J . Phys. Chem. 1990, 94,47 16-4723
4716
though this splitting was more prominent in cyclohexane-d12. Degenerate modes such as u28. 129, 130, and 131 split in phase I1 of both compounds, as did the nondegenerate modes, uI4 and u15. In phase 111, the observed splitting of the nondegenerate a,, modes (such as u I 4 ) and the degenerate e, modes (such as 129) into two and three components, respectively, Table 111, is consistent with the proposed DZhunit cell symmetry. Further splitting was observed in phase 11. In particular, I 2 8 and ~29,both of e, symmetry, split into at least three components, and four weak bands appeared in the 900-750-~m-~ region of the spectrum, including the overtone, 2uI6,and the al, mode, us. The latter mode is forbidden in the free molecule but becomes allowed in the crystal, as the molecule is located on a site of lower symmetry. The spectrum of this phase resembles that reported for a single crystal of phase I1 under pressure in a DAC."' The splittings observed for phases 111 and I1 are further evidence for the proposed unit cell symmetries as, although the predicted number of components were not always observed, no band split into more bands than predicted. The pressure dependences and logarithmic pressure derivatives of selected internal mode frequencies of cyclohexane-d,, are listed in Table 111. For several bands, the d In u/dp values are lower in phase 111 than the other two phases; however, there are several exceptions. This trend is not as pronounced as that observed in cyclohexane. In most cases, the d In u/dp values for the modes in all three phases are lower than those calculated for the corresponding modes in cyclohexane. As the C-D bond is shorter than the C-H bond, cyclohexane-d12is more dense than cyclohexane and, therefore, less compressible, which leads to smaller shifts in frequency with increasing pressure. The vibrations, u15,~ 2 9 and . ~30,which correspond to a mixture of a CD, rock and a CCC bend, a CD, twist, and a ring stretch,%s7
respectively, are among those with the highest d In u/dp values. These vibrations were also among those that exhibited the highest d In u/dp values in cyclohexane. In particular, the ring stretch, u30 (which is 131 in cyclohexane due to the relative positions of the CH2 and CD2 rocks as a result of the isotope effect), is the internal mode with the highest d In u/dp value in both compounds, excluding the C-H stretches. This is in agreement with the trend reported by Ferraro28that stretching vibrations are more pressure sensitive than bending or twisting vibrations. For all the bands that split in phases 111 and 11, the higher frequency component was found to have a higher d In u/dp value than the low-frequency component. This is a result of the increase in factor group splitting due to the increased intermolecular interaction at higher pressure. In most cases, the d In u/dp value for the high-frequency component is much larger than that of the lower frequency component, indicating that the factor group splitting increases significantly due to the reduction in intermolecular distances. In phase 111, in particular, negative values are observed for the lowest frequency components of several split bands, indicating that the increase in factor group splitting makes a significant contribution to the pressure shift. Acknowledgment. This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada. J.H. acknowledges the award of a scholarship from NSERC. (26) Snyder, R. G.; Schachtschneider, J. H. Specrrochim. Acra 1%5,21, 169. (27) Forel, M. T.; Garrigou-Lagrange, C. Ann. Chim. Paris 1973,8,207. (28) Ferraro, J. R. Vibrational Specrroscopy at High Exfernal Pressures: The Diamond Anvil Cell; Academic Press: Orlando, FL. 1984.
Spin-Lattice Relaxation of the Solltonic Defect in Isotopically Substituted trans-Polyacetylene Robert St. Denis,+Eric J. Hustedt,' Colin Mailer,$ and Bruce H. Robinson*.+ Department of Chemistry. University of Washington, Seattle, Washington 98195, and Department of Physics, University of New Brunswick, New Brunswick, Canada E3B 5A3 (Received: October 9, 1989)
We have measured the spin-lattice relaxation rates, R , , for the spin of the electron defect in pristine trans-polyacetylene (t-PA) as a function of temperature for the isotopic forms (CD),, ("CD),, (CH), and ("CH),. The data have been compared with the soliton-phonon-based dynamics model of Maki and the localized metal-phonon based dynamics model of Robinson et at. We concluded that the latter model is preferred; it exactly explains the R I data for (CD), and (I3CD), t-PA. We find that the R1 data of the proton-containing forms of t-PA, (CH), and ("CH),, are complicated by the presence of spin 1/ 2 nuclei, which are strongly coupled to each other, and which provide an additional dynamic path for defectspin relaxation. We show how the dynamic models may be modified to include this additional spin relaxation mechanism. The result suggests that the nature of nuclear relaxation in proton-containing &PA may be more complex than assumed in previous treatments.
Introduction trans-Polyacetylene (t-PA) is the prototypical low-dimensional organic semiconductor. The experimental evidence for the structure of t-PA has been extensively summarized.' The nature of conductance is believed to be related to the ability of t-PA to sustain a defect which acts as a boundary between two domains which show bond alternation of opposite phase. Figure 1 shows the defect delocalized over a few carbons. The defect is thought to be self-localizing: that is, the spin defect will extend over a maximum number of carbons, beyond which bond alteration will start. The spin defect will have very low probability of being in the regions of bond alternation. The early work of Salem and 'University of Washington. *University of New Brunswick.
Longuet-Higgin~~*~ using Huckel theory showed that "the configuration with equal bond lengths is unstable with respect to bond alternationn2when the length of the polymer chain exceeds 55 carbons. The authors then speculate that this number is the maximum range of delocalization of the defect in a region of uniform carbon-carbon bonds. Experimental evidence indicates that the defect is in fact delocalized nearly up to this maximum distance.28 This estimation was based on Huckel theory and on assumptions about the dependence of the so-called Hiickel @ parameter on the bond length. (1) Chien, J. C. W. Polycefylene; Academic Press: New York, 1984. (2) Salem, L. The Molecular Orbital Theory of Conjugated Sysrems; Benjamin: Reading, MA, 1974; p 517. ( 3 ) Longuet-Higgins, H.C.; Salem, L.Proc. R. Soc. (London) 1959, A251, 172.
