Theory of Non-linear Dielectric Effects in Liquids BYJERZYMALECKI Institute of Molecular Physics, Polish Academy of Sciences, 60-1 79 Poznan’, Smoluchowskiego 17/19, Poland Received 11th June, 1975 A general theory of non-linear dielectric phenomena in liquids is proposed with the fundamental assumption, supported by a large body of experimental data, that such phenomena are specifically sensitive to molecular association processes due to electrostatic and chemical interactions. The reIationships derived provide a good interpretation of the effects observed in dipolar liquids with small dipole moments as well as in strongly polar systems. The theory also holds for solutions presenting tautomeric and arbitrary chemical equilibria involving molecular associates and complexes.Most studies of dielectric polarization in liquids are restricted to linear effects. When performed at low electric field strengths, these studies allow the experimental determination of electric permittivity E, molar polarizability and dipole moment. Numerous papers also deal with dielectric polarization and absorption 14-16 in strong fields. Sufficiently sensitive devices record variations in electric permittivity As caused in a liquid by a strong externally applied electric field. The effect, tradi- tionally referred to as dielectric saturation with regard to its formal description in terms of the Langevin function, will be referred to here in general as the non-linear dielectric effect (NDE). A measure of the NDE is provided by the quantity A&/E2,where A&= cE-& is the variation in electric permittivity of the liquid in an intense external electric field E.Obviously the domain of NDE is an extension of that of dielectric polarization to high electric fields. For theoretical considerations, it is convenient to introduce, in place of the experimentally determined quantity A&/E2,the molar NDE constant S, defined as follows : SE-aPp a(F2)’ where Pdipis the molar dipole polarization, and F the local field component in the direction of E. Often, S is expressed in the form of a dimensionless number R,,the NDE correlation factor :2 47rNptR, = S/S1, S, = -~ 45k3T3’ where N is Avogadro’s number, k Boltzmann’s constant, T absolute temperature and pl the dipole moment of the free molecule, or monomer.Hence S1is the value of Sat infinite dilution, and can be computed from eqn (2) once p1 has been determined from polarization studies of dilute solutions. Applying the definitions (1) and (2), R,can be expressed in terms of experimental quantities. For example, from the generally applied local field model of Onsager,17 one obtains the expression : 104 J. MALECKI 105 where c and n denote, respectively, the electric permittivity and refractive index of the liquid for infinite wavelength, and Vits molar volume. A similar expression holds for so1utions.l Further discussion here does not concern macromolecular solutions or liquid crystals. For molecular liquids only, the experimental results given in the work of Piekara et al.,2-9the early work of Kautsch,l and the papers of Thiebaut and Rivail,l0-l2 can be represented in the form of four characteristic curves of R,as a function off, the concentration of the substance under investigation in an inactive solvent.These functions, for the entire mole fraction interval 0 <f< 1 are represented by the full curves of fig. 1. The straight line (dashed) is the hypothetical result for the case of completely free molecules of the liquid, i.e., those occurring only as monomers throughout the concentration range. The experi- mental results closest to this ideal case are those of curve 1 obtained for simple liquids with a small dipole moment.'. This type of R,(f) dependence is accounted for by Debye's theory, which assumes isotropic interaction between the dipolar mole- cule and its surroundings.Ethyl ether is an example. In strongly dipolar liquids such as solutions of nitrobenzene in benzene, A&/E2is positive and R,accordingly negative at high concentrations and for the pure liquid (fig. 1, curve 2). An attempt 2 t F 0.20 Rs -I -2 -3 f FIG.1.