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A numerical model for the influence of a radial electric field on spiral wave dynamics
Physica D 69 (1993) 309-319 North-Holland SDI: 0167-2789(93)E0230-9
A numerical model for the influence of a radial electric field on spiral wave dynamics J . L . F . P o r t e i r o I, V. P 6 r e z - M u f i u z u r i 2, A . P . M u f i u z u r i a n d V. P 6 r e z - V i l l a r Departamento de Fisica de la Materia Condensada, Facultad de Fisicas, E-15706 Universidad de Santiago de Compostela, Spain
Received 21 May 1992 Revised manuscript received 12 April 1993 Accepted 14 June 1993 Communicated by A.V. Holden
The influence of a radial electric field on spiral wave dynamics is numerically investigated using an Oregonator-based model of the Belousov-Zhabotinsky reaction. The electric field is modeled and used to predict wavefront shape and period of rotation as a function of its strength. The period of rotation was found to increase with electric field strength up to a factor of three. No hysteresis effects were found.
1. Introduction
T h e tendency of excitable media to organize themselves into highly structured periodic waves or spirals has been the object of intense interest for the best part of the last twenty years. As a result of this interest there is a significant body of information on the analytical, experimental and numerical aspects of these p h e n o m e n a [1-7]. A m o n g the best known examples of spiral waves in excitable media is that of the chemical waves in the B e l o u s o v - Z h a b o t i n s k y ( B Z ) reagent, involving the propagation of ionic species in the f o r m of traveling waves of concentration disturbances. The B Z reaction has been successfully m o d e l e d by the two variable version of the O r e g o n a t o r [8-12]. T h e effect of an electric field on the dynamics of the B Z reaction and in particular on a model On sabbatical leave from: Mechanical Engineering Dept., University of South Florida, Tampa, FL 33620, USA. 2To whom correspondence should be addressed.
consisting of the same chemical species as the O r e g o n a t o r was studied by Schmidt and Ortoleva using perturbation techniques [13-15]. Sevcikova and M a r e k [16-18] have analyzed both experimentally and computationally the dependence of wave velocity on the intensity of an applied electric field for the case of the one dimensional B Z reaction and discussed a mechanism for the effects of the electric field on the acceleration, deceleration and splitting of the pulse waves. Recent experimental developments have shown that the application of an external electric field to a fully developed spiral can result in the alteration of the fundamental p a r a m e t e r s that characterize spiral wave behavior. Agladze and De K e p p e r [19] investigated the drift caused by an electric field created by two parallel electrodes placed at both sides of a reactor on a rotating spiral wave in a B Z reaction. Using the same technique, Steinbock et al. [20] investigated this drift as well as the interaction of counter rotating spirals. P6rez-Mufiuzuri et al. in
0167-2789/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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J.L.F. Porteiro et al. I Influence o f radial electric field on wave dynamics
their studies of the B Z reaction observed that the application of a radial electric field at the core of a spiral wave resulted in the alteration of its period and wavelength [21]. U n d e r the appropriate conditions they obtained as much as a three-fold increase in the period of rotation as well as an increase in wavelength. Furthermore, as the electric field was removed, the spiral wave returned to its original state, even though hysteresis effects were observed. Through the application of pulses of electrical current locally at the spiral tip, P6rez-Mufiuzuri et al. obtained a spiral wave of larger wavelength and superimposed on the original wave, the so called superspiral wave [22,231 . It is the main objective of this p a p e r to develop an O r e g o n a t o r - b a s e d numerical model capable of reproducing some of the effects of a radial electric field on spiral wave behavior. We will c o m p a r e the results obtained numerically to those obtained experimentally.
Dq = Dfiu. For highly diluted species they are independent of the concentrations and we will assume that this condition is satisfied in our system. The continuity equation can now be written as
o{c,} Ot
- [Dis] V2{Ci} - [Mq] V. {Ci}E -t- {ni} .
(3) The second term on the left hand side of the equation represents the contribution of the electric field to the time rate of change of the concentrations. The electric field E is the sum of the local ionic electric field and the externally applied field. For conditions considered in this work the local electric field is much smaller than than the applied field and its influence in the overall dynamics negligible. For simplicity it can be omitted from the equation without a significant loss in the accuracy of the formulation.
3. Electric field modeling 2. Basic equations Given the concentrations {Ci} of the chemical reactions in the B Z reaction, their change with time can be described through the continuity equation
For the case of a radial electric field applied to a medium in which only the p r o p a g a t o r variable diffuses, the contribution of the electric field to the time rate of change of this variable can be written as
o{c,}
[Mi] ] V ° { Ci}E = M V ( f E r )
Ot
- -V.$+
{Ri},
(1)
where J is the mass flowrate per unit area and {Ri} are the chemical reaction rates, in general a nonlinear function of the concentrations. For the conditions where the fluid is at rest (no convective currents) and in the absence of t e m p e r a t u r e gradients J can be related to the concentrations and electric field E through J = - [Dis] V{Ci} + [Mq]{Ci}E,
(2)
where only linear terms have been related retained [24]. [Mu] an d [Oi]] are the mobility and diffusion coefficient matrices, Mq =Mi6 u and
dr / + dr
,j,
(4)
where M and C are the mobility and concentration of the relevant variable and E, is the radial (and only) c o m p o n e n t of the applied field. U n d e r the experimental conditions reported by P6rez-Mufiuzuri et al. [21], it is the result of a positively charged electrode located at the core of the spiral and a negatively charged circular electrode located at the vertical wall of the Petri dish. The influence of the circular electrode on spiral wave dynamics is negligible except for those spirals developing very close to the wall.
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J.L.F. Porteiro et al. / Influence of radial electric field on wave dynamics
For spirals developing far from the wall the influence of this electrode is several orders of magnitude smaller than that of the central electrode and for the sake of computational expediency may be left out of the overall formulation. Only the portion of the central electrode immersed in the solution is considered to contribute to the electric field strength. This strength evaluated at radial distance r from the electrode tip is functionally given by E~=4,tre =
The 1/r singularity exhibited by E r at the origin is handled numerically by establishing a minimum radius r 0 below which its value is arbitrarily prescribed. It can be seen from eq. (4) that the concentration, C, as well as its radial gradient d C / d r act as multipliers of the effects of the electric field. A comparison of the relative strengths of E r, the gradient multiplier, and Er/r + d E r / d r , the concentration multiplier as a function of r is shown in fig. 1 which was obtained using an electrode length, I, of 1 space unit. Only in regions where high values of C and low values of d C / d r occur simultaneously will the effects of the concentration term be larger than those of the other term. Such a condition will be found mostly at the crest of the wave, where the high value of C at the front of the wave gently slopes down to a lower value at the back of the wave. At the front and back of the wave d C / d r is large and the contribution of the gradient term is several times larger than that of the concentration term.
r2 1+
(5)
Q ~(r,l) 4~re
where Q is the charge per unit length of electrode, e is the dielectric constant of the solution and l is the length of electrode immersed in the solution. Er is scaled to units of ( D k s [ M A ] ) I / 2 / MBr- [15]. For the values of [MA] and k 5 used above and with M B r - = - 4 × 10-4cm2/Vs the unit of electric field strength is - 2 . 4 V/cm. 2.5
d~ S
ds
o
1.5
0.5
1 -0.5
~
2
3
4
5
6
7
8
S (su)
Fig. 1. Variation of the intensity of the electric field components (concentration and concentration gradient multipliers) with radial distance.
