multiple stationary states and hysteresis in a chemical reaction

8/12/2019 Multiple Stationary States and Hysteresis in a Chemical Reaction

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Multiple stationary states and hysteresis in a chemical reactionC. L. Creeland John RossCitation: J. Chem. Phys. 65, 3779 (1976); doi: 10.1063/1.433568View online: http://dx.doi.org/10.1063/1.433568View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v65/i9Published by theAmerican Institute of Physics.Additional information on J Chem PhysJournal Homepage: http://jcp.aip.org/Journal Information: http://jcp.aip.org/about/about_the_journalTop downloads: http://jcp.aip.org/features/most_downloadedInformation for Authors: http://jcp.aip.org/authors

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Multiple stationary states and hysteresis in a chemicalreactionC L Creel and John RossDepartment of Chemistry. Massachusetts Institute of Technology. Cambridge. Massachusetts 02139(Received 22 June 1976)Multiple stationary states have been observed experimentally in a homogeneous chemical reaction which ismaintained far from equilibrium in a closed system by energy flow across the boundaries. The reactionstudied is the gas phase. reversible. dimerization of NO,. 2NO, -= N,04 under the constraint of constantexternal temperature. The system is illuminated with visible light of a wavelength which is absorbed by theNO,. but is not adsorbed by the N,04 Multiple stationary states result from the nonlinear positive feedbackdue to the coupling between the light input. the temperature of the system, and the NO, concentration.These multiple stationary states are evidenced by the hysteresis they produce in NO, concentration withrespect to changing light intensity, which is in good agreement with the predictions of some simpledeterministic equations describing the system. The different states of the system are characterized bydifferent values of NO, concentration as determined from the light absorption by the NO, and the NO,fluorescence emission. Some qualitative information on the fluctuations in the regime of multiple stationarystates can be deduced from the experimental data, which indicates that the system exhibits largefluctuations which induce transitions between stable states prior to the transition predicted by thedeterministic equations.

I INTRODUCTIONChemical reactions with sufficiently complex mecha-

nisms corresponding to feedback may, when maintainedfar from equilibrium, show the interesting phenomenaof multiple stationary states, hystereSiS due to transi-tion among branches of stable stationary states on var-iation of an imposed constraint, and oscillations; andwhen coupled with vectorial processes (diffusion, con-duction) they may show spatial structures. 1-7 The the-ory of chemical systems with such properties due toinstability has received much attention; however, ex -perimental verification of many predictions is yet tocome. Moreover, the systems studied so far experi-mentally in some detail, such as the Zhabotinsky re-action, 8,9 have very complex mechanisms. In thisarticle we investigate experimentally a simple one-step reaction, 2N02= N20 4 , in a closed system underillumination and confirm the previously published the-ory of Nitzan and Ross 1 by showing the existence ofmultiple stationary states and hystereSiS in such sys-tems.

Multiple stationary states have been observed in avariety of systems, 11 especially in open stirred re-actors.12 This includes hysteresis due to conforma-tional changes of polymers and in membranes. Thepresent study reports on hysteresiS due to transitionsbetween branches of stable stationary (nonequilibrium)states of a chemically reacting system which is closedto mass flow across its boundary but open to energy(photon) and heat flow.

In Sec. II we present the deterministic equationswhich describe the instability along with solutions tothese equations. In Sec. III we discuss the experi-mental apparatus and procedures. Section IV de -scribes the measurements which we made and the re-sults obtained from these measurements. In Sec. Vwe discuss the results and show that the shape of theobserved hysteresis loop may contain information on thesystem fluctuations in the region of multiple stationarystates o

II. THEORYThe system chosen for study in the reaction between

nitrogen dioxide and nitrogen tetroxide,kN20 4= 2N02 ,k2 I)

under i l l u ~ i n t i o n o I f an equilibrium mixture of N02and N20 4 is irradiated with visible light, the light isabsorbed only by N0 2 and most of the light is ultimatelyconverted into heat. Thus, the temperature of the re-action system depends on the N0 2 concentration and onthe thermal conduction to the surroundings. However,the N0 2 concentration is coupled to the temperature ofthe system through the temperature dependence of thera.te coefficientso The enthalpy change AH of ReactionI in the forward direction, is 13.9 kcal/mole. Asthe temperature of the system increases by absorptionof light by N0 2 , more N0 2 is produced by Reaction Iwhich causes an increased absorptionand a higher tem-perature, which in turn produces even more N0 2 Hence, this system has a positive feedback mechanism.

