Fick diffusion equation

The crystal surface may be regarded as covered with a layer of saturated solution of a definite thickness through which the products have to diffuse. If the actual solvation of the solid proceeds rapidly in comparison to the process of diffusion the rate of solution wfil he essentially that of diffusion, and can accordingly be expressed by the Fick diffusion equation, the rate of solution per unit area of interface being given by... 

The conditions basic to the Fickian sorption were that (1) D is a function of ct only and (2) a constant surface concentration is maintained during sorption. So long as we wishes to retain the Fick diffusion equation as the basis of the discussion, any attempt for the theoretical interpretation of non-Fickian characteristics must abandon either or both of these conditions. In this section we give a brief account of a theory which involves an alternation of condition (1). It is due originally to Crank and Park (1951). 

If diffusion through pores occurs, the Fick diffusion equation can be solved ... 

To repre.scnt the kinetics of drug release in vitro, a theoretical mixlel wa.s derived from the Fick diffusion equation. Thi.s model is based on u numerical method with finite differences (5). 

We assume that the potential of the electrode is perturbed by a small-amplitude sinusoidal potential given by AA = A exp (/wt). Consequently the faradaic current is perturbed by an amount A/ = 1A/ exp jo)t + ). This perturbation induces a concentration change Ac in the layer that is described by the time-dependent Fick diffusion equation as follows... 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

Liquid phase diffusivities are strongly dependent on the concentration of the diffusing component which is in strong contrast to gas phase diffusivities which are substantially independent of concentration. Values of liquid phase diffusivities which are normally quoted apply to very dilute concentrations of the diffusing component, the only condition under which analytical solutions can be produced for the diffusion equations. For this reason, only dilute solutions are considered here, and in these circumstances no serious error is involved in using Fick s first and second laws expressed in molar units. 

Permeability of an FML is evaluated using the Water Vapor Transmission test.28 A sample of the membrane is placed on top of a small aluminum cup containing a small amount of water. The cup is then placed in a controlled humidity and temperature chamber. The humidity in the chamber is typically 20% relative humidity, while the humidity in the cup is 100%. Thus, a concentration gradient is set up across the membrane. Moisture diffuses through the membrane, and with time the liquid level in the cup is reduced. The rate at which moisture is moving through the membrane is measured. From that rate, the permeability of the membrane is calculated with the simple diffusion equation (Fick s first law). It is important to remember that even if a liner is installed correctly with no holes, penetrations, punctures, or defects, liquid will still diffuse through the membrane. 

We can progress from here provided that we can find expressions for the partial derivatives of equation (2.99). Provided that the concentration of supporting electrolyte is sufficiently high that all the potential difference across the interface is accommodated within the Helmholtz layer, then transport of O and R near the electrode will only take place via diffusion (i.e. we can neglect migration). The equation of motion for either O or R is given by the differential form of Fick s equation, as discussed in chapter I ... 

The Stem-Volmer equations discussed so far apply to solutions of the luminophore and the quencher, where both species are homogeneously distributed and Fick diffusion laws in a 3-D space apply. Nevertheless, this is a quite unusual situation in fluorescent dye-based chemical sensors where a number of factors provoke strong departure from the linearity given by equation 2. A detailed discussion of such situations is beyond the scope of this chapter however, the optosensor researcher must take into account the following effects (where applicable) ... 

In the kinetics of formation of carbides by reaction of the metal with CH4, the diffusion equation is solved for the general case where carbon is dissolved into the metal forming a solid solution, until the concentration at the surface reaches saturation, when a solid carbide phase begins to develop on the free surface. If the carbide has a thickness at a given instant and the diffusion coefficient of carbon is D in the metal and Dn in the carbide, Fick s 2nd law may be written in the form (Figure 8.1)... 

