J. Phys. Chem. 1994,98, 2494-2498
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Magnetic Resonance Imaging of Kinematic Wave and Pacemaker Dynamics in the Belousov-Zhabotinsky Reaction Sunyu Su Institute for Biodiagnostics, National Research Council Canada, 435 Ellice Avenue, Winnipeg, Manitoba, Canada R3B 1 Y6
Michael Menzinger' Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 1Al
Robin L. Armstrong Department of Physics, University of New Brunswick. Fredericton, N B Canada E3B 5A3
Albert Cross and Claude Lemaire Department of Radiology, University of Toronto, Toronto, Ontario, Canada M5S 1Al Received: April 29, 1993; In Final Form: December 2, 1993'
Using a previously described projection technique for magnetic resonance images (MRI), we have studied the dynamics of kinematic waves arising from frequency gradients in a n anisotropically cooling, oscillating BelousovZhabotinsky (BZ) medium. The gradual conversion of purely kinematic waves into diffusion-driven, chemical waves is illustrated by comparing the time dependence of the experimental wave velocity with the velocity calculated from a model for an idealized kinematic wave that excludes diffusive effects. Finally, we have studied the interaction of kinematic waves and trigger (chemical) waves. These measurements provide the explanation for previous observations that had suggested sudden changes of wavefront velocities.
Introduction Two classes of wave phenomena occur in chemically active media.' Chemical (or trigger) waves occur in excitable and oscillating active media, and they are associated with material transport and are driven by diffusion. Kinematic waves2s3(also called phase waves or pseudowaves4), on the other hand, are not associated with matter transport and hence may occur also in the presence of physical barriers, and they arise from phase gradients and/or frequency gradients in oscillating media. Trigger waves propagate at a constant velocity determined by the properties of the medium, while the velocity of kinematic waves, v = (V@)-I, is variable since it depends inversely on the (local) phase gradient. As phase and frequency gradients evolve in time, for instance through the anisotropic cooling of an initially homogeneous oscillating medium, the velocity v is initially infinite and diffusion is unimportant. At later stages, however, when the phasegradient and with it the concentration gradients have grown, diffusion becomes increasingly important. Hence there occurs a continuous transition from kinematic to chemical, diffusion-driven waves. The above phenomena are well-known'-s and easy to observe visually. They have been analyzed*,3 using conventional photography. The MRI technique is well suited for accurately measuring the v e l ~ c i t yof ~ , waves ~ propagating in one effective spatial dimension. The high-resolution displacement versus time images resulting from a stack plot of sequential projections provide a convenient and realistic representation of waves propagating in one effective spatial dimension.6-7 This technique is used here to study thedynamics of kinematic waves, arising from a frequency gradient that evolves in the course of the free cooling of an oscillating BZ medium contained in a vertical tube, and their gradual transformation into diffusion-driven waves. For a quantitative analysis of this phenomenon, the temperature and the frequency distributions in the system must be known. Since an analytic expression for the time-dependent temperature ~~~
~~~
*Abstract published in Aduance ACS Abstracts, February 1, 1994.
