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Spectral distribution of scattered light in polyatomic gases
Physica A 197 (1993) 352-363 North-Holland SDZ: 037%4371(93)E0076-Q
Spectral distribution polyatomic gases
of scattered
light in
W. Marques Jr Znstitut fiir Thermodynamik
und Reaktionstechnik,
TV Berlin,
loo0 Berlin 12, Germany
G.M. Kremer Departamento de Fisica, Vniversidade 81531-970 Curitiba, Brazil
Federal do Paranri, Caixa Postal 19091,
Received 5 November 1992 Revised manuscript received 2 March 1993
The spectral distribution of light scattered by density fluctuations is calculated for a polyatomic gas. The usual hydrodynamic equations are supplemented with a relaxation equation for the dynamic pressure. As an application the spectra of light scattered from CH, is determined.
1. Introduction
In this work we are concerned with the calculation of the spectral distribution of the scattered light by density fluctuations in polyatomic gases consisting of spherical molecules with rotational energy. For simple fluids, it is natural to attempt such a calculation by using the conventional hydrodynamic equations [l]. However, it is well known from light scattering experiments [2] that for molecular fluids these equations do not correctly describe the spectrum of the scattered light. Modifications of the hydrodynamic equations [3-61 have usually been carried out to take into account the internal relaxation processes which occur in molecular fluids. Here, we propose a hydrodynamical model where the usual hydrodynamic equations are supplemented with a relaxation equation for the non-equilibrium part of the pressure, customarily called dynamic pressure. The derivation of the hydrodynamic equations are based on the field equations of a polyatomic gas of rough spherical molecules [7]. The intensity of the scattered light follows from the knowledge of the 0378-4371/93/$06.00 0
1993 - Elsevier Science Publishers B.V. All rights reserved
W. Marques Jr.,
G. M. Kremer
I Scattered light in polyatomic gases
353
density-density correlation function which can be obtained from the linearized hydrodynamic equations. Moreover, the calculation of the correlation function is also based on Onsager’s regression hypothesis which relates the regression to equilibrium of spontaneous microscopic fluctuations to the relaxation of nonequilibrium macroscopic processes. An exact expression for the spectral distribution of the scattered light is derived and applied to methane (CH,). The results are compared with the usual hydrodynamic description [l] and with the translational hydrodynamic theory from Desay and Kapral [5], which includes the effects of diffusion on the internal energy relaxation time. Cartesian notation for tensors is used and angle brackets around two indices denote the symmetric and traceless part of a tensor.
2. The intensity of the scattered light A plane polarized light of intensity I,, , angular frequency o *, wave vector k’ and polarization ni impinges upon a fluid with dielectric constant E(X, t). The intensity of the light, of angular frequency w”, wave vector kS and polarization ns, scattered by the fluid through an angle 0 and observed at a detector located at a distance R from the scattered volume, is given by [8]
Z(q,o,R)=Z,.,
(4” 167r2R2c4
(n’ - a2GL
0) ,
where c is the speed of light in vacuum, o = WI - W’ the shift in angular frequency and q = k’ - kS the scattering vector. The magnitude of the scattering vector is q =
2k’ sin(8/2) .
In eq. (1) the intensity structure factor
(2)
of the scattered
light is related
to the dynamic
m
S(q, w) = &
j dtexp(-iot) --m
(6e*(q, 0) 6e( q, t)) ,
(3)
which is the Fourier transform of the auto-correlation function of the dielectric constant fluctuations. In the above expression 6e( q, t) is the spatial Fourier transform of the dielectric constant fluctuation SE(X, t), defined by
354
W. Marques
6e( 4, t) =
I V
Jr., G. M. Kremer
dx exp(iq . x)
&(x,
I Scattered
t)
light in polyatomic
gases
.
(4)
Direct calculation of &(x, t) is avoided by assuming that fluctuations in the dielectric constant are caused by fluctuations in the mass density and in the temperature. In gases near one atmosphere the fluctuations in the mass density Q(X, t) completely predominate and eq. (3) becomes
dtexp(-i4 @e*(q, 0) Se(q, t)) ,
(5)
--P
where
6e( q, t) is the spatial Fourier
transform
of the density
fluctuation
aeC-5 t). 3. The hydrodynamical
model
The hydrodynamic equations that we adopt to describe the system are based on the balance equations of mass density o, momentum eui, internal energy es and on the relaxation equation for the dynamic pressure a, which is essentially the difference between the translational and the rotational temperatures. For a polyatomic gas of rough spherical molecules they read [7]
ae
aC?ui
at+-=O, axi @vi
at
+
&
(C?uiVj
+
Pij)
=
0
(7)
7
I
(8)
$f+-$-(esui+q:)+p,z=O, I
1
%
+
-&I [WUj+ (y -
l)(qj - hi)] + (y - l)p, 2
= --w,w ) I
where pjj is the pressure tensor, E the specific internal energy, 4: = qj + hi the total heat flux, qj the translational heat flux, hj the rotational heat flux, y = 4/3 the ratio of the specific heat capacities and
(10) is the relaxation
frequency
of the dynamic pressure.
