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MOLECULAR REORIENTATION MODEL FOR DEPOLARIZED RAYLEIGH SCATTERING IN
VISCOELASTIC LIQUIDS
E. Zamir, A. Ben-Reuven
To cite this version:
E. Zamir, A. Ben-Reuven. MOLECULAR REORIENTATION MODEL FOR DEPOLARIZED RAYLEIGH SCATTERING IN VISCOELASTIC LIQUIDS. Journal de Physique Colloques, 1972, 33 (C1), pp.C1-237-C1-240. �10.1051/jphyscol:1972140�. �jpa-00214930�
MOLECULAR REORIENTATION MODEL FOR DEPOLARIZED RAYLEIGH SCATTERING IN VISCOELASTIC LIQUID S
E. ZAMIR and A. BEN-REUVEN 'The Weizmann Institute of Science Rehovot, Israel
Rksum6. - On suggere un modele des mouvements de reorientation dans certains liquides viscoClastiques (principalement les liquides aromatiques a la temperature ordinaire) dans lequel on caracterise les ailes de la raie Rayleigh depolarisee par au moins deux temps de relaxation lar- gement separes, et par une structure fine qui est associk au couplage avec les modes collectifs.
Le temps le plus long est relie & un &tat d'equilibre local dans lequel la molecule non spherique est incapable de tourner a cause de fortes correlations anisotropes spatiales (de position) avec ses voisins. Le temps le plus court correspond
a
une interruption de cet &at pendant lequel la molkule tourne presque librement. Ce modele rend compte des proprietes suivantes : a) On calcule la force du couplage aux modes collectifs en termes de quantitks mol~culaires seulement. b) On attend un rapport de 1,2 entre le temps de relaxation dielectrique et le temps le plus long qui intervient dans la diffusion de la lurnikre. c) Les corrections de champ local dues aux correlations spatiales anisotropes produisent une reduction de I'intensitk de la composante fine par rapport A l'intensite que I'on calcule pour des molkules non correl6es. Cette reduction est en accord avec les observations. d) On fournit une estimation grossiere du rapport des intensites des composantes fine et large.Abstract. - A model is suggested for the reorientational motion of certain viscoelastic liquids (mostly aromatic liquids at room temperature) in which the depolarized Rayleigh wings are cha- racterized by at least two largely separate relaxation times and a fine structure associated with coupling to collective modes. The longer time is related to a local equilibrium state in which the nonspherical molecule is unable to rotate due to strong anisotropic spatial (positional) correlations with its neighbors. The shorter time is associated with a break in this state during which the mole- cule rotates almost freely. This model accounts for the following features : a) The strength of the binding to collective modes is calculated in terms of molecular quantities only. b) A ratio of 1.2 bet- ween the dielectric relaxation and the longer light-scattering time is expected. c) Local-field cor- rections due to the anisotropic spatial correlations result in the observed reduction of the intensity of the sharper component in comparison with the intensity calculated for uncorrelated molecules.
d) A rough estimate is provided for the ratio of intensities of the broader and sharper components.
A) The Model. - We present here a model of orientational motion in certain viscoelastic liquids.
The low-frequency depolarized Rayleigh spectrum of these liquids is characterized by (a) a separation into a sharper and a broader component [I] (disclo- sing a relaxation process with at least two characteris- tic times) and (b) a fine structure caused by coupling to collective motions [2]. Certain aspects of the model, presented here in more detail, have been discussed elsewhere ; namely, the coupling to collective modes [3]
and the ratio of dielectric relaxation to light-scattering time [4].
The class of liquids for which this model applies consists of molecules whose shape is sufficiently anisotropic (and free volume sufficiently small).
The orientational motion of the molecules is then
Consider the case where the relaxation time z, (associated with the sharper component) is sufficiently longer than the other relaxation time 7, (associated with the broader component). We then suggest that for most of the time (over periods of the order of 7,) the molecules are trapped in an anisotropic poten- tial well of their neighbors, being practically unable t o rotate. Thus, a state of rather stable local equilibrium is formed, with the distribution of nearest neighbors largely determined by the shape of the molecule itself.
In between these intervals of local equilibrium, the local correlations break and the molecule rotates almost freely (over periods of the order of 7,). This model is valid only if
7,
s
2,.
(1) severely hindered by their neighbors. Moreover, B) Coupling to collective modes. - This feature these molecules are sufficiently small to have a free- has been studied in detail elsewhere 131, and we only rotation time shorter than the relevant relaxation discuss here some of its consequences. The assumption times. This means that, given the opportunity to underlying this analysis is that in the class of liquids rotate unhindered, they may undergo a large change studied here, the anisotropic dielectric properties of orientation before being stopped again. of the liquid are modulated in a rigid fashion byArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972140
C1-238 E. ZAMIR AND A. BEN-REUVEN collective modes. This idea underlies other studies
of this coupling from a hydrodynamic [5] or a corre- lation-function [6] point of view. With the use of our model it was possible to relate the strength of this modulation to simple molecular quantities [3].
