18740
J. Phys. Chem. 1996, 100, 18740-18748
Ionic Autocatalytic Reaction Fronts in Electric Fields D. Sˇ nita,† H. Sˇ evcˇ´ıkova´ ,† M. Marek,*,† and J. H. Merkin‡ Department of Chemical Engineering, Prague Institute of Chemical Technology, Technicka 5, 166 28 Prague 6, Czech Republic, and Department of Applied Mathematics, UniVersity of Leeds, Leeds, LS2 9JT, U.K. ReceiVed: May 8, 1996; In Final Form: July 23, 1996X
The travelling waves of the front type that are initiated in a system in which there is an ionic autocatalytic reaction, with a quadratic rate law, are considered when a constant current I is applied. These waves are seen to depend on both the current and the parameter δB, the ratio of diffusion coefficients of autocatalyst and substrate. For δB > 1 there is a transition from kinetic front waves to Kohlrausch electrophoretic fronts as I is varied. For δB < 1, both kinetic waves and Kohlrausch fronts are seen as well as an additional type of wave where a much enhanced reaction takes place.
Introduction It is well established that the coupling of chemical reaction with diffusion can, under suitable initiation conditions, give rise to the propagation of stimuli as travelling waves that can be in the form of either fronts or pulses. These reaction-diffusion waves are a fundamental part of many chemical and biological systems, and a detailed consideration of their properties is a necessary prerequisite for understanding the various complex processes involved in such systems (see, for example, refs 1-4). Arguably the most basic of these structures is the propagating front wave, which converts the reacting medium from one (unreacted) state at its front to another (fully reacted) state at its rear. These front waves are usually modeled by some simple autocatalytic kinetics, typically quadratic or cubic or a mixture of the two.5-8 The viability of such models has been clearly demonstrated experimentally for a variety of chemical systems.9-16 Also, autocatalytic reactions have been shown to play an important role in (the kinetically more complex) excitable media such as the BZ reaction,17-20 where pulse waves are observed. Many chemical and biochemical systems particularly, though not exclusively, in the solution phase have ionic components, and it is also well understood that the application of electric fields to such systems can set up electrochemical or Kohlrausch fronts.21,22 These fronts, which, unlike autocatalytic front waves, depend on the different diffusivities of the ionic components for their formation, are used extensively in various electrophoretic separation processes, recently reviewed in detail in ref 23. In the modeling of these electrophoretic fronts, the chemical reactions between the ionic components are usually ignored, or, if included, assumed to be at chemical equilibrium.23 An ionic chemical system where the behavior of front waves in an applied electric field has been investigated experimentally24,25 is one based on the arsenous acid-iodate reaction.26,27 Here the propagation velocity of the front was observed to increase or decrease depending on whether the key ionic species was driven by the electric field in the same direction as or in the opposite direction of the natural direction of propagation of the reaction front. The electroseparation of individual ionic species (due to different diffusivities) across the wave front was found to play a determining role in changing the course of the overall reaction. This either led to another final stationary state (different from the stationary state reached without the electric field) being established at the rear of the wave or could lead to the reaction being inhibited completely as the field strength X
Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01330-5 CCC: $12.00
increased. Also, the application of the electric field was seen to have a profound effect on the shape of the front, in particular on the extent of the reaction zone. This is, as far as we are aware, the only study into the effects of electric fields on reaction-diffusion front waves. No comprehensive theoretical analysis of this aspect has previously been undertaken, though there has been a considerable amount of work (both experimental and numerical simulations) on the effects of electric fields on excitable (BZ) media, where pulse waves arise and where the underlying kinetics are somewhat different and more complex. (For recent work see refs 28-31.) Also, in this medium the enhanced migration of key ionic species gives rise to the acceleration or deceleration of these pulse waves (depending on field orientation), and with further enhanced countermigration the wave can be fully extinguished. The