Solution of the reaction-diffusion equations

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 

The dimensionless diffusion coefficient D can be regarded in some sense as the reaction-diffusion equivalent of the flow rate, or the inverse of the residence time, in a CSTR. In fact, we can interpret D as the quotient of the chemical and diffusional timescales 

The unfolding of the hysteresis loop gives rise to a cusp in the D-f ex parameter plane, as shown in Fig. 9.4(b). Also shown there are the cusps for infinite cylinder and spherical geometries. For the latter, multiple stationary states cease for / ei = 0.1129 and 0.1078 respectively, values still smaller than the 5 for the CSTR. 

---

Numerically, one can take advantage of the matrix method of solution of the reaction-diffusion equation using eigenfunctions of the diffusion operator as a basis [lOld, 103, 138, 143, 161, 301]. This method is computationally... 

Note that the reaction-telegraph equation (2.19) differs from the ad hoc HRDE (2.15) by the additional term —xF p) dp/dt) on the left-hand side. It can be shown that solutions of (2.19) converge to solutions of the reaction-diffusion equation (2.3) as T 0 [494]. Traveling wave front solutions for the reaction-telegraph equation have been investigated by several authors [201, 176, 282, 291, 285, 136, 288, 137, 114, 116, 115, 117]. 

Bifurcations. In many situations the uniform solution of the reaction-diffusion equation exists, but is stable only for certain regimes of the parameters. Near to the transition zone between stability and instability of this uniform solution, other nonuniform, small-amplitude, solutions exist as well. If these solutions are stable, their appearance can be considered as bifurcation phenomena. The equation... 

At this point it is useful to make comparisons to the Euler solution of the reaction-diffusion equation. If we measure time in units of At so that t/At —> t, we can write Eq. [2] (dropping the subscript X) as,... 

If this were the only context in which CML models were used, their utility would be severely limited. For values y beyond the stability limit, the Euler method fails and one obtains solutions that fail to represent the solutions of the reaction-diffusion equation. However, it is precisely the rich pattern formation observed in CML models beyond the stability limit that has attracted researchers to study these models in great detail. Coupled map models show spatiotemporal intermittency, chaos, clustering, and a wide range of pattern formation processes." Many of these complicated phenomena can be studied in detail using CML models because of their simplicity and, if there are generic aspects to the phenomena, for example, certain scaling properties, then these could be carried over to real systems in other parameter regimes. The CML models have been used to study chemical pattern formation in bistable, excitable, and oscillatory media." ... 

Fig. 1. Descriptive Equation (7) or (8) for the expansion speed Vn (Oregonator space unit/time unit) of 66 helices is plotted against numerical observations. The ten helices in which coil-coil separation violated the Keener-T son proscription are excluded. The unit of speed is about 1/6 of wave propagation speed at the parameters used (s = 1/50, / = 1.6, equal diffusion coefficients), so the fastest dilations shown here (negative Vn) are only a few percent of propagation speed. This is by far the best fit found to date with polynomials in curvature and twist if the mathematics exactly described these numerical solutions of the reaction-diffusion equations, then all dots would fall on the line of unit slope through the origin. There is a strong tendency to do so, yet many helices still expand or contract several times faster or slower than predicted . A serious problem with all numerical experiments is that no error bars have been determined around the data points.

Solution of the reaction-diffusion equations that describe the spatial behavior of a chemically reacting system is an extremely demanding computational task. For this reason, it is a great advantage to have a simple model with the smallest possible number of variables. The availability of such a model, the Oregonator [69], is one of the reasons for the popularity of the Belousov-Zhabotinsky reaction for theoretical studies of spatial phenomena. 

For one-dimensional systems the equal stability condition obtained from numerical solutions of the reaction diffusion equations agrees with Schlogl s... 

Burnell, J. G., Lacey, A. A., and Wake, G. C. (1983). Steady-states of the reaction-diffusion equations, part 1 questions of existence and continuity of solution branches. J. Aust. Math. Soc., B24, 374-91. 

The results surveyed in the preceding two sections provide a first clue to the origin of chirality chiral patterns can emerge spontaneously in an initially uniform and isotropic medium, through a mechanism of bifurcations far from thermodynamic equilibrium (see Figs. 4 and 5). On the other hand, because of the invariance properties of the reaction-diffusion equations (1) in such a medium, chiral solutions will always appear by pairs of opposite handedness. As explained in Sections III.B and III.C this implies that in a macroscopic system symmetry will be restored in the statistical sense. We are left therefore with an open question, namely, the selection of forms of preferred chirality, encompassing a macroscopic space region and maintained over a macroscopic time interval. 

Therefore, the net effect of a homogeneous, rapid, reversible reaction is to retard the rate of diffusion of solute through the tissue. Solutions to this equation are identical to solutions of the pure diffusion equation (compare Equation 3-31 with Equation 3-52), except that the diffusion coefficient is reduced by a factor equal to the binding constant plus unity. These same equations can be used to evaluate penetration into tissues when more complicated equilibrium expressions are appropriate, by substituting the non-linear equilibrium expression into Equation 3-50 and solving the resulting equation (see [7]). 

