J. Phys. Chem. 1992,96,8053-8060 time on the CRAY Y-MP/832. Registry No. Ethylarsine, 593-59-9; ethylgallane, 12263 1-95-2.
References .ad Notes (1) Aoyagi, Y.;Doi, A.; Iwai, S.;Namba, S . J . Vac. Sci. Technol. 1985, 135. ..., 1460. (2) Mashita, M.;Tsuda, M.;Oikawa, S.;Morishita, M.Proc. In!. Conf. ~
~~
Elecfron. Muter. Tokyo, 1988. (3) Chm, C. H.; Larsen,C. A.; Stringfellow,G.B. Appl. Le!!. 1981, _ . Phys. . 50, -218. (4) Lum, R. M.;Klingert, J.; Lamont, M.G. Appl. Phys. k!f.1987,50. 284. -. ( 5 ) Larsm,C. A.; Li, S.H.; Buchanan, N. I.; Stringfellow, G.B. J. Cryst. Grow!h 1989,94, 673. (6) Oikawa, S.;Tsuda, M.; Marishita. M.;Mashita. M.;Kuniya, Y.J. C&. Growth 1988.91.47 1. -(7) Omstead, T. R.;VanSiclde, P. M.;Lee, P. W.; Jenscn, K. F. J. Crys!. Growth 1988,93,20. (8) Larsen, C. A,; Buchan, N. I.; Li, S.H.; Stringfellow, G. B. J . Crys!. Grow!h 1989, 94, 673. (9) Pioca, E. A.; Ault, B. S . J. Am. Chem. Soc. 1989, 111, 8978. (10) Schlyer, D. J.; Ring, M. A. J. Orgonomet. Chem. 1976, 9, 114. (11) Schlyer, D.J.; Ring, M.A. J. Elrctrochem. Soc. 1977, 124, 569. (12) Watkins, S.F.; Haacke, J . Appl. Phys. 1991,69, 1625. (13) Nishizawa, J.; Kuryabahi, T. Vacuum 1990, 41, 958. (14) Dobbs, K.;Trachtman, M.;Bock, C. W.; Cowley, A. J. Phys. Chem. 1990, 94, 5210.
8053
(IS) Bock, C. W.; Trachtman, M.Strut!. Chem., in prw. (16) Bock, C. W.; Trachtman, M.;Mains, G. J . Phys. Chcm., in precur. (17) Low, J. J.; Goddard 111, W. A. &ganomc!ullics 1986, 5, 609. (18) Mathey, F. Angew. Chem., In!. Ed. Engl. 1987, 26, 275. (19) Twda, M.;Oikawa, S.;Mori~hita,M. J. Cryst. Growrh 1990,99,545. (20) FrisCh, M. J.; Head-Gordon, M.;Trucks, G.W.;Foreaman, J. B.; Schlcgel, H. B.; Raghavachari, K.; Robb, M.A.; Binkley, J. S.;(3ondw, C.; Dcfreea,D.J.; Fox, D.J.; Whitaide, R. A.; S e e p , R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.; Poplc, J. A. GAUSSIAN 90; Gaussian, Inc.: Pittsburgh, PA, 1990. (21) Huzinap, S.; Andzelm, J.; Klobukowski, M.;Radzio-Andzclm, E.; Sakai, Y.;Tatewaki, H. Gaussian Basis Sets for Molecular Calcula!ionr;
Elsevier: Amsterdam, 1984. (22) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acto 1973, 28, 213. (23) Bock, C. W.; Trachtman, M.;Murphy, C.; Muchert, B.;Mains, G.
J. Phys. Chem. 1991,95,2339. (24) Moller, C.; Plesact, M. S.Phys. Reu. 1934, 46, 678. (25) (a) Bock, C. W.; Dobbs,K. D.;Mains, G.;Trachtman, M.J. Phys. Chem. 1991,95,7668. (b) Binning, R. C.; Curtiss, L. A. J. chmr.Phys. 1990, 92, 1860. (26) Fucno, T.; Bonacic-Koutccky, V.; Koutccky, J. J. Am. Chem. Soc. 1983,105, 5547. (27) Bock, C. W.; Trachtman, M.; Mains, G.J . Phys. Chem. 1985,89, 2283. (28) Schlegel, H. B.; Sosa, C. 1.Phys. Chem. 1985,89, 537. (29) Gordon, M. S.; Gano, D.R. J . Am. Chem. Soc. 1984, 106, 5421. (30) Bock, C. W. Unpublished results. (31) Schlegel, H. B. J . Chem Phys. 1986,84,4530.
