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Many-particle effects in kinetics of bimolecular diffusion-controlled reactions
Volume 117, number 3
CHEMICAL
MANY-PARTICLE EFFECTS IN KINETTCS OF BIMOLEE. KOTOMIN
PHYSICS
14 June 1985
LETTERS
DIFFUSION-CONTROLLED
REACTIONS
and V. ICUZOVKOV
LrruranSrare Uniuersity, Rainrs blvd 19. Riga 226098. USSR
Rece~vexl10 December 1984; in fmal form 3 April 1985
S~~~YIUI~from the generalized theory of blmolwlar diffusion-controlled reactiorus betwatn reagents in soli& and liquids, we + B (exciton annihdation) reaction have performed calculations of Lhe A+B * AB (F’renkel defecL anmhilalion) and A+A kinetics over a wide time interval and for high initial concentrations The prtdickd lowering at long times of the reaction rate +B reaction due to dynamical in Lhc course of the A+B -D AB reaction and acceleration at short times of the A+A aggregation of similar non-interacting reagenls are demonstrated_ The posubility of experimental checks of these effecLs IS
I. Introduction In recent years, there has been considerable theoretical interest in the effects of high reagent concentrations and long reaction times in the tietics uf bimolecular diffusion-controlled reactions. For instance, the kinetics of energy transfer has been analyzed by a number of authors and with a variety of approaches [l-6] _ However most of them contradict each other, includmg the key problem: whether the decay betics at long times remains exponential (but with a concen-
tration-dependent reaction rate K) or not. Probably thrs is due to quite drfferent a priori (and often indirect) approximations. Generalized kioetrc equations have been derived by us 17-91 describing diffusioncontrolled bimolecular reactions, incorporating Frenkel defect anmhilation, A + B + AB, energy transfer, A + B + B, and exciton annihilation, A + A + B. This theory is based on a more complete decoupling of the luerarchy of equa-
tions for many-point reagent densitres. Unlike earlier theories, the spatial correlations of both similar (A-A,
Both our approach [7-91 and theory 1101 show that due to dissimilar reagent recombination, A + B 4 AB, the initial random spa&d reagent distribution is replaced at long tunes by dynamical aggregates of similar reagents (A or B only). It lowers the reaction rate K thus leading to unusual delayed tietics. Thus in the case of u A=ng=n(rzuthemamoscopic concentration) one gets n = rm3j4
(DA,
n 0: t-1’2
(DA
nA=r
-w2
nA = t-l
(DA
= 0, DB # 0),
f
= 0, D,
(la) nB
,
O), # 0).
(lb)
These results differ qualitatively from standard chemical kinetics with a distinctive quasi-steady reaction rate Kg = 4rrDrg (D =D, + DB is the relative diffusion coefficient, r. the mstant recombination radius) [11,12] -
position approtiatron for three-particle densities is used here). The reagent motion is de-bed as inc+
nA a ex~?&-,nBt)
266
0),
(DA,DB
nar-l
effects)_
f
whereas if one of the reagents is in excess, nA &
B-B) and dissimilar (A-B, B-A) reagents are rigorously taken mto account (Kirkwood’s standard super-
herent (wrthout memory
D,
(nA =nB
=n), (nA en,).
(24 (=)
In this Letter we ilhrstrate the general theory [7-91 by actual numerical calculatrons for the reactions of A+B+ABandA+A+Binordertoanal~ehow great the predicted deviation of the rigorous kinetics from the standard one is [l 1] (eq. (2)) and whether it could be checked experimentaIly.
2. Basic equations andresults As earlier [7-g], it is convenient lowing dimensionless units: r’ = r/r*, = 20, fD (u = A. B), n: = 4mr$, , K’ (primes are omitted below.) In these set of kinetic equations reads dnA/dr
= dnB/dt
= -KnAng,
to employ the fol-
r’ = Dt/rg , D: = K/4nDr,-,. variables the basic
K = (CIY/CIr)T=l,
(3)
v’ f v,
(4)
- 2KX&.J(Y-),
aY/at = AY - KY ,=gB
,
hYJ(XY),
(Z(r’) - l)r’dr’,
J(Z) =&j+’
Z=Xv,Y.
