Chemical Physics Xl (19R3) 335-347. North-Hnllnnd Publishing Cnmpang. SOME PROBLEMS OF RECOMBINATION. KINETICS. II. V.N. KUZOVKOV and E.A. ...
Chemical Physics Xl (19R3) 335-347 North-Hnllnnd Publishing Cnmpang
SOME V.N.
PROBLEMS
KUZOVKOV
OF RECOMBINATION and E.A.
KINETICS.
II
KOTOMIN
Lor-iim~ S~urr Unh-ers;[v. Rain;s i9. R&u. USSR
Received
2 hlay 1983
The gcnerd phenomrnological theory ol diffusion-c~~ntroltrd dsfecr rrxomhination. in which the idcntlty crf similar and dissimilar defects are consistently taken intn account and which was presated in part I of this xc&s. is applied hcrc to ths reactions A + B -+ C (Frenkel defect recomhinatiott). A + B -+ B (energy transfer). A i A - B (cxcitutt annihdation). It is Ih-nrtntic 3~rcgrtti’s) xc that during the reaction A + B -+ C at long reaction produced thus leading IO 3 deviation from the quasi-steady rs;lcIion conswnr ~31 known in formal chcmicltl Linctics. (This confirms once more our general idea about the hack-couplin_g of similar and dissimilar rcagnt zplrtial c~vr&ttions.~ The validity range of Kirkwood‘s standard superposition approximation used IO decouple thrcs-point dcnsitis> i\ alw discuz.cd.
1. Introduction
tion
In the first part of paper [1] a new phenomenological
theory
of diffusion-controlled
defect
reac-
tions in solids and liquids was developed and presented. and put into relation with the pre-esisting similar theories (see also refs. 12-41). In paper [I] we discussed mainly its application to the socalled correlatice artrtealirtg (monomolecular stage of the A + B - C reaction) well observed for spatially correlated (e.g. Frenkel) defects at short times. In the preseni paper we shall deal with the effects of long-time reaction asymptotics (bimolecular recombination stage). According to formal chemical kinetics the reaction A + B + C (e.g. Frenkel defect recombination) obeys the phenomenological equation dn,/dt
= dn,/dt
= -iGt,nB.
where ?tA. ltB are the macroscopic (reagent) densities (concentrations) quasi-steady state reaction constant_ after a short transient one obtains from rq.
(1) point defect and K the Assuming that
period K = K, = constant (1) the asymptotic laws:
11.~a I-’ (fr, = na) and II, a exp(-K,n,r) (11.~ -=z nt,). The latter asymptotic coincides \vith that expected for the reaction A + B --+ B (energy transfer). The earlier microscopic approach to the reac0301-0104/83/0000-OOOO/SO3.00
6 1983 North-Holland
A + B -
C [5.6] based on the hierarchy of the many-point reagent densities
equations
for
confirmed
the validity
of cq. (1) thus demonstrat-
ing the c.xistence of a limiting
magnitude
lim K = K. = 0. I--X where K, = 4z_Dr,,. D is the relative diffwinn cnsfficient. r0 being the clear-cut reaction radius. Ho\\-ever the validity of eq. (1) at very long reaction time lx-as questioned bv Zcldovich and Ovchinnikov [7-91 who obtained t-x!\- asymptotic drca! laws: It:, a I- ‘.” ( )I.+ = ?t[%)and tzA a esp( - c-1’ ’ )
(ItA -sz ut3)_ In fact K(r+
x)=0
(since
it nxms that K = K(r) ivith for tt,ar-i’A one gets Ka
t - 1;‘4)The semiqualitative approach [7-91 is based on the follo\ving considerations. Initial spatial distribution of reagents is uniform onI>- on a large scale along kvith the inevitable locsl densi[v fkcttutiot~_s_ In the course of the reaction local excess of the density of one reagent. say A. against the background of the homogeneous density of reagent B has a gwuter chance to survive than the homogeneous density of A. Such aggregation of similar reagents leads to a rerfweci reaction rate. Diffusive smoothing of these fluctuations (aggregates) drtsrmines the reaction rate at long times. Fluctua-
