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Diffusion of a sphere in homogeneous flow
euYSCA ELSEVIER
Physica A 219 (1995) 39-55
Diffusion of a sphere in homogeneous flow Kunimasa Miyazaki a,1, Dick Bedeaux b a Department of Applied Physics, Tokyo InstT"tute of Technology, Meguro-ku, Tokyo 152, Japan b Department of Physical and Macromolecular Chemistry, Gorlaeus Laboratories, P.O. Box 9502, 2300 RA Leiden, The Netherlands
Received 15 December 1994
Abstract We derive the diffusion equation for a sphere immersed in an incompressible fluid in homogeneous flow. The effect of the finite size of the sphere is taken into account. The generalized Green-Kubo formula and the Stokes-Einstein relation are derived. As special cases, we analyze the diffusion coefficient for the cases of the simple shear flow and the pure rotational flow in detail. The relation between nonanalytic dependence on the velocity gradient and on the frequency is discussed.
1. I n t r o d u c t i o n There have been a lot o f attempts to extend the theories for the transport phenomena in a fluid at equilibrium to the system far from equilibrium [ 1 - 6 ] . Most works are dedicated to the analysis o f the nonlinear shear dependence o f the transport coefficients in the presence o f a simple shear flow. Since the nonanalytic dependence o f the stress tensor on the shear rate for a simple fluid was found by Kawasaki et al. [ 1 ] two decades ago, a lot o f theoretical [ 2 - 4 ] and numerical investigations [5,6] to confirm this prediction were presented. They found that the shear viscosity in zero frequency limit in the presence o f shear flow behaves asymptotically, for a small shear rate, as ?7(0,/3) ~ ?70 - r / ' X / ~ ,
(1.1)
where /3 is the shear rate and r/o is the bare shear viscosity. On the other hand, the m o d e - c o u p l i n g theory and the Green-Kubo formula in the equilibrium system predict 1Present address: Department of Applied Physics, Nagoya University, Chikusa-ku, Nagoya 464, Japan. 0378-4371/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4371 (95)00172-7
40
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
the long-retaining memory effect in the transport coefficient which is referred to as the "long-time tail" [7]. In zero shear limit, the shear viscosity for finite frequencies behaves as ~7(o~,0) ~ r}0 - ~7"v/-~.
(1.2)
The long-time tail is responsible for this square dependence on oJ. A similar nonanalytic dependence both on the shear rate and the frequency has attracted attention. Zwanzig [ 8 ] noted the connection between those dependences using the Goddard-Miller rheological model [9]. Though both nonanalytic dependences have their origins in mode-coupling effects, there is no theoretical investigation for the shear viscosity of a simple fluid which takes both effects of finite frequencies and of shear rate into account and explains the intimate connection laid between Eqs. (1.1) and (1.2). Analogous behavior was predicted for the diffusion coefficient of tagged particles in a fluid [ 10-14]. The diffusion coefficient in the absence of shear flow was found to have the same behavior as Eq. (1.2) for finite frequency [10,11]. The extension to the system under shear flow was performed by using the kinetic theory by Marchetti et al. [12] and the mode coupling theory by Dufty [ 13,14]. They derived the diffusion coefficient for finite frequencies and shear rate and found asymptotic behavior analogous to Eq. (1.1) for o~ = 0 and to Eq. (1.2) f o r / 3 = 0. The case where the macroscopic flow is purely elongational was discussed by Dfaz-Guilera et al. [ 15]. They used the fluctuating hydrodynamic approach [ 16]. In the works mentioned above, the size of the particle was assumed to be infinitesimal. Due to this assumption, one has to introduce a cut-off wavevector of the order of the inverse of the size of molecules constituting the fluid in order to avoid divergent behavior of the correlation function. The diffusion of a particle of finite size in the equilibrium system was discussed by Bedeaux et al. [ 17]. They derived the diffusion equation for the spherical particle with radius a immersed in an incompressible fluid within the framework of fluctuating hydrodynamics and showed that the diffusion coefficient is given by the Stokes-Einstein relation generalized to finite frequencies: D ( w ) = kBT [ - i c o m + g~(o)) ] -1 ,
(1.3)
where kB is the Boltzmann's constant, T is the temperature, m is the mass of the sphere and ((o9) is the frequency dependent friction coefficient. The friction coefficient is given by ~:(o~) = 67ra~7 (1 + tea + 91-ce2a2) ,
(1.4)
where ot = ~ / u , R e [ a ] > 0 is the inverse penetration depth. In this expression, r/ is the shear viscosity which corresponds to r/0 in Eqs. (1.1) and (1.2) and ~, ~- rl/p is the kinematic viscosity where p is the density of the fluid. For small w, the diffusion coefficient has the same dependence on the frequency as Eq. (1.2). The aim of this paper is to derive the diffusion equation for a sphere of finite size immersed in an incompressible fluid under arbitrary homogeneous flow. We assume the
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
41
radius of the sphere a to be large compared with the mean free path of molecules constituting the surrounding fluid. The effect of the finite frequency as well as the effect of the velocity gradient of the macroscopic flow are taken into account. In Section 2, the diffusion equation for the sphere is derived. The derivation is along the same line as the one discussed by Bedeaux et al. [ 17] for the equilibrium case. Due to the coupling of rather long memory effect and the macroscopic flow, the form of the diffusion current is quite different from the equilibrium case. The generalized Green-Kubo formula for the diffusion coefficient is obtained. The coefficient is given in terms of the autocorrelation function of the relative velocity of the sphere to the macroscopic flow. In Section 3, the diffusion coefficient will be analyzed by evaluating the correlation function. For this purpose, the Langevin equation for the sphere which we had already derived in two previous papers [18,19] (to be referred to as papers I and II) is employed. Since the fluctuation-dissipation theorem is modified due to the presence of the macroscopic flow, the diffusion coefficient is no more given by the Stokes-Einstein relation Eq. (1.3). As special cases, we shall focus on the simple shear case and the pure rotational flow case in Section 4 and in Section 5, respectively. The asymptotic behavior of the coefficient is shown to have the same form as Eqs. (1.1) and (1.2). Especially, for the pure rotational case, we show the explicit dependence of the coefficient on arbitrary o) and/3. The relation of the diffusion coefficient to that in the comoving frame of reference which moves along the streamline of the macroscopic flow is elucidated. Section 6 is devoted to the conclusions.
2. Derivation of the diffusion equation Consider the spherical particle immersed in an incompressible flow in homogeneous flow which, in the absence of the sphere, is given by Vs(r) = / 3 - r,
(2.1)
where /3 is a constant traceless tensor. We shall start from the density distribution function for the sphere: n(r, t) = 6 ( r - R ( t ) ) ,
(2.2)
where R ( t ) is the position of the sphere at time t which may fluctuate by the thermal agitation of the molecules constituting the surrounding fluid. Differentiating with respect to t, one obtains the continuity equation O n ( r , t) = - d i v { u ( t ) n ( r , t)},
(2.3)
where u ( t ) = d R ( t ) / d t is the velocity of the sphere. We divide the velocity into two parts as u(t) =vs(R(t)
+ By(t),
(2.4)
42
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
where a v ( t ) = u ( t ) -- 13- R ( t ) is the deviation of the velocity around the macroscopic flow vs(r). Then Eq. (2.3) can be rewritten as
{ ° -4- vs(r) . 27} n ( r , t )
=-V-J(r,t),
(2.5)
where J (r, t) = t3v(t)n (r, t) is the diffusive part of the current density. In the derivation of Eq. (2.5), use has been made of v s ( R ( t ) )t3(r - R ( t ) ) = v s ( r ) n ( r , t)
(2.6)
and 27 • Vs(r) = 0. For the sake of convenience, we shall introduce the wavevector representation defined by n ( k , t) =
fdre-ik'rn(r,
t).
(2.7)
In this representation, Eq. (2.5) can be written as (0 0} 07 - k . / 3 . ~-~ n(k, t) = - i k . J ( k , t).
