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Pattern formation in the Schlögl model of nonlinear kinetics
Physica 108A (1981) 63-76 North-Holland
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PATTERN FORMATION IN THE SCHLiiGL MODEL OF NONLINEAR KINETICS Mei Hsu DUNG and John J. KOZAK Department
of Chemistry and Radiation Laboratoryt, University of Notre Dame, Notre Dame, Indiana 46.556. USA
Received 11 September 1980 Revised 28 February 1981 To study the effect of a local, spatial inhomogeneity on the progress of a chemical reaction which may exhibit a nonequilibrium phase transition, we have studied a (slightly) generalized version of a model proposed originally by Schliigl. Focusing on that regime of parameter space where the reaction sequence A +2X%3X and B +X%C allows a first-order transition, we consider the consequences of introducing a spatially-varying diffusion coefficient characterized by a correlation length which calibrates the region over which the spatial inhomogeneity persists. We find that if the inhomogeneity is localized, only small quantitative differences are found between the exact solution reported earlier by Schlijgl and the solutions generated here. However, as the correlation length becomes larger, abruptly at a certain critical value, a qualitative change in the nature of the solutions is found, with apparent oscillations produced in the concentration variable of the problem. We interpret this behavior as indicating the onset of pattern formation, and suggest that this behavior may be of importance in those radiation-induced phenomena where high-energy intermediates are produced.
1. Introduction
Many condensed phase reactions involving high-energy intermediates produced via ionizing-radiation or photo-chemical excitation are thought to alter (at least on certain time scales) the local structure of the medium in which the reaction occurs. This change in the local structure of the host phase may, in turn, affect the progress of subsequent reactions of the intermediate. Perhaps the most striking example of this synergetic effect is found in the radiation chemistry of water and aqueous solutions were the transfer of energy from a reactive intermediate to the host medium may produce a track structure (“spurs” or “blobs”) in the neighborhood of which the subsequent diffusion of the reactants may be modulated significantly (vis a vis the bulk system). Although coorperative phenomena of this type seem not to have been studied in great detail in radiation chemistry, in many areas of physics, chemistry and biology there has been a rather explosive development in tThe research described herein was supported by the Office of Basic Energy Sciences of the Department of Energy. This is Document No. NDRL-2169 from the Notre Dame Radiation Laboratory.
0378-4371/81/D
/$02.50 @ 1981 North-Holland
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MEI HSU D U N G A N D J O H N J. K O Z A K
assessing the effect of inhomogeneities on pattern formation. One thinks immediately of the recent International Conference on Synergetics ~) which dealt with these phenomena, and the earlier, representative work of J.F.G. Auchmuty and G. Nicholis2~), D. Bedeaux, P. Mazur and R.A. Pasmanter2h), the review of B.J. Matkowsky and E.L. Reiss 2c) and the discussion by P. Fife2d). In this paper, we undertake a study of a simple reaction-diffusion model generalized to take into account the consequences of a local inhomogeneity in the structure of the medium. Specifically, we study the SchlSgl model and examine the chemical-kinetic consequences of breaking the symmetry of the space within which the reactive intermediate is produced by introducing a spatially-varying diffusion coefficient characterized by a correlation length l. We focus on that regime of parameter space of the model which exhibits a nonequilibrium chemical phase transition of the first-order, and show that as the correlation length of the problem reaches a certain threshold value, there arise spatial patterns in the concentration variable of the problem. In the following section we restate the SchiSgl model (for first-order transitions), emphasizing those known results which are necessary in our latter development. In section 3 we propose a simple Gaussian function to monitor the effects of a spatial inhomogeneity in the host medium on the diffusion of a reactive intermediate and in section 4 we report the results found in our numerical analysis of the steady-state behavior of the resulting non-linear evolution equation. The "crossover" behavior uncovered in our study is verified analytically in section 5 via a straightforward application of the method of Pad6 approximants. The overall results of our study and their possible implications are discussed in the concluding section of the paper.
2. Restatement of the Schlfgl model
SchlSgl has devised two, chemical reaction models which exhibit the behavior of non-equilibrium phase transitions3). Although the chemical analog of a second-order transition is of great interest, our primary concern in this paper will be nonequilibrium, chemical phase transitions of first order. The model which exhibits first-order behavior, including the coexistence of phases, is kt
A + 2X~,-~-3X, kl
( 1)
k2
B + X~-C. k~
(2)
PATTERN FORMATION IN THE SCHLOGL MODEL
65
If the reactions (1) and (2) occur in a well-stirred system and the concentrations of the reactants A, B and C in the reaction vessel are held constant, only species X (of concentration n) will vary in time. Under these circumstances the time rate of change of the concentration n is dn
d--t = k l a n 2 - k ~ n 3 - k 2 b n + k 2 c ,
(3)
where the lower-case letters {a, b, c} denote the concentrations of species {A, B, C}. By choosing appropriate units of time and concentration, one can set kl -- 1, k l a
=
3,
(4)
and introduce the parameters (5)
[3 = k 2 b , V = k ~ c .
With these simplifications, the reaction (1) may be written dn d--t-= - n3 + 3n2 - [3n + 3'.
(6)
Schl6gl has shown that for certain values of /3 and 3' there exist three steady states, two of which are stable and one of which is unstable. Despite the existence of two possible s t a b l e steady states, coexistence of phases cannot occur in the well-stirred system described by the evolution equation (6). H o w e v e r , Schl6gl has pointed out that coexistence phenomena can arise if one assumes that the species X diffuses much more slowly than the other components A, B and C in the reaction scheme, eqs. (1) and (2); under this constraint the concentrations {a, b, c} will still remain constant (here, in time a n d space) while the variable n will respond to gradients in the concentration of species X. The more general evolution equation describing the reaction and diffusion of species X is iz = V • D V n
+ [ - n 3+ 3n 2-~8n + V].
(7)
Under the assumption that the diffusion coefficient D is independent of the coordinates of the gradient operator and, as well, has no explicit dependence on the concentration variable n, the eq. (7) may be simplified to read h = DV~n
+ [ - n 3+
3n 2 - [3n + V].
(8)
Under steady-state conditions, it was shown in ref. 1 that the two possible states are
stable
nl,2 = 1-7-(3 - [3)~/2,
(9)
66
M E | H S U D U N G A N D J O H N J. K O Z A K
w h e r e a s the unstable state is n3 = 1.
(10)
The condition o f c o e x i s t e n c e was d e t e r m i n e d to be -/=/3-2,
(ll)
and f r o m eqs. (9)-(11), one can easily s h o w that /3 and 7 must satisfy the following b o u n d s : 2~
(12)
0~<7~1.
