Rotational diffusion of a spheroidal Brownian particle

Physica ll0A (1982) 171-187 North-Holland Publishing Co.

ROTATIONAL DIFFUSION OF A SPHEROIDAL BROWNIAN PARTICLE* Boris M. AIZENBUD and IrwinOPPENHEIM Department o[ Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 11 June 1981

The rotational Brownian motion of a large particle is considered in terms of mode-mode coupling theory. A modified Debye law for the rotational diffusion coefficient is obtained.

1. Introduction The purpose of this work is a statistical-mechanical study of rotational Brownian motion. It is well known that the Stokes formula for the resisting force, F, which acts on a sphere of radius R in a stream with velocity V is F = - ~V = - 6~rrlRV

(1.1)

(here, -q is the dynamical viscosity coefficient). Einstein showed that the translational diffusion coefficient, D, is D = kaT _ kBT 67r'0R'

(1.2)

where ka is Boltzmann's constant and T is the absolute temperature. Similarly, the Stokes formula for the resisting torque, M, which acts on a rough sphere of radius R which rotates with angular v e l o c i t y / ~ is M

:

-

~r~'~ :

-

8"/rl"/R31"~.

(1.3)

Debye showed that the rotational diffusion coefficient, Dr, is Dr

kBT

=

kBT

~r = 8"tr'0R3"

(1.4)

Phenomenologically, eq. (1.2) (and similarly (1.4)) can be obtained by using the Langevin equation: mi: = - ~ + y(t),

(1.5)

* A portion of this work was supported by the National Science Foundation under Grant 4CHE-79-23235. 0378-4371182/0000-.00001502.75 © 1982 North-Holland

172

B.M. AIZENBUD AND 1. OPPENHEIM

or with the help of other equivalent methods (e.g. the E i n s t e i n - F o k k e r Planck equation method or the Random Walks method). The important point here is the special stochastic assumptions on f(t) (or some equivalent assumptions in other methods). The correct statistical-mechanical analysis must give expression (1.1) for (and similarly (1.4) for ~'r) and also give the correct behavior of f(t). All this was done for translation diffusion in ref. 1. with the help of generalized hydrodynamics* 17). In this work we shall use generalized hydrodynamics in Mori's formulation for a study of rotational Brownian motion18). We recall now the main points of Mori's theory2). Let A (r, t) be an abstract vector of all the " s l o w " local density variables. The coordinate space Fourier transform of A is

Ak(t) ~ f Mori's

ei~'r A(r, t) dr.

equation

(1.6)

for

the matrix (Ak(t)A~(0)) -= (Ak(t)A~)**), is:

of

the

time

correlation

functions,

d ( A k ( t ) A l ) = -- M ( k ) ( A k ( t ) AI),

(i.7)

dt

where the matrix M is the sum of the Euler and the dissipative matrices:

M = [-

E(k)

+ lim D(k, s)].

(AkA~) -1.

(1.8)

s~0

Here

E(k)

--- (AkAI)

(1.9)

and D(k, s) ~ f e "(e "1 :*'L'(1 - ,°?)~,,[(! -

,°?)Ak]*)dt

0

-= f e "(~i,k(t)A~) '~ dt.

(1.1o)

0

In (1.10), ~ is Zwanig's projection operator which projects any function of phase coordinates onto Ak(0). * Rotational diffusion of "bath" molecules was studied in ref. 7. The present work follows the ideas of refs. 1 and 7. **tdesignates the Hermitian conjugates.

ROTATIONAL DIFFUSION OF BROWNIAN PARTICLE

173

It is obvious that if A(r) is a slow variable, A2(r) will also be a slow variable. The linear theory disregards coupling with such slow variables. It was also shown 1'~2) that in the three-dimensional cases, this coupling leads to small corrections of the transport coefficients of linear hydrodynamics. The situation changes when the translational symmetry of the space is broken in the region of interest. It can happen, for instance, if we introduce some boundary (a study of the behavior of liquid near a wall13), or a study of the dynamics of one particle), or if we consider long range correlations (e.g. in a liquid near the critical point). In this case, we have to consider A2(r), A 3 ( r ) . . . together with A(r) or in terms of Fourier transforms, the set Ak should be completed by Bk,k, =- Ak+k'A k', Tk,k'k" ~ Ak+k'+k"A k'-k"Ak" etc., as was first noted by Fixman 3) and afterwards by Kadanoff and Swift 4) and KawasakiS). We shall follow Kawasaki's presentation of the theory*). In the next section, we shall define a set of slow variables for our system. In section 3 the equations of motion will be obtained. In section 4 we shall solve these equations and find the expression for the rotational diffusion constant. 2. Variables

