Thermalization of low energy electrons in liquid methylcyclohexane studied by the photoassisted ion pair separation technique

Chemical Physics 265 (2001) 87±104

www.elsevier.nl/locate/chemphys

Thermalization of low energy electrons in liquid methylcyclohexane studied by the photoassisted ion pair separation technique Leonid V. Lukin *, Alexander A. Balakin Institute for Energy Problems of Chemical Physics (Branch), Russian Academy of Sciences, Chernogolovka 142432, Moscow Region, Russian Federation Received 24 August 2000

Abstract A spectral dependence of the charge carrier quantum yield in liquid methylcyclohexane was investigated on photoexcitation of geminate electrons at wavelengths between 425 and 867 nm. The electrons were produced by two photon ionization of the anthracene admixture with the UV laser pulse (wavelength 308 nm, duration about 20 ns) and, then, were excited by the dye laser pulse delayed for a time of 25 ns with respect to the UV one. The increase of photocurrent caused by an additional dye laser pulse action was observed as temperature decreased below 230 K. It has been found that the spectrum of the photocurrent enhancement in the range of dye laser radiation wavelengths from 425 to 867 nm is close to that of the trapped electron optical absorption. The increase of photocurrent has been attributed to photogeneration of ``hot'' quasi-free electrons arising from a photon absorption by geminate trapped electrons. Basing on the di€usion model of photostimulated ion pair dissociation, the thermalization length of 4.2±4.7 nm was determined for quasi-free electrons with an initial kinetic energy within the range from about 0.9 to 2.4 eV. The energy dissipation rate and e€ective mean free path were evaluated for the quasi-free electrons with the energy of about 1 eV in hydrocarbon matrix. Ó 2001 Published by Elsevier Science B.V. Keywords: Electron; Thermalization; Localization; Solvation; Geminate recombination

1. Introduction Thermalization and relaxation processes of low energy electrons in molecular media are a central problem of photo and radiation chemistry [1±6] and modern femtochemistry [6±39]. During the last decade, most of the experimental researches on an electron relaxation in liquids have been focused *

Corresponding author. Tel.: +7-902-680-1645; fax: +7-902680-3573/+7-095-137-34-79. E-mail address: [email protected] (L.V. Lukin).

on investigations of the time resolved optical absorption spectrum of solvated electrons produced either by photoionization of solutes and solvent [7±16] or by photodetachment from negative ions [6±8,17±29]. A series of recent experimental works has also been devoted to the investigation of the dynamical changes of the optical absorption spectrum of the equilibrium solvated electron in polar solvents caused by its additional photoexcitation by the femtosecond light pulse [30±38]. These studies have revealed the existence of a nonequilibrium p-like excited state of a localized

0301-0104/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 3 0 1 - 0 1 0 4 ( 0 1 ) 0 0 2 6 0 - 9

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electron which relaxes to the fully solvated electron on a subpicosecond time scale. Despite the advances in the studies of the transient absorption of the non-equilibrium localized electron and its ultrafast reactivity with molecules and ions [6±39], a knowledge of the thermalization processes of a quasi-free ``hot'' electron and mechanisms of its initial localization in molecular media, especially in non-polar liquids, remains limited. It is assumed that after injection of the quasi-free electron with excess energy of several electronvolts into liquid the formation of the equilibrium solvated electron proceeds through fast thermalization stages including energy dissipation due to excitation of surrounding medium, localization in a shallow trap and further solvation due to molecular rearrangement of the solvent. The localization and solvation of electrons photogenerated in molecular liquids occur at short distances from the parent (geminate) cations (1±2 nm in net water [10] and 2± 10 nm in organic liquids [7,8,40]). The thermalization ranges of low energy quasi-free electrons relate to the initial spatial distribution of the ion pair produced by photoionization of a solute or solvent molecule. However, in this case the mechanism of electron thermalization within the geminate electron±ion pair is very complicated by both a fast geminate recombination of a presolvated electron with its parent cation, which can compete with electron localization, and an uncertainty in the initial electron energy because of the possible relaxation of excited states of the electron photodonor molecule prior to its ionization. In addition to the traditional approaches to the excess electron relaxation studies based on the transient optical absorption measurements, in the works [41±51] an experimental method, called the photon assisted ion pair separation (PAIPS) method, has been developed to study thermalization and localization of low energy electrons in organic liquids and glasses. The method is based on an increase of the number of free charge carriers caused by an absorption of light by geminate electrons, i.e. electrons bound by the Coulomb attraction with their parent cations. The observed enhancement of the free charge carrier yield has been attributed [40,41,43±45] to a photoinduced transition from an equilibrium localized state, e ,

(trapped or solvated electron) to a quasi-free state, e0 , with the initial kinetic energy above thermal one. Such the ``hot'' electron ``cools'' down quickly and forms a new trapped state, e± , at a distance from the primary site of the photon absorption: hm ‡ e ! e0 ! e

…1†

The photostimulated ion pair dissociation has been observed, at ®rst, in c-irradiated 3-methylpentane (3MP) glass at 77 K [41] where trapped electrons live long enough. The stationary light from the Xe lamp was used to illuminate the sample in the spectral region of the trapped electron optical absorption band in this ®rst experiment. Basing on the experimental data [41], Yakovlev [43] has determined a dependence of the thermalization distance for the quasi-free electron in 3MP glass on its initial kinetic energy in the range from 0.1 to 1.5 eV. For non-polar liquids, where the lifetime of a geminate electron±cation pair, as a rule, does not exceed 10 6 s even at low temperatures, the two pulse photoconductivity measurements are used in the PAIPS method [44±51]. The pulse of the UV light produces geminate electron±ion pairs …e ; A‡ † due to two photon ionization of a solute molecule A, and then the second (probe) pulse of light irradiates the sample in the electron optical absorption band resulting in photoexcitation of the localized (trapped) electrons. The enhancement of a charge carrier yield caused by additional action of the second pulse is measured. It has been shown [42,46] that measurements of the free ion yield enhancement due to photoexcitation of geminate electrons allow one to determine the thermalization distance of the quasi-free electron and its e€ective temperature. It should be noted that, contrary to the case of photoionization of molecules, the quasi-free electron produced in reaction (1) escapes from the neutral center, and its photogeneration and thermalization occur at a distance away from the Coulomb center. So, the electron thermalization distances obtained by the PAIPS technique may be di€erent from the distances between the charges in the pair …e ; A‡ † determined in conventional