(2.8)
The formal solution of this equation is t
dt' exp ( t - t ' ) k . / 3 . - ~ - k
n(k,t)=no(k,t)+ --00
where no (k, t) is the distribution function in the absence of the diffusive current density J ( k , t). The explicit expression for no in the real space representation is given by no(r, t) = 8 ( r - U ( t - to) • R(to) ) ,
(2.10)
where R(to) is an initial position at an unspecified time to and b/(t) -- exp [fit]
(2.11)
is the time-evolution operator which lets a vector flow away along the streamline of the unperturbed flow. For simplification, one may write Eq. (2.9) in an operator form as (2.12)
n = no + Gn,
where t
dt'exp
Gn=-
(t-t')k./3.~
(2.13)
--OO
The formal solution of Eq. (2.12) is given by n = ( 1 - G)-lno. Taking the ensemble average, one obtains
(2.14)
K. Miyazaki, D. Bedeaux/PhysicaA 219 (1995)39-55 (n) = ((1 -- G ) - l ) n o .
43 (2.15)
On the other hand, the diffusive current density is written as J = ~vn
= 6 v ( 1 - G)-lno.
(2.16)
Then the average is (J) = (By( 1 - G)-a)no.
(2.17)
Substituting Eq. (2.15) into Eq. (2.17) and eliminating no, one obtains (J) = (Sv(1
- G)-I)((1
- G)-l)-l(n).
(2.18)
Expanding Eq. (2.18) in 6 v ( t ) and retaining the lowest order, one arrives at (S} ~ (~vG> (n} t
=-i
dt' (Sv(t)exp ( t - t ' ) k . ~ . ~ - - ~
8v(t')).k(n(k,t')).
(2.19)
Here we shall use the following formula
with the time dependent wavevector k ( t ) - exp [ t ~ t I • k,
(2.21)
where t~ is the transpose o f / 3 . This formula is the same as the one which Onuki et al. have used for the analysis of the critical fluctuations under shear [20]. Using this formula, Eq. (2.19) can be rewritten as t
(J) = - i f dt' ( S v ( t ) 8 v ( t ( ) ) • k ( t - t ' ) ( n ( k ( t - t'), t')) t
= - i [ dt' D'(t, t') • k ( t - t') ( n ( k ( t - t'), t')), d
(2.22)
--o<3
where D ' ( t , t') -= (t3v( t)6v( t') )
(2.23)
is the autocorrelation function for the relative velocity of the sphere. Combining Eqs. (2.8) and (2.22), one arrives at the equation for (n(k, t))
44
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
{'
/ t
- k - ft.
(n(k, t)) = -
d t ' k - D ' ( t , t t ) . k ( t - t ' ) ( n ( k ( t - t ' ) , t')). (2.24)
In the absence of the homogeneous flow, Eq. (2.23) reduces to the well-known GreenKubo formula for the diffusion coefficient for the equilibrium case. It is clearer to see Eq. (2.24) in the real space representation:
{o
~ +v~(r) • V
}
P(r,t) =
/ dt'V.D'(t,t').V(t-t')P(r(t-t'),t'), t
(2.25) where P ( r , t ) ~ ( n ( r , t ) ) is the mean distribution function, r ( t ) = L / - l ( t ) • r is the time dependent position and 0 -- a r ( t ) "
V(t)
(2.26)
Furthermore, using the identity ~7 = ~7(t) •/,/-1 (t), one arrives at
{o
~-~+Vs(r)-V
}
(2.27)
P(r,t)
t
= /
dt' V ( t - t') • D(t, t') • V ( t - t ' ) P ( r ( t
- t'), t'),
(2.28)
--OO
where we defined the diffusion coefficient by D(t, t') =/,/-1 (t - t') • (~v(t)~v(t')}.
(2.29)
This is the Green-Kubo formula generalized to the system in the presence of the homogeneous flow. We must note that, in the presence of the homogeneous flow, the time-dependent position r ( t ) is introduced in the diffusion term in Eq. (2.28). The reason why such a term has appeared may be explained as follows: generally speaking, the diffusion current is proportional to the gradient of the density, V P ( r , t). Due to the rather long memory effect, the diffusion current is affected by all the information of V P ( r , t) in the past. The propagator which connects V P ( r , t) in the past to the diffusion current is the diffusion coefficient D(t, tr). In the presence of the homogeneous flow, however, the value of ~7P at a time t r and at a position r is carried along by the homogeneous flow and will be located at the position r ( t - d) at the present time t. Due to the coupling of the time with the position, we cannot define in Eq. (2.28) the diffusion coefficient in frequency representation which is generally easier to discuss than the time representation.
K. MiyazakL D. Bedeaux/Physica A 219 (1995) 39-55
45
Dufty [ 13,14] has also derived the diffusion equation for tagged particles of infinitesimal size under shear flow. The diffusion term was derived using the cumulant expansions in powers of the moments of the relative velocity. Accordingly, the memory effect which appeared in our analysis was neglected [21 ]. Therefore, the time dependent position did not appear. The diffusion coefficient which he derived has a different form from ours and is given by t
D(t) = / d t t ( S v ( t - t ' ) S v ( 0 ) , t') • t H ( t - t ' ) ,
(2.30)
t /
0
where ( , t t) denotes an average over the nonequilibrium ensemble at the time t/. The t / dependence arises from viscous heating due to the shear flow which he took into account. We do not consider this effect and take the temperature constant.
3. The diffusion coefficient
In this section, we focus on the diffusion coefficient defined in the previous section. In order to analyze the velocity autocorrelation function, we shall make use of the Langevin equation for the sphere which has been already discussed in paper II [ 19]. In paper II, we have shown that the Langevin equation for the sphere, in the frequency representation, is given by -iwmu(w)
= - ~ ( w ) • {u(w) - / 3 . R ( w ) } + KR(o)),
(3.1)
where ~(o~) is the frequency dependent friction coefficient which is given by the matrix form and KR(W) is the random force. ~:(~o) is written in terms of the propagator of the linearized Navier-Stokes equation as -
~:(w)
=
dk
te
sin k(t) a ~ G ( k , t ) - -
i~ot sin k a
k(t)a
-1
(3.2)
0
The derivation of this formula is explained in detail in paper I [18]. In Eq. (3.2), G ( k , t) is the propagator in the wavevector representation which is given by G ( k , t ) = 17",_exp
-
t' g ( k ( t t ) )
"Pk(t)
(3.3)
0
with g(k) = ~,k2 + (1 - 2k[~) •/3,
(3.4)
where f~ ---- k / I k I, Pk -- 1 --kf~ is the transverse projection operator and 7",__ is the time-ordering operator defined by
K. Miyazaki, D. Bedeaux/PhysicaA 219 (1995) 39-55
46 7-~exp
--
t / g(k(t~)) 0 t
t
t
-1-fdrlg(k(rl))+fdrl/dr2g(k(rl)).g(k(r2)) 0 0 7"1
....
(3.5)
On the other hand, the random force KR(W) has the following properties: (KR(og)} = 0
(3.6)
(KR(co)K~(oJ')) = ksT [g(~o) + 8g(w) + (H.C.)] 27r6(o~ - w'),
(3.7)
and
where (H.C.) denotes the Hermite conjugate. The second equation is the fluctuationdissipation theorem generalized to the case in the presence of the homogeneous flow. 8g(co) is the modified friction coefficient which is given by
8~( ~o) =_ ~( w) 8Ix(w). ~:t (~o)
(3.8)
•
with the modified mobility oo
cx3
81x(¢o)--p/(2~/dt/dt'ei'°tsinkaGk
(,t+tt).(flwflt)
o
o •G t ( k ( t ) , t') sin k(t)a k(t)a
(3.9)
Eq. (3.7) shows that the fluctuation-dissipation theorem is violated due to the homogeneous flow and the properties of the thermal fluctuations cannot be written only in terms of the transport coefficient which characterizes a dynamical property of the system. In our analysis, we are not interested in time scales for which the inertial effects come to be dominant. Therefore, we can neglect the inertial term - k o m u ( w ) in the Langevin equation Eq. (3.1). Then, the solution of Eq. (3.1) is given by By(co) = ~ - l (w) • KR(~O).
