Simulations of Polymer Solutions: A Field-Theoretic Approach - ACS

Chapter 17

Simulations of Polymer Solutions: A Field-Theoretic Approach 1

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AlfredoAlexander-Katz ,AndréG.Moreira , and GlennH.Fredrickson 2

Downloaded by UNIV OF LEEDS on June 19, 2016 | http://pubs.acs.org Publication Date: August 17, 2003 | doi: 10.1021/bk-2003-0861.ch017

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Physics Department and Materials Research Laboratory, University of California, Santa Barbara, CA 93106

We used field-theoretic simulations to study the equilibrium behavior of polymers in a good solvent confined to a slit of width L. In particular, we obtained density profiles across the slit for different values of the monomer excluded volume over a wide range of concentrations C. We also obtained mean field results for the profiles. The effective correlation length ξ was calcu lated from the density profiles and compared to the mean field result (valid in the limit of high concentrations). For small ex cluded volume parameters we found that ξ is well described by the mean field result, while for larger excluded volume in teraction the correlation length shows a C-3/4 scaling behavior, which is compatible with the behavior expected for this system in the semi-dilute regime. eff

eff

© 2003 American Chemical Society

Case and Alexandridis; Mesoscale Phenomena in Fluid Systems ACS Symposium Series; American Chemical Society: Washington, DC, 2003.

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Field theory can be a useful tool in the study of condensed matter systems and especially complex fluids. Examples of its application in the latter context include polymer solutions(i,2,3), polyelectrolytes(4), or ionic solutions(5). As usual in such approaches, one often has to invoke approximations that make the calculations amenable to analytical treatment, e.g. perturbation theory, mean field approximation, etc. Although this is some times enough to study the systems un der consideration, it is highly desirable to have a numerical scheme that allows the exact solution of the field theory. On one hand, this would allow a direct compari son with the aforementioned approximations (i.e., one can directly test the validity of the approximations in different limits). On the other hand, it is reasonable to ex pect that under some conditions (e.g. dense systems), a field-theoretic simulation approach can be more effective than a particle-based one. Recently, Fredrickson et. al.(6,7) proposed a new method to simulate the equi librium properties of polymer solutions based on field theory, the so-called FieldTheoretic Polymer Simulation (FTPS). Although there has been previous work on the simulation of polymers using similar techniques^) (see also the article by Sevink and Fraaije in this volume), the new technique is capable of yielding "ex act" results (aside from numerical errors, as well as finite size and discretization effects) that go beyond the mean field approximation. The FTPS method has been successfully applied(6) to the study of the orderdisorder transition for an incompressible two-dimensional diblock copolymer melt. Although incompressible melts are interesting from the scientific (and industrial) point of view, there is a wide range of systems where polymers are in solution. It is then desirable to understand how the FTPS approach works in the latter case. We will focus our attention in this short communication on a homopolymer solu tion under good solvent conditions confined to a slit of width L. This problem has received some attention in the past(9~/6), which provides a theoretical basis for comparison with our simulation results. This article is organized as follows. In section II we provide a brief theoreti cal background on the field-theoretic formulation of a homopolymer solution. In section ΙΠ we introduce the simulation method, where the sampling algorithm is discussed. In section IV we present the simulation results for the homopolymer solution confined to a slit. In particular we discuss the density profile across the slit for several concentrations, ranging from the dilute to the concentrated regime. We compare these results with mean field results obtained both analytically as well as numerically. Finally, section V contains some concluding remarks.

Theoretical Background Polymers can be described in a coarse-grained fashion by the so called Gaus sian thread model(i), which corresponds to the continuum limit of a sequence of beads (monomers) attached by harmonic springs. The chains are represented by continuous curves in space R ( ^ ) , where a = 1 , c o u n t s the different chains and s is a continuous variable between 0 and 1 along each chain contour (R(0) a


281 corresponds to the position of one end of the chain while R ( l ) corresponds to the position of the other end). The energy of a given configuration contains two parts, the first one being an elastic contribution which is given (in units of ksT) by

il/' In the latter expression, all lengths were scaled by R , the ideal radius of gyration, given by R = b N/(2d). The parameter b represents the statistical segment length, d is the dimensionality of space, and Ν is the degree of polymerization. The second contribution to the total energy comes from the interaction be tween non-adjacent monomers in the same polymer and between monomers in different polymers. This interaction can be attractive or repulsive depending on the solvent: in good solvents it is repulsive, in bad solvents it is attractive and in theta solvents it is absent (and the chain behaves accordingly as a Gaussian chain). A typical simple model used for a good solvent is a pairwise repulsive v5(r) potential which prevents the monomers from overlapping. In the case of a poor solvent, ν < 0, one would have to include a repulsive three-body interaction to stabilize the system. Here we will focus only on the case of a good solvent. In terms of the microscopic monomer density, p(r) = ds5(r- R < 4 ff < we expect to find the semi-dilute regime. Scaling arguments(9) predict this regime corresponds to con centrations within the range 1 /2? / < C < B. For the case of Β = 10 (open circles), the semi-dilute regime should occur when 0.4 < C < 10, while for the case Β = 25 (filled squares), it should occur if 0.3 < C < 25. As can be seen from Fig. 4, both systems with Β = 10 and Β = 25 have a that is compatible with the C / scaling law for the semi-dilute regime (cf. dashed lines) in the above ranges. e

