J. Phys. Chem. 1993,97, 13243-13249
13243
Electron Spin Polarization in Radical-Triplet Pairs. Size and Dependence on Diffusion Gert-Hein Goudsmit and Henning Paul' Physical- Chemistry Institute, University Ziirich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland
Anatoly I. Sbushin Institute of Chemical Physics, Academy of Sciences, GSP-1, 1 1 7977 Moscow, Russia Received: June 29, 1993; In Final Form: September 28, 1993'
Using laser flash irradiation and time-resolved E P R spectroscopy, the chemically induced electron polarization (CIDEP) is investigated, which is generated by interaction of persistent 2,2,6,64etramethylpiperidine- l-oxy1 (TEMPO) radicals with triplet excited benzophenone in 1,2-epoxypropane solution. Net emission superposed by a weak emission/absorption multiplet type polarization in the T E M P O radical system is observed and analyzed quantitatively at temperatures between 193 and 298 K. With decreasing temperature the relative diffusion coefficient of the particles decreases from 5.9 X lo-' to 0.9 X cm2/s, and the absolute values of net and multiplet CIDEP increase from 0.6 to 9.0 and from 0.07 to 0.26, respectively, in units of Boltzmann polarization. The CIDEP is attributed to mixing and splitting of doublet and quartet spin states in radicaltriplet pairs. A theoretical model is outlined, which quantitatively describes the size of polarizations as well as their dependence on diffusion. It indicates that the CIDEP is generated predominantly in regions where the exchange interaction is smaller than the Zeeman energy and that the multiplet polarization is diminished effectively by fast TZrelaxation of the triplet spin. For an exchange interaction J = JO exp(-a(r - d)), decaying exponentially with interparticle distance r, IJold/a= 4.4 X 10-6 cm2/s is determined from the net polarization, d being the distance of closest approach. The magnitude of the multiplet polarization allows an estimation of JO = 5 X 109 rad/s and a = 8 X lo7 cm-l, Le., a surprisingly small doubletquartet splitting at distance d.
Introduction
In the 19809, there had been some publications reporting that electron spin systems of free radicals in solution can become polarized in emission if triplet excited molecules are present in the solution.1I2 The phenomenon has been tentatively assigned to spin selective intersystem crossing via an excited quartet state of the radicals' and to transfer of electron spin polarization, present in the triplet sublevels, to the radical spin system.2 Recently, a radical-triplet pair mechanism (RTPM) has been suggested to account for emissive spin polarizations observed for transient radicals in the presence of various triplet-state molecules.3 It was able to explain the emissions without requiring spin polarized triplet states. This mechanism has been extended to cases where it also yields absorptive and multiplet type polarizations4 and has been put on a rigorous theoretical basis.5 Today, it seems to be accepted as an attractive candidate for explaining those polarizations.6 The RTPM resembles the well-known radical pair mechanism (RPM) which, in its basic form, had been proposed by Closs et ai.,' Kaptein and Oosterhoff,* and Adrian? and since has been refined several timesslo In the RPM, electron spin polarization is generated in radical-radical encounters via nonadiabatic transitions between the IT -1) and the reactive IS) spin state of the pairs (yielding essentially net emission) and/or by transitions between their IT 0) state (yielding multiplet type polarization). In the RTPM similar polarizations are created by transitions between quartet IQ) and reactive doublet ID)states in tripletradical pairs.5 Although the RTPM can qualitatively explain the signs of experimentally observed polarization patterns, it has not yet been tested in a real quantitative way. For one system only, a RTPM net polarization of -60 and -24 (in units of Boltzmann polarization) has been estimated from experimenPb and theory,5 respectively. No experimental data are available on the absolute *Ahtract published in Aduance ACS Abstracts, November 15, 1993.
