log plot of time (MCS) versus domain size at late times. The system was assumed to be in the scaling regime if the log-log plot was linear over two different orders of magnitude of time. The simulation results were clearly in the scaling regime for most of the cases. The only exception to linearity over two orders of magnitude of time was the case with 40% of block and 60% of a single homopolymer.

COMP. AND THEOR. POLYMER SCIENCE Volume 7 Number 3/4 1997

A fast king-type

model

for binary polymer Table

1

Scaling

Concentration

blends: exponent

of block

S. Kandasamy versus

block

copolymer

and E. B. Nauman

concentration

(“41)

Scaling

0

block

(b) 4% block

(c)6%

block

(d) 10%

block

(e) 20%

block

(f)-lO%

block

Figure 5 Morphologies block concentrations. while black represents

(n)

0.321 0.232 0.202 0.177 0.123 0.055 0.000

4 6 I0 20 40 IO0

(a)O%

exponent

sizes are a clear function of block concentration at late times, but this dependence is less certain at early times. This is evident from Figure 7 where it can be seen that the curves representing the different block concentrations intersect at various points. Table I shows the scaling exponent as a function of block concentration. The scaling exponent decreases from its classical binary value of 0.33 to near zero at very high block concentrations. The values obtained are fairly consistent with experimental results obtained by Cavanaugh and

at the same late time (IO5 MCS) for different White and grey represent the homopolymers the copolymer

The log-log plots, shown in F&ure 6, yielded the scaling exponents shown in Table I. Semi-log plots of domain size versus time at early times (F@N~ 7) and late times (Figure 8) show that the domain

Figure 7 Semi-log early times

Figure 6 Log-log times. for different

Figure times

plot of domain size versus block concentrations

simulation

time at late

8

Semi-log

plot

of domain

plot of domain

siles versus

sizes versus

simulation

simulation

time

at

time at late

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Nauman*’ on polystyrene/polybutadiene/diblock blends. At 100% block copolymer, a scaling exponent of zero is expected since bulk samples of symmetrical block copolymers form lamella structures which do not ripen with time. The expectation is confirmed in Figure 9. In the picture, the two portions of the block copolymer A and B are represented by black and white. The morphologies at early times appear random, but as time progresses gradually tend to align in a particular direction to form lamellas. At late times, the blocks are ironed out and form nearly perfect thin strips, as expected. All the domain sizes used in estimating the scaling exponents were based on the inclusion of the part of the block that is similar to the minor component. Domain sizes were also found by ignoring the contribution of the block copolymer to the

and E. B. Nauman

Figure 10 Comparison of domain tion is included and excluded

growth

when

the block

contribu-

domains. The results obtained were consistent qualitatively. The scaling exponents obtained either way were very close, but the domain sizes themselves were different. As an example, the domain sizes obtained either way for block-concentration of 6% are plotted in Figure IO. It can be seen that the two lines are almost parallel to each other, suggesting close to identical scaling exponents, with different intercepts. CONCLUSIONS

Figure tration.

9 Morphologies White represents

at different times for 100% block concenmolecules of one type and black. the other

188

COMP. AND THEOR.

POLYMER

SCIENCE

From the results of the simulations, it can be seen that the model predicts lower scaling exponents for systems with surfactants. The results qualitatively agree with those obtained by Kim et al.‘O This also agrees with the results obtained by the quaternary model of Kwak and Nauman’4 who showed a scaling exponent of l/4 at late times for low block copolymer concentrations. Fairly consistent experimental confirmation is available from the results of Cavanaugh and Nauman20 on PS/PB/SB Diblock blends. Formation of lamellar strips at a 100% block copolymer concentration further asserts the validity of the spin-exchange scheme adopted. The model also overcomes the topological constraints as experienced by Kawasaki’s modeli7. Free energy change is the only factor that determines the feasibility of exchange. Also we overcome the extensive computational times required as in the case of Flory-type and other reptation models. The obvious shortcoming of the model is that it directly represents a system with a block copolymer having equal block lengths, each block length being equal to the length of the homopolymer molecule. Extension of the model to more general situations would require complex spin-exchange

Volume 7 Number 314 1997

A fast king-type

model

schemes and would necessitate the presence of voids in the lattice. But even in the more general cases, the size of a lattice site would be in the same order of magnitude as the radius of gyration of the polymer molecules rather that the Kuhn length, as in the case of reptation models. This should still require relatively short computational times. So it can be concluded that this model is a simple and easy-to-use tool to predict polymer phase morphologies and dynamic properties in the presence of block copolymers within reasonable computational constraints. REFERENCES 1. Mazenko, G. F.. Walls, 0. T. and Zhang, F. C., Phys. Rev. B, 1985,31,4453; 32, 5807. 2. Gawlinski, E. T., Grant. M., Gunton J. D. and Kaski, K., Phys. Rev. E, 1985, 31, 281.

for binary polymer

blends:

S. Kandasamy

and E. 6. Alauman

3. Sadiq, A. and Binder, K. J., Stat. Phys., 1984, 35, 517. 4. Amar, J., Sullivan, F. and Mountain. R., Phys. Rev. B, 1988, 37, 196. 5. Roland, C. and Grant, M., Phys. Rev. B. 1989, 39, 11971. 6. Laradgi, M., Guo, H., Grant, M. and Zuckerman, M., J. Phys. Corm’. Mailer, 1992, 4, 6715. 7. Kawakatsu, T., Kawasaki, K., Farusaka, M., Okabayashi, H. and Kanaya, T., J. Chem. Phys., 1993,99, 8200. 8. Vilgis, T. A. and Noolandi, J., Macromolecules. 1990, 23. 2941. 9. Jo, W. H. and Kim, S. H., Macromolecules, 1996, 29. 7204. 1996, 29, IO. Kim. S. H., Jo, W. H. and Kim, J.. ,Macromo/ecules, 6933. Il. Helfand, E., Macromolecules, 1975, 8, 552. 1980. 13, 1602. 12. Lelbler, L., Macromolecules, G. H. and Helfand. E.. J. Chem. Phw, 1987, 87, 13. Fredrickson. 697. 14. Kwak, S. and Nauman, E. B.. J. Po/vm. Sci. Polym. Phys. Ed., 1996, 34, 11715. 15. Widom, B., J. Chem. Phgs., 1986, 84, 6943. 16. Schick. M. and Shih. W. H., Phys. Rev. B. 1986, 34, 1797. 17. Kawakatsu, T. and Kawasaki. K.. J. Coil. Inter/. Sri., 1992, 23, 148. 11. 18. Larson, R. G., J. Chem. Phys., 1992.96, 19. Larson. R. G., J. Chem. Php., 1988, 89, 1642. 20. Cavanaugh, T. J. and Nauman. E. B.. in preparation.

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