Reaction-diffusion equation boundaries

A solution of the reaction-diffusion equation (9.14) subject to the boundary condition on the reactant A will have the form a = a(p,r), i.e. it will specify how the spatial dependence of the concentration (the concentration profile) will evolve in time. This differs in spirit from the solution of the same reaction behaviour in a CSTR only in the sense that we must consider position as well as time. In the analysis of the behaviour for a CSTR, the natural starting point was the identification of stationary states. For the reaction-diffusion cell, we can also examine the stationary-state behaviour by setting doi/dz equal to zero in (9.14). Thus we seek to find a concentration profile cuss = ass(p) which satisfies... 

Up to this point, the treatments have involved reactions for which the discrete form of the reaction-diffusion equations involve only terms in concentration of the species to which the discrete equation applies. That is, if there were two substances involved, O and R as above, then the discrete equation at a point i had terms only in C 0 for species O, and only C R for species R. This made it possible to use the Thomas algorithm to reduce a system like (6.27) to (6.28), treating the two species systems separately. They then get coupled through the boundary conditions. 

The reaction-diffusion equation and boundary conditions for this case are... 

Df = D is the mutual diffusion coefficient, and Uf r) is the interaction potential assumed to be zero for the process (9.117). Equation (9.121) should be solved with the uniform initial condition, p r, 0) = 1, and the reflecting boundary condition at encounter. An equivalent formulation defines P(t) in terms of the pair survival probability [335, 340], a function of time and initial separation, which also satisfies the reaction-diffusion equation but with the adjoint operator (.Sf = in the absence of interaction). 

Extraction of quantitative chemical information from SECM requires a mathematical model of the interaction of the tip and substrate. Such modeling typically involves numerical solution of a reaction-diffusion equation with the boundary conditions appropriate to the interfacial kinetics. Simulation of SECM experiments is computationally much more demanding than for standard electrochemical experiments (discussed in Chapter 1.3). This is because diffusion in at least two dimensions must be considered and the discontinuity in the boundary condition between the tip metal and insulating sheath necessitates a fine mesh. 

We now concentrate attention on the reaction-diffusion system (2.3.1). The system dimension and size, geometrical form, and boundary conditions imposed are not specified at first. In particular, the theory to be developed below applies to situations (A) - (C) equally well. The formulation goes quite in parallel with that in Sect. 2.2. We express the reaction-diffusion equations in terms of u r, t), the space-time dependent deviation from the uniform steady solution. Yq ... 

If one tries to develop a perturbation theory proceeding directly from the reaction-diffusion equations this meets with serious difficulties. They arise because translation and rotation perturbation modes for the spiral wave are not spatially localized. We bypass such difficulties by using the quasi-steady approximation formulated in the previous section. In this approximation the trajectory of the tip motion can be calculated by solving a system of ordinary differential equations which depend only on the local properties of the medium in the vicinity of the tip. The perturbations which originate outside a small neighbourhood of the end point propagate quickly to the periphery and do not influence the motion of the tip. The evolution of the entire curve can then be calculated in the WR approximation using the known trajectory of the tip motion as a dynamic boundary condition. 

All the simulations reported in this chapter are fully resolved. A first-order Euler method is used for time-stepping the equations. The Laplacian terms in the reaction-diffusion equations are approximated with a 9-point finite-difference formula, which to leading order, eliminates the underlying 4-fold symmetry of square grid [13]. (For spiral waves in excitable media, anisotropies in the grid have a far greater effect on solutions than do anisotropies in the boundaries.)... 

As pointed out in the previous section, the spatially extended open Couette flow reactor [27-33] provides a practical implementation of an effectively one-dimensional reaction-diffusion system with an external concentration gradient imposed from the boundaries. With the specific motivation to provide theoretical and numerical support for the recent experimental observations of sustained dissipative structures in the Couette flow reactor, we will consider the standard reaction-diffusion equation ... 

This formulation assumes that the continuum diffusion equation is valid up to a distance a > a, which accounts for the presence of a boundary layer in the vicinity of the catalytic particle where the continuum description no longer applies. The rate constant ky characterizes the reactive process in the boundary layer. If it approximated by binary reactive collisions of A with the catalytic sphere, it is given by kqf = pRGc(8nkBT/m)1 2, where pR is the probability of reaction on collision. 