electroseparation of the ionic species changes their concentrations across the reaction zone and can lead to wave splitting, i.e. to the generation of new waves from the rear of the one originally exposed to the electric field. An important question that needs to be considered is how the separation effects of the electric field on a mixture of reacting ionic chemical species will modify the formation and propagation of reaction-diffusion waves. Here we address this question and present the results of an analysis into the effects of an applied electric field on front waves for a typical prototype chemical system consisting of three ionic components with an autocatalytic reaction, with a quadratic rate expression, taking place between two of the species. Model We assume that we have a substrate A+ and autocatalyst B+ reacting via
A+ + B+ f 2B+
rate kcAcB
(1)
(where cA and cB are the concentrations of A+ and B+, respectively, and k is the rate constant). We also assume that we have a further ionic species C- present in the system, though not taking part in the reaction. (The ionic charges on the three species can be reversed.) In analogy with previous experimental work24,25 we consider a situation where the substrate A+ is present initially in the reactor at uniform concentration cA0 and that some autocatalyst B+ is introduced locally into the system. A reaction-diffusion travelling wave is then allowed to develop, and when this has become fully formed (and the initial transients have died out), the electric field is switched on. We limit our © 1996 American Chemical Society
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J. Phys. Chem., Vol. 100, No. 48, 1996 18741
attention to one-dimensional geometry and take the reactor to be sufficiently long for the effects of its ends to be neglected. The governing equations are derived using local mass balances for the concentrations of each of the three species based on the general treatment given previously in ref 32. Source terms arise in these equations from the chemical reaction, and mass transport is assumed to take place both by molecular (Fickian) diffusion and by the migration of the ionic species in the electric field. The Nernst-Planck equation is used to describe the mass transfer, with Gauss’s law of electrostatics being used for the charge density. The formulation is completed by assuming local electroneutrality and by utilizing the fact that the electric current can be maintained at a constant value i (say) in a spatially one-dimensional system. The first step is to make the equations dimensionless, as this reduces the number of parameters involved. To do so we introduce scales for time, length, electric current, and electric field, t0 ) 1/kcA0, l0 ) (DA/kcA0)1/2, i0 ) F(DAkcA0)1/2cA0, e0 ) (RT/F)(kcA0/DA)1/2, respectively, where DA is the diffusion coefficient of A+, T is the absolute temperature (taken to be constant), R is the gas constant, and F is Faraday’s constant. Then, on writing
cA ) cA0a, cB ) cA0b, cc ) cA0c, X ) x/l0, τ ) t/t0, I ) i/i0, E ) e/e0 (2) and considering only univalent electrolytes, we arrive at the dimensionless equations (following refs 23, 32, 33 for example)
∂a ∂2a ∂ - (aE) - ab ) ∂τ ∂X2 ∂X
(3)
∂2b ∂ ∂b ) δB 2 - δB (bE) + ab ∂τ ∂X ∂X
(4)
The (dimensionless) electric field E is given by
∂a ∂b 1 E ) I + (1 - δC) + (δB - δC) G ∂X ∂X
[
]
(5)
where
G ) (1 + δC)a + (δB + δC)b is the (dimensionless) conductivity, I is the (dimensionless) constant applied electric current, and δB ) DB/DA, δC ) DC/ DA, DB and DC being the diffusion coefficients of B+ and C-, respectively. The concentration of C- can be calculated, using the electroneutrality assumption,23,32,33 from
c)a+b
(6)
The initial condition is the corresponding travelling wave solution to eqs 3-5 without the electric field (i.e. with E set to zero) propagating in the positive X -direction (V > 0). We also assume that the concentrations remain uniform at large distances. A consideration of eqs 3-5 suggests that δB and I are the important parameters to consider, changes in the values of which can make changes in the qualitative nature of the solution; the parameter δC plays a much more passive role (provided it remains of order unity). Furthermore, we expect there to be qualitatively different solutions depending on whether δB > 1 or δB < 1. This led us to obtain numerical solutions to eqs 3-5 for values of δB representative of these two cases and for