W. Marques Jr.,
G. M. Kremer
I Scattered light in polyatomic gases
355
In eq, (10) m, a and K = 4Z/ma* are, respectively, the mass, the diameter and the dimensionless moment of inertia of the spherical molecule. The value of K may range from zero to 2/3 corresponding to a concentration of mass at the center of the molecule and a uniform distribution of mass on the surface of the spherical molecule, respectively. The Boltzmann constant is denoted by k. The system of balance equations (6)-(9) is closed by considering pjj. qT , qi , hi and E as constitutive quantities which are related to the fields o, ui, T and w through constitutive relations. The constitutive relations adopted here are k &=3;T=c,T,
pii=
(
(11)
ekT+w
>
(12)
$-2~2, 1)
(13)
(15) where
(16) h = PC, 650~~ + 2263~~ + 1387~ + 222 -iiF
102K3
+
lolK*
+75K
+
12
’
M = --3 P 650 ~~+287~*+163~+78 10 e AT=
;
102K3+101K2+75K+12 26~~ -I-51~~ f 31~ f 6
pc,
MT=E
102K3 + lOlK*
CL 2
’
+ 75K + 12 ’
26~~+25~*+19~+6
k? 102K3+101K2+75K+12
’
247~~ + 153K + 18 102K3 + lOlK*
+ 75K + 12 ’
169~~ + 156~ +
,dE
5 o
102K3
+
lolK*
+
75K
36 +
12
’
(17)
(18)
(19)
(20)
(21) (22)
W. Marques Jr.,
356
G. M. Kremer
I Scattered light in polyatomic gases
The constitutive equations above were obtained from the field equations of ref. [7] through an iterate scheme by considering Q, ui, T and a as variables. In these equations c, is the specific heat at constant volume, p the shear viscosity, A the thermal conductivity, AT the translational thermal conductivity, AR the rotational thermal conductivity and the coefficients M, MT and MR do not have proper names. We are interested in a linearized theory where the following conditions hold:
e=eo+el
y
T = To + T, ,
ui=u;,
and
w = w,.
(23)
Insertion of eqs. (11)-(E) and (23) into the balance equations (6)-(9) leads to a system of linearized equations for the macroscopic fluctuations el, II,‘, T, and wi. This system, in terms of the spatial Fourier transform of the fluctuations. can be written as follows:
aw(a;7t) +M(q)V(q,t)=O,
(24)
where /
elk2 4
wq, f) = u,(cz,t) T,(q, 4
(25)
w,(q, t> and
e.
0
(uod QOY
M(q) = 0 0
0
WI2 (y-1)
(Y-P, Y
- ;
(uoq)2
DTq2 cod
(Y - l)eoc,4q2
0
- $
%q2 each - 1) Dcq2 + 0, (26)
In (25) and (26) we have introduced the longitudinal velocity u,( q, t> = -iq,ul( q, t), the adiabatic speed of sound uo, the thermal expansion coefficient (Y, the longitudinal kinematic viscosity D,, the thermal diffusivity D, and the quantities D,, D, and D,. They are defined as
W. Marques
Jr., G.M.
D*=(y-l)&),
D,=
Kremer
I Scattered
(G - 0
eoc,
light in polyatomic
gases
2 D, = (y - l)(l@ + M;) .
357
(28)
We close this section by remembering that the eigenvalues of the matrix M( 4) define the so-called hydrodynamic modes. We notice that hydrodynamics is valid only for phenomena varying slowly in space, this means that in the spatial Fourier transform, the only coefficients that are relevant correspond to small values of q. The eigenvalues of the matrix M(q) valid to order q2 are A, = +iv,q
AI, = -
- rq*
,
(29)
‘,D,q* ,
(30)
(31) where
(32) is the acoustic attenuation coefficient. A, represents two longitudinal propagating sound modes that are damped. A,, and A, correspond to a longitudinal heat mode and to a internal relaxation mode, respectively. A,, and A, are purely diffusive, i.e., they do not propagate.
