Linear diffusion equation

Rate equations There are two basic types of kinetic rate expressions. The first and simpler is the case of linear diffusion equations or linear driving forces (LDF) and the second and more rigorous is the case of classic Fickian differential equations. 

Linear diffusion equations This is the simplest case and is used extensively in the related literature (Perry and Green, 1999 Hashimoto et al., 1977 Cooney, 1990, 1993). The equations are the following ... 

Equation (1.57a) implies that in the locally electro-neutral ambipolar diffusion concentration of both ions evolves according to a single linear diffusion equation with an effective diffusivity given by (1.57b). Physically, the role of the electric field, determined from the elliptic current continuity equation... 

For illustration, we present in Fig. 3.3.1a,b the results of a numerical solution of the original system (3.3.19) (Curve 1) for e = 10-2, 10-3, 7=1 together with a plot of the leading term (3.3.48) (Curve 2). We also present for comparison a plot of 1 erfcx (Curve 3), the similarity solution for the linear diffusion equation with the boundary and initial conditions analogous... 

The general treatment for multicomponent diffusion results in linear systems of diffusion equations. A linear transformation of the concentrations produces a simplified system of uncoupled linear diffusion equations for which general solutions can be obtained by methods presented in Chapter 5. 

In other words, the kinetics of smoothing out of Z(r,t) fluctuations is governed by a simple linear diffusion equation which solution and, in particular, the Green function, are well known. Similarly to (2.1.43), (2.1.45) we can now write down... 

The dimensionless distance parameter may be found by substituting the definition of Ax into Equation 20.7. In the case of semi-infinite linear diffusion, Equation 20.17 is used to define Ax this substitution yields... 

Equation (96) is known as the linear diffusion equation since the lowest-order field dependence is linear. Thus we have a microscopic derivation of the Einstein relation, eqn. (98). This relation is normally derived from quite different considerations based on setting the current equal to zero in the linear diffusion equation and comparing the concentration profile C (x) with that predicted by equilibrium thermodynamics. 

If, at places within the oxide layer, the electric field is too large for the exponential expansion utilized above in deriving the linear diffusion equation to be a valid approximation, then the more exact hopping current expression given by eqn. (88) should be utilized instead. Numerical computations are usually easier to carry out in any electric field limit by using the exact microscopic hopping expression (88). 

Consider the limit of thick oxides and moderate values of the electric field where both ion transport and electron transport take place in accordance with the linear diffusion equation [see eqn. (96)]. In essence, this means that we are assuming that the inequality... 

Furthermore if we take the free-energy functional as given by Chrmdler et al.° for molecular(polyatomic) liquids, we obtain from (11) a L-D equation for each of the atomic species constituting the molecules, which is very similar to (20). After liniarization as done in (22), we obtain a linear diffusion equation, which was used by Hirata to investigate the dyntunic structure factors of water. 

Figure 5 Multiscale approach to understand rate of CO2 diffusion into and CH4 diffusion out of a structure I hydrate, (left) Molecular simulation for individual hopping rates, (middle) Mesoscale kinetic Monte Carlo simulation of hopping on the hydrate lattice to determine dependence of diffusion constants on vacancy, CO2 and CH4 concentrations, (right) Macroscopic coupled non-linear diffusion equations to describe rate of CO2 infusion and methane displacement. Graph from Stockie. ...

We will later recognize this as the kernel for the continuum linear diffusion equation once we make the observation that the jump rate and the diffusion constant are related by Z) = Fa. For now we content ourselves by noting that the model of diffusion put forth here is invested with relatively little complexity. All we postulated was the uncorrelated hopping of the particles of interest. Despite the simplicity of the model, we have been led to relatively sophisticated predictions for what one might call the kinematics (which is statistical) of the diffusion field that results once the diffusive process has been set in motion. Indeed, the spreading... 

The result of substituting (6-35) into (6-34) is the classical linear diffusion equation,... 

This solution is valid for all n / 0. We note that it is of a fundamentally different form from the solution to the linear diffusion equation previously obtained. One specific point to note is that / > 0, so that the solution makes physical sense only for i] < 1. In mathematical terminology, the solution is said to have compact support in the sense that/is nonzero only within this region. Thus the value rj = 1 defines the outer boundary of the region in which the pulse of material exists as a function of time. 

