Transport and relaxation phenomena in suspensions of dipolar Brownian particles in an electric field

Physica 1llA (1982) 161-180 North-Holland

TRANSPORT OF DIPOLAR

Publishing Co.

AND RELAXATION BROWNIAN

PHENOMENA

PARTICLES

IN SUSPENSIONS

IN AN ELECTRIC

FIELD

W.E. KijHLER Institut fiir Theoretische Physik der CJniuersitiit Erlangen-Niirnberg,

Erlangen, F.R. Germany

and H. KAGERMANN Lehrstuhl B fiir Theoretische Physik der Technischen Uniuersitiit Braunschweig,

F.R. Germany

Received 29 July 1981

A system of noninteracting nonspherical Brownian particles with dipole moment parallel to the symmetry axis is studied in the presence of an external homogeneous electric field. A generalized Fokker-Planck equation is introduced which takes into account anisotropic translational and rotational friction as well as a translational-rotational coupling via a Magnus force term. From the corresponding kinetic equation for the reduced distribution function for rotationally overdamped particles a system of transport-relaxation equations for macroscopic variables is derived. They are used to treat the particle diffusion under the influence of the electric field and to calculate the spectral functions for infrared absorption and depolarized Rayleigh light scattering.

1. Introduction

In the last years the interest in the theory of suspensions of Brownian particles has mainly developed in two directions: (i) nonequilibrium phenomena in systems of interacting Brownian particles’), and (ii) kinetic description of Brownian particles with nonspherical shape2v3T4).This paper is concerned with the second subject only: The dynamics of a dilute system of nonspherical Brownian particles is described in terms of a one-particle distribution function which depends also on variables characterizing the orientation of the particles in space and their rotational motion. With these additional variables interesting physical phenomena are connected, e.g. diffusio birefringence’) and flow birefringenceq as well as infrared absorption, and depolarized Rayleigh and Raman light scattering by optically anisotropic particles5). In the present paper, a system of noninteracting, nonspherical dipolar Brownian particles is studied which are supposed to possess a symmetry axis u in the direction of the dipole moment d. An external homogeneous electric field shall be present which interacts with the particle dipole 03%4371/82/OOO&OOOO/$02.75 @ 1982 North-Holland

162

W. KijHLER

moments. electric

The

system

is considered

field at the position

may be neglected and angular

to be sufficiently

of a particle

compared

In the first section, linear

AND H. KAGERMANN

with the external

the stochastic

velocity

in ref. 4, in the velocity

caused

of a particle equation

electric

Langevin

so that

dipoles

of motion

In contrast

between

the net

field.

equations

are given.

a coupling

dilute

by the surrounding

for the

to the treatment

angular

(OJ) and linear

(u) velocities is introduced by means of a Magnus force term6) (proportional to w x u), while in the equation for the angular velocity a systematic torque term (which has been used in ref. 4 to describe flow birefringence and which is supposed to be small) is disregarded here. The external electric field enters the angular velocity equation via the torque being exerted on a dipole in a homogeneous field. Translational and orientational friction terms are assumed to be orientation dependent. The pertaining Fokker-Planck equation for the single particle distribution f(t, x, u, u, o) (t = time, x = position of the center of mass, u = velocity, u = orientation vector, o = angular velocity) is stated. In the further treatment, the case of overdamped particle rotation is considered and the Fokker-Planck equation for the reduced distribution p(t, x, IJ, u) is derived. In section 3, from the reduced kinetic equation a system of transportrelaxation equations for macroscopic variables is obtained via the moment method3,‘). In particular, the moment equations for mean particle velocity (u), orientation vector (u), friction pressure tensor (m (5;)), orientational tensor polarization (z), as well as for the fluxes of vector-((uu)) and tensor-((uE)) polarization are studied. The relaxation coefficients occurring are expressed in terms of the parameters characterizing the rotational motion of the particle and its interaction with the solvent medium (friction constants). The electric field is taken

into

account

only

up to terms

of second

order

orientational energy of the dipole, dE, and thermal energy considered as small for room temperature. In the last section (4), the transport-relaxation equations treatment

of several

phenomena

connected

in the ratio

k,T

of

which

can be

are applied

to the

with the orientational

variables.

