Boltzmann equation approximation

The simplified Boltzmann equation can be solved using the Lattice Boltzmann Method (LBM) for the distributed function on a regular lattice. LBM considers each lattice structure as a volume element that consists of a collection of particles in the fluid. This simplified Boltzmann equation approximates the collision term, Q f, / ), in Eq. 37 using a relaxation time, t, providing a linear correlation. The most well-known form of the LBM is the BGKLBM, where the relaxation time is a constant. 

Marmur [12] has presented a guide to the appropriate choice of approximate solution to the Poisson-Boltzmann equation (Eq. V-5) for planar surfaces in an asymmetrical electrolyte. The solution to the Poisson-Boltzmann equation around a spherical charged particle is very important to colloid science. Explicit solutions cannot be obtained but there are extensive tabulations, known as the LOW tables [13]. For small values of o, an approximate equation is [9, 14]... 

Moreover, since the mean free path is of the order of 100 times the molecular diameter, i.e., the range of force for a collision, collisions involving three or more particles are sufficiently rare to be neglected. This binary collision assumption (as well as the molecular chaos assumption) becomes better as the number density of the gas is decreased. Since these assumptions are increasingly valid as the particles spend a larger percentage of time out of the influence of another particle, one may expect that ideal gas behavior may be closely related to the consequences of the Boltzmann equation. This will be seen to be correct in the results of the approximation schemes used to solve the equation. 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

Coefficient Equations.—To determine the coefficients of the expansion, the distribution function, Eq. (1-72), is used in the Boltzmann equation the equation is then multiplied by any one of the polynomials, and integrated over velocity. This gives rise to an infinite set of coupled equations for the coefficients. Only a few of the coefficients appear on the left of each equation in general, however, all coefficients (and products) appear on the right side due to the nonlinearity of the collision integral. Methods of solving these equations approximately will be discussed in later sections. 

Bather than carrying out the calculation for the general case, which yields rather unwieldy expressions, only equations sufficient to obtain certain approximations will be developed. If we multiply the Boltzmann equation, Eq. (1-39), by 1 = i%( 2)3r )) (0.9>)> the resulting equation is simply the equation of conservation of mass, since integrating unity over the collision integral gives zero ... 

Using the lattice model, the approximate value of W in the Boltzmann equation can be estimated. Two separate approaches to this appeared in 1942, one by P. J. Rory, the other by M. L. Huggins, and though they differed in detail, the approaches are usually combined and known as the Rory-Huggins theory. This gives the result for entropy of mixing of follows ... 

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

Third, a further simplification of the Boltzmann equation is the use of the two-term spherical harmonic expansion [231 ] for the EEDF (also known as the Lorentz approximation), both in the calculations and in the analysis in the literature of experimental data. This two-term approximation has also been used by Kurachi and Nakamura [212] to determine the cross section for vibrational excitation of SiHj (see Table II). Due to the magnitude of the vibrational cross section at certain electron energies relative to the elastic cross sections and the steep dependence of the vibrational cross section, the use of this two-term approximation is of variable accuracy [240]. A Monte Carlo calculation is in principle more accurate, because in such a model the spatial and temporal behavior of the EEDF can be included. However, a Monte Carlo calculation has its own problems, such as the large computational effort needed to reduce statistical fluctuations. 

The discrepancy may also be caused by the approximations in the calculation of the EEDF. This EEDF is obtained by solving the two-term Boltzmann equation, assuming full relaxation during one RF period. When the RF frequency becomes comparable to the energy loss frequencies of the electrons, it is not correct to use the time-independent Boltzmann equation to calculate the EEDF [253]. The saturation of the growth rate in the model is not caused by the fact that the RF frequency approaches the momentum transfer frequency Ume [254]. That would lead to less effective power dissipation by the electrons at higher RF frequencies and thus to a smaller deposition rate at high frequencies than at lower frequencies. 

In a fluid model the correct calculation of the source terms of electron impact collisions (e.g. ionization) is important. These source terms depend on the EEDF. In the 2D model described here, the source terms as well as the electron transport coefficients are related to the average electron energy and the composition of the gas by first calculating the EEDF for a number of values of the electric field (by solving the Boltzmann equation in the two-term approximation) and constructing a lookup table. 

In addition to the nearest-neighbor interaction, each ion experiences the electrostatic potential generated by the other ions. In the literature this has generally been equated with the macroscopic potential 0 calculated from the Poisson-Boltzmann equation. This corresponds to a mean-field approximation (vide infra), in which correlations between the ions are neglected. This approximation should be the better the low the concentrations of the ions. 

