- Email: [email protected]
Nonspherical Brownian particles. Kinetic description and application to flow birefringence
Physica 105A (1981) 271-285 © North-Holland Publishing Co.
NONSPHERICAL BROWNIAN PARTICLES. KINETIC DESCRIPTION AND APPLICATION TO FLOW BIREFRINGENCE H. KAGERMANN
Lehrstuhl B fiir Theoretische Physik, Technische Universitiit Braunschweig, F.R. Germany and W.E. KOHLER and S. HESS
lnstitut fiir Theoretische Physik der Universitiit Edangen-Niirnberg, F.R. Germany
Received 1 September 1980
For a system of noninteracting spheroidal Brownian particles a Fokker-Planck equation is derived which takes into account a hydrodynamical coupling between translational and orientational degrees of freedom. With the moment method, transport relaxation equations are obtained from which an expression for the coefficient of flow birefringence is inferred.
1. Introduction
Many physical systems such as aerosols, suspensions, colloids and macromolecules in aqueous solution are characterized by the fact that particles with length scales of the order of about 10-610 -4 cm are suspended in a fluid which consists of relatively small molecules of dimension 10-7-10 -8 cm. This difference in sizes of particles suggests the macromolecules can be described as Brownian particles in a continuous medium1). Quite often the macromolecules have nonspherical shape. Therefore it is not sufficient to describe the dynamics of a system of nonspherical Brownian particles in terms of distribution functions p(t, x, v) (t = time, x = position, v = velocity). New variables must be introduced which characterize the orientation of the particles and their rotational motion. With these additional variables interesting nonequilibrium phenomena are connected, e.g. birefringence and depolarized Rayleigh light scattering if the Brownian particles are optically anisotropic. In the present paper a "mesoscopic" description by means of a suitable Fokker-Planck equation is used: The solvent is characterized by only a few phenomenological macroscopic parameters (density, temperature, viscosity) while the suspended particles are described by their position-, velocity-, 271
272
H. KAGERMANNet al.
orientation-, and angular velocity vectors. Starting points for a kinetic treatment are thus stochastic equations of motion of the Langevin type for these variables. To establish these equations, we assume that the solution is sufficiently dilute to disregard the interaction between the (neutral) Brownian particles. On the other hand, it is stressed that the interaction of a particle with the solvent in general produces a coupling between translational and rotational motion. In our model, we consider the Brownian particles to be rigid homogeneous bodies of revolution. The orientation of such an unaxial particle is denoted by a unit vector u pointing in the direction of its symmetry axis, its rotational motion by the two-dimensional angular velocity ~ ( ~ ± . u--0). This is sufficient, because rotations about the symmetry axis do not change the state of the system. A possible correlation between translational motion and orientation is due to the dependence of the friction on the angle between symmetry axis and particle velocity. This has been used by some authors to treat diffusion problems by means of reduced distributions p(t,x, u)2"3). Recently, a Fokker-Planck equation for a reduced distribution function p(t, x, v, u) has been stated 4) in order to study diffusio-birefringence4). A shortcoming of this kinetic equation, however, was that flow birefringence could not be treated. In the last decade, flow birefringence in gases of nonspherical moleculesfirst proposed by Hess 5)- has been nicely demonstrated experimentally6), but diffusio-birefringence has not yet been demonstrated. Also for colloidal solutions, flow birefringence plays the more important role. One possibility for a kinetic description consists in adding a systematic torque in the equation of motion for g~l. This torque causes a particle in a flow field to orientate one of its principal axes preferentially in the direction of its motion. In contradistinction to different approaches7), here the consequences of a torque are studied which exists even when the flow field is assumed to be practically homogeneous over the extension of the particle. A phenomenological ansatz for this torque M as a function of the vectors u and v is made which is in accordance with all symmetry requirements (positive parity and time reversal behaviour, invariance with respect to the inversion u ~ - u ) and yields M - (u • v) u x v which is quadratic in the particle velocity. The magnitude of this torque, which depends on the density of the solvent and the shape of the particle, can in principle be calculated in a hydrodynamical approach by solving the nonlinear Navier-Stokes equations. The paper proceeds as follows: First, Langevin equations of motion are stated and the corresponding Fokker-Planck equation for the distribution p(t,x, v, u, 1~±) is derived. Due to the systematic torque the equilibrium distribution deviates from a Maxwellian by an additional factor depending on
NONSPHERICAL BROWNIAN PARTICLES
273
u and v, which may be interpreted as a mean field term. The collision operator is even in u and shows rotational-, parity- and time reversal symmetry. Since we are considering the case of overdamped particle rotation (the opposite c a s e - a l m o s t freely rotating particles has been treated in ref. 8), a reduced description with a distribution p(t, x, v, u) is used. Transport relaxation equations necessary for the treatment of flow birefringence are obtained with the moment method4'9). An expression for the pertaining contribution to the coefficient of flow birefringence is derived which is linear in the magnitude of the systematic torque. Finally, some remarks on the effect reciprocal to flow alignment are made.
2. Derivation of a generalized Fokker-Planck equation 2.1. E q u a t i o n s o f m o t i o n
A suspension of Brownian particles in an isotropic fluid is considered which is sufficiently dilute to disregard the particle interaction. The Brownian particles are assumed to be neutral, homogeneous rigid bodies with a symmetry axis characterized by the unit vector u. The particle variables are thus the position of the center of mass, x, the velocity v, the orientation vector u and the angular velocity ~± for rotation about an axis perpendicular to u. The translational motion is described by the usual Langevin ansatz Yc = coV,
(2.1)
~' = - y ~ ( u ) • v + q T ( U ) " ~ ( t ) ,
(2.2)
where V is the particle velocity in units of (2.3)
Co = (2ka T / m ) 112.
Here m is the mass of the particle, T is the temperature of the solution and kB is Boltzmann's constant. Due to the fact that the friction coefficient of a particle depends on the angle between its symmetry axis and its velocity, the friction tensor reads 7T(U) = V~UU + V~(,~ -- UU).
(2.4)
For the conventional sticking boundary condition the hydrodynamic values of the coefficients 7~Tand 3'~ are given by "r~ = 6zr~lR~/m;
a = I[, ±,
(2.5)
where ~ is the viscosity of the solvent and R~ are effective radii of the Brownian particles (cf. ref. 4, p. 284). By introducing the symmetric traceless
274
H. KAGERMANN et al.
tensor uu
-
uu -
(2.6)
,
eq. (2.4) may be rewritten as vT(.) =
+
+
= :7T8 + ~T~u.
(2.7)
As usual, the stochastic force ~T(t) is described by means of a stationary Gaussian process with zero mean and with a white spectrum: ~ " l ~ = 0;
~T(t)~T(t') = 8(t -- t')&
(2.8)
Its magnitude is characterized by the elements of the matrix ~T which has to be determined later by imposing conditions on the equilibrium distribution. For the rotational motion an analogous Langevin description is used: ti = to0//l x u, ~-~± ~- -- "yR~'~±-F
(2.9) ~---~"-- X ' v g ' co00± u
u + ~R~a(t).
(2.10)
Similar to (2.3), ~± is also dimensionless and given in units of (2.11)
Wo = (2ks T/O ±)'/2,
where 0± is the moment of inertia for rotations of the Brownian particle about an axis perpendicular to u. The hydrodynamical value for the rotational friction constant is 3,g = 8w~ (R R)3/01
(2.12)
with R~ defined in ref. 4. It is assumed that the stochastic torque 6R(t) has the properties specified in eq. (2.8). Since we consider bodies of revolution, the coupling between "¢ and ~ l in (2.2) and (2.10) vanishes (cf. Happel and Brenner t°) p. 188), and it is consistent to use the condition (cf. ref. 11) ~T(t)~R(t') = 0.
