Free energy mixing

To see this, consider the mixing free energy expression for a polymer blend with symmetrical chain lengths and with only one crystallizable component (i.e., p = 0 for one component and Ep 0 for the other). In that case the (mean-field) partition function for the liquid mixture is... 

The F may be expressed as a sum of the mixing free energy, Fmix, the free energy due to ions, Fton, and the elastic free energy Fet [1, 6],... 

The mixing free energy of the chains and the solvent molecules for a given chain conformational profile IVC is given by F= kT nQ. The equilibrium conformational profile can be obtained by minimizing the free energy F subject to the following constraints ... 

Equations (14), (15), and (17) will be used to calculate the quantities < >,-, cb lt, and z . The obtained distributions of the segment density and bond densities allow to calculate any structural property of the two interacting polymer brushes as well as the mixing free energy. The latter quantity is used to obtain the interaction force profile. 

At equilibrium, NC=NWC/ R. Introducing the later expression in Eqs. (19) and (20), one finally obtains for the segment-solvent mixing free energy A=—kT ln(Q, the expression,... 

Taking into account the symmetry of the two brushes, the mixing free energy per surface site (f=A/2L) becomes... 

A. Good Solvents. When two surfaces with grafted polymer brushes approach each other, the overlap of the neutral brushes generates an interaction between surfaces. In good solvents, the Flory—Huggins mixing free energy density (eq 3c) increases with the monomer concentration therefore, one expects that the overlap of the brushes would lead always to repulsion. In the present... 

Figure 7. Local volume fractions of the monomers for two configurations of chains which end up at Z and z-i, respectively (a) simple addition of the volume fractions of the brushes (b) configuration that minimizes the Flory— Huggins mixing free energy for the conditions mentioned in the text.

The polymer contribution contains the Rory—Huggins mixing free energy of the segments with the solvent molecules, the van der Waals interaction —UkT between segments and the two plates, and the entropy loss caused by the connectivity of the segments8,23... 

To determine the steric interaction energy for two parallel plates at a separation, h, the mixing free energy per unit area A, G /A, for the approach of two sterically stabilized particles fi om infinite separation to a separation,... 

The mixing free energy as a fimetion of the separation, AG (h), is determined fiom the volume differential, aAG /SV. The Flory-Hug-gins theory for AG is given by [38,39]... 

The quantities pj are the densities of the pure phases at the specified pressure p, that is, with attractive forces operating to achieve experimental densities the partial molar volumes of these pure phases are i)i = 1/pi. Similarly, the densities of the mixture correspond to the same pressure p. Rememhering Eq. (4.39), a way for this mixing free energy, Eq. (4.41), to achieve the form of the initial two terms of Eq. (4.38) is that the excess quantities within the brackets of Eq. (4.41) vanish ... 

This is consistent with Eiq. (4.40) provided the 0)2 introduced first were indeed independent of thermodynamic state. To compose the mixing free energy, note that... 

We now speciahze these general formulae to the common case that this mixing free energy is expressed on a per monomer basis. As a notational convenience we consider specifically the case of a polymer species mixed into a small molecule solvent, and use M = V2/V1 as the empirical polymerization index parameter. Then... 

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