0022-3654/93/2097- 13243$04.00/0
sizeof multiplet polarizations generated by this mechanism. With respect to the dependence on solvent viscosity, the ratio of net to multiplet polarization has been found to be roughly proportional to this quantity;" Le., both polarizations depend on some power of viscosity, being lower by about 1 for the multiplet polarization. Therefore, we have now investigated quantitatively a radicaltriplet system and have determined absolute values for net and multiplet polarizations as well as their dependencies on solvent viscosity. As a model system, suited for accurate measurements of even rather small polarizations, we have chosen persistent TEMPO radicals and triplet benzophenone, a system which has been considered before.2.4b Measurements have been performed in 1,2-epoxypropane solutions over a temperature range 173 5 T 5 298 K. In trying to compare the results with theoretical predictions, the relatively short electron spin relaxation times of triplet molecules in solution were found to be of crucial importance. In addition, it turned out that in the system chosen the polarization is obviously generated in regions of rather small exchange interaction. Since the existing theoretical model does not include relaxational transitionsand also only considers the strong exchange limit, it had to be extended appropriately. Here, we will only outline this extension, taking into account spin relaxation in a semiquantitative way. A more detailed theoretical discussion will be given elsewhere.I2
Experimental Section Our experimental arrangement for time-resolved EPR (TREPR) measurements after laser flash photolytic generation of transients has been described previously.13 It comprises an excimer laser (XeCl, 308 nm, 2 0 4 s pulse width) and a CW EPR detection system without field modulation (80-11s response time). Solutions were freed from oxygen by purging with helium and afterward exposed to laser irradiation (6 mJ per pulse on sample surface) while slowly flowing through a quartz cell (0.5-mmoptical path length) inside the EPR cavity. TEMPO, benzophenone (0
1993 American Chemical Society
Goudsmit et al.
13244 The Journal of Physical Chemistry, Vol. 97,No. 50, 1993
At
Figure 1. TR-EPR spectrum of TEMPO radicals, 1 ps after laser flash. 0.00
0.00
4.02
4.44
4.44 4.w
-0.08 0.0
0.8
f.6
pa
0.0
0.6
1.8
noted that the signals in Figure 1 are exclusively produced by spin polarization and do not contain a Boltzmann contribution, the reason being that our preamplifier is ac coupled. Therefore, no EPR signal can be detected after laser flash irradiation of a pure 1 mM TEMPO solution without BP. We have investigated the polarizations of the low- and highfield EPR line of TEMPO radical in the presence of triplet BP at various temperatures. The change of polarization with time is given in Figure 2 for three different temperatures. From the scaling of the signal height S, measured as voltage at the preamplifier output, it is obvious immediately that both net and multiplet polarization increase with decreasing temperature. To get quantitative data, these time profiles have been carefully analyzed. The parameters obtained fit the experimental time profiles quite well as can be seen from the simulations, also displayed in Figure 2. For simulation of the time profiles we used Bloch equations, modified with additional terms to allow for chemical kinetics and electron spin polarization, as proposed by Fessenden.16 For onresonance measurements, they read in our case
Pa
bs
ifz = -UIU
+ P,[TEMPO] *I
- M,
+
2k,(Pn f P,)[BPT][TEMPO] (2)
WTl (3)
[BPT] = -kq[TEMPO][BPT] - k,[BPTIZ -0.0
-
-1.0
-2.0
0.0
0.8
"k
1.6
line
ua
4 0.0
:
: 0.8
:
: 1.6
i
ua
h@h-t%M line
Figure 2. EPR time profiles and simulationsof TEMPO radical low- and high-field lines at T = 298, 213, and 173 K.
