2 The Motion of Simple Molecules in Liquids By J. S. ROWLINSON and M. EVANS Physical Chemistry Laboratory South Parks Road Oxford OX1 302 1 Introduction In a dilute gas the molecules move freely with a spread of velocities given by the Maxwell-Boltzmann distribution. After they have moved for distances many times their diameters they collide with other molecules and are deflected into new rectilinear paths. Their mean motion over a long time is measured by the coefficient of diffusion which can be expressed in terms of the angles of deflection of the colliding pairs. In liquids matters are not so simple since each molecule is in perpetual interaction with its neighbours; there are no mean-free-paths and no binary collisions. Molecules can also rotate in an irregular manner about one or more axes under the influence of the torques exerted by their neighbours or by an external field.The molecular motions can be studied on a macroscopic scale by measuring for example rates of diffusion or dielectric relaxation but the relation of these crude macroscopic averages to what is happening at a molecular level is a difficult task. The purpose of this review is to describe the progress that has been made in this field in the past ten years. Sections 2 and 3 introduce the statistical language used to describe molecular motion the language of correlation functions and their spectra or Fourier trans- forms. Perhaps the greatest advance of the past ten years has been the systematic use of this language to describe ever-increasing regions of chemical physics statistical mechanics the scattering of light X-rays and neutrons i.r.Raman and n.m.r. spectroscopy are subjects which have gained in precision and unity from these developments. Section 4 describes the measurement and computer simulation of the correlation functions of a monatomic liquid. Sections 5-7 extend the discussion to simple molecular liquids with emphasis on the study of orientational correlation functions by far4.r. spectroscopy and light scattering. We give no derivations or proofs only statements and references. 2 The Velocity Auto-correlation Function A distribution function is an answer to a question of the following form if there is a molecule with a specified position orientation velocity etc.at a certain time t =0 what is the probability that there will be a molecule (the same or different) at a position distant by r with an orientation changed by 4,with a velocity increased by u J. 0.Hirschfelder C. F. Curtiss and R. B. Bird ‘Molecular Theory of Gases and Liquids’ Wiley New York,1954. 5 J. S. Rowlinson and M.Evans etc. at a time t? Such a function can be very complicated but often we are not interested in all these variables. Thus the equilibrium or thermodynamic properties are described by a time-averaged distribution function or what is equivalent by an average taken over an ensemble of systems at a fixed time t = 0. They are functions of r and 4 alone or in a monatomic fluid of r alone.’ The dynamic and transport properties in which we are here interested can be described only if we know the time evolution of these functions but then we can often ignore one of the other variables e.g.r or 4. The term correlation function (which we abbreviate c.f.) is used to describe a distribution function which has been normalized so as to approach zero for large values of the argument r and/or t (as is appropriate). It describes the degree of correlation between two events; such correlation is zero at large times or distances in an isotropic fluid. The most familiar distribution function is probably the radial function g(r) which describes the equilibrium probability of there being two molecules at a separation r and which can be measured from the X-ray diffraction pattern. At large separations this approaches unity (see below Section 3) and the corresponding c.f.is therefore g(r)-1 which is usually called h(r) the total correlation function. We consider below (Section 3) the generalization of h(r) to include time-dependence but start first with a more simple one-molecule c.f. the velocity auto-correlation function. If a molecule has a velocity u(0)at t =0 and a velocity u(t)at time t then a suitable measure of the degree of correlation of these velocities is the scalar product u(0) u(t). After a sufficiently long time e.g. lO-’Os the magnitude and direction of ~(t) will bear no relation to that of u(0) and so the scalar product goes to zero. The velocity auto-correlation function is defined as the average of this product over all molecules in a system at equilibrium and is denoted (u(0) -u(t)).It is convenient to use the symbol $(t) for the normalized function (~(0) u(t))/(u(0)2).The impor- tance of this function is its close relation to the coefficient of diffusion but before discussing this it is useful to examine the behaviour of correlation functions in general and this one in particular as functions of time. Consider first an almost collision-free gas in which a molecule has the same velocity at t as it has at t = 0. Hence the c.f. +(t) is a constant namely unity since (u(0)-u(t)>=(u(O>’>= 3kT/rn (1) where m is the mass of the molecule. A second idealized case is the perfect Einstein solid in which a molecule vibrates about a site at a constant angular frequency a,; here the c.f.is an oscillatory function proportional to cos (aEt).A liquid behaves in an intermediate fashion as is shown in Figure 1 which is based on a computer simulation discussed in Section 4. The c.f. is constant or gas-like at short times (typically t< s) it oscillates weakly and goes to zero at long times. The first negative region is easily explained as the rebound as a molecule reverses its velocity on colliding with a neighbour after travelling on average for the mean molecular separation. (a)H. N. V. Temperley J. S. Rowlinson and G. S. Rushbrooke ‘Physics of Simple Liquids’ North- Holland Amsterdam 1%8;(6)C. A. Croxton ‘Liquid State Physics’ Cambridge University Press 1974; (c) A. F. M. Barton ‘The Dynamic Liquid State’ Longmans London 1974.The Motion of Simple Molecules in Liquids Figure 1 The normalized velocity auto-correlation function for a collision-free gas for an Einstein solid and for a liquid. The last is based on results obtained by computer simulation discussed in Section 4 Each c.f. has a Fourier transform and if the variable of the c.f. is time as here then that of the transform is frequency. We define the transform of $(t) by . rw and conversely we have also the important relation . rw The function $(w) is called the spectrum of $(t) since it can be regarded as the 'sampling' of $(t) by a signal of frequency o.It is often easier to measure the spectra than the functions themselves. Equations (2) and (3) show that we can pass freely from c.f.to its spectrum and vice versa providing we know one of them for all values of its argument. In practice this is a considerable restriction. The spectca of the three cases considered above can be found at once. If $(t) is unity then $(o)is zero unless w = 0 and is infinite if o=0. That is &(o)= I J"OD e-'"' dt= (2~)~ S(o) (4) (2.rrY -OD where 6(o) is Dirac's delta function. This is zero everywhere except where its argument is zero where it is infinite and it is normalized to unity W W f(x)S(x -x*) dx =f(x*) 1-S(x) dx = 1 (5) 00 J. S. Rowlinson and M. Evans *E lo" o/rad s-' Figure 2 The Fourier transforms of the correlation functions of Figure 1. Those for gas and solid are delta-functions whilst that for the liquid has two broad components a difisiue mode centred on o = 0 and an oscillatory mode at frequencies comparable with oEof the solid Thus the spectrum of a collision-free gas is a sharp line at zero-frequency; it will be seen below that the diffusion coefficient is then infinite.For an Einstein solid the c.f. is obtained by integrating the product of e-'"' and cos (oEf), and this integral is zero unless o= wE,so that the spectrum is now a sharp line at the Einstein frequency or $(o)is proportional to S(o-oE).For a liquid we have again both features in the spectrum. The