3804
J. Phys. Chem. B 2000, 104, 3804-3807
Phenomenological Approach to Thermally Assisted Tunneling† Andrzej Plonka* Institute of Applied Radiation Chemistry, Technical UniVersity of Lodz, Wroblewskiego 15, 93-590 Lodz, Poland ReceiVed: September 7, 1999
Most of low-temperature reactions proceed faster than would be predicted by an Arrhenius-type extrapolation from the high-temperature regions. The definitions of low- and high-temperature regions are quite conditional, depending strongly on the properties of the host matrix for a given reactant. The transition from the lowtemperature region to the high-temperature region, manifested by a break in an Arrhenius plot, is accompanied by the marked decrease of reaction dispersivity. In the low-temperature region, the specific reaction rates depend on time. For a time-dependent specific reaction rate, using the concept of the energy profile along the reaction path, one finds the potential energy barrier separating reactants from products to evolve in time during the reaction course. This evolution, in the Gamov picture for a simple rectangular barrier of constant height, is described in terms of a distribution function for the tunneling distance, related directly to the distribution function of logarithms of lifetimes calculable from the kinetic equation with a time-dependent specific reaction rate. In the high-temperature region, classical kinetics with a constant specific reaction rate provides a good approximation. This is rationalized in terms of the stochastic model of reaction kinetics in the renewing environment. In the model, the structural relaxation of the host matrix is included by imposing upon the static disorder model the additional assumption that at random the reinitialization occurs. The reinitialization consists of random reassignment of guest hopping rates with the values having the same initial distribution. Therefore, the breaks in Arrhenius plots might be taken to be an indication of the onset of thermal assistance for tunneling, not necessarily of the change from quantum mechanical under barrier transition to thermally activated over barrier transition. As an example, the reactions of excess electrons in aqueous systems are discussed.
Introduction developing1
We are the kinetics for reactions in condensed media using the specific reaction rate, kRˆ (t), in the form
kRˆ (t)/(dm3/mol) ) (Rˆ /ζˆ Rˆ )(t/ζˆ Rˆ )Rˆ -1
(1)
where t denotes the reaction time, Rˆ is the experimentally observed dispersion parameter (see below), and ζˆ Rˆ scales the time. Depending on the numerical value of Rˆ we have2 (i) 0 < Rˆ < 1, dispersive kinetics for which, like for dispersive transport and relaxation (Scher et al.),3 many time scales coexist (cf. Figure 1); (ii) Rˆ ) 1, classical kinetics, with a constant specific reaction rate k ) 1/ζˆ 1, valid when the rates of internal rearrangements (mixing) in the reaction system exceed markedly the rates of chemical reactions (and, therefore, the disturbance in the reactant reactivity distribution is negligible during the reaction course);1 (iii) Rˆ > 1, strange kinetics, the potential for which exists in any situation where some hierarchical ordering of the random walk occurs (cf. Shlesinger et al.).8 The experimentally observed dispersion parameter Rˆ is given by9
Rˆ ) 1 - (1 - R)(1 - β)
(2)
where R denotes the dispersion parameter for reactant movement * Corresponding author. E-mail:
[email protected]. † Part of the special issue “Harvey Scher Festschrift”.
