- Email: [email protected]
Theoretical investigation and experimental detection of rattling motions in atomic and molecular fluids
Journal Elsevier
of Molecular Structure, 84 (1982) 195-203 Scientific Publishing Company, Amsterdam
THEORETICAL OF RATTLING
S. BRATOS
-
Printed
in The Netherlands
INVESTIGATION AND EXPERLMENTAL MOTIONS IN ATOMIC AND MOLECULAR
DETECTION FLUIDS
and B. GUILLOT
Laboratoire de Physique Thkorique des Liquides Place JLLssieu, 75230 Paris Cedex 05 (France)
*, Uniuersitk
Pierre et Marie Curie, 4
ABSTRACT
The theories
used to investigate
rattling
framework of the Zwanzig-Mori theory motions can be successfully investigated
depolarized experimental
motions
of Brownian by dielectric
light scattering and collision-induced techniques are briefly explored.
in dense fluids are discussed in the motion. It is shown that these relaxation, collision-induced
IR absorption.
The potentials
of these
INTRODUCTION
Considerable interest has been shown over the years in the fast rattling motions in atomic or molecular liquids. These motions consist either of translational oscillations of atoms or of translational-rotational oscillations of molecules in a given site of the liquid. Unfortunately, their experimental detection was found to be unexpectedly difficult. The earliest attempt to solve this problem involved the application of neutron scattering techniques, described in papers by Egelstaff and Schofield [ 1,2] , Rahman et al. [ 3,4] , Sears [ 51 and Damle et al. [ 6] . This method of detection remains controversial; see a recent review by Copley and Lovesey [7 ] . Another method was to use light-scattering techniques and to study the wing of spectral lines of optically anisotropic molecules. These wings were attributed to the collisioninterrupted librations of molecules in liquids. The main exponents of this technique were Starunov [S] , Proti [9], Sechkarov and Nikolaenko [ 101 and Simic-Glavaski et al. [ll, 121. The corresponding theory has remained qualitative, however; for its appreciation, see Keyes and Kivelson [ 131 . The only technique which has so far offered a reliable method of detecting rattling motions in liquids is dielectric relaxation. Here, motions are detectable either by the presence of an excess absorption in the ultra-hertzian spectral range or by a deformation of the Cole--Cole semicircle at high frequencies. The leading workers in this field are Poley [ 141, Davies et al. [ 151, Leroy et al. [lS, 173, Brot and co-workers [18-201, Quentrec and Bezot [21] and Guillot and Bratos [22] . A number of liquids have been successfully examined by this method [23] . *Department
associated
0022-2860/82/0000-0000/$02.75
with the CNRS. o 1982
Elsevier
Scientific
Publishing
Company
196 The purpose of the present paper is to show that two other techniques, collision-induced depolarized light scattering and collision-induced IR absorption, can be employed to study oscillatory motions in atomic liquids; cf. Guillot et al. [ 24-261 . It is helpful to link our discussion with that relative to dielectric relaxation_ AlI three techniques will thus be treated simultaneously here. THEORY
It is convenient to start by presenting the formulae which relate the complex dielectric constant, E(W ), the differential cross-section a2o/&X2, of the depolarized light scattering and the IR absorption coefficient,P(W), to the appropriate correlation functions. Designating the conventional highfrequency dielectric constant by c._,, its static counterpart by eO, the frequency of the incident light by wo, the total dipole moment of the liquid sample by n(t) and its dielectric tensor by g(t) the following equations may be written
(eo+ a=_ 1 -4 (eo -GJ[d4 + 21
s
kb
-=-
a20
awas
1 ~g 4 +F 2n OJc -DD
dt
dc
0
e-‘“t
g
$I(O)gIW
(1)
dt (M(0) M(O)>
e_iOt
w tanh @&)
(2)
kii;O)E.itZ))
jm dt eSiwt <$[R(O), -m
R(t)]
) i-
(3)
These formulae may be found in refs. 27-29. For theoretical convenience the liquid sample subjected to dielectric relaxation is assumed to be a sphere inside a vacuum. The basic dynamic variables entering into this calculation require to be specified in more detail; the following considerations merit attention. (i) The system subjected to a dielectric relaxation experiment is a liquid sample formed by N polar molecules, thus $l = r;E, @Ii where Gi is the dipole moment of the molecule i. (ii) The system subjected to a collision-induced light scattering experiment is a dense rare gas fluid formed by N atoms; its dielectric tensor, 7, is thus given by the sum of $N(N - 1) polarizability tensors, associated wiJh all possible pairs of atoms which may be formed by collisions, 2 = Zy. j =, Zij. (iii) The system subjected to a study by collisioninduced IR absorption is a mixture of rare gas atoms containing N, atoms of species A and N, of species B; its dipole moment fi is thus equal to the sum of dipole moments a, of all possible pairs of atoms of different species which may be formed by collisions, a = zF=~ z,~=T $Iij. The pair dipole ~ij,
