Mean-field theory chemical potential

Fig. 27. Phase diagram of an adsorbed film in- the simple cubic lattice from mean-fleld calculations (full curves - flrst-order transitions, broken curves -second-order transitions) and from a Monte Carlo calculation (dash-dotted curve - only the transition of the first layer is shown). Phases shown are the lattice gas (G), the ordered (2x1) phase in the first layer, lattice fluid in the first layer F(l) and in the bulk F(a>). For the sake of clarity, layering transitions in layers higher than the second layer (which nearly coincide with the layering of the second layer and merge at 7 (2), are not shown. The chemical potential at gas-liquid coexistence is denoted as ttg, and 7 / is the mean-field bulk critical temperature. While the layering transition of the second layer ends in a critical point Tj(2), mean-field theory predicts two tricritical points 7 (1), 7 (1) in the first layer. Parameters of this calculation are R = —0.75, e = 2.5p, 112 = Mi/ = d/2, D = 20, and L varied from 6 to 24. (From Wagner and Binder .)...

For most insertion compounds, the interaction of intercalated ions with each other in the host lattice is not negligible. In order to simply consider the contribution of ionic interaction in Equation (5.8), it is often assumed that each ion experiences a mean interaction or energy field from its neighboring ions, based on a mean-field theory [10]. According to this approximation, the contribution to the chemical potential is proportional to the fraction of sites occupied by the ions 5, and hence the interaction term is introduced into Equation (5.8) as... 

The fraction activated material is given by r] = rN2- In the limit r 0, we have = 0 forX < 1 and r] = 1 - X if X > 1. Hence, CAEPs and TAEPs behave in quite similar ways (see Figure 4 and Figure 8). Indeed, Eqs. (14) and (15) also apply to CAEPs, signifying that CAEPs and TAEPs belong to a different universality class than EPs do. In fact, the formal limits r 0 and Xa 0 are equivalent in that the polymerization transition becomes a true (continuous) phase transition. That this must be so, can be inferred from the behavior of the chemical potential of the monomers nearXp = 1. For both systems the (dimensionless) chemical potential ln(l—A2 ) —N2 exhibits a discontinuity at the critical point Xp = 1. In addition, the heat capacities calculated within the given theoretical framework are typical of mean-field theories near a critical point, that is, their values jump at the critical point [29]. 

Replacement Chemical Potential As x exceeds 1/2 and increases further, A2 becomes negative and its absolute value increases. The unfavorable polymer-solvent interaction can be sufficiently strong to cause the solution to separate into two phases. We will examine the phase diagram of the solution in the mean-field theory for a system of a fixed volume. 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

Equation 6.36 for the adiabatic potential is exact within the framework of the mean field description. However, the structure of the electric part P1 is too complex to disclose its analytic properties. Here we examine the adiabatic potential numerically following the Carlson theory of elliptic integrals [15-21], To proceed with numerical computation, it is necessary to enter a set of parameters designed to describe an experimental situation. It will not surprise the reader who has made it this far that we use values of the chemically fixed parameters specified by the n-butylammonium vermiculite gels [22], namely m+ = 74 mp and m = 36 mp. The average density n0 of the small ions is given by... 

In the second paper of the series [150] the authors extend their treatment with a mean field lattice theory. They found that the adsorbed amount in the pores is smaller and the step in the isotherm shifts to lower chemical potential than in a flat surface in the same conditions. They also established that the influence of the curvature on the phase transtition increases with the length of the headgroup. The shift of the phase transition increases with the adsorption energy. 

---