12117
J. Phys. Chem. 1994, 98, 12117-12124
Temperature-Dependent Rotational Relaxation of Diphenylbutadiene in n-Alcohols: A Test of the Quasihydrodynamic Free Space Model Robert M. Anderton and John F. Kauffman* Department of Chemistry, University of Missouri-Columbia, Columbia, Missouri 65211 Received: June 11, 1994; In Final Form: September 8, 1994@
The rotational correlation times of trans,trans- 1,4-diphenylbutadiene in n-alcohol solvents over a wide temperature range are reported. They are well represented as linear functions of q/T within a single alcohol, but comparison across a series of alcohols indicates that the boundary condition factor in the modified StokesEinstein-Debye equation decreases as the solvent size increases. The experimental results are compared with the Dote-Kivelson-Schwartz (DKS) quasihydrodynamic free space model, which is based on regular solution theory. The model in its original form is able to qualitatively predict the trend toward smaller boundary condition factor with increasing solvent size but is unable to quantitatively predict the observed dependence of the rotational correlation time either on q at fixed temperature across the alcohol series or as a function of temperature within a single alcohol. Substitution of an empirical measure of the solute free space into the DKS theory results in improved agreement between the theoretical and experimental results. This suggests that the general framework of the DKS is appropriate for both regular and associating liquids but that different measures of free volume are required for each solvent type.
I. Introduction Rotational reorientation rates are an important potential source of information regarding the nature of solvent-solute interactions in condensed phases. In the simple case of a spherical solute with stick boundary condition, the solute rotational correlation time ZSED can be related to the solvent-solute friction via the Stokes-Einstein-Debye (SED)
where q is the solvent shear viscosity, V is the volume of the solute molecule, and k and T are the Boltzmann constant and temperature, respectively. The stick boundary condition implies laminar flow, in the sense that the solvent in direct contact with the solute is assumed to stick to the solute and rotate along with it. Clearly such an assumption implies strong attractive interactions between the rotating body and the solvent. Equation 1 is appropriate when the solute is very large compared with the solvent, in which case the solvent acts as a continuous fluid, as would be the case for large, Brownian particles. In this case the solvent-solute interaction is akin to wetting of the surface of a macroscopic body by the solvent. When the solute is an individual molecule, on the other hand, eq 1 is inappropriate due to the molecular nature of the solventsolute interaction^.',^-^ In this case several important assumptions of the SED formulation must be challenged: (1) on a microscopic scale molecules cannot in general be approximated as spheres, ( 2 ) the stick boundary condition may not be appropriate on the molecular scale, and (3) in developing theories for rotational relaxation rates the interstitial spaces which exist in the solvent on the molecular scale must be considered. These three phenomena are strongly interrelated in their effect on solvent-solute friction? For example, whereas a perfect sphere has a hydrodynamic radius of zero under the slipping boundary condition, bumpy spheroids with atomic scale corrugation will experience finite rotational friction. Smooth
* To whom correspondenceshould be addressed. @
Abstract published in Advance ACS Abstracts, October 15, 1994. 0022-3654/94/2098-12117$04.50/0
ellipsoidal molecules are also expected to experience hydrodynamic friction due to geometric effects, even in the absence of attractive solvent-solute interaction^.^^ That is, even if the laminar flow analogy is relaxed, collisional interactions between the nonspherical solute and the solvent will result in a resistance to solute rotation. (We note that such effects result from repulsive solvent-solute interaction and that they are governed by geometric considerations.) Similarly the boundary condition is expected to be strongly influenced by the noncontinuous nature of the solvent on a molecular length scale, so that items 2 and 3 above are intimately related. The SED model can be adjusted to account for these influences by including a factor associated with the molecular shape, denoted byfstick,and a boundary condition factor, denoted as C. For a prolate spheroid' such as diphenylbutadiene(DPB), the subject of the present study, &tick is defined in terms of A, the ratio of the longitudinal solute dimension to the axial solute dimension, according to eq 2, 2(A2 f 1)(A2- l p fstick
=
3A{(2A2 - 1) ln[A
+ (A2 - 1)0,5]- A(A2 - p}
This expression is valid for the stick boundary condition. Hu and Z ~ a n z i gsuggest ~ ~ that on the molecular length scale the slip boundary condition may be most appropriate. They tabulate numerical values of fslidfstick for prolate and oblate spheroids for various values of A, which allows one to adjust the prediction of the SED equation for the case in which repulsive solventsolute interactions are dominant. Thus, researchers are able to predict rotational reorientation rates for both the slip and stick boundary conditions by use of eqs 1 and 2, along with the tabulated values forfsbdfstick. Though this does not allow one to scale solvent-solute interactions between these two limits, it does provide a model against which experimental information can be assessed in order to determine the closeness of the system to each of these limiting conditions. (We note also that the value of &,lip is required in the evaluation of C, as discussed in section IV.)
