ISSN:0301-5696    

DOI:10.1039/FS97711FX001

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Molecular Motions in Liquids Introductory Review and Analysis A Hydrodynamic Slant* BY D. KIVELSON University of California Los Angeles U.S.A. Received 31st December 1976 1. INTRODUCTION Had this conference been held a decade ago the direction followed as well as the symbols and jargon employed would have been very different. Perhaps and even hopefully this shift indicates an expansion in our knowledge of molecular motions in liquids. The study of molecular motions in liquids is in some ways an alarmingly difficult problem; in contrast to the situation in gases a probe molecule in a liquid is always in strong interaction with at least several neighbouring molecules while the simplifica- tions in solids which are associated with long range order are largely absent in liquids.The complexity of the molecular movements are to some degree masked by the averaging and smoothing processes underlying the phenomena observed in the laboratory with the consequence that simplified models can usually be adjusted so as to fit the data; however there is often a non-uniqueness and a lack of fundamental information associated with such modelling largely because of the “coarse-grained ” character of the observations. Nevertheless some progress has been made in our understanding of molecular motions in liquids although to be frank it seems to me that since the pioneering work of Einstein and Debye it may not be commensurate with the effort that has been made. Perhaps at this conference we can evaluate the “ state of the art ” in this discipline.I would like to take this opportunity to present an overview of some aspects of the study of molecular motions in liquids; the view is personal and perhaps somewhat idiosyncratic; its extent is limited by my own interests and learning which have been largely in the area of molecular rotations and I am sure by the reader’s patience. Nevertheless I hope the outline will be helpful in bringing a few common objectives into focus. I will from time to time indicate what I believe to be problems that have as yet eluded explanation; perhaps some of them will be cleared up at this Symposium. The references as well as the discussion are far from complete and those included are presented only as entries to the literature; if I refer to my own work too often it is only because I am probably less prone to misquote it than the work of others.* Supported in part by the National Science Foundation. MOLECULAR MOTIONS IN LIQUIDS 2. EXPERIMENTAL ADVANCES Several new tools have been added in recent years to our experimental workshop. I will mention a few of these. The most dramatic one is undoubtedly the high speed computer and the develop- ment of techniques for carrying out molecular dynamics calculations. Although only one paper on this subject is to be presented at this symposium nearly everyone refers to " computer experiments " because much more detailed data can be assem- bled for comparison with theory than are available from Mother Nature's fluids. The entire subject of " long-time tails " to be discussed at this symposium has I believe still been found only in computer-liquids.Another relatively new technique is her light scattering ; because of the extraordinary monochromaticity and coher- ence of laser light it has opened up new avenues which have been and are being rapidly and extensively exploited-a number of papers to be presented to this audience are connected with light scattering. Thirdly neutron scattering potentially one of the most rewarding of techniques for the study of molecular dynamics in liquids has so far yielded disappointingly little but in part this may be because relatively few people interested in liquids have had the enterprise to associate themselves with laboratories equipped with neutron beams.Fourthly short-pulse spectroscopy a field that has developed greatly in the past few years gives great promise but has not yet added much to our field-perhaps another order of magnitude reduction in pulse times will open up new vistas. Finally the interest in and consequent understanding of resonant Raman and Rayleigh phenomena may be of considerable value to the study of molecular motions-this is a field that is still in its infancy; Dr. Madden will tell us something about its future. 3. CORRELATION FUNCTIONS Today nearly everyone in this field communicates in the language of correlation fun~tions.l-~A correlation function or its Fourier transform is almost always what one measures; all interesting transport and relaxation data particularly at or near equilbrium can be conveniently described in terms of correlation functions.If A(t) is a stochastic process perhaps a molecular velocity q(t) then the corresponding auto-correlation function is GA(t) where { ) indicates an equilibrium ensemble average. Correlation functions are averaged quantities and so they exhibit " smoother " behaviour than do properties of individual molecules ; consequently although an ab initio calculation of a corre- lation function would require the solution to the time-dependent problem of interacting particles many of the salient features of the time-dependence of correlation functions can be obtained from simple approximate theories. In fact as mentioned above one often finds so few salient features that a host of theories are equally applic- able; for example exponential or Gaussian behaviour are the norms for correlation functions.Spectroscopic experiments yield the Fourier transforms IA(co) of correlation functions while correlation spectroscopy and pulse-techniques yield data in the time domain. D. KIVELSON Classical transport experiments are more limited in that they yield transport co- efficients which are zero frequency transforms Za(0) of correlation functions. Al-though in principle it makes no difference whether one carries out experiments in the frequency or in the time domain since the data in either domain are limited the Fourier transforms between them are uncertain; there may therefore be reason to study the same process as a function of both time and frequency.Accurate determination of GA(t)or IA(co) is not trivial. Many of the relevant spectroscopic data are relatively featureless-an almost Lorentzian line with be- haviour in the wings which is often difficult to determine with appreciable confidence because of signal to noise difficulties. Instrumental misalignment can cause apparent spectral asymmetries and the deconvoluting process required to remove the effects of instrumental lineshapes from the observed ones introduces further uncertainties. Background spectra or baseline adjustments often result in non-unique spectral fits even with a single Lorentzian shape function and dramatically so when a two or three Lorentzian shape fit is attempted. Second moments may sometimes be well deter-mined but it is my belief that usually the treatment of the important inaccessible high frequency contributions is biased by a model-based view of the time-dependence of molecular motions.However it is probably true that differential data data represent- ing the variation of parameters from one temperature or pressure to another or from one substance or solution to another are often reasonably dependable. My conclu- sion from these considerations is that by and large it is a simpler matter to study variations in behaviour of different systems within the framework of a given model than it is to determine unambiguously which models are most applicable. The experimental data yield correlation functions or their Fourier transforms but the interpretative study begins by determining what correlation functions have been measured; the next step is the association of a directly measured but perhaps only peripherally interesting correlation function with one of primary interest.To illustrate in dielectric relaxation measurements we first determine a spectrum Z(cr>) and then in a still somewhat uncertain fashion to be discussed at this Symposium,4*' we associate the time-dependent process with dipolar correlation functions and we then usually assume that the dipolar relaxation is totally dependent upon molecular rotation so that we can relate the data to autocorrelation functions of first rank rotational tensors. In magnetic resonance experiments the interpretative uncertainty is less but the " interesting " information is still more indirectly related to the experi- mental data than in the dielectric case.To be specific we associate the relaxation of magnetization which is often exponential with spin auto-correlation functions,6 where coo is the Larmor frequency; kzcan be related to various molecular motions and in each case we must try to determine which molecular motions are important. For example in n.m.r. measurements on nuclei with quadrupole moments & is proportional to a second rank rotational tensor.6 In some cases it is not completely clear which processes are dominant in Raman spectroscopy both vibrational and rotational effects may play significant roles particularly in resonance scattering;' and in proton magnetic resonance experiments molecular reorientation angular momentum and translations may all contribute appreciably.Light scattering yields information on second rank tensors dielectric relaxation on first rank.9 MOLECULAR MOTIONS IN LIQUIDS Some measurements such as those in magnetic resonance experiments yield single particle or molecular correlation functions (A,(t)Ai(O)) where Aiis a property of the ith molecule. Other measurements such as those from light scattering yield many-particle correlation functions Incoherent neutron scattering yields single particle correlation functions and because of the short wave lengths used coherent neutron scattering yields intermediate types of correlation functions i where q is the propagation vector and rj is the position of thejth particle.Both the single (or molecular) correlation functions (A,(t)A,(O)) and the inter-particle correlation function 2 (A,(t)A,(O)) are of physical interest. Few reliable studies i# i of the interrelationship of these quantities have been carried out. Strauss and Gabelnickg have made use of Raman and Rayleigh scattering to study the single and inter-particle correlation functions for second rank rotational functions and infrared spectroscopy and dielectric relaxation for the corresponding functions for first rank rotational functions but relatively little experimental work of this kind has been performed. In neutron scattering work the incoherent scattering yields single particle correlation functions and the coherent scattering many-particle functions of the type indicated in eqn (6) but these experiments though in principle more rewarding than all others are difficult and the results scarce.'' Resonance techniques such as infrared and Raman spectroscopy and magnetic resonance yield single particle correlation functions whereas Rayleigh scattering and dielectric relaxation yield multiparticle correlation functions.I shall have more to say about this separation into single and inter-particle correlation functions. 4. HYDRODYNAMIC THEORY In recent years there have been a number of theoretical developments relevant to the study of molecular motions in liquids. Early in this century the work of Einstein and Debye treated rotating and translating molecules by means of hydrodynamics the molecules were represented by objects moving in a continuous homogeneous fluid.That so simple a viewpoint could be so successful is an indication of the genius of its creators. The more recent literature prior to 1965 indicated a trend away from hydrodynamic approaches and a move towards the introduction of molecular modelsrell models in which the local environment was clearly not continuous and homogeneous; jump models in which details of environmental impact could be obscured by concentrating on the stochastic behaviour of the labelled molecule and extended diffusion or gas-like models in which the probe molecule moves in a field- free manner except when subjected to impulsive collisions. All of these theories have had some success a result which in itself suggests as already indicated the non- uniqueness of these interpretations.Of late there has been a tendency to revert to hydrodynamic theories-generalized hydrodynamic theories-and much of the emphasis today lies in this direction. There are some compelling arguments in favour of the hydrodynamic approach and there are some obvious limitations. Since hydrodynamic or linear transport ap- proaches to molecular relaxation in liquids have been popular and quite successful D. KIVELSON and have been used by almost everyone presenting papers at this Symposium a brief discussion of such considerations would seem to be appropriate. In its simplest form linear transport will indicate an equation of motion where M is independent of time and is often called a transport coefficient or a relaxa- tion frequency.Of course in this case GA(t) is an exponential exp(-Mt) and IA(m) is a Lorentzian with half-width Re(M) and frequency shift Im(M). Interesting examples of such equations are the Bloch equation,6 where A is the spin component S+,and Fick's second law in Fourier transform space where c is the Fourier transform of the concentration. In some cases the time dependence of G,(t) is described not by one but by a set of coupled linear transport equations. We can think of A as a column vector with components {Al Al A3 . . .> in which case {A(t)A(O)) is a square matrix and M is a transport matrix.ll As an example we might consider A = (c e) where e is the Fourier transform of the energy density in which case where Gec(t)= (e(t,q) c(0 -4)) and G,,(t) = (c(t q) c(0 -4)).These are the coupled transport equations for mass and thermal diffusion.* Thus if A represents a set consisting of r-component variables then there are r-coupled transport equations M is an r x r matrix and the solution of GAA(t)can be represented as a sum of r-exponentials of the form exp (-Ant) where the r values of A, are the eigenvalues of M G = Za,,e-h'. n The values of An can be complex. The spectra IAA(m) are then represented by r-generalized Lorentzians i.e. Lorentzians with a "normal " and a " dispersive " contribution :l3 With a few of these generalized Lorentzians one can simulate a wide variety of spectral line shapes.14 The best known set of such linear transport equations are the linearized hydro- dynamic equations of fluid flow the Navier-Stokes equation^.^^ Linear transport equations are valid for "slow " modes of the fluid motions that are slow compared * We choose variables such as energy density on the basis of their molecular significance rather than variables such as temperature which are often more convenient thermodynamically.'2 Ap is the difference in chemical potential of the two interdiffusing species.MOLECULAR MOTIONS IN LIQUIDS to most molecular times. Conserved quantities such as number density concentra- tion in the absence of chemical reaction momentum density energy density and angular momentum density though not conserved in a finite volume of size q4 are almost conserved and fluctuate slowly in large volume elements i.e.for small q. Electron spins and even more so nuclear spins decay slowly since they are only weakly coupled to the rest of the system. It is also found that molecular orientations as described by Wigner functions DgA(sl,) of the Eulerian angles of thejth molecule16 often vary quite slowly. For all these slow variables hydrodynamic descriptions may then be feasible. For individual molecular velocities angular momenta and intermolecular interactions the fluctuations may well be very rapid comparable with those of many other molecular motions and hydrodynamic descriptions of these quantities are likely to be less successful. One turns to hydrodynamic descriptions for a number of reasons. Often one measures the fluctuations of conserved quantities and expects hydrodynamic descrip- tions to be applicable ; for example the Rayleigh-Brillouin light scattering spectrum can be related to the autocorrelation function of the number density whose time dependence is given at low q by the three coupled linear hydrodynamic equations for number momentum and energy densities.*J5*17 Spectra associated with spin resonance or molecular reorientation are often Lorentzian out to several half-widths which suggests that a simple single linear transport equation represents the equation of motion adequately i.e.the Bloch equations for spin resonance and the rotational diffusion equation for molecular reorientation. Whatever molecular property Ai we choose to study the collective quantity CAI associated with that property is more likely to be properly described by a linear transport equation than is the molecular one; the collective quantity gives rise to many-particle correlation functions that are macroscopic (hydrodynamic) in character and more likely to be conserved and slowly varying than are the corresponding molecular properties.For example the behavi- our of (vxj(t) u,,(O)) is thought to be extremely non-e~ponential,~~*~~ whereas that of (vxj(t)zuxk(0)exp (iq[Yi(t) -yk(0)])) where q is in the y-direction is proportional to k the exponential exp(-q2qt/p) which arises from the Navier-Stokes equation for the conserved quantity [~v,,exp(iqyJ].(q is the coefficient of shear viscosity and p the k mass density.)15 Since what we often measure are correlation functions of collective slow variables we are naturally led to hydrodynamic descriptions but we also often find it useful to extend the description to what are hopefully slow molecular quanti- ties.Thus in depolarized Rayleigh scattering we focus on the collective mode 2 DZ,,(aj),*but we often also apply a linear transport (rotational diffusion) descrip- j tion for the molecular variable Df (Qj)which enters into the Raman and magnetic resonance problems.6 Perhaps the most outstanding success of hydrodynamic treatments of molecular motions is represented by the Stokes-Einstein relation; the self-diffusion constant D of a macroscopic sphere of radius r translating in a continuous homogeneous fluid of viscosity q is20 D = kT/Gnqr (1 1) where k is the Boltzmann constant.Although this equation holds rigorously only if r is far greater than the size of the solvent molecules it is found that it holds quite * DP! is a Wigner rotation function of order 2 and represents the three Eulerian angles for the jth molecule. D. KIVELSON well even for small diffusing molecules. One often hears how real molecular systems deviate from the Stokes-Einstein theory but it seems to me that the enormous success of this simple relationship is most striking and argues compellingly for the application of hydrodynamics to molecular problems. It would be wasteful to disregard the simplification in the correlating of data that results from the use of this expression. On the other hand we must be careful not to assume that because the Stokes-Einstein relation works that diffusing molecules behave like ball-bearings.Various modifica- tions to eqn (11) can be introduced within the limits of simple hydrodynamics to account for the shape of the diffusing object and the " stickiness " of its surface. The rotational correlation time 71 defined by represents the time scale for rotational diffusion. Though not as successful as the Stokes-Einstein relationship the Debye expression for r1,*I 6vll 72 = kTZ(Z + 1) where q~ is the molecular volume is eminently useful. Strictly speaking it too is valid only for a macroscopic object in a continuous homogeneous medium but even for molecular rotations it appears to hold at least zlT/qis relatively independent of temperature and pressure for a rotating molecule in a given Corrections can be made for the shape of the rotating object and the stickiness of its surface.Below we shall modify the Debye expression somewhat and discuss its limitations. Why do the Stokes-Einstein and Debye expressions work? This is not completely understood. It does appear that the larger the size of the diffusing molecule relative to the size of the solvent molecules the better the agreement. Bedeaux and Mazur have shown that in the limit r -+ m the diffusion constant of a molecule does indeed approach the Stokes-Einstein value ;26 although the analogous procedure has not been carried out for rotational motion Keyes and Oppenheim have shown that as r -+ m z2is proportional to v.27 Though hydrodynamic approaches have had considerable success and I shall say more about them below it is molecular information that ultimately interests us.5. EXTENSIONS AND CRITIQUE OF HYDRODYNAMIC THEORY As we have seen if eqn (7) holds GA(t) is exponential and IA(m)is Lorentzian. But suppose eqn (7) does not hold; suppose that we have to generalize the equation so that 11*28*5 This equation allows for non-local temporal behaviour i.e. the kernel KA(t-z) is a memory function which relates the behaviour of the system at time t to its past properties at time 2. If the system has a very short memory then KA(t)decays very rapidly KA(f)= MS(t) and eqn (14) reduces to eqn ('7). A possibly more useful way of displaying eqn (14) is in its transformed state IA(m) = W(W)-icor'G(0) (15) MOLECULAR MOTIONS IN LIQUIDS where the frequency dependent transport coefficient M(o) is M(o) =I," K,(t)eiw'dt.If M(o) is frequency independent eqn (15) when transformed reduces to eqn (7). If M(o) is frequency dependent G,(t) will not be describable by a simple exponential and IA(co)will not be Lorentzian. Currently the most popular approach to generalized hydrodynamics is that due to Mori.ll It is systematic and conceptually simple and can be closely related to physically measured quantities and to physical concepts. It should however be emphasized that there are numerous equivalent treatments all with different formula- tions of a similar narrative all with features that recommend themselves strongly to different research groups and the translation from one approach to another is in itself a challenging task.Mori has developed an expression for M(cz>),and hence for K,(z) in terms of the variable A used to define the problem:" M(w) = [{LA*) + (A A)](AA*)-l where (A A) =\mdteiart{(l -P)exp [i(l -P)Lt]& (1 -P)A*) (18) 0 L is the Liouville operator and P a projection operator.* If for a moment we neglect both the thermodynamic quantity (LA*) and the complications introduced by the projection operators in eqn (18) we find that eqn (15) relates the correlation function of A to the correlation function of A. Hydrodynamic approaches while avoiding detailed molecular calculations and lacking detailed molecular information interrelate measurable transport and relaxation coefficients which is to say they interrelate correlation functions.The Stokes-Einstein expression eqn (1 l) is such an interrelationship D is the zero frequency transform of the autocorrelation function of molecular velocity vj and q is proportional to the zero-frequency trans- form of the autocorrelation function of the microscopic stress tensor 0:15928 where xj is the x-position and pyjthe y-component of momentum of the jth particle. The dominant term in the stress tensor is the first moment of the intermolecular force on the molecule and this term is proportional to ij. The Debye expression eqn (13) is also an interrelationship between transport coefficients T~and q an interrelationship which I shall examine more closely below.If the transport coefficient M(o) is frequency dependent a number of different approaches are possible. One can make use of the experimental GA(z)together with eqn (14) or equivalently the experimental I,(cz>) with eqn (15) and (16) to evaluate KA(f). If &(t) has a simple behaviour such as an exponential or Gaussian decay or a sinusoidal oscillation we somehow feel that it is more fundamental than GA(t),and that we can use this KA(t)as the basis for a theoretical development. If however * If G is an arbitrary function PG = <kA><AA)-' A. D. KIVELSON KA(t)is a complicated function then we can push this forward and write Ki(t)is a second order memory function and its experimental form can be determined from GA(t)or IA(u).This process can be continued in the hope that ultimately in some order a simple delta function memory function will be found and that a simple Markovian theory can be built on the basis of this "fundamental " function. Harp and Berne 30 and as reported at this Symposium Ger~chel,~ have cariied out such calcu- lations. Reducing experimental data to memory functions of varying orders can be a very useful way of presenting the observations; the intepretation of such memory functions in terms of the relaxation of physical quantities may however not always be obvious. The determination of KA(t)may not always be possible because of the scarcity of experimental data and an alternative approach is to choose a simple form often a Lorentzian for the frequency dependent transport coefficient M(co).~~ This is the approach of various relaxation theories,32 and a Lorentzian form for M(u)corresponds to an exponential form for KA(t); of course in some cases &(t) is not exponential and M(m) is not L~rentzian.~ A somewhat different and I believe more promising approach is based on an extension of the set of variables A.For example the Rayleigh-Brillouin spectra which arise from density fluctuations do not exhibit one but three Lorentzians which suggest that the appropriate equations of motion consist of three coupled linear transport equations; two variables in addition to the number density. The linearized hydrodynamic equations for the conserved quantities,-number divergence of momentum and energy densities,-are well-known to be the appropriate components of A.I5 We call the number density a primary variable since it is the fluctuations in this variable that give rise to the observed spectrum and the momentum and energy densities are secondary variables which affect the spectrum indire~tly.~~ Whereas M(u) in eqn (15) is obviously strongly frequency dependent if A is taken to be the number density if A is the set of three density variables (number divergence of momentum and energy) then the 3 x 3 transport matrix M is to a much higher degree of accuracy independent of frequency.Thus by extending the number of variables A in the set we may be able to eliminate or at least reduce the frequency dependence of the transport coefficients.In the theory developed by Mori," all the relevant coupled "slow " variables are included in the set A which is treated as a column vector; if the set A contains all the relevant slow variables it is said to be "complete ",and M(m) is independent of frequency i.e. the correlation functions constituting (&A> decay rapidly. I(.>), G(t),G(O),M(m) and K(t) in eqn (15) and (16) are now also matrices and M(m) is given by the expressions in eqn (17) and (18). In some cases such as the Rayleigh- Brillouin example discussed above the choice of a "complete " set1g is obvious in others it is less clear. Even in the Rayleigh-Brillouin problem the set selected above is not complete; the so-called " Mountain line'' constitutes a broad base to the Rayleigh line and this suggests a fourth variable perhaps the internal energy or a rotational variable or alternatively a frequency dependent viscosity.The primary variable D$,@lj) often does an adequate job by itself of describing the reorientational spectrum of molecules since the depolarized light scattering spectrum may be quite Lorentzian. However in the spectral wings i.e. at high frequency Lorentzian behaviour cannot be correct. At high frequency we are MOLECULAR MOTIONS IN LIQUIDS examining the short time or rapidly fluctuating component of Dgi(Qj) so there may be additional relevant coupled variables which must be included in the set A. A reasonable selection of additional variables might include derivatives of the original so that &;(a,)might be a useful secondary variable to help describe the high frequency region.(3:;is closely related to the angular velocity.) If still another variable is required perhaps &@lj) which is closely related to intermolecular torques could be added. By such a procedure i.e. adding derivatives of the original set of variables varied lineshapes can be simulated but the choice of variables is somewhat arbitrary and the significance of such fits may be somewhat obscure.