312
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
4. N u m e r i c a l m o d e l i n g o f the r e a c t i o n - d i f f u s i o n process: the O r e g o n a t o r
In order to simulate the B Z chemical reaction, the two variable version of the O r e g o n a t o r with T y s o n ' s " L o " p a r a m e t e r s was used [8]. The applicable equations, for a spatially uniform, unstirred medium, after scaling, are OU Ot -
_1 [ 2 e u - u
fv
u-q
+ Du
V21A
Ov Ot
- u - v ,
(7)
where u is the ' p r o p a g a t o r ' variable representing the dimensionless concentration [HBrO2] and v, the ' r e c o v e r y ' variable, is the dimensionless concentration [ferriin]. E u is the field strength per unit length of electrode and is equal to - M Q / 4 ~ e . The minus sign is added for convenience as it accounts for the negative value of M. In this way positive applied voltages correspond to positive values of E,. T h e variable s is the scaled radial distance r. T h e p a r a m e t e r s q and e are functions of concentrations and rate constants. A detailed discussion of their values and influence upon the B Z reaction can be found in [11]. D , and D v are the dimensionless diffusion coefficients of u and v and their ratio is important in controlling wave dynamics in general and wave tip meandering in particular. As in our simulation we consider ferriin (or variable v) to be immobilized in a silica gel [21,22,25], we have taken their values to be D u = l and D r = 0 . This has the added benefit of faster execution times as the Laplacian of the recovery variable does not need to be calculated. All our simulations were made with q -- 0.002, e = 1/100 and f = 1.4. For the experimental condition of our simulation [MA] = 0.15 with D u set to ( D / k s [ M A ] ) 1/2 and with k 5 = 0 . 4 M - i s -1 and D = 1.5 x 10 -5 cm2s -1 [8,9] our unit of
space scales to approximately 0.16 mm. Our unit time scaled with 1 / k s [ M A ] is about 17 s. Spirals were generated from the set of initials conditions suggested by Jahnke, Skaggs and Winfree [10] i.e. a narrow wedge of excitation with u = 0.8 and v near equilibrium followed by a smoothly declining circular gradient of refractoriness. This resulted in a counterclockwise rotating wave on grids as small as 60 x 60 and as large as 500 x 500. A uniform time step of 1/ 1000 was used throughout as the differential equations were integrated using the explicit E u l e r m e t h o d with a five point central difference approximation being used for the calculation of the laplacian and no-flux boundary conditions. A grid mesh of 0.125 space units was chosen but runs at grid spacings as large as 0.1875 units did not result in significant changes in wave parameters. E v e n though most of our simulations were carried out on a 200 x 200 grid, the effect of gridsize on wave p a r a m e t e r s was studied on grids as small as 80 x 80 and as large as 500 x 500.
5. C o m p u t a t i o n a l p r o c e d u r e
Special care must be taken in order to guarantee that the discretization of time and space will not significantly influence the resulting spiral wave. It was found that while the condition of D u = 1 and D v = 0 significantly increased the sensitivity to the grid anisotropy introduced by the five point laplacian formulation used, a grid spacing of 0.125 space units together with a 0.001 time step gave rise to spirals without any perceptible anisotropy on 200 x 200 grids. Following Jahnke et al. and Jahnke and Winfree [10,11] u was reset to a value of q whenever it fell below that value in order to increase the allowable time step and to avoid negative values of u that are physically impossible. G e n e r a t e d spirals were allowed to develop f r o m the integration of eqs. (4) and (5) until their rotation periods became constant, at which time the electric field was "turned on".
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
In order to achieve f a s t e r computation times neither the laplacian nor the electric field effects were calculated at those points such that the value of u at the surrounding points (involved in the evaluation of the laplacian) fell below a threshold value. This is similar to Barkley's " b o u n d a r y layer" technique [26,27] for evaluating those points within a small "boundary layer" near the left branch of the u-nullcline. In the present case the fact that all points involved in the evaluation of the laplacian fall inside the boundary layer implies that the values of the laplacian, d u / d r and u will be small. As a result the contribution of the reaction term will be much larger than that of the diffusion and electric field terms and little error is introduced by their elimination. Since a large percentage of all points are within this boundary layer, the savings in computational time are significant. Spirals generated this way are undistinguishable from those generated from the evaluation of all contributing terms at all points of the grid. The electrode was assumed to be located at the geometrical center of the computational grid for all positive electric field cases investigated. When a negative electric field was used it was necessary to alter its location to insure that it was always located at the tip of the spiral.
6. Results
The effects of application of a positive electric field on a fully developed spiral can be observed in fig. 2. In order to better illustrate the process a large grid of 500 x 500 with a grid spacing of 0.1825 was used. The spiral shown on fig. 2a was generated in the absence of an electric field. A total of four rings can be observed with a typical front width of twelve gridpoints. An electric field of intensity E u = 10 except for a radius of 8 gridpoints around the electrode, where E u = 0, and electrode length of 8 gridpoints is then turned on. The evaluation of the spiral after 2.5 time units is shown in fig. 2b. It can be clearly
313
observed that the spiral center is reducing its front curvature and at the same time the front becomes wider, the sections near the spiral tip having now a width of approximately sixteen gridpoints. At the same time it can be observed that the affected section of the spiral alters its rotation rate as it slows down. The process can be observed to continue in figs. 2c-e. The fronts increase to an average width of 28 gridpoints and the spreading out process has reduced the spiral to only one and a half turns. The configuration shown in fig. 2e has reached equilibrium and remains unchanged a further 22.5 time units later, its rotation rate also unchanged. When the electric field was turned off, the spiral returned to its original state without any signs of hysteresis. In order to simulate the influence of the electric field intensity on spiral wave parameters, simulations were carried out on a 200 x 200 grid with a grid spacing of 0.125 space units and an electric field similar to the one described above except for a variable intensity E,. After a steady rotating spiral was generated, the electric field was turned on at a low value. The new spiral was allowed to develop and stabilize a t which time its parameters were measured and then the electric field was increased. Once the maximum value of E u was reached, field strength was decreased using the same procedure. The influence of the electric field strength on the shape and width of the wave for positive fields is shown in fig. 3. It is evident that stronger positive fields result in thicker wavefronts with reduced curvature. The effects of very intense fields such as those in figs. 3e and f result in very complex patterns near the spiral tip. It was observed that while spirals generated with positive electric fields exhibited characteristic tip meandering behavior, they steadily rotated around the positive electrode located at the center of the grid. This however was not true for negative field spirals. The presence of the negative electric field increased tip meandering to the point that after a certain period of time spiral
314
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
Fig. 2. Unfurling of a spiral wave under the influence of an electric field of intensity E u = 10 except for the region r < 1.5 su, where E r = 0. The computational box has 500 x 500 gridpoints and a gridspacing of 0.1875 su. The electrode length is 1.5 su. After 15 tu the spiral has reached steady state as shown in (e) and (f).