(3.10)
Substituting this into Eq. (2.23) and using Eqs. (3.7) and (3.8), one arrives at D'(t, t') = (Sv( t)6v( t') ) oo
o<3
fdt°/da)te-i~oteiw't'(Sv(to)tSv,(tot)} -- oo
-- (x) o(3
do) f d o g ' e - - R o t e i o ~ t t g - - 1 (09). (KR(og)K~(og')} ~ J2~r
= --o(3
--0o
"~'~--l((-0t)
K. Miyazaki, D. Bedeaux/PhysicaA 219 (1995) 39-55
47
O0
= j[d°~e-i~('-") {~-~(~) + ~-~(~). ~(~). et-~(o~)+ (n.c.)} ~ --CO
x kBT O0
= f
+
+ (H.c.)} x kBr
--CO
--D'(t-
t'),
(3.11)
where /x(~o) = ~:-1(o9) is the mobility tensor. Recalling the property of causality for the mobility, we obtain CO
/ ~ - ~ e - i ~ ° t l x ( W ) = { l ~ (t) forf°r t<0t>-0
(3.12)
--OO
and f~_~
•
for t > 0 for t < 0
(3.13)
sinkaG(k,t) sink(t)a k(t)------a-
(3.1Aa
e-'°~t~l~(w) =
{•/x(t) 0
--00
with
dk
(2~) 3 k~-
/x(t) = and
OO
8/x(t)=-p
"Gt(k(t)'/)
t ' - sin - ~ a a G ( ,kt + t ' ) - ( f l + f l t ) 0 sin k (t) a
k(t)a
(3.15)
Using Eels. (3.12), (3.13) and the fact that t _> t ~, Eq. (3.11) reduces to D'(t) =kBT{i~(t) + 6/x(t)}
(3.16)
and Eq. (2.29) to D(t) = kBTlg-l(t) • {/x(t) + 6/x(t)}.
(3.17)
Eq. (3.17) is the Stokes-Einstein relation generalized to the homogeneous flow case. This expression differs from that of the equilibrium case in that the time-evolution operator /,/(t) appears in the expression in addition to the fact that the mobility is replaced by, say, the effective mobility/x(t) -t- ~bt(t). It is possible to write the generalized GreenKubo formula Eq. (2.29) in a different form. Eq. (3.11) shows that the autocorrelation function for the relative velocity has the time translational invariance, i.e. (~v(t)~v(t) > = (Sv(t - t~)~v(0)}. Using this property, Eq. (2.29) can be written as
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
48
I)(t) = u - l ( t )
• (~v(t)~v(0))
= (u ~c~(t)u ~°~(0)),
(3.18)
where u (c) (t) = H -1 (t)- {u(t) - f . R ( t ) } is the velocity of the sphere in the comoving frame of reference which moves along the streamline of the macroscopic flow. We shall come back to this point in Section 5.
4. Simple shear case In this section, we consider the case that the fluid is in simple shear flow along the z-axis for which f is given by
flij = flBix Bjz.
(4.1)
In order to avoid difficulties coming from the presence of the time dependent position, we shall consider the case where the distribution of the Brownian particle is uniform along the direction of the flow, i.e., the distribution function P ( r , t) is a function of only x, y and t. In this case, diffusion only takes place in the x and the y-direction and the diffusion equation reduces to t
dr' VII • D ( t - t') • VIIP(rI[, t'),
~ P ( r l l , t) = --00 t
where rll -- ( x , y ) and VII = (a~ , ~o ) . In the derivation of Eq. (4.2), use has been made of the fact that f 2 = 0 and thus
Lt(t) = 1 + fit.
(4.3)
Now that Eq. (4.2) is in the time convolution form, one may introduce the Fourier transformation with respect to time. In the frequency representation, Eq. (4.2) is written as
-itoP(rll,W) =
Dxx(W)-~x 2 + D y e ( m )
P(rll,~o),
(4.4)
where oc
Dii (co) = f dtei°Jt Dii (t) o = kBT{/,zii(¢_o ) -}- t~ii((-O)}
for i = x, y.
(4.5)
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
49
The closed expressions for Izij(w) is given by the inverse of Eq. (3.2) and 8tzij(oo) by Eq. (3.9). Since it is not possible to give the explicit expression for them for arbitrary frequencies, we shall only discuss two limiting frequency regimes: the large frequency limit w//3 >> 1 and the stationary limit oJ --+ 0. For both cases, the detailed calculations are given in paper II. We shall only summarize the results briefly. If oJ is much larger than/3 then,/x(oJ) and 6/x(oJ) are given up to the first order in /3/w by ~(w)
___6~ra------~ 1 - o~a 1 + 40 - - ~
,]j '
(4.6)
1
81~(o)) ~---6~arlO~a-~-d \ ~
(4.7)
/ .
Therefore, for x x and yy-components, the diffusion coefficient is given up to this order by
kBT
D , ( w ) ~_ - (1 -a~a) 6~-arI
for i = x , y .
(4.8)
This is exactly the same result as the equilibrium case Eq. (1.3) up to the first order in o/a.
On the other hand, in the stationary limit o) ~ 0, one finds, after some numerical analysis, /Zxx(W = 0)
----(
],byy (0.) = O) :
1
6~'a~7
1-0
2
1 (1-0.577~) 67ra----~
(4.9)
and 1 ,//3a 2 c3tZxx(W = o) - 6~ra~ × ° 2 7 7 V u ' I
81~yy ( w = O) - 6~-ar/ 1 × 2.39
/3a 2
~ ,
(4.10)
where we assumed that flV/-fl--~-/~<< 1. These lead to Dxx( O) = O) - 6~arl
D y y ( O) = O) =
kB----~T (1 + 1 . 8 1 ~ - - ) 67ra~7
.
(4.11)
This result exhibits that well known square root dependence of the diffusion coefficient on the shear rate. We will not consider diffusion in the z-direction. The reason for this
K. Miyazaki, D. Bedeaux/PhysicaA 219 (1995) 39-55
50
is that, though Eq. (3.17) gives an explicit expression for Dzz (t), it is not possible to extract a meaningful value of a diffusion coefficient in the more classical sense from it. We have shown in the second paper [ 19] that the mean square displacement in the z-direction is no longer proportional to t and we refer to this paper for a more detailed discussion of these matters.
5. P u r e rotational f l o w case
In this section, we shall consider the case where the macroscopic flow is the pure rotational flow along the y-axis for which fl is given by o
/3 =
0 o90 0
(5.1)
where too is the angular velocity of the macroscopic flow. Since fl is the antisymmetric tensor, 8/x(to) defined by Eq. (3.9) is identically zero. The diffusion coefficient Eq. (3.17) in this case is given by D ( t ) = k~TH( t) • ~ ( t),
(5.2)
where
/./(t) =
cosw0t 0 sino)0t
0 - sin wot ) 1 0
0 cosm0t
(5.3)
is the rotational matrix. In Ref. [22] (to be referred to as paper III), one of the authors has shown that the mobility for a fluid in the rotating flow can be written in terms of that in the comoving frame of reference which rotates with the same angular velocity as the macroscopic flow as /x(t) = H ( t ) •/x(c)(t),
(5.4)
where/x (°) (t) is the mobility tensor in the comoving frame. Substituting Eq. (5.4) into Eq. (5.2), one obtains D ( t ) = kBT/x (c) (t).
(5.5)
This has the same form as the generalized Stokes-Einstein relation in the equilibrium case except that the mobility given is that in the comoving frame. To make a comparison with the equation in the laboratory frame, let us consider the diffusion equation in the comoving frame which moves along with the macroscopic flow. First, we shall consider the macroscopic flow with an arbitrary ft. The mean distribution function in the comoving frame p(c) is defined by
51
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
P(C)(r, t) = P ( H ( t )
.r, t).
