- Email: [email protected]
Influence of solvent permittivity on excited state photoisomerization rates of stilbene in n-alcohols
5 May 1995
CHEMICAL PHYSICS LETTERS Chemical Physics Letters 237 (1995) 145-151
ELSEVIER
Influence of solvent permittivity on excited state photoisomerization rates of stilbene in n-alcohols Robert
M.
Anderton, John F. Kauffman
*
Department of Chemistry, University of Missouri - Columbia, Columbia, MO 65211, USA
Received 13 December 1994; in final form 22 February 1995
Abstract The dependence of the activation energy for trans-stilbene photoisomerization on solvent permittivity has been determined in n-alcohols via fits of the isomerization rates across the alcohol series at constant solvent permittivity to Kramers expression. The results of this procedure indicate that the internal barrier to isomerization in these solvents is considerably larger than that indicated by isoviscosity Arrhenius plots and predicts a strong dependence of the barrier height on temperature and alcohol chain length.
1. Introduction Diphenylpolyene photoisomerization has been widely studied as a prototypical chemical reaction in the gas phase and in solution [1,2]. It is now well established that the rate of the reaction in solution is faster than the rate in the gas phase [3-5]. This result has been interpreted as stemming from the influence o f the solvent on the diphenylpolyene excited state potential energy surface. Much of the previous work in this area has focused on the influence of s o l v e n t solute friction on the reaction rate, and in many cases the failure of theory to predict the observed experimental results has been attributed to the influence of frictional effects on the excited state surface. Such influences as frequency dependent friction and multi-
* Corresponding author. E-mail: [email protected]. edu
dimensional coupling have been invoked to explain these effects [6-9]. In the case of diphenylpolyene isomerization in polar liquids, agreement between theory and experiment has been particularly poor [10]. In recent years a number of groups have acknowledged that polar solvents are likely to influence the excited state potential energy surface, particularly in light of claims that the transition state is zwitterionic, or at least highly polarizable [1,11], suggesting that the transition state should experience a stabilization as solvent polarity is increased. Arguments such as these have been invoked in several cases to explain the deviation of experimental results from theoretical predictions. Hicks et al. have attempted to quantitatively account for the influence of solvent polarity on the rate of stilbene photoisomerization [12]. Their analysis was based on their observation of a quantitative, logarithmic relationship between the rate constant for formation of the twisted intramolecular
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R.M. Anderton, J.F. Kauffman / Chemical Physics Letters 237 (1995) 145-151
charge transfer state of dimethylaminobenzonitrile (DMABN) and the solvent ET(30) parameter, which they attributed to a polarity dependent isomerization barrier height [13]. They then attempted to analyze isoviscosity plots of stilbene in alcohols by assuming that the activation energy for the stilbene photoisomerization reaction can be written as a linear function of a solvent polarity parameter. Though the detailed results of this analysis were not presented in their Letter, the authors stated that this analysis failed for stilbene. They suggest that for the polar DMABN study, dynamic solvent effects may not be very important owing to the polar nature of the solute ground state, while such effects may be very important in the case of the nonpolar stilbene. Zeglinski and Waldeck attempted to use the method of Hicks et al. to correct isoviscosity plots of dimethoxystilbene and reported similar results [14]. The variation of the slopes of the isoviscosity plots with solvent viscosity was taken as evidence of solvent coupling to the reactive surface, but attempts to correct this variation by fitting the data to an activation barrier with a linear dependence on the ET(30) parameter were unsuccessful. These cases illustrate the inherent difficulties in attributing a functional form to the dependence of potential energy surface parameters on solvent polarity, especially when solvent-solute friction is also expected to play a role. In a recent paper, we have demonstrated a method for extracting the influence of solvent polarity on diphenylbutadiene photoisomerization rates which avoids the above difficulties [15]. The method, which we refer to as isodielectric-Kramers-Hubbard (IKH) analysis, rests on two assumptions. The first is that the influence of the solvent-solute friction is well characterized by the Kramers equation when the Hubbard relation is used as the measure of solventsolute friction. The second assumption is that, within a series of homologous solvents such as the n-alcohols, the sole influence of the solvent polarity, as characterized by the bulk solvent permittivity, is to stabilize the transition state for the photoisomerization, and that no other solvent property influences the potential energy surface. Under these assumptions, we argue that barrier heights measured under isodielectric conditions should be correct, since the potential energy surface will be constant for each of
the solvent-temperature pairs. Thus we. calculated isomerization rates and rotational correlation times for solvent-temperature pairs across the alcohol series which resulted in isodielectric conditions, and we did so for eleven different values of the bulk solvent permittivity. Because we accounted explicitly for solvent-solute friction via measured rotational correlation times, the barrier heights resulting from fits of the isodielectric rate constants are correct to the extent that the above assumptions are correct. The results of this analysis showed a barrier height which increased with decreasing solvent permittivity, as is expected for a polar transition state. Furthermore, the barrier height appears to be roughly linear in the bulk permittivity, though over the permittivity range studied (E = 10 to 20) we cannot distinguish between linearity in • and reaction field expressions such as ( • - 1 ) / ( • + 2). In light of the fact that we made no initial assumption regarding the dependence of the activation barrier height on solvent polarity, the above results were particularly gratifying. The purpose of this Letter is to extend the IKH analysis to stilbene, primarily because it has been more widely studied than diphenylbutadiene, in order to examine the breadth of the applicability of this method of analysis.