(6)
Ir-11 The key feature of these equations 1s back-coupling of .similar,Xv, and d&imilar, Y, joint (two-point) correlation functions (densities) of reagents A, B describing spatial correlations of the A-A, B-B and A-B, B-A kinds, respectively. The boundary conditions 1
(7a)
mean that spatial correlations become negligible at long distances, whereas the conditions Y(r < 1) G 0,
Xv(r = 0) - limited
dY/dt
K = @Y/ar),,,,
= AY - 2KYnJ(Y),
(8)
Y(r < 1) = 0.
(9)
there has been great interest in the kinetics of the exciton annihilation (e.g. refs. [13-171) the role of many-particle effects has never been dealt with in detail. In this Letter we are concerned with tlus point, neglecting other previously analyzed effects (exciton hfetime, discreteness of crystalline lattice, the complicated motion of excitons, space dimensionality and traps, dipole+lipole nature of the elementary event of an armihilalron, non-hiarkovian effects at very short times, etc)*The properties we are interested in are (i) the time dependence of the reaction rate, K(t) *, (ii) the critical exponent, y (n a t-7) and (iii) the time development of the correlation functions, Xv, Y. It must be remembered that in the standard approach [11,12] the reaction rate K and the recombination front are stabilized after a short transient period, td = 10&D, smce
Although
where
lim Xy,Y= r--c=
imation) the many-particle effects in question [7-g]The basic equations of the s’andard kinetics [ 11,121 could easily be obtamed from eqs. (3)-(6) omitting these non-linear terms (thus linearizing the equations). It is justified usually either by the smaUness of the reagent concentrations (n, + 0) entering the non-linear terms, or by +he neglect of spatial correlations of similar reagents at all times, Xv E 1. However, these commonly used approximations are not always valid, as is shown below (see also refs. [7-91). If reagents are initiahy distributed at random, the decay kinetics is uniquely descriied by the reagent concentration% nA, ng , and the ratio of the diffusion coefflcrents, DA/DB (for the dimensionlessD, entering eqs. (3)-(6) DA + DB = 2). For exciton anmhilation, A + A + B, one gets [8] dn/dt = -Kn2,
axJat =D,M*
14 June 1985
CHEMICAL PHYSICS LETTERS
Volume 117. number 3
(7b)
describe instant annihilation if reagents approach to within a clear-cut radius r0 (in our units r. = l), assuming their sizes are negligible compared to ‘0. In fact, the continuous description used in the case of crystals means that the annWation sphere, uo = $rri, contains a lot of lattice sites. The second terms on the right-hand side of eqs. (4) and (5) describe (in Kirkwood’s superposition approx-
K = 1 +(~rr)-l’~, Y(r, t% 1) = 1 -
K(t9 l/r.
l)=K,
= 1,
(IO) (11)
To soIve numericaIIy the set of the integrodifferential equations (3)-(5) [or (8) and (9)J, an unplicit conser-
l
Sometimes instead of the the reaction depth z = n(t)/n, will be used. It shows clearly the &axe of rqents survivhg up to instant t.
267
CHEMICAL PHYSICS LETTERS
Volume 117, number 3
14June1985
lations demonstrate clearly [ 181 that for a wide interval of initial concentrations the reaction rate deviates fromKo
= 1 much quicker than for a random distriiu-
tion as shown in fz. 1. The critical exponent at lo* < t < 106 is 033, i.e. considerably Zess than the asymptotic value for a random distribution. All this confums the ideas [7-101 that delayed Ednetlcs takes place at very Iong times, exceeding by transient period td of standard kinetics (eq. (10)). However, one can conclude from curve 4 in fg_ 2 that for large initial reagent concentrations the non-steady part of the reaction rate K(t), even at short times comparable with td, falls with time much faster than expected from stan-
many orders of magnitude the disttct
Dependence of the dimensionlessreacthn rate on the share of survrvingreagents (random initial reagent distribution). Solid lines, DA = Dg; broken lines, DA = 0. hitid di-
Fig 1.
mensionless
concentrationsno
areindicated.