0 1994 American Chemical Society
Anderton and Kauffman
12118 J. Phys. Chem., Vol. 98, No. 47, 1994 The effect of free space on rotational relaxation has taken on great importance recently because of the accumulation of experimental data demonstrating rotational reorientation rates which are faster than predicted by the SED formulation with slip boundary condition.8-'0 This has drawn the attention of several researchers toward theories of rotational relaxation which attempt to correlate the boundary condition factor with the relative sizes of the solvent and solute molecule^.^^^ The expression which predicts the rotational correlation time of a nonspherical solute in a solvent-solute system with boundary condition dependent on the relative sizes of the solvent and solute is given by eq 3,
(3) The term zo has also been added to account for the finite rotational reorientation rate when the viscosity is nearly zero. There is still considerable controversy over the physical significance of this term."J2 Notice that when C = 1, eq 3 is identical to the SED formulation after it has been scaled to account for geometric effects. Similarly, when C = fslidfstick, the predicted value of zr gives the rotational correlation time with slip boundary condition. Thus, the C term allows one to scale between these boundary conditions as well as beyond them, raising the possibility that observed subslip rotational correlation times can be interpreted from a molecular basis. The most widely examined models for C are the Gierer-Wirtz (GW) model6and the Dote-Kivelson-Schwartz (DKS) modeL5 The GW model extends the laminar flow analogy in the stick boundary condition for spherical solute molecules by considering the solvent to be composed of concentric shells surrounding the solute, with each shell following the solute rotation more slowly as the distance from the solute molecule increases. While this model accounts for the molecular nature of the solvent, it does not explicitly account for free space within the solvent. In addition, it is only strictly appropriate for spherical solutes. The DKS model, on the other hand, assumes that free space will play a dominant role in determining the boundary condition when the solvent and solute dimensions are similar and represents an attempt to scale the boundary condition by both the free volume available to the solute and the relative sizes of the solvent and solute molecules. The DKS model correctly predicts the dependence of the rotational correlation time of small to intermediate size solutes in alkane solvents, as well as the dependence of solute rotational correlation times on solvent size in alkanes.' The model qualitatively accounts for the observed apparent change in boundary condition between alkane and alcohol solvents, but it is unable to predict these quantitatively. The DKS model is also unable to predict the dependence of the rotational correlation time on the solute size for large s o l ~ t e s .Thus, ~ while this model has not been able to quantitatively predict all experimental observations, it has been successful in many respects. In this paper we present the results of an experimental investigation of rotational relaxation of the solute diphenylbutadiene (DPB) in a series of n-alcohol solvents. Measurements have been made in each solvent over the -10 to 70 OC temperature range. There are several factors which have motivated us to undertake this investigation. Our initial motivation for this study came from our need to determine corrected activation energies for DPB in n-alcohols in order to examine chemical reactivity in mixed fluid solvents. To our surprise, we have found that very little data exist in the literature regarding the rate of rotational relaxation of diphenylbutadiene in n-alcohol solvents. This represents a significant void in the literature since DPB and stilbene photoisomerization reactions
have been widely studied as prototypical examples of reactivity in solution. Comparison of DPB rotational relaxation rates in alcohols with stilbene rotational relaxation rates cannot currently be made, since the information reported for DPB in alcohols is the result of only a single measurement in a single n-alcohol solvent at a single temperat~re.'~This dearth of information has provided us with additional motivation for the present study. Finally, in reviewing the literature on rotational relaxation in liquids, we have found that the majority of the studies examining the efficacy of the DKS and other models have focused on solute size dependence studies and on studies in which the rotational correlation time of a solute is examined in a series of like solvents in order to examine the influence of solvent viscosity on the rotational r e l a x a t i ~ n . ~These , ~ ~ studies pose an inherent problem in that the boundary condition factor in the modified SED equation is generally expected to depend on the ratio of the solvent and solute molecular volumes, so that the boundary condition is continually varying as the solvent-solute system changes. It has occurred to us that the temperature dependence must also be examined in order to determine the conditions under which the DKS model is appropriate. When only the temperature is varied, the ratio of the solvent and solute molecular volumes is unchanged, and it appears at first glance that the temperature dependence of the rotational relaxation rate provides a better test of free space theories, since it will obviate concerns regarding effects which are peculiar to specific solvent-solute systems. Thus, temperature-dependent rotational relaxation rates are an important, but underutilized, tool for the investigation of models of rotational relaxation. The rest of this paper is outlined as follows. Section I1 describes the experimental apparatus and procedures used to make the rotational correlation time measurements. We present the results of the measurements in section 111. In section IV we describe the methods used to calculate the temperature-dependent boundary condition factors and other parameters needed to compare the measurements with the DKS theory. We also discuss the results of our analysis and their implications for the examination of temperature-dependent rotational relaxation rates and compare our results with previous examinations of the DKS theory. We summarize our findings in section V.
