Infinite boundary condition

At the other boundary, bulk concentration of A must be maintained at some finite distance from the electrode, while the concentration of B will be zero at the same point. This distance may be regarded as the diffusion layer thickness. In terms of the simulation, the establishment of the semi-infinite boundary condition requires the determination of the number of volume elements making up the diffusion layer. This will be a function of the number of time iterations that have taken place up to that point in the simulation. At any time in the physical experiment, the diffusion layer thickness is given by 6(Dt)1/2. This rule of thumb may be combined with Equation 20.7 to calculate Jd, the number of volume elements in the diffusion layer ... 

The methodology developed in section 3.1.2 can be used for semi-infinite boundary conditions, also. The procedure for solving boundary value problems in semi-infinite domain is as follows ... 

If charge diffusion is significantly slower so that the distance of charge transport, L, (=2(Dt) ) is clearly smaller than the thickness of the lamina, 5, the electrochemical response will be equivalent to that recorded when reactants freely diffuse from an infinite volume of solution to the electrode. This situation, often termed as thick-layer behavior, corresponds to semi-infinite boundary conditions, and concentration profiles such as that shown in Figure 2.5c are then predicted. Accordingly, Cottrell-type behavior is observed, for instance, in cyclic voltammetry (CV) and chronoamperometry (CA). In this last technique, a constant potential sufficiently cathodic for ensuring diffusion control in the reduction of Ox to Red is applied. The resulting current-time (i-t) curves should verify the Cottrell equation presented in the previous chapter (Equation (1.3)). 

For a given electrochemical system to be described by equations involving semi-infinite boundary conditions, the cell wall must be at least five diffusion layer thicknesses away from the electrode. For a substance with D = 10 cm /s, what distance between the working electrode and the cell wall is required for a 100-s experiment ... 

The methods described in Chapters 5 to 10 generally employ conditions featuring a small ratio of electrode area, A, to solution volume, V. These allow the experiments to be carried out over fairly long time periods without appreciable changes of the concentrations of the reactant and the products in the bulk solution, and they allow the semi-infinite boundary condition (e.g., Cq(x, 0 = C q as x oo) to be maintained over repeated trials. For example, consider a 5 X 10 M solution of O with V = 100 cm and A = 0.1 cm. Assume that during 1 hour of experimentation an average current of about 100 fiA flows (i.e., current density, 7, of 1 mA cm ). During this time period only 0.36 C of electricity will be passed, and the bulk concentration of electroactive species will have decreased by less than 1%. 

Note that in this case the semi-infinite boundary condition used for the analogous experiment in Section 5.2 has been replaced by the condition for Cq at L Solution of these equations by the Laplace transform method yields (63) ... 

Transient Semi-Infinite Diffusion The simplest transient diffusions problems are generally those that involve semi-infinite or infinite boundary conditions. Consider, for example, the situation illustrated in Figure 4.6, which represents diffusion of a substance from a surface into a semi-infinite medium. 

In the real world, of course, no medium actually extends to infinity. However, infinite or semi-infinite boundary conditions are fully appropriate for many finite situations in which the length scale of the diffusion is much smaller than the thickness of the material. In such cases, the material appears infinitely thick relative to the scale of the diffusion—or, in other words, the diffusion process never reaches the far boundaries of the material over the relevant time scale of interest. Since typical length scales for solid-state diffusion processes are often on the micrometer scale, even diffusion into relatively thin films can often be treated using semi-infinite or infinite boundary condition approaches. Semi-infinite and infinite tfansient diffusion has therefore been widely applied to understand many real-world kinetic processes—everything from transport of chemicals in biological systems to the doping of semiconductor films to make integrated circuits. 

Pulse methods may be applied to both foil and bulk specimens provided that a suitable mathematical model is used in the data analysis. However, most applications of the potentiostatic pulse technique have involved just one surface of a metal specimen. The technique is suitable for bulk specimens since only a single surface need be exposed to the electrolyte, so it offers a practical advantage over permeation methods in that pinholes and sealing problems associated with thin membranes are avoided. Also, diffusion in such specimens can be treated in terms of a semi-infinite boundary condition, which is mathematically appealing. 

The solutions of such partial differential equations require infomiation on the spatial boundary conditions and initial conditions. Suppose we have an infinite system in which the concentration flucPiations vanish at the infinite boundary. If, at t = 0 we have a flucPiation at origin 5C(f,0) = AC (f), then the diflfiision equation... 

We use the sine series since the end points are set to satisfy exactly the three-point expansion [7]. The Fourier series with the pre-specified boundary conditions is complete. Therefore, the above expansion provides a trajectory that can be made exact. In addition to the parameters a, b and c (which are determined by Xq, Xi and X2) we also need to calculate an infinite number of Fourier coefficients - d, . In principle, the way to proceed is to plug the expression for X t) (equation (17)) into the expression for the action S as defined in equation (13), to compute the integral, and optimize the Onsager-Machlup action with respect to all of the path parameters. 

Maxwell obtained equation (4.7) for a single component gas by a momentum transfer argument, which we will now extend essentially unchanged to the case of a multicomponent mixture to obtain a corresponding boundary condition. The flux of gas molecules of species r incident on unit area of a wall bounding a semi-infinite, gas filled region is given by at low pressures, where n is the number of molecules of type r per... 

Example The equation dQ/dx = (A/f/)(3 6/3f/ ) with the boundary conditions 0 = OatA.=O, y>0 6 = 0aty = oo,A.>0 6=iaty = 0, A.>0 represents the nondimensional temperature 6 of a fluid moving past an infinitely wide flat plate immersed in the fluid. Turbulent transfer is neglected, as is molecular transport except in the y direction. It is now assumed that the equation and the boundary conditions can be satisfied by a solution of the form 6 =f y/x ) =j[u), where 6 =... 

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,... 

For axial dispersion in a semi-infinite bed with a linear isotherm, the complete solution has been obtained for a constant flux inlet boundary condition [Lapidiis and Amundson,y. Phy.s. Chem., 56, 984 (1952) Brenner, Chem. Eng. Set., 17, 229 (1962) Coates and Smith, Soc. Petrol. Engrs. J., 4, 73 (1964)]. For large N, the leading term is... 

Figure 2 Periodic boundary conditions in two dimensions. The central simulation cell is replicated infinitely in both directions.

In the typical setup, the lipids are arranged in a bilayer, with water molecules on both sides, in a central simulation cell, or box, which is then replicated by using three-dimensional periodic boundary conditions to produce an infinite multilamellar system (Fig. 2). It is important to note that the size of the central cell places an upper bound on the wavelength of fluctuations that can be supported by the system. 

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