(5.6)
Differentiating with respect to time and using Eq. (2.28), one obtains O--P(C) (r, t) =
P(r',t)
Ot
OH(t) • r
[r'=r(t)-1
- - " V t P ( r t ,
Ot
t)[r/=U(t)-r
t
= f dt' V ' ( t - t') • D ( t -
t') • V ' ( t -
t')P(r/(t
- t ' ) , t') Ir'=U(t)-r -
--OO
(5.7) Furthermore, using Eq. (2.26), one finds a V ' ( t - t') ]r,-_~t(t).r= 0 { H _ 1 ( t
_
t') • H ( t )
• r} = V . U - l ( t ' ) ,
P ( r ' ( t - t l) , t') Ir,-_u(t).r = P ( H -1 (t - t') • H ( t ) • r, t') = P ( L t ( t I) • r, t t) = p(c) (r, t').
(5.8)
Thus, one obtains t
~-~P(C) (r, t) = /
dt'V'D(C)(t't')"
Vp(C)(r,t'),
(5.9)
--00
where D (e) ( t , t') ~ L / - 1 ( t ' ) • D ( t
- t') • tb/-1 ( i t )
(5.10)
is the diffusion coefficient in the comoving frame. Substituting Eq. (2.29) into Eq. (5.10), one obtains D ~c) ( t , / ) =b/-1 (t) • ( 6 v ( t ) B v ( t ' ) ) .
tbl-1 ( t ' )
= (u ~°)(t)u~c~(t')).
(5.11)
Note the difference f f o m E q . (3.18). Altern~ively, if one substitutes Eq. (3.17) into Eq. (5.10), one obtains D(c) (t, tt) = k B T l g - l ( t )
. {/.t(t - t t) + tSIx(t - tt) } • t/g-1 (tt).
(5.12)
B o t h / x ( t ) and 6/~(t) generally do not commute with H ( t ) , so that the velocity autocorrelation function in the comoving frame usually does not satisfy the time translational invariance, i.e., (u(C)(t)u(C)(t')) ~ ( u ( C ) ( t - tt)u(C)(0)). I f / x ( t ) and 6/x (t) commute with H (t) and t H - 1( t ) = U (t), then Eq. (5.12) reduces to the same form as Eq. (3.17). This is the case only when the flow is pure rotational. In the pure rotational case, it follows from invariance under rotation around the y axis that
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
52
tXxx(t) = tXzz(t),
tZxz(t)= -tZzx(t),
l~xy( t) = ]Ct~yx(t) = ]J~yz ( t) = ~ z y ( t)
(5.13)
= 0.
Furthermore, we have 8/x = 0. Using these properties and Eq. (5.3), it can be shown that the mobility and Ll(t) commute. Furthermore, from Eq. (5.3), one may readily conclude/4 -1 (t) = tb/(t). As a result, the diffusion coefficient for both the laboratory frame and the comoving frame is given by the same expression Eq. (5.5). As the velocity autocorrelation function for u (cl (t) has the time translational invariance, Eq. (3.18) and Eq. (5.11) are equivalent. This is a significant result because the diffusion coefficient, even though it is modified by the rotation, does not depend on the choice of frame of reference, while the mobility in the laboratory frame differs from the one in the comoving frame as is shown by Eq. (5.4). It is easily shown that this is true for an arbitrary choice of the rotation of the frame of reference. For this reason, henceforth we shall focus on the description in the comoving frame which rotates with the angular velocity o90. The diffusion equation in this frame is given by t
f d t ' V . D ( t - t ' ) . VP(C~(r,t'),
%P(Cl(r, , ) =
t
=
+Dyy(t-t')~v z P(°~(r,t')
dt' Dxx(t-t')
--00
(5.14) with D(t) = kBTi~(c) (t). Here we have made use of Eq. (5.13). In frequency representation, the diffusion coefficient is given by 0<3
D(w) = /
dtei°~tD(t)
--00
= kBT/x(e) (o9).
(5.15)
The explicit expression for the mobility in the corotating frame was presented in paper III and we shall only show the results: /x(c) ( c ° ) - 6¢rar/1 { 1 - ~
A(c) (w/~°°) }
(5.16)
with
f(w/wo) a
=
g(w/o)o) where
0
o
--g(oJ/o)o)) o
0
f(w/ooo)
,
(5.17)
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
53
lO
logA~(x) I 0.8 0.2 0.
iii
1 log x
Fig. 1. Plot of the real part of Axx(X) (solid line), and its asymptotic behavior for large x, Re[v"Zff] (dashed line).
'{
f(x)
=~
g(x)
= --2-0i { v / _ i ( 2 + x ) 3 ( _ 3 + x) + v/i(2 - x)3(3 + x) }
h(x)
=1
x / - i ( 2 -t- x)3(25 -- 3x + x 2) + ~/i(2 - x)3(25 + 3x + x 2)
},
{ X / - / ( 2 + x)3( 10 + 3x - x 2) + x / i ( 2 - x)3( 10 - 3x - x 2 ) } . (5.18)
These functions have asymptotic behavior as, for x >> 1, A(x) ~-, x/-Z'/-x, and for x << 1, A ( x ) N constant and explicitly given by
A(x=O)=
0
o-}) 74
0
(5.19)
.
In Fig. 1, we have shown, as an example, the behavior of the real part of dxx(X). Thus the diffusion coefficient for o) >> o90 is the same as in the equilibrium case Eq. (4.8). For o) << w0, the diffusion equation will be reduced to
O P(r,t) = {Dxx ( ff-~2-t- ff-~2) + Dyyff-~2} P(r, t)
(5.20)
with
Dxx- 6zrarl 1 - -~ Dyy
-
6~ar/
1- ~
.
(5.21)
This agrees with the result given by Ryskin [23]. He pointed out that, using the fact that Eq. (5.21) differs from the Stokes-Einstein relation, the constitutive equation is not necessarily invariant under the rigid body motion of the whole system.
54
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
6. Conclusions In this article, we have generalized the Green-Kubo formula and the Stokes-Einstein relation for the diffusion coefficient to the system under arbitrary homogeneous flow. The diffusion equation which was derived in Section 2 has quite different form from that in the equilibrium case. The irreversible part of the equation is no more given in the form of the time convolution but a time dependent position vector had to be introduced. The diffusion coefficient is given by the generalized Green-Kubo formula Eq. (3.18) in which the coefficient is given in terms of the autocorrelation function of the velocity in the comoving frame moving along with the macroscopic flow, or the generalized Stokes-Einstein relation Eq. (3.17) which is written in terms of the time evolution operator H ( t ) and the effective m o b i l i t y / t ( t ) + 6 t t ( t ) . It was pointed out that the autocorrelation function of the velocity in the comoving frame does not satisfy the time translational invariance in general. Due to this property, the diffusion coefficient in the comoving frame is different from that in the laboratory frame. The explicit dependence of the diffusion coefficient on both the frequency and the velocity gradient of the macroscopic flow has been examined for the simple shear case and the pure rotational case. For both cases, the asymptotic behavior for small velocity gradients and for small frequencies exhibits the nonanalytic dependence as Eqs. (1.1) and (1.2), though the numerical factors differ. Especially, for pure rotational flow, the diffusion coefficient is found to be given in terms of the mobility in the corotating frame of reference which rotates with the same angular velocity as the macroscopic flow. It is worthwhile to note the fact that, while in the corotating frame the fluid looks as if it is at rest, the diffusion coefficient is not given by the Stokes-Einstein relation with Stokes value 67ra~7 but there is a deviation from it which is proportional to the square root of coo. This was discussed first by Ryskin [23]. He pointed out that this is a counterexample of the principle of material frame invariance (MFI) which asserts the invariance of the constitutive equation under the rigid body motion of the whole system. In paper III [22], one of the authors discussed the mobility (or the friction coefficient) and showed that the expression for the mobility varies depending not only on the rigid body motion of the system but also on the choice of the frame of reference in which the phenomenon is observed. The mobility in the laboratory frame is related to that in the comoving frame by Eq. (5.4). As is discussed in this paper, the diffusion coefficient, though it is dependent on coo, does not depend on the choice of the frame of reference and is therefore always given by the coo dependent mobility in the comoving frame.