First, the influence of the external electric field on the diffusion of the Brownian particles is studied and an expression for the diffusion tensor is given. The Magnus force proves to be decisive for the anisotropic diffusion in the field. Then the autocorrelation functions of the thermal fluctuations of orientational vectorand tensor-polarization are calculated by use of the fluctuation-dissipation theorem and the corresponding spectral functions (which become field dependent second and fourth rank tensors, respectively) are evaluated. They determine the infrared absorption spectrum and, for optically anisotropic particles, Raman light scattering.

the spectra

of the depolarized

Rayleigh-

and

SUSPENSIONS

163

OF BROWNIAN PARTICLES

2. Kinetic equations

2.1. Equations

of motion

We consider neutral Brownian particles of axial symmetry with mass m, tensor of inertia 8 and dipole moment d suspended in an isotropic medium with mass density p and viscosity 7. The suspension should be sufficiently dilute to disregard the interaction of Brownian particles. The particles are assumed to be homogeneous rigid bodies of revolution, the orientation of which with respect to a space fixed coordinate frame is specified by a unit vector u in the direction of the symmetry axis and of the dipole moment d = du. The state of the system is thus determined by the position of the center of mass, x, the velocity u, the orientation vector u, and the angular velocity w. The motion of the dipolar Brownian particle under the influence of an external homogeneous electric field E = Eoe is governed by the equations i = CIJV, ; = -$(I+ Ii = o”,n

(la) v+ $T(U) * p(t)+p(WO*

f-2)x v,

(lb)

x u,

d = -y”(u) f D + +R(~)- eR(t) +t

(lc)

w”,u X e,

(14

where the dimensionless linear and angular velocities introduced by use of the units

V and fl have been

co : = d2kBT/m,

(2)

for the velocity, and o” := o;(S - uu) + wf;uu =: ~2ksT/OL(S - uu) + ~2kJN,,

uu,

(3)

for the angular velocity, respectively, and where T is the temperature of the suspension, ke is Boltzmanns constant and 011and O1 are the moments of inertia for rotations about u and an axis perpendicular to u, respectively. The influence of the surrounding medium has been taken into account by a linear friction law of the Langevin-type and a spin-orbit coupling force of the type w x u which causes the particle to roll aside if its internal angular velocity is not parallel to its velocity (Magnus effecC)). Due to the fact that the friction coefficient depends on the angle between the symmetry axis of the particle, u, and its velocity u, the friction tensor reads ,yT(U)=y~(6_uU)+YifUU:=YT~+YI~,

r:=+r:,

(4)

W. KGHLER

164

AND H. KAGERMANN

where u’; : = uu - $6 denotes the traceless part of the tensor uu. According to the laws of hydrodynamics, the isotropic friction constant is given by -yT:= (2-y: + $)/3 = 67rr@m (the effective radius R depends on size and shape of the particle’), and the correlation factor xT:= -yz/rT is a measure for the effective nonsphericity of the particle. In the case of rigid ellipsoids, e.g., xT has the bounds - 3 s xT s 3. In addition to this correlation between translational and orientational degrees of freedom another coupling between translational and rotational motion has been taken into account by the transverse Magnus force El’:= -pmro

x w,

(5)

which can be derived from the potential

(6) Here, mf is the mass of the fluid contained in a volume equal to that of the Brownian particle, L is the vector of orbital angular momentum and 6 is a dimensionless constant of the order of magnitude unity, which depends on the particle aspect ratio r (e.g. r > 1 for a prolate spheroid; for a sphere, 6 has been calculated by several authorss9p). From reasons of convenience, o has been decomposed into its parts parallel and perpendicular to the symmetry axis according to eq. (3) (both parts being scaled with the respective equilibrium frequency). Effects of higher order in the dynamical variables are neglected4). The externally applied homogeneous electric field E = EOe gives rise to a torque m = d x E, written in a dimensionless notation as M =

(o:OJ’d x E = co”,2

B

uXe=:co-uXe,

Off

(7)

2

which tends to orientate the particle parallel to the field. The magnitude this torque which can be derived from a potential rC,according to

is characterized by the ratio of “orientational” o =: dEo/kBT. The differential operator

corresponds to the operator Brownian particle.

of internal rotational

and

thermal

angular momentum

of

energy,

of the

SUSPENSIONS

OF BROWNIAN PARTICLES

165

The orientation distribution is then determined by the competition of a random disorientating Brownian torque acting against the tendency towards a preferential orientation resulting from the action of the external torque M. As usual, this random torque yR(u) - tR(t) as well as the random force +T(~) - eT(t), is described by means of a stationary Gauss-Markov-process with zero mean

5A(t)= 0; (*(t)~*(t’) = S(t

- t’)6;

A = T, R;

6T(t>5R(t’) = 0.