4. The dynamic structure
factor
Now we apply the results derived in the previous section to calculate the dynamic structure factor. By taking the Laplace transform of eq. (24) it follows that (33)
where
358
W. Marques
Jr., G.M.
Kremer
I Scattered
light in polyatomic
gases
m
@(q,s) =
I
dt exp(-st)
q( q, t)
(34)
0
and I is the identity matrix. If we solve the longitudinal system of eqs. (33) for &( q, s), multiply the result by @T(q, 0) and perform the ensemble average over the initial states of the system, we get G(q, since
4 eT(4,W
the ensemble
(elk
= $j$ averages
0) eT(c 0))
(u,(q,O)eT(q,O)),
(35)
, (T,(q,O)e~(q,O))
h(q,O)
eT(c 0)) vanish, indicating the non-correlation
quantities
P(S) and Q(S) are given by
between them
P(S) = s3 + s2(D,q2 + D,q2 + D,q2 f mr) + s{ D,q2D,q2 + D,q2D,q2
- D,q2D,q2
+ D,q2D,q2Dcq2 + [(Y - l)hl(W2
+ D,q2Dcq2
+ (D,q2 + D,q2)w, + Z[(Y - W~l(~od2~
- D,q2D,q2DBq2
+ D,q2D,q2w,
+ D,q2 - %q2 - D,q2 + dh,d2
Q(s) = s4 + s3(D,q2 + D,q2 + D,q2 + wr) + s2{ D,q2D,q2 + D,q2Dcq2 - D,q2Dsq2
,
(36)
+ D,q2Dcq2
+ (D,q2 + D,q2)w, + IPY - 1) h4uod2~
+ s{ D,q2D,q2D,q2
- D,q2DAq2D,q2
- [(Y - l)bl(D,q2
+ D,q2)(uoq)2 + (D,q2 + D,q2 + 4(uoq)21
+ (W)(D,q2D,q2
and The
- D,q’D,q*
+ D,q2D,q20,
+ D,q24(~oq)2
.
(37)
From the knowledge of the density-density ’ correlation function (&( q, s) eT( q, 0)), the dynamic structure factor S( q, 0) follows from [8]
(38)
In our case the dynamic structure factor can be written as follows: 2
S(q)
N,(w)
D,(o)
+
N2(w)
D;(w) + D;(w) where
D2(w)
’
(39)
W. Marques Jr.,
G.M.
Kremer
4q D*q %I + -__ D,q %I ---_ uo uo uo
359
I Scattered light in polyatomic gases
UO
w,
uo
uo4
(41)
--_D*q %I+ uo uo
(4q
+ i
---__
_ (_D,q+D,q uo uo
w,+2y-1 > uo4
D,q D,q + __ W
D,q
I
Y 0, uo4
1
(42)
’
D2(~,-cy~;(~+~+!$+~) 0 +
0
w ( uoq
&q --_-___ I
--__y-1 Y
In eq. (39) S( 4) = (e,(
uo
4q
D,q uo
Dvq
uo
D,q+D,q (
UO
4,
UO
>
“0
D,q “0
+D,q+D,q __ UO
D,q
+
~ w, uo
Dvq ___
uo
uo4
0) e:( 4, 0)) is the static structure
uo
1 .
D,q uo
w, uo4
(43)
factor.
5. Application
In order to apply the results obtained in the last section, one should observe two conditions that must be followed by the polyatomic gas, namely (i) the molecules of the gas must be spherically symmetrical and (ii) the ratio of the specific heat capacities y must be close to 4/3. These conditions are approximately satisfied by methane (CH,) at ordinary temperatures. In monatomic gases, the hydrodynamic region is determined by the conditions
360
W. Marques
Jr., G. M. Kremer
/ Scattered
light in polyatomic
gases
2 w J -3 -ey, uo4
(44)
where y is a measure of the ratio of the wavelength of the observed fluctuation with the collision mean free path and x is the reduced frequency. For usual hydrodynamics to be valid in molecular gases, we need not only y s 1 and x 4 y but also [9]
(45) where the internal relaxation number z is the ratio of the elastic collision frequency W, = eou~/3/~o to the internal relaxation frequency w,. If only the conditions given in (44) are satisfied, the system is hydrodynamic only with respect to the translational degrees of freedom; the internal degrees of freedom respond too slowly to become thermalized during the period of the wave. Using the characteristic data for methane [lo] we get that the internal relaxation number z is equal to 6.85 at 293 K and 1.013 bar, and we conclude that for the usual hydrodynamic theory the condition x/y <<1 /z is much more restrictive than x 4 y. Figs. 1 and 2 show the scattered light spectrum from CH, gas (293 K and 1.013 bar) for y values equal to 18.27 and 4.46, respectively. In these figures 1.25
,
0.4
0.6
0.6
Reduced Frequency
Fig. 1. The scattered Rayleigh peak.
light spectrum
from
CH,
for y = 18.27.
