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R H Y T H M

JOHN I. G M I T R O A N D L. E. S C R I V E N

Department of Chemical Engineering, Institute of Technology, University of Minnesota, Minneapolis, Minnesota

I N T R O D U C T I O N

A n y general principles governing t h e origins of regular p a t t e r n s in space a n d r h y t h m i c oscillations in time seem v e r y likely to find a p p l i - cation, directly or indirectly, a t m a n y levels of biological science—be- ginning with t h e intracellular, supramolecular level. A basic problem in t h e physical sciences which is increasingly a t t r a c t i n g m a t h e m a t i - cians and engineering scientists is t h e explanation of how specific d y n a m i c p a t t e r n s and r h y t h m s can arise in spatially uniform, s t e a d y - state situations. Of course, macroscopic systems a l w a y s suffer some sort of low-level noise, b u t how can chaotic, w e a k disturbances h a v e no effect in some circumstances, y e t trigger development of strong, regular p a t t e r n and r h y t h m in others?

T h e genesis of d y n a m i c p a t t e r n s depends on t h e coupled effects of t r a n s p o r t processes a n d t r a n s f o r m a t i o n processes. T h e s t u d y of both lies a t t h e h e a r t of engineering science t o d a y . Their application to multicomponent, chemically reactive systems is t h e special concern of chemical engineers. This is one area in which chemical engineering and cell physiology r u n p a r a l l e l ; p r o b a b l y both could profit from closer communication and perhaps even active collaboration. T h e r e - port t h a t follows is offered as an example of c u r r e n t research in engi- neering science which m a y be of interest in connection not only with intracellular t r a n s p o r t b u t also with other biological phenomena t h a t m a y be better known to the reader t h a n the a u t h o r s .

W e begin b y mentioning a few strictly physical examples of d y - namic p a t t e r n a n d r h y t h m . F r o m these we a t t e m p t t o a b s t r a c t t h e key factors a n d to identify a set of specific problems which can be precisely formulated from t h e viewpoint of physical science—that is, in m a t h e m a t i c a l t e r m s . T h e first several of these are then formu- lated and solved for a p r o t o t y p e class of situations involving simul- taneous diffusion and chemical reaction in a v a r i e t y of geometric configurations. T h e bearing of t h e results on signal propagation, p a t -

221

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222 J O H N I . GMITRO AND L . E . S C R I V E N

FIG. 1. Development of Bénard cells in a dish of liquid heated uniformly from

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beneath. (Photographs courtesy of E. L. Koschmieder, Harvard University.)

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224 J O H N I . GMITRO AND L . E . S C R I V E N

tern and rhythm generation, and mechanical movement is discussed in more qualitative terms which we earnestly hope will be informative to the reader who chooses to skip over the mathematical language of the sections on Formulation and Instability and W a v e Propagation.

Figure l ( a - d ) are photographs showing the development of a dy- namic flow structure known as Bénard cells in a shallow dish of ordi- nary liquid that is being uniformly heated over its bottom side. Almost the same flow patterns can be brought about by two different physical mechanisms, one stemming from the dependence of surface tension on temperature, the other from the dependence of density on tempera- ture (Scriven and Sternling [14]). In the latter case, the hotter, buoy- ant fluid at the bottom of the dish tends to rise and the colder, denser fluid at the top to sink. Such a turnover would lower the potential energy of the system and render it more stable until the heating from below reestablished the unstable density profile. In the turnover itself, hot rising columns would necessarily exist somewhere alongside cold sinking columns, and in this situation of lateral velocity and temperature gradients there would be viscous forces opposing the flow and heat conduction reducing the buoyancy differences responsible for the flow. Thus there are two competing tendencies: one toward establishment of dynamic pattern, the other toward its destruction once it is formed (Sani and Scriven [13]). I n fact, the rate of heating from beneath must exceed a certain critical value before the anabolic process can surpass the catabolic process sufficiently to establish flow, which tends to settle down in the steady Bénard-cell pattern if the critical value is not too greatly exceeded. The photographs indicate that the presence of the side of the dish favors concentric ring cells at first; these ultimately break up to give the strikingly hexagonal planform of Bénard cells, provided the dish diameter is much larger than the natural cell size. Flow within a cell is diagramed in Fig. 2, where streamlines are shown. The boundaries between cells are simply symmetry planes across which there is no flow of fluid. They are purely dynamic.

Bénard cells remain fixed in location and the flow within them is steady. They are an example of stationary convection. If a dish of liquid mercury is spun fast enough about its axis, a second type of convection, called oscillatory, occurs. The cellular planform becomes a little more complicated although still basically hexagonal, while the flow within cells m a y diminish and reverse periodically or the cellular pattern itself m a y translate through the liquid. In

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FIG. 2. Schematic diagram of Bénard cells, showing streamlines of flow within a cell.

release. Both heat and solute are transported. There is a tendency for solute to be redistributed in a regular hexagonal pattern a t the freezing face of the solid. This tendency, which stems from local super- cooling in the melt, is opposed by thermal transport and reduced by diffusional transport. When these catabolic processes do not prevail, a regular concentration pattern m a y form over the freezing face;

this results in the tesselated profile pictured in Fig. 3 and a permanent, hexagonal-column concentration pattern frozen into the solid behind as diagramed in Fig. 4 (Chalmers [ 6 ] ) . T h e static structure is merely a partial record of the dynamic processes by which it has been pro- duced—this is the important point.

M a n y other well-studied examples of dynamic structure in physical systems could be cited from fluid mechanics, meteorology, geophysics, either case the net result is an oscillating, or rhythmical, flow at each point in the liquid.

The values of the critical heating rate for the onset of flow have been very successfully predicted by the theory of convective instabil- ity (cf. Chandrasekhar [ 7 ] ) . The analysis below is patterned after t h a t theory.

Another remarkable instance of dynamic structure occurs under certain conditions of freezing of solid out of molten solution. The transformation process is solidification with its accompanying heat

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226 J O H N I . G M I T R O AND L . E . S C R I V E N

a n d astrophysics. As for biological examples, we prefer to leave these a n d t h e biological implications of w h a t follows to the biologically expert reader.

I n all cases the k e y factors a p p e a r to be three. First, t r a n s f o r m a - tion processes: changes in physical state, as by phase transition, or

r

i f •

FIG. 3. Top view of the decanted surface of a solidification front with a hexagonal tessellation. (Photograph courtesy of John Wiley & Sons, Inc.) in chemical state, as by chemical reaction. Second, t r a n s p o r t processes: changes in location, as by convection or diffusion. Third, coupling of the two t y p e s of processes together: both m u s t proceed simultaneously and affect each other. F o r our present purposes chemi- cal reactions p l a y t h e p a r t of t r a n s f o r m a t i o n while simple diffusion plays t h a t of t r a n s p o r t .

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Analysis of these factors can be logically organized around t h e following set of p r o b l e m s :

1. Origin of p a t t e r n and r h y t h m from a uniform and s t e a d y s t a t e of t r a n s f o r m a t i o n in systems in which d e p a r t u r e s from unifor- m i t y give rise to t r a n s p o r t processes (equilibrium systems are a p a r t i c u l a r c a s e ) . A n a t u r a l a d j u n c t of this item t u r n s out to be

FIG. 4. Concentration patterns frozen into the solid as hexagonal columns behind a solidification front. Broken lines indicate boundaries of the unit pattern.

Regions of high solute concentration are shaded.