L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

photoionization studies. The correlation between the dependencies of the thermalization distance on the initial excitation energy, obtained for both approaches, can give a new insight into mechanism of photoionization in condensed media. As to organic liquids, the PAIPS method has been applied so far, mainly, to study the geminate recombination kinetics of the pair …e ; A‡ † [47± 52]. Electron thermalization distances have been determined by the method only for two wavelengths, 694 and 1060 nm (photoexcitation by the pulse ruby and Nd:YAG lasers, respectively) in some solvents [46,47,53]. To determine the thermalization distance of a quasi-free electron in an organic liquid as a function of its initial kinetic energy, in the present work we have applied the PAIPS approach to electrons in liquid methylcyclohexane (MCH) exciting them at di€erent wavelengths of the probe pulse from 425 to 867 nm. So, we were able to change the initial energy of e0 from about 0.9 to 2.4 eV. The second goal of the present work is to continue in studies on optically stimulated charge pair dissociation in organic liquids. The point is that two pulse photoconductivity measurements have been used as a new tool in analytical chemistry of solutions [54±57]. So, a consideration of contribution from geminate electron photoexcitation to the photocurrent enhancement, observed at di€erent wavelengths of a probe pulse light, is necessary for increasing sensitivity of the measurements. It should also be noted that the idea of the e€ect of the probe laser pulse on the d.c. conductivity of a sample has been applied in recent time resolved femtosecond experiments to clarify a mechanism of formation of free charge carriers in p-conjugated polymers [58]. Because of the high concentration of charge carriers created by the UV laser pulse in conductive polymers, the mechanism of the conductivity change caused by the probe pulse action is complicated by inter-pair recombination of charge carriers [58]. In hydrocarbon solutions, choosing small concentrations of the solute molecules, we can consider only a formation of free ions due to dissociation of isolated pairs and study the net e€ect of a photon absorption by a geminate electron on the ion pair dissociation probability.

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Below we present investigations of the two pulse photoconductivity of the solution of anthracene in liquid MCH induced by the cooperative action of a nanosecond excimer laser pulse (308 nm) and a dye laser pulse delayed for a time with respect to the UV one. To extend the lifetime of geminate electron±cation pairs …e ; A‡ † and make them accessible for investigations with the nanosecond light sources used, the experiments were carried out at low temperatures. MCH was chosen as a solvent because the mobility of excess electrons in MCH was known in a wide temperature range [59], and therefore, the lifetime of the geminate pairs could be determined. In addition, the decay kinetics of the PAIPS signal in the methylcyclohexane solutions has been studied previously in the time resolved nanosecond [45] and picosecond [47± 49] experiments. 2. Experimental section MCH was degassed by freeze±pump±thaw cycles and puri®ed from electron acceptors over a metal sodium mirror and molecular sieves as described in Ref. [60]. MCH was distilled into a photoconductivity cell and sealed under vacuum of 10 5 Torr. After the puri®cation a lifetime of excess electrons in the pure solvent with respect to reactions with trace impurities was about 9  10 6 s at room temperature. The time was found from the decay kinetics of an electron photocurrent induced by the powerful pulse of the focused excimer laser beam (308 nm, 20 ns). The light energy in this experiment with the pure solvent was about 100 mJ per pulse. All the further photoconductivity experiments were carried out using the 7:1  10 5 M solution of anthracene in MCH which was prepared by the procedure described in Ref. [60]. A quartz cell for the photoconductivity measurements contained three plane stainless steel electrodes: a high voltage electrode, a collector electrode (1.65 cm long, 0.35 cm wide) and a grounded one. The grounded electrode was made as a frame surrounding the collector one, and both the electrodes were positioned in the same plane. The distance between the high voltage and collector electrodes was d ˆ 0:3 cm.

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Fig. 1. Schematic diagram of the experimental setup for measuring two pulse photoconductivity: QP denotes a splitting quartz plate, QL is a quartz lens, PT is a phototube.

To provide the temperature stability, the photoconductivity cell was placed in a quartz dewar with optical quartz windows and cooled by a ¯ow of the cold nitrogen vapor. A basic diagram of the experimental setup for photoconductivity investigations is schematically shown in Fig. 1. Electrons in the anthracene/MCH solution were produced by two photon ionization of anthracene molecules with the UV pulse of the Xe±Cl excimer laser EMG-101 MSC (Lambda Physik, radiation wavelength 308 nm, pulse duration about 20 ns). Only a small part of the excimer laser beam, re¯ected from a quartz ¯at plate and slightly focused by a quartz lens (focal length 70 cm), was used for ionization of the solution between the collector and high voltage electrodes of the photoconductivity cell. The main part of the 308 nm laser beam had pumped a home-made dye laser consisted of an optical laser cell with a dye solution placed in a non-selective resonator. There were several laser cells with solutions of di€erent dyes. Each laser cell with a dye can be placed, in turn, inside the resonator at the same ®xed position that allowed us to change a dye quickly and irradiate the sample at di€erent wavelengths from 425 to 867 nm in the same experimental conditions, i.e. at the same temperature, the UV light

energy and at the same spatial distribution of the UV light intensity within the sample. The following dyes were used: Stilbene 420 (Exciton, wavelength at the peak of generation k ˆ 425 nm), Coumarin 47 (domestic production, k ˆ 456 nm), Rhodamine 6G (domestic, k ˆ 585 nm), Pyridine 2 (domestic, k ˆ 740 nm) and LDS-867 (Exciton, k ˆ 867 nm). A spectral width of the dye laser radiation was not more than 4 nm. The excimer laser was operated in the single pulse regime. The di€erence in the optical paths (about 7.5 m) provided the 25 ns delay of the dye laser pulse with respect to the ionization one. The dye laser beam was directed along the length of the collector electrode perpendicularly to the direction of the UV beam, see Fig. 2. The cross-sectional areas of both light beams were con®ned by diaphragms. The apertures of the diaphragms were s1 ˆ 0:3  1:65 cm2 and s2 ˆ 0:3  0:3 cm2 for the UV and dye laser beams, respectively. The quartz lens (Fig. 1) provided a homogeneous irradiation by the UV pulse of the anthracene solution over all space of liquid between the collector and high voltage electrodes. The dye laser beam was focused by a lens (with the focal length of 1 m) to the sample so that the spot of the dye laser light, passed through the diaphragm s2 , was about 0:2  0:2 cm2 .

Fig. 2. The spatial arrangement of electrodes within the photoconductivity cell. The arrows hm1 and hm2 show the directions of the UV and dye laser beams, respectively; R is the load resistor.

L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

Fig. 3. The temporal pro®les of the light intensity for the 308 nm laser pulse (d) and the Rhodamine 6G radiation pulse (s). The origin time t ˆ 0 corresponds to the peak of the UV light intensity. Dashed line shows the time pro®le g1 …t† calculated by Eq. (B.4) at tS ˆ 10:1 ns (see the Appendix B). All the curves are normalised to unity at maximum.