in the static, i.e., frozen, host matrix and β denotes the dispersion parameter of host matrix structural relaxation (mixing). Time is scaled by ζˆ Rˆ equal to9 ˆ 1-R/Rˆ ζˆ Rˆ ) ζR/R R τβ
(3)
which is to be recognized as the weighted geometrical mean of the parameter ζR scaling the movement of the reactant in a rigid host matrix and the parameter τβ scaling the host matrix relaxation. If both processes, reactant movement and matrix relaxation, are thermally activated with activation energies given by E(ζR) and E(τβ), respectively, then the activation energy for reactant movement in the relaxing matrix is given by
E(ζˆ Rˆ ) ) (R/Rˆ )E(ζR) + (1 - R/Rˆ )E(τβ)
(4)
In the low-temperature limit, Rˆ f R, one gets from eq 4
E(ζˆ Rˆ ) ) E(ζR)
(5)
while in the high-temperature limit, Rˆ f 1, one gets from eq 4
E(ζˆ Rˆ ) ) RE(ζR) + (1 - R)E(τβ)
(6)
In general, for reactive species, there is E(ζR) < E(τβ) and one has a marked increase of activation energy with temperature.2 In a special case, considered in the present paper, when reactant movement is due to quantum-mechanical tunneling, the increase of activation energy with temperature is dramatic, manifested by the break in the Arrhenius plot of the rate coefficients.10
10.1021/jp9931657 CCC: $19.00 © 2000 American Chemical Society Published on Web 02/16/2000
Thermally Assisted Tunneling
J. Phys. Chem. B, Vol. 104, No. 16, 2000 3805
Figure 1. Dispersive rate processes. Upper part: Traces of transient photocurrent in a material with a well-defined mobility of carries (A) and in disordered systems (B) analyzed quantitatively in terms of the continuous time random walk (CTRW) model of Montroll and Weiss4 by Scher and Montroll.5 The insets show the transiting packets of charge.3 These sequences of “snapshots” represent packets of charge transiting a sample, with time increasing from top to bottom. The normal transport in (A) corresponds to linear evolution in time of the packet’s mean position and gives rise to the current trace shown by the curve A. In (B) the packet’s displacement is controlled by a large dispersion in the arrival time of the carriers at the far electrode. The mean position of the packet, l(t), is a sublinear function of time, l(t) ∼ tβ, 0 < β < 1, and gives rise to the current trace shown by curve B. Middle part: Dielectric relaxation function calculated in Williams laboratory, cf. Cook et al.,6 for poly(ethyl acrylate) (solid line) adequately reproduced by the stretched exponential, eq 14, with R ) 0.38. The Debye, singleexponential relaxation function is shown by the dashed line. Lower part: Decays of trapped electrons observed by Miller7 in pulse radiolysis of 6 M NaOH glass containing 0.1 M NO3-. The absorbance A at each time is divided by the absorbance A0 in the “pure” 6 M NaOH matrix in the same time. At 196 K the decay seems to be single exponential, as expected for classical pseudo-first-order reaction kinetics. At 77 K, cf. inset in Figure 3, the decay is highly dispersive, well reproduced by the stretched exponential, eq 14, with R ) 0.07.
However, we show this break to be an indication of the onset of thermal assistance for tunneling rather than the change from quantum-mechanical tunneling at low temperature to the thermally activated over barrier transition at higher temperature. The reason is that these breaks, contrary to the basic analysis of temperature criteria (cf. Goldanskii),11 in numerous cases2 do depend more on the properties of the host matrix than on the nature of the reactant. Discussion The time scale for reaction at a given temperature is given by
ζˆ Rˆ ) min(ζˆ Rˆ ,ζR)
(7)
which below the intersection point of τβ-1 and ζR-1 in the Arrhenius plot (cf. Figure 2) is equal to ζR. Quite formally, one gets the above result taking β f 0 for T f 0. Then from eq 2 one gets Rˆ f R, and kRˆ (t), given by eq 1,
Figure 2. Arrhenius plot of ζˆ Rˆ -1, eq 3 and 7, scaling the reaction time. The insets (A) and (B) adapted from Phillips12 depict the temperature dependence of the dispersion parameter β. (A) is a normal form of β for a lattice glass such as a spin glass, and to a good approximation β extrapolates to zero for T f 0. (B) is the normal form of β in an offlattice glass such as an inorganic network glass, a polymer, or a fused salt. The range of the temperature dependence of β is narrowed, the ratio of the glass transition temperature (Tg) to the temperature of melting (Tm) is increased, and the plateau region below Tg may vary depending on the probe used to study glass transition. Similar changes of β are expected for a relaxation region other than the glass transition. For electrons trapped in aqueous glasses this is the third relaxation region seen in the thermally stimulated depolarization current in pure ice, cf. Johari and Jones.13 To depict the break in the Arrhenius plot of ζˆ Rˆ -1, the dispersion parameter β was assumed to increase linearly within the limits 0.1-1 in the temperature range 1.1-1.3 in units of T/Tc where Tc is the temperature of intersection of ζR-1 and τβ-1 in the Arrhenius plot.