0022-3654/90/2094-47 16$02.50/0 0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 11. 1990 4717
Spin-Lattice Relaxation of trans-Polyacetylene H I
H
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Figure 1. The structure of trans-plyacetylene showing the delocalization of the defect over several carbons. In this illustration, the defect is distributed over five carbons which have a uniform carbon-carbon bond distance. The defect exists between two regions of t-PA which have pure bond alternation of opposite phase. In the region of phase alternation, the bonding pattern is that of the unperturbed structure characterized by alternating bonds; the double bond is shorter than the single bond. The bond in the uniform region is intermediate in length between the single and double bonds.
More recently Su, Schreiffer, and Heeger4,5carried out a Hiickel calculation similar to that of Salem2 but assumed a form for the nuclear distortion in the region of the defect which led to an estimate of 15 carbons for delocalization-the SSH model. The particular form of the distortion so chosen ensures that the spin density wave will have solitonic properties. The solitonic form of the spin density wave suggests that the wave will move coherently with a very low energy barrier to migration and that the wave will remain self-localized (Le., it will not spread out or dissipate). There is no doubt but that distortions in one dimension will localize, as explained by both the more general Peierls theory (for Jahn-Teller one-dimensional distortions),6 and by Hiickel theory applied to this particular with the caveat that different investigators have made different assumptions about the functional dependence of /3 on nuclear configurations. The implications of the solitonic form as the basis for a diffusive model of spin dynamics have been theoretically investigated by Maki7-8and later summarized and reviewed? He predicted that above 50 K the diffusion coefficient should decrease with increasing temperature according to a power law, unlike a simple activated process. This result is based on the assumption that the diffusion of the soliton, in its SSH form, is driven by three-dimensional acoustic phonons; such phonons are typical of simple solids. We have therefore undertaken a study of the spin-lattice relaxation rates, R 1 ,of t-PA to examine the dependence of the diffusion coefficient on temperature. We have previously reported the dependence of R, on temperature for the (CD), form of ?-PA and compared it to a model in which the defect dynamics arise from phonon scattering like an electron in a simple metal over a limited domain.I5 We now report the measured temperature dependence of R I ,the spin-lattice relaxation rate (the inverse of the spin-lattice relaxation time) of the unpaired or defect electronic spin for the trans form of ?-PA for the isotopically substituted analogues ( I2CD),, (I3CD),, ( 12CH),, and (I3CH),. Experimental Section A suitable technique for obtaining rates of spin-lattice relaxation of the defect is that of saturation recovery (SR) electron paramagnetic resonance (EPR) spectroscopy, in which the spin system is perturbed by a short, high-power microwave pulse and the return to equilibrium monitored. It is a good alternative to electron spin echo (ESE) inversion recovery spectroscopy, being simpler technically, requiring much lower pump powers, and of higher sensitivity. ESE signals are further complicated by the echo envelope modulation in the polyacetylene sample substituted with the I3C isotope. Such modulations are not seen in our SREPR experiments. We performed the experiments on a recently developed, computer-controlled, SR-EPR spectrometer.I0 Cavity
0.0
'.
0
I
I00
I
I
I
200
300
400
TEMPERATURE ( K )
Figure 2. A comparison of spin-lattice relaxation rates ( R l , MHz) obtained from trans-(l2CD), by both saturation recovery (A) and inversion recovery electron spin echo (A)spectroscopy as functions of the temperature. The dashed line indicates the fit to the theoretical form suggested by ref 15, given in eq 5a, where 8 = 80 K,a = 0.22 MHz, and p = 0.
Q s were determined from the decay of power in the EPR cavity with the Zeeman field set away from sample resonance. The temperatures were independently measured with a Bailey BAT 12 thermometer and a thermistor system developed by the Department of Chemistry Electronics Shop. The data were analyzed by fitting a single exponential to the recovery curve. The SR curves fit a single exponential, to within the noise level, over the entire time range of the recovery curve in all cases. None of the recovery curves of any of the samples showed multiple exponential character, which would have been indicative of chemically distinct species or dominant cross relaxation processes; all recovery data fit a single exponential. The time constant for the recoveries, as well as the amplitude and baseline, were simultaneously optimized by a nonlinear least-squares searching algorithm. These fits were compared with multiple exponential fits and, according to standard statistical treatments, the single exponential was always preferred.17 The rate constant of the recovery curve was taken to be the relaxation rate R , . Care was taken to ensure that no artifacts were introduced due to short pump time or too high an observer level." A 1-gs dead time between the end of the pump and the beginning of the observer period was used. The isotopically substituted samples were prepared from starting materials with the following isotopic purities: ( I2CD), contained 98% deuterium substitution for protons, (I3CH), contained 98% I3C,and (I3CD), contained 98% 13Cand 99% D. These samples were prepared either by Dr. G. B. Street of IBM, San Jose, or the group of Dr. L. R. Dalton at the University of Southern California, according to the method of Ito et al.12 Thermal cycling was performed to ensure that the samples were in the fully trans form. CW-EPR and SR-EPR spectra were obtained over the temperature range from 77 to 350 K. Results The spin-lattice relaxation rates for (CD), were measured by SR and compared to those obtained from ESE on the same sample. Figure 2 shows the spin-lattice relaxation rates as a function of temperature obtained by the ESE techniqueI5 for (I2CD), and data obtained by the saturation recovery method. Clearly, the two methods of measuring spin-lattice relaxation rates give essentially identical results. CW-EPR measurements were made to determine the relative number of spins as a function of temperature. The number of spins, N, is given by N = N,,AT/(Q'/2) where No is a constant, A the area of the EPR line, T the absolute temperature, and Q the cavity quality factor.l33l4 To ensure that
(6) Peierls, R. E . Quantum Theory of Solids; Clarendon Press: Oxford,
(10) Mailer, C.; Danielson, J. D. S.;Robinson, B. H. Reu. Sci. Instrum. 1985, 56, 1917. (1 1) Hyde, J. S. In Time Electron Spin Resonance; Kevan, L., Schwarz, R. N.. Eds.: Academic Press: New York. 1979: Chaoter 1. (12)lto. T.; Shirakawa, H.; Ikeda, S. J . Polym. S'ci. Polym. Chem. Ed.
(7) Maki, K. Phys. Rev. B 1982, 26, 2181. (8) Maki, K. Phys. Reu. B 1982, 26, 2187. (9) Andretta, M.; Serra, R.; Zanarini, R.; Pendergast, K. Adu. Chem. Phys. 1985, 53, 1 .