Fig. 3 demonstrates the relative position of the time pro®les for the UV and Rhodamine 6G dye pulses recorded by two phototubes placed near the photoconductivity cell. The optical time delay, td , between maxima of the light intensity of the pulses was about 25 ns. As seen, the pro®les of both pulses contain the strong peak with the full width at half maximum of about 16±18 ns followed by a more weak and smooth peak in 20±25 ns after the former one. The phototubes, operated in the integration regime and calibrated with a laser energy meter, were used also for relative measurements of the incident light energy of each laser pulse. The light energy of the dye laser pulse was about 4 mJ for Stilbene 420 and 8±9 mJ for Pyridine 2 at the excimer laser output energy of 200 mJ per pulse. The density of the UV light energy falling on the sample was, as a rule, not more than W1 ˆ 12 mJ cm 2 . The d.c. photoconductivity of the anthracene solution was measured with a circuit similar to one described earlier [44,45], see Fig. 2. The voltage across the electrodes was V0 ˆ 2 kV. The collector electrode was connected to an electrometer ampli®er with a high input resistance. The pulse

91

photoionization of the sample resulted in a time dependent voltage drop, u…t†, across a load resistor R (from 2:5  109 to 8  1011 X) which was recorded by the time±voltage recorder. The photocurrent was measured in the integration regime. The load resistor R was taken so large that characteristic time Ceff R was much more than the time, tdr , of an ion drift through the distance between electrodes under the electric ®eld applied. Here, the drift time tdr is equal to d=…F lion † where F ˆ V0 =d ˆ 6:7  103 V cm 1 is the electric ®eld strength at V0 ˆ 2 kV, lion is an ion mobility, and Ceff ˆ 1:6  10 10 F is the e€ective capacity of the cable connecting the collector electrode and the ampli®er. After the laser pulse action the transient voltage u…t† across the resistor R increased with a characteristic time tB from zero to an amplitude u0 followed by the decay with the characteristic time Ceff R because of the discharge of the cable through the load resistor. The buildup time tB was close to the ion drift time estimated at known mobilities of organic ions [61], like the anthracene ion. Since Ceff R  tB , the amplitude of the voltage signal is equal to u0 ˆ Q=Ceff where Q is the integral of the photocurrent, i.e. the total photogenerated electric charge passed through the circuit after the laser pulse. The 308 nm light intensity has been chosen so that the recombination of free charge carriers in a bulk of the sample can be neglected, and the condition tdr  trec

…2†

is ful®lled. Here, trec is the bulk recombination time of charge carriers. The criterion for the condition (2) was a small value of the integral of photocurrent [47] Q ˆ Ceff u0  C0 V0  6:7  10

10

C

…3†

in comparison with the electric charge C0 V0 at the electrodes where C0 ˆ eS=4pd  3:4  10 13 F is the geometrical capacity of the photoconductivity cell, e  2 is dielectric constant of liquid, S  0:58 cm2 is the geometrical area of the collector electrode.

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3. Experimental results 3.1. Photoionization by the 308 nm pulse The photocurrent induced by the 308 nm pulse in the pure solvent was about 200 time smaller than that observed in the solution of anthracene at the same laser intensity. The irradiation of the anthracene solution in MCH by the single pulse of the 308 nm light without an additional excitation by the dye laser pulse had induced a transient photocurrent which lasted until photogenerated ions arrived at the electrodes. The build-up time of the photosignal increased with decreasing temperature from tB  0:05 s at room temperature up to the value of 20± 30 s at temperature T ˆ 160 K. This agrees with the increase of viscosity of MCH [62,63] and, hence, with the expected increase of the drift time of ions with cooling. Fig. 4 demonstrates the dependence of the photosignal amplitude u0 on the energy density of the UV light incident on the sample. As seen, the

Fig. 4. The amplitude of the voltage drop across a load resistor versus the density of the 308 nm light energy per pulse on irradiation of the sample with the 308 nm laser pulse acting alone, T ˆ 194 K. The line corresponds to the power law u0 ˆ kW1n at n ˆ 2:05.

photocurrent induced by the UV pulses was quadratic in the laser intensity. As well as in the previous studies [44,45,47±49], the value of Q decreased with cooling as the UV pulse acted alone. In the temperature range between 160 and 240 K, the photogenerated charge reduced to zero electric ®eld, Q=…1 ‡ eFrC =2kB T † [40], has shown the Arrhenius-type temperature dependence with an activation energy of about 0.13 eV (here, e is the electron charge, rC ˆ e2 = ekB T is the Onsager radius, and kB is the Boltzmann constant). 3.2. Photoconductivity on the joint action of the UV and dye laser pulses The irradiation of the anthracene solution by a dye laser pulse, acting alone, did not induce appreciable photoconductivity. At high temperatures T > 230 K the action of a dye laser pulse after the 308 nm one did not result in (at least, within an accuracy of a few percents) a change of photogenerated electric charge in comparison with that produced by the single UV pulse. However, at lower temperatures T < 220 K the electric charge, Q1‡2 , produced by a joint action of the UV (®rst) and dye laser (second) pulses has been markedly larger than the charge, Q1 , arising from the UV pulse acting alone. As the UV laser intensity varied, the enhancement of the charge DQ ˆ Q1‡2 Q1 , caused by the dye laser pulse, was directly proportional to Q1 . For each dye used, the value of DQ increased directly with increasing intensity of the dye laser radiation. Fig. 5 demonstrates such the linear dependence of DQ on the light energy, G2 , of the dye laser pulse for Pyridine 2. As in the previous studies [44,45,47,51], the relative increase of the photocurrent signal DQ=Q ˆ …Q1‡2 Q1 †=Q1 increased sharply with decreasing temperature, Fig. 6. Taking into account that DQ is proportional to both Q1 and G2 , the spectral behavior of DQ obtained at T ˆ 159 K is presented in Fig. 7a as a dependence of the parameter Y ˆ …DQ=Q†  …hm2 s2 =G2 † on the wavelength of the dye laser light (here, hm2 is the photon energy of the dye laser radiation). The parameter Y can be considered to

L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

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Fig. 5. Enhancement of the photocurrent integral as a function of the light energy of the dye laser pulse at 159 K. The dye is Pyridine 2.

Fig. 7. (a) The spectral dependence of the photocurrent enhancement per an incident photon of the dye laser beam at T ˆ 159 K; Y ˆ …DQ=Q†…hm2 s2 =G2 †. (b) Mean square thermalization distance of the photogenerated quasi-free electron as a function of the dye laser radiation wavelength. The upper axis in Fig. 7b shows the initial electron energy calculated by Eq. (15) at ETR ˆ 0:5 eV. The straight lines represent the best ®t of the linear wavelength dependence of Y and L2 to experimental points.

Fig. 6. Temperature dependence of the photocurrent enhancement DQ=Q due to additional irradiation of the sample by the Pyridine 2 laser pulse at G2 ˆ 7:3 mJ (s) and the temperature dependencies of the probability enhancement Dq=q (± ± ±) calculated at di€erent parameters B of the initial Gaussian form of f0 …r†. The numbers near the vertical arrow are the distances B. All the calculated curves have been scaled to the value of 1.873 at T ˆ 158 K corresponding to the DQ=Q signal observed at this temperature.

be proportional to the photoexcitation cross-section for particles, produced by the UV pulses in the anthracene solution, which are responsible for the photocurrent enhancement. The relative slope, b0 =a0 , of the linear ®t, a0 ‡ b0 k, to the experimental curve Y …k† shown in Fig. 7a did not depend on temperature, at least, within the temperature interval 159±190 K. It should be noted that the obtained spectrum of Y is similar to that of the

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L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

excess electron extinction coecient in liquid MCH [64]: the value of Y increases smoothly with increasing wavelength in the range between 425 and 867 nm.