(13)
U n d e r these conditions one finds that n~ and n2 c h a r a c t e r i z e the c o n c e n t r a t i o n of species X in two, well-separated, spatial regimes: in a o n e - d i m e n s i o n a l system, the c o n c e n t r a t i o n n-~n~ w h e n the spatial variable x b e c o m e s large and positive, n ~ n2 w h e n x ~ - ~, and n = n3 = 1 at the origin of the c o o r dinate system. Since n = n3 defines the unstable s t e a d y state, one has here the physical picture of two different, stable, c o n c e n t r a t i o n regimes of species X separated b y an unstable b o u n d a r y layer, at coexistence. F o r p u r p o s e s of our later discussion, it will be c o n v e n i e n t to recast the r e a c t i o n - d i f f u s i o n e q u a t i o n (7) into a slightly different form. U n d e r the c h a n g e o f variables u = l-n,
(14)
eq. (7) for the steady-state p r o b l e m r e d u c e s to d ~DdU'~=u3 dxl. ~x-x/ + (/3 - 3)u + (/3 - 3' - 2), or, given the condition (11), ul ~d- t[ o d"~-J= U3 -l- (/3 - 3 ) u .
(15)
A s s u m i n g the diffusion coefficient D to be a constant, the solution of D u " = u3 + (/3 - 3 ) u ,
(16)
subject to the b o u n d a r y conditions lira u ( x ) = 1 - nl = +(3 - [3) 1/2,
(17a)
lim u ( x ) = 1 - n2 = - (3 -/3)1/2,
(17b)
x~+~
PATTERN FORMATION IN THE SCHL()GL MODEL
67
with lim u(x) = 1 - n3 = 0
(17c)
x=0
is
3 1/2 u ( x ) = ( 3 - / 3 ) l / 2 t a n h [ ( ~ - ~ -) x].
(18)
As is evident, u(x) is an odd function of the variable x; noting this, we remark that the result (17) can also be generated from (16) using the boundary conditions lim u(x) = (3 - / 3 ) '/2,
(19a)
lim u(x) = 0,
(19b)
lim d___uu= 3 - / 3 x~0 dx ~/2--D'
(20a)
lim u(x) = 0.
(20b)
x~0
or
x~0
In what follows, we shall consider the behavior of solutions to the differential equation (15) subject to the choice of boundary conditions, eqs. (19).
3. The diffusion function
We wish to consider the effects of system inhomogeneity on the diffusion and subsequent reaction of the chemical species X at the steady state. For the model under study, i.e. the scheme described by eqs. (1) and (2) with attendant evolution equation (15), it is likely that the consequences of such an inhomogeneity would be felt most significantly in that spatial regime where the concentration n = n3 = 1, i.e., at the origin or, more qualitatively, in its immediate vicinity. Accordingly, in this paper, we introduce a simple Gaussian function to model the spatial variation of the diffusion coefficient; in particular, we write D r = D0[1 + C e-X~/12].
(21)
In this representation, l is a correlation length which specifies how far down
68
MEI H S U D U N G A N D J O H N J. K O Z A K
range the system inhomogeneity persists; C is a (real) constant which controls the peak height of the Gaussian function. Given the structure of the representation (21), C > 0 implies that physical diffusion of species X in the neighborhood of the origin of the reaction space is faster than in bulk solution, whereas the specification C < 0 carries the opposite connotation. When C = 0, the function D~ reduces to the constant value Do. When C is nonzero, the function (21) collapses to a delta function (at the origin) when the correlation length l vanishes; away from the origin, the diffusion of the species X is described by the constant value, D I = Do, in this limit (1 ~ 0 ) . Finally, for C > 0 and in the limit where the correlation length becomes large, the regime of "fast diffusion" of species X propagates outward from the origin until, when l ~ , the asympotic value D I = Do(1 + C) (characteristic of the behavior at the origin only for finite l) is realized over the entire space of the system.
4. Numerical simulations We report in this section the results of our numerical studies on the steady-state behavior of the evolution equation (7). In particular, we study the equation d r dn] ~x-x[Df~xx j = n 3 - 3 n 2 + / 3 n - V, with
,/=/3-2, and D f = D0[1 + C e x2/12], subject to the boundary conditions n(x)[x
0= 1
and n(x)lx=+~
-- 1 - (3
- / 3 ) 1/2.
To examine the coexistence region, we set/3 = 2. Our numerical analysis was performed using a generalized R u n g e - K u t t a method. The differential equation was solved by specifing n - - 1 at x = 0 , and choosing the initial slope ( d n / d x ) [ x ~ o to fit n = 1 - ( 3 - / 3 ) m when x~o~. For the basic Schl6gl model (D l = Do = constant), this procedure recovered the results obtained using the boundary conditions, eqs. (17), as noted previously.
P A T T E R N F O R M A T I O N IN T H E S C H L O G L M O D E L
69
We display in fig. 1 companion results which show at a glance the consequences of specifying different choices of the function D r In the top of this figure is represented the behavior of D s for four representative choices of the correlation length l, with the amplitude C held fixed (at C = 1). The steadystate profies of n(x) are displayed in the bottom of this figure and augmented by the further results reported in fig. 2. It is the remarkable spatial patterns which develop with increase in the correlation length that will be the main focus of the remainder of this paper. Consider first the behavior for the limiting case: C = l, l-~oo. Here, the diffusion coefficient reduces to D f = 2D0, and the expected hyperbolic tangent solution of the steady-state, differential equation for uniform diffusion is realized numerically (the hyphenated line in fig. l). Alternatively, for C = 0 (or C = 1 with 1--+0, conditions which produce delta-function behavior at the origin and D I = Do elsewhere), the pair of solid lines in fig. 1 show the behavior of D i = Do and the consequent (hyperbolic tangent) spatial dependence of n(x) for this case. As is evident, the width of the layer corresponding to the choice D~ = 2/)o is larger than that corresponding to D I -- Do. Consider now the behavior found upon increasing the correlation length from zero through a succession of increasingly larger values. Physically, this corresponds to spreading the region (about the origin) in which the diffusing species X is sensitive to spatial inhomogeneity in the host medium. For values
2.0
m
/i ~
I)f D I
/I
. /7 ~ - , , - - 7/
-~/ f
1,0
I
"i
""
I
I
I
20
n
iI I.C
I
\
\ \
/
\
I
\~ ~
I
II
y
I
/ /llI
II
I
ii
\\\
\x \ t\\
\X,...//
0.(3
_~
_
1
X---*
Fig. 1. Plots of DI versus x, and n(x) versus x for various choices of the correlation length. Here, the choice 1 = + ~ is given by the hyphenated line, the choice I = l0 by the dashed line, the choice I = ! by the dotted line and the choice C = 0 (or, effectively C = l, I = 0) by the solid line.
70
MEI
HSU
DUNG
AND
JOHN
J. K O Z A K
2.0
/~...