a range of values of I (both positive and negative). Throughout we took, for simplicity, δC ) 1, i.e. DC ) DA. Before discussing these solutions, however, we note that in the case when δB ) 1, a + b ) 1 for all X and τ. In this case the system evolves rapidly from the standard Fisher-Kolmogorov wave2,5,34 travelling with (dimensionless) speed V ) 2 to one with the same wave structure but now travelling with the speed V ) 2 + I/(1 + δC). Note that this wave will propagate in the same direction as the original wave if I > - 2(1 + δC), otherwise it will propagate in the opposite direction to it. Travelling Waves Prior to the electric field being switched on, the travelling wave that is initiated is propagating with (dimensionless) speed 2xδB into a region in which a ) 1, b ) 0, leaving behind a region where a ) 0, b ) 1 at its rear35-38 (and see also the recent and very readable review article by Showalter34). For δB * 1 the application of the electric field causes this wave to evolve to a travelling wave that is essentially different from the original one, having both a different wave structure and velocity. To determine what possible new wave structures can arise and to characterize the numerical solutions of eqs 3-5 described below, we need to consider the corresponding travelling wave equations. These are derived by introducing the travelling coordinate y ) X -Vτ, where V is the (constant) wave speed, and then assuming that a and b are now functions only of y. This leads to the ordinary differential equations
a′′ + Va′ - (aE)′ - ab ) 0
(7)
δBb′′ + Vb′ - δB(bE)′ + ab ) 0
(8)
with E being given by (5) and with differentiation now with respect to y. The boundary conditions are that the concentrations are uniform at both the front and rear of the wave, with
a f as, b f 0 as y f ∞
(9)
a f 0, b f bs as y f -∞
(10)
If the wave is propagating into the unreacted part of the system (V > 0), then as ) 1 and bs is some constant concentration of B+ at the rear of the wave to be determined and which will depend on the parameters of the system. If the wave is propagating back into the reacted part of the system (V < 0), then bs )1 and as is some constant concentration of A+ to be determined. It has already been established35-38 that, for front waves in purely quadratic autocatalytic systems (i.e. without the electric field), there is a continuous spectrum of possible wave speeds available from the solution of the corresponding travelling wave equations (eqs 7-10 with E set to zero), bounded below by some minimum value. The wave travelling with this minimum wave speed is stable. Also, for sufficiently local initial conditions, as is the case here, the travelling wave that is initiated is the one that travels with this minimum speed. We can expect a similar situation to hold here for waves propagating into the unreacted part of the system. To determine the minimum wave speed, Vmin, in the present case we need to consider the behavior of the solution of eqs 7-10 near the front of the wave, where b is small and a = 1. Then, by linearizing eq 8 and looking for a solution of the resulting equation in the form e-λy, the requirement of equal roots for the quadratic equation for λ determines Vmin as
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18742 J. Phys. Chem., Vol. 100, No. 48, 1996
Vmin ) 2xδB +
IδB 1 + δC
(11)
The requirement that Vmin > 0, so that the wave is propagating into unreacted medium, then gives the condition that
I > - 2(1 + δC)/xδB
(12)
By adding eqs 7 and 8, integrating once, and applying conditions 9 and 10, we obtain a relation between bs and the wave speed V, namely,
bs ) 1 +
(δB - 1)δCI V(1 + δC)(δB + δC)
(13)
Using expression 11 for V then gives a value for bs for the minimum speed waves as
bs )
2xδB(1 + δC)(δB + δC) + I(δB2 + 2δBδC - δC) (δB + δC) xδB(2(1 + δC) + I xδB)
(14)
Furthermore, if we integrate eq 7 and apply boundary conditions 9 and 10, we obtain the inequality
V-
∞ I - ∫-∞ab dy > 0 (1 + δC)
(15)
Conditions 12 and 15 as well as the requirement that bs > 0 then give, with (14), the conditions for the existence of minimum speed travelling waves as
δB > 1, I > -
δB < 1, -
2xδB(1 + δC)(δB + δC) δB2 + 2δBδC - δC
2(1 + δC)
xδB
1. Here we took δB ) 2 as a representative value for this case. Condition 16 becomes I > -12x2/7 ) -2.4244, which corresponds to the condition bs > 0; note that Vmin > 0 requires I > -2x2 ) -2.8284. The effects of a positive current (I ) 10) on the propagating front is illustrated in Figure 1. The evolution of the concentration profiles a, b and the reaction rate ab in the initial time interval up to τ ) 20, at which point the electric current is switched on, is shown by the broken lines in this figure. The front initially propagates in the positive X-direction and continues to propagate in the same direction after the electric field has been switched on. However, both the shape (now shown by the full lines) and the propagation velocity of the front are changed; this is different from the case with equal diffusivities for A+ and B+ (i.e. δB ) 1), where it is only the propagation velocity that is affected by the application of the electric field. We now consider what happens after the current is applied. In the region to the rear of the wave the conductivity G is higher than it is in the region ahead of the wave, as the mobility of the autocatalyst B+ is twice that of the substrate A+ (since δB )