r o t a t i o n was no longer a r o u n d the electrode located at the center of the grid. As a result of this, the spiral eventually m o v e d across the negatively c h a r g e d electrode. A n e x a m p l e of such a process is shown in fig. 4. It can be o b s e r v e d in fig. 4a that the electrode (located at the g e o m e t r i c center of the figure) interacts with the spiral tip by d e f o r m i n g it. As the spiral continues its rotation, further d e f o r m a t i o n takes place fig. 4b and eventually the spiral is split in two. T h e two spiral sections grow and advance past the electrode, eventually linking up and r e g e n e r a t i n g a single spiral once again (figs. 4e and f). In o r d e r to insure that the electrode was always located at or in the n e i g h b o u r h o o d of the spiral tip, it was necessary to alter its location
f r o m the center-grid position used for positive fields. Fig. 5 shows the effects of a negative electric field on a steady rotating spiral on a 200 x 200 grid with 0.125su grid spacing. D u e to the p r e s e n c e of the negative electric field, slight tip d e f o r m a t i o n can be o b s e r v e d in fig. 5b and significant d e f o r m a t i o n is found in figs. 5c and 5d. As a result of this d e f o r m a t i o n , well established m e t h o d s of defining and locating the spiral tip, such as those described by ] a h n k e et al. [10], failed. N e w m e t h o d s b a s e d on tip t o p o l o g y were d e v e l o p e d in o r d e r to d e t e r m i n e such location [28]. T h e spirals were o b t a i n e d by m o v i n g e v e r y two iterations the position of the e l e c t r o d e so that it would coincide with that of the spiral tip.
315
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics //
I
(b)
A
Y
% (c) Eu =
10 t~A
(~)
(f)
(e) Eu
=
Eu = 500
100
Fig. 3. Influence of field intensity on front curvature for positive values of E,. (200 x 200 grid, 1/8 su gridspacing, E, = 0 for r < 1.5 su; electrode length, 1.5 su.)
It is clear that as the intensity of the negative electric field increases the spirals contract upon themselves and increase the overall wavefront curvature. It should be noted that the spirals shown for negative values of E u above 10 are not stable due to the progressive deformation of the spiral tip. This tip deformation is evident in fig. 5c and d and finally results in the impossibility of obtaining an equilibrium configuration for these values of E,. Fig. 6 illustrates the relationship between the rotation period of the spiral and the intensity of the electric field. As shown on the figure inset, for negative and low positive values of E, (those below a value of 4) the period is weakly affected by the electric field. For higher values rotation periods are dramatically increased and tend to
Fig. 4. Interaction of a negative electric field (E, = - 4 ) with a fully developed spiral. The electrode is fixed and located at the center of the figure. Wavefront deforms (a), (b) and breaks (c) when approaching electrode. Once past the electrode it grows (d) and reattaches (e), (f). (80 x 80 grid, 0.125 gridspacing.)
stabilize for the highest values of E, at a value about three times higher than that of the baseline spiral without an electric field. Rotation periods obtained with E, = 1000 were within 1% of those obtained with E, = 500. When the value of E, was decreased the spirals obtained exhibited the same parameters as those obtained when E u was increasing and no hysteresis effects were observed. An investigation of the influence of the modeling of the electric field in the neighborhood of the electrode on spiral wave parameters was carried out for several values of the electrode
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J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
both models are of the same magnitude as our experimental error.
7. Discussion
Fig. 5. Influence of field intensity on negative values of E computed on a 1 / 8 s u gridspacing. Observe strong tip negative electric in c and d. ( E = 0 for length, 1.5 su.)
front curvature for 200 x 200 grid with deformation due to r < 1.5 su; electrode
length, l, and the electrode radius, r0, below which the strength of the electric field was arbitrarily prescribed. T w o models of the electric field were studied. In one E, was given a zero value for r < r 0 and in the other E r was m a d e to vary linearly with r such that its value was zero at r = 0 and continuous at r = r 0. T h e influence of r 0 on the rotation period of the spiral for the two models is presented in fig. 7, which was obtained on a 200 x 200 grid with 0.125 gridspacing. No difference was observed b e t w e e n the two models. It can be noted from observation of this figure that electric fields effects decrease with r 0 and that spirals with values of r 0 larger than 3.75 su have a period of rotation similar to those obtained in the absence of an electric field. It is clear from this that modeling of the electric field near the electrode is not very important for values of r 0 below 4 su, as the differences observed in the periods for
A comparison of the process illustrated in fig. 2 with the experimental data obtained by P6rezMufiuzuri et al. [21] shows that an O r e g o n a t o r based model is capable of reproducing qualitatively the effects of a radial electric field on spiral waves generated by a B Z reaction. As in the experimental case once the electric field is r e m o v e d , spirals return to their original state. T h e formation of spirals in the O r e g o n a t o r model is the result of the interaction of the reaction and diffusion terms in eqs. (6) and (7). This interaction is altered by the electric field in a complex m a n n e r since a dependency on both u and its radial gradient must be taken into account. For positive values of E u, while the concentration term is always negative, du/dr is negative at the front of the wave and positive at the back and as a result so is the gradient term. In this way the contribution of the electric field to du/dt is negative in the front of the wave and positive at its back. The result is a slowdown in the advance of the front that is proportional to the local field strength. For negative values of E , the effects of the electric field are reversed and wavefront acceleration results. It should be noted, however, that the influence of negative electric fields must be less than that of positive ones. This follows from the fact that, regardless of the intensity of the negative electric field, it is not physically possible to obtain negative values of the concentration u (numerically, as above indicated, values of u below the equilibrium value are restored to equilibrium). This buffering mechanism could explain the markedly w e a k e r influence of negative fields when c o m p a r e d with positive ones that is evident in figs. 3 and 5. It is clear from fig. 2 that the electric field does not act instantaneously throughout the wave. This seems to indicate that the influence of the
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
317
Period
(tu) 4.5
4 J
J
3.5 2,5 ¸ 3
Period
(tu)
2.5
2
2 1.5 1.5
J
1
0.5
I
p
i
-50
-25
1 0
i
i
25
50
Eu
0
-50
0
t
I
I
I
I
I
I
I
I
I
50
100
150
200
250
300
350
400
450
500
EU Fig. 6. R o t a t i o n period of spiral waves as a function of electric field strength. T h e asymptotic value for high value of E u is a p p r o x i m a t e l y three times the rotation period with no field. Inset: B e h a v i o r at low field intensities. (E r = 0 for r < 1.5 su; electrode length, 1.5 su.)