vative difference
scheme has been used. The mtegrals (6) were calculated by means of the mpezoid rule. The co-factors 2Kn,J(y) in eq- (4),KZV=A,B7z2yJ(X,) u-t eq. (5) and 2KnJ(Y) in eq. (9) entering the generalized diffusion equations were calculated in the preceding time step. whereas the other quantities in these equations were calculated in the current time step. The problem of the tidiagonal matrix was solved by using the factorization method- The calculated dependence of the reaction rate on its depth is plotted m fG_1 fortheA+B+ABreactionandvaryinginitial reagent concentrations. (More details will be given elsewhere 1181.) One can conclude that for a given initial concentration the deviation from the quasisteady reaction rate, K. = 1 (see eq. (IO)) is reached quicker if one of the reagents is immobile. For a large initial concenlzation, no = 1 (it is close to the maximal concentration of Frenkel defects which could be accumulated under irradiation [19]),K 1s reduced by almost an order of magnitude when the reaction depth isr= 10S4 anditstimet= l@ (inunitsofr$D) which is believed to be detectable experimentally. The corresponding critical exponents are y = 0.8 (DA,B f 0) and 7 = 0.6 (DA = O.D, # 0) thus approaching the predicted asymptotic limits (7 = 0.75 and 0.5 at r +m), eq. (la)_ At any rate, they differ greatly from the predictions of standard kinetics (7 = 1). Prolonged irradiation results in a non-random reagent distribution. If reagents are mobile, we could approximate the dissimilar reagent distnbution by the quasi-steady one (eq. (11)). The corresponding calcu268
dard chemical kinetics (eq. (10) and broken curve) Similar short-time tiansient kinetics for exciton annihlation,A+A+B,isalsoshowninfig.2.Asseen from eq. (9), the many-particle effect described by the second term on the right-hand side must accelerate
the
armihihtion (see also ref_ [S]) since the general effect of e reagent aggregation here means nothing but ann&ilation itself. However, unlike the A + B + AB reaction, the kinetics of the exciton annhilation at lotlg times could be described by the steady-state reaction rate Ko = 1 (i-e_ Y
_~=&-;~;s[_ firmed by the numen Therefore the many-particle effect for the A + A + B
reaction is most pronounced at sho?7 times t =5 td_ The corresponding data plotted in curves l-3 of fq_ 2 are approxfm~~ed quite well by the following empirical relation (cf. eq_ (10) and ref_ 1131):
Fz_ 2. Nonsteady part of the reaction rate at short dimensionlesst.nnerfortbeA+A+B(awesl--9)andA+B+AB (curve 4) reactio= The broken line 1s the result of standard chemicalldneticsK= 1+ (M)-~~ [11,12].no1,l.O. 2,O.l; 3,001;4.1.0.
Volume 117. number3
CHEMICALPHYSICSLETTERS
14 June1985
tal line X, Y = 1 [7] _) The joint density of simiIar reagents,X,, at small relative distances at long times cz..n exceed several times the Poisson value which could be interpreted as their dynamiml aggrqgatin (clustering) [7-lo]_ (Remember that similar reagents do not interact with each other.) The correlation functions for both similar and dissimilar reagents tend to the Poisson value at a distance L a t1i2 giving a new distinctive reaction scale. The latter may reasonably be used as the mean size of similar reagent aggregates which greatiy exceeds the annihilation radius L = 1 (ro) of the standard chemical kinetics. If reagents of any kind, say A, are immobile, the character of the spatial correlation appears to be quite different (curves b m fg_ 3). For instance, despite a deviation of the correlation function for similar mobile reagents, B, from the Poisson value that is slightly reduced here, the analogous correlation function for immobile reagents, X, , exhibits an abrupt increase at small distances up to X, z= 104s6_ It means formation of aggregates with a very dense core which is no
a
Yoi!! ------se-
__-----
I-
/
/
/
longer smoothed by diffusion of reagents A. It is clear that it affects the reaction kinetics directly (fg. 1).
Fig. 3. Jointcorrelation functionsfor similar.X,. and dissimilar, Y, reagentsfor the largedimensionless time r of the reactiOnA+B~~.(a)DA=Dg.n0=1,n=1_2X 10d,t = 35 X 104 (in unitsof&D>. R&on rateK = 0 19. & exponent y = 0.82. The brokenline showsthe resultof standardchemicalkinetics(I&. [ 11,121 and eq. (11)). (b) Dg # 0. DA = 0. Curve 1 corresponds to mobilereagentsand curve2 to immobdereagents.no = 5, n = 10-4, E= 105, K = 0.076, aiticzl expment 7 = 0.6 _
K = 1 + 2.1(7r:.r)4’2,
r > 102.
