7121
J. Phys. Chem. 1991,95, 7121-7124 difference of decay rates in the triplet sublevels. We analyzed the time developments of the transient EPR signals to determine the decay rate constants. The time evolutions were measured at the magnetic fields of the innermost X and the outermost Z canonical orientations with low microwave power (0.5 mW) at 4.2 K. Thus, it may be assumed that the nutation and the electron spin relaxation are neglected. The signals observed at the Y canonical fields with a low microwave power were too weak for the analysis. The observed time profiles were reproduced by a sum of two exponential functions taking account of apparatus response time ( 7 ) as expressed with eq 1. where PI
It has been reported that flexible monocyclic enones have twisted form in the T,(mr*) We could not observe any evidence for 3mr*-3n** mixing in tropone. Therefore, it can be considered that the short lifetime is due to a large Frank-Condon factor induced by the difference of the potential between the Tland So states. In tropone, the atomic coefficients of HOMO(r4) and LUMO(r5*) are very different each other. The values obtained from a semiempirical PPP type SCF MO calculation are summarized as follows: /7
I(r) = l l d t ' r - l exp(-(t
- t?/7)(P1exp(-klt?
y 0.000
Q)o.530
- P2 exp(-k2t?} (1)
0.417
0.492
and P2 are the relative populating rates of the Zeeman sublevels. The decay rate constants k, ( i = 1,2) of Zeeman sublevels are expressed as follows:
k, = c,&
+ C&y + C,&
(2)
where C, and k, (j= X, Y,Z)are mixing coefficients and the decay rate constants of the zero-field sublevels, respectively. The response time of 0.35ps was used for the deamvolution calculation. We obtained kx = 5 X IOs s-' and &,, N kz N 5 X 10, s-I, which leads to an average decay rate constant of k., = (kx + ky + kz)/3 Y 2 X IO5 s-l. from the analyses of the transient signals at Z canonical points. Similar results were obtained from the decay signals at the X canonical fields. The rate constants and the populating ratios of the zero-field sublevels determined are summarized in Figure 3. The results indicate that the %**state of tropone has very large radiationlessrates from the triplet sublevels. The rates are smaller than those of flexible monocyclic enones.') (13)Yamauchi, S.;Hirota, N.; Higuchi, J. J. Phys. Chem. 1988,92,2129.
Since the C2-C3and C4-C, bonds have an antibonding character in the rs*orbital, the excitation into the TI state would accompany the increase of bond lengths in these bonds and the twisting around these bonds. The difference between the molecular structures of TI and So states may be responsible for the fast radiationless transition rate. Further detailed investigations are in progress by comparing the present results with the characters of several benzotropone derivatives.
Acknowledgment. The present work was partially supported by Grants-in-Aid of Scientific Research No. 02453002 and of Scientific Research on Priority Area No. 02245204 from the Japanese Ministry of Education, Science and Culture. (14) Bonncau, R. J. Am. Chem. Soc. 1980,102, 3816.
Complementary Grating Effect in Forced Rayieigh Scattering Sangwook Park, Jungmoon Sung, Hongdoo Kim? and Taihyun Chang* Department of Chemistry, POSTECH, P.O. Box 125, Pohang, 790-600. Korea, and Division of Organic Materials. RIST, P.O. Box 135, Pohang, 790-600, Korea (Received: June 12, 1991: In Final Form: July 26, 1991)
The problem of non-single-exponential decay profiles of the forced Rayleigh scattering (FRS) is addressed by considering the contributions from complementary phase gratings. Since the technique relies on creating transient optical gratings by means of photolabels and reading the total diffraction signal intensity arising from the complementary gratings, a small difference in the decay time constants of optical fields derived from a complementary pair of such gratings could produce extremely nonexponential FRS decay profiles. By detailed simulation studies, we show how different profiles can be obtained with only minor differences in the decay constants, and how single-exponential analyses of such profiles can mislead in deducing the correct diffusion coefficients.
