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Light scattering from fluids. A semi-phenomenological calculation of the coupling constants between the orientational and translational motion
Physica 107A (1981) 404-422 North-Holland Publishing Co.
L I G H T S C A T T E R I N G F R O M FLUIDS. A S E M I - P H E N O M E N O L O G I C A L C A L C U L A T I O N OF T H E C O U P L I N G CONSTANTS B E T W E E N T H E O R I E N T A T I O N A L AND T R A N S L A T I O N A L M O T I O N Boris M. AIZENBUD* Department of Chemistry, Ben-Gurion University of the Negev, Beer-Shet, a, Israel
Received 10 June 1980 Revised 7 November 1980
The coupling parameters X and Z between translational :tnd orientalional motions, which determine the lineshapes of the VH and HH depolarized light scattering spectra respectively, are calculated. By using a simple model that explains the connection between rotational and translational motions, it is shown that X and Z are proportional to k2~r/p. The correct order of magnitude of the proportionality coefficient is obtained.
I. Introduction
The purpose of this work is to present a phenomenological calculation on the d e p e n d e n c e of some transport coefficients on the shear viscosity in liquids, c o m p o s e d of nonspherical molecules. These coefficients appear in the expression for the lineshape of the low f r e q u e n c y depolarized light scattering. The low f r e q u e n c y depolarized light scattering spectra have been derived by various methods. G e r s h o n and O p p e n h e i m ~) calculated these lineshapes (for the V H and the H H geometries) using the molecular a p p r o a c h of F e l d e r h o f - S e l w y n - O p p e n h e i m theory2). On the other hand Andersen and P e c o r a 3) used Zwanzig 4) and Mori's 5) projection operator method to obtain the V H spectrum. T s a y and Kivelson 6) used also the Z w a n z i g - M o r i method to continue the previous work of K e y e s and KivelsonT). The low f r e q u e n c y depolarized spectra are expressed in terms of f r e q u e n c y independent transport c o e f f i c i e n t s - e . g . , orientational relaxation times, the shear viscosity and coupling coefficients X and Z between the orientational and translational motions. In G e r s h o n and O p p e n h e i m ' s theory ~) these transport coefficients are e x p r e s s e d in terms of molecular quantities which have a regular time dependence. On the other hand the theories based on the Z w a n z i g - M o r i method contain quantities with irregular time d e p e n d e n c e and projection operators. * Present address: Department of Chemistry, M.I.T., Cambridge, MA 02139, USA. 0378-4371/81/0000-0000/$02.50 O North-Holland Publishing C o m p a n y
LIGHT SCATTERING FROM FLUIDS
405
At the present time there is no satisfactory kinetic theory for liquids which can be used to calculate transport coefficients. Therefore it is necessary to rely on phenomenological calculations which are simple although are not extremely accurate. In this fashion Debye 8) and Kivelson, Kivelson and Oppenheim 9) showed the linear dependence of orientational relaxation time on the shear viscosity. As for the coupling parameters X and Z, it was shown experimentally 6'~°-~2)that approximately they also depend linearly on the shear viscosity. In this work we calculate X and Z using a very simple molecular model of liquid, that gives the experimental result. In other words we find that X and Z are proportional t o -k2"l~r/p (where k is a wave vector, fir is the rotational viscosity and p is the density) and moreover we get the correct order for the proportionality coefficients. The work proceeds in the following way. In section 2 the problem of calculation of the macroscopic correlation functions is reduced to the calculation of single molecules correlation functions, in section 3 the force acting on a molecule is calculated with the help of a model, and finally, in section 4 expressions for X and Z are presented. The theoretical results are compared with experimental results in section 5. Some calculations connected with our model appear in appendix A. In appendix B the generalized Stokes' law is represented. Some formulas describing the polarizability tensor are given in appendix C. The following notation is used: a. A subscript k designates the k-component of the space Fourier transform, e.g. if A(r, t) is any arbitrary function of space and time, then
Ak(t) -- f eik'~A(r, t) dr, V
where the integration is performed over the volume of the system. b. A subscript to means the to-component of the one-sided time Fourier transform, e.g.:
A~ --- f ei'~tA(r, t) dt -= F.T. A(r, t). 0
c. Numerical subscripts are used either to designate a particular molecule or a particular axis (1, 2, 3 denote the x, y, z-axes respectively). For example: (Otl.k~,3tOtl,-k,31) is the correlation function of quantities belonging to molecule 1 (denoted by 1 in the first subscript), where 3t denotes the zx-component. d. Matrix superscript "T" denotes the transposition operation.
406
2.
B.M. A I Z E N B U D
Simplification
of the
problem
The c o u p l i n g coefficients b e t w e e n the orientational and m o t i o n s X and Z are given by eqs. (3.8) and (4.8) of ref. i: X = M (n
M ~
Z =~'tc'
~"~
translational
I M m~
(2.1)
/ J v f ~°~
(2.2)
where 1 , 2 , 3 are the c o m p o n e n t s of the l a b o r a t o r y C a r t e s e a n axes x, y, z, respectively, g is the linear m o m e n t u m density, and ~ is the polarizability t e n s o r d e n s i t y (gk and ak are the c o r r e s p o n d i n g s p a c e - F o u r i e r transforms). The reader is referred to ref. 1 for the notations. The M ' s are c o n s t a n t s and are e x p r e s s e d in terms of correlation functions as follows (eqs. (2.5)-(2.7) of ref. 1): (We c h o s e the l a b o r a t o r y s y s t e m of c o o r d i n a t e s so that k = k~0 M Z = (ak~a'[k)(°'[(aka rk) ( ° ' - (ak.o& rk) "'] ' M ('' , .....~, = -[((~k,o>,~"k.~l/~,o)+..(o) . lvl,~,,,;,kak(,,,3j~~'(()) "'"
gl(~31
\(O)r/ O( = k '/ r k , o ,q o ~- kl~/ lkO~k.31 t,.~]/\ ( 0 ) , .
--
(2.3)
k,~I/~(,,j ~
(O~k(o,~l(~
/3
k.t~/\ ( 0 ) 1 J
I
ik,
(2.4)
ik.
(2.5)
H e r e ~" is the stress t e n s o r (see e.g. ref. 13), m is the mass of one molecule, V is the v o l u m e of the s y s t e m a n d / 3 = k , T , w h e r e kB is the B o l t z m a n n c o n s t a n t and T is the t e m p e r a t u r e . M(~°~ ) in eq. (2.3) is a 5 x 5 diagonal matrix. Using the identities n V =- N and gk --= ik • ~'k we find that Ml,,l~,m -
/3 [(6k~>lg kl)ll) + M (°'(,,,,,,,\/ak,o,3,g" k.,)'"]. nan
(2.6)
Using the identity g~ = ik • ~'~, the relation:
and substituting (2.3) to (2.5) we get: M , , ~, ....
• • = (ak,o.13g-k,I)
(,~ M (o7 ....... ,[(c~o,.~c~
~,~r)] , •
(2.7)
Eqs. (2.6) and (2.7) are e x a c t e x p r e s s i o n s for M,n~, and M~, .... up to ()~(k~). Our first simplifying a s s u m p t i o n is to allow the separation of an a v e r a g e of the p r o d u c t of a fast and slow d e c a y i n g quantities into two p a r t s - an a v e r a g e o v e r the fast part and an a v e r a g e o v e r the slow part. As the fast part d e c a y s m u c h faster than the slow one, it is possible to c o n s i d e r the slow part as static. This adiabatic a s s u m p t i o n is a usual one in p h e n o m e n o l o g i c a l calculations (see f o r instance ref. 9). The d a n g e r o u s point in using this a s s u m p tion is the fact that both slow and fast parts of our variable can have different s y m m e t r y f r o m the variable itself, and t h e r e f o r e t h e y can couple to the w r o n g
LIGHT SCATTERING FROM FLUIDS
407
variables. Because of this, the vanishing of a correlation function on the basis of this assumption should be treated with care. As mentioned in section 1, the slow variables of our systems are the hydrodynamic variables and the polarizability density. On the other hand, gk is a fast decaying variable in the discussed system. As is shown later in section 3, gk can be split into an angle dependent slow part and an angular momentum dependent part which decays fast, i.e. g ~ - gsJowgfa~t. Under the above mentioned simplifying assumption it is thus obtained: (6k~o~31g k,I) (1) '¢~ ,wl ]~A(O) /~ ,& a31a31\ttkto,31~
k,I/
\(l) •
The physical meaning of the adiabatic assumption in this situation is as follows: because ot describes the angular position of molecules, it can be influenced only by the antisymmetric part of stress tensor, 7, which can directly couple the angular momentum part of &. So in first approximation: M,)
"~'~'
_
/3 mN
(dko,,31g k,31)(I~.
(2.8)
Using (2.7), (2.8) and definition (2.1) we get after this first approximation: X = ~
r/& kto,3I~-k,l/ .~ \(l)12/r/& J /I.\ kto,31& -k,31/\Io)1J.
(2.9)
I.\
In the same way it can be shown that the expression for Z is: Z = ~
"
'
[(Olkw.33g-k.33)
O)]/[(O/kto.a3a 2 • • k,33) (o)]"
(2.10)
Let us now write down the expressions for ak and gk (eqs. (2.2) of ref. 1 and (3.2) of ref. 13): N
otk(t) = ~, ai(t) exp[ik i-1
• r;(t)],
(2.11)
N
gk(t) = ~ pi(t) exp[ik • ri(t)].