For the present we assume the system to be uniformat all times and hence the deterministic rate equationsare 1

1)dT/dt =a./ox - 3 T - Te) - A dx/dt) , 2)

where x is the N0 2 concentration, C is the concentrationof N20 4 if all the gas were in the form of N20 4 , /0 is theincident light intensity, T is the temperature of the sys-tem, and Te is the temperature of the surroundings.The remaining symbols are

a. =El/C. ,_ _ -=2.:.: 1rK:.: :l:. ..-- --(3-- pC. Vln r2 / r 1

A= V A H / C ~k l=Zlexp -EI /RT) ,

3)4)

(5)(6)

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3780 C. L Creel and J Ross: Multiple stationary states and hysteresis

FIG. 1. System geometry. The system is defined as thatportion of the gas within the cell which is illuminated by thelaser beam.

in which E is the extinction coefficient, the opticalpath length, C the heat capacity at constant volume, Kthe thermal conductivity, p the total density, V thevolume of the system which is illuminated, r 2 the radiusof the cylindrical gas cell, and r 1 the radius of the i lluminated volume as shown in Fig. 1.

Equation 1) is the conventional rate expression for thedifference between the forward and reverse reactionrates. Equation 2) represents the rate of change of thetemperature of the system: the first term is proportional to the light intensity and the N02 concentration,and represents the heating rate due to absorption oflight by N02 the second term describes the coolingrate of the system to its environment we have assumedthe simplest linear form for the thermal conduction),

_

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C. L Creel and J. Ross: Multiple stationary states and hysteresis 3781

THERMOSTATAR LASER r - - - - - ~ ~ - - - - - ; _ ~ ~ ~ - - - - - - - - - - ~ - - - - - - - - ~ - - _ 1

PHOTODIODE

SCATTERPLATE

ABSORPTION

OSCILLOSCOPEPOWER

FIG. 4. Schematic diagram of the experimental arrangement.

system evolves to point E which is another marginalstability point. Further decrease in light intensitybrings about a transition to point F. Thus, the systemdisplays hysteresis on variation of light intensity, oneof the imposed constraints.

The other imposed constraint is the external temperature. The isotherm13 at Te =240 oK Fig. 2 is thecritical one, in that both the first and second derivative of light intensity with respect to concentrationvanish at the point of inflec tion. The analogy of the setof curves in Fig. 2 to isotherms of a van der Waal'sgas has been made in detail. 14III EXPERIMENT L

NOll was obtained from Matheson and purified by arepeated freeze 195 K)-pump-thaw cycle until thesolid was white. Mass spectral analysis indicated noimpurities.

NOll undergoes photodissociation into NO Oll whenirradiated with visible light. 15,16 Due to the very highlight intenSity necessary for this experiment, this dissociation must be reduced to the maximum extent possible.This is accomplished by the addition of a large quantityof COll (> 10 : 1 mole ratio, COz /NOz) to quench the re active NOz excited state and by illumination of the NOzfor a very short period 30 sec) at anyone time.

