Oscillatory Chemical Reaction in a CSTR with Feedback Contro

Oscillatory Chemical Reaction in a CSTR with Feedback Control of Flow Rate

5148J.Phys.Chem.A1997,101,5148-5154

OscillatoryChemicalReactioninaCSTRwithFeedbackControlofFlowRate

MilosDolnik,AlexanderS.Banks,andIrvingR.Epstein*

DepartmentofChemistryandCenterforComplexSystems,BrandeisUniVersity,Waltham,Massachusetts02254-9110

ReceiVed:February27,1997;InFinalForm:April25,1997X

Wepresentanexperimentalandnumericalstudyofthechlorinedioxide-iodide(CDI)reactioninasingle,well-stirred,flow-throughreactorwithfeedbackregulationofflowrateinthreedifferentarrangements.Thecontrolisaccomplishedthroughacomputer-mediatedsigmoidaldependenceofthedynamicflowrate(s)ontheiodideconcentration.NumericalsimulationspredictinterestingdynamicsintheCDIreactionsystem,which,withoutcontrol,displaysonlysteady-stateand/orsimpleperiodicoscillations.Ourexperimentsconfirmthatburstingemergesasaresultofthefeedbackcontrol.Weattributeburstingtothecoexistenceofthestablesteadystateandoscillationsintheuncontrolledsystem.AperioddoublingsequenceleadingtochaosresultsfromthenonlineardependenceoftheoscillatoryperiodontheflowrateinthevicinityofaHopfbifurcationpoint.

Introduction

Additionofafeedbackcontrolmechanismtoachemicaloscillatorcanstabilizeunstablesteadystates1-3andperiodicsolutions.4-6Insomecasesthefeedbackcontrolgivesrisetoquitenewsolutions,e.g.,chaoticorbits.7Anotherbehaviorthatcanariseincontrolledoscillatorysystemsisbursting,i.e.,regularalternatingperiodsofquiescenceandofoscillations.8,9Burstingisacommonbehaviorofneuraloscillatorsandcanoccurinsingleisolatedneuronsorinnetworksofsynapticallyconnectedneurons.Recentdatafromstomatogastricganglionneuronsofspinylobsterssuggestthatneuronsregulatetheirconductancestomaintaintheirstableactivitypatterns.10Abbottandco-workers11-13developedaCa2+-dependentregulationschemetomodeltheeffectsofcellularactivityonconductancesinsingleandmulticompartmentmodelneurons.Theirmechanismallowsmodelneuronstoself-assembleandadjusttheirconductancestoproducestableandrobustpatternsofactivity.

Oscillatingchemicalreactionsexhibitawiderangeofdynamicalphenomenathatarealsoobservedinothersystems.Theyhavethereforebeenconsideredgoodmodelsformorecomplexbiochemicalandbiologicalsystems.Wehaverecentlysuggested14thechlorinedioxide-iodide(CDI)reaction15inacontinuouslystirredtankreactor(CSTR)withfeedbackcontrolasamodelthatmimicsthecontrolmechanismproposedforneurons.11AsanindicatorofactivityoftheCDIreaction,theanalogoftheneuron’sintracellularCa2+concentration,weemployedtheiodideconcentration.Theflowratewastakenasthedynamicalvariabletobecontrolled,inanalogytoaconductanceinaneuron.

ThebehavioroftheCDIreactionwithoutcomputercontrol,i.e.,atconstantflowrate,dependsupontheinputiodideconcentrationandonthefixedflowrate.Fiveseparateregionshavebeenidentifiedforthisreaction(Figure8inref16):twosteadystates,withhigh(HI)andlowiodide(LI)concentrations;aregionofsustainedoscillations;andtwodifferenttypesofbistability,onebetweenHIandLIsteadystates,andthesecondbetweentheHIsteadystateandsustainedoscillations.TheCDIreactiondoesnotdisplayburstingnorchaosifnofeedbackcontrolispresent.

Inthispaper,wefirstdescribetheexperimentalrealizationofthefeedbackcontrolmechanismsuggestedintheprevious

X

work,14andwedescribetheimplementationoftwoadditionalcontrolschemes.Next,wepresentexperimentalandnumericalexamplesofburstingbehaviorandotherregimesobservedinthesesystems.Weanalyzetheconditionsandparametersthatplaythemostimportantrolesinthedynamicsofthecontrolledsystems.Inthediscussion,wedemonstratehowburstingbehaviorandchaosemergeintheCDIreactionwithfeedbackcontrol.

