GENERAL DISCUSSIONDr. Manse1 Davies (Aberystwyth) said: Prof. Pople has reviewed the variousof nuclear spin relaxations in n.m.r. spectra and mentioned also molecular dipolerelaxations. It seems appropriate to offer a reminder of the large body of evidence onmolecular behaviour in liquids provided by dielectric relaxation methods.Three factors have to be distinguished: (i) an inertial effect which persists forthe very short time ( x 10-1 sec) needed for a molecule to achieve constant angularmomentum under the joint influence of an applied torque and viscous damping.It was neglected in Debye’s treatment (1913) and has the effect of pulling the dielectricloss factor E” down to zero at a frequency depending on [1/2kTz2].l As this occurswhere the loss is already very low andthe frequency high ( V ~ 3 0 cm-l) it hasonly recently been directly observed.(ii)The angular reorientation of the (rigid)dipole. For this, Debye introduced thesimple exponential factor exp (- t/z) withan unique time constant z. The singlevalued z-function accurately representsthe behaviour of a considerable numberof polar liquids and of polar solutes inmany different solvents. In the complexcompliance (dielectric) plane the simpleDebye relation leads to a semicircular arc(fig. la). Increasing the inhomogeneityof the local field around a molecule (i.e.,at lower temperatures ; at higher viscosi-ties ; with increased molecular anisotropy ;etc.) an apparent range of relaxationtimes whose incidence is weighed by adistribution function G(z) appears.Thisleads to a “ depressed circular arc ” andthe corresponding distribution functionscan be deduced in alternative forms+ &“FIG. 1.(Cole-Cole ; Fuoss-Kirkwood ; etc.). Further departures are found in some liquidsoften (not always) of complex structural character (di- and trihydroxylic molecules)or in a supercooled (i.e., glassy) state. These lead to “ skewed-arc” plots in thecomplex plane (c). Their analytical representation and the corresponding G(z) hasbeen given by Davidson and Cole, and alternative functions considered by Higasi.Important models for the co-operative molecular relaxation in liquids have beengiven by Kirkwood, Frohlich and others. Most significant is the development byCole and Glarum of a model where re-orientation is partly a localized molecularrotation promoted by the near presence of a “ defect ” that also diffuses through theliquid.Cole has shown how the treatment based on Kubo’s statistical formalismsuffices to represent many of the types of behaviour encountered in liquids.(iii) Some non-rigid polar molecules show distinctly separate dipole relaxations :usually from the essentially well-separated rates of component (orthogonal) dipoleJ. G. Powles, Trans. Farahy Soc., 1948,44,802.23236 GENERAL DISCUSSIONelements (d). These can lead to the direct evaluation of, e.g., energy barriers restrict-ing rotations of hydroxyl groups within a molecular framework. By transferring suchmolecules to a very " viscous '' medium (e.g., a polystyrene matrix) it becomes possiblegreatly to increase the separation of component motions, i.e., an intermolecularrotation can be fully resolved from the whole-molecule re-orientation.The individualactivation energies of these motions can be evaluated.Dipolar relaxations have been characterized from 10f3 to 10-1 sec. Whilstmost have dispersions broader than that for the Debye single relaxation-time, absorp-tions intimately related to the structure in the liquid state have been found in the10-60 cm-1 region of the infra-red which are of a degenerate resonance form : as such,they are appreciably sharper than the Debye type. The approximate positions ofthese absorptions which are characteristic of the liquid state were predicted by Hillin 1963.2 It is suggested that dielectric absorption results offer a major contributionto the appreciation of molecular behaviour in the liquid ~ t a t e .~Prof. A. Bellemans (University of Brussels, Belgium) said : In relation to Pople'spaper, I would mention that the autocorrelation function of the electric dipole momenthas recently been studied by the method of molecular dynamics by Gancberg, Kohlerand myself. We considered a two-dimensional circular array of about 400 rotatorslocated on the sites of a quadratic lattice and provided with a permanent electricdipole. The equations of motion of this system were integrated numerically on aIBM 7040 computer and therefrom the autocorrelation function was obtained.When the interactions between the rotators are negligible, but the angular momentadistribution nevertheless is Maxwellian, the autocorrelation function is of the formexp (--;'/2), where 7 is a reduced time (with the mean period of rotation as unity).When the rotators interact through strictly dipolar forces only, we observed that theautocorrelation function remains nearly the same as for free rotators, even when theabsolute mean potential energy is about equal to the kinetic one.This shows thatdipolar forces play