Chemical potential lattice model

The smoothing terms have a thermodynamic basis, because they are related to surface gradients in chemical potential, and they are based on linear rate equations. The magnitude of the smoothing terms vary with different powers of a characteristic length, so that at large scales, the EW term should predominate, while at small scales, diffusion becomes important. The literature also contains non-linear models, with terms that may represent the lattice potential or account for step growth or diffusion bias, for example. 

Figure 5. The quark number susceptibility for Nf = 2, calculated from the quasiparticle model with the same parameters as in Figs. 3 and 4, for several chemical potentials compared to the lattice data [14] at /u = 0.

The thermodynamic functions of fc-mers adsorbed in a simple model of quasi-one-dimensional nanotubes s adsorption potential are exactly evaluated. The adsorption sites are assumed to lie in a regular one-dimensional space, and calculations are carried out in the lattice-gas approximation. The coverage and temperature dependance of the free energy, chemical potential and entropy are given. The collective relaxation of density fluctuations is addressed the dependence of chemical diffusion coefficient on coverage and adsorbate size is calculated rigorously and related to features of the configurational entropy. 

The necessity of introducing a combinatorial contribution to the chemical potential is a result of the neglect of size effects in the thermodynamics of pairwise interacting surface models. It also appears in lattice models that do not allow for a realistic representation of molecular sizes and are often simplified to models of equally sized lattice objects. The task of the combinatorial contribution is to represent the chemical potential of virtually homogeneous interacting objects of different size in 1 mol of a liquid mixture of a given composition with respect to the size and shape of the molecules. 

Determination of pure component parameters. In order to use the EOS to model real substances one needs to obtain pure component below its critical point, a technique suggested by Joffe et al. (18) was used. This involves the matching of chemical potentials of each component in the liquid and the vapour phases at the vapour pressure of the substance. Also, the actual and predicted saturated liquid densities were matched. The set of equations so obtained was solved by the use of a standard Newton s method to yield the pure component parameters. Values of exl and v for ethanol and water at several temperatures are shown in Table 1. In this calculation vH and z were set to 9.75 x 10"6 m3 mole"1 and 10, respectively (1 ). The capability of the lattice EOS to fit pure component VLE was found to be quite insensitive to variations in z (6[Pg.90]

Krukowski et al. [24] studied the effect of molecular shape in details by performing exact enumerations on lattice models of different molecular shapes. They calculated the entropic component of the chemical potential, i.e.,... 

A partitioning function for a system of rigid rod-like particles with partial orientation around an axis is derived from the use of a modified lattice model. The free energy of mixing is shown as a function of the mole numbers, the axis ratio of the solute particles and a disorientation parameter this function passes through a minimum with increase in the disorientation parameter. The chemical potentials display discontinuities at the concentration at which the minimum appears and then separation into an isotropic phase and a somewhat more concentrated anisotropic phase arises. The critical concentration, v, is given in the form 13) ... 

In the finmewoik of the lattice gas model, the adsorption process is simulated by assuming a square lattice of M=LxL adsorption sites, with periodic boundary conditions, in equilibrium with an ideal gas characterized by chemical potential p and temperature T. The surface as well as the adsorbate are inert upon adsorption. Then, for a given configuration of adparticles, the hamiltonian H of the system is given by... 

The bimolecular reaction rate for particles constrained on a planar surface has been studied using continuum diffusion theory " and lattice models. In this section it will be shown how two features which are not taken account of in those studies are incorporated in the encounter theory of this chapter. These are the influence of the potential K(R) and the inclusion of the dependence on mean free path. In most instances it is expected that surface corrugation and strong coupling of the reactants to the surface will give the diffusive limit for the steady-state rate. Nevertheless, as stressed above, the initial rate is the kinetic theory, or low-friction limit, and transient exp)eriments may probe this rate. It is noted that an adaptation of low-density gas-phase chemical kinetic theory for reactions on surfaces has been made. The theory of this section shows how this rate is related to the rate of diffusion theory. 

The key step in the derivation by Reuter et al. of their lattice model is the use of detailed balance to determine the sticking coefficients for each species on each type of site.31 The total adsorption rate at a particular site can be expressed as Tad = SI(p, T), where S is the local sticking coefficient and I(p,T) is the impingement rate of the species of interest from a gas phase with partial pressure p and temperature T. At steady state, the total adsorption and desorption rates must satisfy the detailed balance condition TdesjTad = exp[(Fb—/j,(T, p))/kT, where Fb is the free energy of the adsorbed species and fi(T, p) is the chemical potential of the gas phase species. The adsorption free energy is well approximated by the adsorption enthalpy, which is simply the adsorption energy calculated by a DFT calculation. This approach provides a direct link between the adsorption and desorption rates and the pressure and temperature of the bulk gas phase. 

Fig. 23. Ternary blend containing two homopolymers A and B and a symmetric AB diblock copolymer within the bond fluctuation model. All chains have identical length, N = 32. (a) Probability distribution at e = 0.054 and system size 48 x 48 x 96 in units of the lattice spacing (i e = 17). Upon increasing the chemical potential S/j of the copolymers the valley becomes shallower, indicating that the copolymers decrease the interfacial tension. One clearly observes a plateau around (f> = 1/2. This assures, that our system size is large enough to neglect interfacial interactions in the measurement of the interfacial tension, (b) Average number of copolymers as a function of the composition. The copolymer number is enhanced in the configuration containing two interfaces. From Muller and Schick [105]...

Figure 9.9 Adsorption isotherms for a lattice gas model consisting of shell and axial sites at the indicated temperatures (reduced by the pair interaction well depth) and various values of the reduced chemical potential. While mean field (MF) results exhibit a discontinuous shell-filling transition at T = 1, essentially exact Monte Carlo (MC) results show a near discontinuity there. (Adapted from Ref. [31, 32])...

The role of the metal in double layer properties can be understood in greater detail when the system is examined on the basis of the jellium model. This model was developed to describe the electron gas within sp metals. It can be used to estimate several properties of interest, including the chemical potential of an electron in the metal, the extent of electron overspill, and the work function of the metal. More recently, it has been extended to describe metal surfaces in contact with polar solvents [26]. In its simplest form, the metal atoms in the metal are modeled as a uniform positive background for the electron gas, no consideration being given to their discrete nature and position in the metal lattice. The most important property of the system is the average electron density, N ), which depends on the number of metal atoms per unit volume and the number of valence electrons per atom, n. Thus, if pjj, is the mass density of the metal, and M, its atomic mass... 

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