Mean-Field Theory of Phase Separation in Thin Films

Mean-Field Theory of Phase Separation in Thin Films 

A quantity of particular interest is the surface excess ( )s induced by the walls, which we define separately for the left wall and right wall in Fig. 5b)-d), to allow for a general asymmetric situation, 

In the limit of large D we have (] (D/2)— ( b, the volume fraction of A in a bulk mixture. In the following, we shall for the most part consider only the situation with symmetric walls, such that ( s = =  

The phenomenological discussion given so far can be made more explicit on the mean-field level, combining a Flory-Huggins free energy density [199-206] with the gradient square -approximation [206-209], supplemented by local surface terms [11,58]. That is, the excess free energy of the mixture (relative to pure B) per unit wall area is written [58] 

Here b is the effective lattice spacing of the underlying Flory-Huggins lattice [199-206], and the coefficient k(( ) of the gradient energy term for a symmetric mixture (effective monomeric units have linear dimensions oA=oB=b, chain lengths are NA=NB=N) is [186,206,207] 

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