Rotational diffusion in aprotic and protic solvents

Rotational diffusion in aprotic and protic solvents

Che&& Physics 30 (1978) 1-8 O.N_orth-HollandPirblishing Company ROTATIONAL DIFFiJSION IN APROTIC AND PROTIC SOLVENTS Kenneth G. SPEARS and Laurenc...

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Che&& Physics 30 (1978) 1-8 O.N_orth-HollandPirblishing Company




Kenneth G. SPEARS and Laurence E. CRAMER Department of Chemistry, _NorthwestemUniversity, Evanston, Illinois 60201, USA Received 28 September


We present the orientational relaxation times in protic and aprotic solvents for rose bengal in its lowest excited singlet state. The method uses a mode locked dye laser for polarized excitation, and thne correlated single photon counting for determination of the time resolved polarized fluorescence_ The observed orientational decay for the dipolar aprotic solvents and the alcohols are in agreement with the values predicted by the Stokes-Einstein diffusion equation. In the latter solvents, volume and shape corrections must be made for attachment of the alcohol to the two anion sites of the dye molecule. The solvent N-methylforrnamfde, however, shows rose bengal reorienting much faster than the alcohols. Our interpretation of this data suggests that agreement with the Stokes-Einstein equation (stickboundary conditions) is coincidental. We propose a solvent torque model in which the solvent interaction at each anion site of rose bengal controls the deviations from an expected slimboundary conditionThis qualitative model is used to correlateour data as well as relevant data in the literature. The values in picoseconds for the observed orientational relaxation times are given in parenthesis; acetone (70), DMF (160), DMSO (420), MeOH (IgO), EtOH (450), isopropanol(840), NMF (500).

I. Introduction The direct observation of molecular orientational decay, as a function of the solvent, can elucidate how solvent-solute forces affect overall molecular motions. In this work we emphasize orientation behavior of a solute larger than the solvent species. In other work, Chuang and Eisenthal [l] directly measured the orientation decay times of rhodamine 6G in a series of straight chain alcohols_ They established that this molecule is isotropic with regard to rotational diffusion and that it follows Stokes-Einstein diffusion without significant effects due to hydrogen bonding. Fleming et al. [2] measured the orientation decay times of rose bengal in three alcohols and qualitatively showed that solvent coordination must be included to achieve approximate agreement with Stokes-Einstein diffusion. Recent work of Mantulin and Weber [3] , compared a series of symmetrical molecules in propylene glycol over a range of temperatures. For those molecules not accepting solvent hydrogen bonds, they found axis specific rotational diffusion rates that were reasonably represented by diffusion with slip boundary conditions [4,5]. Mantulin and Weber also found that isotropic rotational diffusion, in propylene glycol, was described

by the stick boundary conditions of Stokes-Einsiein diffusion only when several hydrogen bonding groups were presen; in the molecule. The goal of this work is to use a variety of solvents with a single solute to refme our understanding of the transition from rotational diffusion in the slip limit to diffusion that is strongly affected by solvent interactions such as hydrogen bonding. In earlier work [6] we studied the solvent dependent fluorescence lifetime of rose bengal, and showed it to be an excellent ROSE BENGAL



C’ / fi

Cl_ Cl Fig. 1. The rose bengal d&anion.


K.6. Spears, L.E. CzarnerfRotational difusn-onin aprotic and protic soknts


tive probe of the hydrogen bond strength at the O- site on the xanthene chromophore. The structure of the rose bengal anion is shown in fig_ 1. As is illustrated by fig. 1, the negatively charged carboxylic acid group provides a second site for hydrogen bonding to the solvent.

