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The Journal of

Physical Chemistry

0 Copyrighr. 1990, by rhe American Chemical Society

VOLUME 94, NUMBER 26 DECEMBER 27, 1990

LETTERS Temperature Dependence of Curvature-Velocity Relationship in an Excitable Belousov-Zhabotinskll Reaction P. Foerster, S. C. Muller,* and B. Hess Max-Planck-lnstitut f u r Ernahrungsphysiologie. Rheinlanddamm 201, 0-4600 Dortmund I , FRG (Received: August 31, 1990)

For the understanding of the geometric shape and the dynamic behavior of chemical waves traveling through an excitable medium, two relationships have to be considered: the dispersion relation in periodic wave trains, and the “eikonal” equation, which expresses a proportionality between speed and curvature of chemical waves. In continuation of previous works, where the dependence of the speed on the shape of waves was investigated for positively and negatively curved wave fronts in the Belousov-Zhabotinskii (BZ) reaction, we report for the same reaction measurements of the curvature-velocity relation for negatively curved wave fronts, obtained at different temperatures ranging from 279 to 300.7 K (6-27.7 “C). In this range, the critical radius for wave initiation varies from 28.2 to 16.4 pm. The activation energy of wave propagation is calculated to be Ev = 34.9 1.2 kJ/M, and that of the diffusion of the autocatalytic species, ED = 13.4 1.0 kJ/M.

*

*

The Belousov-Zhabotinskii (BZ) reaction, in which malonic acid is catalytically oxidized and brominated by acidic bromate, is the most extensively investigated chemical system that displays spatio-temporal patterns, such as temporal oscillations and propagating waves.’ Theoretical models require that the propagating waves, which travel through an excitable medium of the BZ reaction, have to fulfill two relationships. One is the ‘eikonal” equation, which expresses a proportionality between the normal propagation velocity and the curvature of the moving wave,* and the other is the dispersion relation, which specifies the dependence of the propagation velocity on the frequency of wave initiation.) On the basis of previous investigations, in which both relationships were verified e~perimentally,“*~ we report in this paper the tem( I ) Field, R. J., Burger, M., Eds. Oscillations and Traveling Waves in Chemical Systems; Wiley: New York, 1985. (2) Keener, J . P.; Tyson, J . J . Physica D 1986, 21, 307-324. (3) Dockery, J. D.;Keener, J. P.; Tyson, J. J. Physica D 1988,30, 177-191. (4) Foerster, P.; Muller, S. C.; Hess. 6. Science 1988, 2 4 / , 685-687.

perature dependency of the curvature-velocity relationship. A theoretical description of the spatial patterns observed in the BZ system was given in ref 2 following a reaction-diffusion mechanism derived on the basis of the Oregonator modeL6 Treating the reaction-diffusion equations of this model by singular perturbation methods leads to the curvature-velocity relationship

N=c-DK (1) where N is the normal velocity of curved wave fronts, c the velocity of plane waves, D the diffusion coefficient of the autocatalytic species, and K the curvature of the propagating waves. Typical features of this relationship are (a) the increasing normal velocity of the wave fronts with increasing negative curvature, (b) the decreasing normal velocity with decreasing positive curvature, and (c) the existence of a critical minimal radius (&,) (5) Foerster, P.; Muller, S. C.; Hess, B. froc. Nafl.Acad. Sci. U S A . 1989, 86. 6831-6834. (6) Field, R . J.; Noyes, R. M . J . Chem. Phys. 1974, 60, 1877-1884.

0022-3654/90/2094-8859$02.50/00 1990 American Chemical Society

8860 The Journal of Physical Chemistry, Vol. 94, No. 26, 1990

Letters

below which propagating of circular waves will not take place. According to ( I ) , RCri,= D / c . By measuring the curvature-velocity relationship at different temperatures we obtained the influence of temperature on the velocity of the traveling BZ waves, on the diffusion coefficient of the autocatalytic species and, in addition, on the critical radius of wave initiation. On the basis of the Stokes-Einstein relation the diffusion coefficient is given by

D = kBT/6*qa

(2)

where k B is Boltzmann's constant, 7 the viscosity, and a the radius of the diffusing particles. Taking the temperature dependence of the viscosity into account :he diffusion coefficient changes with temperature according to

-

CURVATURE (-I/*

,

1000

1

"4 1

2

D Te-EdRT (3) In ref 7 the propagation velocity u of chemical waves is given approximately by u

-

(kD)'I2

(4)

where k is the rate constant of the rate-determining step of the oscillatory reaction scheme in ref 6, which is the autocatalytic step, and D is the diffusion coefficient of the autocatalytic species. On the basis of the collision theory of reacting particles the reaction constant depends on the temperature according to

-

k T'lZe-ExIRT (5) With (3)-(5) the temperature dependence of the velocity becomes u Or,

N

with E , = '/2(E,

[T~-EDIRTT~I~c-EIIRT]'I~

2

N

:

N

T~/~~-ER/RT

5w

>

2

1

(6a) 0.

c

,

.