Acknowledgements This work is partially supported by the Centennial Memorial Foundation of the Tokyo Institute of Technology for the stay of one of the authors (K. M.) in the Gorlaeus Laboratory. He is grateful to the kind hospitality of the members of the Laboratory. We are indebted to Prof. K. Kitahara for a fruitful discussion.
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
55
References [1] K. Kawasaki and J.D. Gunton, Phys. Rev. A 8 (1973) 2048; T. Yamada and K. Kawasaki, Prog. Theor. Phys. 53 (1975) 111. [2] M.H. Ernst, B. Cichocki, J.R. Dorfman, J. Sharma and H. van Beijeren, J. Stat. Phys. 18 (1978) 237. [3] A. Onuki, Phys. Lett. A 70 (1979) 31. [4] J.W. Dufty, Phys. Rev. A 30 (1984) 622. [5] T. Naitoh and S. Oono, Phys. Lett. A 57 (1976) 448. [6] D.J. Evans, Phys. Lett. A 74 (1979) 229; Phys. Rev. A 23 (1981) 1988. [7] Y. Pomeau and E R6sibois, Phys. Rep. 19 (1975) 63. [8] R. Zwanzig, Proc. Natl. Acad. Sci. USA 78 (1981) 3296. [9] R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. I: Fluid Mechanics (Wiley, New York, 1977), 1st ed. [10] T. Keyes and I. Oppenheim, Phys. Rev. A 8 (1973) 937. [11] D. Bedeaux and E Mazur, Physica 73 (1974) 431; 75 (1974) 79. [12] M.C. Marchetti and J.W. Dufty, J. Stat. Phys. 32 (1983) 255. [13] J.W. Dufty, Phys. Rev. Lett. 51 (1983) 2159. [ 14] J.W. Dufty, Phys. Rev. A 30 (1984) 1465. [ 15] A. Dfaz-Guilera, J.M. Rubi and D. Bedeanx, Physica A 154 (1989) 257. [ 16] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon, New York, 1959). [17] D. Bedeaux and P. Mazur, Physica 76 (1975) 189. [ 18 ] K. Miyazaki, D. Bedeaux and J. Bonet Avalos, preprint. [ 19] K. Miyazaki and D. Bedeanx, preprint. [20] A. Onuki and K. Kawasaki, Prog. Theor. Phys. 63 (1980) 122. [21] N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, 1981) ch. X1V. [22] K. Miyazaki, preprint. [23] G. Ryskin, Phys. Rev. Lett. 61 (1988) 1442.
Physica A 219 (1995) 39-55
Diffusion of a sphere in homogeneous flow Kunimasa Miyazaki a,1, Dick Bedeaux b a Department of Applied Physics, Tokyo InstT"tute of Technology, Meguro-ku, Tokyo 152, Japan b Department of Physical and Macromolecular Chemistry, Gorlaeus Laboratories, P.O. Box 9502, 2300 RA Leiden, The Netherlands
Received 15 December 1994
Abstract We derive the diffusion equation for a sphere immersed in an incompressible fluid in homogeneous flow. The effect of the finite size of the sphere is taken into account. The generalized Green-Kubo formula and the Stokes-Einstein relation are derived. As special cases, we analyze the diffusion coefficient for the cases of the simple shear flow and the pure rotational flow in detail. The relation between nonanalytic dependence on the velocity gradient and on the frequency is discussed.
1. I n t r o d u c t i o n There have been a lot o f attempts to extend the theories for the transport phenomena in a fluid at equilibrium to the system far from equilibrium [ 1 - 6 ] . Most works are dedicated to the analysis o f the nonlinear shear dependence o f the transport coefficients in the presence o f a simple shear flow. Since the nonanalytic dependence o f the stress tensor on the shear rate for a simple fluid was found by Kawasaki et al. [ 1 ] two decades ago, a lot o f theoretical [ 2 - 4 ] and numerical investigations [5,6] to confirm this prediction were presented. They found that the shear viscosity in zero frequency limit in the presence o f shear flow behaves asymptotically, for a small shear rate, as ?7(0,/3) ~ ?70 - r / ' X / ~ ,
(1.1)
where /3 is the shear rate and r/o is the bare shear viscosity. On the other hand, the m o d e - c o u p l i n g theory and the Green-Kubo formula in the equilibrium system predict 1Present address: Department of Applied Physics, Nagoya University, Chikusa-ku, Nagoya 464, Japan. 0378-4371/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4371 (95)00172-7
40
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
the long-retaining memory effect in the transport coefficient which is referred to as the "long-time tail" [7]. In zero shear limit, the shear viscosity for finite frequencies behaves as ~7(o~,0) ~ r}0 - ~7"v/-~.
(1.2)
The long-time tail is responsible for this square dependence on oJ. A similar nonanalytic dependence both on the shear rate and the frequency has attracted attention. Zwanzig [ 8 ] noted the connection between those dependences using the Goddard-Miller rheological model [9]. Though both nonanalytic dependences have their origins in mode-coupling effects, there is no theoretical investigation for the shear viscosity of a simple fluid which takes both effects of finite frequencies and of shear rate into account and explains the intimate connection laid between Eqs. (1.1) and (1.2). Analogous behavior was predicted for the diffusion coefficient of tagged particles in a fluid [ 10-14]. The diffusion coefficient in the absence of shear flow was found to have the same behavior as Eq. (1.2) for finite frequency [10,11]. The extension to the system under shear flow was performed by using the kinetic theory by Marchetti et al. [12] and the mode coupling theory by Dufty [ 13,14]. They derived the diffusion coefficient for finite frequencies and shear rate and found asymptotic behavior analogous to Eq. (1.1) for o~ = 0 and to Eq. (1.2) f o r / 3 = 0. The case where the macroscopic flow is purely elongational was discussed by Dfaz-Guilera et al. [ 15]. They used the fluctuating hydrodynamic approach [ 16]. In the works mentioned above, the size of the particle was assumed to be infinitesimal. Due to this assumption, one has to introduce a cut-off wavevector of the order of the inverse of the size of molecules constituting the fluid in order to avoid divergent behavior of the correlation function. The diffusion of a particle of finite size in the equilibrium system was discussed by Bedeaux et al. [ 17]. They derived the diffusion equation for the spherical particle with radius a immersed in an incompressible fluid within the framework of fluctuating hydrodynamics and showed that the diffusion coefficient is given by the Stokes-Einstein relation generalized to finite frequencies: D ( w ) = kBT [ - i c o m + g~(o)) ] -1 ,
(1.3)
where kB is the Boltzmann's constant, T is the temperature, m is the mass of the sphere and ((o9) is the frequency dependent friction coefficient. The friction coefficient is given by ~:(o~) = 67ra~7 (1 + tea + 91-ce2a2) ,
(1.4)
where ot = ~ / u , R e [ a ] > 0 is the inverse penetration depth. In this expression, r/ is the shear viscosity which corresponds to r/0 in Eqs. (1.1) and (1.2) and ~, ~- rl/p is the kinematic viscosity where p is the density of the fluid. For small w, the diffusion coefficient has the same dependence on the frequency as Eq. (1.2). The aim of this paper is to derive the diffusion equation for a sphere of finite size immersed in an incompressible fluid under arbitrary homogeneous flow. We assume the
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
41
radius of the sphere a to be large compared with the mean free path of molecules constituting the surrounding fluid. The effect of the finite frequency as well as the effect of the velocity gradient of the macroscopic flow are taken into account. In Section 2, the diffusion equation for the sphere is derived. The derivation is along the same line as the one discussed by Bedeaux et al. [ 17] for the equilibrium case. Due to the coupling of rather long memory effect and the macroscopic flow, the form of the diffusion current is quite different from the equilibrium case. The generalized Green-Kubo formula for the diffusion coefficient is obtained. The coefficient is given in terms of the autocorrelation function of the relative velocity of the sphere to the macroscopic flow. In Section 3, the diffusion coefficient will be analyzed by evaluating the correlation function. For this purpose, the Langevin equation for the sphere which we had already derived in two previous papers [18,19] (to be referred to as papers I and II) is employed. Since the fluctuation-dissipation theorem is modified due to the presence of the macroscopic flow, the diffusion coefficient is no more given by the Stokes-Einstein relation Eq. (1.3). As special cases, we shall focus on the simple shear case and the pure rotational flow case in Section 4 and in Section 5, respectively. The asymptotic behavior of the coefficient is shown to have the same form as Eqs. (1.1) and (1.2). Especially, for the pure rotational case, we show the explicit dependence of the coefficient on arbitrary o) and/3. The relation of the diffusion coefficient to that in the comoving frame of reference which moves along the streamline of the macroscopic flow is elucidated. Section 6 is devoted to the conclusions.