2. Fluorescence depolarization method The rotational diffusion times are determined for rose bengal, in the first excited state, from the time dependence ofpolarized fluorescence. The theory of this technique has been described by several groups of workers [7-10,2] .We consider the case of a symmetric top molecule, having three diffusion coefficients 4, = D_”#to,, with an absorption dipole along the D_r or D,, axis. If the absorption and emission dipoles of the molecule are in the same direction, we obtain the following expressions for polarization resolved emission: I,, 0) = 1; f &exp [-(2D, l

+ 4DJtI

& exp(-6DXt)} exp(-krr),

f,_(t) = {& -$


exp [-(7D_Y + 4D,)t]

- $, exp(-6DXt)) exp(--liff).


In these equations kf is the fluorescence decay rate (k = l/r), and exponential fluorescence decay has been emphasized to agree with the exponential nature of the decays observed experimentally. Using the axes dimensions and the appropriate formulas, the diffusion coefficients D, and D, can be calculated for oblate or prolate ellipsoids in terms of the diffusion constant, D, of a sphere. For the case of a classical sphere D, =D,, = D, = D, and the expressions for the intensity contain two exponentials instead of three. We also note that for the case of slip boundary conditions we would have Dz infinite. In this case the resultant decay would appear as two exponentials with relative intensities that are very different from the case of a sphere with stick boundary conditions. This suggests that careful examination of the experimentally determined ratio of preexponential coefficients is important for confirming the type of boundary conditions applicable to a given solute-solvent system.

For spherical particles, and stick boundary conditions, the observed orientational relaxation time, 7ur, is given by hydrodynamic theory (Stokes-Einstein rotational diffusion) as i-or = l/60 = qV/kT.


In this equation n is the shear viscosity of the fluid, k is Boltzmann’s constant, and V is the hydrodynamic volume of the particle. For nonspherical particles, such as eibpsoids, we substitute the particle volume into eq. (3).

3. Experimental technique Experimentally, the sample is excited with a vertically polarized laser, and the fluorescence is detected, at a 90” angle, through a polarizer that is either parallei or perpendicular to the polarization of the incoming laser. Either the parallel or perpendicular fluorescence emission can be analyzed directly for multi-exponential decay. The difference between these decay components can be analyzed for one less exponential than is contained in the separate intensities. Orienting the analyzing polarizer at 54.7” from the vertical, eliiinates the effects of orientational relaxation and provides an independent value for kf, the molecular fluorescence decay rate. The methods used for obtaining fluorescence decay times have been completely described in our previous work [6]. We use the time-correlated single photon counting method, and can deconvolute decay parameters from the observed fluorescence response with excellent precision and accuracy for times as short as 50

PsOptical arrangements and procedures for sample preparation in this work were identical to those described in our previous studies of rose bengal [6]. A thermocouple was used to monitor the temperature of the cell in order to make use of available data relating viscosity and temperature. The observed range of temperature was 24-26OC. Orientation decay parameters may be obtained from the available data by any one, or ail of the three methods described below. Results from analysis by all three methods, provide a consistency cheek. The first method uses the direct deconvolution of

I,,(r) and/orI((r)

to determine two exponential decay

K.G. Specu~. LE. Cnzmer/RotationaI


in aprotic andprotic



Table 1 Experimental and calf-mated parameters for orientation decay Solvent a)

rf b,






TA Acetone


70 f 10





160 = 25





420 t 40





190 * 30





450 + 55




840 + 70

500 * 75




Calc. f)

Viscosity 9)


(CP) 0.309

0.96 0.84























670 ----






a) Solvents listed are: acetone, dimethylformamide (DMF), dimethylsulfoxide propanol (i-PROP), and N-methylformamide (NMF).

