Numerical Studies of Electron Tunnelling in Liquids BY P. ROBIN BUTLER AND MICHAEL J. PILLING Physical Chemistry Laboratory, South Parks Road, Oxford, OX1 342 Received 13th December, 1976 The diffusion equation, derived from Fick‘s second law, with an added exponential sink term to simulate electron tunnelling, is integrated numerically to determine the rate of electron decay at times greater than 1 ps. The effect of a coulomb interaction with a charged scavenger is examined and the steady-state rate constant shown to approximate closely to that obtained by combining the separate effects of tunnelling and charge-affected diffusion, which can be expressed analytically. Diffusion in the presence of a charge-induced dipole interaction is investigated for the case of scavenging of localised electrons in alkanes.The rate constant is shown to be dominated by random diffusion and tunnelling and the bias induced by the interaction is of little consequence. The sensivity of the rate constant to changes in the pre-exponential factor in the sink term is shown to be most favourable at short times. The experimental evidence for tunnelling of electrons to scavengers following irradiation of molecular and aqueous ionic glasses is now well estab1ished.lp2 Theoretical models have been proposed which describe the time dependence of the electron decay sati~factorily.~*~ Tunnelling has also been implicated in electron scavenging reactions in liquids and an analytical steady-state treatment has been presented which agrees well with conventional room temperature experiment^.^ Numerical studies in the time-dependent zone have also found agreement with experimental data obtained on a picosecond time scale at room temperature7 and at longer times at reduced temperatures.* This paper investigates numerically the effect of scavenger charge and polarizability on the reaction rate. It also examines the sensitivity of the rate constant to changes in the pre-exponential factor, a’, in the electron tunnelling term; a’ depends upon the electronic transition moment and the Franck-Condon factors for the reaction and is thus scavenger dependent. A. DIFFUSION EQUATION Pilling and Rice5 proposed that the influence of tunnelling on a reaction of a solvated electron may be determined by including an exponential sink term in the equation derived from Fick’s second law.The equation must be further modified when there are additional long range interactions between the reactants which affect their motion : where S(r, t ) is the ensemble averaged concentration of scavengers around electrons, r is the electron scavenger distance, D the diffusion coefficient and u the inter-reactant potential energy. (2) Zs(r) is the sink term: &) = a’ expi-& - R))P. ROBIN BUTLER AND MICHAEL J . PILLING 39 where a' is a frequency factor and p is a parameter related to the trap depth [fig. l(a)] : p == 2(2172 "( V, - E)>*/tZ (3) where m* is the effective mass of the electron and R is the true encounter distance. Since the electron is produced instantaneously, the initial distribution is assumed random, i.e., the inter-reactant potential is assumed not to influence the electron trapping distribution.The Smoluchowski boundary conditions may consequently be applied: S(r, t ) = 0 r < R, t 2 0 S(r, t ) = So r > R , t = O S(r, t) -+ s o r + co, t > 0. (6) Spur effects, non-random scavenger-scavenger distances and the effects of other ions on u are all neglected. These will all have only small effects at low scavenger concentrations and at the long times associated with steady state reactions (t > 10-los in water at 298 K). The approximations will become less good at the high scavenger concentrations required on picosecond time scales. As will be shown, however, charge effects have only a very small influence on the reaction rate at very short times (< s).B. COULOMB INTERACTION The coulomb potential energy is u = -Ze2/(4m,~r) ( 7 4 where Ze is the charge on the scavenger, e the charge on the electron, E, the vacuum