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distribution is known for the free cooling of a cylinder, the dynamics of pure (and hypothetical) kinematic waves in this system can be predicted. The real BZ system deviates from such purely kinematic predictions due to the increasing involvement of diffusion. The comparison of the experimentally measured wave velocity with the velocity of idealized, purely kinematic wave gives a quantitative description of the increasingly important roleof diffusion in thecontinuous transition from kinematic waves to chemical waves. The imaging technique is also used to illustrate the interaction of kinematic waves with trigger waves emitted from a localized pacemaker center and their convergence to a common dynamics. Last, but not least, these experimental observations provide an explanation of the previously observed apparent sudden changes of wave front velocities.6 Experimental Section An oscillating BZ solution of the following initial composition was prepared at room temperature (21 "C) by mixing stock solutions with a magnetic stirrer in the following order: [H2S04] = 0.2 M, [KBr] = 0.06 M, [H3P04]= 2.8 M, [CH2(COOH)2] = 0.15 M, [NaBrOs] = 0.05 M, [MnS04] = 0.0006 M. After stirring for about 1 min, the solution was poured into a test tube of 7.7-mm inner diameter and 50-mm length, which was subsequently placed into the spectrometer bore where theambient temperature was 17 OC. The temperaturedistribution that evolves in the course of free cooling is reflected by the observed characteristics of the waves in space and time. A General Electric 2T Omega magnetic resonance imager with self-shielded gradient coils was used. The sample was placed in a center-tapped resonator coil 45 mm in diameter, tuned to the proton resonance frequency of 85.5624 MHz. Slices of 2.0-mm thickness were acquired both prallel and perpendicular to the tube axis, and subsequent projections of these slices onto one of the axes were stacked into a displacement versus time format6,' that provides a direct representation of the wave motion. For plane wavefronts propagating quasi-one-dimensionally along the 0 1994 American Chemical Society
Kinematic Wave and Pacemaker Dynamics
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Figure 1. Six consecutive,:Z displacement verus time images taken at (a) 1, (b) 18.03, (c) 36.13, (d) 54.20, (e) 72.23, and (f) 109.67 min after the preparation of the BZ mixture. The abscissa spans 1024 s and the ordinate axis is 80 mm.
axis of the tube, no information is lost by displaying the onedimensional projection onto the tube axis of a 2D slice taken through that axis. The symbol 1; was used7 to denote a projection versus time image, where the subscripts designate the plane in which the slice is imaged, and the superscript is the axis onto which the image is projected. The z axis is parallel to the axis of the tubular reactor. The projections were acquired using a spin-echo sequence with TE= 50 ms and TR= 1 s. The time between individual projections
in a specific image was 4 s. Each image comprises 256 X 256 points, which corresponds to a time-space range of 1024 s X 80 mm. Results and Discussion Transitionfrom Kinematic Waves to Chemical Waves. Figure 1 shows six sequential projection-versus-time fxr images. The images are strongly T2 weighted. The white-to-black contrast reflects the difference in water proton relaxation times in the
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Figure 2. Comparison of the experimental wave velocity (crosses) with the calculated time dependence of wave velocity of idealized kinematic waves (solid line) at z = 16 mm. ( z = 0 is the tube center).
presence of the paramagnetic ions Mn2+ and Mn3+. The white regions correspond to regions of high Mn3+ concentration; the dark regions to regions of high Mn2+ concentration. Acquisition of the first image, Figure la, was started 1 min after preparing the solution and placing it into the spectrometer bore, while the temperature distribution was almost uniform. The initially vertical bright band is followed at increasing time intervals by crescent-shaped wave projections of increasing curvature. After about 3000 s, the horns of these crescent-shaped projections reach a constant finite slope, indicating that the corresponding wave fronts have attained a limiting velocity. The wave front velocity u(z) can be read off directly as the slope of the graph at position z. The qualitative interpretation of these observations is as follows: As a temperature gradient develops due to the more rapid cooling of the free ends of the sample relative to its center, the oscillation frequency7J3Q decreases more rapidly at both ends. The resulting phase gradient V b = JhVQ(z,T ) dT accumulates in time proportional to the frequency gradient, and the local wave velocity initially decreases inversely with V b and hence with time. At long times however, the wave velocity reaches a finitevalue,the speeed utr= (kD) lI2of diffusioncontrolled trigger wavesg (where k is the effective rate coefficient of the autocatalytic step and D the diffusion coefficient of the autocatalyst). The increasing importance of diffusion in the time dependence of the velocity u ( t ) of these initially kinematic waves becomes clear through a comparison with the velocity uKn(t) = (Vb)-l of idealized kinematic waves that exclude diffusive effects. In the presence of a space and time dependent temperature field T(z,t), the local frequency evolves according to
where E, = 50 kJ/mol is the known7 activation energy of bulk oscillations and @ = E , / R = 6 X lo3 K. The kinematic wave propagates2 with a velocity given by
Figure 3. Displacement versus time images taken ca. 10 min after preparing the sample: (a) I‘,image; (b),:Z image. The abscissa spans 1024 s and the ordinate axis is 80 mm for (a) and 20 mm for (b).