A schematic of the experimental apparatus is shownin Fig. 4. The NOll is contained in a cylindrical Pyrexcell, 1.5 cm long by 0.8 cm i. d. and 1.0 cm o. d., withplane windows, which is suspended inside a thermostatcontaining dry air. The NOz is irradiated with the visible light of an argon ion laser capable of 15 W outputin broadband, multimode operation. The light incidenton and transmittedby the NOz is sampled with beamsplitters and diffused by scatter plates. The diffuselight is detected with photodiodes, which have a linearresponse over six decades of light intensity,. The light

incident upon the photodiodes is converted into a voltage, proportional to the light intensity, by amplifiers1 and 2. A one quadrant analog divider forms theratio of these two signals and gives an output voltagewhich is proportional to the ratio of transmitted to in cident light intensity. Amplifier 3 gives an output proportional to the difference between the output of thedivider and a fixed reference voltage. This signal isproportional to the absorption by the NOll Thus, thecircuit computes the absorption A from the relationA =1 IT 0 where IT is the transmitted, and 1 theincident light intensity. This signal, representing theabsorption, is applied to the vertical input of a storageoscilloscope. The output of amplifier 1, which is proportional to the incident light intensity, is applied tothe horizontal input of the oscilloscope. Thus, theoscilloscope records directly the absorption as a function of light intensity.

A small amount of the absorbed radiation is emittedas fluorescence. We detected this NOz fluorescencewith a photomultiplier 8-4 photocathode) through a 10nm bandpass interference filter, centered at 589.6 nmand a 24 in. fiber optic light pipe. This signal can alsobe divided by the incident light intensity and applied tothe vertical oscilloscope input to provide a plot of thenormalized fluorescence intensity as a function of in cident light intensity. Thus we have two independentmeasurements of NOll concentration; absorption andfluorescence.

The laser output is controlled by varying the currentthrough the plasma tube. The scan rate is defined asthe time to go from minimum output to maximum outputand then back to minimium, in a smooth, continuousmanner. As the plasma current changes, the laser output mode structure changes and the output wavelengthdistribution changes, i operating without intracavitywavelength selection. These effects produce slightnonlinearities in the photo diode response under some

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3782 C. L Creel and J. Ross: Multiple stationary states and hysteresis

FIG. 5. Rate of achieving steady state. Vertical scale: 10absorption per division. Horizontal scale: 10 msec per division, Te= 45 oK.

conditions; however, these effects are small and do notproduce any hysteresis.

The thermostat consists of a double walled brasscylinder. Methanol, cooled by a freon refrigerationsystem, is pumped through a heater bank and then be tween the walls of the thermostat, and finally back intothe reservoir. Coarse temperature control is providedby regulation of the freon compressor back pressure ata preset value. Fine control is provided by a feedbackproportional control system which adjusts the currentthrough the heater bank in response to the temperatureof the thermostat. Temperature control of the thermostat is O. 005 oK around a preset value.IV MEASUREMENTS AND RESULTSA Rate of Achieving Steady State

Figure 5 shows an oscilloscope plot of absorption asa function of time. Here, the N02 sample at equilibrium = e was suddenly illuminated with the laserbeam at a power of 2.0 W with the oscilloscope triggered at the instant of illumination. The turn-on timefor the laser was less than 1 msec, A steady state isachieved in approximately 50 msec, which is consistent with that estimated from the values for the heatcapacity and cooling rate given in Sec. II. Figure 6shows a series of similar measurements at differentpower levels. For the measurement at 5 W, note thata quasistationary state is achieved in 30 msec, afterwhich the system undergoes a faster transition to thefinal steady state at a time of about 40 msec.

The rate of achieving steady state can be varied bychanging the system parameters. The addition of helium to the N02 C 02 mixture, with the total pressureconstant, increases the thermal conductivity of the system and decreases the.time required to reach steadystate. Increasing the diameter of the laser beam in creases the system volume and thus increases the heat

capacity of the system. This increases the time required to reach steady state. These effects are shownin Fig. 7. The addition of helium reduces the time toreach steady state from 50 to 20 msec. Passing thelaser beam through a telescope to enlarge the beamdiameter by a factor of three increases the time requiredto reach steady state to 100 msec.B Observation of hysteresis

Figure 8 shows two oscilloscope plots of absorptionas a function of light intensity at two different externaltemperatures. This demonstrates the onset of hysteresis as the external temperature is lowered from253 to 245 OK, in reasonable but not complete accordance with the prediction of the deterministic theory ofSec. II. It should be noted that the magnitude of ab soprtion change with respect to changing the light in tensity is virtually identical for the two curves, thusprecluding any spurious causes for the hysteresis,such as changing the light intensity too rapidly for nearmaintenance of steady state.

