Theory of Dielectric Relaxation of Rigid Polar Fluids* BYJOHNM. DEUTCH Department of Chemistry Massachusetts Institute of Technology Cambridge Massachusetts 02139 U.S.A. Received 26th August 1976 In order to obtain quantitative and unambiguous information on molecular relaxation from di- electric measurements a number of theoretical issues must be resolved. In this article three of the principal issues are discussed (1) the relation of the measured dielectric constant to molecular time correlation functions ; (2) the relation of the many-particle polarization correlation functions which arise in the theory to single-particle orientation correlation functions which are of primary interest; and (3) the ability of dielectric measurements to resolve different models ofsingle-particle reorientation dynamics.1. INTRODUCTION The purpose of this discussion is to present a personal assessment of the present state of the theory of dielectric relaxation of rigid polar fluids and to comment critic- ally on the value of dielectric relaxation measurements for elucidating molecular relaxation mechanisms. The conventional view is that dielectric relaxation measurements in combination with other relaxation measurements in particular n.m.r. can be practically employed to distinguish between models of molecular reorientation in dense fluids. Here we shall argue that this view is optimistic and at best correct only if certain substantial theoretical issues are resolved. The analogous conventional opinion for equilibrium properties is that static dielectric constant measurements can be practically employed to determine equili- brium angular correlations in the fluid.Here also substantial theoretical hurdles must be overcome before one can relate with some degree of precision structural properties of the fluid to the measured quantities. We shall discuss questions that arise in the equilibrium theory of rigid polar fluids' only in so far as these questions bear on the dynamical theory. The subtle issues arise for both the equilibrium and dynamical properties of polar fluids because of the long-range character of the dipole-dipole interaction P(1) ' T(L2) P(2) (1.1) where p(i) is the dipole moment of molecule i and T is the static dipole-dipole tensor The issues involve more than arcane questions of interest only to formal theorists.As we intend to show these issues are a serious barrier to the practical use of dielectric relaxation measurements for learning about molecular relaxation processes in polar fluids. In order to draw the issues as sharply as possible we shall restrict the discussion to * Supported in part by the National Science Foundation. JOHN M. DEUTCH rigid polar fluids where the influence of polarizability is neglected. Clearly in prac- tical applications it is necessary to modify the treatment to include the effects of mole- cular polarizabilities and where necessary hyperpolarizabilities. Needless to say difficult issues are also presented in the treatment of polarizable systems but this topic is outside the scope of this discussion.What are the substantial theoretical issues that arise in the theory of dielectric relaxation? There are three (1) How is the measured frequency-dependent dielectric constant ~(m)related to equilibrium time correlation functions ? (2) Is it possible to relate the many-particle equilibrium time correlation functions that arise in the theory to single-particle equilibrium time correlation functions ? (3) If we assume that questions (1) and (2) have been successfully answered how different are the predictions of different models of molecular reorientation for &(W)? We shall deal with each of these questions in turn below. 2. RELATING E(O) TO MOLECULAR CORRELATION FUNCTIONS The macroscopic relation between the polarization P(o)and the average macro- scopic electric field E(w)is P(@ =x(dE(4 (2.1) where the susceptibility x(o)is related to the dielectric constant by A straightforward application of linear response theory for the response of the polarization to an external field E0(m)leads to the result P(4 = XO(~)EO(4 (2.3) where the "bare " susceptibility2 is related to the many-particle dipole moment equilibrium time correlation function p(t) co x&~) =;P(Mi) dt enp[-iot][ -$?I.0 In eqn (2.4) M is the net dipole moment of the sample We expect the macroscopic susceptibility x(w)to be independent of sample shape. But of course the relation between Eo and E depends on the sample shape and sur- roundings.Hence in comparing eqn (2.1) and (2.3)we see that xo(m)must be shape- dependent in just such a way to compensate for the dependence exhibited by (Eo/E). In order to illustrate the consequences of this situation consider the uniform polar- ization geometry of a sphere with dielectric constant ~(m)embedded in an infinite medium of dielectric constant E~(c~)). For this case one finds3 28 THEORY OF DIELECTRIC RELAXATION OF RIGID POLAR FLUIDS where we have eliminated (Mz2)in eqn (2.5) in favour ofthe static dielectric constants according to the