J. Phys. Chem. 1985,89, 4286-4291
4286
Chemically Induced Dynamic Electron Polarization of the Diethoxyphosphonyl Radical: A Case of Mixed S-T, and S-T- Radical Parr Polarization T. J. Burkey, J. Lusztyk, K. U. Ingold,* Division of Chemistry, National Research Council of Canada,+ Ottawa, Canada K1A OR9
J. K. S. Wan,* Department of Chemistry, Queen’s University, Kingston, Canada K7L 3N6
and F. J. Adrian* Milton S. Eisenhower Research Center, Applied Physics Laboratory, The Johns Hopkins University, Laurel, Maryland 20707 (Received: March 25, 1985)
The electron spin resonance spectrum of the diethoxyphosphonyl radical, (EtO)ZPO, which is characterized by a well-spaced 696-G doublet, exhibits a strong electron polarization when generated by photolysis of a 20% (v/v) solution of diethyl phosphite in di-tert-butyl peroxide. Although the observed emissive-absorptive polarization superficially appears due to the usual S-To radical pair mechanism, a detailed analysis shows that the S-T- radical pair mechanism also contributes to the polarization. The analysis also yields an estimate of the range of the exchange interaction between two (EtO),PO radicals.
Introduction Chemically induced dynamic electron polarization (CIDEP) is a firmly established phenomenon that has been extensively investigated and is now well understood, at least in terms of the general features of the polarization mechanisms.‘ There is an important need, nonetheless, to observe CIDEP in systems which can be analyzed quantitatively in terms of existing theory and, hopefully, to obtain thereby detailed information on the magnetic and valence interactions within and between radicals, their reaction kinetics, and spin-lattice relaxation processes, all of which effects play a role in determining the observed polarizations. This is a difficult task due to the multiplicity of factors which affect the polarization, and it is further complicated by the fact that most CIDEP studies involve complex electron paramagnetic resonance (EPR) spectra with hyperfine splittings (hfs) by many nuclei, predominantly protons. During an investigation of the kinetics of some reactions of the diethoxyphosphonyl radical, (EtO),PO, by EPR spectroscopy,2a it was observed that this radical exhibited a strong polarization, which was apparently of the emissive-absorptive (E-A) type’ as shown in Figure 1 . The linear dependence of the polarization on radical concentration indicated a radical pair mechanism involving encounters between separately produced radicals, Le. random pairs. It was felt that the simplicity of this spectrum, which is characterized by one well-spaced phosphorous hfs doublet (AP = 696 G ) 3 and no other hfs, combined with the relative simplicity of the photochemical reaction which produces the radical, afforded an excellent system for a detailed investigation of the CIDEP mechanism. The polarization of the (EtO),PO radical also represents the first example of CIDEP for a radical having the unpaired electron centered on phosphorous. Although the analysis of CIDEP in this system will turn out to be more complicated than originally envisioned, most notably ’ ~ well ~ as the more because the S-T- radical pair m e ~ h a n i s m as contributes to the polarization, the common S-To me~hanism’.~ analysis to be presented in this paper does yield a reasonably detailed picture of the various factors involved in producing CIDEP in this photochemical system. Experimental Section The (EtO),PO radical was generated by flowing a 20% (v/v) solution of diethyl phosphite in di-tert-butyl peroxide through the cavity of a Varian E-104EPR spectrometer while subjecting the .I_._.
-”
‘Issued as NRCC No. 24478.
0022-3654/85/2089-4286$01.50/0
solution to the UV irradiation from a 1-kW high-pressure mercury lamp.2 The photochemical reaction is Me3COOCMe3-.k2MeCO. MeCO.
