J. Phys. Chem. 1982, 86, 4700-4704
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are many possible combinations of 70 and 73 which are consistent with the experimental results. At early times, the shape of the R(t)curve is determined mainly by the ground-state rotation times, whereas at longer times the effects of the excited-state rotation time becomes more important. The signal-to-noise ratio is greatest at early times, so the acceptable values of TG extend over a relatively narrow range. The variations in 76 are essentially the same as the uncertainties in TROT listed in Table I. A t longer times the noise increases, allowing a wider range in the values of 7E which would be consistent with the observed results. We found that 73 could be as much as 30% greater or less than 7G and still be compatible with our experimental results. The rotational reorientation results presented here are essentially measures of the ground-state rotation time of rhodamine B. Hence, possible differences in ground- a nd excited-state rotation times do not affect the discussion presented above. However, the excited-state rotation times
could vary by as much as 30% from the ground-state values. Since fluorescence depolarization experiments examine only excited-state rotation, a careful study of a particular solvent-solute system using that technique and the transient grating (or transient absorption4)technique would provide a measure of the differences in excited-state and ground-state rotation times. Acknowledgment. We thank Professor A. Acrivos, Chemical Engineering Department, and Professors Hans C. Andersen, R. Pecora, and J. I. Brauman, Chemistry Department, Stanford University, for interesting conversations pretaining to this work. We also acknowledge the National Science Foundation, DMR 79-20380 and PCM 79-26677, and the Department of Energy, DE-FGOB8OCS84006, for supporting this work. S.G.B. acknowledges the Mred P. Sloan Foundation and the Camille and Henry Dreyfus Foundation. M.D.E. thanks the National Science Foundation for a Predoctoral Fellowship.
Correlation Function Theory for Kerr-Effect Relaxation of Axially Symmetric Polar Molecules Robert H. Cole hpattment of Chemistry, Brown University, Providence, Rhode Island 02912 (Received: May 27, 1982; In Final Form: August 2, 1982)
Nonlinear response theory is used to obtain correlation function expressions for time-dependent electric birefringence following removal, application, and reversal of an external field polarizing a system of axially symmetric dipolar molecules. The role of collisions is essential in determining the second-order effects of permanent dipole torques on the field on and field reversal which randomize momenta in times short compared to those for significant reorientations. As for the special case of diffusional reorientation, the field-off response is determined by P2(cosO(t))correlationsonly, otherwise both these correlationsassociated with optical anisotropy and P,(cos O(t)) correlations from dipole torques contribute to the effects. The results thus provide a basis for experimental evaluation of both functions from Kerr-effect measurements.
Introduction Electric birefringence has proved to be a useful probe of changing molecular orientations produced by time-dependent applied fields, notably for studies of polymers in solution’ and for liquids and solutions of smaller molecules at low tempeatures.2J As the effect depends on anisotropy of polarizability, the function studied is an average of second-rank spherical harmonics such as P2(cos e ( t ) )= 1/2(3cos2e(t)- l),where e(t) is the time-dependent angle between a molecular axis and the applied electric field. Because of this, Kerr-effect measurements are a valuable complement to studies of dielectric relaxation, in which the average is of permanent dipole orientation functions with symmetry P,(cos e(t) = cos O(t). For example, one can distinguish between diffusion-like and “large-jump” processes, as Williams and co-workers have shown.2 A second kind of usefulness of the Kerr effect is in distin(1) For reviews, see articles in: “Molecular Electro-Optics, ElectroOptic Properties of Macromolecules and Colloids in Solution”; Krause, S., Ed.;Plenum Press: New York, 1981. See also artices in: “Molecular Electro-Optics”;OKonoki, C. T., Ed.; Dekker: New York, 1976 Vol. 1, Parts 1 and 2. (2) Beevers, M. S.; Crossley, J.; Garrington, D. C.; Williams, G. J. Chem. SOC.,Faraday Symp. 1977,11, 98. (3) Beevers, M.S.; Elliott, D. A.; Williams, G. J. Chem. Soc., Faraday Trans. 2 1980, 76, 112. 0022-3654/82/2086-4700$01.25/0