The width of the hystersis loop is increased by in creasing the rate at which the light intensity is varied.This effect is illustrated in Fig. 9, where the scan rateis varied from 1 to 4 sec, As the scan rate is in creased, a point is eventually reached where the system no longer evolves through a sequence of nearsteady states. This effect, which introduces an additional hysteresis into the system, is likely to be contributing to the fastest scan in Fig. 9. In this scanscan rate = 1 sec) the upward transition of the hysteresis

loop is not as sharp as for slower scan rates. I f i twere not for this effect of departing from steady stateconditions, the width of hysteresis loop should increaseas the scan rate is increased, until a maximum widthdefined by the marginal stability points is reached,This change in the width of the hysteresis loop is probably due to the fact that the rate of transition along astable branch (L e., AFB or DCE in Fig. 3 is comparable to the rate of transition between branches dueto system fluctuations. This effect is detailed in Ref.14.

As the scan rate for the experimental illustrated inFig. 9 is increased beyond 4 sec, the hysteresis loopcloses only slightly more than that of the 4 sec scan.Even at a scan rate of 30 sec, a small amount of hysteresis is still observed. For scan rates slower thanabout 60 sec no hysteresis is observed.C. Fluorescence measurements

Fluorescence emission of N0 2 gives an independentmeasurement of the N02 concentration and should showthe same hysteresis as the absorption measurements.Figure 10 shows two oscilloscope photographs of theN02 fluorescence intensity divided by the incident laserintensity as a function of incident laser intensity, Al-though the signal to noise ratio is considerably reducedfrom the absorption measurements, the similarity in thefluorescence and absorption is easily seen, This figurealso illustrates the effects of focusing the laser beam.A very small diameter beam produces a narrower hysteresis loop with sharper transitions.

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FIG. 6. Rate of achieving steady state for various values of light intensity at e = 235 oK. Vertical scale: 10 absorption perdivision. Horizontal scale: 10 msec per division. Light intensities; left top: 2 W, right top: 7 W, left middle: 3 W, rightmiddle: 8 W, left bottom: 5 W, right bottom; 10 W.

V. DIS USSIONThe results presented in the previous section show

the existence of multiple stationary states in a homo-geneous chemical reaction in a closed system drivenfar from equilibrium by illumination. However, theseresults differ somewhat from those of the deterministicequations presented in Sec. II, in that the deterministicequations predict a sharp transition between the stablesteady states. This lack of sharp transitions may bedue to several effects such as fluctuations departurefrom steady state conditions, critical slowing down,

and inhomogeneities. We will discuss these effects herein some detail. Fluctuations and inhomogeneities areshown to have an appreciable effect on the observedwidth of the hysteresis loop.

In order to see the effects of fluctuations, we mustexamine the stochastic equations which describe thissystem. This approach14 17 predicts that hysteresis isobserved only when the time scale of fluctuations arounda stable steady state the branches AFB and ECD ofFig. 3) is short compared to the time scale of transi-tion between branches induced by fluctuations in the

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3784 C. L Creel and J. Ross: Multiple stationary states and hysteresis

FIG. 7 (a). Rate of achieving steady state with helium addedto the system at Te = 245 oK and 5 W incident power. Verticalscale: 10 absorption per division. Horizontal scale: 10 msecper division. PH. =450 torr, PN02=22 torr, and Pco2=217torr measured at 300 OK). b) Rate of achieving steady statewith laser beam enlarged by a factor of 3 at Te =245 O and7 W incident power. Vertical scale: 10 absorption per di -vision. Horizontal scale: 20 msec per division. PN02 =21torr and PC02 = 711 torr measured at 300 oK).

system. However, i these time scales are comparable,then both branches may be populated at one time or an -other in the region of multiple steady states. Thus thevariation in the width of the experimentally measuredhysteresis loop with the scan rate rate of change of UghtintenSity), see Sec. IV. B, shows that the fluctuationsin the system are of sufficient magnitude in the regionof multiple stationary states, such that transitions be-tween branches of stable states can occur at points con-siderably away from marginal stability points.