* Keyes and Kivelson have studied this problem in some detail.14 The method of expanding the “ slow ” set A is useful provided there are strong reasons for adding specific variables and provided the number of variables needed is not too large. Way out in the spectral wings the relevant fluctuations are very rapid which means that a host of molecular processes may be relevant; the resultant line shape is no longer Lorentzian but probably exponential.An extension of the simple formulation of Mori theory can be made by introducing mode-mode coupling. In this procedure the set of variables A includes bilinear trilinear and higher order combinations of the original set of variables. Thus if A,(q) and A,(q) are variables [A,(q’) A,(q -q’)] may also be a variable and there is an additional variable for each value of q’. (Usually one integrates rather than sums over q’; this is one special case where many in fact an infinite number of variables can be meaningfully included). Mode-mode coupling3’ has been used to study critical diffusion37*27 and the so-called long-time tails to be discussed at this Symposium.29 In conclusion we expect hydrodynamic theory to be useful at long times and low frequencies; at very short times and high frequencies its use is questionable even though apparent fits to experimental data can sometimes be obtained by the indiscriminate addition of variables.6. MOLECULAR ROTATIONS DIFFUSION Both because my own interests have been directed towards the study of molecular rotations and because many of the papers to be presented at this conference are so oriented I would like to focus my remaining comments in the direction of rotations. Taking advantage of the widespread applicability of the Debye expression but noting that the effective hydrodynamic volume p is often much less than that expected from other considerations one can conveniently multiply 97 in the Debye equation by an adjustable factor K.Furthermore recent studies have indicated that though zl is linear in the viscosity q the extrapolated q +0 limit is finite. Thus we have a modified Debye expression for rl First we discuss the factor K~.For a given probe molecule in a given solvent K~ appears to be reasonably independent of temperature and pressure; recent very beautiful and careful work by Jonas and co-workers indicates that K changes with density but not with pressure or temperature at constant density.39 It is found that for a given probe molecule K varies with solvent; although the correlation of K with * This is more or less equivalent to Mori’s continued fraction expansion.” D.KIVELSON particular solvents is not clear-cut it appears that IC increases with increased solute- solvent interaction and with increasing anisotropy of solvent molecules but decreases with increasing size of solvent The absolute values of K are difficult to establish because the molecular volume q? is difficult to determine but for reasonable estimates of 9 u < 1. On the basis of hydrodynamic models effectives 9’s can be determined for non-spherical molecules readily for spheroids 23 and by numerical methods for more complicated The analysis of K in terms of hydrodynamic theory can be interesting. Debye assumed so-called “ stick ” boundary conditions in which the layer of fluid touching the rotating probe has the same angular velocity as the probe; in this case u = 1.Recently Hu and ZwanzigZ4 have studied the problem of spheroids rotating under “slip ” boundary conditions in which the fluid flows freely past a stationary probe; for a spherical probe u = 0 but for anisometric ones the viscous medium impedes the object’s rotation. It is convenient to introduce a factor S,22 where E is the hydrodynamic value of ic in the slip limit for an object of a given symmetry; S is then a factor independent of molecular geometry which is 0 in the slip and 1 in the stick limit and can of course assume intermediate values. The absolute value of S is difficult to determine since the exact hydrodynamic geometry of the probe molecule is not readily available. Pecora and co-worker’s light scattering data taken on relatively weakly interacting probe molecules in solvents composed of quasi-spherical molecules indicate that the slip model appears to work quite well i.e.S E 0 and it is reasonably independent of solvent. Numerical calculations by Acrivos and Youngren for molecules such as benzene taking the non-spheroidal shape into account seem to indicate even more clearly that the slip model works quite well.40 On the other hand larger less “inert ” probes in a variety of solvents of different sizes and shapes indicate that S varies between 0 < S < 1; even though the absolute value of S is uncertain for the complicated geometry of the probe the varia- tion of S equivalent to the variation of u discussed above is clear.22 We can think of S as the boundary condition variable which lies between 0 and 1 i.e.between slip and stick. But though the success of the hydrodynamic approach is encouraging it is really molecular information that we seek a picture which will tell us why the hydrodynamic picture holds. One of the encouraging pictures to emerge is the rough hard sphere model investigated in some detail by Chandler.41 A smooth hard sphere would suffer no intermofecular torques and zl would not have any dependence upon solvent i.e. upon q. However any deviations from sphericity be it elongation as in NH or corrugation as in CH4 would enable the solvent to exert torques on the molecule. A model which should be investigated is what might be called the “ irregular hard core molecular” model a model in which the molecules both probe and solvent are treated as hard objects with shapes which simulate those of actual molecules.This model is the molecular analogue of the “ slip ” hydrodynamic model. Of course in the molecular model the non-continuous structure of the solvent can be sensed. As an extreme example of this solvent-sensing consider a “hard core ” spheroid in a solvent composed of large molecules or in one that forms clathrate structures; the spheroid might rotate freely in a large vacancy and the resultant zzwould be independ- ent of q a result not possible in the hydrodynamic case where the continuous nature of the solvent eliminates the possibility of vacancies. (If this situation occurs MOLECULAR MOTIONS IN LIQUIDS S < 0 a non-hydrodynamic result; in the hydrodynamic case S indicates the nature of the boundary conditions but in the molecular case its significance is still uncertain).The irregular hard core molecular model for “ realistically ” shaped molecules might be treated numerically in much the same way that “ realistically ” shaped molecules have been handled in a slip-hydrodynamic framew~rk.~~ Although the variation of S with solvent might at first seem to argue against a hard-core model much as it argues against a hydrodynamic slip picture it must be remembered that in the molecular case the solvent molecules are also non-spherical hard-core objects and that these hard-cores have different shapes for different solvent molecules and they interact differently with the solute.Therefore it would seem that the hard-core model would have considerable potential for explaining the data. Of course it is possible that attractive forces will also have to be included in the molecular model; but as I have tried to argue above though some ‘‘ stickiness” i.e. S> 0 is necessary to explain the data on the basis of hydrodynamics it may or it may not be necessary in the molecular model. At this point it is perhaps useful to write down a molecular expression for K. Within the rotational diffusion picture a limitation to be discussed K = [V/6Nd (qy) (234 / (090) where 0 is the microscopic stress tensor defined in eqn (19) and 7is defined by the relation (F,T)= NI(I + 1) If the dominant term in Dgi behaves like an intermolecular angular acceleration 7is essentially an intermolecular torque independent of I; furthermore if the main component of0 behaves like a moment of the intermolecular force (xjiYj)in the dif- fusional limit we might expect both the torque and the force to be altered appreciably at each intermolecular “ collision ”.In other words the torque and the force should have similar time dependences and K in eqn (23) can be replaced by the equilibrium expression42 K ;2 [ V/6Nq1l(FF)/(aa). (24) This molecular expression for K illustrates a number of interesting features. (1) Since q is associated with a translational phenomenon K is a coupling parameter linking rotational and translational effects. As written in eqn (24) K is the ratio of torques to forces or anisotropic intermolecular interactions to total intermolecular interactions.For molecules with large anisotropic interactions e.g. water we expect K ;2 1 but for molecules with more spherical behaviour e.g. CH4,K < 1. One must remember that the hard core repulsive forces and torques may well dominate the expression in eqn (24). (2) There is reason to believe that the radial dependence of the isotropic and of the anisotropic parts of the intermolecular potential are reasonably similar to each other; therefore as temperature and pressure change the ratio in eqn (24) remains relatively unchanged. * 22939 Eqn (24) indicates that zl is proportional to 11provided the molecule reorients by a diffusive process.If the reorientation is non-diffusive the correlation times for intermolecular torques may not be similar to those for inter-molecular forces and the * Careful work by J. Jonas and co-workers has shown that IC varies somewhat with density but at fixed density it is independent of pressure and temperature^.^^ This is consistent with the dis- cussion above. D. KIVELSON proportionality between zzand q may no longer hold. When K 1 the torques are weak and the collisions that alter the forces may not have much effect upon the torques; in this case the motion is most likely non-diffusional eqn (24) no longer holds and zl is not proportional to q. There is evidence that this is the case but the conditions under which (zJ’/q) is a constant is still an open question.Recently we have studied the behaviour of a very dilute solution of probe molecules in two-component For a number of solvents we find that eqn (21) holds quite well with the intercept 7; E 0 q being the solution viscosity and K2 [XA7C2(A) + XBK2(B)] (25) where 7~:~) are the IC~’s and 7~;~) for the probe molecule in pure solvent A and pure solvent B respectively and x and x are the corresponding mole fractions of the two components of the solvent. We believe that this expression is valid only for reason- ably ideal solutions i.e. in situations where the concentration of solvent nearest- neighbours to the probe molecules is similar to the bulk concentrations. We have interpreted these data by assuming that rotational relaxation even for a molecule is a Zong wave length phenomenon caused by the interaction of the rotating object and hydrodynamic modes of the fluid.However the coupling of the rotating molecule to the bulk fluid is probably a short-range nearest neighbour interaction; IC~ represents the short range coupling and q represents the bulk hydrodynamic properties. Thus a hard sphere molecule cannot couple to the liquid (IC~= 0) but a non-spherical object with ic2 > 0 does couple to the bulk fluid and its relaxation is then determined by the bulk properties i.e. q. Light scattering measurements by Alms et aL40 can also be interpreted in this way but only if the K;S for all the non-interacting solvents used in these experiments are assumed to be more or less equal and to have their slip limit values ix.S < 1. If the analysis above is useful one must still ponder whether the hydrodynamic modes relaxing angular and linear momentum are the same i.e. whether z1is always proportional to q in the diffusional limit. For example what happens near critical points? Are the wavelengths characteristic of zl similar to those characteristic of q? These are still open questions. 7. ANGULAR VELOCITY AND NON-DIFFUSIONAL MOTION If a molecule can rotate more or less freely in a solvent its classical behaviour is called inertial43 and its rotational correlation function may be more nearly Gaussian rather than exponential as in the diffusional case. If the rotational relaxation is non- exponential the correlation time zl defined by eqn (12) is no longer associated with a decay time.For inertial behaviour one expects zzto be proportional to the square root of the moment of inertia of the rotating molecule; such behaviour has been reported for several small m01ecules.~~*~~ If the spectral shape corresponding to the rotation can be analyzed and this has been done for a few small molecules then detailed information about the non- Lorentzian or inertial behaviour of the rotor can be The dependence of z1on q in intermediate cases is not yet understood. The q -+ 0 intercept zl0 in eqn (21) is an intriguing phenomenon first clearly described by Alms et c~Z.,~Owho suggested that it might be an “ inertial ” contribution but it is not easy to say why such contributions should be additive to the diffusional contributions and independent of the intermolecular interactions which affect v.Keyes and Oppenheim have used mode-mode coupling to study rotational relaxation; 27 they obtain “molecular ” plus “hydrodynamic ” contributions to zl-l and their MOLECULAR MOTIONS IN LIQUIDS results do not appear to fit eqn (21). We have tentatively identified a curious pheno- menon related to this intercept the light scattering data for pure triphenylphosphite (TPP)indicates that z2is proportional to tf/Twith zero intercept but though 72 for TPP in CCl at 24" C is proportional to the solution viscosity q it gives evidence of a substantial q -+0 intercept. Even though these data yield many-particle correlation functions and the results might be different if the single-particle correlation functions were extracted the results suggest that the intercept may not arise from non-diffusional behaviour but from some as yet not well-understood property of mixed solvents.Our e.s.r. data appear to indicate that for dilute probe molecules in pure solvents 72 plotted against q/T does not yield a positive q = 0 intercept but for mixed solvents at constant T,the dependence of z2upon q,though non-linear can sometimes appear to be linear with an q = 0 intercept.22 However data suggesting that the intercept does not vanish even for some directly measured single particle correlation times have been obtained.38 At this Symposium Beer and Pecora are reporting new studies bearing upon the intercept^.^^ Angular velocity autocorrelation functions contain information that in some ways is more fundamental than reorientation correlation functions but they are not as readily available as the latter.Three methods have been used to investigate angular momentum correlation functions (1) analysis of high frequency (short time) behaviour of molecular reorientation (2) spin-rotational interaction analysis of magnetic reson- ance data and (3) computer calculations. We look at the first two in turn. The correlation function (Dgi (t) Dgj) is related to the spectrum Z(o) of Dgi by the expression In the rotational diffusion limit the angular velocity relaxes much more rapidly than does D:; in which case I.$ should relax like the angular velocity. Even if the motion is not strictly diffusional in the high frequency regime the long time behaviour of (& (1) bgi (O)*) is not important and (0&2 (O)*> = 41 + 1)<lD212)(G&)) (27) where G,(t) is the angular momentum correlation time.Eqn (26) and (27) suggest how one can obtain information concerning G,(t); the higher the frequency and the more diffusional the motion the more accurate the approximation in eqn (27).47 Interesting results have been obtained in this way unfortunately for the very non- diffusional motion of N2.48 The experimentally determined left hand side of eqn (27) must oscillate i.e. it must have a negative lobe since its zero frequency transform vanishes [see eqn (26)] ; this must be true even if the motion is Lorentzian. One must therefore be careful in the interpretation of the zero in <by!(t) b$i(O)*) and it is not clear as yet how one determines whether the zero is at a time sufficiently short to validate the approximation in eqn (27).49 On the other hand (t) (O)*), though less well determined than (0%(t) (O)*) is more sensitive to rotational behaviour at short times and so may be a good way of presenting data.Non-diffusional behaviour can be clearly detected at short times (high frequencies) in this way.48*s Magnetic spin relaxation is sensitive to spin-rotational interactions and therefore provides a direct probe of the rotational angular momentum of molecules in liquids although the data are not highly precise because the various relaxation mechanisms 7,; =zscJmotion is diffusional and D.KIVELSON must be sorted out. (Furthermore remember the previous comment about the molecular information being once-removed from the experimental magnetic resonance data). The spin-rotational interaction is of the form S.C*J where S is the spin J the molecular angular momentum and C the interaction tensor. In the rotational diffusion limit we expect J to relax much more rapidly than do Sand C so that S(t)*C(t)* J(t) E S*C*J(t). (28) If this approximation is made the spin-rotational autocorrelation function GscJ(t) is proportional to (J(t)J(O)) which is similar to the angular velocity correlation function G,(t). In the diffusional limit,* where I is the moment of inertia and the angular velocity correlation time z is z = Lrn G,(t)dW,(O).In magnetic resonance experiments one measures zscJ If the left hand side of eqn (29) equals zscJ it seems reasonable to assume that the otherwise we assume the rotation to be non- diffusional. Examples of diffusional and noa-diffusional motion have been detected; non-diffusional motion corresponds to small K.~O (Note that the integral over time in eqn (26) yields zero on the left side and consequently the approximation in eqn (27) is not appropriate for determining z,.) In the diffusional limit we find LWG,,(t)dt is very small perhaps even s so that a negative lobe in GscJ(t),and perhaps in G,(t) might be expected however this interpretation is not unique.48 Depolarized light scattering data give strong evidence of non-Lorentzian and hence non-diffusional behaviour at high frequencie~.~~~~~ Induced high frequency anisotropies in liquids composed of spherical molecules to be discussed at this Symposium appear to depend upon bimolecular collisions.s1 Might the wings of the reorientational spectra of non-spherical molecules depend upon two-particle and not multiparticle correlation functions ? Interferometric experiments on non-spherical molecules indicate broad background contributions to the depolarized sharp spectrum arising from molecular reorientation ; the integrated intensity of the back- ground is often comparable in intensity with that of the Lorentzian peak.This appears somewhat mystifying to me since I do not believe that collision induced aniso- tropies vibrational modulation of the polarizability or non-diffusional wings should have that much intensity.Could the background be due to strong cross-correlation effects14932 As the viscosity of a liquid system is greatly increased its time dependent behaviour begins to alter appreciably. The rotational relaxation time which as we have indi- cated often obeys a modified Debye equation quite well at moderate viscosities becomes appreciably smaller at high vis~osities.~~*~~ Brillouin linewidths (and hyper- sound absorption) which according to hydrodynamic theory increase with viscosity * Eqn (29) is another interrelationship among correlation functions. MOLECULAR MOTIONS IN LIQUIDS actually remain reasonably independent of viscosity. And yet the light scattering data indicate continued Lorentzian line shapes.Magnetic resonance data become anomalous at high viscosity; although the rotational correlation time z2 determined from e.s.r. appears to obey the Debye expression quite well the spin rotational zScJ is no longer equal to z as determined from eqn (29).*54*55 In the non-diffusional case we may ask whether we would expect zl-(the area under the Dg autocorrelation function as defined in eqn (12)) z,-'-the area under the angular velocity autocorrela- tion function) or the apparent exponential decay time of the DgL correlation function at long times to be proportional to q. These are still open questions and the high viscosity anomolies discussed above have not I believe been adequately explained. 8. MODELS We turn briefly to the consideration of a few models that have been used to treat non-diffusional rotations.In the Ivanov modeP the molecules reorient by instantaneous random angular jumps. This model is discussed in the paper by Beevers et al. at this Symposium.57 The orientational correlation function is exponential with decay time zl regardless of the size of the angular jumps but the relationship between zzand zo the time between jumps is dependent upon the size of the angular jumps. In this model molecules are presumably not permitted to rotate except in large random jumps when an occasional free volume cavity forms about the molecule. If the time zo between jumps is associ- ated with translational displacements one might expect this time to be proportional to q-'.For small angular jumps zl 9 zo the model is diffusional 2 = z as specified in eqn (29) and zzis proportional to q as indicated by the Debye relation. But if zI is comparable to z, then we might expect z1 to be inversely proportional to i1 while if zzis only somewhat larger than zo the dependence upon q might be inter- mediate between the cases mentioned above. Note that this model gives rise to On the other hand in the Gordon extended rotational diffusion model a classical free rotor undergoes impulsive randomizing collisions with a Poisson distribution of free flight The distribution of free flight times is characterized by a mean timez, the angular velocity correlation time and if this time is associated with '' translational collisions " then it might be proportional to q-l.In the limit z1 9z, the diffusional limit we once again get zI proportional to q and in the limit zINN z, we get the inertial result with zf independent of q. In this model the reorientational spectrum can be Lorentzian pseudo-Gaussian or over-dampened oscillatory depending upon the value of [kTz&/q;the angular velocity correlation function is close to exponential and is clearly not well approximated by {&;(t)bg(O)*).It would appear that the large-jump Ivanov theory might be a good model for the explanation of high viscosity deviations from the diffusional limit and the Gordon model for low viscosity deviations. A closely related theory has been developed in which the environment of the molecule varies in a stochastic fashion thereby modulating the molecular rotation.59s60 The results to be presented at this Symposium by Van der Elsken and Frenke161 put this general approach on a firmer footing since they have studied in considerable detail * Of course one usually does not deal with spherical molecules and the anisotropy of the magnetic parameters and of the rotational diffusion must be considered.We shall not go into the details of this problem here except to note that for a symmetric top molecule approximated as a spheroid the rotation about the symmetry axis is free (XI = 0) in the " slip " hydrodynamic model. 7,. Lorentzian reorientational spectra but does not specify D. KIVELSON the fluctuations of the relevant interactions between the surroundings and the rotating probe molecule.Several attempts have been made to introduce the principal effects of all these models into a coherent entity which can be used to rationalize some of the observed experimental anomolies; some of these are summarized in ref. (14). The qualitative if not the quantitative features can be obtained by a hydrodynamic model based on the three variables DgA and Egi roughly identical with orientation angular velocity and intermolecular torque re~pective1y.l~ 9. ADDING OF HYDRODYNAMIC MODES Since the advent of the laser the study of collective modes in liquids by means of light scattering has been pursued by numerous investigators. At this Symposium both polarized 62 and de~olarized~~ spectra are discussed. The simple fluid hydro- dynamic equations do not seem to be adequate to explain the results even at moder- ately high viscosities; the fact that Brillouin line widths vary rather little with viscosity and that shear waves are sustained gives liquids at high viscosity a distinctly “ solid-like ’’ appearance.It may be worth mentioning how the depolarized VH light scattering spectrum can be discussed within the hydrodynamic framework outlined previously; this is a particularly simple and reasonably successful application of the linear transport approach. Symmetry arguments indicate that the two variables A * ,9 describe the components of the polarizability tensor that give rise to VH scattering and that for symmetric top molecules there is no hydrodynamic coupling between them.8 The spectrum arising from these variables would consist of two superimposed identical Lorentzians but at high temperatures a doublet structure is observed.The doublet structure varies with 4,i.e. with scattering angle and this suggests not only that an additional variable must be included but also that this variable must be a conserved quantity. A candidate variable with the appropriate symmetry is the curl of the momentum density; the resulting coupled hydrodynamic equations describe the observed low viscosity data very well. At very high viscosity however the character of the VH spectrum changes and the simple description fails. An additional hydrodynamic variable could be added but although we still have symmetry guide- lines the additional variable cannot be uniquely chosen because it is not a conserved quantity.A completely satisfactory explanation of these spectra has yet to be obtained.63 The problem of reducing light scattering and dielectric data to molecular cor- relation functions is one that requires the evaluating of local fields; this is a problem that is currently under intensive study in several laboratorie~.~~*~~* 65 The role played by static correlations (D$(Q,)Dgi(Q,)> and dynamic correlations (b$@l(t)) &@&,)) is also one that is being in~estigated.~~*~~*~~*~’ These problems are men- tioned in some of the papers presented at this Symp~sium.~*~*~~ If the molecules have low symmetry it becomes much more difficult to extract unambiguous informa- tion; it is therefore probably advisable to select symmetric rotor molecules for study unless one is specifically interested in the anisotropy of the motion.MOLECULAR MOTIONS IN LIQUIDS 10. CONCLUSION Each experimental technique used to study molecular motions in liquids appears to be related to a correlation function which is slightly different from that associated with other techniques. In limiting cases e.g. when the rotational motion is dif- fusional the differences are often minor but great care must be exercised in equating the results of different experimental studies. Ideally one would like to study a few well-chosen systems by every conceivable technique; one might choose some big molecules which behave classically and diffusionally and some small ones 61 which are inertial or even quantum mechanical.Unfortunately at present one can often not readily apply many different techniques to a given molecule since the requirements for good candidates for magnetic resonance light scattering infrared and Raman studies may be very different. The relationship of flow birefringence (a steady state method) to VH depolarized light scattering is very close,68 and the two methods complement each other but even here the weighting of the contributions of high frequency motions is different in the two cases. Similarly there are great similarities but some differences between HH-depolarized light scattering and acoustically induced birefringen~e.~’ Kerr effect and acoustical measurements are also closely related. What one seeks is of course a consistent picture which explains a wide range of experiments over a wide range of viscosities.From the experimental viewpoint aside from the difficulties of selecting appropri- ate systems and getting good data the problem of how to present the data so that they are of use to others is non-trivial. Most data have been reduced with a particular model or theory in mind and the results presented to the theoretician are usually a best fit to a theory and not necessarily the best representation of the data. Can model-free data be extracted and can they be used to obtain model-free interpretations? I do not believe that such “abstract ” formulations should replace modelling since a reliable qualitative picture may be the ultimate goal in this field but it should be available for the use of modellers of a different persuasion.In conclusion I believe progress will be made in the next few years perhaps not great strides but some advances. R. G. Gordon in Adv. Mag. Res. (Academic Press 1968 N.Y.) vol. 3 p. 1. ’L. P. Kadanoff and P. C. Martin Ann. Phys. (N.Y.) 1963,24,419. D. Foster Hydrodynamics Fluctuations Broken Symmetry and Correlariort Functions (W. A. Benjamin N.Y. 1975). ‘J. M. Deutch this Symposium. A. Gerschel this Symposium. ti A. Abragam The Principles of Nuclear Magnetism (Oxford Univ. Press London 1961). P. A. Madden this Symposium. B. J. Berne and R. Peoora Dynamic Light Scattering (Wiley-Interscience N.Y. 1976). H. S. Gabelnick and H. L. Strauss J. Chem. Phys. 1967,46,396; also personal communication.lo K. Carneiro and J. P. McTague Phys. Rev. A 1975,11 1749. H. Mori Progr. Theor. Phys. (Kyutu) 1965,33,423; 34 399. l2 P. Madden and D. Kivelson J. Statistical Phys. 1975 12 167. l3 D. Kivelson and K. Ogan Adv. Mag. Res. 1974,7 71. D. Kivelson and T. Keyes J. Chem. Phys. 1972,57,4599. B. U. Felderhoff and I. Oppenheim Physica 1965 31 (1965). l6 M. E. Rose,Elementary Theory of Angular Momentum (John Wiley N.Y. 1957). l7 R. Mountain Rev. Mod. Phys. 1966,38,205. R. Zwanzig J. Chem. Phys. 1960,33 1338. l9 R. Zwanzig and M. Bixon Phys. Rev. A 1970,2,2003. 2o A. Einstein Ann. Phys. 1905 17 549; 1906 19 37. 21 P. Debye Polar Molecules (Dover N.Y. 1928). *’B. Kowert and D. Kivelson J. Chem. Phys. 1976,64 5206. D.KIVELSON 23 F. Perrin J. Phys. Radium 1934,5,497; 1963,7 1. 24 C. M. Hu and R. Zwanzig J. Chem. Phys. 1975,60,4354. 25 G. K. Youngren and A. Acrivos J. Chem. Phys. 1975,63 3846. 26 D. Bedeaux and P. Mazur Physica 1974,76,235. 27 T. Keyes and I. Oppenheim Physica 1974,75 583. '* R. Zwanzig Lectures in Theoretical Physics (Interscience N.Y. 1961) vol. 3 pp. 106. 29 J. L. Jackson and P. Mazur Physica 1964,30,2295. 30 G. D. Harp and B. J. Berne Phys. Rev. A 1970,2,975. 31 R. Zwanzig J. Chem. Phys. 1965 43 714. 32 See for example K. F. Herzfeld and T. A. Litovitz Absorption and Dispersion of Ultrasonic Waves (Academic Press New York 1959). 33 T. Keyes and D. Kivelson J. Chem. Phys. 1972,56,1057. 34 W. Weinberg and R. Kapral Phys. Rev.1971 A4 1127. 35 T. Keyes Principles of Mode-Mode Coupling (preprint). 36 K. Kawasaki Ann Phys. (N.Y.),1970 61 1. 37 T. Keyes and I. Oppenheim Phys. Rev. 1973 A8 937. B. J. Berne this Symposium. 39 M. Fury and J. Jonas J. Chem. Phys. 1976,65,2206. 40 G. R. Alms D. R. Bauer J. I. Brauman and R. Pecora J. Chem. Phys. 1973,59,5310,5321. 41 D. Chandler J. Chem. Phys. 1974,60 3500and 3508; 1974,62 1358. 42 S. Tsay and D. Kivelson MoI. Phys. 1975,29 1. 43 W. A. Steele J. Chem. Phys. 1963,38,2404,2411. 44 P. W. Atkins A. Lowenstein and Y. Margolit Phys. Rev. 1969 17 329. 45 W. G. Rothschild G. J. Rosasco and R. C. Livingston J. Chem. Phys. 1975 62 1253. 46 C. W. Beer and R. Pecora this Symposium. 47 K. Kushick and B. J. Berne J. Chem. Phys. 1973,599,4486.J. F. Dill T. A. Litovitz and J. A. BUWO,J. Chem. Phys. 1975,62 3829. 49 D. Kivelson J. Chern. Phys. 1975,63,5034. 50 R. McClung and D. Kivelson J. Chem. Phys. 1968,49 3380. 51 R. A. Stuckart C. J. Montrose and T. A. Litovitz this Symposium. 52 T. Keyes D. Kivelson and J. P. McTague J. Chem. Phys. 1971,55,4096. 53 I. L. Fabelinskii Molecular Scattering of Light (Plenum Press N.Y. 1968) p. 426. 54 J. S. Hwang R. P. Mason L. P. Huang and J. H. Freed J. Phys. Chem. 1975,79,489. 55 D. Hoe1 and D. Kivelson J. Chem. Phys. 1975,62,4535. 56 E. N. Ivanov Zhur. eksp. teor. Fiz. 1963,45,1509 (Sov. Phys. JETP 1964,18 1041). 57 M. Beevers J. Crossley D. C. Garrington and G. Williams this Symposium. R.G. Gordon J. Chem. Phys. 1966,44 1830. 59 J. E. Anderson J.Chem. Phys. 1967,47,4879. 'O S. H. GIarum and J. H. Marshall J. Chem. Phys. 1967 46,55. 61 J. Van der Elsken and D. Frenkel this Symposium. 62 T. Dorfmiiller G. Fytas W. Mersch and D. Samios this Symposium. 63 G. Searby P. Bezot and P. Sixou this Symposium. 64 A. K. Burnham G. R. Alms and W.H. Flygare J. Chem. Phys. 1975,62,3289. 65 T. Keyes J. Chem. Phys, 1975,63,815. 66 S. L. Whittenburg and C. H. Wang personal communication. 67 D. Kivelson and P. A. Madden Mol. Phys. 1975,30 1749. 68 D. Kivelson T. Keyes and J. Champion MoI. Phys. 1976,31 221 69 R. Lipeles personal communication.

