J . Phys. Chem. 1989, 93, 6246-6250
6246
where p i p E is the excess, noncombinatorial part of the chemical potential . The chemical potential of a hydrophobic molecule in an aqueous solution is normally expressed in terms of the mole fraction, Le., with use of the regular solution theory ( p l w-
p I o ) / R T = In x l w+ w I w E / R T
(A21
This approach is, however, questionable on two points. Firstly, the molecules are assumed to have the same size, and secondly, the molecules are assumed to be randomly mixed. The former point is easily taken into account by the use of Flory-Huggins theory. Even though this theory overestimates the combinatorial entropy, it is probably better than regular solution theory. The latter point has been discussed by Costas and Patterson,8 who point out that not all of the volume is available to the hydrophobic molecule due to the network formation by hydrogen bonding. They propose that a term, In m, should be added to the combinatorial entropy in the expression of the chemical potential, where m is the number of water molecules that form a cage around the hydrophobic molecule. Thus, using the Flory-Huggins expression
for the chemical potential at infinite dilution of the probe, we have the following: (ylw- p l o ) / R T = In
+ ( 1 - V i / V w )+ In m + p l w E / R T (A31
Equating the chemical potentials of the probe in the polymer and in the water gives G t r / R T = (plWE - pIPE)/RT = In 41P/41w- In m
+ Vl/Vw (A41
Hence, the free energy, and thus the entropy, of transfer should be corrected for by including the two last terms in eq A4. These two terms are of opposite signs and cancel if m = exp(Vl/Vw), which in these systems would give a number between 100 and 200 for m. In any case, the correction is not crucial and would at the most be on the order of about 3-6 kJ/mol. That would displace G" and 9' curves by the same amount but not alter the general conclusions. Registry No. H,O, 7732-18-5; PhMe, 108-88-3; PhEt, 100-41-4.
Intrinsic Potential Energy Barrier for Twisting in the trans-Stilbene S, State in Hydrocarbon Solvents Jack Sakiel* and Ya-Ping Sun Department of Chemistry, The Florida State University, Tallahassee, Florida 32306-3006 (Received: January 17, 1989; In Final Form: March 30, 1989)
-
Rate constants for It* Ip* twisting in the SI state of trans-stilbene in the n-alkane solvent series (H(CH2),H, n = 5-16) are treated by transition-state theory. In contrast to claims in the literature, observed activation enthalpies, AH*obad, increase with increasing n and depend linearly on E,, the solvent viscous flow activation energy: AH*,, = AH*, + aE, with AH*t = 2.85 f 0.04 kcal/mol and a = 0.39 f 0.02. This behavior is consistent with a medium-enhanced isomerization barrier model proposed earlier. Entropies of activation are also linearly dependent on E, since an isokinetic relationship exists between them and the activation enthalpies. It is concluded that AH*,, the intrinsic barrier to It* 'p* twisting, is independent of alkane solvent.
-
introduction Extensive research efforts in the laboratories of photochemists and spectroscopists during the past 30 years on the trans s cis photoisomerization of the stilbenes have been The mechanism of this reaction is now well-understood and is often the starting point of discussions on cis-trans photoisomerization generally. The complementarity of trans-stilbene fluorescence and photoisomerization was established relatively early by studies, primarily in Fischer's laboratory, of the temperature dependence cis quantum of fluorescence quantum yields, &, and of trans yields, 4tc,4"and later, with the advent of laser pulse excitation, the temperature dependence of 7f.7-9Comparative quenching studies of excited stilbene singlet and triplet states obtained by
-
( I ) Saltiel, J.; D'Agostino, J.; Megarity, E. D.; Metts, L.; Neuberger, K. R.; Wrighton, M.; Zafiriou, 0. C. Org. Phofochem. 1973, 3, 1. (2) Saltiel, J.; Charlton, J. L. In Rearrangements in Ground and Excited States; de Mayo, P., Ed.; Academic; New York, 1980; Vol. 3, p 25. (3) Saltiel, J.; Sun, Y.-P. Manuscript in preparation. (4) Malkin, S.; Fischer, E. J . Phys. Chem. 1964, 68, 1153, and earlier papers in this series. ( 5 ) Dyck, R. H.; McClure, D. S . J . Chem. Phys. 1962, 36, 2336. ( 6 ) Gegiou, D.; Muszkat, K . A.; Fischer, E. J . Am. Chem. SOC.1968, 90, 12. (7) Sumitani. M.; Nakashima. N.; Yoshihara, K.; Nagakura, S. Chem. Phys. f e l t . 1977, 5 1 , 183. (8) Taylor, J. R.; A d a m , M. C.; Sibbett, W. Appl. Phys. f e l t . 1979, 35, 590. (9) Good, H. P.; Wild, U . P.; Haas, E.; Fischer, E.; Resewitz, E.-P.; Lippert, E. Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 126.