2. Procedure In order to account for frictional influences on the rate of stilbene photoisomerization, the rate constant may be given by Kramers [16] expression,
knr b/[l+t t2105 1) where knr is the nonradiative rate constant and o)a and w b are the initial well frequency and the imaginary barrier frequency, respectively. /3 is the angular velocity correlation frequency, which is a measure of the solvent-solute friction. The isodielectric Kramers-Hubbard fit method involves the assumption that the potential energy surface for the isomer-
R.M. Anderton, J.F. Kauffman / Chemical Physics Letters 237 (1995) 145-151
ization varies as a function of the solvent permittivity. Eq. (1) is fit to the nonradiative rate constants determined for solvent-temperature combinations that maintain a constant solvent permittivity to keep the barrier height constant across the fit. Stilbene nonradiative rate constants in the n-alcohols ethanol through hexanol and octanol were calculated at the appropriate temperatures from the Arrhenius parameters reported previously by Fleming and co-workers [17], representing data taken in the temperature range of - 1 5 to 45°C. The angular velocity correlation frequency for each solvent-temperature pair was calculated by using the rotational correlation times of diphenylbutadiene (DPB) as a probe of the local solvent friction via the Hubbard relation [18], 6kT
/3 =
I
"rr .
(2)
The temperature dependencies of stilbene rotational correlation times in each of the n-alcohols were not available in the literature but /3 values derived from measurements of the rotational correlation times of diphenylbutadiene (DPB) in n-alcohols, which have recently been reported [15], are equally appropriate measures of the local solvent friction, as discussed below. Nonlinear regressions of Kramers equation were performed with the curve-fitting function of the SigmaPlot TM software package (Jandel Scientific). knr , /3, and T were input parameters, and wa, to b, and E a were fitting parameters. The assumption involved in using the rotational correlation time of a diphenylpolyene as a measure of the frictional interactions which oppose the isomerization reaction is that this friction scales with the friction felt by the rotating molecule both as a function of solvent and temperature. When stilbene undergoes an isomerization the phenyl rings change their relative orientations with respect to each other and there is a frictional force from the surrounding solvent which opposes this motion. Thus, while the fl value determined from the rotational correlation time of a DPB or stilbene molecule would not be expected to represent the actual frictional force opposing the relative rotation of the phenyl rings, it may be approximated to scale similarly with changing solvent or temperature. Examination of Eq. (1) reveals that the scaling factor between the friction
147
which opposes the isomerization reaction and the friction which inhibits the rotation of the entire molecule is included in the w b factor, which is a fitting parameter. Thus physical significance can not be attributed to the magnitude of ~ob in these fits since it contains a contribution related to the difference in size between the portion of the molecule that must push solvent molecules aside to undergo isomerization and the entire rotating solute. While the extracted values of E a and 09a are sensitive to changes in fl, they are insensitive to any factor that scales the entire /3 set. Furthermore, /3 for DPB will scale directly with /3 for stilbene except to the extent of the difference in the temperature dependence of the boundary condition, C, in the modified StokesEinstein-Debye equation [19]. We have evaluated C for DPB and stilbene in each of the n-alcohols as a function of temperature using the DKS model [20], modified by using the measured molar volume of the solvent minus the molar van der Waals volume of the solvent as a measure of the free space per solvent molecule. This approach to estimating the free volume is more applicable to associating fluids than the use of Hildebrand Batschinski parameter and the solvent isothermal compressibility, and has been shown to improve the quality of DKS model predictions of diphenylbutadiene rotational correlation times in n-alcohols [21]. These calculations indicate that across the alcohol series and temperature conditions covered in these isodielectric studies, the boundary condition for DPB and the boundary condition for stilbene scale with each other by a constant factor to within one half of one percent, indicating that the rotational correlation times of either molecule are equally appropriate measures of the friction which opposes the isomerization reaction.
