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.