Dilution, free energy integral

The consequences for suspended particles can be understood from either a mechanical or a thermodynamic standpoint. A particle immersed in a polymer solution experiences an osmotic pressure acting normal to its surface. For an isolated particle, the integral of the pressure over the entire surface nets zero force. But when the depletion layers of two particles overlap, polymer will be excluded from a portion of the gap (Fig. 30). Consequently, the pressure due to the polymer solution becomes unbalanced, resulting in an attraction. The same conclusion follows from consideration of the Helmholtz free-energy. Overlap of the depletion layers reduces the total volume depleted of polymer, thereby diluting the bulk solution and decreasing the free energy. 

As we have aheady said, the grand canonical Monte Carlo provides a mean to determining the chemical potential, and hence, die free energy of the system. In other MC and MD calculations a numerical value for the free energy can always be obtained by means of an integration of thermodynamic relations along a path which links the state of interest to one for which the free energy is already known, for example, the dilute gas or the low-temperature solid. Such a procedure requires considerable computational effort, and it has alow numerical stability. Several methods have been proposed and tested. 

Although this particular path of the integration works well only for finding the free energy of dilute and dense liquids, one must seek a reference other than the ideal gas if the system of interest is crystalline solid phase. Let us consider a system of a fixed number of indistinguishable particles. First, one can define an interaction potential function in such a way that a coupling constant is introduced to accommodate the linearity between the reference and target systems as follows ... 

Eichinger s treatment conforms to the limiting behavior of A required by equation (113) at r= 0 and its analog at 0l. To represent behavior at intermediate temperatures by a continuous function, the temperature dependence of the free energy —RTV A 2 c (see equation 111) is explicitly characterized by a partial molar heat capacity of dilution RTV c d [TA 2 )ldT. This is expanded about 0 in a series in non-negative powers of (T—0u)/ u. Two integrations with respect to T between limits 0 and T yield a complicated series expression for This procedure... 

---