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Dispersive recombination in condensed phases
Volume 158, number
CHEMICAL
5
DISPERSIVE Andrzej
RECOMBINATION
16 June 1989
PHYSICS LETTERS
IN CONDENSED
PHASES
PLONKA
Institute ofApplied Radiation
Chemistry,
Techmcai Universily ofLodz,
Wroblewskiego
15, 93-590 Lodz, Poland
Yurii A. BERLIN and Nikolai I. CHEKUNAEV Ilzstituteof Chemical Physics. USSR Academy of Sciences, Kosysina 4, 1 I7977 Moscow, USSR Received 2 February
1989; in final form 24 March 1989
Two interpretations of the parameter dent reaction rate coefficient k(!)ar”-‘,
(Yare presented for a second-order equal-concentration kinetic equation with time-depen0 -Ca< I The first, valid for any system, is in terms of the activation energy distribution
for recombination and follows from the time dependence of the activation energy implied by the form of k(t). The second, valid for systems in which the time dependence of k(t) results from a distribution of reaction rates, is analogous to the interpretation of stretched exponentials as a superposition of exponential decays.
In condensed media numerous radiation-produced species were found [ I] to recombine according to a second-order equal-concentration kinetic equation with a time-dependent reaction rate coeflicient ) o-Car<1
k(t)=&+ i.e. according
)
(1)
“well stirred reactor”. It is possible to obtain a phenomenological interpretation of the parameter Q(following the idea of Hamill [4] that the form ( 1) of the time dependence of the reaction rate coefficient implies a particular time dependence of the activation energy for reaction. Indeed, applying the Arrhenius relation to k(i) one gets [ 5 ]
to
E,+(l--a)RTln(l/~,)
c/co=(l+c~Bta/a)-’
RT
(2)
or
(5)
in which
C/Ca=[l+(t/Zo)“]-’
(3)
for B=a/cor$. For longer
times
relation
(3) reduces to the power-
law time dependence CICOK(tl%-a
(4)
commonly observed [ 2 ] for carrier recombination in amorphous systems. In the modelling of carrier recombination dynamics through random walks, different aspects of disorder were considered [ 3 ] : fractals exemplified the spatial disorder, continuous-time processes the temporal and ultrametric structures the energetic disorder. The dispersion parameter cy was shown to account for deviations from classical kinctics with the implicit underlying assumption of a 380
E=&
+ (1 -a)R7ln(t/-r,)
(6)
has the physical meaning of a time-dependent activation energy. The effective activation energy E. for the second-order kinetics (3) can be obtained by applying the Arrhenius relation to c,r,; s denotes the preexponential factor. In classical kinetics, Q= 1, the activation energy remains constant during the whole reaction. In dispersive kinetics, 0 < a < 1, the activation energy increases with time, i.e. with reaction progress measured by F(f)=1
-c/co.
(7)
Using relation (6) one obtains from F(t) with c/c0 given by eq. (3) the activation energy distri-
0 009-2614/89/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
B.V.
Volume 158, number 5
bution function
CHEMICAL
F(E)
for recombination
[5 ]
F(E)=]-[l+exp(E)]-’ with t=cu(E-Eo)/( given by
PHYSICS LETTERS
(8) 1 -cr)RT.
Its density g(E)
is
(9)
1 cd f(r)dz ’ I+(t/rcJy = s 1+t/r
(12)
0
which we have used previously when considering the kinetics of carbon monoxide rebinding to myoglobin [7]. From eq. (12) sin(wx)
f(r)= ~ and the first two moments
16 June 1989
7rr
(7/7oY
(7/7~)2a+(.S/rg)NCOS(TCCY)+l
are equal to (13)
(E) =&
(10)
and o'=fx2[(l-a)RT/&]*.