(DMSO), methanol (MeOH), ethanol (EtOH), iso-

b, Observed fluorescence

lifetime, measured independently by means of fluorescence collection through a 54.7O polarizer. The of values obtained in the orientation decay experiment agree with these independent measurements. ‘1 Values reported for acetone and DMF are calculated from I,#) - f,Cr), all other values were calculated from direct two exponential deconvolution. Error bounds given far +- one standard deviation. d, The ratio uses values of the preexponential coefficients obtained from deconvolution of fluorescence response data. The values represent the ratio of relative intensities. The theoretical value is 0.8. See text for details. e) ror calculated from Stokes-Einstein diffusion equation for an ablate ellipsoid with no solvent attachment. See text for details. 0 mr calculated from Stokes-Einstein __ for details.

diffusion equation for prolate ellipsoid due to solvent attachment

to rose bengal. See text

g) Values for n. at our temperatures, were obtained from ref. [15]

rates kf and kQ = k, + k,,, as well as their relative intensity. The data precision is only sufficient to calculate a two exponential



is adequate

for the

case of rose bengal. The second method uses a deconvolution of the difference I,,(t) -IL(t)_ The observed decay rate in the resultant difference is 4. This technique is especially useful when the orientational decay rate is very fast or similar to kF The sum of intensities I,,(f) f XL(t) can be used in a similar fashion to obtain the fluorescence decay rate kp The relative intensities of the individual exponential components can be calculated from the ratio of the pre-exponential coefftcients obtained from the above calculations. The continuous pulse train of the dye laser provides a stable average intensity that allows proper matching of It,(t) and I,(r) by merely collecting the data for identical times. Making use of the fact that rose bengal exhibited nearly isotropic rotational relaxation, approximated by a single k,, = 6D, we can also use the assumed theoretical relationship to derive an expression relating rf and ror to the observed integrated intensities I,, and I,. I,,/‘1 = (5rt + 9ror)/(5rt + 3ror).

In this equation the independentIy measured rf is used to compute a 70r_Many factors, such as impurities or dimerization, can severely change integrated intensities so that use of this traditional method is best left as a

check on a direct time resolved measurement.

4. Results

The results are reported in table 1, which lists the fluorescence lifetimes;orientation decay times and the observed pre-exponential ratio (defied in the next

paragraph). The accuracy of our orientation decay lifetimes is + 10%. This accuracy is not adequate to resolve two orientation decays different by only 15% or 20%. Rose bengal rotates nearly isotropically and the difference between 2DX f4Dr and 60x, as determined from the

ellipsoid parameters, is approximately 15% for most cases. Consequently, in the expressions for three exponential decay [eqs. (1) and (2)], the exponent& containing orientation decay rates are nearly equal, so that the observed decay would appear as a two expo-


K.G. Spears, L.E. Cmner/Rotationaldifussian

nential decay with a ratio of coefficients of 0.8. The two nearly equal exponential decays actually appear as an average rate that can be approximated by 30, t 30,. The accuracy of this approximation was checked by computer simulation. The work of Fleming et al. [2] reported orientation lifetimes, for rose bengal, of 180,680 and 890 ps for the solvents methanol, ethanol and isopropanol, respectively.

With the exception

of ethanol, these values

are in good agreement with our results. The results for acetone and diiethylformamide were calculated from 1,,(r) -IL(t) and both solvents showed a dominant short decay in competition with a long decay of = 2-2.5 ns. The long decay has a preexponential of approximately S-IO% of the short decay. The intensity of the long decay component is too small to be associated with a second orientation decay and it may represent the motion of either rose bengal attached to impurity molecules or rose bengal dimers. A small amount of long decay only becomes obvious in competition with a reorientation time that is very short. In fig. 2 we have plotted orientation decay time versus solvent viscosity. The aprotic solvents follow a good straight line of slope 207 ps/cp and zero intercept (within our error bounds). The alcohols follow a monotonic trend that is nearly a straight line. The value for N-methylformamide falls between the aprotic solvents and the alcohols_

in aprotic and protic solvents

5. Discussion and interpretation 5.1. Introduction We will show that orientation decay times in dipolar aprotic solvents as well as the alcohol solvents can be calculated using stick boundary conditions. In the latter case, however, we assume the presence of a solvent cage surrounding

