Picosecond photolysis of azo compounds in liquid alkanes: germinate recombination kinetics for polyatomic free radical pairs

Volume 178, number 1

CHEMICAL PHYSICS LETTERS

15 March 1991

Picosecond photolysis of azo compounds in liquid alkanes: geminate recombination kinetics for polyatomic free radical pairs Thomas W. Scott Departmentofchemistry, New York University,New York,NY 10003, USA and Exxon Corporate Research Laboratories, Rt. 22 East, Annandale,NJ 08801, USA

Charles Doubleday Jr. Departmentof Chemistry, Columbia University,New York,NY 10027, USA Received I2 December 1990

Picosecond optical absorption transients for the photolysis products of azoeumene and the cyclic azo compound 3,8diphenyl1,2diaza-l-cyclooctene have been measured in a series of alkane solvents having different liquid viscosities. The transient intermediate produced from azocumene decays through a diffusion influenced process which is interpreted as secondary recombination of geminate free radical pairs. This assignment is based on the wavelength dependence of the transient absorption signal, the viscosity dependence of the decay kinetics and the complementary decay profdes seen for free and tethered radical pairs. The long time limit of the survival probability for geminate cumyl free radicals follows the reciprocal square root of time decay predicted by the Smoluchowski diffusion equation.

1. Intxoduction Solvent caging of potential reaction partners has been frequently invoked in the analysis of condensed phase chemical reactions [ I]. Following the pioneering work of Frank and Rabinowich [2], who proposed that the solvent dependence of iodine pho-todissociation yields arises from collision-induced atom recombination, the cage concept has been extensively used in discussions of geminate radical pair reactions in liquids 131, as well as in amorphous [ 4 ] and crystalline [ 51 solids. The role of geminate recombination in the kinetic estimation of bond dissociation energies has also been discussed recently in detail [ 61, Cage effects for encounter pairs formed in free radical self-termination reactions have also been explored. Because the rate of diffusive encounter is much slower than the rate of in cage reaction, encounter pair rate processes enter the self-termination rate constant as an effective reaction yield [ 71. Despite the compelling evidence which demon-

strates the importance of solvent cage effects in determining bond dissociation yields, direct time-resolved observations of geminate free radical pairs have rarely been reported and the quantitative description of their dynamics is not well understood. Cage recombination is known to occur in the picosecond time regime and has been studied most extensively for the iodine photodissociation reaction [S-lo]. This work has shown that primary recombination for I2 occurs within 2 ps and is followed by slow vibrational thermalization. The thermalization rate is determined by the efficiency of collisional quenching by the solvent. Recent picosecond Raman scattering experiments have provided direct evidence for the relatively slow vibrational thermalization in this system [ IO,111. The time scales for geminate recombination of polyatomic free radical pairs, however, are expected to be quite different [ 121. Vibrational thermalization occurs more rapidly than in molecular iodine (< 10 ps), while steric constraints on the successful

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4 0

NJ

0

n

Scheme 1.

recombination of anisotropic radicals [ 131 together with the low mobility of large polyatomic fragments combine to make initial pair lifetimes longer. When the electronic spectrum of each radical is well separated from the spectrum of the combination product, recombination kinetics are readily measured by transient optical absorption techniques. Geminate recombination of phenyl thiyl radical pairs formed by photolysis of diphenyl disulfide was recently studied by this method [ 121. It was found that the SOL vent and temperature dependence of the time-dependent survival probability could be described using simple diffusion theory in a homogeneous medium. Here we report picosecond measurements of the photodissociation products of azocumene (I) and 3,8-diphenyl- 1,Zdiaza- I-cyclooctene (II ) shown in scheme 1. Both compounds form carbon centered radical pairs through cleavage of a C-N bond followed by rapid loss of molecular nitrogen. For azocumene the geminate radical pairs may undergo recombination and disproportionation reactions or separate into statistically independent radicals. The azooctene shown in scheme 1 is a cyclic version of azocumene in which the two benzylic carbon atoms are linked by a Cq alkane chain. Photolysis of this compound yields a 1,6-biradical which can be viewed as a tethered geminate radical pair. Transient decays following photolysis of either compound are shown to be solvent viscosity dependent. The recombination of cumyl radical pairs at times greater than 120 ps follows the t-‘I* power law that is expected from macroscopic diffusion theory.