0022-3654/89/2093-6246$01.50/0
direct and by sensitized excitation, respectively, established that intersystem crossing from transoid geometries, It*, is at best a minor process at ambient temperatures.'*I2 The currently accepted mechanism for trans cis photoisomerization is that twisting about the central bond occurs as an activated process in the lowest excited singlet state surface to a twisted intermediate, Ip*, which upon decay partitions itself nearly equally between trans and cis ground-state geometries. This mechanism was proposed by Saltiel following the observation that perdeuteration of the stilbene molecule has essentially no effect on the photoisomerization kinetics.13 It has k e n confirmed by numerous laser pulsed excitation studies, some of which will be considered in detail below. The origin of the barrier to It* Ip* torsional distortion has been the subject of theoretical discussions, and though the notion that it originates in a crossing,I4possibly avoided,I5 between the lowest 'B, state and a higher IA, state enjoys wide popularity, no convincing experimental proof of this hypothesis is available. Other possibilities are that the barrier is primarily inherent in the en-
-
-
( I O ) Hammond, G . s.;Saltiel, J.; Lamola, A. A,; Turro, N. J.; Bradshaw, J. S.; Cowan, D. 0.;Counsell, R. C.; Vogt, V.; Dalton, C. J. A m . Chem. SOC. 1964, 86, 3 197. ( I 1 ) Saltiel, J.; Megarity, E. D. J . Am. Chem. SOC.1972, 94, 2742. (12) Saltiel, J.; Marinari, A,; Chang, D. W. L.; Mitchener, J. C.; Megarity, E. D. J . A m . Chem. SOC.1979, 101, 2982. (13) Saltiel, J. J . A m . Chem. SOC.1967, 89, 1036; 1968, 90, 6394. (14) Orlandi, G . ; Siebrand, W. Chem. Phys. Letr. 1975, 30, 352. (15) Birks, J. B. Chem. Phys. Lett. 1978, 43, 430
0 I989 American Chemical Society
Twisting in the trans-Stilbene SI State
The Journal of Physical Chemistry, Vol. 93, No. 16, 1989 6241
ergetics of twisting in the 'B, state or that it arises through an TABLE I: avoided crossing of two A, states that occurs slightly above the energy of the IB, state.I6 solvent Whatever the origin of the barrier, its internal nature was n-C5 established by Hochstrasser and co-workers using subnanosecond time scale transient absorption It* lifetime measurements in the vapor phase under collision-free conditions." Narrow limits for Ip* process under isolated the height of the barrier for the It* molecule conditions were set by a series of studies by Zewail and others of the variation of fluorescence decay constants18-21and fluorescence quantum yield^^^,^^ of jet-cooled trans-stilbene with E,, the energy in excess to 0,O excitation. Decay rate constants for It* are independent of E, for E , 5 3.4 kcal/mol but increase monotonically as E, is increased beyond this threshold value.18~20-22 n-C, Fluorescence quantum yields are unity for E, 5 2.6 kcal/mol and decrease slightly for 2.6 kcal/mol 5 E , 5 3.7 kcal/mol and much more sharply for higher E , values. Depending on the assumption made, these results have been shown to be consistent with a barrier ~ . ~3.7 ~ f 0.2 k c a l / m 0 1 . ~ ~ - ~ ~ height of 3 . 4 & 0.2 k c a l / m 0 1 ~ or Rate constants for It* 'p* twisting, k o b d , in solution have been inferred from fluorescence quantum yields and fluorescence lifetimes, Tf
