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Theory of delayed geminate recombination. Escape probability
Volume
127, number
THEORY
CHEMICAL
5
OF DELAYED
GEMINATE
PHYSICS
LETTERS
RECOMBINATION.
27 June 1986
ESCAPE
PROBABILITY
M. TACHIYA Divrsron of Basic Research, National Chemrcal Laboratory for Industry, Received
2V March
Yatabe, Ibaraki 305, Japan
1986
The total escape probability of delayed geminate recombination was first derived for a general case in which pairs of different generations have different reaction radii, different reactivity parameters, and different diffusion coefficients. Unlike the case previously studied by Mozumder and Tachiya, the total escape probability depends on scavenging for this general case.
1. Introduction Delayed geminate recombination is the geminate recombination in which the primary pair is converted into a secondary pair by reaction with an added scavenger in the course of recombination. Some years ago Mozumder and Tachiya [l] pointed out that the escape probability of geminate recombination should be independent of scavenging, if scavenging follows the thermalization of the primary pair. Based on this point and the experimental evidence [2] that addition of SF, reduces the free ion yield in neopentane, they concluded that SF, scavenges epithermal electrons in neopentane. In a previous paper [3] we studied the dynamics of delayed geminate recombination. There we pointed out that the statement by Mozumder and Tachiya is correct only for the case in which pairs of all generations have equal reaction radii and undergo totally diffusion-controlled recombinations or the case in which pairs of all generations have equal reaction radii and equal ratios of the reactivity parameter to the diffusion coefficient. For a general case the total escape probability depends on scavenging. In this paper we derive the total escape probability of delayed geminate recombination for a general case in which pairs of different generations have different reaction radii, different reactivity parameters, and different diffusion coefficients.
2. Dynamics of delayed geminate recombination The dynamics of geminate recombination of a pair i with a relative diffusion coefficient Di, a reaction radius Ri, a reactivity parameter pi, and an initial separation r. is described by the Smoluchowski equation: &v;(r, t: r,)/at &r,
= Djv [VW;@, t: q,) + j.@(r,t:
r,,) vu(r)] ,
0: ‘0) = 6(r - ‘0) )
D@v;(r,
t: r,,)/ar +/3&r,
(la) (lb)
c: Q) du(r)/d$.=Ri
=piwf(Ri, t: ro) ,
UC)
where w~(T, f: ro) dr denotes the probability that the pair will have a separation r- r t dr at time c. Throughout this paper the superscript zero denotes the absence of solutes. u(r) is the interaction potential and p = l/kT. The reciprocity theorem of Green’s function allows us to transform eq. (1) into [4]
462
0 009-2614/86/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 127, number 5
awf+,
C: royat
wF(r,0: q))
CHEMICAL PHYSICS LETTERS
= D~[v,~,w!(T,
t: ro) - p7,0u(~o) vrow;(r,
27 June 1986
(24
t: ro)l ,
= 6(r - ‘0) )
oi,[awF(r,t:
(2b)
ro)iaro],,=.i=piW~(~,t:Ri),
(2c)
where the subscript r. indicates that the differentiation is done with respect to ro. The survival propability IVF(ro, t) which is defined as the probability that the pair is still alive at time t is given by
Wi”(ro, t) = Jdr wf(r,t: ro) Integration
.
(3)
of eq. (2) over r leads to [4]
a@(r,, I$(ro,
t)iat = Di]~,2,@Y~o,
0 - 80,,u(r,)
v,e $(ro,
01 ,
(4a)
0) = 1 )
Di[a@(ro,
(4b)
t)/ar,],e=,
i ‘PiI+$r(r()9 t) .
(4c)
Assume that the primary pair consists of species P and Q. In the presence of a scavenger A which reacts with P species to yield S species there coexist in the system two kinds of pairs, viz., P and Q, S and Q, which are referred to as pairs 1 and 2, respectively. Let Wi(ro, t) denote the probability that a pair i will be found at time I, if the primary pair is initially separated by ro. Wi(ro, t) is expressed as [3] w,(%,
t) = I+(),
t, exp(-kAcAf)
,
(5)
t w2@0, f> = Jdtl 0
s&l
@(rl,
f - fl) kAcA#‘ly
cl: ‘0) exp(-kACAtl)
,
(6)
where k, is the rate constant for reaction of P with a scavenger A which has a concentration CA. The differential equation satisfied by FV2(ro, t) is derived in the following way. By transforming tion variable from t1 to t2 through t2 = t - tl, eq. (6) is rewritten as IV,@,, t) = j dr, j-drl
I&$5)
kA c A w % I’1 ,
t -
t2: ‘0)
exp[-kAcA(f
-
5)1 *
the integra-
(7)
0
of eq. (7) with respect to t yields
Differentiation
+ j dt2 0 -
jkl $(rl,
f2) kACA[a&rlF f’: ro)/at’]tt=l_-t2
CA W :( r 1 ,t-
kACA
t2:‘o)
eXP
[-kAcA(t - t2)1
eXp[-kACA(f-
tz)].