In principle, one should solve the Boltzmann equation Eq. (65) in order to arrive at explicit expressions for the pressure tensor p and heat flux q, which proves not possible, not even for the simple BGK equation Eq. (11). However, one can arrive at an approximate expression via the Chapman Enskog expansion, in which the distribution function is expanded about the equilibrium distribution function fseq, where the expansion parameter is a measure of the variation of the hydrodynamic fields in time and space. To second order, one arrives at the familiar expression for p and q... 

As a first approximation, assume that the total entropy change, ASy, is made up solely of configurational entropy. This can be determined by using the Boltzmann equation for the entropy of a disordered system ... 

A major preoccupation of nonequilibrium statistical mechanics is to justify the existence of the hydrodynamic modes from the microscopic Hamiltonian dynamics. Boltzmann equation is based on approximations valid for dilute fluids such as the Stosszahlansatz. In the context of Boltzmann s theory, the concept of hydrodynamic modes has a limited validity because of this approximation. We may wonder if they can be justified directly from the microscopic dynamics without any approximation. If this were the case, this would be great progress... 

The derivation of the mixture-balance laws has been given by Chapman and Cowling for a binary mixture. Its generalization to multicomponent mixtures, as in Equation 5-1, uses a determination of the invariance of the Boltzmann equation. This development has been detailed by Hirschfelderet These derivations were summarized in the notes of Theodore von Karmin s Sorbonne lectures given in 1951-1952, and the results of his summaries were stated in Pinner s monograph. For turbulent flow, the species-balance equation can be represented in the Boussinesq approximation as ... 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

Recent advances in computational power have made it possible to use efficient approximations of the Poisson-Boltzmann equation also to estimate the electrostatic component of protein-ligand interactions [56]. 

For our work, expressions of Ohshlma et. al. (37) obtained from an approximate form of the Poisson-Boltzmann equation were used. These analytical expressions agree with the exact solution for xRp 2. (All of our calculations meet this criterion.) The relation between the surface potential and the surface charge density Is (37)... 

The geometry. It is clear that the geometry of the system is much simplified in the slab model. Another possibility is to model the protein as a sphere and the stationary phase as a planar surface. For such systems, numerical solutions of the Poisson-Boltzmann equations are required [33]. However, by using the Equation 15.67 in combination with a Derjaguin approximation, it is possible to find an approximate expression for the interaction energy at the point where it has a minimum. The following expression is obtained [31] ... 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

The purpose of the present chapter is to introduce some of the basic concepts essential for understanding electrostatic and electrical double-layer pheneomena that are important in problems such as the protein/ion-exchange surface pictured above. The scope of the chapter is of course considerably limited, and we restrict it to concepts such as the nature of surface charges in simple systems, the structure of the resulting electrical double layer, the derivation of the Poisson-Boltzmann equation for electrostatic potential distribution in the double layer and some of its approximate solutions, and the electrostatic interaction forces for simple geometric situations. Nonetheless, these concepts lay the foundation on which the edifice needed for more complicated problems is built. 

The derivation of the Poisson equation implies that the potentials associated with various charges combine in an additive manner. The Boltzmann equation, on the other hand, involves an exponential relationship between the charges and the potential. In this way a fundamental inconsistency is introduced when Equations (26) and (28) are combined. Equation (29) does not have an explicit general solution anyhow and must be solved for certain limiting cases. These involve approximations that —at the same time —overcome the objection just stated. 

The above equation is known as the linearized Poisson-Boltzmann equation since the assumption of low potentials made in reaching this result from Equation (29) has allowed us make the right-hand side of the equation linear in p. This assumption is also made in the Debye-Hiickel theory and prompts us to call this model the Debye-Hiickel approximation. Equation (33) has an explicit solution. Since potential is the quantity of special interest in Equation (33), let us evaluate the potential at 25°C for a monovalent ion that satisfies the condition e p = kBT ... 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

The theoretical inconsistencies inherent in the Poisson-Boltzmann equation were shown in Section 11.4 to vanish in the limit of very small potentials. It may also be shown that errors arising from this inconsistency will not be too serious under the conditions that prevail in many colloidal dispersions, even though the potential itself may no longer be small. Accordingly, we return to the Poisson-Boltzmann equation as it applies to a planar interface, Equation (29), to develop the Gouy-Chapman result without the limitations of the Debye-Hiickel approximation. 

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