integration of the curve in Figure 1 produces a spectrum with two peaks characteristic of diffusional and oscillatory motion but both are now very broad (Figure 2). Thus the c.f.and its spectrum enable us to describe succinctly the essential features of translational motion in a liquid more accurately than was possible with the 'models' of the liquid state that used to be so popular; in which diffusion for example might be treated as an activated jump from one site to another. However if we are to use these correlation functions we must be able to observe them or their spectra to relate them on the one hand to macroscopic properties such as diffusion coefficients and on the other to intermolecular forces by the methods of statistical mechanics. In the rest of this section we say something of the second of these problems and touch on the fringes of the third. The first the measurement of correlation functions we cover in Section 4.The connection between correlation functions and macroscopic properties is a consequence of two broad and related generalizations which lie at the root of our present understanding of transport properties. These are linear response theory and a theorem linking dissipative processes and the regression of fluctuations. Neither is new for specialized versions of both were used for many years by Einstein Onsager and others but the realization of their power and generality is much more recent; it The Motion of Simple Molecules in Liquids stems from the work of Callen Green Kubo and during the years 1955-1965. Linear response theory describes the behaviour of two weakly coupled systems as for example when a beam of light or neutrons interacts with and is scattered by a liquid or when a beam of sound is absorbed and dispersed.Because the coupling is weak it follows that the response of the liquid can be calculated from a knowledge of its behaviour in the absence of the stimulus. This behaviour is described in terms of correlation functions of the appropriate dynamical variables (velocities energies etc.)in which the averages denoted by angle brackets are it is important to note averages over a system at equilibrium. By suitable ingenuity (sometimes called ‘indirect Kubo methods’) the stimulus can be chosen so that the response can include diffusional or viscous motion or transport of thermal en erg^,^'^ and so we are able to relate these transport or non-thermodynamic properties to averages over systems at equilibrium.The theorem on fluctuations stems from a hypothesis due to Onsager which lies behind his reciprocal relations between coupled transport processes. Every system at equilibrium exhibits small departures from the average values of unconstrained dynamical properties (e.g. fluctuations of energy in a system at fixed temperature). These fluctuations regress with time and Onsager’s result is that the average rate of their regression is governed by the usual macroscopic transport coefficients (e.g. thermal conductivity for thermal energy).’ Again we have this direct relation between a transport or dissipation process and a purely equilibrium phenomenon viz.fluctuations. These theorems lead to simple relations between the transport properties and the correlation functions relations which can now be derived in at least six different ways of different assumptions and rig~ur,~ and about whose truth there can now be no doubt.The first relation due originally to Einstein is the one most directly connected with the subject of this review for it shows that the coefficient of diffusion is the time-average of the velocity c.f. D = (~(0)u(t))dt or from equation (3) It is characteristic of hydrodynamic properties such as diffusion and viscosity that they are related to the zero-frequency intercept (Figure 2) of $ and similar functions. In fact the spectrum of a time-dependent c.f. can be regarded as a frequency- dependent transport coefficient. More recent results relate other transport coeffi- cient~~’~ to integrals over other correlation functions.R. Zwanzig Ann. Rev. Phys. Chem. 1%5,16,67. R. Kubo Reports Progr. Phys. 1%6,29 255. W. Marshall and S. W. Lovesey ‘Theory of Thermal Neutron Scattering’ Clarendon Press Oxford 1971 Chap. 11 and Appendix B. H. Mori Progr. Theor. Phys. 1%5 33 423; P. Schofield in ‘Statistical Mechanics’ ed. K. Singer (Specialist Periodical Reports) The Chemical Society London Vol. 2 1975. H. C. Longuet-Higgins Mol. Phys. 1963,6,65. P. A. Egelstaff ReportsProgr. Phys. 1%6,29 333. 10 J. S. Rowlinson and M. Evans The second problem the relationof a c.f. to the underlying molecular behaviour is more difficult and largely unsolved. One useful route which has been much followed recently is to express the c.f.in terms of so-called memory functions which are believed to have more simple structures. This idea has a lot in common with the reduction of the total (static) correlation function h(r)by expressing it in terms of the apparently more simple direct correlation function c(r) of Ornstein and Zernike.’.’’ This reduction which was re-introduced into modern statistical mechanics by Rushbrooke and Scoins,” has proved to be particularly fruitful for it is easier to make intelligent approximations for c(r) than for h(r);for example the Percus- Yevick approximation which lies behind much recent work on the static structure of liquids.2,10 The introduction of memory functions into the time-dependent correla- tion functions is leading to equally fruitful approximations.This approach is best described by discussing first the problem of Brownian motion or the diffusion of a particle of essentially infinite mass m. Its motion may be described by Langevin’s equation6 which separates the total force into two parts a frictional retardation proportional to the velocity (c is a constant) and a randomly fluctuating force K(t)which arises from the impacts of the molecules of the liquid in which the particle is suspended. The two forces are not unrelated for interaction with the molecules of the liquid is also the cause of the frictional retardation. This relation between the random and the systematic components is a very general phenomenon and when put into precise form becomes the mathematical expression of the fluctuation-dissipation the~rern.~ Since the molecules hitting the Brownian particle are light and their impacts frequent and (almost) independent it is usual to assume that K(t) is a Gaussian process with a correlation time negligibly short compared with the time steps of the Brownian motion.That is It follow~~,~,~ that the velocity c.f. of the Brownian particle is an exponential in the magnitude of the time (~(0) u(t))=(u2>exp (-clrl) (10) and where c is the constant in equation (8). Such exponential decay is a valid solution for a massive particle but it will not do if the particle is itself one of the molecules of the liquid. The results in Figure 1 show that $(t) is more complicated than (lo) and moreover it is an even function of time whose derivative vanishes at t =0 for at infinitessimally short times u(t) must be the same as u(0).Langevin’s equation has 9 L. S. Ornstein and F. Zernike Roc. Acad. Sci. Amsterdam 1914 17 793 reprinted in ‘Equilibrium Theory of Classical Fluids’ ed. H. L. Frisch and J. L. Lebovitz Benjamin New York 1964. lo A. Munster ‘Statistical Thermodynamics’ Springer-Verlag Berlin 1%9 Vol. 1 Chap. 10. 11 G. S. Rushbrooke and H. I. Scoins Proc. Roy. SOC.,1953 A216,203. The Motion of Simple Molecules in Liquids therefore been generalized for the discussion of molecular motion by the replace- ment of the constant c by what is in effect a frequency-dependent coefficient of friction. We write -m lo‘ m6(t)= u( t -t’)M,(t’) dt’ +K(t) where Mo(t)is a memory function which describes the past history of the friction which is itself a correlation function and which therefore has in turn a memory function which describes its own evolution.6 That is there exists a function Ml(t) defined by hi,(t)= -6’Mo(t- t’)Ml(t’) dt’ This argument can be extended indefinitely to M2(f) M3(t) etc.If we take the correlation of (12) (13) etc.with u(O),use the fact that this velocity is not correlated with K(t) (v(0)* K(t))=0 (14) and take Laplace transforms* of each expression then Mori showed that we obtain the transform $(p) of the original c.f. $(t) as a continued fraction.6 This approach is useful only if the memory functions are more simple than the original c.f. The first Mo(t)will show a peak at t=0 representing the quasi- Brownian or inertial motion of the molecule and a tail at longer times representing the damped oscillatory motion.