guishing between permanent dipole orientations and induced ionic charge displacements as polarization processes in biopolymers, as in studies by O’Konski and co-workers.2 A continuing difficulty in theoretical treatment of Kerr-effect relaxation, particularly for molecules with permanent dipole moments, arises from its nonlinearity with respect to applied field E(t). The effect is necessarily an even function of E ( t ) ,and even the limiting E 2 ( t )behavior most easily observed presents problems. Most of the progress made has been for rotational diffusion models, and the simplest results of this kind from the classic work of Benoit4p5suffice to illustrate both the kinds of usefulness of Kerr-effect studies and the need for more general treatments. Benoit considered axially symmetric molecules with permanent dipole moments 1 along the axis of symmetry and an anisotropy ACY= aP - a, of polarizabilities CY^ parallel and CY, perpendicular to this axis. Neglecting intermolecular correlation effects, the theoretical quantity derivable from Kerr experiments is the average, ( AaP2(cos O(t))fE(t)), where fE(t)is the time-dependent orientational distribution function for field (E(t). Benoit obtained f E ( t ) by perturbation solution of the classical equation for forced (4) Benoit, H. Ann. Phys. (Paris) 1951, 6, 561. (5) Benoit, H.J. Chim. Phys. Phys.-Chim. Bid. 1952, 49,517.
0 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 86, No. 24, 1982 4701
Kerr-Effect Relaxation
diffusion or other specific dissipative processes.
Theory
0.51dlLJ!L FIELD OFF
FIELD ON
'
I
0 I
3
6
We assume the same molecular symmetry as Benoit did in deriving eq 1 and 2. The appropriate classical Hamiltonian H ( t )for a system of such molecules 1 in an applied field E ( t ) is accordingly H ( t ) = Ho - pE(t)Ccos Bi - hE2(t)C1/2(3cos2 Oi - 1)
9
0
3
6
TIME ,6Dt Figure 1. Normalized diffusion response following application and removal of polarizing field. The dashed curve is the field on response in the absence of dipole torques.
rotational diffusion from permanent dipole and anisotropic polarizability torques proportional to E(t) and F(t).With the resulting expressions for fE(t) to order F as a function of time t after removal of a previously constant field E and after application of E at time t = 0, he obtained the results5 in eq 1 and 2. In these equations, D is the rotational WP,(COS
w t )) = (1/5)PAa(Aa
+ f/3P~2)E2 exp(-6Dt)
E(t)= E
t 0
E(t)= E
i
(3) where Ho is the Hamiltonian for no applied field in terms of sphereical coordinates and conjugate momenta and the sums are over active molecules, assumed identical. The Liouville operator L for evaluation of the time-dependent distribution function f&) from Liouville's equation dfE/dt = - LfE is given by
a
+
a
= Lo pE(t)Csin Oi - AaE2(t)C3 cos O2 sin Bi i aPoi i aPoi
(4) (1)
- exp(-6Dt)] + = (1/5)pAa(Aa + (3/4)(2/15)PAaPp2E2[exp(-6Dt) - exp(-2Dt)] (2)
E(t)= 0
i
0
t
diffusion coefficient and = (kg5'')-l comes from the equilibrium Boltzmann distribution function with the usual meaning of kB and T. For the field-off case, eq 1 predicts a simple exponential decay with time constant T~ = (6D)-' characteristic of P2(cose) relaxation by diffusion. For the field-on case, eq 2 has the counterpart of this relaxation, and for molecules with permanent dipole moments it has the second term with the difference of exponentials with T~ and T~ = (2D)-l for Pl(cos e) relaxation by diffusion. This predicted asymmetry, sketched in Figure 1, provides a basis, if the diffusion model is valid, for establishing the presence or absence of permanent dipole moments, as 1 / 3 ~ pis2 larger than ACYfor even weakly polar molecules. However, a number of cases have been found, especially at low temperatures, of relaxation without detectable asymmetry for molecules which are known, from dielectric constant measurements, for example, to have appreciable dipole moments. Also, substantial deviations from simple exponential time dependence have been observed; these can result from a superposition of as many as five relaxation modes for molecules of sufficiently low symmetry6 but can also indicate shortcomings of the simple diffusion model. These considerations point to the need for more general derivations of at least formal expressions of the response behavior without introducing specific mechanisms at the outset. The work reported here is a step in this direction which leads to a generalization of Benoit's results by an extension of response theory methods to deal with the nonlinear problem in the F limit and obtain formal P1and P2 correlation function expression^'^^ without invoking (6) Wegener, W. A.; Dowben, R. M.; Koester, V. J. J . Chem. Phys. 1979, 70, 622. ( 7 ) Williams, G . Chem. SOC.Rev. 1978, 7 , 89.