ExperimentalSection

(a)ExperimentalArrangement.Athermostatted(T)25.0°C)plexiglasscontinuous-flowwell-stirredtankreactor(CSTR)ofvolume35mLwasusedinallexperiments.Theinputsolutionsofchlorinedioxide,potassiumiodide,andsulfuricacidweredeliveredtothereactorfromseparatetubeswithoutpremixing.RaininRabbitPlusperistalticpumpswereusedtoinfusethesolutions.Chlorinedioxidewaspreparedasdescribedinref17;potassiumiodideandsulfuricacidwerebothFisherACScertifiedreagents.Theconcentrationofchlorinedioxidewasdeterminedspectrophotometrically(??)1260cm-1M-1atλ)358nm)beforeeachexperimentwasstarted.Duringtheexperiments,thestocksolutionwasimmersedina0°CbathtoavoidevaporationofClO2andtomaintainkineticstability.Duringthetransfertothereactor,thetemperatureoftheClO2solutionincreasedtoapproximatelyroomtemperature.Thedynamicvariationoftheflowrateintheexperimentsresultedinsmalltemperaturefluctuationsofthesolutioninsidethereactor.Thelargestobservablefluctuationswerenotlargerthan(0.4°C.Theconcentrationofiodideinthereactorwasmonitoredwithaniodideion-selectiveelectrode(ISE)andacalomelreferenceelectrodeattachedtoadigitalvoltmeter.Theelectrodeswerecalibratedweeklyagainstiodidestocksolutionsranginginconcentrationfrom10-2to10-8M.

Weutilizethreedifferentschemesofdynamicalcontrolbothintheexperimentsandinthesimulations:(i)insetupA(Figure1a),allreagentsaredeliveredintothereactorbyasinglepumpfromseparatetubes.Controlisachievedbythenegativefeedbackmechanism,whichincreasestheflowratewhentheiodideconcentrationinsidethereactorisbelowthetargetconcentrationandviceversa.Thecontrolmechanismdoesnotaffecttheratioofinitialreagents,onlytheoverallflowratek0.(ii)InsetupB(Figure1b),twoindependentpumpsareusedtofeedthereactorwithreagents:ClO2andH2SO4solutions?1997AmericanChemicalSociety

AbstractpublishedinAdVanceACSAbstracts,June1,1997.

S1089-5639(97)00728-7CCC:$14.00

OscillatoryChemicalReactionFigure1.Experimentalsetups.Theiodideconcentrationisestimated

fromthepotentialofaniodideion-selectiveelectrode(ISE)pairedwithacalomelreferenceelectrode.Thepotentialdifferencebetweentheelectrodesismeasuredwithavoltmeterwithanalogtodigitalconverter(A/D).Dataareacquiredbyacomputerwhichcalculatestheflowrateandcontrolsthepump(s).(a)SetupA:reagentsaredeliveredintothereactorbyasinglepump.ControlisachievedbyanegativefeedbackwhichincreasestheflowratewhentheiodideconcentrationinsidethereactorislowandViceVersa.(b)SetupB:twopumpsareusedtofeedthereactor:onepumphasaconstantflowrate,andtheflowrateoftheotherpump(forthepotassiumiodidesolution)iscontrolledbythecomputer.(c)SetupC:twopumpsareusedforfeedingreagents,bothwithdynamicalcontrol.TheflowofKIiscontrolledbyanegativefeedback,whiletheflowrateofanacidicsolutionofClO2iscontrolledbyapositivefeedback.

aredeliveredataconstantflowratek01bythefirstpump,whiletheKIsolutionisdeliveredbythesecondpumpatthevariableflowratek02.Negativefeedbackisemployedfordynamicalcontrolofflowratek02.

(iii)InsetupC(Figure1c),twoindependentpumpsareutilizedtointroducethereagents;botharecomputer-controlled.TheflowofKI(k02)iscontrolledbyanegativefeedback,whiletheflowrateoftheClO2andH2SO4solutions(k01)iscontrolledbyapositivefeedback.

Theconcentrationsofstocksolutionsarethesameforallthreearrangements:[ClO2]S)3×10-4M,[KI]S)11.4×10-4M,and[H2SO4]S)3×10-2M.WithidenticaltubesinsetupAtheinputconcentrationsareequaltoonethirdoftheconcentrationsofstocksolutions.InsetupsBandCtheinputconcentrationsdependupontheratiok01/k02.

(b)ControlMechanism.Weutilizehereacontrolmech-anismsimilartothatusedpreviouslyinsimulations.14Theflowratek0ismadetobeaslowlyvaryingdynamicvariable.Thisregulatorymechanismforcesthesystemtoestablishaspecificaverageconcentrationofiodideinthereactor.ThefeedbackcontrolsandmaintainsthedesiredaverageconcentrationofiodideIT,whichcorrespondstoaparticularpatternofactivity.Inthiswaywebuildasimplechemicalsystemthatmimicsthedynamicregulationinaneuronalsystem.11

Thedynamicflowratecanvarybetweenzeroandamaximumvaluekmaxaccordingtotheconcentrationofiodideinthereactor.Therequiredchangeoftheflowrate?k0isevaluatedaccordingto

τ

?k0

?t)f([I-])-kkmax0)-k1+([I-]/I0T)n

(1)

J.Phys.Chem.A,Vol.101,No.28,19975149

wherenistheexponentofthesigmoidalfunctionfandτisthetimeconstantforflowratechange.

Oncetheemptyreactorhasbeenfilledwithreagents,theexperimentalcontrolmechanismisinitiated.Thecomputerprogramcalculatestheiodideconcentrationinsidethereactor[I-]usingtheelectrodecalibrationcurve.Followingatimeinterval?tofvoltagesampling,theiodideconcentrationissampledagain,anewk0iscalculated,andthecomputeradjuststhepumpspeedaccordingly.

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