/
\\ :
/J//' /
n
:/
5
I0
15
\\~/
20
15
X---~
Fig. 2. P r o f i l e s o f t h e c o n c e n t r a t i o n v a r i a b l e n(x) v e r s u s x c o r r e s p o n d i n g to t h e c o r r e l a t i o n l e n g t h s : I = 10 ~ ( s o l i d line), I = 10 2 ( d o t t e d line), I - 10 ~ ( d a s h e d line) a n d I = 10 4 ( h y p h e n a t e d line).
of l in the range 0 < l < 9.9496 the monotonic decay in the function n(x) for x > 0 follows closely the behavior determined for D l = Do (i.e. C = 0); see the dotted-line profile in fig. 1 for the specific setting C = 1, I = 1. One finds, with increasing l, a very gradual change in the slope of the function N(x) near the origin, with the slopes generated apparently converging toward the one generated when D i is set equal to 2D0; qualitatively, however, one still finds a relaxation to the asymptotic result, n(x)=0, in the range x >5.0. This b e h a v i o r persists until one reaches the threshold value l - 9 . 9 4 9 6 ; a further increase in l by one part in l0 t (to I = 9.9497) causes both a discontinuous change in the slope of the function n(x) near the origin (see table I), and a qualitative change in the downrange behavior of the function n(x). In particular, one finds the onset of oscillations in the concentration variable n(x) as one exceeds the critical correlation length, l0 = 9.9496. Plotted in fig. (1) is the behavior of n(x) for l = 10 and this result is c o m p l e m e n t e d in fig. 2 with the results obtained upon increasing the correlation length through the sequence of values, I = 10n (n = 2, 3, 4). As is seen in figs. 1 and 2 the oscillations in the function n(x) are displaced further and further from the origin with increase in l, with a c o n c o m m i t a n t increase (decrease) in the peak height (trough) and an overall broadening of the peak itself in this limit. Since the function n(x) portrays the steady-state concentration profile corresponding to a particular setting of the correlation length l, we may conclude that the augmented Schl6gl model considered in this p a p e r shows that the d e v e l o p m e n t of spatial (concentration) patterns is a direct consequence of relaxing the continuum diffusion approximation. Having before us the numerical evidence, (at least) two questions arise immediately. Firstly, can one verify analytically the " c r o s s o v e r " behavior
PATTERN FORMATION IN THE SCHLOGL MODEL
71
TABLE I
Values of the slope of n(x) at the origin for values of the correlation length I in the vicinity of the threshold value (results computed in double precision) I
Slope
9.9491 9.9492 9.9493 9.9494 9.9495 9.9496 9.9497 9.9498 9.9499 9.9500
0.49707783 0.49707795 0.49707801 0.49707807 0.49707812 0.49707818 0.44754792 0.44757528 0.44757563 0.44757599
observed numerically when l increases beyond the threshold value l0 = 9.9496? And secondly, since the qualitative change in the structure of the solutions of the underlying differential equation is so reminiscent of a bifurcation, can one prove analytically that l0 is a branch point of the nonlinear problem (10 is suggestively close to 7r2)? The answer to the first question is relatively straightforward and will be discussed in the following section. The answer to the second question demands a more detailed analysis and this will be presented in a subsequent contribution.
5. A n a l y t i c r e s u l t s
Our main concern in this section will be to verify analytically the behavior observed numerically and reported in the preceding section. To do this, we assume series expansions of the functions D r (eq. (21)) and u(x) (in eq. (15)), viz.
Oi=Do
[I+~ .
n.
(- I)"]_1
(22)
and
u(x) = ~ a . x °, rl
(23)
72
MEI
HSU
DUNG
AND
JOHN
J. K O Z A K
where, L =- 1]12. Using these expansions, the eq. (15) b e c o m e s
2Lx
(Lx2)"
l
"
]
n
from which one may develop a recursion relation for the coefficient a,, : it is
1 ~1[ "+" ..... a,,+2 = 2(n + 2)(n + 1 ) ( D [ ~ apa,ar- a,] + ~a,,
2i(n-2t)~-.~ (-1) i n-2i-1 iTi
+2
.
(25)
The first few coefficients for n even are
1 F 1 3 _ ao)], ae = ~ [ ~ ( a i , (3a2a~i + 3aTao- a2) + 6a2L ,
a4 =
1FI , , , a6 = ~6[~(3a~a4 + 6aoa,a3 + 2aoa5 + 3aTa2 - a4)
+ 20a4L - 5a:L2]. Application of the b o u n d a r y condition u(x = O) = 0 reveals that the coefficient a0 = O, from which it follows that all coefficients a, with n even must vanish. Accordingly, the general solution u(x) must be an odd function, written here as
u(x)=
~] a,x".
(26)
n =2i ~ I
N o t e that this identification permits the numerical analysis to be carried out using the b o u n d a r y conditions, eqs. (19). The leading coefficient in the representation (26) is a~, the slope of the function u(x) at the origin. Once the coefficient at is known, the remaining coefficients a, with n odd can be determined explicitly f r o m the recursion relation (25). Then, the resulting series expansion can be analyzed for representative choices of the coefficients (C, l). In particular, starting from the series representation (26), we have constructed [N, N ] Pad6 a p p r o x i m a n t s (through N = 9) to verify the behavior reported in the last section for sensitive values of the coefficient l. When the
P A T T E R N F O R M A T I O N IN T H E S C H L O G L M O D E L
73
correlation length is in the range 0 < / < 9 . 9 4 , the [9,9]-Pad6 approximant successfully reproduces the monotonically-decreasing behavior of the function n(x) displayed in fig. 1; when l exceeds the value l =9.94, the Pad6 approximant reproduces the pattern structure displayed in figs. 1 and 2. To make definite these remarks, we record in table II, values of n(x) determined numerically and via the series-expansion procedure discussed in this section for two representative value of l, viz., a value of l just below the threshold value, l = 9.94, and a value just above the threshold value, I = 9.95. In the latter case, as may be seen, the values of n(x) in the neighborhood of the first turning point (e.g., n(x) = 0.2499 at x = 3.0) are successfully reproduced by the Pad6 method. We conclude that the "crossover" behavior of the concentration function n(x) as the correlation length I exceeds the critical value l0 = 9.9496 is effectively verified.
6. Discussion
The problem of pattern formation in nonequilibrium systems has been the subject of many studies in recent yearsl'2). In chemical kinetics a good deal of research in this area was stimulated by the discovery of chemical reactions TABLE II N u m e r i c a l versus Pad6 a p p r o x i m a n t results for n(x) for two values of the correlation length I (all results c o m p u t e d in double precision) l
x
Numerical soln.