-Experimental NDE results for solutions of molecular liquids : 1, weakly dipolar liquids ; 2, strongly dipolar liquids ; 3, liquids with tautomeric equilibria ; 4, strongly associated liquids. Arrows show the direction in which the molecular interaction energy increases. to explain the inversion in sign of R, was made by Piekara,2 assuming non-rigid antiparallel pairs of dipolar nitrobenzene molecules.Negative values of R,over the entire concentration range (curve 3) have been found for substances exhibiting internal rotation, such as 1,2-dihalogeno derivatives of ethane.4 The effect is well accounted for by the theory of Piekara, Kielich and Chelkowski assuming an influence of the electric field on the trans-gauche equilibrium. Finally, strongly associated liquids, typically represented by solutions of the lower alcohols in hydrocarbons, exhibit very 106 THEORY OF NON-LINEAR DIELECTRIC EFFECTS IN LIQUIDS large negative R,values of about -50 in dilute solutions and positive ones of the order of +50 in concentrated solutions and pure alcohols 6-9 (fig. 1, curve 4). An explanation of these large R,values has also been proposed by Piekara on the assump- tion of influence of external electric field on an equilibrium between two types of the hydrogen bond, a normal bond 0-H .. .0 and an ion pair 0-.. . H-Of. Recent-ly, Rivail and Thiebaut l2 published a NDE theory for systems which involve chemi- cal equilibrium, applying it to chloroform +pyridine mixtures in cyclohexane. The experimental results in this case are well described by curve 1 of fig. 1. As seen from the preceding brief review at least five molecular models have been proposed for interpreting the experimental data. They have played an important role in the development of non-linear studies by stimulating a series of original experi- mental papers. However, it would appear that the difficulties in interpretating the results and the lack of a unique theory are major reasons for the slow progress in the use of this method for studying molecular interaction. The aim here is to derive general relations expressing the experimental quantity R, in terms of molecular structural parameters of the liquid.The fundamental hypothesis of this paper, an extension of that of Rivail and Thiebaut,12 is that NDE are specifically sensitive to molecular association, inherent in the liquid phase. THEORY As an argument in favour of this hypothesis we note (without considering the fore- going suppositions) that, with increasing intermolecular interaction energy, R, decreases (fig. 1). This rule is observed both for increasing concentrations of the dipolar component and for transition from weakly interacting to strongly interacting systems.Generally, intermolecular interactions lead to more or less complex associations, characterized by labile equilibria which are sensitive to variations in the external parameters, and thus to changes in electric field strength E. It is considered here that we are in general dealing with an associated liquid, i.e., strong association, or complexation, e.g. by interaction of the charge-transfer type or hydrogen bond type as in water and alcohols, or much weaker dipole association and states resulting by unspecific electrostatic interactions. Let xidenote the concentration of associates or complexes defined as :I8 iN,xi= -, cxi=l (4)n2 L where Ni is the number of complexes or associates in the solution, each consisting of i molecules of the substance under investigation, and n2 = CiiNithe number of mole- cules of the latter.From definition (1) and without specifying a local field model we can write :11* l6 where summation extends over all associates and complexes present in the solution. (pi)E is the mean value of the projection of the dipole moment of an associate on E. Assume now, as Rivail and Thiebaut l2 did for a similar problem, that the associates undergo no deformation in the electric field, i.e., that we are dealing with rigid associates. This assumption appears to be substantiated by the low interaction energy of dipole moments with strong electric fields amounting to about kT compared to the energy of intermolecular interactions, which is usually 102-104times larger (pi)E can now be expressed simply, by a series expansion of the Langevin J.MALECKI 107 function, and describes usual Debye orientation of a moment in the field F: On similarly introducing to R,a correlation factor of molar polarizability R, = Pdip/ Pyip,we have immediately, with the same assumptions,i8 Assume, albeit as a working hypothesis, that the concentrations xi of the associates are functions of the electric field strength