(11)
Thus for any system, as far as the Arrhenius relation holds, eq. ( 11) can be used to interpret the parameter LYin terms of a dispersion of the activation energy for reaction. Considering the entropy factor the parameter a can be interpreted in terms of free energy of activation in an analogous way. For some systems, such as recombination of ions in pulse irradiated solutions [ 61 or carbon monoxide rebinding following laser photodissociation of carbonmonoxy myoglobin [ 71, it seems reasonable to assume that the increase in activation energy with time is due to depletion from the system of more reactive species. For other systems, such as hydrogen atom recombination in low-temperature acidic glasses [ 8,9], there may be reasons to regard the increase in activation energy as the result of local matrix relaxation making all species less reactive [ lo,1 11. For the first type of system the hypothesis of a rate constant distribution may be valid and one would like to interpret the dispersion parameter a in terms of a reactant reactivity distribution. For a long time this was possible only for first-order processes [l] for which the stretched exponential exp[ - (t/q,)*] was regarded as a superposition of exponential decays with probability density f(z). Then, by definition, exp [ - ( t/rO)N] was the Laplace transform off(z), so that the latter was the inverse Laplace transform of exp [ - ( i/ro)*]. Here we would like to focus on an analogous interpretation of eq. (3), i.e. as a superposition of second-order recombinations with probability density f(T),
fora-0f(7)=6(7-70),andf(7)7hasamaximum at 7=~~ [7]. If the Arrhenius relation holds for any 7 one can express the reactivity distribution given by eq. ( 13 ) in terms of the activation energy sin(ncf) exp (A 1 g(E) = ~ 7cRT exp(2A) +2 cos( Tea) exp(A) + 1 ' (14) where A = ar( E - Eo) /RT. The first two moments (14) are equal to
of
(E) =&I,
(15)
a~=f7LZ(RT/a)2(l-~2),
(16)
where the subscript E was added to distinguish this second moment from that given by eq. ( 11). Fig. 1 illustrates the differences in the respective distribution densities for some numerical values of CY.Den-
-10
0
10
CE-Eo)/RT
Fig. 1.Distributions of activation energy for recombination, calculated according lo eq. (9) (solid lines), and distributions of reactivity, calculated according to eq. ( 14) (dashed lines), for (Y= l/3 and2/3 (from bottom to top).
381
Volume 158, number- 5
CHEMICAL
16June 1989
PHYSICS LETTERS
sities calculated from eq. ( 14) are broader than those calculated from eq. (9); indeed
sitions instead of the Arrhenius barrier transitions.
a;/a’=(l+a!)/(l-LY),
This work was partly supported by contract CPBP 01.19.
(17)
To rationalize this difference we have made a clear distinction between the distribution of reactivity of recombining species in a given system, o:, and the distribution of activation energy for the resulting recombination, (T’. Intuitively it seems correct that the former is wider. Neither the interpretation of LYin terms of an activation energy distribution for recombination nor the intcrprctation in terms of a reactivity distribution for recombining species implies a particular reaction dynamics. Both tend to be universal in visualizing the reactivity dispersion, which should be accounted for by a proper model for a given system. However, to choose such a model more is necessary than simply fitting the kinetic data. The direct-transfer model, the hierarchically constrained dynamic model, and the defect-diffusion model have been shown [ 121 to have common underlying mathematical structure. As a final remark it is worth noting the possibility of extending both interpretations using the Gamov formula for under the barrier tran-
382
relation for over the
References [ 11A. Plonka, Lecture notes in chemistry,
Vol. 40. Timcdependent reactivity of species in condensed media (Springer, Berlin, 1986 ) . [2] J. Klafter, A. Blumen and G. Zumofen, Phil. Mag. B 53 (1986) L29. [ 31 A. Blumen, G. Zumofen and J. Rlafter, in: Fractals. quasicrystals, chaos, knots and algebraic quantum mechanics, eds. A. Amman et al. (Kluwer,Dordrecht, 198X) pp. 2 l-52. [4] W.H. Hamill, Chem. Phys. Letters 77 ( 1981) 467. [ 51A. Plonka, Radiat. Phys. Chem., in press. [ 61A. Plonka, Radtat. Phys. Chem. 30 ( 1Y87) 3 I, [ 71 A. Plonka. J. Kroh and Yu.A. Berlin, Chem. Phys. Letters 153 (1988) 433. [ 81A. Plonka, J. Kroh, W. Letik and W. Bogus, J. Phys. Chem. 83 (1979) 1807. [S] A. Plonka, Radtat. Phys. Chem. 21 ( 1983) 405. [ IO] A. Plonka and W. Bogus, Radiat. Phys. Chem. I6 ( 1980) 365. [ I I] A. Plonka, Radiat. Phys. Chem. I7 ( 1981) 173. [ 121 J. Rlafter and M.F. Shlesinger, Proc. Natl. Acad. Sci. US 83 (1986) 848.
