Relaxation in Simple Liquids by Polarized Light Scattering DORFMULLER FYTAS MERSCHAND DIMITRIOS BYTHOMAS *GEORGE WERNER SAMIOS Fakultat fur Chemie Universitat Bielefeld Morgenbreede 45 4800 Bielefeld 1 W. Germany Received 3rd August 1976 The polarized spectra of liquid CClr CH13 C6H5F and CsH5Cl were obtained by interferometric light scattering analysis over a temperature range of -80°C for each substance. The data obtained from the BriIlouin lines show a hypersound dispersion which could be fitted to a relaxation equation with a single relaxation time. We thus could obtain values for the relaxation time z and for the relaxation strength R as functions of the temperature. The values of R and the temperature dependence of this quantity have yielded an insight into the nature of the observed relaxation processes which appear to correspond to the T-V energy transfer involving the lowest or occasionally the second lowest vibrational level vinit.The temperature dependence of z enables us within the frame of the Schwarz-Slawsky-Herzfeld-Tanczos theory to obtain information about the inelastic collision cross section the steepness of the intermolecular repulsion and the ratio ttv/qs. The outcome of these calculations depends upon a num-ber of assumptions which have been discussed at some length. A macroscopic system is able to dissipate mechanical energy if the state variables are changed at a rate comparable to one of the relaxation times describing the kinetics of energy transfer between any of the degrees of freedom of the system.At high transfer rates the excitation of the process is obtained either by using ultrasonic or hypersonic waves. In the context of this study we shall have to deal with T-V relaxation i.e. with the time dependence of the excitation of molecular vibrations from the thermal pool of the translations. On the molecular level the T-V relaxation is connected with the time-dependent perturbations of the molecular levels by inter- molecular interactions. Specifically when we deal with gases the equation of state enters the calculations via the average time between binary collisions zBCand the duration of the collision zc. Under such simple conditions we may define a relaxation probability P’per second. This can be expressed as a product of a transition probability per collision P and r-lBc P’= P/rBC or P’=Z/Z* where 2 = z-lBC is the number of collisions per second and Z* =P-l the number of collisions which are necessary to induce on the average one transition.On the other hand with a liquid we have to answer the question as to whether the same approximation is physically valid. The isolated binary collision model (IBC)1-6 invokes the rarity of a hard collision event inducing a transition whereas physical intuition about the molecular dynamics in liquids as well as a correlation analysis in liquids seems to substantiate the necessity of a more complex treatment extending to many-molecule correlations. This view is supported by the fact that virial co- efficients up to B5had to be introduced into equation-of-state calculations in order even roughly to describe pVT data of liquids.However the IBC approach has been surprisingly successful in correlating relaxation data with potential parameters and THOMAS DORFMULLER ET AL. models of the liquid The question then arises as to whether this success is due to a mere cancellation of approximation~,~*~* or whether it reflects the fact that the physical content of the IBC approach is a good approximation to the real state of affairs in a simple liquid. This question has been the subject of several investiga- tions but no conclusive answer is as yet available we believe that this is mainly due to the fact that the discussion was necessarily restricted to the few liquids for which reliable relaxation data were available either in a narrow temperature range or even at only one temperature.Although molecular vibrational relaxation data have been obtained mainly by means of ultrasonic methods the frequency range of this method is not appropriate for the study of most simple liquids. Hypersonic data as obtained from Brillouin scattering lie in the more convenient GHz range but are on the other hand less accurate. Brillouin scattering has hitherto not been used as a routine method over large temperature ranges. Our effort has been directed towards improving the accuracy of the Brillouin scattering technique and measuring a class of comparable molecules (CC14 CHC13 CsHSF C6H5Cl C6HI2) over