Otlgh of Spontaneous Wave Generation in Excitable Chemical Systems Eugenia Mori and John Ross* Department of Chemistry, Stanford University, Stanford, California 94305 (Received: February 24, 1992; In Final Form: May 15, 1992)
We investigate the origin of spontaneous chemical wave generation in an excitable Belousov-Zhabotinskii system. We solve one-dimensional reactiondiffusion equations of an Oregonator model with the initial profiles possessing an excitation of varying concentration of either H B Q or Br-, and the excitation occurs within a region of different length. The concentration of the threshold excitation necessary for a wave to propagate depends on the length within which the initial excitation is applied. We further perform an equilibrium stochastic calculation of the recurrence time for a thermal fluctuationto induce a change in concentration of a sufficient magnitude within a suffcient volume for a wave to propagate. The smallcst rtcu~wlcc time calculated is 10'' s. We compare our results with previous experiments and calculations and conclude from all the evidence that an internal thermal fluctuation is highly unlikely to generate a chemical wave in an excitable chemical solution.
I. Introduction Chemical waves may occur in nonlinear reactions far from equilibrium. Various types of waves have been observed experimentally and studied theoretically in unstirred oscillatory reactions and excitablereactions. An excitablechemical system is in a single stable stationary state; upon concentration perturbations of the system in that state, however, with perturbations of appropriate sign and magnitude which comprise a threshold excitation, the system shows large variations (excursions) in concentration space and returns to the stationary state. Chemical waves may be induced in either oscillatory or excitable reaction systems by external perturbations such as the addition of chemical species, light,'J local heating? or electrochemical m e a " (the imposition of a voltage pulse which induces a variation in the concentration of chemical species). Spontaneous generation of chemical waves has also been observed: and here arises the issue of whether such spontaneous generation is due to a heterogeneity in the solution, such as a dust particle, or is due to inherent thermal fluctuations. In this article, we briefly review prior experimental and theoretical studies on spontaneous generation of chemical waves in oscillatory and excitable chemical systems. We address the issue of wave generation in excitable chemical systems with two further contributions. The first is a calculation of the deterministic reactioxdifFuion equations for a modified Oregonator model which 0022-3654/92/2096-8053$03.00/0
we solve in one spatial dimension. We determine the critical radius and critical concentration change which must be achieved in a region of the critical radius so that a wave propagatts, and we compare the results with prior calculations and experiments. The second contribution is a simple calculation, based on equilibrium statistical mechanics, to estimate the probability of achieving a fluctuation of the critical concentration variation within the necessary critical volume so that a wave propagates. We conclude that the probability of a thermal fluctuation initiating a chemical wave in an excitable medium is, for likely areas of observation, negligibly small.
II. Brief Review of Prior Work Showalter, Noyes, and Turnefl study the induction of chemical waves in an excitable ferroin-catalyzed Belousov-Zhabotinskii reaction. They use silver and platinum electrodes and note the time duration of a voltage puke of given amplitude necessary for wave initiation. They find that the critical pulse duration necrssary for wave initiation depends on many factors including the composition of the solution, the pulse voltage, the bias of the electrodes preding the puke, and the time since the last oxidative excursion, that is the time since the last threshold excitation. An estimate of the size of the region depleted of bromide ions by the shortest critical pulse applied to the silver electrode is approximately a Q 1992 American Chemical Society
Mori and Ross
8054 The Journal of Physical Chemistry, Vol. 96, No. 20, I992 layer 2-200 pm surrounding the electrode which is 150 pm. Foerster, MOller, and H e d perform experiments and a thee retical analysis to determine the size of a critical radius of perturbation for wave initiation in an excitable ferroin-catalyzed BZ solution. Silver-coated capillaries of varying radii are dipped into the solution for the duration of the experiment, and whether or not a chemical wave is initiated is noted. They find that a chemical wave is induced when the radius of the capillary is 16 pm or larger and a wave is not generated when the radius of the capillary is less than 16 pm. A diffusion layer of silver ions surrounding the capillary is estimated to be 5 pm. Foerster et al. conclude from the experiments that a critical radius of 21 pm (16 pm from the capillary plus a 5-pm diffusion layer) exists for chemical wave generation. Foerster et a1.6 also obtain an estimate of a critical radius by extrapolating measurements of the chemical wave velocity versus its curvature to zero velocity. The normal velocity, V,, of a chemical wave is predicted to depend linearly on its curvature, K , according to the relation V, = c - DK where c is the velocity of plane waves and D is the diffusion coefficient of the autocatalytic species. The curvature K is equal to 1/R, where R is