(2.12)
It is assumed that the ai-dependence on time is mainly through the variation of the orientation of molecule j relative to the laboratory fixed coordinate system, for example through the Euler's angles of the molecule 4~, ~, 0. We thus obtain for the time derivatives &k(t) = ~ &;(t) • exp[ik • ri(t)] + gk(t)=~
ii~(t).exp[ik, rj(t)]+
ik.m ~'i pj(t)aj(t),
exp[ik • rj(t)],
"~ pi(t)pj(t).exp[ik,
rj(t)l.
(2.13a) (2.13b)
408
B.M. A I Z E N B U D
From now on the tensorial and vectorial signs are dropped, c~k and gk denote either c~k.~ and gk,], or ak.33 and gk.3. We thus can write the correlation function (dk,og k) as (c~k,og k)' ' ' = F.T.( j~ d j ( t ) e x p [ i k , ri(t)] ~, /~,(0)exp[-ik. r,(O)]) - F.T.(~/ d , ( t ) e x p [ i k . r , ( t ) ]
~/ p,(O)pt3(O)exp[-ik, r,(0)])mik
+ F.T.(,~/ pi.~(t)ai(t)exp[ik.r(t)]
~'1 /~t(0)exp[-ik- r,(0)])mik
- F.T.( j~. pj.3(t)aj(t) exp[ik • rdt)] ~ p,(0)p,.,(0)
k × e x p [ - i k • rt(O)] m--~ =- A + B + C + D ,
(2.14)
where A, B, C, D denote the four terms, respectively. Because the l.h.p, of (2.14) is of first order in k, D does not contribute to the sum. Applying the adiabatic hypothesis to C we see that it can be neglected compared to A. In other words, we assume that
( &i( t )fh ) ~>(ai( t )pi( t )pl(O)) -~ (cq( t ))(pi( t )pl(O)). Now, we use the adiabatic hypothesis to compare B and A. Because a depends only on angular coordinates it is possible to represent a(t) in the following manner:
d(t) = a~b(t) + c¢+to(t) + aoO(t),
(2.15)
where (b, tO, 0 are the Euler angles of molecule, defined as in ref. 14. Expressing ~, q~ and 0 in terms of st, s2, s ~ - t h e components of the angular momentum of the molecule at the laboratory system, we find that d(t) = f,[~(t)]s](t) + fz[~(t)]s2(t) + f3[~(t)]s3(t).
(2.16)
Here ~(t) is the set of Euler's angles (b, tO, 0, and fi(~(t)) depends on the time through these angles only (see appendix C). These functions fi change slowly in time and s and p change fast. So we have for B:
(&(t )p(O)p(O)) ~- (f(t )s(t )p(O)p(O)) ~ (f( t ))(s( t )p(O)p(O)).
(2.17)
It can be easily shown (see the end of appendix C) that if(t)) vanishes, so in this approximation the contribution of B can be neglected compared with the contribution of A. The physical idea of applying the adiabatic hypothesis to B
LIGHT SCATTERING FROM FLUIDS
409
is similar to the one, formulated before eq. (2.8). Finally, we have: (Clktag-k): F . T . < ~ t~j(t)exp[ik, rj(t)] ,~ p,(O)exp[-ik, r,(O)]>.
(2.18)
N o w we make our second simplifying assumption, which is: (2.19)
(s( t )s(O)) ~ (s( t )p (O)).
By using this assumption in the same way which lead to (2.18), we get: (~kto~_k) = F . T . ( ~
a , ( t ) e x p [ i k , r,(t)] '~t &,(0)exp[-ik. r,(O)]).
(2.20)
By separating the correlations into sums of self and pair correlations and using the identity of the particles we can write (2.18) and (2.20) in the following form: (&k~g k)II) = F.T.{N(6~(t)/~j(0) exp{ik • [rt(t) - rl(0)]}) "1} + F.T.{N(N - 1)(t~l(t)162(0) exp{ik • [rl(t) - r2(0)]})
(2.21)
(&k~6t k) = F.T.{N(&I(t)&~(0) exp{ik • [r~(t)- rl(0)]})} + F.T.{N(N - 1)(&l(t)&2(0) exp{ik • [rl(t) - r2(0)]})}.
(2.22)
Now we make our third simplifying assumption, namely:
(sl(t)Sl(O)) >>N(sl(t)s2(O)),
(sl(t)gl(0)) >> N(~sj(t)g2(O)).
(2.23)
In general, pair correlation functions cannot be neglected relative to selfcorrelation functions but for those, mentioned in (2.23), this assumption may be right. It also was checked in modelling systems, similar to the model suggested in this work~5). With approximation (2.23), at the limit k ~ 0 we can rewrite (2.21) and (2.22) as: (&k~og_k) ~11= N (F.T.~& l(t)p z(0))tl)),
(2.24)
(dk~& k) ~°)= N (F.T.~d~( t )dl(O))~°~).
(2.25)
After these three simplifications, the expression for X can be written as follows: X-
/3 [F.T.(&l,31(t)/~Ll(0))"~] 2 -- m [F.T.~&L31(t)&L3j(0))~°)]"
(2.26)
Here first subscript of a and p denotes the molecule's number and the subsequent ones are tensorial indexes. Similarly, for Z we have: Z = 13 [F'T'(&I'33(t)/~l'3(0))(1)]2 m [ F . T . ( ~ 1,33(t)& 1,33(0))(°)]"
(2.27)
410
B.M. A I Z E N B U D
3. The model and calculation of p~ and P3
In the preceding section, the calculation of the macroscopic correlation functions was reduced to the calculation of the one-molecule correlation functions (~l,k~,3~Pl, k,1) and (&~,k~,3J&~,k,~a), but the problem is still too complicated for rigorous treatment from first principles. We suggest now a very simple and a very rough model that gives acceptable results (it is nothing more, than a fourth simplifying assumption). From our point of view this model can successfully describe the transfer of rotational motion into translational motion in liquids. As it is shown later, &(t) can be exactly represented in the form: &(f)---fl(~(t))sl(t)+f2(~(t))s2(t)+f3(~(t))s3(t), where fi(ff(t)) depends on time only through angular coordinates of the molecule. Using the procedure of separation of fast and slow variables, we can easily calculate (&k~ k) (see formulas (4.9a) and (4.9b) below). For the calculation of (dk~P k) ~-F.T.
f~(C(t))s,(t) p(O)
(3.1)
it is necessary to find the dependence of p on the angular momentum and on the angular coordinates. This is given in the following model (see fig. 1).
V
Fig. 1. Formation of the additional translational velocity by the angular m o m e n t u m gradient. s2, = -+2R~n grad s. V - additional translational velocity.
LIGHT SCATTERING FROM FLUIDS
411
In our system all gradients are along the z-axis, so the molecules are considered to be in an s field (s Slel+s2e2+s3e3) with grad s~ along the z-axis. Most of the experiments which were done on this problem used planar molecules (pyridine, quinoline)l°'"). Therefore we suppose that our molecules have a planar disk form. Let us focus our attention on a particular molecule (number 1 in fig. 1). As is evident from fig. 1 the effect of spacial gradients in the angular velocity on the translational motion of this rigid molecule is an additional translational velocity V, which is due to the rotation of the surrounding molecules in the medium (numbers 2 and 3 in fig. 1). We consider a more real model, in which molecules are still disks, but situated arbitrary in space (see fig. 2). The additional translational velocity V will be defined by the lowest point of the molecule 2 (point A in fig. 2) and by highest point of the corresponding molecule from a lower layer. As follows from fig. 2 the relative angular momentum in the laboratory system of coordinates is (for the ith component): =
s~ = (al + a2) grad s~ = (a sin 01 + a sin 02) grad s~
(3.2)
and, as it shown in appendix A, the velocity of the lowest point of molecule
Z
0
~
-----~ Y
×
Fig. 2. (X, Y, Z ) - Iaboratory s y s t e m of coordinates. (X', Y ' , Z ' ) - t h e nected with molecule 2. a2 = a cos O.
coordinate s y s t e m con-
412
B.M. AIZENBUD
"2" is
V~ = a(IVsi( sin O sin ch c°s 4a - c ° s
sin+sin0)Ji +
iVs2 [
sin 0 cos 2 J3
sin 0 sin 2 + + l v s 3 ~
cos
Ji
J3
V"=a{lVSl( sinOsin2~~J3 +
'
sinOc°sZcb)+lVs2(
sin 0 sin + cos J~
s i n ~ b c o s ~ b s i n 0 ) + t,,-, v s ~ /cos 0 sin 4))} , JI k j~
V: = -IVs1
cos 4~ cos 0
1•$2
Jl
(3.3)
sin ~b cos 0
Jr
where J~ and J~ are the moment of inertia relative to axis perpendicular and in the plane of the disk respectively. Averaging Vx, V , V. over all possible angular positions of molecule "2" by using the operator rr
( )o.,
'f
- 47r
27r
sin 0 dO
0
f
d+.
(3.4)
0
and substituting to (3.3) l -= a~ + a sin 02 we obtain for V
°
(3.5)
V: = 0. From this point we shall consider molecule l under the action of the current which moves with velocity V, (defined by eq. (3.5)) relative to this molecule. The connection between the translational velocity of the stream and a friction force acting on moving molecule (number l) can be considered to obey the generalized Stokes' law
~6i = ~ fik'oVkReef k
(i = 1, 2, 3).
(3.6)
Here Re, can be taken as any characteristic dimension (a, for instance), and Where fik(O, oh, tO) is a tensorial function of the angular coordinates, which gives a dependence of/~ on the position of the molecule in the space. (The possibility of this representation follows from the linearity of Stokes' law. Indeed, because of the linearity of the Navier-Stokes equation in
fik ==-fofik(O,&, tO).