permittivity and E the relative permittivity of the solvent. Expressing eqn (1) in terms of the reduced variables X = pr/2, Y = XS/S,, L2 = 4a' exp(pR)/p2D, z = P2Dt/4, and RT = IZle2/(4ne,~kT). ( 9 4 R, is the distance at which the coulomb interaction energy is equal to kT. tunnels9 and eqn (2) and (3) become inappropriate. r' from the electron [fig. l(b)] is now given by: The coulomb term also modifies the potential barrier through which the electron The barrier height at a distance V(r') = V, - E + Ze2{(r - re)-' - (Y - r ' ) - 1 ) / 4 x ~ o ~ . Thus Zs(r) = a' exp[-p (1 + b/(r - re) - b/(r - r')} dr'] (10) where b = Ze2/{4n~,~(V0 - E)).The integral in eqn (10) was determined analytically and included in eqn (1) and40 NUMERICAL STUDIES OF ELECTRON TUNNELLING I N LIQUIDS 1b) FIG. l.-(a) Schematic diagram of an electron trap, radius Y, and a scavenger trap, radius rS, with their centres separated by a distance Y. (b) The distortion of the potential energy profile for a charged scavenger. (7b). Eqn (7b) was expressed in incremental form and solved numerically, using the modified Crank-Nicolson method.'' to obtain the time-dependent scavenger profile, The time-dependent rate constant [k(t)] was found by summing the diffusive S(r, 9. [k,(t)] and tunnelling [kT(t)] rate constants: 5 9 6 k,(t) = 4nR2D(ap(R, t)/ar) kT(t) = co 4zr2p(r, t ) I, ( r ) dr R where p(r, t> = S(r, t)/So. A difficulty arises in the choice of the value of the relative permittivity. The aupr term requires the static value whilst it could be argued that the barrier per- meability term requires a high frequency value.Table 1 shows the percentage change (related to the E = co solution) in the rate constant for various values of e. Only the dependence of the barrier permeability on the coulombic interaction has been TABLE ~.-MoDIFJIcATION OF T m TIME DEPENDENT RATE CONSTANT BY A COULOMBIC PER- TURBATION OF THE POTENTIAL ENERGY BARRIER. (a' = s-l, j3 = 1Olo m-l, D = lo-* mz S-l, R = 0.5 ~ n , 2 = - 1) - log(t/s) k,-co/m3 molecule-1 s-' {(k,=m - k,)/k,=,) x 100 e = 80 E = 40 & = 8 12 1.54 x 10-15 2.0 2.5 10.9 10 2.17 x 10-l6 0.9 1.8 7.6 8 1.42 x 10-l6 0.7 1.3 5.9 6 1.36 x 10-l6 0.7 1.3 5.7 4 1.36 x 10-l6 0.7 1.3 5.7P.ROBIN BUTLER AND MICHAEL J . PILLING 41 included i n the calculation; the term in au/ar was excluded. The results show that, even for E = 8, the change in the rate constant is small. Thus %he inclusion of this term is of only minor importance and the exact choice of E is I:ot critical. This conclusion substantiates the qualitative argument proposed by Pilling and Rice.g Eqn (10) was included in the remaining calculations, but the static dielectric constant was employed. Fig. 2 shows the time dependence of the rate constant for a range of values of r -'I -171 ' I I I I J -12 -10 -8 -6 -4 log ( t l s ) FIG. 2.-Time dependence of the scavenging rate constant, k: A, Z/E = 0; X , Z/E = -1.25 x +, Z/E = 1.25 x My Z/E = 2.5 x 0, Z/E = 2.5 x e, Z/E = 5 x a' = 1014 s-l, p = 1O1O m-l, D = m2 s-', R = 0.5 nm.(Trap depth = 1 eV.) Z/e. The charge effect is small at short times (< the steady state. state rate constant is given by: where s) and is most pronounced in For uncharged reactants, Pilling and Rice showed that the steady k = 4nReffD (1 1) and I.V: = 4a'/P2D, y is Euler's constant (0.57721 . . .) and Io(wo) and K,(w,) are the modified first and second kind Bessel functions of zero order and argument w,. The rate constant for a difftision-controlled reaction between charged reactants, exclusive of any tunnelling contribution, is given by;'' where k = 4nRD6/(es - 1) 6 = ZAZ,e"/(4~EoERkT) and ZAe and 2,e are the reactant charges.Fig. 3 shows the percentage difference (based on the analytic value) between the computed steady rate constant (k,) and that obtained from eqn (13) with R replaced by Reff(ka). The analytic expression under- estimates the rate constant for oppositely charged species and overestimates it for42 NUMERICAL STUDIES OF ELECTRON TUNNELLING I N LIQUIDS I I 1-80 - 2.5 0 2.5 102z/c FIG. 3.