Pad6 estimate
9.94
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
0.539908 0.239665 0.946791 0.351652 0.125842 0.437621 0.147621 0.483523 0.152378 0.586189 0.326758 0.249901 0.349845 0.655718 1.12935 1.55958 1.78775 1.82332
0.539908 0.239666 0.94687 × 10 i 0.35301 X 10-I 0.12697 x 10 t 0.4682 × 10 -2 0.2238 × 10 2 0.2196 x 10 z 0.3677 × 10 -2 0.586189 0.586189 0.249905 0.349784 0.654904 1.124579 1.549205 1.789736 1.897046
9.95
x lif t x iff I × lif t × 10-2 × 10 2 × 10 2 X 10-3
74
MEI HSU DUNG AND JOHN J. KOZAK
which exhibit both temporal and spatial oscillations (see, for example, the experimental studies of Belousov4), Zhabotinsky and coworkers 5) and later Pacault and coworkers 6) on the cerium-malonic acid system). Owing to the complexity of oscillating chemical reactions (and, in particular, the difficulty of assigning the correct mechanism) y, a number of models have been proposed to cast light on the possible cooperative phenomena exhibited by simpler networks of coupled chemical reactions. Of these reaction-diffusion models, one of the best known is the Schl6gl model, the model considered in this paper. As Schl6gl showed in 1972, the reaction sequence defined by eqs. (1) and (2) is already rich enough in analytic structure to exhibit a nonequilibrium, a first-order phase transition. What has been demonstrated in this paper is that a relatively simple extension of the Schl6gl model exhibits the onset of pattern formation (here, spatial oscillations in the concentration variable n(x) at the steady state) as a certain correlation length of the problem exceeds a threshold value. This correlation length I was taken to characterize the occurrence of a spatial inhomogeneity in the system; the consequent symmetry breaking was handled functionally by assuming that the diffusion of the reactive intermediate X was sensitive to this inhomogeneity. We have determined numerically and confirmed analytically (via the method of Pad6 approximants) the occurrence of pattern formation as one exceeds the threshold value l0 = 9.9496 for the correlation length. However, we have not proved here that this sensitive value represents, in effect, a bifurcation point of the underlying nonlinear evolution equation. This work is in progress and will be reported in the near future. As it happens, our motivation for considering the relaxation of the continuum diffusion approximation within the context of the Schl6gl model was stimulated not by the experimental studies on oscillating reactions but rather by the now rather extensive literature on effects produced by the passage of charged particles through condensed media. In an important review, Mozumder 8) has identified the physical and chemical factors which play a role in such high-energy, time-dependent processes, particularly with respect to the formation and structure of tracks. As is pointed out in ref. 8, "the successful description of the yields of radiolytic products depends on knowledge of (1) the pattern of energy deposit and (2) a reaction sequence". Whereas in many systems there exists a natural separation of time scales which effectively decouples the physical and chemical events, it is by no means the case that such a separation is universal. In the case of water (and aqueous solutions), for example, the physical stage extends from ~ 10 ~6 to 10 ~ s while the chemical stage extends from ~ 10 ~ to 10 6s, and it is for those systems where the physical and chemical events occur on roughly the same time scale (namely, -~ 10 h~ s) that the considerations of the present paper may be of
PATTERN FORMATION IN THE SCHLOGL MODEL
75
some interest. In particular, in the existing theoretical literature on these problemsS'9), the concept of a "local" macroscopic variable (a "local" temperature, dielectric constant, or diffusion constant) is often introduced in the initial formulation of the problem. In calculations, however, these parameters are set equal to constants (invariably the bulk-system value) with the consequence that effects which arise as a result of a local symmetry breaking may be camouflaged in this approximation. Given the results of this paper, it is seen that a relatively straightforward extension of the simple model introduced by Schl6gl may give a "crossover" in qualitative behavior when a diffusion function is constructed to account for a local, spatial inhomogeneity. Local effects due to the relaxation of the continuum dielectric approximation have been assessed for electrolytes 1°) and charged molecular assemblies"). And, nonlinear temperature effects have been assessed within the context of a simple photoinduced (isomerization) reaction ~2) and a sequence of photochemical reactions13). In each of these studies, the consequence of assuming a "local" variation in the attendent macroscopic variable can be quite dramatic, and it is our hope in later work to examine current theories of radiolytic processes from the more general point of view in which such inhomogeneities are taken explicitly into account.
References 1) Pattern Formation by Dynamic Systems and Pattern Recognition, H. Haken, ed. (Springer, Berlin, 1979). See also, H. Haken, Synergetics (Springer, Berlin, 1977). 2) (a) J.F.G. Auchmuty and G. Nicholis, Bull. MatH. Biol. 37 (1975) 323. (b) D. Bedeaux, P. Mazur and R.A. Pasmanter, Physica 86A (1977) 355; R.A. Pasmanter, D. Bedeaux and P. Mazur, Physica 90A (1978) 151. (c) B.J. Matkowsky and E.L. Reiss, SlAM J. Appl. Math. 33 (1977) 230. (d) P. Fife, Mathematical Aspects of Reacting and Diffusing Systems (Springer, Berlin, 1979). 3) F. Schl6gl, Z. Phys. 253 (1972) 147. 4) B.P. Belousov, Sb. ref. radats, med. Moscow (1959). 5) V.A. Vavilin, A.M. Zhabotinsky and L.S. Yaguzhinsky, Oscillatory Processes in Biological and Chemical Systems (Moscow Science Publ., 1967). 6) A. Pacault, Evolution chimique loin de l'6quilibre: concepts, modeles et r6el, in Synergetics: Far From Equilibrium, A. Pacault and C. Vidal, eds. (Springer, Berlin, 1979), p. 128. 7) For a discussion of the mechanism of the Belousov-Zhabotinsky reaction, see: (i) R.J. Field, E. Kor6s and R.M. Noyes, J. Amer. Chem. Soc. 94 (1972) 8649. (ii) R.J. Field and R.M. Noyes, Nature 237 (1974) 390; (iii) R.J. Field and R.M. Noyes, J. Chem. Phys. 60 (1974) 1877; (iv) R.J. Field and R.M. Noyes, J. Amer. Chem. Soc. 96 (1974) 2001. 8) A. Mozumder, Adv. Rad. Chem. 1 (1969) 1. 9) For a useful survey of the principal results, see: A. Mozumder and J.L. Magee, in Physical Chemistry: An Advanced Treatise, H. Eyring, D. Henderson and W. Jost, eds. (Academic Press, New York, 1975) Vol. VII.
76 10) 11) 12) 13)
MEI HSU DUNG AND JOHN J. KOZAK D.G. Knox and J.J. Kozak, Molec. Phys. 33 (1977) 811. M.H. Dung, D.G. Knox and J.J. Kozak, Bunsenges. Phys. Chem., in press. A. Nitzan, P. Ortoleva and J. Ross, J. Chem. Phys. 60 (1974) 3134. J.J. Kozak, G. Nicholis, J.W. Kress and M. Sanglier, J. Non-equilib. Thermodyn. 4 (1979) 67.