+ (EtO),P(O)H
-
Me3COH 4 (EtO),PO
In this system the light was carefully focused on the 5-mm sample tube to obtain as high a UV light intensity as possible since, for reasons that will be apparent later, the polarization is greatest at high radical concentrations. The radical concentrations were varied by attenuating the intensity of the UV light by inserting wire mesh screens between the lamp and cavity. It is possible that the strong focusing of the light will lead to a nonuniform distribution of the light and, consequently, of the (Et0)2P0 photoproduct over the sample volume. The moderately strong absorption of the light by the di-tert-butyl peroxide can have a similar effect. This can complicate the study of reactions and reactive processes involying second-order kinetics such as recombination of the (EtO),PO radidals. This is not a problem in the present work, however, because, as will be discussed in detail later, the decay of the electron polarization is determined by first-order intraradical relaxation processes rather than radicalradical recombination or spin exchange. An (EtO),PO EPR spectrum exhibiting polarization is shown in Figure 1.2b The spectra used to determine the polarizations reported in this paper were similar to Figure 1 but also contained small amounts of a carbon-centered ”impurity.” radical in concentrations up to ca. 15% that of the (EtO),PO radical. These “impurity” radicals also may have been present in the spectrum of Figure 1 but were not observed because this spectrum was taken at a relatively high microwave power. This would saturate the carbon-centered radicals but not the (EtO),PO radical, which, as later discussion will show, has very short spin-lattice relaxation times. It is to be expected, however, that all radicals will be (1) J. K. S . Wan and A. J. Elliot, Acc. Chem. Res., 10, 161 (1977), and references contained therein. (2) (a) M. Anpo, R. Sutcliffe, and K. U. Ingold, J . A m . Chem. Soc., 105, 3580 (1983); (b),M. Anpo, K. U. Ingold, and J. K. S . Wan, J . Phys. Chem., 87, 1674 (1983). (3) (a) A. D. Davies, D. Griller, and B. P. Roberts, J . A m . Chem. SOC., 94, 1782 (1972); (b) C. M. L. Kerr, K. Webster, and F. Williams, J. Phys. Chem., 79, 2650 (1975). (4) F. J. Adrian and L Monchick, J . Chem. Phys., 71, 2600 (1979); 72, 5786 (1980). ( 5 ) L. Monchick and F. J. Adrian, J . CAem. Phys., 68, 4376 (1978).
0 1985 American Chemical Society
Electron Polarization of Diethoxyphosphonyl Radical
6960
I
The Journal ofPhysical Chemistry, Vol. 89, No. 20, 1985 4281
I
-
Figure 2. Spin levels of the (EtO),PO radical (a,6 denote electron spin states and +, - the nuclear spin states), the EPR transitions (Il,I,,)and the relaxation transitions ( W J .
observed in spectra taken at low microwave powers in order to minimize possible saturation effects on the measured polarizations. The presence of small amounts of "impurity" radicals, which occurred despite careful purification of all reactants, is unlikely to have a significant effect on the results reported here and their analysis. In particular, the possibility that the relatively small number of (EtO)zPO-impurity radical encounters make a disproportionately large contribution to the CIDEP is ruled out by the fact that the very large phosphorous hyperfine splitting of the (EtO),l% radical doqinates the CIDEP mechanism in this system. Nonetheless, (EtO)zPO-impurity radical polarization may contribute to the observed failure of the polarization vs. concentration plot to go through the origin (cf. Figure 3) and to the considerable scatter in the observed polarizations, particularly at high radical concentrations. The observed peak-tepeak intensities of the low- and high-field EPR lines in the various runs at different radical concentrations are given in Table I. These peak-to-peak intensities were related to the signal intensities obtained by double integration of the derivative EPR lines. The radical concentrations can be obtained from these integrated intensities by comparison with the integrated intensity of a solution containing a known concentration of DPPH. The polarization of the lines would not complicate this process if the polarization were of the simple E-A type due to the S-To radical pair mechanism as originally assumed.2b In such a case the polarizations of the high- and low-field lines would be equal and opposite and the total unpolarized intensity would simply be = I, Ih. the sum of the low- and high-field intensities, Le. Itot,O The estimated radical concentrations [R], in Table I were obtained in this way. In the present work, however, the S-Tmechanism was found also to contribute to the polarization. This = I, + I h relation, and the unpolarized invalidates the simple Itot,o intensity and true radical concentration must be determined with the aid of theory, which we next discuss.