LIGHT SCATTERING FROM FLUIDS
413
Stokes' approximation, we can separate the flow on three standard (relatively to the body of molecule) directions, designate their f(O, 05, qs) as ft,kJ%,kf3,~ and take the superposition of the results (see appendix B).) The numerical factor f0 depends on the shape of the molecule. For example, in the classical Stokes' problem "translation of spherical b o d y " fik(O, qS, t~)= 'Sik and f0 = 6~r. Moreover, we include in "f0" the difference between the situations "molecule between molecules" and " b o d y in liquid". Finally, substituting V~ from (3.5) and f(O, 05, ~) from (B.6), (B.9), (B.7) and (B.10) in eq. (3.6) we obtain for p, and p3: /~l=fo~ - a V s 2 ~ 1
at 1
+a 1r
a[f~(1-sin205sinO)+fllsin205sin20]
al)a[(fl_f
3 1 1 1 ~r = f0"0a ( ~ + ~ ) ( ~ + - f f ) { - V s 2 [ f ~ ( l -
) sin2 0 cos 05 sin 05]} (3.7a)
sin: 05 sin 2 0)+ fll sin2 05 sin 2 0)]
+ Vst[(f± -fql ) sin2 0 cos 05 sin 051} and similarly
p3=fo'Oa 3( ~1+ ~ ) ( f ± - . f , i )
1 7r 0)sin0 (~+--ffsin
cos 0
× (Vs2 sin 05 + Vs, cos 4)),
(3.7b)
where the definition a~ = a sin 0t was used. Averaging pj and p3 over the possible directions of molecule number 1 (this corresponds to hydrodynamic limit) we obtain [ 1 + ~3)[f10.415+f, I0.2271. (p,) = - frla 3Vs2\~
(3.8)
As it is expected (p3) vanishes. We shall require now, that (/~) will satisfy the exact hydrodynamic equations for g~ (formula (3.20) of ref. 13):
6ok=_ 2"qrSk:jnik,
gk2 =
2rlrSk, Jn ik,
gk 3 =
0,
(3.9)
where J is the average moment of inertia of a molecule in the principal system of axes, S is the density of angular momentum and fir is the rotational viscosity (see ref. 13). The result (p3) = 0 is in agreement with last equation of (3.9). Comparing eq. (3.8) with the first of eqs. (3.9) and noting that n(pl)k = gk.t and n . (VS)k =
414
B.M. AIZENBUD
ik • Sk, we get the following condition: 2~r
~+
= 0.415f, + 0.227f ii
(3.10)
Finally we have (from (3.7a), (3.7b) and (3.10)) 2 ~ (1/3) + (~r/8) sin 0 p~ = ik Jn 0.415f± + 0 . 2 2 7 / : I x {-s~k:[f:(l - sin 2 4) sin: 0) + f II s in-~ 4' sin 2 0] + s,k,[(f - f I ) sin: 0 × cos 4' sin oh]},
(3.11
2"0 (1/3) + (7r/8) sin 0 ps = ik Jn 0.415f~ + 0.227/,i × (f~ - f I!) sin 0 cos 0 [s,k~ sin 4) + s,~, cos 4)], where s~kj is a space F o u r i e r m o m e n t u m of molecule i.
transform
of the j - c o m p o n e n t
(3.12) of angular
4. Calculation of X and Z
It is seen f r o m (2.26) and (2.27), that for the calculation of X and Z, we have to calculate (&k,0P k) and
(4.1a) + c~).
(4.1b)
Differentiating it with r e s p e c t to time and e x p r e s s i n g & 4), 0, in terms of 0, &, + and s,, s:, s3 ((C.15), (C.17)) we obtain: &3, = f{(0, 4,)Sl + f~(O, 4))s2 + f j ( O , cb)s~,
(4.2a)
(~33 = fl( O, t~)SI-~" f2(O, 4))$2 ~-f~(O, ff))S~.
(4.2b)
We need f ~ , f x . , f ~ , f ~ that are given by (C.16) and (C.18): =
j~
sin 2 0 cos 4) sin 4,, (4.3a)
f~ -- o~1 - or3 (sin 2 0 sin 2 4) - cos 2 0), J,
LIGHT SCATTERING FROM FLUIDS O~32 COS 0 sin 0 cos 4', Jt f~ = or1 - or3 2 cos 0 sin 0 sin 4'. Ji f~
=
415
otl --
(4.3b)
Using equalities F.T.(s,(t)sj(O))-~ (52) 8,j,
(4.4)
(s%
(4.5)
=
J
and eq. (3.18) of ref. 13: M~ -- 4r/~ nJ
(4.6)
(8~j in (4.4) designates the fact that there is no correlation between different Cartesian components of angular momentum), we can calculate F.T.(&k31(t)/5 l(0)) = ik
~n r
1 o t l - ot3 0.415fi + 0.227f II Jt
(s2) ( {[fl(1 -
x~
sin 2 0 sin 2 4')
+ f II sin2 4' sin2 0][ sin2 0 sin 2 4' - cos 2 0] + [(f± - f l l ) sin2 0 cos 4' sin 4'][sin 2 0 cos 4' sin 4']} 1
~r
× (~ +~- sin 4')).
(4.7)
Performing this calculation we finally obtain ik
F.T.(&kl3(t)/5 i(0))
2"or o t l - ot3 J nJ 0.105fi - 0 . 1 4 3 f l 1 7n /3 4"or 0.415fi + 0.227f 11
(4.8a)
and similarly
F.T.(&k33(t)/53(0))
=
ik 2"or ott - or3 n J 2 (.fi - ftl)0.166 Jn ~ 4"or/30.415.fl+0.227fll
(4.8b)
In a similar way we get 9 j2 n (ott-or3) 2 F.T.(&k3~(t)a-k,3(0)) = ~ ~ "O~ /3 ,
(4.9a)
8 j2 n (otl- Or3)2 F.T.(&k33(t)& k33(0)) = ~-~ ~ "Or /3
(4.9b)
416
B.M. AIZENBUD
Substituting (4.8) and (4.9) into the definitions (2.26) and (2.27) of X and Z, we finally obtain: X =
_ [ / 0 . I 0 5 / : - 0.143f ,, )21.667] [\0.415f~ +0.227J'11
Z =-
k2'-Or P ,
(4.10a)
fc - f r l -0.0517 k~rt' 0.415]'~ +0.227fl I p
(4. lOb)
and
Z
(
f ' - f ll
x-= o.I05L -0.143/~
)20.031.
(4.1I)
5. D i s c u s s i o n
Formulas (4.10a), (4.10b) and (4.11) give us expressions for X , Z and trivially, for their ratio. Only two parameters f: and fll which are defined in appendix B, are used. These parameters describe translational properties of a molecule with a cylindrical symmetry. For planar molecules, it is convenient to introduce a dimensionless parameter A =-fL/fll. This parameter changes from zero (for infinitesimally thin molecules) to I (for spherical molecules). With the help of this parameter, the following expression for X, Z and Z / X can be obtained:
X = - A k271r
,
P
(5.1a)
Z = - B k2~)~ --, P
(5.1b)
Z / X = B/A,
(5.2)
where
(0.105
-0.143
2
A --- \0.4-i-~-+ 0.227] × 1.667, B ---
(0
.415-A~-i).22
#
× 0.0517.
(5.3a) (5.3b)
The numerical values of A, B and Z / X as a function of A are represented in table I. The experiments, which were performed to measure the values of X and Z~°-r2), give for A values between 0.5 and 1.1. For B they give values of the same order of magnitude as for A, though there is no reliable information about B. It is seen from table I, that for thin molecules X and Z have acceptable values.
LIGHT SCATTERING FROM FLUIDS
417
TABLE I A
A
B
Z/X
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.662 0.406 0.258 0.168 0.110 0.072 0.047 0.030 0.019 0.011 0.006
1.003 0.581 0.344 0.205 0.121 0.068 0.037 0.017 0.007 0.001 0.000
1.517 1.431 1.334 1.222 1.094 0.947 0.775 0.578 0.356 0.132 0.000
We wish to point out, that for the calculation of X and Z we made a number of strong assumptions, without estimation of their errors. Moreover, the numerical values of parameters [± and fll are sensitive to changes in the form of the molecule from ideal disk to more complicated forms. In principal, similar calculations can be performed for molecules of any form, but because of our assumptions, it will not lead to real improvement of our result.
Acknowledgements I am very grateful to Professor N.D. Gershon who formulated this problem and took a very active part in its solution. Also, I would like to thank Professor V. Volterra for his clarifying discussions and for his help in the final layout of the work and Professor I. Oppenheim for his helpful comments and advice with regard to this manuscript.
Appendix A Derivation of formula (3.3) Let 4~, 0, 0 be a set of Euler's angles, which define the angular position of a molecule relative to the laboratory system of coordinates, (I.s.). Then, the transformation matrix A, from a laboratory system of coordinates to the body system of coordinates, (b.s.), is given as follows (see for instance formula (4.46) of ref. 14): cos ~/, cos tO- sin ~ sin tOcos 0 sin ~ cos q, + cos 6 sin t0 cos 0 sinq, sin0\ ,6,=-cosSsintO-sin$cosqJcos0 -sin$sinO+cos4~cost0cos0 cost0sin0J.(A.1) sin ~bsin 0 -cos ~ sin 0 cos 0
418
B.M.
AIZENBUD
Further, the dynamical equations at the body system of coordinates have the following form: sb~ = db~Ob ....
(A.2)
$"~b.s. = db.~.$b.s.