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PATTERN FORMATION IN THE SCHLiiGL MODEL OF NONLINEAR KINETICS Mei Hsu DUNG and John J. KOZAK Department
of Chemistry and Radiation Laboratoryt, University of Notre Dame, Notre Dame, Indiana 46.556. USA
Received 11 September 1980 Revised 28 February 1981 To study the effect of a local, spatial inhomogeneity on the progress of a chemical reaction which may exhibit a nonequilibrium phase transition, we have studied a (slightly) generalized version of a model proposed originally by Schliigl. Focusing on that regime of parameter space where the reaction sequence A +2X%3X and B +X%C allows a first-order transition, we consider the consequences of introducing a spatially-varying diffusion coefficient characterized by a correlation length which calibrates the region over which the spatial inhomogeneity persists. We find that if the inhomogeneity is localized, only small quantitative differences are found between the exact solution reported earlier by Schlijgl and the solutions generated here. However, as the correlation length becomes larger, abruptly at a certain critical value, a qualitative change in the nature of the solutions is found, with apparent oscillations produced in the concentration variable of the problem. We interpret this behavior as indicating the onset of pattern formation, and suggest that this behavior may be of importance in those radiation-induced phenomena where high-energy intermediates are produced.
1. Introduction
Many condensed phase reactions involving high-energy intermediates produced via ionizing-radiation or photo-chemical excitation are thought to alter (at least on certain time scales) the local structure of the medium in which the reaction occurs. This change in the local structure of the host phase may, in turn, affect the progress of subsequent reactions of the intermediate. Perhaps the most striking example of this synergetic effect is found in the radiation chemistry of water and aqueous solutions were the transfer of energy from a reactive intermediate to the host medium may produce a track structure (“spurs” or “blobs”) in the neighborhood of which the subsequent diffusion of the reactants may be modulated significantly (vis a vis the bulk system). Although coorperative phenomena of this type seem not to have been studied in great detail in radiation chemistry, in many areas of physics, chemistry and biology there has been a rather explosive development in tThe research described herein was supported by the Office of Basic Energy Sciences of the Department of Energy. This is Document No. NDRL-2169 from the Notre Dame Radiation Laboratory.
0378-4371/81/D
/$02.50 @ 1981 North-Holland
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MEI HSU D U N G A N D J O H N J. K O Z A K
assessing the effect of inhomogeneities on pattern formation. One thinks immediately of the recent International Conference on Synergetics ~) which dealt with these phenomena, and the earlier, representative work of J.F.G. Auchmuty and G. Nicholis2~), D. Bedeaux, P. Mazur and R.A. Pasmanter2h), the review of B.J. Matkowsky and E.L. Reiss 2c) and the discussion by P. Fife2d). In this paper, we undertake a study of a simple reaction-diffusion model generalized to take into account the consequences of a local inhomogeneity in the structure of the medium. Specifically, we study the SchlSgl model and examine the chemical-kinetic consequences of breaking the symmetry of the space within which the reactive intermediate is produced by introducing a spatially-varying diffusion coefficient characterized by a correlation length l. We focus on that regime of parameter space of the model which exhibits a nonequilibrium chemical phase transition of the first-order, and show that as the correlation length of the problem reaches a certain threshold value, there arise spatial patterns in the concentration variable of the problem. In the following section we restate the SchiSgl model (for first-order transitions), emphasizing those known results which are necessary in our latter development. In section 3 we propose a simple Gaussian function to monitor the effects of a spatial inhomogeneity in the host medium on the diffusion of a reactive intermediate and in section 4 we report the results found in our numerical analysis of the steady-state behavior of the resulting non-linear evolution equation. The "crossover" behavior uncovered in our study is verified analytically in section 5 via a straightforward application of the method of Pad6 approximants. The overall results of our study and their possible implications are discussed in the concluding section of the paper.
2. Restatement of the Schlfgl model
SchlSgl has devised two, chemical reaction models which exhibit the behavior of non-equilibrium phase transitions3). Although the chemical analog of a second-order transition is of great interest, our primary concern in this paper will be nonequilibrium, chemical phase transitions of first order. The model which exhibits first-order behavior, including the coexistence of phases, is kt
A + 2X~,-~-3X, kl
( 1)
k2
B + X~-C. k~
(2)
PATTERN FORMATION IN THE SCHLOGL MODEL
65
If the reactions (1) and (2) occur in a well-stirred system and the concentrations of the reactants A, B and C in the reaction vessel are held constant, only species X (of concentration n) will vary in time. Under these circumstances the time rate of change of the concentration n is dn
d--t = k l a n 2 - k ~ n 3 - k 2 b n + k 2 c ,
(3)
where the lower-case letters {a, b, c} denote the concentrations of species {A, B, C}. By choosing appropriate units of time and concentration, one can set kl -- 1, k l a
=
3,
(4)
and introduce the parameters (5)
[3 = k 2 b , V = k ~ c .
With these simplifications, the reaction (1) may be written dn d--t-= - n3 + 3n2 - [3n + 3'.
(6)
Schl6gl has shown that for certain values of /3 and 3' there exist three steady states, two of which are stable and one of which is unstable. Despite the existence of two possible s t a b l e steady states, coexistence of phases cannot occur in the well-stirred system described by the evolution equation (6). H o w e v e r , Schl6gl has pointed out that coexistence phenomena can arise if one assumes that the species X diffuses much more slowly than the other components A, B and C in the reaction scheme, eqs. (1) and (2); under this constraint the concentrations {a, b, c} will still remain constant (here, in time a n d space) while the variable n will respond to gradients in the concentration of species X. The more general evolution equation describing the reaction and diffusion of species X is iz = V • D V n
+ [ - n 3+ 3n 2-~8n + V].
(7)
Under the assumption that the diffusion coefficient D is independent of the coordinates of the gradient operator and, as well, has no explicit dependence on the concentration variable n, the eq. (7) may be simplified to read h = DV~n
+ [ - n 3+
3n 2 - [3n + V].
(8)
Under steady-state conditions, it was shown in ref. 1 that the two possible states are
stable
nl,2 = 1-7-(3 - [3)~/2,
(9)
66
M E | H S U D U N G A N D J O H N J. K O Z A K
w h e r e a s the unstable state is n3 = 1.
(10)
The condition o f c o e x i s t e n c e was d e t e r m i n e d to be -/=/3-2,
(ll)
and f r o m eqs. (9)-(11), one can easily s h o w that /3 and 7 must satisfy the following b o u n d s : 2~
(12)
0~<7~1.