where Lois the Liouville operator for Hoand the psiare the conjugate momenta for spherical polar angles Bi. Using standard perturbation expansion methods, the equations satisfied by the linear and quadratic (in E) parts fl(t) and f2(t) o f f ~ ( tare )
afz
- + fof2 = at
afo
a
-AaE2(t)C3 COS Bi sin Bi - - ME(t)Csin Bi -fl(t) 1 aPoi i aPoi (6) Here fo is the equilibrium distribution A exp(-PHo), and an explicit expression for the normalization factor A will not be required. We are interested only in the solution for f2(t)for the Kerr-effect problem, but fl(t) is needed to obtain it. Using afo/ap, = -@foeiand the notation sin eiSi = -Pli for the time derivative of cos Bi, the standard linear response theory result for f l ( t )is
fl(t) = PP S ' d h E(tJ exp[-(t - ~I)LoICPI~~O = P r S t d t l E(tJCPli[-(t - tdlfo i
(7)
with the notation Pli(t) = exp(tL)Plifor the time evolution of Plifrom its value at t = 0, described by Kubog as the natural motion in the absence of applied fields. We note that Pli(t)is a function of all relevant initial coordinates and moments, whereas Pli depends only on Bi. The solution for f2(t)from eq 6 on using eq 7 for f l ( t ) gives an expression for the desired ensemble average.
(AaCPzifi(t)) i = P(Aa)2(CP2iJfdt1 E2(t1)CP2i[-(t i -
(8)Cole, R. H. In 'Physics of Dielectric Solids"; Conf. Series No. 58, Institute of Physics: London, 1980. (9) Kubo, R.J.Phys. .Soc.Jp. 1957, 6 , 570.
The Journal of Physical Chemistry, Vol. 86, No. 24, 1982
4702
We develop here the results for the two cases of relaxation after a previously constant field E is removed at t = 0 and response after a constant field E is applied at t = 0. From these, the result for the experimentally useful effect of field reversal at t = 0 follows immediately. Field-OffSolution. Strictly, the applied field considered here is E ( t ) = E exp(Xt), t < 0, E(t) = 0, t > 0, in the limit % 0 to ensure equilibrium at t = 0 and vanishing of the time integrals in eq 8 for the lower limits t,, t2 = --co. With the proviso, the time integrations present no difficulty and one obtains first
-
( AaCPzifJt)) = P(Aa)zE2(CP,iCPzi(t)fo) + I
1
Cole
persistence of initial angular momentum ps one has P,(t) = cos % ( t )= cos 0 cos pt - sin % (pe/p)sin pt, where p is the total angular momentum, and d cos %(t)/dp,diverges as t at long times. A clue to resolving the problem without invoking specific models of collision processes comes from observing that the last term is multiplied by a factor 0,rather than a factor P2 expected for second-order dipole effects and present in the Benoit and field-off solutions. As an extra factor P can come only from fo, one obtains this by using the product rule on the ps differentiation after shifting the Liouville time displacement operator PZi.For a term ijk in the last average of eq 13, denoted by I I
P2A~pZE2( ?P,iS_odt, exp[-(t - t1)LolCp~if'lif~) (9) We note here that evaluation of the t zintegral gives Pli(t = 0) = Pliand that, since aPli/ap8i= 0, one has only to use af0/ap8,= -PfoOi which gives the second term in eq 9. This simplicity is found only for the limits in the field-off case, and evaluation of the effect of the a/apeioperator is the chief problem for other time dependencies of applied field. The tl integration in eq 9 gives
