The Journal of
Physical Chemistry
0 Copyright. 1992, by the American Chemical Society
VOLUME 96, NUMBER 12 JUNE 11,1992
LETTERS Dynamics of Convective Instability of Waves in the Beiousov-Zhabotinsky Reaction As Measured by Magnetic Resonance Imaging M. Menzinger,* Department of Chemistry, University of Toronto, Toronto, Ontario, Canada, M5S 1 A1
A. Tzalmona,+ Department of Physics, University of Toronto, Toronto, Ontario, Canada, M5S 1Al
R. L.Armstrong,* Department of Physics, University of New Brunswick, Fredericton, New Brunswick, Canada, E3B 5A3
A. Cross, and C. Lemaire Department of Radiology, University of Toronto, Toronto, Ontario, Canada, M5S 1A1 (Received: October 2, 1991; In Final Form: April 16, 1992)
We report the evolution of convective instability of wave fronts in the Mn2+-catalyzedBelow-Zhabotinsky reaction, propagating upwards in a vertical tube. The instability is driven by the, independently measured, isothermal density decrease that occurs during the oxidative phase of the BZ reaction. Use is made of a method, based on magnetic resonance imaging, that was recently developed (Tzalmona et al. (1992)) for the accurate velocity measurement of chemical waves propagating in one effective spatial dimension.
Introduction
The dynamia of chemical waves in initially homogeneous media is traditionally modeled by reaction diffusion equations.’ This description fails, however, when the density gradient caused by the reaction front promotes hydrodynamic flow, in which case the wave velocity may differ dramatically from that of reactiondiffusion waves. Pojman et aL2 have analyzed convective effects ‘On sabbatical leave from the Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel.
on chemical waves in different chemical systems in solution through the influence of gravity on steady-state front velocity, i.e., through the difference in velocity of ascending and descending wave fronts in vertical tubes. Density gradients across reaction fronts may arise from an isothermal density change as well as from thermal expansion or contraction due to the heat of reaction. When both gradients are parallel and point up, “simple convectionn may arise. Antiparallel gradients may give rise to so called “double The present results demonstrate that the Belousov-Zhabotinsky (BZ) reaction3 represents the former case of 0 1992 American Chemical Society
Letters
4726 The Journal of Physical Chemistry, Vol. 96, No. 12, 1992
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Figure 1. Schematic illustration of the construction o f a displacementtime plot of plane waves propagating in one effective dimension.
simple convection, driven by the isothermal decrease of density in the wake of oxidative wave fronts. Hydrodynamic flow may occur not only in vertical geometries, but also in thin horizontal layers: in a Petri dish, convective cells were found4to accompany the well-known target patterns and spiral waves of the BZ reaction. The aim of this paper is twofold. It illustrates the use of a new, nonoptical technique, based on magnetic resonance imaging6 (MRI), that was recently developed5J for the accurate measurement of wave fronts traveling in one effective spatial dimension. The method is applied to demonstrate the sudden and drastic velocity changes that occur to the ascending portion of a spherically expanding chemical wave due to the onset of convection. The reacting medium was the Mn2+-catalyzedBZ system3 in its excitable domain. The isothermal density gradient which drives the convective instability was substantiated by measuring the periodic volume changes associated with the oscillations in a stirred BZ medium. Experimental Section
The experimental method is described el~ewhere,~.~ and we briefly recall here its main features. The primary output of the MRI technique: as applied to chemical waves in the BZ medium, is two-dimensional images7of the proton relaxation time of water, which is strongly affected by the presence of paramagnetic ions. In the present case, Mn2+ ions shorten the transverse relaxation time T2of protons relative to that in the vicinity of Mn3+ionss Experiments were performed with a General Electric 2.0 T Omega magnetic resonance imager. The velocity of onedimensional wave fronts may be accurately measuredSas schematically shown in Figure 1; as long as the wave fronts are essentially planar, as the ones shown at successive times in the left panel of Figure 1, no information is lost by projecting the two-dimensional images onto the (vertical) axis perpendicular to the wave front. By stacking several projections, as illustrated in the right-hand panel of Figure 1,a displacement versus time graph may be generated directly as the output of the imager, whose slope is an accurate measure of the wave velocity. An added advantage of this method is its speed; a single projection may be acquired in 10 ms, while it takes ca. 2 s to record a two-dimensional M R image. The solution, whose initial composition is given in the caption of Figure 2, is excitable. It was first homogenized by stirring and then placed in a test tube of internal diameter 4.3 mm. Bubble formation due to the release of C02 during the reaction3 was minimized by allowing the pressure to build up in the tightly closed tube.