Introduction Since its inception as a tool for the study of mass diffusion, the forced Rayleigh scattering (FRS)technique has been applied to a variety of translational diffusion problems in complex media and self-diffusion such as anisotropic diffusion in liquid and probe diffusion in polymer solutionsc7 and in the bulk state.&I1 The technique requires a photolabel undergoing changes in its optical properties upon irradiation by the optical excitation source, Le., writing beams. If a fringe pattern is imprinted onto a sample To whom correspondence should be addrwaed. 'Polymer Materials Lab.. KIST, P.O.Box 131, Cheongryang, Seoul, 130-650.Korea.
OO22-3654/9 1/2095-7 121$02.50/0
by either crossing two writing beams or relying on a ruled mask, a periodic concentration profile of photochromically shifted or ( 1 ) Hervet, H.; Urbach, W.; Rondelez, F. J. Chem. fhys. 1978.68.2725. ( 2 ) Takezoe, H.; Ichikawa, S.; Fukuda, A,; Kuzc, E. Jpn. J . Appl. fhys.
1983,23,L78. (3)Urbach, W.;Hervet, H.; Rondelez, F. J . Chem. fhys. 1985,83, 1877. (4) Hervet, H.; LCger, L.; Rondelez. F. fhys. Rev. Let?. 1979,42,1681. (5) W w n , J. A.; Noh, 1.; Kitano, T.; Yu, H.Macromolecules 1984,17,
782.
(6)Kim, H.;Chang, T.; Yohanan, J. M.; Wang, L.; Yu, H. Macromolecules 1986,19,2737. (7) Landry, M.R.; Gu, Q.;Yu. H. Macromolecules 1988,21, 1 1 58. 29,(8) 2261. Tran-Cong, Q.; Chang, T.; Han, C. C.; Nishijima. Y. Polymer 1988, Q 1991 American Chemical Society
Letters
7122 The Journal of Physical Chemistry, Vol. 95, No. 19, 1991 photobleached state of the probe is obtained, and this in turn serves as transient optical grating to diffract the optical probe source, i.e., reading beam. For the sake of clear focus, we restrict ourselves to photochromic labels. For the reading beam diffraction to take place, the shifted state of the photolabel must have different optical absorptivity and/or refractive index from that of the unshifted state; the former gives rise to the amplitude grating and the latter to the phase grating.I2-l4 The erasure mechanism of the gratings is then twofold, thermal reconversion to the unshifted state and translational diffusion of the photolabels in both states. Thus the decay profiles of the diffracted optical signal at different fringes are used to deduce the diffusion coefficient and the shifted-state lifetime of the photolabel. In most of the cases, diffracted light intensity decays exponentially, and the decay profile has been analyzed by a model function 1D(t) = [ A exp(-?/r)
+ Bwh12 + Bine
(1)
where T is the decay time constant of the concentration grating, Bwhand Bi, are the coherent and incoherent backgrounds, and A is the amplitude of the diffracted optical field. However, deviations from single-exponential decay of diffracted signal were observed and the origin of such deviations was frequently interpreted as the presence of multiple grating^.".'^*'^ Recently, the presence of the complementary gratings of optically shifted and unshifted state was recognized and the difference in diffusion coefficients of two states is believed to cause the nonexponential decay.I7-l9 Though this is still to be proven directly, it is an entirely plausible explanation. If we start with the double optical grating contribution to the FRS signal, the formation of shifted-state profile, giving rise to the first optical grating, is accompanied by its complementary one of unshifted state. Hence, the time-varying diffraction signal due to the grating erasure kinetics shows the following form Mt)
= [AI e x ~ ( - t / ~ i-) A2 exp(-t/72)
+ &h12 + Binc
(2)
where and T~ are decay time constants of two complementary gratings and A I and A2 are the amplitude of optical fields diffracted from two complementary gratings. The negative sign comes from the fact that two gratings are 180° out of phase; however, this is only true for either the pure phase or the amplitude grating. In principle and in practice we could easily have mixing of the both kinds of grating since photochromically shifted and unshifted states of a photolabel can differ in the optical absorptivity as well as in the refractive index, which leads to a phase shift of the diffracted optical field.Is We pursue the issue by considering only the case of phase grating. Azobenzene derivatives, which have been widely used for FRS study, with negligible absorptivity at 632.8 nm (the wavelength of the commonly used He/Ne laser as the reading beam) would be the appropriate example of the nearly pure phase grating. In fact, a number of cases complying with this situation have been reported where the intensity at the dip position of a decay-growth-decay type signal completely reaches the baseline indicating a complete destructive interference within experimental precision.'*-20 ~~~