The calculation of the diffusion-limited current, and the concentration profile, Co(x, t), involves the solution of the linear diffusion equation ... 

The derivation of (6.2.8) and (6.2.9) employed only the linear diffusion equations, initial conditions, semi-infinite conditions, and the flux balance. No assumption related to electrode kinetics or technique was made hence (6.2.8) and (6.2.9) are general. From these equations and the boundary condition for LSV, (6.2.3), we obtain... 

The Kinetics of Spinodal Decomposition. Cahn s kinetic theory of spinodal decomposition (2) was based on the diffuse interface theory of Cahn and Hilliard (13). By considering the local free energy a function of both composition and composition gradients, Cahn arrived at the following modified linearized diffusion equation (Equation 3) to describe the early stages of phase separation within the unstable region. In this equation, 2 is an Onsager-type... 

Spinodal decomposition (SD) driven by the chemical reaction proceeds isothermally, but the quench depth AT (expressing the temperature difference between LCST and the reaction temperature), increases with time. This situation is quite different from the familiar SD under isoquench depth, where after a temperature jump (or drop) SD proceeds isothermally, and the AT is constant. However, the regular morphology is also obtained in the kinetically driven SD, as in the iso-quench SD. This observation was confirmed by the computer simulation using the Cahn-Hilliard non-linear diffusion equation [Ohnaga et al., 1994]. This should also be the case for the solution casting, described in preceding section and the shear-dependent decomposition in next section. 

Another difficulty arising from this comparison is connected with the mathematical complexity of the corresponding boundary problems even if only linear diffusion equations are used. The mathematical description of the adsorption kinetics from micellar solutions is essentially more complicated in comparison with the case of the adsorption process from sub-micellar solutions. Analytical solutions of the corresponding boundary problems using rather poor approximations have been obtained only for a small number of situations. A sufficiently general solution cannot be obtained analytically and the deficiency of the rather well elaborated numerical methods often compel experimentalists to apply approximate solutions. Therefore, it seems important to consider the main equations proposed for the description of kinetic dependencies of the surface tension and adsorption, and to elucidate the limits of their application before the discussion of experimental results. 

This is also called the linear-diffusion equation, which in its most elementary form is a linear second-order partial differential equation (PDE). The assmnption of a concentration-independent diffusion coefficient is generally true for diffusion in gases, hquids, and solutions. Polymers above the glass-transition temperature and, especially, rubbers such as PDMS can be expected to behave like liquids for small molecular diffusants. Analytical solutions to the Unear-diffusion equation, eq 2, for various... 

Kats, B.M. Kutarov, V.V., and Chagodar, A. A., Applications of non-linear diffusion equations to the kinetics of water vapour adsorption by polymeric fibres. Adsorpt. Sci. Technol.. 9(4), 269-275 (1993). 

The limit conditions are the same as for chronoam-perometry. Solution of the linear diffusion equations leads to the value of the transition time (Sand equation) ... 

Equation (3.3.5) represents a nonlinear phase diffusion equation. It is equivalent to the Burgers equation in the case of one space dimension (Chap. 6). It is known that the Burgers equation can be reduced to a linear diffusion equation through a transformation called the Hopf-Cole transformation (Burgers, 1974), and essentially the same is true for (3.3.5) in an arbitrary dimension. We shall take advantage of this fact in Chap. 6 when analytically discussing a certain form of chemical waves. 

Theorem 2.7.1 can be explained as follows. When Vq is small, the nonlinear source term due to the chemical reaction is a small perturbation on the linear diffusion equation. The state variables are close to their... 

A simple linear diffusion equation for oxygen transport in the GDL leads to the following relation between the oxygen concentration at the CCL/GDL interface Cox,i and the oxygen concentration in the channel... 

Equation (11) is the generalized linear diffusion equation relevant for the early stages of spinodal decomposition. A solution of eq. (11) has sinosodial spatial behavior with an amplitude which grows exponentially in time... 

A step up in complexity from the linear diffusion equation is a nonlinear version of the diffusion equation where the diffusion coefficient depends on the concentration of diffusing species. The equation to be considered is then... 

---