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ods reported in the literature #‘v2,recrystallized from hexane and characterized by NMR, UV and melting point. Azocumene was obtained exclusively as the trans-isomer since the cis-isomer is thermally unstable at room temperature. The cyclic azo compound was obtained as the trans-diphenyl-trans azo isomer. Attempts to detect fluorescent emission from either compound in hexane solution using time-correlated single-photon counting were unsuccessful. These experiments placed an upper limit of 10e4 on the fluorescence quantum yield. Transient absorption measurements were carried out in a series of hydrocarbon solvents at 22°C and a concentration of 0.05-0.10 M. At these concentrations, transient absorption signals from the pure solvents did not interfere with detection of the azo compound photolysis products. Each solution was photolyzed at 354.7 nm using 10 mJ/cm2 from an actively and passively mode-locked Nd :YAG laser with a 35 ps pulse width (fwhm) operating at 10 Hz. The initial photoproducts were monitored by transient absorption at 321.5 nm in anticipation of detecting the benzylic radical chromophore that would be fornied by cleavage of at least one C-N bond in either compound. The benzylic chromophore has a sharp UV absorption spectrum in the 300-330 nm region [ 161 with an extinction coefficient of 8000 M-l cm-’ at 320 nm. The 32 1S nm probe wavelength was generated by stimulated anti-Stokes Raman shifting [ 171 the Nd:YAG third harmonic frequency in methane gas at 350 psi. The kinetic measurements were carried out in a 1 mm path length cell connected to a recirculating flow system with 15 ml total volume. An electromechanical shutter inserted into the path of the excitation light pulse allowed the optical transmission of the solution to be monitored alternately with and without the excitation pulse at each time delay. The shutter motion was automated using the toggled output of a Stanford Research Systems gated integrator to trigger the shutter driver. The combination of this baseline sampling procedure together with the recirculating flow system compensated for PI The ketazine from acetophcnone and hydrazine was treated with Cl2 and the resulting dichloridemethylated withAlMe,

2. Experimental methods and materials Both azo compounds were synthesized by meth10

according to ref. [ 141. m Thepreparation of azocumene 1 by the methodmentionedin ref. [ 151 yields exclusively the trans-diphenyl-transazo isomer.

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small variations in sample transmission which occurred during the time required to record a single decay curve. Without repetitive baseline sampling a small change in the average sample transmission would cause a large error in the apparent transient decay profile. By comparing successive time scans using baseline sampling with brief measurements of the peak and plateau transient absorption strengths for a fresh solution, decay profiles were shown to be free of experimental artifacts which potentially arise from a decrease in the initial azo concentration or the build up of stable photoproducts.

3. Isomerization,bond cleavageand geminate recombination Azocumene has an activation free energy of 24.9 kcal/mol [ 181 for C-N bond dissociation and is susceptible to both thermal and photolytic cleavage. Photodissociation occurs with an effective quantum yield of 0.6 using broad band near UV irradiation [ 19 1. The stable products that are formed consist of molecular nitrogen, the combination and disproportionation products formed by cumyi radical pairs reacting at their benzylic position and small amounts of ortho and para semibenzene combination products [ 201, The photodissociation mechanism in azo compounds may involve two distinct pathways leading eventually to free radical intermediates. Direct cleavage of a C-N bond from the lowest excited singlet state can occur. In addition, an indirect channel arises via photoisomerization of trans-azocumene to form the unstable c&isomer, followed by thermal cleavage of a C-N bond. The relative yields for the two reaction pathways in azocumene are not known, although for many azoalkanes in the gas phase the isomerization channel dominates [ 2 11. A recent nanosecond transient absorption study of azocumene [ 221 showed that primary products from both channels appear in less than 1 ns with comparable transient absorption strengths, These authors also demonstrated that thermal decay of the cis-isomer occurs with a lifetime of 9 us and thus cannot be the source of free radicals seen on subnanosecond time scales. For azocumene, it is reasonable to expect that direct bond cleavage makes a greater contribution to excited state decay than for azoalkanes since the bulky