(10)

Their magnitudes are characterized by the elements of the matrices 9*(u) (A = T, R), which are determined by the fluctuation-dissipation theorem. 2.2. Fokker-Planck

equation

It is well known’@), that the equations of motion (1) together with the properties (10) of the stochastic forces are equivalent to a Fokker-Planck equation for the nonequilibrium distribution f(t, X, V, IL,a) of the Brownian particles:

(~+c~v.&$.

a

[ I +oo,f2,.L+&. [ rR(U).~+14”(u).4R(U).~-~WOlUXel) yT(u)* v+tqT(u)* 9T(u)E- p(wO*n>xv

(11)

x f(t, x, v, u, 0) = 0.

For thermal equilibrium, the distribution function is given by the MaxwellBoltzmann distribution in the presence of an external field, fo( V, u, a) = noC3 e-(V2+R3cx(4~sinh a)-’ eau ‘e,

(12)

which is characterized by the density no and the energy ratio (Y= dEo/kBT. For (Y= 0, the usual Maxwell-Boltzmann distribution is recovered. To determine the covariance matrices of the stochastic force and torque, respectively, we observe, that f. has to be a stationary solution of the Fokker-Planck equation (11). This condition leads to y*(u) = y*(u) - 9*(u),

for A= T, R.

(13)

Because we are interested mainly in overdamped particle rotations (for the other limiting case see Hess”)), i.e. o” * (yR)-’ Q S, it is sufficient, to work with the reduced distribution p(t, x, V, u) defined by p(t, x, V, u):=

J

d3flf(t, x, V, u, 0).

Following the steps made in ref. 4, we multiply eq. (11) successively

(14) with 1

166

W. KiiHLER

and a,

integrate

over

a

AND H. KAGERMANN

and obtain

the following

coupled

differential

equa-

tions: (15) for the probability

density

p, and

D it jit + 4% + WY& * M,a,,),

+ PY” - p(w;;W,,.n,,)n

+ &%fi,,>J

= 03 (164

g

j, + rYjl + w”,% * p(fl~flJn

+ P% * P(wi(a~lflni)n

+ w01(flLJ2,),)

= g wt(u X e)P for the components and perpendicular the abbreviations

(16b)

of the corresponding (jJ

to the symmetry

probability

fluxes parallel

(ill)

In eqs. (W-(17),

D -:=~+cov.&-~.yyu).(“+;$); Dt

(17)

~“=Vx~=uu..r”+(S-UU).~“=:.r~+u;,

(18)

d’fld$$=: p(fi~~>n; jL := p(Oh

jii := are used. equivalent

Then we disregard to the assumption

by their equilibrium (a,,n,,>n

= ;uu;

j~~=-$@ZLp;

ones: (fl,fi,h

= ks - uu);

&-

;r,(p&

+ YU)z- &,p%&

+:rlcl(p6P:+Yu)*(UXe) the

parameters

(a,@,>, to zero.

= (n,a,,>, Then

= 0,

(Y and

(20)

the following

jL-${auxe-%-P_Y+)Pi

which relate jll and jl with p in a linear yields the desired reduced Fokker-Planck

I

(19)

terms of higher than second order in o” which is that the following mean values can be replaced

and set the D/DC-terms in eq. (16) equal stitutive equations result:

Here,

density

axis u, respectively.

con-

(21)

way. Insertion equation

of eq. (21) in eq. (15)

- J&

p=o. I p characterize

(22) the

influence

of the

external

SUSPENSIONS

OF BROWNIAN

electric field and the Magnus force, respectively, are given by

If for the nonequilibrium

distribution

PARTICLES

167

and the quantities rL and I’,

p the usual ansatz (24)

P = po(l+ @) is made with p. being the reduced equilibrium distribution, deviation from equilibrium, @, obeys the kinetic equation

the relative

(25) It should be noted that the influence of the Magnus force modifies the collision operator directly via both components of the operator Z’v, whereas the interaction of the dipole moment with the external electric field enters the collision operator of eq. (25) via the field term in the equilibrium distribution

no

a

-3/2e-V2ea..., POM

P”=47FSinha

[1+au

.e+$(u

.e)2-1)+B(a’)],

where the expansion about the pure Maxwellian pOM:= po(a = 0) is valid for a % 1. It is useful to introduce the collision operator

(27) which can be rewritten by use of 3$+.Y:2=3;; in an alternative w(G) =

I

.&++o,

(28)

way:

- tpo’ &

. yT(u)po * &

- tpo’(P&

+ =mr*po * (WV + %)

Also for (Y,p # 0 the collision operator eq. (27,29) guarantees the conservation of particle number (o(1) = 0) and the increase of entropy ((@w(@))~ > 0 for

W. KiiHLER

168

AND H. KACERMANN

@ # const.) and is invariant with respect to rotations as well as to parity and time reversal operations.