We shall study rotational diffusion of a large particle. It is obvious that the particle must be nonspherical, because we have no adequate way to define a potential which will give rise to the stick boundary conditions which were assumed in a derivation of (1.3). We consider a spheriodal particle with large semi-axis I and small semi-axis a. The volume of this spheroid is (4/3)~ra21. The angular position of this particle can be uniquely given by Euler's angles 0 and q~ which define the position of the large semiaxis. In order to define local variables describing this particle, we must introduce a function analogous to ~ ( r - r) that describes the density of a point-particle; following the spirit of ref. 1 we introduce the function if r is found in the spheroid whose centre is at r, and whose large semiaxis is defined by 0 and q~, otherwise. (2.1)

3 ~ ( r - rl, O, ~p) = O, It is obvious that

f~(r

r,,O,~)dr

1.

(2.2)

V

* A very good account of Mori's and Kawasaki's theories is given in chapters 5 (B. Berne) and 6 (T. Keyes) of ref. 6.

174

B.M. AIZENBUD AND I. OPPENHEIM

W e c o n s i d e r a s y s t e m o f o n e large s p h e r o i d a l p a r t i c l e ( n u m b e r l) w i t h m a s s M a n d N - l i d e n t i c a l light b a t h - p a r t i c l e s ( n u m b e r s 2 . . . . . N ) w i t h m a s s e s m. W e a s s u m e t h a t t h e b a t h p a r t i c l e s are s p h e r i c a l (i.e., t h a t t h e p o t e n t i a l of interaction between them depends only on the distance between them) and that M >> m. W e c h o o s e t h e f o l l o w i n g set o f s l o w v a r i a b l e s . T h e o r i e n t a t i o n d e n s i t y of particle l :

Q(r, t) = 02(O(t), ~o(t))~(r - r~(t), O(t), q~(t)).

(2.3a)

w h e r e Q : ( o , q~) is t h e i r r e d u c i b l e , s e c o n d o r d e r t e n s o r * ) . F r o m n o w o n , it will b e d e s i g n a t e d b y Q(O, q~). T h e n u m b e r d e n s i t y of p a r t i c l e 1 is

n~(r, t) = @ ( r - r l ( t ) , 0(t), ¢ ( t ) )

(2.3b)

a n d the l i n e a r m o m e n t u m d e n s i t y

g(r, t) = p l ( t ) ~ ( r - rt(t), O(t), ~(t)) + ~ pi(t)~(r - rj(t)).

(2.3c)

T h e F o u r i e r t r a n s f o r m s o f t h e s e v a r i a b l e s are, r e s p e c t i v e l y ,

t~k = Q2(o, ~) x ( z ( k , O(t)) e ik''lt~,

(2.4a)

n[ = X(z(k, O(t)) e ikr'"l,

(2.4b)

gk =

Pl( t ) x ( z ( k , O( t )) ei~r'(')+ ~ pj( t ) e ikr~"',

(2.4c)

j>l

w h e r e t h e f u n c t i o n X a n d its d e r i v a t i v e s are d i s c u s s e d in a p p e n d i x A** t). A s w a s d i s c u s s e d in t h e i n t r o d u c t i o n , this set m u s t b e s u p p l e m e n t e d b y b i l i n e a r v a r i a b l e s . T o t h e c h o s e n set o f l i n e a r v a r i a b l e s , we a d d t h e f o l l o w i n g

* A study of rotational diffusion can be performed with the help of many quantities connected with the Brownian particle fixed system of coordinates. For simplification of symmetry considerations, it is convenient to choose irreducible tensors. The first order irreducible tensor is just a unit vector, fi rigidly connected with the particle. In the laboratory system of coordinates its components are (sin 0 cos ~, sin 0 sin q~, cos 0). Because of the absence of coupling with linear momentum, it is more convenient for us to use Q2, whose components are Q,,~ = h . h a - ~ ~5,,~. In general, all the components of QI can be expressed in terms of 21 + 1 functions Y~,.(O,~). ** By introduction of the function ~ into our variables we assume a "hard spheroidal" potential for the Brownian particle. This "non-mechanical" way of description of the interaction does not change the physics of the problem. t We shall write ~(k, O) or even x(k) instead of X(z(k, 0)) but it should be remembered that X is a function of one scalar argument.

