J. Phys. Chem. 1982, 86, 4244-4256
4244
distance in the FHC equation. The results of the analysis of the conductance data according to the above equations are given in Table 111. For all calculations a value of 5.3 8, was selected as the distance parameter except for the Justice method in which the Bjerrum distance of 33.0 8, was used. Also the formation of triple ions was ignored since the mobilities of triple ions are not known. It is clear that the ion association constants obtained from these conductance measurements depend strongly on the conductance equation used and are different from the value obtained from the NMR data. Similar behavior was observed by G i l k e r ~ o nfor ~ ~ion association of cesium tetraphenylborate in acetonitrile, and of silver nitrate and lithium picrate in 2-butanone. The apparent reason for the differences in the ion association constants obtained from various conductance equations is that both the higher-order term in the conductance equation and the terms caused by association have the same concentration dependence in the first a p p r o ~ i m a t i o n .An ~ ~increase ~~~ in the sum ECa log (Ca)+ JICa can be compensated for by a decrease in the ion association constant. Equations which introduce higher-order terms into the conductivity equation produce smaller ion association constants for a given value of the distance parameter. An increase in the distance parameter causes J1to decrease and J2to increase. Both such changes are compensated for by an increase in the ion association constant. One expects association constants obtained from NMR and from conductance measurements to be essentially the same. The equilibria in the solutions are (35) Gilkerson, W. R.; Roberta, A. M. J. Am. Chem. Soc. 1980, 102, 5181. (36) Karl, D. J.; Dye, J. L. J. Phys. Chem. 1962, 66, 477. (37) Kay, R. L.; Dye, J . L. R o c . Natl. Acad. Sci. U.S.A. 1963,49, 5.
M+
+ X- --f K [M+.S.X-] & K M+.X-
(12)
in which M+ and X- are the solvated cation and anion, respectively, and M+.S.X- and M+.X- are solvent-separated and contact ion pairs. Since electrical conductance measures the fraction of uncharged species, then
The second equilibrium is unaffected by changes in concentration except via changes in the relative activity coefficients of the two types of ion pairs. Thus, the observed chemical shift can be written as =
+ 8[XM+SX- + XM+X-]
(14) in which 8 = (6M+.s.x-+ K26M+.X-)/(K2 + 1) is the population-averaged chemical shift of contact and solvent-separated ion pairs. Since the exchange between the two types of ion pairs is fast on the NMR time scale, the relative concentrations of these two species cannot be determined by NMR methods. This leads to 6obsd
6M+xM+
KNMR
= Kcond
Table 111 shows that none of the values of the ion association constant from the conductance measurement is comparable with the NMR value. The discrepancy between the values obtained by the two methods is probably due to the inadequacy of the conductance equations and the inability to measure the conductance of extremely dilute solutions which leads to an uncertainty in A,,. Acknowledgment. We gratefully acknowledge the support of this work by National Science Foundation Grants CHE-80-10808 (A.I.P.) and DMR-79-21979 (J.L.D.)
Dielectric Friction and Molecular Reorientation Paul Madden and Danlel Klvelson’ Department of Theoretical Chemlstry, Cambrklge University, Cambrklge, England and Department of Chemistry, University of California, Los Angeles, California 90024 (Received:Aprll 1, 1982; I n Final Form: April 5, 1982)
We have developed a molecular theory for the dielectric friction on a rotating molecule in a polar medium. The theory accounts for numerous important effects, which are not included in the existing continuum theories, such as molecular translations and the anisotropic relaxation of the polarization induced in the surrounding medium.
1. Introduction
The rate of reorientational motion of a molecule in a liquid is thought to be principally dependent upon the molecular shape and size.’ In liquids of polar molecules appreciable additional contributions might be expected from the slowly relaxing long-range dipolar interactions; for example, it has been observed that the in-plane-reorientation rate of polar heterocycles is slower than that of nonpolar ones.2 Two effects resulting from dipolar interactions are the enhanced two-particle equilibrium orientational correlations and the dielectric friction which
arises from the slow relaxation of the reaction field or, alternatively, of the dipolar torques on a given molecule. The first of these effects is of greatest importance in dielectric relaxation3 where a collective dipole relaxation is observed, but the second should be significant in all observations of the reorientation of polar molecules. Discussions of reorientation of polar molecules have usually been carried out within the framework of dielectric experiments, and the relevant calculations have been based upon models in which single dipoles are introduced into ~
(1) Kivelson, D.; Madden, P. A. Annu. Reo. Phys. Chem. 1980,31,523.