(The memory function of an oscillator of frequency oEis a constant mi.) One might hope that if not the first then one of the low-order memory functions could be adequately approximated by a 6(t),thus truncating the continued fraction of Mori. In Section 4 we discuss the computer simulation of the velocity c.f. and in Section 5 extend the discussion to the rotational velocity c.f. and its memory functions but first introduce in the next section a more general two-particle c.f. 3 The Density-Density Correlation Function The velocity c.f. of the last section describes the motion of one molecule. It is equally important to be able to discuss the motions of pairs for two reasons; first the intermolecular forces in a liquid are to a first approximation the sum of the interactions of the molecules in pairs only and so we must know the static c.f.for pairs even to obtain the thermodynamic properties of internal energy pressure etc. and secondly because the observed scattering of electromagnetic radiation by matter is a coherent interference of the scattering from two different centres. Let the limiting? density on a molecular scale at point r =0 and at t =0 be denoted * A Laplace transform differs from a Fourier transform by the replacement of the oscillating function exp (-iot) by the monotonically decaying function exp (-pt) so that $(p) can be regarded as the result of sampling $(r) by a probe with a relaxation time of p-’.The integration in a Laplace transform is over all t>O. The limit is the ratio (SN/SV) of the number of molecules 6N with centres in a volume GVcontaining the point r = 0 as 6 V goes to zero. J. S. Rowlinson and M. Euans n(O,O) and that at r and t by n(r t). We define a density-density distribution function g*(r t) by g*(r t) = n-'(n(O 0) n(r t)) (15) where n is the mean number density or N/ V. If we average over an ensemble at a fixed time say t =0 then we obtain the static or as it is commonly called the radial distribution'*2g(r) ng(r)= n-'(n(O) n(r)) (r # 0) (16) We have however specified that there is a molecule at r = 0 and so we have there a density described by the delta-function S(r),and we have shown in (16) that the probability that there is a second molecule at r is proportional to ng(r).Hence g*b 0) = S(r)+ngb) (17) The two terms are called the self and the distinct parts of g*. As time passes the first broadens out into a curve as the molecule originally at r = 0 diffuses away and this curve finally collapses to a line gz, = 0. The second term also loses its structures with time and goes finally to the constant value%(r) = 1,(Figure 3). We therefore form the c.f. corresponding to g* by subtracting this long-time limit and the result G(r,t) is called the van Hove c.f." G(r,t) = g*(r t) -n (18) The Fourier transform of this c.f. over the three dimensions of space and one of time is S(k,w) the structure factor 00 S( k,w)=2 G(r,t)exp [i(r * k -of)] dr dt (19) I (2d v -a r Figure 3 The self and distinctparts of van Hove's correlation function G(r t).At zero time the self part is a delta-function at r =0 and the distinct is the (static) pairc.f. h(r) = g(r)-1. At infinite time both parts go to zero 12 L.van Hove Phys. Rev. 1954,95 249. The Motion of Simple Molecules in Liquids 13 It is the structure factor which is measured by the radiation (electromagnetic or neutron) scattered by the liquid. An incoming wave of length A is characterized by a vector k which has the direction of the wave and a magnitude of 27r/A. The wave scattered with vector k has an intensity which is proportional to S(k,a),where k = k,-k,and w is the change of angular frequency.Alternatively we can say that this wave has suffered a change of momentum of hk and of energy ha. Neutrons scattered from a monoenergetic (or monochromatic) beam can be analysed for change of angle and speed and so S(k,w)can be measured as a function of both variables at least over limited ranges. The scattering can be either incoher- ent (from one centre) or coherent (from a pair of centres) according to the nature of the nucleus. Different isotopes of one atom behave differently in this respect. The former arises from the self-part of the c.f. and the latter from the distinct and so in particularly favourable cases both parts of the transforms of G can be studied experimentally. Thermal neutrons have a wavelength of ca. 1A and so k-’ is comparable with the intermolecular spacing and the coherently scattered beam yields useful information on Gdistinct(r, t) on taking the inverse If the analysis by speed (or energy) is omitted then what is obtained is an integral of S(k,o)over all w,which is therefore a function of k only S(k).Information on the time dependence of the c.f. has now been lost and the transform of S(k)yields only the static distribution function g(r). With a beam of X-rays all the scattering is coherent analysis by energy is virtually impossible and so only S(k)and g(r) can be observed. Neutron scattering has therefore told us about the dynamics of liquids in a way which was not possible with X-rays. Visible light is scattered coherently with negligible change of momentum and the spectrum observed is therefore S(0,w).The change of frequency is small but observable if the incident light is from a laser and so highly monochromatic. Measurement of the intensity and angle but not the spectrum of the scattered light tells us only about the static properties. In particular S(k=0) is related to the Such scattering is small in a normal liquid but intense near the critical point where (dn/dP),is infinite. The spectrum of the scattered light is more useful for it has three distinct peaks,14 a Rayleigh line at o = 0 and two Brillouin lines at w =fWs, where W is the speed of sound in the liquid and s the wavenumber of the particular sound wave responsible for the scattering. The Rayleigh line arises from density (or more properly refractive index) fluctuations arising from fluctuations of local entropy at fixed pressure.Such fluctuations do not propagate through the fluid and so the Rayleigh line is centred on o = 0. It may also contain a weak and very broad depolarized component discussed in Section 6. The Brillouin lines arise from fluctuations of density due to fluctuations of pressure at fixed entropy. Such fluctuations propagate as sound waves which are l3 J. G. Powles in ‘Chemical Applications of Thermal Neutron Scattering’ ed. B.T. M. Willis Clarendon Press Oxford 1973. 14 D. McIntyre and J. V. Sengers in ref. 2(a);H. L. Strauss in ‘Chemical Applications of Lasers’ ed. C. B. Moore Academic Press New York 1974. 