3. Analysis and results The best-fit parameters using the IKH method are given in Table 1 along with the standard errors associated with each parameter. The standard errors of parameters between permittivity values are uncorrelated. The large uncertainties associated with the fit parameters for the fits conducted at each permittivity value are a result of fitting a limited data set to an equation with several fitting parameters. When the
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Table 1 Parameters of isodielectric fits to Kramers expression. Uncertainties are the uncorrelated standard errors of the fit parameters at each value of the permittivity E
10 it 12 13 14 15 16 17 18 19 20
~0a (1013 S-1)
£0 b
Ea
(1012 S- I )
(kcal/mol)
20+77 13_+43 8+ 17 5_+10 3.7 -+6.9 2.5 _+4.5 1.5_+2.0 0.86 _+0.84 0.47 + 0.33 0.25 -+0.13 o. 13 + 0.05
7_+ 14 6_+12 6.2-+7.9 6.9_+7.4 7.9 -+7.6 8.8 _+7.4 9.4+5.7 9.8 + 4.2 10.1 + 3.1 10.4 -+2.3 10.6 -+ 1.7
3.7_+4.2 3.4_+3.4 3.1 _+2.3 2.8+2.0 2.6 -+ 1.8 2.4 _+1.5 2.1_+1.1 1.80 -+0.80 1.45 -+0.56 1.11 -+0.40 0.77 -+0.28
s a m e analysis was c o n d u c t e d for D P B w e noted that increasing the n u m b e r o f data points by including m o r e a l c o h o l s o v e r a w i d e r temperature range led to smaller uncertainties for the fitting parameters at each permittivity but did not change the fit values. Thus w h i l e there is significant uncertainty associated with the v a l u e o f each fitting p a r a m e t e r within a single isodielectric fit, there is a clear trend in depend e n c e of E a and o)a on solvent permittivity. Fig. 1 s h o w s the d e p e n d e n c e o f the barrier height on sol-
v e n t permittivity indicated by these fits. The linear regression line s h o w n in Fig. 1 has a slope of - 0 . 2 8 8 6 and an intercept o f 6.600. This line has b e e n included in order to a l l o w interpolation o f E a as a function o f e f r o m this plot, but it important to note that the I K H analysis does not require one to a s s u m e a particular functional f o r m for the dependence of the fitting parameters on solvent permittivity. Further e v i d e n c e of the influence of the solvent permittivity on the potential energy surface c o m e s f r o m the observation that Eq. (1) is able to fit the i s o m e r i z a t i o n rate constants w e l l for s o l v e n t - t e m p e r ature pairs that h a v e identical permittivity across the entire range of /3 but is unable to fit to the rate constants across the alcohol series at constant temperature. The failure of the latter fit is hypothesized to be due to variation of the barrier height as the solvent permittivity changes across the alcohol series at constant temperature. W e have reported similar observations for I K H fits to D P B i s o m e r i z a t i o n rate constants [15]. Fig. 2 s h o w s the temperature d e p e n d e n c e o f the barrier height implied for stilbene in n - a l c o h o l s ob-
40
3.5 40
• 3.0
3.0
--" 2.5
"~
25
i
20
~2
E
1.5 1.5 1.0 I0 0.5 05~
240
~
i
i
260
280
300
~ 320
i 340
360
T (K) O0
[
i
I
10
14
16
20
Permitivitty Fig. 1. The dependence of the stilbene barrier height on solvent permittivity determined from isodielectric Kramers-Hubbard fits in n-alcohols.
Fig. 2. The barrier heights implied by the substitution of the dependence of the barrier height on solvent permittivity into the temperature dependence of the solvent permittivity of the six n-alcohols involved in this study. (O) Ethanol, (N) n-propanol, (zx) n-butanol, ( v ) n-pentanol, ('~) n-hexanol, (O) n-heptanol, (O) n-octanol.
R.M. Anderton, J.F. Kauffman / Chemical Physics Letters 237 (1995) 145-151
tained by the substitution of the dependence of the barrier height on solvent permittivity into the temperature dependence of the solvent permittivity for each alcohol. Since IKH fits involve data taken from a horizontal slice in Fig. 2, this implies a single barrier height for the entire fit. In contrast, fits of isomerization data across a temperature range within a single alcohol would correspond to data taken along one of the diagonal lines in Fig. 2 while fits of isomerization data taken across the solvent series at a single temperature correspond to a vertical line in Fig. 2. Similarly, isoviscosity Arrhenius plots are made up of the isomerization rate constant data in solvent-temperature pairs that cut diagonally across Fig. 2 from upper left to lower right. In addition to the trend in E a with solvent permittivity, which was our primary interest, a trend of decreasing toa with increasing permittivity was observed. This trend is consistent with the trend observed in our previous study of DPB [15]. Qualitatively, a decrease in toa with decreasing E a is physically reasonable since a decrease in E a would be expected to result in a wider initial potential well. to b remains relatively constant across the permittivity range examined in this study, as observed in the DPB study. While the physical significance of to b is ambiguous, the nearly constant value of this fitting parameter is consistent with its role as a scaling parameter, as discussed in Section 2.