ISSN:0301-5696    

DOI:10.1039/FS9771100007

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

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

ISSN:0301-5696    

DOI:10.1039/FS9771100026

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Molecular Relaxation Times from Time-variations of Dispersion and Absorption of Third-order Electric Polarisation in Liquids BY WEADYSEAW JANUSZ BUCHERT ALEXIEWICZ AND STANMAWKIELICH Nonlinear Optics Division Institute of Physics A. Mickiewicz University 60-780 Poznan Poland Received 2nd August 1976 Processes involving an increase in nonlinear 3rd order electric polarisation in dipolar liquids by the action of external electric fields are considered in the approximation of Debye’s rotational dif- fusion model of geometrically spherical molecules. For times comparable with the Debye times of molecular orientation we show that the dispersion and absorption of the above effects are described by the well known Debye factors &(a)and supplementary factors Rlk(m),and that the setting in of steady state polarisation depends on the relaxation times rl z2.We perform a detailed analysis of the case of quadratic changes in electric permittivity in a static electric field. Recently a classical theory has appeared concerning nonlinear processes of mole- cular relaxation in strong electric fields of both low and high frequencies and describing the dispersion and absorption of third-order polarisation of liquid dielectrics in terms of Debye molecular rotational diffusion and Lorentz-Voigt electron dispersion.1*2 For an experimental system in which three electric fields with the frequencies col 02, cog are applied to the dielectric in the same direction i.e. along the z-axis the nonlinear electric polarisation of third order P(2)(co4),is given generally at the fre- quency co4 = col + co2 + co3 by where x is the scalar nonlinear third-order susceptibility of the isotropic dielectric; the dependence of x on the electric field frequencies is considered in ref.(3). The time-variable electric field is given by Ez(t)= 2 Ez(coa)exp(-icoat). a=f(1,2,3) Here by taking into consideration the dependence of the susceptibility tensor x on time we extend the theory to processes of increase and decrease in nonlinear polarisa- tion caused by the imposition and removal of electric fields. In this way we obtain a description of the dynamics of numerous nonlinear processes,such as third-harmcnic generation with the nonlinear susceptibility x(-301 u) co u);t); second-harmonic generation in the presence of a static electric field x(-20 co co 0; t); nonlinear rec- tification of optical and electric frequencies x(-0 co -co 0; t); self-induced changes in electric susceptibility x(-co co co -co; t); and quadratic changes in electric permittivity AE(c~), t)/E2-x(-co co 0 0; t) in a strong static electric field.MOLECULAR RELAXATION TIMES THEORETICAL Consider an isotropic dielectric the N noninteracting axially symmetric molecules of which possess a permanent dipole moment m parallel to the symmetry z-axis the tensor components a, = a, # a, of linear electric polarisability and the tensor components b,,, bxxp= by, of electric hyperpolarisability. We are interested in the time-variations of polarisation due to the switching on of AC electric fields.We calculate these time-variations by determining the probability distribution function of orientation f (6 t) from Debye’s rotational diffusion equation with oJn-the angular part of the spatial derivative operator D -the rotational diffusion coefficient of a molecule with spherical geometry and the potential energy of an axially symmetric molecule in the external field E,(t). Above a = +(a, + 2axx)is the mean linear polarisability of the molecule; y = a, -axxthe anisotropy of its linear polarisability; b = (b,, + 2bx,,)/3 its mean hyperpolarisability; and K = b,, -3bxx,the anisotropy of its hyperpolarisability. The Pl(6)are Legendre polynomials dependent on the polar angle 6. Eqn (3) can be solved by statistical perturbation calculus applying the highly useful method of Racah algebra.5 To this aim we have recourse to the following series expansion in Legendre polynomials and successive powers of 1/RT,the reciprocal of the energy of thermal motion of the molecules The expansion (5) on insertion into eqn (3) yields a set of differential equations of the form -3 2 us(t)&?-’)(t)[1’(Z’ + 1)-Z(l+ 1) -s(s + 1)]r I’ ‘7.(6) l’,s 000 Eqn (6) enables us to find the explicit form of the dynamical coefficients A(?)(t).7 Above denote the Clebsch-Gordan coefficients ; summation over the indices J’ s is rGtricte2 by the triangle ineq~ality.~ In the zero’th approximation of perturbation calculus we obtain from (6) the equation of free rotational diffusion of spherical top molecules with the well known solution WLADYSgAW ALEXIEWICZ ET AL.the zl being rotation relaxation times related to the rotational diffusion coefficient D as follows where zl is Debye's relaxation time. For n > 1 eqn (6) has the solution A(y)(t)= -Dexp(-f/q)21yr + 1) -1(1+ 1) -s(s + 1) S,lt 2 I)(?-I)( t)us( t)exp( + t/zz)dt. ( 10) If integration in (10) is performed from 0 to a,the distribution function thus obtained corresponds to the steady state attained by the liquid dielectric after a sufficiently long time subsequent to the moment at which the external electric fields were switched on or removed. The case in question has been dealt with in full detail in ref. (1)-(3). Standard methods of statistical averaging with the distribution function (5) and dynamical coefficients (10) lead to the result that the increase in polarisation due to the application of the AC electric fields at the moment of time t = 0 is described with accuracy to ES(t) by the temperature-dependent contributions to the tensor of 3rd order susceptibility where MOLECULAR RELAXATION TIMES The latter at k = 0 go over into the Debye factors Rl(Co,bJ = (1 -imabczt)-l dis- cussed in ref.(1)-(4) which define the dispersion and absorption of the effects in steady states (for t = a). APPLICATIONS AND CONCLUSIONS The preceding results provide a coherent description of time-dependent nonlinear relaxation effects in the approximation of Debye rotational diffusion taking into ac- count processes of time-dependent growth of polarisation from the moment of applica- tion of external fields onwards.This is of especial significance in relation to measure- ments of the dielectric and optical properties of molecules by pulse techniques when the times necessary for switching the field on can be comparable with the Debye rotational relaxation times and when difference frequencies between various laser modes can cause nonlinear Debye dispersion in the optical range. Future dielectric studies of the relaxation times of molecules need not be restricted to dispersion and absorption measurements of nonlinear effects but can extend to determinations of the time-variations of dispersion and absorption. For example we have the following formulae for the quadratic variations in electric permittivity in the presence of a strong static electric field (for the sake of clarity we omit the fre- quency-dependence of the molecular parameters assuming electron dispersion to be absent) We have thus resolved each of the temperature-dependent contributions into a part describing the steady state (t = 00) and parts which decrease with z1 and r2and describe WEADYSKAW ALEXIEWICZ ET AL.the increase in polarisation of the medium. The latter are characterised by the oc- currence in addition to factors oscillating with the frequency of the measuring field of non-oscillating factors decreasing in time exponentially and non-exponentially with dispersion given by the function RIk(co). Other effects described by the susceptibility x,also involve these non-oscillating factors e.g.the effects of rectification of dielectric and optical frequencies of the emergence of constant polarisation of isotropic bodies under illumination and of dielectric saturation. The method used by us for the determination of the distribution function from Debye’s kinetic diffusion equation is well adapted to extension by taking intermolecu- lar interactions into a~count~*~ as well as to application within the framework of the more general response function method.lG12 The distribution function determined from eqn (5) can serve as well for calcula- tions of orientation time-autocorrelation tensors in effects of linear and nonlinear light scatteringl3 and for spectral line shape determinations in elastic and inelastic scattering of light by molecules of liquids oriented by the action of an external electric A similar description can be used for pulse electric field-induced effects of birefringence in liquid^.^-^^^^^ B.Kasprowicz-Kielich and S. Kielich Adv. MoZ. ReZaxation Proc. 1975 7 275. B Kasprowicz-Kielich S. Kielich and J. R. Lalanne MoZecuZar Motions in Liquids ed. J. Lascombe (D. Reidel Dordrecht Holland 1974) p. 563. J. Buchert B. Kasprowicz-Kielich and S. Kielich Adv. MoZ. Relaxation Proc. 1976,8 in press. P. Debye PoZare MoZekulen (Hirzel Leipzig 1929). M. E. Rose Elementary Theory of AnguZar Momentum (Wiley New York 1957). L. D. Favro Phys. Rev. 1960 119 53. W. Alexiewicz J. Buchert and S. Kielich J. Polymer Sci. (Polymer Symp.) 1976 in press.R. Zwanzig J. Chem. Phys. 1963,38,2766. R. H. Cole Mol. Phys. 1974 27 1. lo R. Kubo J. Phys. SOC.Japan 1957,12 570. l1 R. Pecora J. Chem. Phys. 1969,50,2650. l2 B. J. Berne Physical Chemistry ed. Eyring Henderson and Jost (Academic Press N.Y. 1971), vol. vm p. 539. l3P. D. Maker Phys. Rev. 1970 Al 923. l4 W,Alexiewicz J. Buchert and S. Kielich Proc. Fifth International Conference on Raman Spectroscopy (Freiburg 1976). l5 I. L. Fabielinskii Usp. Fiz. Nauk 1971 104 77.