(8)
0
Substituting
eqs. (lb) and (2a) in (8), one obtains
463
h,
+ jdf2
27 June 1986
CHEMICALPHYSICS LETTERS
Volume 127, number 5
@(rJ
> f2)
kAC#l
-
[v,2,
v,,v(rO)
vrol
W@j>
t -
t2:
‘0)
exP[-kACA(t
-
t2)1
0
- kACA j- dt, ldrl
w”( 2 ‘1, f2)
kACAWy(‘17
t -
t2:
‘0)
exP[--kAcA(t
- r2)1
.
(9)
0
Eq. (9) is rewritten as follows by changing the order of integration dw,(rO, t)/at = kAcA@(lO,
+
(olIv,2, - V,,u(Q)
and differentiation
in the second term on rhs:
r)
vro] -kACA)
jdf,
j-d@'%'19
f2)
0
x k&&(‘1,
t - t2:
‘0) eXp[-k,&(t
(10)
- t2)] .
With the aid of eq. (7), one obtains from eq. (10) the following differential aw,(Q,
t)/at =D, [0,2,w2(‘0, t) - 0,,~(~o)v,,~2(~09
equation
t)] - k,C,W,(r,,
for W2(ro, t):
t) .
t) + k,C,@(Q,
(11)
On the other hand, one obtains from eq. (6) the following initial condition: W2(‘0, 0) = 0 . The boundary r. yields
condition
(12) on W2(ro, t) is derived in the following way. Differentiation
of eq. (6) with respect to
t a&&p Substitution
t)b,
= j. dt, 1 dr, @(rl, 0 of eq. (2~) in (13) leads to
D1 PW,9 f)l~~olr,=R =pl j dt,
t - tl) kAcA [a+‘,
s
dr, W$(rJ,t0 With the aid of eq. (6), one obtains from eq: (14) 1
Dl[a14’2(‘0,
, tl: r,)/ar,]
exP(--kAcAtL)
.
fJ) kACAW~('l,tl:Rl)eXp(-kACAt1).
O/~~olro=~l=@‘,(4,0.
(13)
(14)
(15)
Eqs. (1 l), (12) and (15) provide the differential equation and the associated initial and boundary conditions satisfied by W2(ro, t). In principle one can obtain W,(r,, t) by solving these equations. However, the inhomogeneous equation (11) is in general difficult to solve. In section 3 we consider the long-time limit of W,(r,, t), hz., the escape probability.
3. Escape probability Let Qi(ro) denote the probability that the escape from recombination is initially separated by ‘0. pi is given by pi
464
= lim Wi(rO, t) . t--r-
will occur via a pair i, if the primary pair
(16)
27June1986
CHEMICAL PHYSICS LETTERS
Volume 127, number 5
With the aid of the final-value theorem of Laplace transformations, the probability of escape via a pair 2 is given by a#~)
(17)
= FiO sc2(r0, s) ,
where Gz(rO, s) denotes the Laplace transform of W2(r 0, t). The Laplace transformation of eqs. (11) and (15) yields s&(Q,
s) = D, [F$,@&,,
s) - V,,u(Q
V,0~2(~0, s)] - k,~,&(‘~,
s) +kAcAG!(‘~, s)
>
(184 (18b)
D1[~~~(‘o,s)/ar,l,o=~l =P&(R~J). With the aid of eq. (17), eq. (18) leads to
D, [=‘~(‘o)/~‘&,=R
1
(19b)
=P1@‘2@1) 3
where @p(r,) E Iim,,, @(rO, t) denotes the escape probability of a pair i in the absence of scavengers. The probability of escape via a pair 2 can be obtained by solving eq. (19). Since the probability of escape via a pair 1 is zero from eq. (S), the total escape probability at(ro) of delayed geminate recombination is given by @&rO)= @‘2(Q). 3.1. Pairs of neutral species If the pairs involved are those of neutral species, one can put v(ro) E 0 to a first approximation. In this case the escape probability of a pair i in the absence of scavengers is expressed as [4]
@f(I,>=
1 - R/jr0 + Qi l+Qi
’
where Qi = Di/p$i. TWOlinearly independent solutions to the homogeneous part of eq. -(19a) with o(rg) E 0 are li2ro]. On the omer hand, eq. (20) with i = 2 given by (l/ro) exp[(kACA/D#‘2Q] and Ulro) exp[-(kAc@l) is a special solution to eq. (19a). Therefore, a general solution to eq. (1 Oa) is given by @2(‘0) = (A/qJ
1 -4,/r,
exP[@AC&)1’2r,]
+ (B/r,) exp[-(kAcA/D1)1’2~0]
+
l+Q2
+ Qz ’
(21)
where A and B are arbitrary constants. Since limrO_,, Q2(ro) < 1, one obtains A = 0. Determination of B through eq. (19b) yields @*C’(j) = @2(‘(J) =
t
&/&)U
+ Q1) - (I+ Q2)
(1 + Q2N1 + Ql
+ (k,@,)1’2/Pl
1 -R2Jro fQ2 1+Q2
(R,/rO)exP[-(kACA/D,)1’2(r
1
0 -&)I
(22)
’
Eq. (22) gives the total escape probability of delayed geminate recombination of neutral species. Several hmiting forms of eq. (22) may be interesting. In the limits of kAcA --, 0 and kACA + 00, it reduces, as expected, to
@,(ro)=
1 -RI/r0 + Q1 1 +Ql
= @y(ro> ,
askAcA -+O,
@,O$=
27 June 1986
CHEMICAL PHYSICSLETTERS
Volume127, number5
1-R2h+Qz=@;('a), askAcA+mq 1+Q2 =p~D2,0neobtains
Ontheotherhand,eitherifR1=R2andpl=p2=””orifR1=R2andpl/Dl
l -Rlh +Ql =(a(l(ro)
@*o))> =
.