a wide temperature range. The data obtained by applying this technique are relaxation strengths R(T) and relaxation times .r(T),both as functions of the temperature T.EXPERIMENTAL The optics and the apparatus used for the experiments in the present work are shown in fig. 1. The light scattering spectrometer consists of a single mode Ar ion laser L usually run at 200 mW on the 5145 A line. The incident beam is focused into a cylindrical cell S. The scattered light is observed at a given scattering angle with a spatial filter (pin holes Bl and lens Ll) a plane piezo scanned Fabry-Perot interferometer and a lens L3 focusing the central ring into a 0.1 mm pin hole. The photomultiplier output is amplified with an electrometer digitized and recorded simultaneously on a chart recorder and on a punch tape.Each spectral triplet is recorded with approximately 300 points. The spectrometer was operated with a finesse of 70-90 due attention having been paid to the divergence of the scattering angle and the stability of the interferometer. The scatter- ing angle could be varied from 20 to 160" by a system of 2 mirrors deflecting the incident beam. The temperature of the cell was controlled with a cold nitrogen stream and an electrical heater. The control thermometer TI was located inside the brass body M of the cell and the temperature was measured at T2 and inside the liquid at T3. M-FIG.1 .-Schematic diagram of experimental apparatus. (fi = 500 mm,fi = 300 mm = 500 mm pin hole Br = 1 mm diam. and Bz= 0.1 mm diam.) 108 RELAXATION IN SIMPLE LIQUIDS BY POLARIZED LIGHT SCATTERING The entire cell was located inside a stainless steel chamber with 18 plane windows situated in the scattering plane at different angles.The chamber could be evacuated for better thermal insulation at low temperatures. At temperatures above the normal boiling point of the liquid the chamber could be pressurized up to 5 atmospheres. LIGHT SCATTERING AND RELAXATION FORMALISM The light scattering spectrum in a relaxing medium can approximately be described by eqn (2) :11*12 The quantities appearing in this equation have the following interpretation The scattering wavevector k = (4nn/A)sin(8/2> The ratio of specific heats Y = CP/G The Rayleigh absorption rR = Kk2/poCp The classical non-relaxational absorption :To= (2~~13 + qv)k2/po The total absorption rB = To+ (r -1)/2rrV The relaxation parameter r = u2/ui The relaxation strength R = U~/U:.The sound velocity at the frequency o The sound velocity at the frequency 0 The sound velocity at the frequency 00 The heat conductivity The Brillouin shift u uo UOO K WB The scattering angle The mean density Index of refraction The vibrational heat capacity The heat capacity at constant volume The heat capacity at constant pressure Adiabatic relaxation time The physical content of the assumption upon which the approximation leading to eqn (2) is based is that the dissipative processes can be considered as small corrections to the elastic hydrodynamic response of the liquid to the dynamic variable. The spectrum expressed in this form contains the central Rayleigh line with the spectrum IR(k,m) the Brillouin stokes and antistokes lines with the spectra Ii(k,o) IA,(k,o) and the relaxation line with the spectrum ZM(k,m) z(k70) = zR(k,m) + zhl(k,m) + Iz(k,w) + I$(k,m) + rcorr(k,o)* (4) The last term represents an additional correction term taking into account the non- Lorentzian form of ZB(k,m).I3 For the present purpose we shall consider only the Brillouin doublet the Rayleigh line being irrelevant for relaxation.A fit of the experimental Brillouin lines gives us an experimental value of Fnand r thus permitting a calculation of rv and R via the relaxation equations THOMAS DORFMULLER ET AL. Additional information can be obtained from the experimental value of the Landau- Placzek ratio with f(t) = (r3-t2-r + R)/[r2-2t + R -y(t -l)'(R -t)]; I and IB are the integrated intensities of the central and the Brillouin lines respec- tively.One can use eqn (5) (6)and (7) to derive the quantities 7v,R,q,/q from the Brillouin spectra. Since however the uncertainty in I is very high and I has to be known with high accuracy to be of any value at all one either has to make some assumption about one of the variables thus reducing the number of variables to two I?"