laxation transitions which for this radical result from a large anisotropic phosphorous hyperfine i n t e r a ~ t i o n .The ~ ~ low- and high-field EPR intensities I , and I,, respectively, will be proportional to @+- a+ and @- - a- where @+, p-, a+, and a- are the radical concentrations in the indicated spin levels. The rate equations for the population and depopulation of these spin levels include the following terms. ( I ) Photolytic Population. The four spin levels are equally populated, each at a rate of QZ/4, where I is the light intensity and Q is the quantum efficiency multiplied by the probability of absorption. ( 2 ) Loss by Radical Recombination. A radical in one electron spin state may recombine upon encountering a radical in the opposite electron spin state. The resulting decay rate of radicals in the a+ state is (1/2)kg,at(@+ + &), and similar expressions hold for the other spin states. Here, k, is the bimolecular rate constant for encounters at a radical separation u, where u is approximately the molecular diameter, the factor 1 / 2 represents the fractional singlet character of an ab pair, and ps is the probability that an encountering singlet pair will recombine. ( 3 ) Spin-Exchange Transitions. During the encounter of two radicals with different electron spins and also different nuclear spins (e.g., an a+@-pair) which does not result in recombination, there are equal probabilities that the pair will separate in the same state or in the spin-exchanged state wherein the two electrons have effectively traded places on the two radicals (e.g., an initial at@pair yields a+@-and The resulting loss rate of a+ and @- states and the corresponding gain rate of a- and @+ states may be estimated as (1/2)[(1/2) + (1/2)(1 -p,)]k,a+@- = [(1/2) (1/4)p,]k,a+P- where the factors and (1/2)(1 - p s ) inside the square brackets represent respectively the triplet fraction and the unrecombined singlet fraction of the separating pair. Similarly, an a_@+encounter depletes the a- and p+ levels and populates the a+ and @- levels at the rate [( 1/2) - (1 /4)p,] kea-@+.In fact, the bimolecular encounter rate for spin exchange is likely to be greater than that for recombination, which is approximately diffusion controlled,' because the spin-exchange process only requires that the encountering radicals reach a point where the valence or exchange interaction becomes large compared to the electron-nuclear hyperfine splitting, which separation is considerably greater than that required for reactive encounters. This point is not considered in the present case because even the larger spin-exchange rate would be too slow relative to the spin-lattice relaxation rates to significantly affect the population of the various spin states. ( 4 ) S-To Radical Pair Polarization. This polarization, which results from hyperfine mixing of the electron spin singlet (S) and
Theory A . Kinetic Equations f o r the Radical Spin States. Figure 2 shows the radical spin levels, the EPR transitions, and the re-
(6) E. M. Purcell and G. B. Field, Asrrophys. J., 124, 542 (1956). (7) D. Griller, B. P. Roberts, A. G. Davies, and K. U . Ingold, J . Am. Chem. SOC.,96, 554 (1974).
Figure 1. EPR spectrum of (EtO),PO showing a strong polarization. TABLE I: Observed Low- and High-Field EPR Intensities and the Corresponding Estimated Radical Concentrations
0.51 0.60 1.16 1.26 1.52 1.75
15.9 19.2 33.1 34.0
12.6 25.2
24.6 29.0 59.5 66.8 109.4 114.7
1.84 2.17 1.76 1.85 1.66 2.25
18.9 30.2 -13.4 -7.9 -26.8 -30.0
128.5 143.6 154.3 155.9 159.9 210.0
+
Burkey et al.
4288 The Journal of Physical Chemistry, Vol. 89, No. 20, 1985
the middle component of the triplet state (To) of an encountering pair of radicals, effectively alters the spin-exchange rates, deby ( 1 / 4 ) k g g ~ ~ , a + P creasing the rate of the process a+@- @+aand increasing the rate of the reverse process by (1/ 4)k&&s-T0‘Y4+. Here, ( 1/2)ps is the excess triplet (over singlet) character of an a@pair which survives a potentially reactive encounter and pS-T is the S-To polarization for a pair initially in a pure To For a simple, if somewhat heuristic, proof of this result, note that the effect of the S-To polarization mechanism will be linear in pS-Toand consider the fates of encountering a+P- and a&+ pairs for the hypothetical case of total S-To polarization, Le. p s = 1 and pS-To = 1. For this case, only the nonreactive triplet components of encountering a+P- and ad+ pairs survive and these both separate into an a+@-pair, thereby leading to complete polarization in which only the a+and @- levels are populated. These results are consistent with our formalism which gives the loss rates of encountering a&, pairs due to the combined effects of spin exchange and S-To polarization as [( 1/2) - (1/4)ps(1 f &-To)]kec&. For p s = 1 and pS-To = 1 these equations give the expected zero and total loss, respectively, of the triplet component of the a+@-and a-@+pairs. ( 5 ) S-T- Radical Pair Polarization. This mechanism