.

.

.3 .I CURVATURE (-l/m )

(1

.2

1

.5

250

0 4 . 0. ,1

.

.

,

.2 .J .4 CURVATURE (-lip )

f

.5

(6b)

For the critical radius, R,,,, = D / c , the expression given in (3) is substituted for the diffusion coefficient D and the expression for v given in (6b) is substituted for the velocity c of plane waves. The temperature dependence of the critical radius is then derived as R,,,,

3

7M

o

T3/4e-Ev/RT

1

loo0

F 750

E

+ Ek) v

1

l-

-

I

(7)

with f ? ~= '/2(Ek - E D ) . The measurements of these temperature effects were done in a reaction mixture that contains 48 mM sodium bromide, 340 mM sodium bromate, 95 mM malonic acid, 380 mM sulfuric acid, and 3.5 mM catalyst ferroin. From this solution, 3.6 mL is poured into an optically flat Petri dish (layer thickness 0.5-0.6 mm).* For our investigations we measured at different temperatures the temporal evolution of cusplike structures that form after the collision of two circular waves. The pair of circular waves was initiated by immersing two silver electrodes (diameter 100 pm) at a distance of ==4 mm with a micromanipulator. The detection of the structures was performed by using spectrophotometric microscope video imaging technique^.^ We extracted isointensity contour lines from the recorded images. Subsequently we fitted hyperbolas at the edge of the cusplike structures at the points of steepest gradients of the catalyst ferroin and determined the curvature in the vertex points as described in ref 4. The propagation velocity of the cusps was calculated along the line connecting the vertex points in the successive contour maps, where it is equal to the normal velocity. The velocity of plane waves was obtained from the outward motion of large circular waves, for which curvature effects can be neglected. These mcasuremcnts weredone at 6.0, 16.0, 17.3, 20.0, 22.5, 25.0, and 27.7 "C (279.0, 289.0, 290.3, 293.0, 295.3, 298.0, 300.7 K). In Figure I the measured dependence of the normal velocity N on curvature K is plotted for seven different temperatures. The (7) Tilden, J. J . Chem. Phys. 1974, 60, 3349-3350. (8) Miiller. S. C.: Plesser, Th.; Hess, B. Science 1985, 230, 661-663.

C U R V A I U R E (-l/m )

Figure I . Normal velocity of wave propagation calculated for the vertex points of hyperbolas fitted to the cusps formed upon wave collision (0) and the velocity c of plane waves ( 0 )plotted vs the curvature of the wave fronts for seven temperatures, as indicated in the graphs. TABLE I: Diffusion Coefficient of the Autocatalytic Species, HBrO,, the Plane Wave Velocity c, and the Critical Radius Itd, for Various Temperatures

temwrature "C 6 16

17.3 20 22.5 25 27.7

K

diffusion coeff, cm2/s x 10'

rmls

rm

279 289 290.3 293 295.5 298 300.7

1.126 f 0.085 1.488 f 0.050 1.443 i 0.066 1.524 f 0.055 I .649 i 0. I36 I .785 f 0.050 1.839 f 0. I22

40 64 68 80 92 102 112

28.2 23.3 21.2 19.1 17.9 17.5 16.4

c,

&it,

graphs of this figure corroborate that a linear relationship between N and K, as expressed in eq 1, holds in the investigated temperature range. From the slope of each regression line, the corresponding diffusion coefficient D was determined according to eq 1 . These values of D together with the values of the velocity c of plane waves and the resulting critical radii Rcritare summarized in Table I as a function of temperature.

J . Phys. Chem. 1990, 94. 8861-8863 4

A

7

t

3.3

8861

3.4

3.5

3.6 .10-J

'

+(K-')

25.54

3.3

3.4

3.5

3,6 a

- 3

'

f(K-1)

Figure 2. Logarithm of the velocity c of plane waves divided by T3I4 plotted vs l / T (T = absolute temperature), according to eq 6b.

Figure 3. Logarithm of the diffusion coefficient D divided by vs l / T (T = absolute temperature), according to eq 3.

