Memory Functions for Angular Motion in Liquids BY A. GERSCHEL Laboratoire de Physique de la Matikre Condensee Universite de Nice Parc Valrose 06034 Nice-France Received 1lth August 1976 Experimental memory functions are calculated corresponding to the orientational correlation function g(t) = <u(O) -u(t)) and to the rotational velocity correlation function gJt) = <i(O) -d(t)>. Analytical expressions are obtained corresponding to the Cole-Glarum and the Fatuzzo-Mason theories relating the spontaneous fluctuations of orientation to the frequency-dependent absorption. The properties of these functions are interpreted in the framework of generalized Langevin theory. Particular attention is given to the damped oscillatory features of the first memory function and to the slowly decaying features of the second.This analysisis achieved by investigating the temperature and density variations in some simple organic liquids (CH3F CH3CI CHF, CHCG and OCS). The occurrence of a plateau in the rotational velocity memory function is discussed. A tentative inter- pretation of this new feature is conducted by analogy with molecular dynamics experiments dealing with translational motion. 1. MOLECULAR RELAXATION IN LIQUIDS ANALYSED IN THE TIME DOMAIN REPRESENTATION Since it has been demonstrated by fluctuation-dissipation theory that current absorption spectra can be readily interpreted in the time domain thereby making accessible the molecular correlation functions this approach has been widely adopted for the analysis of elementary motions in liquids.Indeed spectral characteristics have been consigned to a second rank position because of the sudden appeal of correlation functions. Now after some years of rapid development it is useful to sort out the effective progress that has been made as well as to list the limitations of time domain analysis. As our contribution we aim to analyse the correlation func- tions themselves through their associated memory functions which appear to be very convenient tools for investigating the initial-time motions in relation to the overall dynamics. Molecular relaxation itself is investigated using a variety of experimental methods ; the results may be interpreted in the framework of more or less refined theories the predictions of which conform generally to the description of relaxation rates through one exponential decay time as remarked by Debye or through a combination of exponential decay times for more complicated liquid systems.At this stage only the Markovian nature of the process is characterized not the elementary steps which are responsible for this resultant law. The generality of Markovian behaviour is the ineluctable counterpart of the insensitivity of this law to the actual details of the motion.' This seriously limits the usefulness of relaxation studies for the evaluation of intermolecular potentials or of any genuinely microscopic (molecule-by-molecule) property characterizing the rates and mechanisms of the motion. In itself the derivation of the "microscopic " relaxation time is dependent on the approximations of conflicting theories2 which aim to account for the local reaction fields following 116 MEMORY FUNCTIONS FOR ANGULAR MOTION IN LIQUIDS the molecular displacements.However the differences are generally quite small; certainly of the order of current experimental accuracy. Orientational correlation functions in so far as they depict the statistical overall tumbling motion of the constituent molecules usually show an exponential decay at long times. They may be considered as the interplay of two components auto-correlations reduced to monomolecular properties and crossed correlations from multimolecular interactions. Our concern here will be for the total functions only. Assuming now an analogy with Brownian motion valid if the interactions are frequent compared to the time scale of the motion one may consider the hydrodynamic friction coefficient to be a time-dependent property and thus further analyse this dependence.A general Langevin equation reads u(t) = -lKg(t -z)u(z)dz +f(t) or equivalently g(t) = -I$(t -z)g(z)dz. 0 In these relationships g(t) is a classical correlation function of the molecular property u(t),viz s(t) = (40) u(t)>. For the particular case of angular motion in polar liquids we consider u(t) to be a unit vector attached to the molecular axis supporting the dipole moment. The quantity f(t) has been termed a " random force "by analogy with the Langevin equation (actu- ally the vectorial field is a field of angular velocities); its correlation function is precisely the memory function which has already appeared in eqn (1.1) as a systematic retardation or "frictional " effect.These are well known general properties. They are discussed in the works of Zwanzig4 and M~ri,~ and further analysed by Harp and Berne6 and Pursuing our search for simple functions of the dynamics the structure of Kg(t) itself may be further analysed in terms of a second order memory function K:2)(t) defined by the integro-differential equation Only the computation of these functions for actual physical systems can determine whether the functions deserve attention. We now undertake this task for some selected simple organic liquids. 