the two negative ions in order to calculate

the volume and ellipsoid parameters. In contrast, the solvent coordination model fails to predict the orientation decay of NMF, even though NMF binds to rose bengal as strongly as the alcohols [6]. This failure of the Stokes-Einstein model for NMF supports a proposal that the other agreements are coincidental. A solvent torque mode1 is suggested that correlates the dipolar aprotic solvents, the alcohol solvents and NMF. 5.2. Aprotic solvents ‘The results of a quantitative calculation of orientation decay parameters are shown in table 1 as rA. The T* values for all solvents have been computed with eq. 3 (simpIe stick boundary conditions). We have used the following procedure to account for the shape of rose bengal. Molecular models were used to estimate the dominant shape as an elIipsoid having 4 a and I4 A axes. These parameters were used in calculating the axial diffusion constants from the spherical diffusion constant D. The hydrodynamic volume in eq. (3) was adjusted for the additional protrusions, caused by the C,CldO, group, pependicular to the ellipsoid shape. We measured 123 A3 to be added to the 410 A3 ellipsoid volume. Such a procedure can be considered only an approximate treatment of the problem, but it is justified for the purposes of this interpretation. The orientation decay of rose bengal in aprotic solvents agrees well with the orientation decay calculated by using the Stokes-Einstein equation (table 1). 5.3. Protic solvents

Fig. 2. A plot of observed mean orientation relaxation time in picoseconds, versus so!vent bulk viscosity expressed in centipoise.

The quantitative calculation of orientation decay for rose bengal having solvent coordination layers attached to the two negative ion sites is also shown in table 1 as TB_We used space filling models to simulate hydrogen bonding of four solvent molecules about each anion and

K.G. Spenrs. L.E. Cramer/Rotational &fusion in aprjtic and protic solvents

then measured the dimensions of the resulting structure. For ail of the protic solvents studied, “solvation”, of the rose bengal resulted in a prolate ellipsoid with two nearly equivalent short axes. To simplify the calculation of shapes and volume, we conserved the volume of this ellipsoid while equating the short axes. The axes of this prolate ellipsoid obviously do not ahgn with the original axes of a rose bengal oblate ellipsoid and the effects of such changes might be seen in higher precision data. Two possible effects include changes in the pre-exponential ratio as well as increasing the difference between 60, and 20, t 40,. Our calculation uses the ellipsoid volume to obtain D from the Stokes-Einstein equation (3), and the new shape factors to calculate D, and D,. The orientation decay calculated for the alcohol solvents agrees reasonably well with the observed decay (table 1). A wider range of hydrogen bond strengths would have to be studied in order to see differences between different alcohols. However, the result for NMF suggests that an additional factormight be present for cases of solvent coordination. The orientation decay of N-methylformamide is much faster than expected from the model of a hydrogen bonded solvent cage. Since NMF hydrogen bonds to rose bengal with strength comparable to isopropanol [6] , the faster orientation decay is probably because the coordinated NMF layer cannot interact as strongly with the solvent as the corresponding coordinated alcohols. This idea is developed in the next section. 5!4. Solvent torque model of orientation decay The preceding sections have described our results and the agreement of our results with the Stokes-Einstein model for rotational diffusion (stick boundary conditions). The primary question is whether the agreement with the Stokes:Einstein model is a coincidence or an expected result for orientation decay. The Stokes-Einstein equation with stick boundary conditions is derived by assuming a uniform attachment of solvent to the solute molecule. Behavior in agreement with this condition can also be found when “rough” surfaces are present under solvent-solute slip conditions [I l] _In addition to Stokes-Einstein rotational diffusion

with stick boundary


there is ro-

tation assuming perfect solute slip boundary conditions [4,5]. Under slip boundary conditions, the Stokes-