results from hyperfine mixing of the electron spin singlet and the lowest member of the Zeeman split triplet levels (T-) of an encountering pair. In the present case it actually involves a pair which has separated following an encounter in which the radicals had a chance to recombine and thus has, on the average, a higher probability of being a triplet than a singlet. As in the S-To case such an excess of triplet over singlet character, or vice versa, is required to activate the S-T- mechanism. As the separating radicals pass through the narrow range of separations where the exchange splitting of the singlet and triplet levels equals the triplet Zeeman splitting, so that the S and T- levels cross, the hyperfine interaction can mix these levels, with the resulting probability of a singlet being converted to a triplet or vice versa being4 -+
PS-T
= (*A2/4g/+H) ( r c /2XDo)
(1)
Here, A is the isotropic hyperfine interaction constant of one of the radicals (ps-T_ is doubled if the radicals are identical and both hyperfine interactions can mix the S and T- states), g is the electron g factor, pLBis the Bohr magneton, H is the external magnetic field, 2D0 is the diffusion constant for the relative diffusion of the two radicals, Do being the diffusion constant of an individual radical, r, is the level crossing separation, and X is the range parameter in the exchange interaction, which has the form
Thus, rc is given by gFgH = Finally, it should be noted that, for simplicity, small effects of the hyperfine splitting on the separation of the S and T_ levels have been neglected in eq 1. Since the S-T- mixing is provided by the isotropic hyperfine reinteractions on the individual radicals, A,Il-S, and AZ12-S2, spectively, where A I and A,, I, and I,, and SI and S2 are respectively the isotropic hyperfine constants, the nuclear spins, and the electron spins of the two radicals, the S-T- mechanism depends on both the electron and nuclear spins of the radicals involved. Also for some states the mixing may be affected by the hyperfine interactions on both radicals, while for other states only one hyperfine interaction is active. Thus, it is necessary to consider separately the different possible S and T- states of an encountering radical pair. The T- encounters consist of the @-@-, @+P-, and @+P+ pairs. The first of these cannot undergo T- S crossing because it is impossible to compensate the raising of the electron spin state with lowering of a nuclear spin state as required to conserve total spin angular momentum. The @+@- pair can be converted to the singlet state (a-@-@-a-)/2’/*only through the hyperfine interaction on the first @+
-
(8) F J. Adrian, J . Chem. Phys., 54, 3918 ( 1 9 7 1 ) .
radical. This singlet state can separate into an a& pair effectively converting a P+ radical to a-. The rate of this process is k&s-T.fi+P-, where k,P+fi- is the rate of encounters of a @+and a P- radical at the reaction distance. The P+P+ pair may cross either to the singlet state (KO+ CY+)/^'/^ or the equivalent singlet state (P+cL - CY+@_)/^^/^, these crossings being promoted by the hyperfine interactions on the first and second radicals, respectively. As the radicals comprising these singlet states separate, they evolve with equal probability into the uncorrelated pair states a-@+and @-a+ in the case of the first and a+&for the second singlet. (The evolution singlet and fi+a of a separated singlet pair into an uncorrelated pair occurs via hyperfine mixing of the S and To states, which have the same energy in the separated radical pair.) In each case, the first possibility leads to loss of a P+ state and the gain of an a- state. The second possibility leads to the loss of two @+states and the gain of an a+ and a 0- state. Thus, the net loss rate of @+states from this encounter is (3/2)keps-T_@+2and the gain rates of the a+,a-, and P- states are each (1/2)kepS-T-@+’,where it is to be noted the encounter rate of identical species such as @+ is (1 / 2)keP+’. The S encounters involve the a+@+, a@+,a+@-,and a& pairs, each of which have equal amounts of S and To character. The a+@+ pair cannot undergo S-T- crossing because neither nuclear spin can be raised as required to compensate for the decrease in the electron spin angular momentum. The rates for the S Tcrossings are readily derived from the fact that they are equal to the corresponding T- S rates except for the factor 1 - ps, which is the probability of the singlet pair surviving the reactive encounter. Clearly a nonzero p s is required to produce the needed difference in average singlet vs. triplet character of the separating pairs. The foregoing effects together with the spin-lattice relaxation transitions depicted in Figure 2 lead to the following set of kinetic equations for the four spin states of the radicals:
-
-
da+/dt = j/zkgs(a+P++ a+@-)- Yzke(1 - !&s)(a+P- - .-@+I + + .-P+) + )/&ePS-T-[@+@+ - (1 - ps)a+@-][(We + WJ(1 + Y26) + WnIa+ + We(1 - )/*S)P+ + W2(1 - Y26)@- W,a- (3a)