197
moment ajj has a particular form in the present case and is often assumed to be proportional to the force i?, between these two atoms, fi, = Cp, where C is a constant. It can then be shown that
Ma) 3=(1-x)
F
iTi-x
i=t
“z” $l i’=
(4b)
1
where cj, Gj are the velocities of the atoms i, j,, m A and ll~a the masses of atoms of the species A and B, x the number density of atoms of the species A and 3 the interdiffusion current density fluctuation [25, 261 . It results from this analysis that three dynamic variables, the dipole moment G(t), the dielectric tensor 2 and the interdiffusion current density fluctuation ‘j form the basis of the theoretical description. They are all considered to be classical to a reasonable degree of approximation. Subsequent calculation is based on the Zwanzig-Mori version of the generalized Langevin theory of Brownian motion [30-321. According to this theory, the normalized correlation function G(t) = (A(O)A(t))(A(O)A(O))-' of a dynamic variable A(t)obeys the equation
‘9
=if2G(t)
-
1ds K(s) 0
G(1 -s)
(5)
where 52 is the frequency and h’(s) the memory function; Zwanzig and Mori have shown that this equation is in fact an esact equation of motion. It can be solved by successive approximations, either by truncating the continuous fraction formula established by Mori or by applying the Kivelson-Keyes procedure [32] . The two methods are intimately related and converge, in principle, to an exact solution. In the present problem, the variable A (t) represents either a(t), ES=(t) or j(t); the corresponding Langevin equations were solved by the Kivelson-Keyes method in its three-variable approsimation. The elements of the transport matrix r = --it2 + lim, _ (, j; ds eBzs K(s) which forms part of the theory were expressed in terms of the quantities T = JFds G(s), G”(O), G”“(0). In turn, these latter quantities were estimated, non-empirically, by applying a lattice-gas model of the liquid state. For details of this calculation see refs. 22,24-26. The theory outlined above predicts the existence of two modes of motion in liquids. The first mode is a dissipative mode due to translational diffusion of rare gas atoms or to rotational diffusion of polar molecules. The second mode is a doubly degenerate oscillatory mode associated with translational or rotational rattling motions. These two modes correspond to the real and complexconjugate roots, respectively, of the transport matrix r. The correlation functions have the following characteristic form
198
G(t) = cl emAlt + c2e-ht
cos h3t + c3e-kt
sin X,t
(6)
where XL, A z f C43, c 1, c2, c3 are known functions of the parameters 7, G”( 0), G”“(0) given above. A representative illustration of G(t) is shown in Fig. 1
where the rattling motions are clearly recognizable. As expected, the frequency, h 3, of these oscillations depends on intermolecular forces which determine the value of 7, G”(O), G”“(O)*. The notion of a cage is not introduced explicitly. The frequency dependent quantities E(W ), #CJ/&X2z~(~ ) are obtainable by substituting the correlation function G(t), with A = M, tzXZ, or 3, into eqns. (l)-(3). The results may be expressed in a compact form by denoting the dielectric absorption coefficient, w E”/IZC,the differential scattering crosssection, a’o/aoaQ, and the IR absorption coefficient, Q(W), by a unique symbol I(w ). This leads to
r(d=AWl((1-g+
w29[1-
(l-$)/(l-(&)ll)
(7)
The function A (w ) varies from one technique to another as do the parameters 7, o i and w: ; they are given in Table 1. The fact that the dielectric absorption coefficient varies as w * at small values of w whereas the IR absorption coefficient varies as w4 is due to the particular nature of the collision-induced dipole moment sljj.
1
0
9
0.25
0.50
0.75
1.0
1.25
x 10-Q s Fig. 1. Theoretical correlation function (M(O)M(t))/(M(O)M(O)) Ne-Ar withxAr = 0.989, T = 90 K, P = 728 amagat ( T = 130 K, p = 280 amagat (------).
in dense atomic mixtures; ): Ne-Ar with xAr = 0.989,
*For dielectric relaxation tion is purely kinetic.
of the autocorrelation
in polar
liquids
the second
moment
func-
199 SF-%CTRAL
EFFECTS
OF
Dielectric
relaxation
RATTLING
MOTIONS
Two types of experiment are useful in studying fast rattling motions in polar liquids. The first is to measure the frequency dependence of the dielectric absorption
coefficient
and the second
is to examine
the high-frequency
end of the Cole-Cole diagram. These are confirmed by theoretical calculations illustrated in Figs. 2 and 3. Both diffusive and oscillatory modes affect these diagrams. However, the portions of these curves mainly affected by the oscillatory mode are those associated with the excess absorption and with the high-frequency region of the Cole-Cole diagram. These regions are indicated by a thicker line in Figs. 2 and 3. The conclusions of the present analysis essentially 17,21-23.