lions of the linear size R are smoothed out during the distinctive time f,, = R’/D,.. 0,. (I’ = A, B) is the diffusion coefficient of the r*-kind defects. This permits lo assume that Im-gc aggregates of reagents (which in (he main determine the reaction asymptotics) 111ay be regarded as rhert)rol fluctuations (i.e. Poisson fluctuations of’ (he reagent number in a small selected volume). Note. however. that the dislrihulion of similar reagents at large time is 1101 a Poisson one since [he presence of Poisson fluclualions alone does 11or lead to the dependence of the reaction consIan K on time. Moreover the cmrrgencr of the quasi-steady reaction constant lini , __%A’ = A’,, in rcfs. 15.61 is ttmtc!s due to the Poisson lluctuation spectrum [4] characterizing [he uniform (random) distribution of reagent densities before the rextion. Since the reaction rate is defined by the rate of’ ihe relative diffusive reagent approach. it must be independent of the actual form of the elementary event of the reaction. which W~IS used in refs. [7-9). Some points in refs. [7-91 require clarification. (i) 11 appears that bark reagents are indirectly assumed to be mobile, D,., * 0. since if one of reagents is entirely (or close to) immobile (which is often Ihe case in solids) the estimate of the dislincLiar smoothing time rd = R’/D,. is no longer valid, although the mutual approach of the reagents before Ihc: reaction starts still remains to be diffusion-controlled. (ii) 11 seems 10 be reasonable 10 lrcal large density fluctuations as thermal (Poisson) by the themselves. However. the reaction rate at leas1 at intermediate time is determined by the fofcr! (evidently non-Poisson) spectrum of the reagent density fluctuations produced spotmmeous!r* in rhc course of the rcucriotl and its connection with (he initial (Poisson) spectrum is unknown. Since there is no agreement aboul the validity of eq. (I) at long reaction times (theories [5,6] confirm but [7-91 refute eq. (l)), it could be of interest 10 analyse our generalized kinetic equations [l-4] which are believed to be valid at all reaction times and are based on the truncated hierarchy of equations for many-point reagent densities (distribution functions (DFs)). It would permit (i) to check the results of refs. [7-91 in the framework of the standard approach employing DFs, (ii) 10 bring OUI the shortcomings of the
generally accepted approach of refs. [5.6] al long times. (iii) 10 understand the role of the simiktr reagent correlations and non-Poisson density fluctuations in the decay asymptotics. and (iv) such an analysis could provide an unique estimate at wbith reaction rale the effect acuJally occurs. It is also important in that it gives the possibility to observe experimentally a predicted effect. 2. Long-time
kinetics of the reaction A + B --, C
Let us consider diffusion-controlled which for unequal /I~,) are:
our kinetic equations for the reaction A + B + C [l-4]. reagent concentrations ()I:, =
atl,/at
= atl,,/af
= -4~r~Da~~/at-lr,r,,
a x,./at
= 2 D,.A X,.
= -a. (?a)
-
@X,. rr,tt,.rt,rli, {~~r~~Y(r’)r’
dr’.
P=A.B. (2b)
aY/al=
DAY
t-2
rO_
@I 2rr,,ttAttB
c I’ = h.B
r’)r’
dr’.