Einstein expression, eq. (3), is multiplied by one sixth of a rotational friction coefficient [S] . The rotationa friction cjefficient is defined in terms of an average torque/angular velocity, and it characterizes the drag, or hinderance to rotation, that a non-spherical object is subjected to when fluid must be displaced to achieve rotation. The rotational fraction coefficient of a sphere under slip conditions is zero. Values [5] of the friction coefficient range from 0 to 3.39 for various ellipsoid geometries and axes. Previous data for large uncharged molecules such as benzoic acid diners [ 121 in Ccl4 and anthracene and perylene in propylene glycol [3] have shown that uncharged moIecules, of comparable size to rose bengal, behave reasonably close to rotation with slip boundary conditions [4,5]. The experimental results obtained for rose bengal, in the dianion form, show that the observed rotational behavior approtimately follows the behavior predicted by the StokesEinstein equation_ The next paragraph demonstrates why the agreement with Stokes-Einstein rotational diffusion could be interpreted as coincidental. The two localized charges of rose bengal cannot yield solvent attachment over the entire molecular volume. In addition,


forces with dipolar

aprotic solvents are sufficiently weak that an extensive ion-solvent coordination is unlikely. Nevertheless, if we assumed that “surface roughening” via ion-solvent coordination in dipolar aprotic solvents was the reason for agreement with stick boundary conditions, we then must explain why the alcohol solvents have a larger rotational decay time. One might argue for even more efficient solvent coordination in the alcohols, but with this assumption the result for NMF is quite unexpected when previous work [6] has shown comparable hydrogen bonding of isopropanol and NMF. We believe the data for rose bengal is not well explained by the simple concepts of complete surface coordination or “surface roughness” generated by solvent coordination. Consequently we would like to propose an additional model for the interpretation of our data. The apparent success of slip boundary conditions for uncharged molecules suggests that we might try to understand the results for rose bengal dianion as a deviation from normal decay in the slip limit. We propose a solvent torque model to qualitatively explain our results. The initial premise of the solvent torque model is that uncharged molecules, molecules without other special solvent attractions and molecules without sub-


K.G. Spears, L.E. Cramer/Rotational difunion in oprotic and protic solvents

stantial surface roughness, all exhibit orientation decay in the limit of slip boundary conditions. For molecules with special interaction sites, in which the sites affect only a minor fraction of the total surface volume, we will consider the solvent interaction at the site as a perturbation on the normal friction coefficients established by slip boundary conditions. The special solvent interaction sites of molecules will be considered as providing a solvent torque that retards the average orientation decay. In order to make qualitative comparisons of various molecules, we will use the interaction energy of the special solute site, with the solvent, as an indirect measure of the average solvent torque generated by a particular type of molecular site. While this model is only qualitative, it is consistent with the concept of strong solute-solvent interaction sites providing momentum transfer that slows the average orientation decay times. Such a model might be treated in terms of molecuhu dynamics, but this treatment is beyond the scope of our paper. In our qualitative model, dipolar aprotic solvents can interact with the two negative ions to generate a significant solvent torque. The results for the alcohol solvents show twice the orientational decay of aprotic solvents, and the increased decay (although not the factor of two) is totally expected within our model. Our previous study [6] has established the much stronger interaction of protic solvents with the O-site of rose bengai, and this stronger interaction energy accounts for the reduced orientation decay. The microscopic description of the ion-alcohol interaction is somewhat ambiguous. If there is no long term (with respect to partial rotation) average coordination of the alcohol, we have solvent interactions characterized by very strong attractive forces. On the other hand, a long term coordination about the ions would increase the average size of rose bengal while reducing the solvent torque by the shielding effect of the coordination layer. The data for NMF can be interpreted as supporting the long term coordination model for hydrogen bonding solvents. As mentioned previously, our data showed that NMF can hydrogen bond with strength similar to isopropanol. We expect that without long lived solvent coordination, the alcohols and NMF would have similar orientation decay times. However, if solvent coordination is present, the coordinated NMF molecule will provide a much poorer species for additional solvent attraction than coordinated alcohols, and therefore it will exhibit