Collision-induced
confirm
those reached by previous workers;
cf. refs. 15,
light scattering
The simplest method of studying rattling motions of atoms in liquids is to measure the density dependence of the spectra of depolarized light. The spectral region of interest is that intermediate between the low-frequency component and the high-frequency wing. This suggestion is confirmed by the present calculations and is illustrated in Fig. 4. Although, both the diffusive and oscillatory modes contribute to the spectrum, the latter mode dominates the spectral behavior in the region indicated by a thick line. The theory
further predicts that the spectral band generated by atomic oscillations is weak at low fluid densities where the cage effect is absent. When the density is progressively increased, the oscillatory mode of the fluid becomes better defined and transforms into a genuine lattice mode of the solid. The lowfrequency band is due to the diffusive mode and is not related to the present problem. This is also true for the high-frequency wing due to binary encounters of atoms; cf. refs. 24,33.
Collision-induced The procedure
absorption
suggested to study rattling atomic motions by this tech4n nique involves recording the reduced IR spectrum -wtanh ?I-’ a(w) 13&V and calculating its Fourier transform. The correlation function obtained in this way exhibits oscillations similar to those illustrated in Fig. 1. The oscillations clearly indicate the presence of rattling motions and disappear at sufficiently low densities where the cage effect is absent. It is interesting to note that the rattling motions cannot be seen directly in the spectra. In fact, according to the present calculations, the reduced spectrum is generated by a diffusive mode which creates an inverted lorentzian at low frequencies and
200 TABLE
1
Experiment
A(w)
r
Dielectric relaxation
E,
E.2 + 2
mwat)) s &l(O) 00
-Em
nc
Id2
2
Ea+
dC
G(O),
0
. k=(O)P=(t)) Jdtkx=(o)e=(o)) 0
Light scattering
c
K
_c2
3ficV
IO
I w
0
00
8n
IR absorption
0
0
1.wn4 qExz(0))3
(
l--x mA
+Y mg
>
--2
((J((-J))Zjw
100
Icm-I)
1
tanh
P+
s
do
0
tT(o)Sit)) C~(O,~(O,,
50
25 E’_-ECU
0
Fig. 2. Theoretical dielectric absorption coefficient, we”(w )/nc, of liquid CH,F at T = 133 K. The thickened line indicates the spectral region affected by rattling motions. Fig. 3. The Cole-Cole influenced by rattling
plot, E” = E”(F’), of CH,F motions is very small.
at T=
133
K. The high-frequency
region
by an oscillatory mode which creates a caging peak at high frequencies. The half-width of the inverted lorentzian is of the order of l/7, the correlation time of the interdiffusion current density fluctuations. Theoretical spectra are illustrated in Fig. 5 where the region dominated by the oscillatory mode is indicated by a thick line. The presence of an inverted lorentzian perturbs the absorption band due to the oscillatory mode to an extent which prevents the localization of its peak. Here again, the high-frequency region is of kinetic origin and is not pertinent to the present study; cf. Refs. 25,26,34. DISCUSSION
The existing theories for rattling motion in liquids can be separated, roughly, into five groups. (i) The simplest theories are empirical where the
201
to-3
12 24 36 48 60
72 84 96
I08 I20
w Icm-st
u (cm-‘1
Fig. 4. Theoretical scattered intensity, I(w )Il( Cl), for argon. The curves correspond, from botkom to top, to the following ‘thermodynamic states 150 K and 433 amagat; 120.5 K and 664 amagat; 106.3 K and 723 amagat; 84 and 925 amagat; 84 K and 984 amagat (solid phase). The thick lines indicate the spectral region where the oscillatory mode predominates. Fig_ 5. Theoretical
reduced
absorption
spectra;
Ne-Ar
with Ye,
= 0.989,
‘I’ = 130 K, 0 = - - - -
);Ne-Ar withxAr = 0.989, T- 130 K, p = 280 amagat (The thick lines indicate the spectrai region affected by rattling motions. 597 amagat (
-)