The curves
are normalized
at the
W. Marques
Jr., G.M.
Kremer
I Scattered
light in polyatomic
gases
361
Reduced Frequency
Fig. 2. The scattered Rayleigh peak.
light spectrum from CH, for y = 4.46. The curves are normalized
at the
the solid line represents the spectral prediction derived from the hydrodynamical model proposed in this paper, the dashed line represents the prediction of the usual hydrodynamic theory and the points represent the prediction of the translational hydrodynamic theory from Desai and Kapral. Fig. 1 shows a complete agreement between our hydrodynamical model and the translational hydrodynamic theory for y = 18.27. However, the usual hydrodynamic theory predicts a less-intense and less-displaced Brillouin peak compared to the other two theories. For y = 4.46, as showed in fig. 2, the results of our hydrodynamical model and the translational theory are nearly identical, but the usual hydrodynamic prediction is in complete disagreement. This is to be expected since we are in a region where the condition (45) does not hold and the usual hydrodynamic theory is no longer applicable. Comparisons of experimental results with the translational hydrodynamic theory for y = 18.27, 4.46 and 2.70 were done by Hammond and Wiggins [6]. For the first two values of y the agreement between the experimental results and the translational theory was very good, but for y = 2.70 the translational hydrodynamic theory was not able to predict the spectrum of scattered light from methane. The results of our hydrodynamical model and the translational hydrodynamic theory are compared in fig. 3 for y = 2.70. In this case the theoretical results are also almost identical and we conclude that the hydrodynamical
362
W. Marques Jr.,
G.M.
Kremer
/ Scattered light in polyatomic gases
0.6
0.8
Reduced Frequency Fig. 3. The scattered Rayleigh peak.
light
spectrum
from
CH,
for y = 2.70.
The curves
are normalized
at the
model proposed in this work can be used to analyze those light scattering experiments from methane in the region y > 4 and x/y < 1.
6. Final remarks In this paper we have considered a six-field theory with 9, ui, T and a as basic fields. One could ask why there is a relaxation equation for the dynamic pressure but none for the pressure deviator nor for the translational and rotational heat fluxes. The reason is that for methane the dimensionless moment of inertia is small (K ~0.048) so that the relaxation of the dynamic pressure is more important since it goes much more slowly than the relaxation of the pressure deviator or the relaxations of the translational and rotational heat fluxes. On the other hand, a five field theory with e, ui and T as basic fields does not take into account the exchange of energy between the translational and rotational degrees of freedom. In that theory w becomes a constitutive quantity. The constitutive relation for a can be obtained from an iterate scheme by putting on the left-hand side of eq. (9) the equilibrium values a = 0, pCij, = 0, qi = hi = 0 and getting on the right-hand side
W. Marques Jr.,
G. M. Kremer
I Scattered light in polyatomic gases
363
au,
m=-qdx,’
(46)
where
,=(V-l)QkT w,
m
is the coefficient of volume viscosity. By eq. (47) we conclude that the relaxation frequency of the dynamic pressure is proportional to the inverse of the volume viscosity.
References [l] R.D. Mountain, Rev. Mod. Phys. 38 (1966) 205. [2] G.I.A. Stegeman, W.S. Goumal, V Volterra and B.P. Stoicheff, J. Acoust. Sot. Am. 49 (1971) 979. [3] R.D. Mountain, J. Res. Natl. Bur. Stand. A 70 (1966) 207. [4] R.D. Mountain, J. Res. Natl. Bur. Stand. A 72 (1968) 95. [5] R.C. Desai and R. Kapral, Phys. Rev. A 6 (1972) 2377. [6] C.M. Hammond and T.A. Wiggins, J. Chem. Phys. 65 (1976) 2788. [7] G.M. Kremer, Rev. Bras. Fis. (Brazil) 17 (1987) 370. [8] B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976). [9] C.D. Boley, R.C. Desai and G. Tenti, Can. J. Phys. 50 (1972) 2158. [lo] M.G. Rodbard and G.M. Kremer, Phys. Fluids A 2 (1990) 1269.