a faster decay time. This expectation is based largely on the greater steric hinderance to solvent interaction with the N atom and the resultant reduction in solvent interaction energy. It is possible that a weaker N atom polarizability, compared with 0 atoms, contributes to this effect. In our model, the success of the Stokes-Einstein calculation for both aprotic and alcohol solvents shows that the forces between ions and aprotic solvents are comparable to the forces between the ion-alcohol coordination layer and the alcohol solvent. There is no necessity to assume a rigid coordinated layer in our model, all that is needed is some average coordination that provides a new effective volume and average solvent torque. The average residence time required for a coordinated molecule is open to further defiiition by dynamical models. In summary, we have qualitatively interrelated our results for aprotic solvents, alcoholic protic solvents and NMF. We also can make a few qualitative predictions for positive ions in these systems based on energetic guidelines. For dipolar aprotic solvents such as DMF, DMSO and acetone a positive charge is more ideally suited for close approach to the solvents’ polarizable negative center. This predicts that for equal sterically accessible molecules, positive ions will generate a slightly larger torque than negative ions in dipolar aprotics. For the case of alcoholic solvents a positively charged molecule open to solvent coordination might have an orientation decay similar to dipo&u aprotie solvents. This expectation is based on the fact that positive ions are less polarizable than O- ions and, if solvent coordination does occur, the binding of a coordinated solvent molecule to the solvent is reduced by steric hindrance and blocking of the alcohol proton. In order to support features of the solvent torque model we will briefly examine some aspects of solvation energetics. Data from gas phase ion solvation are available as well as the results for simulation of such data with simple electrostatic and quantum calculations. One of us has published a simple electrostatic model [13,14] that is in quantitative agreement with the data for the interaction of water and closed shell positive and negative ions. One result of these calculations is that the energy needed to change from one solvation arrangement to another can be very small even when the absolute energy of binding for a single solvent molecule is quite large. This means that the “lifetime” of

K. G. Spears. L.E. CramerfRotational

a given solvent molecule coordinated about an-ion is controlled by the barrier for simultaneous insertion of a new molecule and exclusion of the original molecule. The energetics for this process depends on the strength of binding in the first coordination layer. Another result of such models is that binding a molecule to an ion incranes its hydrogen bond strength to the second luyer ofsolvent mofecules. This energy term is useful for understanding the energetics of solvent coordination and the differences between NMF and the alcohol solvents under conditions of coordination. An approximate quantitative comparison of ionsoIvent interaction energies with solvent-solvent interaction energies is possible for most solvents. However, a more quantitative and microscopically based model is necessary to relate dynamical interaction energies to average torques. It is reasonable to assume that both the average solute-solvent interaction energy and the average solvent-solvent interaction energy will be the important parameters in a quantitative model. We feel the quahtative solvent torque model will be usefu1 in correlating data for various molecules and planning additional experiments. Hopefully these ideas can be made more quantitative in the future. 5.5. Comparison with other molecules The orientation decay of large molecules has been little studied as a function of different solvent characteristics. The work of Chuang and Eisenthal [ 1] on rhodamine 6G represents the most comp1ete study of orientation decay for protic solvents, unfortunately, they did not compare the protic solvents with aprotic solvents. The found a good linear correlation of orientation decay with viscosity for a large number of primary alcohols as well as formamide. Such a correlation suggests that no solvent coordination occurs about the positively charged nitrogen atom of rhodamine 6G. We have used models of rhodamine 6G to calculate the volume in a manner similar to rose bengal. We find an oblate ellipsoid of 430 A3 volume with a single protrusion of 126 A3 volume.These calculated volumes yield orientation decay values that agree quite well with the experimental results of Chuang and Eisenthal. The solvent torque model can be used to understand the rhodamine 6G agreement with the Stokes-Einstein mode1 of diffusion. Models show that the single, positively charged nitrogen is quite sterically hindered by its