ROTATIONAL DIFFUSION OF BROWNIAN PARTICLE

175

bilinear variables: Bk,k, ~ n ~÷k'g-k'.

(2.4d)

It is obvious that the number density, nk, and the energy density, ek are also slow variables for the considered system. Roughly speaking, by neglecting them we neglect density and energy fluctuations, that corresponds to Stokes consideration of the problem and hopefully may lead only to a change in the numerical factor in the diffusion coefficient. We also do not consider the variables gk+k'g-k', which are relevant in a study of the hydrodynamics of the whole system. The rotational diffusion coefficient, Dr can be defined with the help of the following equation (see ref. 7):

f e-s'(Qk(t)Q*k) dt = [s + 6Dr]

1

(2.5)

0

Thus, in order to find Dr, we must calculate (QR(t)Q~k). Symmetry considerations show that Qk is coupled only with the bilinear variables, i.e., we must calculate only the first and fourth rows of the matrix M (the fourth row consists of an infinite number of rows, corresponding to different k's), or the same rows of I: and D, and the matrix (AkA~. All this will be done in the next section.

3. Hydrodynamic equations 3.1. Matrix E The first row of this matrix vanishes because of symmetry. In order to calculate the " f o u r t h " row we must have an expression for/~k,~,. It is: d /~k,~, = ~- (n~+k,g-~,)

= [ i ( k + k ' ) . - p] ~ x ( k + k ' ) + ~ x (dk + k ' -

-

x(k + k') e i(k+k')r' ik'- "7-k,,

) ] ei(,+,,)r, g-~, (3.1)

where ~ is the stress tensor which is defined by gk = ik • "~. By using (A.II), (3.1) can be rewritten as*):

* h is the parameter of spheroidality: k --- 121a2- 1.

176

B.M. AIZENBUD AND I. OPPENHEIM

Bk.k.=- {i(k + k')" [ ~ x(Ik + k'l, O) ]

+ k')K(lk + k' I, 0)A sin 0 c o s 0 0 / g _ k

-i(k

J

-ik'.

x(lk + k ' [ , 0) rr k.}C 'k~''r'

(3.2)

N o w , it c a n be easily seen that

(Bk.k,Q~) =

0,

(Bk.k,n ~+) =

i(k +

(3.3)

k')x(k)x(k')x(k + k')/[3 k~O' fl- X'(k

(Bk.k'g~) = (Bk.k,B '~k.,")=

)'

(3.4)

0.

(3.5)

Matrix D In the first r o w we h a v e * )

f

' t ' * ; dt :- F(). e st (Qkrz()Qkyz)

(3.6)

0

T h e s e c o n d t e r m v a n i s h e s . T h e third t e r m ,

f e

st (0k~,,z ( t )

i

( - ik) dt,

(3.7)

0

d i s a p p e a r s in the k ~ 0 limit. T h e f o u r t h t e r m is of m o s t i m p o r t a n c e f o r us. W e c a n r e p r e s e n t it as f o l l o w s : f

" " tk,k').~P dt ~ - ~ l e -st (Qg~.z(t)B

[i(k +

k')AA (k, k') -ik'BA(k, k')

0

- i(k +

k')CA(k,

k')] -----* k~,, - ~ ik' - [AAtk') -

B~(k') -

CAtk')],

(3.8)

where

A~(k,k')=-[3 i e st(Ok.y,(t) 1 P~gk'X(Ik + k'l, O) e -~(k+~')n)~

dr,

(3.9a)

0

B~(k, k') :- [3 ~ e "(Qk.y~(t)'~k,x(Ik + k'l, O) e

i~k+~,),,) ~ dt,

(3.9b)

0

• We shall consider the y - z component of Q-~, The expression for D~ is the same for all elements of Q".