2. Signal transmission by p r o p a g a t i o n of small local disturbances in an initially uniform and s t e a d y - s t a t e system. W i t h solutions to these problems, one can s t u d y

3. Control of p a t t e r n size, r h y t h m i c period, p r o p a g a t i o n velocity, and wavelength, especially t h e dependence of possibilities on the complexity of the system—here, the n u m b e r of p a r t i c i p a t i n g chemical species. Beyond this lie more difficult questions of

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228 J O H N I . GMITRO AND L . E . S C R I V E N

4. Evolution and stability of particular patterns, rhythmic vari- ations, and waveforms, in which so-called nonlinear effects are likely to be dominant. Before answering these questions one can investigate

5. Effect of preexisting pattern on spontaneously developing p a t - tern and r h y t h m and disturbance propagation, as most simply exemplified by these processes in homogeneously compartmen- talized systems. If at the outset chemical effects alone are con- sidered, as is done here, a parallel problem is

6. Coupling of chemical patterns and waves to electrical and me- chanical stress fields in the material and thereby to forces, accel- erations, and movements. Ultimately this coupling and the ac- companying convective transport should be included in the first

problem.

This is a large undertaking and we restrict ourselves here to the first three and last items, focusing on the physicochemical side of the over- all problem. For a variety of reasons, some of which m a y become evi- dent, it has seemed desirable to study pattern and r h y t h m in surfaces or membranes, and lines, or fibers.

FORMULATION

The basic system under consideration is diagramed in Fig. 5. I t consists of a membrane or thread, uniform across its thickness, within

EXCHANGE

ÎÉACTION

DIFFUSION ^REACTION DIFFUSION

EXCHANGE

FIG. 5. Diagram of basic system considered in the text. Reaction and diffusion take place within or on a membrane or thread. There is also exchange with the surrounding media.

which various chemical species are reacting and diffusing along its length. At the same time some or all of the participating species are exchanging with the surrounding media. The number of participat- ing species is left open, for an important question to be answered

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is w h a t influence t h e n u m b e r of species h a s on p a t t e r n a n d r h y t h m . H e r e reaction is t h e only t r a n s f o r m a t i o n process a n d diffusion t h e only t r a n s p o r t process inside t h e s y s t e m ; modifications necessary to account for convective, electrical, a n d other effects can be m a d e subsequently.

Equations of Change

A t every location within t h e system each chemical species obeys t h e conservation equation,

R a t e of ac- n e t r a t e of p r o - n e t r a t e of in- n e t r a t e of c u m u l a t i o n d u c t i o n b y chem- flux b y diffu- i n p u t b y ex-

within t h e ical reaction + sion along + change w i t h (1) s y s t e m w i t h i n t h e sys- t h e s y s t e m t h e sur-

t e m r o u n d i n g s I n m a t h e m a t i c a l symbols t h e equation becomes

^ = Ri +

Ji

+ Qi (2)

where Ci is t h e local concentration of t h e ith species a n d t is time.

T h e system is n o t really described until constitutive relations specifying the r a t e processes present h a v e been s u b s t i t u t e d for t h e r a t e symbols Ri, Ji, a n d Qi.

F o r our present purposes i t is sufficient t o note t h a t t h e r a t e of chemical production of t h e t t h component, Ri, is in general a function of m a n y or all of t h e A^ chemical species p a r t i c i p a t i n g in t h e s y s t e m : Ri = Ri{cx, Co, . . . Cn). I t m a y also depend on other species t h a t do not r a n k as p a r t i c i p a t i n g , as well as on physical factors such as t e m p e r - a t u r e a n d electrical potential. (This is equally t r u e of t h e diffusion a n d mass-transfer coefficients introduced below.)

T h e n e t diffusive flux of t h e ith component, Jiy can be expressed as the surface or line divergence of a flux v e c t o r : Ji = V j V W e shall assume t h a t diffusion in t h e system can be described by a version of F i c k ' s law t h a t accounts for coupled diffusion:

Ν

]i = ^ DijVcj (3)

where Vcj s t a n d s for the gradient in concentration of t h e ; t h com- ponent along t h e system a n d t h e Di ;- are diffusion coefficients giving t h e m a g n i t u d e of t h e flux of t h e ith component caused by a g r a d i e n t in

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230 J O H N I . G M I T R O AND L . E . S C R I V E N

iV

I-

concentration of t h e ; t h component. E a c h diffusivity is in general a function of t h e composition of t h e s y s t e m :

Dij = Dij(ci,c2, . . . cN).

T h u s if F i c k ' s law applies t h e n e t diffusive flux along t h e system is given by

Ν

Ji = V · ^ DijVcj 3 = 1

Ν Ν Ν

3 = 1 j=lk=l

T h e second sum of t e r m s represents t h e effects of composition-de- pendent diffusivities which we shall ignore for the sake of simplicity.

I n the first sum, which contains the familiar t e r m D ^ V 2

^ , t h e L a p l a c i a n operator measures t h e d e p a r t u r e of a concentration profile from l i n e a r i t y — w h a t might be called t h e " b u m p i n e s s " of t h e concentra- tion field in t h e system. Diffusion n o r m a l l y acts to smooth out such

" b u m p i n e s s . "

F o r t h e r a t e of i n p u t of t h e ith component from t h e surroundings to the system, Qh we shall a d o p t an extension of the N e w t o n - N e r n s t t y p e of law, or constitutive relation, t h a t accounts for coupled exchange:

Ν

Qi = 2 Htj(c» - Ci) (5)

3 = 1

H e r e Cj° is t h e concentration of the ; t h component in t h e surroundings.

I t is supposed to be uniform a n d constant. T h e Hij are mass-transfer coefficients giving t h e net i n p u t flux of the ith component caused by a u n i t difference in t h e internal a n d external concentrations of t h e ; t h component. T h e y too m a y depend on composition: Hij = Hn(c1,c2,

. . . cN).

Substituting these constitutive relations into E q . (1) gives the basic equation of change in the system under consideration. F o r the ith p a r t i c i p a t i n g species:

Ν Ν

^ = / ^ ( Γ Χ , Γ , , · · · r.v) + ^ Ι)

ιΊν><Ί + ^ / / ,7( r / - Cj) (6)

3=1 j=l

T h e exchange terms on t h e far right are present because t h e m e m b r a n e or t h r e a d is an open system. T h e y replace, largely or completely,

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b o u n d a r y conditions which are necessary for a complete description of a closed system. And inasmuch as n o n p a r t i c i p a t i n g species are not excluded, t h e equations of change of all Ν p a r t i c i p a t i n g species should be regarded as independent, another w a y in which t h e openness of t h e system shows.

A t s t e a d y s t a t e t h e concentrations of p a r t i c i p a t i n g species are con- s t a n t in time, by definition. If t h e s t a t e is uniform as well, there are no gradients in t h e system, and t h e equation of change reduces to a s t a t e m e n t t h a t r a t e s of production a n d o u t p u t of each component m u s t j u s t cancel:

H e r e cf s t a n d s for t h e c o n c e n t r a t i o n of t h e j t h c o m p o n e n t in t h e s t e a d y s t a t e . T h e corresponding values of t h e r a t e s of chemical p r o d u c t i o n a n d of t h e mass-transfer coefficients are Rf = Ri(ci

s

, c2% . . . cN s ) a n d Η if = Ηί3·(βι% c2

s

, . . . cN s

), respectively.