)/4QI-
f/4k$sPSTo(a+@-
+
da-/dt = Y4QI - Yzkgs(a-@+ + .-E) - f / , k ( l - 1/2ps)(a-P+- a+@-))/4k$sPS-To(a-@+ + a+@-)+ 1/ZkePS-T.[@+2+ 2@+@-- (1 ~s)(a-P+ + 2a-fi-11 - [(We + Wd(1 + )/26) + Wnla- + We(1 - )/26)fi- + Wo(1 - )/26)@+ + Wna+ (3b)
d@-/dt = Y4QI - )/2kgs(a+P- + 4 l/4k$sPS-To(a+@-
- 1 - Y2ke(l - )/,s)(a+P-- a-P+) + + .-@+I + 1/2kePS-T.[PfZ - (1 - ps)a+@-l-
[(We+ WJ(1 -
Y26)
+ W,]@-+ We(l + @)a- +
WZ(1 + l / z 6 ) ~ ++ W,P+ (3c) dD+/dt = 1
!hk~s(a+@+ + a-P+) - !hke(l - )/,s)(a-@+ - a+@-) + a+P-) - %ke~s-T.[3P+~+ 2P+@-- (1 ps)(a-P+ + 2a+P- + 2a-P-)I - [(We + Wd(1 - 728) +
/4QI
-
j/4ke~s~s-To(a-P+
WJP+
+ We(l + yz8)(~++ Wo(1 + f/zS)a_+ W,&
(3d)
Here, 8 is the Boltzmann factor (6 z huM/kTwhere u M is the ESR microwave frequency) which weights the relaxation transitions so they thermally equilibrate the spin levels. It is noteworthy that the S-T- polarization mechanism does not produce a uniform emissive polarization in both the EPR transitions, as had been generally believed on intuitive grounds but, for the present case of a positive hyperfine splitting, primarily polarizes the low-field transition. This can be seen by inspecting the terms containing pS-T-in eq 3, setting p s = 1 , and noting that in zeroth order all quadratic products of radical concentrations
Electron Polarization of Diethoxyphosphonyl Radical
The Journal of Physical Chemistry, Vol. 89, No. 20, 198.5 4289
will be equal, in which case the low-field transition is emissively polarized 3 times faster than is the high-field transition. It also should be noted that the sense of this polarization will be reversed; Le., the high-field transition will have the greatest S-T- polarization, if the hyperfine splitting is changed from positive to negative. It is convenient, and always possible, to linearize eq 3 by writing a+ = [R](y4
of the (Et0,)PO radicaVb rules out possible spin-orbit and spin-rotation relaxation processes.) For this situation the relaxation transitions are9 Wo = 2W,
W e = 3W, W2 = 12W,
W, = 3Wx (8a)
where
a- = [R](y, + Act-); @- = [R1(1/4 + A@-); @+= [R1(1/4 + A@+) (4)
+ Aa+);
(8b)
where [R] is the total radical concentration, given by d[R]/dt = QZ - ~ & J , [ R ] ~
(5)
and the Aa and A@terms are all much less than one and satisfy the equation Aa+
+ Aa- + A@- + A@+ = 0
(6)
and T~ is the rotational correlation time of the radical. For this case the steady-state population differences between the levels involved in the ESR transitions are (A@+- Aa+)/6 =
If we rewrite eq 3 using eq 4 and neglect both the quadratic terms in A and product terms of A times the small polarization factors ps-To and ps-T_and the Boltzmann factor 6, we obtain dAa+ -dt
-
dAa-dt
- --[R]kg,(2Aa+ A@- + A@+)+ 8
1 --[R]kgS(2Aa+ 8
+ A@- + A@+)+
[R1kgS 8W
[
+ + + + l o x + 15 60(2x2 + 5x + 3)
PS-T-
PS-To
6
6
+
6x2 40x 39 60(2x2 5 x 3)
and
1
1 -dA@- - --[R]kg,(Aa+ + Aa- + 2A&) + dt 8
dA@+ - - - - 1[ R ] k g , ( A a + dt
1 4
--- -
8
+ ACU-+ 2A@+)+
B. Steady-State Solution of the Kinetic Equations. Analysis of the present system requires the steady-state solution of eq 7, i.e. da+/dt, da-ldt, etc., are zero. These equations can be solved analytically by changing the variables to A@+- Pa+, A@-- Aa-, and Aa+ - Aa-, where the fourth variable A@+- A@- can be written in terms of these three, and the low- and high-field EPR intensities are proportional to A@+- Aa+ and A@- - Aa-, respectively. The resulting solutions are too cumbersome, however, either to be useful for numerical evaluations or to give significant physical insights. A useful and widely applicable approximate solution may be derived by noting that, first, unless the recombination and spin-exchange processes are significantly faster than the relaxation processes, their effect on the population of the levels is minimal and, second, the spin-lattice relaxation processes here are due solely to the large anisotropic phosphorous hyperfine interaction (Bil = 1 13 G).3b (The very small g factor anisotropy
Here, the population differences are given relative to the Boltzmann factor 6 N 0.0015 at 295 K). The emissive S-T- polarization of the low-field transition @+ a+ is considerably greater than that of the high-field transition whereas the S-To polarizations of the high- and low-field transitions are equal in magnitude but opposite in sign. Note that the foregoing equations predict the polarization is proportional to the radical concentration, which quantity will be the average concentration over the sample volume should the experimental conditions yield a somewhat inhomogeneous distribution of radicals. This simple concentration dependence occurs because the decay of the polarization is dominated by intraradical relaxation processes. If second-order radical-radical processes such as spin exchange and recombination contribute to the polarization decay, then there would be concentration-dependent terms in the denominators of eq 9, and the case of an inhomogeneous radical distribution would be complicated considerably.
