The mean field to Ising crossover in the critical behavior of polymer mixtures : a finite size scaling analysis of Monte Carlo simulations

H. P. Deutsch; K. Binder

doi:doi:10.1051/jp2:1993182

Issue		J. Phys. II France Volume 3, Number 7, July 1993


Page(s)		1049 - 1073
DOI		https://doi.org/10.1051/jp2:1993182

DOI: 10.1051/jp2:1993182
J. Phys. II France 3 (1993) 1049-1073

The mean field to Ising crossover in the critical behavior of polymer mixtures : a finite size scaling analysis of Monte Carlo simulations

H. P. Deutsch and K. Binder

Institut für Physik, Johannes-Gutenberg Universität Mainz. Staudinger Weg 7, 6500 Mainz, Germany

(Received 3 February 1993, accepted 6 April 1993)

Abstract
Monte Carlo simulations of the bond fluctuation model of symmetrical polymer mixtures (chain lengths $N_{\rm A}=N_{\rm B}=N$ ) are analyzed near the critical temperature $T_{\rm c}(N)$ of their unmixing transition. Two choices of interaction range are studied, using a square-well potential with effective coordination number ${z_{\rm eff}}\approx 14$ or ${z_{\rm eff}}\approx 5$ , respectively, at a volume fraction $\phi=0.5$ of occupied lattice sites, and chain lengths in the range $8\leqslant N \leqslant 512$ . A linear relation between N and $T_{\rm c}(N)$ is established, $T_{\rm c}(N)=AN+B$ , where the correction term B is positive for ${z_{\rm eff}}\approx 14$ but negative for ${z_{\rm eff}}\approx 5$ . The critical behavior of the models is analyzed via finite size scaling techniques, paying attention to the crossover from three-dimensional Ising-like behavior to mean-field behavior, using a formulation based on the Ginzburg criterion. It is shown that the location of the crossover does not depend on ${z_{\rm eff}}$ , consistent with the expected entropic origin of the mean-field behavior for long chains. However, despite large numerical efforts only a crude estimation of the crossover scaling function of the order parameter (describing the coexistence curve of the blends) and of the singular chain length dependence of the associated critical amplitude is possible. These results are discussed in the context of pertinent theories and related experiments.