(13) Poole, C. P. Electron Spin Resonance: a Comprehensive Treatise on Experimental Techniques; Wiley: New York, 1983. (14) Mimms, W. B. Electron Paramagnetic Resonance; Geschwind, S . , Ed.; Plenum Press: New York, 1972.
(4) Su,W. P.; Schrieffer,J. R.; Heeger, A. Phys. Reo. Lett. 1979,42,1698. ( 5 ) Su,W. P.; Schricffer, J. R.; Heeger, A. Phys. Rev. B 1980, 22, 2099.
U.K.,1965.
-. --.
1974. 12. 11-20 .. . - - --
4718
The Journal of Physical Chemistry, Vol. 94, No. 1 1 , 1990
St. Denis et al. 0.6
-
0.4
I
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1
1
I
1
1
-
t4
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0.5'
I
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4
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0.004 0.006 0.008 T E M P E RAT UR E - I ( K -I) Figure 3. The relative number of spins, N , of (12CD),(A)and (12CH),
(m) as a function of inverse temperature. (See the text for calculation method.) Since only the relative spin concentration is measured for each sample alone, the true concentrations are not known. The spin density in (I2CD),as compared to (I2CH),is also not known. The relative spin concentrations are arbitrarily set to 1 at 300 K. Each data set is fit to a least-squares straight line. The slope of the line for (I2CD),, and --- for (I2CH),]is indicative of the change in spin concentration with temperature. Within our experimental error, there is no change in spin concentration at different temperatures.
LL
0.2
-
1
0.0 0
[-.-e-
I
1
I
I
I
200 TEMPERATURE ( K )
IO0
1
I
300
The spin-lattice relaxation rates, R, in M H z , for (12CD),(A), ( W H ) , (E),( ' T D ) , (a),and (13CH),(m) are plotted as functions of
Figure 5.
temperature.
0.4
0.2
0.0
c
Ii
1 L
-
0
100
-
P
l
200 300 TEMPERATURE ( K )
400
Figure 4. The spin-lattice relaxation rates, R1in M H z , of trans-(12CD), (A) and truns-(l3CD),(a) (obtained by SR-EPR)plotted as functions
of temperature. The dashed lines are plots of R, calculated from eq 5a (see text). For each line 0 = 80 K, w = 9.5 GHz, a = 0.22 M H z . The dashed line approximating the data of (12CD),was obtained by setting 0 = 0, and the line approximating the data of (I3CD), was obtained by setting 0 = 2.5.
Figure 4 shows the spin-lattice relaxation rates for (12CD), and (I3CD), as a function of temperature. It can be seen that the spin-lattice relaxation rates of these two samples have the same functional dependence on the temperature. At any given temperature, the value of the relaxation rate ( R , ) for (I3CD), is consistently 3.1 times larger than the R1 for (12CD), at the same temperature. The dashed lines are the best fit curves employing the model of Robinson et We have measured the spin-lattice relaxation rates of ( 12CH), and (I3CH), from 77 to 350 K and these are presented in Figure 5, along with the data from the deuterated samples, shown in Figure 4.
Discussion ( 1 2 C D ) and , (13CD),. A model which gave good agreement with ESE spin-lattice relaxation time measurements on t-PA was suggested by Robinson et al.I5 In that model the number of EPR active spins is assumed to be independent of temperature. The model begins with the spin-spin dipolar HamiltonianI8 which is considered in two parts: The first part takes into account the interaction between two defects (S spins):
7f = SI*A,.S2
(la) A second Hamiltonian accounts for the interaction of a single defect with a set of nuclei (I,)
[-.-e-]
N is a valid measure of the relative number of paramagnetic spins at all temperatures the samples were never removed and reinserted into the spectrometer sample cavity in the course of an experiment. Figure 3 is a plot of the relative number of spins as a function of the inverse absolute temperature for (I2CD), and (I2CH),. For both samples the number of spins is normalized to unity at 300 K. These data show that for both systems the number of spins is a constant, consistent with a simple Curie-Weiss behavior of t-PA.l,I6 These measurements were performed to test whether the spin-lattice relaxation rates were proportional to the number of defect spinsI5 and were changing simply because the EPRvisible spin concentration was changing. The temperature dependence of the spin-lattice relaxation rates reported below cannot be attributed to changes in the concentration of defect spins.
where A,, is the tensor describing the defect spin-defect spin interaction and knis the tensor describing the defect spin-nuclear spin interaction.l* This static Hamiltonian is then modulated by a diffusion process which stochastically changes the distance among the spins and leads to relaxation. Abragam18 has shown how the above Hamiltonian leads to a generalized spin-lattice relaxation rate, R1,for the defect. A specific form for Rl is obtained by considering the dynamics to be a one-dimensional modulation of the interaction due to the motion of the S spins only, moving with an effective diffusion coefficient D. The defect spin-defect spin interaction leads to the equation
where (15) Robinson, B. H.; Schurr, J . M.; Kwiram, A. L.; Thomann, H.; Kim, H . ; Morrobel-Sosa, A.; Bryson, P.; Dalton, L. R. J . Phys. Chem. 1985, 89, 4994; Mol. Cryst. Liq. Crysf. 1985, 1 1 7, 421. (16) Schwoerer, M.; Lauterbch, U.; Muller, W.; Wegner, G. Chem. Phys. L e u 1980, 69, 359-361. (17) Provencher, S. W. J . Chem. Phys. 1976, 64, 2 7 7 2 .
(1 8) Abragam, A. Principles of Nuclear Magnetism; Clarendon Press: Oxford, U.K., 1961; Chapter 8.