3. Description

of transport

3.1. Expansion

and relaxation

of the distribution

about

the field direction

e

phenomena

function

The quantity @(t, x, V, u) which characterizes the deviation of the reduced distribution function from thermal equilibrium can be expanded with respect to a complete set of orthonormalized tensors +::I’?!~,,~, .,2(V, u) built up of the Cartesian components of V and u 3.4). Here, I, and I2 characterize the ranks of the expansion tensor in V and u, respectively, and the superscript /Q,,~ distinguishes the various tensors of equal ranks I,, I?. The 4kl!lz are without an external electric field, i.e. assumed to vanish in thermal equilibrium (30) where

(. . .)OM denotes

the

equilibrium

average

performed

with

Maxwellian pOM= (no/47r)Y3’* exp(- V’). The orthonormalization the expansion tensors is expressed by

the

usual

property

of

(31) where the Ajf!. _W,,P~...P~define isotropic tensors which, applied to an arbitrary Ith rank tensor, project out its irreducible part9.“). For I = 1,2, e.g., one has A”‘,= !-W

6

The expansion a,(t, x,

). A (2) = ~(s,& WF’Y’ fiLp9

+ 6,,,8,,*) - ts,,s,,,,.

02)

of @ then reads

v, u)

= y+

i:

i:

%

I,=0

12=0

k=l

a::112!il(l.“, “,Z(f,nM2:!?!~ ,,,l’,...

“,?(V’U). (33)

Note, that in contrast to ref. 3 the expansion tensors used here are not totally irreducible. The tensorial expansion coefficients in eq. (33) can, by help of eq. (31), be inferred as

In the following, terms nonlinear in the deviation from thermal equilibrium are not considered, i.e. the factor (n/no) in eq. (34) can be replaced by 1. The relative deviation (n/no) - 1 of the particle number density n from its

SUSPENSIONS

OF BROWNIAN

PARTICLES

169

equilibrium value no and the expansion coefficients a:$ PI,,“,_. Y*(f) x) are the macroscopic variables describing the nonequilibrium state of the suspension of Brownian particles. For these macroscopic variables a set of coupled differential equations, the so called transport relaxation equations (TRE) can be obtained from the Fokker-Planck equation (22). For all practical purposes it is sufficient to work with a suitably truncated set of expansion tensors. Since heat conduction problems will not be considered in this paper, a set of dimensionless expansion tensors is shown in table I: TABLE Vectors

+(‘O’ = gj

2nd rank tensors

(friction tensor)

3rd rank tensor

VT

tf.l”2’= vi? v G;

pressure

(flux of orientational tensor polarization)

4’3’ = d/2

v

(velocity)

+‘02 = \/F’;;

g@” = d/3” (orientational polarization)

I

vector

(orientational tensor polarization) q#(“’ = uz

vu

(flux of orientational vector polarization)

3.2. Transport

relaxation equations

For the nonequilibrium mean values of the expansion tensors (table I), a set of transport-relaxation equations is derived by multiplicating the FokkerPlanck equation (25) with each of the expansion tensors of table I, performing the integration J p. d3V d*u . . . and observing eqs. (33) and (34). Since the ratio of orientational and thermal energy, (Y= dEo/kBT, in general is very small (for a particle dipole moment d of 1 Debye, an applied electric field E. of the order of 104V/cm and room temperature, (Yis of the order of 10e3; for d = 10 Debye and T = 20 K, cz is still of the order of lo-‘), the expansion of p. in powers of (Y,eq. (26), can be used, and terms in (Y’can, in general, be disregarded unless very low temperatures and huge dipole moments are considered. Because of the applications in section 4, this neglect however, is not made for the relaxation equations of vector- and tensororientational polarization. The particles are considered to have a not too nonspherical shape, i.e. I(q, - rJ/r,l Q 1 and -ya+ yi. Then terms with c@‘(rll- r,) and Y~(Ycan also be neglected compared to terms proportional to cypzTI and ayi, respectively.

W. KbHLER

170

Finally,

the abbreviations

r,:= similar constant For equation

AND H. KAGERMANN

;r,,+ifr,;

I-,:=

T,,- I’,,

(35)

to yi and y,, are used for an average and the corresponding the

dimensionless

isotropic

orientational

relaxation

transport

relaxation

anisotropy.

velocity

(V),

the

following

is then obtained:

+l.&J

(:; 1

A’*‘i(Vur;)=O.