T(r,z,O) and immersed in an infinite bath which is at the reference temperature. The temperature distribution in the tube can be expanded aslo OD
T(r,z,t) =
7,x C n icos(nnz/l)e-AftJo(pir/u) n
(4)
i
where 1is the length of the tube, z is the coordinate along the tube axis with z = 0 in the middle of the tube, r is the radial coordinate, Ai are time constants, JOis the zero-order Bessel function, pi is the solution of the ith node of Jo, and u is the diameter of the tube. The coefficients &can be determined from the initial conditions. For the initial uniform temperature To, eq 3 reduces to W
T(r,z,O) = To =
TaY,Cni cos(naz/l)Jo(pir/a) n
where the phase is
i
A straightforward calculation leads to
(3) and subscripts indicate partial derivatives. For a stationary temperature distribution (Tt = 0), the kinematic wave velocity u(x) = P / t @ T decreases z inversely with time as the phase gradient accumulates. The time dependenceof the distributioncontributes the second term in eq 2. The temperature field T(r,z,t) is calculated by considering the free cooling of a vertical cylinder, initially uniformly at TO =
Hence the time-dependent temperature distribution can be written as
The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 2497
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Figure 4. Four consecutive I", iimages taken (a) 23.00, (b) 47.15, ( e ) 73.02, and (d) 108.27 min after beginning of data acquisition for Figure 3.
The radial function is integrated in the acquisition of the MRI projection:
where Bo is a constant. The time dependence of the temperature at fixed z has been measured previ~usly.~ The experiment showed that the time dependence of temperature measured at the center of the tube can be well represented by a single exponential, suggesting that only the i = 1 term need be considered. Hence the radially averaged temperature distribution reduces to
Substituting ( 5 ) into (1) and (3) yields, after some algebra, the following expression for the wave velocity:
where
Derived for idealized, purely kinematic waves, eq 6 correctly predicts an infinite wave velocity at the origin ( z = 0) and a zero wave velocity at the ends of the tube for t a. The experimental wave velocity however converges to a finite limit, the diffusion controlled velocity ut, = (kd)l/*of pure trigger waves.9 Rather than evaluateE(z) numerically, it was obtained together with A directly from the experimental data u(t1z) as follows: in the limit exp(At) >> 1 , (6) reduces to Ukin(Z,t) D(z) exp(E(z)(exp(At) - 1)). This is recast as l n ( 4 In u/&) = ln(E(z)A) -At. By fitting a straight line to a plot of l n ( 4 In u / d t ) against t, oneobtainsA andE(z) from theslopeand intercept,respectively. The results of this analysis are A = 0.93 X 10-3 and E = 2.18. The experimental velocity of consecutive wave fronts at a fixed z near the top of the tube (ca. 16 mm above the center) is shown in Figure 2 by the circles. The solid line represents the velocity of hypothetical phase waves calculated from (6) using the above values of A and E and normalizing to the first point. The velocity u ~ ,of, the purely kinematic wave is seen to decay to zero extremely
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slice Figure 5. Model of the dynamics of a sphericallyexpandingwave emitted from an off-center pacemaker; d is the eccentricity and dr/dt = u the
velocity. rapidly with the doubly exponential time dependence given by eq 6. The actual wave velocity deviates from this idealized case due to the involvement of diffusion which becomes increasingly prominent with time. Asymptotically the wave velocity approaches that of purely diffusively driven trigger waves, in the present case 0.33 mm/s. Interaction of Kinematic and Trigger Waves. Figure 3 represents a second series of displacement-time images obtained from a solution prepared using the same formula. This figure shows the interaction of phase waves, similar in appearance to those in Figure 1, with trigger waves emitted periodically from a pacemaker near the center of the tube. Parts a and bof Figure 3 represent and It,,. Two other projections, GXand Z& were also obtained; they add no additional information. Acquisition of the first image, Figure 3a, was startedca. lOmin after preparing the solution and placing it into the cool spectrometer bore-a time sufficient for the top and bottom of the tube to have cooled sufficiently for the oscillation period7-*a t the ends of the tube to increase relative to the center of the tube and the crescent shape to