the radius of the chemical wave. The above relation predicts that the velocity of a wave is zero when the radius has a critical value of RcTitwhere Foerster et al. plot measured velocities of a chemical wave versus its curvature and extrapolate to zero velocity. The critical radius obtained in this manner is estimated at 23.1 f 1.O pm, in good agreement with the estimate of 21 pm from the capillary experiments. Vidal and Pagola5 image chemical waves in an oscillatory ferroin-catalyzedBZ system. Waves are seen to generate spontaneously in solution, i.e. nothing is done externally to initiate the waves. The core of the waves is examined with a video camera which has resolution of up to 5.57 pm/pixel, and the center of the wave is termed either "heterogeneous" or "homogeneousn. In the heterogeneous core, a particle, very likely of dust, with a radius on the order of 75 pm is observed which presumably alters the reaction kinetics locally and generates the wave. In the homogeneous core, however, no particle within the resolution of 5.57 pm/pixel is observed. The possibility is mentioned that a local fluctuation of concentration may have generated the wave. Ple%er, Kingdon, and Winters' perform numerical calculations with an Oregonator model of an excitable cerium-catalyzed BZ system. The equations integrated are two partial differential equations describing the evolution of bromous acid and oxidized cerium in time and one-dimensional space. The equations are solved with varying initial profiles in which one or both of the variables contain a region in space which is excited to a higher concentration. Whether or not a wave propagates outward from the initial excitation is noted, and the critical radius is given below which wave propagation is not observed. The authors perform calculations with three types of initial profiles. In the first set of initial conditions, the bromous acid concentration is set at an elevated level within the region of excitation and at its steady-state concentration elsewhere, and the oxidized cerium concentration is set at its steady-state concentration everywhere. For this case, a wave always propagates for every size radius of excitation tried. The smallest radius used is 1.7 X 10" um. The likelv inamlicabdity of the reaction-diffusion equations describing the system for such a small region of excitation is discussed in section III.B.5. In the second set of initial conditions, they set both bromous acid and oxidized cerium at an elevated conantration level within the region of excitation and at their steady state concentrations elsewhere. For this case, the critical radius below which wave propagation is not observed is 4.4pm. In the third set of initial conditions, they set the bromous acid concentration at an elevated level within the region of excitation and at its steady state concentration elsewhere, and the oxidized cerium concentration is
set at an elevated concentration everywhere. For this case, a critical radius of excitation of 51 pm is observed. A calculation in two spatial dimensions for this set of initial conditions agrees well with the calculation in one spatial dimension. Kapral, Lawniczak, and Masiar8 perform cellular automaton (CA) calculations on an excitable Selkov model, a well-known and much studied model9 exhibiting complex behavior. Since the CA model contains dynamics on a molecular level, the question of the possibility of spontaneous wave generation due to internal fluctuations is probed. In the calculations performed, a chemical wave is never initiated due to an internal fluctuation when the parameters in the CA model are chosen to model accurately the physical system. If the effective diffusion of the species involved is slowed by a factor of 3 in the CA model, then traveling Chemical waves do spontaneously generate in the calculations due to internal fluctuations as larger concentrationinhomogeneities are allowed to develop. Chemical waves also propagate if a large enough nucleus of excitation is planted in the model grid. The estimated critical radius for a nucleus of excitation to propagate at a velocity comparable to those observed in experiments is 1-50 pm.lo Walgraef, Dewel, and Borckmans" consider the probability of a thermal fluctuation inducing a chemical wave in an aecillatory system. They use the Brusselator as a model, and the mechanism of wave generation they consider is a local phase shift of the oscillation in the system caused by a fluctuation. Their analysis, however, does not consider a critical volume necessary within which the fluctuation must occur for a wave to propagate. Frankowicz, Kawczynski, and GoreckP perform calculations on the Rovimky-Zhabotinskii model which consists of two partial differential equations describing the evolution of bromous acid and ferriin concentrations in time and onedimensional space. The reaction mixture of their model is in an excitable state. The mechanism of wave generation considered is a fluctuation in a parameter which causes a region of the solution to bacome oscillatory, and chemical waves propagate outward from such a region, While relative probabilities of the generation of chemical waves with varying periods are discussed,the authors do not discuss the actual probability of initiating a chemical wave nor its dependence on the volume excited. T y ~ o nexamines '~ chemical wave generation in excitable and osciUatory reaction systems by using singular perturbation theory; he states that his predictions are in accord with a heterogeneous origin of wave generation.