( 2 ) Pedersen, E. J.; Vold, R. R.; Vold, R. L. Mol. Phys. 1978,35,997. QQ22-3654182/2Q86-4244$0 1.2510
(3) See, for example, Brot, C. “Dielectric and Related Molecular Processes”; Davies, M., Ed.; The Chemical Society: London, 1975; Vol. 2.
0 1982 American Chemical Society
Dielectric Friction
cavities located in a continuous dielectric. The results of continuum electrostatic calculations for static dielectric properties are known to incorporate the long-range dipole-dipole interactions exactly; anything less than an exact treatment of these long-range effects leads to physically inconsistent result^.^.^ These continuum theories combined with cavity constructions have not been very successful in explaining dynamical behavior, and it would seem that this behavior might be more readily understood on a molecular level. Elsewhere we have developed a molecular theory for dielectric relaxation in which we have avoided the introduction of cavities and continuum electrostatic arguments.6 Here we will use similar methods to discuss the influence of slowly relaxing dipolar torques (dielectric friction) on a single molecule orientational correlation function for a spherical tensor of arbitrary rank (1); such a correlation function with 1 = 2 is observed in NMR and Raman scattering spectra, and one with 1 = 1contributes to IR spectra. The relationship of the 1 = 1 single-particle correlation function to the multiparticle one observed in dielectric relaxation experiments is discussed elsewhere.6 Nee and Zwanzigl developed an expression for the dielectric friction on a slowly reorienting dipole in an “Onsager cavity”; the source of the friction is identified with the “lag of the reaction field”. (Fatuzzo and Mason8 had obtained the same result earlier, but the connection of their calculation to ours is less direct.) Hubbard and Wolynesggeneralized the Nee-Zwanzig result by considering the orientational tensor to be of arbitrary rank and by allowing the fluctuating dipolar torques acting on the dipole in the cavity to be dependent upon the rotations of the dipole itself. They envisaged the polar molecule as being subjected to short-range interactions which in itself would lead to the rotational motion being diffusional, and the polar molecule being simultaneouslysubjected to weak, externally applied torques which can be identified with the long-range dipolar interactions between the probe molecule and the solvent. Both continuum theories suggest that in polar liquids there are slowly relaxing torques of sufficiently large amplitude to exert an observable effect. In addition, Hubbard and Wolynes’ theory shows that the dielectric friction coefficient will be dependent on 1; consequently there will be a breakdown of the relationship between the relaxation times (TJ for different spherical harmonics predicted by the rotational diffusion model (e.g., 71/72 = 3). Because the continuum analyses give physically wellfounded expressions for equilibrium properties and account correctly for the dominant long-range interactions, the equilibrium quantities in our molecular theory will be evaluated within a framework consistent with the continuum theory of dielectrics. However, a molecular description of the medium surrounding the “tagged” particle reveals several possible relaxation channels which are not readily identified in a continuum approach, and we shall attempt to evaluate the relative importance and distinctive characteristics of the various relaxation processes which may occur. Our first objective is to obtain a molecular expression which corresponds to that obtained by Nee and Zwanzig from the continuum analysis, so as to show, in
The Journal of Physical Chemistry, Vol. 86,No. 21, 1982 4245