reduces to
kR(t) ) (R/ζR)(t/ζR)R-1 (8)
= tR-1
used for the first time by Hamill and Funabashi14 to fit the tails seen in the experimental data of Miller.7 Furthermore, this particular time-dependent form of the specific reaction rate was shown to be consistent with the long-tail jump-time distribution function in the CTRW model. Because of this consistency, further theoretical treatments of electron decay reactions evolved around two distinct, limiting assumptions. The first was that trapped electrons react directly with scavengers; the second, that electrons hop from one trap to another until they encounter the scavenger with which they will react. Much effort has been devoted to the controversial issue of finding out the correct model. It had to be agreed that both the direct tunneling model and trap-to-trap hopping can fit the experimental data equally well; cf. van Leeuwen et al.15 There is a reason for that. It was overlooked for a rather long time that, in the model of Dainton et al.16 for electron scavenging in irradiated glassy media, based on a tunneling mechanism, k(t) is given by
∫a∞F(a,t) exp(-2qa) da
k(t) ) πΑσ2S0
(9)
s
where F(a,t) is a time-dependent radical density function
[
F(a,t) ) exp -
Aσ2 exp(-2qa)t 4r2
]
(10)
A is a frequency factor, typically of the order 1014-1015 s-1; the factor σ2/4r2 describes the solid angle subtended at the electron by a scavenger of cross section πσ2 at a distance r; S0 is the initial scavenger concentration; q ) 2[2m*(V - E)]1/2/p, where m* is the effective mass of the electron and (V - E) is the total energy; and a ) r - R, where r is the distance between the trap centers and R is the sum of the trap radii. Relation 9
3806 J. Phys. Chem. B, Vol. 104, No. 16, 2000
Plonka
〈ln(t/ζR)〉 ) -γ/R
(15)
where γ denotes the Euler constant; the variance is given by
D2(ln(t/ζR)) ) π2/6R2
(16)
〈a(t) - a(ζR)〉 ) γq-1(1 - 1/R)
(17)
D2(a(t) - a(ζR)) ) (π2/6q2)(1/R - 1)2
(18)
Hence
and
Figure 3. Numerically evaluated specific reaction rate k(t) from the theoretical model of electron scavenging in irradiated glassy media based on a tunneling mechanism. Adapted from Dainton et al.,16 the solid lines are calculated from eq 9 for q ) 1.212 × 1010 m-1, as ) 1.5 nm, R ) 0.6 nm, σ ) 0.5 nm, S0 ) 10-2 M, A ) 1013-1017 s-1. The dashed line depicts the relation (11). The inset illustrates the fit of the experimental results of Miller,7 cf. Figure 1, by a stretched exponential with R ) 0.07.
has been evaluated numerically for typical values of A, q, as, and R; cf. Figure 3. The dashed line in Figure 3 depicts the relation k(t) ∼ t-0.93, which (cf. eq 8) yields R ) 0.07. Using this numerical value of R, one gets an adequate fit of the experimental data of Miller7 for 77 K shown in Figure 1 (cf. inset in Figure 3) with eq 14 in which c(t)/c0 was replaced by the ratio of absorbances. Taking the simple picture of Gamov for rectangular barrier permeability κ
k ) exp(-qa)
(13)
As an example, for a reaction of first or pseudo first order the integrated kinetic equation with k(t) given by eq 8 reads1
c(t)/c0 ) exp[-(t/ζR)R]
(14)
where c(t) and c0 denote the concentration of reactant at time t and 0, respectively. The stretched exponential (14) is identical, in mathematical form, with the function used by R. Kohlrausch17 in 1854 to describe the decay of residual charge on a Leyden jar and by his son F. Kohlrausch18 in 1863 to describe the torque relaxation in glass filaments. For a chequered history of the use of the stretched exponential (14), see Williams;19 for interpretation of the stretched exponential (14) in terms of superposition of the mono exponential decays, see Plonka and Paszkiewicz.20 From eq 14 the mean value of the logarithm of the lifetime is given by
(19)
From eq 19 the mean value of the logarithm of the lifetime is given by20
〈ln(t/ζR)〉 ) 0
(20)
and the variance is equal to