(13)
U n d e r these conditions one finds that n~ and n2 c h a r a c t e r i z e the c o n c e n t r a t i o n of species X in two, well-separated, spatial regimes: in a o n e - d i m e n s i o n a l system, the c o n c e n t r a t i o n n-~n~ w h e n the spatial variable x b e c o m e s large and positive, n ~ n2 w h e n x ~ - ~, and n = n3 = 1 at the origin of the c o o r dinate system. Since n = n3 defines the unstable s t e a d y state, one has here the physical picture of two different, stable, c o n c e n t r a t i o n regimes of species X separated b y an unstable b o u n d a r y layer, at coexistence. F o r p u r p o s e s of our later discussion, it will be c o n v e n i e n t to recast the r e a c t i o n - d i f f u s i o n e q u a t i o n (7) into a slightly different form. U n d e r the c h a n g e o f variables u = l-n,
(14)
eq. (7) for the steady-state p r o b l e m r e d u c e s to d ~DdU'~=u3 dxl. ~x-x/ + (/3 - 3)u + (/3 - 3' - 2), or, given the condition (11), ul ~d- t[ o d"~-J= U3 -l- (/3 - 3 ) u .
(15)
A s s u m i n g the diffusion coefficient D to be a constant, the solution of D u " = u3 + (/3 - 3 ) u ,
(16)
subject to the b o u n d a r y conditions lira u ( x ) = 1 - nl = +(3 - [3) 1/2,
(17a)
lim u ( x ) = 1 - n2 = - (3 -/3)1/2,
(17b)
x~+~
PATTERN FORMATION IN THE SCHL()GL MODEL
67
with lim u(x) = 1 - n3 = 0
(17c)
x=0
is
3 1/2 u ( x ) = ( 3 - / 3 ) l / 2 t a n h [ ( ~ - ~ -) x].
(18)
As is evident, u(x) is an odd function of the variable x; noting this, we remark that the result (17) can also be generated from (16) using the boundary conditions lim u(x) = (3 - / 3 ) '/2,
(19a)
lim u(x) = 0,
(19b)
lim d___uu= 3 - / 3 x~0 dx ~/2--D'
(20a)
lim u(x) = 0.
(20b)
x~0
or
x~0
In what follows, we shall consider the behavior of solutions to the differential equation (15) subject to the choice of boundary conditions, eqs. (19).
3. The diffusion function
We wish to consider the effects of system inhomogeneity on the diffusion and subsequent reaction of the chemical species X at the steady state. For the model under study, i.e. the scheme described by eqs. (1) and (2) with attendant evolution equation (15), it is likely that the consequences of such an inhomogeneity would be felt most significantly in that spatial regime where the concentration n = n3 = 1, i.e., at the origin or, more qualitatively, in its immediate vicinity. Accordingly, in this paper, we introduce a simple Gaussian function to model the spatial variation of the diffusion coefficient; in particular, we write D r = D0[1 + C e-X~/12].
(21)
In this representation, l is a correlation length which specifies how far down
68
MEI H S U D U N G A N D J O H N J. K O Z A K
range the system inhomogeneity persists; C is a (real) constant which controls the peak height of the Gaussian function. Given the structure of the representation (21), C > 0 implies that physical diffusion of species X in the neighborhood of the origin of the reaction space is faster than in bulk solution, whereas the specification C < 0 carries the opposite connotation. When C = 0, the function D~ reduces to the constant value Do. When C is nonzero, the function (21) collapses to a delta function (at the origin) when the correlation length l vanishes; away from the origin, the diffusion of the species X is described by the constant value, D I = Do, in this limit (1 ~ 0 ) . Finally, for C > 0 and in the limit where the correlation length becomes large, the regime of "fast diffusion" of species X propagates outward from the origin until, when l ~ , the asympotic value D I = Do(1 + C) (characteristic of the behavior at the origin only for finite l) is realized over the entire space of the system.
4. Numerical simulations We report in this section the results of our numerical studies on the steady-state behavior of the evolution equation (7). In particular, we study the equation d r dn] ~x-x[Df~xx j = n 3 - 3 n 2 + / 3 n - V, with
,/=/3-2, and D f = D0[1 + C e x2/12], subject to the boundary conditions n(x)[x
0= 1
and n(x)lx=+~
-- 1 - (3
- / 3 ) 1/2.
To examine the coexistence region, we set/3 = 2. Our numerical analysis was performed using a generalized R u n g e - K u t t a method. The differential equation was solved by specifing n - - 1 at x = 0 , and choosing the initial slope ( d n / d x ) [ x ~ o to fit n = 1 - ( 3 - / 3 ) m when x~o~. For the basic Schl6gl model (D l = Do = constant), this procedure recovered the results obtained using the boundary conditions, eqs. (17), as noted previously.
P A T T E R N F O R M A T I O N IN T H E S C H L O G L M O D E L
69
We display in fig. 1 companion results which show at a glance the consequences of specifying different choices of the function D r In the top of this figure is represented the behavior of D s for four representative choices of the correlation length l, with the amplitude C held fixed (at C = 1). The steadystate profies of n(x) are displayed in the bottom of this figure and augmented by the further results reported in fig. 2. It is the remarkable spatial patterns which develop with increase in the correlation length that will be the main focus of the remainder of this paper. Consider first the behavior for the limiting case: C = l, l-~oo. Here, the diffusion coefficient reduces to D f = 2D0, and the expected hyperbolic tangent solution of the steady-state, differential equation for uniform diffusion is realized numerically (the hyphenated line in fig. l). Alternatively, for C = 0 (or C = 1 with 1--+0, conditions which produce delta-function behavior at the origin and D I = Do elsewhere), the pair of solid lines in fig. 1 show the behavior of D i = Do and the consequent (hyperbolic tangent) spatial dependence of n(x) for this case. As is evident, the width of the layer corresponding to the choice D~ = 2/)o is larger than that corresponding to D I -- Do. Consider now the behavior found upon increasing the correlation length from zero through a succession of increasingly larger values. Physically, this corresponds to spreading the region (about the origin) in which the diffusing species X is sensitive to spatial inhomogeneity in the host medium. For values
2.0
m
/i ~
I)f D I
/I
. /7 ~ - , , - - 7/
-~/ f
1,0
I
"i
""
I
I
I
20
n
iI I.C
I
\
\ \
/
\
I
\~ ~
I
II
y
I
/ /llI
II
I
ii
\\\
\x \ t\\
\X,...//
0.(3
_~
_
1
X---*
Fig. 1. Plots of DI versus x, and n(x) versus x for various choices of the correlation length. Here, the choice 1 = + ~ is given by the hyphenated line, the choice I = l0 by the dashed line, the choice I = ! by the dotted line and the choice C = 0 (or, effectively C = l, I = 0) by the solid line.
70
MEI
HSU
DUNG
AND
JOHN
J. K O Z A K
2.0
/~...