-
-
kobsd
=
kf(d'fo
- d'f)/d'fod'f
(1)
where $? is the limiting fluorescence quantum yield at low T and k f and ki, are rate constants for radiative decay and intersystem crossing from It*, respectively. Though the resulting koM values . exhibit Arrhenius behavior in solution, obtention of the intrinsic barrier height is complicated by the observation of solvent-dependent activation parametersz6 The existence of a viscositydependent barrier on the potential energy surface of It* along the twisting coordinate was proposed in 1955 by Becker and Kasha2' and was demonstrated experimentally by Fischer and co-worke r ~ . ~A ~model * ~ for ~ quantitative separation of the activation parameters into intrinsic and medium induced parts was first developed and applied by Saltiel and D'Agostino.26 It accounted nicely for the medium enhanced barrier to photoisomerization in glycerol. Starting with the work of Hochstrasser,30recent extensive research activity in this area has focused on Kramers' p r o p o ~ a l , ~ ' ~ ~ ~ for diffusive barrier crossing in accounting for medium effects on It* decay. The Kramers model adopts the concept of Brownian motion in describing the motion of a molecule over a one-dimensional energy barrier. The Kramers equation for the rate constant of such a process is
(16) Hohlneicher, G.; Dick, B. J . Phofochem. 1984, 27, 215. (17) Greene, B. 1.; Hochstrasser, R. M.; Weisman, R . B. J . Chem. Phys. 1979, 71, 544; Chem. Phys. 1980, 48, 289. (18) Syage, J . A.; Lambert, Wm. R.; Felker, P. M.; Zewail, A . H.; Hochstrasser, R. M . Chem. Phys. Leu. 1982, 82, 266. (19) Syage, J. A.; Felker, P. M.; Zewail, A . H. J . Chem. Phys. 1984,81, 4706. (20) Felker, P. M.; Zewail, A . H . J . Phys. Chem. 1985, 89, 5402. (21) Majors, T. J.; Even, U.; Jortner, J. J . Chem. Phys. 1984, 81, 2330. (22) Amirav, A,; Jortner, J. Chem. Phys. Left. 1983, 95, 295. (23) Sonnenshein, M.; Amirav, A.; Jortner, J. J . Chem. Phys. 1984, 88, 4214. (24) Troe, J. Chem. Phys. Lett. 1985, 114, 241. (25) Schroeder, J.; Troe, J. J . Phys. Chem. 1986. 90, 4216. (26) Saltiel, J.; D'Agostino, J. T. J . A m . Chem. SOC.1972, 94, 6445. (27) Becker, R. S.; Kasha, M . In Luminescence of Biological Systems; Johnson, F. H . , Ed.;AAAS: Washington, DC, 1955. (28) Muszkat, K. A.; Gegiou, D.; Fischer, E. J . Am. Chem. SOC.1967.89, 4814. Gegiou, D.; Muszkat, K. A.; Fischer, E. J . Am. Chem. Soc. 1968, 90, 12. (29) Sharafi, S.; Muszkat, K. A . J . Am. Chem. Soc. 1971, 93, 4119. (30) Hochstrasser, R. M. Pure Appl. Chem. 1980, 52, 2683. (31) Kramers, H. A . Physica 1940, 7, 284. (32) Chandrasekhar, S. Rev. Mod. Phys. 1943, 15, I .