CHEMICAL
5
DISPERSIVE Andrzej
RECOMBINATION
16 June 1989
PHYSICS LETTERS
IN CONDENSED
PHASES
PLONKA
Institute ofApplied Radiation
Chemistry,
Techmcai Universily ofLodz,
Wroblewskiego
15, 93-590 Lodz, Poland
Yurii A. BERLIN and Nikolai I. CHEKUNAEV Ilzstituteof Chemical Physics. USSR Academy of Sciences, Kosysina 4, 1 I7977 Moscow, USSR Received 2 February
1989; in final form 24 March 1989
Two interpretations of the parameter dent reaction rate coefficient k(!)ar”-‘,
(Yare presented for a second-order equal-concentration kinetic equation with time-depen0 -Ca< I The first, valid for any system, is in terms of the activation energy distribution
for recombination and follows from the time dependence of the activation energy implied by the form of k(t). The second, valid for systems in which the time dependence of k(t) results from a distribution of reaction rates, is analogous to the interpretation of stretched exponentials as a superposition of exponential decays.
In condensed media numerous radiation-produced species were found [ I] to recombine according to a second-order equal-concentration kinetic equation with a time-dependent reaction rate coeflicient ) o-Car<1
k(t)=&+ i.e. according
)
(1)
“well stirred reactor”. It is possible to obtain a phenomenological interpretation of the parameter Q(following the idea of Hamill [4] that the form ( 1) of the time dependence of the reaction rate coefficient implies a particular time dependence of the activation energy for reaction. Indeed, applying the Arrhenius relation to k(i) one gets [ 5 ]
to
E,+(l--a)RTln(l/~,)
c/co=(l+c~Bta/a)-’
RT
(2)
or
(5)
in which
C/Ca=[l+(t/Zo)“]-’
(3)
for B=a/cor$. For longer
times
relation
(3) reduces to the power-
law time dependence CICOK(tl%-a
(4)
commonly observed [ 2 ] for carrier recombination in amorphous systems. In the modelling of carrier recombination dynamics through random walks, different aspects of disorder were considered [ 3 ] : fractals exemplified the spatial disorder, continuous-time processes the temporal and ultrametric structures the energetic disorder. The dispersion parameter cy was shown to account for deviations from classical kinctics with the implicit underlying assumption of a 380
E=&
+ (1 -a)R7ln(t/-r,)
(6)
has the physical meaning of a time-dependent activation energy. The effective activation energy E. for the second-order kinetics (3) can be obtained by applying the Arrhenius relation to c,r,; s denotes the preexponential factor. In classical kinetics, Q= 1, the activation energy remains constant during the whole reaction. In dispersive kinetics, 0 < a < 1, the activation energy increases with time, i.e. with reaction progress measured by F(f)=1
-c/co.
(7)
Using relation (6) one obtains from F(t) with c/c0 given by eq. (3) the activation energy distri-
0 009-2614/89/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
B.V.
Volume 158, number 5
bution function
CHEMICAL
F(E)
for recombination
[5 ]
F(E)=]-[l+exp(E)]-’ with t=cu(E-Eo)/( given by
PHYSICS LETTERS
(8) 1 -cr)RT.
Its density g(E)
is
(9)
1 cd f(r)dz ’ I+(t/rcJy = s 1+t/r
(12)
0
which we have used previously when considering the kinetics of carbon monoxide rebinding to myoglobin [7]. From eq. (12) sin(wx)
f(r)= ~ and the first two moments
16 June 1989
7rr
(7/7oY
(7/7~)2a+(.S/rg)NCOS(TCCY)+l
are equal to (13)
(E) =&
(10)
and o'=fx2[(l-a)RT/&]*.