(12)
Applying our approach to the investigation of manyparticle effects in the case of energy transfer, A + B + B, for mobile donors, A, and large concentrations of acceptors (sinks B) cannot be successful in the framework of the standard Kirkwood superposition approximation [8], and wiu be done elsewhere. The time development of the joint correlation f;mctions is grven in fg. 3. As is seen, the initial Poisson fluctuation spectrum, X,, Y = 1 changes qualitatively in the course of reaction. (The measure of the spatial correlation of reagents is a deviation from the horizon-
3. Conclusion It should be stressed that the non-Poisson spatial correlations of reagents arise from the bimolecular reaction itself and take place even at the early (transient) stage of the reaction, but become most pronounced at long times. The effect of the reaction-induced uggregation of similar non-interacting reagents under study can lead at long times and/or large initial reagent concentrations to an appreciable deviation of reaction kinetics from predictions of standard chemical kinetics [11,12] since the latter neglects the reagent density fhrctuations [7-lo]. This aggregation effect reveals a universal character and should take place not only for the diffusioncontrolled reactions considered above but also for static reactions, e.g. for both accumulation of Frenkel defects in solids under irradiation at low temperatures [ 191 and for tunneling recombination of immobile donors and acceptors [20] _ The aggregation is due to back-coupling of spatial correlations of similar and dissimilar reagents: recombination of HISsimilar reagents (A-B) indirectly stimulates the aggre269
Volume
117. number 3
CHEMJCAL
gation of simiIar reagents, which are supposed to be non-interactive.
Probably the aggregation effect could be experimentally detected, not through the deviation of the reation kinetics from standard chemical knetics for large reagent concentiations (eqs. (1) and (lo)), but through the appearance of the non-Poisson spectrum of density fluctuations which has, in fact, been observed earlier in the kinetics of FrenkeI defect accumulation in alkali halide crystals [19] _
References
[13 IX. Kapral, J. Chem. VI B V. Felderhof and
Phys 68 (1978) 1903. JM. Deutch, J. Cbem. Phys. 64
(1976) 4551. LAFI No. 37. International 131 YuH. Kalniu, Aepr& Confaence on Defects in Insulating Solids, Riga (May 1961). Pbys Stat. SOL 1Olb (1980) K139. 143 V. GBsele, J. NucL Mater. 78 (1978) 83. ISI M Boon, J. Chem Phys 75 (1981) 2354. [61 M. Muthukumar. J_ Cbern. Phys 76 (1982) 972; DE. Calefand J.M. Deutch, J. Chem Phys 79 (1983) 203. [‘I V. Kuzovkov and E. Kotomin, Chem. Phys. L.etters 87 (1982) 575.
270
PHYSICS
LETIERS
14 June 1985
E. Kotomin and V. Kuzgvkov, Chem. Phys. 76 (1983) 479; V. Kuzovkov and E. Kotomin. Chern. Phys. El (1983) 335. 191 V. Kuzwkov and E. Kotomin. J. Phys. Cl3 (1980) L499. and YaB. Zel’dovick. Chem. Pbys 28 [W AA. Ovwov (1978) 215 WI TR Waite, Phys Rev. 107 (1957) 463. WI V-V. Antonov-Romanovskii, Kinetics of pbotolumiues cence of crystalphosphors (Nauka, Moszw, 1966); J. Phys. USSR 6 (1942) 120. [131 EP. Ivanova, J. Luminescence 21 (1980) 373; P-V. Elyutin, J. Phys Cl7 (1984) 1867_ 1716. 1141 A. Suna. Phys. Rev. Bl(l970) Liquid 0yst. 1151 Yu. Gaididei and A. Onipko. Mol.-wst. 62 (1980) 213. [161 VXh. Brickenuhtein, VA. Bender&ii and P.G. Fmpov, Phyn stat. Sol. 177lJ (1983) 9; V.M. Agranovich. NA. Efrernov and AA. Zakhidov, In. Akad. Nauk SSSR Fiz. 44 (1980) 759. Z. Physik 843 (1981) 221; in: Exciton 1171 V.M. Kenbe, dynamics 1p molecular a-ystals and aggregates (Springer. B&in, 1982). 1181 E. Kotomin and V. Kuzovkov, Qech. J. Phys. B. to be pUbliShed_ 1191 V. Kuzovkov and E. Kotomin, J. Phys. Cl7 (1984) 2283. WI V. Kuzovkov, Soviet Khirn. Fiz., to be publisbeg