(2.13)
The term quadratic in I1, M:
= &u x 'gg"
u = &(u
• V)u
x V,
(2.14)
describes a systematic torque exerted on the nonspherical particle. From a phenomenological point of view, (2.14) is the simplest ansatz for a torque built up from the particle variables u, V which is consistent with the assumed inversion symmetry of the Brownian particles. In a hydrodynamical approach, the torque is due to the influence of the nonuniform velocity field of the solvent generated by the particle motion. Its magnitude c/ could, in principle, be
NONSPHERICAL BROWNIAN PARTICLES
275
determined by solving the nonlinear Navier-Stokes equation (e.g. in the Oseen approximation). Without detailed calculation one can argue that d is proportional to the solvent density pF and depends on the geometry of the particle. The viscosity, also being important for the mentioned process, should not appear explicitly in the expression for M, which has the same structure as the leading term in the Oseen correction to the friction force. It should be noted that the model equations can easily be extended to include additional couplings between translational and rotational motion, e.g. the transverse forceS): (2.15)
K tr = - "~vt,r X ~'~.
It can be shown, however, that the corresponding term in the kinetic equation does not give a contribution to the coefficient of flow birefringence mainly discussed in this paper. 2.2. Fokker-Planck equation Since the friction tensor ~,T depends on the particle orientation u, eqs. (2.1), (2.2), (2.9) and (2.10) form a system of multiplicative stochastic differential equations 13) ~, = fi(y) + ~ gij(y)6(t);
y = :(x, V, u, I}.).
(2.16)
In the first instance one has to be careful with the interpretation of the involved stochastic integrals. But in the present case, the additional drift term O1
a
g,J(Y)gsi(Y )f ( t,
(2.17)
which occurs if eq. (2.16) is interpreted in the Stratonovitch senset2), vanishes because eq. (2.9) does not contain a stochastic force. As a consequence, one obtains from eq. (2.16) the corresponding Fokker-Planck equation for the nonequilibrium distribution f(t, x, ¥, u, !11) of the Brownian particles in a unique way:
{0~-t "~-COV" ~X0 ~'--V" 0 [ '~T(u)" +tOon±.~_
'T
V"{-~'y
"~T(u)
a .( 1 O 0~± ~/Rn±+2(~/R)20~,
0
oJ__Lo ]} 2k, T M f = 0 .
(2.18)
The differential operator .£P:= u X - -
a
Ou
(2.19)
276
H. K A G E R M A N N et al.
corresponds to the operator of the internal angular momentum of the Brownian particle. Without systematic torque (M = 0) the distribution function for thermal equilibrium is given by the usual Maxwellian fo(& = O) = ~no ,/r-512 e -v2 e -a~
(2.20)
with the constant number density no of the Brownian particles. The existence of the torque (2.14) has to be taken into account by a potential 0 in the equilibrium distribution: - ~0
= M.
(2.21)
From (2.21) and (2.14) the result O _ ka T
~ 'V¢" a uu : ,
a
=
(2.22)
&/2kaT,
can easily be inferred and (2.20) turns into f o ( a # 0) = ~n°1r-S/2(1 + ~a)(l - ~ot)l/2 e -a~ e x p [ - V . (8 - au-uu). V].
(2.23)
The condition that f0 has to be a stationary solution of the Fokker-Planck equation (2.18) determines the magnitude of the stochastic torque (qR)2 = 7R,
(2.24)
and of the stochastic force qT. qT = : FT = (.yi~+ ~avT)~ + (~/T(1 + 3/a) + aViT) ~U = : F~r~ + FaV~U.
(2.25)
If the nonequilibrium distribution f is written as f = f0(1 + to),
(2.26)
the deviation tO obeys the kinetic equation
oo
o
+ ¢Oog~±"~ + 2--~BTM " O a i
,._lO
~Io ~ - ~ "
( ~/'Io
tO = 0.
(2.27)
Note that the systematic torque M influences the collision operator in eq. (2.27) via modified functions f o ( a ) and F T ( u , a ) . Applied to any function independent of V, u,/~l, the collision operator yields zero thus guaranteeing the conservation of particle number.
NONSPHERICAL BROWNIAN PARTICLES
277
2.3. Equilibrium distribution Due to the hydrodynamic torque, f0 becomes a modified Maxweil-Boltzmann distribution taking into account an equilibrium correlation (V--V: ~h-~u)0 (proportional to a for a "~ 1) between the translational and orientational variables. The exponential e x p ( - V 2) is thus replaced by e x p ( - V • D ( u ) . V) with
D(u): = (1 + 31~)(6 - uu) + (1 - 2a)uu.
(2.28)
Hence, for fixed magnitude of velocity and angular velocity, the probability Pfd that the particle velocity V is parallel to its orientation u is proportional to e x p [ - ( 1 - z w t ) ] and therefore different from the probability P± ~ e x p [ (l+31a)] that V is perpendicular to u. Which probability is the larger one depends on the sign of the constant a, i.e. ot ~ 0¢~PII ~ Pl.
(2.29)
Furthermore, lower and upper bounds - 3< a < ~
(2.30)
for o~ exist in agreement with (2.23). Beyond that, (2.29) ensures that the matrix FT(u) is positive definite and that the equilibrium distribution can be normalized. For a ~ 1 (which is to be expected) the equilibrium distribution (2.23) can be expanded about the Maxwellian (2.20) yielding
fo(a) -~ fo(a = 0)[1 + a 'VV': u'--~- ~a 2 + ~a2( 'VV': ~ u )2].
(2.31)
The mean square velocity can be calculated to give
c2
(2.32)
showing a deviation from the ideal gas case due to the existence of a "mean field" term ~b (cf. eq. (2.22)). For the same reason, integration of (2.23) over the internal variables /~1 and u does not result in a Maxwell distribution for the velocity. Note, however, that this mean field, in agreement with isotropy arguments, does not lead to any tensorial alignment of V or u in thermal equilibrium, but only to scalar correlations between tensors of the same rank which are even in both V and u. 2.4. Reduced Fokker-Planck equation For cases where the particle rotation is overdamped, a description with a reduced distribution p(t, x, V, u)
p(t, x, V, u): = f d212ff(t, x, V, u, n ± )
(2.33)
278
H. K A G E R M A N N
et al.
is adequate. Integration of (2.18) over /~± yields D ~ - P + to0~ "ja = O,
(2.34)
where the following abbreviations are used:
D
~+coV
D ~ : --" 0--t
0
0
(
1FT(u). _ff_~) 0 ,
" 0 x - 0--V " , y T ( / / ) , V --I-
Jn: = f / ] j ( t , X, V, u, £/±)d2D..
(2.35) (2.36)
A similar equation for the flux ja is obtained by multiplication of eq. (2.18) by //~ and subsequent integration over d2Ol:
Dj +too . f o;aJ d20 +f (ca
_
too 2ksT M ) f d 2 D l = 0.
(2.37)
By use of the idempotent operator
P~b(t, x, V, u,/~±): = 7r-1 e
d2D±~(t, x, V, u, ~±),
(2.38)
eq. (2.37) can be cast into 1
M
= :toO/.
If the propagator which solves the corresponding homogeneous equation is denoted by Gt, the solution of eq. (2.39) reads t
ja(t) =
G da(t = O)+ tooI d~'Gt-d(r).
(2.40)
0
Because the equations of motion define a Markovian process, the contribution of the initial value drops out in the asymptotic limit. In addition, the assumption is made that the angular velocity relaxes on a characteristic time scale which is sufficiently short to consider the remaining variables of the Brownian particle approximately as constants of motion. Then the solution (2.40) reduces to the stationary value
ja(t) ~ tooI drGd(t)= ~ l(t).