U n s t e a d y states can be represented b y concentrations t h a t are sums of s t e a d y - s t a t e values and excursions from t h e s t e a d y s t a t e :

T h e corresponding reaction rates, mass-transfer, and diffusion coeffi- cients m a y be expanded a b o u t their s t e a d y - s t a t e values by m e a n s of T a y l o r ' s theorem (Sokolnikoff and Redheffer [ 1 5 ] ) . T h u s

Excursions from Uniform Steady State

Ν

(7)

Ci Ci I χ% (8)

k = l

+ (higher-order t e r m s ) (9a)

Ν

Dij(ci,C2, . . . CAT) A / +

k=l

+ (higher-order t e r m s ) (9b) TV

+ (higher-order t e r m s ) (9c)

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232 J O H N I . GMITRO AND L. E. SCRIVEN

Provided t h e excursions, or " p e r t u r b a t i o n s , " x% — o% Ci** are slight, all t h e higher-order t e r m s m a y be neglected. S u b s t i t u t i n g t h e remaining expressions (9) in t h e equation of change ( 6 ) , invoking (7) to cancel certain of t h e t e r m s , a n d neglecting all t e r m s involving Xi's raised to powers higher t h a n t h e first

1

produces a conveniently linear version of t h e equation of change. T h i s version governs small excursions from t h e uniform s t e a d y s t a t e :

3 = 1 k = l j=l

(10) All of t h e q u a n t i t i e s within t h e braces m a y be lumped together under a new symbol Ki3-

S

which s t a n d s for r a t e constants of processes t h a t effectively follow first-order kinetics in t h e concentration p e r t u r b a t i o n s .

~dt

1\ 1\

= ^ Kij'Xj + ^ Dif V%

j=i j=i

( I D

T h e lumping together is merely a m a t h e m a t i c a l convenience; the first-order processes implied by t h e procedure are a c t u a l l y combina- tions of chemical reaction and exchange with t h e surroundings.

T h e s t a t e of t h e system m u s t be t h o u g h t of as t h e set of concentra- tions of all t h e p a r t i c i p a t i n g species. T h e n a t u r a l m a t h e m a t i c a l lan- guage for discussing states t h a t are sets of variables is m a t r i x analysis

(Bellman [ 3 ] ; F r a z e r et al. [ 8 ] ; G a n t m a c h e r [ 9 ] ) . Concentrations are arranged in an ordered set called a column m a t r i x :

Cl Ci

s

Xl

c2 C2 S

X2

+

_cN_ _CN S

_ JX

N_ 1

On the grounds t h a t if the Xi's are sufficiently small, their squares and products are one or more orders of magnitude smaller, and so will be their higher-order powers and products.

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Correspondingly, r a t e constants a n d diffusion coefficients are arranged in square matrices :

Dij s

, i a n d j = 1, 2, . . . Ν [D] =

Kij s

, i a n d j = 1, 2, Ν <-> [Κ] =

O n ' £>12

S

·

# 2 2 * ·

. . DNN\

'Κη· . . Km*

KN1° Km" . . . KNN\ For example, for a system of two species there are two entries in

[x] a n d four each in [ K ] a n d [ D ] , some of which m a y t u r n out to be zero, however .

W i t h m a t r i x n o t a t i o n a n d t h e rules for m a t r i x multiplication t h e e q u a t i o n of change of excursions, E q . (11), t a k e s t h e compact form of a s t a n d a r d linear p a r t i a l differential m a t r i x e q u a t i o n :

= [K][x] + [D]V»[x] (12)

E q u a t i o n (12) s t a t e s t h a t if a s t e a d y s t a t e of t h e system is slightly perturbed, t h e r a t e a t which t h e excursion grows or decays is controlled by t h e competition between (a) first-order processes depending on chemical reaction within t h e s y s t e m a n d mass exchange with t h e sur- roundings, and (b) diffusive processes governed by " b u m p i n e s s " of t h e concentration distributions within t h e system.

W h e t h e r an excursion grows, whether it periodically oscillates, a n d w h a t size of p a t t e r n is likely to emerge as it develops, are questions t h a t can be answered by solving E q . (12).

Solution in Terms of Elementary Patterns

E q u a t i o n (12) can be solved by t h e m e t h o d s of h a r m o n i c analysis, which rest on a r e m a r k a b l e t h e o r e m going b a c k to F o u r i e r : A n y spatial p a t t e r n m a y be expressed as a weighted sum of m e m b e r s of a s u i t a b l y chosen set of elementary patterns (Bell [ 2 ] ; Tolstov [ 1 8 ] ) . M a t h e -

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234 J O H N I . G M I T R O AND L . E . S C R I V E N

m a t i c a l l y , t h e e l e m e n t a r y p a t t e r n s are a complete set of characteristic functions, or eigenfunctions, a p p r o p r i a t e to t h e geometric configura- tion of t h e system of interest. T o solve E q . (12) we express t h e excur- sion [x] in t e r m s of one of t h e sets of eigenfunctions {F0(r), F i ( r ) , . . .

Fn{r), . . . } of the L a p l a c i a n operator. T h a t is, we w r i t e 2

where the m a t r i x [y(t) ]ic is a vector of weighting functions, in general time-dependent, which describe how a given e l e m e n t a r y p a t t e r n of one p a r t i c i p a t i n g species is related in space a n d in time to all of the other e l e m e n t a r y p a t t e r n s . N o t e t h a t t h e [ y ( £ ) U gives t h e t i m e de- pendence and Fk(r) the s p a t i a l dependence of the excursion [ x ] .

Eigenfunctions of the L a p l a c i a n operator satisfy t h e equation

T h e c o n s t a n t k is k n o w n as t h e characteristic p a r a m e t e r , or eigenvalue, corresponding to t h e eigenfunction Fk. Geometrically, k describes t h e m e a n size, o r ^ w a v e l e n g t h , of_the corresponding e l e m e n t a r y p a t t e r n . I n fact, t h e m e a n p a t t e r n size, I, is 2w/k. (If k is complex, as for w a v e p r o p a g a t i o n , t h e n I = 2w/kr; t h e i m a g i n a r y p a r t ki gives t h e r a t e of a t t e n u a t i o n of p a t t e r n or w a v e w i t h distance.)

Eigenfunctions can be so a r r a n g e d as to possess a very useful p r o p e r t y known as orthogonality :

T h e p r o d u c t of two different e l e m e n t a r y p a t t e r n s integrated over the system is zero. Consequently (a) s u b s t i t u t i n g the expansion in eigenfunctions, E q . ( 1 3 ) , i n E q . (12), (b) eliminating t h e L a p l a c i a n operator with (14), a n d (c) removing t h e infinite sum as well as the eigenfunctions through multiplication b y Fj(r) followed by i n t e - gration and application of E q . (15) yield t h e o r d i n a r y m a t r i x differen- tial e q u a t i o n

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V 2

^ ( r ) = -kWk(x) (14)

1, j = k

0,j^k (15)

d[y]dt k = ([K] - k*[O])[y]k (16) 2

Summation must be replaced by integration when the set is continuous rather than discrete (cf. Tolstov [18]).