In order to gain some insight into the effects of fluctua-tions on this system, the deterministic equations wereintegrated under the influence of a random Gaussiannoise R t). Equation 2) takes the form

dT/dt= lox 3 T - Te)+R t) , (7)where the last term, ?t dx/dt), has been neglected dueto its relatively small magnitude. The kinetics, repre-sented by Eq. (1), operate on a much faster time scalethan thermal conduction and can be shown to be atsteady state on the time scale of temperature changes,i .e . , dx/dt=O andx=x T), which represents the low-est order solution of a multiple time scale expansion.The light intenSity 1 is taken to be a function of time,consistent with the experimental values for the scanrate. The results of this integration are shown in Figs.11 for three noise levels. Notice that as < t) I is in-creased, the hysteresis loop becomes more narrow,since a large fluctuation may induce a transition be -tween stable branches prior to the marginal stabilitypoints At each light intenSity in the region of multiple

FIG. 8. Effect of lowering the external temperature. Upperphoto is for Te = 253 O and lower photo is for T = 245 OK.Vertical scale: 10 absorption per division. Horizontal scale:0.25 W per division: PN02 = 22 torr at 300 oK. Laser lightscan time: 2 sec.

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FIG. 9. Effect of changing the scan rate. Scan rates are:upper 4 sec; middle 2 sec lower 1 sec. Vertical scale: 10absorption per division. Horizontal scale: 0.25 W per division.Te =243 oK, PN 2 =20 torr at 300 oK.

stationary states there is a minimum fluctuation amplitude necessary to induce the transition between stablestates. This minimum amplitude decreases as themarginal stability points are approached AB the system approaches the marginal stability points thefluctuations are amplified by the system This is consistent with the marginal stability points being analogous to equilibrium critical points

Figure 11 b) shows the integration carried out threetimes under identical conditions. Since the fluctuationsare random their effect may be different in each ofthe three cases as evidenced in Fig 11 b). I f a sufficiently large fluctuation occur:;; prior to the marginalstability point, the system makes the transition to theother stable branch. However, i no fluctuation happens to occur then the system comes closer to themarginal stability point before making the transition.The magnitude of fluctuations necessary to produce anappreciable effect on the hystereSiS loop in Fig, 11 isquite large. However, it should be noted that we haverestricted the calculation to homogeneous fluctuations.

If the light intensity is varied too quickly, then thesystem does not evolve through a sequence of steadystates. In order to see the effects of a rapid variationin light intensity the deterministic equations were integrated with a time-dependent light intensity. The results are shown in Fig. 12 for two values of Te. A 2.5sec scan rate produces a small hystereSiS in the curvefor T. =250 0 K, where no multiple steady states arepredicted. However, at Te =230 0 K, where multiplesteady states are predicted a rapid variation in 10 hasonly a slight effect on the results predicted by the deterministic equations. Hence, these results show thatin these experiments we scanned slowly enough suchthat any departures from steady state conditions wouldlikely have a minor effect.

AB the marginal stability points are approached thefluctuations in the system show the phenomena ofcritical slowing down, 14 To see this effect Eqs. (1)and (2) are linearized about the steady state. Deviations from steady state ox oT are defined by

x = o + Ox , T = To + 0Twhere Xo and To represent the steady state varues of xand T,

The linearized equations are thend OX) OX)dt oT =M oT

where Mll = - k - rk 2x OM12 = (1/RT2 [2E1 kl C - ixo - 2E2 k2X5l ,M21 =ClIo - AMll ,M 22 = - {3 - AM12

8)

E1 and E2 are activation energies corresponding to kland k2 which are assumed to be of the Arrhenius form.In Eq. 8), kl and k2 are evaluated at To. Solutions toEq. (8) are of the form

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3786 C L Creel and J Ross: Multiple stationary states and hysteresis