(24)
l+Ql
That is, for these cases the total escape probability is independent of scavenging, as already pointed out 131.
If the pairs involved are those of positive and negative ions with a charge ze and -z’e, respectively, one has u(ru) 5 -zz’e2/ero where e is the dielectric constant of the solvent. In this case the escape probability of a pair i in the absence of scavengers is expressed as [4]
exp(-r,lr0)- 4
@‘p(ro)= -
I-Si
(25) ’
where Si = (1 - Dir,,piRF) eXp(+JRi) and rC = zz’e2/ekT. Two linearly independent solutions to the homogeneous part of eq. (19a) with u(ro) = -zz’e2/erg are given by -'I2 '0
exp(-r~~~o~y~~~~/r~: ~ACA) and
rili2
eXp(-r~~~o)Y2~~~~~:
kACA) ,
where Yi(r : s) is given in the literature [ 5J, The former solution diverges as r. + w and has to be discarded. On the other hand, eq. (25) with i = 2 is a special solution to eq. (19a). The same procedure as described in section 3.1 allows us to obtain S2-%
@, 00) = +2(ro) = (I -&)]I
+ +QIO -r~/R1)-(2Dl/Pl'c)y12(uzt/r~: kACA)/Y2(2R,/r,:kACA)]
X CR1/r#/2exP[-- .$r#/ru - WQ)ly~P~/~c: kACA)/Y2(2Rl/re:
~ACA)
q+-r&~)-~2 t
(26) I-S,
'
where y;(r, s) denotes
the derivative of y2(r,s)with respect to r. Eq. (26) gives the total escape probability of delayed geminate recombination of ions. Either if R, = R2 andpI =p2 = m or if R, = R2 and pz/D1 =pz/Dq, it reduces to @,(ru) = @f(ro), indicating that the total escape probability is independent of scavenging. Eq. (26) may be made more transparent by rewriting it in terms of the recombination probab~ity Ki(ru: kAcA) which is defined as the probability that a pair i with an initial separation r. will recombine against scavenging in the presence of a scavenger A and is given by [4] (RJru)1/2 exp[- &(l/ru
-V-$)1 Y2(2r&:
Ki(ro’ kACA)=I t $Qf(l - r,.JRi)- (~i/pir~)y~(~j/r~:
kACA)/Y:!(%%kACA) kftCA)/_Y2(2Ri/rc:
k,CA)
*
(27)
The result is given by Wo) 466
= @?(ru) + W(r0)
-@@u>] [I - KI(ru: kACA)/KL(‘u: o)] .
(28)
Volume 127, number 5
27 June 1986
CHEMICAL PHYSICS LETTERS
Since K1(ru: kACA) vanishes in the limit of kAcA + m, eq. (28) reduces to Q&u) = @(ru) in this limit. On the other hand, in the limit Of kACA --f 0 one obtains a&-u) = Qy(r,$. Eq. (28) also indicates that if the escape probability of a pair 2 is equal to that of a pair 1, the total escape probability is independent of scavenging.
