Volume 136. number7.8 PHYSICSLETTERSA 17 April 1989

SIMULATION OF DIFFUSION CONTROLLED REACTION KINETICSUSING CELLULAR AUTOMATA

H. ScottBERRYMAN and Donald R. FRANCESCHETTIDepartmentof Ph%Sics,MemphisStateUniversity, Memphis. TV38/52, USA

Received13 July 1988; revisedmanuscriptreceived20 January1989: acceptedfor publication 30 January1989Communicatedby AR. Bishop

Cellular automatatechniquesareemployedto simulatetheirreversiblebimolecularreactionkineticsof particlesdiffusing intwo dimensions.The reactioninducesa markedshort-rangeanticorrelationbetweenthe reactingspecies.The simulationresultsarecomparedwith thepredictionsof adiscretestochasticmodel which ignoresinterparticlecorrelations,

1. Introduction theverysimplenatureof therulesgoverningthetimeevolutionof the systemallow its dynamicsandstat-

Fora wide rangeof phenomena,from the quench- ics to be thoroughlyunderstoodand characterized.ing of luminescencein solution [1] to the infection As will be seen,the resultsof two-dimensionalsim-of bacteriaby viruses [2], the ratedeterminingstep ulationsmay be quite different from what might beis thediffusion-controlledapproachof two speciesof expectedin thethree-dimensionalcase.Nonetheless.particleswhich interactstronglyonly at close prox- the two-dimensionalsimulations may be valuableimity. Eventhe simplestbinarycombinationprocess, both in providinginsight into processesalso occur-

ring in threedimensionsand in understandingdif-A+B4-C, (I)

fusion-controlled processes in adsorbed phases.maybe associatedwith quite complexbehavior:on membranes,and in crystalstructureswhich permitthemacroscopicscalebecauseof the nonlinearityof defectdiffusion in two dimensions.the sourceand sink termsintroducedinto the con-tinuity equationsfor theconcentrationsof A, B, andC. on themicroscopiclevel becausethereactionpro- 2. Rate equations and the reaction rate constantcesscaninducedistancecorrelationsinto aninitiallyrandommixture of reactants. Accordingto the conventionalmass-actiontreat-

Thepresentcommunicationreportsthe resultsof ment foundin elementarydiscussionof chemicalki-a seriesof simulationsof the kineticsof reaction (1) netics,therateof productionofspeciesC by reactionoccurringirreversibly amongparticleswhich diffuse (1), treatedas irreversible,is given byby hoppingon a squarelattice. Interestin cellular dCautomatais very highat present,at leastpartially be- = kC5C~. (2)causecellularautomatarulesare very amenabletoparallel computation [3]. As a meansof gaining where the ~ (a=A, B. C) denoteconcentrationsgreaterinsight into diffusion-controlledprocesses, and k is the rateconstant.Fordiffusion-controlledhowever,cellularautomatamodelsare attractivein reactions,the rate constantk is frequently derivedthat cellularautomatatechniquesare known to be [1.6] using a model first introduced by Smolu-capableof generatingthemacroscopicstructures(e.g. chowski [7] for the growth of colloidal particles. InBelousov—Zhabotinski contours [4,5], diffusion this approachparticlesof typeA are each assumedlimited aggregates[31)producedby diffusion,while to be initially surroundedby a uniform distribution

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Volume 136. number7,8 PHYSICSLETTERSA 17 April 1989

of type B particles,exceptfor a sphereof radiusR, 3. The cellular automaton rulethe distanceat which A andB react.If the reactionbetweenA and B is assumedto berapid and instan- A cellularautomatonconsistsof a latticeof equiv-taneous,thenthe concentrationof B particlesis set alentpointsor cells, to eachof which is assignedoneequalto zero at R andthe rateof disappearanceof of a small numberof integral values.The valuesas-A calculatedfrom the flux of B particlesto the sur- signedareupdatedin discretestepsby a rule inwhichfaceat R. Assumingthe motions of A andB to be thevaluesassigneddependson thevalueassignedtogovernedby Fick’s first and secondlaws, with dif- it and a limited numberof neighboringcells in thefusion coefficientsDA andDB, one thusobtains

previousiteration.The set of cells which determinek=41cR(DA+DB){l +R[lt(DA+DB)t] i/2} (3) the stateofagiven cell at the next timestepis called

the neighborhoodof the cell. Margolus [3] has in-In most discussions,the transientcomponentis troduceda neighborhoodassignmentschemewhich

consideredof little importanceandthe t—~cclimit, permitsa convincingsimulationof randomwalk dif-which canbe deriveddirectly from the steadystate fusion. In theMargolusprocedure,the latticeof cellsdistribution is dividedat eachtimestepinto blocks of size2x2.