11. Experimental Section The rotational correlation times of DPB were determined from steady state fluorescence anisotropies in a series of n-alcohols including the C1 through cg alcohols as well as n-decanol. For polarized excitation the anisotropy, r, is defined as15,16
(4) where Zpl is the emission intensity collected parallel to the excitation, I@ is the emission intensity collected perpendicular to the excitation, and G is an instrumental factor which accounts for the dependence of the optical throughput on the polarization. Steady state anisotropies were measured on a SLM-Aminco 8100 spectrofluorometer with 1 cm quartz cuvettes in a sample holder whose temperature was controlled by a recirculating temperature bath. The instrument G factor was measured before each measurement, and each reported anisotropy is the average of 10 such measurements. A typical standard deviation for these measurements was 0.002 unit. The anisotropies were in the range 0.1180-0.2091. If both the anisotropy r(t) and the fluorescence lifetime are single exponentials, then the rotational correlation time, z, of the fluorescent solute is given by the Perrin e q u a t i ~ n ~ ~ J ~
J. Phys. Chem., Vol. 98, No. 47, 1994 12119
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Figure 1. Arrhenius plot for DPB isomerization in ethanol.
z, = zd(rdr - 1) (5) where ro is the limiting anisotropy and zfis the fluorescence lifetime of the solute. Both of these parameters must also be measured in order to calculate the rotational correlation time. The value of ro was calculated from the anisotropy of DPB measured in glycerin at -20 "C under the assumption that z, is infinite under these conditions and was found to be 0.39 ic 0.01. Fluorescence lifetimes were measured by the time correlated single photon counting method. Details of the experimental apparatus are described in a companion paper. Steady state anisotropies and fluorescence decay times were measured in each alcohol at 10 K temperature increments between 263 and 343 K. Heptanol, obtained from Eastman Kodak, was refluxed over CaH2 and fractionally distilled then kept over type 4A molecular sieves. Optima grade methanol was obtained from Fisher. The other n-alcohols were obtained from Aldrich (spectrophotometric grade) and used as received. Diphenylbutadiene was obtained from Aldrich and used without further purification. 111. Results Diphenylbutadieneis known to undergo a trans-to-cis isomerization in the excited electronic state, which is the dominant nonradiative relaxation pathway in low-viscosity solvents.l8 The large nonradiative rate results in a short fluorescence lifetime, so that very short rotational correlation times can be determined from steady state anisotropies. The rate of the isomerization reaction is dependent on viscosity as well as temperature. Thus, in the analysis of the steady state anisotropies it is necessary to use values for the fluorescence lifetime determined at the same temperature as the anisotropy measurement. Figure 1 shows an Arrhenius plot for the rate of the isomerization reaction calculated from the measured fluorescence lifetimes of DPB in ethanol as a function of temperature. The natural log of the rate is linear in T', and the linear correlation coefficient is R2 = 0.999. This plot is typical of the results obtained for each of the alcohols examined in this study. Analysis of the isomerization rates is the subject of a companion paper and will not be discussed further here. The interested reader is referred to the companion paper for further details of the lifetime measurements and isomerization kinetics. Table 1 lists the values for zf,r, and tr of DPB at 20 "C for each solvent, along with the solvent viscosities. The 20 O C
0
5
10
15
Viscosity (cp) Figure 2. Plot of the rotational correlation time versus viscosity at 20 "C for the n-alcohol series methanol through decanol.