(2.41)
0
From the equation for ( 1 - P ) I , which is similar to (2.34), the estimate ( I - P ) f = O(~o0) can be inferred. Therefore, the contribution of ( 1 - P)[ to
NONSPHERICAL BROWNIAN PARTICLES
279
(2.34) is of the order O(to03) and can be neglected within a weak coupling approximation. Insertion of eq. (2.41) into (2.34) yields the desired reduced Fokker-Planck equation
yR~
-~uX'VV.u+
~
o(t,x,V,u)=O.
(2.42)
Again, the nonequilibrium distribution is written as o = p0(1 + ,/,(t, x, v, u));
no Ir_3/2(1 + ~a)(1 - ~a)l/2 e- v. t~). v P0 = 4Ir
(2.43)
and the kinetic equation for the relative deviation d~ is
0x - ~po 1 _,~0 {~+coV • ~where
.(rr(u).po~)_l
~po , ~ .(FRp0~)Id ~ = 0,
(2.44)
the a b b r e v i a t i o n
FR: = to0~/yR
(2.45)
has been introduced. In the case of a = 0, p0 becomes a Maxwellian and FT(u) = ~'T(u), i.e. eq. (2.44) reduces to the Fokker-Planck equation recently used in ref. 4. If, on the other hand, one starts with the reduced equations of motion (2.1), (2.2) and fi = ~
(M x u) + to0~R(t) X U,
(2.46)
eq. (2.44) is obtained immediately. Again, the result is independent of the interpretation of the stochastic differential equations because the differences are proportional to u • O/au and vanish in consequence of [u I = 1. 2.5. Properties of the collision operator If the collision operator is denoted by 1
to(t~):=--~po
-I
{O~'(rT(u)po'O~)+~'(rRpo~dp)},~-V
(2.47)
it is evident that to( 4, = 1) = 0 which guarantees the conservation of the particle number. Since to is rotationally invariant, the (Cartesian) Wigner-Eckart Theorem may be applied to factorize matrix elements of to taken with two
280
H. K A G E R M A N N et al.
arbitrary irreducible expansion tensors ¢k,,...,,(V, u) and toy,..... t(V, u) into a product of a reduced matrix element and an isotropic tensor of rank 2l:
l f f d3V d2udp~i...~,(V, u)poto(to,,l.... i(V, u)) no J ) D
(~bu,...,,to(0,, .... i))o = ~
(~bl °to(Ot))0A.,...~l,vl .... ,.
(2.48)
Here, the dot o denotes a contraction over all tensor indices and zi ~1...,~,,~ .... is an isotropic tensor which, applied to an arbitrary lth rank tensor, projects out its irreducible part~4). Since the collision operator to remains unchanged in a parity transformation ¥ -o - V, u ~ - u, L# ~ 5~, it can only connect expansion tensors of the same parity. Because F R> 0 holds and FT(u) is a positive definite symmetric tensor, it is obvious from eq. (2.47) that for any ~b(V, u) the inequality (6to(6))0 ~>0
(2.49)
is valid. Thus the collision operator is positive definite if restricted to a space of expansion tensors ~b¢ const. This guarantees the decay of any initial nonequilibrium state to thermal equilibrium or, equivalently, the increase of entropy. Finally to is invariant with respect to time reversal (indeed, both terms of to (eq. (2.47)) are separately invariant), i.e. 03T(V, u): = o3(- V, u) = to(V, u).
(2.50)
This implies that for arbitrary functions 6(V, u), to(V, u) the symmetry property (cbto(to))0 = (toTto(&r))0
(2.51)
holds, where 6 r ( ¥ , u) = 4)( - V, u) etc. If both 4~ and to show definite behaviour under time reversal, i.e. &T = T~b, tOT= T,to with T , , T , = - 1 , Onsager relations (~to(to))o =
T~T,(toto(6))o
(2.52)
exist.
3. Application to flow birefringence
3.1. Transport relaxation equations The quantity ~b(t, x, u, V) characterizing the relative deviation of the reduced distribution p(t, x, u, V) from its equilibrium value p0 can be expanded with
NONSPHERICAL BROWNIAN PARTICLES
281
respect to a complete set of irreducible expansion tensors .~,,htk')...~,,(U, V), where l is the tensor rank running from zero to infinity and where the superscript kt distinguishes various tensors of equal rank. The expansion coefficients (moments) are then lth rank tensors depending on t, x (moment method, cf. refs. 4, 9). For physical reasons it is convenient to choose a basis which is orthogonal with respect to a Maxwellian p0M, p0M = ~ 1' h " -3/2exp(_ V2),
(3.1)
viz. ,h(kt) ,h(kl,) "r ,l . . . m'e ~'~ . . .
,,v)o~
=
~lt'~
',
kf/k~ A
(3.2)
, l . . . m, vl . . . "V
For the isotropic tensor A , , . . ~/.vI. . . . l see refs. 14. The expansion tensors have the property that their mean values taken with the true equilibrium function p0(u, V) vanish for l # 0, i.e. (d~!.. ~)0 = 0,
l ~ 0.
(3.3)
For a kinetic description of flow birefringence, only a few moments prove to be sufficient, namely 4~(l°) = X/2V;
~(20) = X/2'VV ;
¢~(02) = X/~- u~--u.
(3.4)
According to notations used earlier, in (3.4) the first superscript means the tensor rank in V, the second superscript the tensor rank in u. The deviation ~b used for the description of flow birefringence in colloidal solutions is thus given by t~) = A ° ~(io)_~_ BI-" ~(2o)_~_ B2: ~(02),
(3.5)
where the vector A and the second rank tensors B~ and B2 depend on t and x. To find their respective meanings, we multiply p = p0(1 + ~b) with ~(10), d}(20) and d}(°2), integrate over u and V and use (3.3) and the definition of a nonequilibrium average
n(~'): = f f p~ d~V d2u.
(3.6)
Since we assume the number density n to be almost homogeneous (we are not interested in diffusion processes here) we may put n ~ no. Thus we obtain (d~(l°)) -~ A,
(3.7)
but to determine B1 and B2, we must observe that due to the "internal field" a correlation between 4}(2°) and 4~(°2) exists. Expanding p0 in powers of the
282
H. KAGERMANN et al.
coupling constant a and retaining only terms linear in a, we find ( 6 (20))
BI + ~ ' O/ ~ B 2,"
( 0 (02)) ~ ~ l S B I
+ f12.
(3.8)
In linear approximation in a one has thus ~ l ~ (4[~(20)) -- ~'-1~(6(02));
~2 = ( 6 (°2,)- ~15i6(20)).
(3.9)
Transport-relaxation equations adequate for the treatment of flow birefringence are obtained by multiplying the linearized reduced Fokker-Planck equation (2.44) with 6 (20), 6 (°2), respectively, integrating over u, V and observing eqs. (3.6)-(3.9). An equation for i6"°)) - iV) is not needed, since the mean velocity iV) is considered to be determined by the experimental arrangement, e.g. Couette flow between rotating cylinders6). Defining reduced collision frequencies of the linearized Fokker-Planck operator by 0(20): = -]i6 (20)'.0(0(20)))0; 0(02): = ls(6 (°2)'. 0(0(02)))0 o(°g) = to(02b=-~(6(°2)"to(6(20)))0,.