-Percentage difference ({(k, - k,)/k,) x 100) between the computed steady state rate constant, k,, and the analytic rate constant, k,, based on eqn (12) and (13), as a function of Z/E (a). Percentage difference ({(kEZm - k,)/k,=,) x 100) between the rate constant with Z/E = 0 and with Z/E non-zero (0). u' = 1014 s-', p = 10" m-', D = lo-* m2 s-', R = 0.5 nm. (Trap depth = 1 eV.) similarly charged reactants.Fig. 3 demonstrates that the analytic estimate is generally quite good except for Z/E > 2.5 x The correction to the Z/E = 0 rate constant (fig. 4), arising from the combined effects of charge and tunnelling, is con- siderably greater than the error involved in using the analytic expression [eqn (12), (13)]. The accuracy of the computational estimates of the steady state rate constant may be assessed by comparing the value for Z/E = 0 with that from eqn (1 l), which is exact; the error is only 0.5%. 2.0 0. 0.81 I I I I I I -12 -10 -8 -6 l o g ( t / s ) FIG. 4.-Time dependence of the rate constant, k, for reaction with a scavenger of polarizability up. 0, polarization potential included; 0, polarization potential excluded. u' = 4 x 1014 s-l, /3 = 0.7 x 10" m-', D = 7.2 x nm3,12 f = 0.69.11 (Trap depth = 0.5 eV.) m2 s-', R = 0.592 nni, aI, = 6.55 xP .ROBIN BUTLER AND MICHAEL J . PILLING 43 Quite crude approximations have been made regarding the structure of the dielectric medium; no allowance has been made, for example, for the effects of di- electric saturation. The range of values of Z/E used is also quite limited. The major aim of this study was to examine the effects of charge and tunnelling on electron reactions in water at 298 K. The results show that a good estimate of the rate constant for a diffusion-controlled reaction may be obtained by combining eqn (1 1)- (14). At very short times, the reaction is dominated by static tunnelling6 and the charge effect is small. C. CHARGE-INDUCED DIPOLE INTERACTION Baird l2 recently suggested that the charge-induced dipole interaction between an electron and a neutral scavenger is important in low polarity solvents.He examined the reaction between e; and SF6 and solved the steady state equation [cf. (1) with as/& = 01 for reactant diffusion, but did not include a tunnelling term. He found that, for delocalized electrons, ie., those with a velocity correlation length, I, >0.1 nm, the rate constant is less than the diffusion-controlled value, confirming that tunnelling from distances >R is indeed unimportant. For localised electrons ( I < 1 A, e.g., in ethane) the rate constant (k) is greater than the diffusion-controlled value (kdiff) (table 2). This is reminiscent of several reactions in aqueous solution, TABLE 2.-A COMPARISON OF EXPERIMENTAL RATE CONSTANTS FOR ELECTRON SCAVENGING BY SF6 IN AJXANES WITH THOSE CALCULATED FROM EQN (13) (R = 0.592 nm) hexane13 297 2.1 x 10-7 3.3 x 10-15 273 1.0 x 10-7 1.6 x 1045 248 4.0 x 7.5 x 217 2.6 x 10-6 5.3 x 10-14 175 7.2 x 10-7 1.3 x 10-14 142 4.9 x 10-8 1.7 x 10-15 110 1.2 x 10-9 8.3 x 10-17 ethane l4 200 1.6 x 2.5 x 2.1 2.2 2.5 2.7 2.1 2.4 4.7 9.4 2.Ob 2.2b 2.5b 2.3d 2Sd 2.7d 3.P 5.0d 2.1' 2.3" 2.5" 2.3" 2.5' 2.6" 3.3" 4.2" where large rate constants have been rationalized by invoking t~nnelling.~ In particular, it should be noted that k/kdiff increases as D falls, a property associated with a long-range rea~tion.~ The diffusion equation is insoluble even in the steady state if the tunnelling term is included, and numerical techniques must be resorted to.The interaction potential is now: where up is the polarizability (assumed isotropic) of the scavenger andfis the screening function12 given by f = ( E + 2)/3~.44 NUMERICAL STUDIES OF ELECTRON TUNNELLING I N LIQUIDS Thus is defined as before and Ri = a,e2f/(8m0kT). The interaction potential is of such short range (RT - 0.7 nm) that, in the light of the results discussed in section C, the effect of the polarization potential on the barrier permeability was neglected. Because of the strong dependence