(2c)
Here the DFs X,.(r). Y(r) are thejoint (two-point) for similar and dissimilar reagents respectively and describe spatial correlations within A-A. B-B. and A-B (= B-A) pairs of reagents. Their asymptotics can be expressed through single densities: X,.(r --, ~0) = tt:. Y(r - co) = tzAttB. In eqs. (2) Y(r 6 ro) = 0 (instant annihilation), D = DA + D,,. whereas the second terms in eqs. (2b) and (2~) describe (in the Kirkwood’s superposition approximation) the many-particle correlations of the reagents. Eqs. (2) are reduced 10 the basic equations of earlier theories 15.61 provided the spatial correlations of similar reagents (A-A, B-B) are neglected (see refs. [l-4]) and this leads directly to the quasi-steady rate constant K, = 4mDro. However. such an approximation though generally accepted fuils since at long times similar reagent fluctuacietdies
tions arc produced by the reaction in t\vo ways. l%st. disappearance during the reaction of close A-B pairs of reagents leads also to the disnppcarance of reagents from otherwise ho~nogc~icous spatial distribution_ On the other hand. local c~ccss of the reagent densities (aggregates) present in the initial random distribution against ;I hackground of the l~on~ogeneous density of the second reagent hwc a greater chance of surviving at the COSt of the relilti\:ely smitll share of their particles. The emergence of the new reaction-induced fluctuation spectrum with a distinctive “drop” (aggrcgate) structure (since their diffusive smoothing 413)
is rather slow) must result in an ~‘scas of the joint densities X:. at small distances when compared to their :isymptotic values. i.e. X8. 2 elf_ Therefore even the incorporation of joint (and indirectly higher) correlations of similar reagents in eqs. (2) permits. in principle. to obtain a rww reaction asymptotic law. However. the spectrum of reagent density fluctuations is yielded by the CWIpkte set of the DFs p,-,,._ The great aggregates which according to refs. [7-91 affect reaction kinetics at very long times can formally be described by p,,.,,. with large ~z.~z’. On the other hand. applying Kirkwood’s superposition approximation with a validity range which is yet poorly studied (cf. statistical theory of liquids [lo]). the i+~i,tire hierarchy of coupled equations for the (II + II’)point densities p,,.,,. is reduced to several eqs. (2). All the same even a rough truncation of the hierarchy of equations for the DFs does WI mean neglect of the many-particle spatial correlations. II lvould be justified if these correlations could be approximated through functionals (,I -t- )I’) < 2. E.g. this is the case
of P,,.,,. \vith for the energy
transfer (A + B --, 9). where. given immobile donors (D.., = 0) and an initial Poisson distribution of 9. the superposition approximation yields an L:\-CI~Iresult [ 1 l] (see also section 4). The study of the reaction A + B --, C given below argues also that many-particle correlations of a high order are not very important. The long-time kinetics of the reaction _A+ B C in the particular case of equal concentrations and diffusion coefficients. 11:. = 11,~= )?(I 1. D.., = D!,, tl;is been briefly investigated in ref. [4] where [he asymptotic estimate 11, 0: I-‘-‘~ 17-91 is con-
firmed. Further \ve consider the problem in more detail and illustrate more rigorously the analytical estimates obtained by computer calculations of eqs. (2). Following ref. [4]_ let us employ dimensionless variables (dashes are omitted in the next variables on the left-hand side): I’ = T/T,,_ I =. T)i/J;f. A-,, = A-,./,I f. I- = l-/t~.,n~,. 0,. = 2 D,./D (a* = 1. 2). Now ‘I, = 1. D, i D,, = 2. lim~._,S,..l-= 1. Let US introduce also dimensionless concsntrurions 11,. = 4zr,:tzr. II is convenknl 10 pass from the DF l-(r) to Z(r) via the relation 1-(r)=[(r1 )/~]ZO. - 1) xvhich permits us to express the reaction rate “constant” I\’ through Z(O) = K rather than the gradient al-/‘& at r = 1. Note that the quasi-steady joint profile l- = 1 - I/r_ r 2 1 [5.6] transforms into Z = I Hnd noxv K = K(, = 1 instead of h’,, = 3z;L)r,, in ‘ausual” units. Therefore the deviation of Z from unity indicates considerable spatial correlation of dissimilar reagents resulting
(Al)
I’.
(3)
F(r) LI
= 1- $ =
0.
u=r--2.
/;(r*)r*d,‘_ - L1 r
(4a)
1.
1.