The Journal of Physical Chemistry, Vol. 94, No. 1 I , I990 4719
Spin-Lattice Relaxation of trans-Polyacetylene and the defect spin-nuclear spin interaction leads to an analogous relaxation rate
where asn
= 0.46(h~eyn)~p,bl-~
(24
ye and yn are the defect and nuclear gyromagnetic ratios, h is Planck's constant, w is the spectrometer frequency, ps and pi are
the number of S and I spins per unit length along the chain, and bs and biare the distances of closest approach for the defectdefect interactions and the defect-nuclear interactions, respectively. Because these Hamiltonians do not cross correlate, the relaxation rate, R 1 ,is the sum of the two terms defined by eqs 2a and 2c: as
Rl=-+(Dw)
asn
(3)
(Dw)
The model of Robinson et al.I5 further assumes that the diffusion properties of the defect are similar to those of an electron near the Fermi surface in a simple metal, in which the electron is scattered by acoustic phonons. This assumption, using the Gruneisen-Bardeen formula for the diffusion of such specie^,^^^^^ leads to the following expression for the diffusion coefficient DI9 D-1
= qz-/e)c(e/n
(4)
which depends on 8,a constant defined by eq 22 in ref 15, and on the Debye temperature, 8. G(8/7') is the Gruneisen functionrg which approaches unity at infinite temperature. Substitution of (4) into (3) gives the relationship of the spin-lattice relaxation rate R I to the temperature, T
L
where a and
J
are defined as follows
w0 is a
reference frequency which gives a the units of megahertz. This functional form is superimposed on the experimental data in Figure 4. The quantity a is a measure of the defect-defect interaction and P is a measure of the relative contribution to relaxation of the defect-nuclear interaction. For (I2CD), a is 0.225 MHz and we neglect the deuteron contribution to relaxation; i.e., /?is zero. This assumption will be justified later. The (I3CD), are fitted with the same a and a value of P of 2.5. In all cases, the value of 8 is taken to be 80 K.lS The agreement of eq 5a with the deuterated polyacetylene data shown in Figure 4 is excellent. The equation predicts that the spin-lattice relaxation rate has a square root temperature dependence when the temperature is above the Debye temperature, where G(8/7') is nearly unity. When the temperature is well below the Debye temperature, the equation predicts that the spin-lattice relaxation rate will have a PI2temperature dependence, in reasonable agreement with the results of Clark et aI.*l who found that the spin-lattice relaxation rate was approximately proportional to TZ in the 35-500 mK temperature range. Theories for relaxation rates of one-dimensional conduction in the low-frequency (or Redfield) limit show that the rate of relaxation should be proportional to N / ( D u ) ' / ~ where , N is the number of spins, D is the Einstein diffusion coefficient of the motion, and w is the spectrometer frequency during the experi(1 9) Haug, A. Theoretical Solid State Physics; Pergamon Press: Elmsford, N Y , 1980; Volume 2, p 95 ff. (20) Bardeen, J. Phys. Reu. 1940, 50, 1098. (21) Clark, W. G.; Glover, K.; Mozurkewich, G.; Murayama, C. T.; Sanny, J.; Etemad, S.: Maxfield. M. J . Phys. 1983, 44, C33-239.
ment.18*22The inverse square root frequency dependence has been amply confirmed by experimental measurements on t-PA.'5*22*23 At any given frequency it is found that the relaxation rate increases as the temperature increases; hence to explain the observed spin-lattice relaxation rate dependence, the diffusion coefficient must decrease with increasing temperature. Maki,7-8using the SSH structure for the solitonic defect, developed a mean free path model for the dynamics of the defect. Maki's model, assuming three-dimensional phonons, and onedimensional soliton dynamics, predicted that the diffusion coefficient would decrease as the temperature increased, which is qualitatively correct. In the high-temperature limit Maki's model predicts a T1l4dependence of R, on temperature. While this model has qualitative validity it is quantitatively incorrect. Nechtschein22 and Holzcer et al.24attempt to explain the rate dependence by assuming that both N a n d D increased with increasing temperature in such a way that N / ( D ) l i 2 gave the experimentally observed temperature dependence. They postulated two types of spins, mobile and fixed, with only the mobile spins contributing to the relaxation. The diffusion coefficient was assumed to increase with temperature in an Arrhenius manner, but the number of mobile solitons was assumed to increase even faster, the net result being a square root temperature dependence of the relaxation rate. It is difficult to see how this can be consistent with the results in Figure 3 which show that the spin concentration of the EPR active species remains constant with temperature. Furthermore, the spin-lattice relaxation rates were measured at various points across the EPR line and no differences in R , were found, implying only one species was present. All recoveries fit a single exponential, also implying a single species system. As seen in Figure 3 the relative number of spins does not change, within experimental error, over the temperature range 100 to 300 K. From this result we conclude that for these samples of t-PA, the dynamics model of Nechtschein and others22-24is inapplicable because that model requires that the number of EPR visible spins change as a function of temperature. The data presented here put the fast, mobile soliton p i ~ t ~ r e into ~ , some ~,~~,~~ question for t-PA. The model presented here also provides a reasonable explanation based on conventional solid-state physics. Comparison of R Ifor (12CD),and (12CH)x.As is shown in Figure 5, substitution of protons for deuterons in (I2CD), t-PA caused a slight increase in the relaxation rate of the defect. The magnitude of the increase is 1.7 times at 100 K, 1.6 at 150 K, and 1.3 at 300 K, Le., about 1.5 with a slight dependence on temperature. If all of the spin-defect relaxation were due to the interaction with deuterons or protons then this ratio would be 16 (as contrasted to the measured value of 1.5); this is estimated from the ratios of the gyromagnetic ratios for protons and deuterons.I5J8 The small change in the rates upon going from (I2CD), to (I2CH), suggests that, for these two isotopic forms of [-PA, the major contribution to R , comes from defect spin-defect spin interactions. The interaction with both the protons and deuterons is weak. We can approximate the observed relaxation rate of (I2CH), by the addition of the spin defect-proton relaxation rate to the electron spin-spin relaxation rate. Figure 6 shows that the (I2CH), rates can be fitted with a staying at 0.225 and P set to 0.7 (indicating a defect spin-proton interaction weaker than the defect-defect interaction) in eq 5. This value of 0 may be compared to = 2.5 for the case of (I3CD),. Since the densities of carbon and hydrogen are identical, the ratio of the p's are equal to the ratio of the distances to the 4th power times the ratio of the gyromagnetic ratios to the 2nd power. It follows that the protons are effectively 3 times further from the defect than the carbons. This would suggest that the on-chain carbon and hydrogen atoms are the ones most responsible for relaxation of the defect spin. As can be seen, the behavior of (I2CH), is reasonably well explained (22) Nechtschein, M.; Devreux, F.; Genoud, F.; Guglielmi, M.; Holczer, K. J . Phys. 1983, 44, C3-209. (23) Mizogouchi, K.; Kume, K.; Shirakawa, H. Solid State Commun. 1984, 50, 213. (24) Holczer, K.; Boucher, J. P.; Devreux, F.; Nechtschein, N . Phys. Reu. 1981, 823, 1051.