molecular terms, what approximations are implicit in this theory. Our second objective will be to improve upon the limited description of the molecular motion which occurs in the continuum theory. In particular, we note that existing continuum theories relate the dielectric friction only to the orientational relaxation rates, rD-land D, (where 7 D is the Debye time and D, the single molecule rotational diffusion coefficient), even though the dipolar torque on a molecule depends on positional as well as orientational variables. (Both orientational and positional variables enter in a similar way into the description of interactioninduced spectra of molecular fluids,1° and in the latter it is found that translational relaxations usually dominate.l0) In addition, the continuum theories assume that the polarization of the material surrounding the dipole relaxes with a single decay time, whereas it is known that the polarization induced by an inhomogeneous external field relaxes with at least two decay even when the field varies slowly on the characteristic intermolecular distance scale. [The molecular description thus reveals additional relaxation channels.] Our final result for the dielectric friction is expressed entirely in terms of wellspecified and readily determined molecular and collective properties: the molecular moment of inertia (0,the molecular rotational (D,) and translational (Dt) diffusion constants, and the dielectric permittivity 40). In the next section we construct a formal framework for the problem by making use of the Mori procedure.12 Following that we look in some detail at the dipolar torques and their inclusion into the theory. We next evaluate the molecular expressions for the dielectric friction for a model fluid chosen to be similar to that used by Nee and Zwanzig in their continuum calculations.I In the succeeding section we discuss the extension of the molecular theory to include molecular translations, rotations of the probe molecule, and anisotropic relaxations of the induced polarization in the dielectric surrounding the probe. In the sixth section we obtained results for this extended theory, and we conclude with a summary and discussion of the implications of this work for the interpretation of experimental data. Many of the detailed calculations are relegated to the appendices. 2. General Formulation We will consider the half Fourier transform Sl(o)of the correlation function of an lth rank spherical harmonic,l3,l4 Clm(Ql(t)),of the orientation, Ql(t),of the probe molecule denoted by 1. We set Cl,(Q,(t)) = C;,(t) and Cim(0) Cim; then
where ( ) indicates an equilibrium ensemble average. The probe molecule is the analogue of the one which resides within a cavity in the continuum theories. A convenient form for Sl(w)is obtained by expanding Sl(w)to second order in Mori’s continued fraction12
Here &(a) is a rotational friction coefficient given by (4) Onsager, L. J. Am. Chem. SOC.1936, 58, 1486. (5) A good review of the classical theories of the static permittivity has been given by Buckingham, A. D. “M. T. P. Physical Chemistry”;Buckingham, A. D., Ed.; Butterworth London, 1972; Vol. 2, p 241. (6) Madden, P. A.; Kivelson, D. To be published. (7) Nee, T-W.; Zwanzig, R. J . Chem. Phys. 1970, 52, 6353. (8) Fatuzzo, E.; Mason, P. R. h o c . Phys. SOC.1967, 90,741. (9) Hubbard, J. B.; Wolynes, P. G . J. Chem. Phys. 1978, 69, 998.
(10) Cox, T. I.; Madden, P. A. Mol. Phys. 1981, 43, 287. (11) Fulton, R. L. Mol. Phys. 1975, 29, 405. Berne, B. J. J . Chem. Phys. 1975, 62, 1154. (12) Mori, H. h o g . Theor. Phys. 1965, 33, 423. 1965, 34, 399. (13) Steele, W. Ado. Chem. Phys. 1976, 34. (14) Brink, D. M.; Satchler, G . R. “Angular Momentum”: Oxford University Press: Oxford, 1967; 2nd ed.
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Madden and Kivelson
The Journal of Physical Chemlsiry, Vol. 86, No. 27, 1982
where the rounded brackets are specified by the relation
(B(o),B)+ = Jmdt eiut((exp[i(l- P + ) L t ] B ) B t )
(4a)
in which B is a column vector of variables {Bl,B2, ...I,L is the LiouviUe operator, P+ is the Mori-Zwanzig projection operator defined by
P + B ( t ) = ( B ( t ) A * )( IAI2)-’A
(4b)
and the superscript (+) indicates the variables A = {Al, A2, ...) along which the projections are taken; in the case above A = (Cim,Cim). In order to make use of eq 2 we must evaluate averages of Cim and its derivatives. For simplicity, we shall restrict our attention to linear molecules for which13
C:, = iwl.L1cim
(5)
where bJis the dipole moment of the j t h molecule and F the field at molecule 1 due to all other dipoles in the system. We shall restrict our attention to unpolarizable molecules for which F is given by
F = CT(rlj).pj j#l
(14)
where T(rlj) is the dipole-dipole interaction tensor
T(rlj) = ( ~ T ~ , , ) - ~ V V ~ ~ ; ~
(15)