rattling motions are described by appropriate ad hoc formulae, e.g. by an expression indicating a damped oscillation. The theories of Egelstaff and Schofield [1,2] and Leroy et al. [17] are of this type. (ii) In quasi-crystalline theories, a molecule of a liquid is believed to be trapped in a potential wall and its motion perturbed by a random force. The proposals of Rahman et al. [3] , Brot and co-workers [ 18-2001 and Starunov [S] belong to this group. (iii) The third group of theories is of the Langevin type which allows for diffusive and oscillatory motions simultaneously. The itinerant oscillator model proposed by Sears [5] and the parametric Zwanzig-Mori theory by Quentrec and Bezot [21] as well as the present theory are of this type. (iv) The fourth category employs the mode-mode coupling theory. The papers of this group are due to Gijtze et al. [ 37, 381 and consider the effect of collective excitations on the velocity correlation function. (v) Finally, molecular dynamics techniques are described in papers by Rahman [5] and Posch et al. 1351. Moreover, the latter authors also compare the results of molecular dynamics calculations and those derived from Langevin-type theories. The aforementioned reports give a reasonable understanding of each process and show why the experimental detection of rattling motions is so difficult. In future, the collision-induced light-scattering techniques will probably be employed to examine rattling motions in molecular liquids. Only limited modifications to the technique are required as long as the molecules under study are optically isotropic. However, the major problem arising when the experiment refers to optically anisotropic molecules, is the presence of and the interference between two dynamic variables, one describing collective reorientations of the molecules and the second associated with collisioninduced processes. Progress in this field and the future of the Starunov proposal depend on the ability to disentangle these two variables. Finally, study of the collision-induced absorption in atomic fluids will probably be replaced by that of the far-IR absorption in non-polar molecular liquids. A review by Davies 1361 illustrates the potential of this latter technique. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12
P. A. Egelstaff and R. Schofield, Nucl. Sci. Eng., 12 (1962) 260. P. A. Egelstaff and R. Schofield, Adv. Phys., 11 (1962) 203. A. Rahman, K. S. Singwi and A. Sjiilander, Phys. Rev. A, 126 (1962) 997. A. Rahman, J. Chem. Phys., 45 (1966) 2585. V. F. Sears, Proc. R. Sot. London, 86 (1965) 953. P. S. Damle, A. SjGlander and K. S. Singwi, Phys. Rev., 165 (1968) 277. J. R. D. Copley and S. W. Lovesey, Rep. Prog. Phys., 38 (1975) 461. V. S. Staxunov, Sov. Phys. DokI., 8 (1964) 1205. A. I. Prorvin, Opt. Spektrosk., 27 (1968) 46. A. V. Sechkarov and P. N. Nikolaenko, Sov. Whys. Dokl., 13 (1969) 888. B. Simic-Glavaski, D. A. Jackson and J. G. Powles, Phys. L&t., 33A (1970) 328. B. Simic-Glavaski and D. A. Jackson, J. Phys. (Paris), Colloq., 33 (1972) 183. 13 T. Keyes and D. Kivelson, J. Chem. Phys., 56 (1972) 1057. 14 J. Ph. Poley, J. Appl. Sci. Res. B, 4 (1955) 337.
203 15 M. Davies, G. W. F. Pardoe, J. E. Chamberlain and H. A. Gebbie, Trans. Faraday Sot., 64 (1968) 847. 16 Y. Leroy and P. Desplanques, CR. Acad. Sci., 265 (1967) 1111. 17 Y. Leroy, E. Constant, C. Abbar and P. Desplanques, Adv. Mol. Relax. Interact. Processes, 1 (1967-1968) 273. 18 C. Brot, J. Phys. Radium, 28 (1967) 789. 19 B. Lassier and C. Brat, Discuss. Faraday Sot., 48 (1969) 39. 20 C. Brot and I. Darmon, Mol. Phys., 21 (1971) 785. 21 B. Quentrec and P. Bezot, Mol. Phys.. 27 (1974) 879. 22 B. Guillot and S. Bratos, Phys. Rev. A, 16 (1977) 424. 23 A. Gerschel, in A. D. Buckingham, E. Lippert and S. Bratos (Eds.), Organic Liquids; Structure, Dynamics, and Chemical Properties, John Wiley, New York, 1978, p. 195. 24 B. Guillot, S. Bratos and G. Birnbaum, Phys. Rev. A, 22 (1980) 2230. 25 B. Guillot, S. Bratos and G. Birnbaum, Mol. Phys., 44 (1981) 1021. 26 B. Guillot, S. Bratos and G. Birnbaum, Adv. Chem. Phys., 51 (1982) 19. 27 B. J. Berne and R. Pecora, Dynamic Light Scattering, Wiley, New York, 1976. 28 J. Van Kranendonk, Can. J. Phys., 46 (1968) 1173. 29 C. Brot, in M. Davies (Ed.), Dielectric and Related Molecular Processes, Specialist Periodical Reports, The Chemical Society, London, 1975, Vol. II, p. 1. 30 R. Zwanzig, Phys. Rev., 124 (1961) 983. 31 H. Mori, Prog. Theor. Phys., 33 (1965) 423. 32 D. Kivelson and T. Keyes, J. Chem. Phys., 57 (1972) 4599. 33 A. J. C. Ladd, T. A. Litovitz and C. J. Montrose, J. Chem. Phys., 51 (1979) 4242. 34 J. Van Kranendonk, Chem. J. Phys., 46 (1968) 1173. 35 H. Posch, F. Vesely and W. Steele, J. Mol. Phys., 14 (1981) 241. 36 M. Davies, in J. Lascombe (Ed.), Molecular Motions in Liquids, R. Reidel, Dodrecht, 1974, p. 615. 37 W. Gijtze and M. Liicke, Phys. Rev. A, 11 (1975) 2173. 38 J. Bosse, W. Gijtze and A. Zippelius, Phys. Rev. A, 18 (1978) 1214.