The long-lived modulation by a collective mode affects the time derivative of the dielectric tensor and is expressed in the spectra1 distribution as sharply- frequency-dependent terms in the so-called (( memory function N.
The spectral distribution function can be written a s [31
where A , is the integrated intensity, and
is the memory function. Here
r,
is the low-frequency contribution of short-lived ((( collisional D) relaxation processes, and the sum over m is over all collective modes affecting the spectral distribution. Each term has a coupling-strength parameter B, and a complex shape function fk(o) determined by the temporal behavior of the collective mode.Modes propagating with a wave vector k, and having a well-defined dispersion law (relating to k a frequency
a3
contribute to (3) two terms, centered around the two frequency shifts $. w,.In broad outlines, the position and half-width of the fine-structure features in the case of longitudinally polarized modes (case HH of ref. 2b) closely resemble the corresponding parameters derived from the Bril- louin spectrum. In the case of transverse modes (case VH), to which no analogous Brillouin spectrum is observed, the experimental data [2] indicate the presence of overdamped collective ((( shear D) modes.
These modes have a practically vanishing resonance frequency and a half-width comparable in magnitude to that of the longitudinal modes.
The temporal behavior of the collective modes is outside the reach of the present model. One is usually tempted to assume a simple Lorentz dispersion law,
A refined study of the experimental line shape should disclose in more detail the manner of propagation of the collective modes.
The coupling-strength parameter is given by this model as [3]
Here k, T is the Boltzmann constant times the tempe- rature ; m is the mass of the molecules ; g ( 2 ) a struc- ture factor resulting from orientational and spatial correlations in the second-rank dielectric tensor (to be
discussed below) ; @ is a dimensionless factor depend- ing on the experimental geometry. Expression (5) is obtained under the condition of rigid coupling and can therefore be valid only in the low-temperature limit where (1) holds. It can hardly be expected to give the right temperature dependence. At higher temperatures this rigidity conidition should relax and the value of B, decrease. On phenomenological grounds we may have to multiply Bk by a (( rigidity factor )) of the form
or a similar form, with an activation energy E somehow related to zl/z2, and approaching unity asymptotically at low temperatures. At finite temperatures, the correct value of Bk may be therefore somewhat smaller than the calculated one. Here again, a careful study of the fine structure should yield m.ore information on this temperature dependence.
C) Ratio of dielectric to light-scattering reorien- tation times. - As stated in the model, we require the molecules to have a free-rotation time shorter than both z, and 7,. Under this condition, each new equilibrium state (formed after a break in the former one) will reorient the molecule in a practically random fashion. According to the Ivanov [7] theory, such a relaxation by random studden jumps should give a ratio of 1.2 between the relaxation times of a first- rank (dipole) and a second-rank (anisotropic-polariza- bility) tensor. In a comparisoil of z, with the dielectric relaxation times [4] this ratio is indeed closely observed.
It should be noted here that both z, and the dielectric relaxation time z, represent macroscopic relaxation times, affected by correlations between the mole- cules
[a],
and by local-field corrections. The micro- scopic (( tumbling >) time measured by nrnr techniques is indeed appreciably shorter [9] than 7,. However, so far as inequality (1) holds for the macroscopic as well as the microscopic times, the ratio of 1.2 still holds.Of course, it would not make sense to compare the macroscopic dielectric time with the microscopic nmr tumbling time.
D) Spatial Correlations. -- In the absence of local-field corrections (and assuming short-range correlations only) the integrated intensity of the Rayleigh wings differs from that expected for uncor- related molecules, as result of orientational pair correlations [lo], [Ill. This disparity is quantitatively expressed by a (( structure factor )> [l 11
where
ZF)
is the anisotropic part of the polarizability tensor of molecule A, and the brackets denote a thermal average. An experimental value of gC2) can be obtained for each component of the Rayleigh wingsby taking the ratio of its intensity to that of the isotropic Rayleigh-Brillouin triplet 191, [Ill. The latter intensity can be calculated by the well-known Einstein formula relating it to the isothermal compres- sibility. A difficulty arises here since the separation of the triplet from the depolarized wings necessitates a knowledge of the exact line shape of the triplet. In previous works [9] we have used the Mountain expres- sion [12], neglecting the broad component associated with the dispersion of the sound velocity [13]. This would give an upper limit gk2'. A lower value gi2) is obtained by imposing the theoretical depolarization ratio of 3 : 4 for 900 scattering. The isotropic scattering does not contribute at perpendicular (VH or HH) polarizations. Therefore one can get a value of g ( 2 )
by multiplying the VH (or H H ) intensity by 413 and subtracting it from the parallel-polarized (VV) inten- sity to obtain the isotropic-scattering intensity. The two values of g ( 2 ) thus obtained are listed in Table I.