/% = 1 R = (C*-CMCP -YCJ or use more equations to determine the three unknown quantities. The most con- venient way to obtain more equations is to vary the wave vector k by varying the scattering angle 0. In doing this we obtain the frequency dependence of the sound velocity u(k) or of the relaxation parameter r(k).These data can then be fitted to eqn (5). A fit of the angle-dependent values of u(u) to the theoretical dispersion eqn (5) for CC14 at 20 "C is given as an example in fig. 2. The description of relaxation data is usually based on the Schwartz-Slawsky- Herzfeld theory (SSH) and the extensions introduced by Tanc~os.~ The basic ideas of this approach can be summarized as follows A linear head-on non-reactive collision model is applied where the molecular vibrations are considered to be harmonic and the rotational transitions are neglected. The T-V relaxation process takes place via collisional perturbation of the molecular states. This perturbation is essentially of the same nature in the liquid and in the gas (isolated binary collision or IBC approach).The molecular interaction enters the calculations through the repulsive inter- molecular exponential potential V(r,x) which is a function of the translational co-ordinate r and of the vibrational coordinate x. The r and x dependence of V(r,x) are separable as a product V(r)V(x). 1100 c I * 1000 E \ 6 900 1 10 100 -0xlO-'/ rad s-1 FIG.2.-Dispersion curve of the velocity of CCl at 20 "C. The resulting parameters are R = 1.370 7" = 51 PS. 110 RELAXATION IN SIMPLE LIQUIDS BY POLARIZED LIGHT SCATTERING The transition probability Pi + can be evaluated through the quantum mechani- cal expression for the perturbation integral Q =1tp 7V(r)tp,d.c whereiy and yj are the wave functions of the perturbed harmonic oscillator in the vibrational states i and j respectively.The solution of the Schrodinger equation of the system involves the use of the first-order disordered wave approximation. The average transition probability (Pi+j> per collision can be obtained by the integration over the Boltzmann-Maxwellian distribution of the relative collision velocities making the assumptions (,u/2)vY2 9 h and exp(4z2v/a*uF) 9 1. The SSH-Tanczos formula for (Pl -o) can then be given by the equation with In eqn (8) p is the reduced mass of the collision pair C the Sutherland constant ur the most effective initial relative velocity to induce a quantum jump with the energy hv Bo the potential well depth and a* a parameter in the used intermolecular potential V(r).To distinguish wave vector k from Boltzmann's constant the latter is printed kg. DISCUSSION OF THE RESULTS As already mentioned the experiments at an angle of go" when using eqn (6) enable us to calculate two of the three quantities R,zv and nv/vs. The temperature dependence of R,however is extremely helpful in obtaining additional information. Fig. 3 shows the temperature dependence of R compared with the theoretical Planck- Einstein results represented by two curves for each substance (indexed A-E). Curve (a)gives R for the whole of the internal vibrations and curve (6)gives R-for all but the vibration with the lowest frequency vmin. The experimental points have been calculated for CCl, and C6H5CI with the assumption vv/qs= 1 which is known to be approximately true for many Kneser liquids.For CC14 the points seem to fit the curve (a) and for C,H,CI they seem to fit curve (b). In order to obtain a fit of the experimental points for the other sub- stances we have to use different values for qv/qs. Table 1 gives the results for all the liquids which will be discussed here. Since the calculations of R and qv/qsdepend upon the absorption measurements given with an accuracy of lo%,the error in qv/qsis rather high; therefore the values are given without decimal figures. The decision however as to which vibrational levels have to be included in R is facilitated by the fact that any other choice than the one given here leads to quite unreasonable values for qv/qs(negative values for in- stance).One can see furthermore that this decision cannot be obtained from measurements at one temperature only at least with presently feasible experimental accuracies. The 4th column of table 1 shows the frequency of the lowest relaxing vibrational level which is the second lowest level in C6H5F C6H5CI and C6H12. The 5th column shows the isothermal relaxation times z at 25 "C. In the range of temperature where the relaxation process seems to have within THOMAS DORFMULLER ET AL. 1.5 r 1.1 2 1.0' 0 40 80 -Tloc 1 FIG.3.