little part in dielectric relaxation and do not lead to the Debyepicture.On the contrary, simple angular potential other than the dipolar one may bemuch more efficient in this respect and transform the free rotator autocorrelationfunction into a function much more like the exponential decaying function of Debye.This already occurs for rather weak coupling, e.g., an absolute ratio of about 20 %between mean potential and kinetic energies.Frof.H. G. Hertz (Technical University, Karlsruhe) said : Dr. Dwek introducedthe concept of a distribution of correlation times. In addition to the reference toa paper by Waugh given by Dr. Dwek, I would point out that the formulation of adistribution of correlation times should not be used too easily in future n.m.r. work.First, we almost never have a precise knowledge of the time correlation functions ofthe spherical harmonics of second order-or other functions-entering in the theoryof nuclear relaxation. Then, usually one starts from a simple model and the devia-tion of the experimental results from those predicted by this model are ascribed toa distribution of correlation times.However, the intramolecular and intermolecular relaxation rates are generally ofequal magnitude ; they clearly have different dispersion behaviour.The intra-molecular relaxation rate may be caused by anisotropic rotational motion, the latterbeing determined in a compIicated way by two or three microdynamic time constantsG. W. Chantry and H. A. Gebbie, Nature, 1965, 208, 398 ; and later publications.N. E. Hill, Proc. Physic. Soc., 1963, 82, 723.for further references, see C. P. Smyth, Ann. Rev. Physic. Chem., 1966, 17,433GENERAL DISCUSSION 237even if one uses the most simple rotational diffusion model. The intermolecularrelaxationrate depends as well on the special form of translational motion.Theremay be a finite jump-or infinitesimal jump-mechanism, there may be a superpositionof molecular rotation on the translation or partial rotational contribution to theintermolecular rate caused by association effects. These more detailed models arecharacterized by a small number of well-defined microdynamic time constants-withdifferent temperature dependence. All these facts produce a dispersion behaviourwhich generally is not explicitly known. Thus, if one uses the concept of adistributionof correlation times too easily one may veil the information buried in the dispersionbehaviour of the relaxation time in the form of a small number of microdynamictime constants.Moreover, even if one considers the name '' distribution of correlation times "to be another word for the more precise formulation of non-exponential decay of thecorrelation functions, the distribution of correlation times thus determined is notnecessarily the distribution of correlation times in the various environments in thesystem under consideration.The latter correlation times are those one would obtainif the particle is trapped for a sufficiently long time in a given environment. The twodistributions are no longer equal if the residence times of a molecule in the variousenvironments are shorter than the corresponding correlation times. Thus, in pureliquid, a distribution of correlation times should not be ascribed to different environ-ments.Finally, even for a purely translational intermolecular relaxation rate in a highlyfluid liquid we have a " distribution of correlation times " because the correlation timefor a spin-spin vector depends on the length of this vector.Writing z = d2/D in thecase of the most simple diffusion-determined intermolecular rate is only a formal ab-breviation; what determines the relaxation mechanism is the macroscopic self-diffusioncoefficient D and the distance of closest approach d. The relation z = d2/D onlytranscribes the diffusion coefficient into a quantity of dimension time and " micro-scopic '' magnitude.Another comment-which is related to the first one-concerns the separationof the motion of toluene in solutions of free radicals into translational and rotationalmotion of a solvation complex formed by the free radical and the toluene as reportedby Kruger, Muller-Wahrmut and van Steenwinkel. One should be sceptical aboutsuch a procedure if it yields an activation energy of the rotational motion 2+ timesas great as the one for translational motion. The correlation time z, of the rotationalmotion of the complex is roughly given by z,-4nqa3/3kT, a being the radius of thecomplex, thus we would expect to find the activation energy for the rotation to beabout equal to that of the viscosity q which is not very different from the one fortranslational motion.Furthermore, if for acetonitrile we find the correlation time for the rotationalmotion of the CN-bond to be equal to the correlation time of the rotational motionof the proton-proton vectors, then we must conclude that the rotation of the wholemolecule is isotropic.This is so because for an isotropic rotational motion of a rigidbody the correlation