D2(ln(t/ζR)) ) π2/3R2
(21)
From eqs 13, 20, and 21
〈a(t) - a(ζR)〉 ) 0
(22)
D2(a(t) - a(ζR)) ) (π2/3q2)(1/R - 1)2
(23)
and
(12)
There are moments of ln(t/ζR), the logarithms of lifetimes, for reaction of any order, and one can calculate the moments of a(t) from those of ln(t/ζR)
〈(a(t) - a(ζR))n〉 ) q-n(1 - R)n〈lnn(t/ζR)〉
c(t)-1 - c0-1 ) (t/ζR)R
(11)
in the present notation, one finds that the decrease in time of k(t) given by eq 8 corresponds to the increase in time of the tunneling distance a(t)
a(t) ) a(ζR) + q-1(1 - R) ln(t/ζR)
From Miller’s data (cf. inset in Figure 3) and the numerical value of q ) 2 × 10-10 m-1 estimated by Dainton et al.,16 〈a(t) - a(ζR)〉 ) 0.37 and D2(a(t) - a(ζR)) ) 0.84 nm2 for a(ζR) ) 1.23 nm for the frequency factor 1015 s-1. For the second-order reaction, equal concentration case, the integrated kinetic equation with k(t) given by eq 8 reads1
In the high-temperature limit, for β f 1, Rˆ f 1 (cf. eq 2), and kRˆ (t) given by eq 1 reduces to the form
k1 ) 1/ζˆ 1
(24)
ζˆ 1 ) ζRR τ1-R β
(25)
with
The activation energy for ζˆ 1 is equal to
E(ζˆ 1) ) (1 - R)E(τβ)
(26)
for E(ζR) ) 0. This is the activation energy in the hightemperature limit for thermally assisted tunneling. For comparison, for reaction of hydrated electrons with nitrate ions, followed at low temperatures by Miller7 (cf. Figure 1), the hightemperature limit, at around room temperature, of activation energy is reported to be in the range 10-16 kJ/mol (cf. Buxton et al.21). Interestingly, in this range of activation energies one finds about 70% of the reported values for a large group of solutes, with a few exceptions such as cobalt(II) or manganese-
Thermally Assisted Tunneling (II) ions, for which the activation energy for reaction with hydrated electrons around room temperature exceeds 20 kJ/mol. Concluding Remarks We have given the phenomenological interpretation of the temperature dependence of reaction patterns in condensed media in terms of the stochastic model of reaction kinetics in the renewing environments. In the model, the thermally activated structural relaxation is imposed upon the static disorder model of reactant movement in a disordered system due to quantummechanical tunneling. The main result is that the transition from the low-temperature region of the activationless movement to the high-temperature region of thermally activated movement is accompanied by a marked decrease of reaction dispersivity. In the high-temperature limit the classical kinetics might be valid with a constant specific reaction rate given by eqs 24 and 25. This is the result valid for reactive species, i.e., for reactions with high local reaction probability, P, when two reactants collide. In general, for P f 1 there is diffusion-limited reaction kinetics, while for P , 1 one has reaction-limited kinetics. Only for P f 1 is eq 24 is valid. The lower the numerical value of P, the closer to 1 is the apparent numerical value of R seen in computer simulations performed by Shi and Kopelman22 for bimolecular reactions in one dimension, on a two-dimensional square lattice, on a two-dimensional critical percolation cluster, and in three-dimensional cubic lattices with various local reaction probabilities. This is the way to get k ) 1/ζR from eqs 24 and 25, which covers the result of usual theories. As an example, the reactions of excess electrons in aqueous systems were discussed. However, the idea of thermally assisted tunneling is involved for numerous systems2 and breaks in Arrhenius-type plots, like the one presented in Figure 2, are nowadays quite common.2 Acknowledgment. This work was partly supported by the State Scientific Research Committee (KBN). References and Notes (1) Plonka, A. Prog. React. Kinet. 1991, 16, 157. (2) Plonka, A. Annu. Rep. Prog. Chem., Sect. C 1988, 85, 47; 1992, 89, 37; 1994, 91, 107; 1998, 94, 89.
