The Journal of
Physical Chemistry
0 Copyright, 1990, by the American Chemical Society
VOLUME 94, NUMBER 21 OCTOBER 18, 1990
LETTERS Turing Structures in the Ferroin-Catalyzed Bromate Oscillating System A. M. Zhabotinsky* National Scientific Centre of Hematology, Novozykovsky 4, Moscow 125167, U S S R
and A. B. Rovinsky* Institute of Biological Physics, Puschino, Moscow Region 142292, U S S R (Received: March 28, 1990;
In Final Form: June 12, 1990)
Mathematical models of the Belousov-Zhabotinsky reaction carried out in gel-filtration or anion-exchange media are proposed. The analysis shows that conditions for the appearance of the Turing structures can be fulfilled in the media commonly used for liquid chromatography.
Stationary periodic structures in reaction diffusion systems predicted by Turing' have been thoroughly studied in numerous theoretical works.2J However, these structures were not experimentally observed until now whereas analogous convective patterns have been found in various system^.^ Several attempts have been made to analyze the possibility of the Turing structures arising in the Belousov-Zhabotinsky (BZ) reaction-diffusion It has been shown that the diffusion coefficient of the catalyst must be greater than the diffusion coefficients of some other essential species. Apparently this condition is not fulfilled in the homogeneous water solution which is the standard medium for the BZ r e a ~ t i o n . ~ ( 1 ) Turing, A. Philos. Trans. R. SOC.London, Ser. B 1952, 37, 237. (2) Nicolis, G.; Prigogine, 1. Se/jWrganization in Non-Equilibrium Systems; Wiley: New York, 1977. (3) Vastano, J. A.; Pearson, J . E.; Horsthemke, W.; Swinney, H. L. J . Chem. Phys. 1988,88. 6175. (4) Borckmans. P.; Dewel, G.; Walgraef, D.; Katayama, Y. J . Stat. Phys. 1988, 48, IO3 1. ( 5 ) Becker, P. K.; Field, R. J. J . Phys. Chem. 1985, 89, 118. (6) Zhabotinsky, A . M.; Rovinsky, A. B. J . Star. Phys. 1987, 48, 959. (7) Rovinsky, A. B. J . Phys. Chem. 1987, 91, 4606. (8) Pearson, J. E.; Horstemke, W . J . Chem. Phys. 1989, 90, 1588.
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In this paper the Turing instability is analyzed in the model that takes into consideration the diffusion of three principal species of the BZ reaction: Fe(phen),, H B Q , and Br-. It has been shown that the Turing instability can take place only if the diffusion coefficient of Ferriin exceeds the diffusion coefficients of the other two species. The conditions can be fulfilled in a medium commonly used for gel filtration or anion-exchange liquid chromatography.
The Model The model of the bromate-ferroin-bromomalonic acid system was described earlier.'0-" The simplified scheme is as follows:
- + + - + + -
H+ + HBr02 + H B r 0 3 Br02*
HBr02+
H+
BrO,'
+ H20
HBr02+
(1)
(2)
Fe(phen)32++ HBr02+
Fe(phen)33++ HBr02
(3)
H+ + 2HBr02
HBr03 + H+
(4)
H+ + Br-
HOBr
HBr02
2HOBr
(9) Ruff, I.; Zimony, M . Magy. Kem. Foly. 1973, 79, 135.