Spectral distribution polyatomic gases
of scattered
light in
W. Marques Jr Znstitut fiir Thermodynamik
und Reaktionstechnik,
TV Berlin,
loo0 Berlin 12, Germany
G.M. Kremer Departamento de Fisica, Vniversidade 81531-970 Curitiba, Brazil
Federal do Paranri, Caixa Postal 19091,
Received 5 November 1992 Revised manuscript received 2 March 1993
The spectral distribution of light scattered by density fluctuations is calculated for a polyatomic gas. The usual hydrodynamic equations are supplemented with a relaxation equation for the dynamic pressure. As an application the spectra of light scattered from CH, is determined.
1. Introduction
In this work we are concerned with the calculation of the spectral distribution of the scattered light by density fluctuations in polyatomic gases consisting of spherical molecules with rotational energy. For simple fluids, it is natural to attempt such a calculation by using the conventional hydrodynamic equations [l]. However, it is well known from light scattering experiments [2] that for molecular fluids these equations do not correctly describe the spectrum of the scattered light. Modifications of the hydrodynamic equations [3-61 have usually been carried out to take into account the internal relaxation processes which occur in molecular fluids. Here, we propose a hydrodynamical model where the usual hydrodynamic equations are supplemented with a relaxation equation for the non-equilibrium part of the pressure, customarily called dynamic pressure. The derivation of the hydrodynamic equations are based on the field equations of a polyatomic gas of rough spherical molecules [7]. The intensity of the scattered light follows from the knowledge of the 0378-4371/93/$06.00 0
1993 - Elsevier Science Publishers B.V. All rights reserved
W. Marques Jr.,
G. M. Kremer
I Scattered light in polyatomic gases
353
density-density correlation function which can be obtained from the linearized hydrodynamic equations. Moreover, the calculation of the correlation function is also based on Onsager’s regression hypothesis which relates the regression to equilibrium of spontaneous microscopic fluctuations to the relaxation of nonequilibrium macroscopic processes. An exact expression for the spectral distribution of the scattered light is derived and applied to methane (CH,). The results are compared with the usual hydrodynamic description [l] and with the translational hydrodynamic theory from Desay and Kapral [5], which includes the effects of diffusion on the internal energy relaxation time. Cartesian notation for tensors is used and angle brackets around two indices denote the symmetric and traceless part of a tensor.
2. The intensity of the scattered light A plane polarized light of intensity I,, , angular frequency o *, wave vector k’ and polarization ni impinges upon a fluid with dielectric constant E(X, t). The intensity of the light, of angular frequency w”, wave vector kS and polarization ns, scattered by the fluid through an angle 0 and observed at a detector located at a distance R from the scattered volume, is given by [8]
Z(q,o,R)=Z,.,
(4” 167r2R2c4
(n’ - a2GL
0) ,
where c is the speed of light in vacuum, o = WI - W’ the shift in angular frequency and q = k’ - kS the scattering vector. The magnitude of the scattering vector is q =
2k’ sin(8/2) .
In eq. (1) the intensity structure factor
(2)
of the scattered
light is related
to the dynamic
m
S(q, w) = &
j dtexp(-iot) --m
(6e*(q, 0) 6e( q, t)) ,
(3)
which is the Fourier transform of the auto-correlation function of the dielectric constant fluctuations. In the above expression 6e( q, t) is the spatial Fourier transform of the dielectric constant fluctuation SE(X, t), defined by
354
W. Marques
6e( 4, t) =
I V
Jr., G. M. Kremer
dx exp(iq . x)
&(x,
I Scattered
t)
light in polyatomic
gases
.
(4)
Direct calculation of &(x, t) is avoided by assuming that fluctuations in the dielectric constant are caused by fluctuations in the mass density and in the temperature. In gases near one atmosphere the fluctuations in the mass density Q(X, t) completely predominate and eq. (3) becomes
dtexp(-i4 @e*(q, 0) Se(q, t)) ,
(5)
--P
where
6e( q, t) is the spatial Fourier
transform
of the density
fluctuation
aeC-5 t). 3. The hydrodynamical
model
The hydrodynamic equations that we adopt to describe the system are based on the balance equations of mass density o, momentum eui, internal energy es and on the relaxation equation for the dynamic pressure a, which is essentially the difference between the translational and the rotational temperatures. For a polyatomic gas of rough spherical molecules they read [7]
ae
aC?ui
at+-=O, axi @vi
at
+
&
(C?uiVj
+
Pij)
=
0
(7)
7
I
(8)
$f+-$-(esui+q:)+p,z=O, I
1
%
+
-&I [WUj+ (y -
l)(qj - hi)] + (y - l)p, 2
= --w,w ) I
where pjj is the pressure tensor, E the specific internal energy, 4: = qj + hi the total heat flux, qj the translational heat flux, hj the rotational heat flux, y = 4/3 the ratio of the specific heat capacities and
(10) is the relaxation
frequency
of the dynamic pressure.