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The Journal of Physical Chemistry, Vol. 94, No. 1 1 , 1990 0.6
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I
I
I
I
I
I
0.4
N
I
2. w I-
a a 0.2
0.c 0
100 200 TEMPERATURE (K)
300
Figure 6. The spin-lattice relaxation rates, R , in MHz, simulating the rates for (I2CD), (A), (I2CH), (W), ( W D ) , (O), and (13CH), (0) calculated from eq 9, are plotted as functions of temperature. In all four cases a = 0.22 MHz, and 8 = 80 K. The spectrometer frequency, W , is 9.5 GHz, and the reference frequency, w,,, is 1 GHz. The two deuterated forms are calculated for A = 0 (in such cases eq 14 reduces to eq 5 ) where @ = 0 for (I2CD), and 2.5 for ("CD),. (See Figure 4.) For (I2CH), and (I3CH), &A = 8 and A = 400 K. For (I2CH), 6 = 0.5 and for ("CH), 6 = 3.5.
within the framework of the model for relaxation developed above. Comparison of R Ifor (I3CD), and (13CH),. We now examine the effect on the spin-lattice relaxation rate, R l , of substituting protons for deuterons in ?-PA while using the I3C form of carbon (see Figure 5). Deuterons are not expected to be important in the relaxation of ("CD), because of the small change in R I between (CD), and (CH),. One would anticipate that R1 of (I3CH), would be the simple sum of the relaxation rate of (I3CD), and the proton contribution to the relaxation of ('*CH),. The data are clearly inconsistent with this assumption. The relaxation rates in the region below 150 K are qualitatively correct, but not quantitatively. For example, inspection of Figure 5 shows that at 100 K the relaxation contributed by I3C to ("CD), is 2.5 times that contributed by the defect spin-defect spin interaction, comparing (l2CD), with (13CD),. The contribution from protons is 0.7 times that for defect spin-defect spin, comparing (12CD), with (I2CH),. From eq 5 the total relaxation of (I3CH), should be 4.3 times that of (I2CD),, but the experimentally determined ratio is 6.0. Thus addition of the rates does not account for the relaxation rate of (I3CH), t-PA below 150 K. Above 150 K the observed ("CH), rates decrease with temperature and even fall below those of ( I3CD),, in complete disagreement with simple relaxation rate addition. The relaxation rates of (13CH), clearly demonstrate that the presence of both spin 1/2 species, "C and H, relaxes the electron in a way that cannot be the simple sum of the effects of each and these spin 1 /2 nuclei separately. This implies that not only is there a defect-carbon coupling, SA,&, and a defect-proton coupling, S.AsH.p,but there must be a proton-carbon coupling, F.AcH.P. [The superscript C or H labels the I spin as belonging to a carbon or proton, respectively. The coupling F.ACH# is a formal term since there is no single carbon-proton coupling but many different types of coupling, e.g., on chain vs interchain couplings.] It is very difficult to see how this additional coupling would yield a contribution to the relaxation of ( S z )within the framework of a Redfield form for the relaxation operator.I8 At second order in a Redfield type time-dependent perturbation expansion, which is the Redfield form described in Abragam,I8 there is no direct contribution to ( S , ) from F.AcH.fH. The effect of such a term is seen only via cross relaxation rates. However, at third order in the same expansion one can obtain terms that lead to direct (S,) relaxation. Such terms are interesting because they show
St. Denis et al.
GI!
that an electron flipflop term is necessary to make the pairs and GI: contribute to (Sz) relaxation. As a result the fluctuation of the electron spin will couple the proton-carbon relaxation directly to the electron relaxation. The mechanism of protoncarbon relaxation need not be the same as that of the electron relaxation. The relaxation of (13CH), cannot be simply explained by adjusting rates in the Redfield relaxation matrix; an additional diffusion mechanism must be considered. A possible mechanism is the system of an electron spin delocalized over N carbons, to which are attached 8 protons, where N is around 50. These are the on chain or local nuclei strongly coupled to the electron spin. The nuclei, labeled i and j , are coupled to each other by couplings of the form I(i)vli;I(j). Clearly the coupling of one electron to approximately 100 nuclei which are coupled to each other is a very challenging problem. However, we can obtain insight into how so many strongly coupled nuclei might relax the electron by using some of the arguments used to construct the theory of spin diff~sion.~ Consider that the I+(i).I&) operators will transfer magnetization from one spin to another, thus leading to a system which might be characterized by a magnetization which can move from one nuclear site to another by an effective diffusion mechanism. What processes might explain the (I3CH), relaxation rates? Because it is known that spin diffusion is a common process in solids, which is particularly efficient for protons, it is tempting to consider a mechanism similar to spin diffusion. Spin diffusion traditionally has been used to explain the relaxation of nuclei that are distant from a d e f e ~ t .Scott ~ and Clarke25and Clark et aL2' analyzed the relaxation of protons in t-PA which are distant from the defect and found that spin diffusion to a paramagnetic center is indeed a valid model to understand proton NMR relaxation rates. In their experiments the NMR relaxation rates of protons were measured in two samples: pure (I2CH), and in (12CDo,92H0.08)~. In the second sample the presence of a large excess of deuterons removes the spin diffusion process but does not disturb the direct electron-proton relaxation. Substitution of deuterons did lead to an observed reduction in proton relaxation rate, from which it was concluded that some of the total relaxation to the protons in (12CH), was due to spin diffusion to a paramagnetic center. The rates of diffusion in classical spin diffusion are typically around 10-8-10-12cm2/s. In our cases the rates of spin migration along the chain, either from carbon to carbon, proton to proton, or carbon to proton, must be on the order of 10-4-10-s cm2/s.15*21,22 We suggest that spin diffusion within the field of the electron, enhanced by the presence of the electron, could provide efficient energy transfer between carbons and protons. This is consistent with the possibility that proton-carbon coupling can directly relax (Sz)at third order, via electron flipflop terms. The problem then is that of an unpaired electron strongly coupled to several hundred nuclei, which are strongly coupled to each other. Dynamic fluctuations of the electron and nuclear coordinates are responsible for the spin-lattice relaxations of the electrons. Direct solution of such a problem is numerically daunting. Therefore we make the ansatz that all nuclei may be replaced by an effective nuclear spin which may diffuse in the vicinity of the electron. From this approach we will gain some insight into the effective rates of spin migration and the dependence of this process on temperature and the number of nuclei. We consider a model in which the electron may diffuse in one dimension, along the z axis, up to distance L with a diffusion coefficient D. The effective nuclear particle may also diffusion along z up to distance L with a diffusion coefficient D, and also along x and y (which is perpendicular to the motion along the chain), up to distance R , with a diffusion coefficient D,. The Hamiltonian is the simple dipolar Hamiltonian S.A.I, where t r A = 0. Additional scalar terms, which do not markedly effect the outcome of the discussion, are neglected for the sake of simplicity. From this model, the rate of spin-lattice relaxation (as given by A b r a g a d ) may be directly determined: (25) Scott, J . C.; Clarke, T. C. J . Phys. 1983, 44. C3-365. (26) Op. cit. 18, p 295, eq 87.