~
~
~
(9) Nemoto, N.: Landry, M. R.; Noh, I.; Yu, H. Polym. Commun. 1984, 25, 141. (10) Antonietti, M.; Coutandin, J.; Grutter, R.; Sillucu, H. Macromolecules 1981, 17, 798. ( 1 1 ) Wang, C . H.; Xia, J. L. Macromolecules 1988. 21, 3519. (12) Kogelnik, H. BellSysr. Tech. J . 1969, 48, 2909. (13) von Jena, A.; h i n g , H. E. Opr. Quanrum Electron. 1979,11,419. (14) N e h , K. A.; Caaalegno, R.; Miller, R. J. D.; Faya. M. D. J . Chem. Phys. 1982,77, 1144. (1 5) Rhee, K. W.; Gabriel, D. A.; Johnson, Jr., C. S.J. Phys. Chem. 1984, 88, 4010. (16) Zhang, J.; Wang, C. H. J . Phys. Chem. 1986, 90,296. (17) Johnnon, Jr., C. S.J . Opr. Soc. Am. B 1985, 2, 317.
(18) Kim, H. D. Ph.D. Thais, University of Wisconsin, Madison. 1987. (19) Huang, W. J.; Frick, T. S.;Landry, M. R.; Lee, J. A.; Lodge, T. P.; Tirrel, M. AICHE J . 1987, 33, 573. (20) Lodge, T. P.; Lee, J. A.; Frick, T. S.J. Polym. Scf.Polym. Phys. Ed. 1990, 28, 2607.
I
t/
; i
Figure 1. Simulated FRS profiles according to eq 2. The parameters were chosen as A , / A z = 9.4/8.4 and the relative difference of tl and t z is varied. Each line corresponds to ( r 1- ~ ) / =f 0 (---.-); 10% (-20% (-); -10% (---) -20% where t = ( q + r 2 ) / 2 . (e-),
.-.e);
Due to the mutually destructive interference, the observed FRS signal intensity is much weaker than the diffraction intensity from each grating. In this circumstance, a minute difference in decay time constants can lead to a dramatic change in FRS signal, which has been partially recogni~ed.'~"~ Nonetheless, we are not sure that the seriousness of this effect has been fully appreciated and propwe to establish the point that the double grating contribution to the FRS signal must be taken into account correctly in order to deduce accurate diffusion Coefficients. In this Letter, we present our simulation studies which have been carried out to make the case.
Results and Discussion Diffraction efficiency of a weak phase grating is known to be proportional to (An)2, where An is the peak - null difference of refractive index in the grating.I3-l5 This is equivalent to the diffraction efficiency of each grating being proportional to (dn/dc)2, the differential refractive index increment as in the scattering contrast in ordinary Rayleigh light scattering. As a first-order approximation, AI and A2 may be approximated to be proportional to (b,,, - nmdium)and (PI&- nmcdi,,,,,),respectively, while the observed optical field is the difference of the two, i.e, (b - h).Here the medium represents the common background for both complementary optical gratings. The refractive index of azobenzene is available only for trans form which is 1.6266 at sodium D line2' while the refractive indexes of stilbene are reported for both isomers, 1.6264 for trans and 1.6130 for the cis form, at the same wavelength.21From the isoelectronicstructure of stilbene and ambenzene, ye may use these values to get a rough idea of the relative diffraction amplitudes of azobenzene derivatives. For instance, if we are to perform an FRS experiment in a medium whose refractive index is 1.5, the ratio of AI:A2:(AI - A2)is 9.4:8.41 which would be used to predict that the observed diffraction intensity should barely exceed 1% of the diffracted intensity from each complementary grating; (9.4)2:(8.4)2:1would be the intensity ratios of the complementary gratings to the observed. In this circumstance, a minute difference in decay time constant can result in a significant change in signal shape as displayed in Figure 1, where FRS profiles simulated according to eq 2 are shown. The abscissa is normalized by the mean decay 2 , the central dash-dot line represents constants, 7 = (T' ~ ~ ) /and . lines correspond the single-exponential case, Le., rI = T ~ Other to 1 1 0 and 120% differences in decay time constants in terms of (rl - T 2 ) / 7 . As the difference increases, a bump (decay) an growth-decay) starts to develop in one hand ( T , < T ~ while after pulse rise becomes evident on the other hand (71 > Q).