2. Derivation of the diffusion equation Consider the spherical particle immersed in an incompressible flow in homogeneous flow which, in the absence of the sphere, is given by Vs(r) = / 3 - r,
(2.1)
where /3 is a constant traceless tensor. We shall start from the density distribution function for the sphere: n(r, t) = 6 ( r - R ( t ) ) ,
(2.2)
where R ( t ) is the position of the sphere at time t which may fluctuate by the thermal agitation of the molecules constituting the surrounding fluid. Differentiating with respect to t, one obtains the continuity equation O n ( r , t) = - d i v { u ( t ) n ( r , t)},
(2.3)
where u ( t ) = d R ( t ) / d t is the velocity of the sphere. We divide the velocity into two parts as u(t) =vs(R(t)
+ By(t),
(2.4)
42
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
where a v ( t ) = u ( t ) -- 13- R ( t ) is the deviation of the velocity around the macroscopic flow vs(r). Then Eq. (2.3) can be rewritten as
{ ° -4- vs(r) . 27} n ( r , t )
=-V-J(r,t),
(2.5)
where J (r, t) = t3v(t)n (r, t) is the diffusive part of the current density. In the derivation of Eq. (2.5), use has been made of v s ( R ( t ) )t3(r - R ( t ) ) = v s ( r ) n ( r , t)
(2.6)
and 27 • Vs(r) = 0. For the sake of convenience, we shall introduce the wavevector representation defined by n ( k , t) =
fdre-ik'rn(r,
t).
(2.7)
In this representation, Eq. (2.5) can be written as (0 0} 07 - k . / 3 . ~-~ n(k, t) = - i k . J ( k , t).
(2.8)
The formal solution of this equation is t
dt' exp ( t - t ' ) k . / 3 . - ~ - k
n(k,t)=no(k,t)+ --00
where no (k, t) is the distribution function in the absence of the diffusive current density J ( k , t). The explicit expression for no in the real space representation is given by no(r, t) = 8 ( r - U ( t - to) • R(to) ) ,
(2.10)
where R(to) is an initial position at an unspecified time to and b/(t) -- exp [fit]
(2.11)
is the time-evolution operator which lets a vector flow away along the streamline of the unperturbed flow. For simplification, one may write Eq. (2.9) in an operator form as (2.12)
n = no + Gn,
where t
dt'exp
Gn=-
(t-t')k./3.~
(2.13)
--OO
The formal solution of Eq. (2.12) is given by n = ( 1 - G)-lno. Taking the ensemble average, one obtains
(2.14)
K. Miyazaki, D. Bedeaux/PhysicaA 219 (1995)39-55 (n) = ((1 -- G ) - l ) n o .
43 (2.15)
On the other hand, the diffusive current density is written as J = ~vn
= 6 v ( 1 - G)-lno.
(2.16)
Then the average is (J) = (By( 1 - G)-a)no.
(2.17)
Substituting Eq. (2.15) into Eq. (2.17) and eliminating no, one obtains (J) = (Sv(1
- G)-I)((1
- G)-l)-l(n).
(2.18)
Expanding Eq. (2.18) in 6 v ( t ) and retaining the lowest order, one arrives at (S} ~ (~vG> (n} t
=-i
dt' (Sv(t)exp ( t - t ' ) k . ~ . ~ - - ~
8v(t')).k(n(k,t')).
(2.19)
Here we shall use the following formula
with the time dependent wavevector k ( t ) - exp [ t ~ t I • k,
(2.21)
where t~ is the transpose o f / 3 . This formula is the same as the one which Onuki et al. have used for the analysis of the critical fluctuations under shear [20]. Using this formula, Eq. (2.19) can be rewritten as t
(J) = - i f dt' ( S v ( t ) 8 v ( t ( ) ) • k ( t - t ' ) ( n ( k ( t - t'), t')) t
= - i [ dt' D'(t, t') • k ( t - t') ( n ( k ( t - t'), t')), d
(2.22)
--o<3
where D ' ( t , t') -= (t3v( t)6v( t') )
(2.23)
is the autocorrelation function for the relative velocity of the sphere. Combining Eqs. (2.8) and (2.22), one arrives at the equation for (n(k, t))
44
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
{'
/ t
- k - ft.
(n(k, t)) = -
d t ' k - D ' ( t , t t ) . k ( t - t ' ) ( n ( k ( t - t ' ) , t')). (2.24)
In the absence of the homogeneous flow, Eq. (2.23) reduces to the well-known GreenKubo formula for the diffusion coefficient for the equilibrium case. It is clearer to see Eq. (2.24) in the real space representation:
{o
~ +v~(r) • V
}
P(r,t) =
/ dt'V.D'(t,t').V(t-t')P(r(t-t'),t'), t
(2.25) where P ( r , t ) ~ ( n ( r , t ) ) is the mean distribution function, r ( t ) = L / - l ( t ) • r is the time dependent position and 0 -- a r ( t ) "
V(t)
(2.26)
Furthermore, using the identity ~7 = ~7(t) •/,/-1 (t), one arrives at
{o
~-~+Vs(r)-V
}
(2.27)
P(r,t)
t
= /
dt' V ( t - t') • D(t, t') • V ( t - t ' ) P ( r ( t
- t'), t'),
(2.28)
--OO
where we defined the diffusion coefficient by D(t, t') =/,/-1 (t - t') • (~v(t)~v(t')}.
(2.29)
This is the Green-Kubo formula generalized to the system in the presence of the homogeneous flow. We must note that, in the presence of the homogeneous flow, the time-dependent position r ( t ) is introduced in the diffusion term in Eq. (2.28). The reason why such a term has appeared may be explained as follows: generally speaking, the diffusion current is proportional to the gradient of the density, V P ( r , t). Due to the rather long memory effect, the diffusion current is affected by all the information of V P ( r , t) in the past. The propagator which connects V P ( r , t) in the past to the diffusion current is the diffusion coefficient D(t, tr). In the presence of the homogeneous flow, however, the value of ~7P at a time t r and at a position r is carried along by the homogeneous flow and will be located at the position r ( t - d) at the present time t. Due to the coupling of the time with the position, we cannot define in Eq. (2.28) the diffusion coefficient in frequency representation which is generally easier to discuss than the time representation.
K. MiyazakL D. Bedeaux/Physica A 219 (1995) 39-55
45
Dufty [ 13,14] has also derived the diffusion equation for tagged particles of infinitesimal size under shear flow. The diffusion term was derived using the cumulant expansions in powers of the moments of the relative velocity. Accordingly, the memory effect which appeared in our analysis was neglected [21 ]. Therefore, the time dependent position did not appear. The diffusion coefficient which he derived has a different form from ours and is given by t
D(t) = / d t t ( S v ( t - t ' ) S v ( 0 ) , t') • t H ( t - t ' ) ,
(2.30)
t /
0
where ( , t t) denotes an average over the nonequilibrium ensemble at the time t/. The t / dependence arises from viscous heating due to the shear flow which he took into account. We do not consider this effect and take the temperature constant.