The Journal of Physical Chemistry, Vol. 94, No. 11. 1990 4721
Spin-Lattice Relaxation of trans-Polyacetylene Rlen = (YeYnh)'f(f + I)."
- wI)
This six-dimensionalintegral may be reduced to a four-dimensional integral by defining A = Z - 2, and A,, = 2, - Z,o and integrating over Z I and Zlo:
3
+ ; J ' ( w ~ )+
P ( w ) is the spectral density function, or Fourier transform of the correlation function, and Jv is the number of effective nuclei. We can assume that ws >> wl, and average over all possible chain orientations. The result is
R/2
71
d2,,0(6) d2,,o(60)
~ - ( A - A O ) ~ / ~ E1I
.)
[4~Et]1/2
Rlen= ( ~ , r , Jv , hreal ) ~ (Se-iwstG(t) ~ dt)
(7a)
where G(t) is the averaged correlation function for the electronnuclear dipolar interaction 2+
'(')
= m--2 I3
(
1
)
D i , o ( 4 4 ( d ) D:,0(~,4(0)) rJ(t)
r3(O)
(7b)
where D i , o ( 6 , 4 )are the Wigner rotational matrix elements and r is the electron-nuclear distance. The statistical averaging is performed by using the conditional probabilities (W)and averaging over all dynamic coordinates
+
where r2 = A2 p2, ro2 = A,,2 + po2,cos 6 = A/r, and cos 6, = & / r 0 , and E = D D, is the sum of the diffusion coefficients along 2. While the general solution to the remaining four dimensional integral remains a numerical problem, one may obtain some insight into the rate, RIe,, by taking the Fourier transform at this time (substitution of eq 10 into eq 7)
+
X m=-2
ker ({nEl'/2)b[ppo]1/2dA d& dp dpo ( 1 1) where 52 = w b 2 / D , Di,o(~odJo)
P,P, dX/dY, dZI d Z dXIo dYl0 d2, (sa)
rO3
where r2 = X t
D< = min I D , E )
+ YF + (2, - Z)2
This particular problem is more easily treated by using cylindrical coordinates: z, p, 4, in which p2 = X t Y? and 9 = p2 + (2, - 2)2.The conditional probabilities are
+
and "ker" is the ker f ~ n c t i o which n ~ ~ ~is ~the ~ real part of the modified Bessel function of the second kind for which asymptotic analytical formulas and numerical approximations are available.37-38The functional dependence of Rlcnis rather forbidding but there are some limiting cases that provide insight. If either E >> D, or D, >> E then the rate goes as
and
and the equilibrium distributions are 1
1
= L.2n(R2 - b2)/4
P
and P% = L
where b is the closest distance of either nucleus. Since D:,o(6,$) = d;,,(6)ein+as defined by Wigner,36 the integration over 4 and 4omay be done explicitly, reducing the problem to a six-dimensional integral:
R/2
d2,.o(e) d2,.0@0) -X rJ
8 ( R 2 - b2)
r03
[PPol'/2 dP dP, d Z dZ0 dZ, dZ,
where p, is the density of nuclei, p, = Jv/2?rLR2, and D, = max ( E , D ). This is the one-dimensional diffusion result;12 the more rapi8 diffusion coefficient controls the relaxation rate, while the slower diffusion rocess makes no contribution. Notice that the rate El, l/ol as is characteristic of one-dimensional diffusion and goes as either 1/E1I2or 1/DP1l2.Therefore, if difffusion away from the defect to neighboring chains were to dominate, then the relaxation would still appear to be one-dimensional. If along chain diffusion is most efficient Rlen0: 1 / [ D + D , ] 1 / 2 . When E and D, are of similar magnitude
(9)
wherep then is the effective dimensionality of the diffusion process which can be from one-dimensional to three-dimensional, and 1 Ip I2. Therefore, in the region from 200 to 250 K, the rate RI may not obey a simple integer p dependence, and the frequency dependence could be temperature dependent. This would certainly be one avenue for testing whether interchain hopping significantly contributed to the defect dynamics. Another limiting form of RIe,, that is somewhat accessible, is when E = D,; the rates of diffusion along the chain are on the
4722
The Journal of Physical Chemistry, Vol. 94, No. 11, 1990
St. Denis et al. rhenius temperature dependence for D,
order of those perpendicular to the chain.
D, = Ae-(A/n
where Q = -wb2
E
r'
and E = ( TP )-~Po+ ( T )
- To 2
A-A0
- r2 + ro2- 2rro cos (e - 0,) b2
The result is
(15)
where A is the activation energy, a barrier to migration, and A is the temperature-independent proportionality constant. Figure 6 shows the simulation to the data of Figure 5 using eqs 14,4, and 15. The values of the constants are given in the legend. The value of A = 400 K may have significance because it represents the barrier to spin migration, and coincidentally, has the same value as that determined from ENDOR experiments on cis- and t r a n ~ - ( l ~ C H ) > 'and - ~ ~from an analysis of spin-spin relaxation rates for both (12CH), and (12CD),.30 We have also assessed a model in which D, is taken to be proportional to (based on a theory of Wada and Ishiuchi31 in which chain fluctuations at domain walls are driven by a process with that temperature dependence). The results of this model fit the data just as well as the model of eq 15 (results not shown).