2). Thus the intensity of the electric field E for a constant current I is higher behind the front than ahead of it. In the present case, the ratio of conductivities G behind and ahead of the front is 3/2, and thus the ratio of the electric fields is 2/3. (This follows from eq 5 with ∂a/∂X ) 0 and ∂b/∂X ) 0.) Now the migration fluxes of reactants A+ and B+ are proportional to aE and δBbE, respectively, and therefore, the migration flux of B+ behind the front is higher than the migration flux of A+ ahead of it, the ratio of these fluxes being 4/3 in the present case. As a result, component B+ accumulates in the region behind the front, and so its concentration increases there. The extent of the region where there is a higher concentration of B+ increases with time as the front propagates away from the position it was in when the current was switched on. In general, the increase in the concentration bs above unity is directly proportional to the difference between the flux of B+ behind the front and the flux of A+ ahead of it and inversely proportional to the velocity of propagation. The velocity of the propagating wave increases after the current is switched on because both components A+ and B+ have the same (positive) charge and both move to the right in the positive electric field. The change in shape of the front does not affect this basic feature. As the more mobile component B+ moves into the region where the less mobile component A+ is present, a mixing of both components occurs, leading to an increased reaction rate and a wider reaction zone with the front becoming more dispersed. Figure 2 illustrates the effects of a high negative current on the propagating front (here I ) -10). Again the concentration profiles a and b and the reaction rate ab are shown by the broken lines prior to the electric field being switched on (again at τ ) 20) and by the full lines afterward. The front is propagating in the positive X -direction before the electric field is switched on. The positively charged A+ and B+ will tend to migrate to the left in the negative field after this is switched on. This migration effect becomes more important than the tendency of the autocatalytic front to propagate to the right into the unreacted medium, and thus the direction of propagation of the wave is reversed (as can be seen by the full lines in Figure 2). The conductivity to the left of the front (now ahead of the propagating front) is higher than in the region to the right of the front (now behind the front), and consequently, the flux of B+ ahead of the front is higher than the flux of A+ behind it. Thus the concentration of A+ falls in the region at the rear of the front, and this region spreads out to the left from the initial position where the current was applied. Because the mobility of B+ is higher than that of A+, both components tend to separate. However, a negative charge in the region of the partially separated A+ and B+ is created (see Figure 5, where profiles of ∂E/∂X are shown) and the electrostatic forces prevent further separation. The reaction zone (where both reactants A+ and B+ are present) becomes narrower, the reaction rate decreases, and a focusing of the front occurs. The separation of A+ and B+ increases with increasing values of the negative current, and the reaction between A+ and B+ becomes less significant. A front still propagates and for high values of |-I| approaches the frontal structure that would propagate in a nonreacting system. Such fronts, which exist in systems of nonreacting mixtures of ionic components where two of them have the same charge and different mobilities, are wellknown and are usually referred to as Kohlrausch fronts.21-23 To obtain the velocity of such a front for our system (and as, the concentration of A+ at the rear of the wave), we note that, with the reaction terms neglected, eqs 7 and 8 can be integrated once. Then, the application of boundary conditions 9 and 10
Ionic Autocatalytic Reaction Fronts in Electric Fields
J. Phys. Chem., Vol. 100, No. 48, 1996 18743
Figure 1. Concentration profiles for a and b and reaction rate ab for I ) 10, δB ) 2.0. Profiles before the electric current is switched on at τ ) 20 are shown by the broken lines, and profiles afterward are shown by the full lines.