(3.10)
we obtain the following systems of moment equations: 3(6'2°))0t + (t°(20)- ~q--15t°(°22°))(6(20)) + (to (20)- ~ 1 5 to(20))(6 (02))= _ Co~7i6"°);;
(3.1 l)
(3.12) 3.2. Relaxation coeHicients In this section, detailed expressions are given for the relaxation-and coupling coefficients (3.10) and for the relaxation coefficient of the particle velocity. In order to do this, the equilibrium distribution p0 is approximated by its expansion (cf. eq. (2.31)) up to terms quadratic in a. For the calculation of the integrals extensive use is made of Cartesian tensor formulae, e.g. 14) ~ u~-~ = 2E]~v,x, ~u~u~",
(3.13)
.~C~2'U~Uv'= -- 6 'U/zUv',
(3.14)
where [Z,v,~,~ is an isotropic 5th rank tensor which describes the rotation of an irreducible 2nd rank tensort4). The relaxation frequencies are calculated up to quadratic order in a. For the relaxation coefficients of velocity and friction
N O N S P H E R I C A L BROWNIAN PARTICLES
283
pressure tensor one obtains ,o(lO) = H , to(20) =
(3.15)
2H(1 + ]a 2) + ar a(1 + ]a).
(3.16)
The relaxation coefficient of the tensor polarization ( T u ) is given by to(02) = 3FR(1 + ]a2).
(3.17)
The coupling coefficient of friction pressure tensor and tensor polarization turns out to be zero: 02 to(20) = ¢o(02~= 0.
(3.18)
For a = 0, the expressions (3.15)-(3.18) are in agreement with the results in ref. 4, if the identifications riT(a = 0) = wt, r R = 2Wu are made. From expression (3.18) it is obvious that the systematic torque does not give a direct collisional coupling between friction pressure tensor and tensor polarization. This is also true for all powers of a. While in the case a = 04) this would mean that flow birefringence cannot be treated in this theory, it is not true here, because via the correlation between u~-~ and 'VV' an effective coupling is introduced. 3.3. C o e n i c i e n t o f f l o w bire[ringence To obtain an expression for the coefficient of flow birefringence, 1SEa, the constitutive relation for the anisotropic part of the dielectric tensor, ~', stated in ref. ~5) has to be observed: (3.19)
~" = -- 2flFB ~7 W - 2~DB ~TVD.
Here, W = (Or + IOFVF)/(p+ OF) is the mean mass velocity of the system Brownian particles + fluid (F) and Vo -- v - vF is the diffusion velocity. On the other hand, a connection between ~' and the tensor polarization (~(02)) exists for a dilute system 4'5) "~ = e'(ck(°2)),
e': = X/~4~rn(arf- a~),
(3.20)
where n is the particle number density of the Brownian particles and all, a . are their effective polarizabilities for light polarized parallel and perpendicular to the symmetry axis u, respectively. A connection between the tensor polarization (d~t°2)) and the gradient of the mean particle velocity ~7(Vi can be inferred from the stationary transportrelaxation equations (3.11) and (3.12). For lowest order in a, this relation is (6to~)) ~
- ~
1
rRct
~
'
'
CoV(V).
(3.21)
284
H. K A G E R M A N N et al.
Now, one has Co(V) = W + (PF/(P + pF))VD, but the occurrence of ~ would lead to diffusio-birefringence which will not be treated here. (Otherwise, corrections to the coefficient of diffusio-birefringence obtained in ref. 4 are found.) In any case, by combining eqs. (3.19)-(3.21) the contribution to the coefficient of flow birefringence due to the systematic torque eq. (2.14) is inferred as /3FB---- ~/]~ -~-~r.
(3.22)
Notice, that a depends on the shape of the particle; in particular, it has opposite signs for prolate and oblate particles. Furthermore, a should be independent of the viscosity, analogously to the term quadratic in V in the friction force (Oseen correction). It is well known 7'~6) that a torque due to the spatial variation of the flow velocity over the extension of the particle leads to an orientational alignment (~(02)) which is also proportional to the velocity gradient. While in the present treatment one has (~f~(02)) OCOLTtrVD,
with the different approach in refs. 7, 16 one obtains (1~ (02)) OCKTalVl0
where K is a number describing the nonsphericity of the particle. The time coefficients ~'tr= (F~) -t and ~'al= (3FR) -I determine hereby the relaxation of the velocity and orientation anisotropies, respectively. Since *tr ~ moard67r~lv]~, the contribution (3.22) to flow birefringence is expected to be inversely proportional to the viscosity of the solvent. This is different for the other contribution where one has ~',1-~8~r~vR3/6kT, thus the alignment being proportional to the viscosity in that case. Furthermore, the relaxation times ~'tr and ~',r differ in their dependences on the size of the particles. Therefore the term eq. (3.22) is expected to be important if the viscosity of the fluid is small (e.g. for gases) and the particles become smaller. Note, however, that the particles have still to be large enough to guarantee that the rotational motion is overdamped (X/k---~ < ~.~i). Finally, however, it should be mentioned that, in contrast to the standard approach of refs. 7, 16, the present theory accounts also for the treatment of the effect reciprocal to flow alignmenttT), viz. a tensorial anisotropy in the velocity distribution caused by an orientational alignment under nonequilibrium conditions. This can be seen by solving the differential equations (3.1 1), (3.12) for a situation where one has (~02))# 0 but (~(20))= 0 at time t = 0.
NONSPHERICAL BROWNIAN PARTICLES
285
Then, for a ~ 1, one finds otto(20)
(3.23)
Thus a transient anisotropic velocity distribution, viz. (~-'v)~0 of the Brownian particles is produced. For to(20)>> to(02) and to(20)t >> 1, eq. (3.23) reduces to (~(2°))(t) ~ ~
ot
e x p ( - ~o(02)t) • (~(°2))(t = 0),
i.e. the anisotropy in the velocity distribution decays with the decay constant of the orientational anisotropy.
References 1) G.K. Batchelor, in Theoretical and Applied Mechanics, W.T. Keuler ed. (North-Holland, Amsterdam, 1976). 2) R. Gans, Ann. Phys. 86 (1928) 628. 3) S.J. Prager, J. Chem. Phys. 23 (1955) 2404. 4) S. Hess, Physica 74 (1974) 277. 5) S. Hess, Phys. Letters 30A (1969) 239. 6) F. Baas, Phys. Letters 36A (1971) 107; F. Baas et al., Physica $8A (1977) 1, 34, 44. 7) A. Peterlin und H.A. Stuart, in Hand-und Jahrbuch der Chemischen Physik, A. Eucken und K.L. Wolf, eds. (Akad. Verlagsgesellschaft, 1943) p. 113; L.G. Leal and E.J. Hinch, J. Fluid Mech. 55 (1972) 745. 8) S. Hess, Z. Naturforsch. 23a (1968) 597. 9) S. Hess and L. Waldmann, Z. Naturforsch. 21a (1966) 1529; H.H. Raum and W.E. K6hler, Z. Naturforsch. 25a (1970) 1178. 10) J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, (Prentice Hall, Englewood Cliffs, N.Y., 1965). 11) S. Kim and I. Oppenheim, Physica 54 (1971) 593. 12) R.E. Mortensen, J. Stat. Phys. I (1969) 271. 13) R.L. Stratonovich, Topics in the Theory of Random Noise I (Gordon and Breach, New York, 1963) pp. 100ft. 14) S. Hess, Z. Naturforsch. 23a (1968) 1095; S. Hess und W.E. K6hler, Formeln zur Tensorrechnung (Palm and Enke, Erlangen, 1980). 15) W.E. K6hler and J. Halbritter, Physica 74 (1974) 294; J. Halbritter and W.E. K6hler, Physica 76 (1974) 224. 16) S. Hess, Z. Naturforsch. 31a (1976) 1034. 17) S. Hess, Z. Naturforsch. 25a (1973) 1531.