TABLE 1: Lifetimes, Anisotropies, Rotational Correlation Times, and Viscosities at 20 "C for DPB in n-Alcohols methanol ethanol propanol butanol pentanol hexanol heptanol octanol decanol
35.2 70.6 107.9 147.8 204.6 252.1 297.4 366.5 457.8
0.201 0.1667 0.1711 0.1672 0.1579 0.1577 0.1634 0.1576 0.1705
36.5 50.4 83.8 110.0 119.1 151.9 205.4 238.7 342.7
0.554 1.144 2.185 2.978 4.315 5.022 6.827 8.494 14.298
TABLE 2: Linear Regression Slope of rr vs q for Several Solutes in Alkanes and Alcohols with Zero Intercept at Room Temperature solute solvent slope (ps/cP) ratio ref stilbene n-alkanes 38 0.34 8 n-alcohols 13 8 DPB n-alkanes 66 0.41 13 n-alcohols this work 27 9,lO-diphenylanthracene n-alkanes 0.64 19 78 n-alcohols 50 19 181 0.80 26 POPOP n-alkanes n-alcohols 145 26 430 1 14 BTBP n-alkanes 430 14 n-alcohols rotational correlation times have been plotted versus viscosity in Figure 2 for the n-alcohol series. The line through the data is the result of a linear least-squares analysis. Values for the slope and y intercept of the plot are 23 ps/cP and 34 ps, respectively. Table 2 presents the parameters of a line forced to a rotational reorientation time of zero at vanishing viscosity, for the purpose of comparison with previously published re~u1ts.l~ Importantly, the slope determined from our measurements is consistent with slopes determined for solutes of similar size in n-alcohols, indicating that the rotational correlation times determined from steady state anisotropies of DPB in n-alcohols give results which are similar to those determined from both time-correlated photon counting measurements1 ~ 1 9and absorption anisotropy experiments.l3 Temperature-dependentrotational correlation times measured in each of the nine n-alcohols are plotted versus v / T in Figure 3. Also shown in Figure 3 are the anticipated results for both stick and slip boundary conditions. The data for each solvent
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Figure 3. Temperature dependence of rotational correlation times for DPB in (a) (0)methanol, (0)ethanol, (V) n-propanol, (V)n-butanol, (0) n-pentanol and (b) (0) n-pentanol, (W) n-hexanol, (A) n-heptanol, (A)n-octanol, and (0) n-decanol. TABLE 3: Parameters for the Lines Describing the Dependence of the Rotational Correlation Time of DPB on t]lT in Each of the n-Alcohols alcohol C” slope ( io3ps WcP) intercept (ps) R2 16.9 f 0.9 3.6 f 2 0.986 methanol 0.409 f 0.02 0.982 12.3 f 0.6 5.0 f 3 ethanol 0.300 f 0.01 10.1 f 0.1 5.8 f 1 0.999 propanol 0.245 f 0.002 0.993 9.5 f 0.3 8.3 f 4 butanol 0.230 f 0.006 6.7 f 0.1 19.7 f 0.1 1.00 pentanol 0.162 f 0.002 0.998 7.9 f 0.1 15.5 f 3 hexanol 0.191 f 0.002 6.9 f 0.2 23.1 8 0.994 heptanol 0.167 f 0.004 1.55 f 3 0.999 7.8 f 0.1 octanol 0.189 f 0.002 22.0 f 4 0.999 6.5 f 0.1 decanol 0.157 f 0.002 The C parameter is the measured value of the boundary condition factor, calculation from the tabulated slopes and eq 3. have been fitted to a linear function describing the dependence of zr on v/T, and the slopes, y intercepts, and linear correlation coefficients are given in Table 3. Viscosities of the alcohols at each temperature are calculated from an expansion of the viscosity in temperature given by In 9 = A
+ B/T + CT + D?
(6)
where A, B, C, and D are empirical coefficients?O It is interesting to note that the observed data are all well represented by linear relationships between z, and v/T but that the slopes of the lines are observed to decrease with increasing solvent size. According to eq 3 , this suggests that the boundary condition factor, C, is decreasing with increasing solvent size. This is consistent with the notion that the boundary condition should become more sticklike as the solute size increases relative to the solvent. Values of C calculated from the slopes and eq 3 have also been tabulated in Table 3. The value of fstick for DPB using eq 2 is 2.739. Note that the value Offshdfstick given by Hu and Zwanzig for a prolate, ellipsoidal molecule with an axial ratio of 3.4 is 0.508, while the value of C determined from the slopes varies from 0.409 in methanol to 0.157 in decanol, assuming that the molecular volume of DPB is 208 A3,the van der Waals v01ume.l~ Thus, the observed behavior is subslip even in the smallest of the solvents. In addition, we observe that the y intercept of the line describing the dependence of z, on qlT increases with increasing solvent size.