phenyl groups hinder isomerization and also provide resonance stabilization for the formation of free radical products. Fig. 1 shows picosecond transient absorption measurements of the azocumene photolysis products in a series of alkane solvents. The probe wavelength was selected as 32 1.5 nm in anticipation of detecting cumy1 free radicals formed by direct photodissociation. The initial decay and the long time plateau ab sorption are seen to depend strongly on solvent viscosity. The viscosity dependence is consistent with geminate recombination of free radical pairs for which the initial decay reflects the competition between recombination and diffusive separation and the long time plateau arises from the fraction of radicals which diffuse into the bulk solution. The separated radicals are expected to undergo bimolecular reactions on a time scale which is longer than that depicted in fig. 1. The viscosity dependence seen in fig. 1 is qualitatively similar to that reported for geminate recombination of phenyl thiyl free radical pairs [ 121 in a series of alkane liquids. To test a free radical assignment for the azocumene photolysis

I ’

I

I

I

400

600

% a

0

200 Time (pe)

Fig. 1. Transient absorption decay at 32 1.5nm following 354.7

nm photolysisof azoeumenein hexane,hexadeeaneand paraffin oil. Smoothsolid lines are generated from eq. (5) of the text using~=1.3,~=0.24and~~=13 ps in hexane; r=1.3,1=1.8 and q,=BO ps in hexadecane and K= 1.3, A=20.0 and ro=900 ps h para!Xn oil.

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products, transient absorption measurements at three discrete wavelengths were performed. The relative absorption intensities at 309, 321 and 355 nm following 266 nm photolysis of azocumene in butanol were found to be 0.6/ 1.0/O.1, which is in agreement with the values 0.7/1.0/0.2 previously reported [ 161 for the cumyl free radical. Previous studies of N2 evolution, azocumene consumption and the yield for free radical scavenging have provided useful information on geminate radical recombination following dissociation of azocumene. The yield of free radical scavenging per dissociation event, regardless of whether thermal of photochemical means are used for bond breaking, was found to depend on solvent viscosity and was interpreted as a viscosity-dependent geminate recombination yield. A recombination yield of 26% has been reported [ 191 for azocumene in benzene solution using this technique. Geminate recombination yields have also been shown to respond to changes in viscosity caused by varying the solvent composition [ 191, the solvent temperature or through the application of high external pressures [ 241. It is revealing that both the rate of nitrogen evolution and the rate of azo disappearance are independent of viscosity. This has been used to argue for a rapid two bond dissociation mechanism which forms molecular nitrogen and a geminate cumyl free radical pair. Since geminate recombination of cumyl free radicals does not reform the parent azo compound or reincorporate extruded nitrogen, solvent properties would have no effect on the evolution of NZ or the loss of azocumene. The cyclic azo compound shown in scheme 1 may be expected to exhibit the same primary photochemistry as its acyclic analogue, with the exception that the radical pairs will be tethered by a C4 alkyl chain. Diffusive separation of the two radical centers is impossible and all centers must eventually recombine. The absorption transients displayed in fig. 2 are consistent with this interpretation. The decays can be approximately described as exponential with lifetimes of 0.55 ns in hexane, 0.93 ns in tetradecane and 1.7 ns in paraffin oil. Biradical cyclization reactions such as those responsible for the transient absorption decays in fig. 2 have been discussed by several authors [25-271. In the model discussed by Doi [27] and Szabo et al. [ 261 the relative motion 12

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CHEMICAL PHYSICS LETTERS

z

a

WI lo

I

I

I

1

2

3

Time (ns)