PAR =PAR(l —R/r) (4) The four membranesof each2x2 arrayare rotatedeither clockwise or counterclockwiseon a random

is used.The transientcomponent,which enhances basis.At the next time stepthealternative2 x 2 par-the reactionrateat short times,describesthe rapid titioning of the lattice is chosenand the newblocksreactionofthoseA—B pairswhichby chancearequite are randomlyrotated.The partitioningschemeis il-closetogetherwhenthe reactionbegins. Overtime lustratedin fig. 1.the A—B correlation function evolvesfrom a step In additionto generatinga realisticrandomwalkfunction in r to that given in eq. (4). of isolatedparticles,theMargolusprocedure,and in-

In the caseof particlesdiffusing in two dimen- deedany cellular automaton,would not sharethesions, the rate “constant” k computedin this way featureof Fick’s differential equationsthat permitscan have no time-independentcomponent sinceFick’s secondlaw,underisotropicconditions,hasno 0000 ®®®®non-trivial steadystate solution. The time depen- — — _______ —

denceof k is expectedto be mostpronouncedfor 0 0 0 0 ® 0 0 0equal initial concentrationsof A and B, while with t

onespeciesin greatexcess,PAR would be expectedto 0 0 ‘ 0 0 0 0 0 ©changeless drastically with time and the time de- ~5••~5~~ -~ ._.~E; o o opendenceofk maybefar lessmarked.Fordiscussionpurposes,we note that after direct integration,eq. (a) (b)

(2) yields [8] for the initial conditionsCA=C~,,(-R=CB, ~C=O,

___ ___ 00:00 0000______ C?A(C~-Gc) — ____ —

k=(COlCO)lnCO(CQC), © ®,® ® ® © ®

C°A�C~, (5a) © ®~©© © © ® ©=C(/C°A(C°A—C(’)t, C~=C~. (5b) ® ©‘:® © ® ® © ©

The “integral” rate constant k, defined by these)c) (d)

equations,provides a convenientmeansof sum-marizingthe simulationresults. Fig. I. (a) Alternating partition scheme(Margolus neighbor-

hood)for thesimulationofdiffusion. (b)—(d) Threesuccessivestepsin thesimulationof adiffusion-controlledreactionona lat-tice of 16 Sites.

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Volume 136. number7,8 PHYSICSLETTERSA 17 April 1989

a localized concentrationperturbationto propagate (over intervals including many time steps),twicethroughtheentiresystemin anarbitrarily smalltime, that of the standardrandomwalk.On the other hand, requiring that all particlessi-multaneouslybedisplacedby one lattice constantisnot realistic for eitherthe solid or liquid stateand 4. A stochasticmodelsuggestscautionmaybe requiredin simulatingsomediffusional processes. For comparisonpurposes,a discrete stochastic

To simulatethediffusion-controlledreactionoftwo model which maintainsa random distribution ofdistinct species,the Margolus diffusion procedure particlesat all timeswas also developed.In this cal-was generalizedin a straightforwardway. Eachcell culationwe first found the numberof 2 x 2 arraysofwas assigneda value0 (vacantor speciesC), 1 (spe- each type that would exist if all the particleswerecies A), or 2 (speciesB). Since the reactionproduct randomlydistributed.Thenumberof A’s and B’swaswas assumednot to dissociateit was not necessary then decreasedby the numberof cells containingato distinguishbetweenvacantsitesand sites occu- singleA—B pairplustwicethe numbercontainingtwopied by the product species.An initial 2x2 parti- A—B pairs. This process,beginning with a randomtiontng was chosenand then in eachblock contain- redistributionof the A’s and B’s at each time steping both l’s and 2’s, the siteswere resetto zero in was repeatedat eachtime step.randomlyselectedpairsproducingin effect onepar-ticle of C for eachA eliminated.The valuesassignedto eachof the 2 x 2 blocks in the systemwasthen 5. Resultsand discussionrandomly permuted.The alternative2 x 2 partitionwas then chosen and the processrepeated.The Foursetsof tensimulationsof 200 time stepseachmethod,as it might apply to a systemof 16 sites, is wereperformedon a 128x 128 squarelattice withillustrated in thelast threepartsof fig. 1. periodic boundaryconditions.For concisenesswe

At this point, it might beusefulto briefly compare designatetheaverageinitial siteoccupanciesby (N /the diffusional behaviorgeneratedby the modified N, N,/N), where N= 16384 is the total numberofMargolus procedureadoptedhere with that of the sitesand N1 andN2 are respectivelythe numberofstandardMargolusprocedureandthe randomwalk l’s and 2’s. The initial statesstudiedwere (0.5, 0.5),with fixed stepsize. For theunbiasedrandomwalk (0.05,0.05), (0.9, 0.1) and (0.09, 0.01). Most ofof an isolatedparticleon a one-dimensional,two-di- the (0.9, 0.1) simulationshadachievedcompletere-mensional (square)or three-dimensional(simple actionafter threetime steps,while only oneor twocubic) lattice,onehasthewell-establishedresult for of the minority particlessurvivedthirty time stepsthe meansquaredisplacement. in the (0.09, 0.01) simulations.