W. Marques Jr.,
G. M. Kremer
I Scattered light in polyatomic gases
355
In eq, (10) m, a and K = 4Z/ma* are, respectively, the mass, the diameter and the dimensionless moment of inertia of the spherical molecule. The value of K may range from zero to 2/3 corresponding to a concentration of mass at the center of the molecule and a uniform distribution of mass on the surface of the spherical molecule, respectively. The Boltzmann constant is denoted by k. The system of balance equations (6)-(9) is closed by considering pjj. qT , qi , hi and E as constitutive quantities which are related to the fields o, ui, T and w through constitutive relations. The constitutive relations adopted here are k &=3;T=c,T,
pii=
(
(11)
ekT+w
>
(12)
$-2~2, 1)
(13)
(15) where
(16) h = PC, 650~~ + 2263~~ + 1387~ + 222 -iiF
102K3
+
lolK*
+75K
+
12
’
M = --3 P 650 ~~+287~*+163~+78 10 e AT=
;
102K3+101K2+75K+12 26~~ -I-51~~ f 31~ f 6
pc,
MT=E
102K3 + lOlK*
CL 2
’
+ 75K + 12 ’
26~~+25~*+19~+6
k? 102K3+101K2+75K+12
’
247~~ + 153K + 18 102K3 + lOlK*
+ 75K + 12 ’
169~~ + 156~ +
,dE
5 o
102K3
+
lolK*
+
75K
36 +
12
’
(17)
(18)
(19)
(20)
(21) (22)
W. Marques Jr.,
356
G. M. Kremer
I Scattered light in polyatomic gases
The constitutive equations above were obtained from the field equations of ref. [7] through an iterate scheme by considering Q, ui, T and a as variables. In these equations c, is the specific heat at constant volume, p the shear viscosity, A the thermal conductivity, AT the translational thermal conductivity, AR the rotational thermal conductivity and the coefficients M, MT and MR do not have proper names. We are interested in a linearized theory where the following conditions hold:
e=eo+el
y
T = To + T, ,
ui=u;,
and
w = w,.
(23)
Insertion of eqs. (11)-(E) and (23) into the balance equations (6)-(9) leads to a system of linearized equations for the macroscopic fluctuations el, II,‘, T, and wi. This system, in terms of the spatial Fourier transform of the fluctuations. can be written as follows:
aw(a;7t) +M(q)V(q,t)=O,
(24)
where /
elk2 4
wq, f) = u,(cz,t) T,(q, 4
(25)
w,(q, t> and
e.
0
(uod QOY
M(q) = 0 0
0
WI2 (y-1)
(Y-P, Y
- ;
(uoq)2
DTq2 cod
(Y - l)eoc,4q2
0
- $
%q2 each - 1) Dcq2 + 0, (26)
In (25) and (26) we have introduced the longitudinal velocity u,( q, t> = -iq,ul( q, t), the adiabatic speed of sound uo, the thermal expansion coefficient (Y, the longitudinal kinematic viscosity D,, the thermal diffusivity D, and the quantities D,, D, and D,. They are defined as
W. Marques
Jr., G.M.
D*=(y-l)&),
D,=
Kremer
I Scattered
(G - 0
eoc,
light in polyatomic
gases
2 D, = (y - l)(l@ + M;) .
357
(28)
We close this section by remembering that the eigenvalues of the matrix M( 4) define the so-called hydrodynamic modes. We notice that hydrodynamics is valid only for phenomena varying slowly in space, this means that in the spatial Fourier transform, the only coefficients that are relevant correspond to small values of q. The eigenvalues of the matrix M(q) valid to order q2 are A, = +iv,q
AI, = -
- rq*
,
(29)
‘,D,q* ,
(30)
(31) where
(32) is the acoustic attenuation coefficient. A, represents two longitudinal propagating sound modes that are damped. A,, and A, correspond to a longitudinal heat mode and to a internal relaxation mode, respectively. A,, and A, are purely diffusive, i.e., they do not propagate.