ISSN:0301-5696    

DOI:10.1039/FS9771100033

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Dielectric and Dynamic Kerr-effect Studies in Liquid Systems BY MARTIN S. BEEVERS CROSSLEYt, * JOHN DAVID AND WILLIAMS C. GARRINGTONGRAHAM Edward Davies Chemical Laboratories University College of Wales Aberystwyth Dyfed SY23 1NE Received 21st July 1976 Studies have been made of the dielectric relaxation and dynamic Kerr-effect of supercooled fluorenone + o-terphenyl and tri-n-butyl ammonium picrate + o-terphenyl solutions in the frequency range lo4 to Hz and over a range of temperature. One process the o! process was observed and is primarily due to the reorientational motions of the dipolar solute molecules. The dielectric relaxation times zD,and the Kerr-effect decay (relaxation) times 7K.d were found to be the same for the fluorenone + o-terphenyl system being consistent with a " fluctuation-relaxation" model for re- orientation and inconsistent with the small-angle rotational-diffusion model.The same result applies to the ion-pair + o-terphenyl system at the lower temperatures but as the temperature is raised TD and tK,d become different with at higher temperature zDN 3TK.d being indicative of a small- angle rotational diffusion situation. These data are discussed in terms of models for cooperative reorientation in viscous liquids. Reorientational motions of molecules in the liquid state may occur and be studied in the time-range 10-13s < t < 103s. While most small molecules reorientate in the short-time part of this range (<10-los) their rate of reorientation may be made as slow as is desired by working in the supercooled state.l For many systems notably viscous molecular liquids and amorphous polymers,2 motions occur in the range 10-los < t < 103s the rate being strongly dependent on temperature and applied pressure.The fact that " slow " reorientational motions lead to an absorption curve in the frequency domain which is at least one decade of frequency in half-width-or a transient decay function which is at least as slow as an exponential decay in time makes it very difficult to identify the mechanism for the reorientation process. Vari-ous mechanisms with their adjustable parameters may be applied to the results of a given experiment. Increasing the accuracy of the data from a given experiment may not lead to a substantially improved identification of the mechanism.The reasons for the difficulty in identifying reorientation mechanisms are made clear if we consider the function that describes the reorientation$ of a dipole vector in a molecule. We define @(a,tl0,O)dQ as the field-free conditional probability that the dipole vector points into the element of solid angle dQ around i2 at time t given its direction was along the z axis at t = 0. The probability function may be expanded in terms of Legendre polynominals of u = cos 8 where 8 is the polar angle and decay functionsly,(t) k # 0. 1" @(Q tl0,O) = 4n 2 (2k + 1)%4V&(~) (1) k=O * Present address Department of Physical Chemistry University of Sydney Sydney Australia. t On leave from Department of Chemistry Lakehead University Thunder Bay Ontario Canada.$ For simplicity isotropic reorientation with respect to an initial orientation is assumed. MARTIN S. BEEVERS ET AL. yo(t)= 1. The dielectric experiment measures the frequency dependent permittivity ~(co),and this is related to yl(t)in eqn (1) by the relation3 p(u)is a local field factor E~ and cooare the limiting low and high frequency pemittivi- ties respectively cr) is angular frequency and 9indicates the Fourier transformation. Thus the dielectric experiment probes only yl(t)in eqn (1). This is insufficient in- formation to define O(Q,tJ0,0) and hence the mechanism for the reorientation pro- cess. If a model for relaxation is assumed then yl(t) may be fitted using the adjust- able parameters of the model.However several models may be used in this way and this gives rise to a rather unsatisfactory situation. In order to improve the situa- tion it is essential to conduct at least two experiments which reflect different aspects of the reorientational motions. This may be achieved by making dielectric relaxation and dynamic Kerr-effect measurements over a wide range of time (or frequency) and temperature. It may be shown* that the dynamic Kerr-effect decay transient ex- cluding internal field problems is given4v5 by ty2(t)of eqn (1). If yl(t)and y2(t)are determined experimentally then O(Q,t J0,O)is partially characterized. If yz(t)decays faster thanyl(t) thenty,(t),y,(t) etc. may be estimated fromy,(t) oryz(t) usinga method based on information theory due to Berne and coworkers.6 Hence O(Q t l0,O) is then fairly well characterized at each (T,P)condition and models may be applied at that stage.Alternatively yl(t)andy2(t) may be considered in terms of models for reorien- tation since a knowledge of both acts as a considerable constraint on the applicability of a given model. Two extreme cases should be noted. (1) Reorientation by small angle steps (rotational diffusion) then4 ty,(t) = exp[-n(n + l)Dt] where D is the rotational diffusion coefficient which predicts that both tyl(t) and yz(t)are exponential in time and the relaxation time for the Kerr-effect experiment is just one third of the dielectric relaxation time. (2) Reorientation occurring by large-angle steps of ar- bitrary size (" fluctuation-relaxation " model)5 for which yl(t)= y2(t)=y,(t) = c(t) say and the relaxation function for the rise and decay Kerr-effect transients are the same as t(t).The present work describes a study of the dielectric relaxation and the dynamic Kerr-effect on two systems exhibiting slow reorientational motions and is aimed at clarifying mechanisms for reorientation in supercooled and other viscous liquids l and in solid amorphous polymers.2 Despite the obvious potential of the combined approach for cooperative low-frequency processes we have noted only one previous comparison of dielectric relaxation and Kerr-effect data. Results are given here for fluorenone + o-terphenyl and tri-n-butyl ammonium picrate + o-terphenyl in the supercooled liquid state. The former system was chosen as being representative of several solute + o-terphenyl systems1 in which the motions of solute and solvent are wholly cooperative while the latter system is non-typical being one in which the ion-pairs relax on average more slowly than the so1vent.l EXPERIMENTAL o-Terphenyl (Koch-Light Puriss Grade) was dried over zeolite fluorenone was recrystal- lized from ethanol and tri-n-butyl ammonium picrate (TBP) was prepared by the usual method." Dielectric measurements were made using a coaxial cell together with a General The pn(t)of eqn (1) are the field-free orientational correlation functions and may be defined as P"(0 = t)>= J~(Q,tIO,O)P,(u)~.40 DIELECTRIC AND DYNAMIC KERR-EFFECT STUDIES IN LIQUID SYSTEMS Radio 1620-A Assembly (lo2to lo5Hz) and a Scheiber bridge (0.6 to lo2Hz).Dynamic Kerr-effect measurements were made using an apparatus constructed in this laboratory which operated with step-pulses (<3 kV) whose plateau was variable over the range 1O-j s to lo2s. The step-pulses were applied to a Kerr-cell of optical path-length 7.45 cm and an inter-electrode spacing of 0.138 cm. The optical-transients (square-law detection) were displayed on a storage oscilloscope (Tektronix Type 7313) and photographed or recorded directly on a pen recorder. The Kern-cell was cooled using methanol circulated from a Lauda Ultra-Kryostat. Super-cooled solutions were obtained by rapidcooling of the Kern-cell from 340 K down to about 258 K. In contrast with several supercooled solute + 0-terphenyl systems,l the fluorenone + o-terphenyl and TBP + 0-terphenyl supercooled solutions were very stable and no recrystallization problems were encountered over several hours.Major problems with such low-temperature optical measurements include (i) strain- birefringence in the cell-windows (ii) water-condensation on the cell windows. Both difficulties were encountered; (i) was eliminated using well-annealed windows and checking that good optical extinctions were possible over a reasonable volume of the cell (ii) was eliminated by passing a stream of dry-air over the windows. RESULTS A. FLUORENONE IN 0-TERPHENYL Three concentrations were studied using the Kerr-effect method namely 15.1 % 22.5%and 30.4%(wt %) fluorenone in o-terphenyl. The static Kerr-constant B12= An/(hE2)for these solutions is positive and B12is far greater than the solvent Kerr- constantBl.(For example I?, for the 15.1 % solution at 260.9 K is 9.1 x VV2m and Blat 259.1 K is 1.41 x VW2m.) Fig. 1 shows representative Kerr-effect tls . tls FIG.1 .-Normalized birefringence (An/Anmax)against linear time (t) for 22.5% fluorenone in 0-terphenyl at the temperatures (K) indicated. transients at four temperatures; the fact that the rise and decay transients form reasonable normalized master curves is illustrated in fig. 2 for the 15.1 % and 30.4% solutions. Average relaxation times derived from the decay-transient (rkSd) and the rise-transient (rk,J are given in table 1. Plots of the birefringence against time may not be fitted by an exponential function of time but are fitted with reasonable a_ccuracy by the William-Watts empirical function exp -(t/.ro)B.Table 1 gives the values MARTIN S. BEEVERS ET AL. FIG.2.-Normalized birefringence(An/AnmaX)against normalized time (t/zo)for (a) 30.4% fluoren-one in o-terphenyl at 255.0 K-0 ;243.8 K-•;240.6 K-• and (b)15.1% fluorenone in o-terphenyl at 260.2 K-0 ; 254.9 K-•; 244.6 K-0. TABLE IN 0-TERF'HENYL 1.-FLUORENONE Kerr dielectric* rise decay ~~~~ ~~ ~~ 15.1 260.2 0.28 0.83 0.34 0.69 253.6 5.98 0.56 254.9 4.50 0.69 3.40 0.65 251.2 18.9 0.56 249.4 69 0.62 72 0.58 250.0 42.4 0.56 246.4 410 0.58 510 0.53 244.6 1400 0.57 1650 0.45 22.5 259.6 0.16 0.71 0.12 0.69 253.7 1.69 (0.45) 257.7 0.37 0.72 0.24 0.69 251.2 6.71 0.54 246.9 95 0.68 75 0.61 249.9 10.6 0.56 241.2 2720 0.65 2300 0.60 248.3 22.5 0.58 239.9 6200 0.59 7600 0.52 30.4 256.3 0.18 0.73 0.13 0.68 255.0 0.41 0.73 0.26 0.59 246.4 40 0.72 -243.8 175 0.70 98 0.55 240.6 960 0.60 820 0.55 239.2 2000 0.61 2020 0.50 * Shears and William~.~ 42 DIELECTRIC AND DYNAMIC KERR-EFFECT STUDIES IN LIQUID SYSTEMS for the experimental transients.Fig. 3 shows* (logf,,, T-I) plots for the data of table 1. Also included in table 1 are the dielectric relaxation results of Shears and Williams9 for this system. Inspection of table 1 and fig. 2 and 3 show that (a) the rise and decay transients (Kerr-effect) have approximately the same average relaxation time at each temperature; (b) that z (Kerr) N-z (diel.) where a comparison can be made (Le for the 15.1% and 22.5% solutions); (c) that the p values for the rise and decay transients (Kerr-effect) are similar at a given temperature and are similar to but slightly larger than the corresponding dielectric relaxation values.FIG.3.-log (fmax/Hz)against lo3KIT for (1) 15.1%; (2) 22.5% and (3) 30.4%fluorenone in o-ter- phenyl. dielectric-A ; Kerr-effect rise-0 ; Ken-effect decay-0. The dashed line represents di-electric and Kerr-effect results for pure o-terphenyl. Clearly the same motional process is being detected by the two techniques being the cooperative Brownian motions of the (dipolar) fluorenone molecules ; this process is characterized by the large apparent activation energy (Q -250 kJ mol-l) and by rather broad transients (or broad asymmetric loss curves).Note from fig. 3 that the fluorenone molecules "plasticize " the solvent implying that the glass-transition temperature T for fluorenone is lower than that of the solvent. We emphasize that the data of table 1 and fig. 1-3 relate to the supercooled liquid state and not to the glassy state. When logf,, < -4 the material may be regarded as a glass. B. TRI-N-BUTYL AMMONIUM PICRATE IN O-TERPHENYL Dynamic Ken-effect data for 1.33% 2.25% 2.91% 3.29% and 7.12% TBP in o-terphenyl (wt %) are given in table 2. In all cases B12was positive and was rather larger than the solvent value. For example B12for the 1.33% solution was 4.52 x 10"" (V-2 m) at 261.7 K. Representative transients are shown in fig.4 and in normalized form for two concentrations in fig. 5. The curves of fig. 4 are qualitatively different from those of fig. 1. At the higher temperatures the rise-transient is far * The Kerr-effect data are in the time-domain and yield to. In fig. 1 (Kerr) = 1/(2nr0). The average dielectric relaxation time (z> is obtained from the maximum loss condition in the fre- quency domain. fmar (diel). = l/(Zn<r>). We may readily show that log [fmax (Kerr)/fmaX (Gel.)] = F@) >0 and F(B) = 0.06 for = 0.75. MARTIN S. BEEVERS ET AL. TABLE Z.-nRR-EFFECT RELAXATION DATA FOR Bu3NHPiIN 0-TERPHENYL rise decay 1.33 260.6 9.6 0.83 3.95 0.90 257.1 46.0 0.78 16.7 0.96 0.86 253.8 200 0.75 108 0.71 251.7 860 0.74 400 249.0 3 100 0.68 3100 0.69 2.25 260.4 15.5 0.84 6.0 0.85 257.2 68.0 0.83 28.0 0.80 251.4 1730 0.81 1350 0.65 250.3 3100 0.69 2300 0.66 249.4 5700 0.73 3900 0.63 2.9 1 261.4 12.7 0.87 4.7 0.93 258.1 53.0 0.83 22.2 0.97 255.2 240 0.78 86.0 0.84 251.1 2900 0.78 880 0.60 250.0 5200 0.76 3100 0.73 3.29 260.7 13.3 0.91 5.6 0.90 257.7 57.0 0.90 19.0 0.96 256.4 110 0.88 46.0 0.86 254.5 260 0.86 112 0.81 251.7 1420 0.76 930 0.78 250.8 2400 0.74 1630 0.70 249.9 3950 0.75 2550 0.69 249.3 5600 0.74 3600 0.70 7.12 257.3 53.0 0.77 33.0 0.79 254.5 190 0.74 220 0.66 250.6 2200 0.59 2200 0.50 tlms tls FIO.4.Normalized birefringence (An/An,,,) against linear time (t)for 2.91%Bu3NHPi in a-terphenyl at the temperatures (K) indicated.44 DIELECTRIC AND DYNAMIC KERR-EFFECT STUDIES IN LIQUID SYSTEMS tlTg FIG.5.Normalized birefringence (An/An-) against normalized time (t/zo)for (a)3.29% BuSNHPi in o-terphenyl at 260.7 K-0 ;256.4 K-0 ;250.8 K-0 and (6) 2.25% Bu3NHPi in o-terphenyl at 260.4 K-0 ;257.2 K-0 ; 249.4 K-0.slower than the decay transient and as the temperature is lowered the rise and decay transients become more symmetrical. The data of table 2 and fig. 6 show that (zK,~/zK,~) is significantly greater than unity at the higher temperatures and tends towards unity at the lower temperatures. Table 3 gives dielectric data for this and earlier work;1° a comparison with Kerr-effect results as shown in fig. 6 reveals that the zDand z~,~ are quite similar in the region of overlap and that (Z&~,J is approxi- (a) (6) 103~1~ FIG.6.-10g(fm,,/Hz) against lo3K/Tfrom Kerr-effect rise (0) transients and Kerr-effect decay (0) for (a)1.33% and (b)3.29% Bu3NHPi in o-terphenyl and from dielectric loss curves for 1.52”/,-a ; 2.140/,4 and 2.91%-A Bu3NHPi in o-terphenyl.The dashed Iine represents Kerr-effect and dielectric results for pure 0-terphenyl. MARTIN S. BEEVERS ET AL. mately 3. Note in fig. 6 that log tfmax) for the solutions is in all cases less? than that for pure o-terphenyl; it has been suggestedlO that this arises since the cooperative mechanism applicable for fluorenone + o-terphenyl and other systems is coupled to an intrinsic relaxation mechanism for the (large) ion-pair solute. Table 2 shows that the p values obtained from the rise and decay transients are not only similar but TABLE3.-DIELECTRIC RELAXATION DATA FOR BujNHPi IN U-TERPHENYL 1.52%* 2.14%* 2.91% ~~~~~~~~ ~ ~~ ~ ~~~ ~ 266.8 0.90 0.95 268.3 0.48 0.88 268.9 0.47 0.95 262.2 5.65 0.88 263.1 5.03 0.86 267.9 0.58 0.95 258.8 20.0 0.78 259.5 21.5 0.76 266.5 1.45 0.95 260.8 12.1 0.91 259.8 24.7 0.90 257.4 63.4 0.88 255.7 130 0.85 * Davies Hains and Williams.l0 as the temperature is reduced decrease from near single relaxation time values towards values comparable with that for the fluorenone system.This behaviour closely parallels the dielectric results for this system.1° DISCUSSION While the contribution of the solvent to the dielectric and Kerr-effect relaxation in the present systems is not negligible the observed behaviour arises primarily from the dipolar solute molecules.These act as a probe on the cooperative motions between solute and solvent molecules and whereas there is little doubt that re- orientational and translational motions of the solute molecules occur on a comparable time-scale it is only the reorientational motions of the dipole vector which are being detected in both experiments. It is reasonable to neglect cross-correlation terms for the solutions studied here so wl(t) and lyz(t) are being determined; their characteristic features are summarized in tables 1 and 2 and ref. (9) and (10). The observation for the fluorenone $-o-terphenyl system (fig. 3) that z~,~ lirK,d N zDrules out a mechanism of reorientation through small-angle steps (rotational diffusion) which would require ZK,d = (1/3) Z and the Kerr-effect rise transient to be slower in time than the decay transient.The results are entirely consistent with the "fluctuation-relaxation " model5 mentioned in the Introduction. In this model a reference molecule only moves when that molecule and the local environment are subjected to a fluctuation (in local volume or energy say) exceeding a critical size. After the molecule has moved it is assumed that there is no orientational correlation on average between the initial and final directions of the dipole vector and this means that all angular correlation fuctions (P,(u,t)) have the same time-dependence c(t)say where c(t)is determined by the temporal evolution of the fluctuations.The realistic nature of this model for cooperative (or " structural ") relaxation in viscous liquids is supported by observation (using photon-correlation spectroscopy) that the These data for pure o-terphenyl refer to both dielectric relaxation and Kerr-effect relaxation. Beevers and coworkers" studied a-terphenyl and its mixtures with tri-tolyl phosphate and found z~,~ = 7K.d = Z~ for both pure liquids over a wide range of temperature. 46 DIELECTRIC AND DYNAMIC KERR-EFFECT STUDIES IN LIQUID SYSTEMS correlation time for density fluctuations is the same as that measured by dielectric or viscoelastic methods in certain viscous 1iq~ids.l~~'~ We shall consider below models which lead to the form of k(t). The observations for the TBP +o-terphenyl solutions are extremely interesting since they show that the mechanism for relaxation is changing as the temperature is varied.Fig. 6 and table 2 show that at the lowest temperatures studied z~,~ -+rK,d and although we do not have dielectric data at the lower temperatures an extra- polation of the high temperature data suggests that rD*z(Kerr) in this range. This suggests that at the lowest temperatures the ion-pairs reorientate by the "fluctuation-relaxation "mechanism applicable to the fluorenone +o-terphenyl systems. This is supported by the fact that the observed relaxation times for these fairly dilute solutions approach that of the solvent in the lower temperature region indicating that the motions of the solute and solvent on average have the same correlation time and are part of the same cooperative process.Additional confirmation of this mechanism comes from the variation of B with temperature. Tables 2 3 and the dielectric data of ref. (10) show that tends to -0.55 at lower temperatures similar to that for fluorenone +o-terphenyl and several other systems,l being a characteristic of wholly cooperative relaxation. A similar conclusion had been inferred lo from the dielectric data for TBP +o-terphenyl; the additional constraint of the Kerr-effect data makes it fairly certain. Fig. 6 shows that as Tis increased (rD/rK,d) increases to a value near 3 and that z~,~ zDat higher temperatures. This is an obvious departure from the "fluctuation-relaxation "model towards the rotational diffusion model in which the solute molecule moves through smaller angular steps.The fact that zDin this temperature range is longer than that for pure solvent implies that many "impacts "of the solvent are required in order to randomize the dipole-vector of the ion-pair-this being a condition for small-angle motions (rotational diffusion). We now briefly consider models for the relaxation in both systems. The "fluctua-tion-relaxation "mechanism may be considered to arise from the diffusion of "de-fects "through the liquid where the arrival of a "defect "at a molecule relaxes it completely on a time-scale far shorter than that required for the diffusion of "defects ". This process has been evaluated by Glarum14 and by Phillips and co-workers15 and leads to a function t(t)which has an average relaxation time governed by the defect- diffusion time 7dSdefect,and a functional form which is numerically the same as the Williams-Watts functions with D =0.514 if first and second nearest neighbour defects are inc1~ded.l~ Clearly this simple model gives an adequate representation of the "fluctuation-relaxation "mechanism for the fluorenone +o-terphenyl system and for TBP +o-terphenyl at lower temperatures.The observation that the same function c(t)characterizes both dielectric relaxation and the dynamic Ken-effect may also be rationalized* in terms of the jump-model of Anderson1'*18 for the special case18 where the probability function for reorientation through a jump-angle y is given by P(y)=C exp -(?/yo).For yo >n/4 zD fi zK,d. In physical terms this corresponds to jumps through angles of variable size and the probability of jumping through a small angle is greater than that for a large angle. The results for the TBP +o-terphenyl system at lower temperatures may be interpreted in the same manner as those for the fluorenone +o-terphenyl system. At higher temperatures a small-angle rotational diffusion situation is implied but it is essential to include the "fluctuation-relaxation "mechanism as a component to the *Ivanov16 has deduced relations for rDand ZK,d for rotational motion through steps of variable sizes and the form of these relations has been discussed by Beevers and co-workers" and by Ander- s0n.17 We note that Ivanov predicts (z&~,~)30.6 for discrete jumps through y =7r whereas the ratio should be zero in this case.17 MARTIN S.BEEVERS ET AL. overall process since there is a continuous variation in mechanism as the temperature is raised. A simple coupled scheme was previously suggested by Davies and co-workers lo for the dielectric relaxation being A 7+A * -+ C. At lower temperatures A -+ A* is the rate determining step leading to the " randomized " state C. At higher temperatures A* -+C becomes the rate determining step in the intrinsic re- orientation by the solute. Alternatively we may say that at low temperatures the solute molecules suffer infrequent but large fluctuations in their local environment leading to large-angle jumps on average when they move.At higher temperatures the frequency of the local fluctuations increases and with the increased thermal energy coupled with the large size of the solute there is a tendency to move through smaller angles (yo decreases) leading to z, == 32K,d,the small-angle rotational diffusion case. CONCLUSIONS The combined dielectric relaxation and dynamic Kerr-effect study of two super- cooled liquid systems has shown that the mechanism of dipole reorientation may be classified in a way that neither technique on its own would be able to achieve. For fluorenone + o-terphenyl reorientational motions occur by a " fluctuation-relaxation " mechanism in which the solute moves through large-angle steps. A similar mechan- ism applies to TBP + o-terphenyl at low temperatures but as the temperature is raised the mechanism appears to change to small-angle rotational diffusion.These systems may typify the behaviour of other viscous liquids and of amorphous solid polymers. We gratefully acknowledge the award of a studentship (to D. C. G.) and an equip- ment grant from the Science Research Council. G. Williams in Dielectric and Related Molecular Processes ed. M. Davies (Spec. Period. Rep. Chem. SOC. London 1979 vol. 2 p. 151. N. G. McCrum B. E. Read and G. Williams Anelastic and Dielectric Efects in Polymeric Solids (J. Wiley London 1967). G. Williams Chem. Rev. 1972 72 55. E. Fredericq and C. Houssier Electric Dichroism and Electric Birefringence (Oxford U.P. Lon-don 1974). M. S. Beevers J. Crossley D. C. Garrington and G.Williams J.C.S. Faraday ZZ 1976 72 1482. B. J. Berne P. Pechukas and G. D. Harp J. Chem. Phys, 1968,49 3125. R. Coelho and D. K. Manh Compt. rend. Cy1967,264 641. a M. M. Davies and G. Williams Trans. Favaday SOC.,1960 56 1619. M. F. Shears and G. Williams J.C.S. Faraday IZ 1973 69 608. lo M. Davies P. J. Hains and G. Williams J.C.S. Faraday IZ 1973 69 1785. l1 M. Beevers J. Crossley D. C. Garrington and G. Williams J.C.S. Faraday I& 1976 72 1482. l2 C. Demoulin C. J. Montrose and N. Ostrowsky Phys. Rev. A 1974 9 1740. C. Lai P. B. Macedo and C. J. Montrose personal communication. l4 S. H. Glarum J. Chern. Phys. 1960,33,639. l5 M. C. Phillips A. J. Barlow and J. Lamb Proc. Roy. SOC. A 1972 329 193. l6 E. N. Ivanov J. Exp. Theor. Phys. 1963,45,1509 (Sov. Phys. JETP 1964 18 1041). l7 J. E. Anderson Faraday Symp. Chem. SOC. 1972,6,82. la J. E. Anderson Faraday Symp. Chem. SOC. 1972 6 91.

ISSN:0301-5696    

DOI:10.1039/FS9771100038

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Rotational Relaxation in Liquids Molecular Hydrodynamics and Long Time Tails BY BRUCEJ. BERNE Department of Chemistry Columbia University New York N.Y. 10027 U.S.A. Received 24th September 1976 1. MOLECULAR HYDRODYNAMICS It is remarkable that hydrodynamic models can be applied with great accuracy at the molecular level. The evidence for this comes from several quarters. For example Alder et d.,'using computer experiments have shown that the translational friction constant c of a sphere in a one-component dense smooth hard sphere fluid is c = 4nqa where 11 is the shear viscosity and a is the radius of the sphere. This result is identical to Stoke's calculation of c for asphere with slip boundary conditions. Recently a variation principle has been used to evaluate the rotational friction co- efficient of a uniformly rotating slippery spheroid and has shown that this can differ from the friction coefficient of a sticky spheroid by as much as an order of magnitude or more.Assuming that benzene is a slippery ellipsoid Zwanzig and Hu's calculation is in excellent agreement with rotational diffusion coefficients determined by de- polarized light ~cattering.~ It would appear from the foregoing that the details of the intermolecular potential are unimportant when it comes to the determination of translational and rotational friction constants except insofar as these details effect the value of the viscosity. The success of hydrodynamics does not stop with the determination of static friction constants. Several years ago Zwanzig and Bixon calculated the normalized velocity correlation function by solving the Navier Stokes equation in the low Reynolds number limit for a particle executing non-uniform translatory motion in a visco-elastic continuum fluid with boundary conditions intermediate between pure stick and pure slip boundary condi- tions.This model has the general characteristic that at long times the correlation function decays asymptotically as a,t-3/2 where the coefficient a and the exponent are in excellent agreement with the asymptotic time dependence observed by Alder and Wainwright in their computer experiments on smooth hard sphere fluids. More- over Verlet et a1.' have shown that the hydrodynamic model gives an excellent fit to the detailed structure of the velocity correlation function determined by them for a dense Lennard-Jones fluid.Several years ago Ailawadi and Berne using hydrodynamic equations containing the angular momentum variables showed that the normalized angular velocity correla- tion function BRUCE J. BERNE decays asymptotically as C"(t) = badt-'d +2)/2 (103) where and where I my n v D,and d are respectively the molecular moment of inertia molecular mass number density kinematic shear viscosity (v = q/mn) self diffusion coefficient and dimensionality of the system. This should be compared with the asymptotic prediction for the linear velocity cv(t) = avdf-d'2 (1 5) where -1 avd = 4nd -[4n(D + v)]"'2. The latter result has been confirmed by computer experiment for d = 2 3.Berne9 subsequently calculated the full time dependence of Cm(t)for a sticky sphere executing non-uniform rotations in a viscous continuum fluid and showed that in the limit t -+ 00 it reduces to eqn (1.3). Eqn (1.3) has also been confirmed by mode-mode calculations. Assuming the validity of a generalized rotational diffusion equation Berne conjectured that the orientational correlation functions G(t>=(Pl(U(0) u(t))> (1 -7) also behave asymptotically as This asymptotic behaviour also follows directly from the "hydrodynamic equa- tions " that have been used to calculate the so called " shear doublets "in depolarized light scattering and thus appears to be quite reasonable.lo Subsequent mode-mode calculations l1 have taken issue with eqn (1.7) and there exist several other predictions for the asymptotic form of Cl(t).Recently Hill and Deutch,12 using Faxen's theorem have shown that eqn (1.7) is valid. They conjecture that the velocity fields arising from finite sized particles give eqn (1.8) whereas the mode-mode coupling theory calculations correspond to velocity fields arising from point sources. Recently Montgomery and Berne13 have computed Cco(t)for a sticky sphere executing non-uniform rotations in a viscoelastic continuum fluid. This model gives the asymptotic result stated in eqn (1.3) no matter what model is used for the visco- elasticity. It is shown nevertheless that the short time behaviour is strongly depend- ent on the details of the viscoelastic model.For example (1.9a) L (1 -nl/r t + o(t2). viscoelastic (1.9b) It is interesting to compare this hydrodynamic model for Co(t)with the rough ROTATIONAL RELAXATION IN LIQUIDS sphere molecular dynamics of O’Dell and Berne. l4 First note that the initial decay of Co(t)in a rough sphere fluid is given by (1.10) where K is the dimensionless moment of inertia of a sphere of diameter 0 (K = 41/ma2) and tEis the Enskog relaxation time. Comparison of eqn (1.10) and (1.9) shows that if the hydrodynamic model is to describe the rough sphere fluid a viscoelastic theory must be used instead of the viscous theory. Eqn (1.10) can then be used to eliminate one of the two parameters in the model. Fig 1.1 gives a comparison between the “best O00,2 4 6 .8 10 12 *1 ~ time in mean collision times FIG.1.1 .-The dots ...indicate the values of Cw(t)determined from computer experiments and the solid line indicates the “ best fit ” theory.fit ” single relaxation time viscoelastic model and a computer experiment on a rough sphere fluid containing 108 spheres and undergoing 250 OOO collisions. The results look very encouraging and we might conclude that the hydrodynamic model is excellent. Nevertheless the comparison involves the choice of a single parameter. A further check can be made. We note from eqn (1.3) that the model gives a long time tail which depends on D and v thus we can relate the parameter to be determined to v and D. We find that the best fit parameter differs dramatically from the value calculated for it using the Enskog values of v and D.Thus there is no internal consistency. Putting it another way if we determine the parameter from the Enskog values of v and D,we can determine Co(t)on the basis of our hydro- dynamic model. Fig. 1.2 shows a comparison between this procedure [curve @)] and experiment [curve (a)]. The troublesome thing about the hydrodynamic theory is that if it is forced to give the supposedly correct initial and long time decay it predicts strong oscillations in Co(t). Although oscillations like this sometimes appear in fluids containing structured particles they do not appear to occur in rough sphere fluids. Because the spheres are microscopically rough we made the plausible assumption that stick boundary conditions should apply.In hydrodynamics the rotating particle creates a boundary layer. This boundary layer can be accounted for by introducing a BRUCE J. BERNE FIG.1.2.-Curve (a)is a result of the computer experiment and curve (b)is C,(t) as determined from the model by fitting the initial decay to eqn (1.10)and long time decay to eqn (1.3) with Y,D calculated from Enskog theory. parameter of slip p. To allow for arbitrary slip at the surface of the rotating sphere the boundary condition is generalized so that the tangential component of the force is proportional to the velocity of the fluid relative to the velocity of the particle surface. a,. a.a = B [u -S2 x R]. a,. Here p = 00 corresponds to the complete stick limit discussed above and /? = 0 corresponds to pure slip (i.e.free rotation). This is discussed in ref. (13b) with ap- propriate references. A straightforward calculation gives the following short time behaviour 1-2Pt + O(P) viscous (1.lla) ~w(0 tz t + o(t2)viscoelastic one relaxation time. (1.1 1b) Comparison of eqn (1.1 la) with eqn (1.9~)shows that slip boundary conditions drastic- ally alter the short time behaviour so that the viscous model is now consistent with the exact short time decay expected from kinetic theory. Since L is known and q is given by kinetic theory it is possible to compute p by forcing eqn (1.1 la) to agree with eqn (1 .lo). On the other hand since we do not know y we cannot evaluate B from eqn (l.llb)andeqn(l.lO). An alternative viscoelastic model is given by Theodosoplu and Dahler from the kinetic theory of a dense rough sphere fluid with q yl and y2 expressed in ROTATIONAL RELAXATION IN LIQUIDS terms of microscopic quantities.The short time dependence of Co(t)for this model is identical to eqn (l.lla) and again B can be expressed in terms of microscopic quantities. As expected the introduction of partially slippery boundary conditions together with various models of viscoelasticity does not alter the long time behaviour of Co(t)which is given by eqn (1.4) (with D = 0). In fig. 1.3 molecular dynamics (dots) are compared with various hydrodynamic models with arbitrary slip. One curve corresponds to the purely viscous model in which the viscosity is chosen to give the best fit to experiment.Another curve corresponds to the purely viscous model in which the viscosity is computed using the Enskog viscosity. The last curve corresponds to the viscoelastic model of Theo-dosoupolu and Dahler in which all the parameters are computed by kinetic theory. In all of these models the coefficient of slip j3 is computed from the initial slope and the values are listed in the figure. It should be noted that in all cases the value of p is much closer to slip than stick. It should be noted that there are three distinct time regimes. First Co(t)decays exponentially with a rate determined from binary collisions. This is followed by a slower exponential decay with a renormalized rate constant and finally the Cw(t) decays with the asymptotic tail.O.OOI~-1 I I i 1 i I I I 0 4 8 12 16 20 24 30 time m mean collision times FIG.1.3 -A comparisonof the angular velocity correlation function for varioushydrodynamic models with molecular dynamics (dots). The reduced density is fi = 0.333. The slip parameter is denoted 8. The reduced viscosity is denoted $(qE = 0.855). Stick (best fit) $ = 0.103 B = 00 y = 0.0909. Slip (best fit) $ = 0.267 B = 1.97 y1 = y2. Slip (ab initiu) = 0.855 B = 0.465 y1 = 0.792 72 = 0.495. It is useful to determine when the best fit model reaches its asymptotic decay. This is given in fig. 1.4. BRUCE J. BERNE 53 Iin mean collision times 30 20’ 13 10 7 0.10 1 1 I 0.08 d 2 0.06 / Q3 0.04 0.02 0 1 0.005 0.010 i’2 in mean collision times FIG.1.4.-A plot of the best fit C&) against t -5/2.This curve looks fairly linear for times between 7 and 13 mean collision times but it is clear from the inset that this is not its time-asymptotic behaviour. The asymp- to tic behaviour becomes observable only beyond approximately 30 mean collision times. It is clear from this that although the preasymptotic behaviour may appear to behave like some power law it is dangerous to come to any conclusion based on data found for times shorter than 30 or 40 mean collision times. In concluding this section it is worth noting that for arbitrary slip boundary condi- tions and for all viscoelastic models the rotational diffusion coefficient is 2. COMPUTER EXPERIMENTS ON ROTATIONAL LONG TIME TAILS Given the remarkable agreement between the hydrodynamic prediction of the linear velocity correlation function [as embodied in eqn (1.5)] and various computer experiments it is natural to assume that the hydrodynamic prediction of the angular velocity correlation function [as embodied in eqn (1.3)] will likewise be in agreement with experiment.ROTATIONAL RELAXATION IN LIQUIDS O’Dell and Berne l5 have performed molecular dynamics experiments on fluids composed of 512 rough discs subject to periodic boundary conditions. Typical trajectories consisted of anywhere between 2 x lo6 and 5 x lo6collisions. The independent binary collision approximation for rough discs yields the short time Enskog exponential results for the linear and angular velocity correlation func- tions respectively,15 Cv(z)= exp -[::3 Cm(z) = exp -[1Y KIT where IC = 41/ma2,0is the diameter of a disc and z is in units of the mean collision time hereinafter abbreviated to rnct.FIG.2.1.-Log plots of the normalized linear velocity correlation functions computed for rough discs. Notice the arrow at approximately 22 rnct. This is the general area where we expect to see the sound “ kinks ”. K specifies the dimensionless moment of inertia and 2 specifies the density in units of the closest packed density. Fig. 2.1 shows a log plot of the normalized linear velocity autocorrelation functions (VCF) for three different svstems. For short times (<4 rnct) there is excellent agree- BRUCE J. BERNE ment with eqn (2.1).At longer times all three curves show a positive deviation from the exponential. At about 20 rnct there is a characteristic “ sound kink” in the VCF’s. This kink arises from the arrival of a sound wave from adjacent periodic cells and consequently illustrates gross interference effects due to the boundary conditions. We expect to find long time tails between 4 or 5 rnct up to a maximum of 20 rnct. For longer times the results are obscured by the boundary conditions. In fig. 2.2 corresponding plots are given for the angular velocity correlation func- tions (AVCF). These functions start deviating from eqn (2.2) at relatively short times when compared with the VCF’s. Since the same kind of hydrodynamic argu- ments apply to the long time tails in both Cu(z)and Cw(z),we expect that these tails will become manifest in the same time regimes.I .O c u \I I I I \. I I1 I 8‘ 12 16 20 24 28 T (in mean collision times 1 FIG.2.2.-Log plots of the normalized angular velocity correlation functions computed for rough discs. In fig. 2.3 C,(z) is plotted against 7-l. If there are “long time tails ” as predicted by eqn (1.5) for d = 2 these plots should be linear functions of z-l as z-l -f 0 as indeed they are. The coefficient a is determined from the slope and is presented in table 2.1. In fig. 2.4 Cm(z)is plotted against r2.If Cm(z)decays as r2at long times as predicted by eqn (1.3) with d = 2 these curves should also possess a definite linear region as they do. The coefficient aa as determined from the slope is presented in table 2.1.ROTATIONAL RELAXATION IN LIQUIDS Before discussing these results it is very important to note that as z -+ 00 the linear region Cm(z)does not extrapolate to zero. We think that this arises from the periodic boundary conditions. Our analysis of this effect is presented in a forthcoming manuscript. Had Alder and Wainwright not made their (N-1) correction their asymptotic linear velocity correlation functions would also extrapolate to a negative value. In their studies on smooth discs Dorfman and Cohen16have derived a formula TABLE 2.1.rALCULATED AND MEASURED VALUES OF THE DECAY COEFFICIENTS FOR THE LINEAR AND ANGULAR VELOCITY CORRELATION FUNCTIONS iia K a,(meas) a,(calc) a,(meas) a,(calc) runb 0.500 0.01 0.18 0.21 0.22 4.1 x 10-4 2.5 0.500 1.00 0.06 0.11 3.9 1.2 x lo-* 5.0 0.289 0.40 0.07 0.10 0.71 2.1 x 10-3 5.0 The density is measured relative to closest packed density.The number of collisions in the run in millions. 0.040L 16-5rcT 0.030-c. e c 0 .-Y 0 t 7 r 0 .-r K =0.4 -. 0.020-z aJ L 0 x c. 0 L Q) > L g 0.010-C .-d K = 1.0 n" = 0.500 n I I "0 0.1 0.2 I/T (in inverse mean collision times) FIG. 2.3.Plots of linear velocity correlation function against reciprocal time to display the long time tails on these functions for the fluid of rough discs. BRUCE J. BERNE from kinetic theory for the coefficient a,,which is identical to eqn (1.5) with the one crucial difference that v and D are replaced by their Enskog values vE and DB-a result that agrees well with the computer experiments of Wood and Erpenbecklg on hard discs.Unfortunately vE and DEare unknown for rough discs. O'Dell and Berne15 have performed this calculation. The values of a predicted on the basis of -0.020 16-5rcT ICT x h w u 2 C .-U V c 3 rc C .-L z 0.010-Q L V x e .-u 0 d Q) L b d 7 CI) 5 1/r2 ( in inverse mean collision times squared 1 FIG.2.4.-Plots of the angular velocity correlation function against reciprocal time squared to show the long time tails for rough disc fluids. this calculation are listed in column 4 of table 2.1.The agreement between GC determined from computer experiment and kinetic theory should be noted. The value of a determined from the kinetic theory is listed in column 5 of table 2.1. The most dramatic result of this study is that the predicted and "measured " values of ola differ by more than two orders of magnitude. The measured angular velocity long time tail is much larger than expected on the basis of all the theories. Were it not for the hydrodynamic theory presented in section 1 this result would be very disconcerting. Fig. 1.4 clearly shows that we are seeing a preasymptotic decay. Inthis connection Nady and Bernel' have completed a study of two dimen- sional fluids of ellipses containing 1296ellipses in which each pair of ellipses interacts according to a Lennard-Jones 12-6 potential with parameters e and Q which are dependent on the relative orientations of the ellipses.The form of E and 0 are given analytically by Berne and Pechukas.18 The dynamics are studied at different reduced ROTATIONAL RELAXATION IN LIQUIDS temperatures T* and different axial ratios a(=length/width). Cv(t)is shown in fig. 2.5. Again we see the sound kinks. The corresponding angular velocity correlation functions are presented in fig. 2.6. In addition the correlation function of the molecular orientation I 1 1 I L 010 0.20 0.30 I /t (inverse time in reduced units) FIG.2.5.-The linear velocity correlation function for Lennard-Jones ellipses for various reduced temperatures T*and various axial ratios a.(a) T*= 0.805 a = 2.0; (6) T*= 2.415 a = 1.0; (c) T*= 2.415 a = 1.3. is determined. This is shown for T* = 0.805 and a = 2.0in fig. 2.7. In these studies it is assumed that at long time C,(t) -+ aVt-l Cw(t)-t awt-2 and C,(t)3 CC~~-~. If the measured value of a is used to compute or it is once again found that the theoretical value of ucodiffers from the experimental value by two orders of magnitude-a result totally consistent with the foregoing. Thus we are observing a preasymptotic region here. In addition we see that the orientational correlation function behaves pre- asymptotically as does Co(t). In all of the foregoing studies it would appear that we are observing a preasymp- totic behaviour of the rotational correlation functions.This behaviour seems to last for an astoundingly long time and should have some important consequences. BRUCE J. BERNE I- t 1/t2 (inverse squaretime in reduced units) FIG.2.6.-The angular velocity correlation function for Lennard Jones ellipses for various reduced temperatures T*,and various axial ratios. (a) T* = 2.415 a = 1.3; (b) T*= 0.805 a = 2.0. Recently Desai and co-workers20 have calculated Cw(t)and C,(t)for the rough sphere fluid by mode-mode coupling theory and obtain reasonable agreement with our rough sphere calculations. In addition to this our own calculations indicate that periodic boundary conditions cannot be responsible for the large preasympto tic correlations. In closing we mention that in one run of 40 000 steps on a system of ellipses of low axial ratio (a = 1.3) and very high reduced temperature (T* = 2.415) there is clearly a region that looks asymptotic (see fig.2.8). The deviation from the exponen- tial is plotted against l/t2 in fig. 2.9. We have not yet completed data analysis so that fig. 2.9 should be regarded as tentative. Nevertheless it should be noted that the slope at long time is within a factor of 10 of what is predicted. ROTATIONAL RELAXATION IN LIQUIDS 0.040 I -0.030 - Y h-0.020-0.010 -I I ' O ' I 0.05 ' ' ' 0.10 ' ' ' 1 /t2 (inverse squaretime in reduced units) Fro.2.7.-The orientational correlation function defined in eqn (2.3) for Lennard-Jonesellipses at T*= 0.805 and a = 2.0. BRUCE J.BERNE S O .-c V t Y .e I= .-Y -L L 0 ....... . -0. .-0.OOll I 1 I I I I I f 1 I I I I I I 0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 reduced time It/t,J FIG.2.8.-CV(t) and Cw(t)for a fluid of ellipses of axial ratio a = 1.3 and reduced temperature T*= 2.415. At = 5 X 10-3t,. to = (ma"/&)+. *.* ROTATIONAL RELAXATION IN LIQUIDS t units of t/f c0.008 - 0.006- a .. 1 - *. 0.004-a0 -0.002--0 lll’1llllllll roughly indicates small amplitude of long time tail in Co(t). T* = 2.415 a = 1.3 p/po-= 0.3 N = 1296. We have benefited from several very useful conversations with Prof. R. Kapral and C. Hynes. These investigators have independently developed a hydrodynamic model.B. J. Alder D. M. Gass and T. E. Wainwright J. Chem. Phys. 1970 53 3813. C. M. Hu and R. Zwanzig J. Chem. Phys. 1974,60,4354. G. K. Youagren and A. Acrivos J. Chem. Phys. 1975 63 3846. D. R. Bauer J. I. Brauman and R. Pecora J. Amer. Chem. Sac. 1974,96,6840. R. Zwanzig and M. Bixon Phys. Rev. A 1970,2,2005. B. J. Alder and T. Wainwright Phys. Rev. Letters 1967 18 968. L. Verlet D. Levesque and G. Weiss personal communication. N. K. Ailawadi and B. J. Berne IUPAP Conference on Statistical Physics Chicago March 1971. B. J. Berne J. Chem. Phys. 1972,56,2164. lo B. J. Berne unpublished manuscript. l1 F. Garisto and R. Kapral Phys. Rev. A 1974,10,309; Y. Pomeau and J. Weber Phys. Rev. A 1973 8 1422; T. Keyes and I. Oppenheim Physica 1974,75 583.l2 B. Hills and J. M. Deutch Physica 1976 in press. l3 J. Montgomery and B. J. Berne J. Chem. Phys. 1976 (2 papers in press). l4 J. O’Deil and B. J. Berne J. Chem. Phys. 1975 63 2376. l5 J. O’Dell and B. J. Berne The Persistence of Linear and Angular Velocity in a Fluid of Rough Discs Manuscript in preparation Sept. 1976. See also J. O’Dell Ph.D. Dissertation (Columbia University 1976). l6 J. R. Dorfman and E. G. D. Cohen Phys. Rev. Letters 1970 25 1257. l7 L. Nady and B. J. Berne Persistence Effects in Anisotropic Fluids manuscript in preparation Sept. 1976. l8 B. J. Berne and P. Pechukas J. Chem. Phys. 1972,56,4213. l9 See for example W. W. Wood Fundamental Problems in Statistical Mechanics IIZ,ed. E. G. D. Cohen (North Holland Amsterdam 1975).2o J. R.Mehaffrey R. Desai and R Kapral J. Chem. Phys. 1977 66 1665.