of Molecular Structure, 84 (1982) 195-203 Scientific Publishing Company, Amsterdam
THEORETICAL OF RATTLING
S. BRATOS
-
Printed
in The Netherlands
INVESTIGATION AND EXPERLMENTAL MOTIONS IN ATOMIC AND MOLECULAR
DETECTION FLUIDS
and B. GUILLOT
Laboratoire de Physique Thkorique des Liquides Place JLLssieu, 75230 Paris Cedex 05 (France)
*, Uniuersitk
Pierre et Marie Curie, 4
ABSTRACT
The theories
used to investigate
rattling
framework of the Zwanzig-Mori theory motions can be successfully investigated
depolarized experimental
motions
of Brownian by dielectric
light scattering and collision-induced techniques are briefly explored.
in dense fluids are discussed in the motion. It is shown that these relaxation, collision-induced
IR absorption.
The potentials
of these
INTRODUCTION
Considerable interest has been shown over the years in the fast rattling motions in atomic or molecular liquids. These motions consist either of translational oscillations of atoms or of translational-rotational oscillations of molecules in a given site of the liquid. Unfortunately, their experimental detection was found to be unexpectedly difficult. The earliest attempt to solve this problem involved the application of neutron scattering techniques, described in papers by Egelstaff and Schofield [ 1,2] , Rahman et al. [ 3,4] , Sears [ 51 and Damle et al. [ 6] . This method of detection remains controversial; see a recent review by Copley and Lovesey [7 ] . Another method was to use light-scattering techniques and to study the wing of spectral lines of optically anisotropic molecules. These wings were attributed to the collisioninterrupted librations of molecules in liquids. The main exponents of this technique were Starunov [S] , Proti [9], Sechkarov and Nikolaenko [ 101 and Simic-Glavaski et al. [ll, 121. The corresponding theory has remained qualitative, however; for its appreciation, see Keyes and Kivelson [ 131 . The only technique which has so far offered a reliable method of detecting rattling motions in liquids is dielectric relaxation. Here, motions are detectable either by the presence of an excess absorption in the ultra-hertzian spectral range or by a deformation of the Cole--Cole semicircle at high frequencies. The leading workers in this field are Poley [ 141, Davies et al. [ 151, Leroy et al. [lS, 173, Brot and co-workers [18-201, Quentrec and Bezot [21] and Guillot and Bratos [22] . A number of liquids have been successfully examined by this method [23] . *Department
associated
0022-2860/82/0000-0000/$02.75
with the CNRS. o 1982
Elsevier
Scientific
Publishing
Company
196 The purpose of the present paper is to show that two other techniques, collision-induced depolarized light scattering and collision-induced IR absorption, can be employed to study oscillatory motions in atomic liquids; cf. Guillot et al. [ 24-261 . It is helpful to link our discussion with that relative to dielectric relaxation_ AlI three techniques will thus be treated simultaneously here. THEORY
It is convenient to start by presenting the formulae which relate the complex dielectric constant, E(W ), the differential cross-section a2o/&X2, of the depolarized light scattering and the IR absorption coefficient,P(W), to the appropriate correlation functions. Designating the conventional highfrequency dielectric constant by c._,, its static counterpart by eO, the frequency of the incident light by wo, the total dipole moment of the liquid sample by n(t) and its dielectric tensor by g(t) the following equations may be written
(eo+ a=_ 1 -4 (eo -GJ[d4 + 21
s
kb
-=-
a20
awas
1 ~g 4 +F 2n OJc -DD
dt
dc
0
e-‘“t
g
$I(O)gIW
(1)
dt (M(0) M(O)>
e_iOt
w tanh @&)
(2)
kii;O)E.itZ))
jm dt eSiwt <$[R(O), -m
R(t)]
) i-
(3)
These formulae may be found in refs. 27-29. For theoretical convenience the liquid sample subjected to dielectric relaxation is assumed to be a sphere inside a vacuum. The basic dynamic variables entering into this calculation require to be specified in more detail; the following considerations merit attention. (i) The system subjected to a dielectric relaxation experiment is a liquid sample formed by N polar molecules, thus $l = r;E, @Ii where Gi is the dipole moment of the molecule i. (ii) The system subjected to a collision-induced light scattering experiment is a dense rare gas fluid formed by N atoms; its dielectric tensor, 7, is thus given by the sum of $N(N - 1) polarizability tensors, associated wiJh all possible pairs of atoms which may be formed by collisions, 2 = Zy. j =, Zij. (iii) The system subjected to a study by collisioninduced IR absorption is a mixture of rare gas atoms containing N, atoms of species A and N, of species B; its dipole moment fi is thus equal to the sum of dipole moments a, of all possible pairs of atoms of different species which may be formed by collisions, a = zF=~ z,~=T $Iij. The pair dipole ~ij,