A further check on the contribution of the broad Mountain line is obtained by evaluating its maximum value in the high-frequency limit. Its relative contri- bution to the isotropic scattering is thus smaller than this limiting value [13],
where Vo and V , are, respectively, the low and high- frequency limits of the sound velocity, and y is the specific-heat ratio C,/C,.
The structure factor for the sharp component of the depolarized Rayleigh spectrum at room temperature.
g:' is an experimental upper limit obtained by neglecting the Mountain peak, g{2' is a lower value obtained by requiring a 3 : 4 depolarization ratio.
gb2'
is calculated by a model of IocalJield correction due to anisotropic spatial correlations and neglecting orientational correlations.g?'
d2'
-
Quinoline .28 .28
Nitrobenzene .68 .37
Bromobenzene .64 .57
Chlorobenzene .42 .37
Aniline .30
Benzonitrile .91 .73
Pyridine .2
In the liquids listed in Table I, the calculated value of (8) does not exceed 16
%.
Allowing for an approxi- mate error of 5%
in the intensity measurements, all these estimates lead to quite consistent values of g(2) - with the exception of nitrobenzene. In nitrobenzeneis almost twice as large as g{'). The evaluated upper limit for the Mountain line is only 7
%.
Appa- rently the value of gl(2) is too low, for unknown reasons.All these values of g(2) are appreciably smaller than unity, the would-be value for uncorrelated molecules.
In contrast, simple calculations show that reasonably conceivable stackings of neighboring molecules yield a
g ( 2 ) factor greater than unity. For example, clustering into parallel (or antiparallel) pairs would require a g ( 2 ) = 2. Obviously orientational correlations can not satisfactorily account for the intensities. A more satisfactory account can be obtained by the introduc- tion of anisotropic spatial distributions and their resulting nonvanishing local-field correction. As the molecule stays in the nonrotating equilibrium state, the neighboring molecules form an anisotropic (( cage )) around it ; i. e., the positional distribution of their centers is not isotropic, but roughly follows the shape of the surrounded molecule. Together with each molecule, its neighbors are also polarized by the radiation field. This polarization, in turn, induces a Coulomb field over the original molecule. The resul- tant field produced by all neighbors would have vanished if their positions were distributed isotropically.
The anisotropy in the spatial distribution, which roughly coincides with the polarizability anisotropy of the original molecule, produces a resultant local field. This field reduces the applied field component along the larger axis of the molecule and increases the applied field component along the smaller axis, thus effectively decreasing the anisotropy in the polariza- tion of the molecule. If we still wish to express the polarization as proportional to the applied field, we have to modify instead the molecular polarizability to take into account the local-field effect. A method of evaluating these corrections was described recently by Hellwarth, making a point-dipole approxima- tion [14]. According to this method (assuming again only short-range correlations, or the k + 0 limit) we can define a modified polarizability tensor
7,
by solving the equationHere
is the second-rank tensor determining the depen- dence of the dipole field on the radius-vector r, with? being the unit tensor. Given a model distribution of neighboring molecules, we can solve (8) for
FA,
and subtract from it its spherically-symmetric part t o obtain the modified second-rank polarizability,This modified tensor should replace
z2'
in the nume- rator of (7). Since we obtain g ( 2 ) empirically by compar- ing with the isotropic Rayleigh-Brillouin spectrum, we must also correct for the variation of a, from the unmodified cr,. This is usually a small correction, ofC1-240 E. ZAMIR AND A. BEN-REUVEN
the order of 5
%.
The modified structure factor will quantities, it becomes nonsensically dependent on thetherefore be parameters when it is much smaller than unity. It
a
"
(F Gy)
:Z p )
would therefore be futile to try and get better values= unless a much more detailed account of the molecular
o;
< ~ 2 ) : &a >
' (I1) distributions can be provided.
In the calculation of g:", shown in the last column of table I, we have used a model of six near neighbors lying along the three principal polarizability axes of the molecule, and having no orientational correlations (that is, we averaged over the orientation of each neighbor) [15]. As the intermolecular distances we have taken
ai
+
max (ax, a,, az),
(12) where ai (i = x, y, or z) are the molecular radii along the principal axes, calculated from known Lennard- Jones radii and interatomic distances [16].The calculated numerical values should be regarded with due reservation, owing to the crudeness of the model distribution. In particular, the use of the maxi- mum of the three radii in (12), rather than (say) their average, sounds somewhat arbitrary. The latter choice would seem more realistic, and should lead to even lower values of g('), below the lower experimental limits. This reduction can be compensated for by orientational correlations. Parallel stacking, for exam- ple, would tend to increase the local-field reduction of the self correlations, but this would be more than compensated by the positive contribution of pair correlations.