CHEMICAL
MANY-PARTICLE EFFECTS IN KINETTCS OF BIMOLEE. KOTOMIN
PHYSICS
14 June 1985
LETTERS
DIFFUSION-CONTROLLED
REACTIONS
and V. ICUZOVKOV
LrruranSrare Uniuersity, Rainrs blvd 19. Riga 226098. USSR
Rece~vexl10 December 1984; in fmal form 3 April 1985
S~~~YIUI~from the generalized theory of blmolwlar diffusion-controlled reactiorus betwatn reagents in soli& and liquids, we + B (exciton annihdation) reaction have performed calculations of Lhe A+B * AB (F’renkel defecL anmhilalion) and A+A kinetics over a wide time interval and for high initial concentrations The prtdickd lowering at long times of the reaction rate +B reaction due to dynamical in Lhc course of the A+B -D AB reaction and acceleration at short times of the A+A aggregation of similar non-interacting reagenls are demonstrated_ The posubility of experimental checks of these effecLs IS
I. Introduction In recent years, there has been considerable theoretical interest in the effects of high reagent concentrations and long reaction times in the tietics uf bimolecular diffusion-controlled reactions. For instance, the kinetics of energy transfer has been analyzed by a number of authors and with a variety of approaches [l-6] _ However most of them contradict each other, includmg the key problem: whether the decay betics at long times remains exponential (but with a concen-
tration-dependent reaction rate K) or not. Probably thrs is due to quite drfferent a priori (and often indirect) approximations. Generalized kioetrc equations have been derived by us 17-91 describing diffusioncontrolled bimolecular reactions, incorporating Frenkel defect anmhilation, A + B + AB, energy transfer, A + B + B, and exciton annihilation, A + A + B. This theory is based on a more complete decoupling of the luerarchy of equa-
tions for many-point reagent densitres. Unlike earlier theories, the spatial correlations of both similar (A-A,
Both our approach [7-91 and theory 1101 show that due to dissimilar reagent recombination, A + B 4 AB, the initial random spa&d reagent distribution is replaced at long tunes by dynamical aggregates of similar reagents (A or B only). It lowers the reaction rate K thus leading to unusual delayed tietics. Thus in the case of u A=ng=n(rzuthemamoscopic concentration) one gets n = rm3j4
(DA,
n 0: t-1’2
(DA
nA=r
-w2
nA = t-l
(DA
= 0, DB # 0),
f
= 0, D,
(la) nB
,
O), # 0).
(lb)
These results differ qualitatively from standard chemical kinetics with a distinctive quasi-steady reaction rate Kg = 4rrDrg (D =D, + DB is the relative diffusion coefficient, r. the mstant recombination radius) [11,12] -
position approtiatron for three-particle densities is used here). The reagent motion is de-bed as inc+
nA a ex~?&-,nBt)
266
0),
(DA,DB
nar-l
effects)_
f
whereas if one of the reagents is in excess, nA &
B-B) and dissimilar (A-B, B-A) reagents are rigorously taken mto account (Kirkwood’s standard super-
herent (wrthout memory
D,
(nA =nB
=n), (nA en,).
(24 (=)
In this Letter we ilhrstrate the general theory [7-91 by actual numerical calculatrons for the reactions of A+B+ABandA+A+Binordertoanal~ehow great the predicted deviation of the rigorous kinetics from the standard one is [l 1] (eq. (2)) and whether it could be checked experimentaIly.
2. Basic equations andresults As earlier [7-g], it is convenient lowing dimensionless units: r’ = r/r*, = 20, fD (u = A. B), n: = 4mr$, , K’ (primes are omitted below.) In these set of kinetic equations reads dnA/dr
= dnB/dt
= -KnAng,
to employ the fol-
r’ = Dt/rg , D: = K/4nDr,-,. variables the basic
K = (CIY/CIr)T=l,
(3)
v’ f v,
(4)
- 2KX&.J(Y-),
aY/at = AY - KY ,=gB
,
hYJ(XY),
(Z(r’) - l)r’dr’,
J(Z) =&j+’
Z=Xv,Y.