197
moment ajj has a particular form in the present case and is often assumed to be proportional to the force i?, between these two atoms, fi, = Cp, where C is a constant. It can then be shown that
Ma) 3=(1-x)
F
iTi-x
i=t
“z” $l i’=
(4b)
1
where cj, Gj are the velocities of the atoms i, j,, m A and ll~a the masses of atoms of the species A and B, x the number density of atoms of the species A and 3 the interdiffusion current density fluctuation [25, 261 . It results from this analysis that three dynamic variables, the dipole moment G(t), the dielectric tensor 2 and the interdiffusion current density fluctuation ‘j form the basis of the theoretical description. They are all considered to be classical to a reasonable degree of approximation. Subsequent calculation is based on the Zwanzig-Mori version of the generalized Langevin theory of Brownian motion [30-321. According to this theory, the normalized correlation function G(t) = (A(O)A(t))(A(O)A(O))-' of a dynamic variable A(t)obeys the equation
‘9
=if2G(t)
-
1ds K(s) 0
G(1 -s)
(5)
where 52 is the frequency and h’(s) the memory function; Zwanzig and Mori have shown that this equation is in fact an esact equation of motion. It can be solved by successive approximations, either by truncating the continuous fraction formula established by Mori or by applying the Kivelson-Keyes procedure [32] . The two methods are intimately related and converge, in principle, to an exact solution. In the present problem, the variable A (t) represents either a(t), ES=(t) or j(t); the corresponding Langevin equations were solved by the Kivelson-Keyes method in its three-variable approsimation. The elements of the transport matrix r = --it2 + lim, _ (, j; ds eBzs K(s) which forms part of the theory were expressed in terms of the quantities T = JFds G(s), G”(O), G”“(0). In turn, these latter quantities were estimated, non-empirically, by applying a lattice-gas model of the liquid state. For details of this calculation see refs. 22,24-26. The theory outlined above predicts the existence of two modes of motion in liquids. The first mode is a dissipative mode due to translational diffusion of rare gas atoms or to rotational diffusion of polar molecules. The second mode is a doubly degenerate oscillatory mode associated with translational or rotational rattling motions. These two modes correspond to the real and complexconjugate roots, respectively, of the transport matrix r. The correlation functions have the following characteristic form
198
G(t) = cl emAlt + c2e-ht
cos h3t + c3e-kt
sin X,t
(6)
where XL, A z f C43, c 1, c2, c3 are known functions of the parameters 7, G”( 0), G”“(0) given above. A representative illustration of G(t) is shown in Fig. 1
where the rattling motions are clearly recognizable. As expected, the frequency, h 3, of these oscillations depends on intermolecular forces which determine the value of 7, G”(O), G”“(O)*. The notion of a cage is not introduced explicitly. The frequency dependent quantities E(W ), #CJ/&X2z~(~ ) are obtainable by substituting the correlation function G(t), with A = M, tzXZ, or 3, into eqns. (l)-(3). The results may be expressed in a compact form by denoting the dielectric absorption coefficient, w E”/IZC,the differential scattering crosssection, a’o/aoaQ, and the IR absorption coefficient, Q(W), by a unique symbol I(w ). This leads to
r(d=AWl((1-g+
w29[1-
(l-$)/(l-(&)ll)
(7)
The function A (w ) varies from one technique to another as do the parameters 7, o i and w: ; they are given in Table 1. The fact that the dielectric absorption coefficient varies as w * at small values of w whereas the IR absorption coefficient varies as w4 is due to the particular nature of the collision-induced dipole moment sljj.
1
0
9
0.25
0.50
0.75
1.0
1.25
x 10-Q s Fig. 1. Theoretical correlation function (M(O)M(t))/(M(O)M(O)) Ne-Ar withxAr = 0.989, T = 90 K, P = 728 amagat ( T = 130 K, p = 280 amagat (------).
in dense atomic mixtures; ): Ne-Ar with xAr = 0.989,
*For dielectric relaxation tion is purely kinetic.
of the autocorrelation
in polar
liquids
the second
moment
func-
199 SF-%CTRAL
EFFECTS
OF
Dielectric
relaxation
RATTLING
MOTIONS
Two types of experiment are useful in studying fast rattling motions in polar liquids. The first is to measure the frequency dependence of the dielectric absorption
coefficient
and the second
is to examine
the high-frequency
end of the Cole-Cole diagram. These are confirmed by theoretical calculations illustrated in Figs. 2 and 3. Both diffusive and oscillatory modes affect these diagrams. However, the portions of these curves mainly affected by the oscillatory mode are those associated with the excess absorption and with the high-frequency region of the Cole-Cole diagram. These regions are indicated by a thicker line in Figs. 2 and 3. The conclusions of the present analysis essentially 17,21-23.