/
\\ :
/J//' /
n
:/
5
I0
15
\\~/
20
15
X---~
Fig. 2. P r o f i l e s o f t h e c o n c e n t r a t i o n v a r i a b l e n(x) v e r s u s x c o r r e s p o n d i n g to t h e c o r r e l a t i o n l e n g t h s : I = 10 ~ ( s o l i d line), I = 10 2 ( d o t t e d line), I - 10 ~ ( d a s h e d line) a n d I = 10 4 ( h y p h e n a t e d line).
of l in the range 0 < l < 9.9496 the monotonic decay in the function n(x) for x > 0 follows closely the behavior determined for D l = Do (i.e. C = 0); see the dotted-line profile in fig. 1 for the specific setting C = 1, I = 1. One finds, with increasing l, a very gradual change in the slope of the function N(x) near the origin, with the slopes generated apparently converging toward the one generated when D i is set equal to 2D0; qualitatively, however, one still finds a relaxation to the asymptotic result, n(x)=0, in the range x >5.0. This b e h a v i o r persists until one reaches the threshold value l - 9 . 9 4 9 6 ; a further increase in l by one part in l0 t (to I = 9.9497) causes both a discontinuous change in the slope of the function n(x) near the origin (see table I), and a qualitative change in the downrange behavior of the function n(x). In particular, one finds the onset of oscillations in the concentration variable n(x) as one exceeds the critical correlation length, l0 = 9.9496. Plotted in fig. (1) is the behavior of n(x) for l = 10 and this result is c o m p l e m e n t e d in fig. 2 with the results obtained upon increasing the correlation length through the sequence of values, I = 10n (n = 2, 3, 4). As is seen in figs. 1 and 2 the oscillations in the function n(x) are displaced further and further from the origin with increase in l, with a c o n c o m m i t a n t increase (decrease) in the peak height (trough) and an overall broadening of the peak itself in this limit. Since the function n(x) portrays the steady-state concentration profile corresponding to a particular setting of the correlation length l, we may conclude that the augmented Schl6gl model considered in this p a p e r shows that the d e v e l o p m e n t of spatial (concentration) patterns is a direct consequence of relaxing the continuum diffusion approximation. Having before us the numerical evidence, (at least) two questions arise immediately. Firstly, can one verify analytically the " c r o s s o v e r " behavior
PATTERN FORMATION IN THE SCHLOGL MODEL
71
TABLE I
Values of the slope of n(x) at the origin for values of the correlation length I in the vicinity of the threshold value (results computed in double precision) I
Slope
9.9491 9.9492 9.9493 9.9494 9.9495 9.9496 9.9497 9.9498 9.9499 9.9500
0.49707783 0.49707795 0.49707801 0.49707807 0.49707812 0.49707818 0.44754792 0.44757528 0.44757563 0.44757599
observed numerically when l increases beyond the threshold value l0 = 9.9496? And secondly, since the qualitative change in the structure of the solutions of the underlying differential equation is so reminiscent of a bifurcation, can one prove analytically that l0 is a branch point of the nonlinear problem (10 is suggestively close to 7r2)? The answer to the first question is relatively straightforward and will be discussed in the following section. The answer to the second question demands a more detailed analysis and this will be presented in a subsequent contribution.
5. A n a l y t i c r e s u l t s
Our main concern in this section will be to verify analytically the behavior observed numerically and reported in the preceding section. To do this, we assume series expansions of the functions D r (eq. (21)) and u(x) (in eq. (15)), viz.
Oi=Do
[I+~ .
n.
(- I)"]_1
(22)
and
u(x) = ~ a . x °, rl
(23)
72
MEI
HSU
DUNG
AND
JOHN
J. K O Z A K
where, L =- 1]12. Using these expansions, the eq. (15) b e c o m e s
2Lx
(Lx2)"
l
"
]
n
from which one may develop a recursion relation for the coefficient a,, : it is
1 ~1[ "+" ..... a,,+2 = 2(n + 2)(n + 1 ) ( D [ ~ apa,ar- a,] + ~a,,
2i(n-2t)~-.~ (-1) i n-2i-1 iTi
+2
.
(25)
The first few coefficients for n even are
1 F 1 3 _ ao)], ae = ~ [ ~ ( a i , (3a2a~i + 3aTao- a2) + 6a2L ,
a4 =
1FI , , , a6 = ~6[~(3a~a4 + 6aoa,a3 + 2aoa5 + 3aTa2 - a4)
+ 20a4L - 5a:L2]. Application of the b o u n d a r y condition u(x = O) = 0 reveals that the coefficient a0 = O, from which it follows that all coefficients a, with n even must vanish. Accordingly, the general solution u(x) must be an odd function, written here as
u(x)=
~] a,x".
(26)
n =2i ~ I
N o t e that this identification permits the numerical analysis to be carried out using the b o u n d a r y conditions, eqs. (19). The leading coefficient in the representation (26) is a~, the slope of the function u(x) at the origin. Once the coefficient at is known, the remaining coefficients a, with n odd can be determined explicitly f r o m the recursion relation (25). Then, the resulting series expansion can be analyzed for representative choices of the coefficients (C, l). In particular, starting from the series representation (26), we have constructed [N, N ] Pad6 a p p r o x i m a n t s (through N = 9) to verify the behavior reported in the last section for sensitive values of the coefficient l. When the
P A T T E R N F O R M A T I O N IN T H E S C H L O G L M O D E L
73
correlation length is in the range 0 < / < 9 . 9 4 , the [9,9]-Pad6 approximant successfully reproduces the monotonically-decreasing behavior of the function n(x) displayed in fig. 1; when l exceeds the value l =9.94, the Pad6 approximant reproduces the pattern structure displayed in figs. 1 and 2. To make definite these remarks, we record in table II, values of n(x) determined numerically and via the series-expansion procedure discussed in this section for two representative value of l, viz., a value of l just below the threshold value, l = 9.94, and a value just above the threshold value, I = 9.95. In the latter case, as may be seen, the values of n(x) in the neighborhood of the first turning point (e.g., n(x) = 0.2499 at x = 3.0) are successfully reproduced by the Pad6 method. We conclude that the "crossover" behavior of the concentration function n(x) as the correlation length I exceeds the critical value l0 = 9.9496 is effectively verified.
6. Discussion
The problem of pattern formation in nonequilibrium systems has been the subject of many studies in recent yearsl'2). In chemical kinetics a good deal of research in this area was stimulated by the discovery of chemical reactions TABLE II N u m e r i c a l versus Pad6 a p p r o x i m a n t results for n(x) for two values of the correlation length I (all results c o m p u t e d in double precision) l
x
Numerical soln.
