Free-energy concentration expansion

Often the most important effective cluster interaction of the concentration fluctuation expansion of the configurational free energy is the EPI which is defined as... 

The mass action variable X can be rendered into an experimentally more meaningful form. To do this, let X be a reference value for given values of the concentration and temperature T. Then, by a linear Taylor expansion of the dimensionless free energy f3g around the reference temperature T, we have... 

The degree of ionization/ and the scaled expansion parameter l as a function of the various parameters that enter the model are obtained by minimizing the total free energy. Analysis revels that the degree of ionization decreases if the chain flexibility or the salt concentration increases [48]. This decease is also observed if the chain length or the dielectric constant is decreased [48],... 

If the monomer is a good solvent for the polymer, the latex particles might be assumed to expand indefinitely beeause of inhibition of monomer. An equilibrium monomer concentration and swelling equilibrium is reached, however, because the free energy decrease due to mixing of polymer and monomer is eventually balanced by the increase in surface free energy which accompanies expansion of the particle volume. 

The FCEM expressions for binary alloys were obtained using the Ising model Hamiltonian and an expansion of the partition function and free energy in terms of solute concentration [1,3,19]. The free energy of a binary alloy (A -solute, S - solvent) reads. 

When concentrations of substitutional atoms are high, the presumption generally has been that transition temperatures will vary in a simple linear fashion with dopant concentration. This supposition can be rationalized by a Landau-Ginzburg excess Gibbs free-energy expansion (Salje et al. 1991). A simple second order phase transition for phase A with the regular free energy expression... 

A second caveat concerns the dependence of the numerical values of the free energies and entropies of complexation on the concentration scale used. AH" must be calculated by applying the van t Hoff equation to or values, the complexation constants on the mole fraction or molal concentration scales, respectively. If one uses Kc (molar concentration), enthalpies mnst be corrected for the thermal expansion of the solvent. 

In Pitzer s model the Gibbs excess free energy of a mixed electrolyte solution and the derived properties, osmotic and mean activity coefficients, are represented by a virial expansion of terms in concentration. A number of summaries of the model are available (i,4, ). The equations for the osmotic coefficient (( )), and activity coefficients (y) of cation (M), anion (X) and neutral species (N) are given below ... 

Examination of the terms to O(k ) in the SL expansion for the free energy show that the convergence is extremely slow for a RPM 2-2 electrolyte in aqueous solution at room temperature. Nevertheless, the series can be summed using a Pade approximant similar to that for dipolar fluids which gives results that are comparable in accuracy to the MS approximation as shown in figure A2.3.19(a). However, unlike the DHLL + i 2 approximation, neither of these approximations produces the negative deviations in the osmotic and activity coefficients from the DHLL observed for higher valence electrolytes at low concentrations. This can be traced to the absence of the complete renormalized second virial coefficient in these theories it is present... 

Here the bare surface free energy / = — Hi quadratic function of the local surface concentration < >i, Hi g are phenomenological coefficients. This ansatz can be justified as a Taylor expansion in the case where 4>i 1 or where 1 — < )i 1, respectively [125], or alternatively by... 

Finally, the Landau-Ginzburg approach can be applied to microemulsions. In this approach the thermodynamic behavior of a system in the neighborhood of critical points is studied. For microemulsions this approach uses the expansion of the free energy in (spatially varying) order parameter fields that are identified as the local concentration differences of oil, water, and surfactant. We will not elaborate this approach but refer to a recent review by Gompper [25]. 

This expression indicates that eqn (2.87) is the first term in the virial expansion of the free energy with respect to the local concentration c(r). Therefore the excluded volume parameter v can be regarded as the virial coefficient between the segments. 

This approximation is justified if the concentration of the polymers is low enough. Systematic improvement of this approximation can be done by the virial expansion of the free energy. " However, for polymers of long aspect ratio L/b 1), the higher order terms are not needed because the transition is shown to occur at very low concentration. 

The concentration fluctuations are related to fluctuations in free energy F, where for small fluctuations the Taylor series expansion is... 

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