4. Discussion In order to put the results of this study in context it is necessary to compare this method with other approaches that have been applied to this problem, which have primarily been isoviscosity plots in various solvent systems. Based on an isoviscosity plot at 2.55 cp in n-alcohols Sundstrom and co-workers [22,23] concluded that stilbene isomerization proceeds on a potential energy surface lacking a barrier. Kim and co-workers [24] report that isoviscosity plots of stilbene over a broader range of viscosities in n-alcohols yield decreasing values for the barrier height with increasing viscosity. This phenomenon has been attributed to solvation dynamics but it has been noted that other indications of solvation dynamics such as time-dependent spectral shifts and non-
149
exponential decays are not present. The slopes obtained from these isoviscosity plots at different viscosities are the result of two competing deviations from the assumptions inherent in making the plots. The first is the change in the solvent permittivity and its effect on E a and toa. Isoviscosity plots utilize data taken in cold short chain alcohols and hot long chain alcohols, the combination that yields the greatest change in solvent permittivity over a single plot. This indicates E a and o)a will change over the span of the plot, contrary to the assumptions implicit in making such a plot. The competing effect is the large deviations from Stokes-Einstein-Debye behavior similar to that noted for stilbene in alkanes [25] and DPB in alcohols [21] in which the boundary condition decreases as the alcohol molecule size increases to approach the size of the solute. The IKH method obviates this concern by utilizing the rotational correlation times as a microscopic measure of solventsolute friction. Qualitatively it is clear that these effects may be expected to lead to inconsistent values in the slope when progressing from isoviscosity plots at one value of the viscosity to another, though it is difficult to predict the observed trends because of the interplay between these competing effects. Isoviscosity plots of DPB [10] and stilbene [26] in alkanes have been shown to yield constant slopes at all viscosities. Park and Waldeck [27] have noted a slight trend toward increasing barrier at higher viscosity for stilbene in alkanes when the chain length range of the alkane series was changed to achieve changes in the viscosity, but it was noted that in plots where the solvent series did not change there was no apparent trend. This observation lends support to the premise that the leading causes of the failure of isovisocity plots in n-alcohols are changing boundary conditions and solvent permittivity which are not strong effects in alkanes. The rotational correlation time in alkanes is expected to scale linearly with bulk viscosity across the alkane series, since the alkane chain length is on the order of or larger than the solute [21]. Furthermore, there is only a slight dependence of the solvent permittivity on temperature or chain length for alkanes. Isoviscosity plots of stilbene in n-alkyl nitriles have been demonstrated to yield consistent slopes at several different values of viscosity [28] which would seem to indicate that over a wide permittivity range
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the barrier height is constant. One explanation for the observation that isoviscosity plots yield a consistent slope in the presence of both of these competing effects is that the nitriles exhibit a weaker temperature dependence of the solvent permittivity since hydrogen bonding is not extensive [29]. Isoviscosity Arrhenius plots of stilbene isomerization conducted in a single alcohol by applying pressure to the alcohol to maintain constant viscosity over a temperature range [30] would be expected to suffer from similar effects of changing solvent permittivity on the potential energy surface. Influences resulting from a changing boundary condition may be diminished in this case since there is no change in the relative size between solvent and solute, although applying pressure would decrease the free space between solvent molecules. As a final point we note that the barrier height determined by the IKH analysis of stilbene at the lowest solvent permittivity is similar to the barrier height which has been in a cold jet [5]. Further insight into the applicability of the IKH method of analysis to the study of unimolecular isomerization reactions may be gained by examination of other diphenylpolyenes whose barrier heights have been measured under isolated molecule conditions.
5. Conclusions The IKH method presented here does not depend on the assumption of any functional form for the dependence of either the barrier height or other parameters of the potential energy surface on permittivity. The only assumption inherent in the method is that any changes in the potential energy surface with changing temperature or alcohol chain length are a result of the changing solvent permittivity. The resuits presented in this Letter indicate the importance of the solvent permittivity in determining the height of the barrier to isomerization of stilbene in n-alcohols. The barrier height increases with increasing temperature within a single alcohol solvent and increases with increasing alcohol chain length at constant temperature. Analyses such as isoviscosity Arrhenius plots or fits to Kramers expression in a single solvent across a temperature range, which assume a constant barrier height, are inappropriate if
the solvent permittivity undergoes a large change across a range of solvents or temperatures. The IKH analysis also indicates that the magnitude of the barrier to stilbene photoisomerization in n-alcohols is considerably larger than the aforementioned studies [22,24] have indicated.
Acknowledgement Acknowledgement is made to the Donors of The Petroleum Research Fund, administered by the American Chemical Society, for the support of this research.
References [1] M. Allen and D. Witten, Chem. Rev. 89 (1989) 1691. [2] G. Fleming, S. Courtney and M. Balk, J. Stat. Phys. 42 (1986) 83. [3] J. Syage, Laser Focus World 30 (1994) 73. [4] R. Hochstrasser, Pure Appl. Chem. 52 (1980) 2683. [5] J. Syage, P. Felker and A. Zewail, J. Chem. Phys. 81 (1984) 4706. [6] N. Park and D. Waldeck, Chem. Phys. Letters 168 (1990) 379. [7] B. Bagchi and D. Oxtoby, J. Chem. Phys. 78 (1983) 2735. [8] J. Schroeder, Ber. Bunsenges. Physik. Chem. 95 (1991) 233. [9] R. Grote and J. Hynes, J. Chem. Phys. 74 (1981) 4465. [10] S. Velsko and G. Fleming, J. Chem. Phys. 76 (1982) 3553. [11] D. Waldeck, Chem. Rev. 91 (1991) 415. [12] J. Hicks, M. Vandersall, E. Sitzmann and K. Eisenthal, Chem. Phys. Letters 135 (1987) 413. [13] J. Hicks, M. Vandersall, Z. Babarogic and K. Eisenthal, Chem. Phys. Letters 116 (1985) 18. [14] D. Zeglinski and D. Waldeck, J. Phys. Chem. 92, (1988) 692. [15] R. Anderton and J. Kauffman, J. Phys. Chem. 98 (1994) 12125. [16] H. Kramers, Physica 7 (1940) 284. [17] S. Courtney, S. Kim, S. Canonica and G. Fleming, Chem. Soc. Faraday Trans. II 82 (1986) 2065. [18] P. Hubbard, Phys. Rev. 131 (1963) 1155. [19] P. Perrin, J. Phys. Radium 5 (1934) 497. [20] J. Dote, D. Kivelson and R. Schwartz, J. Phys. Chem. 85 (1981) 2169. [21] R. Anderton and J. Kauffman, J. Phys. Chem. 98 (1994) 12117. [22] V. Sundstrom and T. Gillbro, Chem. Phys. Letters 109 (1984) 538. [23] E. Akesson, H. Bergstrom, V. Sundstrom and T. Gillbro, Chem. Phys. Letters 126 (1986) 385.