4. Numerical results and discussion In the Onsager model [6] in which the reaction radius is set equal to zero the escape probability of an ion pair depends only on the initial separation. However, for a general case in which both the reaction radius and the reactivity parameter are finite the escape probability depends not only on the initial separation but on the reaction radius, the reactivity parameter, and the diffusion coefficient (see eq. (25)). It has been shown experimentally [7,8] that certain pairs of positive and negative ions have extremely low reactivity parameters and therefore have very high escape probabilities compared with Onsager theory. In delayed geminate recombination, if the primary pair is converted into a pair with a low reactivity parameter, the total escape probability is expected to be increased by scavenging. Figs. 1 and 2 show the total escape probability of delayed geminate recombination of ions as a function of scavenger concentration for initial separations r&e = 0.5 and 0.25, respectively. The curves were calculated on the basis of eq. (28). In both figures the numbers attached to the curves denote the assumed values of the escape probability of the secondary pair with the same initial separation as that of the primary pair. The total escape probability increases or decreases with an increase in the scavenger concentration, depending on the magnitude of the escape probability of the secondary pair as compared with that of the primary pair. So far a few experimental studies have been published on the effect of scavengers on the free ion yield [2,9]. In these studies neopentane was used as a solvent and electron scavengers were added. In all examples studied addition of scavengers reduced the total free ion yield. As already stated, this result was interpreted in terms of epithermal electron scavenging, on the assumption that if electrons are scavenged after thermalization, the total free ion yield is independent of scavenging. Let us consider the validity of the above assumption. In the case of non-polar liquid hydrocarbons the Onsager lengthr, is a few hundreds A at room temperature. On the other hand, the reaction radius Ri is usually several A. Therefore, unless the reactivity parameter pi is unrealistically low, Si in eq. (25) practically vanishes because of
1.0 -
rdk -0.25 0.8
1.0
0.0
2
a
0.6 0.6
0.4.
Fig. 1. The total escape probability @t(ro) of delayed geminate recombination of ions as a function of scavenger concentration kAcA for re/rc = 0.5. The reaction radius R 1 of the primary pair is assumed to be R 1 = 0. The unit of time is ryD 1. The numbers attached to the curves denote the values of ai(
0.4
k,
k,
Fig. 2. Total escape probability @t(ro) for re/rc = 0.25. Same asfii. 1.
467
Volume 127, number 5
CHEMICAL PHYSICS LETTERS
27June1986
the factor exp(-rJRi), and the escape probability @p(ru) depends only on the initial separation. Then in eq. (28) the difference G$(~,) - cPr)(ru)vanishes and one comes to the conclusion that the total free ion yield is independent of scavenging. That is, the above assumption is practically valid for non-polar liquid hydrocarbons, although strictly speaking it is not correct. So the previous interpretation of the reduction in the total free ion yield seems still rational in the light of the present theory. In the case of polar solvents the Onsager length is only several times as large as the reaction radius. Therefore, Sj in eq. (28) is not negligibly small for realistic values of pi, and the escape probability depends significantly on the reaction radius, the reactivity parameter, and the diffusion coefficient. Then it is very likely that in eq. (28) @(mu) differs appreciably from @(ru) and that as a result the total free ion yield depends on scavenging. The present theory may be of some relevance to photoinduced charge separation which has recently received much attention in connection with solar energy conversion. Photoinduced charge separation occurs in the following way [lo]. When donors or acceptors are excited with light, electron transfer occurs from donors to acceptors, and pairs of a cation and an anion are produced. Part of pairs become free ions and contribute to charge separation, but a considerable part of them undergoes geminate recombination. Thus the efficiency of charge separation is suppressed by geminate recombination. If the pair has a low reactivity parameter, the charge separation should occur with a high efficiency. This has been demonstrated experimentally by Ohno et al. [S] . According to the present theory, even if the primary pair has a high reactivity parameter, one may enhance the charge separation efficiency by converting the primary pair into a secondary pair with a low reactivity parameter through scavenging. However, this enhancement by scavenging is possible only in polar solvents. In non-polar solvents the efficiency of charge separation should be practically independent of scavenging, as already pointed out.
5. Concluding remarks
We have derived for the first time the total escape probabilities of delayed geminate recombination of neutral species and of ions for a general case in which pairs of different generations have different reaction radii, different reactivity parameters, and different diffusion coefficients. It has been shown that the total escape probability generally depends on scavenging. However, we have concluded that in the case of non-polar solvents the total escape probability of ions should be practically independent of scavenging because of the large Onsager length compared with the reaction radius. In the case of polar solvents it is very likely that the total escape probability of ions depends appreciably on scavenging.
References [l] A. Mozumder and M. Tachiya, J. Chem. Phys. 62 (1975) 979. [2] W.F. Schmidt, Radiat. Res. 42 (1970) 73. [ 31 M. Tachiya, J. Chem. Phys., to be published. (41 H. Sano and M. Tachiya, J. Chem. Phys. 71 (1979) 1276. [5] K.M. Hong and J. Noolandi, J. Chem. Phys. 68 (1978) 5163. [6] L. Onsager, Phys. Rev. 54 (1938) 554. [7] J. Belloni, F. Billiau, P. Cordier, J. Delaire, M.O. Delcourt and M. Magat, Faraday Discussions Chem. Sot. 63 (1977) 55. [8] T. Ohno, S. Kato, A. Yamada and T. Tanno, J. Phys. Chem. 87 (1983) 775. [9] W.F. Schmidt and A.O. AIlen, J. Chem. Phys. 52 (1970) 2345. [lo] D. Mauzerali and S.G. Ballard, Ann. Rev. Phys. Chem. 33 (1982) 377.
