Helmholtz free energy blend

Fig. 6.31 Results from SCFI calculations for diblock/homopolymer blends (Matsen 1995b). (a) The dimensionless Helmholtz free energy Fu() as a function of homopolymer volume fraction at y X = 12, / = 0.45 and /3 = The dashed line shows the double tangent construction used to locate the binodal points denoted with dots. The dotted line is the free energy of non-interacting bilayers, (b) Phase diagram obtained by repeating this construction over a range of %N. The dots are the binodal points obtained in (a), and the diamond indicates a critical point below which two-phase coexistence does not occur. The disordered homopolymer phase is labelled dis, and the lamellar phase lam.

The equilibrium thermodynamics controls the PVT behavior of any system, its thermal expansion coefficient, compressibility, bulk modulus, hardness, etc. The thermodynamic pressure can be defined as a partial derivative of the Helmholtz free energy (see Eq 2.5). For multicomponent blends the latter function comprises two interac-... 

The S-S theory describes the structure of a liquid by a lattice model with cells of the same size and a coordination number of z = 12. The disordered structure of the liquid is modeled by allowing an occupied lattice-site fraction y=y(V,73 of less then 1. The configurational or Helmholtz free energy, F, is expressed in terms of the volume V, temperature 7] and occupied lattice-site fraction y = y(K7), F=F(V,T,y). The value of y is obtained through the pressure equation P = —(9F/9V)r and the minimization condition (dFldy)v,T = 0. The hole fraction is given by the fraction of unoccupied lattice sites (holes or vacancies), which is denoted by h, h(P,T) = —y(P,T). This theory provides an excellent tool for analyzing the volumetric behavior of linear macromolecules but was also applied successfully to nonlinear polymers, copolymers, and blends. Several universal relationships where found which allow an approximate estimation of the fraction of the hole (or excess) free volume h and the total or van der Waals free volume/ [Simha and Carri, 1994 Dlubek and Pionteck, 2008d]. For more details, see Chapters 4, 6, and 14. 

Sanchez and Lacombe supposed that in a binary polymer blend, free volume occupied Nq lattice sites, and the bulk polymer density p = NKN + Nq), where N = ILNiri and r, was the chain length of /th fraction, then they developed the lattice fluid theory to calculate Helmholtz free energy (Sanchez and Lacombe 1974 Sanchez 1978), as given by... 

Equilibrium thermodynamics controls the PVT behavior of any system and its thermal expansion coefficient, compressibility, bulk modulus, hardness, etc. The thermodynamic pressure, which can be defined as a partial derivative of the Helmholtz free energy (see Eq. 2.5), for multicomponent systems, comprises of two interaction parameters, e.g., (e )(v ) k = 2,4. These values can as easily be determined from dilatometric measurements as from the phase diagram (Jain et al. 1982). With the advance of other methods, dUatometry has been largely neglected It is still being used to characterize the compressibility of neat resins, but rarely nowadays to study the behavior of polymeric blends (Plochocki 1982, 1983, 1986 Zoller 1989 Steller and Zuchowska 1990 Zoller and Walsh 1995). 

The starting point is the reduced Helmholtz free energy of the blend in the standard charging parameters form ... 

Thermodynamics of Polymer Blends and Solutions. Fiory-Huggins Theory 247 With the Helmholtz free energy... 

The specific Helmholtz free energy of mixing / for a binary incompressible kx. x/Cy i-y copolymer blend emerges from the BLCT in the apparently familiar form [95]... 

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