14 J. S. Rowlinson and M. Evans present in all liquids at equilibrium and which diffract the light at the appropriate Bragg angle.The frequency shift is a Doppler effect of the moving 'grating' and since the sound wave of appropriate length and orientation can be moving in either direction a pair of lines is produced one on each side of the incident frequency. The Rayleigh and Brillouin lines provide a wealth of information even for a monatomic liquid. The total intensity yields the compressibility (20) the ratio of intensities yield CJC, the width of the Rayleigh line yields the thermal diffusivity and the displacement and width of the Brillouin lines yield the speed and coefficient of absorption of sound at frequencies above 10"Hz that is above the range accessible by mechanically generated sound waves.l5 4 The Simulation and Measurement of Correlation Functions in Monatomic Liquids The study of the dynamics of liquids by computer simulation started with the work of Alder and Wainwright16 in 1959 who solved Newton's equations of motion for 32 hard spheres moving in a cubical box. It has progressed rapidly hand-in-hand with the advances in computer speed and capacity but even now it is clearly impossible to handle systems of molecules; the present practicable limit is about lo3 or perhaps up to lo4for particularly simple systems. In a sample of liquid of this size many molecules would be near a wall and so not representative of those in a bulk liquid. This problem is solved by surrounding the cubical sample on all sides by replicas of itself so that even molecules at a side or edge interact only with molecules in a similar en~ir0nment.l~ In these conditions even a sample of 1000molecules is amply large enough to study the dynamics and thermodynamics of a liquid since correlation functions decay virtually to zero over lengths of the order of 10molecular diameters except for liquids near their critical points.Before we can solve the equations of motion we must choose an intermolecular potential and the most popular for simulating the liquefied inert gases has been the Lennard-Jones (1 2,6) potential,' which is a reasonable compromise between simp- licity and realism. Geometrically more complicated potentials are now being used to simulate diatomic molecules;'8 one of the most complicated that has so far been used is that chosen by Rahman and Stillinger" for a simulation of water.The first and still perhaps the most informative simulation of the properties of a monatomic liquid was Rahman's study2' of 864 Lennard-Jones (12,6) particles confined to a cubic cell of side 10.2 c at a reduced temperature of kT/&= 0.786 where u and E are the collision diameter and depth of the Lennard-Jones potential. The density and temperature were chosen to simulate argon at 1.374 g cm-3 and 94.4 K. The velocity c.f. and its transform are shown in Figures 1 and 2. From the area under the curve in Figure 1 or equivalently from the intercept at zero frequency in Figure 2 we get a diffusion coefficient of 2.43 X lop9m2s-' which is the D. Sette in ref. 2(a).l6 B. J. Alder and T. E. Wainwright J. Chem. Phys. 1959 31 459. l7 B. J. Alder and W. G. Hoover and W. W. Wood in ref. 2(a). l8 J. Barojas D. Levesque and B. Quentrec Phys. Rev. 1973 A7,1092;P. S. Y. Cheung and J. G. Powles Mol. Phys. 1975 30 921. I9 A. Rahman and F. H. Stillinger J. Chem. Phys. 1971,55 3336. 20 A. Rahman Phys. Rev. 1964 136 A405. The Motion of Simple Molecules in Liquids Figure 4 The mean-square displacement as a function of time (schematic) same as that of liquid argonz1 at a temperature of 90 K. However the full curves give much more detailed information on the molecular motion than the value of the macroscopic or hydrodynamic coefficient of diffusion and in particular show the inadequacy of the unmodified Langevin equation.It is instructive to calculate the mean-square displacement (r2) of any one molecule as a function of time (Figure 4).This was computed directly by Rahman or in principle could have been obtained from the self-term of van Hove’s c.f. (r2(t)>= Jo r2Gs(r,t) dr At short time (r2)grows quadratically with time; that is the motion of the molecule is unretarded as in a perfect gas. For a crystal in which the average is taken from a fixed zero time over the unrelated phases of the oscillators (r‘) settles down to a constant value. For a liquid the initially unretarded motion quickly passes into a linear dependence of (r2)on time which corresponds to a constant rate of diffusion. For ‘argon’ this linear or hydrodynamic regime is reached after ca. 2.5 x 1O-l2s a time in which a molecule has moved on average through a distance of about 0.The hydrodynamic regime is thus reached surprisingly quickly. At high temperatures or at densities substantially lower than those of a typical liquid the negative region of the velocity c.f. disappears and there is a monotonic fall with this increasing time. In this region the dissection of the c.f. into memory functions has proved Much recent interest has centred about this decay to zero at long times of this positive c.f. in a fluid of moderate density. The decay is slow 21 J. Naghizadeh and S. A. Rice J. Chem. Phys. 1962,36 2710. z2 D. Levesque and L. Verlet Phys. Rev. 1970 A2,,2514. 16 J. S. Rowlinson and M. Evans (k7 not exponential) and the consensus of opinion23 is that at long times it goes as t-$.Such a slow decay gives rise to an anomalously large coefficient of diffusion at these densities. The cause of the tail is probably to be found in a weak vortex pattern which a moving molecule apparently generates. If a molecule is moving along say the x-axis at a particular time then its very motion will tend to establish a pattern of motion in the neighbouring molecules which resembles a vortex ring with cylindrical symmetry about the x-axis. The motion of the molecules in this ring gives an impetus to the first molecule along the x-axis thus tending to prolong its motion in that Such long tails in the c.f. and the complicated molecular motions which give rise to them are clearly going to make it difficult to develop a statistical theory of transport for fluids of densities between those of the dense liquid and the dilute gas.That is no early truncation of the memory function expansion is likely to do justice to the complexity of the motions. The measurement of correlation functions for a real liquid is more difficult than their computer simulation. In Section 3 we saw that their spectra can be obtained from scattering experiments but these rarely cover a sufficiently complete range of k or o for their successful Fourier inversion. If we want to study the one-molecule correlation functions then we must use incoherent neutron scattering and so are restricted to substances containing atoms at least one of whose isotopes has a large incoherent cross-section.The best is the proton with an incoherent cross-section of 79.7 barn and a coherent of 1.8 barn and after that the best is apparently sodium for which both areas are 1.7 barn. For argon (incoherent 0.4 barn and coherent 0.5 barn) Dasannacharya and Ra024 have obtained Gs(r,t) at 85 K but only with an accuracy of ca. 15%. From this result we could go to the diffusion coefficient by calculating (r2)from equation (21) and then obtaining D from the limiting slope of Figure 4 or we can use the fact that if G,at time t has a Gaussian shape (as they aver) then its width w(t)is related to the velocity c.f. by 5.13.25 w(r)=$[ (u(O)-u(t))(t-t’)dt’ For liquid sodium the results are more extensive although judging by the agreement between different not necessarily more accurate.Figure 5 shows a spectrum of the velocity c.f. which resembles that for a liquid of Lennard-Jones molecules (Figure 2) more closely than either resemble the Lorentzian form predicted by a simple Langevin equation. The coefficient of diffusion calculated from the intercept of the spectrum at zero frequency is ca. 2.8 x mz s-’ which is in only rough agreement with the experimental valuez6 of 4.3 x mz s-’. 23 B. J. Alder and T. E. Wainright Phys. Rev. 1970 Al 18; T. E. Wainright B. J. Alder and D. M. Gass Phys. Rev. 1971 A4 233; R. Zwanzig in ‘Statistical Mechanics-New Concepts New Problems New Applications’ ed. S. A. Rice K. F. Freed and J. C. Light University of Chicago Press Chicago 1972 p. 241; Papers by B. J. Alder and J. M. Deutch and the discussion on them in ‘Transport Phenomena- 1973’.ed. J. Kestin American Institute of Physics 1973. z4 B. A. Dasannacharya and K. R. Rao Phys. Rev. 1965,137 A417. 25 B. J. Berne and G. D. Harp Adv. Chem. Phys. 1970 17 63; B. J. Berne in ‘Physical Chemistry an Advanced Treatise’ ed. D. Henderson Academic Press New York 1971 Vol. 8B; B. J. Berne and D. Forster Ann. Rev. Phys. Chem. 1971 22 563; R. T. Bailey in ‘Molecular Spectroscopy’ ed. R. F. Barrow D. A. Long and D. J. Millen (Specialist Periodical Reports). The Chemical Society London 1974 Vol. 2 p. 173. 