R.M. Anderton, J.F. Kauffman / Chemical Physics Letters 237 (1995) 145-151 [24] S. Kim, S. Courtney and G. Fleming, Chem. Phys. Letters 159 (1989) 543. [25] S. Kim and G. Fleming, J. Phys. Chem. 92 (1988) 2168. [26] V. Sundstrom and T. Gillbro, Ber. Bunsenges. Physik. Chem. 89 (1985) 222. [27] N. Park and D. Waldeck J. Chem. Phys. 91 (1989) 943.
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[28] N. Sivakumar, E. Hoburg and D. Waldeck, J. Chem. Phys. 90 (1989) 2305. [29] S. Goldman and C. Joslin, J. Phys. Chem. 97 (1993) 12349. [30] J. Schroeder, D. Schwarzer, J. Troe and P. Vohringer, Chem. Phys. Letters 218 (1994) 43.
CHEMICAL PHYSICS LETTERS Chemical Physics Letters 237 (1995) 145-151
ELSEVIER
Influence of solvent permittivity on excited state photoisomerization rates of stilbene in n-alcohols Robert
M.
Anderton, John F. Kauffman
*
Department of Chemistry, University of Missouri - Columbia, Columbia, MO 65211, USA
Received 13 December 1994; in final form 22 February 1995
Abstract The dependence of the activation energy for trans-stilbene photoisomerization on solvent permittivity has been determined in n-alcohols via fits of the isomerization rates across the alcohol series at constant solvent permittivity to Kramers expression. The results of this procedure indicate that the internal barrier to isomerization in these solvents is considerably larger than that indicated by isoviscosity Arrhenius plots and predicts a strong dependence of the barrier height on temperature and alcohol chain length.
1. Introduction Diphenylpolyene photoisomerization has been widely studied as a prototypical chemical reaction in the gas phase and in solution [1,2]. It is now well established that the rate of the reaction in solution is faster than the rate in the gas phase [3-5]. This result has been interpreted as stemming from the influence o f the solvent on the diphenylpolyene excited state potential energy surface. Much of the previous work in this area has focused on the influence of s o l v e n t solute friction on the reaction rate, and in many cases the failure of theory to predict the observed experimental results has been attributed to the influence of frictional effects on the excited state surface. Such influences as frequency dependent friction and multi-
* Corresponding author. E-mail: [email protected]. edu
dimensional coupling have been invoked to explain these effects [6-9]. In the case of diphenylpolyene isomerization in polar liquids, agreement between theory and experiment has been particularly poor [10]. In recent years a number of groups have acknowledged that polar solvents are likely to influence the excited state potential energy surface, particularly in light of claims that the transition state is zwitterionic, or at least highly polarizable [1,11], suggesting that the transition state should experience a stabilization as solvent polarity is increased. Arguments such as these have been invoked in several cases to explain the deviation of experimental results from theoretical predictions. Hicks et al. have attempted to quantitatively account for the influence of solvent polarity on the rate of stilbene photoisomerization [12]. Their analysis was based on their observation of a quantitative, logarithmic relationship between the rate constant for formation of the twisted intramolecular
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-2614(95)00295-2
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R.M. Anderton, J.F. Kauffman / Chemical Physics Letters 237 (1995) 145-151
charge transfer state of dimethylaminobenzonitrile (DMABN) and the solvent ET(30) parameter, which they attributed to a polarity dependent isomerization barrier height [13]. They then attempted to analyze isoviscosity plots of stilbene in alcohols by assuming that the activation energy for the stilbene photoisomerization reaction can be written as a linear function of a solvent polarity parameter. Though the detailed results of this analysis were not presented in their Letter, the authors stated that this analysis failed for stilbene. They suggest that for the polar DMABN study, dynamic solvent effects may not be very important owing to the polar nature of the solute ground state, while such effects may be very important in the case of the nonpolar stilbene. Zeglinski and Waldeck attempted to use the method of Hicks et al. to correct isoviscosity plots of dimethoxystilbene and reported similar results [14]. The variation of the slopes of the isoviscosity plots with solvent viscosity was taken as evidence of solvent coupling to the reactive surface, but attempts to correct this variation by fitting the data to an activation barrier with a linear dependence on the ET(30) parameter were unsuccessful. These cases illustrate the inherent difficulties in attributing a functional form to the dependence of potential energy surface parameters on solvent polarity, especially when solvent-solute friction is also expected to play a role. In a recent paper, we have demonstrated a method for extracting the influence of solvent polarity on diphenylbutadiene photoisomerization rates which avoids the above difficulties [15]. The method, which we refer to as isodielectric-Kramers-Hubbard (IKH) analysis, rests on two assumptions. The first is that the influence of the solvent-solute friction is well characterized by the Kramers equation when the Hubbard relation is used as the measure of solventsolute friction. The second assumption is that, within a series of homologous solvents such as the n-alcohols, the sole influence of the solvent polarity, as characterized by the bulk solvent permittivity, is to stabilize the transition state for the photoisomerization, and that no other solvent property influences the potential energy surface. Under these assumptions, we argue that barrier heights measured under isodielectric conditions should be correct, since the potential energy surface will be constant for each of