Period (tU) 1.9
1.8
1.7
•
Er = kr f o r r < r O
•
Er = 0 f o r r < r O Eu = 10
I = 1.5 su 1.6
1.5
1.4
1.3 0
I
I
I
I
0.5
1
1.5
2
rO
I
I
[
I
2.5
3
3.5
4
(su) Fig. 7. Influence of electric field core modeling on the rotation period of the spirals. Periods are identical for both models. For spirals where with values of r 0 above I su the period decreases rapidly until it reaches the no field value for a core radius of about 3.5 su.
318
J.L.F, Porteiro et al. / Influence o f radial electric field on wave dynamics
electric field is only felt, at least for moderate field intensity values, by the sections of the spiral closest to the tip and that diffusion plays an important role in propagating the disturbance to the rest of the spiral. As a result the front straightens out and slows down locally while the rest of the spiral "catches up" with it and also straightens out and slows down. Further support for this mechanism can be found in fig. 7 which shows that spirals with a radius r 0 at or above 3.75 su exhibit the same rotation period as a spiral with no electric field at all. For so large electrode diameters the electric field acts only upon the outer sections of the spiral, while the tip and inner sections are unaffected. The fact that under these conditions both wavefront shape and rotation periods are identical to those obtained with E u = 0 seems to indicate that the radial electric field effects on the outer section of the spiral are overwhelmed by the diffusion processes. From the results shown in fig. 3 it is clear the straightening process slows down dramatically for high values of E u, as larger values of the electric field are required to generate smaller effects on both front curvature and rotation period. This can be explained since the influence of the du/dr term decreases steadily with high values of E , as a consequence of the straightening of the wave front. It can be seen that for high values of the field such as those in fig. 3f, the spiral tip is almost radially aligned. In this condition the value of du/dr is very small and large values of Eu are needed to make the term significant. In addition it must be remembered that at the front of the spiral wave the contribution of the electric field to du/dt is negative. Since the value of u is never allowed to fall below that of q, this further reduces the influence of very high values of E,. /t is remarkable that the present model can so faithfully reproduce the experimental values obtained for the ratio of the maximum rotation period (high electric field) to the baseline rotation period (no field) and this seems to be a product of the
underlying reaction-diffusion characteristics of the B Z reaction. The absence of hysteresis effects is most likely a consequence of neglecting the local ionic field in eq. (4). The presence of an externally imposed field on the B Z reagent leads to a redistribution of ionic charges throughout the solution in such a way that equilibrium is reached for the new conditions. As the external electric field is withdrawn this distribution will no longer be in equilibrium and will relax through a certain period of time to a new equilibrium state. This period, for the experimental case, will depend on the initial charge distribution and the diffusion coefficient. In our numerical simulation the absence of a local ionic field results in a relaxation to the original (no electric field) state that is only controlled by the interrelationship between the diffusion and reaction terms in eq. (4). The failure to generate stable, steadily spirals for even moderately low negative values of E~ raises the question of whether if it is indeed possible to obtain such spirals. In sharp contrast, much larger positive fields cause significant tip deformation but the resulting spirals are stable and steadily rotate around a fixed electrode. While such a failure could be attributed to the difficulties associated with the localization of the spiral tip, it is quite likely that those difficulties arise from tip deformation caused by the strong electric fields present in the neighborhood of the tip. It is also likely that the mechanisms responsible for wavefront breakup shown in fig. 3 are quite capable of altering tip dynamics in a similar manner.
8. Conclusion
The effects of an externally applied electric field on spiral wave dynamics have been successfully modelled. While the results obtained are in good agreement with the experimental data, much remains to be done in order to understand how some aspects of the modelling, such as the
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
d e s c r i p t i o n o f t h e e l e c t r i c field n e a r t h e spiral tip, affect m o r e c o m p l e x w a v e b e h a v i o r such as m e a n d e r i n g a n d w a v e f r o n t stability for m o d e r a t e a n d l a r g e n e g a t i v e v a l u e s o f t h e electric field. It c a n also b e c o n c l u d e d t h a t t h e O r e g o n a t o r m o d e l has p r o v e n to b e r e m a r k a b l y r o b u s t a n d flexible t o o l c a p a b l e o f r e p l i c a t i n g s o m e o f t h e most intriguing aspects of the BelousovZhabotinsky reaction.
Acknowledgements T h i s w o r k was s u p p o r t e d in p a r t b y the C o m i s i 6 n I n t e r m i n i s t e r i a l d e C i e n c i a y Tecnologia (Spain) under grant DGICYT-PB910660.
References [1] A.T. Winfree, Science 175 (1972) 634; 181 (1973) 937. [2] S.C. Miiller, T. Plesser and B. Hess, Physica D 24 (1987) 71. [3] E. Meron, Physica D 49 (1991) 98. [4] J.P. Keener, SIAM J. Appl. Math. 46 (1986) 1039. [5] V.S. Zykov, Biofizika 31 (1986) 862; 32 (1987) 337. [6] G.S. Skinner and H.L. Swinney, Physica D 48 (1991) 1. [7] J.J. Tyson and J.P. Keener, Physica D 32 (1988) 327. [8] J.P. Keener and J.J. Tyson, Physica D 21 (1986) 307.