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T h e solution is a v a i l a b l e (Bellman [ 3 ] ; F r a z e r et al. [ 8 ] ) a n d can usually be p u t in t h e form

Ν

[y]k = ^ a *) W[ A kne x

* . n < (17)

71 = 1

where λ^η are t h e Ν eigenvalues 3

a n d [ A ]f c jW are t h e corresponding Ν eigenvectors of t h e m a t r i x [ K ] — / c

2

[ D ] , and a^n are constants dic- t a t e d by initial conditions. S t a n d a r d m e t h o d s are a v a i l a b l e for determining eigenvalues a n d eigenvectors of a given m a t r i x ( F r a z e r et al. [8] ) . T h e eigenvalues are themselves t h e roots of the d e t e r m i n a n - tal equation.

d e t ( [ K ] - k 2

[O] - λ[Ι]) = 0 (18) where [ I ] is the i d e n t i t y m a t r i x . F o r every v a l u e of fc there are as

m a n y eigenvalues, λ/ν%1, A/C)2? etc., as p a r t i c i p a t i n g chemical species, in general.

F r o m E q s . (13) and (17) a solution of t h e equation of change in an excursion, E q . (12), is

[ x ] =

Σ (Σ

a

^[Mk^) F

k

(r) (19)

k=Q n=l

Before t a k i n g u p t h e i n t e r p r e t a t i o n of this result it should be pointed out t h a t there m a y be other solutions, solutions t h a t a m o u n t to p r o p a - gating waves in p a r t i c u l a r . F o r example, t h e function representing simple h a r m o n i c waves p r o p a g a t i n g in t h e x-direction, v i z .

4

[x] = [a]e i(k

r x

-^e- k

^ x

(20) is found by substitution to satisfy E q . (12) provided t h e w a v e - n u m b e r

k (= kr -f- iki) a n d frequency factor ω ( = 2ττ/) together satisfy t h e d e t e r m i n a n t a l E q . (18) with λ replaced by — ίω.

E q u a t i o n (18) for t h e eigenvalues of t h e t r a n s f o r m a t i o n - a n d - t r a n s - port m a t r i x is t h e p i v o t on which the whole theoretical analysis t u r n s .

I N S T A B I L I T Y A N D W A V E P R O P A G A T I O N

T h e formal solution, E q u a t i o n (19), indicates t h a t t h e ability of a system to support p a t t e r n formation and r h y t h m i c oscillation is

3

Equation ( 1 7 ) must be modified in special circumstances when two or more eigenvalues are equal. T h e exponential time-dependence persists, however. 4

The excursion [x] is a set of real quantities which may be taken equally well as the real part or as the imaginary part of the right-hand side.

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236 J O H N I . G M I T R O AND L . E . S C R I V E N

governed by t h e eigenvalues of t h e m a t r i x [ K ] — fc 2

[D], which depend in t u r n on t h e reaction-and-exchange coefficients Kif a n d diffusion coefficients Di ;

s

(discussed above) a n d on t h e p a t t e r n size factor k . These eigenvalues, denoted b y λ^Μ, are in general complex, t h e real p a r t being a growth factor t h a t determines whether p a t t e r n can de- velop spontaneously, a n d t h e i m a g i n a r y p a r t being an oscillatory frequency (which m a y be zero) associated with t h e p a t t e r n .

If a n y of t h e Ν values, λ^,ι, λ ^ , . . . \k,N, h a v e real p a r t s t h a t a r e positive t h e s y s t e m is said to be u n s t a b l e with respect to p a t t e r n of size I = 2w/k, because a n y t r a c e of such p a t t e r n t h a t h a p p e n e d to b e p r e s e n t in a t i n y d i s t u r b a n c e of t h e otherwise uniform a n d s t e a d y s t a t e

STATIONARY I 2

OSCILLATORY I 2

FIG. 6. Stationary and standing oscillatory chemical instability in a one- dimensional system. Distributions of two participating species, 1 and 2, are shown at three successive times, with pattern size i indicated.

would grow exponentially in t h e s u b s e q u e n t excursion. I n m o r e m a t h - e m a t i c a l t e r m s , a n y c o n t r i b u t i o n of Fk(r) to a n initial, r a n d o m l y con- s t i t u t e d [x] would increase as exp (\k,

nt). T h e converse s i t u a t i o n in which all t h e eigenvalues h a v e n e g a t i v e real p a r t s is stable, according to a celebrated t h e o r e m of L y a p u n o v (Bellman [3]; G a n t m a c h e r [9]).

If t h e \k,

n h a v i n g t h e largest positive real p a r t is entirely real, t h e s y s t e m is said to exhibit s t a t i o n a r y instability w i t h respect to p a t t e r n size i = 2ir/k. I n t h i s case c o n c e n t r a t i o n s a t each p o i n t in t h e s y s t e m simply increase a n d p a t t e r n r e m a i n s in place, as depicted in Fig. 6a.

On t h e o t h e r h a n d , if t h e d o m i n a n t \k,n is complex, t h e s y s t e m is said to exhibit oscillatory i n s t a b i l i t y w i t h respect to p a t t e r n of size i = 2w/k.

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C o n c e n t r a t i o n s a t each p o i n t r h y t h m i c a l l y cycle w i t h frequency equal to t h e i m a g i n a r y p a r t of \k,n divided b y 2π, t h e a m p l i t u d e of t h e excur- sions increasing w i t h t i m e as shown in Fig. 6b. P a t t e r n m a y t r a n s l a t e along t h e s y s t e m in t h e m a n n e r of t r a v e l i n g waves, or a l t e r n a t e l y wax a n d w a n e while r e m a i n i n g in place, like s t a n d i n g waves. I n d e e d , t h e d o m i n a n t p a t t e r n size can b e viewed as a n a t u r a l " c h e m i c a l wave- l e n g t h ' ' belonging to t h e original s t e a d y - s t a t e s y s t e m .

5

Because t h e real p a r t of λ&,η is an exponential growth factor it is clear t h a t t h e p a t t e r n size for which t h e growth factor is largest will d om i na te an excursion sooner or later, provided the competition continues to be among independent, exponentially growing p a t t e r n s . Strictly speaking this is t h e case only as long as t h e excursion r e m a i n s sufficiently small so t h a t t h e linearized equation of change, E q . (10) or (11), a c c u r a t e l y describes it. Thereafter t h e neglected nonlinear effects come into p l a y and eventually h a l t t h e exponential r u n a w a y process.

T y p i c a l l y t h e nonlinear effects bring t h e u n s t a b l e system t o a new s t e a d y s t a t e or oscillatory limit cycle in which p a t t e r n size a n d r h y t h - mic frequency differ only slightly from those t h a t d o m i n a t e d t h e early linear stages of i n s t a b i l i t y — a t least in convectively u n s t a b l e systems ( C h a n d r a s e k h a r [ 7 ] ) ; there is no reason to expect otherwise from the t y p e of system u n d e r consideration here.

T h e special situation in which t h e largest growth factor is precisely zero is t e r m e d m a r g i n a l instability. I t w a r r a n t s a t t e n t i o n because it d e m a r c a t e s stable situations from u n s t a b l e ones and, in t h e u n s t a b l e ones, those p a t t e r n sizes t h a t c a n n o t develop from those sizes t h a t can. M a r g i n a l i n s t a b i l i t y m a y be s t a t i o n a r y or oscillatory; in either case t h e contribution of the corresponding Ffc(r) to t h e initial distur- bance is constant with time, ne i t h e r growing nor decaying.