-
Results and Discussion Although it is still convenient to relate the polarization to the difference between the high- and low-field EPR intensities, it is clear from eq 9 that the unpolarized intensity, and hence the radical concentration, is not proportional to the sum of these intensities if the S-T- mechanism contributes significantly to the polarization. For this situation, eq 9 must be treated as a pair of transcendental equations relating the observed EPR intensities to the radical concentration and one other quantity appearing in this equation. We take this quantity to be the range of the interradical exchange interaction, X in eq 2, which quantity is of fundamental interest and is important in determining the polarization efficiencies pS-To and P S - ~ . . The remaining parameters appearing either directly or implicitly in eq 9 must be determined from other experiments or from theory. These parameters include the radical diffusion constant. Following current practice,I0 we assumed this to be the same as the diffusion (9) (a) K. H. Hauser and D. Stehlik, Adu. Magn. Reson., 3, 79 (1968); (b) A. Carrington and A. D. McLachlan, 'Introduction to Magnetic Resonance", Harper and Row, New York, 1967, pp 231-233. (10) (a) C. Huggenberger, J. Litscher, and H. Fischer, J . Phys. Chem., 84, 3467 (1980); (b) M. Lehni and H. Fischer, Int. J. Chem. Kinet., 15, 733 (1983); (c) J. Litscher and H. Fischer, J . Phys. Chem., 88, 255 (1984).
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The Journal of Physical Chemistry, Vol. 89, No. 20, 1985
Burkey et al.
TABLE II: Radical Concentration and the Calculated Polarizations of the Individual Low- and High-Field Lines for the Different Experimental Runs" p,,,
pS-T.
pS-To
[Rl, rmol/L
1
h
1
h
1
h
0.19 0.62 0.73 1.50 1.69 2.80 2.92 3.26 3.28 3.66 3.96 4.00 4.11 5.40
-0.018 -0.060 -0.071 -0.146 -0.164 -0.272 -0.284 -0.317 -0.319 -0.356 -0.385 -0.389 -0.400 -0.525
0.000 -0.001 -0.002 -0.004 -0.004 -0.007 -0.007 -0.008 -0.008 -0.009 -0.010 -0.010 -0.010 -0.013
-0.01 3 -0.043 -0.051 -0.105 -0.118 -0.195 -0.204 -0.227 -0.229 -0.255 -0.276 -0.279 -0.287 -0.377
-0.006 -0.019 -0.022 -0.046 -0.051 -0.085 -0.089 -0.099 -0.099 -0.1 11 -0.120 -0.121 -0.125 -0.164
-0.005 -0.017 -0.020 -0.041 -0.046 -0.077 -0.080 -0.089 -0.090 -0.101 -0.109 -0.110 -0.1 13 -0.148
0.005 0.017 0.020 0.042 0.047 0.078 0.082 0.091 0.092 0.102 0.1 11 0.112 0.115 0.151
P,,,,Ps-T., and
PS-TO are
respectively the total and S-T- and S-To contributions to the total polarization, which polarizations are calculated as (I is the unpolarized ESR intensity given by 80[R].