NONSPHERICAL BROWNIAN PARTICLES. KINETIC DESCRIPTION AND APPLICATION TO FLOW BIREFRINGENCE H. KAGERMANN
Lehrstuhl B fiir Theoretische Physik, Technische Universitiit Braunschweig, F.R. Germany and W.E. KOHLER and S. HESS
lnstitut fiir Theoretische Physik der Universitiit Edangen-Niirnberg, F.R. Germany
Received 1 September 1980
For a system of noninteracting spheroidal Brownian particles a Fokker-Planck equation is derived which takes into account a hydrodynamical coupling between translational and orientational degrees of freedom. With the moment method, transport relaxation equations are obtained from which an expression for the coefficient of flow birefringence is inferred.
1. Introduction
Many physical systems such as aerosols, suspensions, colloids and macromolecules in aqueous solution are characterized by the fact that particles with length scales of the order of about 10-610 -4 cm are suspended in a fluid which consists of relatively small molecules of dimension 10-7-10 -8 cm. This difference in sizes of particles suggests the macromolecules can be described as Brownian particles in a continuous medium1). Quite often the macromolecules have nonspherical shape. Therefore it is not sufficient to describe the dynamics of a system of nonspherical Brownian particles in terms of distribution functions p(t, x, v) (t = time, x = position, v = velocity). New variables must be introduced which characterize the orientation of the particles and their rotational motion. With these additional variables interesting nonequilibrium phenomena are connected, e.g. birefringence and depolarized Rayleigh light scattering if the Brownian particles are optically anisotropic. In the present paper a "mesoscopic" description by means of a suitable Fokker-Planck equation is used: The solvent is characterized by only a few phenomenological macroscopic parameters (density, temperature, viscosity) while the suspended particles are described by their position-, velocity-, 271
272
H. KAGERMANNet al.
orientation-, and angular velocity vectors. Starting points for a kinetic treatment are thus stochastic equations of motion of the Langevin type for these variables. To establish these equations, we assume that the solution is sufficiently dilute to disregard the interaction between the (neutral) Brownian particles. On the other hand, it is stressed that the interaction of a particle with the solvent in general produces a coupling between translational and rotational motion. In our model, we consider the Brownian particles to be rigid homogeneous bodies of revolution. The orientation of such an unaxial particle is denoted by a unit vector u pointing in the direction of its symmetry axis, its rotational motion by the two-dimensional angular velocity ~ ( ~ ± . u--0). This is sufficient, because rotations about the symmetry axis do not change the state of the system. A possible correlation between translational motion and orientation is due to the dependence of the friction on the angle between symmetry axis and particle velocity. This has been used by some authors to treat diffusion problems by means of reduced distributions p(t,x, u)2"3). Recently, a Fokker-Planck equation for a reduced distribution function p(t, x, v, u) has been stated 4) in order to study diffusio-birefringence4). A shortcoming of this kinetic equation, however, was that flow birefringence could not be treated. In the last decade, flow birefringence in gases of nonspherical moleculesfirst proposed by Hess 5)- has been nicely demonstrated experimentally6), but diffusio-birefringence has not yet been demonstrated. Also for colloidal solutions, flow birefringence plays the more important role. One possibility for a kinetic description consists in adding a systematic torque in the equation of motion for g~l. This torque causes a particle in a flow field to orientate one of its principal axes preferentially in the direction of its motion. In contradistinction to different approaches7), here the consequences of a torque are studied which exists even when the flow field is assumed to be practically homogeneous over the extension of the particle. A phenomenological ansatz for this torque M as a function of the vectors u and v is made which is in accordance with all symmetry requirements (positive parity and time reversal behaviour, invariance with respect to the inversion u ~ - u ) and yields M - (u • v) u x v which is quadratic in the particle velocity. The magnitude of this torque, which depends on the density of the solvent and the shape of the particle, can in principle be calculated in a hydrodynamical approach by solving the nonlinear Navier-Stokes equations. The paper proceeds as follows: First, Langevin equations of motion are stated and the corresponding Fokker-Planck equation for the distribution p(t,x, v, u, 1~±) is derived. Due to the systematic torque the equilibrium distribution deviates from a Maxwellian by an additional factor depending on
NONSPHERICAL BROWNIAN PARTICLES
273
u and v, which may be interpreted as a mean field term. The collision operator is even in u and shows rotational-, parity- and time reversal symmetry. Since we are considering the case of overdamped particle rotation (the opposite c a s e - a l m o s t freely rotating particles has been treated in ref. 8), a reduced description with a distribution p(t, x, v, u) is used. Transport relaxation equations necessary for the treatment of flow birefringence are obtained with the moment method4'9). An expression for the pertaining contribution to the coefficient of flow birefringence is derived which is linear in the magnitude of the systematic torque. Finally, some remarks on the effect reciprocal to flow alignment are made.
2. Derivation of a generalized Fokker-Planck equation 2.1. E q u a t i o n s o f m o t i o n
A suspension of Brownian particles in an isotropic fluid is considered which is sufficiently dilute to disregard the particle interaction. The Brownian particles are assumed to be neutral, homogeneous rigid bodies with a symmetry axis characterized by the unit vector u. The particle variables are thus the position of the center of mass, x, the velocity v, the orientation vector u and the angular velocity ~± for rotation about an axis perpendicular to u. The translational motion is described by the usual Langevin ansatz Yc = coV,
(2.1)
~' = - y ~ ( u ) • v + q T ( U ) " ~ ( t ) ,
(2.2)
where V is the particle velocity in units of (2.3)
Co = (2ka T / m ) 112.
Here m is the mass of the particle, T is the temperature of the solution and kB is Boltzmann's constant. Due to the fact that the friction coefficient of a particle depends on the angle between its symmetry axis and its velocity, the friction tensor reads 7T(U) = V~UU + V~(,~ -- UU).
(2.4)
For the conventional sticking boundary condition the hydrodynamic values of the coefficients 7~Tand 3'~ are given by "r~ = 6zr~lR~/m;
a = I[, ±,
(2.5)
where ~ is the viscosity of the solvent and R~ are effective radii of the Brownian particles (cf. ref. 4, p. 284). By introducing the symmetric traceless
274
H. KAGERMANN et al.
tensor uu
-
uu -
(2.6)
,
eq. (2.4) may be rewritten as vT(.) =
+
+
= :7T8 + ~T~u.
(2.7)
As usual, the stochastic force ~T(t) is described by means of a stationary Gaussian process with zero mean and with a white spectrum: ~ " l ~ = 0;
~T(t)~T(t') = 8(t -- t')&
(2.8)
Its magnitude is characterized by the elements of the matrix ~T which has to be determined later by imposing conditions on the equilibrium distribution. For the rotational motion an analogous Langevin description is used: ti = to0//l x u, ~-~± ~- -- "yR~'~±-F
(2.9) ~---~"-- X ' v g ' co00± u
u + ~R~a(t).
(2.10)
Similar to (2.3), ~± is also dimensionless and given in units of (2.11)
Wo = (2ks T/O ±)'/2,
where 0± is the moment of inertia for rotations of the Brownian particle about an axis perpendicular to u. The hydrodynamical value for the rotational friction constant is 3,g = 8w~ (R R)3/01
(2.12)
with R~ defined in ref. 4. It is assumed that the stochastic torque 6R(t) has the properties specified in eq. (2.8). Since we consider bodies of revolution, the coupling between "¢ and ~ l in (2.2) and (2.10) vanishes (cf. Happel and Brenner t°) p. 188), and it is consistent to use the condition (cf. ref. 11) ~T(t)~R(t') = 0.