T o sum u p with regard to i n s t a b i l i t y : A n y uniform, s t e a d y - s t a t e system of chemical reaction a n d diffuson suffers low-level noise, either t r a n s m i t t e d from surroundings or arising i n t e r n a l l y in molecular fluc- t u a t i o n s . A p p e a r i n g more or less continually a n d r a n d o m l y t h r o u g h o u t t h e system as small concentration p e r t u r b a t i o n s , t h e disturbances are w i t h o u t effect or trigger development of p a t t e r n a n d r h y t h m , according to t h e n a t u r e of sets of coefficients t h a t characterize reaction, exchange, a n d diffusion. W h a t m a t t e r s are t h e eigenvalues of m a t r i c e s comprising these coefficients; in p a r t i c u l a r , t h e eigenvalue containing t h e largest

5

This is the point of view taken by Turing in his pioneering paper, cited below.

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238 J O H N I . G M I T R O AND L . E . S C R I V E N

growth factor. Because t h e eigenvalues through t h e coefficients m a y depend, as r e m a r k e d above, on factors other t h a n t h e p a r t i c i p a t i n g species, it is entirely possible t h a t a previously stable s t e a d y s t a t e m a y t u r n unstable owing to changes in such factors. F a c t o r s other t h a n t h e p a r t i c i p a t i n g species can also control w h e t h e r t h e ensuing instability is s t a t i o n a r y or oscillatory, how r a p i d l y it develops, and t h e p a t t e r n size and r h y t h m i c frequency. Once t h e n u m b e r of p a r t i - cipating species is known and values of the coefficients are avail- able, one can determine all of these features from t h e solutions of E q . (18) for every admissible value of t h e size factor, k (see section on E x a m p l e s ) .

T h e simple-harmonic-wave solution, E q . (20), implies t h a t t h e wavelength, a t t e n u a t i o n with distance, a n d speed of t r a v e l i n g chemical w a v e s excited b y a r h y t h m i c a l l y v a r y i n g local c o n c e n t r a t i o n source are also governed b y E q . (18), t h e e q u a t i o n for t h e eigenvalues of t h e t r a n s f o r m a t i o n - a n d - t r a n s p o r t m a t r i x . I n this situation, however, t h e frequency / = ω/2π of t h e source is k n o w n ; i.e., t h e i m a g i n a r y p a r t of λ is given. F u r t h e r m o r e , if t h e a m p l i t u d e of oscillation a t t h e source is c o n s t a n t , as is c u s t o m a r i l y t h e case, t h e real p a r t of λ is zero a n d λ = —τω as n o t e d a t E q . (20). T h e wavelength I a n d exponential a t t e n - u a t i o n factor ki of w a v e s excited b y a given frequency are u n k n o w n a priori. B u t if t h e n u m b e r of p a r t i c i p a t i n g species, t h e r a t e c o n s t a n t s a n d diffusion coefficients are given, E q . (18) can b e solved for t h e com- plex variable k, t h e real p a r t of which gives t h e w a v e l e n g t h i = 2w/kr a n d t h e i m a g i n a r y p a r t of which is t h e a t t e n u a t i o n factor in exp ( — kix). T h e velocity of p r o p a g a t i o n , or w a v e speed, is c = ji.

F o r given matrices [ K ] a n d [ D ] , it m a y t u r n out t h a t there is a p a r t i c u l a r frequency which excites waves t h a t are least a t t e n u a t e d or even amplified as t h e y t r a v e l o u t w a r d from their source; i.e., k\ m a y pass through a m i n i m u m as a function of ω. If so, the system m a y be regarded as having a n a t u r a l "chemical frequency."

U n a t t e n u a t e d wave propagation, which occurs a t frequencies for which ki = 0, is of special interest for tw

T

o reasons. I t m a y , of course, separate conditions of wave a t t e n u a t i o n from those of amplification with distance. I t is also t h e bridge with instability a n a l y s i s : t h e con- ditions for u n a t t e n u a t e d w a v e propagation are exactly those for m a r g i n a l oscillatory instability, inasmuch as λ is a pure i m a g i n a r y n u m b e r and k is a real n u m b e r in both. Figure 7 shows a chemical wave being t r a n s m i t t e d u n a t t e n u a t e d by an otherwise uniform and s t e a d y - s t a t e system.

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concentration excursion

FIG. 7. Unattenuated chemical wave propagation, or marginal traveling chemical instability in a one-dimensional system. Distributions of one partici- pating species are shown a t three successive times, with wavelength I indicated.

T h e wave is moving t o t h e right.

Analysis of instability a n d of wave p r o p a g a t i o n m a y be contrasted as shown in t h e following t a b u l a t i o n :

Wave-

length Atten-

Growth or p a t - uation Ultimate factor Frequency tern size factor goal

λΐ = ω = 2Τ Γ / / k

r = 2ττ/€ ki

Instability Find Find Given 0 P a t t e r n size having m a x i m u m growth r a t e

Propagation 0 Given Find Find Frequency having

m i n i m u m a t t e n u a t i o n

E X A M P L E S

I n t h e preceding sections we have shown how t h e origin of regular p a t t e r n s a n d r h y t h m s in t h e t y p e of system under consideration, a n d chemical signal transmission as well, can all be t h o u g h t of as i m m e - diate consequences of E q . (18), which involves sets of p a r a m e t e r s t h a t describe chemical reaction, exchange with t h e surroundings, a n d internal diffusion in small excursions from t h e uniform s t e a d y state originally present. W e also outlined the derivation of t h e pivotal e q u a - tion from basic physicochemical principles. O u r object now is t o show b y m e a n s of examples w h a t some of t h e implications of t h e analysis are, p a r t i c u l a r l y in regard t o control of p a t t e r n size, r h y t h m i c period or frequency, propagation speed, and wavelength.

T o do this we examine cases of one, two, a n d three p a r t i c i p a t i n g species. I n t h e first t w o we can write t h e characteristic equation for the eigenvalues, X

k>n, explicitly in t e r m s of t h e system p a r a m e t e r s

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240 J O H N I . G M I T R O AND L . E . S C R I V E N

a n d t h e p a t t e r n factor k; in t h e t h i r d case it is scarcely worthwhile a t t e m p t i n g to do so (a cubic equation m u s t be s o l v e d ) . F r o m t h e characteristic equation, w h e t h e r or n o t in explicit form, t h e relation- ships can be found which t h e system p a r a m e t e r s m u s t satisfy, as functions of k, in order for a given t y p e of i n s t a b i l i t y to occur—sta- t i o n a r y , oscillatory, m a r g i n a l . T h e R o u t h - H u r w i t z criteria for eigen- values of matrices a n d R o u t h ' s algorithm for polynomials with real coefficients are v e r y useful ( G a n t m a c h e r [9] ) .

F o r our purposes it is not necessary t o know or assign values of t h e system p a r a m e t e r s Kij

8

a n d D j j s

. T h e relationships a m o n g these q u a n - tities are more i m p o r t a n t t h a n their individual m a g n i t u d e s , a fact t h a t is brought o u t b y their m a t r i x representation. A t first glance it a p p e a r s t h a t where t h e r e are Ν p a r t i c i p a t i n g species, t h e eigenvalues are influenced by all Λ

72

r a t e constants plus N 2

diffusion coefficients as well as t h e size factor, fc—a t o t a l of 2N

2

+ 1 p a r a m e t e r s (of which some m a y be zero). I n a c t u a l i t y t h e y can depend on no more t h a n

(N 2

-\-3N)/2 independent quantities, according to m a t r i x theory.

These quantities are i n v a r i a n t s of t h e matrices [ K ] a n d [ D ] and their p r o d u c t ; t h e y are independent combinations, or functions, of t h e system p a r a m e t e r s of which t h e matrices are composed. F o r a q u a l i t a - tive examination of examples we need specify no more t h a n ranges of values for the i n v a r i a n t s .