relation In eqn (2.7) we use the notation 3[F(t)]to denote the Laplace transform of F(t) with Laplace transform variable z = ico. The shape dependences of ~(t), (Mi) and hence according to eqn (2.5),x,,(Co> are clearly indicated by the two relations eqn (2.7) and (2.8).The values for the correla- tion functions on the left hand side of eqn (2.7) and (2.8) depend on the nature of the external medium; thus even in the thermodynamic limit one finds that the correlation function of the embedded system depends on the surroundings. It should be noted that in this simple case which involves an embedded sphere the issue that emerges is dependence of certain correlation functions on the externaZ sur- roundings. The question of shape dependence arises when one considers more com- plicated situations for the geometry of the embedded system e.g. an ellipsoidal sys- tem. Then one would find that the correlation functions of the embedded system would depend on the ratio of major to minor axes of the ellipsoid i.e.its shape as well as the surroundings. These more complex situations need not concern us here. In the special case of a sphere in vacuum the result eqn (2.7) [cl(o) = I] reduces to where the subscript " sp " has been added to p(t) to emphasize that the result holds for this particular geometry. The other special case of interest a sphere embedded in its ownmedium [E~(c~))= &(a)]yields the result &(O)-1 240) + 1 (2.10) 9C-vim(t)J = [E(O) -1 ][2&(O)+ 1][3] where the subscript " co " has been added to V(t) to indicate the embedded sphere geometry. The original application of linear response theory to dielectric relaxation was car- ried out by Glar~m,~ who obtained the result eqn (2.9) for a sphere in vacuum.However for the embedded sphere geometry Glarum obtained4 (2.1 1) in contrast to eqn (2.10). The result displayed in eqn (2.10) was put forward by Fatuzzo and Mason,5 and later by Klug Kranbuehl and Vaughan,6 and by Greffe et aL7 The controversy over the relationship of the measured transport coefficient e(o) to the molecular correlation function VcO(t)is one of the theoretical issues mentioned in the introduction. The issue is of significance because of the different predictions that arise for pm(t) from the two alternative expressions. However perhaps more importantly the issue exhibits a shortcoming in our understanding of the consequences of long-range dipolar forces for transport properties. Titulaer and Deutch3 have recently presented a detailed analysis of the conflicting theories of dielectric relaxation and the reader is referred to that paper for many of the relevant references.These authors conclude on the basis of linear response theory JOHN M. DEUTCH and on the basis of fluctuation theory that the Fatuzzo-Mason expression eqn (2.lo) is correct. Nevertheless it is possible to harbour legitimate reservations. The theoretical arguments that lead to eqn (2.10) are based on the use of fictitious external “cavity ”electric fields that could not in principle be constructed in nature i.e. there is no set of charges that gives rise to these fields and the fields do not satisfy Maxwell’s equations. Only recently have arguments been put forward based on a microscopic theorys and on a more sophisticated fluctuation theory9 that lend support to the result eqn (2.10) without explicit use of fictitious external cavity fields.While present opinion appears to favour the Fatuzzo-Mason result it is important to recognize that a fundamental issue must be resolved before the measured e(w) is related to a time correlation function. At least for the present the prudent researcher is advised to employ eqn (2.10). 3. THE RELATIONSHIP BETWEEN MANY-PARTICLE AND SINGLE-PARTICLE CORRELATION FUNCTIONS Dielectric measurements strictly speaking yield information on many-particle equilibrium time correlation functions ~(t).However our primary interest is in the properties of the single-particle equilibrium time correlation function ql(t) Cpl(t>=(PA1 9 t)PZ(l Y O>XP2(1Y O)PZ(L 0)) (3.1) which can be directly related to the correlation function of the first spherical harmonic of the representative particle’s orientation.Spherical harmonic correlation functions of single-particle orientation are important ingredients for characterizing molecular reorientation mechanisms. Many theories of dielectric relaxation are semi-macroscopic and adopt a single-particle view from the outset. These theories focus attention on a single representa- tive dipole and treat its surroundings at one level of approximation or another as a dielectric continuum. For example Debye originally employed a single-particle cavity model to obtain the expression where z is identified with the relaxation time of the single-particle correlation function ql(t).This result eqn (3.2) can be obtained equally well by asserting that qsp(t)= exp[-t/z] in eqn (2.9). An alternative single-particle mode is obtained by the identification of yco(t)with ql(t)in eqn (2.10); in the Onsager-Cole modello the choice qa(t) = exp[-t/z] is made. More