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[9] J.J. Tyson, in: Oscillations and Traveling Waves in Chemical Systems, R.J.. Field and M. Burger, eds. (Wiley, New York, 1985 p. 93. [10] W. Jahnke, W.E. Skaggs and A.T. Winfree, J. Phys. Chem. 93 (1989) 740. [11] W. Jahnke and A.T. Winfree, Int. J. Bif. Chaos 1 (1991) 445. [12] A.T. Winfree, Chaos 1 (1991) 303. [13] S. Schmidt and P. Ortoleva, J. Chem. Phys. 67 (1977) 3771. [14] S. Schmidt and P. Ortoleva, J. Chem. Phys. 71 (1979) 1010. [15] S. Schmidt and P. Ortoleva, J. Chem. Phys. 74 (1981) 4488. [16] H. Sevcikova and M. Marek, Physica D 9 (1983) 140. [17] H. Sevcikova and M. Marek, Physica D 13 (1984) 379. [18] H. Sevcikova and M. Marek, Physica D 21 (1986) 61. [19] K.I. Agladze and P. De Kepper, J. Phys. Chem. 96 (1992) 5239. [20] O. Steinbock, J. Schiitze and S.C. Miiller, Phys. Rev. Lett. 68 (1992) 248. [21] V. P6rez-Mufiuzuri, R. Aliev, B. Vasiev and V.I. Krinsky, Physica D 56 (1992) 229. [22] V. P6rez-Mufiuzuri, R. Aliev, B. Vasiev, V. P~rez-Villar and V.I. Krinsky, Nature 353 (1991) 740. [23] V. P~rez-Mu~uzuri, M. Gomez-Gesteira and V. P6rezVillar, Physica D 64 (1993) 420. [24] J. O'M. Bockris and A.K.N. Reddy, Modern Electrochemistry (Plenum, New York, 1974). [25] R.R. Aliev and K.I. Agladze. Physica D 50 (1991) 65. [26] D. Barkley, M. Kness and L.S. Tuckerman, Phys. Rev. A 42 (1990) 2489. [27] D. Barkley, Physica D 49 (1991) 61. [28] J.L.F. Porteiro, V. P6rez-Mufiuzuri, A.P. Mufiuzuri and V. P6rez-Villar, in preparation.
A numerical model for the influence of a radial electric field on spiral wave dynamics J . L . F . P o r t e i r o I, V. P 6 r e z - M u f i u z u r i 2, A . P . M u f i u z u r i a n d V. P 6 r e z - V i l l a r Departamento de Fisica de la Materia Condensada, Facultad de Fisicas, E-15706 Universidad de Santiago de Compostela, Spain
Received 21 May 1992 Revised manuscript received 12 April 1993 Accepted 14 June 1993 Communicated by A.V. Holden
The influence of a radial electric field on spiral wave dynamics is numerically investigated using an Oregonator-based model of the Belousov-Zhabotinsky reaction. The electric field is modeled and used to predict wavefront shape and period of rotation as a function of its strength. The period of rotation was found to increase with electric field strength up to a factor of three. No hysteresis effects were found.
1. Introduction
T h e tendency of excitable media to organize themselves into highly structured periodic waves or spirals has been the object of intense interest for the best part of the last twenty years. As a result of this interest there is a significant body of information on the analytical, experimental and numerical aspects of these p h e n o m e n a [1-7]. A m o n g the best known examples of spiral waves in excitable media is that of the chemical waves in the B e l o u s o v - Z h a b o t i n s k y ( B Z ) reagent, involving the propagation of ionic species in the f o r m of traveling waves of concentration disturbances. The B Z reaction has been successfully m o d e l e d by the two variable version of the O r e g o n a t o r [8-12]. T h e effect of an electric field on the dynamics of the B Z reaction and in particular on a model On sabbatical leave from: Mechanical Engineering Dept., University of South Florida, Tampa, FL 33620, USA. 2To whom correspondence should be addressed.
consisting of the same chemical species as the O r e g o n a t o r was studied by Schmidt and Ortoleva using perturbation techniques [13-15]. Sevcikova and M a r e k [16-18] have analyzed both experimentally and computationally the dependence of wave velocity on the intensity of an applied electric field for the case of the one dimensional B Z reaction and discussed a mechanism for the effects of the electric field on the acceleration, deceleration and splitting of the pulse waves. Recent experimental developments have shown that the application of an external electric field to a fully developed spiral can result in the alteration of the fundamental p a r a m e t e r s that characterize spiral wave behavior. Agladze and De K e p p e r [19] investigated the drift caused by an electric field created by two parallel electrodes placed at both sides of a reactor on a rotating spiral wave in a B Z reaction. Using the same technique, Steinbock et al. [20] investigated this drift as well as the interaction of counter rotating spirals. P6rez-Mufiuzuri et al. in
0167-2789/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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J.L.F. Porteiro et al. I Influence o f radial electric field on wave dynamics
their studies of the B Z reaction observed that the application of a radial electric field at the core of a spiral wave resulted in the alteration of its period and wavelength [21]. U n d e r the appropriate conditions they obtained as much as a three-fold increase in the period of rotation as well as an increase in wavelength. Furthermore, as the electric field was removed, the spiral wave returned to its original state, even though hysteresis effects were observed. Through the application of pulses of electrical current locally at the spiral tip, P6rez-Mufiuzuri et al. obtained a spiral wave of larger wavelength and superimposed on the original wave, the so called superspiral wave [22,231 . It is the main objective of this p a p e r to develop an O r e g o n a t o r - b a s e d numerical model capable of reproducing some of the effects of a radial electric field on spiral wave behavior. We will c o m p a r e the results obtained numerically to those obtained experimentally.
Dq = Dfiu. For highly diluted species they are independent of the concentrations and we will assume that this condition is satisfied in our system. The continuity equation can now be written as
o{c,} Ot
- [Dis] V2{Ci} - [Mq] V. {Ci}E -t- {ni} .
(3) The second term on the left hand side of the equation represents the contribution of the electric field to the time rate of change of the concentrations. The electric field E is the sum of the local ionic electric field and the externally applied field. For conditions considered in this work the local electric field is much smaller than than the applied field and its influence in the overall dynamics negligible. For simplicity it can be omitted from the equation without a significant loss in the accuracy of the formulation.
3. Electric field modeling 2. Basic equations Given the concentrations {Ci} of the chemical reactions in the B Z reaction, their change with time can be described through the continuity equation
For the case of a radial electric field applied to a medium in which only the p r o p a g a t o r variable diffuses, the contribution of the electric field to the time rate of change of this variable can be written as
o{c,}
[Mi] ] V ° { Ci}E = M V ( f E r )
Ot
- -V.$+
{Ri},
(1)
where J is the mass flowrate per unit area and {Ri} are the chemical reaction rates, in general a nonlinear function of the concentrations. For the conditions where the fluid is at rest (no convective currents) and in the absence of t e m p e r a t u r e gradients J can be related to the concentrations and electric field E through J = - [Dis] V{Ci} + [Mq]{Ci}E,
(2)
where only linear terms have been related retained [24]. [Mu] an d [Oi]] are the mobility and diffusion coefficient matrices, Mq =Mi6 u and
dr / + dr
,j,
(4)
where M and C are the mobility and concentration of the relevant variable and E, is the radial (and only) c o m p o n e n t of the applied field. U n d e r the experimental conditions reported by P6rez-Mufiuzuri et al. [21], it is the result of a positively charged electrode located at the core of the spiral and a negatively charged circular electrode located at the vertical wall of the Petri dish. The influence of the circular electrode on spiral wave dynamics is negligible except for those spirals developing very close to the wall.