26 p. A. Egelstaff ‘Introduction to the Liquid State’ Academic Press London 1967 p. 4. The Motion of Simple Molecules in Liquids 1 2 3 0/10” rad s-’ Figure 5 The experimental Fourier transform of the velocity c.f.for liquid sodium’ compared with the form of this function predicted by Langevin’s equation 5 Absorption in Molecular Liquids In this Section and the next two we describe how we can study the rotary motion of molecules by means of the bandshapes of their spectra which are linked to orientational auto-correlation functions. Let u be a unit vector along a convenient axis of the molecule (usually along the permanent dipole moment if any) and J the angular velocity vector which is perpendicular to u in a diatomic molecule. We shall use the following correlation functions; the first in this section the second in Section 6 and the third in Section 7. UR)$(t) =(u(0) dt)> (R) +(t) =;(~[u(o)u(~)I~ -1) (23) ‘J’rL(t) =(J(0) J(f)>/(J2(O)> Here (IR) stands for infra-red and (R) for Rayleigh rather than Raman since we shall be considering scattered light that is symmetrically disposed about the exciting line and not about a line displaced from it.The Fourier transform of (IR)+(?) is related to the dielectric abs~rption~’*~~ that arises from the attempt of dipolar molecules to respond to a small perturbing electric field which may oscillate over a wide range of frequencies. Energy is absorbed because they cannot follow the applied field F instantaneously. More than fifty years ago Debye discussed this phenomenon in terms of a rotational Langevin equation in which the torque on a dipole p at an angle 8 to the field F is opposed by a frequency-independent microscopic coefficient of friction { which arises from the force-fields of neighbouring molecules.pXF=-l9 (24) 27 N. E. Hill A. G. Price W. E. Vaughan and M. Davies ‘Dielectric Properties and Molecular Behaviour’ Van Nostrand Reinhold London 1969. 28 S. Kielich in ‘Dielectric and Related Molecular Processes’ ed. Mansel Davies (Specialist Periodical Reports) The Chemical Society London 1973 Vol. 1 p. 192. 18 J. S. RowEinson and M. Evans The Langevin equation per unit moment of inertia is then (cf. equation 8) e(t)= -&t)+r(t) (25) where r(t)is the random torque imposed on a molecule by the motion of its neighbours. If 4(t)is the angle between u(0)and u(t)then25’29 (40) 4t)>=(cos 4(t)) (26) These equations describe adequately the rotational dynamics and so the absorption up to field frequencies of ca.10” Hz. As we move into the far4.r. region3’of lOI3 Hz (i.e.,for molecular motion at times of s)then the same limitations apply to these equations as applied before to the translational Langevin equation. An obvious weakness appears if the absorption is expressed in terms of a(o),the absorption coefficient per unit path length. Integrati~n~~ of equation (25) gives an absorption coefficient a(@)proportional to 02(1 +02)-’,which means that at high frequencies a(@)has a plateau and spectral transparency is not regained. The trouble lies as before in the neglect of molecular inertia and so the assumption that the random torque r(t)has an infinitely small correlation time. Only then is l independent of time.Equation (25) is the truncation of the rotational equivalent of Mori’s continued fraction6 (Section 2) so that the memory function is a peak at t = 0; (IR)M( t) =D6 (t) (27) where D is a rotational diffusion coefficient equal to kTl/1. If r(t)is to be non-Gaus~ian,~~ and if l is to be a function of time then we must generalize Langevin’s equation in the same way as we went from equation (8)to equation (12); This equation has been used recently to describe the far4.r. absorption of furan and chlor~form.~~ t) which replaces the delta-function of The memory function ‘IR’Mo( the simple Langevin equation is the c.f. of the random torque (IRkO(t)=(r(o)r(t)) (29) Since ‘IR)Mo(t) defined by the is itself a c.f. it has its own memory function (IR)Ml(t) analogue of equation (13) and we can again extend the series indefinitely.Mori’s continued fraction for the Laplace transform starts Table 1summarizes how this series can be used as a framework into which to fit some of the widely used models for molecular rotation. The rotational like the transla- 29 G. Wyllie in ref. 28 p. 21; G. Williams Chem. Rev. 1972 72 55. 30 C. Brot in ‘Dielectric and Related Molecular Processes’ ed. Manse1 Davies (Specialist Periodical Reports) The Chemical Society London 1975 Vol. 2 p. 1. 31 B. Quentrec and P. Bezot Mol. Phys. 1974 27 879. The Motion of Simple Molecules in Liquids tional velocity c.f. is necessarily an even function of time,32 and has a Taylor expansion t2 t4 -(u(0) * u(t))= 1-(ri2(0))+-(ii2(0))-' (31) 2! 4! The mean angular velocity (u2(0))of a linear molecule is 2kT/I.The mean square acceleration (u'(0)) comprises two terms; a radial or centripetal acceleration due to the fact that the vector u is of fixed length and a tangential acceleration. The first is independent of the molecular interactions and is 8(kT/1)' and the second is (02( V))/I where O(V) is the torque that the environment exerts on the molecule via the intermolecular potential V. The substitution of these expressions into equation (31) gives the equation of motion of an ensemble of interacting molecules. The equation shows the value of the c.f. representation in isolating the short-time behaviour from that at long times. The former can be followed analytically but the latter can be obtained only by invoking statistical arguments which usually lead to an exponential tail in the c.f.Gordon3* and othersz5 have calculated the first few terms of the expansions of (IR)$(t) '"'$(t) and ")+(t);each contains the torque a functional of the intermolecu- lar potential V (0(V)) or its time derivative. The true c.f. is even in time and Table 1 shows the degree to which particular models satisfy this condition. Gordon's M and J diffusion models are zeroth-order truncations of Mori's series and so have a term in t3,and all higher odd terms. In these models the torque is not defined at the moment of impact (it?.,it becomes instantaneously infinite) and so although transparency is regained in the far4.r.(as o-'),it is regained more slowly than is found experimen- tally33 for many dipolar liquids and for those solid phases in which translational freedom is lost but rotational is retained. More the truncation at M,(t) shown in the last section of Table 1 has been used to ensure evenness to t4. The torque (0(V)) is now well-defined at all times although its derivative (6(V)) which is part of the term in t6,is not. This is because the truncation at M,(t)implies that the angular acceleration is randomized in direction at each impact so that its derivative has an infinite singularity. The absorption spectrum a(w)is the Fourier transform of (IR)+(t) and behaves asymptotically as 0-4at high frequency. It reduces3' to a Debye curve when 04<< w2,as is shown in Figure 6 where it is compared with experimental results33 for liquids and rotationally-free solids.Further comparisons with experi- ment can be found in the papers of Quentrec and Bezot3' and Evans and Evans.35 It is remarkable that although the agreement of a(@) with the experimental results is good over several decades of frequency the overall memory function 'IR'M(f),is even only to t2. This function is another equilibrium property and so should be even in time. One possible remedy would be to truncate Mori's series at R. G.Gordon J. Chem. Phys. 1%6,44,1830; Ado. Magn. Resonance 1%8,3,1;R. E. D. McClung J. Chem. Phys. 1972,57 5478. I. W. Larkin J.C.S. Faraday Symposia 1972,6 112; R. Haffmanns and I. W. Larkin J.C.S. Faraday ZZ 1972,68,1729;M.Evans M. Davies and 1. W. Larkin ibid. 1973,69,1011;I. W. Larkin ibid. 1973,69 1278;I. W. Larkin and M. Evans ibid. 1974,70,477;I. W. Larkin ibid. 1974,70,1457;M. Evans ibid. 1975,71,2051. F. Bliot C. Abbar and E. Constant Mol. Phys. 1972 24 241; F. Bliot and E. Constant Chem. Phys. Letters 1973,18,253; 1974,29,618. G.J. Evans and M. Evans J.C.S. Faraday II 1976,72 in press. Table 1 Some dynamical models in Mori’s formalism ‘IR’Mn(t) (IR)Mn(PI ‘IR)Mo(t) =D6(t) (IR)MOW =D (IR) Mo( t) =MFR(t)e-YJltl where MFR( t) is the