+
(21) Weest, R. C. Ed. Handbook of Chemistry and Physics, 70th ed.;CRS Press: Boca Raton, FL, 1989.
Letters
The Journal of Physical Chemistry, Vol. 95, No. 19, 1991 7123
n
1
cd
Y
h
Y
v CI
C
Figure 2. Results from nonlinear regression analyses according to eq 1 of simulated decay profiles such as shown in Figure 1. The same parameters are used for A I / A 2as in Figure l and the ratios of determined decay constants to t are plotted against ( T ~ ~ ) / The t . arrows indicate the boundary beyond which the deviation from the model function becomes evident judging from residual plots.
Furthermore, the plots in Figure 1 make it clear that T~ and r2 must be extremely close to each other in order to observe the single-exponential decay. We carried out nonlinear regression analyses of the simulated FRS profiles according to eq 1, which is a common practice to analyze a singleexponential looking decay. The errors in the determination of decay time constant from such analyses relative to the mean decay constant, 7, are displayed in Figure 2. For this analysis, we deliberately added random noises to the simulated decay, which amounts to 2% of initial signal intensity at time 0. The arrows indicate the boundaries beyond which the deviation from the model function becomes evident judging from the residual plot. Figure 2 clearly illustrates the problem associated with the FRS technique that diffusion coefficients determined from a perfectly normal looking single-exponential may have an uncertainty more than 50% easily. If the signal to noise ratio is not much better, one would end up with an erroneous result without even recognizing a deviation from the single exponential. Paradoxically speaking, the condition beyond the boundaries may be better since at least one can notice the deviation. Some years ago, Lever et a1.22 reported that the diffusion coefficient of cis-azobenzeneis lower than that of the trans form by 5%. If azobenzene has a similar contrast to stilbene, one would obtain a single-exponential-looking decay profile and a usual analysis according to eq 1 would result in an apparent diffusion coefficient 30% smaller than the true mean value. We note a few experimental reports where the diffusion coefficients obtained by FRS were compared with those measured by other means. One system regarded the diffusion of polystyrene (PS) labeled with an azobenzene moiety in o-fluorotoluene (0-FT) solution of poly(viny1 methyl ether) (PVME).23 Since eFT and PVME are isorefractive, dynamic light scattering (DLS) is able to monitor the dynamics of PS only as FRS does for the labeled PS. Also, the concentration change of PVME does not alter the background refractive index, and thus AI and A2 remain constant. The diffusion coefficients measured by two techniques are similar, but D m is consistently smaller than hlsby about 15% at low PVME concentration. Another system is the polystyrene diffusion in toluene where we can find that DFRsis again consistently smaller than D D by~ about 109b.6 Although this much error does not affect the validity of the studies, it is likely that the consistent deviation indicates the problem associated with the FRS technique and such a magnitude of deviation can be brought about by only (22) Lever, L.S.;Bradley, M.S.;Johnson, Jr., C. S.J. Magn. Reson. 1986,
68. ~.335.