Results and Discussion C m d v e Instabilityof Waves. Accelerating wave fronts were routinely observed in solutions of low viscosity, but convection could be suppressed by adding agarose. Figure 2 is a sequence
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I Figure 2. Successive displacement-time graphs n a solution of the following starting composiiion: [NaBrO,]= 0.05 M, [KBr] = 0.05 M, [MA] = 0.2 M, [MnSO,] = 0.001 M, [H2S04] = 0.95 M, (H3P04] = 1.5 M. Panels A-G were acquired at intervals of 268 s, followed by panels H and I after 540 and 1404 s, respectively.
of nine displacement versus time plots. Each panel is built up from 256 vertical projections, with each projection consisting of 256 spatial steps covering an 80-mm vertical range, and with adjacent projections separated in time by 1 s. The cusp which recurs at the same spatial location in every panel represents a pacemake? which periodically emits spherical waves. The two branches of the cusp are the projections of the upper and lower boundaries of the spherical wave. Both wave fronts travel at the same speed of 3.5 f 0.1 mm/min. When the fronts have diverged by 7.7 mm, ca. 66 s after the cusp, the upper front suddenly becomes unstable and develops into a rapidly rising plume which travels ca. 10 times faster than the initial wave. Given the tube diameter of 4.3 mm, this instability evolves a short time after the spherical wave has collided with the wall and has separated into two distinct wave fronts. It should be noted that, in the meantime, the descending waves travel unperturbed and at constant velocity. Only ascending wave fronts are subject to convective instability. These results illustrate the reasons for the previously noted’ difficulty in obtaining three-dimensional chemical structures that are not distorted b y convection, in tubes that exceed a certain diameter, unless the viscosity of the medium is enhanced through the addition of agarose. The successive plots, Figure 2.A-I, of the recurrent waves reveals also the evolution of a zigzag pattern of subsidiary pacemakers*(cusps) within the convective plumes. These recur with slightly smaller periods than the original pacemaker, thus moving toward earlier time. It is apparent that these pacemakers evolve from localized gradients at irregular locations within the convective plume. The physical nature of the pacemakers-be it heterogeneous or homogeneous9-remains elusive, as usual.* The instability of the ascending wave front may be understood in terms of hydrodynamic stability criteria2.**I0 and of a consideration of the density changes accompanying the oxidative fronts of the BZ reaction. The Raleigh number, a measure of the hydrodynamic stability of a fluid, is defined for a vertical cylinder of radius t aslo Ra = (PS/PC) dP/dZ where g is the gravitational acceleration, p is the kinematic viscosity, and C is the transport coefficient of the quantity causing the density gradient in the vertical ( 2 ) direction. The critical value
J. Phys. Chem. 1992, 96,4121-4130 Ra, above which convection develops is known1° to be Ra, = 61.9. In the exothermally reactive system considered here, both chemical and thermal gradients are present. However, thermal gradients may be neglected because the diffusivity K lo-, cm2/s of heat is about 100 times greater than the molecular diffusion coefficients D = cm2/s. From the equation it is evident that for a given concentration gradient there is a critical tube radius below which the wave front will propagate by a pure reactiondiffusion mechanism. Our results indicate that this critical radius is comparable to the tube radius. To substantiate the presence of sufficiently strong density gradients we have measured the density changes in an oscillating BZ medium as a function of time as follows. The bubble-free glucuse/acetone/Mn2+ version12of the BZ system was employed, and control experiments were performed with pure water. The deaerated solution was contained in a 280-cm3 glass flask fitted with a glass capillary of 0.65 mm inner diameter. The flask was magnetically stirred and was immersed in a water bath at 24.5 OC. The volume change was measured through visual observation of the position r(t) of the meniscus. The initial concentrations were [glucose] = 0.05 M, [acetone] = 0.025 M, [NaBrO,] = 0.01 M, [H2S04] = 1.0 M, and [MnZ+] = 0.001 M. The volume always increased in a stepwise fashion, with periodic jumps at the oscillation frequency. Decreases in volume were not observed. On a plot of &/At as a function of time one discerns periodic spikes of =0.002 cm/s that rise by a factor of ca. 1.5 above the reading error. These sudden expansions correspond to fractional changes of volume and density of AV/V = Ap/p = Ap = 2.4 X lo-'. They occur within the 10-s reading intervals at the oscillation period of the medium. Assuming that the density change across a propagating front is equal to that occurring in time in the stirred oscillating reaction, the density gradient across the wave front is estimated to be (dpldz), = 4.8 X 10" g/cm4. This value exceeds the critical gradient (dp/dz),, = 3.2 X 1odg/C" calculated from Ra, and suffices for instability. To the extent that the density changes of the bubble-free reaction are representative of those of the malonic acid BZ system, these measurements confirm the conclusions reached from the MRI measurements: the density decreases in the wake of the oxidative wave front, and the measured gradients are sufficient to induce convective instability. Consequently, only ascending waves are destabilized. An oxidative wave of the BZ reaction' may be viewed as the rapid autocatalytic oxidation of the metal ions, followed by the
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oxidative degradation of the organic substrate. During the first phase, the density is claimed" to increase due to electrostriction of the solvent molecules by the Mn3+ ions. However, the subsequent breaking up of the organic substrate into molecular fragments (including COz in the malonic acid version) causes the density of the medium to decrease and to outweigh any initial increase. Thus, in ascending waves, a region of decreased density develops below the wave front, which exerts a buoyant force that eventually destabilizes the system when the magnitude of the density gradient, the lateral size of the region, and the curvature of the front have evolved to their critical values. In their analysis, Pojman et al.2Bhave predicted a convective effect acting in the opposite direction than that observed here, based on their assumption of a density increase behind the wave front. The present results are clear evidence for simple convection, dominated by the isothermal density gradient which accompanies the chemical wave front, rather than for the cases of double and thermally induced convection.