Fig. 2. Transient absorption de-cayat 321.5 nm following 354.7 nm photolysis of 3,8diphenyl-l,2diaza-lqxlooctene in hexane, tetradecane and paraffln oil. Smooth solid lines are fits to a single-exponential decay and include a constant background ab sorption on the order of 2-S% of the maximum signal.

of the reactive chain ends is described as the diffusion of two harmonically bound particles. When the mean particle separation is large compared to their hard sphere collision diameter, the biradical survival probability is approximately described by a singleexponential decay. The decay lifetime depends on the rate of diffusive encounters between the two chain ends, which is the source of the viscosity-dependent lifetime, and on the reaction yield per collision. The solvent dependence of the biradical lifetime seen in fig. 2 is similar to that observed for intramolecular fluorescence quenching and the rate of intramolecular exciplex formation when the two active groups are separated by an alkane chain [ 28 ] H3.

4. The diffusion model of geminatepair recombination The decay of the transient intermediate seen in fig. 1 extends over a few hundred picoseconds in para3 The influence of viscosity on the lifetimes of triplet biradicals has been reported in ref. [ 29 1.

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affrn oil and is much longer than would be expected for primary recombination within the initial solvent cage. Molecular dynamics simulations [ 301 and pi-

cosecond studies of I2 recombination [S-IO] have shown that cage lifetimes are on the order of l-10 ps, while secondary recombination of iodine atoms occurs over several hundred picoseconds. For polyatomic radicals the initial cage pair may persist for a longer time than for two iodine atoms due to steric constraints on the recombination reaction and the relatively slow rates of rotational and translational motion, The’ recombination signals seen here for azocumene and those reported previously for diphenyl disulfide [ 121 arc more reasonably assigned to diffusive secondary recombination of geminate pairs than to primary cage recombination. Seccndary geminate recombination has much in common with the initial time-dependent rate seen in fluorescence quenching experiments at high quencher concentrations [ 3 11. Diffusion theory has been used to describe both geminate recombination and fluorescence quenching kinetics for both neutral and electrically charged species. Its application to fluorescence quenching has been critically discussed [ 32 1. The application of diffusion theory to geminate recombination [ 311, describes the average survival probability for a large number of free radical pairs which were created with the same initial conditions and subsequently move independently of one another in response to fluctuations in collisions with the surrounding fluid. The fraction of the original number of pairs which on average will be found within a given volume element measured radially outward from one radical center is described by the laws of probability using the pair distribution funo tion p( r, t) . When each radical exhibits only a spherically symmetric interaction potential w(r) with its geminate partner and when the rotational motion of each radical is much more rapid than its translational motion, the probability distribution obeys the diffusion equation [ 33 ]

ap

Da

at

l-2ar

-=--

(?;+p$p),

(1)

where D=D, +D2 is tht sum of the diffusion coefficients for the individual radicals and B= ( kB2”)- ’ is the reciprocal thermal energy. The diffusion equa-

tion can also be interpreted in terms of a hopping

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model for molecular transport with D=A2/q where A is the average jump length and r is the average time between jumps. Eq. (1) is obtained from the hopping model in the limit of a large number of steps, although the mean displacement and the total elapsed time may remain quite small [ 341, Recombination and disproportionation reactions are introduced into the diffusion model using the radiation boundary condition [ 351. This approach balances the diffusive flux and the reactive tlux when the radical pairs reach their collision diameter u

(2) The reaction velocity V, (cm/s) is assumed to include a steric factor for the recombination of chemically anisotropic free radicals [ 361. Rotational dynamics will enter eqs. ( 1) and (2) unless the translational time scale is long and rotations are effectively averaged [ 361. ,Even in the limit of fast rotational motion, the time-averaged reaction velocity will in general retain a temperature and solvent viscosity dependence which reflects the competition between rotational motion, chemical reaction and diffusive separation. The quantity measured in this work is thi time-dependent radical pair survival probability q(t) which is described within the diffusion model by integrating the pair distribution function over all radicat pair separations m