(BP), and the solvent 1,2-epoxypropanewere obtained from Fluka and Aldrich in their purest available forms and used as purchased. Results After laser flash irradiation of a 1,2-epoxypropanesolution of BP (0.3 M) and TEMPO (1 mM), an EPR spectrum, composed of three nearly Lorentzian lines, is observed as shown in Figure 1. Thegvalue, g = 2.0060, as well as the hyperfine coupling with the nitrogen nucleus, a = 1.56 mT, agrees with the EPR parameters of the TEMPO radical. The spectrum exhibits a net emissive and a smaller, superimposed E/A multiplet polarization. Both are thought to be due to the RTPM, Le., to encounters of TEMPO radicals and benzophenone triplets (BPf). Latter species are generated via laser excitation of BP and fast intersystem crossing. Due to the very short relaxation time of benzophenone (T2 < TI= 0.9 ns at room temperature (RT)14) the EPR spectrum of the triplets is not detectable. All these characteristics are consistent with the literature data.4b It is emphasized that, over the whole temperature range considered, the three TEMPO EPR lines always have exactly the same line width; i.e., their different heights are not caused by hyperfine-dependent line widths, which are often observed for nitroxide radicals in solutions of higher viscosity.lS It is also
Here, u represents they magnetization in the rotating frame, w1 the microwave field amplitude, PCp the equilibrium polarization, P,,and P, the polarization factor for net and multiplet polarization, and T1and T2 the relaxation times. BP triplet decay kinetics of order as well as triplet quenching by first ( r l ) and second (km) TEMPO (kq)is accounted for in eq 3. Spin polarization, generated in BPf via ISC and transferred to TEMPO radicals by spin exchange,2is not considered because of fast TIrelaxation in BPT and low radical concentration. The polarization term in eq 2 appears with a factor 2k,, assuming that doublet pair states of BPTand TEMPO lead to triplet quenching and quartet encounters to spin polarization. After convolution with the response function of the spectrometer, the EPR signal is proportional to u. Not all the parameters in eqs 1-3 were varied. wI= 5.3 X 105 rad/s at an incident microwave power of 1 mW was known for our EPR cavity. Spin-spin relaxation times T2 were measured separately by taking steady-state EPR spectra of an oxygen-free 1 mM TEMPO solution at different temperatures. The Tzvalues obtained from the EPR line widths are listed in Table I. The equilibrium polarization Pq was calculated from the Curie relation 2 2
P, = SB&',t,VS(S
+ 1)
(4)
where NA represents Avogadro's number, V the volume of the irradiated part of thecell, BOthe magnetic field, andS the electron spin quantum number. As intersystem crossing in BPis extremely fast (ki, GZ 10" s-I 17), this process was not considered in eq 3 but accounted for in the initial condition. The initial triplet concentration after the laser flash (0.9 mM) was determined from laser power, volume V,and the absorption coefficient of BP (c = 62 dm3 mol-' cm-I). The rate constant k, for BPTquenching by TEMPO radicals is known for benzene solution (k, = 2.0 X lo9 M-I s-l at RTl*). The corresponding rate constant for 1,2epoxypropanecould be calculated from the viscosity ratio of both solvents. The rate constant for triplet-triplet annihilation was obtained in a similar way from km = (1.9 f 0.7) X 1010 M-1 s-I
Electron Spin Polarization in Radical-Triplet Pairs
The Journal of Physical Chemistry, Vol. 97, No. 50, 1993 13245
TABLE I: Temperature Dependence of Diffusion Coefficient, Spin Polarizations, Relaxations, and Correlation Time T (K) 4 cmz s-I) pn (pq) lpml (pq) pi (ns) T2 (ns) rc (ps) 298 273 253 233 213 193 173
5.86 4.30 3.20 2.26 1.52 0.92 0.50
-0.6 -1.5 -2.7 -4.6 -6.8 -9.0 -9.1
0.06 0.11 0.10 0.12 0.16 0.26 0.23
346 388 446 512 541 625 977
29 31 34 38 45 46 49
6.4 8.7 11.7 16.5 24.5 40.5 74.6
in freon at RT.I9 Here, however, we chose as reference value the lower end of the error range, i.e., km = 1.2 X 1010 M-1 s-1, which led to much better fits than the literature value of 1.9 X 1OloM-1 SI.