Pad6 estimate
9.94
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
0.539908 0.239665 0.946791 0.351652 0.125842 0.437621 0.147621 0.483523 0.152378 0.586189 0.326758 0.249901 0.349845 0.655718 1.12935 1.55958 1.78775 1.82332
0.539908 0.239666 0.94687 × 10 i 0.35301 X 10-I 0.12697 x 10 t 0.4682 × 10 -2 0.2238 × 10 2 0.2196 x 10 z 0.3677 × 10 -2 0.586189 0.586189 0.249905 0.349784 0.654904 1.124579 1.549205 1.789736 1.897046
9.95
x lif t x iff I × lif t × 10-2 × 10 2 × 10 2 X 10-3
74
MEI HSU DUNG AND JOHN J. KOZAK
which exhibit both temporal and spatial oscillations (see, for example, the experimental studies of Belousov4), Zhabotinsky and coworkers 5) and later Pacault and coworkers 6) on the cerium-malonic acid system). Owing to the complexity of oscillating chemical reactions (and, in particular, the difficulty of assigning the correct mechanism) y, a number of models have been proposed to cast light on the possible cooperative phenomena exhibited by simpler networks of coupled chemical reactions. Of these reaction-diffusion models, one of the best known is the Schl6gl model, the model considered in this paper. As Schl6gl showed in 1972, the reaction sequence defined by eqs. (1) and (2) is already rich enough in analytic structure to exhibit a nonequilibrium, a first-order phase transition. What has been demonstrated in this paper is that a relatively simple extension of the Schl6gl model exhibits the onset of pattern formation (here, spatial oscillations in the concentration variable n(x) at the steady state) as a certain correlation length of the problem exceeds a threshold value. This correlation length I was taken to characterize the occurrence of a spatial inhomogeneity in the system; the consequent symmetry breaking was handled functionally by assuming that the diffusion of the reactive intermediate X was sensitive to this inhomogeneity. We have determined numerically and confirmed analytically (via the method of Pad6 approximants) the occurrence of pattern formation as one exceeds the threshold value l0 = 9.9496 for the correlation length. However, we have not proved here that this sensitive value represents, in effect, a bifurcation point of the underlying nonlinear evolution equation. This work is in progress and will be reported in the near future. As it happens, our motivation for considering the relaxation of the continuum diffusion approximation within the context of the Schl6gl model was stimulated not by the experimental studies on oscillating reactions but rather by the now rather extensive literature on effects produced by the passage of charged particles through condensed media. In an important review, Mozumder 8) has identified the physical and chemical factors which play a role in such high-energy, time-dependent processes, particularly with respect to the formation and structure of tracks. As is pointed out in ref. 8, "the successful description of the yields of radiolytic products depends on knowledge of (1) the pattern of energy deposit and (2) a reaction sequence". Whereas in many systems there exists a natural separation of time scales which effectively decouples the physical and chemical events, it is by no means the case that such a separation is universal. In the case of water (and aqueous solutions), for example, the physical stage extends from ~ 10 ~6 to 10 ~ s while the chemical stage extends from ~ 10 ~ to 10 6s, and it is for those systems where the physical and chemical events occur on roughly the same time scale (namely, -~ 10 h~ s) that the considerations of the present paper may be of
PATTERN FORMATION IN THE SCHLOGL MODEL
75
some interest. In particular, in the existing theoretical literature on these problemsS'9), the concept of a "local" macroscopic variable (a "local" temperature, dielectric constant, or diffusion constant) is often introduced in the initial formulation of the problem. In calculations, however, these parameters are set equal to constants (invariably the bulk-system value) with the consequence that effects which arise as a result of a local symmetry breaking may be camouflaged in this approximation. Given the results of this paper, it is seen that a relatively straightforward extension of the simple model introduced by Schl6gl may give a "crossover" in qualitative behavior when a diffusion function is constructed to account for a local, spatial inhomogeneity. Local effects due to the relaxation of the continuum dielectric approximation have been assessed for electrolytes 1°) and charged molecular assemblies"). And, nonlinear temperature effects have been assessed within the context of a simple photoinduced (isomerization) reaction ~2) and a sequence of photochemical reactions13). In each of these studies, the consequence of assuming a "local" variation in the attendent macroscopic variable can be quite dramatic, and it is our hope in later work to examine current theories of radiolytic processes from the more general point of view in which such inhomogeneities are taken explicitly into account.
References 1) Pattern Formation by Dynamic Systems and Pattern Recognition, H. Haken, ed. (Springer, Berlin, 1979). See also, H. Haken, Synergetics (Springer, Berlin, 1977). 2) (a) J.F.G. Auchmuty and G. Nicholis, Bull. MatH. Biol. 37 (1975) 323. (b) D. Bedeaux, P. Mazur and R.A. Pasmanter, Physica 86A (1977) 355; R.A. Pasmanter, D. Bedeaux and P. Mazur, Physica 90A (1978) 151. (c) B.J. Matkowsky and E.L. Reiss, SlAM J. Appl. Math. 33 (1977) 230. (d) P. Fife, Mathematical Aspects of Reacting and Diffusing Systems (Springer, Berlin, 1979). 3) F. Schl6gl, Z. Phys. 253 (1972) 147. 4) B.P. Belousov, Sb. ref. radats, med. Moscow (1959). 5) V.A. Vavilin, A.M. Zhabotinsky and L.S. Yaguzhinsky, Oscillatory Processes in Biological and Chemical Systems (Moscow Science Publ., 1967). 6) A. Pacault, Evolution chimique loin de l'6quilibre: concepts, modeles et r6el, in Synergetics: Far From Equilibrium, A. Pacault and C. Vidal, eds. (Springer, Berlin, 1979), p. 128. 7) For a discussion of the mechanism of the Belousov-Zhabotinsky reaction, see: (i) R.J. Field, E. Kor6s and R.M. Noyes, J. Amer. Chem. Soc. 94 (1972) 8649. (ii) R.J. Field and R.M. Noyes, Nature 237 (1974) 390; (iii) R.J. Field and R.M. Noyes, J. Chem. Phys. 60 (1974) 1877; (iv) R.J. Field and R.M. Noyes, J. Amer. Chem. Soc. 96 (1974) 2001. 8) A. Mozumder, Adv. Rad. Chem. 1 (1969) 1. 9) For a useful survey of the principal results, see: A. Mozumder and J.L. Magee, in Physical Chemistry: An Advanced Treatise, H. Eyring, D. Henderson and W. Jost, eds. (Academic Press, New York, 1975) Vol. VII.
76 10) 11) 12) 13)
MEI HSU DUNG AND JOHN J. KOZAK D.G. Knox and J.J. Kozak, Molec. Phys. 33 (1977) 811. M.H. Dung, D.G. Knox and J.J. Kozak, Bunsenges. Phys. Chem., in press. A. Nitzan, P. Ortoleva and J. Ross, J. Chem. Phys. 60 (1974) 3134. J.J. Kozak, G. Nicholis, J.W. Kress and M. Sanglier, J. Non-equilib. Thermodyn. 4 (1979) 67.