The results of the (0.05, 0.05) simulationsareKr-> = N/, (6) shown in fig. 2, plottedasthetotalnumberof C par-

ticlesproducedby thereactionas a functionof time.whereN is the numberof stepstakenand1 the lattice Forcomparison;urposestheresult of the stochasticconstant. Comparisonwith the Einstein—Smolu- model is also plotted. As expected.the simulationchowski relations, resultsyield a smallernumberof productparticlesat

2 —2D1 (7 any time, correspondingto an anticorrelationof AK ‘ > — ‘ and B positions inducedby the reaction.Also in-

then yields a diffusion coefficientD=12/2 per time eludedin fig. 2 is the productionof C particlesfor

step[9]. For boththe standardMargolusprocedure the (0.5, 0.5) case,beyondthe point at which 90%andthemodification usedhere one finds of the A and B particleshavereacted.In theabsence

of any correlation in particle position this curveKr2> = (2N—I )/2 , (8) would be identical to that just discussed,displaced

aboutten time stepsto the right. The relativeslow-leading to a diffusion coefficient /2 per time step nessof the reaction at this point indicatesa strong

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Volume 136, number7,8 PHYSICSLETTERSA 17 April 1989

900 —

20 -

800 — STOCHASTIC — — —(0.05, 005) STOCHASTIC

U, — — — . S •~700— •,.. e0

SIMULATION

°~ 600 —1.5

(05 08) 0::~IISoc~, U,

10

200

(0.5. 2 8)

0.5is: -

SO 100 180TIME STEP

Fig.2. Totalnumberofproductparticlesgeneratedstartingfromthe (0.05, 0.05) initial state,andafter 90% reactionof a (0.5. I

50 100 1550.5) initial state.The solid lines are the averageof 10 simula- TIME STEP

tions.Theplottedpointsarefor asingle simulationona 128 X 128lattice.The dashedline is theresultof thestochasticcalculation Fig. 3. Integral rateconstantk (eq. (5)) for the (0.01,0.09),for uncorrelatedreactants. (0.05,0.05) and(0.5,0.5) simulationsandthe (0.05,0.05)sto-

chasticcalculation.The unitsare(particletime step)—

anticorrelationof A—B pairs. It shouldbenotedthatanticorrelationeffects are greatestfor the caseof 6. Summaryrapid and irreversiblereactionsconsideredhere. Ifthe reversereaction(C—~A+B)occurredwith finite A cellularautomatonrule was used to simulateprobabilityor if thecombinationof neighboringA’s controlledreaction among particlesmoving in twoand B’s occurredwith less thanunit probability at dimensions.The resultsare consistentwith the ex-each time step the correlation function PAB would pectationthat particleanticorrelationeffects wouldvary lessrapidly at small distancesandthe simula- be far more pronouncedin two dimensionsthan intion resultswould fall closerto the stochasticresult. three.The dependenceof the reactionrateasa func-The (0.5, 0.5) simulationthus showsthe maximal tion of time on initial reactantratio andconcentra-effect of anticorrelationfor a reactionoccurringin tioncanbequalitativelyexplainedin termsof anan-two dimensions. ticorrelationof reactantparticlepositionscreatedby

Fig. 3 showsthe integral rateconstant,definedby the reactionof nearneighborpairs.eq. (5), asa function of time for the (0.01, 0.09),(0.05, 0.05) and (0.5, 0.5) simulationsand for the(0.05, 0.05) stochasticcalculation.The stochastic Acknowledgementresult showsonly a slight decreasewith time. Theinitial rateconstantfor the threesimulationsareall Acknowledgementis madeto the Donorsof theclose to the stochasticresult, but k decreasessub- Petroleum ResearchFund, administeredby thestantially with time, most rapidly for the (0.5, 0.5) AmericanChemicalSociety for the supportof thiscase,least rapidly for the (0.01,0.09)casefor which research.the minority reactingparticlesarewidely separated.

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Volume 136, number 7.8 PHYSICSLETTERSA 17 April 1989

References [5] B.F. Madorc andW.L. Freedman,Science22 (1983) 615.[6] Si-I. Lin, K.P. Li andH. Evring. in: Physical chemistry,an

1] Yguerabide.MA. Dillon andM. Burton. J. (‘hem. Phys.40 advancedtreatise.Vol. 7. ed.H. Evring (AcademicPress..New1964) 3048. . York) pp. 1ff.

[2] MV. Vol’kcnshtein.Moleculesandlife (Plenum,New York. [7] M. Smoluchowski.Ann. Ph~s.(Leipzig) 18 (1915) 1103.

19701 p. 47. [81Wi. Moore. Physical chemistry. 4th Ed. (Prentice Hall.[3] T. Toffoli and N. Margolus, Cellular automatamachines EnglewoodCliffs, 1972) p. 335.

(MIT Press.Cambridge,1987). [9] F. Rcif Fundamentalsof statisiical and thermal physics[4] S.Wolfram.ed., Theoryandapplicationsofcellularautomata (McGraw-Hill. New York, 1965).

(World Scientific.Singapore.1986).

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