468
127, number
THEORY
CHEMICAL
5
OF DELAYED
GEMINATE
PHYSICS
LETTERS
RECOMBINATION.
27 June 1986
ESCAPE
PROBABILITY
M. TACHIYA Divrsron of Basic Research, National Chemrcal Laboratory for Industry, Received
2V March
Yatabe, Ibaraki 305, Japan
1986
The total escape probability of delayed geminate recombination was first derived for a general case in which pairs of different generations have different reaction radii, different reactivity parameters, and different diffusion coefficients. Unlike the case previously studied by Mozumder and Tachiya, the total escape probability depends on scavenging for this general case.
1. Introduction Delayed geminate recombination is the geminate recombination in which the primary pair is converted into a secondary pair by reaction with an added scavenger in the course of recombination. Some years ago Mozumder and Tachiya [l] pointed out that the escape probability of geminate recombination should be independent of scavenging, if scavenging follows the thermalization of the primary pair. Based on this point and the experimental evidence [2] that addition of SF, reduces the free ion yield in neopentane, they concluded that SF, scavenges epithermal electrons in neopentane. In a previous paper [3] we studied the dynamics of delayed geminate recombination. There we pointed out that the statement by Mozumder and Tachiya is correct only for the case in which pairs of all generations have equal reaction radii and undergo totally diffusion-controlled recombinations or the case in which pairs of all generations have equal reaction radii and equal ratios of the reactivity parameter to the diffusion coefficient. For a general case the total escape probability depends on scavenging. In this paper we derive the total escape probability of delayed geminate recombination for a general case in which pairs of different generations have different reaction radii, different reactivity parameters, and different diffusion coefficients.
2. Dynamics of delayed geminate recombination The dynamics of geminate recombination of a pair i with a relative diffusion coefficient Di, a reaction radius Ri, a reactivity parameter pi, and an initial separation r. is described by the Smoluchowski equation: &v;(r, t: r,)/at &r,
= Djv [VW;@, t: q,) + j.@(r,t:
r,,) vu(r)] ,
0: ‘0) = 6(r - ‘0) )
D@v;(r,
t: r,,)/ar +/3&r,
(la) (lb)
c: Q) du(r)/d$.=Ri
=piwf(Ri, t: ro) ,
UC)
where w~(T, f: ro) dr denotes the probability that the pair will have a separation r- r t dr at time c. Throughout this paper the superscript zero denotes the absence of solutes. u(r) is the interaction potential and p = l/kT. The reciprocity theorem of Green’s function allows us to transform eq. (1) into [4]
462
0 009-2614/86/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 127, number 5
awf+,
C: royat
wF(r,0: q))
CHEMICAL PHYSICS LETTERS
= D~[v,~,w!(T,
t: ro) - p7,0u(~o) vrow;(r,
27 June 1986
(24
t: ro)l ,
= 6(r - ‘0) )
oi,[awF(r,t:
(2b)
ro)iaro],,=.i=piW~(~,t:Ri),
(2c)
where the subscript r. indicates that the differentiation is done with respect to ro. The survival propability IVF(ro, t) which is defined as the probability that the pair is still alive at time t is given by
Wi”(ro, t) = Jdr wf(r,t: ro) Integration
.
(3)
of eq. (2) over r leads to [4]
a@(r,, I$(ro,
t)iat = Di]~,2,@Y~o,
0 - 80,,u(r,)
v,e $(ro,
01 ,
(4a)
0) = 1 )
Di[a@(ro,
(4b)
t)/ar,],e=,
i ‘PiI+$r(r()9 t) .
(4c)
Assume that the primary pair consists of species P and Q. In the presence of a scavenger A which reacts with P species to yield S species there coexist in the system two kinds of pairs, viz., P and Q, S and Q, which are referred to as pairs 1 and 2, respectively. Let Wi(ro, t) denote the probability that a pair i will be found at time I, if the primary pair is initially separated by ro. Wi(ro, t) is expressed as [3] w,(%,
t) = I+(),
t, exp(-kAcAf)
,
(5)
t w2@0, f> = Jdtl 0
s&l
@(rl,
f - fl) kAcA#‘ly
cl: ‘0) exp(-kACAtl)
,
(6)
where k, is the rate constant for reaction of P with a scavenger A which has a concentration CA. The differential equation satisfied by FV2(ro, t) is derived in the following way. By transforming tion variable from t1 to t2 through t2 = t - tl, eq. (6) is rewritten as IV,@,, t) = j dr, j-drl
I&$5)
kA c A w % I’1 ,
t -
t2: ‘0)
exp[-kAcA(f
-
5)1 *
the integra-
(7)
0
of eq. (7) with respect to t yields
Differentiation
+ j dt2 0 -
jkl $(rl,
f2) kACA[a&rlF f’: ro)/at’]tt=l_-t2
CA W :( r 1 ,t-
kACA
t2:‘o)
eXP
[-kAcA(t - t2)1
eXp[-kACA(f-
tz)].
