Comparison of LB Fluid Mixture Model to Polymer Blends

It is apparent from the examination of the LB mixture model above that the critical properties of this fluid are described by mean-field theory and that the critical exponents predicted by this model are inconsistent with those measured for many real mixtures. This limits, of course, the comparison of LB model calculations to certain qualitative trends in the equilibrium properties of near critical fluid mixtures. (The theory should become more reliable, however, away from the critical point.) Such inconsistencies can also be expected for certain dynamic properties near the critical point. For example, the shear viscosity of a near critical Ising-type fluid mixture diverges near the critical point while no divergence occurs in a model mean-field mixture [88]. Mode-coupling effects due to compositional flucuations have an even larger effect on the collective diffusion coefficient [30,31]. Although mean-field models of fluid mixtures are idealized there is a class of real liquids whose behavior approaches this ideal type critical behavior. The phase separation of polymer blends in the theoretical limit of infinite molecular weight of the homopolymer components has been argued [45,46] to be described by mean-field theory, so that the phase separation of high molecular weight blends should be reasonably approximated by this idealized model. Monte Carlo calculations support these theoretical arguments, although the chain length must generally be rather high for this approximation to apply [89,90]. There have also been recent reports of apparent mean-field critical behavior in fluid mixtures with salts [90] and in ionic fluid mixtures [91].

It is common practice in polymer science to fit the critical properties of polymer blends to the Flory-Huggins (FH) mean-field lattice model of phase separation where all inaccuracies of the modeling (monomer structure, chain architecture, compressibility, ....) are absorbed into the phenomenological "$\chi-$ parameter" in Eq. (16) [92]. A virtue of the FH lattice model is that it often allows the prediction of qualitative trends in the scattering properties of polymer blends. We can retain this advantage and avoid the conceptual pitfalls of interpreting $\chi$ too literally as a "molecular" parameter by establishing a formal correspondence between the parameters of this model and the LB fluid mixture model.

In the FH model of polymer blend phase separation, the reduced temperature variable is [46,92]

$$\tau_{FH}=\vert\chi-\chi_c\vert/\chi =\vert T_c-T\vert/T_c$$ (29)

where the critical interaction $\chi_c$_c is defined by the condition,

$$\chi_c = (2N_A\phi_c)^{-1}+ [2N_B(1-\phi_c)]^{-1}$$ (30)

and N_A and N_B are homopolymer polymerization indices. Symmetric blends are defined by the idealized condition N_A = N_B = N so that $\phi_c = 1/2$_c = ½ and
$\chi_c =2/N$_c = 2/N. The critical composition $\phi_c$_c of the blend no longer equals 1/2 when the blends are not symmetric ( (N_A $N_A \ge N_B$ N_B)[21,46,74],

$$\phi_c= \frac{N_B^{1/2}}{N_A^{1/2}+N_B^{1/2}}$$ (31)

As mentioned above, the incorporation of asymmetry into the LB model requires adjusting the mass asymmetry or volume asymmetry to give a variation in the critical composition. We can then mimic the asymmetric phase of phase boundary of polymer blends by varying the mass asymmetry $\delta_M$_M or $\lambda$ and formally replacing $\tau _G$_G by $\tau_{FH}$_FH .

The correlation length $\xi$ of the FH model in conjunction with the random phase approximation [46,74] yields a scaling relation for $\xi$ in the two phase region,

$$\xi^- = \xi^-_0{\tau_{FH}}^{-1/2} , \xi^-_0 = R_g/\sqrt6,$$ (32)

where R_g is the chain radius of gyration. We see from a comparison of Eqs. (32) to the LB expression Eq. (23) that the lattice spacing in a coarse-grained model of polymer blends must be large since the lattice spacing is on the order of R_g. This implies that the lattice spacing must be taken to depend on chain molecular weight in comparison with measurements. Moreover, the predictions of the LB model must be considered with caution when physically relevant scales in physical problems become smaller than this coarse-graining scale (lattice spacing) of the LB lattice model. This limitation is natural since the LB model is a mesoscopic description of a fluid rather than a microscopic model.

There are a number of points to be drawn from our discussion of polymer blend critical properties in comparison with the LB model of fluid phase separation:

(1) Polymer blends are reasonable candidates for comparison with the LB mixture model.

(2) The mean-field model gives rise to universal scaling relationships that should allow fixing the parameters of the LB model according to the blend molecular characteristics. This gives some insight into the qualitative variation of the LB parameters with the variation of molecular structure.

(3) Comparison of the LB model with parameters fixed by the FH model with Monte Carlo calculations of the lattice model of polymer blends should provide some insight into the mean-field approximation in the case of properties not tractable using analytic mean-field theory. For example, we can compare LB calculations of the interfacial tension to Monte Carlo calculations for polymer blends that avoid the mean-field approximation.

(4) Fixing the LB model parameters through "matching" to the FH model then allows a comparison with dynamical properties (transport coefficients) and processes (phase separation, wetting, dewetting) of blends calculated using the mean-field approximation.

(5) The expression of the results of measurements in terms of general and universal scaling relations (when they exist) offers advantages to representations involving the phenomenological $\chi$ interaction parameter. Expressions between large scale observable properties deduced from mean-field theory often have greater applicability than expressions between observables and temperature-like variables such as $\chi$ and $\tau$.