ISSN:0301-5696    

DOI:10.1039/FS9771100048

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Collective Molecular Motions in Liquids from Depolarized Light Scattering SEARBY AND PIERRE BY GEOFFREY PIERREBEZOT SIXOU Laboratoire de Physique de la Matiere Condens6e,t Parc Valrose 06100-Nice France Received 6th August 1976 The basic hypotheses used by some of the different theories (viscoelastic and microscopic) that have been proposed to explain the lineshapes observed in quasi-elastic depolarized light scattering are summarized and the results predicted by these theories compared and contrasted in different situations. These theoretical results are in turn compared with the experimental results obtained from different physical systems liquids with low viscosity composed of relatively small and simple molecules; liquids composed of more cumbersome but rigid molecules ; supercooled liquids ; liquid crystals in the isotropic phase.The coupling parameters obtained from VH and HH spectra ix. the coupling between molecular orientation and transverse or longitudinal waves in the liquid are assessed whereby it is shown that the transverse (shear) waves can be strongly propagative at low temperatures in a supercooled liquid but that as the viscosity is decreased these waves become pro- gressively softer and for low viscosities they have a purely diffusive character. Although most theories are adequate to explain the diffusive shear waves observed for low viscosities a theory using generalised hydrodynamics is necessary to explain the results observed in supercooled liquids. It has been known for over fifty years now that the spectral density of light scat- tered inelastically from a fluid reproduces the spectral density of polarizability fluc- tuations in the fluid and thus contains information on molecular movements.Light scattered with the same polarization as the incident light contains informa- tion about isotropic fluctuations in the fluid whereas depolarized scattering contains information about anisotropic fluctuations arising from molecular reorientations. The intensity of the depolarized spectrum generally weak is proportional to the optical anisotropy of the molecule. For simple liquids the depolarized spectrum is frequently Lorentzian with an inverse half-width which is equal to a correlation time for molecu- lar orientation.2 Leontovich3 first predicted that the depolarized line may contain a central dip corresponding to propagative but highly overdamped shear (transverse) waves in the liquid.This effect was first confirmed by Fabelinskii and co-w~rkers.~ Since then there has been a considerable amount of work in this field both theoretical and experimental. DISCUSSION OF DIFFERENT THEORETICAL APPROACHES The visco-elastic theory of Rytov5 assumed a priori the existence of a frequency- dependent shear modulus which couples the stress and strain tensors. Fluctuations in the strain tensor produce light scattering via a photo-elastic coefficient. The depolarized spectrum has a width l/z where z is the common relaxation time for the stress and strain tensors. The lineshape is non-Lorentzian because of modulation of the strain tensor produced by the damped shear (and longitudinal) waves.More iLaboratoire associ6 au C.N.R.S. no. 190. COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS recently a number of schools have developed a rather different approach which we shall call microscopic. It is not possible to list all the authors who have contributed to this work but amongst the most frequently cited we may mention the schools of Keyes and Kivelson,”s Pecora,9~10Gershon and Oppenheim,’l-13 Ailawadi and Although outwardly these theories appear quite different there is a great deal of similarity in the type of approach. Briefly one chooses one (or more) “pri-mary ” variable(s) which is directly coupled to the local dielectric constant generally an orientation or polarizability tensor this (these) primary variable(s) is allowed to couple to one (or more) secondary variable(s) generally a momentum density tensor via transport equations.A secondary variable by definition does not couple directly to the local dielectric constant. Except where discussed explicitly we shall be con- cerned in the rest of this paper with the << two variable ”theories in which one primary variable couples to one secondary variable. The various treatments mentioned above differ in their choice and exact definition of the primary and secondary variables and also in the mathematical tools used. However with the exception of the results of Ailawadi Berne and Forster,15 the above treatments eventually predict the same form for the shape of the VH scattering spectrum.The physical interpretation of the para- meters in the various theories is of course different. The theory of Ailawadi Berne and Forster is somewhat different in that it contains two primary variables angular momentum and stress. Their published lineshape is fundamentally different from that of the other theories but does not reproduce the experimental spectra . It is important to note that these treatments were formulated for liquids composed of spherical top molecules. Nevertheless the application to molecules of a more general shape seems to be verified experimentally for liquids well above their freezing point.10.’6,17 At this point we mention an approach used by De GenneP in 1971 to calculate the spectrum of depolarized light scattered by a liquid crystal in the isotropic phase in the VH geometry.It does not seem to have been widely recognized that De Gennes’ theory predicts the same shape for the spectrum of the scattered light (with the exception of a Lorentzian term which he explicitly neglects) as the above mentioned “two variable ” theories. In fact Fleury and have remarked that the iso- tropic phase of a liquid crystal should not be very different from other liquids com- posed of very anisotropic molecules but fail to bring out the correspondence of De Gennes’ expression with that of the other theories. De Gennes’ approach is simple and physical and we feel it is worthwhile recalling the main steps in his argument. His primary variable is a traceless symmetric second rank tensor Qap describing the local orientational order in the liquid i.e.the tensor orientational order parameter. The secondary variable is the hydrodynamic velocity Y from which he constructs the shear rate tensor cap = apYp + abVa. He then writes down two coupled linear equations linking the fluxes Qap and cap with their conjugate forces. The force con- jugate to the shear rate tensor is the stress tensor and the force conjugate to the rate of change of the order parameter is obtained by expanding the free energy as a func- tion of powers of Qab and then differentiating with respect to Qa~. It is now sufficient to add Newton’s acceleration equation to obtain a closed system for the equations of motion. The flow birefringence effect and depolarized scattered light spectrum can be calculated directly.Moreoever De Gennes’ theory is valid not only for tempera- tures well above the nematic-isotropic transition of a liquid crystal but also close to this transition temperature where it is possible to include the effects of the critical divergence of certain parameters. One should note however that De Gennes has neglected the relaxation of the GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU elements of Qa~ which contribute to the scattered spectrum but which do not couple to the hydrodynamic modes. These elements decay with a single relaxation time and add an undisplaced Lorentzian to the spectrum. DISCUSSION OF VH SCATTERING The two-variable of theories Pecora,l0 Gershon and Oppenheim13 lead to a VH lineshape which may be written in the following form The intensity factor B is related to correlation functions of two elements of a sym- metric second rank tensor (defined below) the TIand I'2 are inverse correlation times of elements ofthe same tensor.The other parameters are defined below. The authors argue that they expect B = 1 and Tl= r2,in the limit of small k for Pecora and for reasons of symmetry according to Gershon and Oppenheim. The expression for the VH spectrum then reduces to the expression given by Keyes and Kivelson6 x (1 -R) + CO~/~~) I' cc sin2(e/2) 1 + co2/r2 (l/r)~0~~(9/2)((k~q~/p~I'~) 1/r + (k2qlpr-co2/r2)2 + (co/Q2 x (1 + (1 -R)k2y/pr)2' (2) Here 8 is the scattering angle; k is the scattering wave vector; q is the normal shear viscosity; p is the density; r (= 117) is an inverse correlation time.For Keyes and Kivelson Gershon and Oppenheim z is the correlation time for the anisotropic part of the polarizability density for De Gennes it is the correlation time of the orientational order parameter for Ailawadi and Berne (in their equivalent expression for I:) it is the correlation time of the angular momentum density for Pecora it is the correlation time of a second rank tensor introduced in a general fashion and identified first with the stress tensorg and later with the orientation density tensor.1° R is a dimensionless parameter expressing the coupling between the hydrodynamic modes and the anisotropy of the liquid. In the VH scattering geometry only the transverse hydrodynamic modes (shear waves) are able to couple to the primary vari- able.The results for the HH scattering geometry will be discussed in a later section. The expression for I has been written here in such a way that it can be seen that the shape of the spectrum is a function of two and only two dimensionless parameters R and k2q/pT. The first is a coupling parameter and the second is the ratio of two characteristic times. The first time l/T has already been discussed the second time p/k2qis the characteristic time for a viscous disturbance in the liquid. The normal- ized variable is co/F. We shall not repeat the results of Rytov's theory here but simply remark that if we put R 31 in eqn (2) then we find the same spectral shape as pre- dicted by Rytov.' However experimental evidence suggests that R N" 0.4.10*17921 Moreover in Rytov's theory z is identified with the relaxation time of the shear strain tensor whereas more recent arguments associate z with a correlation time for molecular orientation,* We now wish to discuss in detail the predictions of the microscopic theories under different physical conditions and compare these results with experimental evidence.COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS To do this we shall discuss the lineshape as a function of k2q/pI'. We divide the range of k2q/pI'into four rather arbitrary but characteristic regions (1) k2ullPI' < 1 (2) k2V/Pr< 1 (3) k2rllPI'> 1 (4) k2q/pT 9 1' The spectral shapes calculated using expression (2) are shown in fig.1 for these four values of k2q/pI' and at a scattering angle of z/2. For R -+0 or k2q/pI' -+0 the spectrum becomes a simple Lorentzian of half-width r. REGION k2q/pI'< 1 This situation corresponds to that of liquids composed of small simple molecules which have very short reorientation times (large r)and low viscosity. Such molecules are interesting because they can be very good approximations to a symmetric top as required by theory. Moreoever it should be possible to perform numerical model calculations for them. However from an experimental point of view the fine struc- ture arising from coupling to the hydrodynamic modes is almost impossible to resolve. As an example we show the depolarized spectrum of cyanopropyne CH,-C-C-C-N (a "rigid rod " molecule) in fig.2. The calculated value of k2q/pTat this temperature is 3 x The central dip as expected from fig. 1 is not observed. This is FIG.1.-VH lineshapes according to the microscopic theories [eqn (2)] for four characteristic values of kzq/pI'. The first three columns are normalised to constant F. probably because the instrumental resolution is 1.5 GHz. whereas the expected width of the dip is only 1 GHz. At present work is in progress to improve instrumental resolution and verify the existence of the dip. REGION k2q/pI'< 1 This zone corresponds to a large number of " normal "liquids and also " viscous " liquids far above their freezing point. It is the region which has received the most attention.The agreement between theory and experiment is always very good despite the fact that few of the molecules studied can really be approximated to symmetric tops. As an example we have chosen tolane DC-C Fig. 3 showsa comparison between an experimental spectrum and a computer fit (convoluted with GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU 1 I I I -50 0 +50 frequency/GHz FIG.2.-VH spectrum of cyanopropyne at 16 "C. k2q/pr= 3 x -12.5 0 *12.5 frequency/GHz FIG.3.-VH spectrum of tolane at 83 "Ctogether with a best fit using expression (2). The fitted curve has been displaced in order to be visible. k2q/pr= 0.099 R = 0.39. the instrumental function). The fitted curve has been displaced in order to be visible. Fig. 4 shows the evolution of the lineshape as a function of temperature.Note that the width of the dip [ ~c(k~qI'/p)~ see ref. (7)] is nearly independent of temperature. This is not evident from fig. 1 since the plotted curves have been normalized to con- stant r. The values of the coupling parameter R obtained from best fits are shown in fig. 5. The full circles are two-parameter fits (R,q) the crosses are one parameter fits where is given its static measured value. Contrary to the findings of Enright and Stoicheff for salo1,22 there is no significant difference between the two results. The reason for this is probably that tolane is a rigid reasonably symmetric molecule whereas salol is certainly not. The values of R are closely grouped around 0.4. COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS I 1 I -12 0 +12 f requencyl GHz FIG.4.42) VH and (6)HH spectra of tolane showing the evolution with temperature.0.t -& ** +t +.+ + ++ .+ t 0.3 R 0.2 0.1 I 1 I I I I 1 1 0.0 FIG.5.-The coupling parameter R for tolane as a function of temperature 0 two parameter fits (R q) + one parameter fits (R). GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU Fig. 6 shows the values of R for a number of liquids studied in our laboratory as a reduced function of k2q/pI'. Although the scatter for some liquids is rather large the grouping of the different results around R E 0.4 is quite striking. This same value for R has been observed by many author^^^*^^*^^*^^^^ but it is not yet obvious from present day treatments why molecules as diverse as CS2 and ethyl benzoate should have very similar values for the coupling parameter.Much smaller values of R have been reported by Champion and Jacksonu for a number of n-alkanes. These mole- cules however are quite flexible which may well have an effect on the coupling and moreoever the values of R were deduced less directly from measurements of flow birefringence. ~~~ ,~ * 0.51 I 1 1 1 1 1 1 -$#no1 ll,...T ~ 0 0 10 -2 10 -1 1 EQ FIG.6.-Values of R measured in our laboratory as a reduced function of k2q/pr. pyridine; 0 acetophonone; 0ethyl benzoate; b tolane. REGIONk2q/pI'> 1 This region corresponds typically to the case of viscous liquids just above their freezing point. The width of the depolarized line is now roughly equal to the width of the dip that was observed at higher temperatures.The dip has disappeared but the lineshape is very non-Lorentzian. From a mathematical viewpoint the shear disturb- ances in the liquid are propagative in the sense that they are less than critically ~lamped.l~*~~ A typical example is shown for benzyl alcohol in fig. 7. The I and IE spectra at 12 "C are quite different the I spectrum is Lorentzian. Different authors have fitted expression (2) to experimental spectra in the range 30> k2q/pI'> 1 with varying degrees of success. Rouch et aZ.24claim good agreement in quinoline with k2qlpI' = 20. However Enright and Stegeman,22 Bezot et ~2Z.l~ find increasingly poor fits as k2q/pI' increases.Tsay and KivelsonZ5 have recalculated Keyes and Kivelson's expression for I retaining terms to higher order in k. They findz6 Expression (3) reduces to expression (2) in the limit k2q/pI' -g 1. Tsay and Kivelson state that expression (3) provides a good fit to the experimental spectra of super-cooled triphenyl phosphite up to at least k2q/pI'= 50. COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS +12°C -28" c A-38" c -1'2 +i2 I I I -12 0 +12 frequencyIGHz FIG.7.-(a) VH and (b)HH spectra of benzyl alcohol showing the transition from k2q/pr> 1 to k2q/pr& 1. The VH spectra at -28 "Cand at -38 "Chave been amplified to show the appearance of the propagating shear waves. REGION k2y/pI' 1 This region corresponds typically to supercooled liquids approaching the glass transition temperature.Expression (2) predicts for R 1 a triplet spectrum and for R < 1 a central Lorentzian of width I' superposed on a wider Lorentizian of width r/(l-R),see fig. 1. In this latter case the shear waves have lost their propagative character and have become diffusive once more. In either case the integrated inten- sity of the central narrow Lorentzian is equal to the intensity of the rest of the spectrum. The modification proposed by Tsay and Kivels~n~~ [eqn (3)] does not substantially change the general features of the predicted spectrum. The experimental spectra however are quite different.17s22*27*28 At sufficiently low temperatures a very weak depolarized doublet appears on the wings of the central line indicating the presence of well defined propagating shear waves.Fig. 8 shows that it is not possible to fit an experimental spectrum using expression (2). Fig. 7and 9 show the way in which the depolarized doublet detaches itself from the wings of the centre line for supercooled benzyl alcohol and supercooled triphenyl phosphite. The corresponding I2 spectra show no sign of this structure. Tsay and Kivels~n~~ say that they have investigated the spectra of triphenyl phosphite down to -20 "Cbut do not report the appearance of the doublet. This structure is however very weak about 1% of the total de- polarized intensity and it is quite possible that it would pass unnoticed. This weak doublet was first observed in supercooled salol by Fabelinskii et aZ.,4and has been studied in more detail by Vaucamps et aZ.,27Enright and Stoicheff,22 and by Bezot Searby and S~XOU.~~.~~ These latter authors have studied the spectra of ethyl ben- GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU zoate and of benzyl benzoate.They show that the intensity of the depolarized doub-let decreases as the temperature is lowered and the corresponding transverse sound velocity increases. The large decrease in the intensity of the doublet is the main cause of the dis-agreement with expression (2). Both the intensity and the velocity tend to limiting values at very low temperatures as would be expected if a relaxation process were at work. They also show that in the case of benzyl benzoate the transverse sound velocity obtained from their spectra is in good agreement with ultrasonic measurements made by Barlow and Ergin~av.3~ Y I ii; ..... 2: . .I.I - .I i: . .I * :* s i; % c .-ln c al .c. .c I I -12.5 0 + !.5 frequency /GHz FIG.8.-Attempts to fit expression (2) to the VH spectrum of benzyl benzoate at -60 "C,k2q/pI' = 5.5 x lo1*(value obtained by extrapolation). * R = 1.0; -R = 0.9999 97; --R= 0.999995. The experimental and fitted curves have been normalised to the same intensity for the central line. Vaucamps et aZ.,27interpret the spectra of supercooled salol in terms of a modified viscoelastic theory in which there are two relaxation processes contributing to the frequency dependence of the shear viscosity and the shear modulus.They find that this phenomenological approach gives an accurate description of the experimental spectra. Enright and Stoicheff22use expression (l) the more general expression given by Andersen and Pecora and again find good agreement provided TI# T2and B # 1 in the supercooled region. The parameter B is related to the relative intensity of the de-polarized doublet. The fact that Vaucamps et al. have to assume the existence of two relaxation pro-cesses or that Enright and Stoicheff need to let elements of the polarizability tensor be strongly temperature (frequency) dependent suggestthat a third (at least) variable is necessary in the microscopic theories in order to completely describe the behaviour of the liquid i.e.one of the so-called "fast " variables can no longer be averaged out COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS tto"c -9.4"c d&-1 7.5"c -f2 0 +112 f requency/GHz FIG.9.4~) VH and (b)HH spectra from supercooled triphenyl phosphite showing the appearance of propagating shear waves as the temperature is lowered. The width of the central component is dominated by the instrumental width as is always the case for supercooled liquids. of the equations of motion. These difficulties almost certainly arise as the transla- tional motions of the molecules slow down into the hydrodynamic region and we suggest that a pertinent third variable would be one describing the translational correlations of the molecules. Keyes Kivelson and McTague 29 have calculated the spectra arising when an unspecified "fast " variable couples to the orientation tensor (a "slow " variable) and show that this coupling could account for the relative in- tensity of the depolarized broad background frequently observed.However they do not calculate the effect of coupling the hydrodynamic modes into the system. Andersen and Pecorag have formulated a three-variable theory in which the variables are a velocity tensor the symmetric part of the stress tensor and a molecular orienta- tion tensor. Again this theory was proposed to account for the broad background of the depolarized spectra and apparently no attempt has been made to compare the results with the spectra of supercooled liquids. More recently Quentrec30 has con- structed a new set of hydrodynamic equations for a dense fluid using a more general- ized version of De Gennes' approach.18 He introduces two microscopic variables into his equations of motion.The first is the tensor used by De Gennes and which de- scribes the local orientational order the second is an analogous tensor describing the local translational order of the molecules. However at present the calculations have not been applied to depolarized light scattering. LIQUID CRYSTALS Liquid crystals are particularly interesting because of the strong orientational correlations that may exist between molecules even in the isotropic phase.31 Stinson GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU Litster and Clark32 have examined the small angle scattering spectra of a liquid crystal MBBA.They do not show the observed I spectra but report good agreement between their experimental data and De Gennes' theory. It is curious to note that the values that they deduce for the coupling parameter R (G2p2/vq in De Gennes' theory) range from 0.2 at high temperatures to 0.4 close to the transition temperature. These values are lower than those observed for " normal "liquids (Region k2q/pI' < 1 above). Fig. 10 shows the I and I2 spectra of another liquid crystal 4-cyano pentyl bi- A FIG.10.-(a) VH and (6) HH spectra of a liquid crystal 4-cyan0 pentyl biphenyl in the isotropic phase 140 "Cabove the transition point. The flat top to the VH spectrum is indicative of an un- resolved " dip ".phenyl. The experimental conditions are different from those of Stinson in that the temperature is here far above the transition point. The flat top to the I spectrum suggests the presence of a shear wave dip but at the time of writing instrumental resolution was not sufficient to allow accurate lineshape analysis. It is interesting to remark that in the isotropic phase of a liquid crystal the re- orientation time z is frequently very much longer than for a simple liquid whereas the viscosity has the normal value of a few centipoise. This means that in such a case it is possible to obtain values of k2q/pI'& 1 under low viscosity conditions contrary to the case of supercooled liquids. In other words the translational correlations are always negligible compared with the orientational correlations and the simple two-variable theories should be adequate to explain the depolarized spectra even for k2q/pI' & 1.One should thus observe that in the low temperature limit the shear waves become increasingly diffusive in accordance with eqn (2). DISCUSSION OF HH SCATTERING The HH scattering geometry has received much less attention than has VH scat-tering. In the HH geometry the transverse velocity gradients cannot contribute to the scattering for reasons of symmetry; however coupling with the longitudinal gradients is now permitted. At scattering angles other than n/2 there is of course a further contribution from the normal isotropic Brillouin scattering. Keyes and Kivelson,6 Gershon and Oppenheim13 have calculated the shape of the 1; spectra using their microscopic theories.However in this geometry it is not COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS possible to express their results in an equivalent form. Keyes and Kivelson6 obtain the following expression where E = k2q/pI'; mB is the frequency of the longitudinal sound waves with wave vector k (Brillouin shift) ; S is the half width of the Brillouin lines; r has the same definition as before and R is a coupling constant between the longitudinal sound waves and the primary variable. We have assumed 0 = n/2. Using equivalences analogous to those used in re-writing Gershon's expressions for the VH spectra we may re-write his expres~ion'~ for Ig 1; cc [cos2(8/2)+ &sin2(8/2)I2 S S + m)2 + s2+ (mB -m)2 + s2 The intensity factors a1and a2are related to time independent correlation functions of elements of the polarizability tensor.The spectral shapes predicted by these expres- sions are shown in fig. 11 as a function of mB/r. We have set al = a2 in expression FIG.11.