(4b)
Note 11lat eqs. (3) and (4) do ~IOIyield steadystate solutions and the correspondins quasi-steadv reaction constant K = Z(0) is time dependent. since the second terms therein play the role of sources. In terms of mathematics the lowering of the recombination rate \vith time. earlier predicted in rcfs. [7-91 has to consist in the dependence of i\ on tI provided lim ,2_oK(n)=0. To obtain this relation one has to analyse the character of the reagent spatial correlations. A closer look at cqs. (2) [or (3) and (4)] reveals that they describe ;1 positive back-coupling of similar and dissimilar densities X2. and 2; at long reaction time for an arbitrary initial distribution their deviations from the asymptotics Xv. Z = 1. caused by the reaction. have opposire signs (_\t. >
1. Z < 1) and contribute to one another. Note the singularity of Fin eq. (4a) at small r < 2 leading to I-‘=- 0. X > 1. It means that the recombination leading to a lowering in the concentration of pairs of close dissimilar reagents A-B (Y c 1 at small r. negative correlation) results in a positive correlation of similar reagents X,.. The bimolecular recombination process can be divided into two stages [3.4]. The first one leads to formation (after a transient period) of a nearly quasi-steady profile with X,.. Z = 1. The second stage of the reaction is characterized by the nonPoisson density fluctuations XV >> 1. Z 10 the formally obtained rate constant is an order of mugnitude less than K, = 45;Dr,,. The time dependence K = K(r) makes attempts at finding K, in the abovementioned way rather doubtful. Bearing in mind that K at long time behaves as a small power of t. the curve
reagent distribu-
X = Z = 1 corresponding to the K ( Y = 1 - I/r) in the generally accepted [kg]. One can conclude from fig. 1 that the “reaction rate” constant K introduced with time (or depth of reaction ~=n,-Jn) tion
ing monotonously
___---_-__-
K )r---
from the quasi-steady
= K, = 1
approach formally decreases deviat-
constant
K, = 1. Defining
the distinctive time rrl at which K decreases by a factor of three, one obtains I~ = Sr,f/D (in usual units). Therefore I,, considerably exceeds the distinctive transient time r,‘/D (cf. ref. [3]) after which the quasi-steady state is reached in the standard theory [5,6]. The less the initial concentratron or,, the greater the I*_ This effect is important in clarifying the experimental condi-
Fig. 2. The inverse reagenl concentration versus time (in units curve I - earlier theories [X6]. curve 2 - the present
r,f/D)
theory.
As is shown below. the restriction hy the asymptotic coefficient a essentially simplifies the problem. At I -+ x the earlier scale of spatial correlation R = q, characterizing an elementary event elf 111~ reaction is replaced by a new one. K a r’ ‘. ;II \vhich considerable spatial correlations. S,. >> I_ Z CC 1. are observed (see section 2.1). Due IO thi. it is convenient instead of the old variable> I and tIn these vari:thk.\ the to employ f and 5 = rr -I/‘. kinetic equations (3) at f -+ x: read as follows. FCV
II = IIT, = tlH:
1/11(t) may be approximated by a straight line over a limited tinle interval but formally the obhas nothing to do wi-ith tained “rate constant” K,, = 4nDr,, (or 1 in dimensionless units). And finally. the time development of the reaction-induced spatial correlatioru of similar and dissimilar reagents is presented in fig. 3. The measure of the correlation is a deviation from the horizontal line X. Z = 1 (the quasi-steady approximation). The emergence with time of a new scale of spatial correlation R a t’/2 B- 1 is Lvell seen. The positive correlation of similar reagents. X> 1. observed at small r. may be reasonably interpreted as their oggt-egares (dynamical clusters). One can also conclude that in the course of the reaction the initial Poisson fluctuation spectrum (X. Z = 1) is replaced by an essentially rlo,l-Poisson one. Small aggregates of similar reagents are dissolved quicker than large aggregates and this is why the latter determine the reaction rate at long times \vhich was postulated in ref. [8].
22
Attn!vticnl
Let the asymptotics
lim a In ~?,/a
r-r)
Strictly satisfies contain
In t = -a.
In I = J,z -a(
+ gaz/ag ~z,‘rz~,)z( .I-, -
1).
In deriving
sqs. (6) and (7) srarting follo\ving limiting relations obtained and (5) are used
Since the DFs _YPand Z on large scale R a I’ ’ dwvn rqs. (6) and (7) right-hand side. Besides.
of
the
long-time
(5)
speaking. not only the trivial II:, a f-” eq. (5) but also ~1.~a r-“ln I. etc.. \vhich co-factors slower than the power of I.
(721 from (3) ~hrt from e+;_ ( 1 )
change considrrably only WC could ;lt I - x xvrite \vith IWUI tm-ns on ths s~nall unimportant terms
are also omitted in rqs. (7) since and therefore the B-B corrslations
estitwtes critical exponent be defined as
az/a
)lH = constan
can bc II+ glwted. Eqs. (6) and (7) arc non-linear hut their non-linearity (due to thr last terms) enters in a rather simplt way through deviations of the DFs .Y. Z from their asymptotic values. ix. ( _Y- 1) and (Z1).
exponent is based on the proof of the scaling behaviour of the DFs at long times with the asymptotically separated variables. X,.=
-t- . . . .