where as,is given in eq 2b for a single-chain model or in eq 12b for a many parallel chains model, and E = D + 0,. The equation may be rearranged into the form of eq 5
Conclusions The spin-lattice relaxation rates of the paramagnetic electron spin defect in ?-PA have been measured by saturation recovery as a function of temperature for the various isotopically substituted forms of t-PA. The results show that for the deuterated forms (Le., (12CD), and (13CD),) the temperature dependence is essentially T112(over the temperature range from 80 to 300 K). This is consistent with the model previously developed by Robinson et a l l 5 The relaxation rates of (I2CH), and (I3CH), do not fit the simple model as well. It does not seem possible to reconcile the discrepancies in terms of a simple phenomenological relaxation matrix after the Redfield form and still use the relaxation rates from the (12CD), and (I3CD), systems. However, postulating an additional diffusion mechanism, associated with spin 1 /2 nuclei leads to a model for the spin-lattice relaxation rates in quantitative agreement with the experimentally determined rates. This study highlights the importance of the nuclear spin system which contributes another dynamic mechanism to the overall relaxation, and illustrates the difficulty in interpreting relaxation rates in terms of defect diffusion. Our conclusion is that all of the dynamic interactions seen in the spin-lattice relaxation rates are due either to spin-phonon interactions (which explains the rates in the deuterated materials) and an additional relaxation mechanism associated with the presence of spin 112 nuclei strongly coupled to each other. Neither of these processes is necessarily associated with actual migration of the mean of the defect in the solid and the diffusion rates determined in this work suggest a basis for conduction but provide no direct measure of conductance. The SR-EPR studies showed no evidence of two components to the relaxation for any of the isotopic forms of ?-PA at any temperature. The CW-EPR studies showed no change in the relative number of defects as a function of temperature. This suggests that the data may be characterized by a single defect
using the definitions of CY,/3, and 8 . If we assume that D, has the same temperature dependence as D, then this model predicts that the relaxation rate for (I3CH), and (13CD), would differ only by a constant. If D, were temperature independent then the spin-lattice relaxation rate would increase with increasing temperature and at higher temperatures become temperature independent as D, >> D there would be no local maximum in the rate as seen in the experimental data. D, is usually temperature independent for the case of spin diffusion of nuclei distant from the defect.I8 This model can explain the data only if D, increases with increasing temperature. Furthermore, the rate of D, must be on the order of, or greater than, D for strong one-dimensional nuclear coupling to be an efficient relaxing mechanism. Therefore D, IO4 cm2/s is required based on previous estimates of D.Is To illustrate that it is possible to model R , in quantitative agreement with the experimental data, we have chosen an Ar-
(27) Thomann, H.; Dalton, L. R.; Tomkiewicz, Y.; Shiren, N. S.; Clarke, T. C. Mol. Crysr. Liq. Cryst. 1982, 83, 33. (28) Thomann, H.; Cline, J. F.; Hoffman, B. M.; Morrobel-Sosa, A,; Robinson, B. H.; Dalton, L. R. J. Phys. Chem. 1985, 89, 1994. (29) Cline, J. F.; Thomann, H.; Kim, H.; Morrobel-Sosa, A.; Dalton, L. R.; Hoffman, B. M. Phys. Rev. B 1985, 31, 1605. (30) Shiren, N. S.; Tomkiewicz, Y . ;Thomann, H.: Dalton, L. R.; Clarke, T. C . J. Phys. 1983,44, C3-223. (31) Wada, T.; Ishuchi, H. J. Phys. SOC.Jpn. 1985, 51, 372. (32) Blumberg, W. E. Phys. Rev. 1960, 1 1 9, 79. (33) de Gennes, P.-G. J. Phys. Chem. Solids 1958, 7, 345. (34) Bloembergen, N. Physica 1949, 25, 386. (35) Abragam, A.; Goldman, M. Nuclear Magnetism: Order and Disorder; Clarendon Press: Oxford, U.K., 1982. (36) Wigner, E. P. Group Theory; Its Application to the Quantum Mechanics of Atomic Spectra; Academic Press: New York, 1959. (37) Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Trio", F. The Eateman Project Manuscript: Tables of Transforms, Z; McGraw-Hill: New York, 1953. (38) Oliver, F. W. J. Bessel Functions of Integer Order. In Handbook of Mathematical Functions; Abramowitz, ., Stegun, J., Eds.;Dover Press: New York, 1970: Chapter 9.
This is the ideal case of three-dimensional nuclear diffusion, where a l/w. Notice that since the defect has only one-dimensional diffusion Rlcna 1/ w and not 1/d2. The dependence of Rlcnon E is no longer a power law, but In (Rlen)is linearly related to Thus, the temperature dependence of the experimentally determined relaxation rates for (I3CH), could provide insight into the dimensionality of the diffusion. If the diffusion process in the nuclear pool remains one-dimensional, then 0,and D must have a different temperature dependences. On the other hand interchain nuclear coupling or linear on-chain carbon-hydrogen coupling could contribute a relaxation that would have an effective diffusion coefficient D,. In this case as well, D, must be on the order of or greater than D. Such a process could be experimentally observed by measuring the frequency dependence of ale,for various fixed temperatures. One possible explanation for Rlenof (I3CH), lies in the application of the one-dimensional diffusion equation, eq 12, where D is the diffusion coefficient for the defect and D, is the effective diffusion coefficient for on-chain migration of magnetization through the nuclear spin system. Including R1, the total relaxation for R I becomes
ale,
Substitution of
given in eq 12 results in Rl=-+-
-
ass
ffsn
[EW]'/~
J . Phys. Chem. 1990, 94, 4123-4121 with a temperature-independent concentration, in contrast to previous model^.^^-^^ The relaxation of ("CH), is unusual. In the case of ("CD), the I3C are strongly coupled to the defect but are not strongly coupled to each other. Whereas in the case of (CH), the protons are not strongly coupled to the defect (and, presumably, are strongly coupled to each other), in (I3CH), the nuclei are not only strongly coupled to the defect but are strongly coupled to each other. In this case, and only this case, is the strong nuclear-nuclear coupling an efficient mechanism for defect spin relaxation. The nature of diffusion of magnetization through nuclei strongly coupled to an electron remains a largely unexplored subject. We have outlined how both one-dimensional and three-dimensional nuclear diffusion could contribute to R I for the electron defect. It appears that nuclear spin diffusion within the region of the defect
4723
has markedly different effects on the defect relaxation than distant nuclei. We suggest that 13Cand H nuclei are quasi-iso-spins in the presence of the electron; the energy mismatch between them is accommodated by the defect, and diffusion for matrix nuclei can be a very efficient process. Acknowledgment. We are most grateful to Dr. L. R. Dalton and Dr. Street for the polyacetylene samples used in these studies and for many helpful discussions. We are also grateful to Drs. H. Thomann, G. Drobny, and J. M. Schurr for many helpful suggestions and stimulating discussions. We acknowledge the support of the National Science Foundation DMB-87-06175, the Research Foundation, IBM, Exxon Educational Foundation, and the National Science and Engineering Research Council of Canada.