W h e n Ν is two, t h e r e are five i n v a r i a n t s in place of nine p a r a m e - ters. As Ν becomes large there is an A^

2

/2 versus 2N 2

, or fourfold, saving in t h e n u m b e r of factors whose influence should be studied.

Moreover, t h e stability or instability of a system is unchanged if the r a t e constants a n d diffusion coefficients are changed b u t in such a w a y as to leave t h e i n v a r i a n t s unchanged. H e r e m a y be seen some a d v a n t a g e s of a m a t r i x formulation when sets of variables and sets of p a r a m e t e r s are involved.

T h e simplest conceivable situation is of course t h a t of JV = 1.

T h e matrices contain b u t one element e a c h : [ Κ ] = Κ and [ D ] = D.

T h e characteristic E q . (18) is simply

One Participating Species

Κ — k 2

D — λ = 0 R e a r r a n g e d , this gives t h e single eigenvalue,

(21)

(22)

(21)

Stability behavior is i m m e d i a t e l y obvious: there is no possibility of spontaneous oscillation because \jCA can never t a k e on complex v a l u e s ; if Κ is n e g a t i v e — a u t o c a t a l y t i c d i s a p p e a r a n c e of t h e compound—the system is a l w a y s stable since t h e diffusion coefficient is positive; if Κ is p o s i t i v e — a u t o c a t a l y t i c a p p e a r a n c e — t h e system is stable for all

I

< 2ττ

\/D/K,

m a r g i n a l l y u n s t a b l e a t

i

= 2?r

Λ/Ό/Κ,

a n d is u n s t a b l e with respect to all larger p a t t e r n sizes. G r o w t h r a t e increases m o n o t o n i - cally with p a t t e r n size, a p p r o a c h i n g Κ a s y m p t o t i c a l l y ; there is no domi- n a n t size a p a r t from t h e largest t h a t can fit into t h e physical system.

Stability with respect to the smallest sizes is assured by the action of diffusion—the t r a n s p o r t process provides some stabilization when t h e t r a n s f o r m a t i o n processes are destabilizing.

NO OSCILLATION POSSIBLE

FIG. 8. Stability behavior with one participating species. Growth factor λ

Γ is

plotted versus pattern size t for cases t h a t would be stable (left) or unstable (right) if there were no diffusion. T h e solid point denotes marginal instability, stationary mode.

T h e s e findings can b e c o n v e n i e n t l y s u m m a r i z e d b y d i a g r a m s as in Fig. 8, w h e r e we h a v e s k e t c h e d g r o w t h factor ( t h e real p a r t of λ) versus p a t t e r n size I = 27r/fc. N o t e t h a t a d j u s t m e n t s of Κ a n d D give crude control a t b e s t a n d v i r t u a l l y no selection of t h e features of t h e m o s t r a p i d l y developing p a t t e r n s .

W a v e - p r o p a g a t i o n c h a r a c t e r i s t i c s are o b t a i n e d b y solving E q . (21) for real a n d i m a g i n a r y p a r t s of k, given λ = —τω= —

i2wf :

< = % = J /

8Τ2

° (23) 1-+ 2ττ VD/K

a n d c - » 0 a s / - > 0 (23a)

2 VvD/f^O a n d c - » 2 V ^ a s / ^ oo (23b)

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242 J O H N I . G M I T R O AND L . E . S C R I V E N

yz 2

+ 4ττ 2

/ 2

- κ 2D

(24) h -> irf/\/\K\ D -> 0 as / -> 0 (24a) (24b)

00

T h e r e is no possibility of u n a t t e n u a t e d p r o p a g a t i o n because ki c a n n o t t a k e on t h e v a l u e zero (except in t h e limit of no excitation, i.e., as / •—> 0). T h e r a t e of a t t e n u a t i o n w i t h distance,/^-, increases as exciting frequency, / , increases. T h e r a t e of a t t e n u a t i o n diminishes as t h e r a t e c o n s t a n t Κ increases, being least when t h e s y s t e m is highly u n s t a b l e , i.e., Κ » 0. T h e largest w a v e l e n g t h t h a t can b e excited is i = 2ΤΓ

Λ/D/K,

which is also t h e m a r g i n a l l y u n s t a b l e p a t t e r n size. T h e p r o p a g a t i o n velocity, c{=Jl), can always be increased b y raising t h e frequency b u t always a t t h e cost of greater a t t e n u a t i o n . Possibilities for control of w a v e transmission are v e r y limited. T h e r e is, incidentally, a close m a t h e m a t i c a l analogy with electrical signal t r a n s m i s s i o n by s u b m a r i n e cables a n d w a v e guides (Carson [5]; Adler et al. [1]).

One of the curious aspects of chemical wave p r o p a g a t i o n is the role of diffusion. If diffusion were either impossible (D = 0) or in- finitely rapid ( D- * o o )— a s in d r e a m s of chemical kineticists—simple harmonic waves of t h e sort under consideration would be impossible;

either t h e y would be halted a t t h e source or t h e y would never t a k e shape in the system. T h e larger t h e diffusion coefficient, t h e lesser t h e r a t e of a t t e n u a t i o n , t h e greater the wavelength and the faster the wave velocity, all else being equal.

Two Participating Species T h e matrices have four entries e a c h :

6

[K] = KH κ

κ» κ.

22

12 [D] = Du D

û21 D 22

12

The characteristi c Eq . (18 ) reduce s t o λ

2

+ d{K + cl

2 = 0 (25)

where dt and d2 are functions of t h e i n v a r i a n t s : di = /c

2

/i -

h

d2 = k*h

- m, + h

6

The superscript s is suppressed hereafter.

(26)

(23)

FIG. 9. Stability behavior with two participating species. Growth factor λΓ is

plotted versus pattern size I. Open points denote oscillatory and solid points stationary marginal instability. Broken curves indicate oscillatory instability and heavy solid curves stationary instability.

and t h e i n v a r i a n t s are defined as follows:

Ii = t r a c e [D] = Du + Ώ22, h = t r a c e [K] = Ku + K22

Iz = d e t [D] = DUD22 - D12D21, h = d e t [ K ] = KnK22 - K12K21 h = ( t r a c e [ K ] ) ( t r a c e J D ] ) - t r a c e J [ K ] [ D ] ) = KnD22 + K22Dn

— K2iDi2 — K\2D2\ E q u a t i o n s (25) a n d (26) yield t w o eigenvalues :

λ*,» = ill* - k^h

± VkVi

2

~ 4 /3) + 2 &

2

( 2 /4 - hl2) + /2 2

- 4 /5} (27)

λ . , η - > - ^ {/1 ± V/1

2

- 4/

3

} 0 as 0 (27a)

λ*.» - > i { /2 + V/2 2

- 4 / 5 } as 00 (27b)

T h e various t y p e s of instability according to these equations a r e shown in F i g . 9, a n d t h e corresponding restrictions on i n v a r i a n t s are listed in T a b l e I . B o t h of t h e possibilities with a single species also a p p e a r in F i g . 9 ; indeed, t h e t y p e s of i n s t a b i l i t y possible w i t h a

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244 J O H N I . G M I T R O A N D L . E . S C R I V E N

Restrictions on invariants

Case ———

Ii h h h h

1 > ( ) < 0 >() <() > 0

2 > o <kSh

b

d2 = 73(/b 2

- ki

2

){k

2

+ c), a n y 73 > 0, c > 0

3 > o = d2 > 0 when k

2

> ki

2

, a n d di

2

- 4d

2 < 0

when k

2

< ki

2

4 > 0 <0 d2 = h(k

2

- ki

2

)

2

, a n y 7

3 > 0

5 > o <0 d

2 = I3(k 2

- k^ik

2

- k2 2

), a n y 7

3 > 0

6 > o = ki

2

Ii d2 > 0 when k

2

> k

2 2

, a n d - 4d2 = Cl(k

2

- k2 2

)(k

2

- c2)

a n y Ci > 0, c

2 > k\

2

7 > 0 = /C2

2

/l d

2 = h(k 2

- kS)(k

2

- k2 2

), a n y 7

3 > 0, and 7i

2

(A:

2

- k2 2

) - 4:h(k

2

- ki

2

) > 0, k

2

< k

2 2

a

T h e invariants are defined a t E q . (26).