F. In other words, we assume that the field causes a slight perturbation of the association equilibrium. A similar assumption restricted, however, to one case of tautomeric equilibrium, was made by Piekara,3* and was applied to chemical equilibrium by Rivail and Thiebaut.12 With the absence of a privileged direction in the liquid, the function has to be symmetric with respect to the field direction F.Thus, we can express our working hypothesis in the form of the expansion : On rejecting in the expansions (6) and (8) all terms higher than those quadratic in F, and using eqn (5), we obtain directly : From (2), we have : -15k2T2A).ax.R, = P14f: xi a(F2) The problem now reduces to that of obtaining expressions allowing one to deter- mine the derivatives dxijd(F2)numerically. To do this, assume quite generally that the associates (complexes) are formed as the result of n equilibria of the following type : K, m,A,+nz,A,+ --* .+m,A,+n~,+,A,+,+ -* (11) As usual, assume that the right-hand coefficients m2in (11) are positive and the left hand ones negative.Denoting by cz the respective molar concentrations of the substrates and products of the reaction described by eqn (1 1) : = NI/NV (12) we express simply the logarithm of the equilibrium constant as : In K, = 1 rnl In c,, (13) where the summation extends over all complexes of the given equilibrium (1 I). Obviously, the system of equilibria (1 1) must be a complete system, i.e., the n constants KE,a = 1, . . . n, uniquely determine the concentrations of all the associates and complexes. Note that, on omitting the small correction for electrostriction, we have : 108 THEORY OF NON-LINEAR DIELECTRIC EFFECTS IN LIQUIDS From the Boltzmann distribution, we have : Nl = exp (-UJRT), where Uzis the energy of the Zth state, and can be expressed as the sum of the energy V,"at F = 0 and the energy of interaction of the dipole p1and field F: Uz = Ur+~~FcosQl, Ql being the angle between the direction of pz and field F.We find the population of the Zth state by summation of those of all possible orientations Q1: N~ = A exp( -U;/~T)1:exp( -$cos Q,) sin Q, dQl. The constant A is determined from the normalization condition. On integration and using the expansion of the exponential function bearing in mind that plF/kT < 1, we obtain from eqn (12), (13) and (16) : In Ka(F2)= In Ka(0)+xini~ P12F2 6k2T2' from which, with (14), we obtain : Hence, taking into account the dipole-field interaction energy in the total energy of the complex, our working hypothesis is justified.We now introduce, for brevity, the notation : y, = 6k2T2 8x1 XI a(F2)' Hence, the complete set of equations of linear and non-linear polarizability for the case of an arbitrary number n of simultaneous equilibria (1 1) is, finally, of the form : Furthermore, from (10) and (18) : R, = 4 Tp;(pf -2.5~~) f whence we have to find n+ 1 quantities yi by solving a set of equations, in which we have n relations of the type : c m,y1 = c md 1 1 resulting from (17) and (18), and one equation : xxiyi = 0 i resulting directly from definition (4) with the notation (18). Summation over i in eqn (19), (20) and (22) includes all associates and complexes in the solution ; in the equations of the form (21), summation over Z includes all complexes forming in an equilibrium of the type (12).J. MALECKl 109 DISCUSSION The relations (19)-(22) do not contain explicit expressions describing the local field, nor did we specify a local field model when deriving them. They are of a general nature, allowing their application to the description of non-linear effects within a wide class of liquid solutions. In particular, eqn (19)-(22), with the assumption of only one tautomeric equili- brium of the type A’ 21 A” yield for R, and R,relationships identical to the equations of Piekara, Kielich and Chelkowski for trans-gauche tautomerism as well as Piekara’s equations for the process of proton shift in the hydrogen bond bridge.In the more general case of a chemical equilibrium (provided the process is fully described by a single equilibrium constant) the set of equations (19)-(22) gives a relation very similar to the Rivail- Thiebaut equation for isotropically polarizable molecules. As an example, we shall apply eqn (19)-(22) to the case of self-associating liquids, such as alcohols. Hence, we particularize the reactions (11) by assuming that