(6)
Ir-11 The key feature of these equations 1s back-coupling of .similar,Xv, and d&imilar, Y, joint (two-point) correlation functions (densities) of reagents A, B describing spatial correlations of the A-A, B-B and A-B, B-A kinds, respectively. The boundary conditions 1
(7a)
mean that spatial correlations become negligible at long distances, whereas the conditions Y(r < 1) G 0,
Xv(r = 0) - limited
dY/dt
K = @Y/ar),,,,
= AY - 2KYnJ(Y),
(8)
Y(r < 1) = 0.
(9)
there has been great interest in the kinetics of the exciton annihilation (e.g. refs. [13-171) the role of many-particle effects has never been dealt with in detail. In this Letter we are concerned with tlus point, neglecting other previously analyzed effects (exciton hfetime, discreteness of crystalline lattice, the complicated motion of excitons, space dimensionality and traps, dipole+lipole nature of the elementary event of an armihilalron, non-hiarkovian effects at very short times, etc)*The properties we are interested in are (i) the time dependence of the reaction rate, K(t) *, (ii) the critical exponent, y (n a t-7) and (iii) the time development of the correlation functions, Xv, Y. It must be remembered that in the standard approach [11,12] the reaction rate K and the recombination front are stabilized after a short transient period, td = 10&D, smce
Although
where
lim Xy,Y= r--c=
imation) the many-particle effects in question [7-g]The basic equations of the s’andard kinetics [ 11,121 could easily be obtamed from eqs. (3)-(6) omitting these non-linear terms (thus linearizing the equations). It is justified usually either by the smaUness of the reagent concentrations (n, + 0) entering the non-linear terms, or by +he neglect of spatial correlations of similar reagents at all times, Xv E 1. However, these commonly used approximations are not always valid, as is shown below (see also refs. [7-91). If reagents are initiahy distributed at random, the decay kinetics is uniquely descriied by the reagent concentration% nA, ng , and the ratio of the diffusion coefflcrents, DA/DB (for the dimensionlessD, entering eqs. (3)-(6) DA + DB = 2). For exciton anmhilation, A + A + B, one gets [8] dn/dt = -Kn2,
axJat =D,M*
14 June 1985
CHEMICAL PHYSICS LETTERS
Volume 117. number 3
(7b)
describe instant annihilation if reagents approach to within a clear-cut radius r0 (in our units r. = l), assuming their sizes are negligible compared to ‘0. In fact, the continuous description used in the case of crystals means that the annWation sphere, uo = $rri, contains a lot of lattice sites. The second terms on the right-hand side of eqs. (4) and (5) describe (in Kirkwood’s superposition approx-
K = 1 +(~rr)-l’~, Y(r, t% 1) = 1 -
K(t9 l/r.
l)=K,
= 1,
(IO) (11)
To soIve numericaIIy the set of the integrodifferential equations (3)-(5) [or (8) and (9)J, an unplicit conser-
l
Sometimes instead of the the reaction depth z = n(t)/n, will be used. It shows clearly the &axe of rqents survivhg up to instant t.
267
CHEMICAL PHYSICS LETTERS
Volume 117, number 3
14June1985
lations demonstrate clearly [ 181 that for a wide interval of initial concentrations the reaction rate deviates fromKo
= 1 much quicker than for a random distriiu-
tion as shown in fz. 1. The critical exponent at lo* < t < 106 is 033, i.e. considerably Zess than the asymptotic value for a random distribution. All this confums the ideas [7-101 that delayed Ednetlcs takes place at very Iong times, exceeding by transient period td of standard kinetics (eq. (10)). However, one can conclude from curve 4 in fg_ 2 that for large initial reagent concentrations the non-steady part of the reaction rate K(t), even at short times comparable with td, falls with time much faster than expected from stan-
many orders of magnitude the disttct
Dependence of the dimensionlessreacthn rate on the share of survrvingreagents (random initial reagent distribution). Solid lines, DA = Dg; broken lines, DA = 0. hitid di-
Fig 1.