time for the rotation of one selected vector is identical with thecorrelation time of any other vector within the rigid body (the molecule). If we hadvery rapid motion about the C-CN bond, then we would find a smaller correlationtime for the proton relaxation. With acetonitrile one other possibility is that theproton relaxation is partly due to spin-rotational interaction. This would meanthat the correlation time as determined for the proton-proton vector considering onlymagnetic dipole-dipole interaction is not correct. This question seems not to besettled experimentally as yet238 GENERAL DISCUSSIONProf.A. D. Buckingham (University of Bristol) (partly communicated): In reply toMagat and Davies, it is true that if a larger value for 8, were used in the approximateequation for the dielectric polarization of a polar liquid (see eqn. (20) of my paper),the corresponding value of g would be smaller. The equation is based on a separationof the " distortion " and " orientation '' polarizations, and for water there may besome ambiguity in this separation. The " distortion " polarization is the meanpolarization induced in the system for fixed molecular orientations, and the conceptdepends on the molecule retaining its integrity when the field is applied; if protonexchange were induced by the field, the resulting polarization could not properly beincluded as distortion polarization. The librational motion of a molecule in inter-action with its neighbours does not contribute to the distortion polarization whicharises from electronic and nuclear motion at optical frequencies.This is the reasonthat, in my table 1, E , for water is only about 1.8. The fact that the resulting valuesof g are in agreement with the Pople model for water lends support to this interpreta-tion of the dielectric polarization of the liquid.The difficulty in assigning a correct value to E , for water has been discussed byHi1l.l She concluded that the infra-red dispersion in water should be attributed todistortion polarization and that the hydrogen bonding raises the dielectric constantby enhancing the polarizability rather than by modifying the dipole distribution.The ambiguity stems from simplifications inherent in the Onsager model in which themolecular dipole in the liquid is related to the permanent moment p and polarizabilitydo) of the molecule :where R is the " reaction " field.The high-frequency dielectric constant enters thetheory through the equationrn = p+a'O'R,Normally a(') is approximately independent of state, but in water it may not be.However, if the interaction is so strong that do) is increased by a factor of about 2.5,it is difficult to believe that the gas value of p is relevant.In reply to Stecki, it is true that the Kerr constant of a dipolar fluid is dependenton the three-particle distribution function.Studies of the relaxation of the Kerrconstant would give information about the time dependence of both P,(cos y12) and($ cos y12 cos 713 - cos 723), and might be a useful source of information about cor-relation times in liquids. It would be interesting to compare the results with studiesof dielectric relaxation (which depends on Pl(cos y1 J) and nuclear magnetic relaxationwhich is also related to P,(cos y12).The three-particle distribution function is also involved in dielectric relaxation inmonatomic fluids, for in this case, an interacting pair of molecules has no dipolemoment, whereas a trio does. However, we have little knowledge of the dependenceof the dipole moment on the positions of the three atoms, and the effect may be largelydetermined by short-range interactions.Dr.G. H. Findenegg (Bristol University) said: I wish to ask Dr. Matheson whatassumptions are made as to hindered rotation around the C-C bonds of the n-alkanemolecules in the calculation of the residual heat capacities given in fig. 2 of theirpaper. In the liquid state these internal librational modes are affected not only bythe intramolecular potential barriers but also by surrounding molecules ; this mayN. E. Hill, Trans. Faraday SOC., 1963, 59, 344GENERAL DISCUSSION 239alternatively be described in terms of a great number of conformational isomers ofdifferent potential energy. By raising the temperature, conformations of low energywill be converted into others of higher energy.The corresponding contribution tothe heat capacity cannot be calculated at present, and this forms a major problem foran interpretation of C, of these compounds. However, it is improbable that chainmolecules can rotate as one unit in the liquid state.Dr. A. J. Matheson (University of Essex) said: In “ simple ” liquids such aschloroform or toluene it is unlikely that the vibrational heat capacity will be signi-ficantly different from that of the substance in the ideal gas phase at the same tempera-ture. In such liquids a structural contribution to the residual heat capacity existsonly below the Arrhenius temperature. We have assumed that in the n-alkanes also,the vibrational