Figure 2. Concentration profiles for a and b and reaction rate ab for I ) -10, δB ) 2.0. Profiles before the electric current is switched on at τ ) 20 are shown by the broken lines, and profiles afterward are shown by the full lines.
(with bs ) 1) gives
V≈
δB + δC IδB , as f as I f -∞ (18) δB + δC δB(1 + δC)
Figures 1 and 2 illustrate the two limiting cases where the propagation of the reacting front is accelerated to the right for large positive currents and where the direction of propagation is reversed for high negative currents (see also Figure 4a). For intermediate values of the applied current both these effects can compete, and such a situation is illustrated in Figure 3. As before, the evolution of the concentration profiles and reaction rate before the current is switched on is shown by the broken lines and after the constant current (I ) -2.5) is switched on (again at τ ) 20) by the full lines. After the electric field is switched on, two counterpropagating fronts emerge in the system. The “kinetic” front continues to propagate to the right while a “Kohlrausch” front is initiated, which propagates to the left. These two fronts subsequently separate, and a widening region where the concentrations of both reactants are low is created, with these concentrations decreasing as the width of this region increases.
Here the reaction rate profiles have two local maxima (corresponding to the “kinetic” and “Kohlrausch” fronts) that become increasingly separated. The conductivity G in the region between these two fronts (where there are low concentrations of both A+ and B+) is low, giving a high electric field at constant current. The potential difference between the boundaries of this low concentration region increases with time. Hence these two fronts that are seen in the present case cannot be regarded as fronts that approach a permanent form as time increases. This is different from the cases shown in Figures 1 and 2. The dependence of the velocity of the propagating fronts on the applied electric current I for δB ) 2 is shown in Figure 4a. For I > -12x2/7 the numerical solutions (shown by the symbol +) approach the analytic solutions given by eq 11 (which is shown by the full line). For I < -2x2 the numerical solutions (shown by the symbol ]) approach the analytic solutions for the equivalent nonreacting system (Kohlrausch front speeds), as given by eq 18 and shown by the broken line in Figure 4a. The existence of both the kinetic and Kohlrausch fronts for I in the range -2x2 < I < -12x2/7
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Figure 3. Concentration profiles for a and b and reaction rate ab for I ) -2.5, δB ) 2.0. Profiles before the electric current is switched on at τ ) 20 are shown by the broken lines, and profiles afterward are shown by the full lines.
Figure 4. (a) Dependence of the wave velocity V on the applied electric current I for δB ) 2.0. Velocity Vmin of minimum speed waves, given by expression 11, is plotted by the full line, and the velocity of Kohlrausch fronts, as given by (18), is plotted by the broken line. Numerical solutions of eqs 3-5 are shown by + and ]; + represents minimum speed waves and ] other travelling wave structures. (b) Dependence of bs and as on the applied electric current I for δB ) 2.0. The full line shows values of bs for minimum speed waves as given by (14), and the broken line denotes the constant value of as as given by (18); + represents numerically calculated values of bs for minimum speed waves travelling with positive velocities; ] represents numerically obtained values of as for the waves travelling with negative velocities.