(8)
0
Substituting
eqs. (lb) and (2a) in (8), one obtains
463
h,
+ jdf2
27 June 1986
CHEMICALPHYSICS LETTERS
Volume 127, number 5
@(rJ
> f2)
kAC#l
-
[v,2,
v,,v(rO)
vrol
W@j>
t -
t2:
‘0)
exP[-kACA(t
-
t2)1
0
- kACA j- dt, ldrl
w”( 2 ‘1, f2)
kACAWy(‘17
t -
t2:
‘0)
exP[--kAcA(t
- r2)1
.
(9)
0
Eq. (9) is rewritten as follows by changing the order of integration dw,(rO, t)/at = kAcA@(lO,
+
(olIv,2, - V,,u(Q)
and differentiation
in the second term on rhs:
r)
vro] -kACA)
jdf,
j-d@'%'19
f2)
0
x k&&(‘1,
t - t2:
‘0) eXp[-k,&(t
(10)
- t2)] .
With the aid of eq. (7), one obtains from eq. (10) the following differential aw,(Q,
t)/at =D, [0,2,w2(‘0, t) - 0,,~(~o)v,,~2(~09
equation
t)] - k,C,W,(r,,
for W2(ro, t):
t) .
t) + k,C,@(Q,
(11)
On the other hand, one obtains from eq. (6) the following initial condition: W2(‘0, 0) = 0 . The boundary r. yields
condition
(12) on W2(ro, t) is derived in the following way. Differentiation
of eq. (6) with respect to
t a&&p Substitution
t)b,
= j. dt, 1 dr, @(rl, 0 of eq. (2~) in (13) leads to
D1 PW,9 f)l~~olr,=R =pl j dt,
t - tl) kAcA [a+‘,
s
dr, W$(rJ,t0 With the aid of eq. (6), one obtains from eq: (14) 1
Dl[a14’2(‘0,
, tl: r,)/ar,]
exP(--kAcAtL)
.
fJ) kACAW~('l,tl:Rl)eXp(-kACAt1).
O/~~olro=~l=@‘,(4,0.
(13)
(14)
(15)
Eqs. (1 l), (12) and (15) provide the differential equation and the associated initial and boundary conditions satisfied by W2(ro, t). In principle one can obtain W,(r,, t) by solving these equations. However, the inhomogeneous equation (11) is in general difficult to solve. In section 3 we consider the long-time limit of W,(r,, t), hz., the escape probability.
3. Escape probability Let Qi(ro) denote the probability that the escape from recombination is initially separated by ‘0. pi is given by pi
464
= lim Wi(rO, t) . t--r-
will occur via a pair i, if the primary pair
(16)
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With the aid of the final-value theorem of Laplace transformations, the probability of escape via a pair 2 is given by a#~)
(17)
= FiO sc2(r0, s) ,
where Gz(rO, s) denotes the Laplace transform of W2(r 0, t). The Laplace transformation of eqs. (11) and (15) yields s&(Q,
s) = D, [F$,@&,,
s) - V,,u(Q
V,0~2(~0, s)] - k,~,&(‘~,
s) +kAcAG!(‘~, s)
>
(184 (18b)
D1[~~~(‘o,s)/ar,l,o=~l =P&(R~J). With the aid of eq. (17), eq. (18) leads to
D, [=‘~(‘o)/~‘&,=R
1
(19b)
=P1@‘2@1) 3
where @p(r,) E Iim,,, @(rO, t) denotes the escape probability of a pair i in the absence of scavengers. The probability of escape via a pair 2 can be obtained by solving eq. (19). Since the probability of escape via a pair 1 is zero from eq. (S), the total escape probability at(ro) of delayed geminate recombination is given by @&rO)= @‘2(Q). 3.1. Pairs of neutral species If the pairs involved are those of neutral species, one can put v(ro) E 0 to a first approximation. In this case the escape probability of a pair i in the absence of scavengers is expressed as [4]
@f(I,>=
1 - R/jr0 + Qi l+Qi
’
where Qi = Di/p$i. TWOlinearly independent solutions to the homogeneous part of eq. -(19a) with o(rg) E 0 are li2ro]. On the omer hand, eq. (20) with i = 2 given by (l/ro) exp[(kACA/D#‘2Q] and Ulro) exp[-(kAc@l) is a special solution to eq. (19a). Therefore, a general solution to eq. (1 Oa) is given by @2(‘0) = (A/qJ
1 -4,/r,
exP[@AC&)1’2r,]
+ (B/r,) exp[-(kAcA/D1)1’2~0]
+
l+Q2
+ Qz ’
(21)
where A and B are arbitrary constants. Since limrO_,, Q2(ro) < 1, one obtains A = 0. Determination of B through eq. (19b) yields @*C’(j) = @2(‘(J) =
t
&/&)U
+ Q1) - (I+ Q2)
(1 + Q2N1 + Ql
+ (k,@,)1’2/Pl
1 -R2Jro fQ2 1+Q2
(R,/rO)exP[-(kACA/D,)1’2(r
1
0 -&)I
(22)
’
Eq. (22) gives the total escape probability of delayed geminate recombination of neutral species. Several hmiting forms of eq. (22) may be interesting. In the limits of kAcA --, 0 and kACA + 00, it reduces, as expected, to
@,(ro)=
1 -RI/r0 + Q1 1 +Ql
= @y(ro> ,
askAcA -+O,
@,O$=
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1-R2h+Qz=@;('a), askAcA+mq 1+Q2 =p~D2,0neobtains
Ontheotherhand,eitherifR1=R2andpl=p2=””orifR1=R2andpl/Dl
l -Rlh +Ql =(a(l(ro)
@*o))> =
.