n-C, n-Cs
n-C9 n-Clo
Rate Constants for It*
-
Ip* Twisting
T. K
koM x 10-9, s-I
310.7 3 19.0 328.2 338.3 349.0 304.0 294.5 294.5 287.5 296.0 305.9 3 15.9 327.8 282.6 291.6 296.2 304.6 314.5 303.0 294.5 294.5 289.7 296.4 304.5 310.9 280.3 285.7 292.5 299.3 309.3 324.4 346.1 305.0 294.6 294.5 287.9 295.5 305.3 311.9 322.7 333.3 291.2 295.2 296.5 304.6 315.2 303.0 294.5
13.02 15.91 18.67 23.03 28.98 11.45 9.73 9.00 6.83 8.31 10.74 13.49 14.94 4.97 6.97 7.95 10.01 12.75 9.60 8.12 7.52 6.60 8.09 8.53 10.09 4.43 5.30 6.37 7.91 10.30 14.63 22.19 9.01 7.21 6.74 6.20 6.64 8.35 8.73 11.27 11.97 5.42 6.17 6.38 8 .oo 10.92 7.81 5.77
kobrd
T. K 223.2 233.2 243.2 253.2 263.2 273.2 283.2 293.2 303.2 302.0 294.5 220.0 223.1 232.0 240.4 250.2 259.7 266.2 269.3 272.7 272.9 278.2 281.0 285.9 290.8 296.6 301.9 304.0 294.5 294.5 294.1 291.6 281.7 272.7 304.0 294.5 294.5 247.0 250.9 254.9 258.6 263.2 268.1 272.9 280.2 288.9 295.9 301.2
s-I
ref
1.52 2.24 2.96 4.21 5.70 7.93 10.46 13.16 16.51 15.79 13.72 1.33 1.51 1.94 2.57 3.68 4.79 5.79 6.51 7.00 7.57 8.41 8.75 10.24 11.44 13.66 16.62 15.03 12.66 11.74 10.97 9.20 6.97 6.22 12.91 11.28 10.69 2.11 2.50 2.79 2.96 3.51 4.08 5.04 5.85 7.44 9.18 10.45
12
solvent
37 38 36
37 38 38 39
37 38 38 39
I5 n-C16
ref -
37 38 38 35
39
31 38 38 35
39
37 38 38 35
39
37 38
where w and w' are the frequency of the initial potential energy well and the curvature a t the top of the barrier, respectively, E, is the intrinsic barrier height, and /3 is the reduced friction coefficient. In applications of eq 3 fi is usually taken as proportional to the solvent's shear viscosity, q s , in accord with the hydrodynamic model. Use of the hydrodynamic approximation Ip* rate of the Kramers equation to fit experimental It* constants in low-viscosity alkanes leads to underestimation of rate constants for higher viscosity alkanes.33 Several explanations for this apparent failure of the Kramers equation have been suggested. In this paper we address the proposal advanced by Troe and c o - w o r k e r ~ ,and ~ ~ supported, in part, by measurements by Sundstrom and GiIlb1-0,~~ that the barrier height Eo decreases as the alkane chain length is increased. By use of rate constants reported independently by several laboratories, it will be shown that, on the contrary, observed enthalpies of activation for the twisting process, a H * o b s d , increase with increasing alkane chain length, in agreement with a slightly modified version of the model proposed by Saltiel and D'Agostino,26and that the intrinsic barrier height Eo remains essentially constant in the n-alkane solvent series.
-
(33) Rothenberger, G.; Negus, D. K.; Hochstrasser, R. M. J . Chem. Phys. 1983, 79, 5360.
(34) Maneke, G.; Schroeder, J.; Troe, J.; Voss, F.Ber. Bunsen-Ges. Phys. Chem. 1985,89, 896.
6248
The Journal of Physical Chemistry, Vol. 93, No. 16 '989 18.6,
t
TABLE 11: Activation Parameters for the It* n-Alkane Solutions'
17.4-
alkane n-C, n-C,
2
0
fn m
*o
v
-c
Saltiel and Sun
n-C8 n-C,o
16.2-
n-c,, CE
15.0
n-c,, n-C16
glycerol'
-
'p* Process in
AS'IR + In ( K k / h ) obsd' calcdd obsd' calcd' 3.54 (4) 3.44 23.70 (8) 23.44 3.48 (4) 3.51 23.52 (8) 23.49 3.63 (49) 3.66 23.60 (86) 23.58 3.85 (7) 3.85 23.80 (12) 23.72 3.87 (18) 4.02 23.71 (31) 23.89 3.95 ( 1 1 ) 4.10 23.75 (18) 23.91 4.11 (27) 4.27 23.84 (45) 24.03 9.06 (12) 8.97 30.80 (20) 27.45 3.H ', kcal/mol
E,,b kcaljmol 1S O
1.69 2.09 2.56 3.01 3.20 3.64 15.7
9 0.998 0.999 0.965 0.997 0.968 0.989 0.973 0.998
'Values in parentheses indicate standard deviation in last significant digit(s) shown. From ref 43, except for n-C16which is from viscosities in ref 45. CBest fit values based on eq 1 and 2. dAHF,,lcd= 2.85 0.39E, kcal/mol; see text. 'Obtained by fitting data with AH* fixed at AHFcalcd value. 'From ref 26.
+
15.5
2.1
-
1
I
I
3.I
3.5
3.9
T-I
io3(
I
'
c5
4.3
~-9
Figure 1. Transition-state plots of the rate constants in Table 1. Suspect points from ref 35 are designated by A; suspect point from ref 39 is designated bq 0.