2. COMPUTATION OF EXPERIMENTAL MEMORY FUNCTIONS In what follows we use the notation 9g(t) =\me-i'utg(t)dt 0 to represent a Laplace transform and we write complex quantities as 9g(t) = RePg(t) -i ImZg(t) and E* = E' -id' (the dielectric constant).A. GERSCHEL 117 We will make use of the relati onship $4g(t) = im9g(t) -g(O) (2.2) and $4g(t) = -029g(t) -icog(0) -g(O) (2.3) where the odd derivatives at time zero e.g. the last term in (2.3) vanish for our classical molecular correlation functions and analytical intermolecular potentials. We have now to select a relationship linking the experimental data [here the values of ~'(o), ~"(m),eo and g(O) the Kirkwood-Frohlich correlation parameter (2.6)] with the molecular correlation function g(t). In a recent analysis3 we have shown that some discrepancies in calculating g(t) with different theories could arise with highly polar liquids which have a large zero-frequency value c0 and a very intense absorption in the FIR (far-infrared) and microwave frequency ranges.This does occur with some liquids considered here,9910 so for these we shall have recourse to the F-M expression (2.4) first established by Fatuzzo and Mason,ll and re-examined later by Zwanzig and Nee.12 Titulaer and Deutch2 indicate that the F-M analysis is preferred for liquids having a high static dielectric constant and intense absorption-dispersion features. For liquids of medium or low polarity13 the functions calculated using either F-M or C-G (Cole and Glarum) l4theories are indistinguishable; therefore we here adopt the simpler expression (2.5) of C-G which has the advantage of being linear in the complex quantity E* and which leads to a single relaxation time for an exponential decay of the correlation function.The expressions are with K-F do) = 9kTV where p is taken as the dipole moment in vacuum and the other symbols have their usual significance. Now we come back to eqn (l.l) which after taking the Laplace transforms yields which after combination with eqn (2.1) can be written Since by definition Re9.Kg(t) =LaKg(t)cosot dt by inversion for t > 0 one has K'(t) = Re9Kg(t)coscot do ikw 118 MEMORY FUNCTIONS FOR ANGULAR MOTION IN LIQUIDS In this expression we insert the values of Regg(t) and ImZg(t) corresponding to eqn (2.4) or (2.5) as required that is F-M ReZg(t) = EOEtt[2(E12+ Ett2) + EW] (2.10) 0 (go -EW)(2EO + EW)(Et2 + Ett2) IrnZg(t) =g*){ 1 -EOIEt(E' -Ew)(2E' + Ew) + Et'2(2E' -Em)]} (2.11) 0 (Eo -Em)(2Eo + e,)(d2 + E"2) (2.12) (2.13) The resulting memory functions K,(t) possess very similar features to those of the functions grdf) = -g(t) = {d(O) u(t)> (2.14) the so-called rotational velocity correlation function~.'~ These in turn are compar- able under the same conditions with the angular velocity correlation functions.8 A systematic comparison of experimental functions KJt) and grv(t)has already been presented in ref.(3) where it was concluded that a slow decay of g(t) compared to grv(t) or the condition < (E~)~, are sufficient requirements for an acceptable identification.Turning now towards the second order memory function defined in (1.4) a very similar calculation applies in which one makes use of the analogue of (2.7) i.e. together with (2.8). This results in an expression for LPK(;)(t) similar to (2.8) the real part of which may be inverted thus leading to the analogue of (2.9) viz (2.15) with (2.16) According to Rahman,' such a second order memory function is the same as the memory function of the second derivative of g(t) defined in (2.14) provided a con- venient consistent definition of the memory function is used since gr,(t)is an " oscil-latory " function-i.e. of null integral 6grv(t)dt-whose memory function would otherwise incorporate a Heaviside step. Following the argument of ref.(7) such a well-behaved Krv(t)would be defined through We may now evaluate (2.15) by inserting the values of Re9Kg(co) and Im9Kg(co) computed from (2.8) and eqn (2.10) to (2.13). A. GERSCHEL A particularisation of (2.15) in the framework of C-G theory has already been obtained,ls it reads (2&0+ &')2 + &"2 o &t'2(2&(3 6" + Em)2 + [(2&,+ &')(&I -Em) + & ] cos cot do. (2.17) A somewhat more elaborate expression corresponded to the F-M relationship otherwise the analogue of (2.17). 