3. The diffusion coefficient
In this section, we focus on the diffusion coefficient defined in the previous section. In order to analyze the velocity autocorrelation function, we shall make use of the Langevin equation for the sphere which has been already discussed in paper II [ 19]. In paper II, we have shown that the Langevin equation for the sphere, in the frequency representation, is given by -iwmu(w)
= - ~ ( w ) • {u(w) - / 3 . R ( w ) } + KR(o)),
(3.1)
where ~(o~) is the frequency dependent friction coefficient which is given by the matrix form and KR(W) is the random force. ~:(~o) is written in terms of the propagator of the linearized Navier-Stokes equation as -
~:(w)
=
dk
te
sin k(t) a ~ G ( k , t ) - -
i~ot sin k a
k(t)a
-1
(3.2)
0
The derivation of this formula is explained in detail in paper I [18]. In Eq. (3.2), G ( k , t) is the propagator in the wavevector representation which is given by G ( k , t ) = 17",_exp
-
t' g ( k ( t t ) )
"Pk(t)
(3.3)
0
with g(k) = ~,k2 + (1 - 2k[~) •/3,
(3.4)
where f~ ---- k / I k I, Pk -- 1 --kf~ is the transverse projection operator and 7",__ is the time-ordering operator defined by
K. Miyazaki, D. Bedeaux/PhysicaA 219 (1995) 39-55
46 7-~exp
--
t / g(k(t~)) 0 t
t
t
-1-fdrlg(k(rl))+fdrl/dr2g(k(rl)).g(k(r2)) 0 0 7"1
....
(3.5)
On the other hand, the random force KR(W) has the following properties: (KR(og)} = 0
(3.6)
(KR(co)K~(oJ')) = ksT [g(~o) + 8g(w) + (H.C.)] 27r6(o~ - w'),
(3.7)
and
where (H.C.) denotes the Hermite conjugate. The second equation is the fluctuationdissipation theorem generalized to the case in the presence of the homogeneous flow. 8g(co) is the modified friction coefficient which is given by
8~( ~o) =_ ~( w) 8Ix(w). ~:t (~o)
(3.8)
•
with the modified mobility oo
cx3
81x(¢o)--p/(2~/dt/dt'ei'°tsinkaGk
(,t+tt).(flwflt)
o
o •G t ( k ( t ) , t') sin k(t)a k(t)a
(3.9)
Eq. (3.7) shows that the fluctuation-dissipation theorem is violated due to the homogeneous flow and the properties of the thermal fluctuations cannot be written only in terms of the transport coefficient which characterizes a dynamical property of the system. In our analysis, we are not interested in time scales for which the inertial effects come to be dominant. Therefore, we can neglect the inertial term - k o m u ( w ) in the Langevin equation Eq. (3.1). Then, the solution of Eq. (3.1) is given by By(co) = ~ - l (w) • KR(~O).
(3.10)
Substituting this into Eq. (2.23) and using Eqs. (3.7) and (3.8), one arrives at D'(t, t') = (Sv( t)6v( t') ) oo
o<3
fdt°/da)te-i~oteiw't'(Sv(to)tSv,(tot)} -- oo
-- (x) o(3
do) f d o g ' e - - R o t e i o ~ t t g - - 1 (09). (KR(og)K~(og')} ~ J2~r
= --o(3
--0o
"~'~--l((-0t)
K. Miyazaki, D. Bedeaux/PhysicaA 219 (1995) 39-55
47
O0
= j[d°~e-i~('-") {~-~(~) + ~-~(~). ~(~). et-~(o~)+ (n.c.)} ~ --CO
x kBT O0
= f
+
+ (H.c.)} x kBr
--CO
--D'(t-
t'),
(3.11)
where /x(~o) = ~:-1(o9) is the mobility tensor. Recalling the property of causality for the mobility, we obtain CO
/ ~ - ~ e - i ~ ° t l x ( W ) = { l ~ (t) forf°r t<0t>-0
(3.12)
--OO
and f~_~
•
for t > 0 for t < 0
(3.13)
sinkaG(k,t) sink(t)a k(t)------a-
(3.1Aa
e-'°~t~l~(w) =
{•/x(t) 0
--00
with
dk
(2~) 3 k~-
/x(t) = and
OO
8/x(t)=-p
"Gt(k(t)'/)
t ' - sin - ~ a a G ( ,kt + t ' ) - ( f l + f l t ) 0 sin k (t) a
k(t)a
(3.15)
Using Eels. (3.12), (3.13) and the fact that t _> t ~, Eq. (3.11) reduces to D'(t) =kBT{i~(t) + 6/x(t)}
(3.16)
and Eq. (2.29) to D(t) = kBTlg-l(t) • {/x(t) + 6/x(t)}.
(3.17)
Eq. (3.17) is the Stokes-Einstein relation generalized to the homogeneous flow case. This expression differs from that of the equilibrium case in that the time-evolution operator /,/(t) appears in the expression in addition to the fact that the mobility is replaced by, say, the effective mobility/x(t) -t- ~bt(t). It is possible to write the generalized GreenKubo formula Eq. (2.29) in a different form. Eq. (3.11) shows that the autocorrelation function for the relative velocity has the time translational invariance, i.e. (~v(t)~v(t) > = (Sv(t - t~)~v(0)}. Using this property, Eq. (2.29) can be written as
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
48
I)(t) = u - l ( t )
• (~v(t)~v(0))
= (u ~c~(t)u ~°~(0)),
(3.18)
where u (c) (t) = H -1 (t)- {u(t) - f . R ( t ) } is the velocity of the sphere in the comoving frame of reference which moves along the streamline of the macroscopic flow. We shall come back to this point in Section 5.
4. Simple shear case In this section, we consider the case that the fluid is in simple shear flow along the z-axis for which f is given by
flij = flBix Bjz.
(4.1)
In order to avoid difficulties coming from the presence of the time dependent position, we shall consider the case where the distribution of the Brownian particle is uniform along the direction of the flow, i.e., the distribution function P ( r , t) is a function of only x, y and t. In this case, diffusion only takes place in the x and the y-direction and the diffusion equation reduces to t
dr' VII • D ( t - t') • VIIP(rI[, t'),
~ P ( r l l , t) = --00 t
where rll -- ( x , y ) and VII = (a~ , ~o ) . In the derivation of Eq. (4.2), use has been made of the fact that f 2 = 0 and thus
Lt(t) = 1 + fit.
(4.3)
Now that Eq. (4.2) is in the time convolution form, one may introduce the Fourier transformation with respect to time. In the frequency representation, Eq. (4.2) is written as
-itoP(rll,W) =
Dxx(W)-~x 2 + D y e ( m )
P(rll,~o),
(4.4)
where oc
Dii (co) = f dtei°Jt Dii (t) o = kBT{/,zii(¢_o ) -}- t~ii((-O)}
for i = x, y.
(4.5)
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
49
The closed expressions for Izij(w) is given by the inverse of Eq. (3.2) and 8tzij(oo) by Eq. (3.9). Since it is not possible to give the explicit expression for them for arbitrary frequencies, we shall only discuss two limiting frequency regimes: the large frequency limit w//3 >> 1 and the stationary limit oJ --+ 0. For both cases, the detailed calculations are given in paper II. We shall only summarize the results briefly. If oJ is much larger than/3 then,/x(oJ) and 6/x(oJ) are given up to the first order in /3/w by ~(w)
___6~ra------~ 1 - o~a 1 + 40 - - ~
,]j '
(4.6)
1
81~(o)) ~---6~arlO~a-~-d \ ~
(4.7)
/ .
Therefore, for x x and yy-components, the diffusion coefficient is given up to this order by
kBT
D , ( w ) ~_ - (1 -a~a) 6~-arI
for i = x , y .