Photochemistry In Polymers. Photoinduced Electron Transfer between Phenosafranine and Triethylamine in Perfluorosulfonate Membrane K. R. Gopidas and Prashant V. Kamat* Notre Dame Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 (Received: July 25, 1989; In Final Form: January 15, 1990)
Absorption and emission properties of protonated and unprotonated forms of phenosafranine dye are investigated in the perfluorosulfonate ion-exchange membrane (Nafion) and polymer solution. Only the unprotonated form of phenosafranine is photoactive and undergoes electron transfer with triethylamine upon excitation with visible light. The triplet excited state of phenosafranine and the electron-transferproducts in the Nafion film are characterized by laser flash photolysis with 532-nm excitation. In a dry evacuated film the triplet lifetime of phenosafranine is greater than 1.5 ms. During the steady-state photolysis, the semireduced dye undergoes disproportionation to accumulate a colorless two-electron-reduction product.
Introduction Photoactive polymers have important applications in photoresists, xerography, photocuring of paints and resins, and solar energy conversion systems.' These polymeric systems can be broadly classified into two categories: (1) in which chromophores are directly attached to the backbone of the polymer and ( 2 ) in which the polymer film acts as a host to the photosensitizing molecules. Photoactive guest molecules can be incorporated in a polymer film via electrostatic or hydrophobic interactions. The type and degree of interaction between the dye molecule and the polymer determines the course of a photochemical reaction. Various aspects of photochemical and photophysical processes in polymers have been discussed in detail elsewhere.I4 Nafion (manufactured by Du Pont Corp.) is a perfluorosulfonate ionic polymer with a wide variety of applications in electrochemistry and photochemistry. For example, an electrode surface modified with Nation film containing electroactive species exhibits interesting electrocatalytic proper tie^.^,^ Nafion membrane has been used in integrated chemical systems for the purpose of solar enegy c o n ~ e r s i o n . The ~ ~ ~solid polymer matrix has been found useful in the preparation of quantized semiconductor particles.*I2 The strong acidic microenvironment of the protonated form of Nafion facilitates organic rearrangement^'^*'^ and isomerization^.'^ The fluorocarbon network of the swollen Nafion film consists of solvated -SO3- head groups and counterionsolvent clusters (-40 A in diameter) that are interconnected by short channels (- 10 A).ls This biphasic structure has been compared with the structure of reverse micelles.16 Luminescent molecules such as pyreneI7 and Ru(bpy),Z+'6,18q19have been used to probe the excited-state interactions with the polymer and to characterize To whom correspondence should be addressed.
various sites in such clusters. The effect of acidic environment on the excited triplet of xanthone and benzophenone20 and the kinetic properties of singlet oxygen generation in Nafion film2' (1) Guillet, J. Polymer Photophysics and Photochemistry; Cambridge University Press: New York, 1985. (2) Farid, S.;Martic, P. A.; Daly, R. C.; Thompson, D. R.; Specht, D. P.; Hartman. S. E.: Williams. J. L. R. Pure Aool. Chem. 1979. 51. 241. (3) Kalyansundaram, K.Photochemistry'h MicroheterogeneousSystems; Academic Press: New York, 1987; p 255. (4) Kamat, P. V.; Fox, M. A. In Applicationr of Lasers in Polymer Science and Technology; CRC Press, in press. (5) (a) Rubinstein, I.; Bard, A. J. J . Am. Chem. Soc. 1981, 103, 5007. (b) Henning, T.P.; White, H. S.;Bard, A. J. J. Am. Chem. Soc. 1981, 103, 3937. (c) Martin, C. R.; Rubinstein, I.; Bard, A. J. J . Am. Chem. SOC.1982, 104, 4817. ( 6 ) (a) Buttry, D. A.; Anson,
F. C. J . Am. Chem. Soc. 1982, 104,4824. (b) Tsou, Y.-M.; Anson, F. C. J . Phys. Chem. 1985.89, 3818. (7) Krishnan, M.; White, J. R.; Fox, M. A.; Bard, A. J. J. Am. Chem. SOC. 1983, 105, 7002. (8) Meisner, D.; Memming, R.; Kastening, B. Chem. Phys. Leu. 1983,96, 34. (9) Hilinski, E. F.; Lucas, P. A.; Wang, Y. J . Chem. Phys. 1988,87,3435. (IO) Gopidas, K. R.; Kamat, P. V . Mater. Lett., in press. (11) Bard, A. J. Eer. Bunsen-Ges. Phys. Chem. 1988, 92, 1187. (12) Olah, G. A,; Meidar, D. Nouu. J. Chim. 1979, 3, 269. (13) Demuth, M.; Mikhail, G.; George, M. V. Helu. Chim. Acta 1981,64, 2759. (14) Childs, R. F.;Mika-Gibala, A. J. Org. Chem. 1982, 47, 4204. (15) Yeo, S.C.; Eisenberg, A. J . Appl. Polym. Sei. 1977, 21, 875. (16) Lee, P. C.; Meisel, D. J . Am. Chem. Sot. 1980, 102, 5477. (17) Lee, P. C.; Meisel, D. Photochem. Photobiol. 1985, 41, 21. (18) Prieto, N. E.; Martin, C. R. J. Electrochem. SOC.1984, 131, 751. (19) Szentirmay, M. N.; Prieto, N. E.;Martin, C. R. J . Phys. Chem. 1985, 89, 3017. (20) Weir, D.; Scaiano, J. C. Tetrahedron 1987, 43, 1617. (21) Lee, P. C.; Rodgers, M. A. J. J . Phys. Chem. 1984,88, 4385.
OQ22-3654/90/2094-4123$02.50/0 0 1990 American Chemical Society