3. PROPERTIES OF THE 1s-r AND ~ND ORDER EXPERIMENTAL MEMORY FUNCTIONS Taking advantage of very complete experimental information available for a few simple organic liquids at densities and temperatures ranging from near the triple point up to the vicinity of the critical point and all along the liquid-gas coexistence curve a systematic investigation of the properties of these functions is now possible.The selected liquids are at one extreme a triatomic linear molecule OCS and a symmetric top molecule CHCl,; these are medium polarity liquids. At the other extreme for highly polar liquids we have the three symmetric top molecules CH3F CH&l and CHF,. A fully detailed discussion of the experimental data is to be found in the appropriate referen~es.l~~l~*~~ Here we only specify very briefly the method employed and some correlative limitations. The experimental data consist of absorption-dispersion spectra over an extended range of frequencies.Zero-frequency determinations of the relative permittivity c0 and consequently of the K-F angular correlation factor g(O) are derived from classical capacitance measurement^.^ Dielectric relaxation studies involved deter- mination by waveguide techniques of both E'(CO) and ~"(o) at a few selected frequencies in the microwave range (namely 10 GHz 35GHz and 70 GHz). The fit of the data to Debye relationships was acceptable within the experimental accuracy. However a certain gap persists between the higher microwave frequencies and the lower fre- quencies of absorption spectroscopy in the FIR. Measurements are lacking between the wave numbers 3 cm-I on the lower side and 18 cm-l on the higher side though the higher frequencies in the FIR do range up to 180 cm-l.The correlative disper- sions in this range have been evaluated according to a distribution-of-resonances model.I7 For the time being spectral characteristics in the FIR provide the most complete and accurate source of information on short-time angular motions provided pure dipolar absorption is actually the main phenomenon responsible for the band profile. Limitations are mainly due to other than purely dipolar absorption features super- imposed on the raw experimental spectra in the FIR. A careful analysis of their origin is always necessary which implies that adequate corrections to the spectra are worked out before Fourier transformation into molecular correlation and memory functions. For the selected liquids having simple enough constituent molecules we obtained reasonably reliable frequency spectra of the neat dipolar absorption.Thus to begin with we reproduce in fig. 1 a comparison of &(t) with the cor- relation functions grv(t)defined in (2.14); perfect agreement is obtained close to the triple point but gradual departures develop as one moves towards expanded liquid 120 MEMORY FUNCTIONS FOR ANGULAR MOTION IN LIQUIDS 0.5 0.3 0.2 0. 0. 0. 0. 0. FIG.1 .-The first memory functions K,(t) (open circles) compared with the rotational velocity correlation functions g,,(t). Both functions are in normalized representation. Three temperatures Tt are shown for liquid fluoroform (a) -156 "C (TR =-T-= 0,01) (6) -80 "C (TR= 0.42) Tc -Tt (c) and -3 "C (TR= 0.85).conditions. Similar conclusions are valid for the other liquids mentioned in this paper. The characteristics of the second order memory functions or rotational velocity memory functions K,,(t) are displayed in fig. 2 where they are compared with their associated correlation functions g,,(t). Striking features of KJt) are (i) an initial decay as sharp as for g,,(t) but certainly not sharper; (ii) a small amplitude oscillation rapidly dying out; (iii) an appreciable positive correlation effect persisting in the dense liquid for the time of damping of the librations [as represented by the oscillations of grv(t)] and thereafter falling off gradually; (iv) a progressive disappearance of such positive correlation as the dynamics evolve at the lower liquid densities obtained at higher temperatures.-08 0.7-06 - 05-0.5-0.5 -t 0.4 -$ 0.3-FIG.2.-The second memory function K,,(t) (upper curves) compared with the rotational velocity correlation functions. Same liquid as in fig. 1. (a) -156 "C (b) -80 "C (c) -3 "C. A. GERSCHEL The generality of these observations is made plain in fig. 3 and 4 where data over a set of temperatures are illustrated for each liquid. FIG.3.-The second memory functions normalized for two highly polar liquids (a)methyl fluoride at -140 "C TR = 0.01 (curve l) -80 "C TR = 0.32 (curve 2) and 0 "C TR= 0.76 (curve 3) and (b) methyl chloride at -97 "C TR = 0.01 (curve l) -60 "C TR = 0.15 (curve 2) and -3 "C TR = 0.40 (curve 3). -0.6 0.6-0 --I -0 20.5 --z k-0.L 0.3 0.3-0.2 0.1 0 0.1 [bl FIG.4.