In the actual calculations, a use of the average instead of the maximum radius leads in some cases to computational troubles. Since g(2) is derived with the help of (lo), which is a difference of two larger
E) Ratio of intensities of sharp and broad compo- nents. - The two times z, and z2 provide a crude estimate of the relative durations of the hindered and rapid-rotation states, respectively. Assuming that, in the latter state, all correlations leading to a g") different from unity practically vanish, we can write
I (broad) -2.-
-
z 1 I (sharp) z, g(2)for the ratio of intensities of the broad and sharp components.
The data available for testing (13) are very scant. For nitrobenzene, with z, = 40 ps [9], z2 = 6 ps 1171, and g(2) of about 0.6, the estimated intensity ratio is about 0.25. The experimental room-temperature ratio is 0.28. In writing (13) we did not consider the effect of the difference between microscopic and macroscopic relaxation times, and of possibly remaining correlations during the rapid-rotation phase. It should therefore be considered only as a crude latimate of the intensity ratio.
A better understanding of the shorter-time processes would require a more detailed study of the line shape in the 10 to 100 cm-I range. Unlike the sharp compo- nent which (aside from the coupling to collective modes) discloses a Markoffian single-time relaxation owing to condition (I), the situation at the more rapidly varying broad component may be quite different.
References [1] a) STARUNOV (V. S.), TIGANOV (E. V.) and FABELINS-
KII (I. L.), Soviet Phys.-JETP Letters, 1966,4, 176.
b) SZOKE (A.), COURTENS (E.) and BEN-REUVEN (A.), Chem. Phys. Letters, 1967,1,87.
C) CRADDOCK (H. C.), JACKSON (D. A.) and Pow-
LES (J. G.), Molec. Phys., 1968, 14, 373.
[2] a) STARUNOV (V. S.), RGANOV (E. V.) and FABELINS-
KII (I. L.), Soviet Phys.-JETP Letters, 1967,5,260.
b) STEGEMAN (G. I. A.) and STOICHEFF (B. P.), Phys.
Rev. Letters, 1968, 21,202.
[3] a) BEN-REUVEN (A.) and GERSHON (N. D.), J. Chem.
Phys., 1971, 54, 1049.
b) BEN-REUVEN (A.) and GERSHON (N. D.), Ber. Bun- senges. Physik. Chem., 1971,75,340.
[4] BEN-REUVEN (A.) and ZAMIR (E.), J. Chem. Phys., 1971,55,475.
[5] a) LEONTOVICH (M. A.), J. Phys. (USSR), 1941,4,499.
b) Rrrov (S. M.), SovietPhys.-JETP, 1957,6,130 ; 6, 401, 6, 513.
c) VOLTERRA (V.), Phys. Rev., 1969, 180, 156.
[6] KEYES (T.) and KIVELSON (D.). J. Chem. Phys., 1971, 54,1786.
[7] IVANOV (E. N.), Soviet Phys.-JETP, 1964, 18, 1041.
[8] KEYES (T.) and KIVELSON (D.), J. Chem. Phys., in press.
[9] a) ZAMIR (E.), GERSHON (N. D.) and BEN-REUVEN (A.), J. Chem. Phys., 1971, 55, 3397.
6) GERSHON (N. D.), ZAMIR (E.) and BEN-REUVEN (A.), Ber. Bunsenges. Physik. Chem., 1971, 75, 316.
[lo] PECORA (R.) and STEELE (W. A.), J. Chem. Phys., 1965,42, 1872.
1111 B E N - R E U V E ~ (A.) and GBRSHON (N. D.), J. Chem.
Phys., 1969,51,893.
[12] a) MOUNTAIN (R. D.), Rev. Mod. Phys., 1966,38,205.
6) MONTROSE (C. J.), SOLOVYEV (V. A.) and LITO-
WTZ (T. A.), J. Acou.~t. Soc. Am., 1968,43, 117.
[13] MOUNTAIN (R. D.), J. Res.. Natl. Bur. Std., 1966,70A,
207.
[14]
HELLWARTH
(R. W.), J. Chem. Phys., 1970, 52, 2128.[15] ZAMIR (E.), Doctoral Thesis, The Weizmann Institute of Science, Rehovot, Ilsrael, May, 1971.
[16] a) SUTTON (L. E.), (Ed.), Tables of Interatomic Distances and Configurations in Molecules and Ions, Chem.
Soc. (London), Spec. Publ., 1958, 11 ; 1965, 18.
[16) b) ROTH (V. G.), 2. Natiirforsch., 1963, 18A, 516.
[17] L u c ~ s (H. C.), JACKSON (D. A.), POWLES (J. G.) and SIMIC-GLAVASKI (B.), Molec. Phys., 1970, 18, 505.