- ~ ~ ~ ~ , , , t , O ) / where ~ , O t ~ IOis the line intensity and
coefficient for (EtO),P(O)H in the reaction medium and measured it as described previously," obtaining Do = cm2/s. Other required parameters are the product of the bimolecular radical encounter rate constant and the probability of a singlet radical pair recombining upon encounter ( k g , ) ,the rotational correlation - p~~ - ~ which ., theory gives in terms of the time TR, and p ~ and former parameters and the isotropic hfs constant. The quantity k g , may be obtained from the bimolecular reaction constant via the relation k g , = 4k2, the factor of four appearing because only I/., of the encountering radicals are in the reactive singlet state. Experiment gives kz = 3.3 X lo9 M-'s-l for the self-reaction of (EtO),PO radical^;^ hence, p,k, = 1.32 X I O i o M-l SKI,which rates are diffusion controlled. There are two hydrodynamic models for estimating 78 of which the slipping-boundary-conditionmodel12appears better to represent the rotation of small molecules in liquids13J4 than does the sticking-boundary-condition model.15 (Using the sticking model in the present analysis yields very fast spin-lattice relaxation rates and polarizations which are somewhat less than the observed ones for reasonable values of the other parameters, which is consistent with the idea that the "slip" model is more accurate.) In the "slip" modeli2J3rR = (4ra377/3kT){= (a2/3Do){, by using the relation Do= kT/4aaq,I6where a is the molecular radius, 7 is the viscosity, and { is the ratio of the friction coefficient calculated by using the slipping boundary condition to that calculated using the sticking boundary condition. The conformation of the (EtO)zPO radical is not known. However, estimates based on a space-filling model gave (i) for a highly compact configuration of approximate oblate spheroidal shape a mean radius a = 2.8 A, a short/long axis ratio of 0.6, and { = 0.15412and (ii) for a fully extended chain configuration of approximate prolate spheroidal shape a mean radius of 2.8 A, a short/long axis ratio of 0.5, and { = 0.240. The corresponding rotational correlation times, calculated by using cm2/s, are 4 X and 6 X the measured value of Do = lo-'' s, respectively, with the average being 7 R = 5 X s. To check the consistency of these estimates, we note that the calculated encounter rate is k, = 8 ~ ~ D d ? o / 1 0 0where 0 Q is the encounter separation and No is Avogadro's nurnber.l7 Taking u = 2a gives k, = 8.5 X lo9 M-' s-l which agrees well with the value 13.3 X lo9 M-'s-I calculated from the observed' k2 by using (1 1) T. J. Burkey, D. Griller, D. A. Lindsay, and J. C. Scaiano, J . A m . Chem. SOC.,106, 1983 (1984). (12) C. Hu and R. Zwanzig, J . Chem. Phys., 60, 4354 (1974). (13) L. M. Jackman and N. M. Szeverenyi, J . Am. Chem. SOC.,99,4954 (1977). (14) D. R. Bauer, J. I. Brauman, and R. Pecora, J. A m . Chem. Soc., 96, 6840 (1974). (15) A. Einstein, 'Investigations on the Theory of the Brownian Movement", Dover, New York, 1956, pp 19-34. (16) J. C. M. Li and P. Chang, J . Chem. Phys., 23, 518 (1955). (17) M. V. Srnoluchowski,2. Phys., 17, 557, 583 (1916).
0.5
t I
/
j
Figure 3. Electron spin polarization vs. concentration in the (EtO),PO radical: (e),experimental points; (-), least-squares fit to the experimental points; (---), theory.
the relation psk, = 4kz and assuming p , = 1. Although analytic solutions exist for the stochastic Liouville they are not apmodel of the S-To radical pair me~hanisrn,~J* plicable to the present case of very strong hyperfine mixing. Thus, psTomust be determined from numerical solutions of this equation. Extrapolation of the results of Pedersen and Freed19to the present hyperfine splitting (Q = 61.6 in their notation) and interpolation for D = 2D0 = 2 X lo5 cm2/s at various values of X give ~ s - T=~ 0.0015
+ 0.137X-' - 0.0227X-'
(10)
where X is in A-1. The analytic solution gives pSWTo = 0.21/X, a considerable overestimate as expected for a very large hfs constant. The S-T- polarization is given by eq 1 where the S-T- level crossing separation is determined by the relation gwaH = JocbC. Thus, determination of rc requires an estimate of Jo, but this is not a critical factor in the calculation because r, depends only on In Jo. It is reasonable to assume that J will be comparable to the thermal energy at the reactive encounter separation u = 2a, that N kT, and thus r, N 2a + X-' In (kT/gwaH). Thus, is, (18) F. J. Adrian, Chem. Phys. Letr., 80, 106 (1981). (19) J. B. Pedersen and J. H. Freed, J . Chem. Phys., 58, 2746 (1973).