4. The dynamic structure
factor
Now we apply the results derived in the previous section to calculate the dynamic structure factor. By taking the Laplace transform of eq. (24) it follows that (33)
where
358
W. Marques
Jr., G.M.
Kremer
I Scattered
light in polyatomic
gases
m
@(q,s) =
I
dt exp(-st)
q( q, t)
(34)
0
and I is the identity matrix. If we solve the longitudinal system of eqs. (33) for &( q, s), multiply the result by @T(q, 0) and perform the ensemble average over the initial states of the system, we get G(q, since
4 eT(4,W
the ensemble
(elk
= $j$ averages
0) eT(c 0))
(u,(q,O)eT(q,O)),
(35)
, (T,(q,O)e~(q,O))
h(q,O)
eT(c 0)) vanish, indicating the non-correlation
quantities
P(S) and Q(S) are given by
between them
P(S) = s3 + s2(D,q2 + D,q2 + D,q2 f mr) + s{ D,q2D,q2 + D,q2D,q2
- D,q2D,q2
+ D,q2D,q2Dcq2 + [(Y - l)hl(W2
+ D,q2Dcq2
+ (D,q2 + D,q2)w, + Z[(Y - W~l(~od2~
- D,q2D,q2DBq2
+ D,q2D,q2w,
+ D,q2 - %q2 - D,q2 + dh,d2
Q(s) = s4 + s3(D,q2 + D,q2 + D,q2 + wr) + s2{ D,q2D,q2 + D,q2Dcq2 - D,q2Dsq2
,
(36)
+ D,q2Dcq2
+ (D,q2 + D,q2)w, + IPY - 1) h4uod2~
+ s{ D,q2D,q2D,q2
- D,q2DAq2D,q2
- [(Y - l)bl(D,q2
+ D,q2)(uoq)2 + (D,q2 + D,q2 + 4(uoq)21
+ (W)(D,q2D,q2
and The
- D,q’D,q*
+ D,q2D,q20,
+ D,q24(~oq)2
.
(37)
From the knowledge of the density-density ’ correlation function (&( q, s) eT( q, 0)), the dynamic structure factor S( q, 0) follows from [8]
(38)
In our case the dynamic structure factor can be written as follows: 2
S(q)
N,(w)
D,(o)
+
N2(w)
D;(w) + D;(w) where
D2(w)
’
(39)
W. Marques Jr.,
G.M.
Kremer
4q D*q %I + -__ D,q %I ---_ uo uo uo
359
I Scattered light in polyatomic gases
UO
w,
uo
uo4
(41)
--_D*q %I+ uo uo
(4q
+ i
---__
_ (_D,q+D,q uo uo
w,+2y-1 > uo4
D,q D,q + __ W
D,q
I
Y 0, uo4
1
(42)
’
D2(~,-cy~;(~+~+!$+~) 0 +
0
w ( uoq
&q --_-___ I
--__y-1 Y
In eq. (39) S( 4) = (e,(
uo
4q
D,q uo
Dvq
uo
D,q+D,q (
UO
4,
UO
>
“0
D,q “0
+D,q+D,q __ UO
D,q
+
~ w, uo
Dvq ___
uo
uo4
0) e:( 4, 0)) is the static structure
uo
1 .
D,q uo
w, uo4
(43)
factor.
5. Application
In order to apply the results obtained in the last section, one should observe two conditions that must be followed by the polyatomic gas, namely (i) the molecules of the gas must be spherically symmetrical and (ii) the ratio of the specific heat capacities y must be close to 4/3. These conditions are approximately satisfied by methane (CH,) at ordinary temperatures. In monatomic gases, the hydrodynamic region is determined by the conditions
360
W. Marques
Jr., G. M. Kremer
/ Scattered
light in polyatomic
gases
2 w J -3 -ey, uo4
(44)
where y is a measure of the ratio of the wavelength of the observed fluctuation with the collision mean free path and x is the reduced frequency. For usual hydrodynamics to be valid in molecular gases, we need not only y s 1 and x 4 y but also [9]
(45) where the internal relaxation number z is the ratio of the elastic collision frequency W, = eou~/3/~o to the internal relaxation frequency w,. If only the conditions given in (44) are satisfied, the system is hydrodynamic only with respect to the translational degrees of freedom; the internal degrees of freedom respond too slowly to become thermalized during the period of the wave. Using the characteristic data for methane [lo] we get that the internal relaxation number z is equal to 6.85 at 293 K and 1.013 bar, and we conclude that for the usual hydrodynamic theory the condition x/y <<1 /z is much more restrictive than x 4 y. Figs. 1 and 2 show the scattered light spectrum from CH, gas (293 K and 1.013 bar) for y values equal to 18.27 and 4.46, respectively. In these figures 1.25
,
0.4
0.6
0.6
Reduced Frequency
Fig. 1. The scattered Rayleigh peak.
light spectrum
from
CH,
for y = 18.27.
