Energy of mixing

We may also define a free energy of mixing [240]. The alternative (and equally acceptable) definition of G given in Eq. 111-87 is... 

Fig. IV-22. Excess free energy of mixing of condensed films of octadecanol-docosyl sulfate at 25°C, at various film pressures. Top curve t = 5 dyn/cm bottom curve ir = 50 dyn/cm intermediate curves at 5-dyn/cm intervals. The curves are uncorrected for the mixing term at low film pressure. (From Ref. 246.)...

Physically, is a measure of the difference in the energies of vaporization of the two species (roughly a difference in nomial boiling point), and L is a measure of the energy of mixing. With these definitions equation (A2.5.8) can be rewritten as... 

The molar Helmholtz free energy of mixing (appropriate at constant volume) for such a synnnetrical system of molecules of equal size, usually called a simple mixture , is written as a fiinction of the mole fraction v of the component B... 

Figure A2.5.15. The molar Gibbs free energy of mixing versus mole fraetionxfor a simple mixture at several temperatures. Beeause of the synuuetry of equation (A2.5.15) the tangent lines indieating two-phase equilibrium are horizontal. The dashed and dotted eiirves have the same signifieanee as in previous figures.

Figure C2.1.10. (a) Gibbs energy of mixing as a function of the volume fraction of polymer A for a symmetric binary polymer mixture = Ag = N. The curves are obtained from equation (C2.1.9 ). (b) Phase diagram of a symmetric polymer mixture = Ag = A. The full curve is the binodal and delimits the homogeneous region from that of the two-phase stmcture. The broken curve is the spinodal.

The entropy of a solution is itself a composite quantity comprising (i) a part depending only on tire amount of solvent and solute species, and independent from what tliey are, and (ii) a part characteristic of tire actual species (A, B,. ..) involved (equal to zero for ideal solutions). These two parts have been denoted respectively cratic and unitary by Gurney [55]. At extreme dilution, (ii) becomes more or less negligible, and only tire cratic tenn remains, whose contribution to tire free energy of mixing is... 

Solubility Parameter. CompatibiHty between hydrocarbon resins and other components in an appHcation can be estimated by the Hildebrand solubiHty parameter (2). In order for materials to be mutually soluble, the free energy of mixing must be negative (3). The solubiHty of a hydrocarbon resin with other polymers or components in a system can be approximated by the similarities in the solubiHty parameters of the resin and the other materials. Tme solubiHty parameters are only available for simple compounds and solvents. However, parameters for more complex materials can be approximated by relative solubiHty comparisons with substances of known solubiHty parameter. 

Basic Thermodynamics. Equilibrium-phase behavior of mixtures is governed by the free energy of mixing and how this quantity, consisting of enthalpic... 

One proposed approach (75) to modeling the phase behavior for hydrogen bonding pairs uses the following expression for the free energy of mixing (eq. 7). 

Practical Solubility Concepts. Solution theory can provide a convenient, effective framework for solvent selection and blend formulation (3). When a solute dissolves in a solvent, a change in free energy occurs as a result of solvent—solute interactions. The change in free energy of mixing must be negative for dissolution to occur. In equation 1,... 

A.ctivity Coefficients. Activity coefficients in Hquid mixtures are directiy related to the molar excess Gibbs energy of mixing, AG, which is defined as the difference in the molar Gibbs energy of mixing between the real and ideal mixtures. It is typically an assumed function. Various functional forms of AG give rise to many of the different activity coefficient models found in the Hterature (1—3,18). Typically, the Hquid-phase activity coefficient is a function of temperature and composition expHcit pressure dependence is rarely included. 

Fig. 25. Relationship between the measured interfacial strength and the (negative) Gibbs free energy of mixing, (-AG )o5, for glass beads treated with various silane coupling agents embedded in a PVB matrix. Error bars correspond to 95% mean confidence intervals. Redrawn from ref. [165].

Most polymer pairs are thermodynamically incompatible, in the sense that their free energy of mixing is positive. This does not mean that there is absolutely no interdiffusion at all at the interface between them adjacent to the interface limited interdiffusion occurs, which can be seen as an increasing of the low surface entropy implied by a smooth surface [30-33]. This nanoscale roughening of an interface can increase the adhesion between the polymers. 

Next we differentiate Eq. (8-44) with respect to 2, obtaining the partial molar energy of mixing of 2 in 1 ... 

The well-known Flory treatment [50-52] of the en-thropic contribution to the Gibbs energy of mixing of polymers with solvents is still the simplest and most reliable theory developed. It is quite apparent, however, that the Flory-Huggins theory was established on the basis of the experimental behavior of only a few mixtures investigated over a very narrow range of temperature. Strict applications of the Flory-Huggins approach... 

In a thermodynamic sense, the compatibility of polymers is similar to the dissolving solute in a solvent. The thermodynamic standard of solubility is the free energy of mixing Ga. If AGm < 0, then two components are soluble to each other. According to the definition ... 

The free energy of mixing Umix for the fee Cu-Zn alloys is shown as a function of concentration in Fig. 1. It is obtained from the usual formula... 

Figure 1. The free energies of mixing of fee disordered alloys. The filled eireles eonneeted with a solid line are the energies ealeulated with the LSMS. The erosses eonneeted with a dotted line are the energies calculated with the CPA-LSMS without the Conlomb energy, while the open circles connected with dotted lines include the Conlomb contribution. The plusses connect with a dashed-dotted line are the energies calculated with the SCF-KKR-CPA without the Coulomb energy, while the squares connected with dashed-dotted lines include the Coulomb contribution.

Considering an (incompressible) polymer mixture with volume fraction (j)A = (j) of A-monomers and volume fraction (j)B = 1 - (j) of B monomers, the mean-field expression for the excess-free energy of mixing is given by the well-known Flory-Huggins expression " ... 

The contribution that (46) makes to the free energy of mixing is — kT In Wc and it will be noticed that, if the right-hand side of (47) is multiplied by kT, it becomes identical with (45), which is the total change in the free energy, when an ideal solution is formed from its components. 

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