Results and Discussion Isomerization rate constants, calculated via eq 1 and 2 from data in the l i t e r a t ~ r e , ' are ~ . ~listed ~ ~ ~in Table I. In some instances the data had to be extracted graphically by measuring the points with a ruler after enlargement of the figure on which they appeared using an opaque overhead projector. As is common practice, the kobsd values in Table 1 are based on the assumptions that dfo = 1 .O, kf = 6 X IO' s-I, and kis = 0. The negligible value of k,, in hydrocarbon media can be based on independent measurements of = 0.95 in a 1 : 1 mixture of methylcyclopentane/methylcyclohexane at 83 K29 and in n-pentane at 77 K.I2 Experimental measurements of 4; and 7; in the 77-298 K range in a 3:2 methylcyclohexane/isohexane solvent mixture give kf = (6.2 0.7) X IO8 s-I, which agrees well with kf = 5.9 X IO'S-' calculated40 via the Birks-Dyson r e l a t i ~ n s h i pwith ~ ~ spectra measured in n-pentane a t 30 'C. Variation in kf due to changes in the index of refraction42were neglected. Experimental values of kf = (3.7 f 0.1) X IO' and kf = (4.0 f 0.1) X 10' s - I , ~ ~ for trans-stilbene in the gas phase under collision-free conditions provide a measure of the maximum index of refraction correction that can be expected. Small changes in kf and kis will not affect the koW values in Table I since, under the experimental conditions employed, these rate constants do not contribute significantly to 71.
-
Transition State Theory Plots. The temperature dependence Ip* rate constants in Table I in each solvent is of the It* described well by the transition state theory equation
(35) Sundstrom, V . ; Gillbro, T. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 222. (36) Courtney, S . H.; Fleming, G . R. J . Chem. Phys. 1985, 83, 215. (37) Courtney, S. H.; Kim, S. K.; Canonica, S.; Fleming, G. R. J . Chem. Soc., Faraday Trans. 2 1986,82, 2065. (38) Lee, M.; Bain, A . J.; McCarthy, P. J.; Han, C. H.; Haseltine, J. N.; Smith, A . B.; Hochstrasser, R. M. J . Chem. Phys. 1986, 85, 4341. (39) Kim, S. K.; Fleming, G. R. J . Phys. Chem. 1988, 92, 2168. (40) Marinari, A.; Saltiel, J. Mol. Phorochem. 1976, 7 , 225. (41) Birks, J. B.; Dyson, D. J. Proc. R. SOC.London, Ser. A 1963, 275, 135. (42) Charlton, J . L.; Saltiel, J . J . Phys. Chem. 1977. 8 / , 1940.
0
Angle of Twist
-
TI2
Figure 2. The medium-enhanced barrier model for stilbene photoisomerization.
where K , AS*oM, and AHtobsdare the transmission coefficient, the entropy of activation, and the enthalpy of activation, respectively, and the other symbols have their usual meanings. As expected, plots of In (kobsd/T)vs T'are linear (Figure I), with slopes yielding the AH',bsd values and intercept yielding the [(AS*,,/R) + In ( K k l h ) ] values listed in Table 11. In order to balance the contribution of data from different laboratories, the single temperature data from ref 33 were given a statistical weight of 5 in the least-squares fitting treatment. Examination of the values in Table I 1 reveals a systematic increase with increasing n-alkane chain length. The low value obtained by Sundstrom and Gillbro for n-hexadecane is obviously due to the two extreme points in their data set which deviate signficantly from the line derived from data from other laboratories. These points were excluded from our least-squares treatment. Medium Enhanced Barrier Model. The model employing a medium enhanced barrier to rotation has been used successfully to treat the temperature dependence of +f of trans-stilbene and its analogue, trans- I,l'-biindanylidene, in glycerol.26 A slightly modified form of this model, Figure 2, follows. In terms of transition state theory, the rate constant, k,, for the medium unimpeded crossing of the barrier along the twisting coordinate is determined by the inherent enthalpy and entropy of activation, respectively, and by the transmission coefficient, AH*t and AS*,, K , associated with this process. The slowing down of the isomerization rate by the medium can be regarded as resulting from the addition of a medium-characteristic viscosity imposed enthalpy barrier, AHv, to the intrinsic energy barrier, AH,. The observed rate constant is furthermore proportional to a medium-characteristic entropy-related factor a, and can be expressed as kobsd = ktav exp(-AH,/RT)