(11)
Thus for any system, as far as the Arrhenius relation holds, eq. ( 11) can be used to interpret the parameter LYin terms of a dispersion of the activation energy for reaction. Considering the entropy factor the parameter a can be interpreted in terms of free energy of activation in an analogous way. For some systems, such as recombination of ions in pulse irradiated solutions [ 61 or carbon monoxide rebinding following laser photodissociation of carbonmonoxy myoglobin [ 71, it seems reasonable to assume that the increase in activation energy with time is due to depletion from the system of more reactive species. For other systems, such as hydrogen atom recombination in low-temperature acidic glasses [ 8,9], there may be reasons to regard the increase in activation energy as the result of local matrix relaxation making all species less reactive [ lo,1 11. For the first type of system the hypothesis of a rate constant distribution may be valid and one would like to interpret the dispersion parameter a in terms of a reactant reactivity distribution. For a long time this was possible only for first-order processes [l] for which the stretched exponential exp[ - (t/q,)*] was regarded as a superposition of exponential decays with probability density f(z). Then, by definition, exp [ - ( t/rO)N] was the Laplace transform off(z), so that the latter was the inverse Laplace transform of exp [ - ( i/ro)*]. Here we would like to focus on an analogous interpretation of eq. (3), i.e. as a superposition of second-order recombinations with probability density f(T),
fora-0f(7)=6(7-70),andf(7)7hasamaximum at 7=~~ [7]. If the Arrhenius relation holds for any 7 one can express the reactivity distribution given by eq. ( 13 ) in terms of the activation energy sin(ncf) exp (A 1 g(E) = ~ 7cRT exp(2A) +2 cos( Tea) exp(A) + 1 ' (14) where A = ar( E - Eo) /RT. The first two moments (14) are equal to
of
(E) =&I,
(15)
a~=f7LZ(RT/a)2(l-~2),
(16)
where the subscript E was added to distinguish this second moment from that given by eq. ( 11). Fig. 1 illustrates the differences in the respective distribution densities for some numerical values of CY.Den-
-10
0
10
CE-Eo)/RT
Fig. 1.Distributions of activation energy for recombination, calculated according lo eq. (9) (solid lines), and distributions of reactivity, calculated according to eq. ( 14) (dashed lines), for (Y= l/3 and2/3 (from bottom to top).
381
Volume 158, number- 5
CHEMICAL
16June 1989
PHYSICS LETTERS
sities calculated from eq. ( 14) are broader than those calculated from eq. (9); indeed
sitions instead of the Arrhenius barrier transitions.
a;/a’=(l+a!)/(l-LY),
This work was partly supported by contract CPBP 01.19.
(17)
To rationalize this difference we have made a clear distinction between the distribution of reactivity of recombining species in a given system, o:, and the distribution of activation energy for the resulting recombination, (T’. Intuitively it seems correct that the former is wider. Neither the interpretation of LYin terms of an activation energy distribution for recombination nor the intcrprctation in terms of a reactivity distribution for recombining species implies a particular reaction dynamics. Both tend to be universal in visualizing the reactivity dispersion, which should be accounted for by a proper model for a given system. However, to choose such a model more is necessary than simply fitting the kinetic data. The direct-transfer model, the hierarchically constrained dynamic model, and the defect-diffusion model have been shown [ 121 to have common underlying mathematical structure. As a final remark it is worth noting the possibility of extending both interpretations using the Gamov formula for under the barrier tran-
382
relation for over the
References [ 11A. Plonka, Lecture notes in chemistry,
Vol. 40. Timcdependent reactivity of species in condensed media (Springer, Berlin, 1986 ) . [2] J. Klafter, A. Blumen and G. Zumofen, Phil. Mag. B 53 (1986) L29. [ 31 A. Blumen, G. Zumofen and J. Rlafter, in: Fractals. quasicrystals, chaos, knots and algebraic quantum mechanics, eds. A. Amman et al. (Kluwer,Dordrecht, 198X) pp. 2 l-52. [4] W.H. Hamill, Chem. Phys. Letters 77 ( 1981) 467. [ 51A. Plonka, Radiat. Phys. Chem., in press. [ 61A. Plonka, Radtat. Phys. Chem. 30 ( 1Y87) 3 I, [ 71 A. Plonka. J. Kroh and Yu.A. Berlin, Chem. Phys. Letters 153 (1988) 433. [ 81A. Plonka, J. Kroh, W. Letik and W. Bogus, J. Phys. Chem. 83 (1979) 1807. [S] A. Plonka, Radtat. Phys. Chem. 21 ( 1983) 405. [ IO] A. Plonka and W. Bogus, Radiat. Phys. Chem. I6 ( 1980) 365. [ I I] A. Plonka, Radiat. Phys. Chem. I7 ( 1981) 173. [ 121 J. Rlafter and M.F. Shlesinger, Proc. Natl. Acad. Sci. US 83 (1986) 848.