VOLUME 65, NUMBER 15 PHYSICAL REVIEW LETTERS 8 OCTOBER 1990 Static and Dynamic Crossover in a Critical Polymer Mixture Frank S. Bates and Jeffrey H. Rosedale Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455 Petr Stepanek and Timothy P. Lodge Department of Chemistry, University of Minnesota, MinneapolisM, innesota 55455 Pierre Wiltzius A T& T Bell Laboratories, Murray Hil/, New Jersey 07974 Glenn H. Fredrickson Department of Chemical and Nuclear Engineering and Materials Department, University of California, Santa Barbara, Santa Barbara, California 93106 Rex P. Hjelm, Jr. Los Alamos Neutron Scattering Center, Physics Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Received 23 April 1990) A model low-molecular-weight polymer mixture was studied as a function of temperature near the critical point by small-angle neutron scattering (SANS) and dynamic light scattering (DLS). SANS measurements reveal a crossover in the static susceptibility from mean-field to non-mean-field behavior at T„=-T,+30 C, in quantitative agreement with the Ginzburg criterion. This crossover behavior is also reflected in the DLS experiments, which reveal mode-coupled critical dynamics far into the mean- field regime. blends, while largely unexplored experimentally, is po- tentially very rich. ' Ultimately, on close approach to the critical point, the behavior should be that of model H, although the crossover to the nonclassical critical properties of model H can be quite complex. To discuss this crossover we focus on the collective (mutual) diffusion coefficient D„which can generally be written as the ratio of an Onsager transport coefficient A to the concentration susceptibility S(0): D, =A/S(0). The simplified description of a symmetric binary fluid em- bodied in model H highlights critical fluctuations of the order parameter (concentration field), as well as non- linear interactions (mode couplings) of these fluctuations with fluctuations in the momentum density. Proper treatment of these fluctuations' leads to renormaliza- tions of both A and S(0). At high temperatures (e& 1), however, such renormalizations are negligible and mean-field (Van Hove) theory applies. For polymer mixtures, the bare transport coefficient scales as ' ' Ap N /lV, where N, is the number of segments between entanglements. ' Hence, in the case of polymers sufficiently large to be well entangled, we have Ap-N ', while for shorter chains obeying Rouse dy- namics one sets N — N, to obtain Ap — 1. Because the mean-field susceptibility of a blend scales as Ne ', the expected behavior for t. ~ 1 is D, p — N N t. , which has recently been observed in high-molecular-weight mix- tures. ' As the critical point is approached, singular contribu- tions to A and S(0) emerge, but not necessarily in the same range of reduced temperature. In particular, for e — N the susceptibility is expected to cross over to the Ising expression' S(0) — N "e ", but the fluctuation contribution (Kawasaki approximation ' with reptation 1893 PACS numbers: 61.25. Hq, 05.70.3k, 64.60.Fr, 64.60. Ht It has long been appreciated that the equilibrium criti- cal properties of binary mixtures of simple liquids fall into the Ising universality class. The dynamical critical properties have only been more recently established, with extensive theoretical and experimental evidence support- ing the hypothesis that such mixtures belong to the same dynamical universality class as model H of Hohenberg and Halperin. ' Key to the maturation of this field was the advent of mode-coupling theories and the dynamical renormalization group. As was first appreciated by de Gennes, binary mix- tures (blends) of high-molecular-weight, flexible poly- mers also exhibit Ising-like static critical properties, al- though the long-ranged nature of the polymer-polymer interactions shrinks the Ising (i.e. , non-mean-field) re- gime to a narrow range of reduced temperatures about the critical point, t. &N '. Here, N is the degree of polymerization of a symmetric polymer mixture and e = (g, — g)/g„where g =AT '+8 represents the segment-segment interaction energy; A and 8 are system-specific constants, and g, corresponds to the ther- modynamic stability limit. The existence of a Ginzburg parameter -N ' consistent with numerous experimen- tal observations of mean-field static exponents in high- molecular-weight polymer mixtures (N + 100). Recent- ly, however, Schwahn, Mortensen, and Yee-Madeira have succeeded in observing the crossover in the static susceptibility from mean-field to Ising-like temperature dependence in a relatively high-molecular-weight (N — 10 ) polystyrene-polyvinylmethylether (PS-PVME) mixture, by conducting small-angle neutron-scattering (SANS) measurements near the critical point, 5x10 KE&3x10 The dynamical critical behavior of binary polymer 1990 The American Physical Society