Todetermine kqand kmfor the whole rangeof our investigation, we have measured the diffusion coefficients of BP (as a model system for BPT) and TEMPO in the temperature range from 298 to 21 3 K, using a chromatographic broadening technique.20 Both diffusion coefficientswere found to be well described by Arrhenius functions. Assuming that both triplet quenching and triplettriplet annihilation are diffusion-controlled processes,21 we thus obtained for kq and km frequency factors of 1.3 X 10" and 3.4 X 1011 M-1 s-l as well as activation energies of 8.6 and 8.7 kJ/ mol. To have as few adjustable parameters as possible, we also determined separately the proportionality factor between m a g netization and the monitored EPR signal. Since, as mentioned eariler, the TR-EPR experiment is only sensitive toward ac signals, the steady-state EPR absorption of a 1 mM solution of TEMPO was measured by modulating the magnetic field with 40 Hz over a range of approximately 2 mT. The proportionality factor of this experiment was obtained from the signal amplitude received and the y magnetization of the TEMPO sample in thermal equilibrium as calculated from W I , T2, and M A via steady-state Blcch equations. The actual proportionality factor for the TREPR experiment could then be evaluated by taking into account the irradiated volume and the sensitivity distribution in the EPR cavity. Measurements a t different temperatures showed this proportionality factor to be temperature independent. With respect to the TEMPO concentration we also considered the dependence of solvent density upon temperature. The solution, prepared as 1 mM in TEMPO radicals at 298 K, in fact becomes about 1.2 mM at 173 K. Thus, the only adjustable parameters were T I ,P,, and P,. To separate multiplet P, and net polarization P,, the time profiles of high- and low-field line were simulated with free parameters Ph and PI,respectively. The final polarization factors could then be obtained from Pn = (PI+ Ph)/2 and P, = (PI- Ph)/2. As can be seen from Figure 2, the simulations with only three adjustable parameters are in good agreement with the experimental traces. The small deviations on the early time scale may be caused by still slightly too large values for km. The spinlattice relaxation times (averages from the high- and low-field time profiles) and polarization factors, obtained for the TEMPO resonances at various temperatures, are listed in Table I, together with Tz, the relative diffusion coefficient Dr of TEMPO and BP, and correlation times re, which will be discussed later. The polarization factors are all specified in units of the temperaturedependent Boltzmann factor Pq, given in eq 4. There is a between T = 193 and T = 173 K. remarkable increase of However, T = 173 K is rather close to the melting point of the solution, and therefore, the results obtained a t this temperature might not be very reliable and will not be taken into further consideration. Before we discuss the size of the polarizations and their dependence on the diffusion coefficient in the next section, it should be noted that especially the multiplet polarization P, is rather weak. Therefore, detection of P, might remain limited
to systems consisting of triplet molecules and persistent radicals. Only in those systems is pure polarization of a relatively large radical concentration monitored, which allows measurement of such small polarizations. It will be rather difficult to detect P, in systems of transient radicals, the more so since it will be always superimposed by the strong multiplet polarization, which is generated by the radical pair mechanism. It is not unexpected that typical multiplet polarizations generated by the RPM are at least 1-2 orders of magnitude larger than the weak P, created by the RTPM in our system. In both mechanisms P, stems from mixing of nearly degenerate spin states of different multiplicity in spin-correlated radical-radical and radical-triplet pairs, respectively. In the latter, however, effective spin relaxation in the triplet molecules will often destroy spin coherence faster than mixing occurs. Thus, the influence of triplet spin relaxation on RTPM polarization will have to be considered in the next section. For the net polarization, P,,, the existing theory5 predicts proportionality to 1 /Drat small Dr values, provided Pn is generated by nonadiabatic quartet4oublet transitions in regions of strong exchange interaction. As Table I shows by going from T = 298 to 193 K, P,, in our system increases by roughly a factor of 22, whereas Dr is diminished by only a factor of 6. This feature can be understood if P,, is essentially created in regions of weak exchange interaction, as will be also shown in the next section.