rlj is the distance between the probe molecule and the jth molecule in the liquid, and to is the permittivity of free space. We shall assume that the short-range torques (VR) are strong and rapidly relaxing, but that the dipolar torques (AIDD) are long-range and relax slowly. In the spirit of this model we will ignore the cross-correlation between VR and NDD,in which case we may rewrite eq 2 as
where w1 is the angular velocity of molecule 1, and Lk is the a t h spherical tensor component of the angular momentum operator which acts on the angles Q1.14 1
^L&C!, = [ I ( I t 1 ) ( 2 1 + l ) l l y l ) ~ + ” ( m , - m
1 -a )%‘
where tSRand tDD are the friction constants arising from the short-range (PR) and long-range (NDD) torques:
(6)
The orthogonality relationship for spherical harmonics yields14
(c:,c~;!,= 4$“,/(21 + 1)
in an isotropic ensemble. It readily follows from eq 5-7 and the Boltzmann average for ( 1ukI2)that
(CjmC)kt)= 6&,,,JzBT1(l
+ 1)/1(21 + 1)
(8)
where I is the molecular moment of inertia, kB is the Boltzmann constant, and T is the absolute temperature. The Ci, quantity that appears in eq 3 contains terms bilinear in the angular velocity u1of molecule 1, which we shall call the “kinetic terms”, and terms dependent upon the angular acceleration d,which we shall denote as the “torque terms”. The torque terms contain contributions from short-range intermolecular interactions and others from long-range dipole-dipole interactions between molecule $ and the other molecules in the fluid. From eq 3 and 4a we see that the relevant “effective torque” is (1 P+)Ci,, and we write (1 - P+)Cf, = M$
+ NgD
(9)
N:
where contains the kinetic terms and the torque terms due to the short-range intermolecular potential ( USR), and N E contains torque terms arising purely from the dipolar interaction energy (PD) of particle 1. More specifically
where = -iLl@R/I
&SR
(11)
flz = ~ - l ( - l ) a ( ~ ~ P D ) ( L ~ ~ C ~ , ) (12) Throughout we will sum over repeated Greek indices. The first term in eq 10 is the kinetic term. The dipolar energy PDon particle 1, which appears in eq 12, can be written as P
D
= -bl.F
(18)
(7)
(13)
and (+) indicates the projections along C;, and Ci (see eq 3,7-9). We shall assume that the frictional term @(w) is independent of all long-range contributions and that it can be evaluated in the rotational diffusion limit, i.e., for frequencies of interest p ( o )=
>> w
(19)
It is customary to introduce a rotational diffusion constant D,, defined as
D, = k B T / I t S R ( o ) Equation 16 can then be rewritten as
(20)
This is the form of Sl(w) obtained by Hubbard and Woly n e ~and , ~ it is the form that we shall use as the basis of all further calculations. I t should be noted that eq 16 contains the correct high-frequency limiting behavior, including inertial and other nondiffusional effects, but that eq 21 does not; generalizations of the continuum theory of dielectric friction corresponding to the retention of inertial contributions have been sought by Lobo et al.15
Dipolar Interactions Our task is the evaluation of the dipolar frictional term tDD(w) in eq 21. We wish, insofar as it is feasible, to relate &‘(LO) to independently observable properties such as the dielectric permittivity €(a).In this section we introduce some general considerations and approximations which apply to all our further work. From eq 18 we see that we must evaluate the static average, ( INEl2). In Appendix A it is shown that in order (15) Lobo, R.; Robinson, J. E.; Rodriguez, S. J. Chem. Phys. 1973,59, 5992.
Dielectric Friction
The Journal of Physical Chemistty, Vol. 86, No. 21, 1982 4247
to evaluate ( INfjnDI2), only that part of the pair distribution
tp replace the tensor T(rv) in eq 14 by the cut dipole tensor
function which transforms under rotations of the coordinate system in the same way as does the dipole-dipole interaction (p1-T'2-p2)is needed. This distribution func.tion, often called hD in the dielectric literature,16 can be approximated as
T(r1j)
hD(T12,fii1a2) = E-'p1*F2*ji2/kBT
(224
for r12> ro and
hD(riz,fii,Qz)= 0
Qrlj) = T(rij)
9(rlj) = o
( r l j I ro)
(22c)
and g is the Kirkwood g factor17 which describes shortrange dipolar correlation. In a continuum analysis, pi is the "apparentn dipole17of a molecule viewed from a distance in a solvent with dielectric constant e. That eq 22a-c give the exact form of hD as r12 m has been shown by a number of workers;lBJ8it also correctly describes the impenetrability of molecules at very short range (r12