(2.13)
The term quadratic in I1, M:
= &u x 'gg"
u = &(u
• V)u
x V,
(2.14)
describes a systematic torque exerted on the nonspherical particle. From a phenomenological point of view, (2.14) is the simplest ansatz for a torque built up from the particle variables u, V which is consistent with the assumed inversion symmetry of the Brownian particles. In a hydrodynamical approach, the torque is due to the influence of the nonuniform velocity field of the solvent generated by the particle motion. Its magnitude c/ could, in principle, be
NONSPHERICAL BROWNIAN PARTICLES
275
determined by solving the nonlinear Navier-Stokes equation (e.g. in the Oseen approximation). Without detailed calculation one can argue that d is proportional to the solvent density pF and depends on the geometry of the particle. The viscosity, also being important for the mentioned process, should not appear explicitly in the expression for M, which has the same structure as the leading term in the Oseen correction to the friction force. It should be noted that the model equations can easily be extended to include additional couplings between translational and rotational motion, e.g. the transverse forceS): (2.15)
K tr = - "~vt,r X ~'~.
It can be shown, however, that the corresponding term in the kinetic equation does not give a contribution to the coefficient of flow birefringence mainly discussed in this paper. 2.2. Fokker-Planck equation Since the friction tensor ~,T depends on the particle orientation u, eqs. (2.1), (2.2), (2.9) and (2.10) form a system of multiplicative stochastic differential equations 13) ~, = fi(y) + ~ gij(y)6(t);
y = :(x, V, u, I}.).
(2.16)
In the first instance one has to be careful with the interpretation of the involved stochastic integrals. But in the present case, the additional drift term O1
a
g,J(Y)gsi(Y )f ( t,
(2.17)
which occurs if eq. (2.16) is interpreted in the Stratonovitch senset2), vanishes because eq. (2.9) does not contain a stochastic force. As a consequence, one obtains from eq. (2.16) the corresponding Fokker-Planck equation for the nonequilibrium distribution f(t, x, ¥, u, !11) of the Brownian particles in a unique way:
{0~-t "~-COV" ~X0 ~'--V" 0 [ '~T(u)" +tOon±.~_
'T
V"{-~'y
"~T(u)
a .( 1 O 0~± ~/Rn±+2(~/R)20~,
0
oJ__Lo ]} 2k, T M f = 0 .
(2.18)
The differential operator .£P:= u X - -
a
Ou
(2.19)
276
H. K A G E R M A N N et al.
corresponds to the operator of the internal angular momentum of the Brownian particle. Without systematic torque (M = 0) the distribution function for thermal equilibrium is given by the usual Maxwellian fo(& = O) = ~no ,/r-512 e -v2 e -a~
(2.20)
with the constant number density no of the Brownian particles. The existence of the torque (2.14) has to be taken into account by a potential 0 in the equilibrium distribution: - ~0
= M.
(2.21)
From (2.21) and (2.14) the result O _ ka T
~ 'V¢" a uu : ,
a
=
(2.22)
&/2kaT,
can easily be inferred and (2.20) turns into f o ( a # 0) = ~n°1r-S/2(1 + ~a)(l - ~ot)l/2 e -a~ e x p [ - V . (8 - au-uu). V].
(2.23)
The condition that f0 has to be a stationary solution of the Fokker-Planck equation (2.18) determines the magnitude of the stochastic torque (qR)2 = 7R,
(2.24)
and of the stochastic force qT. qT = : FT = (.yi~+ ~avT)~ + (~/T(1 + 3/a) + aViT) ~U = : F~r~ + FaV~U.
(2.25)
If the nonequilibrium distribution f is written as f = f0(1 + to),
(2.26)
the deviation tO obeys the kinetic equation
oo
o
+ ¢Oog~±"~ + 2--~BTM " O a i
,._lO
~Io ~ - ~ "
( ~/'Io
tO = 0.
(2.27)
Note that the systematic torque M influences the collision operator in eq. (2.27) via modified functions f o ( a ) and F T ( u , a ) . Applied to any function independent of V, u,/~l, the collision operator yields zero thus guaranteeing the conservation of particle number.
NONSPHERICAL BROWNIAN PARTICLES
277
2.3. Equilibrium distribution Due to the hydrodynamic torque, f0 becomes a modified Maxweil-Boltzmann distribution taking into account an equilibrium correlation (V--V: ~h-~u)0 (proportional to a for a "~ 1) between the translational and orientational variables. The exponential e x p ( - V 2) is thus replaced by e x p ( - V • D ( u ) . V) with
D(u): = (1 + 31~)(6 - uu) + (1 - 2a)uu.
(2.28)
Hence, for fixed magnitude of velocity and angular velocity, the probability Pfd that the particle velocity V is parallel to its orientation u is proportional to e x p [ - ( 1 - z w t ) ] and therefore different from the probability P± ~ e x p [ (l+31a)] that V is perpendicular to u. Which probability is the larger one depends on the sign of the constant a, i.e. ot ~ 0¢~PII ~ Pl.
(2.29)
Furthermore, lower and upper bounds - 3< a < ~
(2.30)
for o~ exist in agreement with (2.23). Beyond that, (2.29) ensures that the matrix FT(u) is positive definite and that the equilibrium distribution can be normalized. For a ~ 1 (which is to be expected) the equilibrium distribution (2.23) can be expanded about the Maxwellian (2.20) yielding
fo(a) -~ fo(a = 0)[1 + a 'VV': u'--~- ~a 2 + ~a2( 'VV': ~ u )2].
(2.31)
The mean square velocity can be calculated to give
c2
(2.32)
showing a deviation from the ideal gas case due to the existence of a "mean field" term ~b (cf. eq. (2.22)). For the same reason, integration of (2.23) over the internal variables /~1 and u does not result in a Maxwell distribution for the velocity. Note, however, that this mean field, in agreement with isotropy arguments, does not lead to any tensorial alignment of V or u in thermal equilibrium, but only to scalar correlations between tensors of the same rank which are even in both V and u. 2.4. Reduced Fokker-Planck equation For cases where the particle rotation is overdamped, a description with a reduced distribution p(t, x, V, u)
p(t, x, V, u): = f d212ff(t, x, V, u, n ± )
(2.33)
278
H. K A G E R M A N N
et al.
is adequate. Integration of (2.18) over /~± yields D ~ - P + to0~ "ja = O,
(2.34)
where the following abbreviations are used:
D
~+coV
D ~ : --" 0--t
0
0
(
1FT(u). _ff_~) 0 ,
" 0 x - 0--V " , y T ( / / ) , V --I-
Jn: = f / ] j ( t , X, V, u, £/±)d2D..
(2.35) (2.36)
A similar equation for the flux ja is obtained by multiplication of eq. (2.18) by //~ and subsequent integration over d2Ol:
Dj +too . f o;aJ d20 +f (ca
_
too 2ksT M ) f d 2 D l = 0.
(2.37)
By use of the idempotent operator
P~b(t, x, V, u,/~±): = 7r-1 e
d2D±~(t, x, V, u, ~±),
(2.38)
eq. (2.37) can be cast into 1
M
= :toO/.
If the propagator which solves the corresponding homogeneous equation is denoted by Gt, the solution of eq. (2.39) reads t
ja(t) =
G da(t = O)+ tooI d~'Gt-d(r).
(2.40)
0
Because the equations of motion define a Markovian process, the contribution of the initial value drops out in the asymptotic limit. In addition, the assumption is made that the angular velocity relaxes on a characteristic time scale which is sufficiently short to consider the remaining variables of the Brownian particle approximately as constants of motion. Then the solution (2.40) reduces to the stationary value
ja(t) ~ tooI drGd(t)= ~ l(t).