mensionless
concentrationsno
areindicated.
vative difference
scheme has been used. The mtegrals (6) were calculated by means of the mpezoid rule. The co-factors 2Kn,J(y) in eq- (4),KZV=A,B7z2yJ(X,) u-t eq. (5) and 2KnJ(Y) in eq. (9) entering the generalized diffusion equations were calculated in the preceding time step. whereas the other quantities in these equations were calculated in the current time step. The problem of the tidiagonal matrix was solved by using the factorization method- The calculated dependence of the reaction rate on its depth is plotted m fG_1 fortheA+B+ABreactionandvaryinginitial reagent concentrations. (More details will be given elsewhere 1181.) One can conclude that for a given initial concentration the deviation from the quasisteady reaction rate, K. = 1 (see eq. (IO)) is reached quicker if one of the reagents is immobile. For a large initial concenlzation, no = 1 (it is close to the maximal concentration of Frenkel defects which could be accumulated under irradiation [19]),K 1s reduced by almost an order of magnitude when the reaction depth isr= 10S4 anditstimet= l@ (inunitsofr$D) which is believed to be detectable experimentally. The corresponding critical exponents are y = 0.8 (DA,B f 0) and 7 = 0.6 (DA = O.D, # 0) thus approaching the predicted asymptotic limits (7 = 0.75 and 0.5 at r +m), eq. (la)_ At any rate, they differ greatly from the predictions of standard kinetics (7 = 1). Prolonged irradiation results in a non-random reagent distribution. If reagents are mobile, we could approximate the dissimilar reagent distnbution by the quasi-steady one (eq. (11)). The corresponding calcu268
dard chemical kinetics (eq. (10) and broken curve) Similar short-time tiansient kinetics for exciton annihlation,A+A+B,isalsoshowninfig.2.Asseen from eq. (9), the many-particle effect described by the second term on the right-hand side must accelerate
the
armihihtion (see also ref_ [S]) since the general effect of e reagent aggregation here means nothing but ann&ilation itself. However, unlike the A + B + AB reaction, the kinetics of the exciton annhilation at lotlg times could be described by the steady-state reaction rate Ko = 1 (i-e_ Y
_~=&-;~;s[_ firmed by the numen Therefore the many-particle effect for the A + A + B
reaction is most pronounced at sho?7 times t =5 td_ The corresponding data plotted in curves l-3 of fq_ 2 are approxfm~~ed quite well by the following empirical relation (cf. eq_ (10) and ref_ 1131):
Fz_ 2. Nonsteady part of the reaction rate at short dimensionlesst.nnerfortbeA+A+B(awesl--9)andA+B+AB (curve 4) reactio= The broken line 1s the result of standard chemicalldneticsK= 1+ (M)-~~ [11,12].no1,l.O. 2,O.l; 3,001;4.1.0.
Volume 117. number3
CHEMICALPHYSICSLETTERS
14 June1985
tal line X, Y = 1 [7] _) The joint density of simiIar reagents,X,, at small relative distances at long times cz..n exceed several times the Poisson value which could be interpreted as their dynamiml aggrqgatin (clustering) [7-lo]_ (Remember that similar reagents do not interact with each other.) The correlation functions for both similar and dissimilar reagents tend to the Poisson value at a distance L a t1i2 giving a new distinctive reaction scale. The latter may reasonably be used as the mean size of similar reagent aggregates which greatiy exceeds the annihilation radius L = 1 (ro) of the standard chemical kinetics. If reagents of any kind, say A, are immobile, the character of the spatial correlation appears to be quite different (curves b m fg_ 3). For instance, despite a deviation of the correlation function for similar mobile reagents, B, from the Poisson value that is slightly reduced here, the analogous correlation function for immobile reagents, X, , exhibits an abrupt increase at small distances up to X, z= 104s6_ It means formation of aggregates with a very dense core which is no
a
Yoi!! ------se-
__-----
I-
/
/
/
longer smoothed by diffusion of reagents A. It is clear that it affects the reaction kinetics directly (fg. 1).
