Linear diffusion equations, solution

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... 

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 ... 

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. 

Upon substitution of an appropriate kinetic expression for the rate of generation or consumption of solute within the tissue space, Equation 3-50 can be solved to determine concentration as a function of time and position. Full analytical solutions are generally difficult to obtain, unless both the kinetic expression and the geometry of the system are simple. For example, consider the linear diffusion of solute from an interface where the concentration is maintained constant (as in Figure 3.4d). If the diffusing solute is also eliminated from the tissue, such that the volumetric rate of elimination is first order with a characteristic rate constant k, Equation 3-51 can be reduced to ... 

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... 

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) ... 

The partial differential equation for linear diffusion and solution chemical kinetics is ... 

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... 

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

For illustration, consider the simplest type of diffusion, described by the partial differential equation (2.5.3), also called linear diffusion. The system will be represented by an infinite tube closed at one end (for x = 0) and initially filled with a solution with concentration c°. Diffusion is produced by very fast removal (e.g. by precipitation or an electrode reaction) of the dissolved substance at the x = 0 plane (the reference plane). The initial concentration c° is retained at large distances from this reference plane (x— < >). The initial condition is thus... 

The calculation becomes more difficult when the polarization resistance RP is relatively small so that diffusion of the oxidized and reduced forms to and from the electrode becomes important. Solution of the partial differential equation for linear diffusion (2.5.3) with the boundary condition D(dcReJdx) = —D(d0x/dx) = A/sin cot for a steady-state periodic process and a small deviation of the potential from equilibrium is... 

In all considered above models, the equilibrium morphology is chosen from the set of possible candidates, which makes these approaches unsuitable for discovery of new unknown structures. However, the SCFT equation can be solved in the real space without any assumptions about the phase symmetry [130], The box under the periodic boundary conditions in considered. The initial quest for uy(r) is produced by a random number generator. Equations (42)-(44) are used to produce density distributions T(r) and pressure field ,(r). The diffusion equations are numerically integrated to obtain q and for 0 < s < 1. The right-hand size of Eq. (47) is evaluated to obtain new density profiles. The volume fractions at the next iterations are obtained by a linear mixing of new and old solutions. The iterations are performed repeatedly until the free-energy change... 

Substrate transport through the film may be formally assimilated to membrane diffusion with a diffusion coefficient defined as12 Ds = Dch( 1 — 9)/pjort. In this equation, the effect of film structure on the transport process in taken into account in two ways. The factor 1—0 stands for the fact that in a plane parallel to the electrode surface and to the coating-solution interface, a fraction 9 of the surface area in made unavailable for linear diffusion (diffusion coefficient Dcj,) by the presence of the film. The tortuosity factor,, defined as the ratio between the average length of the channel and the film thickness, accounts for the fact that the substrate... 

Rp) ] /Dg can be considered a relaxation time for diffusion accompanied by linear sorption. Three solutions to Equation 38 corresponding to values of F equal to 0.2, 0.5, and 0.8 are illustrated in Figure 7. 

Sometimes, there is no linear portion to a Randles-Sevdik graph, and the data yield a curved plot. The derivation of equation (6.13) assumes that diffusion is the sole means of mass transport. We also assume that all diffusion occurs in one dimension only, i.e. perpendicular to the electrode, with analyte arriving at the electrode solution interface from the bulk of the solution. We say here that there is semi-infinite linear diffusion. 

The diffusion equation with constant diffusivity (Equation 3-8) is said to be linear, which means that if fand g are solutions to the equation, then any linear combination of f and g, i.e., u = af+ bg, where a and b are constants, is also a solution. To show this, we can write... 

The above derivation has not made use of the initial and boundary conditions yet, and shows only that A may take any constant value. The value of A can be constrained by boundary conditions to be discrete Ai, A2,..., as can be seen in the specific problem below. Because each function corresponding to given A is a solution to the diffusion equation, based on the principle of superposition, any linear combination of these functions is also a solution. Hence, the general solution for the given boundary conditions is... 

The superposition principle can be used to combine solutions for linear partial differential equations, like the diffusion equation. It is stated as follows ... 

It is seen that the predicted linear relationship is indeed realized. However, it can be shown that the values for the (B) term from the Knox equation curve fit also give a linear relationship with solute diffusivity so the linear curves shown in figure 4 do not exclusively support the van Deemter equation. 

As the first term of the right-hand side of Equation 11.20 is independent of fluid velocity and is proportional to the radius fg (m) of particles packed as the stationary phase under usual conditions in chromatography separation, Hs (m) will increase linearly with the interstitial velocity of the mobile phase u (m s ), as shown in Figure 11.9. With a decrease in the effective diffusivities of solutes (m s ), Hs for a given velocity will increases, while the intercept of the straight lines on the y-axis, which corresponds to the value of the first term of Equation 11.20, is constant for different solutes. The value ofthe intercept will depend on the radius of packed particles, but does not vary with the effective diffusivity of the solute. 

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

The local stability of a given stationary-state profile can be determined by the same sort of test applied to the solutions for a CSTR. Of course now, when we substitute in a = ass + Aa etc., we have the added complexity that the profile is a function of position, as may be the perturbation. Stability and instability again are distinguished by the decay or growth of these small perturbations, and except for special circumstances the governing reaction-diffusion equation for SAa/dr will be a linear second-order partial differential equation. Thus the time dependence of Aa will be governed by an infinite series of exponential terms ... 

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). 

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