(5)
At "zero" or low viscosities, AHv and a, approach zero and unity, respectively, and k&sd approaches k,. A possible relationship
The Journal of Physical Chemistry, Vol. 93, No. 16, 1989 6249
Twisting in the trans-Stilbene SI State
5.0-
between the parameters AH, and a, to thermodynamic properties of the solute/solvent cage system can be derived as follows. We assumed that solvent organization around the solute, It, is random with respect to the geometric requirements of the twisting in It* and that, upon excitation, a distinct population of It* is produced in solvent cages, S’, which do not restrict rotation while the rest occupies solvent cages, S, which are unfavorable to isomerization. The two populations are related by equilibrium constant Ks,the overall twisting process can be represented by (‘t*S)
K,
z=
(It*S’)
k,
4.5 -
/
4.0 -
(‘p*S’)
-t
and the effective twisting rate constant is kobd = K,k,. It follows that a, = exp(AS,/R) and AH, = AH,, where ASs and AH, are the entropy and enthalpy changes required for the solvent cages to achieve configurations favorable to the geometry change associated with the twisting process. The temperature dependence of the shear viscosity, qs, of a solvent can be described by a variety of empirical functions; see, e.g., ref 43 and 44, which can be used to express kow as a function of 7,. In the Saltiel and D’Agostino treatment this was achieved by employing the Andrade equation43 In qs = In A,, + E,,/RT
(a 1
I
I
(b)
h
*I-
t
Y
-c +
+t
- A e 22.50.0
E,, (kcal/mol)
(7)
where E,, is the activation energy for viscous flow. Solving for (RT)-’ in eq 7 and substituting in eq 5 give
2.0
1.0
-
3.0
4.0
Figure 3. Dependencies of (a) AHIobsd and (b) AS*obd on E,,.
t
h
h ‘0
t
which predicts the often observed linear dependence of In kobsd When medium viscosity is varied a t constant T on In by varying the medium, plots of In koW vs In 7, may deviate from linearity since each medium has its characteristic E, and A , values which, in turn, affect the magnitude of a, and AH,.26 However, for small medium changes, a linear relationship will persist provided that k , remain constant and changes in E,, and A,, are compensated by changes in a, and AH,. Under such conditions eq 8 can be written as 7,.26928929936
In
kobd
= In avkt + ( A H , / E , ) In A,, - ( A H , / E , , ) In
7,
(9)
and the plot of In kobsd vs In 7, will be linear with slope -(AHv/E,,).26 Equation 9 is a more complete version of equations employed by Fischer and ~ o - w o r k e r s and ~ ~ ,by ~ ~Fleming and c o - ~ o r k e r with s ~ ~ a = AHv/E,,.3 In the special case of a = 1, eq 9 is equivalent to the Smoluchowski-Stokes-Einstein limit of Kramers’ equation,33 which is supposed to be approached in media of high viscosity. This limit of Kramers’ equation requires that the entire activation energy for viscous flow, rather than a fraction of it as is usually observed, be added to the intrinsic barrier height. The availability of the activation parameters in Table I1 allows a direct test of the dependence of a on medium within the n-alkane solvent family. As shown in Figure 3, the plot of AH*obsdvs E,, gives a reasonably good straight line, implying that a is constant, or nearly so, throughout the alkane series. Perhaps fortuitously, the point for glycerol falls close to the alkane line. In principle, a = 0.28 f 0.03 could be based on the slope and AH, = 3.1 f 0.1 kcal/mol on the intercept of the least-squares line in Figure 3. However, these values are somewhat unreliable because they reflect the relatively large uncertainties of the individual AH*obsd values. We have therefore employed a global treatment of all the rate constants in Table I to obtain more reliable a and AH, parameters. Since AH*oM= AH*, aE,,has been demonstrated empirically, plots of [In ( k o b s d / T ) aE,,/RT] vs T ishould
+ +
(43) Kierstead, H. A,; Turkevich, J. J . Chem. Phys. 1944, 12, 24. (44) Litowitz, T. A. J . Chem. Phys. 1952, 20, 1088. (45) (a) Timmermans, J. Physico-Chemical Consfants of Pure Organic Compounds; Elsevier: New York, 1950. (b) American Insfitute of Physics Handbook, 3rd ed.; McGraw-Hill: New York, 1972. (c) Tables of Physical and Chemical Consfants, 5th ed.; Longman: London, 1986.