-Normalized memory functions KrY(f)for the two liquids OCS and CHC13.Carbonyl sulphide (a)is shown at temperatures -138 "C TR = 0.01 (l) -30 "C TR = 0.44 (2) and +90 "C TR= 0.90 (3). Chloroform (b) is at -60 "C TR = 0.01 (l) +25 "C TR = 0.27 (2) and +260 "C TR = 0.90 (3). In fig. 5 are illustrated the power spectra Re9Kr,(t) for the sake of comparison with the corresponding features of ~"(m)and Re9gr,(t) since for a better under- standing of the physical features we find it advisable to use both time-domain and frequency-domain representations. At low frequencies the behaviour mimics that of e"/m as is obvious from (2.17). The high frequency end of the spectrum on the other hand resembles an ~"(m) spectrum shifted towards higher frequencies.An additional shoulder can be noticed before the final decrease as a consequence of the rapid fall off of E" on the high frequency side of the FIR band [since this quantity contributes substantially to the denominator of the kernel in eqn (2.17)]. 122 MEMORY FUNCTIONS FOR ANGULAR MOTION IN LIQUIDS For purposes of comparison the significant parts of the spectrum are therefore the high frequency characteristics of orientational relaxation including the region between microwave and FIR measurements (3-1 8 cm-l) together with the higher frequencies of the absorption in FIR. . From this observation we conclude that some features in the power spectrum of KrY(t)are less reliable than are the direct experi- mental spectra E"(CO) and ReA?grv(t)-the last being simply a(u) corrected for internal field effects.vlcm" v /cm" FIG.5.Normalized frequency spectra for two temperatures of liquid fluoroform left -140 "C TR= 0.10 right -3 "C,TR = 0.85. Curves (1) are the e"(i) representation. Curves (2) are the frequency spectra of the second memory function. Curves (3) are the frequency spectra of the rotational velocity correlation function-i.e. the absorption spectra a(i) as measured in FIR cor-rected for internal field effects. To clarify this situation we have simulated erroneous spectra within the limits of uncertainties in our experimental data with deliberate exaggeration of the distortion at the high frequency ends of both dielectric and FIR spectra.The distorted Krv(f) functions exhibit general behaviour quite similar to the original observations except that the amplitudes or elevation of the positive tails are shifted upwards or down- wards by amounts of 0.02 (in our normalized representation). Accordingly when the total amplitude above or below the axis exceeds 0.05 it may certainly be considered as significant. The steepness of the initial decay of Krv(t),as well as the period of the small damped oscillations is rather dependent on the decay of the FIR absorption equated at high frequencies and on the correlative dispersion. Finally the limit E~ to nIR2,suffers from being an estimate via the Lorentz-Lorenz relationship with measured polarizabilities.It would be much more satisfactory to have a true experimental determination of the refractive index variation and its high frequency limit in view of the presence of an (E' -term in the denominator of (2.17). To sum up a scrutiny of the uncertainties in the power spectrum leads one to be confident of the positive correlation features and their variations to within &2%,but at the present stage of our experimental knowledge quantitative assessments of the initial time decay of the function and the period of the superimposed oscillations can- not be made to better than &15% of the observed features. A. GERSCHEL 4. DISCUSSION Features of KJt) merit scant discussion since they are so close to the functions g,,(t) described in current FIR work.The oscillating tail is characteristic of strongly angular-dependent intermolecular potentials entailing correlating collisions the consecutive alternance being associated with reversal of the angular velocity within the local structure. As the density decreases the oscillations increase in period and become less pronounced evolving closer to the free rotation function yet still retaining some structure even in the vicinity of the critical temperature. Such evolution expresses in fact the weakening of the oscillator strength in the expanding structure. At long times the difference of behaviour between K,(t) and grV(t) lies in the fact that the first function maintains slightly higher (more positive) values especially at low densities. This is obvious from the properties where zD is the dielectric microscopic relaxation time.We now examine some newer aspects of the dynamics with