0 1990 American Chemical Society
(5)
8002 The Journal of Physical Chemistry, Vol. 94, No, ZI, 1990 H+ + BrH+ + Br-
+ HOBr
+ HBrO,
Fe(phen)33++ CHBr(COOH), H20
+ 'CBr(COOH)2
-
- + H+
+ CHBr(COOH)2
-
+ H20 HBrO, + HOBr F e ( ~ h e n ) ~ ,++
Br-
(6)
Br,
'CBr(COOH),
+ H+ (8)
+ 'COH(COOH),
(9)
(12)
+ k7hoAY - 2k4hd(Z + D , M
qk8k95Z
Y , = k-,ho(C - Z )
+ k9 - k,h,,XY - k,hoAY + k I 2 B+ D,SY
Z , = 2klhoAX-
kgkgBZ k-,ho(C - 2 ) + k9
+ D,AZ
(13)
Here X = [HBrO,], Y = [Br-1, Z = [Fe(phen)3)+], A = [HBrOJ, E = [CHBr(COOH),], C = [ F e ( ~ h e n ) ~+ ~ +[Fe(~hen),~+], ] ho is the Hammett acidity function used instead of proton concentration,I2 and 9 is the empirical stoichiometric factor equal to the ratio of Br- production rate to catalyst reduction rate.13 The scheme (I)-( 12) gives 9 = 0.5. However, q can vary significantly, taking into consideration numerous reactions with different bromo derivatives and other products of malonic acid o x i d a t i ~ n . ' ~A~ ' ~ is the Laplacian operator, and Di is the diffusion coefficient of the ith species. In the most cases the rate constant k, is very small in comparison with the term k8ho(C- Z ) and it can be neglected. The essential rate constants have been estimated for 40 OC as f0llows:'OJ~ k , = 100 M-, s-l; k , = 1.7 X IO4 M-2 s-l; k , = 1 X IO7 M-' s-'; k7 = 15 M-' s-'; kgo = kgkg/k-, = 2 X M S-I; k I 2= 1 X s-l
At room temperature (20-25 "C) the above rate constants must be reduced about 10 times coinciding as a result with the "Lo" set by Tyson.I6 Introduce the following scaling: 2k4k7
X = -k2k4 , Ax, r=(s) t , y = - k,k5
cy=-
k4k80B '1
(k,hoA)2'
- Z
+ ZEE
(15)
where d , = D J D , and dz = D y / D z . Preliminary analysis of the system (1 5) shows that the Turing instability can arise only if D,/D, = d , and D,,/D, = d, are less than I . This condition is not fulfilled in a homogeneous water solution where the diffusion coefficient of ferriin ion is small in comparison with the diffusion coefficients of Br- and HBr02.9 However, as the following analysis shows, this condition can be realized in heterogeneous modification of the BZ system. The BZ reaction should proceed in a chromatographic gel-filtration medium or in a layer of anion exchanger with narrow pores which permit the free diffusion of the large cation of catalyst and strongly retard the effective diffusion of relatively small HBrO, molecule and Br- ion." Consider an anion exchanger. In this case there are three compartments: the interparticle volume with free diffusion ( Vf), the volume in pores with restricted diffusion ( Vr), and the surface adsorbing anions (S). Suppose that the catalyst is located only in the free diffusion volume ( Vr) between the particles, HBrO is in Vf and V,, and Br- is in all three compartments. Assume that the exchange between the compartments is in the state of equilibrium. The partition coefficient between Vf and V, equals 1, and that between S and Vr equals K . Then the following equations can be written on the basis of the homogeneous system 15:
z , = x - -yc
1-z
+ ZEF
where p = Vf/(Vf + V,), a = Vf + V, + KS, and b , and 6, are relations of rate constants of the same heterogeneous and homogeneous reactions. In the case of the pure gel filtration K = 0 and (16) is simplified: X,
112
y = - -ky, A , k5
Z
z,=x--(Y- 1
CBr,(COOH)
The three-component system of PDE has been deduced from the above scheme:
X,= k , h d X - kSh&Y
Then system 13 becomes
(7)
+ H20 (IO) Br, + CHBr(COOH)2 CBr2(COOH), + H+ + Br- ( 1 1) H 2 0 + CHBr(COOH), CHOH(COOH), + H+ + BrHOBr
Letters
1 = -(x - x2 - XY
+ ~ y +) pdlxet
el
klA
=k4C
Z
z,=x-a- 1- z
+
ztt
It is seen that (17) can be obtained from (1 5) by multiplication of d , , d,, and 9 by the same coefficient p . System 17 has been studied numerically, and the results are presented below.