4. Discussion 4.1. Nature of the photocurrent enhancement The observed enhancement of photocurrent caused by an additional action of the dye laser can naturally be attributed to one photon excitation of species produced by the UV light. Such the particles responsible for the photocurrent enhancement are not the excited singlet and triplet states of the anthracene molecules. This conclusion is based on the estimates of the ionization potential of singlets and triplets made in the Appendix A and on the obvious distinction between the photocurrent spectrum observed (Fig. 7a) and photoionization spectra of neutral particles in non-polar liquids [65±67]. As shown in the Appendix A, the photon energies hm2 6 2:92 eV used in our experiments are not sucient for one photon ionization of the anthracene singlets and triplets. So, we can ignore one photon ionization of the excited anthracene states by the probe pulses although both triplets and a part of singlets live to the dye laser pulse because its temporal pro®le, as seen from Fig. 3, overlaps slightly with the ``tail'' of the 308 nm pulse. The same conclusion that photoionization of the anthracene singlets by the probe pulse can be neglected was made in the work [45] in which DQ was measured as a function of the time delay td . It has been found that the half-decay time of the dependence of DQ on td (about 30 ns at T ˆ 177 K and more than 60 ns at T ˆ 150 K [45]) is much longer than the anthracene singlet lifetime. Following the concept developed in the previous studies [45±49], the observed enhancement of photocurrent due to the dye laser action is associated with photoexcitation of geminate trapped electrons produced initially by the UV light: A ‡ 2hm1 ! …e ; A‡ † ! free charge carriers

…4†

…e ; A‡ † ‡ hm2 ! …e0 . . . A‡ † ! …e ; A‡ †NEW ! free charge carriers

…5†

It is assumed [44±46] that just after thermalization of the quasi-free electron e0 a new geminate pair …e ; A‡ †NEW is formed which has a larger probability of dissociation than that of the ``old'' pair …e ; A‡ † by the time the trapped electron e absorbs the photon hm2 . The proposed mechanism of the dissociation probability enhancement is based on the assumption that the initial kinetic energy of e0 remains in excess above thermal one for sometime when it thermalizes and forms the new pair …e ; A‡ †NEW [41,46]. The model allows one to explain all the experimental data on the PAIPS signals observed [44±53]. Speci®cally, both the rise of DQ=Q with cooling at a ®xed time delay td (Fig. 6) and the increase of the decay time of the curve DQ…td † with decreasing temperature [45,47] can be naturally attributed to the increase of the recombination time of the geminate pairs …e ; A‡ † because the electron mobility goes down with cooling. 4.2. Comparison with the di€usion model: electron thermalization distances In the absence of the bulk recombination of charge carriers the relative increase of photogenerated charge DQ=Q is equal to Dq=q where q is the dissociation probability of the geminate pair …e ; A‡ † without an additional electron photoexcitation, and Dq is the average enhancement of the pair dissociation probability caused by the photoexcitation of electrons by the dye laser pulse. In the framework of the di€usion model of the photostimulated ion pair dissociation [46,47] the mean squared thermalization distance, L, of the electron e0 produced in the reactions (1) or (5) is related to Dq=q by the equation   Dq L2 T ˆN 2 1 …6† K…td † q Teff 6rC where N ˆ …G2 =s2 hm2 †cr, r is the optical absorption cross-section of the trapped electron (in cm2 ), c is the quantum yield of the quasi-free electron generation in reaction (1) and, thus, cr is the

L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

photoionization cross-section of the trapped electron. Here, L is the mean squared distance between the position of the trapped electron by the moment it absorbs a photon and the new site of electron trapping just after the full electron thermalization in the absence of electric ®eld, and Teff is the e€ective temperature of the quasi-free electron e0 given by [46] Teff ˆ eL2 =6kB De

…7†

where De is the drift displacement of e0 in an unit electric ®eld (in cm2 V 1 ). The value of Teff shows how much greater an average energy of e0 during thermalization is than the thermal energy [46]. The high e€ective temperature 800±1000 K reported for excitation by the 694 and 1060 nm light [46±49] has been attributed to the formation of e0 with a large initial kinetic energy much more than kB T followed by fast localization of e0 as soon as its energy drops down to a value of the order of kB T . The displacement De was obtained from the independent measurements of the photocurrent transients arising from photoexcitation of free (i.e. non-geminate) trapped electrons in MCH by nanosecond laser pulses: 5  1  10 13 cm2 V 1 at 694 nm from Ref. [68] and 7  2  10 13 cm2 V 1 at 1060 nm from Ref. [69]. The parameter K in Eq. (6) does not depend on the wavelength k. It depends on temperature, on the interpulse optical delay td and time pro®les of the laser pulses. As seen from Eq. (6), at Teff  T the dependence of Dq=q on k is determined, mainly, by the spectral dependence of rL2 . Thus, taking into account that Dq=q ˆ DQ=Q, the electron thermalization distance can be derived from DQ=Q by the following equation:   DQ 6rC2 6kB T De : ‡ L2 ˆ …8† Q NK e The drift displacement De may depend on k. However, the measurements of De at 694 and 1060 nm have shown that the wavelength dependence of De , if it exists, is rather weak. In addition, the second term in the r.h.s. of Eq. (8), as mentioned above, is far less than the ®rst one. Fig. 7b shows the square of thermalization distance L2 as a function of k derived from the ratio …DQ=Q†=r by

95

Eq. (8) with the use of c ˆ 1, De ˆ 5  10 13 cm2 V 1 and the known optical absorption spectrum r…k† for the trapped electron in liquid MCH [64]. It is suggested that the absorption of a photon by the trapped electron in liquid MCH, as for 3MP glass [43], is a direct transition from the bound state to the quasi-free electron state of continuum spectrum with c ˆ 1. As to the value of K required for calibration of the absolute thermalization distances, it has been found in the following manner. Let the geminate electron±ion pairs …e ; A‡ †, produced initially at time t ˆ 0, be illuminated at time t ˆ td by a short light pulse with a pulse duration much less than td . In this case, K is equal to [46] Z 1 1 K0 ˆ 4pq rC4 r 2 f …td ; r†p…r† dr …9† A0

where f …t; r† is the probability density for a trapped electron e to be in the elemental volume d3 r at distance r from its sibling cation A‡ , and A0 is the recombination radius of electrons (of the order of the molecular size). In the di€usion model the distribution function f …t; r† satis®es the Smoluchowski equation   2   of of 2 rC of ˆD ‡ …10† ‡ ot or2 r r2 or with the initial condition f …t; r† ˆ f0 …r† at t ˆ 0 where D is the di€usion coecient of trapped electrons. In the limit of small external electric ®eld the dissociation probability of the pair in nonpolar liquids is given by [4,5,40] Z Z q ˆ f0 p d3 r ˆ f …t; r†p d3 r …11† where p ˆ exp … rC =r† is the escape probability of the single pair with the distance r between charges [70]. Using the dimensionless time s ˆ t=tsep and radius x ˆ r=rC , the parameter K0 can be presented in a form [46] Z 1 K0 …sd † ˆ 4pq 1 x 2 F …sd ; x† exp … 1=x† dx n