1 +/,.(I)w#.([)
The function cal exponent y=,limll
(9)
/,.( I ) is characterized
h[/;.(l)]/a
1 +/(r)w(t)+
In I.
(10)
._. .
(11)
For y > 0 we have /(I) a fy lvhereas for y = 0 not only /= constant satisfies eq. (11) but also functions f( I ) lvhich are slo\ver than polvers of I. e-g. (In I )“. We assume (and prove below) that w(c) is restricted at 5 = 0 and decreases monotonously do\vn approaching zero at 5 + x. This behaviour is in agreement with the derivation of .I’,. at small I from its asgniptotics as is well seen in fig. 3. Eq. (6b) contains the large parameter iz’ = Zaf(r) B 1. Since h’ is an increasing function of f (y 2 0) the solution of eq. (6b) can be obtained from the quasi-steady state condition for eq. (6b) neglecting the small left-hand-side term i3Z/d In I -CK IJ’w( OZ. The substitution z = 5-‘u X exp( -=$‘/S) leads to the equation d’U/d I the many-particle correlations cannot be reduced due to non-Poisson density to joint densities fluctuations of the reagents A (see below). If donors are mobile. D.4 = 0, relative distances between donors and acceptors become correlated (the approach of a given donor A to any B affects its reaction with other B particles) which affects the behaviour of many-particle correlations. Now the densities p,.,,,. )I* >, 2 can no longer be reduced to the joint densities and the problem is not capable of an exact solution as before. The Kirkwood’s superposition approximation does not bear out the difference between the kinetics in the cases of immobile and mobile donors A and thus cannot confirm the result of ref. [13]. However. the fact itself that for DA = 0 there exists an exact solution coinciding with the superposition approximation. argues for the rutio DA/D as a measure of the accuracy of our theory_ When applying other methods [17.19] for large acceptor concentrations 12~ and mobile donors, DA = 0, the concentration-dependent quasi-steady rate constant (in units 4mDrO) is obtained (see also ref. [15]) l+t~lri/~+~t~~ln,~,.
(24)
Such a rate constant ously contradicts the
independent of time obviresults obtained in refs.
K,=
.i
[t;. 13. IS] and this is probably due to approsimations (e.g. the existence of the quasi-steady statr) made. The solution of the approsimatc eq. (721) cimlonstrates that at long reaction time the spatial distribution of reagents A is JKN the Poisson OIW. i.e. the donors exhibit spatial dgnamicai aggrcgation with time similar to the reaction A + B - C [3]. However. mlike the latter reaction. thcrc is IIU back-coupling betlveen the DF -Y:, and the rcaction rate [Z(O) in eq. (Zk)]. Therefore the spatial fluctuations of similar reagents do IIOI affect the kinetics of the A I B - B reaction as it does the kinetics of the A + B - C reaction and. what is more important. this divcrgence is tkwcred by the superposition approsinmtion. As a conclusion of section 3.1 it should he noted that for the A + B --, B cxccmz~rlr~rior~kinctits spatial correlations described by the DFs of [Zl]. The superposiof/-orders p ,.,,. are important tion approximation yields here a very rough result. Thus for the accumulation of immobile reagents instead of the esact kinetics d tl.,/dr
= p exp( -II
(p is the reagent volume. oC,= 4/3 the expansion of was taken by us exact one.) ? Rcucrim _3.-_
,,L+,
)
creation rate per unit time and 7ir,f) it retains only t\vo terms in the right-hand side. (This result in ref. [I I] by mistake to be an
A + A -+ B
The general formalism for describing this reaction similar to ours was presented by Suna [22] but the equations for the DFs \vere not decoupled. Recently in ref. [33] this has been done along xvith incorporation of trapping of mobile escitons. The “interval method” has also been used in ref. [34] for deriving the basic equations being based on an approximation simpler than the superposition one. Since similar reagents are involved now their spatial densities are described by the .sitI_~&~DF X( I-) only. In the superposition approsimations the basic equations analogous to eqs. (2) read
(25a)
--
as
I
r-r..
rr,,,,.: - !r-r.J
dr-‘.
_I-( r’)r’
I’ :, ,;,.