In Figure 2 the logarithm of the plane wave velocity divided by T3I4is plotted versus 1 / T. As expected from eq 6b, one can derive the activation energy of wave propagation by a linear fit to the data, yielding 37.10 f 1.26 kJ/mol. This value is in close agreement with data given in the literature (34.9 f 1.2 kJ/mol in ref 9) as obtained for planar waves in a similar reaction mixture. Figure 3 shows the logarithm of the diffusion coefficient D divided by the temperature T as a function of I / T . According to eq 3 we calculated from the slope of the regression line the activation energy of diffusion to be 13.4 f 1 .OkJ/mol. This value is in the same order as the activation energy for the diffusion of CuS04 ( 1 5.3 f 1.3 kJ/molI0), which we use for comparison because of the similar size of this molecule. With E, = ' / z ( E D Ek) the activation energy Ek is determined to be 60.8 kJ/mol, and with E R = '/Z(Ek - ED), ER becomes 23.7 kJ/mol. The corresponding values for the critical radius, Rcrit(Table I), show that with increasing temperature smaller initiation radii are necessary so that propagation of circular waves will take place. In previous work5 we have obtained RCri,directly at room temperature by using thin, silver-coated electrodes. It should be of interest to perform such direct measurements also at different

temperatures, so that the temperature dependence of Rcritcould be obtained directly and the lower limit for this quantity determined. The results are further experimental proof for the theoretically predicted relationship between the normal velocity and the curvature of wave fronts. In the case where the assumptions of the model in ref 2 concerning the predominant influence of the autocatalytic species HBrOZin eq 1 are correct, we obtain in this way details about the diffusion coefficient of an important intermediate of the BZ reaction. For this intermediate, so far no direct data are available because its concentration in the BZ reaction is too low for direct measurements. Usually, the data documented for substances of comparable size have to be used for numerical calculations. Our results for the activation energies have also some practical use, because they allow the comparison of velocity measurements reported by different groups at different temperatures. The measured temperature effects show the increasing reactivity of the reaction solution and the enhancement of the diffusion processes with increasing temperature. In this context, it would be interesting to vary the relevant diffusion coefficients in this reaction, e.g., by increasing the viscosity of the solution. One could expect a critical transition point when the process of wave propagation becomes diffusion-limited, which needs future experimentation.

+

(9)Kuhnert, L.; Krug, H.-J.; Pohlmann, L. J . Phys. Chem. 1985,89, 2022-2026. (IO) Lundo/r-B6rnstein; Springer: New York. 1969; Vol. 11, part 5, p 631.

T plotted

Zero-Field Splitting of the First Excited Triplet State in Biradicals Estimated from Magnetic Effects on the Fluorescence Decays V. Lejeune, A. Despres, and E. Migirdicyan* Laboratoire de Photophysique Moliculaire du CNRS, Bdtiment 21 3, UniversitC Paris-Sud. 91405 Orsay, France (Received: October 31, 1990)

The fluorescencedecay of matrix-isolated m-xylylene biradicals is nonexponential and attributed to the emission from different sublevels of the first excited triplet state. In the presence of a magnetic field, the lifetime of the slow decay component decreases. Its dependence as a function of a weak magnetic field can be caiculated for different values of the zero-field splitting parameter D. The best fitting value is 101= 0.04 0.01 cm-'. This D value is found to be significantly larger in the first excited triplet state than in the ground state of the m-xylylene biradicals.

*

Introduction The conventional electron paramagnetic resonance (EPR) method has commonly been used to determine the zero-field splitting (zfs) parameters D and E of the ground triplet state in matrix-isolated biradicals and carbenesl,z and of the lowest m* ( I ) Murray, R. W.; Trozzolo, A. M.; Wasserman, Chem. SOC.1962,84, 3213.

E.;Yager, W. A. J . Am.

excited triplet state of aromatic hydrocarbons having lifetimes of several second^.^ Unfortunately, this method cannot be used for halogenated aromatic molecules and aromatics with lowest n r * triplet states that have triplet lifetimes Of the order Of mil(2) Brandon, R. W.; Closs, G. L.;Davoust, C. E.;Hutchison, Jr., C. A.; Kohler, B. E.; Silbey, R. J . Chem. Phys. 1965,43. 2006. (3) Hutchison, Jr., C. A.; Mangum, B. W. J . Chem. Phys. 1961,34,908.

0022-3654/90/2094-886 1 %02.50/0 0 I990 American Chemical Society