memory function of a free rotator (IR)M1 (t) =(‘R)Ml(o) e-71‘ Model and references Debye 25,27-30 Gordon M-diffusion 30 32 33 Gordon J-diffusion 32,34 25 31 35 N Description and comment (IR)#(t) where T is the Debye Molecular inertia neglected; infini- exp (-t/~~) tesimal angular displacement in relaxation time.infinitely short time. Used to Not an even function of t describe dielectric absorption at low frequencies but leads to a ‘Debye plateau’ in the far4.r. Instantaneous elastic collisions This has a Taylor expansion perturb the rotation at random 1-ut2+O(t3)and so starts as an times. Angular velocity vector is even function of t randomized in direction; torque is infinite at impact. A slow return to transparency in the far-ix. As for M-diffusion except that A complicated function whose angular velocity is randomized Taylor expansion has a term in both direction and magnitude. in t3 An ‘inertia-corrected’ Debye model Torqueis always finite.Taylor expansion is even to t4 Describes both low-frequency and but contains a term in t5. far-i.r. absorption more Exponential at long times satisfactorily than previous models. oscillatory at short -a The past history of molecular reflection of molecular movements and interactions libration and the origin of the influences future behaviour far-i.r. Poley absorption i.e. a non-Markovian model The Motion of Simple Molecules in Liquids 21 higher and higher order but this introduces an inacceptably large number of averages (IR)M2(0), . . . ,"R)M,(0),which cannot be obtained analytically and which would therefore remain as phenomenological coefficients. A limitation of Mori's approach is that it does not give a natural picture of the long-time or hydrodynamic tail of the c.f.of angular velocity (J)t,b(t),as emphasized in many papers at a conference in Paris.36 The long-time behaviour of (J)t,b(t)appears to go as t-* as for the translational case discussed in Section 4.Such a limit is expected also for (IR)t,b(t) at least for spherical tops but this tail would distort the spectrum only on the low-frequency side of the Debye absorption. It is natural to expect i.r. absorption associated with the rotation of a dipolar molecule in a liquid but not so obvious that non-dipolar molecules also absorb in the far-i.r. and high microwave region.37 This arises from the small temporary dipole induced in a normally non-polar molecule by the fluctuating fields of moving neighbours.The reciprocal of the half-width of the absorption band is of the order of the lifetime of the induced dipole which is generally ca. 0.2 X s. The associated c.f. is one of orientation coupled with interaction and it falls to zero much more rapidly than its purely orientational dipolar counterpart (IR)t,$(t). Mori's approximation injects a unity into the description of both permanent and induced dipolar absorption as can be seen by the ease with which the truncation at M, which was successful in reproducing the permanent dipole absorption (Figure 6) also reproduces the induced dipole absorption in a range of liquids from nitrogen to benzene (Figures 7 and 8). For these liquids both Mo(0) and M,(O) are multi- molecular in origin since an isolated molecule would not absorb.Both averages are related to (02(V)) although not in a simple way and so can be used as rough probes for the change with pressure and temperature of the mean-square 6 Depolarized Rayleigh Scattering-a Study of ("'Nt) We saw in Section 3 that light scattered from a monatomic liquid had two compo-nents a Rayleigh line of the same frequency as the incoming light and surrounding it on either side a pair of Brillouin lines shifted by an amount proportional to the speed of sound in the liquid. In a molecular liquid there is usually also a weak very broad depolarized band centred on the incident frequency which leads to the so-called Rayleigh wings. It is now generally agreed that this band arises from the re-orientation of single molecules an interpretation which differs from that of collective (shear-wave) modes which prevailed39 before the c.f.formalism was introduced about ten years ago. However both mechanisms may be involved in the long-time tails on the c.f. '")t,b(t)which molecular dynamic studies have shown to be present. As with the corresponding tails in the translational velocity c.f. (Section 4) these probably arise from a coupling of the motion in this case orientational of a single molecule with the hydrodynamic transverse velocity gradient^.^' j6 'Molecular Motions in Liquids' ed. J. Lascornbe Reidel Dordrecht 1974. 37 M. Davies Ann. Reports 1970,67,65;M. Davies G. W. F. Pardoe J. Chamberlain and H. A. Gebbie Trans. Faraday SOC.,1970,66,273; G.W. F. Pardoe ibid. p. 2699; G.J. Davies J. Chamberlain and M. Davies J.C.S. Faraday ZZ 1973,69 1223; G. J. Davies and J. Chamberlain ibid. 1973,69 1739; G.J. Davies and M. Evans ibid. 1975,71 1275. 38 G. J. Davies and M. Evans J.C.S.Faraday ZZ 1976,72 in press. 39 I. L. Fabelinskii 'Molecular Scattering of Light' Plenum Press New York 1968. 40 J.-L.Greffe J. Goulon J. Brondeau and J.-L. Rivail in ref. 36 p. 151. 35r T 'r T I I I I I 20 40 60 80 100 120 140 OO 1 2 3 4 5 140-141-120 -40 -20 -' ?I I I I I I I Oh 20 40 60 80 100 I 126 140 Figure 6 The absorption (top left) and dielectric loss (top right) of t-butyl chloride in the rotator solid phase at 238 K. The abscissae are wavenumbers (cm-') and the ordinates are a in neper cm-1 and d'.The dashed lines are the experimental absorption33 and the points the experimental measurements of dielectric loss ,in the microwave region.33 The full lines are calculated from the truncation in the last line of Table 1. '7kc lower figures are similar resultsfor the absorption ad loss in liquid Me2CCIN02at 296 K. The arrows in the lower right-hand figure mazk tk?pasit'rox and knW%Tityof the ptdk in he cfiebcmc bss .-. 'r Fwe7 The absorption3" a in neper cm-'for non-dipolar substances as a function of wavenumber in cm-'. Liquid N2at 76.4 K (top left) liquid Cot at 273 K (top right) liquid CH4 at 98 K (bottom left the broader band) solid CH4 at 77 K (bottom left the narrower band) and liquid (CN) at 301 K (bottom right). The experimental results are shown by the full curves and the theoretical calculations from the truncation in the last line of Table 1 by the dashed cuws.J. S. Rowlinson and M Evans Figure 8 The absorption^^^ a in neper cm-' as a function of wavenumber in cm-' for the non-dipolar liquids CC14 (left) and GH6 (right) If the incoming light is travelling in the x-direction if it is polarized in the z-direction (vertically) and is observed in the y-direction polarized in the x-direction (horizontally) then the scattered spectrum is Im(o) and it arises from a dipole induced in the x-direction by a field along the z-direction. Such a dipole is proportional to the xz-element of the electric polarizability of the scattering volume. For a system with isotropically polarizable molecules (e.g.CCl,) this element is zero. Gordon3* showed in 1966 that this depolarized component is the Fourier transform of the average motion of the polarizability tensor and so for self-correlation in a linear or symmetric top molecule of the c.f. '"'ll/(t). The depolarized light scattered from liquids with anisotropic polarizability arises from local fluctuations of the orientation from the random isotropic average. If it is assumed that the movement of neighbouring molecules is uncorrelated then the scattered intensity at frequency o from the exciting line is This expression is similar to the quanta1 equation32 for the rotational absorption in the microwave and far-i.r. bands; Im(o) and A(@)have quantitatively the same features.The low-frequency Lorentzian4' of the scattered light corresponds to the low-frequency Debye relaxa- tion in dipolar absorption. This Lorentzian is imposed on a broader background which extends to ca. 100-150cm-' corresponding to the far-i.r. Poley absorp- ti~n.~'?