Polymer simulations, field-theoretic

Keywords Polymer blends Self-consistent field theory External potential dynamics Field-theoretic polymer simulations Polymeric microemulsion Polymer dynamics... 

By the Hubbard-Stratonovich transformation we have rewritten the partition function of the interacting multi-chain systems in terms of noninteracting chains in complex fluctuating fields, il7 -i- W and il7 - W. hi field theoretical polymer simulations, one samples the fields U and W via computer simulation using the above Hamiltonian (cf. Sect. 4.4). 

G. H. Fredrickson, V. Ganesan, and F. Drolet (2002) Field-theoretic computer simulation methods for polymers and complex fluids. Macromolecules 35, pp. 16-39... 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

In the field-theoretic SCMF method [46,47], the external mean field acting on a polymer chain is calculated self-consistently with the composition profile. The utilization of the SCMF simulation scheme demands several important assumptions. Polymers are represented by phantom ehains (thread-like space curves) of statistical segments that pass through the spatial domain in the mean field produced by other components of the system (polymers and solvents). The interactions between pairs of segments are determined by phenomenological Flory-Huggins parameters x-... 

At d = 1 one has a completely stretched chain with ly = 1. At d = 2 the exact result v = 3/4) [13] is obtained. The upper critical dimension is d = 4, above which the polymer behaves as a random walker. The values of the universal exponents for SAWs on d - dimensional regular lattices have also been calculated by the methods of exact enumerations and Monte Carlo simulations. In particular, at the space dimension d = 3 in the frames of field-theoretical renormalization group approach one has (v = 0.5882 0.0011 [11]) and Monte Carlo simulation gives (i/ = 0.592 0.003 [12]), both values being in a good agreement. 

To summarize, the example of homopolymer/copolymer mixtmes demonstrates nicely how field-theoretic simulations can be used to study non-trivial fluctuation effects in polymer blends within the Gaussian chain model The main advantage of these simulations is that they can be combined in a natural way with standard self-consistent field calculations. As mentioned earlier, the self-consistent field theory is one of the most powerful methods for the theoretical description of polymer blends, and it is often accurate on a quantitative level hi many regions of the parameter space, fluctuations are irrelevant for large chain lengths (large Jf) and simulations are not necessary. Field-theoretic simulations are well suited to complement self-consistent field theories in those parameter regions where fluctuation effects become important. 

If the film becomes so thin that the chains cannot cross through each other, the polymer conformations become truly two-dimensional and belong to a different imiversality class -namely, two-dimensional self-avoiding walks. In this Umit, an exact field-theoretic description demonstrates that the chains again become Gaussian, Rf N, without any logarithmic corrections. This behavior has been studied in great detail by extensive computer simulations. ... 

Popov YO, Lee J, Fredrickson G (2007) Field-theoretic simulations of polyelectrolyte complexation. J Polym Sci Pol Phys 45 3223-3230... 

Polymer Alloys. Perturbation of meltlike conformation upon transfer to a multicomponent environment is not understood. The influence of proximity to phase boundaries, coupled density and concentration fluctuations, and mixture composition on both single-chain dimensions and miscibility are problems that have begun to be addressed within the PRISM formalism for the simple symmetric blend model by Singh and Schweizer and symmetric diblock copolymer model by David and Schweizer,"" and other more coarse-grained field-theoretic approaches. Comparisons with the few available simulations " have also... 

Some of the coarse-grained parameters, i e and can be easily measured by experiments or in simulations. The other two parameters, %N and the suppression of density fluctuations, XqN, are thermodynamic characteristics, which are not directly related to the structure (i.e., they cannot be simply expressed as a function of the molecular coordinates). If density fluctuations of the polymeric liquid are small on the length scale of interest (e.g., width of an interface between domains), then the value of the compressibility has only a minor relevance and decreasing it even further will not significantly affect the behavior of the system. Thus, field-theoretic calculations often take the idealized limit of strict incompressibility. In particle-based simulations, however, one often softens the constraint in order to facilitate the motion of the interaction centers and, thereby, reduces the viscosity of the polymer liquid. The Flory-Huggins parameter, in turn, is a crucial coarse-grained parameter and different methods have been devised to extract it from experiments or simulations [16, 20-25]. We shall briefly discuss this important issue in Section 5.2.3, and further refer the reader to the literature, where computer simulations have been quantitatively compared with mean field predictions and where the role of fluctuations on the coarse-grained parameters is discussed [16, 22]. 

Similar attempts to include fluctuations beyond one-loop in SC FT have also been exercised in the context of neutral polymers using field theoretical simulations [55,90] or by bridging SCFT with Monte Carlo techniques [91]. However, these techniques have not been applied for the case of polyelectrolytes with counterions and added salt ions due to very high computational cost. 

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