Boltzmann diffusion

WolfE and more recently other investigators have applied the Boltzmann diffusion equation to a description of the secondary-electron cascade. This approach is quite satisfying because it has a clearly defined foundation which seems to encompass all of the basic physical processes needed to describe the situation. It also yields an approximate solution in analytic form which is given by Equation 22. 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

As the particles in a coUoidal dispersion diffuse, they coUide with one another. In the simplest case, every coUision between two particles results in the formation of one agglomerated particle,ie, there is no energy barrier to agglomeration. Applying Smoluchowski s theory to this system, the half-life, ie, the time for the number of particles to become halved, is expressed as foUows, where Tj is the viscosity of the medium, k Boltzmann s constant T temperature and A/q is the initial number of particles. 

This result comes from the idea of a variational rate theory for a diffusive dynamics. If the dynamics of the reactive system is overdamped and the effective friction is spatially isotropic, the time required to pass from the reactant to the product state is expected to be proportional to the integral over the path of the inverse Boltzmann probability. 

Figure 13. Voltage relaxation method for the determination of the diffusion coefficients (mobilities) of electrons and holes in solid electrolytes. The various possibilities for calculating the diffusion coefficients and from the behavior over short (t L2 /De ) and long (/ L2 /Dc ll ) times are indicated cc h is the concentration of the electrons and holes respectively, q is the elementary charge, k is the Boltzmann constant and T is the absolute temperature.

Ludwig Boltzmann (1844-1906) was born in Vienna. His work of importance in chemistry became of interest in plastics because of his development of the kinetic theory of gases and rules governing their viscosity and diffusion. They are known as the Boltzmann s Law and Principle, still regarded as one of the cornerstones of physical science. 

Gas density Propellant density Boltzmann constant A factor to account for temperature oscillations ignition delay time Diffusion time... 

The hydrodynamic radius reflects the effect of coil size on polymer transport properties and can be determined from the sedimentation or diffusion coefficients at infinite dilution from the relation Rh = kBT/6itri5D (D = translational diffusion coefficient extrapolated to zero concentration, kB = Boltzmann constant, T = absolute temperature and r s = solvent viscosity). 

The outer layer (beyond the compact layer), referred to as the diffuse layer (or Gouy layer), is a three-dimensional region of scattered ions, which extends from the OHP into the bulk solution. Such an ionic distribution reflects the counterbalance between ordering forces of the electrical field and the disorder caused by a random thermal motion. Based on the equilibrium between these two opposing effects, the concentration of ionic species at a given distance from the surface, C(x), decays exponentially with the ratio between the electro static energy (zF) and the thermal energy (R 7). in accordance with the Boltzmann equation ... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

In order to calculate the rates for electron impact collisions and the electron transport coefficients (mobility He and diffusion coefficient De), the EEDF has to be known. This EEDF, f(r, v, t), specifies the number of electrons at position r with velocity v at time t. The evolution in space and time of the EEDF in the presence of an electric field is given by the Boltzmann equation [231] ... 

The ideal conductor model does not account for diffuseness of the ionic distribution in the electrolyte and the corresponding spreading of the electric field with a potential drop outside the membrane. To account approximately for these effects we apply Poisson-Boltzmann theory. The results for the modes energies can be summarized as follows [89] ... 

Routh and Russel [10] proposed a dimensionless Peclet number to gauge the balance between the two dominant processes controlling the uniformity of drying of a colloidal dispersion layer evaporation of solvent from the air interface, which serves to concentrate particles at the surface, and particle diffusion which serves to equilibrate the concentration across the depth of the layer. The Peclet number, Pe is defined for a film of initial thickness H with an evaporation rate E (units of velocity) as HE/D0, where D0 = kBT/6jT ir- the Stokes-Einstein diffusion coefficient for the particles in the colloid. Here, r is the particle radius, p is the viscosity of the continuous phase, T is the absolute temperature and kB is the Boltzmann constant. When Pe 1, evaporation dominates and particles concentrate near the surface and a skin forms, Figure 2.3.5, lower left. Conversely, when Pe l, diffusion dominates and a more uniform distribution of particles is expected, Figure 2.3.5, upper left. 

This theory of the diffuse layer is satisfactory up to a symmetrical electrolyte concentration of 0.1 mol dm-3, as the Poisson-Boltzmann equation is valid only for dilute solutions. Similarly to the theory of strong electrolytes, the Gouy-Chapman theory of the diffuse layer is more readily applicable to symmetrical rather than unsymmetrical electrolytes. 

In the treatment so far we have considered the /c-step to be irreversible. This will not be true for Case III systems, where the exchange at k is much more rapid than diffusion along the x-coordinate. This situation is considered in more detail in the Appendix, where we show that, in keeping with our general derivation, the concentration of transition states is one half of the Boltzmann concentration and that the fraction committed to reaction is also one-half. 

Displacements of lattice members are determined by energy factors and concentration gradients. To a considerable extent, diffusion in solids is related to the existence of vacancies. The "concentration" of defects, N0, (sites of higher energy) can be expressed in terms of a Boltzmann distribution as... 

---