IV. Analysis and Discussion Before comparing the observed change in the slopes in Figure 3 to the boundary condition predicted by the quasi-hydrodynamic model, we will present a brief description of the DKS model for the calculation of the C parameter. In the DKS model? the boundary condition is assumed to depend on the ratio of the solute volume to the total volume available for solute rotation. Thus
(7) where V, is some measure of the solute (probe) molecular volume, 6 is a factor describing the ratio of the “effective solute volume” to V,, and the term yV, is the free space surrounding the solute molecule. This expression can be written
c = (1 + y/f#))-l
(8)
and calculation of C now requires a determination of q5 and y. Dote, Kivelson, and Schwartz suggest that if the solvent-solute interactions are “primarily geometric”, it may be appropriate to use
4J=hip
(9)
since fsfip is a function of the solute shape only. We now consider the calculation of y. Dote, Kivelson, and SchwartzZ1 suggest the use of eq 10,
where Vs is a measure of the volume of the solvent molecule and AV is the “smallest volume of free space per solvent molecule”. It is the calculation of the latter term which requires the most attention. In the DKS model this free volume parameter is interpreted in terms of the FrenkeI hole theory of liquids,21 resulting in the expression
AV = BK,vkT
(11)
where B is the Hildebrand-Batschinski parameter of the
J. Phys. Chem., Vol. 98, No. 47, 1994 12121
Rotational Relaxation of Diphenylbutadiene in n-Alcohols
TABLE 4: Slope and y Intercept Describing the Linear Dependence of the Isothermal Compressibility on Temperature for Each of the Alcohols Examined in This Study" alcohol methanol ethanol n-propanol n-butanol n-pentanol (I
A1
MPa) 7.997 7.836 7.584 7.261 6.887
MPa) -1.080 77 -1.174 58 -1.215 08 -1.211 26 -1.172 11
A0
The values given are parameters of the equation KT = A0
alcohol n-hexanol n-heptanol n-octanol n-decanol
A1 (
MPa) 6.482 6.067 5.663 4.968
MPa)
A0
-1.106 63 -1.023 79 -0.9325 9 -0.7610 7
+ AlT, where T is the temperature in kelvin.
TABLE 5: Smallest Volume of Free Space per Solvent Molecule, AV, Calculated Using Eq 11 for Each of the n-Alcohols at Nine Temperatures (A3) T, "C
-10 0 10
20 30 40 50 60 70
methano1 73.49 68.37 64.78 62.26 60.49 59.19 58.19 57.35 56.34
ethanol 124.55 112.53 102.26 93.39 85.67 78.91 72.93 67.61 62.84
propanol 235.61 200.84 173.35 151.30 133.36 118.58 106.24 95.81 86.91
butanol 294.79 245.27 206.80 176.52 152.38 132.88 116.96 103.82 92.32
and KT is the solvent isothermal compressibility. The Frenkel hole theory and the Hildebrand treatment of solvent viscosity were developed for regular solutions. Thus, eq 11 may not be a valid measure of the free space per solvent molecule in alcohols. However, this formulation of the DKS model has been used in the analysis of a variety of solvent types, including alcohols, nitriles, and alkanes.'J4 We have therefore made an extensive comparison of the DKS model as presented above with the results of our experimental measurements in order to explore the limits of applicability of this model. In calculating AV, we assume that B is independent of temperature, and 11 can be calculated at each temperature by use of eq 6. We must now determine the temperature dependence of the isothermal compressibility in order to calculate C. KT has been shown to exhibit a linear dependence on temperature for methanol and ethanol and for the C5 through C9 n-alcohols as well as dodecan01.*~~*~ Garg et aLZ5have also demonstrated that KT for the Cs through Clz alcohols exhibits a smooth dependence on carbon number at several temperatures within the range of interest in the present study. We make use of these experimental observations in order to develop empirical correlations which we will use to determine the temperature dependence of the isothermal compressibility. First we fit the known values of KT for the C1, CZ, C5 through C9, and C12 n-alcohols at 50 and 100 "C to cubic equations in the carbon number, following Garg? The plots of the known values are shown in Figure 4,along with the function to which the data are fit. We use the fitted functions to calculate the values for KT at 50 and 100 "C for each alcohol, and the resulting values are used to determine the linear function which describes the temperature dependence of the isothermal compressibility for each solvent. Table 4 lists the parameters of these linear functions for each alcohol. Values calculated from the linear correlations are consistent with measured values found in the literat~re.~~ Having determined 11 and KT versus temperature for each of the alcohols, we now present values of A V calculated for each alcohol at several temperatures in Table 5. The most remarkable result of these calculations is that the free volume per solvent molecule in the DKS formulation decreases as temperature increases. We note that the isothermal compressibility of the solvents always increase with increasing temperature, and therefore the increase in A V reflects the decrease in viscosity with increasing temperature. This can be interpreted as a kinetic