Theory and Discussion Formulation of the Model. To calculate the electronic spin polarization, which can be generated in radical-triplet pairs in presence of weakexchange interaction and electron spin relaxation, we consider the same pair Hamiltonian as in previous works
+
+
(5) H = H, Hhf + H,, Hex It includes Zeeman (Hz) and hyperfine interaction (Hhf) as well as zero-field splitting in the triplet molecule (Hzfs) and the exchange interaction (Hex)between triplet (T) and radical (R). For simplicity, we assume isotropic T-R interaction, take equal g values for both particles ( g = ~ = g = 2), neglect any hfi in the T molecule, and use the conventional exponential approximation for the exchange in dependence on interparticle distance, r. Thus, we have
(7)
(9)
J ( r ) = Jo exp[-cy(r - d)]
(10)
where d is the distance of closest approach, ({,[,9) specify the eigenaxes of Hzf, in the molecular frame, and wo = gal?,,. In contrast to our previous work,S we here assume weak exchange in form of the condition
-
which will be discussed below. At large distance r m the eigenstates of H a r e the spin states of noninteracting particles. In the limit gSB0 >> IlHhfll, llHdsll these are eigenstates of Hz: At small distances r zi d, where J(r) >> llHzrSll,the eigenstates of the spin Hamiltonian Hcoincide with eigenstates of the total spin S = ST SR,i.e.
+
Goudsmit et al.
13246 The Journal of Physical Chemistry, Vol. 97, No. 50, 1993
IQf3/2) = lfl)lf1/2)
+
(14) 1D*1/2) = --10)1&1/2) -1*1)171/2) for S = 3/2 and S = 1.2, respectively. A qualitative scheme of terms E Hof the spin Hamiltonian H is shown in Figure 3. For r > d there is a region where levels approach, located at r z r,,, (J(r) z AR). In addition, there are three level crossings, at r z rl and r cz rlr. The main consequence of assumption11 is that they are located within the region r < d, which is inaccessible for T-R pairs. It is emphasized that Figure 3 sketches the termsof thespin Hamiltonian only; Le., the splitting of doublet and quartet terms with decreasing r. The total energy of interaction between triplet and radical, of course, has to increase drastically at short distance r, so that the region r < d cannot be penetrated. For d < rlrrlr the dependence of the net polarization Pn on diffusion coefficient should be P, a Dr-1,5in obvious disagreement with the experimental finding for the BPT-TEMPO system. Thus, we here consider the opposite limiting case d > rl, rl’. The relative T-R motion induces transitions between those terms. All information about them is contained in the T-R pair spin density matrix p satisfying the stochastic Liouville equationZZ
= D,Ap - i[H,p] - Wp (15) The first term on the right-hand side of eq 15 corresponds to free relative diffusive T-R motion with diffusion coefficient D,; the second term describes quantum evolution due to the spin Hamiltonian H.Theoperator Wrepresents spin-lattice relaxation induced by the zfs interaction strongly fluctuating due to stochastic rotational motion of the T molecule. Quenching of triplet excitation by radicals is modeled by the radiative boundary condition p
d p / d r - k,{P,,P)l,=d = 0, k,
-
(16) where { , I is the anticommutator and PD = Cm,mt=lpJDm) (Dml is the projection operator on the T-R pair’s D states. Condition 16 corresponds to nonreactive Q and diffusion-controlled reacting D states. As initial condition the T-R pairs are assumed to be created at a distance d of closest approach with equal population of the four Q states: p(r,O) = (1/4)( 1/4?rd2)6(r- d)Z (17) where I is the unity matrix in the space of Q states. Now a few words about the relaxation operator W. In general, the detailed analytical structure of W is rather complex, but for our semiquantitative analysis it is sufficient to know some general properties of Wwhich are discussed below. The only assumption we would like to introduce from the very beginning is that Wcan be described by the short correlation time a p p r o x i m a t i ~ n in ,~~ which the matrix elements of W a r e well-known,
where li) and 1)are eigenstates between which relaxation occurs and uti is the splitting of these states. Expression 18 is derived assuming an exponential decay of the autocorrelation function ofHzfcwith correlation time T,. Thelineof average means average over orientations of the T molecule with respect to the external magnetic field B. The short correlation time approximation (1 8 ) is valid if D, E < 7;‘ (19) In what fo!lows we will make another serious assumption that
rl
fl
d
r
7,
Figure 3. Schematics of terms of the spin Hamiltonian H(r).