(A.3)
or
(Here Db.~. is the angular velocoty of a molecule and d is the tensor of inertia.) Using the fact that Sb.s,--
ASI....
(A.4)
we find that /~b~ = db~As] ....
(A.5)
or, in coordinate form sl cos ¢b + ~ sin ~b
t
\
s,
J
/ -- -~s' sln rb cos O + ~ cos 4) cos O + ~ sm O •
\~/
SI
•
•
S~
sm ~ sm 0 - ~ c o s
.
S~
¢b sm 0 + • c o s
(A.6)
0]
The velocity of the lowest point of molecule "2" in fig. 2 (its body coordinates are ( 0 , - a , 0)) is Yb.s.= ,Ob.s.× rb.~ = a(O3el-J~le3). Returning to the laboratory system of coordinates: VL~ = AVb .... we get
/gt3a cos ¢b -- O,a sin 4~ sin 0 / Vl.~. = "~O3a sin ¢h - g21a cos 4' sin 0~ .
(A.7)
\g21a cos 0 Substitution into (A.7) the values of g2~ and g23 from (A.6) leads to formula (3.3). Appendix B
Translation of bodies in a liquid (Stokes' approximation) Let us consider any b o d y , moving in a liquid. We can characterize its motion by the following two statements (Generalized Stokes' law): (i) The resistance of the liquid to a body motion is proportional to the velocity of motion, V, to the viscosity of the liquid, 7, and to the charac-
LIGHT SCATTERING FROM FLUIDS
419
teristic dimension of the body, a, i.e.: 3
Fi = ~--1 SikTIVka
(i = 1, 2, 3).
(B.1)
(ii) If the coefficients Sik for any three arbitrary directions of the flow are known, the resistance force for any position of the" body in space can be found. Indeed, connecting with the body the coordinate system x', y', z', and denoting the coefficients f.~k for the flows directed by axis x', y', z' a s Sl,ik, S2,ik, [3,~k, respectively, we can calculate all of the components of the resistance force, and after it return to laboratory system of coordinates. Consider a flow with velocity "¢ along the x-axis of the laboratory system of coordinates (V = V~j) and consider any body situated at the origin of the laboratory system of coordinates. If this body has a cylindrical symmetry axis (let it be the z'-axis of the body coordinate system), then we have for f~k:
f i,ik =
f±
(B.2)
.
0 If A is a transformation matrix (see (A.1)), then at the body system the flow has the following velocity: / V [ c o s 4> cos qs - sin ~b sin ~, cos 0] \ /
!
Yb.s. = A(Vdl) = I V [ - c o s 4b sin Ik - sin ~b cos ~ cos 0] I .
/
/
\ V [ s i n ~b sin 0]
(B.3)
Using (B.1), (B.2) and (B.3) we find that the resistance force in the body system of coordinates has the following components:
/foSl ~lVa[cos cb cos * - sin cb sin tk cos O] Fb.s.
-----
Ifofl'oVa[-cos /
41 sin ~b- sin 6 cos tk cos 0] I .
\fo[ IIrlVa [sin ~b sin 0]
/
(B.4)
In the laboratory system of coordinates we have: Ft.s. = ArFb ....
(B.5)
or, in coordinate form:
F~ = forlVa[f±(1 - sin 2 4~ sin 2 0) + S II sin2 ~b sin 2 0],
(B.6)
Fy = .forlVa[ql - f. II) sin ~b cos 4b sin 2 0],
(B.7)
F~ = - f o ' o V a [ ( f i - [ 1 1 ) sin ~b sin 0 cos 0].
(B.8)
420
B.M. AIZENBUD
S i m i l a r l y f o r t h e flow a l o n g t h e y - a x i s ( V = V~2) we h a v e :
Fx = fo~Va[(fz - f II) sin 4' c o s ~b s i n 2 0],
(B.9)
Fy = fo~Va[f±(l - c o s : 4~ s i n : 0) + f II c o s : 4~ sin 2 0],
(B. 10)
Fz = f o ~ V a [ ( f i - . f l l ) c o s ~b sin 0 c o s 0]
(B.11)
a n d f o r t h e flow a l o n g t h e z - a x i s ( V = V~3) w e h a v e :
F, = f o ~ V a [ ( f ~ - f l l ) sin d~ s i n 0 c o s 0],
(B.12)
Fy = f o ~ V a [ ( f ± - f l l ) c o s ~ s i n 0 c o s 0],
(B.13)
F~ = fo~Va[f± sin 2 0 + f l l cos2 0].
(B.14)
Appendix C S o m e properties of the polarizability tensor L e t t h e p o l a r i z a b i l i t y t e n s o r a in t h e m o l e c u l a r b o d y s y s t e m o f c o o r d i n a t e s have the following representation:
ab.s. =
/i0:) a2
0
,
(C. 1)
a3
i.e. x ' , y', z ' a r e t h e p r i n c i p l e a x e s o f t~. T h e n in t h e l a b o r a t o r y s y s t e m o f coordinates: ~l.s. = ATtltb.s.A.
(C.2)
If a j = a2 (the c a s e o f c y l i n d r i c a l s y m m e t r y , the s y m m e t r y axis is z') w e h a v e : a, - (al - a3) sin2 4~ sin2 0 a~.~=
(aT a3) sin4, cos~bsin zO \-(a~ - ~3) sin ~ sin 0 cos 0
(a, - a3) sin 6 cos 4, sin 2 0 ctl - (a~ - a0 cos 2 d~ sin2 0 ( a l - a3) cos 4, sin 0 cos 0
- ( a , - a 3 ) sin 6 sin 0 cos 0 \ (a~-a3) cos 6 sin O cos O ) . ctl - (a~ - ct~)cos 2 0
(C.3) If t~l # a2, t h e e x p r e s s i o n s f o r t h e c o m p o n e n t s o f o0.s. a r e m o r e c o m p l i c a t e d . For example: ats..3j = ( a l - a2) c o s ~ sin ~ c o s ~ sin 0 + (a3 - a l sin 2 ~O- a2 c o s 2 ~ ) x sin 6 s i n 0 c o s 0, aLs..33 = ( a j sin 2 ~ + a2 c o s 2 ~ ) sin 2 0 + a3 c o s 2 0.
(C.4) (C.5)
LIGHT SCATTERING FROM FLUIDS
421
In order to obtain the traceless form of polarizability tensor we have to subtract ](a~ + a2 + a3) from the diagonal terms. Differentiating corresponding components of (C.3) with respect to time, we get &3~ = (a3 - al) cos ~b sin 0 cos 0 ~ + (a3 - al) sin 4) cos 20 0,
(C.6)
•33
(C.7)
~---
2(Otl - -
or3)
sin 0 cos 00.
Now we want to express ~ and 0 as functions of s~, Sz and s 3 - t h e components of angular momentum at the laboratory system of coordinates and also through ~b, 4, 0. To do it, we define (i) =
,
ll=
(sintksin0 sin0cos~ cos 0
0 0
-
co s t p ) n~ .
(C.8)
1
Let J~, J2 and J3 be the principal values of the inertia tensor, and assume that its principal axes are the same as the principal axes of the polarizability tensor. At the molecular system of coordinates Sb.s. = db.s. " ,Ob.s. = db.s.II~.
(C.9)
Then at the laboratory system of coordinates sl.s. = ATsbs = ATJ 1:1~.
(C. 10)
Because the matrices A, d and II are not singular, and because A - l = A T, we get from (C.10). : Ob.'s.dg.'~.As~
....
(C.11)
Substituting to (C.11) the matrices in an explicit form we get: 1
(~ = fj (-s~ ctg 0 sin ~b + s2 ctg 0 cos ~b + s3), [lsin,cos0 ~b = JI sin 0 +
=
(C.12)
1 . ] t - ~ s m ~b sin 0 s,
[ J,lcos20cos4, sin 0
1 .
]
t- ~33sin 0 COS t~ S2
cosd) +sin(/) Jl sl jj s2.
(C.14)
Substituting (C.12) and (C.14) into (C.6) and (C.7) we obtain:
Or3, = f ~'( O, ~ )S, + f ~'( O, ~)$2 + f ]l( o, ~)$3,
(C.15)
B.M. AIZENBUD
422
where fifO,
4,) - c~,~(~3(sin 4' cos 4, sin 20),
f2(O, 4,)-aJ
a 3 ( s i n 24,sin 2 0 _ c o s 2 0 ) ,
f3(O, 4,) --
Otl f~ -- ~3 ( c o s 4, sin 0
~33 = f~3(4,,
O)SI +f~3(O, 4,)$2,
(C.16)
cos 0),
and (C.17)
where
f?(4,, 0 ) -
a~ j, - c~32 cos 4, sin 0 cos 0,
y?(4,, 0)=
a~ J, - a3 2 sin 4, sin 0 c o s 0.
(C.18)
Similarly, formulas for other f~ can be obtained. Using the averaging operator (defined b y (3.4)) it c a n be e a s i l y s e e n that (.f~) = 0.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)
N.D. Gershon and 1. Oppenheim, Physica 64 (1973) 247. P.A. Selwyn and 1. Oppenheim, Physica 54 (1971) 161. H.C. Andersen and R.J. Pecora, Chem. Phys. 54 (1971) 2584. R. Zwanzig, J. Chem. Phys. 33 (1960) 1338. H. Mori, Progr. Theor. Phys. 33 (1965) 423. S.J. Tsay and D. Kivelson, Mol. Phys. 29 (1975) I. T. Keyes and D. Kivelson, J. Chem. Phys. 54 (1971) 1786. P. Debye, Polar Molecules (Dover, New York, 1929). D. Kivelson, M.G. Kivelson and I. Oppenheim, J. Chem. Phys. 52 (1970) 1810. J. Rouch, J.P. Chabrat, L. Letamendia, C. Vaucamps and N.D. Gershon, J. Chem. Phys. 63 (1975) 1383. G.R. Alms, D.R. Bauer, J.T. Brauman and R.J. Pecora, Chem. Phys. 59 (1973) 5304. G.D. Enright and B.P. Stoicheff, J. Chem. Phys. 64 (1976) 3658. N.D. Gershon and I. Oppenheim, Physica 62 (1972) 198. H. Goldstein, Classical Mechanics (Addison-Wesley, New York, 1972). B.M. Aizenbud, A study of relaxation of rotation, Unpublished.