Diffusion is quantified by measuring the concentration of the diffusing species at different distances from the release point after a given time has elapsed at a precise temperature. Raw experimental data thus consists of concentration and distance values. The degree of diffusion is represented by a diffusion coefficient, which is extracted from the concentration-distance results by solution of one of two diffusion equations. For one-dimensional diffusion, along x, they are Fick s first law of diffusion ... 

This is known as Fick s second law of diffusion or more commonly as the diffusion equation. In these equations, J is called the flux of the diffusing species, with units of [amount of substance (atoms or equivalent units) m2 s-1], c is the concentration of the diffusing species, with units of [amount of substance (atoms or equivalent units) m-3] at position x (m) after time t (s) D is the diffusion coefficient, units (m2 s 1). 

This is Fick s second law of diffusion, the diffusion equation. 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

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. 

For convenience, let the flux of A within the ash layer be expressed by Fick s law for equimolar counterdiffusion, though other forms of this diffusion equation will give the same result. Then, noting that both (2a dCJdr are positive, we have... 

The diffusion equation in an anisotropic medium is complicated. Based on the definition of the diffusivity tensor, the diffusive flux along a given direction (except along a principal axis) depends not only on the concentration gradient along this direction, but also along other directions. The flux equation is written as F = —D VC (similar to Fick s law F= -DVC but the scalar D is replaced by the tensor D), i.e.. 

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. 

In order to calculate the density of reactant B about A, it is necessary to know by what means the reactants migrate in solution. Under most circumstances, diffusion is a very adequate description (the limitations of and complications to diffusion are discussed in Sect. 6, Chap. 8 Sect. 2 and Chap. 11). In this simple analysis of diffusion, Fick s laws will be used with little further justification, save to note that Fick s second law is identical to the equation satisfied by a random walk function. Hardly a surprising result, because diffusion is a random walk with no retention of information about where the diffusing species was before its current location. In Chap. 3 Sect. 1, the diffusion equation is derived from thermodynamic considerations and the continuity equation (law of conservation of mass). 

The right-hand side of eqn. (9), which is the diffusion equation or Fick s second law, involves two spherically symmetric derivatives of p(r, t). In the general case of three-dimensional space, lacking any symmetry, it can be shown that the Laplacian operator... 

In Chap. 2 and 3, the motion of two reactants was considered and a diffusion equation was derived based upon the equation of continuity and Fick s first law of diffusion (see, for instance, Chap. 2 and Chap. 3, Sect. 1.1). When one reactant (say D) can transfer energy or an electron to the other reactant (say A) over distances greater than the encounter separation, an additional term must be considered in the equation of continuity. The two-body density n (rj, r2, t) decays with a rate coefficient l(r, — r2) due to long-range transfer. Furthermore, if energy is being transferred from an excited donor to an acceptor, the donor molecular excited state will decay, even in the absence of acceptor molecules with a natural lifetime r0. Hence, the equation of continuity (42) becomes extended to include two such terms and is... 

For computing the diffusion parameter, D /ro, Fick s diffusion equation was assumed to be applicable to the system, with D independent of concentration of the diffusing species. Solving Fick s law for a spherical particle, where the external gas pressure is constant gives ... 

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... 

If Fick s equation is used and the diffusivities of the ion pair DAb and DBa are assumed to be equal, as for binary gas diffusivities, predicted exchange rates are the same, regardless of direction of diffusion. Thus, the newer theory taking into account the electric potential is clearly an improvement over the Fick s law approach. However, if an empirical view of this simple theory is used, allowing each diffusivity to assume a... 

The diffusion equation is the partial-differential equation that governs the evolution of the concentration field produced by a given flux. With appropriate boundary and initial conditions, the solution to this equation gives the time- and spatial-dependence of the concentration. In this chapter we examine various forms assumed by the diffusion equation when Fick s law is obeyed for the flux. Cases where the diffusivity is constant, a function of concentration, a function of time, or a function of direction are included. In Chapter 5 we discuss mathematical methods of obtaining solutions to the diffusion equation for various boundary-value problems. 

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