311
J.L.F. Porteiro et al. / Influence of radial electric field on wave dynamics
For spirals developing far from the wall the influence of this electrode is several orders of magnitude smaller than that of the central electrode and for the sake of computational expediency may be left out of the overall formulation. Only the portion of the central electrode immersed in the solution is considered to contribute to the electric field strength. This strength evaluated at radial distance r from the electrode tip is functionally given by E~=4,tre =
The 1/r singularity exhibited by E r at the origin is handled numerically by establishing a minimum radius r 0 below which its value is arbitrarily prescribed. It can be seen from eq. (4) that the concentration, C, as well as its radial gradient d C / d r act as multipliers of the effects of the electric field. A comparison of the relative strengths of E r, the gradient multiplier, and Er/r + d E r / d r , the concentration multiplier as a function of r is shown in fig. 1 which was obtained using an electrode length, I, of 1 space unit. Only in regions where high values of C and low values of d C / d r occur simultaneously will the effects of the concentration term be larger than those of the other term. Such a condition will be found mostly at the crest of the wave, where the high value of C at the front of the wave gently slopes down to a lower value at the back of the wave. At the front and back of the wave d C / d r is large and the contribution of the gradient term is several times larger than that of the concentration term.
r2 1+
(5)
Q ~(r,l) 4~re
where Q is the charge per unit length of electrode, e is the dielectric constant of the solution and l is the length of electrode immersed in the solution. Er is scaled to units of ( D k s [ M A ] ) I / 2 / MBr- [15]. For the values of [MA] and k 5 used above and with M B r - = - 4 × 10-4cm2/Vs the unit of electric field strength is - 2 . 4 V/cm. 2.5
d~ S
ds
o
1.5
0.5
1 -0.5
~
2
3
4
5
6
7
8
S (su)
Fig. 1. Variation of the intensity of the electric field components (concentration and concentration gradient multipliers) with radial distance.
312
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
4. N u m e r i c a l m o d e l i n g o f the r e a c t i o n - d i f f u s i o n process: the O r e g o n a t o r
In order to simulate the B Z chemical reaction, the two variable version of the O r e g o n a t o r with T y s o n ' s " L o " p a r a m e t e r s was used [8]. The applicable equations, for a spatially uniform, unstirred medium, after scaling, are OU Ot -
_1 [ 2 e u - u
fv
u-q
+ Du
V21A
Ov Ot
- u - v ,
(7)
where u is the ' p r o p a g a t o r ' variable representing the dimensionless concentration [HBrO2] and v, the ' r e c o v e r y ' variable, is the dimensionless concentration [ferriin]. E u is the field strength per unit length of electrode and is equal to - M Q / 4 ~ e . The minus sign is added for convenience as it accounts for the negative value of M. In this way positive applied voltages correspond to positive values of E,. T h e variable s is the scaled radial distance r. T h e p a r a m e t e r s q and e are functions of concentrations and rate constants. A detailed discussion of their values and influence upon the B Z reaction can be found in [11]. D , and D v are the dimensionless diffusion coefficients of u and v and their ratio is important in controlling wave dynamics in general and wave tip meandering in particular. As in our simulation we consider ferriin (or variable v) to be immobilized in a silica gel [21,22,25], we have taken their values to be D u = l and D r = 0 . This has the added benefit of faster execution times as the Laplacian of the recovery variable does not need to be calculated. All our simulations were made with q -- 0.002, e = 1/100 and f = 1.4. For the experimental condition of our simulation [MA] = 0.15 with D u set to ( D / k s [ M A ] ) 1/2 and with k 5 = 0 . 4 M - i s -1 and D = 1.5 x 10 -5 cm2s -1 [8,9] our unit of
space scales to approximately 0.16 mm. Our unit time scaled with 1 / k s [ M A ] is about 17 s. Spirals were generated from the set of initials conditions suggested by Jahnke, Skaggs and Winfree [10] i.e. a narrow wedge of excitation with u = 0.8 and v near equilibrium followed by a smoothly declining circular gradient of refractoriness. This resulted in a counterclockwise rotating wave on grids as small as 60 x 60 and as large as 500 x 500. A uniform time step of 1/ 1000 was used throughout as the differential equations were integrated using the explicit E u l e r m e t h o d with a five point central difference approximation being used for the calculation of the laplacian and no-flux boundary conditions. A grid mesh of 0.125 space units was chosen but runs at grid spacings as large as 0.1875 units did not result in significant changes in wave parameters. E v e n though most of our simulations were carried out on a 200 x 200 grid, the effect of gridsize on wave p a r a m e t e r s was studied on grids as small as 80 x 80 and as large as 500 x 500.
5. C o m p u t a t i o n a l p r o c e d u r e
Special care must be taken in order to guarantee that the discretization of time and space will not significantly influence the resulting spiral wave. It was found that while the condition of D u = 1 and D v = 0 significantly increased the sensitivity to the grid anisotropy introduced by the five point laplacian formulation used, a grid spacing of 0.125 space units together with a 0.001 time step gave rise to spirals without any perceptible anisotropy on 200 x 200 grids. Following Jahnke et al. and Jahnke and Winfree [10,11] u was reset to a value of q whenever it fell below that value in order to increase the allowable time step and to avoid negative values of u that are physically impossible. G e n e r a t e d spirals were allowed to develop f r o m the integration of eqs. (4) and (5) until their rotation periods became constant, at which time the electric field was "turned on".
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
In order to achieve f a s t e r computation times neither the laplacian nor the electric field effects were calculated at those points such that the value of u at the surrounding points (involved in the evaluation of the laplacian) fell below a threshold value. This is similar to Barkley's " b o u n d a r y layer" technique [26,27] for evaluating those points within a small "boundary layer" near the left branch of the u-nullcline. In the present case the fact that all points involved in the evaluation of the laplacian fall inside the boundary layer implies that the values of the laplacian, d u / d r and u will be small. As a result the contribution of the reaction term will be much larger than that of the diffusion and electric field terms and little error is introduced by their elimination. Since a large percentage of all points are within this boundary layer, the savings in computational time are significant. Spirals generated this way are undistinguishable from those generated from the evaluation of all contributing terms at all points of the grid. The electrode was assumed to be located at the geometrical center of the computational grid for all positive electric field cases investigated. When a negative electric field was used it was necessary to alter its location to insure that it was always located at the tip of the spiral.