-HH lineshapes calculated from the expressions given by Keyes and Kivelson (K-K) expression (4) and by Gershon and Oppenheim (G-0) expression (5). The arrows in the last line show the position of the fine structure for K-K. Note that R runs horizontally and not vertically as in fig. 1. (5). The common features of these spectra are an intense central Lorentzian of half- width I'plus fine structure at the Brillouin frequencies. The calculated intensity of the fine structure tends towards zero as wB/r -+0 and also as mB/r -+ CQ.The strongest effect is expected when mB/r E 1. In the limit mB/F< 1the two expressions become equivalent and predict a dip in the spectrum at the Brillouin frequencies. The upper curve in fig. 12 shows that this dip is observed experimentally. In the opposite limit ws/r > 1 both expressions predict a structure whose intensity diminishes rapidly with GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU -50 0 +50 frequency /GHz FIG.12.-HH spectra of pyridine. The fitted spectrum (smooth line) calculated using expression (4) with only R as a variable parameter has been displaced in order to be visible. The small triangle at the centre of the line represents the contribution from elastic parasitic scattering.increasing wB/T. However Keyes and Kivelson predict a hump at the Brillouin frequencies whereas Gershon and Oppenheim predict a dip. Experimental spectra indeed show structure in this case but since the features are commonly very weak one understands why the IE structure has not often been analysed. Fig. 13shows that for ethyl benzoate the Zg fine structure at large wB/I',is certainly a hump not a dip in agreement with Keyes and Kivelson. Enright and Stoicheff 21 have analysed the Zg and Z$ spectra of CS2using the expressions given by Keyes and Kivelson. They find that the Zg spectra are well described by expression (4) over the temperature range 1 I I -8 0 +0 frequency / GHz FIG.13.-HH spectrum of ethyl benzoate at -102 "Cshowing the fine structurearising from coup- ling to the longitudinal sound waves.The position of the Brillouin lines in the VV spectrum is indicated by the arrows. There is an overlap because the Brillouin shift is greater than the free spectral range of the instrument. COLLECTIVE MOLECULAR MOTIONS IN LIQUIDS investigated. They also find that within experimental error the values of R obtained from the I and I; spectra are equal with an average value of R z 0.37. Searby Bezot and Sixou20 have undertaken a similar analysis for the case of pyridine. The IF fine structure in this case is more intense than for CS2and they also find good agree- ment between experiment and expression (4). The lower curve in fig. 12 shows a spectrum obtained with oB/T= 1.08 together with the corresponding best fit.The fitted curve has been displaced horizontally in order to be visible. However unlike Enright and Stoicheff Searby et al. do not find agreement between the values of R obtained from the I; and Ig spectra but find RVH= 0.4 and RHHz 0.65. The reason for this difference is not satisfactorily explained at the time of writing. CONCLUSION In conclusion we see that according to the value of k2q/pr(and therefore to a certain degree according to the type of liquid considered) the models proposed explain the experimental depolarized spectra with varying degrees of success. In the case of ordinary organic liquids (k2q/prz 1) the two-variable microscopic theories work very well perhaps deceptively well since the observed spectra do not seemat allsensitive to the detailed molecular structure of the liquid.The molecules may be symmetric tops or not rigid or partly flexible; even the coupling parameter R seems to have a near-universal value of around 0.4 in this viscosity region. The spectra of supercooled liquids (k2q/pI’9 1) are still difficult to explain quanti- tatively and probably need to be studied in the light of a more generalized theory. In the special case of liquid crystals in the isotropic phase the non-existence of a propa- gating doublet at high k2q/pI’ needs to be confirmed by a more systematic study. For liquids composed of small simple molecules and having a very low viscosity it is necessary to verify that the non-observation of the fine structure is due to lack of instrumental resolution and not to more fundamental reasons such as the impossi- bility of separating “fast ” and ‘‘slow ” variables.Lastly there remain a number of questions concerning the exact significance of the parameters RyHand RHH(=RvH?) and the calculation of their values as a function of molecular shapes and interactions. l R. D. Mountain Rev. Mod. Phys. 1966,38,205. R. Pecora J. Chem. Phys. 1968,49,1036. M. A. Leontovich Izvest. Akad. Nauk S.S.R. Ser. Fiz. 1941 5 148 (J. Phys. USSR,1941 4,499). V. S. Starunov E. V. Tiganov and I. L. Fabelinskii Zhur Eksp. Teor. Fiz. Pis’ma Red. 1967 5 317 (JETP Letters 1967 5,260). ’S. M. Rytov Zhur eksp teor. Fiz. 1957,33 514 (Sov. Phys. JETP 1958 6,401). T. Keyes and D.Kivelson J. Chem. Phys. 1971,54,1786. ’T. Keyes and D. Kivelson J. Chem. Phys. 1972,56,1876. T. Keyes and D. Kivelson J. Physique 1972 Colloque C1,33,231. H. C. Andersen and R. Pecora J. Chem. Phys. 1971,54,2584. loG. R. Alms D. R. Bauer J. I. Brauman and R. Para J. Chem. Phys. 1973,59,5304. l1 N. D. Gershon and I. Oppenheim Physica 1973,64,247. l2 N. D. Gershon and I. Oppenheim J. Chem. Phys. 1973,59,1337. l3N. D. Gershon and I. Oppenheim MoZecuZar Motions in Liquids ed. J. Lascombe (Reidel Dordrecht 1974) p. 553. l4 N. Ailawadi and J. Berne J. Physique 1972 Colloque C1 33,22 1. l5 N. Ailawadi J. Berne and D. Forster Phys. Rev.,1971 A3 1462. l6 P. Sixou P. Bezot and G. M. Searby MoZ. Phys. 1975,30 1149. l7 P. Bezot G. M. Searby and P. Sixou,J. Chem.Phys. 1975,62,3813. l8 P. G. De Gennes MoZ. Cryst. Liq. Cryst. 1971 12 193. l9 P. A. Heury and J. P. Boon Adv. Chem. Phys. 1973 XXZY 1. *O G. M. Searby P. Bezot and P. Sixou J. Chem. Phys. 1976,64 1485. GEOFFREY SEARBY PIERRE BEZOT AND PIERRE SIXOU *l G. Enright and B. P. Stoicheff J. Chem. Phys. 1974,60,2536. 22 G. Enright and B. P. Stoicheff J. Chem. Phys. 1976,64,3658. 23 J. V. Champion and D. A. Jackson MoZ. Phys. 1976,31,1169. 24 J. Rouch J. P. Chabrat L. Letamendia and C. Vaucamps J. Chem. Phys. 1975,63,1383. 25 S. J. Tsay and D. Kivelson MoZ. Phys. 1975,29 1. 26 There is a typographical error in the expression given in ref. (25). D. Kivelson personal communication. 27 C. Vaucamps J. P. Chabrat L. Letamendia G. Nouchi and J. Rouch Opt.Comm.,1975 15 201* '* P. Bezot G. M. Searby and P. Sixou Opt. Comm.,1976,16,276. 29 T. Keyes D. Kivelson and J. P. McTague J. Chem. Phys. 1971,55,4096. 30 B. Quentrec J. Physique 1976 in press. 31 T. D. Gierke and W. H. Flygare J. Chem. Phys. 1974,61,2231. 32 T. W. Stinson J. D. Litster and N. A. Clark J. Physique 1972 Colloque C1,33 69. 33 A. J. Barlow and A. Erginsav J.C.S. Faraday 11 1974,70 885.

ISSN:0301-5696    

DOI:10.1039/FS9771100063

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Depolarized Rayleigh Spectroscopy Studies of Orientational Relaxation in Solutions of the Xylenes W. BEER AND ROBERT BY CHARLES PECORA" Department of Chemistry Stanford University Stanford California 94305 U.S.A. Received 17th September 1976 Depolarized Rayleigh spectroscopy measurements of orientational relaxation of the three xylenes in solution are presented. The viscosity dependences of the single molecule reorientation times are obtained from the directly measured collective reorientation times by varying the solution concentra- tion. It is found that within experimental error all three xylenes have the same single particle re- orientation time viscosity dependence Values of the effective static and dynamic correlation factors are however quite different for the three xylenes.Studies of the para-xylene-CBr4 complex are also presented. Measurements of the intensity and spectrum of depolarized scattered light may be used to obtain information about the dynamics of molecular reorientation of optically-anisotropic molecules in the liquid state. Among the quantities that may be studied are the collective reorientation times and their relation to single-particle reorientation times as well as the viscosity dependence of the single-molecule reorien- tation time~.l-~ In this article we present measurements of these quantities for the three xylenes in solution and as neat liquids. We also discuss complexes of para-xylene with CBr as an illustration of how information about chemical interactions of molecules in solution may be obtained from light scattering.Previous measurements have shown that the reorientation times determined by light scattering and various other techniques such as carbon-1 3 nuclear magnetic relaxation nuclear quadrupole relaxation and Raman scattering do not agree even when the time being measured is about the same molecular axis. The difference is attributable to the fact that light scattering measures a different reorientation time from these other techniques. If 5'" is a given component of a laboratory-fixed second rank tensor associated with the orientation of molecule i in the fluid then light scattering measures quantities associated with the time-correlation function of where the sum is over all relevant molecules in the fluid.Thus in light scattering information is obtained about the time correlation function Clearly if light scattering experiments are performed in dilute solution and the results extrapolated to infinite dilution the second term on the right hand side of t Present address E. I. Dupont De Nemours Experimental Station Wilmington Delaware 19898. CHARLES W. BEER AND ROBERT PECORA eqn (1) will be negligible and only the first term will contribute to the light scattering. The other techniques mentioned above usually measure quantities associated with (c(l)(t)c(l)(0))even in concentrated solutions and neat liquids. Thus only in the limit of infinite dilution would reorientation times obtained from light scattering results be expected to agree with those obtained from the other techniques.We call the relaxation time obtained from the light scattering measurement the “collective ” reorientation time and those of (c(l)(t)c(l)(O))the “single molecule ” reorientation times. The differences between these times are as may be seen from eqn (l) due to the time-displaced correlations between the orientations of different molecules (c(1)(oc(2)(o)). One experimental strategy for studying both the single-molecule and collective orientation times is to perform light scattering measurements to measure the collective time and other types of experiments to measure the single-molecule The difficulty with this procedure however is that for non-spherical molecules the different techniques often also measure reorientation rates about different molecular axes or more often differently weighted combinations of reorientation rates.In favourable circumstances this difficulty may be turned to advantage and several experiments may be performed to obtain reorientation rates about separate molecule- fixed axes.3 Recent observations have shown however that for certain classes of solutions of optically anisotropic molecules both the collective and single molecule reorientation times are linear functions of solution viscosity if the solute concentration is kept constant.1*2 The particular solvent used makes little difference to the measured reorientation times. Thus by performing light scattering measurements of the collective reorientation time as a function of viscosity for a series of concentrations and then extrapolating to infinite dilution the viscosity dependence of the single- molecule reorientation time may be obtained.Since the single molecule time depends very weakly on solute concentration the single-molecule reorientation time in the neat liquid (or concentrated solutions) could then be estimated from knowledge of the solution viscosities. In order to interpret the results of our experiments on the xylenes we shall use a simplified language first developed to interpret experiments on reorientation of sym- metric top molecules.6 The collective reorientation time zLs of the molecular sym- metry axis is related to the single molecule time z, about this same axis by the relation wherefand g are respectively known as the static and dynamic orientation correlation factors.The quantityfis a measure of the tendency of the molecular axes of a pair of molecules to be aligned parallel (f positive) or perpendicular (fnegative). It is proportional to (~c1)(0)~(2)(O)) mentioned in eqn (1) above. The quantity f may be measured independently of zLsby studying the integrated depolarized intensity IHv as a function of concentration. I& is given by where n is the refractive index of the solution and p is the molecular optical aniso-tr~py.~ Spectral studies at infinite dilution then give z, (since zLs= z,J and integrated intensity studies [eqn (3)] give$ The dynamic correlation factor g may be estimated from eqn (2). DEPOLARIZED RAYLEIGH SPECTROSCOPY STUDIES Since the xylenes are not symmetric top molecules the values of f and g obtained from this procedure should be treated as experimental parameters that describe differences between zLs and qP.A more complete theoretical treatment for asymmetric tops is simple to perform but contains many parameters and is thus difficult to compare with experiment. Our main object however is to obtain the viscosity dependences of the single molecule reorientation times. This objective is not affected by uncertainties in the physical interpretation off and g. EXPERIMENTAL All spectra were recorded using a 488081 laser source and a piezoelectrically scanned Fabry-Perot interferometer described previously.2 The xylenes used in this study were either obtained from Aldrich (0-and m-xylene 99%pure) or MCB (p-xylene).The xylenes were distilled prior to use. The solvents used in this study were isopentane (reagent grade) cyclohexane and carbon tetrachloride (spectroquality grade) and cyclo-octane (distilled prior to use). The solutions were prepared on a volume-to-volume basis and fdtered through 0.2 Alpha Metricel filters to remove dust. Viscosities were measured using a Cannon-Ubbelohde viscometer thermostatted in a constant temperature bath. The tempera- tures at which the individual spectra were obtained were recorded and the viscosities of the solutions were determined at the same temperature. All measurements were made at 21.5 f2 "C. Total intensities were measured for the neat xylenes and their solutions using a method described previ~usly.~ Carbon tetrachloride which gives a much lower depolarized back- ground than most hydrocarbon solvents was used as the solvent in the intensity measurements.The measured intensities were corrected to remove the contribution from collision-induced s~attering.~ The dependence of the total intensity upon the refractive index of the solution was also taken into account using eqn (3).' The p-xylene-CBr4 complex was prepared by heating equimolar portions of CBr (MCB sublimed) and p-xylene until the CBr dissolved. The solution solidified upon cooling yielding the solid complex which was used to prepare the solutions used in this study. Since CBr4 is unstable to sunlight solutions were prepared and used as quickly as possible. RESULTS The recorded spectra were digitized and fitted to a single Lorentzian plus a base-line with a nonlinear least squares program.8 The half-width at half-height found by the program was then corrected for instrumental broadening and used to deter- mine zLs.Plots of zLs against solution viscosity for 0- m-,and p-xylene are shown in fig. 1,2 and 3. The line-width and zLsfor all three xylenes were found to depend upon the concentration of the xylene in solution as well as the viscosity of the solution. This is especially apparent for p-xylene as shown in fig. 3. TABLEME MEASURED AND DERIVED QUANTITLESFOR THE NEAT XYLENES u-x ylene rn-x ylene p-xylene dCP" 0.769 0.594 0.638 %SIPS 10.10 7.70 1 1.06 single particle slope C,,/ps cp-l single particle intercept z',od/ps single particle reorientation time %,IPS fN gN 6.57 rt 0.13 2.05 & 0.09 7.10 rt 0.14 0.03 f0.13 -0.27 & 0.09 6.62 f0 33 2.05 f0.03 5.98 f0.20 -0.31 f0.10 -0.46 -f 0.09 6.79 f0.30 1.99 f0.09 6.32 f0.21 0.68 f 0.15 -0.04 & 0.11 a Measured at 23.5 "Cfor 0-and m-xylene 21 "C for p-xylene.* zLsmeasurement error estimate is &S%. CHARLES W. BEER AND ROBERT PECORA The total intensity data (corrected for collision-induced scattering and refractive index dependence) were fitted to a parabolic equation as a function of concentration [eqn (3)]. The regression coefficients were then used to determine.fN for the neat xylenes and their solutions. The total intensity data for the three xylenes are shown in fig. 4 and the values offlv are given in table 1.The quantity z, has been found to be a linear function of viscosity q and is given by1 =sp - G,V 3-z\?* (4) The single molecule slope Csp,for each xylene was found from the experimentally determined slopes of fig. 1 2 and 3 as follows for p-xylene where most of the concentration dependence of the slope was believed to arise from fN,the experi-I 1 8 1" I "'1 I. $1 0 0.5 1.0 1.5 q/cP FIG.1.-Collective reorientation time 7 against solution viscosity for ortho-xylene. 0 15% (v/v) 0-xylene 0 30% o-xylene 0 50% o-xylene A 70% o-xylene x 85% o-xylene + neat o-xylene. 0 0.5 1.o 1.5 q/cP FIG.2.-Collective reorientation time 7 against solution viscosity for rneta-xylene. Symbolsdenote the same concentrations as in fig. 1. DEPOLARIZED RAYLEIGH SPECTROSCOPY STUDIES ,-~.~--,-,-..,-,-I-I* 0 05 1 .o 1.5 IpP FIG.3.-ColIective reorientation time z against solution viscosity for para-xylene.010%p-xylene 0 20%p-xylene 030%p-xylene V 45%p-xylene 0 60%p-xylene A 80%p-xylene x 90%p-xylene, +neat p-xylene. l,,,,,,,,, 20 4 0 60 80 100 % xylene (v/v) FIG.4.4ntegrated intensity against % xylene (v/v) Elp-xylene 0o-xylene and A m-xylene. mentally determined slopes were fitted as a function of concentration to a straight line 11. -1 * . .1 1 1 .1 1 111 1. PY least squares. ine intercept was taKen to pe tne single molecule slope ana is given in table 1. For 0-and m-xylene,fN is either nearly zero or negative. However the slopes of the lines in fig. 1 and 2 increase with concentration indicating that gN is fairly large and negative.Hence a' linear extrapolation to zero concentration is not justified. To obtain an accurate extrapolation for Cspfor o-and rn-xylene zLs/(l + fN)was calciilated and the Tz,,/ll 4-fN\. viscnsitvl data were fitted tn R straight line CHARLES W. BEER AND ROBERT PECORA by least squares. Plots of zLs/(l +fN) against viscosity for m-and p-xylene are shown in fig. 5 and 6. All of the p-xylene data regardless of concentration fit a single line of slope 6.97 0.19 ps cP-l to within experimental error. This is in good agreement with the value of 6.79 5 0.30 ps cP-l given in table 1 thus reinforcing our earlier hypothesis that most of the concentration dependence of Csp for p-xylene arises from 1 +flv.For 0-and m-xylene the reciprocals of the slopes of zLs/(1 +flv) against q predicted by least squares were plotted as a function of concentration. The reciprocals of the intercepts predicted by least squares analyses are the single molecule slopes which are given in table 1 for 0-and rn-xylene. The values of the single molecule slopes for the three xylenes agree with one another to within experimental error. The values of the zero viscosity intercepts z(O) in fig. 1,2 and 3 do not exhibit much concentration dependence hence we took the variance-weighted average as z:? for 0 0.5 1.0 1.5 q/cP FIG.S.-z/( 1 +fN)against solutionviscosity for meta-xylene. Symbols denote the concentrations asin fig. 2. i,,,,,,,,,,,,,,,~,,, 0 0.5 1.0 1.5 2.0 ?) /cP FIG.6.-2 /(1 +fN)against solution viscosity forp-xylene symbols denote the same concentrations as in fig.3.DEPOLARIZED RAYLEIGH SPECTROSCOPY STUDIES each xylene. These values which are listed in table 1 agree to within experimental error. Using the values of Cspand of table 1 and measured values of the viscosi- ties allows one to calculate z, for the neat xylenes and their solutions. The zsp values for the neat xylenes are listed in table 1 ; differences between the values reflect differences in the viscosities of the neat xylenes. The gN for the neat xylenes and their solutions are calculated from zsp, fN and measured values of zLs,using eqn (2). The values of gN for the neat xylenes are listed in table 1. The value of gN for neat p-xylene is nearly zero while the value for neat rn-xylene is negative and has the largest magnitude yet observed for a neat liquid.The spectra of the p-xylene-CBr complexes were fitted to a single Lorentzian. The solutions used in this study contained either 11 %p-xylene (v/v) at a 1 :1 p-xylene/-CBr molar ratio or 23% p-xylene (v/v) at a 2 1 p-xylene/CBr molar ratio. A plot of zLs against y for the 11 % p-xylene (1 :1 molar ratio) and 23% p-xylene (2 1 molar ratio) solutions is shown in fig. 7. For comparison the least squares lines for 0 0.5 1.0 1.5 2.0 q fCP FIG.ir.-Collective reorientation time z against solution viscosity for p-xylene-CBr solutions and p-xylene solutions. 0 1 :1 p-xylene-CBr4 (1 1% (v/v) p-xylene) 0 2:1 p-xylene-CBr4 (23% (v/v) p-xylene,--10%p-xylene,-.--20% p-xylene.the 10% and 20%p-xylene solutions of fig. 3 are also included (dashed lines). The slopes and intercepts of the 11 % and 23 %p-xylene-CBr solutions and the 10% and 20%p-xylene solutions are given in table 2. The slopes and intercepts of thep-xylene- CBr solutions clearly differ from those of the p-xylene solutions at comparable concentration. The difference is particularly striking in the case of the 11 % (1 :1) p-xylene-CBr solution as compared with the 10% p-xylene solution. As more p-xylene is added the slope and intercept of the 23 % (2 1)p-xylene-CBr solution become quite similar to those of the 20%p-xylene solution. These observations suggest the pre- sence of a long-lived 1 :1 complex formed between p-xylene and CBr in solution.* * In fitting the 1 :1 p-xylene-CBr4 solutions to a single Lorentzian we assumed that the complex remains 100%associated in solution.This assumption may not be unreasonable since it is possible quantitatively to sublime the solid complex indicating that it remains associated in the gas phase. CHARLES W. BEER AND ROBERT PECORA 2.-Vrscosm DEPENDENCEOF SOLUTIONS OF p-XYLENE TABLE AND OF p-m~m-CBr,COMPLEXES slope/ps cP1 intercept /ps ~ ~ ~~ ~ ~~~ ~~ ~~ 1 :1 p-xylene-CBr [11% p-xylene (v/v)] 2:1 p-xylene-CBr [23% p-xylene (v/v)]10%p-xylene (vlv) 20%p-xylene (v/v) 4.80 0.31 7.35 f 0.11 8.57 f.0.36 7.60 f 0.22 6.80 f 0.28 3.26 f 0.16 1.78 f 0.14 1.99 f 029 When a second mole of p-xylene is added as in the 2 1p-xylene-CBr case the slope and intercept take on values intermediate between those of the 1:1p-xylene-CBr case and the 20%p-xylene solution.This implies that the second mole of p-xylene does not interact appreciably with the complex already in solution and we simply observe the average linewidth. Had we performed a two Lorentzian fit at the proper viscosity we might have observed two Lorentzians a narrow one corresponding to the 1:1 complex in solution and a broader one corresponding to the free xylene. DISCUSSION The collective reorientation times of the three xylenes in concentrated solution or the neat liquid have very different viscosity dependences while the single molecule times have identical viscosity dependences within experimental error. We note that the collective time is always greater than the single molecule time.The molecular shapes and sizes are apparently not different enough to give rise to appreciable differences in the single molecule viscosity dependence^,^ but the inter- actions between the xylenes are sufficiently different to give rise to different fand g values. Since the xylenes are not symmetric tops and there are uncertainties as to the proper refractive index correction to the depolarized intensities these fand g values must be cautiously interpreted. More data are needed to clarify these points. The fact that the xylene with no electric dipole moment para-xylene has the largest value offN while that with the largest dipole moment has fN N” 0 probably means that dipolar forces are not the dominant influence in determining the orienta- tional correlations in these liquids.It is probable that the molecular shape and hence the short range repulsive interactions are the dominant influence. Para-xylene the most symmetrical molecule has the greatest tendency to orient with the axes passing through the methyl substituents parallel. This work was supported by a grant from the National Science Foundation (USA). We wish to thank Mr. Jason Chang for performing the viscosity measurements and Dr. Q.-H. Lao for performing some of the integrated intensity measurements. G. R. Alms,D. R. Bauer J. I. Brauman and R. Pecora J. Chem. Phys. 1973,58,5570. G. R. Alms D. R. Bauer J. I. Brauman and R. Pecora J. Chem. Phys. 1973,59,5310,5321. D.R. Bauer G. R. Alms J. I. Brauman and R. Pecora J. Chem. Phys. 1974,61,2255. ‘D. R. Bauer J. I. Brauman and R. Pecora J. Amer. Chem. SOC.,1974,96 6840. ’D. R. Bauer J. I. Brauman and R. Pecora J. Chem. Phys. 1975,63 53. T.Keyes and D. Kivelson J. Chem. Phys. 1972,56 1057. ’The refractive index contribution to IHvis currently a matter of great controversy. In eqn (3) we have used an especially simple form for this factor so that caution must be exercised in interpreting the values off obtained. See for example A. K. Burnham G. R. Alms and W. H. Flygare J. Chem. Phys. 1975,62,3289; T. Keyes J. Chem. Phys. 1975,63,815; G. D. Patterson J. Chem. Phys. 1975 63,4032. R. I. Shrager MODELAIDE A Computer Graphics Program for Evaluation of Mathematical Models Tech. Rep. No. 5 (U.S.Department of Health Education and Welfare 1970).