III. Estimate of Critical Radius and Critical Concentration Changa for Wave Initiation from Numerical S o l u t i ~ of~ Deterministic Reaction-(Diffusion) E q ~ r t i o ~ For the calculations indicated in the title of this section, we choose a modified Oregonator model14 for an excitable Bclousov-Zhabotinskii reaction and solve numerically the reaction diffusion equations for the system. We do so for chemical wave propagation in one spatial dimension and wish to determine both the critical radius and the critical concentration change necespary for chemical wave propagation to proceed. Two types of perturbations are consided perturbations in the initial concentration of HBrO, are discussed in section 1II.B and perturbations in the initial concentration of Br- are discussed in section 1II.C. The model is reviewed in section 1II.A. IllA Tbe cbemicrlReadion Model. The modified Oregonator model used in this paper has been presented in ref 14,and further details of the modeling can be found there. The chemical reactions comprising the model are
+ H+ 2Br0,' + H 2 0 + H+ F= Fe3+ + HBr02 HOBr + Br03- + H+
HBr02 + Br03Fez++ BrO,' 2HBr02
-
HBr02 + BrBr03- + BrFe3++ BrMA
-
+ H+
+ 2H+
2HOBr
HBr02
+ HOBr
(Rl) (R3) (R4)
(R5) (R7)
+ MA * Fez++ H+ + BrMA' + MA' (R8)
The Journal of Physical Chemistry, Vol. 96, No.20, 1992 8055
Wave Generation in Excitable Chemical Systems BrMA'
+ MA' + H 2 0
-
+ MA + product(s)
h Br-
(R9)
MA and BrMA are shorthand for malonic acid and bromomalonic acid, respectively, Fe2+and Fe3+are shorthand for ferroin and ferriin, respectively, and the factor h in reaction R9 is a stoichiometric parameter. For all calculations performed, the initial concentrations of the reactants are [H+Io= 0.333 M, [BrO3-Io = 0.199 M, [MA], = 0.043 M, [BrMA], = 0.0662 M, and [Fe2'lO + [Fe3+Io= 0.00294 M, and the diffusion coefficient, D, of all species is taken to be 1.5 X l(rscm2/s. The concentrationschosen are the same as those used in calculating chemical waves in an excitable medium in ref 14. IILB. IdthlExcitrtloain[HBr021. IILB.1. Reaction-Equrtioas for the Two-Variable Modified oresonator Model. For the calculations considering an initial excitation in [HBr02], two reaction-diffusion equations are obtained from the chemical model which describes the evolution of [HBr02] and [ferriin] in time and one-dimensional space. Five ordinary differential equations are initially written describing the time evolution of [HBr02], [ferriin], [Br-1, [BrO,'], and [BrMA'] [MA'] due to the reaction kinetics given by the model. The steady-state approximation is made for the concentrations of Br-, BrO,', and BrMA' MA'. After adding the diffusion term, and converting to the dimensionless concentration variables, the resulting reaction-diffusion equations to be solved are
+
+
ax
a2x
aT
as2
hz
c
p(c-z)+1
(=)
x+q
-K-,u']
(3)
where
2X(2 U =
(C
- Z) + [(C
- Z)'
+ K-gZ) + 8 ~ - ~ x +( 2K-gZ)]'/2
(5)
In the above equations, x and z are the dimensionless concentration variables related to the actual concentrations by 2k4k8B x = - 2k4 [HBr02] z = -[Fe3+] (6) klHA (kJW2
s and T are dimensionless space and time related to the dimensional variables by s = (k8B/D)1/2(space)
T
= k8B (time)
(7)
and A, 8, C, and Hare the following initial reactant concentrations assumed to be constant:
A = [BrO,-] C = [Fe"]
The parameters
C,
B = [MA]
+ [BrMA]
+ [Fe2+]
c, q, p,
K4=-=-
H = [H+l K - ~ , and K~ are given by
klk-3HA 2k4k8B
k-3 2k4t
(8)
(9)
(12)
where (13) & = k8 k9 / k-g The values of the parameters used for all calculations model an excitable system and are given in Table I.
parameter
value
h
1.3 1.65 X lo-* 7.31 X lo-* 2.00 x 10-4
t
C
Q
parameter P K-5 ILg U
value 5.32 x 103 3.92 X l P 6.04 X 10-I 1.20 x 10-2
"The parameter u is used only in the calculations presented in section 1II.C.