(36)

In eq. (36), the following relaxation Planck collision operator occur: relaxation coefficient of the velocity:

and coupling

coefficients

of the Fokker-

W( 10) = yi + P’ri; coupling

(37)

coefficient

between

velocity

and

flux

of

orientational

tensor

polarization:

and field induced coupling coefficients orientational vector polarization:

between

velocity

and

the

flux

of

(39) For the orientational vector polarization quadratic in (Y (to be used in section 4):

$ (14)+ c0V - (Vu) with the equilibrium

+ w(Ol)((u)

(u),

one

obtains

- (u)~) - &(Ol)ee . (u) = 0,

up to terms

(40)

polarization

(u)o= +ae,

(41)

and the relaxation

coefficients 2

w(ol)=r,

( ) l+%

)

(42)

SUSPENSIONS

OF BROWNIAN PARTICLES

171

2 S(Ol)

=

;

(43)

l-1.

Next, the equation for the dimensionless Brownian particles is considered:

friction

pressure

tensor

~(~)+c~v~)+w(zoKW)=o, with the relaxation

of the

(44)

coefficient

O(20) = 2yi + 3p*ri.

(45)

For the second rank orientational tensor polarization (which is important for the depolarized Rayleigh scattering by the optically anisotropic Brownian particles) one obtains up to terms quadratic in (Y: ~(U”)+C~v.(VUU~+W(02~((UCU)-(~)O)+G(02)ke(uuj=o. Here, the equilibrium tensor polarization

(u”u)o is given by

CY2 (uu)O=-i-Jee. The irreducible (e-)),.

(46)

(47)

tensor ke(uuj is defined by := A$?&e,e,(u,u,)

=

‘Giuru,),

(48)

where A E$&, : = A $sA !!,&A ?& is an isotropic 6th rank tensor irreducible in the 3 index pairs12). The relaxation coefficients are calculated as w(O2) = 3r1,

(49)

$(02) = &ly2.

(50)

Notice, that there is no direct collisional coupling of (u”u) and (‘vvj which would be needed for flow birefringence13). Finally, the transport relaxation equations for the fluxes of orientational vector and tensor polarization are derived. For their respective relaxation constants the “spherical approximation” <~#$‘y’w(~lfr?))o=00(11)6,,&~;

(~~,%@!%))o=

~0(12)&A!?,

will be made. Then the equation for the flux of vector polarization

1 +@

l1 (V)e=O, C10) 2

(5 1) is given by

(52)

W. KiiHLER

172

AND H. KAGERMANN

with CIJ,(ll)=~i+t~+P*r~,

(53)

and (54) For the flux of tensor

polarization

one obtains

(55) with W()(12)=yi+31,+p2ri,

(56)

and (57)

4. Applications In this section, the transport relaxation equations (36), (40), (46), (52) and (55) are applied to the treatment of diffusion of Brownian particles in an external electric field and to the calculation of spectral functions of the dipole moment and the orientational tensor polarization. Due to the electric field, the diffusion coefficient and the dipole spectral function become field dependent second rank tensors while the spectral function of the orientational tensor polarization becomes a 4th rank tensor. Because of the general smallness (Y= dEo/kBT, it is sufficient to study the effects in the lowest order in cx.

4.1. Diflusion

in an external

The constitutive

homogeneous

law for stationary

jn& = nv, = -D,,(E)V,n.

of

electric field

diffusion

in an applied

electric

field is (58)

Here, v,, = cO(V,) is the average velocity of the Brownian particles. Starting point for the calculation of D,,(E) is the TRE for the average velocity, eq. (36). Neglecting first in (36) the electric field as well as the orientation effects and gradients of nonconserved quantities (hydrodynamical approach), one obtains the simple relation w( 10)nOu = -ic:Vfl,

SUSPENSIONS

OF BROWNIAN PARTICLES

173

from which the isotropic diffusion coefficient in crudest approximation inferred as 2

keT0

(59)

mw(l0)’

DO=&=

can be

Taking into account the occurrence of the flux of tensor polarization, (Vu';;), but still neglecting the electric field contributions, we obtain from eqs. (36) and (55) the two coupled equations

2 vn+0(10)(v)+

A'2'.(VE)=0,

&(;;)

0

:o’ (V).A'2'=0. C 1

wo(12)( Vu’;) + L&J

(60) (61)

(Vu"u), after eq. (61), in terms of (V)and using

Expressing

A’,2AVPM Ear = W,), we find the constitutive

law

(62) from which (with the Onsager symmetry coefficient can be inferred as 5 44’ ‘-LJ(10)oo(12)

I

D=Do

~(13 = @(ii)) the isotropic diffusion

-I

(63)

1 ’

with Do from eq. (59). The result eq. (63) coincides with the result obtained by Hess’) if one observes that w(&~~ = 4/5/3w(t$ holds. Since

44’ W(lO)Wo(l2)= 15(yi+

2(Ya- Q3’r.J’ 3rl+

P’ri)(yi+

(64)