of u on Y, the barrier is quite strongly perturbed in the immediate vicinity of the scavenger, but our lack of a detailed knowledge of the electron or scavenger trap limits the utility of incor- porating such an effect.Fig. 4 shows a plot of the rate constant against time both with and without the polarization term. The rate constants are designed to simulate roughly Bakale et aZ’s14 data for e- + SF6 in ethane at 175 K. Tunnelling has a marked effect, increasing the rate constant by 90% (5.36 x m3 s-l to 1.02 x m3 s-l). The polarization effect is, however, quite small, enhancing k by only a further 10%. This arises because the effective encounter distance, Reff, is greater than RT, the distance at which the polarization energy is comparable with thermal energies. Thus, for localized electrons in alkanes, the charge-induced dipole interaction may be neglected for rapid reactions and the analytic eqn (1 l), for a diffusion-controlled reaction with a tunnelling contribution, used to estimate the rate constant.Table 2 contains estimates of the rate constants on this basis. The calculated rate constants reproduce the experimental variation reasonably well, with realistic tunnelling para- meters, given the probable errors associated with the experimental data and the simplifying assumptions of constant a’ and p. p = 7 and 5 nm-l correspond to trap depths of 0.5 and 0.25 eV respectively. The mechanism of diffusion of electrons in alkanes is thought to involve thermal promotion to the continuum followed by retrapping.13 For liquids with Z < 1, this mechanism closely approaches the Brownian motion assumed in Fick’s second law. The treatment outlined in this section assumes that reaction takes place from the electron trap; it presumes that there is a low probability of reaction from the con- tinuum.An alternative mechanism may be proposed in which the electron reacts during the time that it is in the continuum. The alternatives could only be distinguished if the trapping frequency were known. D. THE DEPENDENCE OF THE ENSEMBLE-AVERAGED RATE CONSTANT ON a’ Using Fermi’s golden rule, the distance dependent scavenging probability may be expressed in the form2’l6 where W(r) is the electronic matrix element for electron transfer at a distance Y, p is the vibrational overlap integral for the scavenger ion and the scavenger vibrational levels, Ef and El are the total h a 1 and initial energies of the reacting system and q R is the vibronic partition function of the reactants. l h e electronic matrix element contains both the electronic wavefunctions for the scavenger and scavenger anion and the vibronic wavefunctions for the occupied (initial) and unoccupied (final andP .ROBIN BUTLER AND MICHAEL J . PILLING 45 unrelaxed) electron trap. The assumed exponential distance dependence of Zs(r) is contained in I W(r)I2. The vibrational wavefunctions involved in the overall Franck- Condon factor refer to both high frequency molecular modes and low frequency medium modes? Some attempts have been made to calculate the ensemble averaged rate constant, k, on a somewhat similar An important preliminary step, however, is an assessment of the sensitivity of k to changes in a'. Dainton et al.4 examined the dependence of the electron decay kinetics on a' for scavenging in rigid media.They showed that the decay is very insensitive to a' for t > s, and is determined primarily by p. This conclusion presents difficulties in accommodating the variation in scavenger efficiency found under these conditions. Miller ascribed the variation entirely to the Franck-Condon factors, but very large changes in these were required to explain the experimental results. A further un- resolved problem arises since the decay curve is expected to deviate2p3 from the linear form proposed by Dainton et aL4 if a' is small. There are, in consequence, difficulties which remain in a pure tunnelling description of electron scavenging in glasses. It has been suggested, although not proved, that these problems