Chemical potential equilibration

The establishment of chemical potential equilibrium (with respect to either a setpoint or phase coexistence) is the central component of most Monte Carlo schemes for simulation of the phase behavior and stability of molecular systems. Simulation of the chemical potential (or chemical potential equilibration) in a polymeric system requires more effort than the corresponding calculation for a simple fluid. The reason is that efficient conformational sampling of the polymer is implicitly required for a free-energy calculation and, in fact, the ergodicity problems described in earlier sections are often exacerbated. 

The main advantage of the expanded ensemble approach is that it provides a natural framework for inserting and removing chain molecules from a system. By facilitating such insertion-deletion moves, chemical potential equilibration can be effectively attained in grand canonical and Gibbs... 

Escobedo and de Pablo have proposed some of the most interesting extensions of the method. They have pointed out [49] that the simulation of polymeric systems is often more troubled by the requirements of pressure equilibration than by chemical potential equilibration—that volume changes are more problematic than particle insertions if configurational-bias or expanded-ensemble methods are applied to the latter. Consequently, they turned the GDI method around and conducted constant-volume phase-coexistence simulations in the temperature-chemical potential plane, with the pressure equality satisfied by construction of an appropriate Cla-peyron equation [i.e., they take the pressure as 0 of Eq. (3.3)]. They demonstrated the method [49] for vapor-liquid coexistence of square-well octamers, and have recently shown that the extension permits coexistence for lattice models to be examined in a very simple manner [71]. 

The osmotic pressure becomes equivalent to a mechanical pressure in a two phase fluid model where the solvent is treated as a pure phase. Under these circumstances, a mechanical pressure increases the solvent chemical potential and it flows to a lower pressure (or lower osmotic pressure), hi a polymer solution, the pressures of the polymer and the water must sum to the applied (often atmospheric) pressure. As the polymer concentration is increased, the chemical potential of the water is reduced. In the two phase fluid models, this is equivalent to reducing the pressure of the solvent (indeed it can go negative). When exposed to suspensions where the solvent is at atmospheric pressure (higher chemical potential) the solvent flows from the suspension to the polymer solution until the pressures (chemical potentials) equilibrate. As a result, osmotic consolidation can be used to densify aggregated suspensions while keeping them saturated and with the application of no external pressure. These same ideas can be applied to drying and... 

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

Situations that depart from thermodynamic equilibrium in general do so in two ways the relative concentrations of different species that can interconvert are not equilibrated at a given position in space, and the various chemical potentials are spatially nonuniform. In this section we shall consider the first type of nonequilibrium by itself, and examine how the rates of the various possible reactions depend on the various concentrations and the lattice temperature. 

Under what conditions can experiments yield data relevant to the goal we have just described Two conditions have to be fulfilled the various dissolved hydrogen species have to have had time to get equilibrated with each other before the surface boundary conditions have changed appreciably, and the surface chemical potential /x must be, if not known, at least reproducible in experiments involving different bulk dopings. At the present writing, there have been no experiments that are entirely beyond question in either of these respects, but several experiments, which we shall presently discuss, can plausibly be argued to satisfy both criteria. 

For the reasons we have just been discussing, we shall focus attention on the uptake of hydrogen by samples hydrogenated by exposure to plasma products for times of the order of an hour at 300°C and shall analyze the data on the assumption that the surface chemical potential / for given external and surface conditions is roughly independent of donor or acceptor doping. However, our conclusions will be tentative, since presently available data are limited and both the assumption of local equilibration and that of constant surface p need further checking. 

We have seen that the cell potential is generated at the interfaces between the electrodes and the electrolyte. Therefore, the composition of the electrode at this interface is important and this does not have to be identical with the bulk composition. In fact, large deviations have been observed due to segregation of some of the components of the electrode and especially due to impurities at the surface. If the surface of the electrode is equilibrated with the bulk, both have the same chemical potential of the electroactive component if that is sufficiently mobile in... 

THL.5. R. Defay et I. Prigogine, Les potentiels chimiques lateraux d une phase superficielle non en equilibre (The lateral chemical potentials of a nonequihbrium superficial phase). Bull. Cl. Sci. Acad. Roy. Belg. 32, 176-184 (1946). 

When it comes to the equilibration of water concentration gradients, the relevant transport coefficient is the chemical diffusion coefficient, Dwp. This parameter is related to the self-diffusion coefficient by the thermodynamic factor (see above) if the elementary transport mechanism is assumed to be the same. The hydration isotherm (see Figure 8) directly provides the driving force for chemical water diffusion. Under fuel-cell conditions, i.e., high degrees of hydration, the concentration of water in the membrane may change with only a small variation of the chemical potential of water. In the two-phase region (i.e., water contents of >14 water molecules... 

Instead of the dilute solution approach above, concentrated solution theory can also be used to model liquid-equilibrated membranes. As done by Weber and Newman, the equations for concentrated solution theory are the same for both the one-phase and two-phase cases (eqs 32 and 33) except that chemical potential is replaced by hydraulic pressure and the transport coefficient is related to the permeability through comparison to Darcy s law. Thus, eq 33 becomes... 

Let us now analyze osmotic flow from the viewpoint of the Gibbsian equilibration conditions (Chapter 5). Because volume V cannot be freely exchanged between chambers, the pressure does not equalize P(L) P(R). Similarly, solute nB cannot freely exchange, so /4P p However, solvent nA is free to exchange through the semipermeable membrane, so its chemical potential must equalize between left and right chambers ... 

Since the state of a crystal in equilibrium is uniquely defined, the kind and number of its SE s are fully determined. It is therefore the aim of crystal thermodynamics, and particularly of point defect thermodynamics, to calculate the kind and number of all SE s as a function of the chosen independent thermodynamic variables. Several questions arise. Since SE s are not equivalent to the chemical components of a crystalline system, is it expedient to introduce virtual chemical potentials, and how are they related to the component potentials If immobile SE s exist (e.g., the oxygen ions in dense packed oxides), can their virtual chemical potentials be defined only on the basis of local equilibration of the other mobile SE s Since mobile SE s can move in a crystal, what are the internal forces that act upon them to make them drift if thermodynamic potential differences are applied externally Can one use the gradients of the virtual chemical potentials of the SE s for this purpose ... 

If several ( ) charged species i equilibrate across the phase boundary, the set of Eqns. (4.116) has to be solved simultaneously for i = 1,2,..This does not lead to an over-determination of Atpb but ensures that the chemical potentials of the electroneutral combinations of the ions (= neutral components of the system) are constant across the interface. The electric structure (space charge) of interfaces will be discussed later. 

At the liquid solution/membrane feed interface, the chemical potential of the feed liquid is equilibrated with the chemical potential in the membrane at the same pressure. Equation (2.7) then gives... 

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