heat capacity is little affected by the transition from the gas to theliquid : it is then found that, as in the “simple ” liquids, a structural heat capacityexists only below the Arrhenius temperature.Dr.R. A. Dwek (Oxford University) said: Hertz comments on the use by some auth-ors of a distribution of correlation times to interpret results which cannot be explainedon the Kubo and Tomita theory with a single value of 7,. It is true that there mustbe other models which could equally well be made to fit the experimental results. Allthat can be said is, that in many of the systems concerned, the postulate of a randomdistribution about a mean correlation time is a plausible one on which to base aninterpretation. Some examples are to be found under ref. (3) of our paper.Dr. I. Henderson (University of Southampton and Defence Research Board, Ottawa)(partly communicated). Smith has noted the importance of determining the effectof volume changes on transport properties.Recently developed techniques for high-pressure studies have already given rise to many data relating to isothermal pressurecoefficients and isochoric temperature coefficients, the interpretation of which islikely to lead to further insight into the structural features of liquids. The transportprocess most amenable to precise measurement is that of conductance, and over thelast few years, Hills and his co-workers have studied the variation over a wide rangeof pressure and temperature of the conductance of a number of aqueous and non-aqueous systems.Presented here are some preliminary results of a numerical analysis in progress atSouthampton of these and other data relating to transport processes.The influenceof isothermal volume changes, and in particular of changes of “ free volume ”, onfluidity, conductance and diffusion, has been the subject of several semi-empiricaltheories, one of the most recent being that of Macedo and Litowitz who relatedthe jump probability to the product of a free volume term and a Boltzmannfactor. Following Cohen and Turnbull,2 they expressed the free volume term asexp ([Y- Vg]/Yg), where V is the molar volume and Yg a corresponding limitingvolume characteristic of a glassy state. This relationship has not been found to beappropriate for ionic transport in a wide range of systems in which the free volumedefined as (Y- Yo) (where Yo is the low temperature volume of the solid) representeda large fraction (8-20 %) of the total volume.With solutions for which Walden’srule is a good approximation, namely those for which the solute ions are large inP. B. Macedo and T. A. Litowitz, J. Chem. Physics, 1965,42.245.M. H. Cohen and D. Turnbull, J . Chem. Physics, 1959,31, 1164240 GENERAL DISCUSSIONcomparison with the probable size of solvent voids, a simple relation similar to that ofBatchinskishould be valid. This model assumes that concurrently with ionic displacement thereoccurs a co-operative motion of solvent molecules and that the hydrodynamic pro-perties of the system are therefore involved.1/11 = A(V-VO)/V (1)Og2O t/ 0.001 1 I I I0 5 0 I00&IFIG.1 .-Limiting equivalent conductance of potassium picrate in dimethyl formamide againstIn the present analysis, values of a parameter V' were determined to yield the best(V- V')/V. *, 25°C; 0, 35°C; 0,45"C ; x , 55°C ; 0, 65°C.fit to the isothermal relations.where A is the equivalent conductance and K(T) a density-independent proportionalityfactor. The resultant values of V' were substantially temperature-independent, sothat K(T) could be further expressed aswhere E,, is an isochoric energy of activation. Fig. 1 shows the isothermal data ofBrummer for potassium picrate in dimethyl formamide plotted as a function of(Y-67*34)/V (corresponding to a pressure range of 1000 atm). Fig. 2 shows thesame data computer-plotted against values calculated from the equationSimilar analyses for tetra-methyl and tetra-butyl ammonium picrates in dimethylformamide and for four tetra-alkyl ammonium picrates in nitrobenzene (using thedata of Barreira 3), showed that all of the data for each system could be fitted by asimilar single relationship.When the solute ionic species are small in comparison with the dimensions ofsolvent vacancies, Walden's rule is a poor approximation.Ionic motion can thenA = K(T)(V- V')/V, (2)K(T) = K' exp (- EJRT), (3)A(calc) = 3.33 x lo3[( Y- 67-34)/V] exp (- 1100/RT). (4)A. J. Batchinski, 2. physik. Chem., 1913, 84, 643.* S. B. Brummer, J. Chem. Physics, 1965,42, 1636.F. C. Barreira, D.Z.C. Thesis (Imperial College, London, 1964)GENERAL DISCUSSION 241occur without concurrent rearrangement of solvent molecules, although such solventmotion must take place subsequently.In such a situation, the ionic jump probabilityshould be proportional to the probability that a vacancy exists adjacent to an ionin the plane normal to the direction of the applied field. Thus the overall conductanceequation should contain a factor ([ V - Y'] /V)', correspondingto the pv of MacedoandLit0witz.l have shown thatthis analysis is applicable to a number of ionic solutes in methanol.Preliminary calculations using the data of Howard5 0 I00A0FIG. 2.