(2.41)
0
From the equation for ( 1 - P ) I , which is similar to (2.34), the estimate ( I - P ) f = O(~o0) can be inferred. Therefore, the contribution of ( 1 - P)[ to
NONSPHERICAL BROWNIAN PARTICLES
279
(2.34) is of the order O(to03) and can be neglected within a weak coupling approximation. Insertion of eq. (2.41) into (2.34) yields the desired reduced Fokker-Planck equation
yR~
-~uX'VV.u+
~
o(t,x,V,u)=O.
(2.42)
Again, the nonequilibrium distribution is written as o = p0(1 + ,/,(t, x, v, u));
no Ir_3/2(1 + ~a)(1 - ~a)l/2 e- v. t~). v P0 = 4Ir
(2.43)
and the kinetic equation for the relative deviation d~ is
0x - ~po 1 _,~0 {~+coV • ~where
.(rr(u).po~)_l
~po , ~ .(FRp0~)Id ~ = 0,
(2.44)
the a b b r e v i a t i o n
FR: = to0~/yR
(2.45)
has been introduced. In the case of a = 0, p0 becomes a Maxwellian and FT(u) = ~'T(u), i.e. eq. (2.44) reduces to the Fokker-Planck equation recently used in ref. 4. If, on the other hand, one starts with the reduced equations of motion (2.1), (2.2) and fi = ~
(M x u) + to0~R(t) X U,
(2.46)
eq. (2.44) is obtained immediately. Again, the result is independent of the interpretation of the stochastic differential equations because the differences are proportional to u • O/au and vanish in consequence of [u I = 1. 2.5. Properties of the collision operator If the collision operator is denoted by 1
to(t~):=--~po
-I
{O~'(rT(u)po'O~)+~'(rRpo~dp)},~-V
(2.47)
it is evident that to( 4, = 1) = 0 which guarantees the conservation of the particle number. Since to is rotationally invariant, the (Cartesian) Wigner-Eckart Theorem may be applied to factorize matrix elements of to taken with two
280
H. K A G E R M A N N et al.
arbitrary irreducible expansion tensors ¢k,,...,,(V, u) and toy,..... t(V, u) into a product of a reduced matrix element and an isotropic tensor of rank 2l:
l f f d3V d2udp~i...~,(V, u)poto(to,,l.... i(V, u)) no J ) D
(~bu,...,,to(0,, .... i))o = ~
(~bl °to(Ot))0A.,...~l,vl .... ,.
(2.48)
Here, the dot o denotes a contraction over all tensor indices and zi ~1...,~,,~ .... is an isotropic tensor which, applied to an arbitrary lth rank tensor, projects out its irreducible part~4). Since the collision operator to remains unchanged in a parity transformation ¥ -o - V, u ~ - u, L# ~ 5~, it can only connect expansion tensors of the same parity. Because F R> 0 holds and FT(u) is a positive definite symmetric tensor, it is obvious from eq. (2.47) that for any ~b(V, u) the inequality (6to(6))0 ~>0
(2.49)
is valid. Thus the collision operator is positive definite if restricted to a space of expansion tensors ~b¢ const. This guarantees the decay of any initial nonequilibrium state to thermal equilibrium or, equivalently, the increase of entropy. Finally to is invariant with respect to time reversal (indeed, both terms of to (eq. (2.47)) are separately invariant), i.e. 03T(V, u): = o3(- V, u) = to(V, u).
(2.50)
This implies that for arbitrary functions 6(V, u), to(V, u) the symmetry property (cbto(to))0 = (toTto(&r))0
(2.51)
holds, where 6 r ( ¥ , u) = 4)( - V, u) etc. If both 4~ and to show definite behaviour under time reversal, i.e. &T = T~b, tOT= T,to with T , , T , = - 1 , Onsager relations (~to(to))o =
T~T,(toto(6))o
(2.52)
exist.
3. Application to flow birefringence
3.1. Transport relaxation equations The quantity ~b(t, x, u, V) characterizing the relative deviation of the reduced distribution p(t, x, u, V) from its equilibrium value p0 can be expanded with
NONSPHERICAL BROWNIAN PARTICLES
281
respect to a complete set of irreducible expansion tensors .~,,htk')...~,,(U, V), where l is the tensor rank running from zero to infinity and where the superscript kt distinguishes various tensors of equal rank. The expansion coefficients (moments) are then lth rank tensors depending on t, x (moment method, cf. refs. 4, 9). For physical reasons it is convenient to choose a basis which is orthogonal with respect to a Maxwellian p0M, p0M = ~ 1' h " -3/2exp(_ V2),
(3.1)
viz. ,h(kt) ,h(kl,) "r ,l . . . m'e ~'~ . . .
,,v)o~
=
~lt'~
',
kf/k~ A
(3.2)
, l . . . m, vl . . . "V
For the isotropic tensor A , , . . ~/.vI. . . . l see refs. 14. The expansion tensors have the property that their mean values taken with the true equilibrium function p0(u, V) vanish for l # 0, i.e. (d~!.. ~)0 = 0,
l ~ 0.
(3.3)
For a kinetic description of flow birefringence, only a few moments prove to be sufficient, namely 4~(l°) = X/2V;
~(20) = X/2'VV ;
¢~(02) = X/~- u~--u.
(3.4)
According to notations used earlier, in (3.4) the first superscript means the tensor rank in V, the second superscript the tensor rank in u. The deviation ~b used for the description of flow birefringence in colloidal solutions is thus given by t~) = A ° ~(io)_~_ BI-" ~(2o)_~_ B2: ~(02),
(3.5)
where the vector A and the second rank tensors B~ and B2 depend on t and x. To find their respective meanings, we multiply p = p0(1 + ~b) with ~(10), d}(20) and d}(°2), integrate over u and V and use (3.3) and the definition of a nonequilibrium average
n(~'): = f f p~ d~V d2u.
(3.6)
Since we assume the number density n to be almost homogeneous (we are not interested in diffusion processes here) we may put n ~ no. Thus we obtain (d~(l°)) -~ A,
(3.7)
but to determine B1 and B2, we must observe that due to the "internal field" a correlation between 4}(2°) and 4~(°2) exists. Expanding p0 in powers of the
282
H. KAGERMANN et al.
coupling constant a and retaining only terms linear in a, we find ( 6 (20))
BI + ~ ' O/ ~ B 2,"
( 0 (02)) ~ ~ l S B I
+ f12.
(3.8)
In linear approximation in a one has thus ~ l ~ (4[~(20)) -- ~'-1~(6(02));
~2 = ( 6 (°2,)- ~15i6(20)).
(3.9)
Transport-relaxation equations adequate for the treatment of flow birefringence are obtained by multiplying the linearized reduced Fokker-Planck equation (2.44) with 6 (20), 6 (°2), respectively, integrating over u, V and observing eqs. (3.6)-(3.9). An equation for i6"°)) - iV) is not needed, since the mean velocity iV) is considered to be determined by the experimental arrangement, e.g. Couette flow between rotating cylinders6). Defining reduced collision frequencies of the linearized Fokker-Planck operator by 0(20): = -]i6 (20)'.0(0(20)))0; 0(02): = ls(6 (°2)'. 0(0(02)))0 o(°g) = to(02b=-~(6(°2)"to(6(20)))0,.