(24)
l+Ql
That is, for these cases the total escape probability is independent of scavenging, as already pointed out 131.
If the pairs involved are those of positive and negative ions with a charge ze and -z’e, respectively, one has u(ru) 5 -zz’e2/ero where e is the dielectric constant of the solvent. In this case the escape probability of a pair i in the absence of scavengers is expressed as [4]
exp(-r,lr0)- 4
@‘p(ro)= -
I-Si
(25) ’
where Si = (1 - Dir,,piRF) eXp(+JRi) and rC = zz’e2/ekT. Two linearly independent solutions to the homogeneous part of eq. (19a) with u(ro) = -zz’e2/erg are given by -'I2 '0
exp(-r~~~o~y~~~~/r~: ~ACA) and
rili2
eXp(-r~~~o)Y2~~~~~:
kACA) ,
where Yi(r : s) is given in the literature [ 5J, The former solution diverges as r. + w and has to be discarded. On the other hand, eq. (25) with i = 2 is a special solution to eq. (19a). The same procedure as described in section 3.1 allows us to obtain S2-%
@, 00) = +2(ro) = (I -&)]I
+ +QIO -r~/R1)-(2Dl/Pl'c)y12(uzt/r~: kACA)/Y2(2R,/r,:kACA)]
X CR1/r#/2exP[-- .$r#/ru - WQ)ly~P~/~c: kACA)/Y2(2Rl/re:
~ACA)
q+-r&~)-~2 t
(26) I-S,
'
where y;(r, s) denotes
the derivative of y2(r,s)with respect to r. Eq. (26) gives the total escape probability of delayed geminate recombination of ions. Either if R, = R2 andpI =p2 = m or if R, = R2 and pz/D1 =pz/Dq, it reduces to @,(ru) = @f(ro), indicating that the total escape probability is independent of scavenging. Eq. (26) may be made more transparent by rewriting it in terms of the recombination probab~ity Ki(ru: kAcA) which is defined as the probability that a pair i with an initial separation r. will recombine against scavenging in the presence of a scavenger A and is given by [4] (RJru)1/2 exp[- &(l/ru
-V-$)1 Y2(2r&:
Ki(ro’ kACA)=I t $Qf(l - r,.JRi)- (~i/pir~)y~(~j/r~:
kACA)/Y:!(%%kACA) kftCA)/_Y2(2Ri/rc:
k,CA)
*
(27)
The result is given by Wo) 466
= @?(ru) + W(r0)
-@@u>] [I - KI(ru: kACA)/KL(‘u: o)] .
(28)
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Since K1(ru: kACA) vanishes in the limit of kAcA + m, eq. (28) reduces to Q&u) = @(ru) in this limit. On the other hand, in the limit Of kACA --f 0 one obtains a&-u) = Qy(r,$. Eq. (28) also indicates that if the escape probability of a pair 2 is equal to that of a pair 1, the total escape probability is independent of scavenging.