the solvent-temperature pairs. Thus we. calculated isomerization rates and rotational correlation times for solvent-temperature pairs across the alcohol series which resulted in isodielectric conditions, and we did so for eleven different values of the bulk solvent permittivity. Because we accounted explicitly for solvent-solute friction via measured rotational correlation times, the barrier heights resulting from fits of the isodielectric rate constants are correct to the extent that the above assumptions are correct. The results of this analysis showed a barrier height which increased with decreasing solvent permittivity, as is expected for a polar transition state. Furthermore, the barrier height appears to be roughly linear in the bulk permittivity, though over the permittivity range studied (E = 10 to 20) we cannot distinguish between linearity in • and reaction field expressions such as ( • - 1 ) / ( • + 2). In light of the fact that we made no initial assumption regarding the dependence of the activation barrier height on solvent polarity, the above results were particularly gratifying. The purpose of this Letter is to extend the IKH analysis to stilbene, primarily because it has been more widely studied than diphenylbutadiene, in order to examine the breadth of the applicability of this method of analysis.
2. Procedure In order to account for frictional influences on the rate of stilbene photoisomerization, the rate constant may be given by Kramers [16] expression,
knr b/[l+t t2105 1) where knr is the nonradiative rate constant and o)a and w b are the initial well frequency and the imaginary barrier frequency, respectively. /3 is the angular velocity correlation frequency, which is a measure of the solvent-solute friction. The isodielectric Kramers-Hubbard fit method involves the assumption that the potential energy surface for the isomer-
R.M. Anderton, J.F. Kauffman / Chemical Physics Letters 237 (1995) 145-151
ization varies as a function of the solvent permittivity. Eq. (1) is fit to the nonradiative rate constants determined for solvent-temperature combinations that maintain a constant solvent permittivity to keep the barrier height constant across the fit. Stilbene nonradiative rate constants in the n-alcohols ethanol through hexanol and octanol were calculated at the appropriate temperatures from the Arrhenius parameters reported previously by Fleming and co-workers [17], representing data taken in the temperature range of - 1 5 to 45°C. The angular velocity correlation frequency for each solvent-temperature pair was calculated by using the rotational correlation times of diphenylbutadiene (DPB) as a probe of the local solvent friction via the Hubbard relation [18], 6kT
/3 =
I
"rr .
(2)
The temperature dependencies of stilbene rotational correlation times in each of the n-alcohols were not available in the literature but /3 values derived from measurements of the rotational correlation times of diphenylbutadiene (DPB) in n-alcohols, which have recently been reported [15], are equally appropriate measures of the local solvent friction, as discussed below. Nonlinear regressions of Kramers equation were performed with the curve-fitting function of the SigmaPlot TM software package (Jandel Scientific). knr , /3, and T were input parameters, and wa, to b, and E a were fitting parameters. The assumption involved in using the rotational correlation time of a diphenylpolyene as a measure of the frictional interactions which oppose the isomerization reaction is that this friction scales with the friction felt by the rotating molecule both as a function of solvent and temperature. When stilbene undergoes an isomerization the phenyl rings change their relative orientations with respect to each other and there is a frictional force from the surrounding solvent which opposes this motion. Thus, while the fl value determined from the rotational correlation time of a DPB or stilbene molecule would not be expected to represent the actual frictional force opposing the relative rotation of the phenyl rings, it may be approximated to scale similarly with changing solvent or temperature. Examination of Eq. (1) reveals that the scaling factor between the friction
147
which opposes the isomerization reaction and the friction which inhibits the rotation of the entire molecule is included in the w b factor, which is a fitting parameter. Thus physical significance can not be attributed to the magnitude of ~ob in these fits since it contains a contribution related to the difference in size between the portion of the molecule that must push solvent molecules aside to undergo isomerization and the entire rotating solute. While the extracted values of E a and 09a are sensitive to changes in fl, they are insensitive to any factor that scales the entire /3 set. Furthermore, /3 for DPB will scale directly with /3 for stilbene except to the extent of the difference in the temperature dependence of the boundary condition, C, in the modified StokesEinstein-Debye equation [19]. We have evaluated C for DPB and stilbene in each of the n-alcohols as a function of temperature using the DKS model [20], modified by using the measured molar volume of the solvent minus the molar van der Waals volume of the solvent as a measure of the free space per solvent molecule. This approach to estimating the free volume is more applicable to associating fluids than the use of Hildebrand Batschinski parameter and the solvent isothermal compressibility, and has been shown to improve the quality of DKS model predictions of diphenylbutadiene rotational correlation times in n-alcohols [21]. These calculations indicate that across the alcohol series and temperature conditions covered in these isodielectric studies, the boundary condition for DPB and the boundary condition for stilbene scale with each other by a constant factor to within one half of one percent, indicating that the rotational correlation times of either molecule are equally appropriate measures of the friction which opposes the isomerization reaction.