can be resolved if trap to trap tunnelling is incorporated.19 This uncertainty in the overall mechanism is absent from descriptions of liquid phase reactions, since diffusion, by whatever mechanism, is catered for.A reaction in solution which takes place by a contact interaction, and which does not involve a long range mechanism, may be described by the scheme: kd kr A + B P- {AB} + products k-d where kd is the rate constant for formation of the encounter pair, (AB} and k, and k-d are the first order rate constants for reaction of (AB) and for diffusive separation respectively. Since the reaction does not involve tunnelling, p = co and k, = Z,(R) = a', [provided the description contained in eqn (15) remains valid]. The dependence of the overall rate constant, k, on a' is shown in fig. 5. For a' < k-d, k is linear in a' (k = a'kd/k -d) and provided kd/k -d, the equilibrium constant for formation of the encounter pair, can be estimated,20 a' can be deduced.When a' 9 k - d , k = kd and log I o('/s- ' 1 FIG. 5.-The dependence of the rate constant, k, on u'. - -, /? = 00, t = co; -.- , B = m , D = 1 0 - 8 mz s-1, t = 10-1' s; 0, /j = loLo m-l, t = co; 0, B = 1O1O m-l, t = lo-" s. R = 0.5 nm.46 NUMERICAL STUDIES OF ELECTRON TUNNELLING I N LIQUIDS the reaction is diffusion-controlled with a rate constant independent of a’. At short times, k is larger in the diffusion-controlled region, because reaction takes place from an unrelaxed reactant distribution. s for a reaction involving tunnelling (j3 = lolo m-l). The steady-state rate constant con- tinues to increase even when the reaction is diffusion-controlled, because the effective encounter distance [eqn (12)] increases with a’.k remains, however, comparatively insensitive to variations in a’, a five-decade increase in a’ leading to only a four-fold increase in k. The sensitivity is increased somewhat at shorter times; at s the same increase leads to a ten-fold change in a’. The determination of the rate constant for an electron reaction can lead to accurate information on a’, the frequency term in the distance dependent scavenging probability, only provided a’ < loll s-’, in water at 298 K, i.e., provided the reaction rate is limited by the rate of reaction in the encounter pair. This could arise if the Franck- Condon factor were small, or, alternatively, if the reaction were endoergic.For larger values of a’, the rate constant is most sensitive to variations in a’ at short times. k still contains no information on a’. Fig. 5 also shows plots of k against a’ in the steady-state and for t = We thank Dr. Stephen A. Rice for helpful discussion and the S.R.C. for a student- ship to P. R. B. J. R. Miller, J. Phys. Chem., 1975, 79, 1070. M. J. Pilling and S. A. Rice, Chem. Reu., to be published. M. Tachiya and A. Mozumder, Chem. Phys. Letters, 1974, 28, 87. F. S. Dainton, M. J. Pilling and S. A. Rice, J.C.S. Faraday 11, 1975, 71, 1333. M. J. Pilling and S. A. Rice, J.C.S. Faraday 11, 1975, 71, 1563. P. R. Butler, M. J. Pilling, S. A. Rice and T. J. Stone, Cunad. J. Chem., to be published. C. D. Jonah, J. R. Miller, E. J. Hart and M. S. Matheson, J. Phys. Chem., 1975,79,2705. G. V. Buxton, F. C. R. Cattell and F. S. Dainton, J.C.S. Faruday I, 1975, 71, 115. M. J. Pilling and S . A. Rice, J . Phys. Chem., 1975, 79, 3035. lo J. Crank and P. Nicolson, Proc. Canzb. Phil. Soc., 1947, 43, 50. l1 P. Debye, Trans. Electrochem. Soc., 1942, 8, 265. J. K. Baird, Canad. J. Chem., to be published. l3 A. 0. Allen, T. E. Gangwer and R. A. Holroyd, J. Phys. Chem., 1975,79,25. l4 G. Bakale, U. Sowada and W. F. Schmidt, J. Phys. Chem., 1975,79,3041. l5 B. Brocklehurst, Chem. Phys., 1972, 2, 6. l6 J. Ulstrup and J. Jortner, J. Chem. Phys., 1975, 63, 4358. A. Henglein, Ber. Bunsenges. phys. Chem., 1974, 78, 1078. A. Henglein, Ber. Bunsenges. phys. Chem., 1975, 79, 129. l9 G. V. Buxton and K. Kemsley, J.C.S. Faraday I, 1975, 71, 115. 2o A. M. North, The Collision Theory of Chemical Reactions in Liquids (Methuen, London, 1964), p. 37.