III.B.2. Initial h f h for tbe ReactiowDiffusion Equation& The initial pmfde of [HBrO,] contains a region of excitation, while the initial profile of [ferriin] is always flat in space with its concentration set at its steady-statevalue of 2.46 mM. The initial excitation is contained only in [HBr02] because in the chemical system, the autocatalytic generation of HBr02 is turned on by a depletion of bromide ions, and in that process, ferroin is oxidized to ferriin. The initial concentration of HBr02 is set at its steady-state value of 0.298 pM everywhere in space except in the middle of the profile, where in a given region, [HBr02] is set at a larger constant value. The resulting initial profile of [HBr02] in space then resembles a single square wave pulse (see Figure la). The concentration of HBr02 in the excited region for the various calculations varies from 0.331 to 166 pM, where 166 pM is approximately the maximum HBr02 concentration of a traveling wave for the concentratim chosen. The spatial width of the region in which [HBr02] is excited varies from 0.1 pm to 1 mm. This corresponds to studying regions of excitation with radii 0.05-500 ctm. llI.B.3. Method of Integration. The coupled partial differential equations (3 and 4) with added periodic boundary conditions are transformed to 200 coupled ordinary differential equations in time by the method of lines. The resulting ordinary differential equations describe the time evolution for 100 points in space for each variable and are integrated with the RungtKutta-Merson method. The time step chosen by the integrator is approximately 0.01 s. The space grid contains 100 points, and the distribution of t h e points is not uniform in space but changes as the calculation continues, in a manner developed by Dwyer.I5 A much denser distribution of points is used near the point of initial perturbation and the emerging wavefront, where the concentrations are changing most rapidly. The length represented by the 100 grid points also increases as the calculation continues so that an initial excitation with a radius on the order of micrometers can be sped% and followed as it grows into a wave and propagates over several millimeters while a high degree of spatial accuracy is maintained throughout. At the beginning of calculation, the chosen length represented by the grid is small so the narrow initial profile is specfied with a high degree of spatial accuracy with the 100 spatial grid points. As the calculation proceeds and the wave fronts spread outward, the calculation is stopped before the wave fronts reach the ends of the space grid. The length represented by the grid is incrtased at this time by adding grid points to each side of the terminal spatial grid points. At the same time, internal grid points are deleted from the currently existing space grid so the total number of grid points remains 100. The concentration values at the added grid points are the same as those at the formerly terminal grid points next to which they were added, and these values are the steady-state concentrations since the wave has not reached the end of the grid yet. The calculation is then restarted on this longer space grid. Repeating this process as necessary allows for good spatial accuracy throughout an entire calculation. In all calculations, the wave fronts never reach the boundary of the spatial grid. III.B.4. sdutiom of Reaction-Diffmion Equrtiom with Initinl Excitation in HBrO2. Calulations are performed with the initial profile of [HBrOz] containing an excitation of varying concentration which occurs within regions of varying radii. Whether or not a wave propagates for a given initial profile is noted. In the cases where wave propagation is observed from the initial region of excitation in [HBr02],the initial spike in [HBrO,] grow
Mori and Ross
8056 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
:=Do
200
50
distonce (mm)
0
distance (mm)
-
1. Calculated sequence of profiles of [HBr02] when an initial excitation in [HBIO,] propagates. Profile a is the initial profile, and profiles b, c, and d occur 3.43,9.16,and 22.9 s later, respectively. The region excited initially is 50 pm in radius and has an HBr02 concentration of 0.994 rrM.
distance (mm)
0.75
0.2
0.2
1
1
1
distance (mm)
distance (mm)
Figure 3. Calculated sequence of profiles of [HBr02] when an initial excitation in [HBrOZ]d m not propagate. Profile a is the initial profile, and profiles b, c, and d occur 1.14,4.58,and 22.9 s later, respectively. The region excited initially is 5 pm in radius and has an HBr02 concentration of 0.994pM.
r J , i ,w l I
2.44
Y
2.51-1--
0
2
4
6
8
1
0
22.51-.-
2.
0
distance (mm)
-52.2
0
2
1
6
8
distance (mm)
2
4
6
8
1
0
distance (mm)
1
0
-52.2
I-1 0
2
4
6
8
1
0
distance (mm)
Figure 2. Calculated sequence of profiles of [ferriin] which correspond to the profiles of [HBr02] in Figure 1.
to the height at which it travels, [HBrOz]-, and then propagates outward. A region of excitation is generated in ferriin as well, and a ferriin wave of maximum concentration [ferriin],, prop agates. An example of a series of profiles demonstrating this is shown in Figure 1 for [HBrOz] and in Figure 2 for [ferriin]. Profile a is the initial profile used, and profiles W follow chronologically. In Figure la, the initial region of excitation in [ H B Q ] has a radius of 25 pm,and the concentration of HBrOz within that region is 0.994 pM. The initiation of traveling waves in both variables is apparent. The maximum concentration each variable attains in a traveling wave is always the same regardless of the initial concentration of HBrOz in the region of excitation. Thus,in all cases,if a wave propagates, [ H B Q ] and [ferriin] grow to their maximum heights of [HBrO2],, and [ferriin],, and propagate outward. For the conccntmtions of reactants chosen for all calculations, [HBrOz], = 166 pM and [ferriin],,, = 2.98 mM. In the cases where no traveling wave is produced, the initial spike of [HBr02]either grows and then decays to its steady-state value, or the initial spike simply decays through diffusion and relaxes to its steady-state value. An example in which no wave propagates is shown in Figure 3, w h m again profde a is the initial profile used and profiles W follow chronologically. Figure 4 displays the accompanying ferriin profiles and demonstrates that a region of ferriin may be excited slightly, but not to the height at which it propagates. The region of ferriin excitation also eventually decays back to its steady-state value. The region of excitation in the initial p f d e of [HBrOz]in Figure 3a has a radius of 5 pm,and the concentration of HBr02 in that region is 0.994 PM. In the initial profiles of [HBrOJ, several different concentrations of excitation in [HBr02] are used, and for each concentration of excitation, several different radii are used within
2.4
distance (mm)
~
distance (mm)
27----7
”L
2.
distance (mm)
Figure 4. Calculated sequence of profiles of [ferriin] which correspond to the profiles of [HBr02] in Figure 3.