pentanol 433.60 345.91 279.28 228.72 189.82 159.47 135.51 116.36 100.89
hexanol 470.30 370.82 297.02 241.45 198.98 166.08 140.25 119.73 103.24
2.0
0.6
heptanol 650.73 494.67 383.07 301.80 241.58 196.23 161.55 134.67 113.55
octanol 772.47 584.49 447.83 349.43 277.27 223.41 182.58 151.15 126.64
decanol 1375.0 969.14 706.05 521.95 393.98 303.07 237.21 188.62 152.17
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Figure 4. Plot of the isothermal compressibility, K r , versus carbon number for the n-alcohols at (0)50 and (0)100 "C. influence on the free volume, in the sense that as temperature increases, each solvent molecule is able to occupy more space due to its thermal motion. The results shown in Table 5 imply that this effect is stronger than the influence of thermal expansion on the free volume. We note that while it has been suggested that the parameters of the C factor are only mildly dependent on temperature,' we find that the calculated values of C based on the DKS model can vary by a factor of 2 or more over the temperature range examined in this study as discussed below. The values given in Table 5 along with suitable measures of the solvent and solute volumes can be used to predict the rotational correlation time for DPB in the n-alcohol solvents. We have performed these calculations using both the van der Waals volumes and room temperature molar volumes of the solvent and solute and find similar trends with both measures of molecular volume. The ratio of C values calculated using each measure of solute volume is essentially constant with varying temperature for a given alcohol, varying by less than 5 % across the temperature range examined in this study. The calculated rotational correlation times using the molar volumes
12122 J. Phys. Chem., Vol. 98, No. 47, 1994 250
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Figure 5. Plot of rotational correlation times predicted from the DKS free space model using molar volumes as discussed in the text versus q/T in n-alcohols: (a) (0)methanol, (0)ethanol, (V) n-propanol, (V)n-butanol, (0)n-pentanol; (b) (0)n-pentanol, (m)n-hexanol, (A) n-heptanol, (A) n-octanol, (0)n-decanol. ~~
are closer to the experimental values at low temperature, while the values predicted with the van der Waals volumes are closer to the experimental values at high temperature. Representative plots of the temperature dependence of the rotational correlation time of DPB predicted by the DKS theory in several n-alcohols are shown in Figure 5 . The model is not able to quantitatively predict the temperature dependence of observed rotational correlation times, though both experiment and theory exhibit similar trends. Note that the calculated temperature dependence exhibits a marked downward curvature in cases where the calculation is carried out over a broad range of q/T values, such as in the case of decanol. The measured values do not appear to have the same degree of curvature, though a slight curvature is apparent in some of the longer chain alcohol data in Figure 3. The observed rotational correlation times also exhibit a steeper dependence on temperature than predicted by the DKS theory. We note a general trend for both experimental and theoretical results, that as the data for a given alcohol span a greater range of viscosities, curvature becomes more evident, suggesting that experimental studies over a broader range of viscosities may tend to bring out the curvature in the data. However, in the current study there appears to be no statistical justification for favoring a second-order fit to the experimental data over a first-order fit. In contrast, a second-order fit to the rotational correlation times calculated from the DKS model provides a much better representation than a first-order fit. It is interesting to note that the curvature observed in the rotational correlation times predicted by the DKS model arises as a result of a boundary condition factor which increases with increasing temperature. This implies that in each alcohol a more sticklike condition is approached as the temperature is increased. Both the experimental and predicted values of the rotational correlation times can be plotted versus viscosity at fixed temperature across the series of alcohols. Figure 2 presents the experimental rotational correlation times measured at 20 "C in this way, and we have generated similar plots for the - 10 and 50 "C data in Figure 6. The best fit parameters for the linear dependence of the experimental values on the viscosity at each temperature are given in Table 6. Also included in Figure 6 are plots of the values for the rotational correlation time predicted by the DKS model for similar temperatures. Note