P2ri)’

and ya < yi, r, < ri, rl, the orientation effects are supposed to give only small corrections to the diffusion coefficient of not too nonspherical Brownian particles. To treat the influence of the applied electric field, we start from eq. (36) but with having (VE) already eliminated:

In order

to eliminate

also (Vu),we observe

that from

eq. (52) in the

174

W. KiiHLER

AND H. KAGERMANN

in hydrodynamical (v,~,)=

-&wO(ll)-’

approximation

(6 (:~),(e,,(V,)-(V

follows

*e)L)

+ 6 (:~)2(Vl)e,r)Insertion

(66)

of (66) into (65) yields

(;;)f- ;(:;)~)P;~

+ (?3

(::,Y-

G ($:,

PIA,.].

(67)

where P!,

= eAek;

PAL@ = & - eAe,

(68)

are second rank projection tensors, which, applied to an arbitrary vector, project out its components parallel and perpendicular to e, respectively. Since the change of the diffusion coefficient due to an electric field is small for a ~0.1, the mean particle velocity on the r.h.s. of eq. (67) may be expressed in terms of Vn according to eq. (62). Then we have again a linear connection between (V) and Vn. The diffusion tensor in the presence of an external E = Eoe can then be written as DGy = D$,,

+ D,e,e,.

(6%

with D

CD

]_&m:-;t145

I

D,, w(lO)wa(ll)

’

(70)

and D2 G3: Da= -&J(lo)wo(ll)~ Notice, that D, C 0 is always valid. For the difference of the diffusion coefficients D’ = D,,PL,. and D,,PL,, describing diffusion parallel and perpendicular to the electric respectively, AD -=pzz D

(71)

D” = field,

one finds D’--D” D

d3: a 3 2c w(10)w0(11) = 12(x+r,+Pr~,)(Y,+P?rl).

which means that D’> D” and that the anisotropy in the diffusion Brownian particles is caused by the Magnus force (p) and is quadratic particle dipole moment and in the field strength.

(72) of the in the

SUSPENSIONS

4.2. Calculation

OF BROWNIAN PARTICLES

175

of spectral functions

4.2.1. General remarks The frequency and wave vector dependences of certain physical phenomena, e.g. light absorption and scattering, are determined by spectral functions S(O, k) 5). A spectral function can be obtained as Fourier transform of a time correlation function C(t, k): S(o, k) = d Re/ e’“‘C(t, k) dt. The time (auto-)correlation defined by the relation13) a(& k) = @(t,

function

k)a(O, k),

(73) of a physical observable

a(t, x) can be

(74)

where a(t, k) is the spatial Fourier transform of a(t, x). In the following, two kinds of spectral and time correlation functions, respectively, are studied which require only the knowledge of the one particle distribution function: (i) Correlation function of the particle dipole moment d = du: C’,“?(r,k) = @u(r),WWo,

(75)

where 6u=u-(u)o

(76)

is the fluctuation of the orientation vector about equilibrium and the bracket ( . . .). denotes an equilibrium mean value. While for isotropic systems the tensor C$(t, k) is isotropic, i.e. it can be replaced by C(“)(t, k)6,,, in the presence of an external electric field E it becomes a field dependent second rank tensor. This correlation tensor is important for the description of the spectrum connected with infrared absorption. (ii) Autocorrelation function of the fluctuations of the polarizability tensor a&L,, . CjP?,,&, k) = @c+(r)6cu,,(o)>o,

(77)

where SajLv= apu - &.l”)O.

(78)

This 4th rank correlation tensor (in the presence of an electric field) determines for a dilute system of particles the light scattering spectra. In particular, if the polarizability tensor is split into an isotropic and an anisotropic part

W. KijHLER

176

AND H. KAGERMANN

according to %u = CY’&,+z:, the corresponding

correlation

C:u!e&, k) = @&h&i

(79) tensor (O))o

VW

is important for the description of dipolarized Rayleigh and Raman light scattering. For Brownian particles with symmetry axis u, the anisotropic part of the polarizability tensor is given by’) &L = (“II- aL)Z.

(81)

where (~11 and (Yeare the polarizabilities for light with the electric field vector parallel and perpendicular to the symmetry axis, respectively. Thus, for our

E# 0, becomes a 4th rank tensor. For the evaluation of the correlation tensors, the fluctuation-dissipation theorem is used, which allows their calculation by use of kinetic theory, in particular from the transport relaxation equations.

4.2.2. Dipole moment autocorrelation and spectral functions The autocorrelation function of the thermal fluctuations of the particle orientation vector u (and thus of the dipole moment d = du) can -according to the fluctuation-dissipation theorem -be obtained by solving the relaxation equation for the nonequilibrium quantity a(t, x) = (u) - (u)o = (6u).