J. Phys. Chem. 1985, 89, 4291-4302 from eq 1 with H = 3400 G at 295 K we have psT_
0.055X-'
+ 0.064X-2
(11)
where X is in A-'. For the expected values of A, ca. 1-2 A-1, psr, and pS-T_are of comparable magnitude. By use of the foregoing parameters, a "best fit" of the experimental data to eq 9 gave X = 1.74 A-i and the polarization vs. concentration results shown by the individual points in Figure 3. Here, the polarization is defined as pt = (1, - Il)/ztot,O
(12)
where ZtOt,, is the total unpolarized EPR intensity which is determined by using theory to aportion the observed polarization represented by Z, - Zl to the individual lines and thus determine the unpolarized intensities. Ztot,o is related to the radical concentrations given in Table I1 by N (rmol/L) = o.01251t,,0. The solid line in Figure 3 is a linear least-squares fit to the experimental points, and the dashed line is the theoretical polarization vs. concentration dependence. Theory and experiment agree except for a small nonzero experimental polarization at zero radical concentration. It is not known whether this zero-concentration polarization is real, which would imply that a small fraction of the (EtO),PO radicals are formed as part of a geminate radical pair, or is an artifact. In either event, however, it can be shown by numerical experimentation that the determination of the concentration dependence of the polarization from the experimental data is not significantly affected by the presence of this apparent zero-concentration polarization. Table I1 gives the radical concentrations for the various experimental runs and the corresponding calculated polarizations of the individual ESR lines, denoted Pi and Ph for the low- and high-field lines, and their decomposition into the S-T- and S-To contributions. The calculated polarizations are all proportional to the radical concentration and are related to the polarization
4291
in Figure 3 by the relation Pt = Ph - PI. Clearly both the S-To and the S-T- mechanisms contribute to the polarization and the two polarizations cancel almost completely in the high-field line. The range of the exchange interaction between two ethoxyphosphonyl radicals, X = 1.74 AT', is reasonable and in accord with other estimates of this q ~ a n t i t y . ~ It , ~ must ~ be noted, however, that the present analysis determines X only roughly since the value obtained for X depends on the other parameters in the calculation, especially the radical encounter rate and the relaxation rates, with the latter depending on the rotational correlation time which itself could only be roughly estimated. In summary, this diethoxyphasphonyl radical system has proved amenable to a rather detailed analysis using the radical pair theory of CIDEP which has resulted in an improved picture of the polarization process. It is likely that further exploration of this and related systems will be profitable. Time-resolved investigations using either a pulsed laser or a rotating sector to produce the radicals should be especially useful, giving among other things a better estimate of the relaxation rates. Such studies present especially challenging problems in this system, however, because of the rapidity of these relaxation processes (from eq 8, W = 2.3 X lo5 s-'), thereby requiring high-intensity photolysis sources to produce the high radical concentrations needed for observable polarization in the face of rapid relaxation loss. Acknowledgment. F.J.A. thanks Professor Jeffrey Wan and his colleagues in the Chemistry Department of Queen's University for their hospitality during a visit when part of this work was done. Thanks are also due Professor Samuel I. Weissman for several helpful suggestions. This work was supported by the U S . Naval Sea Systems Command under Contract N00024-85-C-5301. Registry No. (EtO)#O, 3 1682-65-2; (Et0)2P(0)H,762-04-9. (20) F. J. Adrian, J . Chem. Phys., 57, 5108 (1972).
Structure and Dynamics of Perylene Complexes: Comparisons of Atomic and Molecular Complexation Mark M. Doxtader and Michael R. Topp* Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104 (Received: March 26, 1985)
Following recent work in low-temperature crystals, fluorescence studies on isolated perylene complexes with Ar, CHI, C2H,, C3H8,and C3H6(cyclopropane)have been used to show the effect of increased perturbation on radiationless relaxation processes. It is shown that internal rotation of the complexes provides an efficient means for suppressing resonance fluorescence characteristic of the perylene internal modes. It is further shown that characteristic patterns of resonances may be correlated with different degrees of aggregation, so that distinct positional isomers of various complexes may be confidently assigned. Vibrational predissociation was observed in all cases, and showed dynamical differences between the different complexing species. Where possible, the experimental dissociation energies are in agreement with computed binding energies.
1. Introduction 1. I . Vibrational Relaxation in the Condensed Phase. Much
attention has been devoted in recent years to study the effects of external perturbations on fast molecular processes. Recent work in this laboratory has studied vibrational relaxation of polycyclic aromatic hydrocarbons in the condensed phase.'** The main aims of this work have been to develop a quantitative picture of the (1) Choi, K.J.; Topp, M. R. Chem. Phys. Lett. 1980, 69, 441. Boczar, B. P.; Choi, K. J.; Topp, M. R. Chem. Phys. 1981, 57, 415. (2) Boczar, B. P.; Topp, M. R. Chem. Phys. Lett. 1984, 108, 490.
0022-3654/85/2089-4291$01 .50/0
dependence of the rate of vibrational energy relaxation on the internal vibrational energy of the molecule concerned, and on the actual vibrational levels excited. It is also valuable to study the influence of matrix perturbations on these rates. Thus, transient fluorescence lifetimes were measured for crystalline solutions of perylene in heptane and neon at 4 KS2s3 The excitation wavelength was varied from -445 to 355 nm, so that S1 vibrational energies from 0 to at least 4500 cm-' could be generated in each case. An important conclusion to be derived (3) Boczar, B. P. Ph.D. Thesis, University of Pennsylvania, 1984
0 1985 American Chemical Society