3. Analysis and results The best-fit parameters using the IKH method are given in Table 1 along with the standard errors associated with each parameter. The standard errors of parameters between permittivity values are uncorrelated. The large uncertainties associated with the fit parameters for the fits conducted at each permittivity value are a result of fitting a limited data set to an equation with several fitting parameters. When the
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Table 1 Parameters of isodielectric fits to Kramers expression. Uncertainties are the uncorrelated standard errors of the fit parameters at each value of the permittivity E
10 it 12 13 14 15 16 17 18 19 20
~0a (1013 S-1)
£0 b
Ea
(1012 S- I )
(kcal/mol)
20+77 13_+43 8+ 17 5_+10 3.7 -+6.9 2.5 _+4.5 1.5_+2.0 0.86 _+0.84 0.47 + 0.33 0.25 -+0.13 o. 13 + 0.05
7_+ 14 6_+12 6.2-+7.9 6.9_+7.4 7.9 -+7.6 8.8 _+7.4 9.4+5.7 9.8 + 4.2 10.1 + 3.1 10.4 -+2.3 10.6 -+ 1.7
3.7_+4.2 3.4_+3.4 3.1 _+2.3 2.8+2.0 2.6 -+ 1.8 2.4 _+1.5 2.1_+1.1 1.80 -+0.80 1.45 -+0.56 1.11 -+0.40 0.77 -+0.28
s a m e analysis was c o n d u c t e d for D P B w e noted that increasing the n u m b e r o f data points by including m o r e a l c o h o l s o v e r a w i d e r temperature range led to smaller uncertainties for the fitting parameters at each permittivity but did not change the fit values. Thus w h i l e there is significant uncertainty associated with the v a l u e o f each fitting p a r a m e t e r within a single isodielectric fit, there is a clear trend in depend e n c e of E a and o)a on solvent permittivity. Fig. 1 s h o w s the d e p e n d e n c e o f the barrier height on sol-
v e n t permittivity indicated by these fits. The linear regression line s h o w n in Fig. 1 has a slope of - 0 . 2 8 8 6 and an intercept o f 6.600. This line has b e e n included in order to a l l o w interpolation o f E a as a function o f e f r o m this plot, but it important to note that the I K H analysis does not require one to a s s u m e a particular functional f o r m for the dependence of the fitting parameters on solvent permittivity. Further e v i d e n c e of the influence of the solvent permittivity on the potential energy surface c o m e s f r o m the observation that Eq. (1) is able to fit the i s o m e r i z a t i o n rate constants w e l l for s o l v e n t - t e m p e r ature pairs that h a v e identical permittivity across the entire range of /3 but is unable to fit to the rate constants across the alcohol series at constant temperature. The failure of the latter fit is hypothesized to be due to variation of the barrier height as the solvent permittivity changes across the alcohol series at constant temperature. W e have reported similar observations for I K H fits to D P B i s o m e r i z a t i o n rate constants [15]. Fig. 2 s h o w s the temperature d e p e n d e n c e o f the barrier height implied for stilbene in n - a l c o h o l s ob-
40
3.5 40
• 3.0
3.0
--" 2.5
"~
25
i
20
~2
E
1.5 1.5 1.0 I0 0.5 05~
240
~
i
i
260
280
300
~ 320
i 340
360
T (K) O0
[
i
I
10
14
16
20
Permitivitty Fig. 1. The dependence of the stilbene barrier height on solvent permittivity determined from isodielectric Kramers-Hubbard fits in n-alcohols.
Fig. 2. The barrier heights implied by the substitution of the dependence of the barrier height on solvent permittivity into the temperature dependence of the solvent permittivity of the six n-alcohols involved in this study. (O) Ethanol, (N) n-propanol, (zx) n-butanol, ( v ) n-pentanol, ('~) n-hexanol, (O) n-heptanol, (O) n-octanol.
R.M. Anderton, J.F. Kauffman / Chemical Physics Letters 237 (1995) 145-151
tained by the substitution of the dependence of the barrier height on solvent permittivity into the temperature dependence of the solvent permittivity for each alcohol. Since IKH fits involve data taken from a horizontal slice in Fig. 2, this implies a single barrier height for the entire fit. In contrast, fits of isomerization data across a temperature range within a single alcohol would correspond to data taken along one of the diagonal lines in Fig. 2 while fits of isomerization data taken across the solvent series at a single temperature correspond to a vertical line in Fig. 2. Similarly, isoviscosity Arrhenius plots are made up of the isomerization rate constant data in solvent-temperature pairs that cut diagonally across Fig. 2 from upper left to lower right. In addition to the trend in E a with solvent permittivity, which was our primary interest, a trend of decreasing toa with increasing permittivity was observed. This trend is consistent with the trend observed in our previous study of DPB [15]. Qualitatively, a decrease in toa with decreasing E a is physically reasonable since a decrease in E a would be expected to result in a wider initial potential well. to b remains relatively constant across the permittivity range examined in this study, as observed in the DPB study. While the physical significance of to b is ambiguous, the nearly constant value of this fitting parameter is consistent with its role as a scaling parameter, as discussed in Section 2.