J. Phys. Chem. B, Vol. 104, No. 16, 2000 3807 (3) Scher, H.; Shlesinger, M. F.; Bendler, T. Phys. Today 1991, Jan., 26. (4) Montroll, E. W.; Weiss, G. H. J. Math. Phys. 1965, 6, 167. (5) Scher, H.; Montroll, E. W. Phys. ReV. B 1975, 12, 2455. (6) Cook, M.; Watts, D. C.; Williams, G. Trans. Faraday. Soc. 1970, 66, 2563. (7) Miller, J. R. J. Phys. Chem. 1975, 79, 1070. (8) Shlesinger, M. F.; Zaslavsky, G. M.; Klafter, J. Nature (London) 1993, 363, 31. (9) Plonka, A.; Paszkiewicz, A. J. Chem. Phys. 1992, 96, 1128. In the stochastic model of reaction kinetics in renewing environments, the structural reorganization of the host matrix was included by imposing upon the static disorder model the additional assumption that at random instants the reinitialization occurs. The reinitialization consists of random reassignment of guest hopping rates with the values having the same initial distribution. Renewals are described by the fractal set of renewal moments following the use of Kohlrauschs’ function in the form Φ(t) ) exp[-(t/τβ)β], 0 < β e 1, to describe the internal rearrangements in the system. The number of distinct sites visited, S(t), by the walker in time t in the static, disordered system was taken to sublinear in time S(t) ) (t/ζR)R. Averaging this number of distinct sites visited over the reneval sequences following from the Kohlrauschs’ relaxation function, one gets 〈S(t)〉 ) (t/ζˆ Rˆ )Rˆ with Rˆ given by eq 2 and ζˆ Rˆ given by eq 3. The time derivative of 〈S(t)〉 yields the timedependent specific reaction rate kRˆ (t) given by eq 1. (10) In the Arrhenius equation k ) A exp(-E/RT) k denotes the constant specific reaction rate in s-1, E is the activation energy, and R is the gas constant. This equation predicts that the specific reaction rate, and therefore the reaction rate, vanishes in the limit T f 0. Generally, it became clear, cf. ref 11, that most low-temperature reactions actually proceed faster than would be predicted on the basis of an Arrhenius-type extrapolation. The problem is that at low temperatures there is no constant specific reaction rate. The specific reaction rate is time dependent, cf. relation 1 with 0 < Rˆ < 1. Furthermore, there is no mean value 〈k(t)〉. One may use any kind of rate parameter to get the qualitative picture like that displayed for ζˆ Rˆ -1 in Figure 2 but for quantitative purposes one needs ζˆ Rˆ or, alternatively, 〈ln k(t)〉. (11) Goldanskii, V. I. Annu. ReV. Phys. Chem. 1976, 27, 85. (12) Phillips, J. C. Rep. Prog. Phys. 1996, 59, 1133. (13) Johari, G. P.; Jones, S. J. J. Chem. Phys. 1975, 62, 4213. (14) Hamill, W. H.; Funabashi, K. Phys. ReV. B 1977, 16, 5523. (15) Van Leeuwen, J. W.; Heijman, M. G. J.; Nauta, H.; Castelijn, G. J. Chem. Phys. 1980, 73, 1483. (16) Dainton, F. S.; Pilling, M. J.; Rice, S. A. J. Chem. Soc., Faraday Trans. 2 1975, 71, 1311. (17) Kohlrausch, R. Pogg. Ann. 1854, 91, 179. (18) Kohlrausch, F. Pogg. Ann. 1863, 119, 337. (19) Williams, G. J. Non-Cryst. Solids 1991, 131/133, 1. (20) Plonka, A.; Paszkiewicz, A. Chem. Phys. 1996, 212, 1. (21) Buxton, G. V.; Grennstock, C. L.; Helman, W. P.; Ross, A. B. J. Phys. Chem. Ref. Data 1988, 17, 513. (22) Shi, Zh. Y.; Kopelman, R. J. Phys. Chem. 1992, 96, 6858.