~' Beyond this the intensity falls exponentially with frequency. The Lorentzian behaviour at low frequencies implies that the long-time behaviour of both (IR)t,b(t) and (")ll/(t) is exponential with relaxation times of T~ and rR respectively the reciprocals of the half-widths of the Lorentzians. Details of the 41 D. A. Pinnow S. J. Candau and T. A. Litovitz,J. Chem. Phys. 1%8,39,347; H. Dardy V. Volterra and T.A. Litovitz J.C.S. Faraday Symposia 1972,6,71; J Chem. Phys. 1973,59,4491. The Motion of Simple Molecules in Liquids 25 molecular motion are reflected in the deviations from the exponentials at short times and give rise to an added background or shoulder in the scattered light and to the Poley absorption in the i.r.A simple comparison of TM with TR tells us something of the mechanism of re-orientation; for example whether it is by large rotational jumps or Thus if motion about an axis can be described by a model of rotational diffusion with individual time steps of a short time T then Hubbard's equation43 links T,TM and TR r = I/6kTrR= I/2kTrM (34) If the Rayleigh scattering is observed as a function of temperature4' then in principle the different contributions to the scattering can be assigned different energies of activation which can be compared with those obtained from viscosity and dielectric measurements.For the substituted benzenes41 good agreement with experiment can be obtained by attributing the spectrum of linearly anisotropic molecules to a single mechanism of re-orientation. However the far wing of the Rayleigh scattering contains also information about intermolecular properties since a weak exponential scattering is observed here even with molecules with scalar p~larizabilities~~ used a simple binary-collision such as CCI,. Bucaro and Lit~vitz~~ approach to model this scattering for spherically polarizable molecules. However the spectra of anisotropic liquids show the same quasi-exponential tail and also a shoulder around 50-90cm-' which cannot be accounted for by this simple distortional binary mechanism.After subtracting the collision-induced component of m)t+b(t) Dardy et al.41 found that a molecule such as benzene behaves much as a free-rotator at short times. There is an average rotation of ca. 15" between collisions. The long-time behaviour of (")&(t)is found to be exponential reflecting the ultimate diffusional behaviour. It can be that the short-time behaviour of a c.f. is revealed in greater detail by an analysis of its second derivative; for example (R)$(t) is related to the correlation of angular momenta and angular orientation. This c.f. s~~ws~~,~~ that there is not a complete loss of memory during collisions in molecules like benzene and so the simple diffusion model which lies behind the Hubbard equation is inadequate.The Rayleigh scattering has been observed also as a function of pressure47 and from these measurements the change with density of (R)t,$(t) and (R)q$(t)can be obtained. The mechanism of re-orientation appears to involve rotation which is randomly affected by collisions but it is a process which is not described accurately by the J-diffusion model. The c.f. ("'q$(t)has a negative region and oscillates [as is well for (lR)$(t)] thus showing that there is not a complete randomiza- tion of the angular velocity at each collision. There is also evidence that memory of 42 F. J. Bartoli and T. A. Litovitz J. Chem. Phys. 1972,56 413. 43 P. S. Hubbard Phys. Reu. 1%3,131 1155. 44 J. P. McTague and G. Birnbaum Phys.Reu. Letters 1%8,21,661; W. S. Gornall H. E. Howard-Lock and S. P. Stoicheff Phys. Reu. 1970 Al 1288. 45 J. A. Bucaro and T. A. Litovitz J. Chem. Phys. 1971,553846. 46 T. Keyes and D. Kivelson,J. Chem. Phys. 1Y72,56 1057; ibid. 1972,57,4599; A. G. St. Pierre and W. A. Steele J. Chem. Phys. 1975 62 2286. 47 J. F. Dill T. A. Litovitz and J. A. Bucaro J. Chem. Phys. 1975,62,3839; P. van Konynenburg and W. A. Steele ibid.,p. 2301; M. Perrot J. Devaure and J. Lascombe Mol. Phys. 1975,30 97. J.S. Rowlinson and M. Evans one impact is carried through to the next and beyond. The oscillations in (“)$(t) are most pronounced in strongly anisotropic molecules such as benzene and suggest that such strong anisotropic forces lead towards molecular librati~n.~~ This mechanical anisotropy is accentuated at high molecular densities.The J-diffusion model is qualitatively adequate only for the simplest molecules such as N and CO. From the Mori series for depolarized Rayleigh scattering we see that the trunca- tion (R) MO(t)= DR s(t) (35) is equivalent to the Debye model of dielectric absorption and leads to the Loren tzian which is adequate only at long times. The ~houlder~~,~~ found at higher frequencies in moderately and highly anisotropic liquids can neither be described by a Lorentzian nor fitted by the distortional mechanism of Bucaro and Lit~vitz.,~ The truncation yields on Fourier transformation of the corresponding (R)$( t) the scattering func- tion of the M-diffusion model which is proportional to o-,at high frequencies and reduces to a Lorentzian when o4<< 02.The equilibrium average (R)Mo(0) is still for this model a property of a single molecule (no torque involved); for a linear molecule it is 3kT/I.The limitations of the M-and J-diffusion models in their treatment of torque (Section 5) and the false assumption of complete randomization of angular velocity at each collision have been revealed by the measurements at high pressure. Thus the J-diffusion model is adequate for the Rayleigh scattering in liquid CO,up to 100bar but fails at higher pressures when the index n of the frequency dependence (w-”) changes to higher n at intermediate and high frequencies. This sharper asymptotic fall-off with osuggests that the truncation might lead to a suitable function for IVH(u).A finite torque is now implied in ‘R’Ml(0); for a linear molecule The function IvH(w)is again Lorentzian at low frequencies but now has a peak2* around 50 cm-’ near the shoulder found by Dardy et al.,’ in anisotropic liquids.The complete spectrum of depolarized scattered light can be written as the sum of three terms; I(@)= 140)+ &mL(m) + IvH-coL(o) (40) where ICoL(o) is the collisional part of the intensity observable in molecules with isotropic polarizability such as CCI,. The theoretical expressions from which the sign and magnitude of the cross-term can be calculated are only crude but its The Motion of Simple Molecules in Liquids 27 does not seem to affect the consistency of the subsequent analysis of the results.In anisotropic molecules Iw(w) accounts for almost all the intensity and IcoL(o) is restricted to the wings. Bucaro and Lit~vitz~~ predict lCoL(o)012/7 exp (-w/oo) (o>oo) (41) where wo is calculated from a chosen intermolecular potential usually of the Lennard-Jones form. IcoL(w) is a Lorentzian if w <wo. The equivalent expression in the far-i.r.41 is acoL(w)~ w26/7exp (-o/oo) (o>wo) (42) but this fits the results much less satisfa~torily~~ than the anisotropic term shown in Figures 7 and 8. Bucaro and Lit~vitz~~ emphasize the similarity between IcoL(o) and the population-corrected dielectric loss factor in non-polar liquids viz. They have shown that both are fairly well described by semi-empirical equations such as (41)and (42).Both have separate high- and low-frequency portions whereas the generalized Langevin theory of Kubo Mori and others stresses that these peaks in the i.r. (or shoulders in the Rayleigh spectrum) are to be treated as part of the lower frequency orientational processes -a unifying formalism for both dipolar (aniso- tropic) and non-dipolar (isotropic) molecules. It would therefore seem to be fruitful to treat IcoL(o)by invoking Mori's truncation at first-order with both (R)Ml(0) and (R'Mo(0)as torque dependent parameters. 7 N.M.R. Spin-Rotation Relaxation-a Study of "'Ht) The relaxation of nuclear spins is determined by their coupling with the translational and rotational motions of the molecule. For a nucleus of spin i,the spin-rotation interaction of a linear molecule has a Hamiltonian of the form -cI J where Iis the angular momentum of the nucleus J is that of the molecule and c is the spin- rotation coupling con~tant.