L I G H T S C A T T E R I N G F R O M FLUIDS. A S E M I - P H E N O M E N O L O G I C A L C A L C U L A T I O N OF T H E C O U P L I N G CONSTANTS B E T W E E N T H E O R I E N T A T I O N A L AND T R A N S L A T I O N A L M O T I O N Boris M. AIZENBUD* Department of Chemistry, Ben-Gurion University of the Negev, Beer-Shet, a, Israel
Received 10 June 1980 Revised 7 November 1980
The coupling parameters X and Z between translational :tnd orientalional motions, which determine the lineshapes of the VH and HH depolarized light scattering spectra respectively, are calculated. By using a simple model that explains the connection between rotational and translational motions, it is shown that X and Z are proportional to k2~r/p. The correct order of magnitude of the proportionality coefficient is obtained.
I. Introduction
The purpose of this work is to present a phenomenological calculation on the d e p e n d e n c e of some transport coefficients on the shear viscosity in liquids, c o m p o s e d of nonspherical molecules. These coefficients appear in the expression for the lineshape of the low f r e q u e n c y depolarized light scattering. The low f r e q u e n c y depolarized light scattering spectra have been derived by various methods. G e r s h o n and O p p e n h e i m ~) calculated these lineshapes (for the V H and the H H geometries) using the molecular a p p r o a c h of F e l d e r h o f - S e l w y n - O p p e n h e i m theory2). On the other hand Andersen and P e c o r a 3) used Zwanzig 4) and Mori's 5) projection operator method to obtain the V H spectrum. T s a y and Kivelson 6) used also the Z w a n z i g - M o r i method to continue the previous work of K e y e s and KivelsonT). The low f r e q u e n c y depolarized spectra are expressed in terms of f r e q u e n c y independent transport c o e f f i c i e n t s - e . g . , orientational relaxation times, the shear viscosity and coupling coefficients X and Z between the orientational and translational motions. In G e r s h o n and O p p e n h e i m ' s theory ~) these transport coefficients are e x p r e s s e d in terms of molecular quantities which have a regular time dependence. On the other hand the theories based on the Z w a n z i g - M o r i method contain quantities with irregular time d e p e n d e n c e and projection operators. * Present address: Department of Chemistry, M.I.T., Cambridge, MA 02139, USA. 0378-4371/81/0000-0000/$02.50 O North-Holland Publishing C o m p a n y
LIGHT SCATTERING FROM FLUIDS
405
At the present time there is no satisfactory kinetic theory for liquids which can be used to calculate transport coefficients. Therefore it is necessary to rely on phenomenological calculations which are simple although are not extremely accurate. In this fashion Debye 8) and Kivelson, Kivelson and Oppenheim 9) showed the linear dependence of orientational relaxation time on the shear viscosity. As for the coupling parameters X and Z, it was shown experimentally 6'~°-~2)that approximately they also depend linearly on the shear viscosity. In this work we calculate X and Z using a very simple molecular model of liquid, that gives the experimental result. In other words we find that X and Z are proportional t o -k2"l~r/p (where k is a wave vector, fir is the rotational viscosity and p is the density) and moreover we get the correct order for the proportionality coefficients. The work proceeds in the following way. In section 2 the problem of calculation of the macroscopic correlation functions is reduced to the calculation of single molecules correlation functions, in section 3 the force acting on a molecule is calculated with the help of a model, and finally, in section 4 expressions for X and Z are presented. The theoretical results are compared with experimental results in section 5. Some calculations connected with our model appear in appendix A. In appendix B the generalized Stokes' law is represented. Some formulas describing the polarizability tensor are given in appendix C. The following notation is used: a. A subscript k designates the k-component of the space Fourier transform, e.g. if A(r, t) is any arbitrary function of space and time, then
Ak(t) -- f eik'~A(r, t) dr, V
where the integration is performed over the volume of the system. b. A subscript to means the to-component of the one-sided time Fourier transform, e.g.:
A~ --- f ei'~tA(r, t) dt -= F.T. A(r, t). 0
c. Numerical subscripts are used either to designate a particular molecule or a particular axis (1, 2, 3 denote the x, y, z-axes respectively). For example: (Otl.k~,3tOtl,-k,31) is the correlation function of quantities belonging to molecule 1 (denoted by 1 in the first subscript), where 3t denotes the zx-component. d. Matrix superscript "T" denotes the transposition operation.
406
2.
B.M. A I Z E N B U D
Simplification
of the
problem
The c o u p l i n g coefficients b e t w e e n the orientational and m o t i o n s X and Z are given by eqs. (3.8) and (4.8) of ref. i: X = M (n
M ~
Z =~'tc'
~"~
translational
I M m~
(2.1)
/ J v f ~°~
(2.2)
where 1 , 2 , 3 are the c o m p o n e n t s of the l a b o r a t o r y C a r t e s e a n axes x, y, z, respectively, g is the linear m o m e n t u m density, and ~ is the polarizability t e n s o r d e n s i t y (gk and ak are the c o r r e s p o n d i n g s p a c e - F o u r i e r transforms). The reader is referred to ref. 1 for the notations. The M ' s are c o n s t a n t s and are e x p r e s s e d in terms of correlation functions as follows (eqs. (2.5)-(2.7) of ref. 1): (We c h o s e the l a b o r a t o r y s y s t e m of c o o r d i n a t e s so that k = k~0 M Z = (ak~a'[k)(°'[(aka rk) ( ° ' - (ak.o& rk) "'] ' M ('' , .....~, = -[((~k,o>,~"k.~l/~,o)+..(o) . lvl,~,,,;,kak(,,,3j~~'(()) "'"
gl(~31
\(O)r/ O( = k '/ r k , o ,q o ~- kl~/ lkO~k.31 t,.~]/\ ( 0 ) , .
--
(2.3)
k,~I/~(,,j ~
(O~k(o,~l(~
/3
k.t~/\ ( 0 ) 1 J
I
ik,
(2.4)
ik.
(2.5)
H e r e ~" is the stress t e n s o r (see e.g. ref. 13), m is the mass of one molecule, V is the v o l u m e of the s y s t e m a n d / 3 = k , T , w h e r e kB is the B o l t z m a n n c o n s t a n t and T is the t e m p e r a t u r e . M(~°~ ) in eq. (2.3) is a 5 x 5 diagonal matrix. Using the identities n V =- N and gk --= ik • ~'k we find that Ml,,l~,m -
/3 [(6k~>lg kl)ll) + M (°'(,,,,,,,\/ak,o,3,g" k.,)'"]. nan
(2.6)
Using the identity g~ = ik • ~'~, the relation:
and substituting (2.3) to (2.5) we get: M , , ~, ....
• • = (ak,o.13g-k,I)
(,~ M (o7 ....... ,[(c~o,.~c~
~,~r)] , •
(2.7)
Eqs. (2.6) and (2.7) are e x a c t e x p r e s s i o n s for M,n~, and M~, .... up to ()~(k~). Our first simplifying a s s u m p t i o n is to allow the separation of an a v e r a g e of the p r o d u c t of a fast and slow d e c a y i n g quantities into two p a r t s - an a v e r a g e o v e r the fast part and an a v e r a g e o v e r the slow part. As the fast part d e c a y s m u c h faster than the slow one, it is possible to c o n s i d e r the slow part as static. This adiabatic a s s u m p t i o n is a usual one in p h e n o m e n o l o g i c a l calculations (see f o r instance ref. 9). The d a n g e r o u s point in using this a s s u m p tion is the fact that both slow and fast parts of our variable can have different s y m m e t r y f r o m the variable itself, and t h e r e f o r e t h e y can couple to the w r o n g
LIGHT SCATTERING FROM FLUIDS
407
variables. Because of this, the vanishing of a correlation function on the basis of this assumption should be treated with care. As mentioned in section 1, the slow variables of our systems are the hydrodynamic variables and the polarizability density. On the other hand, gk is a fast decaying variable in the discussed system. As is shown later in section 3, gk can be split into an angle dependent slow part and an angular momentum dependent part which decays fast, i.e. g ~ - gsJowgfa~t. Under the above mentioned simplifying assumption it is thus obtained: (6k~o~31g k,I) (1) '¢~ ,wl ]~A(O) /~ ,& a31a31\ttkto,31~
k,I/
\(l) •
The physical meaning of the adiabatic assumption in this situation is as follows: because ot describes the angular position of molecules, it can be influenced only by the antisymmetric part of stress tensor, 7, which can directly couple the angular momentum part of &. So in first approximation: M,)
"~'~'
_
/3 mN
(dko,,31g k,31)(I~.
(2.8)
Using (2.7), (2.8) and definition (2.1) we get after this first approximation: X = ~
r/& kto,3I~-k,l/ .~ \(l)12/r/& J /I.\ kto,31& -k,31/\Io)1J.