(3) (I Analytical solutions for the survival probability are available for two cases: non-interacting geminate pairs, w(r) =O, and for pairs interacting by the Coulomb force [ 371. Onsager originally considered [ 38 ] the recombination of ion pairs in solution in terms of eq. ( 1). His predictions for how the escape yield depends on temperature, applied electric field and ion mobility ‘have been confirmed for a .variety of reaction systems [39]. More recently the timedependent Onsaget problem has been shown to give good agreement with experimental recombination kinetics for electron-cation pairs formed by the photoionization of liquid solutions [ 401. In the appertdix we consider the effect of rotationally averaged 13

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van der Waals forces on the escape probability for electrically neutral radical pairs and find than at room temperature they enhance the recombination yield over non-interacting pairs by no more than 10-l 5%. Considering the severe approximations inherent in using macroscopic diffusion theory for this problem, this weak pair interaction energy will be neglected in discussing the main features of the experimental results. The pair distribution function for non-interacting radicals has been elegantly derived using the method of Laplace transforms for a mathematically equivalent problem of heat conduction in solids [ 4 11. The instantaneous source function

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where q(m) = 1-A/K( 1+A) is the pair escape prob ability. Eq. (6) is a reasonable approximation to the fullsolutionfortimest~[j(lt~)-2tj(rc-1)2]r~ The functional form of the solution at long times is independent of the initial pair separation and therefore does not depend on assumptions concerning the spatial distribution of initial pairs. The combination of parameters I-& ( 1i-1) can be determined from the ratio of the slope to the infinite time intercept in a plot of q(f) versus l/4_ Fii. 3 illustrates this treatment of the data for azocumene and for data ob tamed previously [ 12 ] for phenyl thiyl radical pairs formed by photolysis of dipheayl disulfide. Linear

x{exp[ - (r-ro)‘/4Dt] -exp[ - (r+ro)2/4Df]}

(4)

is a solution to eq. (1) in an infinite medium with the initial condition u(r, 0) =O when r is not equal to ro, the initial pair separation. The full solution is obtained by setting p(r, t) =u(I; t) +v(r, t) where v( r, t ) is a second solution to eq. ( 1) which vanishes at t=O and causes p(r, t) to satisfy the boundary condition in eq. (2). The solution is obtained as a Laplace transform and then inverted using standard tables. The resulting distribution function can be integrated over all separations to give the timadependent pair survival probability [ 421 q(t)=l-

-exp(l

1.01

2.0,

2.0 II v$Gj

2.5

I

I

I

I 1.5

I

I

2.0

2.5

1.5

&[4S) +A) (K-1)

xexp(lt1)2rerf

(

(1t~)r’/2+~

)I, (5)

where x=r,/a is the ratio of the initial pair separation r. to the hard sphere collision diameter a, A=kJkD is the ratio of the reaction rate constant ~=4n02v, to the diffusion rate constant k,=hlraD and r= t/r,, is the ratio of the time elapsed since pair formation t to the diffusive time constant r. = a2/D. At long times the survival probability in eq. (5) approaches a reciprocal square root of time decay

(6) I4

Hexadeeane

I.01 1.0

II /to Fig. 3. Upper: Plot of the azocumene transient absorption decays from fg. 1 versus I-‘? The time range extends from 120 ps to 1.2 ns after photolysis. The slope-to-intercept ratios predicted by eq. (6) of the text are 3.3 ps’” in hexadecane and 20 ps”’ in parat% oil. Lower: Same treatment of the data for geminaterecombination of phenyl tbiyl free radical pairs reported in RX [ 121. The slope&-intercept ratios are IO ps”* in decalin and 4.1 ps”* in dodecane.