can also be seen in Figure 4a (more clearly in detail). The dependence of bs on I as given by eq 14 is shown in Figure 4b (by the full line) as well as the value of as as given by eq 18, which in the present case gives as ) 0.75. The results from the numerical simulations are shown by the symbol + (for V > 0) and by the symbol ] (for V < 0). Figure 5 shows profiles for the concentrations (a, b, and c), the reaction rate (ab), the conductivity (G), the electric field (E), and the electric field gradient (∂E/∂X) for the constantform propagating fronts that are generated at different values of the applied current I. The gradient of the electric field (∂E/∂X) is proportional to the charge density, and we can see
that either positive or negative charges can arise in the fronts caused by the spatial gradients of the ionic species, see in more detail ref 32. We can see that spatial gradients of the species A+, B+, and C become larger with increasing absolute values of the negative current I, while they become smaller with increasing values of the positive current I. Concentration profiles of a, b, and c for I ) 4, 0, and -2, respectively, correspond to kinetic fronts travelling from left to right with the minimum wave speed for which the value of as ahead of the front is equal to 1. The profiles of a, b, and c for I ) -10 approach those for the Kohlrausch front that travels from right to left with V given by eq 18. The value of a behind this front is also given by eq 18. The reaction rate ab increases for positive I and decreases for negative I. Very sharp gradients of a and b in the case I ) -10 result in a very narrow region over which the reaction between both species occurs. The difference in diffusion coefficients DA and DB gives rise to a different conductivity G ahead and behind the travelling front even when no electric field is applied. The changes of species concentration evoked by an applied field are manifested in changes of the conductivity. The conductivity ahead of the kinetic front does not change when the electric field is applied, while it increases behind the front in positive fields and decreases in negative fields. When electroseparation processes take over and a Kohlrausch front is created, the conductivity increases ahead of the front and decreases behind the front (note that in this case the front travels from right to left). Spatial gradients of the conductivity G and of the reaction species give rise to nonhomogeneous spatial profiles of electric field intensity E (following from eq 5). These are relatively small for kinetic fronts but can become very large when electroseparation occurs and the Kohlrausch front is established in the system. The spatial gradient of E across the front gives rise to a nonzero electric charge (see the shapes of the ∂E/∂X profiles in Figure 5). A large negative charge arises at the Kohlrausch front, while small positive charges arise at the kinetic fronts when I * 0. A nonzero electric charge also arises across the front when I ) 0 due to nonequal diffusion coefficients of A+ and B+, which again follows from eq 5. Figures 4 and 5 show the existence of three qualitatively different situations that can arise for δB > 1 as the magnitude and direction of the applied electric current is varied. The results
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J. Phys. Chem., Vol. 100, No. 48, 1996 18745
Figure 5. Concentration profiles a, b, and c, reaction rate ab, conductivity G, electric field E, and electric field gradient ∂E/∂X for the permanentform propagating fronts for δB ) 2.0 and for I ) 4, 0, -2, -10.
from our numerical simulations confirm the analytic predictions for the minimum speed waves and support the predicted ranges of their validity. (b) δB < 1. Here we took δB ) 0.5 as a representative value, with condition 17 giving values of I in the range -4x2 < I < 4x2 for the existence of minimum speed waves. The effect of a positive current on the propagating front is illustrated in Figure 6. Again the concentration profiles of a and b and the reaction rate ab before the electric field is switched on (at τ ) 20) are shown by the broken lines and by the full lines after a current of I ) 10 has been switched on. In this case the application of a positive electric field causes the positively charged components A+ and B+ to migrate in the positive X-direction. The more mobile component A+ (here DA ) 2DB) tends to move away from the less mobile component B+. A relatively small decrease in the concentrations of the positively charged ions due to their electroseparation results in a negative charge (see also the spatial profiles of ∂E/∂X given in Figure 9), and the electrostatic forces prevent the full separation of A+ and B+. However, the reaction zone, where both A+ and B+ are present, narrows and the importance of the chemical reaction decreases (Figure 6, the full lines). The concentration gradients in the front increase, and the front become more focused. The conductivity in the region at the rear of the front is lower and the electric field higher than in