the time T, is shorter than the time of passing through the characteristic distance a-l of change of the exchange interaction J(r), that is
DrrcC CY-’ (20) This inequality means that a t each distance, corresponding to a certain value of J(r), we can introduce the relaxation operator W as obtained in the short correlation time approximation. Relation 20 is more restrictive than eq 19. However, in the considered limit of relatively weak interaction (see inequality ( l l ) ) , J(r) can be treated perturbatively (see below). In this case, the effect of J ( r ) is given by an integral of J(r)over distances and, strictly speaking, no relations like (20) are really needed. More rigorous and detailed consideration of this problem will be given in another work.I2 To avoid this discussion here we confine ourselves to the limit (20). Net Polarization. Net polarization results from nonadiabatic transitions between terms of the T-R pairs at short distances where the exchange interaction is effective. These transitions are induced by the fluctuating H ~ finteraction. i In reality, Hzf~ causes transitions a t all distances. This is nothing else but zfsinduced spin-lattice relaxation. At short distances, however, for some pairs of terms transitions are reduced by the exchange interaction due to an increase of splitting of terms, while for other pairs transitions are accelerated by this interaction because the splitting of terms diminishes. This effect can be easily understood in the short correlation timeapproximation (eq 18). For example, for the transitionIQ+3/ 2) 1D+1/2) we have
-
whereas for the transition 1+3/2)
-
ID-1/2)
(Q-3/21MD-1/2) =
In deriving expressions 21 and 22, we took into account the inequality (20) and treated the exchange interaction as static at times t < 7,. Simple analysis shows that only transitions between six pairs of states contribute to the net polarization
In (23) the transitions are combined into pairs, differing only in
The Journal of Physical Chemistry, Vol. 97, No. 50, 1993 13247
Electron Spin Polarization in Radical-Triplet Pairs sign of spin projection S,and thus making contributions of opposite sign to the net polarization. Because of rapid reaction at r = d (seeeq 16),weassume thepopulationofbothDstates((Df1/2)) to be negligibly small, so that there are no back-transitions from these two states to all others. Combining all rates (with corresponding sign) into the total net polarization generation rate W,,we get
In deriving eq 24, we used (1) the fact that the absolute values of transition matrix elements remain the same under simultaneous change of all spin orientations and ( 2 ) that in the limit of weak exchange [1
+ (oo- 2J)2r,Z]-' - [ 1 + (oo+ 2J)27,21-' = z
8Jw07;/ [ 1 + wo rC] (25) The coefficient 1/3 in the second term of expression 24 is due to the presence of both + 1 / 2 and -1/2 radical spin orientations in the state 1Q+1/2) with weights 2/3 and 1/3, respectively. After calculation of average transition matrix elements we have 2 2
Wn(r)= (32/45)J(r)(D/~,)~x[l/(l + x2)2+ 4/(4
+ x ~ ) (26) ~ ]
where x = (war,)-*. Substituting this transition rate into the diffusion equation of relative T-R motion (with reflective boundary condition in the Q states), one gets a simple expression for the net polarization
pn = ( 1 / 4 ) ( d / ~ r ) J d = dWn(r) r = pO+(x)
(27)
where JodxD2
Po = (8/45)-
aD,wl
and @ ( x )=
l + with x = (007,)-l (29) (1 x2)2 ( 4 x2)2 We will have to compare the experimental net polarizations, given in Table I, with eqs 27-29. Multiplet Polarization. The multiplet polarization is generated by nonadiabatic transitions in the region r,,, of approaching of terms of the spin Hamiltonian ( 5 ) (see Figure 3). It can be described by solving thestochastic Liouville equation (1 5 ) , similar to the calculation of multiplet polarization generated in radical pairs.22 The pattern of terms, however, differs from those for radical pairs, and that is why the formulas obtained for radical pairs cannot be used directly but have to be derived separately for T-R pairs. This has already been discussed and presented rather thoroughly elsewhere.5 The general expression for the amplitude P, of the multiplet polarization, obtained without taking into account fast spin relaxation in the T molecule, is rather comp1ex.s However, with reasonable accuracy (- 10%) we can neglect complicated terms and cast this expression into a form similar to that which has been derived for radical pair recombination p r e v i o u ~ l y . The ~~ generalization of that form to the case of presence of fast spin