v
-+ C
Q =
16.21 2.7
I
I
3.5
3. I
T-I x
103 ( ~ - 1 )
I
3.9
. I
4.3
4.7
Figure 4. The global transition state plot of the rate constants in Table I extrapolated to the E, = 0 limit (see text). Suspect points A and 0 as in Figure 1.
generate a family of parallel lines with common slope AH*,. The procedure employed to obtain best fit parameters consisted of selecting a value of a and then merging the resulting lines into a single line by arbitrarily moving all points by the separation a t 20 O C . Stepping the value of a to minimize standard deviations of individual points from the common line by using linear least-squares fitting gave a = 0.39 f 0.02 and AH*, = 2.85 f 0.04 kcal/mol as best fit parameters (Figure 4). The line drawn through the Wow value in Figure 3 is based on these parameters. The [(a’,b,d/R) + In ( ~ k / h )values ] obtained from the intercepts of the transition state plots in Figure 1 reveal no clear dependence on E,,, and hence on AH,,in conflict with the expected isokinetic relationship between medium imposed entropy and enthalpy increments.26 To eliminate possible error compensation between slopes and intercepts in the plots in Figure 1, we used the independently obtained AH*, and a values. Averaging [In ( k o b d / g + ( A H , + aE,,)/RT] values for each solvent generates ] in Table 11as more the calculated [AStObd/R+ In ( ~ k / h )listed reliable measures of this quantity. As shown in Figure 3, the plot of calculated [AstObsd/R+ In ( ~ k / h )vs] E,, is linear with slope b = 0.284 f 0.010 and intercept = [ ( A S * , / R )+ In ( ~ k / h )=] 23.00 f 0.03, 3 = 0.997. The true uncertainties in the intercept and slope of the isokinetic plot are larger than indicated because the error range given does not reflect the uncertainty of each individual point in the plot. The isokinetic plot in the Saltiel and D’Agostino paper was based on experimental activation parameters for radiationless decay rate constants of several substrates, including stilbene, in solvents ranging from hydrocarbons to alco-
J . Phys. Chem. 1989, 93, 6250-6257
6250
hols.26 The slope, cy = 0.7 I , is related to the slope of our plot by the factor a In 10 and is equivalent to 0.64, hich is more than twice the value obtained here. This accounts for the considerable deviation of the glycerol point from the alkane isokinetic plot. The significance of this observation remains to be established. I n Figure 4, the constants c, where n is the number of carbon atotns in the alkane, shift the global line so that it gives the intercept of the isokinetic plot. Examination of eq 5 shows that c, = -In a",, = -bE,. Therefore, the line in Figure 4 represents the temperature dependence of the medium unrestricted It* 'p* twisting process because all rate constants have been extrapolated to a hypothetical alkane medium with E , = 0. Since the PAVcontribution is expected to be negligible, AH*,, the slope of the line provides an excellent estimate of the internal potential energy barrier to twisting, Eo = 2.85 0.04 kcal/mol, in an alkane medium. Previous estimates of Eo = 3.5 kcal/mol were improperly based on activation energies of Arrhenius plots2,26,33.36-39 and are therefore larger than our value by -RT. Fitting rate constants for twisting a t a constant temperature to the Kramers equation using the larger E,, value leads to an overestimation of the parameter w by the factor e. The most successful applications to date of the Kramers equation to the rate constants in the n-alkane solvent series have employed rotational reorientation times of the whole stilbene molecule as a measure of microscopic friction, thus avoiding the use of shear v i s c ~ s i t i e s . ~Two ' ~ ~ ~independent tests of the resulting Kramers-Hubbard relationship have led to remarkably similar fitting parameters. Of interest to us are the values of the parameter w = 3.64 X lOI3 s-l (ref 37) and 3.69 X s-' (ref 38) which is related to the intercept in Figure 4.