(25h)
provided S( I- < I;, ) = 0. In dimsnsionlsss units used /Q~u irr &riving qs. (3) and making the substitution .Y = I( r 1 )/r]Z(r - 1) one xrivss a1 ths set of cquatinm nil/al
= - fir1 2.
az/ar = _!lz
+ _,h-Zlf.
(XXI ) )’
2
0.
(Xh
!
where /{(t-j
(‘7
= 1 - 2(1
xvith (1 = 0 for 1’ < 1 and 1. (In the particular case H = 0 cys. (76) and (27) ZI). ix. the lasI term in cq. (26h) acts as a p~~sir~t-c*corrzlation source. thus increasing Z. Hoxvsvcr &ics If decreases as Z approaches IO iIs upper limiting magnitude 1 i l/r therms II dtxmxmx \vith time. this source of positive correlation is ~vsal; and cannot compete xvith the main diffusion WI-111 in q. (26b). ieading IO Z = 1. Therefore the fluctuation effects can only or-eiivmr this reaction_ rtxuiting in a slightI\- j~~c~.~t~_wd r:tte constmt K > K,, = but C&OI affccI its asymptotic II c: I- ’ ;::$. (26a)]. This conclusion can easily be explained in tern1.i of qualit_v. Indeed. the aggreg:tte (drop) structure is unstable for the reucrion .A -+ X - B since due IO ths only kind of interacting rexgents the rcacrion inside 1hr3 drops grrs accelerated. The homogcneous distribution appears here IO br niorr .VICJ~~
obe_vs the formal chernicr~l kinetic kt\vs. The conclusion that the fluctuation effects xc not important for the 4 P _A - B reaction is also confirmed in ref. 1251 in studying the smric- decav kinetics with dipole-dipole interactions. It is sho\vk that use of the joint densities only describe 111s kinetics accurately even for the great initial rcasent dimensionless densities II = lo_ Note ako Ihat the kinetics of Ihis reaction was used in deriving rhe clegmt formalism of diffusion-controlled reactions based on rhr zecandxx and
yuttntization
techniqtw
[9.X-31.
344
V.N. Kuzovkov, I:'.A. Kotomin /Some problem~ of recombination kineticv. !!
4. Accuracy of the superposition approximation
densities are
The accuracy and reliability of the involved approximations are the key problem of the phenomenological kinetic equations. As is well known from the analytical theory of liquids [10] dealing with problems similar to those in the stochastic theory of chemical reactions, it is impossible to determine the range of applicability of the theory in the framework of the analytical methods. It c a n n o t be done by a comparison of theory with actual experimental data since diffusion coefficients, initial reagent distribution and other parameters are not exactly known. For the analytical theory of liquids this difficulty can be reduced by comparing theory with computer simulations [ 10]. Unfortunately. in the theory of chemical reactions this cannot be done since in the course of reaction the number of particles greatly decreases. which complicates the determination of the longtime asymptotics. The computer simulations can be informative only in the 0 ---, A + B a c c u m u l a t i o n kinetics. The initial spatial correlation of reagents in the A + B---, C reaction is unknown. The joint densities X. Y analogous to the radial distribution functions in the analytical theory of liquids could, in principle, be determined by diffraction scattering methods [I0] but this involves great difficulties, however. The optical absorption method permits. e.g. to distinguish a single vacancy in a crystal (F centre) and its small aggregates consisting o f two, three and four nearest vacancies, but there is no hope of obtaining information about other manyparticle densities p,.,., n + n ' > 3 with the exception of another limiting case of very large aggregates seen in a microscope [29]. This is why the time development of the single reagent densities (concentrations) n A, n n remains the most reliable a n d available characteristics. The self-consistency of the diffusion-controlled theory could be checked by reproducing known results for the model limiting cases, e.g. monomolecular (correlative) annealing [1,6.7] and, probably, the asymptotic decay laws. If the initial distribution function within Frenkel pairs is f ( r ) [ 4 ~ r f ~ : f ( r ) r z d r = 1] and the reagent concentrations are n.~, = n B = n 0, the joint
);,.=no.
) =no+nor(r).
(28)
It is obviously affected by prolonged recombination (see refs. [3,21]) but for an instant excitation it mostly yields Y(r