+
+
relaxation is knownz4 and can be transferred to the T-R pair process to yield
where k~ = Im[(Wz + ~ A R ) / D , ] ~and / ~ kR = Re[(W2 + iAR)/ Dr]1/2. WZ= 1/T2 is the triplet spin-spin relaxation rate (the rate of dephasing), A R the hfi constant in the TEMPO radical, and Le a radius of spin exchange relaxation of radical magnetization by the T molecules. In our particular case of weak exchange interaction24
Le z d (31) The characteristic length parameter X also depends on the exchange interaction. In previous work the expression for X has been obtained in the limit of strong exchange interaction Jo >> D,aZ.s Here, we consider weak exchange in the limit JO< 4 2 < D,&. The method of calculation of X in this case has already been presented.24 For weak interaction the value of X essentially depends on the boundary condition of nondiagonal matrix elements of the spin density matrix in the basis of the total spin. If the condition is absorptive (rapid dephasing a t contact), then
= ha z (4/(u)(Jo/D,cY2) If the condition is reflective (no dephasing at contact)
(32)
h = A, z 2d(Jod/D,a)
(33) It is difficult to choose in advance which of these two formulas is more realistic, but from a general physical point of view the absorptive condition seems to be more reasonable, so that in what follows we will use X = A,. The derivation of expression 30 implicitly assumes T2 > 2.7 X lo8 rad/s = A, and obtain
k, ' / 2 ( A 2 / W 2 D r ) ' / 2as Well as k R ( W 2 / D r ) 1 / 2 (34) where the phase relaxation rate Wz is given byz3
W 2 = ( 1 / 1 5 ) ( D 2 i , ) [ 3+-1 5x2 +x2
+*I
4+x2
(35)
Comparison withExperiment. The main goal of the theoretical model is ( 1 ) to understand the unexpected strong dependence of the net polarization on diffusion and ( 2 ) to examine, a t least qualitatively, the size of the multiplet polarization in view of the fast, coherence destroying Tz relaxation of triplet-state molecules in solution. The net polarization P, isdescribed by eqs 27-29. Theprefactor POshould be independent of the diffusion coefficient D, because ~ ~ a- D, 1 (and thus x/Dr = 1/o0rcDr) does not depend on D,).
Goudsmit et al.
13248 The Journal of Physical Chemistry, Vol. 97, No. 50, 1993
they are about 1 order of magnitude smaller than the simple estimation T , = &/D, with d 7 X 10-8 cm. Thus, we can conclude that the theoretical model describes quantitatively the experimental dependence of the net polarization on the diffusion coefficient for our system B F / T E M P O . Consequently, a t least in BPT/TEMPO radical-triplet pairs, the polarization is generated in a region where 21Jol g@&, the regions of term crossing a t r = rl and r = rl' (see Figure 3) become accessible for the radical-triplet pairs, resulting in a dependence P, a Dr-l,5926J7quite different from what is observed here. Finally, we turn to the multiplet polarization P,. The experimental results, given in Table I, are plotted versus D, in Figure 5 (top). As lPml is extremely small, amounting to only a fraction of the Boltzmann polarization Pq in thermal equilibrium, the limited accuracy of our measurements (about fO. 1P,) leads to considerable scattering of data. Thus, we hesitate to draw any conclusions with respect to specific values of a,Jo, and d, which, in principle, would be accessible via eqs 30-32, 34, and 35. However, the data have been obtained in a region where war, is close to unity (multiply T~ in Table I with wo = 6 X 10'0 rad/s). Right there, eq 30 might be rather inaccurate because it is based on the assumption TI >> T2. As has been mentioned earler, neglect of T1processes in the pair results in an overestimation of lP,l as calculated by eq 30, the effect being