6. Results
The effects of application of a positive electric field on a fully developed spiral can be observed in fig. 2. In order to better illustrate the process a large grid of 500 x 500 with a grid spacing of 0.1825 was used. The spiral shown on fig. 2a was generated in the absence of an electric field. A total of four rings can be observed with a typical front width of twelve gridpoints. An electric field of intensity E u = 10 except for a radius of 8 gridpoints around the electrode, where E u = 0, and electrode length of 8 gridpoints is then turned on. The evaluation of the spiral after 2.5 time units is shown in fig. 2b. It can be clearly
313
observed that the spiral center is reducing its front curvature and at the same time the front becomes wider, the sections near the spiral tip having now a width of approximately sixteen gridpoints. At the same time it can be observed that the affected section of the spiral alters its rotation rate as it slows down. The process can be observed to continue in figs. 2c-e. The fronts increase to an average width of 28 gridpoints and the spreading out process has reduced the spiral to only one and a half turns. The configuration shown in fig. 2e has reached equilibrium and remains unchanged a further 22.5 time units later, its rotation rate also unchanged. When the electric field was turned off, the spiral returned to its original state without any signs of hysteresis. In order to simulate the influence of the electric field intensity on spiral wave parameters, simulations were carried out on a 200 x 200 grid with a grid spacing of 0.125 space units and an electric field similar to the one described above except for a variable intensity E,. After a steady rotating spiral was generated, the electric field was turned on at a low value. The new spiral was allowed to develop and stabilize a t which time its parameters were measured and then the electric field was increased. Once the maximum value of E u was reached, field strength was decreased using the same procedure. The influence of the electric field strength on the shape and width of the wave for positive fields is shown in fig. 3. It is evident that stronger positive fields result in thicker wavefronts with reduced curvature. The effects of very intense fields such as those in figs. 3e and f result in very complex patterns near the spiral tip. It was observed that while spirals generated with positive electric fields exhibited characteristic tip meandering behavior, they steadily rotated around the positive electrode located at the center of the grid. This however was not true for negative field spirals. The presence of the negative electric field increased tip meandering to the point that after a certain period of time spiral
314
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
Fig. 2. Unfurling of a spiral wave under the influence of an electric field of intensity E u = 10 except for the region r < 1.5 su, where E r = 0. The computational box has 500 x 500 gridpoints and a gridspacing of 0.1875 su. The electrode length is 1.5 su. After 15 tu the spiral has reached steady state as shown in (e) and (f).
r o t a t i o n was no longer a r o u n d the electrode located at the center of the grid. As a result of this, the spiral eventually m o v e d across the negatively c h a r g e d electrode. A n e x a m p l e of such a process is shown in fig. 4. It can be o b s e r v e d in fig. 4a that the electrode (located at the g e o m e t r i c center of the figure) interacts with the spiral tip by d e f o r m i n g it. As the spiral continues its rotation, further d e f o r m a t i o n takes place fig. 4b and eventually the spiral is split in two. T h e two spiral sections grow and advance past the electrode, eventually linking up and r e g e n e r a t i n g a single spiral once again (figs. 4e and f). In o r d e r to insure that the electrode was always located at or in the n e i g h b o u r h o o d of the spiral tip, it was necessary to alter its location
f r o m the center-grid position used for positive fields. Fig. 5 shows the effects of a negative electric field on a steady rotating spiral on a 200 x 200 grid with 0.125su grid spacing. D u e to the p r e s e n c e of the negative electric field, slight tip d e f o r m a t i o n can be o b s e r v e d in fig. 5b and significant d e f o r m a t i o n is found in figs. 5c and 5d. As a result of this d e f o r m a t i o n , well established m e t h o d s of defining and locating the spiral tip, such as those described by ] a h n k e et al. [10], failed. N e w m e t h o d s b a s e d on tip t o p o l o g y were d e v e l o p e d in o r d e r to d e t e r m i n e such location [28]. T h e spirals were o b t a i n e d by m o v i n g e v e r y two iterations the position of the e l e c t r o d e so that it would coincide with that of the spiral tip.
315
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics //
I
(b)
A
Y
% (c) Eu =
10 t~A
(~)
(f)
(e) Eu
=
Eu = 500
100
Fig. 3. Influence of field intensity on front curvature for positive values of E,. (200 x 200 grid, 1/8 su gridspacing, E, = 0 for r < 1.5 su; electrode length, 1.5 su.)
It is clear that as the intensity of the negative electric field increases the spirals contract upon themselves and increase the overall wavefront curvature. It should be noted that the spirals shown for negative values of E u above 10 are not stable due to the progressive deformation of the spiral tip. This tip deformation is evident in fig. 5c and d and finally results in the impossibility of obtaining an equilibrium configuration for these values of E,. Fig. 6 illustrates the relationship between the rotation period of the spiral and the intensity of the electric field. As shown on the figure inset, for negative and low positive values of E, (those below a value of 4) the period is weakly affected by the electric field. For higher values rotation periods are dramatically increased and tend to
Fig. 4. Interaction of a negative electric field (E, = - 4 ) with a fully developed spiral. The electrode is fixed and located at the center of the figure. Wavefront deforms (a), (b) and breaks (c) when approaching electrode. Once past the electrode it grows (d) and reattaches (e), (f). (80 x 80 grid, 0.125 gridspacing.)
stabilize for the highest values of E, at a value about three times higher than that of the baseline spiral without an electric field. Rotation periods obtained with E, = 1000 were within 1% of those obtained with E, = 500. When the value of E, was decreased the spirals obtained exhibited the same parameters as those obtained when E u was increasing and no hysteresis effects were observed. An investigation of the influence of the modeling of the electric field in the neighborhood of the electrode on spiral wave parameters was carried out for several values of the electrode
316
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
both models are of the same magnitude as our experimental error.
7. Discussion
Fig. 5. Influence of field intensity on negative values of E computed on a 1 / 8 s u gridspacing. Observe strong tip negative electric in c and d. ( E = 0 for length, 1.5 su.)