R O T A T I O N A L D I F F U S I O N OF B R O W N I A N P A R T I C L E

177

C~(k, k') -~ [3 I e ~'(Qk.yz(t) sin 0 cos 00gh,K(lk + k'[, 0) 0

× e i(k+k')rl) ~ )t i(k + k') dt.

(3.9c)

As the k ~ 0 limit is taken, A, B and C b e c o m e functions of k' only. In the " f o u r t h " row, the first element (in k ~ 0 limit) is just give by' (3.8) but with opposite sign. The s e c o n d term vanishes b e c a u s e of spatial s y m m e t r y . The third e l e m e n t disappears as k ~ 0. The f o u r t h term can be r e p r e s e n t e d as follows* (see ref. 1):

f e-S'(Bk,k,(t)Bh.k,,) ~ dt = - N m [(k + k') 2 D2(A)+ k'2vo(h)] ~k'k".

(3.10)

r'0

It must be noted that D2(h) and u0(h) contain x2(k + k', O) a v e r a g e d o v e r 0. 3.3, A few words about the matrix (AkA~) -~ We shall not need its explicit expression, so we just note that it consists of three diagonal blocks.: (QkQ~) -~, "~nknk i J,,-i ~ , and the block U. Its elements will be d e n o t e d by the n a m e s of the variables which g e n e r a t e the c o r r e s p o n d i n g elements of the matrix (AkAtk). 3.4. Equations Substituting (3.6) and (3.8) into (1.8), and then substituting the final expression for M into (1.7) we have d -dt- (Qk,yz(t) Q~,yz) = - 15F~(Qkyz(t) , Qk,yz) * 1 ~

~k' ~ [- ik'~(AAk")- B~(k") - C~(k"))] UBrBk,(B~.k(t)Q*k,yz) ,

.

+ [- ik"(A~(k") - B~(k") - CAk"))] UBk.ak,(B~,k'(t) Q*k,yz).

(3.1 1)

We see that we also need equations for (B~:~,(t)Qk.y~), i.e. the " f o u r t h " r o w in the matrix e q u a t i o n (1.7). It is: d v,~ "~ d~ (B k,k,(t) Qk,r~) = --

[Ax(k') - Ba(k') - Ca(k')l(Qk.,z(t)Q*k.yz)

Nm [3 ~k' k'2[D:(h) + v0(h)] Ua~,ok,(B ~.k"(t)Qk,yz). '~ *

(3.12)

• In ref. 1. rI is used for the kinetic coefficient of viscosity. We use u for this purpose, our "q ~-- v p .

178

B.M. AIZENBUD AND I. OPPENHEIM

4. S o l u t i o n of s y s t e m ( 3 . 1 1 ) - ( 3 . 1 2 )

We designate:

K(t) =- (Qk,yz(t) Qtk.yz); C~,":(t)=-(B~,:~,(t) Q,.~z)*

(4.1)

I((s)~ f

(4.2)

and

e ~'K(t)dt;

C~;~(s)~ f e stC~;Z(t)dt.

0

0

T a k i n g i n t o a c c o u n t t h e f a c t t h a t C~;Z(t = 0 ) = 0, b y p e r f o r m i n g t h e L a p l a c e t r a n s f o r m o f (3.12) w e o b t a i n 1

sCU(s) = - ~ ik'~,y[aA(k') - BA(k') - C ~ ( k ' ) ] / ( ( s ) - --1 [3

N m ~ k ' 2 [ D 2 ( h ) + uo(h)] UBk ~k,,C~,, ~ .z(s).

(4.3)

k"

A s s ~ 0, w e find t h a t * )

UBk'Bk'C~",Z(S) = -

ik'~ y[A~(k') - Bh(k')CA(k')] l((s) ' Nmk'2[D2(h ) + u0(h)]

(4.4)

W e r e t u r n to (3.11). C h a n g i n g t h e o r d e r o f s u m m a t i o n s in its r i g h t - h a n d p a r t , w i t h t h e h e l p o f t h e d e f i n i t i o n (4.1) w e h a v e

15F;K(t)-~

dot K ( t ) = -

~k', ~,, {ik~tAx(k")- Bx(k")-C,(k")]

+ ik"~[Ax(k") - Bx(k") - Cx(k")] UB~,,B~,,Ck'.