we have first, as shown in the Appendix
and then
(AaCPzifz(t)) = i
P(Aa)2E'( CPziCPzi(t)fo)+ '/2P2AaE2 ( CPziCPli2(t)fo) I
i
1
i
(10) In eq 9 and 10, time reversibility has been used to write (PziPzj(-t)fo) = (PziPzj(t)fo)). If joint correlations are neglected, we may introduce the simple normalized correlation function +,(t) defined by
(CPZiPZi(t)fo)= (CPziPzi)o@z(t) = (1/5)@'z(t) i
I
(11)
as for fluids and isotropic solids the time evolution is independent of initial orientation in the applied field direction. Equation 10 then takes the simple form for a single molecule (Aap2fi(T))
(1/5)PAa(Aa
+ ' / 3 P p 2 ) @&t) ~ (12)
where (P'jpli2)8 = (2/3)(p,i2)s = 2/15 has been used. Equation 11 reduces immediately to Benoit's field-off result for rotational diffusion dynamics, as the "equation of motion" for P2(t)is then dPz/at = DQzPz,where Q2P,= 1(1 + l)Pl,and P2(t)= Pzexp(-6Dt). We defer further discussion of the more general results until the solution for the field-off case has been obtained. Field-On Solution. For a constant field E applied at t = 0, the limits on tl in eq 8 are 0 and t and on t2 are 0 and t,. The first time integrals give
- CPzi(t))fo)+ (AaCPzifz(t))= P(Aa)2E2(CP2i(Cf'~i i
i
i
p2Aap2E2(~ P z i ~ t d exp[-(t tl - P,i(-tl))fo)
+
dt, exp[-(t - t l ) f0]
where
X
as the derivative of the entire term vanishes in the ensemble averages from ps = --03 to +m. The average in eq 15 is of the expected form and immediately integrable with respect to tl but is secured at the expense of the equally awkard Iz expression defined by eq 16. Comparison with eq 14 for I,, however, shows a suggestive similarity. The conjecture that Iz is simply a multiple of I , proves to be correct, at least for persistence of momentum short in comparison with the time scale of reorientations, as expected in condensed phases. The basic assumption made is that the effect of an increment Aps in initial angular momentum ps is to produce an increment A%in the trajectory %(t).In dilute gases with infrequent collisions, this would clearly be a poor assumption, but it should be well satisfied for liquids with orientational times longer than a few picoseconds. Thus, if the correlation time for persistence of angular velocity w is r, and of orientation % is ro,a relation of the form r, (1/w2) gives r, 0 is then simply the sum of the solutions for the field-off and field-on cases, and the transient part of the response is given by the last term in eq 19 or, for self-correlations only, the last term of eq 21. These are clearly the second-order effect of dipole torques only, with the observed transient effect a measure of the difference of Pl and P2 correlation functions in the absence of applied fields. Permanent dipoles are thus required, but there can be no net effect even so if these correlation functions are the same, as in the extreme case of strong collisions which completely randomize orientations.
T
Flgure 2. (a) Molecular axis orientations 8. (b) SurBce triangle of orientations 0 and 8(t), with the effect of increment A8 for increment ApPein initial angular momentum. (c) Surface triangles for orlentations of a pair (i, j) of molecules.