When the reaction rate depends on the concentrations of several species, or when more than one reaction is involved, analytical solutions of the pore diffusion equations are impossible or too complicated to be useful. The equations for simultaneous diffusion and reaction of several species can be solved numerically if concentrations at the center are specified, but then many cases must be solved to match given external concentrations. For such cases, a simplified method can be used instead to show the approximate effect of gradients for each species. 

So far we have discussed the probabilistic solution of the convection-diffusion equation only. There are various directions in which a probabilistic approach to PDFs can be extended and generalized. The first direction is to extend it to the case where chemical reactions are taken into account. The next direction would be to allow the velocity field v and the diffusion matrix D to depend on both space x and time t. Another direction for generalization is to analyze initial-boundary problems. 

This electrode is uniformly accessible [4], in that during a reaction the flux (and hence the current density) is the same across the entire disc surface. Also, the system possesses circular symmetry about the z-axis. This greatly simplifies the mathematical description of the hydrodynamics, and allows an analytical solution of the convective-diffusion equation [5]. 

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. 

DAt/ Axf < Ijld, where d is the dimension of the system, this scheme will yield a faithful solution to the reaction-diffusion equation. The grid points need not lie on a cubic lattice and the size of the neighborhood used to... 

By "Chemical waves", we mean a mode of propagation of difference(s) in concentration(s) in which a chemical reaction and the transport of matter by diffusion take part. In a more mathematical statement, we state that a chemical wave is a solution written C(r,t) - which depends on space and time( ) - of the reaction-diffusion equation ... 

The concentrations of the reactants and reaction prodncts are determined in general by the solution of the transport diffusion-migration equations. If the ionic distribution is not disturbed by the electrochemical reaction, the problem simplifies and the concentrations can be found through equilibrium statistical mechanics. The main task of the microscopic theory of electrochemical reactions is the description of the mechanism of the elementary reaction act and calculation of the corresponding transition probabilities. 

Photosensitization of diaryliodonium salts by anthracene occurs by a photoredox reaction in which an electron is transferred from an excited singlet or triplet state of the anthracene to the diaryliodonium initiator.13"15,17 The lifetimes of the anthracene singlet and triplet states are on the order of nanoseconds and microseconds respectively, and the bimolecular electron transfer reactions between the anthracene and the initiator are limited by the rate of diffusion of reactants, which in turn depends upon the system viscosity. In this contribution, we have studied the effects of viscosity on the rate of the photosensitization reaction of diaryliodonium salts by anthracene. Using steady-state fluorescence spectroscopy, we have characterized the photosensitization rate in propanol/glycerol solutions of varying viscosities. The results were analyzed using numerical solutions of the photophysical kinetic equations in conjunction with the mathematical relationships provided by the Smoluchowski16 theory for the rate constants of the diffusion-controlled bimolecular reactions. 

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

Another derivation has been given by Resibois and De Leener. In principle, eqn. (287) can be applied to describe chemical reactions in solution and it should provide a better description than the diffusion (or Smoluchowski) equation [3]. Reaction would be described by a spatial- and velocity-dependent term on the right-hand side, — i(r, u) W Sitarski has followed such an analysis, but a major difficulty appears [446]. Not only is the spatial dependence of the reactive sink term unknown (see Chap. 8, Sect. 2,4), but the velocity dependence is also unknown. Nevertheless, small but significant effects are observed. Harris [523a] has developed a solution of the Fokker—Planck equation to describe reaction between Brownian particles. He found that the rate coefficient was substantially less than that predicted from the diffusion equation for aerosol particles, but substantially the same as predicted by the diffusion equation for molecular-scale reactive Brownian particles. 

However, the calculation of NAO( f)> which is the matrix composition at explicit solution to the coupled diffusion equations of the components before and behind the reaction front. Since the transport coefficients in these mixed crystals depend on local composition, one therefore cannot find analytical solutions. Only if the A2+ ions are almost immobile (DA< DB) do we have NAO( F) =N 0. This specific case has been discussed in the literature [H. Schmalzried (1984)]. 

We want now to see how this state of affairs is affected by the chiral perturbation of our reaction-diffusion equations [term in M in equation (29)]. To this end we follow the lines of imperfection theory (Section II.C) and expand the variables and parameters in series around X = X,. We also set the frequency fl of the solution to be identical to the external frequency w, and assume that o> is close to the linearized intrinsic frequency, ft, in the absence of the field ... 

When van der Waals and double-layer forces are effective over a distance which is short compared to the diffusion boundary-layer thickness, the rate of deposition may be calculated by lumping the effect of the particle-collector interactions into a boundary condition on the usual convective-diffusion equation. This condition takes the form of a first-order irreversible reaction (10, 11). Using this boundary condition to eliminate the solute concentration next to the disk from Levich s (12) boundaiy-kyersolution of the convective-diffusion equation for a rotating disk, one obtains... 

---