difiission in aprotic and protic solvents


attached ethyl group, proton and xanthene framework. It is unlikely that more than two alcohol molecules could be coordinated to this site. The strength of binding is probably weaker than for O- to alcohols because the Nf ion is sterically hindered and less polarizable. modamine 6G, in comparison with rose bengal, has one half of the charged sites, the positive charge can interact with fewer solvent molecules and the interaction energy is probably weaker. In the solvent torque model, these effects will cause a much faster orientation decay time than exists for rose bengal. The agreement with the Stokes-Einstein equation implies that the net torque for rhodamine 6G in protic solvents is similar to rose bengal in aprotic solvents. We note that formamide shows no unusualbehavior for rhodamine 6G because strong solvent coordination is not present for rhodamine 6G. The work of Mantulin and Weber [3] is qualitatively . in agreement with our concept of solvent torque as the controlling factor in orientation decay. They used only propylene glycol as a solvent, which presents a hydrogen bonding solvent molecule that has two active positions. This type of solvent structure adds another factor to the solvent coordination discussed in our previous sections. They find that uncharged molecules with zero or one hydrogen bonding site exhibit anisotropic rotation with a decay time that is much faster than given by a StokesEinstein calculation. This result supports our assumption that molecules without strong interaction sites behave in the limit of pure slip boundary conditions. Additional quantitative work in more solvents is necessary to further study this limit of rotational diffusion. Mantulin and Weber also show that negatively charged sulfonate or carboxylic acid groups decrease the rotational diffusion times as well as change the behavior to an isotropic difsusion. They do not attempt to distinguish the differences between charged and uncharged interaction sites. The possible unusual properties of propylene glycol prevent us from further correlating their data with our solvent torque model. There is some data available for carboxylic acids in aqueous solutions [12] . These workers use light scattering to obtain the orientation decay times for formic and acetic acid and their anions. A puzzling feature of these data is the larger orientation decay time for the neutral acid than the corresponding anion However, a model for the coordinated anion with slip boundary conditions does agree with the experiment. It is worth



Spearf, L.E. CmnerjRotational

n&g that these molecules are dominated by the anion site, u&e our molecules in which the anion site is a small paa of the total molecular volume. This aspect of the geotietry and the use of water as a solvent restricts comparisons with our data.

Acknowledgement We wish to thank the National Science Foundation for their support of this research (CHE-7305256). K.G.S. would like to thank the Alfred P. Sloan Foundation for a fellowship. L.E.C. would like to thank the DuPont Corporation for a graduate fellowship.

References [ 11 TJ. Chuang and K.B. Eisenthal, Chem. Phys. Letters 11 (1971) 368.


in aprotic aizd prot(c solvents

(21 G.R. Fleming,J.M. Morris, G.W. Robinson,ehem. Phis. 17 (1976) 91. [31 W-W. Mantutin anrl G. Weber, I. Chem. Phys. 66 (1977) 4092. [41 C.M. Hu and R. Zwanzig, I. Chem. Phys..60 (1974) 4354. [5] G.K. Youngren and A. Acrivos, J. Chem. Phys. 63 (i975) 3846. .._ [6] L.E.Cramerand K.G. Spears, 3. Am.Chem. Sot. 100 (1978) 221. [7] T. Tao, Biopolymers 8 (1969) 609. [8] T.J. Chuang and K.B. Eisenthzl, J. Chem. Phys. 57 (1972) 5094. [9] J. Yguerabide, Meth. Enzymology 26 (1972) 498. [lo] G. Weber, I. Chem. Phys. 55 (1971) 2399. [ll] S. Richardson, J. Fluid Me& 59 (1973) 707. [12] D.R. Bauer, J.I. Brauman and R. Pecora, J. Ani. Chem. Sot. 96 (1974) 6840. [I31 K.G. Spears and S.H. Kim. J. Phys. Chem. 80 (1976) 673. [14] KG. Spears, J. Phys.Chem. 81 (1977) 186. [ 151 J.A. Riddick and W.B. Bunger, Organic solvents, physical properties and methods of puritication, 3rd Ed., Techniques of chemistry, vol. 2, ed. A. Weissberger (Wiley-Interscience. New York, 1970).