recently Nee and Zwanzig” have developed a single-particle dipole relaxation model that includes the effects of “ dielectric friction ” and derive the result eqn (2.10)with (3.3) which was previously obtained by Fatuzzo and Mason,3 and Scaife.12 The essential feature of all single-particle dielectric relaxation theories (with one exception discussed below) is the physical assumption at some stage of the calcula- tion that the representative dipole of interest is in a cavity which is embedded in a dielectric continuum of specified properties.Physically such a picture can never be entirely correct. Each dipole will have its local surroundings distort as its neighbours adjust to the reorientation. To replace this reality on the molecular scale with a 30 THEORY OF DIELECTRIC RELAXATION OF RIGID POLAR FLUIDS "cookie cutter "cavity surrounded by a homogeneous fluid is to do some violence to the actual situation and hence inevitably to introduce an approximation. Thus at present one believes that in principle perfect knowledge of pl(t) is insuffi- cient to determine e(co). Conversely perfect measurements of ~(m) yield information on the many-particle correlation functions and additional analysis is required to relate these correlation functions to ql(t).Kivelson and Maddenx3 have directly confronted the issue of the relationship between p(t) and pl(t). These authors employing projection operator techniques,14 have developed exact formal relationships between ~(t), for various sample geometries and pl(t). A representative result obtained by these authors relates the " relaxation times "for p(t) denoted by TM,to the relaxation time for pl(t) denoted by 2,as follows Til = [1 + Nfl7lfl + Ns1. (3.4) Here [l + Nl] is a factor which measures the correlation between neighbouring dipoles and f is a dynamic orientation time correlation function for the factor :):(')]c" of two different dipoles. The time dependence involved in the definition off is determined by a "projected " Liouville operator L which makes evaluation of this quantity essentially impossible.The important contribution of Kivelson and Madden is that they have provided a systematic framework for introducing approximations that relate ql(t)to p(t) so that the contribution of cross correlations to p(t) may be evaluated. However an un- fortunate feature of the Kivelson-Madden analysis is that it involves the factor 11 3-WI = P-2 2 MI) PW (3.5) i which depends upon sample shape. Thus eqn (3.4) must always be identified with a particular geometry e.g. for a sphere in vacuum [l+Nf=-E(0) -1 4nppp2 E(0) + 2[ -9 1 ' An alternative procedure is to seek to identify the ql(t)and q(t)with a formal cor- relation function which is shape-independent.Sullivan and Deutch l5 have developed a molecular theory of dielectric relaxation that introduces a projection operator 9 9G = M. <MM)-l. {MG) (3.7) where M is the net dipole moment of the sample that assumes shape-independent correlation function. The formal result of this molecular theory is where T;~(w) = dt exp[-icot]k(t) (3.9) [el with k(t)= p'2(,uz(l)exp[i(l -99Lt]&z). (3.10) While k(t)is a multi-particle correlation function it is a local quantity that does not depend on sample and surroundings because the projection operator removes this disagreeable feature of long-range dipole-dipole forces. 3OHN M. DEUTCH Following Kivelson and Madden one can now seek to relate tp-'(w) to single- particle correlation functions.This can be done at various levels of approximation employing either continued fraction expansions or one two and three variable theories. Sullivan16 has presented an extensive discussion of this approach and provided a valuable critique of Kivelson and Madden's approach. At the lowest level of approximation Sullivan finds in the low frequency limit l]T (3.11) If Onsager's equation for the static dielectric constant is employed to eliminate p2in eqn (3.1 1) then one obtains (3.12) which in conjunction with eqn (3.8) gives the Powles-Glarum relation.'' Of course any number of other approximations can be formulated within the framework of the theory. We conclude this section by noting the importance of assessing the various approxi- mations that are introduced to inter-relate 9(t),ql(t),pp(t).Two approaches are possible. First one might employ molecular dynamics to directly determine pl(t) and p(t). To our knowledge this has not been done. An important question would arise as in the equilibrium calculation about which sample geometry to associate with the numerical simulation. The second approach is to adopt specific dynamical models and evaluate the pertinent correlation functions for each model. One candidate model is the lattice model introduced by Zwanzig'* and examined by ColeI9 where dipoles on a discrete lattice undergo rotational Brownian motion and in addition interact via dipole-dipole forces. A second model that can be adopted is a hydrodynamic one.Here the cross correlation effects between impurity dipoles in a solvent are computed exclusively on the basis of hydrodynamics; the rotation of a dipole sets up a velocity perturbation that places a torque on other dipoles. This hydrodynamic effect certainly is present in actual systems and as it turns out the effect has a long-range character. Preliminary calculations20 indicate that in this model one cannot justify neglecting the dynamical factor of Kivelson and Madden fin eqn (3.4). 