TABLE 11: R d t a Prrdietcd by the Reaction-Mmh Equations 3 and 4 for Whether a Wave Is Initiated or Not Givea M Initial Profile in Which tbe RePioa of Excitation Is Contained in WIBr0,P excitation concn ~~~~
~
~
_
of [HBr02],pM 0.05 pm 0.5 pm 5 pm 25 pm 50pm 500pm 165.7 X x x x x X 16.57 0 x x x x X 1.657 o x x X 0.9941 o x x X 0.6627 o x x X 0.4970 o o x X 0.3976 0 0 0 X 0.3314 0 0 0 0
“The initial region of excitation has a concentration as indicated by the row and a radius indicated by the column. If a wave propagates for the given initial excitation, an X is marked, if no wave propagates for the initial excitation used, an 0 is marked. which the excitation is applied. Table I1 summarizes the results of the calculations performed. If a traveling wave is generated for an excitation of a particular HBrOz concentration and within a particular radius, an X is placed in the appropriate space in the table. If no wave travels and the system relaxes back to its steady state, an 0 is marked. The larger the concentration of the initial excitation, the smaller the radius of excitation sufficient for wave initiation. For an excitation concentration of 0.3314 pM, the bottom row of Table 11, even exciting a radius of 500 pm does not generate a wave. Yet, for an excitation concentration of 16.57 pM, the second row in Table 11, a wave propagates if a region of only 0.5 pm in radius is excited. IILB.5. Regia of Validity of the Defenniaistic EqmtioaS. The initial profiles used in the calculations presented in section 1II.B contain a region in which the concentration of HBrOz is excited. The radius of this region for which the calculations are meaningN
_
_
Wave Generation in Excitable Chemical Systems is limited by the consideration of the number of particles contained within that region. The deterministic equations describe the evolution in time and space of concentrationsof species due to chemical reactions and diffusion. The equations presume that concentration is a meaningful macroscopic variable, and it only has meaning when a large enough number of particles reside within a volume so that the fluctuations in the number of particles do not change the average concentration significantly. Likewise, the process of diffusion describes a macroscopic average motion of particles due to concentration gradients. A large enough number of particles must again exist within a large enough volume in order for the concept of a concentration gradient to be meaningful and for diffusion to describe the evolution of concentration. When [HBrO,] is at its steady-state concentration of 0.298 pM, an average of only 0.094 HBr02 molecule is in a sphere of radius 0.05 pm, and when [HBrO,] is at the maximum concentration considered, 165.7 pM, still only an average of 52.2 molecules are in a sphere of radius 0.05 pm. When deterministic equations are used to model systems which contain tens of particles, errors due to fluctuations are significant;I6hence, deterministicdifferential equations cannot adequately model the system. The steady-state concentration of femin is several orders of magnitude larger than that of HBr02 and is not the limiting concern. The minimum radius of excitation for which the two deterministic reaction4flusion equations are a meaningful description of the system is a radius on the order of 0.5 pm. An average of 94.0 HBrOz molecules are contained in a sphere with this radius if [HBrO,] is at its steady-state concentration, and 522.4 or 5224 molecules are contained within the sphere if the excitation concentration is 1.657 or 16.57 fiM, which are the concentrations near the transition from a propagating to a nonpropagating wave as seen in Table 11. Such numbers of particles are sufficient for a deterministic description. The calculations of Plesser involved radii of excitation down to 1.7 X pm. The concentrations of the system modeled in that work are of the same order of magnitude as those discussed in this work. Hence, the deterministic description is not a useful description for the radii of excitation investigated below approximately 0.5 pm. m.C. Initial Excitation in [Br-1. In section III.B, the initial excitation is in the concentration of HBr02. However, a decrease in the bromide ion concentration can induce the autocatalytic production of HBr02. In this section, calculationsare presented to determine how much of a decrease in bromide concentration in a given volume is necessary to increase the HBr02 concentration in that volume to values in Table I1 where wave initiation occurs. Three coupled ordinary differential equations in time for the concentrationsof Br-, HBrO,, and ferriin are integrated for this purpose. III.C.1. Reaction Equrrtions for the Three-Variable Modified Oregodor Model. For the calculations considering an initial excitation in [Br-1, three reaction equations are solved which describe the time evolution of the concentrations [HBrOJ, [ferriin], and [Br-1. Five ordinary differential equations are initially written describing the time evolution of [HBrO,], [ferriin], [Br-1, [BrO;], and [ B r W ] [MA’] due to the reaction kinetics given by the model. The steady-state approximation is then made only for the concentrationsof BrO,’ and BrMA‘ MA’. After converting to the dimensionless concentrationvariables x, y, and z representing dimensionless [HBrOJ, [Br-1, and [ferriin], respectively, the resulting reaction equations are t dx/d7 = U(C - Z) - K+XZ - x + K - ~ u ’ - X’ - xy + qy (14)