These suffice to evaluate all the pederivatives of Pl(t)and P2(t)in terms of d8/dp,. when these are used in eq 14 and 16 for self-correlations i = j = k, one obtains after averaging over initial orientations 8, 4
I2= -311 = f/s((a8/ape)[3sin (-tl) cos ( t - tl) sin ( t ti) COS P + COS (-t,)(3 COS2 (t - tl) - 1)])e,# (18) The same result I, = -31, is obtained for joint correlations i # j = k after taking the extra spherical triangle of Figure 2c into account and averaging over orientations 8, with fixed relative orientations of Bi and 8, to obtain a more complicated function of d8/ape, corresponding to the increment A8 shown in Figure 2b and the relative orientations. Three-particle correlations with i # j # k vanish in the average over initial orientations. After I2= -31, is used in eq 15, substitution in eq 13 gives ( AaCP2ifdt)) = P(Aa)2E2(f~CP2i[CPi - CPzi(t)l)+ i
i
1
Doing the tl integrations and combining terms gives the final result (AaCP2if*(t)) = P(Aa)2E2(f~CP~i(C2i - CPZi(t))) + i
Discussion For the special case of rotational diffusion without joint correlations, our results confirm those of B e n ~ i t .They ~ go further by showing the essential role of collisions in determining the effects of dipole torques on the polarizability response in the field-on cases and in providing formal expressions of these effects for both self-correlations and joint correlations. If permanent dipoles are present, the results provide one a means for obtaining both dipole (5) and polarizability (P2) correlation functions from the field-on and field-off Kerr-effect experiments. The analysis given here has for simplicity been limited to axially symmetric molecules with dipole moment along the symmetry axis. Solutions for molecules with three different principal axes of polarizability and components of dipole moment along each have been obtained by Rosato and Williams'O for the field-off case using the perturbation approach developed here. These authors were unable to obtain complete solutions for the field-on response in terms of angular correlation functions by the rotation matrix methods used, and this part of the more general problem requires further study. The solutions should be valuable for dynamics of large molecules in nonpolar solvents, for example, but it seem clear that major deviations from simple exponential decay ("Debye relaxation") are not to be accounted for in such terms, and for even moderately polar fluids one has the problems of dipole interaction effects as well. As in other treatments of Kerr-effect relaxation, we have ignored the ever troublesome internal or local field problem. For the case of equilibrium in neat polar fluids, Ramshaw et al. l1 have derived a correction factor for the permanent dipole contribution to (Aaf,,) of the form [3t/(2t + tm)I2[(tm + 2)/312 (ref 12) if the correlations are evaluated for short-range interactions of dipoles imbedded in an infinite medium of permittivity E , with t, the permittivity for induced dipole polarization. A proper evaluation of the relation between macroscopic and molecular correlation times would require detailed molecular response theory, and we forego here even discussion of whether a ratio [3t/(2t + of macroscopic to molecular time scales is appropriate at the continuum level of description. Some light could be thrown on such questions by comparison of experimentally derived Kerr and dielectric functions. It may be noted also that, for nonpolar molecules, Keyes and Ladanyi13have rigorously related the (10)Rosato, V.; Williams, G. J. Chem. SOC.,Faraday Trans. 2 1981,
77 . . , 1767 - .- ..
(11)Ranshaw, J. D.; Schaefer, D. W.; Waugh, J. S.; Deutch, J. M. J. Chen. Phys. 1971,54, 1239. (12) Bottcher, C. J. F.; Bordewijk, P. "Theory of Electric Polarization"; Elsevier: Amsterdam, 1978;Vol. 11, Chapter 13.
4704
J. Phys. Chem. 1082, 86,4704-4708
internal problem in the Kerr effect to the same problem in depolarized light scattering. There appear to be possibilities of extending the methods used here to several related problems. The principal uses of Kerr-effect relaxation have been to study relatively slow motions as of biopolymers or of smaller molecules in highly viscous states. Better time resolution for faster processes should be possible by various combinations of sine wave and pulse excitations. Analyses of such situations by our method will be discussed in another paper, as will possibilities of extending the analysis to higher-order field effects for both electrooptic and dielectric responses. Acknowledgment. I was introduced to the Kerr-effect problem by Professor G. Willams and thank him for sev~
~~
~
(13)Keyes, T.;Ladanyi, B. Mol. Phys. 1979,37, 1643.
eral subsequent discussions. The work was supported by NSF Grant CHE-7822209.