front curvature for 200 x 200 grid with deformation due to r < 1.5 su; electrode
length, l, and the electrode radius, r0, below which the strength of the electric field was arbitrarily prescribed. T w o models of the electric field were studied. In one E, was given a zero value for r < r 0 and in the other E r was m a d e to vary linearly with r such that its value was zero at r = 0 and continuous at r = r 0. T h e influence of r 0 on the rotation period of the spiral for the two models is presented in fig. 7, which was obtained on a 200 x 200 grid with 0.125 gridspacing. No difference was observed b e t w e e n the two models. It can be noted from observation of this figure that electric fields effects decrease with r 0 and that spirals with values of r 0 larger than 3.75 su have a period of rotation similar to those obtained in the absence of an electric field. It is clear from this that modeling of the electric field near the electrode is not very important for values of r 0 below 4 su, as the differences observed in the periods for
A comparison of the process illustrated in fig. 2 with the experimental data obtained by P6rezMufiuzuri et al. [21] shows that an O r e g o n a t o r based model is capable of reproducing qualitatively the effects of a radial electric field on spiral waves generated by a B Z reaction. As in the experimental case once the electric field is r e m o v e d , spirals return to their original state. T h e formation of spirals in the O r e g o n a t o r model is the result of the interaction of the reaction and diffusion terms in eqs. (6) and (7). This interaction is altered by the electric field in a complex m a n n e r since a dependency on both u and its radial gradient must be taken into account. For positive values of E u, while the concentration term is always negative, du/dr is negative at the front of the wave and positive at the back and as a result so is the gradient term. In this way the contribution of the electric field to du/dt is negative in the front of the wave and positive at its back. The result is a slowdown in the advance of the front that is proportional to the local field strength. For negative values of E , the effects of the electric field are reversed and wavefront acceleration results. It should be noted, however, that the influence of negative electric fields must be less than that of positive ones. This follows from the fact that, regardless of the intensity of the negative electric field, it is not physically possible to obtain negative values of the concentration u (numerically, as above indicated, values of u below the equilibrium value are restored to equilibrium). This buffering mechanism could explain the markedly w e a k e r influence of negative fields when c o m p a r e d with positive ones that is evident in figs. 3 and 5. It is clear from fig. 2 that the electric field does not act instantaneously throughout the wave. This seems to indicate that the influence of the
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
317
Period
(tu) 4.5
4 J
J
3.5 2,5 ¸ 3
Period
(tu)
2.5
2
2 1.5 1.5
J
1
0.5
I
p
i
-50
-25
1 0
i
i
25
50
Eu
0
-50
0
t
I
I
I
I
I
I
I
I
I
50
100
150
200
250
300
350
400
450
500
EU Fig. 6. R o t a t i o n period of spiral waves as a function of electric field strength. T h e asymptotic value for high value of E u is a p p r o x i m a t e l y three times the rotation period with no field. Inset: B e h a v i o r at low field intensities. (E r = 0 for r < 1.5 su; electrode length, 1.5 su.)
Period (tU) 1.9
1.8
1.7
•
Er = kr f o r r < r O
•
Er = 0 f o r r < r O Eu = 10
I = 1.5 su 1.6
1.5
1.4
1.3 0
I
I
I
I
0.5
1
1.5
2
rO
I
I
[
I
2.5
3
3.5
4
(su) Fig. 7. Influence of electric field core modeling on the rotation period of the spirals. Periods are identical for both models. For spirals where with values of r 0 above I su the period decreases rapidly until it reaches the no field value for a core radius of about 3.5 su.
318
J.L.F, Porteiro et al. / Influence o f radial electric field on wave dynamics
electric field is only felt, at least for moderate field intensity values, by the sections of the spiral closest to the tip and that diffusion plays an important role in propagating the disturbance to the rest of the spiral. As a result the front straightens out and slows down locally while the rest of the spiral "catches up" with it and also straightens out and slows down. Further support for this mechanism can be found in fig. 7 which shows that spirals with a radius r 0 at or above 3.75 su exhibit the same rotation period as a spiral with no electric field at all. For so large electrode diameters the electric field acts only upon the outer sections of the spiral, while the tip and inner sections are unaffected. The fact that under these conditions both wavefront shape and rotation periods are identical to those obtained with E u = 0 seems to indicate that the radial electric field effects on the outer section of the spiral are overwhelmed by the diffusion processes. From the results shown in fig. 3 it is clear the straightening process slows down dramatically for high values of E u, as larger values of the electric field are required to generate smaller effects on both front curvature and rotation period. This can be explained since the influence of the du/dr term decreases steadily with high values of E , as a consequence of the straightening of the wave front. It can be seen that for high values of the field such as those in fig. 3f, the spiral tip is almost radially aligned. In this condition the value of du/dr is very small and large values of Eu are needed to make the term significant. In addition it must be remembered that at the front of the spiral wave the contribution of the electric field to du/dt is negative. Since the value of u is never allowed to fall below that of q, this further reduces the influence of very high values of E,. /t is remarkable that the present model can so faithfully reproduce the experimental values obtained for the ratio of the maximum rotation period (high electric field) to the baseline rotation period (no field) and this seems to be a product of the
underlying reaction-diffusion characteristics of the B Z reaction. The absence of hysteresis effects is most likely a consequence of neglecting the local ionic field in eq. (4). The presence of an externally imposed field on the B Z reagent leads to a redistribution of ionic charges throughout the solution in such a way that equilibrium is reached for the new conditions. As the external electric field is withdrawn this distribution will no longer be in equilibrium and will relax through a certain period of time to a new equilibrium state. This period, for the experimental case, will depend on the initial charge distribution and the diffusion coefficient. In our numerical simulation the absence of a local ionic field results in a relaxation to the original (no electric field) state that is only controlled by the interrelationship between the diffusion and reaction terms in eq. (4). The failure to generate stable, steadily spirals for even moderately low negative values of E~ raises the question of whether if it is indeed possible to obtain such spirals. In sharp contrast, much larger positive fields cause significant tip deformation but the resulting spirals are stable and steadily rotate around a fixed electrode. While such a failure could be attributed to the difficulties associated with the localization of the spiral tip, it is quite likely that those difficulties arise from tip deformation caused by the strong electric fields present in the neighborhood of the tip. It is also likely that the mechanisms responsible for wavefront breakup shown in fig. 3 are quite capable of altering tip dynamics in a similar manner.
8. Conclusion
The effects of an externally applied electric field on spiral wave dynamics have been successfully modelled. While the results obtained are in good agreement with the experimental data, much remains to be done in order to understand how some aspects of the modelling, such as the
J.L.F. Porteiro et al. / Influence o f radial electric field on wave dynamics
d e s c r i p t i o n o f t h e e l e c t r i c field n e a r t h e spiral tip, affect m o r e c o m p l e x w a v e b e h a v i o r such as m e a n d e r i n g a n d w a v e f r o n t stability for m o d e r a t e a n d l a r g e n e g a t i v e v a l u e s o f t h e electric field. It c a n also b e c o n c l u d e d t h a t t h e O r e g o n a t o r m o d e l has p r o v e n to b e r e m a r k a b l y r o b u s t a n d flexible t o o l c a p a b l e o f r e p l i c a t i n g s o m e o f t h e most intriguing aspects of the BelousovZhabotinsky reaction.
Acknowledgements T h i s w o r k was s u p p o r t e d in p a r t b y the C o m i s i 6 n I n t e r m i n i s t e r i a l d e C i e n c i a y Tecnologia (Spain) under grant DGICYT-PB910660.
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