Ua,,.,.C~, (4.5)

L a p l a c e t r a n s f o r m a t i o n o f (4.5) as s -->0 g i v e s :

- K(t = 0) = - 1 5 F 6 / ( ( 6 ) - ~

a=y,z

~,, ik"[Ax(k") - Bx(k")

^nota (s). - Cx (k")] ~ UBrBk, Ck,

(4.6)

k'

S u b s t i t u t i n g (4.4) i n t o (4.6) w e g e t (for s ~ 0 ) : ,

- (Qk,yzQk, yz) = - 15F(~/((s) - k~~

k~2+ ki2 [ A x ( k " ) - BA(k")- C~(k")] 2 g(s). [D2(h) + P 0 ( ~ ) ] (4.7)

* Solving an infinite system of linear equations, or, in other words, inverting a linear operator in Hilbert space, we must be sure that this operator has continuous inversion. It is enough to show that the operator is bounded (see e.g. chapter IV of ref. 8.). With the help of additional physical assumptions, the analogous problem was studied in ref. 1. Because all these assumptions do not influence our result (at k ~ 0 , s ~ 0 ) we shall not study this problem here, i.e. we just assume that our further manipulations are correct.

ROTATIONAL DIFFUSION OF BROWNIAN PARTICLE

179

By comparison of (4.7) with (2.5) we find that Dr=~F;+12

1 ,~< k"2+k'~2[A~(k")-B~(k")-Cx(k")]2 63 [3Nm Ik kc ' ' y k"2 [D2(h)+ v0(h)]

(4.8)

We perform a h-expansion of the second item in (4.8) and we consider only the first nonvanishing term of this expansion. Our discussion can be easily applied to higher terms in h, but it will not lead to any additional interesting information. We first note that lim A~(k") = lira B~(k") = lim C~(k") = 0, ,~0

~.~0

h~0

(4.9)

because of symmetry. The first nonvanishing term is of first order in h and the contribution of the denominator is of zero order in h. But lim[D2(h) + v0(h)] = [D2 + v0] gE(ak"), ,~0

(4.10)

where D2 and v0 are defined by eq. (39) of ref. 1. Assuming 1) v0 >>D2, replacing Ek,, by W(2~r) 3 f dk" and changing the dummy variable k" to k we can rewrite (4.8) as follows: 2 V f k~ + k 2 [A~(k) - B~(k) - CA(k)]2 Dr = 2.5F6 + ~ Nm[Jv8~r 3 k2 x2(ak) dk.

(4.11)

It is shown in appendix B, that the first order (in h) terms of A, B and C contain factors xE(ak), then we must calculate only the integral: kc

x2(ka) dk.

k~

0

(4.12)

For a large Brownian particle, the integration will be cut off by x2(ka) before kc is reached, and the upper limit in (4.14) can be replaced by oo: kc f

oo 2

2

k~

. ~ f 2~ x2(ka) dk J k~

0

2

4k 2 x2(ka) dk = a---r.

(4.13)

0

Finally, with the help of (4.13), (4.11) can be rewritten in the following fashion:

or= ro + rl-

kBT

h,

(4.14)

where F1, is a constant and F0 --- 2.5F~ is the molecular diffusion coefficient. We could not estimate Fo in terms of the mode-mode coupling theory, but

180

B.M. AIZENBUD AND I. OPPENHEIM

with the help of kinetic theory one can show that F0~ I;l-4 and then it disappears for large Brownian particles by comparison with the hydrodynamic part. Indeed, at low densities, the bilinear variables (which are responsible for the appearance of the hydrodynamic part of Dr) are not important 10,11), and then the rotational diffusion coefficient, which is calculated for a low density gas with the help of the Boltzmann equation, will give an estimate for f'0. D.W. Condiff and J.S. Dahler showed 9) that for a large Brownian particle in a dilute gas, there is no coupling between translational and rotational motion, and that

Fo~ n il--[~/k-~T l 4

(4.15)

(see in ref. 9 the discussion after formulas (39) and (59), where m2= m m~ = M; K~= K2= 0.4; (r~ -- 2a; ~r = a). The replacement of smooth spheroids by rough spheres changes only the numerical factor in Dr. Finally, for a large Brownian particle we have

kBT A.