1.5
I
4.5
3.0
f
I
Figure 3. Simulated FRS profiles according to eq 2. The parameters / -5% t and the relative difference of A , and were chosen as ( T , - ~ ) = A, is varied. Each line corresponds to ( A , - A2)LA (-----); -5% (-. .-..); -20% (-); 5% (----I; 20% (-), where A = ( A , + A2)/2. 5.0
,
4.0
-
3.0
-
It\ t-
-
2.0
-
5
Figure 4. Results from nonlinear regression analyses according to eq 1 of simulated decay profiles such as shown in Figure 3. The same parameters are used for ( T ~- T 2 ) / 7 as in Figure 3 and the ratios of determined decay constants to T are plotted against (Al - A 2 ) / A . The arrows indicate the boundary between which a deviation from the model
function becomes evident judging from residual plots. 1-2s difference in decay rates of the complementary gratings as can be seen in Figure 2. The last system we note is the labeled bovine serum albumin (BSA) in aqueous buffers.*' It was reported that azobenzene-labeled BSA yielded the same diffusion coefficient as measured by DLS at high ionic strength; however, the FRS signal shape changed at low ionic strength substantially to develop double-exponential behavior. The signal shape also sensitively depends on pH of the solution. In 8 mM phosphate buffer, it was found that the apparent decay constant decreases as pH increases eventually developing a bump.'* All of these observations appear to be associated with the complementary grating effect and can be explained by our simulation results. The contrast of azobenzene derivatives should be different from that of stilbene which also influences the signal shape. The contrast effect of the complementary gratings was examined in a similar way. In Figure 3 are shown the simulated FRS profiles, where the difference of two decay time constants is fixed at 5% and the relative amplitudes are varied. As in Figure 1, the signal shape changes substantially with a small variation of relative amplitude. In order to examine the contrast effect systematically, nonlinear regression fit to eq 1 was carried out for simulated data. ~
~~
~
~~~
(23) Chang, T.;Hen, C. C.; Whecler, L. M.;Lodge, T.P.Macromolecules
1988, 21, 1870.
(24) Arunyawongsakorn, U.; Johnson, Jr., C. S.;Gabriel, D. A. Anal. Biochem. 1985, 146, 265.
7124
J. Phys. Chem. 1991,95,7124-7127
The results are shown in Figure 4, where the ratio of determined decay constant to t is plotted against relative difference of AI and A2. The arrows have the same meaning as before, but, in this case, a deviation from single exponential is evident between the arrows. It is clear that a high contrast is necessary for a reliable determination of the diffusion coefficient. With the 5% difference in decay time the contrast of complementary gratings must exceed 40% to claim an accuracy better than 10%. Taking ntrlns= 1.6266 and nmcdium= 1.5, this means that ncishas to be as small as 1.584. Although it has to be confirmed experimentally, this large difference of refractive index does not seem very likely. Even more deviation from single exponential will come about as the refractive index of the medium gets further from that of dye. Considering the clean single-exponential decays observed with ~ - suppose 7J~~~ azobenzene derivatives in a number of s o l v e n t ~ , ~ we that the difference in diffusion coefficients of the two isomers may be smaller than reported.22 Also we note a report that no measurable difference in diffusivities of the two isomers of stilbene was found by a similar NMR experiment.20 The problem illustrated so far may not be a serious drawback depending on the system. For an example, FRS has been most widely employed for the study of slow diffusion where a change of diffusion coefficients over several orders of magnitude was often In this case, a few tens of % error may be quite acceptable. It would also not be crucial for the systems in which a relative change of diffusion coefficients is of interest while the contrast does not change ~ U C ~ . ~ ’ J However, ~ - ~ J ~ if one wishes (25) Chang, T.; Kim, H.; Yu, H. Chem. Phys. Lett. 1984,111,64. (26) Lee, J. A,; Lodge, T. P. J . Phys. Chem. 1987, 91, 5546. (27) Kim, H.; Chang, T.; Yu, H. J . Phys. Chem. 1984, 88, 3946.
to measure diffusion coefficients with accuracy, one has to know a good deal of the optical properties of probe dyes. The measurement of refractive index increments of cis- and trans-azobenzene derivatives is in progress in this laboratory. But the complication brought about by complementary grating may not necessarily be a disadvantage. From Figure 1, we note a remarkable sensitivity of FRS in distinguishing a small difference in diffusion coefficients. Therefore, if we know the contrast of the system showing nonexponential decay, we can extract the information about the minute differences in the diffusion coefficients of diffusants. For an example, the anomalous after pulse rising signals found in azobenzene-labeled polystyrenes in 0 solvents5 were possibly due to the chain shrinkage upon photoisomerization of the labeled dye.29 If this is indeed the case, detailed information on the optical properties of photolabels could be exploited to infer photoinduced chain dimensions. We conclude this report by taking note that the causal relationship between the nonexponential FRS decays and the complementary grating effect is yet to be established; however, the effect can clearly be manifested in producing a wide variety of nonexponential decay profiles. Thus, it is rather apparent that photochemical details of the photoprobes in use for FRS experiment are indispensable for further exploitation of the technique.