Collision-induced
confirm
those reached by previous workers;
cf. refs. 15,
light scattering
The simplest method of studying rattling motions of atoms in liquids is to measure the density dependence of the spectra of depolarized light. The spectral region of interest is that intermediate between the low-frequency component and the high-frequency wing. This suggestion is confirmed by the present calculations and is illustrated in Fig. 4. Although, both the diffusive and oscillatory modes contribute to the spectrum, the latter mode dominates the spectral behavior in the region indicated by a thick line. The theory
further predicts that the spectral band generated by atomic oscillations is weak at low fluid densities where the cage effect is absent. When the density is progressively increased, the oscillatory mode of the fluid becomes better defined and transforms into a genuine lattice mode of the solid. The lowfrequency band is due to the diffusive mode and is not related to the present problem. This is also true for the high-frequency wing due to binary encounters of atoms; cf. refs. 24,33.
Collision-induced The procedure
absorption
suggested to study rattling atomic motions by this tech4n nique involves recording the reduced IR spectrum -wtanh ?I-’ a(w) 13&V and calculating its Fourier transform. The correlation function obtained in this way exhibits oscillations similar to those illustrated in Fig. 1. The oscillations clearly indicate the presence of rattling motions and disappear at sufficiently low densities where the cage effect is absent. It is interesting to note that the rattling motions cannot be seen directly in the spectra. In fact, according to the present calculations, the reduced spectrum is generated by a diffusive mode which creates an inverted lorentzian at low frequencies and
200 TABLE
1
Experiment
A(w)
r
Dielectric relaxation
E,
E.2 + 2
mwat)) s &l(O) 00
-Em
nc
Id2
2
Ea+
dC
G(O),
0
. k=(O)P=(t)) Jdtkx=(o)e=(o)) 0
Light scattering
c
K
_c2
3ficV
IO
I w
0
00
8n
IR absorption
0
0
1.wn4 qExz(0))3
(
l--x mA
+Y mg
>
--2
((J((-J))Zjw
100
Icm-I)
1
tanh
P+
s
do
0
tT(o)Sit)) C~(O,~(O,,
50
25 E’_-ECU
0
Fig. 2. Theoretical dielectric absorption coefficient, we”(w )/nc, of liquid CH,F at T = 133 K. The thickened line indicates the spectral region affected by rattling motions. Fig. 3. The Cole-Cole influenced by rattling
plot, E” = E”(F’), of CH,F motions is very small.
at T=
133
K. The high-frequency
region
by an oscillatory mode which creates a caging peak at high frequencies. The half-width of the inverted lorentzian is of the order of l/7, the correlation time of the interdiffusion current density fluctuations. Theoretical spectra are illustrated in Fig. 5 where the region dominated by the oscillatory mode is indicated by a thick line. The presence of an inverted lorentzian perturbs the absorption band due to the oscillatory mode to an extent which prevents the localization of its peak. Here again, the high-frequency region is of kinetic origin and is not pertinent to the present study; cf. Refs. 25,26,34. DISCUSSION
The existing theories for rattling motion in liquids can be separated, roughly, into five groups. (i) The simplest theories are empirical where the
201
to-3
12 24 36 48 60
72 84 96
I08 I20
w Icm-st
u (cm-‘1
Fig. 4. Theoretical scattered intensity, I(w )Il( Cl), for argon. The curves correspond, from botkom to top, to the following ‘thermodynamic states 150 K and 433 amagat; 120.5 K and 664 amagat; 106.3 K and 723 amagat; 84 and 925 amagat; 84 K and 984 amagat (solid phase). The thick lines indicate the spectral region where the oscillatory mode predominates. Fig_ 5. Theoretical
reduced
absorption
spectra;
Ne-Ar
with Ye,
= 0.989,
‘I’ = 130 K, 0 = - - - -
);Ne-Ar withxAr = 0.989, T- 130 K, p = 280 amagat (The thick lines indicate the spectrai region affected by rattling motions. 597 amagat (
-)