(83)

From eq. (41) follows, that a(t, x) obeys the equation a z a + c0V - (Vu) + w(Ol)a - 5(01)ee . a = 0.

(84)

To obtain a relaxation equation for a alone, the spatial inhomogeneity term v - (Vu) has to be eliminated. This is achieved in lowest approximation, neglecting higher order coupling terms, by solving eq. (52) under neglection of the time derivative of (Vu). This is possible, since wo(11) S ~(01) holds in general. The result co(Vu)= -&00(11)-‘Vu

(85)

SUSPENSIONS

OF BROWNIAN PARTICLES

is inserted into eq. (84) thus yielding the desired relaxation

177

for a(t, x):

-$ a - D,Aa + o(Ol)a - &(Ol)ee *a = 0, with the diffusion coefficient for the orientational -=---

4

(86) vector polarization

ksTo

Do = 2w0(11)

(87)

moo(ll)’

which, in general, is different from the particle diffusion coefficient Do defined by eq. (59). The corresponding relaxation equation for the spatial Fourier transform b(t, k) then reads 5 ii(t, k) + D,k*ci(t, k) + w(Ol)ii(t, k) - i(Ol)ee

- ti(t, k) = 0.

(88)

Using the projection tensors Pm and P* (cf. Eq. (68)), we define the components of ci(t, k) parallel and perpendicular to the field direction by $l=plI.~.

a‘i=Pl.(j

,

and obtain two decoupled

(89)

?

relaxation

equations

$8” + D,k*$ + (~(01) - &(Ol))til= 0,

Wa)

$ ii’+ D,k*$ + o(Ol)iil = 0.

(9Ob)

Eqs. @a, b) have the solutions G”(t, k) = Cl(t, k)iil(O, k); cE’(t, k) = C’(t, k)BI(O, k); The field-dependent CJt,

S'(,, k)

=

D,k*)t.

+ C’(t, k)Pi,(e).

spectral tensor SJo,

S” and S’ are calculated

Mb)

(92)

k) has the same structure:

according

~(01) + D,k*- &(Ol) (w(O1) + Dak*- d(Ol))* + a*’

~(01) + D,k* s’(O’ k, = (w(O1) + D,kp* + oz.

(91a)

tensor can then be written as

S,Jw, k) = S’(W, k)Pe”(e) + S*(w, k)PL,(e). The quantities functions

$01) - D,k%,

C’(t, k) = exp(-w(Ol)--

autocorrelation

k) = C”(t, k)P$(e)

The corresponding

Cs(t, k) = exp(-w(Ol)+

(93) to eq. (73) as Lorentzian

CW Wb)

178

W. KOHLER

Due to the external width

is smaller

perpendicular

static

AND H. KAGERMANN

electric

field EOe(d(O1) m Ei), the absorption

for polarization

of the incoming

radiation

responds

to e than

to e.

For C$C*4 w(Ol)wO( 11), i.e. for large wavelengths, is essentially

parallel

line-

determined

to the case

by the relaxation

of “pressure

field induced anisotropy absorption line has been 42.3. Autocorrelation polarization

broadening”

of the “diffusional disregarded here.

and spectral

the absorption

coefficient in dilute

This

cor-

gases13). A possible

broadening”

functions

linewidth

~(01) = rl. (term

of the orientational

D,k’)

of the

tensor

The autocorrelation of the thermal fluctuations of the tensor polarization 6 of the particle symmetry axes is of crucial importance for the depolarized Rayleighand Raman-light scattering in dilute solutions or suspensions of optically anisotropic Brownian particles in an optically isotropic medium. The corresponding spectral function S(w, k) (which, in the presence of an external electric field, becomes a 4th rank tensor) determines the spectrum, if o is identified with the difference of scattered and incoming the photon momentum transfer in scattering.

frequency

and k with

The autocorrelation function of the fluctuation of tensor polarization calculated from the relaxation equation for the macroscopic quantity A(t, x) = (ur;)-(E,o,

is

(95)

which, analogously to the derivation of eq. (86), is obtained from eqs. (47) and relax(55) and the approximation (c0/2)V( u”u ) = - OO(12)( Vu”u ). The resulting ation equation for the spatial Fourier transform &t, k) is then ~A(r.k)+Dak’di(t,k)+w(02)A(t,11~+:(02)~~)=0 with the diffusion

coefficient

of the tensor

(96) polarization

D/4=&&j. Now three

mutually

(97)

orthogonal

A;‘@.= scO’(e).&;&&& A”‘:=

(9”‘(e)+

A”‘:=

(W2’(e) + 9’-*‘(e)):

components .