~~,~~ When this is the only part of the total energy which leads to relaxation of the spins then the spin-relaxation time TI is 1 c2 OD -=-(J(0).J(t))e-'%' dt (42) Tl 3H2 -oo where (')$(t) is the third c.f.defined in equation (23) and wois the Larmor precession frequency. In a typical liquid w;' is of the order of s and this is so much longer than the time for the angular momentum c.f. to decay to zero (say lO-"s) that the exponential term in (42) can be put equal to unity. For a linear molecule the expan~ion*~.Of (')t+h(t)in powers of t2 resembles those of the ratios (IR)$( t)/('")$(O) and (R)$(t)/(R)$(0).Gerschel Darmon and Brat"showed that the (IR)ratio oscillated at short times and measurements of light scattering from 48 M.Evans J.C.S. Faraday II 1975,71,71. 4y J. A. Bucaro and T. A. Litovitz J. Chern. Phys. 1971 55 3585; T. A. Litovitz in ref. 36 p. 613. 50 A. Gerschel I. Darmon and C. Brot Mol. Phys. 1972,23 317. 28 J. S. Rowlinson and M.Evans liquids at high pressures4’ have shown that the (R) ratio does also. There has been little if anything reported from n.m.r. studies of the full time dependence of (J)t,!t(t) but Berne and Harp25 have made computer calculations which show that it is negative for an interval of time if the pair potential is anisotropic.It remains positive and changes little over an interval of s if the pair potential is of the Lennard- Jones form. If there is no molecular interaction then (02( V)) is zero the c.f. (J)t,!t(t)is unity and its memory function (J’M(t) is zero. Hence “’M(t)can be looked upon as a molecular memory of the interactions. In contrast (IR’M(t)and ‘R’M(t)are non-zero decaying functions even in the absence of torque since there is always a distribution of frequencies of rotation. If ‘J)J/(t) is at any time negative then ‘J’M(t)must be non-zero; that is a molecule retains some memory of its interactions. Berne and Harp25 find that in simulated CO ‘J’M(t) goes almost to zero within 0.3 X s and that there is then a much slower decay with a final positive tail.The time of 0.63 x s is roughly that taken by a molecule to move from the centre of its ‘cage’ and meet one of its neighbours at the ‘wall’. An example of an n.m.r. study of spin-rotation relaxation is the work of Rigny and Virlet51 on the relaxation of the fluorine nuclei in UF, WF, and MoF, all of which are liquid at temperatures little above room temperature. They are unusual in that contrary to the behaviour of most the spin-rotation interaction is the dominant mechanism of relaxation even at temperatures well below the critical. The molecules retain their angular momentum for a long (correlation) time but the rotation is not free since they can move only about 1 radian at 343 K before their angular momentum changes. These conclusions are revealed by using the generalized Langevin equation which now takes the form .f(f)=-c “’M(f-f‘)J(f’)df’+rj(f) (43) where rJ(t) is the random torque and M the memory function.The associated spectrum is then O‘ ‘“J(io) (J(0) J(f))e-’”O‘dr O>= (44) = ‘J’tj(~)[ioo + (J)~(oo)l-’ where a,,is the Larmor frequency of spin precession. This is again sufficiently small to be neglected so the exponential term is unity. That is rJ(t) fluctuates so rapidly that its c.f. which is related to “’M(w)by is a delta function. The simple Langevin equation is thus regained with (J)M independent of frequency. Therefore 51 P. Rigny and J. Virlet J. Chem. Phys. 1%7,47 4645. 52 J. G. Powles in ‘Molecular Relaxation Processes’ ed. M. Davies Academic Press New York 1%6.The Motion of Simple Molecules in Liquids 29 where ?J is the angular momentum correlation time inversely proportional to TI,the spin-rotation relaxation time. As the temperature is raised the torque fluctuates more rapidly the ‘friction’ is reduced and so Tl decreases. It is found in practice that this decrease follows an Arrhenius law.27 For spherical tops such as CH4 CF4 SF, and the hexafluorides above the contribution of spin-rotation interaction to the n.m.r. lineshape is comparable with and greater than at high temperatures that of spin-spin magnetic dipole interaction. The latter is the contribution to spin resonance relaxation from the interaction of a pair of identical nuclei in the same molecule of spin quantum s and a separation defined by the vector u and the scalar distance b.The spin-spin relaxation time is where y is the gyromagnetic ratio. This relaxation time provides therefore a method of measuring the area under the ‘R’JI( t) curve which is a correlation time and is to be compared with a diffusion coefficient in the translational case equation (6). It does not however tell us anything about ‘”’Jl(t) itself as a function of time. Such n.m.r. results can be compared usefully3o with those of dielectric meas~rements,~”~~ particularly at temperatures near the triple point where rotational diffusion might be a useful concept. At the boiling point and at higher temperatures the mechanism of reorientation is generally interpreted with the help of the spin-rotation component of TI thus taking advantage of the increased periods of rotation.The principal interest of such work is to study any anisotropy of motion; sometimes the rotation is almost free about one axis and diffusional about another perpendicular to it.53 The integral of ‘“’+(t) over all time defines a correlation time ?R. In the limit of rotational diffusion this is related to ?J by Hubbard’s TRrJ = I/6kT (48) and in the limit of a rarely perturbed free spherical-top rotation by For the intermediate region M~Clung~~ has discussed the relation between ?R and 75 for spherical tops in terms of the M-and J-diffusion models. These approach the limits of equations (48) and (49) when ?J is very small and very large. Sillescu has extended54 the Debye model of Brownian motion and the random jump of rotation to take account of temporal fluctuations in their rates.Further develop- ments in the use of n.m.r. in this field can be found in the comprehensive reports edited by We conclude by applying the formulation in terms of memory functions to spin resonance relaxation. From the equipartition of kinetic energy over two degrees of 53 D. K. Green and J. G. Powles Proc. Phys. SOC.,1965,85,87; T. T. Bopp J. Chem. Phys. 1967,47,3621; D. E. Woessner B. S. Snowden and E. T. Strom Mol. Phys. 1%8 14 265; J. Jones and T. M. Di Gennaro J. Chem. Phys. 1%9,50,2392;A. A. Marryot,T. C. Farrar and M. S. Malmberg ibid. 1971 54,64. 54 H. Sillescu J. Chem. Phys. 1971 54 2110. 55 ‘Nuclear Magnetic Resonance’ ed.R. K. Harris (Specialist Periodical Reports) The Chemical Society London 1971-1975 Vols. 1-4. J. S. Rowlinson and M.Evans rotational freedom we have kT=(J2(0)>/21 so that cn T~= I (J)$(r) dt When ‘J’Mand ‘R)Mare both delta functions we have and TJ and rR are linked by Hubbard’s relation. The mean field of force due to the neighbours tends to hold a given molecule in a fixed orientation for a variable time while superimposed on this time-smoothed field is a rapid fluctuation due to the actual molecular motions. This is approximated in Brownian theory by a random torque of simple character superimposed on a steady orientating field. If the constraining field is strong the molecule moves as a damped gyrostatic pendulum. If the constant is highly anisotropic the motion about one axis may approach free rotation.If the molecule is a spherical top its components of rotation behave independently. All these different modes are not describable by a simple exponential c.f. and the introduction of memory functions will probably lead to more satisfactory descriptions. Thus =‘J’~o(~) ‘J)Mo(t) exp (-t/T;) (53) is likely to lead to a better account of the dependence of the correlation time TJ on temperature. In the M-diffusion model the product TJTRgoes through a minimum dependent on (02(V)) which is more realistic than Hubbard’s relation equation (48). With the truncation (J)Ml(t)=“’~~(0) exp (-t/T;) (54) we have that depends both on (02(V))and its time-derivative so that the product TJTR might behave ever more realistically.These applications have how- ever still to be explored.