'ΖΤΓ

b

kn = — , where l

n are t h e points of marginal stability in Figure 9.

given n u m b e r of p a r t i c i p a t i n g species a l w a y s include t h e t y p e s possi- ble with fewer species.

A fair degree of control c a n now b e exerted over s t a t i o n a r y insta- bilities. I n C a s e 4 of Fig. 9, for example, w e can select a n y single p a t t e r n size a n d , b y adjusting p a r a m e t e r s m a k e t h e s y s t e m m a r g i n a l l y u n - stable w i t h respect t o it while stable w i t h respect t o all o t h e r sizes. I n Case 5 a n entire b a n d of sizes from Λ t o l2 could b e excited, w i t h 1 = i\i2 \/2/(ti

2

+ l2 2

) d o m i n a n t . Finer control, for example, over g r o w t h r a t e , would r e q u i r e m o r e a d j u s t a b l e p a r a m e t e r s a n d hence m o r e p a r t i c i p a t i n g species.

P a r t i c i p a t i o n of t w o species brings possibilities of s p o n t a n e o u s oscillation, as in Cases 3, 6, a n d 7 of Fig. 9. I n these t h e g r o w t h r a t e increases w i t h p a t t e r n size, a s y m p t o t i c a l l y a p p r o a c h i n g (Kn + K22)/2

in C a s e s 3 a n d 7, (Ku + K22)/2 + V ( # n - K22) 2

/4: - K12K21 in Case 6; consequently t h e r e is again n o d o m i n a n t size a p a r t from t h e largest t h a t c a n fit i n t o t h e physical system. T a b l e I indicates t h a t t h e restrictions on i n v a r i a n t s for t h e oscillatory cases a r e stringent. I n a sense t h e s t a b i l i t y of Case 1 a n d s t a t i o n a r y i n s t a b i l i t y of Case 2 a r e easier t o achieve.

T h e behavior a t vanishingly small p a t t e r n sizes is governed b y TABLE I . Conditions for Instability Behavior Shown in Fig. 9

ft

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t h e i n v a r i a n t s of t h e diffusion m a t r i x alone, while t h a t a t v e r y large p a t t e r n sizes—approaching u n i f o r m i t y — d e p e n d s exclusively on t h e i n v a r i a n t s of t h e r e a c t i o n - a n d - e x c h a n g e m a t r i x . T h i s conclusion fol-

lows here from t h e limiting forms of E q . ( 2 7 ) , b u t in fact holds regardless of t h e n u m b e r of p a r t i c i p a t i n g species.

T h e solution of E q . (25) for w a v e - p r o p a g a t i o n characteristics is too cumbersome t o display here. I t does reveal, we find, t h a t with j u s t t w o p a r t i c i p a t i n g species t h e system p a r a m e t e r s c a n be so a d - justed as to give u n a t t e n u a t e d w a v e p r o p a g a t i o n (fcj = 0 ) a t v i r t u a l l y a n y desired wavelength or exciting frequency.

Three Participating Species

T h e m a t r i c e s h a v e nine entries each, a n d t h e characteristic E q . (18) becomes

λ 3

+ diX 2

+ d2\ + dz = 0 (28)

where now

d1 == k 2

Ii - h d2 == /c

4 /3 - k

2

h + h dz = k«h - h*I7 + k

2

I8 - U

and t h e i n v a r i a n t s a r e traces, seconds, a n d d e t e r m i n a n t s of [ K ] , [ D ] , a n d their product. Since these expressions a n d those for t h e eigenvalues are algebraically cluttered we omit t h e m here.

W e find t h e p a r t i c i p a t i o n of a t h i r d species brings m a n y additional t y p e s of instability behavior, among t h e m t h e cases shown in F i g . 10. Of these p e r h a p s t h e most significant a r e those in which oscillatory i n s t a b i l i t y d o m i n a t e s , for these represent spontaneous development of r h y t h m i c processes in systems originally in s t e a d y s t a t e . I n Case 1, for example, t h e system will m a i n t a i n an oscillation of one p a r t i c u - lar wavelength, or p a t t e r n size, a n d frequency while d a m p i n g all others. I n Case 2, a whole b a n d of wavelengths a n d t h e i r associated frequencies can a p p e a r spontaneously, b u t there is one h a v i n g t h e largest growth factor a n d i t would be expected t o d o m i n a t e t h e others even after t h e linearization t h e o r y fails—quite possibly its dominance would persist in a limit cycle established after nonlinear effects come into p l a y .

M o r e involved s t a t i o n a r y a n d oscillatory behavior is i l l u s t r a t e d by Cases 4 - 6 a n d 7-9, respectively. All of t h e cases shown can be realized u n d e r t h e restriction t h a t t h e d e t e r m i n a n t of [ D ] be positive,

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246 J O H N I . G M I T R O AND L . E . S C R I V E N

FIG. 10. Selected instability behavior with three participating species. T h e various modes are indicated as in Fig. 9.

which is sometimes imposed on certain t y p e s of systems close to equi- librium; if this restriction is removed, still more involved dependence on p a t t e r n size can occur. I n some of the cases shown, as well as others, it is possible to h a v e p a t t e r n s of widely different sizes in close competition for dominance, and as a result t h e y combine to give new a n d more elaborate p a t t e r n s .

D I S C U S S I O N

W i t h o u t examining thoroughly all of t h e new possibilities of con- trol of p a t t e r n size, r h y t h m i c period, growth factor, wave p r o p a g a t i o n speed, a t t e n u a t i o n factor, and wavelength which a c c o m p a n y a third p a r t i c i p a t i n g species, we can see t h a t richer ranges of possibilities come with each additional compound a n d t h e reactions into which it enters. I n s t a b i l i t y behavior j u s t possible with fewer compounds can be realized in a v a r i e t y of w a y s ; entirely new behavior can be produced. T h e same is t r u e with regard to p r o p a g a t i o n of chemical waves when t h e n u m b e r of p a r t i c i p a t i n g species is increased.

Chemical concentration waves could provide large n u m b e r s of p a r - allel signal-transmission channels. I n small-scale systems these might for some purposes be competitive with electrical transmission m e a n s . A s t e a d y - s t a t e reaction system can be arranged to p r o p a g a t e signals a t far faster speeds t h a n diffusional movements, and w i t h o u t t h e a t -

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t e n u a t i o n t h a t also limits p u r e l y diffusional processes, as shown above.