The curves
are normalized
at the
W. Marques
Jr., G.M.
Kremer
I Scattered
light in polyatomic
gases
361
Reduced Frequency
Fig. 2. The scattered Rayleigh peak.
light spectrum from CH, for y = 4.46. The curves are normalized
at the
the solid line represents the spectral prediction derived from the hydrodynamical model proposed in this paper, the dashed line represents the prediction of the usual hydrodynamic theory and the points represent the prediction of the translational hydrodynamic theory from Desai and Kapral. Fig. 1 shows a complete agreement between our hydrodynamical model and the translational hydrodynamic theory for y = 18.27. However, the usual hydrodynamic theory predicts a less-intense and less-displaced Brillouin peak compared to the other two theories. For y = 4.46, as showed in fig. 2, the results of our hydrodynamical model and the translational theory are nearly identical, but the usual hydrodynamic prediction is in complete disagreement. This is to be expected since we are in a region where the condition (45) does not hold and the usual hydrodynamic theory is no longer applicable. Comparisons of experimental results with the translational hydrodynamic theory for y = 18.27, 4.46 and 2.70 were done by Hammond and Wiggins [6]. For the first two values of y the agreement between the experimental results and the translational theory was very good, but for y = 2.70 the translational hydrodynamic theory was not able to predict the spectrum of scattered light from methane. The results of our hydrodynamical model and the translational hydrodynamic theory are compared in fig. 3 for y = 2.70. In this case the theoretical results are also almost identical and we conclude that the hydrodynamical
362
W. Marques Jr.,
G.M.
Kremer
/ Scattered light in polyatomic gases
0.6
0.8
Reduced Frequency Fig. 3. The scattered Rayleigh peak.
light
spectrum
from
CH,
for y = 2.70.
The curves
are normalized
at the
model proposed in this work can be used to analyze those light scattering experiments from methane in the region y > 4 and x/y < 1.
6. Final remarks In this paper we have considered a six-field theory with 9, ui, T and a as basic fields. One could ask why there is a relaxation equation for the dynamic pressure but none for the pressure deviator nor for the translational and rotational heat fluxes. The reason is that for methane the dimensionless moment of inertia is small (K ~0.048) so that the relaxation of the dynamic pressure is more important since it goes much more slowly than the relaxation of the pressure deviator or the relaxations of the translational and rotational heat fluxes. On the other hand, a five field theory with e, ui and T as basic fields does not take into account the exchange of energy between the translational and rotational degrees of freedom. In that theory w becomes a constitutive quantity. The constitutive relation for a can be obtained from an iterate scheme by putting on the left-hand side of eq. (9) the equilibrium values a = 0, pCij, = 0, qi = hi = 0 and getting on the right-hand side
W. Marques Jr.,
G. M. Kremer
I Scattered light in polyatomic gases
363
au,
m=-qdx,’
(46)
where
,=(V-l)QkT w,
m
is the coefficient of volume viscosity. By eq. (47) we conclude that the relaxation frequency of the dynamic pressure is proportional to the inverse of the volume viscosity.
References [l] R.D. Mountain, Rev. Mod. Phys. 38 (1966) 205. [2] G.I.A. Stegeman, W.S. Goumal, V Volterra and B.P. Stoicheff, J. Acoust. Sot. Am. 49 (1971) 979. [3] R.D. Mountain, J. Res. Natl. Bur. Stand. A 70 (1966) 207. [4] R.D. Mountain, J. Res. Natl. Bur. Stand. A 72 (1968) 95. [5] R.C. Desai and R. Kapral, Phys. Rev. A 6 (1972) 2377. [6] C.M. Hammond and T.A. Wiggins, J. Chem. Phys. 65 (1976) 2788. [7] G.M. Kremer, Rev. Bras. Fis. (Brazil) 17 (1987) 370. [8] B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976). [9] C.D. Boley, R.C. Desai and G. Tenti, Can. J. Phys. 50 (1972) 2158. [lo] M.G. Rodbard and G.M. Kremer, Phys. Fluids A 2 (1990) 1269.