…12† rC2 =D

where n ˆ A0 =rC . Here, tsep ˆ and sd ˆ td =tsep are the di€usion separation time of the pair …e ; A‡ † and dimensionless delay time, respectively,

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L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

and F …s; x† ˆ rC3 f …t; r† is the dimensionless distribution function. (The functions K0 …sd † have been calculated previously [47,52] for di€erent initial distributions. It has been found that K0 decreases as sd increases.) To ®nd the needed di€usion coecient D ˆ lkB T =e, the experimentally determined mobility of excess electrons in liquid MCH l ˆ 105 exp… 2160=T † cm2 V 1 s 1 was used [59]. Dielectric constant at low temperatures was found from the relationship e ˆ 2:02 ‡ 1:161  10 3 (293 T ) [62]. Parameters used in calculations are presented in Table 1 for some temperatures. In the case of ®nite pulse durations when the ®rst (UV) laser pulse overlaps with the second (dye laser) one, the probability enhancement Dq, as shown previously [47], is determined by Eq. (6) where the coecient K is given by the convolution Z ‡1 Z ‡1 K…sd † ˆ g1 …s1 † ds1 g2 …s2 † s 1 sd

1

 K0 …sd ‡ s2

s1 † ds2 :

…13†

Here, s1 and s2 are the dimensionless times in the units of tsep , sd ˆ td =tsep , td ˆ 25 ns, g1 …s† is the rate of the geminate trapped electron photogeneration on the UV light irradiation and g2 …s† is the rate of electron photoexcitation on the dye laser irradiation. are normalized to unity: R 1 Both functions R1 g …s† ds ˆ g …s† ds ˆ 1: 1 2 1 1 The photoexcitation rate g2 in Eq. (13) is directly proportional to the dye laser light intensity. As to g1 …t†, it should be noted that the temporal pro®le g1 …t† does not coincide with that of the square of the 308 nm light intensity because at low temperatures the lifetime of the anthracene singlets is comparable to the laser pulse duration. In the

Appendix B a simple model of the two photon stepwise ionization of anthracene by the 308 nm pulse is considered which permits one to calculate the temporal pro®le g1 …t† and ®nd K for each temperature. Although the value of DQ=Q increases with cooling, the absolute electron thermalization distance was determined at elevated temperatures in order to avoid uncertainty in the initial spatial distribution f0 …r† of the geminate pairs …e ; A‡ †. The point is that at long times t > 0:1rC2 =D the spatial distribution f …t; r†=q and, hence, the parameter K0 , as calculations show [40,46,47,52], do not depend practically on f0 …r†. As may be seen from Table 1, the delay td D=rC2 reaches 0.1 at temperatures T > 200 K. On the other hand, as seen from Fig. 6, at T > 200 K the signal DQ=Q is dicult to measure accurately at the dye laser intensities used in our experiments. So, the temperature range 180±190 K is the best interval for ®nding thermalization lengths by Eq. (8) because at these temperatures the signals DQ=Q still remain to be large enough, and an uncertainty in K due to an uncertainty in the initial distribution f0 …r†, as calculations show, is less than 20%, i.e. not more than the experimental errors of the DQ=Q measurements. Basing on the measurements of DQ=Q at T ˆ 185±190 K and the numerical solution of Eq. (10) to calculate K…T †, the magnitudes of L2 ˆ …19  4†  10 14 cm2 and Teff ˆ 735  130 K have been obtained for k ˆ 740 nm (the dye Pyridine 2). This value of L2 has been used to calibrate absolute thermalization distances for other wavelengths plotted in Fig. 7b. The bars near experimental points in Fig. 7 correspond to the errors of the measured signals

Table 1 Parameters used for calculations of the electron thermalization distances at some temperatures T (K)

l (10

160 170 180 190 200 220

1.44 3.19 6.45 12.1 21.4 57.2

4

cm2 V

1

s 1)

rC (nm)

rC2 =D (ns)

td D=rC2

tS (ns)

tgem (ns)

46.74 44.31 42.16 40.23 38.5 35.53

11000 4208 1776 814.7 401.5 116.4

2:27  10 3 5:94  10 3 1:4  10 2 3:07  10 2 6:23  10 2 0.21

10.1 9.32 8.67 7.96 7.37 6.41

4.5 2.0 1.0 0.5 0.29 0.1

rC2 =D is the di€usion separation time of the pair …e ; A‡ †, td ˆ 25 ns is the interpulse optical delay, tgem ˆ eB3 =…3el† is the geminate recombination time at the initial distance B ˆ 5 nm.

L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

DQ with respect to other wavelengths obtained at the same experimental conditions. As to absolute values of L and Teff , the main source of the error seems to be the non-homogeneous intensity distribution in the cross-sectional area of laser beams [47]. Nevertheless, it is worthwhile noting that L ˆ 4:5  0:3 nm and Teff ˆ 735  130 K obtained for k ˆ 425±867 nm in the present work are rather close to magnitudes of L ˆ 4:8  0:4 nm and Teff ˆ 730  60 K estimated previously at k ˆ 1060 nm [47]. 4.3. Initial separation of the charges in the geminate pair (e± ,A‡ ) It is interesting to compare the observed temperature dependence of the PAIPS signal (Fig. 6) with predictions of the di€usion model regarding spatial evolution of the electron±cation distribution f …t; r†. In ionization studies the Gaussian form (14) of the initial distribution function is often used, as one of the probe one-parameter functions [4,5,7,40]: f0 …r† ˆ …pB2 †

3=2

exp… r2 =B2 †

…14†

For two photon ionization of anthracene by the 308 nm light the distance B would be expected to lie between 4:5  1 and 7  1 nm. These lengths were obtained for photoionization by the laser pulses at 360 and 270 nm, respectively [47±49]. We have tried to ®nd the parameter B of the initial distribution by correlating the temperature dependence of DQ=Q observed with the curves Dq=q calculated at di€erent B. The point is that, as seen from Table 1, the dimensionless delay sd ˆ td D=rC2 becomes less than 0.01 as temperature decreases below 180 K. The calculations of Dq=q as a function of sd have shown [47,52] that at small sd < 0:01 the dependence of Dq=q on sd (and, hence, on temperature at a ®xed time td ) carries an information on the width of the initial distribution. The greater is B, the more gentle is the decay of Dq=q as a function of sd or, as a function of T at a ®xed time td . Fig. 6 demonstrates such the comparison. The probability enhancements Dq=q in Fig. 6 were calculated by Eqs. (6) and (13) basing on numerical solutions of Eq. (10) at di€erent dis-