(IO) Rovinsky, A. B.; Zhabotinsky, A. M . J . Phys. Chem. 1984,88,6081. ( I 1) Rovinsky. A . B.; Zhabotinsky, A. M. In Fundamental Research in Chemical Kinetics; Shilov, A. E., Ed.; Gordon and Breach: London, 1986. (12) Hammett, L. P. Physical Organic Chemisrry; McGraw-Hill: New York, 1970. (13) Field, R. J.; Noyes, R. M. J . Chem. Phys. 1974, 60, 1877. (14) Jwo, J. J.: Noyes, R . M . J . Am. Chem. SOC.1975, 97 5422. (IS) Noyes, R. M.; Jwo, J. J . J . Am. Chem. SOC.1975, 97, 5431. ( I 6) Tyson, J . In Oscillations and Traveling Waves in Chemical Systems: Field, R. J., Burger, M., Eds.; Wiley: New York, 1985.
Results The lines of Hopf and Turing bifurcations in ( A J ) plane are shown in Figure 1 for the catalyst concentration equal to 1 X IO4 M is shown in Figure 2. M. The same plane for C = 1 X The supercritical Turing instability takes place in the regions between lines of Turing and Hopf. One can see that with diminishing C the oscillation region decreases and relative area of Turing instability region increases. ( 1 7) Snyder, L. R.; Kirkland, J. J., Eds. Introduction to Modern Liquid Chromatography: Wiley: New York, 1979.
The Journal of Physical Chemistry, Vol. 94, No. 21, 1990 8003
Letters
Figure 1. Lines of Turing and Hopf bifurcations in the bromate (A)bromomalonic acid ( E ) concentration plane for model 17. C = [Fe(phen),] = 1 X IO4; ho = I ; q = 0.5; DHBd2/DFe(,,= d , ; DE,-/&+) = d2;dl = d2 = I . -, Hopf line; - - -,Turing line ( p = 0.625, q = 0.8); -, Turing line ( p = 0.3, 9 = 1.667); Turing line ( p = 0.1, 9 = 5 ) . - e
-.e-
r-7-
-
---
-
il i
0. L
0 :0
0: 1
0: 2
0: 3
q ! 0
Figure 3. Turing structure. Z = [Fe(phen)F] in IO4 M; X = [HBrO,] in 10-4M; Y = [Br-]in10-5M;C= 1 X 1 0 ” ; A = 0 . 1 1 ; E = 0 . 1 8 5 ; h 0 = 1; q = 1.667; p = 0.3; abscissa in millimeters.
4 Fefphen)? 3. ?o4 9.
Figure 2. Lines of Turing and Hopf bifurcations in the ( A , B ) plane. C = 1 .O X IO-,. Designations and parameters as in Figure I .
Figures 3 and 4 demonstrate the character of the calculated Turing structures. Calculations have been made in the points placed far from the borders of the Turing regions. As a result, the nonlinear effects are pronounced, especially in the shape of the Br- profile.