Fig. 3. Jointcorrelation functionsfor similar.X,. and dissimilar, Y, reagentsfor the largedimensionless time r of the reactiOnA+B~~.(a)DA=Dg.n0=1,n=1_2X 10d,t = 35 X 104 (in unitsof&D>. R&on rateK = 0 19. & exponent y = 0.82. The brokenline showsthe resultof standardchemicalkinetics(I&. [ 11,121 and eq. (11)). (b) Dg # 0. DA = 0. Curve 1 corresponds to mobilereagentsand curve2 to immobdereagents.no = 5, n = 10-4, E= 105, K = 0.076, aiticzl expment 7 = 0.6 _
K = 1 + 2.1(7r:.r)4’2,
r > 102.
(12)
Applying our approach to the investigation of manyparticle effects in the case of energy transfer, A + B + B, for mobile donors, A, and large concentrations of acceptors (sinks B) cannot be successful in the framework of the standard Kirkwood superposition approximation [8], and wiu be done elsewhere. The time development of the joint correlation f;mctions is grven in fg. 3. As is seen, the initial Poisson fluctuation spectrum, X,, Y = 1 changes qualitatively in the course of reaction. (The measure of the spatial correlation of reagents is a deviation from the horizon-
3. Conclusion It should be stressed that the non-Poisson spatial correlations of reagents arise from the bimolecular reaction itself and take place even at the early (transient) stage of the reaction, but become most pronounced at long times. The effect of the reaction-induced uggregation of similar non-interacting reagents under study can lead at long times and/or large initial reagent concentrations to an appreciable deviation of reaction kinetics from predictions of standard chemical kinetics [11,12] since the latter neglects the reagent density fhrctuations [7-lo]. This aggregation effect reveals a universal character and should take place not only for the diffusioncontrolled reactions considered above but also for static reactions, e.g. for both accumulation of Frenkel defects in solids under irradiation at low temperatures [ 191 and for tunneling recombination of immobile donors and acceptors [20] _ The aggregation is due to back-coupling of spatial correlations of similar and dissimilar reagents: recombination of HISsimilar reagents (A-B) indirectly stimulates the aggre269
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117. number 3
CHEMJCAL
gation of simiIar reagents, which are supposed to be non-interactive.
Probably the aggregation effect could be experimentally detected, not through the deviation of the reation kinetics from standard chemical knetics for large reagent concentiations (eqs. (1) and (lo)), but through the appearance of the non-Poisson spectrum of density fluctuations which has, in fact, been observed earlier in the kinetics of FrenkeI defect accumulation in alkali halide crystals [19] _
References
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LETIERS
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E. Kotomin and V. Kuzgvkov, Chem. Phys. 76 (1983) 479; V. Kuzovkov and E. Kotomin. Chern. Phys. El (1983) 335. 191 V. Kuzwkov and E. Kotomin. J. Phys. Cl3 (1980) L499. and YaB. Zel’dovick. Chem. Pbys 28 [W AA. Ovwov (1978) 215 WI TR Waite, Phys Rev. 107 (1957) 463. WI V-V. Antonov-Romanovskii, Kinetics of pbotolumiues cence of crystalphosphors (Nauka, Moszw, 1966); J. Phys. USSR 6 (1942) 120. [131 EP. Ivanova, J. Luminescence 21 (1980) 373; P-V. Elyutin, J. Phys Cl7 (1984) 1867_ 1716. 1141 A. Suna. Phys. Rev. Bl(l970) Liquid 0yst. 1151 Yu. Gaididei and A. Onipko. Mol.-wst. 62 (1980) 213. [161 VXh. Brickenuhtein, VA. Bender&ii and P.G. Fmpov, Phyn stat. Sol. 177lJ (1983) 9; V.M. Agranovich. NA. Efrernov and AA. Zakhidov, In. Akad. Nauk SSSR Fiz. 44 (1980) 759. Z. Physik 843 (1981) 221; in: Exciton 1171 V.M. Kenbe, dynamics 1p molecular a-ystals and aggregates (Springer. B&in, 1982). 1181 E. Kotomin and V. Kuzovkov, Qech. J. Phys. B. to be pUbliShed_ 1191 V. Kuzovkov and E. Kotomin, J. Phys. Cl7 (1984) 2283. WI V. Kuzovkov, Soviet Khirn. Fiz., to be publisbeg