(2.9)
I.\
In the same way it can be shown that the expression for Z is: Z = ~
"
'
[(Olkw.33g-k.33)
O)]/[(O/kto.a3a 2 • • k,33) (o)]"
(2.10)
Let us now write down the expressions for ak and gk (eqs. (2.2) of ref. 1 and (3.2) of ref. 13): N
otk(t) = ~, ai(t) exp[ik i-1
• r;(t)],
(2.11)
N
gk(t) = ~ pi(t) exp[ik • ri(t)].
(2.12)
It is assumed that the ai-dependence on time is mainly through the variation of the orientation of molecule j relative to the laboratory fixed coordinate system, for example through the Euler's angles of the molecule 4~, ~, 0. We thus obtain for the time derivatives &k(t) = ~ &;(t) • exp[ik • ri(t)] + gk(t)=~
ii~(t).exp[ik, rj(t)]+
ik.m ~'i pj(t)aj(t),
exp[ik • rj(t)],
"~ pi(t)pj(t).exp[ik,
rj(t)l.
(2.13a) (2.13b)
408
B.M. A I Z E N B U D
From now on the tensorial and vectorial signs are dropped, c~k and gk denote either c~k.~ and gk,], or ak.33 and gk.3. We thus can write the correlation function (dk,og k) as (c~k,og k)' ' ' = F.T.( j~ d j ( t ) e x p [ i k , ri(t)] ~, /~,(0)exp[-ik. r,(O)]) - F.T.(~/ d , ( t ) e x p [ i k . r , ( t ) ]
~/ p,(O)pt3(O)exp[-ik, r,(0)])mik
+ F.T.(,~/ pi.~(t)ai(t)exp[ik.r(t)]
~'1 /~t(0)exp[-ik- r,(0)])mik
- F.T.( j~. pj.3(t)aj(t) exp[ik • rdt)] ~ p,(0)p,.,(0)
k × e x p [ - i k • rt(O)] m--~ =- A + B + C + D ,
(2.14)
where A, B, C, D denote the four terms, respectively. Because the l.h.p, of (2.14) is of first order in k, D does not contribute to the sum. Applying the adiabatic hypothesis to C we see that it can be neglected compared to A. In other words, we assume that
( &i( t )fh ) ~>(ai( t )pi( t )pl(O)) -~ (cq( t ))(pi( t )pl(O)). Now, we use the adiabatic hypothesis to compare B and A. Because a depends only on angular coordinates it is possible to represent a(t) in the following manner:
d(t) = a~b(t) + c¢+to(t) + aoO(t),
(2.15)
where (b, tO, 0 are the Euler angles of molecule, defined as in ref. 14. Expressing ~, q~ and 0 in terms of st, s2, s ~ - t h e components of the angular momentum of the molecule at the laboratory system, we find that d(t) = f,[~(t)]s](t) + fz[~(t)]s2(t) + f3[~(t)]s3(t).
(2.16)
Here ~(t) is the set of Euler's angles (b, tO, 0, and fi(~(t)) depends on the time through these angles only (see appendix C). These functions fi change slowly in time and s and p change fast. So we have for B:
(&(t )p(O)p(O)) ~- (f(t )s(t )p(O)p(O)) ~ (f( t ))(s( t )p(O)p(O)).
(2.17)
It can be easily shown (see the end of appendix C) that if(t)) vanishes, so in this approximation the contribution of B can be neglected compared with the contribution of A. The physical idea of applying the adiabatic hypothesis to B
LIGHT SCATTERING FROM FLUIDS
409
is similar to the one, formulated before eq. (2.8). Finally, we have: (Clktag-k): F . T . < ~ t~j(t)exp[ik, rj(t)] ,~ p,(O)exp[-ik, r,(O)]>.
(2.18)
N o w we make our second simplifying assumption, which is: (2.19)
(s( t )s(O)) ~ (s( t )p (O)).
By using this assumption in the same way which lead to (2.18), we get: (~kto~_k) = F . T . ( ~
a , ( t ) e x p [ i k , r,(t)] '~t &,(0)exp[-ik. r,(O)]).
(2.20)
By separating the correlations into sums of self and pair correlations and using the identity of the particles we can write (2.18) and (2.20) in the following form: (&k~g k)II) = F.T.{N(6~(t)/~j(0) exp{ik • [rt(t) - rl(0)]}) "1} + F.T.{N(N - 1)(t~l(t)162(0) exp{ik • [rl(t) - r2(0)]})
(2.21)
(&k~6t k) = F.T.{N(&I(t)&~(0) exp{ik • [r~(t)- rl(0)]})} + F.T.{N(N - 1)(&l(t)&2(0) exp{ik • [rl(t) - r2(0)]})}.
(2.22)
Now we make our third simplifying assumption, namely:
(sl(t)Sl(O)) >>N(sl(t)s2(O)),
(sl(t)gl(0)) >> N(~sj(t)g2(O)).
(2.23)
In general, pair correlation functions cannot be neglected relative to selfcorrelation functions but for those, mentioned in (2.23), this assumption may be right. It also was checked in modelling systems, similar to the model suggested in this work~5). With approximation (2.23), at the limit k ~ 0 we can rewrite (2.21) and (2.22) as: (&k~og_k) ~11= N (F.T.~& l(t)p z(0))tl)),
(2.24)
(dk~& k) ~°)= N (F.T.~d~( t )dl(O))~°~).
(2.25)
After these three simplifications, the expression for X can be written as follows: X-
/3 [F.T.(&l,31(t)/~Ll(0))"~] 2 -- m [F.T.~&L31(t)&L3j(0))~°)]"
(2.26)
Here first subscript of a and p denotes the molecule's number and the subsequent ones are tensorial indexes. Similarly, for Z we have: Z = 13 [F'T'(&I'33(t)/~l'3(0))(1)]2 m [ F . T . ( ~ 1,33(t)& 1,33(0))(°)]"
(2.27)
410
B.M. A I Z E N B U D
3. The model and calculation of p~ and P3
In the preceding section, the calculation of the macroscopic correlation functions was reduced to the calculation of the one-molecule correlation functions (~l,k~,3~Pl, k,1) and (&~,k~,3J&~,k,~a), but the problem is still too complicated for rigorous treatment from first principles. We suggest now a very simple and a very rough model that gives acceptable results (it is nothing more, than a fourth simplifying assumption). From our point of view this model can successfully describe the transfer of rotational motion into translational motion in liquids. As it is shown later, &(t) can be exactly represented in the form: &(f)---fl(~(t))sl(t)+f2(~(t))s2(t)+f3(~(t))s3(t), where fi(ff(t)) depends on time only through angular coordinates of the molecule. Using the procedure of separation of fast and slow variables, we can easily calculate (&k~ k) (see formulas (4.9a) and (4.9b) below). For the calculation of (dk~P k) ~-F.T.
f~(C(t))s,(t) p(O)
(3.1)
it is necessary to find the dependence of p on the angular momentum and on the angular coordinates. This is given in the following model (see fig. 1).
V
Fig. 1. Formation of the additional translational velocity by the angular m o m e n t u m gradient. s2, = -+2R~n grad s. V - additional translational velocity.
LIGHT SCATTERING FROM FLUIDS
411
In our system all gradients are along the z-axis, so the molecules are considered to be in an s field (s Slel+s2e2+s3e3) with grad s~ along the z-axis. Most of the experiments which were done on this problem used planar molecules (pyridine, quinoline)l°'"). Therefore we suppose that our molecules have a planar disk form. Let us focus our attention on a particular molecule (number 1 in fig. 1). As is evident from fig. 1 the effect of spacial gradients in the angular velocity on the translational motion of this rigid molecule is an additional translational velocity V, which is due to the rotation of the surrounding molecules in the medium (numbers 2 and 3 in fig. 1). We consider a more real model, in which molecules are still disks, but situated arbitrary in space (see fig. 2). The additional translational velocity V will be defined by the lowest point of the molecule 2 (point A in fig. 2) and by highest point of the corresponding molecule from a lower layer. As follows from fig. 2 the relative angular momentum in the laboratory system of coordinates is (for the ith component): =
s~ = (al + a2) grad s~ = (a sin 01 + a sin 02) grad s~
(3.2)
and, as it shown in appendix A, the velocity of the lowest point of molecule
Z
0
~
-----~ Y
×
Fig. 2. (X, Y, Z ) - Iaboratory s y s t e m of coordinates. (X', Y ' , Z ' ) - t h e nected with molecule 2. a2 = a cos O.
coordinate s y s t e m con-
412
B.M. AIZENBUD
"2" is
V~ = a(IVsi( sin O sin ch c°s 4a - c ° s
sin+sin0)Ji +
iVs2 [
sin 0 cos 2 J3
sin 0 sin 2 + + l v s 3 ~
cos
Ji
J3
V"=a{lVSl( sinOsin2~~J3 +
'
sinOc°sZcb)+lVs2(
sin 0 sin + cos J~
s i n ~ b c o s ~ b s i n 0 ) + t,,-, v s ~ /cos 0 sin 4))} , JI k j~
V: = -IVs1
cos 4~ cos 0
1•$2
Jl
(3.3)
sin ~b cos 0
Jr
where J~ and J~ are the moment of inertia relative to axis perpendicular and in the plane of the disk respectively. Averaging Vx, V , V. over all possible angular positions of molecule "2" by using the operator rr
( )o.,
'f
- 47r
27r
sin 0 dO
0
f
d+.