Acknowledgment. This study was supported in part by grants from the Korean Science and Engineering Foundation and the Ministry of Education. We also gratefully acknowledge Prof. Hyuk Yu at the University of Wisconsin for his help. (28) Chang, T.; Kim, H.; Yu, H. Macromolecules 1987, 20, 2629. (29) Irie, M.; Menju, A.; Hayashi. K. Macromolecules 1979, 12, 1176.
Does a Dlssoclatlng Molecule Sample the Available Phase Space? F. Remade and R. D. Levine* The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (Received: June 19, 1991; In Final Form: July 26, 1991) We argue that dissociating molecules of realistic size will not be a representativesample of the (quasi)bound phase space, particularly so for optical excitation. Analytical considerations and numerical results are presented. Attention is drawn to the faster sampling of phase space by quasibound molecules due to coupling via the continuum. It is shown that on the time scale of bond breaking a moderate amount of mixing in the initially prepared state suffices to ensure a more representative sampling of phase space. The familiar RRKM theory of unimolecular dissociation can therefore remain valid for a non very state selective initial activation. Theories of intramolecular vibrational energy redistribution (see refs 1 and 2 for up to date reviews) continue to be of central interest in molecular reaction dynamics. The key question is how fast is the rate of transfer on the time scale of chemical interest. If the rate is faster, the molecules will dissociate independently of their mode of preparation;’A if not, modeselective unimolecular reactionss are possible. There is extensive experimental evidence in support of this rate being quite fast. Typically, this evidence is derived from chemical activation’ or IR multiphoton dissociation: More recently, both theory7**and expmiments9J0for direct (1) 5357. (2) (3) (4) 6717.
Stuchebrukhov, A.; lonov, S.; Letokhov, V. J . Phys. Chem. 1989,93,
Uzer, T. Phys. Rep. 1991. 199, 73. Oref, I.; Rabinovitch, B. S.Ace. Chem. Res. 1979, 12, 166. Marcus, R. A.; Haw, W. L.; Swamy, K. N. J . Phys. Chem. 1984,88.
(5) Manz, J.; Parmenter, C. S. Eds. Mode Selectivity in Chemical Reactions. Chem. Phys. 1989, 139(1). (6) Lupo, D. W.; Quack, M. Chem. Reu. 1987, 87, 181. (7) Engel, V.; Schinke, R.; Staemmler, V. J . Chem. Phys. 1988,88, 129. ( 8 ) Zhang, J.; Imre, D.; Frederick, J. J . Phys. Chem. 1989, 93, 1840. (9) Van der Wal, R. L.; Scott, J. L.; Crim, F. F. J . Chem. Phys. 1990,92, 803. Crim, F. F.Science 1990, 249, 1387.
optical excitation show that selectivity can be obtained. We suggest that, in general, highly excited polyatomic molecules prepared in well-defined initial states will not have enough time, prior to their dissociation, to sample most of the energetically accessible phase space. If, however, the initial excitation has already populated a moderate fraction of the (quasi)bound phase space then an RRKM-like behavior” is to be expected. We are interested in molecules of realistic size and energy content. The detailed theories1V2available for simpler systems cannot therefore be readily applied. We choose therefore to work with the molecular eigenstates. The process of sampling of the available phase space is then to be interpreted as a dephasing12 of the initially prepared nonstationary state. Heller’’ has discussed the computation of the rate of sampling of phase space of a bound system from this point of view. Much additional discussion and concrete applications were provided by Lorquet and Pavlov(IO) Luo, X.; Rizzo, T. R. J . Chem. Phys. 1991, 94, 889.
(11) Wardlaw. D. M.; Marcus, R. A. Adu. Chem. Phys. 1988. 70, 231. (1 2) Mukamel, S. Chem. Phys. 1978,31,327. Delory, J. M.; Tric, C. fbid. 1974, 3, 54. (13) Heller, E. J. Phys. Reu. 1987, ,435, 1360.
0022-365419112095-7 124$02.50/0 @I 1991 American Chemical Society