Dr = C, ~ -

(4.16)

Of course, Dr disappears when a spherical particle (A = 0) is considered. Formula (4.16) is the Debye law. The value F I = l/8~r was obtained by Debye for a rough sphere. Light scattering 13) and NMR jl) showed that relaxation times evaluated with the help of (1.4) are smaller than observed by 5-20 times. The attempts to explain this disagreement (mostly by considering Edwardes' solution for the creeping rotation of ellipsoid 15) also failed. Thus Hu and Zwanzig 16) considered slip or almost slip conditions in hydrodynamic calculations for molecules; these calculations strongly decrease the friction (for not very oblate molecules) and correspondingly increase the relaxation time. Our consideration corresponds to the slip conditions. In this case, the Dr disappears for spherical molecules; for nonspherical molecules there is coupling with the translational motion. The actual computation of F~, is complicated and strongly modeldependent.

Acknowledgments We are grateful to Dr. A. Onuki and to Dr. I. Goldhirsch for helpful and interesting discussions. One of the authors (BMA) would like to acknowledge the support of a Chaim Weizmann Postdoctoral Fellowship.

ROTATIONAL DIFFUSION OF BROWNIAN PARTICLE

181

Appendix A In this a p p e n d i x , w e p r e s e n t a c a l c u l a t i o n of t h e F o u r i e r t r a n s f o r m o f the f u n c t i o n ~ ( r - rt, 0, ~ ) w h i c h is d e f i n e d b y (2.1). W e first n o t e t h a t

f ~(r rl, 0,~) eik.r d r

eik.rl C

~ ~(r, 0,~) ei~.r dr.

V

(A.1)

V

It is m o r e c o n v e n i e n t to c a l c u l a t e the last integral in t h e b o d y - f i x e d s y s t e m of c o o r d i n a t e s (b.s.), w h i c h is c o n n e c t e d with the l a b o r a t o r y - f i x e d s y s t e m o f c o o r d i n a t e s (1.s.), b y the f o l l o w i n g e q u a t i o n :

(A.2)

zh,s/

zl,s/

where _cos ~ sin ~ c o s 0 sin q~ sin 0

A=

sin q~ cos ~ cos 0 - c o s q~ sin 0

0 / sin 0 cos 0 /

(A.3)

and

~=A x.

A

(A.4)

In the b o d y s y s t e m of c o o r d i n a t e s , the last integral in (A.1) is:

j ~ f ~)(r,O, go)eik.rdr_

3

f

.... O)dx d y dz. e 'k(y sin0+

(A.5)

V

R e p l a c i n g k sin 0 --- k,, k c o s 0 ~- kl, a n d using c y l i n d r i c a l c o o r d i n a t e s (x = r c o s ~, y = r sin q~, z = z) we c a n r e w r i t e J as f o l l o w s : 3

u

2~

t ~

J = ~-~q f r [ f e'k°"~'"" d,~ l [ 0

0

f

e'k'~dz ] dr

~

a

3

2

r[27rJo(k~r)][~sin(kli~/1 r2/a~)]dr

-

0

= 3 sin z - z c o s z z2 z4 z3 --=X(Z) = 1 - - T 6 + ~ - ~ 6 + • • •

(A.6)

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B.M. AIZENBUD AND I. OPPENHEIM

where z = X / ( a k . ) = + (lk/) 2 = k a X / 1 + h cos 2 0

(A.7)

A = 12/a 2 - 1.

(A.8)

and

Finally f~(r

-

rl,

0, ¢) e ik'" dr = ei~r'X(z(O)).

(A.9)

We use also the fact that dx 3 2 (i z 2 d-~-=~(z sin z - 3 sin z + 3z c o s z ) = - z ~ - ~ +

.....

)

zK(z),

(A.10)

where K(z) =

3(3 s i n z - 3 z c o s z - z : s i n z ) _ z5

I

z~

-~-7~+....

(A. II)

Now: dg_ -ik dt

---

• i k a 2 K ( z ) h sin 0 cos O0

(A.12)

and dx _ ik • ka2K(g) c°s2 0 d~ -

~

(A.13)

Appendix B

In this appendix we analyse the h-expansions of A~, B~ and CA which appear in (3.9). We begin with A~, which in the k ~ 0 limit is the Laplace transform of --~ ( O ( t ) [ p l g k , x ( z ( k ' , 0)) e-i~>']) ~.