-Conductance calculated from eqn. (4) against limiting equivalent conductance ofDr. F. W. Smith (N.R.C., Ottawa) said: I would like to raise a general questionof the physical interpretation of volume parameters in liquids, notably activationvolume.These parameters are introduced into equations of thermodynamic type asempirical scalars with the dimensions of volume, e.g., 20 cm3/mole. A molecularliquid is an aggregate of closed surfaces moving in euclidean space, and the questionis " what is the geometric property or function of this aggregate that can be said tohave this value of 20 cm3/mole under the conditions of the experiment?" Theanswer evidently involves integral or statistical geometry, but a crude analogy suggeststhat an activation volume function may resemble a deformation tensor, at least tothe extent of being characterized by more than one invariant or eigenvalue. Differentexperiments on the same liquid may therefore yield different scalar " activationvolumes " which may be different invariants, or different functions of the invariants,of the underlying geometrical function.Dr.B. Cleaver (Southampton University) (communicated): Dr. F. W. Smith hasraised the question of the physical meaning of the activation volume for transportprocesses in liquids. It is wrong to ascribe to experimental activation volumes thesimple meaning which the term imp lie^.^ However, the empirical meaning of thisparameter is unambiguous. The relative importance of the isochoric activationenergy and the activation volume in determining the temperature dependence of atransport process at atmospheric pressure may be distinguished in the following way.The variation of the transport parameter with temperature at atmospheric pressuremay be represented by an Arrhenius equation.Using the self-diffusion coefficientpotassium picrate in dimethyl formamide. *, 25°C ; 0, 35°C ; 0,45"C ; x , 55°C ; 0, 65°C.l P. B. Macedo and T. A. Litowitz, J. Chem. Physics, 1965,42,245.B. Howard, Ph.D. Thesis (University of London, 1963).see for e.g., D. Lazarus and N. H. Nachtrieb, Solids under Pressure (McGraw Hill, 1963), p. 47242 GENERAL DISCUSSIONas an example,D1 atm = Do ~ X P (-EpIRT)*Defining the activation volume asand the isochoric activation energy asthenAV* = - RT[a In D/dpIT,E, = -R[d In D/d(l/T)lV,Ep = E,,+(aT/P)AV*.(This is strictly true in the limit of zero pressure, but is a sufficiently good approxima-tion for pressures up to about 1 kb.)Substituting (2) into ( l ) ,D1 atm = Do exp (- E,/RT) exp (- nAV’ jRT), (3)where n = aT/P, the “ internal pressure ” of the liquid.Similar equations applyfor the fluidity 1 / q , and for the equivalent conductance A for ionic liquids.The table shows data for different types of liquid. On substituting these valuesof E, and nAV* into (3), the relative importance of the two exponential terms changesdramatically as we pass down the table, in a way which one could not have predictedin the absence of experimental data. This reflects real differences in the microscopictransport EP E V nAV *process (kcal/moie) (kcal/mole) (kcal/mole) ref* liquidCC14 fluidity 3.02 1 -22 1.80 2benzene fluidity 3.24 1.12 2.12 2chlorobenzene fluidity 2.83 0.95 1-88 2ethanol fluidity 3.83 2.10 1 -73 2butanol fluidity 6.40 4-25 2-15 2mercury diffusion 1-00 0.96 0.04 3gallium diffusion 1.12 0.98 0.24 3fused LiN03 conductance 3-42 3.31 0.11 ownworkfused CsN03 conductance 3-69 1 -09 2.60 (unpublished)mechanisms of transport, but the exact nature of these is obscure until an adequatetheory of the liquid state is available.In the meantime, I suggest that the transportprocesses in these liquids be described as “ energy restrained ” or “ volume restrained ”according as E, is greater, or less, than zAV*. This distinction is based directly onempirical data, and can be applied to any type of liquid.The terms ‘‘ energy limited ” and ‘‘ volume limited ” have been used by Macedoand Litovitz in a slightly different sense to that proposed here. Their usage isbased on a model for the transport process and on an expression for the size distribu-tion of holes in the liquid, so it breaks down if these are incorrect. The purelyempirical terms suggested above are more appropriate at this stage, when satisfactorytheories are lacking for some types of liquid and experimental data are relativelysparse. There seems little chance that successful theories of transport will be developeduntil experimental investigations have been extended to cover a wider range of liquids.Transport measurements made at atmospheric pressure give no indication that thedistinction drawn above even exists, and there is need for more work on pressuredependence of transport processes.P. B. Macedo and T. A. Litovitz, J. Chem. Physics, 1965,42,245.M. K. Nagarajan and J. O’M. Bockris, J. Physic. Chern., 1966,70, 1854