A related technique is the current-step method The current is zero for t < 0, and then a constant current density j is applied for a certain time, and the transient of the overpotential 77(f) is recorded. The correction for the IRq drop is trivial, since I is constant, but the charging of the double layer takes longer than in the potential step method, and is never complete because 77 increases continuously. The superposition of the charge-transfer reaction and double-layer charging creates rather complex boundary conditions for the diffusion equation only for the case of a simple redox reaction and the range of small overpotentials 77 [Pg.177]

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 ... 

For modeling of the electrode reaction coupled with adsorption of both forms of the redox couple (2.146), the diffusion equations (1.2) and (1.3) have to be solved for conditions given by (2.148) to (2.152) completed with the following boundary conditions ... 

The mathematical difficulty increases from homogeneous reactions, to mass transfer, and to heterogeneous reactions. To quantify the kinetics of homogeneous reactions, ordinary differential equations must be solved. To quantify diffusion, the diffusion equation (a partial differential equation) must be solved. To quantify mass transport including both convection and diffusion, the combined equation of flow and diffusion (a more complicated partial differential equation than the simple diffusion equation) must be solved. To understand kinetics of heterogeneous reactions, the equations for mass or heat transfer must be solved under other constraints (such as interface equilibrium or reaction), often with very complicated boundary conditions because of many particles. 

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. 

Figure 22.3 One-dimensional concentration profiles at steady-state calculated from the diffusion/advec-tion/reaction equation (Eq. 22-7) for different parameter values D (diffii-sivity), x (advection velocity), and kr (first-order reaction rate constant). Boundary conditions at x = 0 and x - L are C0 and CL, respectively. Pe = 7. vx ID is the Peclet Number, Da = Dk/v] is the Damkohler Number. See text for further explanations.

Equation (10) is a diffusion equation which applies equally well for matter or heat. The solution of this equation has been studied by many workers. As with differential equations in general, one arbitrary constant is required for each derivative. Since the diffusion equation has partial derivatives, the arbitrary constants have to be functions of the variable which is not involved in the derivative. The diffusion equation requires two functions of time at fixed values of space coordinates (the boundary conditions) and one function of distance at a given time (the initial condition). These have already been established [eqns. (3)—(5)]. It is possible to proceed to solve the diffusion equation for p(r,t) now and then calculate the particle current of B towards each A reactant and so determine the rate of reaction. 

When the activation process is comparable with or slower than the rate of approach of reactants to form encounter pairs, it is no longer satisfactory to say that the reactants can not co-exist within a distance R of one another. Because the rate of reaction, /eact, of the activation process is finite, so too is the lifetime (and hence concentration) of encounter pairs non-zero. The inner boundary condition, which describes reaction of A and B together in the diffusion analysis, is unsatisfactory. Collins and Kimball [4] suggested an alternative boundary condition and the remainder of this section analyses their work following Noyes [5]. Firstly, the boundary condition is developed and then included in the diffusion equation analysis to obtain the density distribution. Finally, the rate coefficient is obtained. 

Wilemski and Fixman [51] have suggested that it is intrinsically more satisfactory to treat chemical reaction by means of a sink term included in the diffusion equation than as a boundary condition imposed on the density distribution. They recommended writing the diffusion equation as... 

When there are two or more reactants diffusing throughout space, the motion of each reactant influences that of all the others due to the solvent being squeezed from between the approaching reactants. The effect of this hydrodynamic repulsion on the rate of a diffusion-limited reaction was discussed in Chap. 8, Sect. 2.5. In this section, this discussion is amplified. First, the nature of the hydrodynamic repulsion is discussed further and then a general diffusion equation for many particles is derived. The two-particle diffusion equation is selected and solved subject to the usual Smoluchowski initial and boundary conditions to obtain the rate coefficient. Finally, this is compared with the rate coefficients in the absence of hydrodynamic repulsion and from experiments. 

Zwanzig s diffusion equation [444], eqn. (211), can be reduced to the stochastic equation used by Clifford et al. [442, 443] [eqn. (183)] to describe the probability that N identical reactant particles exist at time t (see also McQuarrie [502]), Let us consider the case where U — 0, with a static solvent, for a constant homogeneous diffusion coefficient. This is a major simplification of eqn. (211). Now, rather than represent the reaction between two reactants k and j by a boundary condition which requires the... 

In order to solve the diffusion equations for the species taking part in the electrode reaction, it is first necessary to specify (a) the initial conditions, (b) the boundary conditions and (c) the relationship between C0/CR at the electrode surface and the electrode potential. 

---