rattling motions are described by appropriate ad hoc formulae, e.g. by an expression indicating a damped oscillation. The theories of Egelstaff and Schofield [1,2] and Leroy et al. [17] are of this type. (ii) In quasi-crystalline theories, a molecule of a liquid is believed to be trapped in a potential wall and its motion perturbed by a random force. The proposals of Rahman et al. [3] , Brot and co-workers [ 18-2001 and Starunov [S] belong to this group. (iii) The third group of theories is of the Langevin type which allows for diffusive and oscillatory motions simultaneously. The itinerant oscillator model proposed by Sears [5] and the parametric Zwanzig-Mori theory by Quentrec and Bezot [21] as well as the present theory are of this type. (iv) The fourth category employs the mode-mode coupling theory. The papers of this group are due to Gijtze et al. [ 37, 381 and consider the effect of collective excitations on the velocity correlation function. (v) Finally, molecular dynamics techniques are described in papers by Rahman [5] and Posch et al. 1351. Moreover, the latter authors also compare the results of molecular dynamics calculations and those derived from Langevin-type theories. The aforementioned reports give a reasonable understanding of each process and show why the experimental detection of rattling motions is so difficult. In future, the collision-induced light-scattering techniques will probably be employed to examine rattling motions in molecular liquids. Only limited modifications to the technique are required as long as the molecules under study are optically isotropic. However, the major problem arising when the experiment refers to optically anisotropic molecules, is the presence of and the interference between two dynamic variables, one describing collective reorientations of the molecules and the second associated with collisioninduced processes. Progress in this field and the future of the Starunov proposal depend on the ability to disentangle these two variables. Finally, study of the collision-induced absorption in atomic fluids will probably be replaced by that of the far-IR absorption in non-polar molecular liquids. A review by Davies 1361 illustrates the potential of this latter technique. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12
P. A. Egelstaff and R. Schofield, Nucl. Sci. Eng., 12 (1962) 260. P. A. Egelstaff and R. Schofield, Adv. Phys., 11 (1962) 203. A. Rahman, K. S. Singwi and A. Sjiilander, Phys. Rev. A, 126 (1962) 997. A. Rahman, J. Chem. Phys., 45 (1966) 2585. V. F. Sears, Proc. R. Sot. London, 86 (1965) 953. P. S. Damle, A. SjGlander and K. S. Singwi, Phys. Rev., 165 (1968) 277. J. R. D. Copley and S. W. Lovesey, Rep. Prog. Phys., 38 (1975) 461. V. S. Staxunov, Sov. Phys. DokI., 8 (1964) 1205. A. I. Prorvin, Opt. Spektrosk., 27 (1968) 46. A. V. Sechkarov and P. N. Nikolaenko, Sov. Whys. Dokl., 13 (1969) 888. B. Simic-Glavaski, D. A. Jackson and J. G. Powles, Phys. L&t., 33A (1970) 328. B. Simic-Glavaski and D. A. Jackson, J. Phys. (Paris), Colloq., 33 (1972) 183. 13 T. Keyes and D. Kivelson, J. Chem. Phys., 56 (1972) 1057. 14 J. Ph. Poley, J. Appl. Sci. Res. B, 4 (1955) 337.
203 15 M. Davies, G. W. F. Pardoe, J. E. Chamberlain and H. A. Gebbie, Trans. Faraday Sot., 64 (1968) 847. 16 Y. Leroy and P. Desplanques, CR. Acad. Sci., 265 (1967) 1111. 17 Y. Leroy, E. Constant, C. Abbar and P. Desplanques, Adv. Mol. Relax. Interact. Processes, 1 (1967-1968) 273. 18 C. Brot, J. Phys. Radium, 28 (1967) 789. 19 B. Lassier and C. Brat, Discuss. Faraday Sot., 48 (1969) 39. 20 C. Brot and I. Darmon, Mol. Phys., 21 (1971) 785. 21 B. Quentrec and P. Bezot, Mol. Phys.. 27 (1974) 879. 22 B. Guillot and S. Bratos, Phys. Rev. A, 16 (1977) 424. 23 A. Gerschel, in A. D. Buckingham, E. Lippert and S. Bratos (Eds.), Organic Liquids; Structure, Dynamics, and Chemical Properties, John Wiley, New York, 1978, p. 195. 24 B. Guillot, S. Bratos and G. Birnbaum, Phys. Rev. A, 22 (1980) 2230. 25 B. Guillot, S. Bratos and G. Birnbaum, Mol. Phys., 44 (1981) 1021. 26 B. Guillot, S. Bratos and G. Birnbaum, Adv. Chem. Phys., 51 (1982) 19. 27 B. J. Berne and R. Pecora, Dynamic Light Scattering, Wiley, New York, 1976. 28 J. Van Kranendonk, Can. J. Phys., 46 (1968) 1173. 29 C. Brot, in M. Davies (Ed.), Dielectric and Related Molecular Processes, Specialist Periodical Reports, The Chemical Society, London, 1975, Vol. II, p. 1. 30 R. Zwanzig, Phys. Rev., 124 (1961) 983. 31 H. Mori, Prog. Theor. Phys., 33 (1965) 423. 32 D. Kivelson and T. Keyes, J. Chem. Phys., 57 (1972) 4599. 33 A. J. C. Ladd, T. A. Litovitz and C. J. Montrose, J. Chem. Phys., 51 (1979) 4242. 34 J. Van Kranendonk, Chem. J. Phys., 46 (1968) 1173. 35 H. Posch, F. Vesely and W. Steele, J. Mol. Phys., 14 (1981) 241. 36 M. Davies, in J. Lascombe (Ed.), Molecular Motions in Liquids, R. Reidel, Dodrecht, 1974, p. 615. 37 W. Gijtze and M. Liicke, Phys. Rev. A, 11 (1975) 2173. 38 J. Bosse, W. Gijtze and A. Zippelius, Phys. Rev. A, 18 (1978) 1214.