Discussion The results presented above show that the Turing structures arise under fairly large values of p ( p L 0.3) which can be obtained with standard chromatographic gel-filtration media. The parametric diagrams (Figures 1 and 2) and concentrational profiles (Figures 3 and 4) have been calculated with a value of p q = 0.5. The value of p q must exceed 0.3 for the Turing structures to be stable. Therefore, it is desirable to have q as large as possible. The stoichiometric factor q is a poorly defined parameter. Estimates show that q can be close to 2.14J5 The comparison of Figures 3 and 4 shows that the amplitude of Turing structure increases with the enlargement of the concentrations of bromate and bromomalonic acid. The calculations made with the rate constants estimated for 40 O C give the space period about which is rather short‘ At room temperature the rate constants are IO-fold less as has been stated above. Expressions 14 show that it leads to en-
o.o&---
-+
+--
0.12 Oll8 .%%I Figure 4. Turing structure. C = 1 X IO-,; A = 0.18; E = 0.9;ho = I; q = 1.667; p = 0.3; other designations as in Figure 3. 0.00
0.06
largement of the space period by factor of lo1/*. The estimates made above’ give hope to obtaining the Turing structures experimentally in the heterogeneous modification of the ferroin-catalyzed BZ system. It seems that these estimates are also valid for the dipyridyl complexes of ruthenium and iron ions as catalysts. The situation is less clear with cerium and manganese ions because the smaller size of these ions leads to poorer conditions of chromatographic separation. Gels were frequently used for studies of the BZ reaction-diffusion systems to prevent convection and immobilize the metal ion catalyst.’*-*’ In the present paper chromatographic gels are (18) Kuhnert, L. Natunuissenschaften 1983, 70, 464. (19) Kuhnert, L.: Yamaguchi, T.; Nagi-Ungvarai, Z.; Mueller, S.C.; H a , B. In Preprints of Lectures and Abstracts of Posters, International Conference on Dynamics of Exotic Phenomena in Chemistry, Hajduszobozlo, Hungary, 1989; p 289.
8004
J , Phys. Chem. 1990, 94, 8004-8006
proposed to diminish effective diffusivity of HBrOz autocatalyst and Br- inhibitor while the catalyst ion is supposed to be free with the unchanged diffusion coefficient. The above model supposes that the chromatographic medium
is chemically resistant and not reacting with any significant species in the oscillating chemical reaction. Adsorption of the initial compounds like bromate and bromomalonic acid plays no significant role in the pattern formation. Recentlv a communication on the observation of the Turing structures‘in the chlorite-iodide system appeared.zz
I
(20) Maselko, J.; Reckly, J. S.; Showalter, K. J . Phys. Chem. 1989, 93, 2774. (21) Maselko, J.; Showalter. K. Nature 1989, 339, 609
(22) Castets, V.; Dulos, E.; Boissonade, J.; De Kepper, P. Preprint, 1990.
Effect of Divalent Electrolyte on the Hydrophobic Attraction Hugo K. Christenson,* Jiafu Fang,+ Barry W. Ninham, and John L. Parker Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, G.P.O. Box 4, Canberra A.C.T. 2601, Australia (Received: May 16, 1990)
Measurements of the attraction between hydrophobic surfaces (chemically modified mica and mica coated with Langmuir-Blodgett films) across water, 0.01 M magnesium sulfate solution, and 0.1 M magnesium sulfate solution (equivalent to concentrations of free 2 2 electrolyte of up to 0.01 M) have been carried out. The magnitude of the hydrophobic attraction is reduced with increasing electrolyte concentration but remains much larger than any van der Waals force. The apparent exponential decay length of the attraction is 9-13 nm for the monolayer surfaces at all concentrations and 5.4 nm for the chemically hydrophobed surfaces at 0.01 M MgSO.,. These results cannot be reconciled with recent theoretical models that ascribe an electrostatic origin to the hydrophobic attraction.
