Solution of Fick’s second law

Use of Computer Simulation to Solve Differential Equations Pertaining to Diffusion Problems. As shown earlier (Section 4.2.11), differential equations used in the solutions of Fick s second law can often be solved analytically by the use of Laplace transform techniques. However, there are some cases in which the equations can be solved more quickly by using an approximate technique known as the finite-difference method (Feldberg, 1968). 

Sketch qualitatively the variation of concentration profiles with time that you would expect from the solution of Fick s second law. Assume suitable boundary and initial conditions. 

The Normal Distribution An Important Solution of Fick s Second Law Advanced Topic)... 

In Box 18.3 the normal distribution solution of Fick s second law is extended to three dimensions. 

We now discuss two additional solutions of Fick s second law (Eq. 18-14) for particular boundary conditions. The first one deals with diffusion from a surface with fixed boundary concentration, C0, into the semi-infinite space. The second one involves the disappearance ( erosion ) of a concentration jump. Both cases will be important when dealing with the transport through boundaries (Chapter 19). No derivations will be given below. The interested reader is referred to Crank (1975) and Carslaw and Jaeger (1959) or to mathematical textbooks dealing with particular techniques for solving Eq. 18-14. 

In the case of mass transport by pure diffusion, the concentrations of electroactive species at an electrode surface can often be calculated for simple systems by solving Fick s equations with appropriate boundary conditions. A well known example is for the overvoltage at a planar electrode under an imposed constant current and conditions of semi-infinite linear diffusion. The relationships between concentration, distance from the electrode surface, x, and time, f, are determined by solution of Fick s second law, so that expressions can be written for [Ox]Q and [Red]0 as functions of time. Thus, for... 

We have seen already that the general solution of Fick s second law in the Laplace domain can be formulated in various convenient ways. Here, we choose the notation of eqn. (147)... 

The solution of Fick s second law with the following initial and boundary conditions 303... 

This equipment can be used for the study of a single-component diffusion, and the measurement of the corresponding Fickean diffusion coefficient made using a solution of Fick s second law for a geometry appropriate for the experimental setup [87-92], In this case, the flow rates were adjusted to get the desired partial pressure (6.7 Pa, P/Pn = 0.006) [90],... 

For a spherical geometry, which is the case if the zeolite crystals which conform the wafer are approximately spherical with a variable surface concentration and the initial concentration inside the sphere equals zero, the solution of Fick s second law is given by an equation very similar to Equation 5.86 [5]... 

The solution of Fick s second law gives the variation of flux, and thence diffusion-limited current, with time, it being important to specify the conditions necessary to define the behaviour of the system (boundary conditions). Since the second law is a partial differential equation it has to be transformed into a total differential equation, solved, and the transform inverted1. The Laplace transform permits this (Appendix 1). 

Equation (4.2) reveals that the fraction of drug released is linearly related to the square root of time. However, (4.2) cannot be applied throughout the release process since the assumptions used for its derivation are not obviously valid for the entire release course. Additional theoretical evidence for the time limitations in the applicability of (4.2) has been obtained [10] from an exact solution of Fick s second law of diffusion for thin films of thickness S under perfect sink conditions, uniform initial drug concentration with cq > cs, and assuming constant diffusion coefficient of drug T> in the polymeric film. In fact, the short-time approximation of the exact solution is... 

Entire books are devoted to the solution of Fick s second law subject to different boundary and initial conditions, one of the most notable being Crank [22]. Consider the case when a HOP, initially confined to a narrow region between -h and h, is allowed to diffuse away in one dimension to infinity. Formally we can represent these initial and boundary conditions as follows ... 

The most widely used unsteady state method for determining diffusivities in porous solids involves measuring the rate of adsorption or desorption when the sample is subjected to a well defined change in the concentration or pressure of sorbate. The experimental methods differ mainly in the choice of the initial and boundary conditions and the means by which progress towards the new position of equilibrium is followed. The diffusivities are found by matching the experimental transient sorption curve to the solution of Fick s second law. Detailed presentations of the relevant formulae may be found in the literature [1, 2, 12, 15-17]. For spherical particles of radius R, for example, the fractional uptake after a pressure step obeys the relation... 

To show that Equation 1.5 is a possible solution of Fick s second law, it can be substituted into Equation 1.4 and the differentiations performed (M and D are constant daX1 /3x = de /dx = atv3l leaxn, and duv/dx = udv/dx + vdu/dx). The solution of Equation 1.4 becomes progressively more difficult when more complex conditions or molecular interactions (which cause variations in D ) are considered. Indeed, many books have been written on the solutions to Equation 1.4, where Equation 1.5 actually represents thefirst term of a power series [note that (nD t)lfZ can be replaced by fnDji). 

Oxygen transport measurements were conducted at 25°C, 0% and 50% relative humidity RH, 1 atm partial oxygen pressure difference using the commercially manufactured diffusion apparatus OX-TRAN 2/20 (Modem Control Inc.). This apparatus employs a continuous-flow method (ASTM-D 3985-81) to measure oxygen flux, J(t), through polymer films or thin sheets. In order to obtain the diffusion coefficient and to accurately determine the permeability coefficient, the data, flux, J(t), were fitted to the solution of Fick s second law ... 

The effective diffusion coefficients were calculated from the experimentally observed data (time, amount of cation exchanged, temperature), using Paterson s solution of Fick s second law, or published approximate solutions (8, 16). Taking into consideration particle shape and particle size distribution, the differential coefficients of internal diffusion in ion exchange can be ascertained by a method previously described (9). 

With this analytical solution of Fick s Second Law of Diffusion, the various curves in Figure 3.3 can now be ealenlated. On doing this, one will find that the onteome is exactly the same as in the corresponding ealculations with the different numerical solutions presented in Chapter 15. Yet, the components of Figure 3.4, with the multiple change of the concentration in bottom water and the memoiy of which is preserved over several cycles in the pore water fraction, is not accessible with this rather simple analytical solution. 

By way of example, consider the numerical procedure for the solution of Fick s second law of diffusion in two dimensions... 

A common technique is to deposit a very thin film of radioactive isotopes on a plane surface of a sample, and, after subsequent diffusion anneal, determine the activity of diffusion species as a function of distance from the plane surface. If the thickness of the sample is very much larger than the penetration depth of the tracers, the solid can be considered semi-infinite. Furthermore, if the diffusion is homogenous (e.g. taking place by lattice diffusion), the concentration of the diffusing tracers normal to the plane is through solution of Fick s second law with appropriate boundary conditions given by... 

Lag-time method. Figure 11.18 shows an apparatus for measuring the permeation and sorption of oxygen in polymers flhns. The solution of Fick s second law when the initial concentration of permeant in the film is zero is ... 

In order to calculate D, the equation based on the solution of Fick s second law for the model of very thin film (tracer) diffusing into semi-infinite layer was used ... 

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