ISSN:0301-5696    

DOI:10.1039/FS9771100078

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Resonance Raman Lineshape Studies of Vibrational and Rotational Relaxation in Solution BY PAULA. MADDEN University Chemical Laboratory Lensfield Road Cambridge CB2 1EW Received 19th August 1976 The theory of resonance scattering of light is reviewed to show how the lineshape is determined by various relaxation processes. Possible applications of linewidth measurements are then discussed. The object of this paper is to clarify the relationship of the shape of a Raman line at resonance to the parameters which characterise the rotational and vibrational relaxation of molecules in their electronic ground states. The type of information which can be obtained will then be discussed. The reason for extending the study of Raman bands into the resonance region is that the scattering changes in three potentially useful ways (a) The scattering from very low concentrations of scatterers becomes detectable leading to the possibility that the vibrational relaxation in the neighbourhood of the chromophores in biological macromolecules may be investigated.(b) In addition to zero’th and second rank tensors which contribute to the ordinary Raman scattering a first rank tensor may also contribute giving the possibility of studying two moments of the reorientational distribution function in the same experimental conditions. (c) Long overtone progressions of totally symmetric vibrations of small molecules may be enhanced the relationship between their linewidths gives a way of probing the vibrational dephasing processes for small molecules and ions in solution.Before discussing further these possibilities it must be shown that these relaxation processes are indeed observable in the experiment. The Raman lineshape off-reson- ance is now well characterised and we shall use it as a reference point but at resonance the situation is confused because in addition to Raman scattering (RR) the resonance fluorescence(RF) will begin to contribute. RF is envisaged as a process of absorption followed by emission; the characteristic bandshape of the scattered light is thus expected to be that of an emission band and to reflect the relaxation behaviour of the excited electronic state. It will be seen that the theory of the next section includes both terms and that the predicted lineshape includes contributions from each; this means that ground state relaxation parameters are only recovered under certain limiting conditions.THEORY OF RESONANCE SCATTERING To show how the scattering process is related to resonance fluorescence and resonance Raman we present a simplified version of a theory which has been treated in detail e1sewhere.l We consider the time evolution of a density operator p which describes the occupation of states of the scattering molecule (a) and the radiation PAUL A. MADDEN field (x). In the absence of the field molecule interaction the density operator factorises po = tToxo. (1) p evolves subject to a Liouville operator where 9O A = [Hmol+ Hrad, A]-BA (3) where Hmolis the time independent part of the Hamiltonian for a molecule in a medium and Hradthe Hamiltonian for the radiation field.The time dependent parts of the molecule-medium interaction cause relaxation of the populations of molecular energy levels and this is described by g2.9’ is the part of the Liouville operator arising from the field-molecule interaction and for our purposes may be written as 9’A = (.9+ 9”A K [/mi A] (4) A] + [/lSES where pi and /is are the components of the molecular dipole operator along the polarisation vectors of the incident and scattered radiation and Eiand Es are the electric field operators for the modes of the radiation field which are populated by the incident and scattered beams. In the interaction representation the equation of motion of p is which is solved iteratively to obtain To understand the scattering process it is convenient to proceed via a description of absorption and emission.Experimentally in an absorption or emission experiment we observe the rate of change of the population of a final state of the molecule- radiation field system (called o,) when the system was prepared in an initial state Q. The molecular state in o is different from that in cc and the final radiation field state of o has one fewer photon in the mode i of the radiation field (for absorption) or one more in the mode s (for emission) than has the initial state cc.’ These processes arise from the second term in (8) Since we never explicitly observe the initial and final states of the radiation field only that one differs from the other by the presence of one photon we average over the radiation field (k= trXoA) dtl(ppl.@(r)&i(tl) + &s(t).@(tl)]aCC)02a (10) RESONANCE RAMAN LINESHAPE STUDIES the first term on the right represents the absorption process and the second the emission.We have omitted the cross terms L?i(t)@s(tl) etc. because the radiation field operators in the two modes are statistically independent (i.e. such products as Ei(t)Es(tavanish). In the scattering experiment the final molecular state differs from the inital molecular state (for a Raman process) and the number of photons in the final state is increased by one for the scattering mode and decreased by one for the incident mode from the occupancies of the initial state.The fourth order terms from (9) contribute to this process when by averaging the term over the radiation field we obtain where T1 = .@'(t)gs(t Yi(t2)2'(tJ + gi(t) Pi(tJ Ys( t2) L@ (tJ T2 = @(t)PS( (ti) =Pi( t2)Pi t3) + &( tl) LPs(t3)gi(t)Y'(t,) t3)Yi(ti)gi t2) gi(t)gi T3 = .@(t)gs( (t2) + .S@(rl)ps( (t3). By comparison with eqn (10) it can be seen that the term T1 corresponds to an absorp-tion process over the interval tz 3t3followed (after a delay note the time ordering of t3,tz tl and t)by emission from tl to t (and vice versa). It thus corresponds to the term normally called resonance fluore~cence.~ T2 and T3 on the other hand involve simultaneous " absorption " and " emission " processes and thus describe the Raman scattering process.This conclusion is borne out by the dispersion relationship' for the differential scattering cross section (a2C/~sZ8ws)(oi, us)for the two classes of terms. TI gives the classical RF dispersion relationship (see fig. 1) showing separate absorption and emission lineshapes; the T's are relaxation matrix elements thus rtlx Rqz ria (1 3) w+d 1 Ad FIG.1.-Explanation of the symbols used in eqn (12) and (IS). PAUL A. MADDEN is the linewidth for the a -+ q transition and is the inverse lifetime of the molecular level q. Away from resonance the T2 (and T3) cross sections reduce to the usual Raman scattering expressions where 6 is the shift from the peak of the Raman line and is the transition polarisability tensor. Note that eqn (15) identifies the parameter Tpa as the normal Raman width.Since all the terms in eqn (1 1) contribute to the scattering we must include them all in our expression for the lineshape. For the case of resonance with a single level we obtain' Frequently we expect to encounter the situation where the linewidth terms involving the intermediate state q are large compared to those involving only a and q because electronic relaxation and dephasing processes contribute only to the former. In this case and for Raman shifts not too far from the band centre (6 -r,,&the lineshape of the scattered light appears Lorentzian and the width at half height determines the desired parameter rpa. EXPERIMENTAL The experimental difficulties with these lineshape studies arises from sample heating by absorption of the laser light.This local heating tends to defocus the beam reducing the scattering intensity as well as making the temperature of the scattering medium an uncertain parameter. An additional diflticulty is that since we are dealing with solutions the solvent Raman bands will interfere with lineshape measurements. Rotating sample techniques have been developed by a number of workers and extensive reviews of this work have ap- ~eared.~*~ For lineshape studies the equipment must satisfy two basic requirements (i) the polarisation state of the incident and scattered radiation must be well defined (so that iso- tropic scattering may be separated). Consequently the polarisation vectors must always be tangent to (or avoid altogether) any curved surface through which the incident or scattered radiation passes.(ii) There must be some means of removing the solvent bands from the spectra. With these criteria in mind we have designed the system shown schematically in fig. 2. The cell consists of a piece of precision Suprasil tubing with windows fused to the ends so as to make a right angle at the corners (from which the scattering will be observed). The cell is divided into two halves one half contains the solution under study and the other pure solvent. The cell is rotated at -1800 r.p.m. by a hysteresis synchronous motor so that the position of the two halves is in phase with the sinusoidal voltage which drives the motor. A reference is taken from this voltage to control the gates of a two-channel photon counter (with a variable phase and duty cycle).We are thus able to measure the radiation scattered from the solution minus that from pure solvent. Complete elimination of the solvent bands RESONANCE RAMAN LINESHAPE STUDIES PC osc I ref I' S amp verticalT I \-pF-l FIG.2.-Schematic diagram of the spinning cell system; m is the hysteresis synchronous motor which is driven by a variable (0-100 c.P.s.) frequency oscillator (osc.) from which a reference signal (ref.) is taken to control the two channels of the photon counter (pc.) spec. is the spectrometer. is difficult because of noise and because the laser intensity in the (absorbing) solution is lower. The medium temperature in the laser focus may be determined by measuring the Stokes to anti-Stokes line intensities for the solvent bands.DISCUSSION OF APPLICATIONS The possible applications of the techniques labelled (a)and (b)in the Introduction have as yet received no attention and although the third is now fairly well character- ised more work is needed before significant results are obtained. (a)Many resonance Raman spectra of biological systems have been published particularly of the heme-pr~teins,~.~ but these studies have been concerned with the band positions and excitation profiles. The bands which are enhanced at resonance are those whose force field is appreciably altered by the electronic excitation. For example in the much studied cytochrome-c which consists of an iron atom bound to a porphyrin ring surrounded by protein the electronic excitation which is resonant is of the porphyrin and only ring modes are observed.An interesting systematic variation in the linewidths of these modes may be observed. The porphyrin has a distorted C, structure and we may label the modes according to their parentage in this group7 as Al A2 and B2 (there being several of each type). It is observed that the widths of these lines is such that r(AJ <r(A2)<r(B2).* Since these widths must arise from vibrational relaxation such a systematic variation suggests that specific information on the porphyrin-protein interaction could be learnt from a line- width study of these bands. (b)The possibilities posed by the presence of a first rank part in the scattering tensor for the study of reorientational motions may be more difficult to realise.McClain9 has published the group theoretical rules to be fulfilled by a band to give antisymmetric scattering. If a band were to give pure antisymmetric scattering (such as an A2 band of a C4 molecule) only a depolarised component would appear in the spectrum whose shape would be determined by both vibrational and rotational relaxation. We require instead a band which will induce both symmetric and anti- symmetric components in the polarisability tensor (e.g. the nominally A2 bands of cytochrome c) this can occur for the A2 vibrations of molecules of symmetry lower than C3,. We must then measure as well as the polarised and depolarised spectra the backscattering ~pectra,~*~~*~ in order to separate the contributions from the zero'th first and second rank polarisability tensors.It is even then unlikely that for the laser lines currently available and for the small molecules for which the re- PAUL A. MADDEN orientational motion is of interest antisymmetric scattering will be observed as it appears that the most stringent requirement is that the mode under study should vibronically mix two resonant excited states.11-13 It seems unlikely that there will be many molecules for which these criteria are met. (c) The overtone progressions of the totally symmetric vibrations of a variety of small molecules have been studied by a number of workers and the variation of half band width with the order of the overtone pubIished.l"-l6 The width of the isotropic part of these lines arises from vibrational relaxation and to understand the nature of the results we will briefly review a recent theory of this 1ine~idth.l~ The medium-molecule interaction is represented by a perturbation where the time dependence of Bl and B2arises from molecular motions in the medium and q is the oscillator coordinate.The theory of the relaxation rnatri~~*~' shows that rmn = ?/mn + rm/2 + rnl2 (19) where m and n label the vibrational levels and where ymnis the dephasing contribution to the linewidth and r,,,and the energy relaxation terms with rmand I'n involve the power spectrum of the correlation function of a medium operator at a vibrational frequency (of the order of say lo2 cm-l).Consider as an example the case where the term B,q arises from the vibrationally induced dipole- dipole interaction. in medium where Timis the dipole-dipole interaction tensor between a medium molecule j and the molecule under study. In this case which is a spectrum very similar to that of the dipole-induced dipole light scattering ZD~D(CO).'~ For small molecules this decays exponentially IDID(co> cc e-"/"O (25) with coo a frequency of the order of 10 cm-l. In this case we have Jl,(co = 100 cm-I) = e-I0 J,,(co) and we may neglect Trnand in eqn (19). Similar conclusions have been reached from a different point of view by 0thers.l' RESONANCE RAMAN LINESHAPE STUDIES If we assume that the oscillator is harmonic then qnn= 0 and q2, oc n.We then predict the width of the nth overtone Fnocc n2 (26) which is qualitatively what is observed (fig. 3) note that for this harmonic model I’ varies as n. We can then see by reference to eqn (20) that the power spectrum at zero frequency of the medium operators which cause the relaxation may be obtained from the experiments. FIG.3.The variation of overtone bandwith I‘, with 2.(a) is for the Add’ ion [ref. (1511 (b)is for Ti14in cyclohexane [ref. (16)] and (c) is for I2in CCl. [ref. (14)]. (d) shows the variation of (&)2 with n2 for a Morse oscillator basis the Morse potential is chosen to fit the observed frequencies for I2/CCI4. However as the figures show the experimental results tend to deviate from quadratic behaviour for high order overtones.To see if this is a defect of the har- homic model a Morse potential was determined from the measured anharmonicity constants its eigenvectors found and the matrix elements q, and q2, evaluated. For the Morse oscillator both qnnand q2, vary as n (q2, actually increases slightly more rapidly) and so the agreement with experiment is not improved. It should be noted that the experimental linewidths are subject to error for these cases as the bands are overlapping of low intensity and the solvent bands were not removed. An over-estimate of the linewidth would result if the observed spectra had a background contribution. The other possibility is that because rVcL is very large for these high overtones the approximate form of the lineshape from eqn (17) does not hold.This however seems unlikely as the resonant level for I2with the 5145 A laser line is t = 43 (in the gas phase)20 and the broadening of this level should be much higher than that of Y = 14 involved in the highest measured overtone. We have tried to give a survey of the possibilities for resonance Raman lineshape studies. It can be seen theoretically that the lineshape has too much information in the wings and that only the width is an important parameter (and then only under certain assumptions). Experimental studies of sufficiently high quality can now be accomplished and promise to yield very interesting information on the vibrational dephasing processes of bio-macromolecules and of small molecules.PAUL A. MADDEN P. A. Madden and H. Wennerstrom MoZ. Phys. 1976,31,1103. See e.g. C. P. Slichter Principles of Magnetic Resonance (Harper and Row N.Y. 1961). J. Behninger J. Raman Spectr. 1974,2,275. ‘W. Kiefer and H. J. Bernstein J. AppZ. Spectr. 1971,25 500. A very thorough review by W. Kiefer will appear in Adv. I.R. Raman Spectr. vol. 3 ed. R. J. H. Clark. See e.g. T. C. Strekas A. J. Packer and T. G. Spiro J. Raman Spectr. 1973,1 197. ’J. R. Nestor and T.G. Spiro J. Raman Spectr. 1973,1,539. a L. D. Barron and P. A. Madden unpublished. W. M. McClain J. Chem. Phys. 1971,55,2789. lo M. Pezolet L.A. Nafie and W. L. Peticolas J. Raman Spectr. 1973 1,455. l1 L. D. Barron MoZ. Phys. 1976,31,129. l2 0.Sonnich Mortensen Chem. Phys.Letters 1975 30,406. l3 J. Friedman and R. M. Hochstrasser Chem. Phys. Letters 1975 32,414. l4 W. Kiefer and H. J. Bernstein J. Raman Spectr. 1973 1,417. l5 Y.Bosworth and R. J. H. Clark J.C.S. Dalton 1975 381. l6 R. J. H. Clark and P. D. Mitchell J. Amer. Chem. Soc. 1973,95 8300. *’ P. A. Madden and R. M. Lynden-Bell Chem. Phys. Letters 1976,38,163. la T. I. Cox and P. A. Madden Chem. Phys. Letters 1976 41 188. l9 D. Oxtoby and S.A. Rice,preprint received. 2o R. F. Barrow and K.K. Yee,J.C.S. Faraday 11 1973,69,684.