+
+
The notation of all variables, parameters, and constants in the above equations is identical to that given in section II.B.l; x represents dimensionless [HBrOJ, z represents dimensionless
The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 8057 [ferriin], and the additional dimensionless concentration variable y is related to [Br-] by
The additional parameter u is given by the following expression: u = 2k4/ksH (18) All values of the parameters used in all calculations again model an excitable system and are given in Table I. IILC.2. I n l t i p 1 C I ” S e a d M e t b o d o f ~ ~Theinitial concentrationsof HBr02 and femin are set at their steady-state values for the threavaiable system of equations. The steady-state concentration of HBr02 is 0.298 pM and the steady-state concentration of ferriin is 2.44 mM. The initial concentration of bromide is put at various concentrations below its steady-state concentration [Br-],t.st.of 14.3 pM. The steady-state conccntrations [HBr021fim.and [ferriin],t,t, for the threevariable model and for the two-variable model are within 1% of each other. The three ordinary differential equations (14, 15, and 16) for the evolution of [HBr02], [ferriin], and [Br-] are solved by the routine DO2NBF in the NAG fortran library for stiff coupled ordinary differential equations. III.C.3. Soloti- of the Deterministic Reactioo Equations witb Initinl Excitation in [Br-k When the initial bromide concentration is 11.9 pM or lower, the autocatalytic production of HBr02 is turned on, and [HBrO,], [ferriin], and [Br-] undergo a large positive excursion in concentration space, or a complete excursion, in which both HBrOz and ferriin attain their maximum values of [HBr02], and [femin], and eventually relax back to their steady-state values. Only a 27% change in [Bf], from its steady-state concentration of 14.3 pM to a critical concentration [Br-I,, of 11.9 pM, is necessary to induce a 555-fold increase of [HBrOJ from 0.298 to 166 pM. Hence, a smaller change in [Br-] away from its steadystate concentration can produce a much larger change in [HBrO’] from its steady-state concentration. If the initial bromide concentration is below its steady-state value but not below the threshold value [Br-Idt, then [HBr02] and [ferriin] undergo excursions but of considerably smaller magnitude. If, for example, the initial bromide concentration is 12.0 pM, the maximum concentration attained by [HBrOz] before it returns to its steady-state concentration is then only 0.41 1 pM, representing just a 38% increase from its steady-state value. Within a range of just 0.1 pM in initial bromide ion concentration, from 12.0 to 11.9 pM, the maximum concentration attained by [HBrO,] during its excursion spans almost 3 orders of magnitude. The bromide ion concentration therefore acts as a switch for the autocatalytic production of HBr02. When [Br-] is below the critical concentration [Br-Icdt of 11.9 pM, the autocatalytic production of HBr02 proceeds, and when [Br-] is above [Br-]&t, the production of HBrOz is far less extensive. For the concentration of HBrOz to attain its critical concentration necessary for wave generation within a given volume, the bromide ion concentration must dip below its critical concentration of 11.9 pM in that volume. III.C.4. R e g h of Validity of the DetermWtic JIqmtio~~ To check the validity of the deterministic calculations with regard to the number of Br- ions, the number of bromide ions in a sphere of radius 0.05 pm is considered. An average of only 4.52 bromide ions is contained in that sphere when [Br-] is at its steady-state concentration of 14.3 pM, and only 3.77 ions are in the sphere when [Br-] has reached its critical concentration of 11.9 pM. There are too few Br- ions as well as too few H B d 2 molecules involved when considering a radius of excitation of 0.05 pm or smaller to justify the integration of the deterministic equations as a realistic representation of the system. But, for a radius of excitation of 0.5 pm, the smallest radius for which the number of HBr02 molecules warrant a deterministicdescription, several thousand bromide ions are present. The deterministic equations of the three-variable model are then considered to be a realistic model for systems in which the radius of excitation is 0.5 pm or larger.