C o n c e n t r a t i o n s of p a r t i c i p a t i n g species can be exceedingly low; in- deed, concentration could be r e i n t e r p r e t e d as t h e p r o b a b i l i t y density of finding a molecule of the species in a given locale.

I n a n engineering sense these observations m e a n t h a t more species broaden t h e range of a l t e r n a t i v e s a n d m a y p e r m i t a better o p t i m u m solution to a n y p a r t i c u l a r design problem. Indeed, a new a l t e r n a t i v e m a y provide a solution where no acceptable solution w a s available before. T h e point is t h a t increased n u m b e r s of species in a m u l t i c o m - p o n e n t system of t r a n s p o r t and t r a n s f o r m a t i o n m a y h a v e decisive influence on t h e a p p e a r a n c e and n a t u r e of p a t t e r n a n d r h y t h m .

Linearized s t a b i l i t y t h e o r y illuminates t h e origin of p a t t e r n a n d r h y t h m as well as t h e p r o p a g a t i o n of small disturbances. B u t although its predictions of d o m i n a n t p a t t e r n size or wavelength are likely t o be close to t h e values for r e s u l t a n t s t e a d y states or limit cycles, t h e p a r t i c u l a r p a t t e r n shape a n d v a r i a t i o n of a m p l i t u d e with t i m e or waveform which evolve are i n v a r i a b l y determined by nonlinear effects t h a t h a v e n o t been considered here. U n t i l an analysis of these effects has been completed, m a n y p a t t e r n s can be p u t forward as c a n d i d a t e s .

Representative Geometric Configurations and Patterns T h e analysis a n d examples t h a t h a v e been presented a c t u a l l y p e r - t a i n to a whole g a m u t of line-like a n d surface-like configurations.

All t h a t is necessary is to place a p p r o p r i a t e i n t e r p r e t a t i o n s on t h e e l e m e n t a r y p a t t e r n functions, Fk(r), and t h e p a t t e r n - s i z e factor, fc.

T h e most i m p o r t a n t cases are listed in T a b l e I I ; r e p r e s e n t a t i v e e x a m - ples are shown in Fig. 11. Some mechanical deformations t h a t m i g h t be produced by t h e chemical p a t t e r n s are also shown, and will be discussed.

If t h e system is unbounded, or a t least v e r y extensive compared to significant p a t t e r n sizes, t h e n continuous ranges of p a t t e r n size i and the factor k m u s t be considered. If, on the other h a n d , t h e system is closed, as are loops, rings, a n d spheres, t h e n these factors can t a k e on only certain discrete values. T h e reason is t h a t t h e basic u n i t of p a t t e r n , w h a t e v e r it is, m u s t r e p e a t itself in t h e system a n integral n u m b e r of times. R a r e l y can t h e fastest-growing a m o n g all p a t t e r n sizes be so accommodated, in which case t h e nearest admissible p a t t e r n size is likely to be d o m i n a n t .

A cylindrical surface is two dimensional a n d can s u p p o r t circum- ferential as well as longitudinal w a v e p a t t e r n s . A rich a s s o r t m e n t

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248 J O H N I . GMITRO AND L . E . S C R I V E N

Elementary pattern

Configuration Coordinates functions k

2

Line X ei2TTxlL 4T T

2

/ L

2

Circle Φ βϊτηφ m

2

/R

2

Cylinder Χ,φ ez(2irx/L+m<£) m

2

4ττ

2

R

2 +

~JJ

Ring θ, φ βΐ(ηθ+πιφ) m

2

η

2

'R?

+

R?

Plane Rectangular: χ , y βΐ ( 2 7ΓΧ / Lx+2TTy 1 Ly )

Polar: r,0 Jn(ar)e im

* a

2

Sphere θ, φ P„

w

(cos Q)e

im

* n(n + 1)

^ , η <

R

2 a

Position is measured by standard length and angle coordinates.

of combinations can be made, e.g., the helically wound concentration pattern diagramed in the fourth row of Fig. 11. Moving patterns might provide means for facilitating diffusion of selected species along a thread, or along a membrane for t h a t matter.

Of all the concentration patterns t h a t can exist on flat surfaces (and certain others) only three are strictly regular—those in which the basic unit is a square, triangle, or hexagon, as sketched in the sixth row of Fig. 11. P a t t e r n functions t h a t are combinations of ele- mentary pattern functions are known for all three (Chandrasekhar

[7]). There is no way of predicting from the linearized analysis whether one of these three or some less regular candidates having the same pattern size will be established when a large expanse becomes unstable. The presence of a boundary or edge nearby m a y favor one pattern over another in a fairly predictable way however. The con- figuration of the source in cases of wave propagation m a y also favor one pattern over another. Figure 12 serves as a reminder t h a t a point source produces circular waves on a homogeneous and isotropic plane ; a line source, lineal waves.

7

M a n y , m a n y other wave patterns can of course be generated by interference of multiple sources. 7

These are described by functions listed for the plane in Table I I , with m = 0 and Ly = 0, respectively.

TABLE I I . Elementary Pattern Functions for Representative Geometric Configurations'

1

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I n t h e case of t h e closed spherical surface, linear t h e o r y provides a little more information. Before describing it we should point o u t some of t h e p a t t e r n s possible on a sphere: t w o axially s y m m e t r i c zonal p a t t e r n s (n = 1 a n d η = 4 ; m = 0) a n d one sectorial p a t t e r n

(n = 10, m = 10) a r e shown in t h e seventh r o w of F i g . 1 1 . T h e r e

UINIFORM, S T E A D Y - CHEMICAL POSSIBLE MECHANICAL STATE CONFIGURATION PATTERNS DEFORMATIONS

2 dVddddQOWar ddzraaaaaaazr ^ ^ Ί ^ - ^ · ^ ^ ^

3

FIG. 11. Representative geometric configurations with some regular chemical patterns t h a t can arise spontaneously. Concentration dependent stress could produce the corresponding mechanical deformations.

are only five strictly regular p a t t e r n s : these correspond t o t h e five regular p o l y h e d r a , t h e P l a t o n i c solids, as shown in F i g . 13. Of these, t h e cube a n d octahedron a r e a conjugate pair, a n d so a r e t h e dodecahe- dron a n d icosahedron; for if t h e concentration m a x i m a form one p a t - tern, t h e m i n i m a m u s t form t h e other ( t h e same is t r u e for t h e t r i -

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250 J O H N I . G M I T R O AND L . E . S C R I V E N

angle-hexagon pair on simple surfaces). F u r t h e r distinctions can be drawn on t h e basis of multicomponent concentration p a t t e r n s b u t we need not go into these here. P a t t e r n functions t h a t are combinations of elementary spherical harmonics are known for all five.

Because t h e surface of a sphere is closed, t h e p a t t e r n - s i z e factor, k, can t a k e on only certain discrete values, which h a p p e n to be given

LINE SOURCE

POINT SOURCE

FIG. 12. Lineal and circular waves propagating on a plane.

OCTAHEDRAL ICOSAHEDRAL

FIG. 1 3 . Strictly regular patterns on the surface of a sphere, corresponding to the five regular polyhedra.

Ábra

FIG. 1. Development of Bénard cells in a dish of liquid heated uniformly from
FIG. 2. Schematic diagram of Bénard cells, showing streamlines of flow within  a cell
FIG. 3. Top view of the decanted surface of a solidification front with a  hexagonal tessellation
FIG. 4. Concentration patterns frozen into the solid as hexagonal columns  behind a solidification front
+7

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