the region ahead of the front. The migration flux of B+ behind the front is less than the flux of A+ ahead of it, and hence the concentration of B+ in the region behind the propagating front decreases. This region of decreased concentration of B+ spreads out from the location of the front at the time of the initial current application. With increasing values of the positive current the separation of A+ and B+ becomes more pronounced, the reaction becomes less significant, and the front approaches the corresponding Kohlrausch front. The appropriate Kohlrausch front velocity can again be obtained from eqs 7 and 8 with the reaction terms neglected. Then, integrating these equations and applying boundary conditions 9 and 10 (with as ) 1) gives
V≈
δB(1 + δC) I , bs f as I f ∞ 1 + δC δB + δC
(19)
The effects of a negative current (I ) -10) are shown in Figure 7. As previously, the evolution of the concentration profiles and reaction rate is shown by the broken lines before the application of the electric field (at τ ) 20) and afterward by the full lines. The front is propagating to the right as the field is switched on. The negative current and the consequent negative electric field support the migration of both positively charged ions to the left and tend to reverse the direction of
18746 J. Phys. Chem., Vol. 100, No. 48, 1996
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Figure 6. Concentration profiles for a and b and reaction rate ab for I ) 10, δB ) 0.5. Profiles before the electric current is switched on at τ) 20 are shown by the broken lines, and profiles afterward are shown by the full lines.
Figure 7. Concentration profiles for a and b and reaction rate ab for I ) -10, δB ) 0.5. Profiles before the electric current is switched on at τ ) 20 are shown by the broken lines, and profiles afterward are shown by the full lines.
propagation of the front. However, the inflow of the more mobile component A+ from the right into the front is higher than the outflow of the less mobile component B+ to the left. Thus a mixing of A+ and B+ occurs and the accumulation of B+ in the front causes a large increase in the reaction rate (see Figure 7, full lines) with more B+ being produced. The increase in the concentration of B+ in the front increases the diffusional flux of B+ into the unreacted region, and thus the propagation of the front to the right is enhanced by the autocatalytic reaction mechanism. This effect is more pronounced than the migration of the components to the left caused by the electric field and results in the slow propagation of the front to the right. The slow outflow of B+ to the left allows this component to accumulate further behind the front. Also, the negative charge arising at the front (Figure 9, spatial gradients of ∂E/∂X) attracts the component A+, and thus its concentration ahead of the front is increased. Both these processes act together to greatly increase the reaction rate in the front. The dependence of the velocity of propagation of the front on the applied electric current I for δB ) 0.5 is depicted in Figure 8a. For I > 4x2 the values obtained from the
Figure 8. (a) Wave speeds V calculated from the numerical solution of eqs 3-5 for δB ) 0.5, shown by the symbol ], Vmin, given by (11), is shown by the full line, and the Kohlrausch front velocity (19) by the broken line. (b) The concentration of B+ at the rear of the wave obtained from the numerical solutions, shown by the symbol ]. Values of bs given by (14) are shown by the full line, and the Kohlrausch front values (19) by the broken line.
numerical solutions (shown by the symbol ]) approach the analytical solution for the Kohlrausch front given by eq 19 (shown by the broken line), and for I in the range -4x2 < I < 4x2 they approach the analytical solution for the minimum
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J. Phys. Chem., Vol. 100, No. 48, 1996 18747
Figure 9. Concentration profiles a, b, and c, reaction rate ab, conductivity G, electric field E, and electric field gradient ∂E/∂X for the permanentform propagating fronts for δB ) 0.5 and for I ) 10, 2, 0, -2, -10.