-Relaxation strength against temperature. Curve (1) gives R and curve (2) gives R-1. (a) CCL; (b)CHC13; (c) C~HSF;(d)C,H4C1 and (e) cyclohexane C6H12. the experimental error a single relaxation time z the temperature dependence of z can be described by eqn (8).In order to apply this equation we have to use the following assumptions (1) The IBC model is a good approximation for the situation leading [eqn (S)] to the relation (p1-0)1iquid = (pl-O)gas(l +-)e-"o/k~T. (9) C T The IBC approach has been subjected to numerous ~bjections.'~-'~ However these objections although theoretically sound could not disprove the validity of the IBC approach as a satisfactory approximation for the relaxation dynamics of liquids. It seems that at present the experimentalist does not have the choice of another approach. (2) The analytical form of the intermolecular potential V(r)is approximated by a Lennard-Jones potential 112 RELAXATION IN SIMPLE LIQUIDS BY POLARIZED LIGHT SCATTERING This potential is used mainly because of the availability of the parameters E and 0.This has to be fitted to an exponential repulsion potential V(r)= Voexp(-ar) + po at the classical turning point rc as the evaluation of the perturbation integral using the exponential form for V(r)gives an analytical form for Pl -o. TABLE 1.-vALUES OF THE RELAXATION STRENGTH R RATIO qv/qs,VIBRATIONAL MODE Itinst. AND ISOTHERMAL RELAXATION TIMES z FOR THE MEASURED LIQUIDS AT 25 "C substance R VVhS v1crn-l t"/PS CS2 R = 1.180 0 397 2640 CCl R = 1.378 1 218 124 CHCl3 R = 1.240 0 262 91 C6H5F R1 = 1.277 2 368 107 C6HsCl R1 = 1.217 1 296 58 CsH12 R1 = 1.297 2 384 63 a Calculated from the formula z = zv(Ry -l)/(y -1).It is well known that the Lennard-Jones potential especially its repulsive com- ponent is not satisfactory. In our case this is a serious drawback as T-V relaxation is mainly determined by the steepness of the repulsive potential at small values of the translational coordinate. On the other hand the virial coefficients are more depend- ent on the attractive form of the potential while the transport properties are sensitive to the repulsive part up to thermal energies. This means that Lennard-Jones para- meters evaluated from transport coefficients are to be preferred. Neglecting the attractive component we have to either use literature data for E Q rn or evaluate some of these quantities on the basis of other suppositions. We have used ccritvalues obtained from critical data.(3) The liquid structure can be described by means of a cell model. In connection with relaxation processes the models with movable walls either with an fcc structure (model A) or with a simple cubic structure C (model B) have been used quite success- fully. The resulting expressions for zBCare model A zBC= (21/6p0-1J3-o)/a model B:zBC = -a)/; where the average velocity 17is d = (8RT/nA4)'J2. The transition probabilities (PI -+ o) for the case of a harmonic oscillator can thus be obtained by using eqn (12) <Pl-+ 0) = %C/Z I1 -exP(-@v/T)l (12) and the calculated values of the isothermal relaxation times z. On the basis of the above mentioned three assumptions we have calculated the parameters m and 0for the liquid models A and B.The results are shown in table 2 Eqn (8) can be used to represent the results in a reduced form the reduction pertaining to the constants E 0 rn vinit for each substance. Fig. 4 represents a plot of log 2 against for various liquids at 25 "Cwhere 2 = (@o/~v)z~1~6exp(0v/2T) V~-o/((Pl,,)T1/6). Under the assumption that the choice of R and qv/qsis correct the plot should be linear and this result should confirm the interpretation of the relax- ation measurements as given in tables 1 and 2. This is indeed roughly the case for all THOMAS DORFMULLER ET .AL. TABLE Z.-LENNARD-JONES PARAMETERS CS2 A 434 23 4.82 4.44 4.5 1 B 22 4.22 CClJ A 335 25 5.38 5.88 5.79 B 24 4.83 CHCl A 427 22 5.17 5.43 5.04 B 20 4.5 1 C6H5F A 299 17 5.40 -5.99 B 15 4.69 CbHsCI A 346 17 5.51 -6.20 B 15 4.91 C6H12 A 329 24 5.69 6.09 6.04 B 21 5.05 substances but CS,.The point representing CS lies at a value of P which is smaller by an order of magnitude. The value of P given in the literature which was calculated applying a different is greater by a factor of 3. 6.5 I I 25 35 45 FIG.4.-A plot of log Zl against @i3. In conclusion one can say that the values of R,qv/qs,0 and m can be deduced from the hypersonic measurements. The conclusion however is not unique and has to be substantiated by plausibility arguments about the possible values of these quantities and about the validity of the liquid models chosen. 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