most severe around W ~ =T 1. The solid line drawn for IP,,,l(D,) in Figure 5 has been obtained from eqs 30-32,34, and 35 by arbitrarily setting JO= 5.0 X 109 rad/s, a = 8.0 X lo7 cm-l, and d = 7 X 10-8 cm (estimated from molecular volumesz8),so that condition 4 2 > JO(eq 11) is met and Jd/a = 4.4 X 1od cm2/s as has been found from P,(D,). Qualitatively, this theoretical curve somewhat overestimates lPml a t higher and underestimates it a t lower D, values. This feature is also found for other reasonable combinations of Jo, d, and a. It might indeed indicate the influence of T I relaxation and, a t low D, values, deviations from the short correlation time approximation as DT, < 1 is increasingly violated. However, theory and experiment are a t the limit of their accuracy. At least we can conclude that the theoretical model not only correctly describes size and diffusion dependence of the net polarization but also explains the multiplet polarization, its small size, and its slight increase with decreasing D,. Therefore, the exchange interaction in BPT/TEMPO radical-triplet pairs in 1,2-epoxypropane solution indeed seems to be smaller than the Zeeman interaction at X-band frequencies ( ~ 0 . 3cm-1). This result might well be true also for other pairs of triplets and nitroxyls. At least it is not unparalleled; Porter et aLZ9studied quenching of aromatic triplets by N O and nitroxyl radicals and concluded that the exchange interaction between those triplets and N O must be smaller than 1 cm-1. Finally, it is worth mentioning that there is no contradiction between a very small exchange interaction in radical-triplet pairs and a, nevertheless, diffusion-controlled reaction rate for triplet quenching, because these reactions are not necessarily mediated by the exchange interaction. At least for sufficiently energetic triplets and nitroxyl radicals the quenching process is known to be dominated by charge-transfer interaction or just energy transfer, since nitroxyl radicals possess a low-lying first excited doublet state.18.29
=
In 0,
-
-
-10
-12
-14
Figure 4. Dependenceof net polarization P, on diffusion coefficient D,.
'7 I
t A. I
0 ' 0
\
*
2
4
e 8 lO4cm2/s
Figure 5. Comparison of experimental polarizations (dots) with theory (solid lines). Simulation of 1Pl. (bottom) with J&/a = 4.4 X 1V cm2/s, wor& = 2.2 X 1W5cm2/s and of lpml(top) with JO = 5.0 X 109 rad/s, d = 7 X 10-8 cm, and a = 8.0 X lo7 cm-1.
Therefore, the diffusion dependence of P, is given solely by @ ( x ) in which x 0: D,. It is immediately seen that for large D, (when x = (woT,)-~ > 1) theory predicts a steep diffusion dependence PJD,), up to P, a Or4 for x >> 1. In the opposite limit of slow diffusion ( x < 1) the theoretical dependence @(x) flattens out, so that both @ ( x ) and P, approach constant values as D, 0. This dependence on D, will be always the same for net polarizations created in freely diffusing radical-triplet species in the region of weak exchange interaction. To compare the theoretical and experimental P,(D,), it is convenient to plot In Pn = In PO+ In @(XIversus In D, = In C + In x , where c characterizes the proportionality D, = Cx. Then, theoretical and experimental Pn(Dr) can be brought to coincide by parallel shifting against each other. The result is shown in Figure 4, where the values for lPnl are those of Table I, but all based on P,(298 K). The agreement is nearly perfect, as can also be seen from the more sensitive nonlogarithmic plot in Figure 5 (bottom). Thorough evaluation yields C = 2.2 X 10-5 s/cmz and PO= 1.1 X 1P2, which allows calculation of IJdd/a = 4.4 X 1od cmz/s (D(BPT) = 3.4 X 1010 rad/sZs) as well as the correlation times T , which are listed in Table I. As expected,
-
Acknowledgment. G.-H.G. and H.P.gratefully acknowledge the financial support by the Swiss National Foundation for Scientific Research. References and Notes (1) Thurnauer, M. C.; Meisel, D. Chem. Phys. Lett. 1982, 92, 343.
(2) (a) Imamura, T.; Onitsuka, 0.;Obi,K.J . Phys. Chcm. 1986, 90, Obi,K.;Imamura, T. Rev. Chem. Inrcrmed. 1986, 7, 225.
6741. (b)
~
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