Free Energy of an Inhomogeneous System

The free-energy density should not depend on the choice of coordinate system [i.e., /( , VO should not depend on the gradient s direction] and therefore L = 0 and K will be a symmetric tensor.5 Furthermore, if the homogeneous material is isotropic 

4There are expansions that contain higher-order spatial derivatives, but the resulting free energy is the same as that derived here [1, 4], 

5If the homogeneous material has an inversion center (center of symmetry), L is automatically zero. 

The free-energy density is thus approximated as the first two terms in a series expansion in order-parameter gradients the first term is related to homogeneous molar free energy and the second is proportional to the gradient squared. 

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Once the free energy of an inhomogeneous system is given, one can calculate by standard methods the properties of the interface—for example, the interfacial tension or the density profile perpendicular the interface [285]. Weiss and Schroer compared the various approximations within square-gradient theory discussed earlier in Section IV.F for studying the interfacial properties for pure DH and FL theory [241, 242], In theories based on local density approximations the interfacial thickness and the interfacial tension were found to differ by up to a factor of four in the various approximations. This contrasts with nonionic fluids, where the density profiles and interfacial... 

By a derivation of the free energy of an inhomogeneous system somewhat analogous to that of Cahn and Hilliard, Debye (16) has shown ... 

Finally, recalling eqns [68]-]70] we obtain the free energy of an inhomogeneous system of long ideal chains (with c(r) 0 for... 

Non-Random Systems. As pointed out by Cahn and Hilliard(10,11), phase separation in the thermodynamically unstable region may lead to a non-random morphology via spinodal decomposition. This model is especially convenient for discussing the development of phase separating systems. In the linearized Cahn-Hilliard approach, the free energy of an inhomogeneous binary mixture is taken as ... 

The most elegant and clear exposition of the ideas of Ornstein and Zernike is that of Klein and Tisza. The ideas of Ornstein and Zernike will be understood to mean generally the assumption of a linear direct (or short-range) coupling between fluctuations of density in different volume elements, in an expression for the free energy in an inhomogeneous system. 

A detailed derivation of the square gradient term relies on the use of the random phase approximation, which is discussed in the appendix to this chapter. By this approach we find that we can write the free energy per segment of an inhomogeneous system as... 

This fomi is called a Ginzburg-Landau expansion. The first temi f(m) corresponds to the free energy of a homogeneous (bulk-like) system and detemiines the phase behaviour. For t> 0 the fiinction/exliibits two minima at = 37. This value corresponds to the composition difference of the two coexisting phases. The second contribution specifies the cost of an inhomogeneous order parameter profile. / sets the typical length scale. 

Here f(c) is the free energy density of a hypothetical completely homogeneous system with volume fraction c. It is understood that c=c(r). For small fluctuations, one can expand f(c) about c, the bulk composition of the mixture. Keeping only terms to second order and invoking conservation of mass, one may obtain an expression for AF, the difference between the free energy of homogeneous and inhomogeneous mixtures ... 

In such cases a local-equilibrium structure may be obtained theoretically by minimization of the free energy of the system under the constraint of a fixed alloy composition in the surface region [8,17-24]. Although this approach is very similar to the one used for bulk systems, it should be modified due to the specific features introduced by the surface. First of all, since the structure of the underlying bulk system is fixed, it acts as the source of an external field for the surface alloy, creating, for instance, epitaxial strain. Secondly, since the surface is an open system, it allows the formation of a great variety of different structures, which may not have any connection at all to the crystal structure of the substrate. Finally, the surface is a spatially inhomogeneous system, and thus different alloy components have their own... 

To proceed, then, let us attempt to calculate the free energy associated with a surface with an arbitrary density profile the principles of equilibrium will then assert that the equilibrium surface profile will be the one with the lowest value of the free energy. Let us assume, then, that near the surface the density p(z) is a function of the depth z. For a xmiform system we can write a free energy density as a function of density and temperature, ipip, T) if we were to assume that the same function describes the free energy density in an inhomogeneous system then we could write the surface tension as follows ... 

Here FsoivCO is the solvation free energy (or equivalently the excess chemical potential) of the solute pair with fixed distance r. In some work, above equation refers to potential distribution theorem (PDT) wherein the one-body direct correlations (namely, the excess chemical potentials) are present. As demonstrated above, the gcznd potential for an inhomogeneous system, or equivalently the solvation free energy for a small individual solute molecule can be calculated accurately by using DFT. While for the calculation of PMF between a particle and a macroscopic obstacle such as a wall, a MC-DFT or MD-DFT algorithm can be used. 

We shall see that the disjoining potential plays a crucial role in the contact line motion. Before we go on to dynamics, it should be noted that even static problems, such as computing an equilibrium shape of a liquid droplet, with due account for intermolecular interactions near the contact line is very much non-trivial. The overall shape minimizing the free energy of the system is influenced by forces operating on widely separated scales - from molecular distances to the drop size, and, due to intermolecular interactions, the conditions near the three-phase line are very sensitive to surface inhomogeneities, both geometric and chemical. No wonder that the dynamics of the contact line is not yet well understood after decades of effort. 

Let us consider a multicomponent two-phase system with a plane interface of area A in complete equilibrium, and let us focus on the inhomogeneous interfacial region. Our approach is a point-thermodynamic approach [92-96], and our key assumption is that in an inhomogeneous system, it is possible to define, at least consistently, local values of the thermodynamic fields of pressure P, temperature T, chemical potential p, number density p, and Helmholtz free-energy density xg. At planar fluid-fluid interfaces, which are the interfaces of our interest here, the aforementioned fields and densities are functions only of the height z across the interface. 

When the two phases separate the distribution of the solvent molecules is inhomogeneous at the interface this gives rise to an additional contribution to the free energy, which Henderson and Schmickler treated in the square gradient approximation [36]. Using simple trial functions, they calculated the density profiles at the interface for a number of system parameters. The results show the same qualitative behavior as those obtained by Monte Carlo simulations for the lattice gas the lower the interfacial tension, the wider is the interfacial region in which the two solvents mix (see Table 3). 

Polymers don t behave like the atoms or compounds that have been described in the previous sections. We saw in Chapter 1 that their crystalline structure is different from that of metals and ceramics, and we know that they can, in many cases, form amorphous structures just as easily as they crystallize. In addition, unlike metals and ceramics, whose thermodynamics can be adequately described in most cases with theories of mixing and compound formation, the thermodynamics of polymers involves solution thermodynamics—that is, the behavior of the polymer molecules in a liquid solvent. These factors contribute to a thermodynamic approach to describing polymer systems that is necessarily different from that for simple mixtures of metals and compounds. Rest assured that free energy will play an important role in these discussions, just as it has in previous sections, but we are now dealing with highly inhomogeneous systems that will require some new parameters. 

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