Ising critical behavior

Sariban et al. [101, 107, 276, 277] were the first to emphasize that the Ising critical behavior can be seen in polymer mixtures for not too long chains and verified it by their simulations. A consequence of Ising behavior that is easily verified by experiment is that the spinodal temperature (or mean field critical temperature T F, respectively) which is defined for 4>A = < >Acrit from a linear extrapolation of the inverse scattering intensity S fq = 0) with temperature to the point where S, n(q = 0) = 0 must be offset from the actual critical temperature Tc (Fig. 32). This phenomenon has been seen in simulations [92,101,107] as well as in various experiments [69, 71, 215, 216, 278], A detailed analysis of the non-mean field critical behavior has allowed the estimation of critical exponents y = 1.26 0.01 [215-217,69], v = 0.59 + 0.01 [215] or v as 0.63 [71], and also the exponent describing the decay of correlations at Tc has been estimated [215], 0.047 0.004. These numbers are in fair agreement with... 

Schwahn, D., Mortensen, K., and Madeira, H. Y. (1987) Mean-field Uid Ising criticed behavior of a polymer blend, Phys. Rev. Lett. 58, 1544-1546. 

A field-theory based on this simple expansion will already yield three-dimensional Ising critical behavior. Note that the non-trivial critical behavior is related to fluctuations vaW or a- b- Fluctuations in the incompressibility field U or the total density a + are not important. 

Both the statics and the collective dynamics of composition fluctuations can be described by these methods, and one can expect these schemes to capture the essential features of fluctuation effects of the field theoretical model for dense polymer blends. The pronounced effects of composition fluctuations have been illustrated by studying the formation of a microemulsion [80]. Other situations where composition fluctuations are very important and where we expect that these methods can make straightforward contributions to our understanding are, e.g., critical points of the demixing in a polymer blend, where one observes a crossover from mean field to Ising critical behavior [51,52], or random copolymers, where a fluctuation-induced microemulsion is observed [65] instead of macrophase separation which is predicted by mean-field theory [64]. 

Another possibility is that the crossover between mean field and Ising critical behavior, which is spread out over many decades in 1 — TITc (68,69), also causes the exponents xi, X2, X3 in equation 10 to be effective exponents, which show a significant variation when one studies B N),cQ ), etc. over many decades in N. Usually experiments and simulations have only 1 to 2 decades in N at their disposal, and therefore all conclusions on the validity of equations 10 and 11 are still preliminary. However, experiments do allow a study of enough decades in 1 - T/Tcl to confirm the theoretical expectation that the critical exponents etc. take the values of the Ising universality class. 

The polydispersity was estimated from the observed values of Z and was very small (i.e. a few percent). The value of Qo indicates that an inter-droplet correlation exists with a spacing of about 20 nm. From these results, we conclude that the system consists of relatively monodis-perse droplets (radius 5 nm), which scatter uniformly. Therefore, the general features of this system are quite similar to those of a simple fluid and may be expected to exhibit 3D-Ising critical behavior. 

Monte Carlo simulations very early demonstrated the effect of thermal composition fluctuations in low molecular blends. Studies by Sariban et al. [16] exclusively found Ising critical behavior in blends of molar volume up to about 16000 cm /mol and no indications of a crossover to mean field behavior. Such a mean field crossover was later detected by Deutsch et aL [ 17] in blends with an order of magnitude larger chains. These results and the techniques of Monte Carlo simiflations have been extensively reviewed by Binder in [4]. 

Thermal fluctuations can be described within the Gaussian approximation at sufficiently high temperatures above the critical temperature. For these situations, the system fulfills the conditions of mean field approximation [9]. On the other hand, thermal composition fluctuations become strong near the critical temperature, leading to non-Hnear effects which asymptotically close to the critical temperature imply that the system obeys the universality class of 3D-Ising critical behavior. Thermal fluctuations are described by the Ginzburg-Landau Hamiltonian which is written as a fimctional of the spatially varying order parameter

[Pg.21]

Crossover to the renormahzed Ising critical behavior was observed in two symmetric ( a = b) PB/PS polymer blends without any solvent but not in the corresponding blends mixed with the non-selective solvent oDCP up to a concentration of 20vol%. It might be instructive to compare the observations in both blends with results from polymer blend-solution mixtures obtained by the group of Nose [44,45]. In their experimental work no crossover was found in symmetric blend-solvent samples in accord with... 

Jacob J, Kumar A, Anisimov M A, Povodyrev A A. and Sengers J V 1998 Crossover from Ising to mean-field critical behavior in an aqueous electrolyte solution Phys. Rev. E 58 2188... 

Only in the limit Ixj/Gi 1 does one expect Ising-like critical behavior in Eq (7) due to the universality principle of critical phenomena, "... 

Note that for = 2 both Eqs. (17), (18) essentially reduce again to the Ising Hamiltonian, Eq. (9), with nearest neighbor interaction only. The latter model is described by the following critical behavior for its order parameter if/, ordering susceptibility and specific heat C ... 

In the literature there have been repeated reports on an apparent mean-field-like critical behavior of such ternary systems. To our knowledge, this has first been noted by Bulavin and Oleinikova in work performed in the former Soviet Union [162], which only more recently became accessible to a greater community [163], The authors measured and analyzed refractive index data along a near-critical isotherm of the system 3-methylpyridine (3-MP) + water -I- NaCl. The shape of the refractive index isotherm is determined by the exponent <5. Bulavin and Oleinikova found the mean-field value <5 = 3 (cf. Table I). Viscosity data for the same system indicate an Ising-like exponent, but a shrinking of the asymptotic range by added NaCl [164],... 

The transition to the continuum fluid may be mimicked by a discretization of the model choosing > 1. To this end, Panagiotopoulos and Kumar [292] performed simulations for several integer ratios 1 < < 5. For — 2 the tricritical point is shifted to very high density and was not exactly located. The absence of a liquid-vapor transition for = 1 and 2 appears to follow from solidification, before a liquid is formed. For > 3, ordinary liquid-vapor critical points were observed which were consistent with Ising-like behavior. Obviously, for finely discretisized lattice models the behavior approaches that of the continuum RPM. Already at = 4 the critical parameters of the lattice and continuum RPM agree closely. From the computational point of view, the exploitation of these discretization effects may open many possibilities for methodological improvements of simulations [292], From the fundamental point of view these discretization effects need to be explored in detail. 

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