TABLE 6: Parameters for the Best Fit Lines of the Experimental Rotational Correlation Times versus Viscosity at Constant Temperature across the Series of n-Alcohols temp, "C slope (103 ps/cP) intercept (ps) R2 -10 25.029 62.171 0.984 0 26.250 36.610 0.993 10 22.535 42.661 0.990 20 22.681 34.168 0.984 0.982 30 2 1.085 28.340 40 21.485 20.593 0.992 50 22.685 15.633 0.987 60 19.121 17.794 0.953 0.882 IO 18.452 14.242 again that the low-temperature experimental data which spans the broadest range of viscosities exhibits curvature, though less than that predicted by the DKS model. Thus, from both the temperature-dependent data in each alcohol and the viscositydependent data at fixed temperature, it is clear that a broad range of q/T values are required to observe the curvature in the experimental data which is predicted by the DKS model. At a given fixed temperature, the boundary condition term C in the DKS model decreases with increasing chain length, owing to the strong dependence of hV on solvent viscosity. The inverse dependence of the free volume surrounding a solute molecule on solvent volume tends to offset this effect to some extent, but the effect of the viscosity on C is dominant in the DKS model. This explains the predicted curvature at fixed temperature across the alcohol series as arising from the decrease in C with increasing chain length. Attempts to fit the DKS predictions of the rotational correlation times at fixed temperature across the alcohol series result in best fit lines whose correlation coefficients are poor at low temperature, but quite acceptable at higher temperatures, as would be expected from the above discussion. However, the linear slopes thus predicted are observed to increase with increasing temperature, in contrast to the experimental results. The results of the above comparison indicate that while trends in the experimental results are often consistent with trends predicted by the DKS model, quantitative agreement between experiment and theory is poor. We have suggested that this may result from the failure of eq 11 to adequately represent the
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n / T ( l o - ' cp K - l ) n / T (lo-' cp K - l ) Figure 6. Plot of experimentalrotational correlation times vs viscosity at (0)- 10 and (0)50 "C. The solid lines represent the rotational correlation times predicted by the DKS free space model at these temperatures.
0.0
0.2
0.4
0.6
0.8
1 .a
n/T ( l o - * c p K - ' )
Figure 7. Plots of experimental rotational correlation times vs y/T where the solid line is the DKS prediction using eq 11 for A V and the dashed line is the DKS prediction using eq 12 for AV for (a) ethanol and (b) heptanol. solute free space in alcohols. We now suggest that AV may be calculated empirically using the expression AV = V, - Vvdw
(12)
where V, is the solvent molar volume divided by the Avagadro constant and Vvdw is the van der Waals volume of the solvent molecule. This measure of AV may be more appropriate for polar or associating fluids. Figure 7 presents a comparison of our experimental results with the predictions of the DKS model using eq 12 for AV for ethanol and heptanol. The molar volume as a function of temperature for each alcohol was estimated from the molar volume at 293 "C using the modified Rackett technique.'' Importantly, this approach to the determination of AV leads to values which increase upon increasing temperature. Table 7 presents the parameters of linear regression analyses of the predictions of the DKS model for each alcohol over the range of temperatures examined experimentally using eq 12 to
TABLE 7: Parameters for the Linear Regression Lines of the Rotational Correlation Times Predicted from the DKS Model Using Eq 12 To Calculate AV vs qlT for Each IZ -Alcohol alcohol slope (lo3ps WcP) intercept (ps) R2 methanol 18.5 -3.7 1.oo ethanol 14.6 -4.5 0.999 propanol 12.5 -6.3 0.999 butanol 10.8 -6.8 0.999 pentanol 9.4 -6.2 0.999 hexanol 8.6 -7.9 0.999 heptanol 7.8 -9.0 0.999 octanol 7.1 -7.9 0.999 decanol 6.0 -7.5 0.999 calculate AV. This approach correctly predicts the linear dependence of the rotational correlation time with temperature within a given alcohol. Though the slopes of the lines of RCT vs v/T given by the DKS model in Table 7 are slightly higher
12124 J. Phys. Chem., Vol. 98, No. 47, 1994 than the slopes of the lines through the experimental rotational correlation times given in Table 3, use of this empirical measure of AV results in very good agreement between experiment and theory. We note that eq 11 predicts that within a given alcohol the free space per solvent molecule in the fluid will decrease as the temperature is increased, resulting in an increase in the boundary condition factor, while eq 12 predicts the opposite trends, with the free space per solvent molecule increasing and the boundary condition decreasing with increasing temperature.