AAt)

(B. 1)

Because o f r o t a t i o n a l s y m m e t r y lira A~(t) = 0.

(B.2)

X-,O

Then for small h .

A~(t)--~

dAx(t) dA ~o" h + (7(A2).

(B.3)

ROTATIONAL DIFFUSION OF BROWNIAN PARTICLE

183

Now we shall calculate dAx(t)/dh]x=o. Remembering that h is found in gv, X, in the evolution operator and in the averaging operator, we can write down the following equation*): ddt(t) ~=0 =/3M {"

+" uX e-i,.r,] [ dh J I~=o/ + f

[eitl-~)Lt( 1 - ~)0]lx=0 [Plgvx e

_ik,.rllt

d

ll;,=o~-~(f~quil))[h=odr

phase space

_~ [3 (A + B + C + D).

(B.4)

M

Before analysing these terms, we note that X[~=0= x(ka), ~[~=0= r(ka) and L[A=0 is the Liouvillian of the system with a spherical (and then rotating free) large Brownian particle.

B.1. Analysis of B By using the definition of g~ (2.4c) and (A.13) we see that dgv dh A=o=-Pl

eik,.r,

k,2a2K(k,a)co; 0

(B.5)

and then B = - (Q(t) cos 2 0)

k'2aZK(k'a)x(k'a) 2

(plpO = 0,

(B.6)

where the fact of free evolution of rotation and then the possibility of separation of angular and other variables were used.

B.2. Analysis of C By using (A.13), C can be rewritten as follows:

C = - (Q(t) cos 2 0) *

k'Ea2•(k'a)

~

.

~ptgv

e-ik.~)

= o.

(B.7)

fequil means equilibrium distribution function; dF is the differential of the volume in phase

space.

184

B.M. AIZENBUD AND I. OPPENHEIM

B.3. Analysis of D We can use the canonical ensemble averaging operator ( Then D can be rewritten in the following form:

D = - ~(Q(t)l:o[plgk,x(k'a)

e

ik'r ,, dH 1 A=~~- A-0/"

)=- Z ~f e ~" dU.

(B.8)

But the part of H, containing h is independent of linear nomentum, and because Q is a function of 00, ¢0 and conserved angular momentum, the averaging on linear momentum is separated, and we have:

(

d.

D = Q(t)~ ~

M

A :o ~ X"(k'a) =- - D , ~ - x2(k'a).

(B.9)

B.4. Analysis of A First we write down the Hamiltonian of our system: 9

H = ~ +pi

~ 2 L + U ( r N ) + Trot+&(rl . . . . rN, O,q~). .. 2m

Here U(r N) ==-Ei.j. 1

U(Ir,~l)

(B.10)

is the potential energy of the interaction between

bath particles, 4~ is given by

&==-~ Utj(rl, ri, O, cp)

(B.I I)

i>1

and Tro, = ~(J,`0 ~+ J2`0~ + J3.O]),

(B. 12)

where 12~, 12:, .Os are the angular velocities in the Brownian particle fixed system of coordinates (the axis z is along the large axes of the spheroid). By using the fact that J~ = J:-= J, 41 = 0 and expressing `0 in terms of Euler's angles we can rewrite (B.12) as follows: Trot = ~[J(q52 sin: 0 + ~J:) + J3q52cos 2 01 = ~ [ ~ b J ( s i n 2 0 + ~ +2 c o s 2 0)+02].

(B.13)

Before performing the expansion of qS, we give it a more concrete form. We represent U~i as - X 2 q_

U,i = rlUo 1 - 1\ ~(12

2

2-

+ L12~]

/ /,

b.s.J

(B.14)

ROTATIONAL DIFFUSION OF BROWNIAN PARTICLE

185

where

0, x < 0 , ~u°(x)=- Uo, x >-O,

(B.15)

(U0 is some large energy). Here the subscript "b.s." means: "Brownian particle system of coordinates" (or " b o d y system"). Expressing xlj, ytj and z~j with the help of (A.3) we can perform the A expansion of H : 2

2

2

2

/

2

_ Pt +S'-~--+ U(rN)+P~+P°+~f rluo~l -r;j~ H-~ ~2m 2J a:} +A

cos2 0 + U o ~ ~ 1 J

~p sin 0 + zij cos 012~ × [xli sin q~ sin 0 - yq cos a2 J ~- H0 + ,XH~.