97

tances B of the initial distribution (14). To compare the rates of decay of the curves Dq=q with increasing temperature, each calculated curve has been scaled to coincide with magnitude of DQ=Q at 158 K. As can be seen, the distance B ˆ 5:5  0:5 nm is consistent with experiment. Such the initial separation in the pair …e ; A‡ † is close to the value of B ˆ 5:4 nm reported for two photon ionization of anthracene in n-hexane by the same excimer laser pulses at 308 nm [71]. These ®ndings may be considered as a veri®cation of the di€usion model depicted by Eqs. (6)±(13) because both the solvents (MCH and n-hexane) are close each other with respect to excess electron properties (conduction band energy and electron mobility). A more rigorous treatment of the model requires a consideration of the temperature dependence of f0 …B; r†. The width B is thought to decrease with cooling [5,48,49]. However, within a narrow temperature interval DT ˆ 40 K the estimates of oB=oT  0:015±0:025 nm K 1 [48,49] give the change of the width DB ˆ 0:6±1 nm that is far less than B. It should be emphasized that an uncertainty in B, as mentioned above, has no e€ect on the obtained distances L because they were determined at the elevated temperatures 185±190 K at which K did not depend practically on f0 …B; r†. It is worthwhile noting that for the single pulse laser regime the laser heating a sample due to dissipation of the absorbed 308 nm light is insigni®cant and cannot markedly a€ect the separation of charges and their motion within the Coulomb well. In fact, the local temperature increase within the volume vC ˆ 4prC3 =3 around an anthracene molecule after it absorbed a photon, estimated as 1 DTL  …1 /F †hm1 …qcp vC † < 1:5  10 3 K, is too small to be taken into consideration. Here, hm1 ˆ 4:026 eV is a photon energy of the 308 nm light, /F ˆ 0:36 [72] is the ¯uorescence quantum yield of anthracene, q  0:8 g cm 3 and cp  1:9 J g 1 K 1 [62] are the density and heat capacity of MCH, respectively, and rC > 35:5 nm at T < 220 K. 4.4. Dependence of thermalization distance on the electron energy Mean squared thermalization distances L ˆ 4:2±4:7 nm found (Fig. 7b) are in agreement with

98

L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

magnitudes obtained by the PAIPS technique in 3MP glass [43], squalane [53] and with other data on the electron runs in low electron mobility hydrocarbon liquids, like n-hexane, derived from various experiments: measurements of free ion radiation yields [5] and photoconductivity of solutions due to photoionization of solute molecules [40,71,73], photoelectron energy loss spectroscopy of thin organic ®lms [3,74±76] and photoemission of electrons from metals to liquid [77]. The thermalization distances L have been obtained above basing on the assumption that in reaction (1) the quantum yield of the quasi-free electron is c ˆ 1 as for the direct bound-free transition. It has been found [43,47,78±84] that such the mechanism of the trapped electron optical absorption in organic liquids and glasses is consistent, at least, at photon energies hm > 1 eV with measurements of photoconductivity and photoassisted free ion radiation yield as well as with optical absorption studies. As noted previously [47], phenomenon of the photoassisted separation of the electron±cation pairs with the high e€ective temperature Teff  103 K is itself an evidence in support of the concept of the electron transition from a trapped state directly to the continuum spectrum of quasi-free states. In this case, the initial kinetic energy of e0 is equal to E0 ˆ hm2

ETR

…15†

where the threshold energy, ETR , of the trapped electron ionization can be estimated as 0:5  0:1 eV as for 3MP glass [43,79,80,84]. So, at ETR ˆ 0:5 eV the range of wavelengths studied corresponds to the change in E0 from about 0.93 eV at k ˆ 867 nm up to 2.42 eV at k ˆ 425 nm. As seen from Fig. 7b, the thermalization distance only slightly decreases by about 10% as the wavelength increases from 425 to 870 nm. A similar behavior of the spectrum L…k† at k < 1030 nm has been found by Yakovlev [43] for 3MP glass at 77 K. In the glass, the value of L begins to decrease sharply with decreasing photon energy only at hm2 < 0:9 eV when E0 becomes less than about 0.5 eV. Such the dependence of L on E0 has been attributed to a decrease in the electron thermalization rate when the electron energy becomes less than the threshold energy (0.4 eV) for excitation of C±H stretch-

ing vibrations, the most active infrared vibration modes of alkane molecules. The obtained dependence of L2 on k and, hence, on E0 carries information on the electron energy loss rate. In a simple thermalization model, an electron with energy E and velocity v undergoes a large number, M, of quasi-elastic scattering events during thermalization process. The rate of energy dissipation can be evaluated as follows [43]: zˆ

dE dE dE ˆv ˆ vkEL dt dLTOT d…L2 †

…16†

where LTOT ˆ MkEL is the total (straightened) thermalization path of the electron and kEL is the elastic scattering mean free path. It is assumed that kEL is independent of energy, and so, L2 ˆ Mk2EL . The evaluation of kEL for low energy electrons in organic liquids is a challenging theoretical task. If kEL is of the order of molecular size kEL  0.5 nm, insertion of the ratio DL2 =DE ˆ 2:3  1 nm2 eV 1 , derived from the rather small slope of the plot of L2 versus k in Fig. 7b, into Eq. (16) gives z ˆ 1:3  1014 eV s 1 at E ˆ 1 eV. As noted previously [43], such the value of z is much more than the rate of energy loss in indirect (long distance) collisions caused by the transient electric ®eld induced in dielectric by the motion of electron. Basing on the theory of Fr olich and Platzman [85] and experimental spectrum of the complex dielectric constant, Raitsimring [86] has found that for n-hexane z ˆ 7  1012 eV s 1 at E ˆ 1 eV. The close value 4:3  1012 eV s 1 was obtained for electrons in polyethylene [87]. The results show that in the energy range of 1±2.5 eV the main channel of the electron energy dissipation is the direct collisions caused by shortrange interaction with solvent molecules. The high resolution electron energy loss spectra of long chain alkanes and polyethylene [74±76] have shown that at E ˆ 0:1±5 eV electrons lose the energy, mainly, in inelastic scattering collisions by excitation of intramolecular vibrations with the loss quanta dEi equaled the quanta of the vibration modes of alkane molecules. The loss quanta 0.37, 0.18 and 0.09 eV appropriate to the C±H stretching and bending modes were observed. Assuming that the electron can be scattered also in quasielastic collisions with small …<0.01 eV† loss

L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

quanta, the enhancement of the square of electron distance after every inelastic collision is shown [88] 1 to be dri2 ˆ 2‰kIN1 …kEL1 ‡ kIN1 †Š where kEL and kIN are the elastic and inelastic mean free paths, respectively. So, for the energy range of 1±2.5 eV the parameter keff , called the e€ective scattering free path, can be determined from DL2 =DE as follows: k2eff ˆ 2‰kIN1 …kEL1 ‡ kIN1 †Š

1

ˆ hdEi i…DL2 =DE†

…17†

where hdEi i  0:2 eV is the average energy loss [88]. The relationship (17) gives keff ˆ 4:8 nm and kIN ˆ 1:2 nm (at DL2 =DE ˆ 2:3 nm2 eV±1 and kEL ˆ 0:5 nm). Further investigations of the spectrum L…k† with better wavelength resolution appropriate to the energy interval of dEi  0:1 eV

