Solution of a diffusion equation

The solutions of a diffusion equation under the transient case (non-steady state) are often some special functions. The values of these functions, much like the exponential function or the trigonometric functions, cannot be calculated simply with a piece of paper and a pencil, not even with a calculator, but have to be calculated with a simple computer program (such as a spreadsheet program, but see later comments for practical help). Nevertheless, the values of these functions have been tabulated, and are now easily available with a spreadsheet program. The properties of these functions have been studied in great detail, again much like the exponential function and the trigonometric functions. One such function encountered often in one-dimensional diffusion problems is the error function, erf(z). The error function erf(z) is defined by... 

The role of thermal fluctuations for membranes interacting via arbitrary potentials, which constitutes a problem of general interest, is however still unsolved. Earlier treatments G-7 coupled the fluctuations and the interaction potential and revealed that the fluctuation pressure has a different functional dependence on the intermembrane separation than that predicted by Helfrich for rigid-wall interactions. The calculations were refined later by using variational methods.3 8 The first of them employed a symmetric functional form for the distribution of the membrane positions as the solution of a diffusion equation in an infinite well.3 However, recent Monte Carlo simulations of stacks of lipid bilayers interacting via realistic potentials indicated that the distribution of the intermembrane distances is asymmetric 9 the root-mean-square fluctuations obtained from experiment were also shown to be in disagreement with this theory.10... 

Details of the neutralization process following radiation-induced primary charge separation may be examined via the medium of ultrafast techniques now employed in studies of luminescence decay processes. As an example, the form of luminescence decay curves of dilute organic scintillator in aliphatic hydrocarbon solution excited by x-ray pulses of about 0.5-1.0 nsec, duration is attributed (in previous papers) to neutralization processes involving ions. The relation, t cc r3, for the time required for neutralization of an ion pair of initial separation r, when applied to such curves, leads to a distribution function of ion-pair separations. A more appropriate and desirable approach involves solution of a diffusion equation (which includes a Coulomb interaction term) for various initial conditions. Such solutions are obtained by computer techniques employed in analogy to corresponding electrical networks. The results indicate that the tocr3 law affords a fair description of the decay if the initial distribution can be assumed to be broad. 

A powerful insight into polymer chain conformations is an analogy with the diffusion equation. When polymers are subject to complex geometric constraints, the conformational free energy can sometimes be found as a solution of a diffusion equation. 

Appendix solution of a diffusion equation in cylindrical coordinates... 

Example The equation 3c/3f = D(3 c/3a. ) represents the diffusion in a semi-infinite medium, a. > 0. Under the boundary conditions c(0, t) = Cq, c(x, 0) = 0 find a solution of the diffusion equation. By taking the Laplace transform of both sides with respect to t,... 

A situation which is frequently encountered in tire production of microelectronic devices is when vapour deposition must be made into a re-entrant cavity in an otherwise planar surface. Clearly, the gas velocity of the major transporting gas must be reduced in the gas phase entering the cavity, and transport down tire cavity will be mainly by diffusion. If the mainstream gas velocity is high, there exists the possibility of turbulent flow at tire mouth of tire cavity, but since this is rare in vapour deposition processes, the assumption that the gas widrin dre cavity is stagnant is a good approximation. The appropriate solution of dre diffusion equation for the steady-state transport of material tlrrough the stagnant layer in dre cavity is... 

The first and second integrals have their coordinate systems centered on the catalytic C and noncatalytic N spheres, respectively. The local nonequilibrium average microscopic density field for species a is pa(r) = [Y = 5(r - ( )) The solution of the diffusion equation can be used to estimate this nonequilibrium density, and thus the velocity of the nanodimer can be computed. The simple model yields results in qualitative accord with the MPC dynamics simulations and shows how the nonequilibrium density field produced by reaction, in combination with the different interactions of the B particles with the noncatalytic sphere, leads to directed motion [117],... 

The last two assumptions are the most critical and are probably violated under field conditions. Smith et al. (3) found that at least a half-hour was required to achieve adsorption equilibrium between a chemical in the soil water and on the soil solids. Solution of the diffusion equation has shown that many volatile compounds have theoretical diffusion half-lives in the soil of several hours. Under actual field conditions, the time required to achieve adsorption equilibrium will retard diffusion, and diffusion half-lives in the soil will be longer than predicted. Numerous studies have reported material bound irreversibly to soils, which would cause apparent diffusion half-lives in the field to be longer than predicted. 

For small colloidal particles, which are subject to random Brownian motion, a stochastic approach is more appropriate. These methods are based on the formulation and solution of the diffusion equation in a force field, in the presence of convection... 

The trouble is now that the source term does not include the sum of sines, so we will use a trick resting on the Leibniz s rule for differentiating integrals. A particular solution of the diffusion equation with radiogenic accumulation is... 

Finally, as described in Box 4.1 of Chapter 4, an exact numerical solution of the diffusion equation (based on Fick s second law with an added sink term that falls off as r-6) was calculated by Butler and Pilling (1979). These authors showed that, even for high values of Ro ( 60 A), large errors are made when using the Forster equation for diffusion coefficients > 10 s cm2 s 1. Equation (9.34) proposed by Gosele et al. provides an excellent approximation. 