8058 The Journal of Physical Chemistry, Vol. 96, No. 20, I992
IV. Estimate of the Probability of Chemical Wave Initiation by Thermal Fluctuations In order to make an estimate of the probability of a thermal fluctuation of sufficient magnitude in concentration occurring within a sufficient volume of critical size, we use the simplest approach of an equilibrium calculation. Limitations of such an estimate in a nonequilibrium system are recognized, but we think it unlikely that the estimate is incorrect by orders of magnitude. N.A. Forrrmhtion of the Calculation. The theory of fluctuations in ideal solutions at equilibrium is well formulated and discussed in Chandrasekhar’s classic article” and other We consider a chemical reaction system in an excitable stationary state which is at constant temperatureand spatially homogeneous. The volume of the entire system is V, and we select a small volume, u, which is in equilibrium with the larger volume surrounding it, equilibrium being taken with respect to concentration fluctuations. The number of small volumes comprising the entire system is N. Consider measurement of the number, n, of particles in u which are made in regular time intervals, T , and let the average number of particles in the volume u be Y. The frequency, W(n), with which n independent particles will be observed in a volume u is given by the Poisson distribution e-”v“ W(n)= n! The root mean square fluctuation, 6, from the average number of particles v is 6= (20) When measurements are made at a regular time interval T apart, the probability, P,that a particle somewhere within a volume u at time t = 0 will have moved outside the volume u at time t = T is given by
$5
P = 1 - ( 4 u D ~ ) ~ / ~ u exp(
F)
drl drz (21)
where D is the diffusion coefficient of the particle, and rl and r2 are vectors integrated over the volume u. The probability that if n particles are observed in a volume u at a certain time and then, after a time interval T , n particles are again observed within the same volume u is given by w(n,n) where Ci”P(1 - P)”‘(vP)’ W(n,n) = e-vpE n
I
i-0
1:
(22)
In the above equation, C; is the combinatorial coefficient
n! Ci, I: i!(n - i)!
W(#n,n) represents the probability that a volume u, initially in a state in which there are not n particles in it, undergoes a transition in a time interval T to a state in which n particles are in u. At equilibrium,the number of transitions within a volume u from states # n to the state n must equal the number of transitions occurring within the volume from the state n to states # n in the time interval T . The following balance hence holds true [ l - W(n)JW(#n,n)= W(n)[l - W(n,n)] (24) where [ 1 - W(n)] is the occurrence frequency of the states # n in a volume u, W(#n,n) is the probability of a volume u which is in the state # n becoming the state n, W(n)is the Occurrence frequency of the state n in a volume u, and [ 1 - W(n,n)] is the probability that a volume u in the state n becomes #n. Solving for W(#n,n) yields
W(#n,#n) is the probability that a transition Occurs within a volume u from a state # n to a state also # n and is simply W(#n,#n) = 1 - W(#n,n) (26)
Mori and Ross An expression for +,,(kT), the probability that an arbitrary state of the entire system in which all of the N small volumes are in a state # n is observed to remain in a state in which all N small volumes remain in any state # n on k - 1 successive intervals of observation and is observed to have at least one small volume in the state n on the kth occasion is $,(kT)
Wfk-’)N(#n,#~)[l - FP(#n,#n)]
(27)
Finally, the recurrence time, e,,,defined here as the amount of time necessary to wait before observing the state n in any volume u after the state n has been observed previously in any volume u, can be expressed as
8, =
5knCn(k7)
(28)
k= 1
After performing the back substitutions and evaluating the infinite geometric series, the recurrence time takes the form
e, =
T
(29)
1 - FP(#n,#n)
and, finally 7
1 - W(n,n)
1
(30)
1 - W(n) For all calculations in this paper, the value of the diffusion coefficient D is 1.5 X los5 cm2/s, the value of the observation interval T is 1 s, and the total volume Vis 6 mL. The calculated values of both W(n) and w(n,n) are very small numbers for every calculation. The largest calculated value of W(n)or W(n,n)in this paper is of the order Hence, the above expression for 8,can be simplified by expanding the term in the square bracket in powers of W(n)or W(n,n) and keeping terms to first order in W,where W represents either W(n)or W(n,n). The resulting expression for 8, becomes
e, =
T
+ O(W)]N
1 - [ 1 - W(n)
(31)
Expanding the denominator again yields T
e,, = 1 - [l - NW(n) + O(N2W)I
(32)
If N W ca. 10-i M). Moreover, we have revealed that the dynamic behaviors of IPS including the chemical reactivity are strongly dependent on their production pathways in the cases of BP-N,N-dimethylaniline (DMA) and -N,N-diethylaniline (DEA) systems in a c e t ~ n i t r i l e . ' ~ J ~ . ~ ~ The elucidation of the different reactivity of the IP depending on the mode of its production provides very important information not only for the comprehensive understanding of the hydrogen abstraction mechanism of BP* from AH but also for the elucidation of the mechanisms of the ET reaction and related prin general?' since the intermolecular PT process is much more strongly dependent on the donoracceptor mutual geometry such as intermolecular distance and orientation than the ET process and, hence, the information on the structure of the IP could be obtained from the PT dynamics. In the present paper, we report the reduction process of BP* with N-methyldiphenylamine (MDPA). The reactivity of IPS depending on the mode of their production will be discussed on the basis of the femtoseumd and picusecondtimeresohred transient absorption spectra. In addition, integrating the present experimental results with previous 0nes17J6,20 obtained by using DMA, DEA, and Nfldiethyl-ptoluidine (DET) as hydrogen donors and with a number of accumulated data on photoinduced ET and
0022-3654/92/2096-8060$03.00/00 1992 American Chemical Society