P”-“(e)):A=

of &t, k) are introduced:

,

(98a) 2(X--

Z(A;:Z)),

A = A - 2LC+:Z(A:

Here, the gDcm’(e) are fourth rank projection introduced in ref. 14. Since in an electric

(98b) Z).

(98~)

operators with XL=_* 9’“) = A’*‘, field the odd-in-the-field-terms

SUSPENSIONS

OF BROWNIAN PARTICLES

119

cannot occur because of parity reasons, the combinations i(B”‘- 9(-I)), i(p”“@-‘)), which in general are needed for a complete system of projection tensors do not occur here. With the relations gm(e):

‘z . d = @p’(e):

A,

(994

*ii = &W’)(e)

(@‘j(e) + 9(-‘)(e)): (W2)(e) + 9(-‘)(e)):

+ B’-“(e))* * A

(99b)

9

(99c)

7

the following system of three decoupled relaxation

equations is found:

$ ii(O)+ DAk%@)+ (w(O2) + 5QO2))A’O = 0,

(100a)

$ Ai + DAk2A(‘)+ (~(02) + tb(O2))A;“’ = 0,

(1OOb)

$ A’” + DAk2&” + o(02)A’” = 0.

(1OOc)

The solution of the three differential equations yields the autocorrelation tions

func-

C”‘(t, k) = exp(-w(02)

- i&(02) - DAk’)t,

(101a)

C”‘(t, k) = exp(-o(02)

- i&(02) - DAk%,

(101b)

C”‘(t 9k) = exp(-o(02)

- DAk2)t.

(101c)

The total correlation then

tensor in the presence

CPY+A(f,k) = C”‘(t, k)P!%~(e)

of the external field CC+(f, k) is

+ C”‘(t, k)(9F’:?,,A(e) + Pzh(e))

+ Cc2’(t,k)(S” rv,rr(e) + $%%(e)).

Notice, that for vanishing field the correlation &v,.k(f, k) = C(t, k)A$,&

(102)

tensor is given by

C(t, k) = C’“(t, k).

For the spectral tensor SPy,KA(o,k) a d ecomposition with the Lorentzians

(103) analogous to (102) holds

S’“‘(w, k) =

~(02) + $5(02) + DAk2 (w(02) + f&(02) + DAk2)2+ 6~”

(104a)

S”‘(o, k) =

~(02) + $(02) + DAk2 1 (~(02) + t&(02) + DAk2)2+ o 2’

(lwb)

180

W. KiiHLER

AND H. KAGERMANN

~(02) + Djjk’ s(2)(W’k, = (~(02) + DAk2)*+ w*’

(104c)

A possible field influence of the “diffusional broadening” (CCk*) as well as its anisotropy in k-space (cf. ref. 15) has been disregarded here. Further interesting phenomena in a system of Brownian particles, viz. neutron scattering and dielectric relaxation, are in general connected with two-particle correlations and cannot be described with the single particle Fokker-Planck equation for noninteracting Brownian particles.

References 1) W. Hess and R. Klein, Physica 94A (1978) 71; 99A (1979) 463; 105A (1981) 552. B.U. Felderhof, J. Phys. All (1978) 929; Physica 89A (1977) 373. A.R. Altenberger and J. M. Deutsch, J. Chem. Phys. 59 (1978) 894. 2) L.G. Lea1 and E.J. Hinch, J. Fluid Mech. 55 (1972) 745. 3) S. Hess, Physica 74 (1974) 277. 4) H. Kagermann, W.E. Kohler and S. Hess, Physica 1OSA (1981) 271. 5) B.J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976). 6) A. Magnus, Poggendorfs Ann. Phys. 88 (1853) 1. 7) H. Lamb, Hydrodynamics (Cambridge Univ. Press, Cambridge, 1932). 8) S. Rubinow and J.B. Keller, J. Fluid Mech. 11 (1961) 447. 9) S. Hess, Z. Naturforsch. 23a (1968) 191. 10) R.L. Stratonovich, Topics in the Theory of Random Noise I (Gordon and Breach, New York, 1963). 11) S. Hess, Z. Naturforsch. 23a (1968) 597. 12) S. Hess and W.E. Kiihler, Formeln zur Tensorrechnung (Palm und Enke, Erlangen. 1980). 13) S. Hess, Springer Tracts in Modern Physics 54 (1970) 136. 14) S. Hess and L. Waldmann, Z. Naturforsch. 26a (1971) 1057. 15) S. Hess and R. Miiller, Optics Comm. 10 (1974) 172.