V. Summary In this paper we have presented a study of DPB rotational relaxation as a function of temperature in (9) n-alcohol solvents. The experimental rotational correlation times are well represented as linear functions of q/T within a single alcohol. Comparison of the alcohols indicates that the boundary condition decreases as the solvent size increases. The DKS model is unable to quantitatively predict the observed dependence of the data on q/T when the Frenkel-Hildebrand formulation is used as a measure of the free space per solvent molecule. One explanation for the failure of the DKS theory (as it was originally proposed) to quantitatively model behavior in alcohols arises from the fact that both the Frenkel hole theory of liquids and the Hildebrand treatment of solvent viscosity were derived for regular solutions, which explicitly excludes the alcohols.21,22 The apparent ability of the DKS model to quantitatively predict rotational relaxation in alkanes supports this e ~ p l a n a t i o n . The ~J~ predictions of the DKS model are greatly improved, however, when the solvent molar volume is used as a measure of the free volume per solvent molecule. Agreement between experiment and theory in this case suggests that the general framework of the DKS model (Le., eqs 7-10) is appropriate for both regular and associating solvents but that different measures of free volume are required for each solvent type.
Anderton and Kauffman Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. References and Notes (1) Roy, M.; Doraiswamy, S. J. Chem. Phys. 1993, 98, 3213. (2) Stokes, G. Trans. Cambridge Philos. Soc. 1956, 9, 5. (3) Einstein, A. Ann. Phys. 1906, 19, 371. (4) Debye, P. In Polar Molecules; Dover: New York, 1929. (5) Dote, J. L.; Kivelson, D.; Schwartz, R. N. J. Phys. Chem. 1981, 85, 2169. (6) Gierer, A,; Wirtz, K. Z. Naturforsch. A 1953, 8, 532. (7) (a) Hu, C. M.; Zwanzig, R. J. Chem. Phys. 1974, 60, 4354. (b) Zwanzig, R. J. Chem. Phys. 1978, 68,4325. (c) Zwanzig, R.; Harrison, A. K. J. Chem. Phys. 1985, 83, 5861. (8) Courtney, S. H.; Kim, S. K.; Canonica, S.; Fleming, G. R. Chem. Soc., Faraday Trans. 2 1986, 82, 2065. (9) Canonica, S.; Schmid, A. A,; Wild, U.P. Chem. Phys. Lett. 1985, 122, 529. (10) Barkky, M. D.; Kowalczyk, A. A.; Brand, L. J. Chem. Phys. 1981, 75, 3581. ( 1 1 ) Alicki, R.; Alicka, M.; Kubicki, A. J. Phys. Chem. 1993,97,3668. (12) Evans, G. T.; Kivelson, D. J. Chem. Phys. 1986, 84, 385. (13) Waldeck, D. H.; Lotshaw, W. T.; McDonald, D. B.; Fleming, G. R. Chem. Phys. Lett. 1982, 88, 297. (14) Ben-Amotz, D.; Drake, I. M. J. Chem. Phys. 1988, 89, 1019. (15) Lakowicz, J. R. In Principles of Fluorescence Spectroscopy; Plenum Press: New York, 1983. (16) O’Connor, D. V.; Phillips, D. In Time-correlated Single Photon Counting; Academic Press: New York, 1983. (17) Perrin, F. J. Phys. Radium 1934, 5, 497. (18) Velsko, S. P.; Fleming, G. R. J. Chem. Phys. 1982, 76, 3553. (19) Ben-Amotz, D.; Scott, T. W. J. Chem. Phys. 1987, 87, 3739. (20) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. In The Properties of Gases and Liquids; McGraw-Hill: New York, 1987. (21) Frenkel, J. In Kinetic Theory of Liquids; Dover: New York, 1955. (22) Hildebrand, J. H. Science 1971, 174, 490. (23) Freyer, E. B.; Hubbard, J. C.; Andrews, D. H. J. Am. Chem. SOC. 1929, 51, 759. (24) Moriyoshi, T.; Inubushi, H. J. Chem. Thermodyn. 1977, 9, 587. (25) Gag, S. K.; Banipal, T. S.; Ahluwalia, J. C. J. Chem. Eng. Data 1993, 38, 227. (26) Lakowicz, J. R.; Maliwal, B. P. Biophys. Chem. 1985, 21, 61.