suc- cessive open multimers (Aio)and cyclic multimers (Ai,) arise in the reactions : Kio Kic iA, +Aio and iA, +Aic Assuming that pic = 0, eqn (21) becomes : 2 Yio-iY1 = Pi2 -iP1 and yic-iyl = -ip:. Denoting the respective multimer concentrations by xi, and xicwe have from (22) : i= 1 and can determine all the other yio,yicfrom eqn (23) and (24). Too avoid introducing too many parameters Kio,Kicwe put the free energy of a bond in the linear multimers as AFo and that of a bond in cyclic dimers as AFZc= 0.5AF0,in trimers as AFSc = 0.67AF0,and in a higher multimer as AF, = 0.9AFo.Assuming moreover that the dipole moments of multimers pi have the values resulting from the geometry of the latter, the only parameter in our calculations if AFo, which allows determination of all the constants KiO, Kic and hence xio,xic and yio, yic. In turn, eqn (19) and (20) enable us to calculate the quantities R, and R,. Fig. 2 shows the results, calculated for three values of the parameter AFo.For comparison, in fig. 3 are given the experimental curves for R, and R,for solutions of three alcohols in hexane.6p ’9 From fig. 2 and 3 it can be seen that eqn (19)-(22) provide an accurate description of the experimental facts as observed in associating liquids (the present paper is not aimed at giving a detailed picture of the association processes occurring in these alcohols). Eqn (19)-(22) provide an interpretation both of the very large positive and negative values of R,without the necessity of invoking the improbable process of proton tran~fer.~ We shall deal with the application of non-linear effects to the study of alcohol association, as well as to the question of proton transfer in hydrogen bonds, in separate papers. Along similar lines, with appropriately smaller interaction energies AF for dipolar 110 THEORY OF NON-LINEAR DIELECTRIC EFFECTS IN LIQUIDS liquids one obtains from eqn (19)-(22), curves of R,(f) similar to curves 1 and 2 of fig.1. As an example, we give theoretical curves in fig. 4. This problem, too, will be the subject of further papers. -40 -20 0.2 0.: 0.6 0.3 f f FIG.2.-Theoretical Rp(f)and Rdf) curves for strongly associating liquids for three values of the free energy of the intermolecular bond in open multimers. AF, = -0.5 kcal mol-' (l), -1.8 kcal mol-1 (2), -4.0kcal mol-' (3) (1 cal = 4.184 J). F 1 4 3 RP 2 L Qz 0.1, 0.6 Q8 f f FIG.3.-Experimental Rp(f)and Rdf curves for solutions of n-butanol ( 0),6 t-pentyl alcohol ( x ) ' in n-hexane.t-butyl alcohol (0) J. MALECKJ 111 Curve 3 of fig. 1, which was obtained for trans-gauche tautomerism, is adequately explained by the theory of Piekara, Kielich and Chelk~wski.~ For this case (as already noted) the equations (1 9)-(22) give the relations derived by these authors. The theory proposed provides a simple explanation of all non-linear dielectric effects hitherto observed in molecular liquids. It is interesting that this can be achieved with only the simplest assumption of a Debye effect of reorientation of rigid complexes proportional to pf, whereas the expressions in the form -2.5 p:j)i [eqn (20)], signifi-cantly affecting the value of R,,result from the dependence of the energy of the complex on the electric field strength favouring complexes with a larger dipole moment.The 1 0 Rs -1 -2 f FIG.4.-Theoretical &(f) curves for weakly associating liquids. AF,, = 1.70 kcal mol-' (l), 0.85 kcal mol-' (2), 0.6 kcal mol-' (3). present approach has moreover the important advantage of not requiring the intro-duction of a specific molecular model for each particular case. The foregoing dis- cussion leads to the conclusion that nonlinear dielectric effects are specifically sensitive to very broadly defined molecular association and conformational equilibria in liquids and hence these effects should find application as an accurate method of investigating such processes. Available experimental data indicate that molecular association occurs to some degree in all liquid dipolar systems.F. Kautzsch, Phys. Z., 1928,29, 105. A. Piekara, Acta Phys. Polon., 1950, 10, 37, 107. A. Piekara, S. Kielich and A. Chelkowski, Arch. Sci. (fasc. spec.), 1959, 12, 59. A. Chelkowski, Acta Phys. Polon., 1963, 24, 165. A. Piekara, J. Chem. Phys., 1962, 36, 2145. J. Malecki, Acta Phys. Polon., 1962, 21, 13 ; 1963, 24, 107 ; J. Chem. 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