4. Numerical results and discussion In the Onsager model [6] in which the reaction radius is set equal to zero the escape probability of an ion pair depends only on the initial separation. However, for a general case in which both the reaction radius and the reactivity parameter are finite the escape probability depends not only on the initial separation but on the reaction radius, the reactivity parameter, and the diffusion coefficient (see eq. (25)). It has been shown experimentally [7,8] that certain pairs of positive and negative ions have extremely low reactivity parameters and therefore have very high escape probabilities compared with Onsager theory. In delayed geminate recombination, if the primary pair is converted into a pair with a low reactivity parameter, the total escape probability is expected to be increased by scavenging. Figs. 1 and 2 show the total escape probability of delayed geminate recombination of ions as a function of scavenger concentration for initial separations r&e = 0.5 and 0.25, respectively. The curves were calculated on the basis of eq. (28). In both figures the numbers attached to the curves denote the assumed values of the escape probability of the secondary pair with the same initial separation as that of the primary pair. The total escape probability increases or decreases with an increase in the scavenger concentration, depending on the magnitude of the escape probability of the secondary pair as compared with that of the primary pair. So far a few experimental studies have been published on the effect of scavengers on the free ion yield [2,9]. In these studies neopentane was used as a solvent and electron scavengers were added. In all examples studied addition of scavengers reduced the total free ion yield. As already stated, this result was interpreted in terms of epithermal electron scavenging, on the assumption that if electrons are scavenged after thermalization, the total free ion yield is independent of scavenging. Let us consider the validity of the above assumption. In the case of non-polar liquid hydrocarbons the Onsager lengthr, is a few hundreds A at room temperature. On the other hand, the reaction radius Ri is usually several A. Therefore, unless the reactivity parameter pi is unrealistically low, Si in eq. (25) practically vanishes because of
1.0 -
rdk -0.25 0.8
1.0
0.0
2
a
0.6 0.6
0.4.
Fig. 1. The total escape probability @t(ro) of delayed geminate recombination of ions as a function of scavenger concentration kAcA for re/rc = 0.5. The reaction radius R 1 of the primary pair is assumed to be R 1 = 0. The unit of time is ryD 1. The numbers attached to the curves denote the values of ai(
0.4
k,
k,
Fig. 2. Total escape probability @t(ro) for re/rc = 0.25. Same asfii. 1.
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the factor exp(-rJRi), and the escape probability @p(ru) depends only on the initial separation. Then in eq. (28) the difference G$(~,) - cPr)(ru)vanishes and one comes to the conclusion that the total free ion yield is independent of scavenging. That is, the above assumption is practically valid for non-polar liquid hydrocarbons, although strictly speaking it is not correct. So the previous interpretation of the reduction in the total free ion yield seems still rational in the light of the present theory. In the case of polar solvents the Onsager length is only several times as large as the reaction radius. Therefore, Sj in eq. (28) is not negligibly small for realistic values of pi, and the escape probability depends significantly on the reaction radius, the reactivity parameter, and the diffusion coefficient. Then it is very likely that in eq. (28) @(mu) differs appreciably from @(ru) and that as a result the total free ion yield depends on scavenging. The present theory may be of some relevance to photoinduced charge separation which has recently received much attention in connection with solar energy conversion. Photoinduced charge separation occurs in the following way [lo]. When donors or acceptors are excited with light, electron transfer occurs from donors to acceptors, and pairs of a cation and an anion are produced. Part of pairs become free ions and contribute to charge separation, but a considerable part of them undergoes geminate recombination. Thus the efficiency of charge separation is suppressed by geminate recombination. If the pair has a low reactivity parameter, the charge separation should occur with a high efficiency. This has been demonstrated experimentally by Ohno et al. [S] . According to the present theory, even if the primary pair has a high reactivity parameter, one may enhance the charge separation efficiency by converting the primary pair into a secondary pair with a low reactivity parameter through scavenging. However, this enhancement by scavenging is possible only in polar solvents. In non-polar solvents the efficiency of charge separation should be practically independent of scavenging, as already pointed out.
5. Concluding remarks
We have derived for the first time the total escape probabilities of delayed geminate recombination of neutral species and of ions for a general case in which pairs of different generations have different reaction radii, different reactivity parameters, and different diffusion coefficients. It has been shown that the total escape probability generally depends on scavenging. However, we have concluded that in the case of non-polar solvents the total escape probability of ions should be practically independent of scavenging because of the large Onsager length compared with the reaction radius. In the case of polar solvents it is very likely that the total escape probability of ions depends appreciably on scavenging.
References [l] A. Mozumder and M. Tachiya, J. Chem. Phys. 62 (1975) 979. [2] W.F. Schmidt, Radiat. Res. 42 (1970) 73. [ 31 M. Tachiya, J. Chem. Phys., to be published. (41 H. Sano and M. Tachiya, J. Chem. Phys. 71 (1979) 1276. [5] K.M. Hong and J. Noolandi, J. Chem. Phys. 68 (1978) 5163. [6] L. Onsager, Phys. Rev. 54 (1938) 554. [7] J. Belloni, F. Billiau, P. Cordier, J. Delaire, M.O. Delcourt and M. Magat, Faraday Discussions Chem. Sot. 63 (1977) 55. [8] T. Ohno, S. Kato, A. Yamada and T. Tanno, J. Phys. Chem. 87 (1983) 775. [9] W.F. Schmidt and A.O. AIlen, J. Chem. Phys. 52 (1970) 2345. [lo] D. Mauzerali and S.G. Ballard, Ann. Rev. Phys. Chem. 33 (1982) 377.
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