Acknowledgment. This work is supported by the NSERC of Canada. We express our thanks to Prof. Mikhailov for communicating his work prior to publication.
References and Notes (1) Fife. P. C. In Non-Eauilibrium Dvnamics in Chemical Svstems: Vidal. Ch.', Pacaht, A., Eds.; Sphnger Verlag Berlin, 1984. (2) (a) Pojman, J. A.; Epstein, I. R. J . Phys. Chem. 1990, 94,4966-4972. (bl Poiman. J. A,; Epstein, I. R.; McManus, T. J.; Showalter K. J . Phys. Chem.-1991, 95, 1299-1306. (c) Pojman, J. A.; Nagy, I. P.; Epstein, I:R. J . Phys. Chem. 1991, 95, 1306-1311. (3) Oscillatiom and Travelling Waves in Chemical Systems; Field, R.J., Burger, M., Eds.; Springer Verlag: Berlin, 1984. (4) Miike, H.; Mueller, S. C.; Hess, B. Phys. Lett. A 1989, 141A, 25-30. ( 5 ) Tzalmona, A.; Armstrong, R. L.; Menzinger. M.; Cross, A.; Lemaire, C. Chem. Phys. Lett. 1992, 188, 457-460. (6) (a) Bottomley, P. A. Rev. Sci. Instrum. 1982, 53, 1319-1327. (b) Kuhn, W. Angew. Chem., Int. Ed. Engl. 1990, 29, 1-19. (7) Tzalmona, A.; Armstrong, R. L.; Menzinger, M.; Cross, A.; Lemaire, C. Chem. Phys. Lett. 1990, 174, 199-203. (8) Winfree A. T. In ref 1, pp 441-472. (9) (a) Walgraef, D.; Dewel, G.; Borckmans, P. J . Chem. Phys. 1983, 78, 3043-3048. (b) Mikhailov, A. Physica D, submitted for publication. (10) Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems; Cambridge University Press, Cambridge, MA, 1984; Chapter 12. (11) Welsh, B. J.; Gomatam, J.; Burgess, A. E. Nature 1983; 304, 611-614. (12) Ouyang, Q.; Tam, W. Y.; DeKepper, P.; McCormick, W. D.; Noszticzlus, Z.; Swinney, H. J. Phys. Chem. 1987, 91, 2181.
Polymorphism of (Quasi) Two-Dimensional Micelles' Jiayi Zhu, R. Bruce Lennox,* and Adi Eisenberg* Department of Chemistry, McGill University, 801 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6 (Received: December 17, 1991; In Final Form: April 7, 1992)
Langmuir films of block polyelectrolyte surface aggregates formed at the air-water interface have been visualized as Langmuir-Blodgett films using transmission electron microscopy. The morphologiesadopted by these (quasi) 2-D aggregates are critically dependent upon the ratio of the hydrophobic and hydrophilic block sizes. Three morphologies are observable (starfish, rod, and planar) in polystyrene/decylated poly(viny1pyridinium) block polyelectrolytes. Each particular morphology is observed within a well-defined composition range. Parallel polymorphism in 3D aggregates of block copolymers and small molecular weight amphiphiles suggest that the rules governing self-assembly in 2D and 3D and both small and large molecular weight molecules a r e qualitatively quite similar.
The existence of novel two-dimensional surface micelles was very recently reportede2 We have found, for example, that AB block copolymers, composed of polystyrene (260 units) and 4vinylpyridinium decyl iodide (240 units), can readily spread on
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should be addressed.
0022-3654/92/2096-4127%03.00/0
a water surface. The resulting monolayer exhibits distinctive surface pressure (r)-molecular area ( A ) relationships and can be visualized as a Langmuir-Blodgett film using both transmission electron microscopyZand atomic force microscopy.' A striking feature of the Langmuir-Blodgett films of this block copolymer is the existence of a highly uniform array of circular structures 0 1992 American Chemical Society