(B.16)

Here P~

O~b =J~b sin20

h+2cos20

,

0 T~ot_ j0.

(B.17a) (B. 17b)

Po- aO

Now we can write

(B.18)

L = L o + AL~,

where L0 is the Liouvillian of the system with a spherical Brownian particle, and Li ~ [Hi, ].

(B.19)

With the help of the identity l

e (iL°+ixL')f = e iL°t + h I elL°U-')iLl eiL°" d r +

(B.20)

J

0

we can rewrite A as follows: t

A = ~ ([e iL°('-T)iLl

eiL°~o(o)][plgk,x(k'a) e ik'rt])dr

0

----i ([eiL°U ~) iLtQ(T)][Plgk'x(k'a) e-ik'" r,]) dr, 0

(B.21)

186

B.M. AIZENBUD AND I. OPPENHEIM

but iLtQ = -

sin 20 c o s 20 c o s ¢ -

2 j 2 c o s - 0 c o s 20 sin ¢ + M

c o s 2 0 sin 20 c o s ¢

6 1-

x 2[xli sin q~ sin 0 - Yli c o s ~0 sin 0 + z~i c o s 0]

Ja 2

I'

× ~(x u c o s ¢ sin 0 + Yu sin ¢ sin 0) sin2 0 sin - ( x usinq~cosO- y u c o s ¢ c o s O - z usin0)cos20cos¢}. Because

(B.22)

a v e r a g i n g o n a n g l e s is s e p a r a t e d , t h e p a r t o f iL~O c o n t a i n i n g

disappears and then averaging on linear momentum

r~i

is s e p a r a t e d , w h i c h finally

l e a d s to

A = A~ ~ x2(k'a).

(B.23)

Finally

AA(t) = (A~ - DO x2(k'a)A + ¢)'(A2). BA c a n b e r e p r e s e n t e d

(B.24)

in a s i m i l a r w a y , a n d CAt) = t'~,'(X:).

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)

T. Keyes and I. Oppenheim, Phys. Rev. A8 (1973) 937. H. Mori, Progr. Theor. Phys. 33 (1965) 423. M. Fixman, J. Chem. Phys. 36 (1962) 310. L.P. Kadanoff and J. Swift, Phys. Rev. 165 (1968) 310. K. Kawasaki, Ann. Phys. (N.Y.) 61 (1970) I. B.J. Berne (editor), Modern Theoretical Chemistry, V.6 (Plenum Press, New York, 1977). T. Keyes and I. Oppenheim, Physica 75 (1974) 583. A.N. Kolmogorov and S.V. Fomin, Elements of the Theory of Functions, Functional Analysis (Nauka, Moscow, 1976). D.W. Condiff and J.S. Dahler, J. Chem. Phys. 44 (1966) 3988. T. Keyes and I. Oppenheim, Physica 81A (1975) 241. J.R. Dorfman, W.A. Kuperman, J.V. Sengers and C.F. McClure, Phys. Fluids 16 (1973) 2347. T. Keyes and I. Oppenheim, Phys. Rev. A7 (1973) 1384. B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley-lnterscience, New York, 1976). H.W. Spiess, D. Schweitzer, U. Haeherlen and K.H. Hausser, J. Magn. Res. 5 (1971) 101. D. Edwardes, Quart. J. Pure Appl. Math. 26 (1892) 70. C.M. Hu and R. Zwanzig, J. Chem. Phys. 60 (1974) 4354.

ROTATIONAL DIFFUSION OF BROWNIAN PARTICLE

187

17) Recently, R.I. Cukier, R. Kapral, J.R. Lebenhaft and J.R. Mehafley, J. Chem. Phys. 73 (1980) 5244, considered this problem by using kinetic theory and generalized hydrodynamics of hard spheres. They also compared these approaches. 18) Translational and rotational Brownian motion of large particle was considered by A.J. Masters and P.A. Madden, J. Chem. Phys. 74 (1981) 2450, 2460. They also used a mode-mode coupling approach but used a different set of variables and introduced some phenomenological assumptions.