Appendix The manipulations to obtain eq 15 and 16 from Z both involve the rule for differentiating a product: D(uu) = UDV + UDU,where D is either the Liouville operator Lo= d/dt for u and u not explicit functions of time or d/dps. As the equilibrium ensemble average of d(uu)/dt vanishes and f do= - Pfo(dH0/dt)= 0, repeated application of the rules gives
(fouLo"u) = (fo[(-1)"~c,"ulu, from which eq 14 follows on remembering the series expansion of the exponential. Using the rule with D = d/dpo then gives eq 15 and 16 as the integrated part vanishes at the limits p e = --m and p s = m.
Self-Ionlzation of Water at High Temperature and the Thermodynamic Properties of the Ions Kenneth S. Piker Department of Chemism and Lawrence Eerkeley Laboratory, University of California, Eerkeley, California 94720 (Received: June 2, 1982)
It is shown that gas-phase data on hydrated H+and OH- ions from mass spectrometry can be used to calculate the ionization product for water at high temperature and at high enough pressure to allow relating these results with those directly measured near 1000 K and 0.5 g ~ m - The ~ . thermodynamic properties of the hydrated H+ and OH- are discussed and the heat capacity is compared with results calculated from the Born equation for an appropriate region of temperature and pressure.
The ionization product for water has been investigated over a wide range of temperature, and Marshall and Franck' (MF) have recently proposed an empirical equation after a careful review of the various direct measurements. This equation appears to be an excellent representation of these data which are for densities above 0.4 g ~ m - ~It . is interesting to consider the behavior of the ionization product at lower density than 0.4 g in the range near or above the critical temperature. The Marshall and Franck equation can be extrapolated into this region; indeed Gates, Wood, and Quint2have used such extrapolated values to discuss the behavior of the partial molal heat capacity of ions near the critical point. But long extrapolations are dangerous. In this paper the properties of gaseous, water-related ions3are used to calculate the ion product for steam. These results appear to be valid at pressure gradually increasing with temperature until the curve can be connected with the directly measured exAt. perimental values of Quist4 at 1073 K and 0.5 g ~ m - ~ lower temperatures one can interpolate between these ~
(1)W.L. Marshall and E. U. Franck, J. Phys. Chem. Ref. Data, 10, 295 (1981). (2) J. A. Gates, R. H. Wood,and J. R. Quint, J. Phys. Chem., in preaa. (3)P. Kebarle, Annu. Rev. Phys. Chem., 28, 445 (1977);"Modern Aspects of Electrochemistry", Vol. 9,B. E. Conway and J. OM. Bockris, Ed!., Plenum Press, New York, (1974,p 1, and references cited in these reviews. (4)A. R.Quiet, J. Phys. Chem., 74,3396 (1970). 0022-3854/82/2086-4704$01.25/0
newly calculated curves at lower density and the equation of Marshall and Franck for the high-density region. These interpolated curves, although still somewhat uncertain, should be much more reliable than extrapolations considering only the high-density data. The thermodynamic properties, including the partial molal heat capacities, of the ions can be calculated from the ion product equation. These properties are discussed in comparison with calculations from the Born equation and the earlier results of Gates et a1.2
Gaseous Ion Equilibria The thermodynamic properties of gaseous H+ and OHand their hydrated ions are known. The equilibria between the successive hydrate ions were measured, primarily by Kebarle and his a s s o ~ i a t e s , with ~ ~ ~ a' mass spectrometer having a relatively high pressure of H20 in the ion source. The first hydration of the proton is complete under all conditions of present interest; hence, we consider as the initial reaction 2H20 = H30++ OH-
K1 = P H s 0 + P O H - / P H 2 0 '
(1)
( 5 ) Y. K. Lau, S. Ikuta, and P. Kebarle, J. Am. Chem. SOC.,104,1462 (1982). (6) M. Arshadi and P. Kebarle, J. Phys. Chem., 74,1483 (1970). (7)J. D.Payzant, R. Yamdagni, and P. Kebarle, Can. J. Chem., 49, 3308 (1971).
0 1982 American Chemical Society
