Mean-field approximation binary mixture

As we are again interested in determining the phase behavior of the binary mixture in confinement and near solid interfaces, we are essentially confronted with the same problem already discussed in Section 4.5, namely finding minima of the grand potential for a given set of thermodynamic (T, /x) and model parameters [see Eqs. (4.125)]. To obtain expressions for u> that are tractable, at least numerically, we resort again to a mean-field approximation. That... 

Here the gradient square term describes the extra free energy cost due to concentration inhomogeneities. Boltzmann s constant is denoted as and the parameter r then has dimensions of length (in microscopic models, e.g., lattice models of binary mixtures treated in mean-field approximation, r has the meaning of the range of pairwise interactions among the particles). 

One of the most powerful methods to assess such phenomena theoretically is the self-consistent field (SCF) theory. Originally introduced by Edwards [8] and later Helfand et al. [9], it has evolved into a versatile tool to describe the structure and thermodynamics of spatially inhomogeneous, dense polymer mixtures [ 10-13]. The SCF theory models a dense multi-component polymer mixture by an incompressible system of Gaussian chains with short-ranged binary interactions and solves the statistical mechanics within the mean-field approximation. 

As has been shown in Section 3.4, COSMO-RS provides access to the thermodynamic parameters of solvents as well as of solute molecules. It provides the knowledge of the chemical potential of any compound in any fluid. Thus COSMO-RS enables the handling of almost any partition problem of compounds between liquid and gaseous phases. In this context it is important that COSMO-RS handles multicomponent mixtures without additional complications, because it does not make any mean-field approximation. Therefore multicomponent mixtures are described almost as reliably as binary mixtures or pure liquids. 

A schematic phase diagram of a symmetrical binary mixture is shown in Fig. 1 in a temperature versus composition representation. A symmetric polymer blend is characterized by two polymer components of the same molar volume and, therefore, with a 50% critical composition. Within mean field approximation the binodal and spinodal phase boundaries of a binary (A/B) incompressible polymer mixture are described by the Gibbs free energy of mixing AG according to... 

At the outset we emphasize, however, that F c) is not a well-defined quantity thermodynamic potentials are well defined for thermal equilibrium states only states with (d F (c)ldc )T < 0 violate the basis laws of statistical thermodynamics. For Ccoex < c < CcoeJ- the only well-defined free energy is the free energy which corresponds to the lever rule, i.e., F(c) = F(cil, T))X + F(c ,(7))(l - X), i.e., a linear function of the concentration. The double-well shape of F(c) obtained from mean-field theories, such as the Bragg-Williams approximation of binary mixtures, is an artifact of an uncontrolled approximation. So the singular behavior resulting at the spinodal should not be taken seriously. 

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