99

dence of L on hm at hm > 1 eV obtained in the present work and in Ref. [43] for 3MP glass. To clarify possible reasons of discrepancy between the behavior of B and L as functions of the initial excitation energy, it is necessary to consider the unresolved problem for electron thermalization and trapping within the Coulomb well provided that the dependence of L on the initial electron energy without the Coulomb ®eld is known. This is a challenging task even in the framework of a classical approach because both kEL and kIN are thought to depend on energy [3,74±76]. However, the e€ect of the Coulomb ®eld on the yield of thermalized and trapped electrons can be considered qualitatively in the following scheme:

…18†

can determine kEL and kIN and open fresh opportunities for detailed study of electron thermalization in liquids. It is interesting to compare the obtained spectrum L…k† with the spectral dependence of thermalization distance for electrons produced by photoionization of solute molecules in liquid solutions. In the latter case, it is assumed that the initial spatial distribution over the distances, r, between thermalized photoelectron and its parent cation A‡ may be given by a simple one-parameter probe function f0 …B; r†, like the Gaussian form (14). It has been found [89] that for one photon ionization of N,N,N 0 ,N 0 -tetramethyl-p-phenylenediamine (TMPD) in liquid hydrocarbons the distance B increases monotonically with increasing photon energy, hm, above the TMPD ionization threshold, IPliq . Since the initial kinetic energy of photoelectrons is thought to increase with increasing excess excitation energy hm IPliq , the observed monotone increase of B with increasing hm [89] di€ers essentially from rather weak depen-

where the geminate recombination of the non-fully thermalized mobile electron, em , precursor of the trapped electron e , is taken into account. Such the mobile electron state em may be a quasi-free electron with energy of 0.01±0.1 eV near the edge of the conduction band. It is assumed that the mobile electron em can recombine geminately before full electron thermalization and trapping. For the sake of simplicity, we accept here that the e€ect of the Coulomb electric ®eld on the motion of nonthermalized electron em can be taken into account, semi-quantitatively, as a drift of a thermalized charge with the mobility l0 ˆ 1±10 cm2 V±1 s±1 as for quasi-free electrons near the conduction band edge in disordered materials. Such a di€usion description for the electron motion in the prethermal regime can be justi®ed for small elastic scattering mean free paths kEL  L, kEL  B. In deed, for similar organic matrices with high eciency of electron localization the data on the photoelectron passage through organic ®lms [74±76] well agree with small kEL of the order of molecular size.

100

L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

Let the electron em be thermalized at a distance, r0 , in the absence of the Coulomb ®eld. The probability for em to recombine geminately prior to being thermalized and trapped in the Coulomb well may be estimated crudely as follows: 1 tgrm ˆ …1 ‡ er03 =3eDe † 1 ‡t 1 tgrm M

1

 0:6

…19†

at r0 ˆ L  4 nm and l0 tM ˆ De ˆ 5  10 13 cm2 V±1 [68,69]. Here, tM is the full thermalization and solvation time of em , and the geminate recombination time of em is estimated to be tgrm ˆ er03 =3el0 . As seen, the geminate recombination of electrons before thermalization and solvation is rather essential. The result suggests that the yield of thermalized and trapped electrons in the Coulomb well and, hence, the yield of free charge carriers increases with increasing r0 . So, the expected dependence of q and B on the initial excitation energy is thought to be more strong than the dependence of L on the initial electron energy without the Coulomb ®eld.

5. Conclusion The following main inferences can be drawn from the present investigation. (1) The spectral dependence of the photoelectron thermalization distance, L, derived from the measurements of two pulse photoconductivity of the anthracene solution in liquid MCH has con®rmed the electron relaxation picture in which the quasi-free electron with the energy above 1 eV quickly loses its energy in inelastic collisions with solvent molecules resulting in a weak dependence of L on the initial electron energy E0 . The electron energy dissipation rate of 1:3  1014 eV s 1 was estimated within the interval 0:9 < E0 < 2:5 eV. (2) Considering the anthracene solution as a model electron photodonor system for the study of the photostimulated migration of electrons in nonpolar liquid, the present work has demonstrated a rather wide range of the initial quasi-free electron energies accessible to the PAIPS technique: from near-zero initial energy, when a photon energy …hm2 † of the probe pulse radiation approaches to

photoionization threshold for the trapped electrons …0:5  0:1 eV†, up to, at least, the energy about 2.5 eV at hm2  3 eV. Further researches on photoionization of other solute molecules with a more wide gap between the ionization potential and long-lived excited singlet level can extend essentially the electron energy range under investigation. Acknowledgements The research described in this publication was made possible in part by grant no. 97-03-032018 from the Russian Foundation for Basic Researches. Appendix A The threshold energy of the ionization of the ®rst excited singlet state of anthracene in liquid MCH can be estimated as IPSliq ˆ IPliq ES where IPliq is the ground-state ionization potential of anthracene in MCH and ES is the relaxed singlet energy of anthracene molecule. The energy IPliq at room temperature was found basing on the wellknown relationship [90] IPliq ˆ IPG ‡ V0 ‡ P‡ where IPG is the ground-state ionization potential of anthracene in gas phase, V0 is the energy of the conduction band bottom for an excess electron in liquid with respect to the vacuum level and P‡ is the polarization energy of the anthracene cation. Using the experimentally determined value of IPliq ˆ 6:14 eV [66] for anthracene in 2,2,4-trimethylpentane (TMP) and the Born formula P‡ ˆ …e2 =2R‡ †…1 1=e†, the ionization potential IPliq in MCH at room temperature was evaluated by the equation IPliq …MCH† ˆ IPliq …TMP† ‡ V0 …MCH† V0 …TMP†   e2 1 1 ‡ 2R‡ e…MCH† e…TMP†  6:42 eV:

…A:1†

Here, we used the known values of V0 …MCH† ˆ 0:08 eV [91] and V0 …TMP† ˆ 0:24 eV [92] for MCH and TMP, respectively, the radius of the anthracene positive ion R‡ ˆ 0:325 nm [66]

L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

and dielectric constants e…MCH† ˆ 2:02 and e…TMP† ˆ 1:943 at T ˆ 293 K. The ionization potential IPliq in liquid is known to increase with decreasing temperature [40]. So, if for anthracene oIPliq =oT ˆ 0:004 eV K 1 as for TMPD in MCH [93], the threshold energy IPSliq is expected to be 3.27 and 3.55 eV at temperatures 230 and 160 K, respectively, and at the singlet energy ES  3:3 eV. As to photoionization of triplets, their ionization potential in liquid, IPTliq , should be still more than IPSliq because the triplet state energy for the anthracene molecule (1.8 eV) is less than ES . Thus, the values of IPSliq and IPTliq are much more than photon energies of the dye radiation used. Appendix B Below we present the calculation of temporal pro®le of the geminate electron photogeneration rate g1 …t† needed for the convolution (13). Although the value of K in Eq. (6) does not a€ect the relative spectrum L…k†, a knowledge of g1 …t† and K is necessary for determination of the absolute thermalization distance L. Let us consider, at ®rst, the ionization channel through the ®rst excited singlet of anthracene, S1 , when the second photon of the 308 nm light is absorbed by the vibrationally relaxed S1 state. In this case, g1 …t† / I308 …t†n1 …t† where I308 is the intensity of the 308 nm light and n1 is the time dependent concentration of the anthracene singlets. To determine n1 …t†, it is necessary to know the lifetime, tS , of the relaxed anthracene singlets. The singlet lifetime at low temperatures was derived from the known ¯uorescence quantum yield, /F , related to tS by the equation [94,95] /F ˆ kF =…kF ‡ kST † ˆ kF tS