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w
= 2~e(kT/h)~e~*t/~
(10)
Substitution of the w values into eq 10 gives K exp(AS*,/R) = 0.34 and 0.35 (ref 37 and 38, respectively), which are in satisfactory agreement with K exp(AS*,/R) = 0.47 obtained from the intercept of our isokinetic plot. The demonstration that E , remains constant, within experimental uncertainty, in the n-alkane solvent series, should put to rest attempts34 to use this parameter as a variable in forcing twisting rate constants to fit the Kramers equation in the hydrodynamic theory limit. Data points from Gillbro and S ~ n d s t r O m(SG) ~ ~ which seemed to support this approach are clearly off the global line in Figure 4.46 The much better fits of rate constants to the Kramers-Hubbard equation have provided ample demonstration that it is the use of the shear viscosity as a measure of the friction coefficient that is the major culprit for early failures of the Kramers e q ~ a t i o n . ~ W ~ ~e ~have * * employed ~~ the translational diffusion coefficient of toluene as a measure of Ip* twisting the microfriction experienced by stilbene in the It* process. This approach, which will be described in a separate paper, has allowed the definition of an effective viscosity and has led to an excellent fit of the rate constants in Table I to the Kramers equation.
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Acknowledgment. This research was supported by N S F Grant C H E 87-1 3093. Registry No. frons-Stilbene, 103-30-0 (46) The two rejected SG points deviate from the line in Figure 4 by more than 40,where u is the overall standard deviation of experimental points from the global line. (47) Bowman, R. M.; Eisenthal, K. B.; Millar, D. P.J . Chem. Phys. 1988, 89, 7 6 2 .
Photophysical Studies of Molecular Mobiflty in Polymer Films and Bulk Polymers. 2. Quenching of Pyrene Fluorescence by Phthalic Anhydride in Bulk Poly(dimethylslloxanes) Deh Ying Chu and J. K. Thomas* Department of Chemistry, University of Notre Dame, Notre Dame, Indiana 46556 (Receiued: August 25, 1988; In Final Form: March 31, 1989)
Molecular mobility in bulk poly(dimethylsiloxanes), PDMS, has been monitored by measuring the quenching rate constant k , of pyrene fluorescence by phthalic anhydride, PA. Increasing PDMS molecular weight resulted in a marked change in viscosity but did not impose significant restriction on solute mobility in PDMS at room temperature and at viscosities >50 cP. The simple relationship k 0: (9)*,04 is obeyed. However, a relationship k, a ( 9 ) * 0 . holds ~ in PDMS at low viscosity to 2.2 X IO" cm2 s-I and are larger than those from 1 to 50 cP. Diffusion coePficients of reactants in PDMS are 1.4 X in squalane, the room temperature viscosity of which is only 35.4 cP. It is concluded that no simple relationship between k , and 9 can be applied for all viscous systems. A significant effect of temperature on the quenching rates was observed, and an empirical relation between k,, T , 7, and activation energy, k , 0: ( T/7)Ea/2.3+E* was derived in agreement with the experimental data, where E , and E , are the activation energies of quenching and viscous flow, respectively. The data are explained by utilizing concepts of macroviscosity, microviscosity, chemical structure of the polymer chains, and also free volume theory. High yields of excited pyrene triplets were observed in the laser photolysis of pyrene-phthalic anhydride in bulk PDMS and in cyclohexane, while the yield in acetonitrile was reduced. This is attributed to a fast back electron transfer process from the product anion to the pyrene cation which yields excited triplets of pyrene in alkane solution but not in polar solvents.
Introduction Concepts of molecular motion of low molecular weight solutes in polymer films and bulk polymers are of immediate importance to many areas of science. There has been a steady increase in the application of polymers to many fields of research based on an understanding of diffusion or movement of molecules of small size in polymer matrices. Prime examples are surface coatings, drug delivery systems, replacement parts for biosystems, etc. The measurement of permeability and diffusion of oxygen and other 0022-3654/89/2093-6250$01.50/0
gas molecules in polymer systems has been studied previously by the time-lag method'-3 or photophysical and photochemical method^.^-^ Laser photolysis studies utilizing T,-T, absorption ( 1 ) Comyn, J . Polymer Permeability; Elsevier: New York, 1985.
(2) Felder, R. M.; Huvard, G. S . In Methods and Experimental Physics; Marton, L., Marton, C., Eds.; Academic: New York, 1980; Vol. 16C, Chapter 17. (3) Meares, P. Polymers: Structures and Bulk Properties; Van Nostrand: London, 1965; Chapter 12.
0 1989 American Chemical Society