(3.4)
0
and substituting to (3.3) l -= a~ + a sin 02 we obtain for V
°
(3.5)
V: = 0. From this point we shall consider molecule l under the action of the current which moves with velocity V, (defined by eq. (3.5)) relative to this molecule. The connection between the translational velocity of the stream and a friction force acting on moving molecule (number l) can be considered to obey the generalized Stokes' law
~6i = ~ fik'oVkReef k
(i = 1, 2, 3).
(3.6)
Here Re, can be taken as any characteristic dimension (a, for instance), and Where fik(O, oh, tO) is a tensorial function of the angular coordinates, which gives a dependence of/~ on the position of the molecule in the space. (The possibility of this representation follows from the linearity of Stokes' law. Indeed, because of the linearity of the Navier-Stokes equation in
fik ==-fofik(O,&, tO).
LIGHT SCATTERING FROM FLUIDS
413
Stokes' approximation, we can separate the flow on three standard (relatively to the body of molecule) directions, designate their f(O, 05, qs) as ft,kJ%,kf3,~ and take the superposition of the results (see appendix B).) The numerical factor f0 depends on the shape of the molecule. For example, in the classical Stokes' problem "translation of spherical b o d y " fik(O, qS, t~)= 'Sik and f0 = 6~r. Moreover, we include in "f0" the difference between the situations "molecule between molecules" and " b o d y in liquid". Finally, substituting V~ from (3.5) and f(O, 05, ~) from (B.6), (B.9), (B.7) and (B.10) in eq. (3.6) we obtain for p, and p3: /~l=fo~ - a V s 2 ~ 1
at 1
+a 1r
a[f~(1-sin205sinO)+fllsin205sin20]
al)a[(fl_f
3 1 1 1 ~r = f0"0a ( ~ + ~ ) ( ~ + - f f ) { - V s 2 [ f ~ ( l -
) sin2 0 cos 05 sin 05]} (3.7a)
sin: 05 sin 2 0)+ fll sin2 05 sin 2 0)]
+ Vst[(f± -fql ) sin2 0 cos 05 sin 051} and similarly
p3=fo'Oa 3( ~1+ ~ ) ( f ± - . f , i )
1 7r 0)sin0 (~+--ffsin
cos 0
× (Vs2 sin 05 + Vs, cos 4)),
(3.7b)
where the definition a~ = a sin 0t was used. Averaging pj and p3 over the possible directions of molecule number 1 (this corresponds to hydrodynamic limit) we obtain [ 1 + ~3)[f10.415+f, I0.2271. (p,) = - frla 3Vs2\~
(3.8)
As it is expected (p3) vanishes. We shall require now, that (/~) will satisfy the exact hydrodynamic equations for g~ (formula (3.20) of ref. 13):
6ok=_ 2"qrSk:jnik,
gk2 =
2rlrSk, Jn ik,
gk 3 =
0,
(3.9)
where J is the average moment of inertia of a molecule in the principal system of axes, S is the density of angular momentum and fir is the rotational viscosity (see ref. 13). The result (p3) = 0 is in agreement with last equation of (3.9). Comparing eq. (3.8) with the first of eqs. (3.9) and noting that n(pl)k = gk.t and n . (VS)k =
414
B.M. AIZENBUD
ik • Sk, we get the following condition: 2~r
~+
= 0.415f, + 0.227f ii
(3.10)
Finally we have (from (3.7a), (3.7b) and (3.10)) 2 ~ (1/3) + (~r/8) sin 0 p~ = ik Jn 0.415f± + 0 . 2 2 7 / : I x {-s~k:[f:(l - sin 2 4) sin: 0) + f II s in-~ 4' sin 2 0] + s,k,[(f - f I ) sin: 0 × cos 4' sin oh]},
(3.11
2"0 (1/3) + (7r/8) sin 0 ps = ik Jn 0.415f~ + 0.227/,i × (f~ - f I!) sin 0 cos 0 [s,k~ sin 4) + s,~, cos 4)], where s~kj is a space F o u r i e r m o m e n t u m of molecule i.
transform
of the j - c o m p o n e n t
(3.12) of angular
4. Calculation of X and Z
It is seen f r o m (2.26) and (2.27), that for the calculation of X and Z, we have to calculate (&k,0P k) and
(4.1a) + c~).
(4.1b)
Differentiating it with r e s p e c t to time and e x p r e s s i n g & 4), 0, in terms of 0, &, + and s,, s:, s3 ((C.15), (C.17)) we obtain: &3, = f{(0, 4,)Sl + f~(O, 4))s2 + f j ( O , cb)s~,
(4.2a)
(~33 = fl( O, t~)SI-~" f2(O, 4))$2 ~-f~(O, ff))S~.
(4.2b)
We need f ~ , f x . , f ~ , f ~ that are given by (C.16) and (C.18): =
j~
sin 2 0 cos 4) sin 4,, (4.3a)
f~ -- o~1 - or3 (sin 2 0 sin 2 4) - cos 2 0), J,
LIGHT SCATTERING FROM FLUIDS O~32 COS 0 sin 0 cos 4', Jt f~ = or1 - or3 2 cos 0 sin 0 sin 4'. Ji f~
=
415
otl --
(4.3b)
Using equalities F.T.(s,(t)sj(O))-~ (52) 8,j,
(4.4)
(s%
(4.5)
=
J
and eq. (3.18) of ref. 13: M~ -- 4r/~ nJ
(4.6)
(8~j in (4.4) designates the fact that there is no correlation between different Cartesian components of angular momentum), we can calculate F.T.(&k31(t)/5 l(0)) = ik
~n r
1 o t l - ot3 0.415fi + 0.227f II Jt
(s2) ( {[fl(1 -
x~
sin 2 0 sin 2 4')
+ f II sin2 4' sin2 0][ sin2 0 sin 2 4' - cos 2 0] + [(f± - f l l ) sin2 0 cos 4' sin 4'][sin 2 0 cos 4' sin 4']} 1
~r
× (~ +~- sin 4')).
(4.7)
Performing this calculation we finally obtain ik
F.T.(&kl3(t)/5 i(0))
2"or o t l - ot3 J nJ 0.105fi - 0 . 1 4 3 f l 1 7n /3 4"or 0.415fi + 0.227f 11
(4.8a)
and similarly
F.T.(&k33(t)/53(0))
=
ik 2"or ott - or3 n J 2 (.fi - ftl)0.166 Jn ~ 4"or/30.415.fl+0.227fll
(4.8b)
In a similar way we get 9 j2 n (ott-or3) 2 F.T.(&k3~(t)a-k,3(0)) = ~ ~ "O~ /3 ,
(4.9a)
8 j2 n (otl- Or3)2 F.T.(&k33(t)& k33(0)) = ~-~ ~ "Or /3
(4.9b)
416
B.M. AIZENBUD
Substituting (4.8) and (4.9) into the definitions (2.26) and (2.27) of X and Z, we finally obtain: X =
_ [ / 0 . I 0 5 / : - 0.143f ,, )21.667] [\0.415f~ +0.227J'11
Z =-
k2'-Or P ,
(4.10a)
fc - f r l -0.0517 k~rt' 0.415]'~ +0.227fl I p
(4. lOb)
and
Z
(
f ' - f ll
x-= o.I05L -0.143/~
)20.031.
(4.1I)
5. D i s c u s s i o n
Formulas (4.10a), (4.10b) and (4.11) give us expressions for X , Z and trivially, for their ratio. Only two parameters f: and fll which are defined in appendix B, are used. These parameters describe translational properties of a molecule with a cylindrical symmetry. For planar molecules, it is convenient to introduce a dimensionless parameter A =-fL/fll. This parameter changes from zero (for infinitesimally thin molecules) to I (for spherical molecules). With the help of this parameter, the following expression for X, Z and Z / X can be obtained:
X = - A k271r
,
P
(5.1a)
Z = - B k2~)~ --, P
(5.1b)
Z / X = B/A,
(5.2)
where
(0.105
-0.143
2
A --- \0.4-i-~-+ 0.227] × 1.667, B ---
(0
.415-A~-i).22
#
× 0.0517.
(5.3a) (5.3b)
The numerical values of A, B and Z / X as a function of A are represented in table I. The experiments, which were performed to measure the values of X and Z~°-r2), give for A values between 0.5 and 1.1. For B they give values of the same order of magnitude as for A, though there is no reliable information about B. It is seen from table I, that for thin molecules X and Z have acceptable values.
LIGHT SCATTERING FROM FLUIDS
417
TABLE I A
A
B
Z/X
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.662 0.406 0.258 0.168 0.110 0.072 0.047 0.030 0.019 0.011 0.006
1.003 0.581 0.344 0.205 0.121 0.068 0.037 0.017 0.007 0.001 0.000
1.517 1.431 1.334 1.222 1.094 0.947 0.775 0.578 0.356 0.132 0.000
We wish to point out, that for the calculation of X and Z we made a number of strong assumptions, without estimation of their errors. Moreover, the numerical values of parameters [± and fll are sensitive to changes in the form of the molecule from ideal disk to more complicated forms. In principal, similar calculations can be performed for molecules of any form, but because of our assumptions, it will not lead to real improvement of our result.
Acknowledgements I am very grateful to Professor N.D. Gershon who formulated this problem and took a very active part in its solution. Also, I would like to thank Professor V. Volterra for his clarifying discussions and for his help in the final layout of the work and Professor I. Oppenheim for his helpful comments and advice with regard to this manuscript.