speed waves as given by eq 11 (shown by the full line). For I < -4x2 the behavior depicted in Figure 7 is observed. The dependence of bs on I is shown in Figure 8b. The full line denotes the values obtained from eq 14 and the broken line the values obtained from eq 19. The values obtained from the numerical simulations are shown by the symbol ]. Hence, for δB < 1, three qualitatively different types of behavior can arise as the current I is varied in magnitude and direction. Figure 9 shows spatial profiles of the concentrations (a, b, and c), the reaction rate (ab), the conductivity (G), the electric field (E), and the electric field gradient (∂E/∂X), proportional to the charge density, corresponding to permanent-form propagating fronts for specific values of I shown in Figure 8. The structure of Figure 9 is similar to Figure 5, but now δB ) 0.5. All the wave fronts represented in this figure propagate from left to right (V > 0). Large gradients of the electric field and small values of the reaction rate arise for sufficiently large positive values of the current (I ) 10) where the observed front approaches the Kohlrausch front. When large absolute values of negative current are used (I ) -10) the front does not reverse its direction of propagation (as might be expected) but the concentration of B+ behind the front increases. The local increase of a ahead of the front is caused by the local decrease of the charge density (∂E/∂X < 0). This negative charge density
is caused both by the negative current and by the negative value of the conductivity gradient. Conclusions The differences between the cases δB > 1 and δB < 1 in the travelling fronts that result as the applied current is varied are now clear. With δB > 1 for I positive the waves are essentially chemical front waves determined primarily by their autocatalytic kinetics. This situation persists for I slightly negative. As |I| is increased, there is a transition region where two waves, propagating in opposite directions, are initiated, one being essentially still the kinetic wave and the other an electrophoretic, Kohlrausch front. As |I| is increased further, the kinetic wave is no longer initiated and only negatively propagating Kohlrausch fronts are formed. For δB < 1 there is a bounded range of I over which the kinetic waves are initiated. For the larger positive values of I, it is Kohlrausch fronts that are formed. However, for sufficiently large negative values of I a new sort of travelling wave structure is initiated that appears to have little similarity with either purely kinetic or electrophoretic waves. In this case the waves travel in the positive direction for all values of the applied current and sustain a much enhanced reaction rate. The model treated here is arguably the simplest one in which
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18748 J. Phys. Chem., Vol. 100, No. 48, 1996 nontrivial effects of the application of an electric field on autocatalytic front waves can be manifested. The simplicity of our model has enabled considerable progress to be made in understanding all the various mechanisms that can result as the electric current is altered for different diffusional mobilities of substrate and autocatalyst. Other extended models involving more ionic species and different autocatalytic kinetics can be envisaged. These may be required to obtain quantitative agreement with specific experimental observations. However, the insights gained from our simple model provide the framework for understanding these extended models and form a basis for their comprehensive treatment. These extended models as well as an experimental program based on the iodate-arsenous acid reaction system to test the various theoretical predictions obtained above are being actively pursued and will be reported in future communications. References and Notes (1) Field, R. J., Burger, M., Eds. Oscillations and traVelling waVes in chemical systems; Wiley, New York, 1985. (2) Murray, J. D. Mathematical biology, Springer: Berlin, 1989. (3) Holden, A. V., Markus, M., Othmer, M. G., Eds. Nonlinear waVe processes in excitable media; Plenum: New York, 1990. (4) Kapral, R., Showalter, K., Eds. Chemical waVes and patterns; Kluwer: Dordrecht, 1995. (5) Merkin, J. H.; Needham, D. J. J. Eng. Math. 1989, 23, 343. (6) Billingham, J.; Needham, D. J. Phil. Trans. R. Soc. London 1991, A334, 1. (7) Needham, D. J.; Merkin, J. H. Nonlinearity 1992, 5, 413. (8) Gray, P.; Merkin, J. H.; Needham, D. J.; Scott, S. K. Proc. R. Soc. London 1990, A430, 509. (9) Horvath, D.; Petrov, V.; Scott, S. K.; Showalter, K. J. Chem. Phys. 1993, 98, 6332; 1989, 85, 3871. (10) Nagy, I. P.; Pojman, J. A. J. Phys. Chem. 1993, 97, 3443. (11) Keresztessy, A.; Nagy, I. P.; Bazsa, G.; Pojman, J. A. J. Phys. Chem. 1995, 99, 5379.
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