A solution of the diffusion equation for an electrode reaction for repetitive stepwise changes in potential can be obtained by numerical integration [44]. For a stationary planar diffusion model of a simple, fast, and reversible electrode reaction (1.1), the following differential equations and boundary conditions can be formulated ... 

In many treatments of free diffusion the propagator is immediately written as a Gaussian function with the argument that it fulfils the diffusion equation. Equation (27) shows the relation with the normal mode solution of the diffusion equation. For diffusion in a bounded region the propagator is no... 

The general solution of the diffusion equation for a system with spherical symmetry can therefore be written as ... 

The DICTRA programme is based on a numerical solution of multi-component diffusion equations assuming that thermodynamic equilibrium is locally maintained at phase interfaces. Essentially the programme is broken down into four modules which involve (1) the solution of the diffusion equations, (2) the calculation of... 

Sorption relates to a compound sticking to the surface of a particle. Adsorption relates to the process of compound attachment to a particle surface, and desorption relates to the process of detachment. Example 2.2 was on a soluble, nonsorptive spiU that occurred into the ground and eventually entered the groundwater. We will now review sorption processes because there are many compounds that are sorptive and subject to spills. Then, we can examine the solutions of the diffusion equation as they apply to highly sorptive compounds. 

The solution of Example 7.3 will be compared with an analytical solution of a diffusive front moving at velocity U, with D = 1/2U Az. First, we must derive the analytical solution. This problem is similar to Example 2.10, with these exceptions (1) convection must be added through a moving coordinate system, similar to that described in developing equation (2.36), and (2) a diffusion gradient will develop in both the +z-and -z-directions. 

Infinite fluid volume and solid diffusion control Practically, infinite solution volume condition (w 1) amounts to constant liquid-phase concentration. For a constant diffu-sivity and an infinite fluid volume, the solution of the diffusion equations is (Helfferich, 1962 Ruthven, 1984)... 

These equations, for the case of solid diffusion-controlled kinetics, are solved by arithmetic methods resulting in some analytical approximate expressions. One common and useful solution is the so-called Nernst-Plank approximation. This equation holds for the case of complete conversion of the solid phase to A-form. The complete conversion of solid phase to A-form, i.e. the complete saturation of the solid phase with the A ion, requires an excess of liquid volume, and thus w 1. Consequently, in practice, the restriction of complete conversion is equivalent to the infinite solution volume condition. The solution of the diffusion equation is... 

In a transient method one creates an initial distribution of labels in the plane of the membrane that is nonuniform, c(x, y, 0), where c is the mole % of the spin-label lipid (e.g., at point x, y in the membrane, at time t = 0), For times f O the lateral distribution of label c(x, y, r) is then determined from a solution of the diffusion equation... 

Fick s second law (Eq. 18-14) is a second-order linear partial differential equation. Generally, its solutions are exponential functions or integrals of exponential functions such as the error function. They depend on the boundary conditions and on the initial conditions, that is, the concentration at a given time which is conveniently chosen as t = 0. The boundary conditions come in different forms. For instance, the concentration may be kept fixed at a wall located atx0. Alternatively, the wall may be impermeable for the substance, thus the flux at x0 is zero. According to Eq. 18-6, this is equivalent to keeping dC/dx = 0 at x0. Often it is assumed that the system is unbounded (i.e., that it extends from x = - °o to + °°). For this case we have to make sure that the solution C(x,t) remains finite when x -a °°. In many cases, solutions are found only by numerical approximations. For simple boundary conditions, the mathematical techniques for the solution of the diffusion equation (such as the Laplace transformation) are extensively discussed in Crank (1975) and Carslaw and Jaeger (1959). 

Now, everything falls into place We set out to study the laws of random walk by using the simple model of Fig. 18 and found the Bernoulli coefficients. We then saw that for large n (which is equivalent to large times), the Bernoulli coefficients can be approximated by a normal distribution whose standard deviation, a, grows in proportion to the square root of time, tm (Eq. 18-3). And now it turns out that the solution of the Fick s second law for unbounded diffusion is also a normal distribution. In fact, the analogy between Eqs. 18-3b and 18-17 gave the basis for the law by Einstein and Smoluchowski (Eq. 18-17) that we used earlier (Eq. 18-8). The expression (2Dt)U2 will also show up in other solutions of the diffusion equation. 

To solve the diffusion equation and obtain the appropriate rate coefficient with these initial distributions is less easy than with the random distribution. As already remarked, the random distribution is a solution of the diffusion equation, while the other distributions are not. The substitution of Z for r(p(r,s) — p(r, 0)/s) is not possible because an inhomogeneous equation results. This requires either the variation of parameters or Green s function methods to be used (they are equivalent). Appendix A discusses these points. The Green s function g(r, t r0) is called the fundamental solution of the diffusion equation and is the solution to the... 

Comparison with eqn. (A.10) of Appendix A, shows that p is the fundamental or Green s function solution of the diffusion equation, eqn. (10). 

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