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Conclusions Although the results presented above are quite encouraging, one should men tion that the C P U time required to obtain sensible averages for the density profile for concentrations in the semi-dilute regime is still relatively long. In our simple explicit integration scheme, the Langevin time step has to be kept small to avoid numerical instabilities. It seems likely that semi-implicit schemes might alleviate


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Figure 4: Effective correlation length for Β = 1 (filled diamonds), Β = 10 (open circles), and Β = 25 (filled squares) as a function of the parameter BC. The solid line denotes the Gaussian correlation length 1^—1/ \/2BC. The dashed curves have a slope of — 0.75 (reproduced with permission from reference (21). Copyright 2003 American Institute of Physics).

this difficulty. Because the solution is not incompressible, the field has to generate local correlations that account for the monomer excluded volume, which tends to make the convergence process slower. On the other hand, as the overall density fluctuations were reduced by increasing C, the method presented here did con verge quite rapidly, which is a promising feature since there exist a wide variety of important problems that involve concentrated solutions. In this respect, we expect that this method should prove to be very useful in the study of dense poly mer phases and melts, which tend to be more difficult to treat with particle-based simulation methods. In summary, we have presented a field theoretic simulation method to calculate in an essentially exact way the equilibrium properties of polymer solutions. As an example, we performed a simulation for a homopolymer solution under good solvent conditions confined to a slit. The density profiles and the correlation length were calculated for this particular case, and we showed that the simulations lead to the expected deviations from mean field solution. It is also possible to use this method to calculate other thermodynamic quantities like the chemical potential, the density-density correlations and the osmotic pressure. This work was partially supported by the NSF through grant DMR-98-70785. Acknowledgment is also made to the donors of the Petroleum Research Fund, administered by the ACS. Extensive use of the UCSB-MRSEC Central Computing Facilities is also acknowledged. A A K would like to thank C O N A C Y T for partial support through the U C Mexus program.


289 References 1. 2. 3.


4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

Edwards, S. F. Proc. Phys. Soc. 1966, 88, 265. Freed, K. F. Adv. Chem. Phys. 1972, 22, 1. Doi, M . ; Edwards, S. F. The theory of polymer dynamics; Clarendon Press: Oxford, 1986. Netz, R. R.; Andelman, D., to appear in Phys. Reports. 2003. Moreira, A . G.; Netz, R. R. in Electrostatic Effects in Soft Matter and Biophysics-Nato Science Series, edited by Holm, C.; Kékicheff, P.; Podgornik R.; Kluwer: Dordrecht, 2001. Ganesan, V.; Fredrickson, G. H. Europhys. Lett. 2001, 6, 814. Fredrickson, G. H.; Ganesan, V.; Drolet, F. Macromolecules 2002, 35, 16. Fraaije, J. G. Ε. M. J. Chem. Phys. 1993, 99, 9202. Daoud M.; de Gennes P. G. J. Physique (Paris) 1977, 38, 85. Wang, Y.; Teraoka, I. Macromolecules 1997, 30, 8473. Wang, Y.; Teraoka, I. Macromolecules 2000, 33, 3478. Teraoka, I.; Cifra, P. J. Chem. Phys. 2001, 115, 11362. Yethiraj Α.; Hall, C. K.Macromolecules 1990, 23, 1865. Milchev, Α.; Binder, K. J. Comput.-Aided Mater. Des. 1995, 2, 167. Milchev, Α.; Binder, K. Macromolecules 1996, 29, 343. Milchev, Α.; Binder, K. J. Phys. II (France) 1996, 6, 21. Schoenmaker, W. J. Phys. Rev. D 1987, 36, 1859. Helfand, E. J. Chem. Phys. 1975, 62, 999. Gausterer, H. Nucl. Phys. A 1998, 642, 239c. Lee, S. Nucl. Phys. Β 1994, 413, 827. Alexander-Katz, Α.; Moreira, A. G.; Fredrickson, G. H. to appear inJ.Chem. Phys. 2003. de Gennes, P. G . Scaling concepts in polymer physics; Cornell University Press: Ithaca, 1979.


Simulations of Polymer Solutions: A Field-Theoretic Approach - ACS

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