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plots are obtained for both experiments and are consistent with a diffusion mode1 for secondary geminate pair recombination. It can bc seen that the experimental slopes are steeper in high viscosity liquids than at low viscosities. If a Stokes-Einstein fotmulation of the diffusion coefficient is assumed, for which D scales as 1/of, the slope to intercept ratio has a viscosity dependence of #/’ for fast reactions and q31zfor slow reactions. The slope to intercept ratios observed for cumyl radical recombination is 3.3 ps”* in hexadecane (q =2.21 cP) and 20 p@ in paraffin oil (t/w25 cP). The solid lines in fg. 1 were generated using the full solution to the diffusion equation. Eq. (5) was convoluted with two 35 ps (fwhm) Gaussian functions to represent the pump and probe light pulse time profiles. To reduce the number of independent parameters that appear in the diffusion model, the Stokes-Einstein form of the diffusion coefficient was assumed. This requires Aand r. to scale linearly with solvent viscosity when recombination kinetics are measured in a series of different liquids #4. In addition it was assumed that x is independent of the molecular nature of the solvent and has the same value in all three liquids. Using this approach, a single choice of the three independent parameters is used to describe all three decays. The values of K, 1 and r. that give simulated decays which closely match the experiments are given in the caption to fig. 1. Ao cording to these parameters, the long time limit of eq. (5) is valid for times greater than 7 ps in hexadecane and 22 ps in paraffin oil. The time scale for secondary geminate recombination can be recovered from the convoluted decays by substituting the fitting parameters from the cap tion to fig. 1 into eq. (5 ). This procedure gives first half-lives of 75 ps in paraffin oil and 23 ps in hexadecane. Each successive half-life for geminate recombination is longer than the previous one and with the time resolution of the current apparatus, only recombination events beyond the first half-Me are resolved in hexadecane. In liquid hexane recombination was not observed, indicating a high radical

mobility in this solvent. A first half-life of no more than 11ps is indicated by the data. A similar result was obtained previously for geminate recombination of phenyl thiyl radicals in hexane [ 12] _The yield for geminate recombination according to the diffusion mode1 is given by eq. (5) as ~-CCLUsing the values of K, 1 and r. that described the kinetic measurements, we estimate recombination yields of 58% in paraffin oil, 44% in hexadecane and no more than 14% in hexane at room temperature. The yields cannot be taken directly from fig. 1 because early time recombination is not fully resolved. The diffusion coefficient for a single cumyl radical (Dl ) can be estimated from the data in fig. 1 using the relation T~=o*/(D, tD2). Assuming a=6 A, we obtain diffusion coefficients of 1.4~ 10T4 cm2/s in hexane, 2.3 x 10e5 cm2/s in hexadccane and 2.0~ 10m6cm*/ s in paraffin oil. The diffusion coefficients estimated in this manner are 3-4 times larger than the Spernol-Wirtz [43] estimates of diffusion with microfriction corrections. This latter model gives diffusion coefficients of 5.6x 10D6 cm2/s for cumene in hexadecane and 4.1 x lo-’ cm*/s in hexane. Part of this discrepancy may lie in our use of a single initial pair separation instead of a distribution of radical pair thermalization lengths.

5. Conclusions Geminate radical pair recombination following direct C-N bond cleavage from the lowest excited singlet state of azocumene has been studied by transient optical absorption. Cumyl free radical pairs are formed more rapidly ( Q 10 ps) than the laser pulse width and subsequently undergo geminate recombination which, in viscous alkane liquids, extends over several hundred picoseconds. Primary cage recombination is thought to occur more rapidly than can be followed with the time resolution of the current apparatus. The decays of the transient absorption signals seen here are interpreted as secondary geminate recombination and follow the reciprocal square root of time behavior

M Relativevaluesof d and r, at diierent solventviscositiesq are given by I’ /k=rb/o =q’ /q based on the Stokes-Einsteinde scription of the mutual diffusioncoeffkient D.