ISSN:0301-5696    

DOI:10.1039/FS9771100086

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Comparison of Interaction Induced Light Scattering and Infrared Absorption in Liquids BY R. A. STUCKART AND T. A. LITOVITZ C. J. MONTROSE Catholic University of America Washington D.C. U.S.A. Received 21st September 1976 The spectra of scattered depolarised light and infrared (IR) absorption in Ne Ar Kr Xe CH4and CF are investigated. The spectra were analysed in terms of a two component induction mechanism for the light scattering-a long range DID and a short range mechanism-and a single short range interaction for the IR. It is concluded that at liquid densities the high frequency portions of both spectra arise from isolated binary collisions at liquid densities. It is argued that the short range interaction mechanisms in the molecular and atomic liquids are the same if one considers interactions as occurring between the individual atoms of neighbouring molecules.1. INTRODUCTION Light scattering studies even though wavelengths of 21 5000 A are used can under certain conditions be used to investigate translational motions over the 0.1 to 1 A range. To do this one takes advantage of interaction induced anisotropies in the molecular polarisability of molecules. Depolarised scattering from fluids composed of optically isotropic particles results from anisotropy in the pair (and perhaps higher order) contributions to the polarisability caused by interactions between the particles. The anisotropic part of the polarisability tensor p for a pair of spherically symmetric particles is given by p =/3(3uu -1) (1) where 1 is the unit tensor and u is a unit vector along the line joining the interacting particles; B is the difference in the pair polarisabilities parallel and perpendicular to this line.Generally the anisotropy depends on the separation of the particles R consequently we shall write /?@)-and thus the spectral shape of the depolarised Rayleigh line reflects the molecular dynamics of pairs as characterised by the time variation of their relative positions. Specifically the spectrum is determined as the Fourier transform of the time correlation function '$1) = 2 2 (/3[Rij(o)lP[Rkl(t)lP[u,j(o) ukl(t>l>* (2) k#l i#j This is of especial interest since it means that depolarised Rayleigh spectra can reflect translational motions of molecules.The range of /3((R) determines which aspect of the translational motion is observed (e.g. collision details diffusion etc.). Both short and long range types of anisotropy functions exist. For the case of monatomic fluids the short range interaction is generally thought of as arising from overlap of electronic wave functions (EWO) and has been empirically given by P(R) = Ae-R/A (3) R. A. STUCKART C. J. MONTROSE AND T. A. LITOVITZ where A is characteristically of the order of a/lO,where 0 is the Lennard-Jones dia- meter. For atomic separations greater than 0 the EWO is negligible and the longer ranged dipole-induced dipole (DID) mechanism dominates. Here the incident light induces an oscillating dipole on a given atom which in turn induces a dipole on a neighbouring atom-but generally not aligned with the incident polarisation and thus yielding depolarised scattering.The DID mechanism can be described by an anisotropy P(R) = 60tO2/R3 (4) where is the isolated atom polarisability. Recently Oxtoby and Gelbartl have shown that when interacting atoms of finite size approach each other closely the point dipole approximation must be dropped and the finite size of the atoms taken into account. They have determined that a more appropriate expression is where A is 0.130 for argon. Because this is essentially the same as the range parameter for EWO it seems improbable that one can experimentally separate the EWO and finite size contribu- tions. For our purposes we shall simply use the fact that p(R) contains both short and long range components.Elsewhere2 we deal with the use of the long range component to study quasi- diffusional translational motions. It is the purpose of this work to study the high frequency wings of depolarised Rayleigh spectra to obtain an understanding of short ranged interactions and the molecular motions they reflect. We also include a comparable analysis of the data on interaction induced infrared absorption where only short range interactions are present. This will allow us to isolate the role of the short range process in the light scattering data. We compare the light scattering spectrum I(w) with the reduced infrared line shape where cc(co) is the absorption coefficient per cm co the incident IR frequency h Planck's constant k the Boltzmann constant and T the temperature.A(@)is equal to the Fourier transform of CIR(t),where where pkl is the induced dipole moment on the kl pair caused by their interaction. 2. MONATOMIC LIQUIDS A. DEPOLARISED RAYLEIGH SPECTRA We begin with a comparison of the depolarised Rayleigh spectra of liquid Ne Ar Kr and Xe. The spectra in different systems at different temperatures are com- pared by scaling the frequencies according to the prescription proposed by B~ontempo.~ Specifically a reduced variable x is introduced x = co'r where COMPARISON OF INTERACTION INDUCED LIGHT SCATTERING where m is the reduced mass p the range of the interaction and u)the frequency in cm-l which compensates for differences in the interaction dynamics from one system to another.The reduced spectra are shown in fig. 1. The main feature deserving of comment is that to within experimental accuracy the high frequency spectra (x >z 1.8) are the same. The spectra of argon at three densities are shown in fig. 2. It is clear that 1.0 0.1 0 * Y 4 0.01 0.00' 0 1 2 3 4 x FIG.1.-Depolarised light scattering spectrum plotted against scaled wavenumber for liquid Ne (open circles) Ar (open squares) Kr (open triangles) and Xe (closed circles) at temperatures of 300 84 116 and 161 K respectively. Data are from ref. (8). The value of x = 1 corresponds to 95,29 22.3 and 18.6 wavenumbers for Ne Ar Kr and Xe respectively. The range parameter p is assumed equal to 0.10.the spectral form is density dependent broadening considerably as the density is increased from that of a relatively dilute gas (200 amagat) to that of the dense liquid (784 amagat). We propose that these observations can be explained in terms of a relatively simple picture. Recall that the pair anisotropy arises from two processes. The DID contribution is characterized by a range of roughly 0.40; the short range contribution has a range 210.la. In a relatively dilute gas intimate collisions between pairs are rather infrequent and consequently the depolarised scattering comes mainly from the longer ranged DID process. Except for the exceedingly rare collisions in which r < l.lcr the DID anisotropy is overwhelmingly dominant. In the dilute gas three- and four- particle correlations are negligible and only the pair terms in eqn (2) i.e.the i = k R. A. STUCRART C. J. MONTROSE AND T. A. LITOVITZ t . =*i i 0 1 2 3 X FIG.2.-Depolarised light scattering spectrum plotted against scaled wavenumber for liquid argon at three different densities. Upper curve is a density of 784 amagat (84 K) the middle curve 664 amagat (120 K) and the bottom curve is 200 amagat (300 K). Data taken respectively from ref. (8); personal communication S-C.An; and P. Fleury W. Daniels and J. Worlock Phys. Rev. Letters 1971 27 1493. The value of x = 1.0 corresponds to 29 cm-' (84 K) 35 cm-' (120 K) and 55 cm-' (300 K). The range parameter was assumed to equal 0.1~. j = I terms are important. The intensity will vary as the density squared and the dynamics will be density independent since during the interaction of two atoms the probability that a third will be sufficiently close to modify the trajectories is quite low.The situation is illustrated schematically in fig. 3(a) and (b). At moderately high gas densities the DID mechanism still contributes to the spectral intensity at all frequencies. However the contribution of the short range mechanism is more important than in the dilute gas because with the atoms being on the average closer to one another there are relatively more intimate (small impact parameter) collisions and thus a greater number of interactions now involve the short range induction mechanism. It is clear that the time duration of the short range induction process is less than that of the long range mechanism.As a result the short range contribution to the spectrum contains substantially more high frequency content; their increased importance thus has the effect of broadening the spectrum. The effects of the increased density on the longer range DID scattering spectrum are rather more complicated. This stems from the fact that the three-particle and perhaps four-particle correlations become important contributors to the correlation function of the long range process. One effect of these terms on the correlation function is a reduction of the intensity below what would be expected on the basis of pair correlations alone;4 the intensity variation is less than the density sq~ared,~ as cancellation of the overall pair anisotropy (described by the three- and four-particle terms) becomes ~ignificant.~.~ From a spectral standpoint the modifications brought about by the triplet and quadruplet correlations occur principally at the lower frequencies.' Basically this simply reflects the fact that the average time between COMPARISON OF INTERACTION INDUCED LIGHT SCATTERING interactions (which roughly characterizes the three- and four-particle correlations) is rather longer in a gas than the time duration of a single pair interaction.Of course the three- and four-particle correlations also affect the short range interaction induced scattering. However at a given density the cancellation effects are considerably less important here than for the DID scattering; moreover what contributions there are will be as in the case of DID characterized by the time (a) IC) FIG3.-Schematic diagram of path of an atom in a fluid at various densities.The dashed circles indicate the range of a short and long range interaction. (a)pictures an isolated binary interaction typical of the dilute gas situation; (b)is the moderately dense gas and (c)is the liquid. between interactions (which for the short range induction process will be at least as long as for the long range process) and therefore modifies only the low frequency spectrum. Thus it follows that at moderately high densities the high frequency region of the spectrum is determined by contributions from two sources (1) the short time behav- iour of the many long range interactions occurring at this density; and (2) the dynamics of the relatively fewer short range interactions.Due to the short time duration of the latter they make a disproportionately large contribution to the high frequency region of the scattering spectrum. In the high density liquid the relative importance of the DID and short range processes is rather different. Because of the increased symmetry in the local liquid structure the intensity is rather severely reduced owing to cancellation in the three- and four-particle terms and is found to be a decreasing function of density.6 Moreover as illustrated in fig. 4(c) the dynamics governing the spectrum associ- ated with the long range mechanism are primarily of the stochastic random-walk type and therefore are considerably slower than simple binary central force motion.In comparison with the situation in a gas a given pair of atoms will spend on the average a considerably greater time within the range of the DID mechanism and thus the correlation time for this process will be considerably longer for the dense liquid. R. A. STUCKART C. 1. MONTROSE AND T. A. LITOVITZ W/cm-' FIG.4.-Depolarised light scattering spectrum in liquid and solid argon. The open squares are the solid data. Data are taken from ref. (8). Consequently the DID spectrum should be enhanced at low frequencies* (at the expense of the high frequencies) and we might anticipate that the high frequency spectrum that is observed derives primarily from the short range anisotropy induction mechanism.The number of these short range interactions that must be considered is rather large in a liquid relative to a gas because the average atomic separation in a liquid is sufficiently small that many intimate interactions (collisions) occur. For example in liquid Ar at the triple point the first peak in the radial distribution function occurs at about 3.7 A or about 1.10. Because the short range anisotropy induction occurs for pairs separated by distances between about 0 and 1.10 (recall A 21 0.10) there can be little doubt about the importance of this mechanism. The question of the contribution of the three- and four-particle correlations is difficult to assess in an a priori fashion. Generally we expect these terms to affect primarily the low frequency spectral range; however because the duration of and time between intimate collisions are probably not terribly disparate for dense liquids the question of a clean separation of the spectrum is unclear.However if we recall that the experimental high-frequency spectra were reduced to a single curve by a density independent parameter z then it follows that in this spectral region the pair correlation term @SR [rll(o)l/%R [rll(f)lp2 [ulJ(o) ulJ(f 11) (9) fi is dominant. 100 COMPARISON OF INTERACTION INDUCED LIGHT SCATTERING In addition it follows that insofar as the short range induction mechanism is concerned the dynamics governing the shape of the spectral wings are equivalently just isolated binary collisions such as one would expect in a dilute system.That such dynamical behaviour is observed at liquid densities is simply a reflection of the short range character of the anisotropy induction mechanism coupled with the absence of significantly correlated intimate ternary collisions. In terms of the picture just presented of interaction induced scattering in noble gas fluids we can draw several conclusions based on our earlier observations concern- ing the experimental depolarized Rayleigh spectra. (1) At liquid densitities the high frequency portion of the depolarized spectra arises chiefly from a short range anisotropy induction mechanism. At these frequencies only the pair correlation terms in C(t) are significant and the atomic motions that determine this part of the correlation function are equivalent to the isolated binary collision dynamics that are more usually associated with dilute gas interactions.(2) The lower frequency parts of the spectra contain the contributions from the DID process as well as from the three- and four-particle correlation terms in the short range correlation function. Moreover the atomic motions represent a many- body superposition and are thus characterized by rather strongly density dependent relaxation times and transport coefficients (e.g. the diffusion constant). As a consequence the low frequency scattering spectra in the different systems (at different temperatures and densities) are not reduced to a single “universal” curve via a density-independent frequency scaling parameter.(3) The density dependence of the argon spectra presented in fig. 2 is similarly compatible with the ideas presented above. The spectrum of the dilute gas at 200 amagats density is dominated by the long range DID induction at all frequencies; the contribution of the short range mechanism is simply too weak to be observable. The intermediate density (664 amagats) spectrum contains significant contributions at all frequencies from both the long and short range mechanisms. At liquid densi- ties each mechanism again contributes to the spectrum but their importance in different frequency regions is not the same. The DID mechanism and the three- and four-particle short range correlations influence primarily the low frequencies while at the high frequencies the short range induction mechanism and isolated binary collision dynamics are predominant.An interesting comparison of these liquid spectra with similar data obtained in crystalline solids can be made. Measurements have been made by Fleury et aZ.,* in liquid and solid samples of Ar Kr and Xe which were in phase equilibrium at their respective triple points. The spectra for Ar are shown in fig. 4 where we note the rather striking result that at high frequencies the spectra are identical to within experimental error. Apparently the same short range induction mechanism pair (only) correlations and effectively isolated binary dynamics are applicable to the description of short time processes in the solid just as they are in the liquid.This comes as a surprise because one might expect that the highly cooperative pheno- mena that lead to strong three- and four-particle correlations in the liquid would be much more important in the solid and probably even dominate the dynamics of the atoms in this phase. Moreover solid dynamics are traditionally described in terms of phonons rather than the motions of individual atoms. However such des- criptions are appropriate only for time scales longer than roughly l/mD (aDis the Debye cut offto the phonon spectrum). At times shorter than these the dynamics of interest are just the motion of a single atom relative to its near neighbours and as can be seen in fig. 4 the spectra agree only for frequencies above roughly 1.5 mD. The same result is true for Kr and Xe.R. A. STUCKART C. J. MONTROSE AND T. A. LITOVlTZ B. INFRARED ABSORPTION Up to this point in the discussion it has been difficult to isolate the short range induction mechanism from the DID. It has been necessary to restrict the discussion carefully to liquid densities and to the high frequency portion of the spectra. In dilute gases it was not possible to study the short range mechanism at all. Clearly it would be advantageous if it were possible to " switch off "the long range mechanism and deal only with the short range one. It is possible to approximate this by consider- ing the far infra-red absorption of noble gas mixtures where the spectrum results from interaction induced dipoles for which the only induction mechanism is of short range.9 To do this we have in fig.5 compared the IR line shape A(w) with light scattering spectra by scaling the frequencies as described above for the light scattering. Fig. 5 shows the scaled A(o)of neon + argon gas mixtures at 295 K and of the liquid neon + argon mixtures at 90 "C. The shaded band includes the three liquid light scattering spectra (Ar Kr Xe). There is a striking agreement between the high frequency behaviour of the IR and the light scattering spectra. We consider the implications of this agreement. As was the case with collision 0.0011 I I I 0 1 2 3 1 X Fro. S.-Depolarised light scattering spectra and reduced IR line shape plotted against scaled wave- number. The shaded area summarizes data in the four liquids in fig.1. The closed circles are IR absorption data in Ne + Ar gas at 295 K taken from ref. (12). The crosses are IR absorption in data in liquid Ne + Ar taken at 90 K. Data taken from ref. (3). x = 1 .O corresponds to 73.5 cm-' for Ne + Ar gas and 40 cm-I for Ne + Ar liquid. In each case p was assumed equal to O.lu the L-J diameter appropriate for a mixture. 102 COMPARISON OF INTERACTION INDUCED LIGHT SCATTERING induced light scattering eqn (7) shows that CIR(t)is determined both by the dynamics of the atomic motions through &(t) and ukl(t),and the functional form of the dependence ofthe dipole p on Rkl(t). Therefore the agreement of the high frequency line shapes implies that the functional form of the dependence of p on R,,is essentially the same as the dependence of j3 on Rkl.In general the induced dipole in molecular systems will be made up of two components. The first is a long range part that results when the field of a permanent multipole moment of one molecule induces a dipole in its neighbours. The second is an electron overlap (EO) part produced in a manner analogous to that of the EWO mechanism discussed above. It is clear that for noble gases there are no non-zero multipole moments of the constituent particles so that only the EO mechanism need be considered. Several calculationsgJ0 show that the form of eqn (3) applies and the range of the EO mechanism is also short (i.e. A 2i 0.1 A). The agreement of the liquid IR and light scattering spectra confirms our earlier assertion that the short range anisotropy induction mechanism dominates the high frequencies of the scattering from systems at liquid densities.The difference observed at low frequencies is to be expected because the light scattering includes contributions from the long range DID mechanism as well as the short range while there is no corresponding long range mechanism in the IR. What would not be so completely expected is the remarkable agreement at high frequencies between the Ne + Ar absorption spectra in the dense liquid and the dilute gas at high frequencies. The reduction of the low frequency absorption in the liquid compared to the gas is attribut- able to the von Kranendonk interference effect." Note that a low frequency depres- sion is also visible in the gas spectrum.This three-body spectrum characterizing the correlation between successive collisions of an atom is confined to lower frequen- cies in the gas than in the liquid because of the greater time between collisions in the dilute gas. Therefore the interference effect will not affect the high frequency region of the gas spectrum beyond about x = 1.8. Four body correlation functions can reasonably be expected to decay even slower than the two and three body functions and so their spectra will be even narrower than the three body. We are thus reason- ably confident that the absorption spectrum of Ne + Ar gas beyond x = 1.8 is due almost entirely to two body correlations. There can be little doubt that at dilute gas densities the dynamics of the atomic interactions are those of isolated binary collisions.Beyond x 21 1.8 the gas and liquid line shapes are the same. Even though the three-body spectrum is distinctly wider in the liquid than the gas we can infer from the agreement in the high frequency wings that even at liquid densities the high frequency wing of the short range spectrum in the IR arises from two body correlation functions and isolated binary collision dynamics. The excellent agreement between the high frequency light scattering spectra in liquids and the IR spectrum in the dilute gas supports the idea that in liquids the dynamics that control the scattering due to the short range anisotropy are those of essentially isolated binary collisions. At the same time the disagreement at low frequencies is consistent with our contention that the important mechanism for the induction of anisotropy changes as the density changes with the short range being im- portant at high densities and the long range being the most important in dilute systems.3. LIQUIDS COMPOSED OF TETRAHEDRAL MOLECULES We consider now the quasispherical tetrahedral molecuIes with chemical formula CX,. In particular we will examine the data in liquid CCl and CF,. We address the question of what role short range interactions play in the spectra of these liquids. R. A. STUCKART C. J. MONTROSE AND T. A. LITOVITZ The polarisability of a molecule with tetrahedral symmetry is isotropic. The DID mechanism should occur in these molecules in the same way it does with the noble gases-at large molecular separations (i.e./I= 6a,2/R3where R is now the distance between molecular centres). Therefore we might expect many similarities in the light scattering in liquids composed of monatomic or tetrahedral molecules. The situation as regards infrared absorption is not quite so simple. There is a dipole induction mechanism in the tetrahedral molecules that is not present in the noble gases. While these molecules have no permanent dipole moment they do possess a non-vanishing octupole moment. The octupole-induced dipole (OID) mechanism has the angular dependence of an octupole moment and varies as l/R5. This classes it as a relatively long range induction mechanism compared with EO. 0 40 80 120 160 w/cm-’ FIG.6.-Depolarised Rayleigh scattering and IR absorption plotted against frequency for liquid CCI4(295 K) and liquid CF4 (194 K).Upper curve is CF4. Closed triangles are IR data closed circles are light scattering. IR data are taken from a personal communication from G. Birnbaum. The light scattering and IR line shapes of liquid CC14 and CF4 are presented in fig. 6. They are in reasonably good agreement at high frequencies indicating that similar ranges exist for the short range induction mechanism in the two phenomena. We can compare the high frequency line shapes of the molecular liquids with the noble gases by using the frequency reduction in eqn (8). All of the spectra decay exponentially at high frequencies I(m) oc e-wiwo. We determined mooand scaled it in table 1 to the form 104 COMPARISON OF INTERACTION INDUCED LIGHT SCATTERING and for the IR data A(x) oc e-x/xo (1 1) where x = 007 and x = or.The values of xo (calculated assuming p 2 0.1 a) are consistently higher for the molecular liquids than for the monatomic ones. We propose that the reason for TABLE REDUCED EXPONENTIAL PARAMETER Xo FOR SOME ATOMIC AND MOLECULAR LIQUIDS tTa m T 00 rb ref. /A /(10-~~8) /K /cm-' /PS XO CF4 C 4.70 66.8 194 23.7 0.234 1.05 CCI 4 d 5.88 122 295 20.0 0.321 1.21 Ar e 3.405 33.4 84 18.9 0.183 0.65 Kr e 3.610 70 116 15.4 0.239 0.69 Xe e 4.055 110 161 12.8 0.285 0.69 Ne-Ar f 3.10 21.9 295 50.3 0.072 0.69 (gas) Ne-Ar 3.10 21.9 90 27.7 0.131 0.68 C 2.79 66.8 194 23.7 0.139 0.63 d 3.40' 122 29 5 20.0 0.186 0.70 a J.0.Hirschfelder C. Curtiss and R. Bird Molecular Theory of Gases and Liquids (John Wiley and Sons New York 1954). b 7 = (a/lO)dM/kT). c Present work. d S-C. An personal communication. e ref. (8). f D. R. Bosomworth and H. Gush Camd. J. Phys. 1965,43 751. g ref. (3). h L-J diameter of Ne used to approximate ionic diameter of F. i L-J diameter of Ar used to approximate ionic diameter of C1. this is that the range of the induction mechanism in CX is smaller than 0.10 the molecular diameter. This can be understood by assuming that the DID mechanism (or EO) operates between the individual atoms in each of the colliding molecules. At large separations the point dipole approximation is valid and each molecule can be regarded as if it were a single perfectly spherical atom.But when two CX4molecules approach so closely that two of the X atoms are much closer than the molecular centres this approximation breaks down. This situation is illustrated in fig. 7. Molecules i andj are separated by a distance R,,,and the two X atoms are separated by a distance rap. If r,,g < 1.10~ where a is the "diameter " of the bonded X atom then the Oxtoby-Gelbart finite size mechanism may well operate between the FIG.7.Cchematic diagram of two CX molecules in close contact. Rl is the separation between molecular centres raB is the separation between X atoms of two adjacent molecules. R. A. STUCKART C. J. MONTROSE AND T. A. LITOVITZ X atoms in the same way as it does between a pair of rare gas atoms.In this case the correct value of p to be used in eqn (8) would be 210. la and not 0.1acx4. The results using this assumption are summarized in table 1 where it can be seen that xofor CF4and CC14 are now in good agreement with the noble gas atoms. We conclude that the induction mechanisms in the tetrahedral molecules are indeed the same as those in the noble gases and there is no need to introduce any new mechanisms to account for the observed line shapes either in Rayleigh scattering or in infrared absorption. D. W. Oxtoby and W. M. Gelbart Mol. Phys. 1975,29 1569. * S-C. An C. J. Montrose and T. A. Litovitz J. Chem. Phys. 1976,64 3717. U. Buontempo S. Cunsolo and G. Jacucci Canad. J. Phys. 1971,49,2870. J. P McTague W.D. Ellinson and L. H. Hall J. Physique 1972 33 C1-241. V. Volterra J. A. Bucaro and T. A. Litovitz Phys. Rev. Letters 1971 26 55. B. Alder H. Strauss and J. Weiss J. Chem. Phys. 1973 59 1002. 'I M. Thibeau G. C.Tabisz B. Oksengorn and B. Vodar J. Quant. Spectr. Radiation Tramfer 1970,10 839. * P. Fleury J. M. Worlock and H. L. Carter Phys. Rev. Letters 1973,30 591. 'V. F. Sears Canad. J. Phys. 1968 46,1163. lo R. Matcha and R. Nesbitt Phys. Rev. 1967 160,72. J. von Krandendonk and Z. J. Kiss Canad. J. Phys. 1965,43,751. l2 D. R. Bosomworth and H. P. Gush Canad. J. Phys. 1965,43 751.

ISSN:0301-5696    

DOI:10.1039/FS9771100094

出版商:RSC    

年代:1977

数据来源: RSC