the help of our findings in the Krv(t) representation. Considering fig. 2 to 4 it is evident that the important feature is the positive plateau extended over the time of damping out of the librations. This is the time that the molecule spends probing its environment before a local orientational rearrangement takes place. It may be termed a “ time of residence ” still keeping in mind that throughout this time the motion may be quite smooth or alternatively impulsive according to the details of the structural fluctuations. It is therefore some kind of generalized friction that is manifested by this plateau showing the effects of time-varying local torques imposed by the motion of the anisotropic surroundings about any definite molecule while the molecule performs correlated oscillations accompanying its reorientation.Such observations are reminiscent of those reported in earlier molecular dynamics work (computer simulation) dealing with translational velocities.ls Levesque and Verlet found a “ slowly varying part of the memory function positive for states not far from the triple point and negative if the temperature is substantially higher or the density lower ” interpreting the long-time part of this generalized coefficient “ as produced by an accompanying motion of the medium which according to the state reduces or enhances the self-motion ”. The authors taking advantage of their microscopic insight into the molecular motion were able to give a detailed interpreta- tion “In the general vicinity of the triple point where owing to strong cohesive effects the motion of the distinguished particle is on the average reversed after a short time the current gives rise to a retarded positive friction; that is the memory function is positive for large times.At higher temperatures or lower densities on the other hand where the initial motion of the distinguished particle tends to persist for large times the current pattern of the neighbouring particles enhances this forward motion. The memory function interpreted as a generalized friction coefficient is in that case negative for large times.” It is not unreasonable to transpose such a mechanism to rotational motion; our finding of a very similar behaviour of the angular memory functions does favour such an interpretation.It remains to be noted however that those currents revealed by our analysis of the rotational dynamics are possibly themselves subject to mutual interactions through some kind of dragging effect that shows up in the long-time tail of 124 MEMORY FUNCTIONS FOR ANGULAR MOTION IN LIQUIDS the memory functions K,,(t). Although a tentative explanation this is as much as we can say at present lacking as we do microscopic information as detailed as that provided by numerical molecular dynamics experiments. Furthermore one must realize that the negative portions sometimes encountered at the lower density range of the liquids were not sufficiently pronounced in the rotational dynamics studies to be certainly significant.Generally speaking we may conclude that if one reason for considering the second order memory functions is the search for simpler analytical objects than the rotational velocity correlation functions themselves one must admit that this expecta- tion is not met. More precisely any expected trend of successive memory functions towards a delta function is quite hopeless being in complete contradiction with experimental evidence. On the contrary the observed structure of the functions of higher order becomes increasingly involved and it is likely that the underlying physical processes so implied are correspondingly intricate in nature. In the meantime we have gained some physical evidence of the occurrence of coupling mechanisms between the fast and slow angular motions currently described as librations and reorientations respectively both kind of motions appearing as specialised aspects of the more general diffusion process.Furthermore we can ascertain that a detailed analysis of the motion does not reveal a white noise of molecular interactions indeed it has led us to propose the advent of some microscopic dynamic organization establishing local circular currents. Here memory functions have been a useful tool. They also have shown how general this structure is irrespective of the specific molecular shapes and polarities of some simple liquids though the structure is dependent on their physical state (density temperature).Finally it is also satisfactory that analogies have been revealed with the findings of molecular dynamics experiments for translational motion. R. I. Cukier and K. L. Lindenberg J. Chem. 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