Boundary conditions planar

X and vectors are the actual and the running coordinates over the planar contact area respectively. The boundary conditions are ... 

Computer simulations of bulk liquids are usually performed by employing periodic boundary conditions in all three directions of space, in order to eliminate artificial surface effects due to the small number of molecules. Most simulations of interfaces employ parallel planar interfaces. In such simulations, periodic boundary conditions in three dimensions can still be used. The two phases of interest occupy different parts of the simulation cell and two equivalent interfaces are formed. The simulation cell consists of an infinite stack of alternating phases. Care needs to be taken that the two phases are thick enough to allow the neglect of interaction between an interface and its images. An alternative is to use periodic boundary conditions in two dimensions only. The first approach allows the use of readily available programs for three-dimensional lattice sums if, for typical systems, the distance between equivalent interfaces is at least equal to three to five times the width of the cell parallel to the interfaces. The second approach prevents possible interactions between interfaces and their periodic images. 

Simulations of water in synthetic and biological membranes are often performed by modeling the pore as an approximately cylindrical tube of infinite length (thus employing periodic boundary conditions in one direction only). Such a system contains one (curved) interface between the aqueous phase and the pore surface. If the entrance region of the channel is important, or if the pore is to be simulated in equilibrium with a bulk-like phase, a scheme like the one in Fig. 2 can be used. In such a system there are two planar interfaces (with a hole representing the channel entrance) in addition to the curved interface of interest. Periodic boundary conditions can be applied again in all three directions of space. 

The interface free energy per unit area fi,u is taken to be that of a planar interface between coexisting phases. Considering a solution v /(z) that minimizes Eq. (5) subject to the boundary conditions vj/(z - oo) = - v /coex, v /(z + oo) = + vj/ oex one finds the excess free energy of a planar interface ... 

Darrieus and Landau established that a planar laminar premixed flame is intrinsically unstable, and many studies have been devoted to this phenomenon, theoretically, numerically, and experimentally. The question is then whether a turbulent flame is the final state, saturated but continuously fluctuating, of an unstable laminar flame, similar to a turbulent inert flow, which is the product of loss of stability of a laminar flow. Indeed, should it exist, this kind of flame does constitute a clearly and simply well-posed problem, eventually free from any boundary conditions when the flame has been initiated in one point far from the walls. 

Stationary (i.e. for dA/ dz = 0) localized solutions to Eq.(3.2) represent nonlinear modes in the planar waveguide and may be found in an analytical form via matching the partial solutions of Eq.(3.2) at the core/cladding boundary. The partial solutions are Jacobi elliptical function in the core and 2l rccosh — )E]/E in the cladding (the functional dependence similar to a fundamental soliton in a uniform nonlinear medium). Here is a parameter which depends on the boundary conditions. Contrary to the modes of a linear waveguide, the transverse profile of a nonlinear mode depends on the power in the mode. 

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 ... 

As in the previous chapter, the semi-irrfinite diffusion at a planar electrode is considered, where the adsorption is described by a linear adsorption isotherm. The modeling of reaction (2.173) does not require a particular mathematical procedure. The model comprises equation (1.2) and the boundary conditions (2.148) to (2.152) that describe the mass transport and adsorption of the R form. In addition, the diffusion of the O form, affected by an irreversible follow-up chemical reaction, is described by the following equation ... 

E and E are identical always, so we will call them E from now on. One correction for the planar boundary condition will transform this curve to the proper RTD. 

When we do the step experiment by switching from one fluid to the other we obtain the curve (see Chapter 11), from which we should be able to find the F curve. However, this input always represents the flux input, while the output can be either planar or flux. Thus we only have two combinations, as shown in Fig. 15.8. With these two combinations of boundary conditions their equations and graphs are given in Eq. 6 and Fig. 15.9. 

While a proper aiming of the atom-probe can be experimentally determined, information on field lines and on equipotential lines is difficult to derive with an experimental method because of the small size of the tip. Yet this information is needed for interpreting quantitatively many experiments in field emission and in field ion emission. We describe here a highly idealized tip-counter electrode configuration which may be useful for describing field lines at a short distance away from the tip surface but far enough removed from the lattice steps of the surface. The electrode is assumed to consist of a hyperboloidal tip and a planar counter-electrode.30 In the prolate spheroidal coordinates, the boundary surfaces correspond to coordinate surfaces and Laplace s equation is separable, so that the boundary conditions can be easily satisfied. 

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... 

Note that we have assumed the vacancies to be ideally diluted. We can then introduce a perturbation of the planar boundary, z = A +0(x,y,t), and define °(x,y) = Cartesian coordinates perpendicular to z. In this way, the morphological stability becomes a two-dimensional problem. Since we also assume that local equilibrium prevails at both interfaces (surfaces), the boundary conditions are... 

Three conditions are required for a complete solution to the problems illustrated in Figs. 3.10 and 16.1. If the grain boundary remains planar, dL/dt in Eq. 16.2 must be spatially uniform—the Laplacian of the normal surface stress under quasisteady-state conditions must then be constant ... 

The planar grain-boundary condition given by Eq. 16.18 is satisfied if... 

The planar, cylindrical, and spherical forms of Fick s second law, and combinations of those forms, are sufficient to describe diffusion to most microelectrode geometries in use today. Just as was illustrated in Chapter 2, the appropriate form of Fick s second law is solved, subject to the boundary conditions that describe a given experiment, to provide the concentration profile information. The sought-after current-time or current-voltage relationship is then obtained by evaluating the flux at the electrode surface. 

In single step voltammetry, the existence of chemical reactions coupled to the charge transfer can affect the half-wave potential Ey2 and the limiting current l. For an in-depth characterization of these processes, we will study them more extensively under planar diffusion and, then, under spherical diffusion and so their characteristic steady state current potential curves. These are applicable to any electrochemical technique as previously discussed (see Sect. 2.7). In order to distinguish the different behavior of catalytic, CE, and EC mechanisms (the ECE process will be analyzed later), the boundary conditions of the three processes will be given first in a comparative way to facilitate the understanding of their similarities and differences, and then they will be analyzed and solved one by one. The first-order catalytic mechanism will be described first, because its particular reaction scheme makes it easier to study. 

Asymptotic Conditions. As a model problenj we consider a single cell containing the origin of coordinates r = 0 and subject to a uniform external electric field set up by a pair of parallel planar electrodes at a large distance from the cell. Letting z denote the direction perpendicular to the electrodes and E be the electric field set up by the electrodes in the absence of the cell, we have the asymptotic boundary conditions ... 

This equation is the same as for a planar electrode and the boundary conditions are of the same form. Thus the method of solution is the same. The result is... 

Cyclic voltammetry at planar electrodes The boundary conditions are... 

The equations for a planar electrode with only O present initially are the same as for the potential step, (10.3), as well as boundary conditions (10.4). The differences reside in the choice of the last boundary condition. To this end we need to know the concentrations of O and R on the electrode surface, [O] and [R] respectively. 

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