4. Discussion In order to put the results of this study in context it is necessary to compare this method with other approaches that have been applied to this problem, which have primarily been isoviscosity plots in various solvent systems. Based on an isoviscosity plot at 2.55 cp in n-alcohols Sundstrom and co-workers [22,23] concluded that stilbene isomerization proceeds on a potential energy surface lacking a barrier. Kim and co-workers [24] report that isoviscosity plots of stilbene over a broader range of viscosities in n-alcohols yield decreasing values for the barrier height with increasing viscosity. This phenomenon has been attributed to solvation dynamics but it has been noted that other indications of solvation dynamics such as time-dependent spectral shifts and non-
149
exponential decays are not present. The slopes obtained from these isoviscosity plots at different viscosities are the result of two competing deviations from the assumptions inherent in making the plots. The first is the change in the solvent permittivity and its effect on E a and toa. Isoviscosity plots utilize data taken in cold short chain alcohols and hot long chain alcohols, the combination that yields the greatest change in solvent permittivity over a single plot. This indicates E a and o)a will change over the span of the plot, contrary to the assumptions implicit in making such a plot. The competing effect is the large deviations from Stokes-Einstein-Debye behavior similar to that noted for stilbene in alkanes [25] and DPB in alcohols [21] in which the boundary condition decreases as the alcohol molecule size increases to approach the size of the solute. The IKH method obviates this concern by utilizing the rotational correlation times as a microscopic measure of solventsolute friction. Qualitatively it is clear that these effects may be expected to lead to inconsistent values in the slope when progressing from isoviscosity plots at one value of the viscosity to another, though it is difficult to predict the observed trends because of the interplay between these competing effects. Isoviscosity plots of DPB [10] and stilbene [26] in alkanes have been shown to yield constant slopes at all viscosities. Park and Waldeck [27] have noted a slight trend toward increasing barrier at higher viscosity for stilbene in alkanes when the chain length range of the alkane series was changed to achieve changes in the viscosity, but it was noted that in plots where the solvent series did not change there was no apparent trend. This observation lends support to the premise that the leading causes of the failure of isovisocity plots in n-alcohols are changing boundary conditions and solvent permittivity which are not strong effects in alkanes. The rotational correlation time in alkanes is expected to scale linearly with bulk viscosity across the alkane series, since the alkane chain length is on the order of or larger than the solute [21]. Furthermore, there is only a slight dependence of the solvent permittivity on temperature or chain length for alkanes. Isoviscosity plots of stilbene in n-alkyl nitriles have been demonstrated to yield consistent slopes at several different values of viscosity [28] which would seem to indicate that over a wide permittivity range
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the barrier height is constant. One explanation for the observation that isoviscosity plots yield a consistent slope in the presence of both of these competing effects is that the nitriles exhibit a weaker temperature dependence of the solvent permittivity since hydrogen bonding is not extensive [29]. Isoviscosity Arrhenius plots of stilbene isomerization conducted in a single alcohol by applying pressure to the alcohol to maintain constant viscosity over a temperature range [30] would be expected to suffer from similar effects of changing solvent permittivity on the potential energy surface. Influences resulting from a changing boundary condition may be diminished in this case since there is no change in the relative size between solvent and solute, although applying pressure would decrease the free space between solvent molecules. As a final point we note that the barrier height determined by the IKH analysis of stilbene at the lowest solvent permittivity is similar to the barrier height which has been in a cold jet [5]. Further insight into the applicability of the IKH method of analysis to the study of unimolecular isomerization reactions may be gained by examination of other diphenylpolyenes whose barrier heights have been measured under isolated molecule conditions.
5. Conclusions The IKH method presented here does not depend on the assumption of any functional form for the dependence of either the barrier height or other parameters of the potential energy surface on permittivity. The only assumption inherent in the method is that any changes in the potential energy surface with changing temperature or alcohol chain length are a result of the changing solvent permittivity. The resuits presented in this Letter indicate the importance of the solvent permittivity in determining the height of the barrier to isomerization of stilbene in n-alcohols. The barrier height increases with increasing temperature within a single alcohol solvent and increases with increasing alcohol chain length at constant temperature. Analyses such as isoviscosity Arrhenius plots or fits to Kramers expression in a single solvent across a temperature range, which assume a constant barrier height, are inappropriate if
the solvent permittivity undergoes a large change across a range of solvents or temperatures. The IKH analysis also indicates that the magnitude of the barrier to stilbene photoisomerization in n-alcohols is considerably larger than the aforementioned studies [22,24] have indicated.
Acknowledgement Acknowledgement is made to the Donors of The Petroleum Research Fund, administered by the American Chemical Society, for the support of this research.
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