FIG. 10. Fluorescence measurements and the effect of laser beam diameter. Beam diameters: upper 2 mm; lower 34 { lmVertical scale: fluorescence emission in arbitrary units. Horizontal scale: 0.25 W per division. e= 35 oK PN02 =23 torr at300 oK. Laser light scan time: 2 sec

where l and 2 are constants and YI and Y are theeigenvalues of M The time behavior of 6x can be seenqualitatively by examining the eigenvalues. Figure 13shows a plot of Yl as a function of 1 along with the cor-responding absorption curves. Yl goes to zero at themarginal stability points which is of course neces-sary in order for the system to make a transition to a

new stable state. Thus the time response of the sys-tem becomes longer as the marginal stability pointsare approached. This slowing down may prevent thesystem from evolving through a sequence of steadystates near the transition region; even though the scanrate is such that the system is at steady state awayfrom this region.

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z0fILn0f)CD


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3788 C L Creel and J. Ross: Multiple stationary states arid hysteresisAll of the theory presented here has been based onthe assumption that we have a homogeneous system.

This is, of course, only an approximation. Under theconditions of our experiments the time scale for diffusion across the 2mm laser beam is approximately 1sec. Since the laser beam has a Gaussian profile, thegas in the center of the beam is at a substantially highertemperature than the gas at the edge of the laser beam.Thus, the gas at the center of the beam makes thetransition from the lower branch to the upper branchprior to the gas at the edge of the beam. Therefore,the experimental observations represent an averageover a series of "systems" undergoing sequential transitions between the branches. This is probably a primaryfactor for not observing the sharp transitions predictedby the deterministic equations. This effect is furthersubstantiated by the observation that the transitionsbecome sharper when the laser beam is focussed to acylinder of small radius see Fig. 10), In this Situation,diffusion across the laser beam is on a short enoughtime scale to average out the inhomogeneities and makethe system appear more homogeneous.

The transition between branches represents a changein the temperature of the system of the order of 100 OKThe system cannot undergo this change instantly, andtherefore the transitions do not appear to be sharp if10 is changing .on a time scale comparable to the timescale for this transition.

We have reported here a very simple chemical system which displays an instability and this instabilitycan be well characterized by simple relations amongknown parameters. This, or Similar, systems are thuslikely candidates for further study of chemical instabilities, such as detailed investigations of fluctuationsin the regime of multiple steady states. Such systems

SCAN RATE I~ = 10sec. I--- = 2.5 sec I30 IIII 4 I0 I Ioro I I: : '20 I I

z I IQr I IL I I0 I I(f) fIJ4 I

f//;.-

Te=230 -- ---2 I 2 3 4 5

POWER OF INCIDENT LIGHT (Walls)FIG. 12. Integration of the deterministic equations with a timedependent light intensity, for two choices of scan rate. Results are presented for cases with and without multiple steadystates.

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C L Creel and J. Ross: Multiple stationary states and hysteresis 3789

HH. Gerreno, K. KLlchner, and G. Ley, Chemie-Ing. Techn.43, 693 1971); R. Cox, Biophys. Acta 68, 401 1963); A.KatchaIsky and R. Spangler, Q. Rev. Biophys. 1, 127 1968);M. Chang and R. A. Schmitz, Chern. Eng. Sci. 30, 8371975)

D D. Perlmutter, Stability of hemical Reactors PrenticeHail, Englewood, N. J . 1972); S. F. Bush, Proc. R. Soc.London) Ser. A 309, 1 1969).

1aWe use the term isotherm to refer to a constant external temperature.

14A Nitzan, P. Ortoleva, J . Deutch, and J . Ross, J . Chern.Phys. 61, 1056 1974), and references therein.

15: T. N. Jones and K. D. Bayes, J . Chern. Phys. 59, 48361973).

16C. L. Creel and John Ross, J . Chern. Phys. 64, 3560 1976).17G Nicolis, Adv. Chern. Phys. 19, 209 1971).

J. Chern. Phys. Vol. 65 No.9 1 November 1976

multiple stationary states and hysteresis in a chemical reaction

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