Appendix A Derivation of formula (3.3) Let 4~, 0, 0 be a set of Euler's angles, which define the angular position of a molecule relative to the laboratory system of coordinates, (I.s.). Then, the transformation matrix A, from a laboratory system of coordinates to the body system of coordinates, (b.s.), is given as follows (see for instance formula (4.46) of ref. 14): cos ~/, cos tO- sin ~ sin tOcos 0 sin ~ cos q, + cos 6 sin t0 cos 0 sinq, sin0\ ,6,=-cosSsintO-sin$cosqJcos0 -sin$sinO+cos4~cost0cos0 cost0sin0J.(A.1) sin ~bsin 0 -cos ~ sin 0 cos 0
418
B.M.
AIZENBUD
Further, the dynamical equations at the body system of coordinates have the following form: sb~ = db~Ob ....
(A.2)
$"~b.s. = db.~.$b.s.
(A.3)
or
(Here Db.~. is the angular velocoty of a molecule and d is the tensor of inertia.) Using the fact that Sb.s,--
ASI....
(A.4)
we find that /~b~ = db~As] ....
(A.5)
or, in coordinate form sl cos ¢b + ~ sin ~b
t
\
s,
J
/ -- -~s' sln rb cos O + ~ cos 4) cos O + ~ sm O •
\~/
SI
•
•
S~
sm ~ sm 0 - ~ c o s
.
S~
¢b sm 0 + • c o s
(A.6)
0]
The velocity of the lowest point of molecule "2" in fig. 2 (its body coordinates are ( 0 , - a , 0)) is Yb.s.= ,Ob.s.× rb.~ = a(O3el-J~le3). Returning to the laboratory system of coordinates: VL~ = AVb .... we get
/gt3a cos ¢b -- O,a sin 4~ sin 0 / Vl.~. = "~O3a sin ¢h - g21a cos 4' sin 0~ .
(A.7)
\g21a cos 0 Substitution into (A.7) the values of g2~ and g23 from (A.6) leads to formula (3.3). Appendix B
Translation of bodies in a liquid (Stokes' approximation) Let us consider any b o d y , moving in a liquid. We can characterize its motion by the following two statements (Generalized Stokes' law): (i) The resistance of the liquid to a body motion is proportional to the velocity of motion, V, to the viscosity of the liquid, 7, and to the charac-
LIGHT SCATTERING FROM FLUIDS
419
teristic dimension of the body, a, i.e.: 3
Fi = ~--1 SikTIVka
(i = 1, 2, 3).
(B.1)
(ii) If the coefficients Sik for any three arbitrary directions of the flow are known, the resistance force for any position of the" body in space can be found. Indeed, connecting with the body the coordinate system x', y', z', and denoting the coefficients f.~k for the flows directed by axis x', y', z' a s Sl,ik, S2,ik, [3,~k, respectively, we can calculate all of the components of the resistance force, and after it return to laboratory system of coordinates. Consider a flow with velocity "¢ along the x-axis of the laboratory system of coordinates (V = V~j) and consider any body situated at the origin of the laboratory system of coordinates. If this body has a cylindrical symmetry axis (let it be the z'-axis of the body coordinate system), then we have for f~k:
f i,ik =
f±
(B.2)
.
0 If A is a transformation matrix (see (A.1)), then at the body system the flow has the following velocity: / V [ c o s 4> cos qs - sin ~b sin ~, cos 0] \ /
!
Yb.s. = A(Vdl) = I V [ - c o s 4b sin Ik - sin ~b cos ~ cos 0] I .
/
/
\ V [ s i n ~b sin 0]
(B.3)
Using (B.1), (B.2) and (B.3) we find that the resistance force in the body system of coordinates has the following components:
/foSl ~lVa[cos cb cos * - sin cb sin tk cos O] Fb.s.
-----
Ifofl'oVa[-cos /
41 sin ~b- sin 6 cos tk cos 0] I .
\fo[ IIrlVa [sin ~b sin 0]
/
(B.4)
In the laboratory system of coordinates we have: Ft.s. = ArFb ....
(B.5)
or, in coordinate form:
F~ = forlVa[f±(1 - sin 2 4~ sin 2 0) + S II sin2 ~b sin 2 0],
(B.6)
Fy = .forlVa[ql - f. II) sin ~b cos 4b sin 2 0],
(B.7)
F~ = - f o ' o V a [ ( f i - [ 1 1 ) sin ~b sin 0 cos 0].
(B.8)
420
B.M. AIZENBUD
S i m i l a r l y f o r t h e flow a l o n g t h e y - a x i s ( V = V~2) we h a v e :
Fx = fo~Va[(fz - f II) sin 4' c o s ~b s i n 2 0],
(B.9)
Fy = fo~Va[f±(l - c o s : 4~ s i n : 0) + f II c o s : 4~ sin 2 0],
(B. 10)
Fz = f o ~ V a [ ( f i - . f l l ) c o s ~b sin 0 c o s 0]
(B.11)
a n d f o r t h e flow a l o n g t h e z - a x i s ( V = V~3) w e h a v e :
F, = f o ~ V a [ ( f ~ - f l l ) sin d~ s i n 0 c o s 0],
(B.12)
Fy = f o ~ V a [ ( f ± - f l l ) c o s ~ s i n 0 c o s 0],
(B.13)
F~ = fo~Va[f± sin 2 0 + f l l cos2 0].
(B.14)
Appendix C S o m e properties of the polarizability tensor L e t t h e p o l a r i z a b i l i t y t e n s o r a in t h e m o l e c u l a r b o d y s y s t e m o f c o o r d i n a t e s have the following representation:
ab.s. =
/i0:) a2
0
,
(C. 1)
a3
i.e. x ' , y', z ' a r e t h e p r i n c i p l e a x e s o f t~. T h e n in t h e l a b o r a t o r y s y s t e m o f coordinates: ~l.s. = ATtltb.s.A.
(C.2)
If a j = a2 (the c a s e o f c y l i n d r i c a l s y m m e t r y , the s y m m e t r y axis is z') w e h a v e : a, - (al - a3) sin2 4~ sin2 0 a~.~=
(aT a3) sin4, cos~bsin zO \-(a~ - ~3) sin ~ sin 0 cos 0
(a, - a3) sin 6 cos 4, sin 2 0 ctl - (a~ - a0 cos 2 d~ sin2 0 ( a l - a3) cos 4, sin 0 cos 0
- ( a , - a 3 ) sin 6 sin 0 cos 0 \ (a~-a3) cos 6 sin O cos O ) . ctl - (a~ - ct~)cos 2 0
(C.3) If t~l # a2, t h e e x p r e s s i o n s f o r t h e c o m p o n e n t s o f o0.s. a r e m o r e c o m p l i c a t e d . For example: ats..3j = ( a l - a2) c o s ~ sin ~ c o s ~ sin 0 + (a3 - a l sin 2 ~O- a2 c o s 2 ~ ) x sin 6 s i n 0 c o s 0, aLs..33 = ( a j sin 2 ~ + a2 c o s 2 ~ ) sin 2 0 + a3 c o s 2 0.
(C.4) (C.5)
LIGHT SCATTERING FROM FLUIDS
421
In order to obtain the traceless form of polarizability tensor we have to subtract ](a~ + a2 + a3) from the diagonal terms. Differentiating corresponding components of (C.3) with respect to time, we get &3~ = (a3 - al) cos ~b sin 0 cos 0 ~ + (a3 - al) sin 4) cos 20 0,
(C.6)
•33
(C.7)
~---
2(Otl - -
or3)
sin 0 cos 00.
Now we want to express ~ and 0 as functions of s~, Sz and s 3 - t h e components of angular momentum at the laboratory system of coordinates and also through ~b, 4, 0. To do it, we define (i) =
,
ll=
(sintksin0 sin0cos~ cos 0
0 0
-
co s t p ) n~ .
(C.8)
1
Let J~, J2 and J3 be the principal values of the inertia tensor, and assume that its principal axes are the same as the principal axes of the polarizability tensor. At the molecular system of coordinates Sb.s. = db.s. " ,Ob.s. = db.s.II~.
(C.9)
Then at the laboratory system of coordinates sl.s. = ATsbs = ATJ 1:1~.
(C. 10)
Because the matrices A, d and II are not singular, and because A - l = A T, we get from (C.10). : Ob.'s.dg.'~.As~
....
(C.11)
Substituting to (C.11) the matrices in an explicit form we get: 1
(~ = fj (-s~ ctg 0 sin ~b + s2 ctg 0 cos ~b + s3), [lsin,cos0 ~b = JI sin 0 +
=
(C.12)
1 . ] t - ~ s m ~b sin 0 s,
[ J,lcos20cos4, sin 0
1 .
]
t- ~33sin 0 COS t~ S2
cosd) +sin(/) Jl sl jj s2.
(C.14)
Substituting (C.12) and (C.14) into (C.6) and (C.7) we obtain:
Or3, = f ~'( O, ~ )S, + f ~'( O, ~)$2 + f ]l( o, ~)$3,
(C.15)
B.M. AIZENBUD
422
where fifO,
4,) - c~,~(~3(sin 4' cos 4, sin 20),
f2(O, 4,)-aJ
a 3 ( s i n 24,sin 2 0 _ c o s 2 0 ) ,
f3(O, 4,) --
Otl f~ -- ~3 ( c o s 4, sin 0
~33 = f~3(4,,
O)SI +f~3(O, 4,)$2,
(C.16)
cos 0),
and (C.17)
where
f?(4,, 0 ) -
a~ j, - c~32 cos 4, sin 0 cos 0,
y?(4,, 0)=
a~ J, - a3 2 sin 4, sin 0 c o s 0.
(C.18)
Similarly, formulas for other f~ can be obtained. Using the averaging operator (defined b y (3.4)) it c a n be e a s i l y s e e n that (.f~) = 0.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)
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