…B:1†

where kF and kST are the radiative and intersystem crossing rate constants, respectively. For anthracene it has been found [94,95] that the radiationless internal conversion …S1 ! S0 † is negligible, and kF is independent practically of temperature. The times tS were obtained from Eq. (B.1) using tS ˆ 4:9 ns at room temperature [72] and experimental temperature dependence of /F for anthracene in

101

liquid 3MP shown in Fig. 1 of Ref. [95]. The times tS found are presented in Table 1. It is worth while noting that these lifetimes are close to ¯uorescence lifetimes of anthracene in polymethylmethacrylate [96]. The probability for anthracene molecule to absorb a photon during the 308 nm laser pulse is rather small r01 W1 =hm1 ˆ 0:074 at the typical density of a light energy W1 ˆ 10 mJ cm 2 where r01 ˆ 4:8  10 18 cm2 is the optical absorption cross-section of anthracene at 308 nm [72,97]. So, the concentration, n0 , of the anthracene ground states can be considered to be constant during the 308 nm light pulse and be equal to anthracene concentration in the solution n0 ˆ 4:3  1016 cm 3 . Rate equations for formation and decay of the S1 singlets and geminate electrons during the 308 nm pulse may be written as follows: dn1 =dt ˆ r01 I308 n0

tS 1 n1

bi r1n I308 n1 ‡ kgem ngp ; …B:2†

dngp =dt ˆ bi r1n I308 n1

kgem ngp ;

…B:3†

where ngp is the concentration of the geminate pairs …e ; A‡ †, r1n is the cross-section of the ®rst singlet optical absorption resulting in the high excited singlets …S1 ! Sn †, bi is the autoionization probability for the Sn states to eject electron and form the geminate electron±ion pairs. Here, for the sake of simplicity, the geminate electron recombination is considered to be the ®rst order process with the rate constant kgem . It is assumed that on a time scale of 10 ns the geminate pair recombination results in an excited singlet state of anthracene followed by its fast relaxation to the S1 state. The value of kgem can be estimated as kgem ˆ 1=tgem where tgem ˆ eB3 =3el is the recombination time of the pair …e ; A‡ † with the initial distance B between the partners. Table 1 shows the times tgem calculated at di€erent temperatures for the typical distance B ˆ 5 nm. As seen, at T > 160 K the time tgem is less than both the singlet lifetime and pulse duration (about 20 ns). This allows one to use the quasi-stationary approximation dngp =dt ˆ 0. In this case, from Eq. (B.2) it follows that

102

L.V. Lukin, A.A. Balakin / Chemical Physics 265 (2001) 87±104

Z g1 …t† ˆ aI308 …t†

t 0

I308 …t1 † exp ‰…t1

t†=tS ŠtS 1 dt1 …B:4†

where a is a normalizing coecient. Here, a time t ˆ 0 corresponds to the time origin of the 308 nm laser pulse, i.e. n1 ˆ 0 and I308 ˆ 0 at t ˆ 0: Moreover, the experimental dependence of the photosignal amplitude u0 on W1 shown in Fig. 4 allows us to make an estimate of bi r1n and conclude that at the light intensity used Eq. (B.4) holds at any geminate recombination rate. Really, the integral of photocurrent induced by the 308 nm pulse is equal to Q1 ˆ Ceff u0 ˆ e Dv n0gp q…1 ‡ eFrC =2kB T †

…B:5†

where Dv ˆ 0:15 cm3 is the volume of the sample between the high voltage and collector electrodes irradiated with the 308 nm light, n0gp is the total concentration of the geminate pairs …A‡ ; e † produced during the 308 nm pulse, and the factor 1 ‡ eFrC =2kB T takes into account the increase of the pair dissociation probability, q, in the electric ®eld F applied [4,5,40]. On the other hand, at bi r1n I308  tS 1 from Eq. (B.2) it follows that n0gp can be estimated as n0gp ˆ n0 …r01 bi r1n tS =tP †…W1 =hm1 †

2

…B:6†

where tP  20 ns is the laser pulse duration. As can be seen from Fig. 4, the signal u0 observed at W1 ˆ 10 mJ cm 2 corresponds to the charge Q1 ˆ u0 Ceff ˆ 5:6  10 11 C. Inserting these values and tS ˆ 7:7 ns at 194 K to Eqs. (B.5) and (B.6), we obtain the evaluation bi r1n q  6:9  10

23

cm2 :

…B:7†

On the other hand, the calculation of q for the Gaussian form of f0 …r† gives q > 2:5  10 4 at B > 3 nm and T ˆ 194 K. This yields the upper limit of bi r1n < 2:8  10 19 cm2 . Since at the used laser intensity bi r1n I308 < 2:2  105 s 1  1=tS , the devastation of the n1 concentration due to conversion of a part of singlets S1 to the electron± cation pairs during the laser pulse can be neglected (here, the laser light intensity equals I308 ˆ W1 = …hm1 tP †  0:8  1024 photon cm±2 s±1 at W1 ˆ 10 mJ cm 2 ). This justi®es Eq. (B.4) even if tgem > tP .

However, it should be noted that at tgem < 5 ns the ratio ngp …t†=n1 …t† ˆ bi r1n I308 tgem < 1:1  10 3 during the laser pulse action, i.e. the kinetics of the S1 formation is determined, mainly, by the singlet lifetime tS at small laser intensities used. Fig. 3 shows an example of the time pro®le g1 …t† calculated by Eq. (B.4) at tS ˆ 10:1 ns as for T ˆ 160 K. As seen, at this temperature the time interval between the peaks of g1 …t† and g2 …t† is somewhat below the actual optical delay td ˆ 25 ns. It should be noted that the coecient K in Eq. (8), obtained by the convolution (13), can di€ers markedly from K0 . For example, at the distance B ˆ 4 nm of the initial distribution the values of K0 ˆ 35:03, K ˆ 46:7 and K0 ˆ 8:58, K ˆ 13:1 were found at 185 and 200 K, respectively. As seen, the distinction between K0 and K is essential for ®nding the absolute value of thermalization distance at elevated temperatures. It is worth while mentioning that the anthracene photoionization through the ®rst triplet T1 is unlikely to be accessible for the 308 nm light. As is shown in the Appendix A, the ionization potential of T1 in MCH at T < 230 K is expected to be IPTliq ˆ IPSliq ‡ ES ET > 4:6 eV that is less than the photon energy 4.03 eV of the 308 nm light (here ET ˆ 1:8 eV is the energy of the anthracene triplet T1 level).

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