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that is expected for

macroscopic diffusion. Secondary recombination is described in this model as diffusive motion over a few molecular diameters to form encounter pairs whose fate is determined by more rapid time scale 15

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events. The rapid recombination and separation events during each pair collision are collectively represented in the diffusion model by an effective reaction velocity. The direct observation of primary geminate pair recombination and collision pair dynamics in liquids remains an unfulfilled goal for polyatomic free radicals.

where Z(z) is defined by the integral I(z)= yr-2 exI@w(r)] dr.

q(m)=l-

Appendix It is interesting to estimate the survival probability for geminate free radical pairs which recombine under the influence of van der Waals interactions. For uncharged pairs with non-vanishing dipole moment and pohuizability, the long range van der Waals forces responsible for solute-solute interactions include the electromagnetic induction and orientation forces and the London dispersion force. All three interactions give rise to a potential of the form w(r) = - (y/r6 for rotationally averaged radicals where IYdepends on both the radical and solvent properties [ 441. It has been shown previously [45] that the survival probability q(m) for geminate pairs subject to any spherically symmetric interaction potential can be obtained without solving the corresponding timedependent boundary value problem. In the notation used here the survival probability is given [45] by a(cn)=l16

aJ-exp[ -Ma) 1I(& l+aAexp[ -pw(a)] I(a) ’

(A.11

(A-2)

For an interaction potential of the form w( r ) = - ac/ r6, eq. (A.2) can be expressed as an incomplete gamma function and the survival probability takes the form

Acknowledgement The authors acknowledge the Exxon Corporate Research Laboratories for use of their facilities. CD acknowledges the National Science Foundation for support (CHE-8722164) and Professor N.J. Turro for use of his facilities. TWS wishes to thank Professor C.L. Braun for helpful discussions concerning alkyl chain dynamics in liquids. The authors also thank Dr. Michael Drake for attempting single-photon counting fluorescence lifetime measurements on azocumene and Dr. James A. Franz for performing Spemol-Wirtz calculations of the microfiiction coefficients for diffusion in hexane and hexadecane.

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MKU)

4 1+w4 1

xexrWWW-WNl~.

(A.3)

When the van der Waals interaction energy for a radical pair separated by one collision diameter is less than the thermal energy, w (a)lkeT< 1, x(z) is given by the expansion [ 46 ] (A.4) For non-interacting pairs eq. (A.4) gives x= 1 and we recover the long time survival probability predicted by eq. ( 5 ) . For a finite pair interaction energy x> 1, the survival probability is reduced by an amount depending on the initial pair separation KU and the relative rates of reaction and diffusion. An estimate of q(m) for cumyl radical recombination can be made using the interaction potential determined [47] from the second viral coefficient for benzene vapor, cy= 4.1 x 10’ A”cal/mol. The van der Waals interaction in a liquid medium will bc lower than in gas phase due to the self-interaction energy of each radical with the surrounding solvent [ 44 1.Assuming a refractive index of 1.43 for the solvent and 1,59 for the radical, the dispersion energy in a solvent is reduced from its value in free space by a factor of approximately 20. Using K= 1.3 and a=6 A, the van der Waals interaction between radical pairs reduces the survival probability by 7% in hexadecane and 13%in paraffm oil compared to the calculated value for non-interacting pairs.

References [ 1 ] J.W. Moore and RG. Pearson, Kinetics and mechanism ( Wiley-Interscience, New York, 1981) pp. 237-244.

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J. Bullot, P. Cordier and M. Gauthier, I. Chem. Phys. 69 (1978) 4908; M. Yokoyama, Y. Endo, A. Matsubara and H. Mikawa, J. Chcm. Phys. 75 ( 1981) 3006. (401 C.L. Braun and T.W. Scott, Rad. Phys. Chem. 32 (1988) 315; J. Phys. Chem. 87 (1983) 4776; 91 (1987) 4436; T.W. Scott and CL. Braun, Chem. Phys. Letters 127 (1986) 501; Can. J. Chem. 63 (1985) 228. [41] H.S. Carslaw and J.C. Jaeger, Conduction of heat in solids (OxfordUniv. Press, Oxford, 1959) pp. 357-359.

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