with increasing monovalent salt concentration found in a later, Introduction more comprehensive study may well have been entirely due to a A very long range attraction acts between hydrophobic surfaces reduced surface hydrophobicity, as the surfaces were observed across water. This interaction is up to 2 orders of magnitude to charge up and the ease with which cavitation occurred destronger than any conceivable van der Waals force over most of creased.” The effect of high electrolyte (> concentrations its range.’-s Early indications of such a force were obtained by has so far not been investigated. using surfaces of surfactants adsorbed from s o l ~ t i o n ,but ~ , ~the In general, the results of measurements to date suggest that attraction measured in such cases is dwarfed by that obtained later the nature of both the surface and of the ions in solution has an with other, more well defined hydrophobic surfaces. Attractive effect on the hydrophobic attraction and its dependence on forces with an apparently exponential decay length of 12-16 nm electrolyte concentration. The influence of electrolyte is obviously at separations beyond about 20 nm and a measurable range of of prime importance if we are to gain more insight into the hyup to 90 nm have been found between a variety of hydrophobic drophobic attraction. It is also the key test of the validity of several surfaces. Langmuir-Blodgett layers of both hydrocarbon and theoretical model^.^^-'^ In this Letter we present results of an fluorocarbon surfactants adsorbed to methylated silica,3 investigation of the effect of divalent electrolyte on the hydrophobic and chemically modified mica6 all give similar results. attraction, using differently prepared hydrophobic surfaces. Despite a growing number of investigations, very little is Symmetrical divalent electrolytes are not completely dissociated presently known about the hydrophobic attraction. Its origin is even at moderate M) concentrations, but the possible not understood. The influence of the degree of surface hydrocomplication of ion pairing giving rise to species of differing charge phobicity has so far not been properly rationalized. It is clear does not occur. We therefore chose magnesium sulfate, which only that it is not simply related to the conventional measure of has the added advantage of being thermally stable so that it can surface hydrophobicity-the advancing contact angle of a sessile be purified by roasting. water droplet.’ Even such a simple quantity as the adhesion between these surfaces in water appears to defy e x p l a n a t i ~ n . ~ ~ ~ ( I ) Christenson, H. K.; Claesson, P. M. Science (Washington, D.C.) 1988, On the other hand cavitation, or the spontaneous formation of 239, 390. a vapor bubble between contacting hydrophobic surfaces, is a (2) Claesson, P. M.; Christenson, H. K. J . Phys. Chem. 1988, 92, 1650. phenomenon, theoretically well understood, that follows directly (3) Rabinovich, Ya. 1.; Derjaguin, B. V. Colloids Surf. 1988, 30. 243. (4) Israelachvili, J. N.; Pashley, R. M. Nature (London) 1982,300, 341. from the fact that the energy of a hydrophobic surface is larger (5) Pashley, R. M.; McGuiggan, P. M.; Ninham, B. W.; Evans, D. F. in water than in vapor.* However, the possible relation between Science (Washington, D.C.) 1985, 229, 1088. the cavitation that is often observed in the experiments with very (6) Parker, J. L.; Cho, D. L.; Claesson, P. M. J. Phys. Chem. 1989, 93, hydrophobic and the long-range hydrophobic attraction 6121. (7) Kekicheff, P.; Christenson, H.K.; Ninham, B. W. Colloids Surf. 1989, remains unclear. 40, 31. Studies of the effect of electrolyte have led to some confusing ( 8 ) Yushchenko, V. S.; Yaminsky, V. V.; Shchukin, E. D. J . Colloid results. The problem throughout has been to separate actual lnterface Sci. 1983, 96, 307. changes in the surface structure caused by adsorption of ions or (9) Rabinovich, Ya. I.; Derjaguin, B. V.; Churaev, N. V. Adu. Colloid Inrerface Sei. 1982, 16. 63. exchange of surfactant molecules for ions from those due to the ( I O ) Claesson, P. M.; Blom, C. E.; Herder, P. C.; Ninham, B. W. J. presence of ionic species in solution. One early study showed only Colloid Interface Sei. 1986, 114, 234. M 1:l electrolyte compared to pure a very minor influence of ( I 1) Christenson, H. K.; Claesson, P. M.; Berg, J.; Herder, P. C. J. Phys. water.I0 The reduction in range of the hydrophobic attraction Chem. 1989, 93, 1472. ‘Present address: Pennzoil Products Corporation, P.O. Box 7569, The Woodlands, TX 77387.
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(12) Eriksson, J. C.; Ljunggren, S.; Claesson, P. M. J. Chem. Soc., Faraday Trans. 2 1989,85, 163. (13) Attard, P. J . Phys. Chem. 1989. 93, 6441. (14) Podgornik, R. J . Chem. Phys. 1989, 91, 5840.
0 1990 American Chemical Society