develop. The limbs of the cones that merge with the phase wave crescents represent the projections of the upper and lower boundaries of spherical trigger waves emitted from a periodic pacemaker that is located near the apex of the cone (the fact that this cone is rounded off is explained below). The images of Figure 4 show that these apices appear a t successively earlier times relative to the arrival of the kinematic wave crescents, indicating that the pacemaker operates at a frequency that is slightly higher than that of the bulk oscillations. This difference in frequency accounts for the gradual displacement of the phase were crescents by the pacemaker cones.9 The series of images also reflect the increasing oscillation periods due to anisotropic cooling. As the trigger wave gradually displaces the kinematic wave, the hornsofthelatter decreaseinslopeuntil they reach thevelocity of the former and become indistinguishable from it. These images are another illustration of the gradual conversion of a kinematic wave into a trigger wave propagating by the classical reactiondiffusion mechanism. In addition to the two waves described above, the plots of Figures 3a and 4 show that the bottom of the tube emits a trigger wave of a slightly lower frequency than that emitted by the pacemaker near the center. Consequently it is gradually displaced by the latter. The Z:y plot of Figure 3b is consistent with the above interpretation of Figure 3a: the asymmetric cone that precedes the leading edge of the crescents shows that the pacemaker is located near the periphery of the tube and that the spherical symmetry of the chemical wave is perturbed by interaction with the wall. The facts that the pacemaker is off-center and that the
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”(1 Figure 6. Comparison of experimental displacement versus time data with a calculated curve based on the wave equation derived from the model of an off-center pacemaker. xz imaging plane passes near the center of the tube causes the apices of the cones to be rounded off. Simple geometric considerations, as illustrated in Figure 5 , lead to the equation for the wavefront arising from a pacemaker located a t a distance d from the imaging plane:
a2(t) = uZt2- dZ
(7)
where t, z, and u are the time, wavefront position, and wavevelocity, respectively. This equation provides a near-quantitative fit of the blunt pacemaker cones of Figures 3a and 4 as shown in Figure 6. In Figure 4 of an earlier paper6 we presented data showing two wavefronts that first converge linearly and then apparently accelerate rapidly before they annihilate upon collision. This sudden acceleration appeared paradoxical a t the time;6 the present results provide a natural explanation of the phenomenon. The sequence of Figure 4a-i in ref 6 shows two trigger waves, emitted periodically by some pacemakers at the bottom and top of the oscillating medium which propagate with constant velocities toward the center. In addition, the bulk oscillation of the phasedistributed medium and some dephasing mechanism give rise to crescent shaped, rapidly moving “kinematic” waves, not unlike those of Figure 1 of the present paper. These kinematic wavefronts colloide with the ascending and descending trigger waves and give rise to the unusual wave front projection plot reported earlier? Due to their higher frequency, the trigger waves slowly displace the kinematic waves, not unlike the scenario shown in Figure 3 of the present paper. In reality, there is no sudden change of wavefront velocities. Acknowledgment. This work was supported in part by the NSERC of Canada and by the Faculty of Arts and Sciences at theUniversityofToronto. We thankFathei Ali for computational assistance. References and Notes (1) Murray, J. D. Mathematical Biology; Springer-Verlag: Heidelberg, 1989. ( 2 ) Beck, M. T.; Varadi, Z . B. Nature 1973,235,15. Thoenes, D. Nature 1973, 243, 18. (3) Kopell, N.; Howard, L. N. Science 1973, 180, 946. (4) Winfree, A. T. Science 1972, 175, 634. (5) Haken, H. Dynamics of Synergetics Systems; Springer-Verlag: Heidelberg, 1980; p 80. (6) Tzalmona, A,; Armstrong,R.L.;Menzinger, M.; Cross, A,; Lemaire, C. Chem. Phys. Lett. 1992, 188,457. (7) Su, S.; Armstrong, R.L.; Menzinger, M.; Cross, A.; Lemaire, C. J .
Chem. Phys., in press. (8) Koros, E. Nature 1974, 251, 703. (9) Mikhailov, A. S. Foundations of Synergetics I: Distributed Active Systems; Springer-Verlag: Berlin, 1990. (10) Goertzel, G.; Tralli, N. Some Mathematical Methods of Physics; McGraw-Hill: New York, 1960.