4. RESOLVING MODELS OF MOLECULAR MOTION THROUGH DIELECTRIC MEASUREMENTS The ultimate question about dielectric relaxation particularly at this conference must be " How useful are dielectric relaxation measurements for distinguishing be- tween various models of molecular motion in dense fluids? " On the basis of present evidence one cannot give an optimistic answer to this question.Quite generally one finds that the differences among the predictions of molecular models of the correlation functions that enter into dielectric measurements become more pronounced at short-times. Accordingly roughly speaking one would anticipate that predicted differences in E(O)between different models would occur at moderately high frequency. But at these frequencies (> lo6Hz)E(W) is difficult to measure with precision. Moreover many of the theoretical issues and underlying assumptions become more uncertain at higher frequencies-e.g. the effects of intramolecular mo- tion and polarizabilities. 32 THEORY OF DIELECTRIC RELAXATION OF RIGID POLAR FLUIDS Theories for the single-particle correlation function q~,(t)are based on variations of the Debye rotational diffusion model or variations of this model that include finite step size for reorientation,21 in one way or another.These models are based on continuum-hydrodynamic ideas that are not entirely valid on the scale of molecular distances and times. Moreover itis knownZ2 that a wide variety of models can lead to single exponential behaviour for ql(t)so that this feature may not be used as an unambiguous signature of any particular type of model. One might hope that the combination of dielectric measurements and other types of measurements say n.m.r. would be useful for discriminating among various models of single-particle reorientation.However there are few examples of systems which have been sufficiently studied to permit this kind of analysis. The experiments that are required are n.m.r. relaxation time measurements and dielectric constant measure- ments at various temperatures and densities. Hopefully several carefully selected systems will be studied in detail in the next few years in order to determine precisely how much may be learned quantitatively by combining different measurements. The assessment presented above indicates the theoretical and practical difficulties that are encountered when one seeks to employ dielectric relaxation measurements to obtain quantitative information on molecular relaxation. These difficulties should encourage workers either to seek qualitative information from their measurements or to undertake combinations of experimental measurements in order to elucidate molecular motion in dense fluids.For example see the recent reviews J. M. Deutch Ann. Rev. Phys. Chem. 1973 24 301; S. A. Adelman and J. M. Deutch Adv. Chem. Phys. 1975,31,103. P. Mu in Cargise Lectures in Theoretical Physics 1964 Statistical Mechanics ed. B. Janco-vici (Gordon and Breach New York 1966). See U. M. Titulaer and J. M. Deutch J. Chern. Phys. 1974,60,1502 for the derivation of this result and pertinent prior references. 'S.H. Glarum (a)J. Chem. Phys. 1960,33 1371; (b) see also Mol. Phys. 1972,24,1327. E. Fatuzzo and P. R. Mason Proc. Phys. Soc. 1967,90,741. ti D. D. Klug D.E. Kranbuehl and W. E. Vaughan J. Chem. Phys. 1969,50,3904. J.-L. Greffe J. Gouloz J. Brondeau and J.-L. Rivail J. Chem. Phys. 1973 70,282. * See the work of R. L. Fulton e.g.,J. Chem. Phys. 1975,62,4355 and Mol. Phys. 1975,29,405 and references cited therein. B. U. Felderhof personal communication. lo R. H. Cole J. Chem. Phys. 1938 6 385. T. W. Nee and R. W. Zwanzig J. Chem. Phys. 1970,52,6353. B. K. P. Scaife Proc. Phys. SOC., 1964 84 616 and section (4) of B. K. Scaife in Complex Permittivity ed. J. H. Calderwood (English U.P. London 1971) for a review of other calcula- tions of this type. l3D. Kivelson and P. Madden Mol. Phys. 1975,30 1749. l4 See for example H. Mori Prog. Theor. Phys. Kyoto 1965,33,423. l5 D. E. Sullivan and J. M. Deutch J.Chem. Phys. 1975 62,2130. l6D. E. Sullivan,Ph.D. dissertation (Department of Chemistry M.I.T. Cambridge Mass. 1976). chap. 111. l7 See N. E. Hill W. E. Vaughan A. H. Price and M. Davies in Dielectric Properties and Molecular Behaviour (Van Nostrand London 1969); ref. (4a) and J. G. Powles J. Chem. Phys. 1953,21 633. R.W. Zwanzig J. Chem. Phys. 1963,38,2776. l9 R. H. Cole Mol. Phys. 1973,26,969; 1974,27 1. 2o P. G. Wolynes and J. M.Deutch unpublished calculations. 21 See for example E. N. Ivanov Zhur. eksp. theor. Fiz. 1963,45 1509; R. G cordon J. chew. Phys. 1966,44 1830. z2 R. I. Cukier and K.Lakatos-Lindenberg J. Chem. Phys. 1972 57 342