(3.10)
we obtain the following systems of moment equations: 3(6'2°))0t + (t°(20)- ~q--15t°(°22°))(6(20)) + (to (20)- ~ 1 5 to(20))(6 (02))= _ Co~7i6"°);;
(3.1 l)
(3.12) 3.2. Relaxation coeHicients In this section, detailed expressions are given for the relaxation-and coupling coefficients (3.10) and for the relaxation coefficient of the particle velocity. In order to do this, the equilibrium distribution p0 is approximated by its expansion (cf. eq. (2.31)) up to terms quadratic in a. For the calculation of the integrals extensive use is made of Cartesian tensor formulae, e.g. 14) ~ u~-~ = 2E]~v,x, ~u~u~",
(3.13)
.~C~2'U~Uv'= -- 6 'U/zUv',
(3.14)
where [Z,v,~,~ is an isotropic 5th rank tensor which describes the rotation of an irreducible 2nd rank tensort4). The relaxation frequencies are calculated up to quadratic order in a. For the relaxation coefficients of velocity and friction
N O N S P H E R I C A L BROWNIAN PARTICLES
283
pressure tensor one obtains ,o(lO) = H , to(20) =
(3.15)
2H(1 + ]a 2) + ar a(1 + ]a).
(3.16)
The relaxation coefficient of the tensor polarization ( T u ) is given by to(02) = 3FR(1 + ]a2).
(3.17)
The coupling coefficient of friction pressure tensor and tensor polarization turns out to be zero: 02 to(20) = ¢o(02~= 0.
(3.18)
For a = 0, the expressions (3.15)-(3.18) are in agreement with the results in ref. 4, if the identifications riT(a = 0) = wt, r R = 2Wu are made. From expression (3.18) it is obvious that the systematic torque does not give a direct collisional coupling between friction pressure tensor and tensor polarization. This is also true for all powers of a. While in the case a = 04) this would mean that flow birefringence cannot be treated in this theory, it is not true here, because via the correlation between u~-~ and 'VV' an effective coupling is introduced. 3.3. C o e n i c i e n t o f f l o w bire[ringence To obtain an expression for the coefficient of flow birefringence, 1SEa, the constitutive relation for the anisotropic part of the dielectric tensor, ~', stated in ref. ~5) has to be observed: (3.19)
~" = -- 2flFB ~7 W - 2~DB ~TVD.
Here, W = (Or + IOFVF)/(p+ OF) is the mean mass velocity of the system Brownian particles + fluid (F) and Vo -- v - vF is the diffusion velocity. On the other hand, a connection between ~' and the tensor polarization (~(02)) exists for a dilute system 4'5) "~ = e'(ck(°2)),
e': = X/~4~rn(arf- a~),
(3.20)
where n is the particle number density of the Brownian particles and all, a . are their effective polarizabilities for light polarized parallel and perpendicular to the symmetry axis u, respectively. A connection between the tensor polarization (d~t°2)) and the gradient of the mean particle velocity ~7(Vi can be inferred from the stationary transportrelaxation equations (3.11) and (3.12). For lowest order in a, this relation is (6to~)) ~
- ~
1
rRct
~
'
'
CoV(V).
(3.21)
284
H. K A G E R M A N N et al.
Now, one has Co(V) = W + (PF/(P + pF))VD, but the occurrence of ~ would lead to diffusio-birefringence which will not be treated here. (Otherwise, corrections to the coefficient of diffusio-birefringence obtained in ref. 4 are found.) In any case, by combining eqs. (3.19)-(3.21) the contribution to the coefficient of flow birefringence due to the systematic torque eq. (2.14) is inferred as /3FB---- ~/]~ -~-~r.
(3.22)
Notice, that a depends on the shape of the particle; in particular, it has opposite signs for prolate and oblate particles. Furthermore, a should be independent of the viscosity, analogously to the term quadratic in V in the friction force (Oseen correction). It is well known 7'~6) that a torque due to the spatial variation of the flow velocity over the extension of the particle leads to an orientational alignment (~(02)) which is also proportional to the velocity gradient. While in the present treatment one has (~f~(02)) OCOLTtrVD,
with the different approach in refs. 7, 16 one obtains (1~ (02)) OCKTalVl0
where K is a number describing the nonsphericity of the particle. The time coefficients ~'tr= (F~) -t and ~'al= (3FR) -I determine hereby the relaxation of the velocity and orientation anisotropies, respectively. Since *tr ~ moard67r~lv]~, the contribution (3.22) to flow birefringence is expected to be inversely proportional to the viscosity of the solvent. This is different for the other contribution where one has ~',1-~8~r~vR3/6kT, thus the alignment being proportional to the viscosity in that case. Furthermore, the relaxation times ~'tr and ~',r differ in their dependences on the size of the particles. Therefore the term eq. (3.22) is expected to be important if the viscosity of the fluid is small (e.g. for gases) and the particles become smaller. Note, however, that the particles have still to be large enough to guarantee that the rotational motion is overdamped (X/k---~ < ~.~i). Finally, however, it should be mentioned that, in contrast to the standard approach of refs. 7, 16, the present theory accounts also for the treatment of the effect reciprocal to flow alignmenttT), viz. a tensorial anisotropy in the velocity distribution caused by an orientational alignment under nonequilibrium conditions. This can be seen by solving the differential equations (3.1 1), (3.12) for a situation where one has (~02))# 0 but (~(20))= 0 at time t = 0.
NONSPHERICAL BROWNIAN PARTICLES
285
Then, for a ~ 1, one finds otto(20)
(3.23)
Thus a transient anisotropic velocity distribution, viz. (~-'v)~0 of the Brownian particles is produced. For to(20)>> to(02) and to(20)t >> 1, eq. (3.23) reduces to (~(2°))(t) ~ ~
ot
e x p ( - ~o(02)t) • (~(°2))(t = 0),
i.e. the anisotropy in the velocity distribution decays with the decay constant of the orientational anisotropy.
References 1) G.K. Batchelor, in Theoretical and Applied Mechanics, W.T. Keuler ed. (North-Holland, Amsterdam, 1976). 2) R. Gans, Ann. Phys. 86 (1928) 628. 3) S.J. Prager, J. Chem. Phys. 23 (1955) 2404. 4) S. Hess, Physica 74 (1974) 277. 5) S. Hess, Phys. Letters 30A (1969) 239. 6) F. Baas, Phys. Letters 36A (1971) 107; F. Baas et al., Physica $8A (1977) 1, 34, 44. 7) A. Peterlin und H.A. Stuart, in Hand-und Jahrbuch der Chemischen Physik, A. Eucken und K.L. Wolf, eds. (Akad. Verlagsgesellschaft, 1943) p. 113; L.G. Leal and E.J. Hinch, J. Fluid Mech. 55 (1972) 745. 8) S. Hess, Z. Naturforsch. 23a (1968) 597. 9) S. Hess and L. Waldmann, Z. Naturforsch. 21a (1966) 1529; H.H. Raum and W.E. K6hler, Z. Naturforsch. 25a (1970) 1178. 10) J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, (Prentice Hall, Englewood Cliffs, N.Y., 1965). 11) S. Kim and I. Oppenheim, Physica 54 (1971) 593. 12) R.E. Mortensen, J. Stat. Phys. I (1969) 271. 13) R.L. Stratonovich, Topics in the Theory of Random Noise I (Gordon and Breach, New York, 1963) pp. 100ft. 14) S. Hess, Z. Naturforsch. 23a (1968) 1095; S. Hess und W.E. K6hler, Formeln zur Tensorrechnung (Palm and Enke, Erlangen, 1980). 15) W.E. K6hler and J. Halbritter, Physica 74 (1974) 294; J. Halbritter and W.E. K6hler, Physica 76 (1974) 224. 16) S. Hess, Z. Naturforsch. 31a (1976) 1034. 17) S. Hess, Z. Naturforsch. 25a (1973) 1531.