(4.8)
This is exactly the same result as the equilibrium case Eq. (1.3) up to the first order in o/a.
On the other hand, in the stationary limit o) ~ 0, one finds, after some numerical analysis, /Zxx(W = 0)
----(
],byy (0.) = O) :
1
6~'a~7
1-0
2
1 (1-0.577~) 67ra----~
(4.9)
and 1 ,//3a 2 c3tZxx(W = o) - 6~ra~ × ° 2 7 7 V u ' I
81~yy ( w = O) - 6~-ar/ 1 × 2.39
/3a 2
~ ,
(4.10)
where we assumed that flV/-fl--~-/~<< 1. These lead to Dxx( O) = O) - 6~arl
D y y ( O) = O) =
kB----~T (1 + 1 . 8 1 ~ - - ) 67ra~7
.
(4.11)
This result exhibits that well known square root dependence of the diffusion coefficient on the shear rate. We will not consider diffusion in the z-direction. The reason for this
K. Miyazaki, D. Bedeaux/PhysicaA 219 (1995) 39-55
50
is that, though Eq. (3.17) gives an explicit expression for Dzz (t), it is not possible to extract a meaningful value of a diffusion coefficient in the more classical sense from it. We have shown in the second paper [ 19] that the mean square displacement in the z-direction is no longer proportional to t and we refer to this paper for a more detailed discussion of these matters.
5. P u r e rotational f l o w case
In this section, we shall consider the case where the macroscopic flow is the pure rotational flow along the y-axis for which fl is given by o
/3 =
0 o90 0
(5.1)
where too is the angular velocity of the macroscopic flow. Since fl is the antisymmetric tensor, 8/x(to) defined by Eq. (3.9) is identically zero. The diffusion coefficient Eq. (3.17) in this case is given by D ( t ) = k~TH( t) • ~ ( t),
(5.2)
where
/./(t) =
cosw0t 0 sino)0t
0 - sin wot ) 1 0
0 cosm0t
(5.3)
is the rotational matrix. In Ref. [22] (to be referred to as paper III), one of the authors has shown that the mobility for a fluid in the rotating flow can be written in terms of that in the comoving frame of reference which rotates with the same angular velocity as the macroscopic flow as /x(t) = H ( t ) •/x(c)(t),
(5.4)
where/x (°) (t) is the mobility tensor in the comoving frame. Substituting Eq. (5.4) into Eq. (5.2), one obtains D ( t ) = kBT/x (c) (t).
(5.5)
This has the same form as the generalized Stokes-Einstein relation in the equilibrium case except that the mobility given is that in the comoving frame. To make a comparison with the equation in the laboratory frame, let us consider the diffusion equation in the comoving frame which moves along with the macroscopic flow. First, we shall consider the macroscopic flow with an arbitrary ft. The mean distribution function in the comoving frame p(c) is defined by
51
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
P(C)(r, t) = P ( H ( t )
.r, t).
(5.6)
Differentiating with respect to time and using Eq. (2.28), one obtains O--P(C) (r, t) =
P(r',t)
Ot
OH(t) • r
[r'=r(t)-1
- - " V t P ( r t ,
Ot
t)[r/=U(t)-r
t
= f dt' V ' ( t - t') • D ( t -
t') • V ' ( t -
t')P(r/(t
- t ' ) , t') Ir'=U(t)-r -
--OO
(5.7) Furthermore, using Eq. (2.26), one finds a V ' ( t - t') ]r,-_~t(t).r= 0 { H _ 1 ( t
_
t') • H ( t )
• r} = V . U - l ( t ' ) ,
P ( r ' ( t - t l) , t') Ir,-_u(t).r = P ( H -1 (t - t') • H ( t ) • r, t') = P ( L t ( t I) • r, t t) = p(c) (r, t').
(5.8)
Thus, one obtains t
~-~P(C) (r, t) = /
dt'V'D(C)(t't')"
Vp(C)(r,t'),
(5.9)
--00
where D (e) ( t , t') ~ L / - 1 ( t ' ) • D ( t
- t') • tb/-1 ( i t )
(5.10)
is the diffusion coefficient in the comoving frame. Substituting Eq. (2.29) into Eq. (5.10), one obtains D ~c) ( t , / ) =b/-1 (t) • ( 6 v ( t ) B v ( t ' ) ) .
tbl-1 ( t ' )
= (u ~°)(t)u~c~(t')).
(5.11)
Note the difference f f o m E q . (3.18). Altern~ively, if one substitutes Eq. (3.17) into Eq. (5.10), one obtains D(c) (t, tt) = k B T l g - l ( t )
. {/.t(t - t t) + tSIx(t - tt) } • t/g-1 (tt).
(5.12)
B o t h / x ( t ) and 6/~(t) generally do not commute with H ( t ) , so that the velocity autocorrelation function in the comoving frame usually does not satisfy the time translational invariance, i.e., (u(C)(t)u(C)(t')) ~ ( u ( C ) ( t - tt)u(C)(0)). I f / x ( t ) and 6/x (t) commute with H (t) and t H - 1( t ) = U (t), then Eq. (5.12) reduces to the same form as Eq. (3.17). This is the case only when the flow is pure rotational. In the pure rotational case, it follows from invariance under rotation around the y axis that
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
52
tXxx(t) = tXzz(t),
tZxz(t)= -tZzx(t),
l~xy( t) = ]Ct~yx(t) = ]J~yz ( t) = ~ z y ( t)
(5.13)
= 0.
Furthermore, we have 8/x = 0. Using these properties and Eq. (5.3), it can be shown that the mobility and Ll(t) commute. Furthermore, from Eq. (5.3), one may readily conclude/4 -1 (t) = tb/(t). As a result, the diffusion coefficient for both the laboratory frame and the comoving frame is given by the same expression Eq. (5.5). As the velocity autocorrelation function for u (cl (t) has the time translational invariance, Eq. (3.18) and Eq. (5.11) are equivalent. This is a significant result because the diffusion coefficient, even though it is modified by the rotation, does not depend on the choice of frame of reference, while the mobility in the laboratory frame differs from the one in the comoving frame as is shown by Eq. (5.4). It is easily shown that this is true for an arbitrary choice of the rotation of the frame of reference. For this reason, henceforth we shall focus on the description in the comoving frame which rotates with the angular velocity o90. The diffusion equation in this frame is given by t
f d t ' V . D ( t - t ' ) . VP(C~(r,t'),
%P(Cl(r, , ) =
t
=
+Dyy(t-t')~v z P(°~(r,t')
dt' Dxx(t-t')
--00
(5.14) with D(t) = kBTi~(c) (t). Here we have made use of Eq. (5.13). In frequency representation, the diffusion coefficient is given by 0<3
D(w) = /
dtei°~tD(t)
--00
= kBT/x(e) (o9).
(5.15)
The explicit expression for the mobility in the corotating frame was presented in paper III and we shall only show the results: /x(c) ( c ° ) - 6¢rar/1 { 1 - ~
A(c) (w/~°°) }
(5.16)
with
f(w/wo) a
=
g(w/o)o) where
0
o
--g(oJ/o)o)) o
0
f(w/ooo)
,
(5.17)
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
53
lO
logA~(x) I 0.8 0.2 0.
iii
1 log x
Fig. 1. Plot of the real part of Axx(X) (solid line), and its asymptotic behavior for large x, Re[v"Zff] (dashed line).
'{
f(x)
=~
g(x)
= --2-0i { v / _ i ( 2 + x ) 3 ( _ 3 + x) + v/i(2 - x)3(3 + x) }
h(x)
=1
x / - i ( 2 -t- x)3(25 -- 3x + x 2) + ~/i(2 - x)3(25 + 3x + x 2)
},
{ X / - / ( 2 + x)3( 10 + 3x - x 2) + x / i ( 2 - x)3( 10 - 3x - x 2 ) } . (5.18)
These functions have asymptotic behavior as, for x >> 1, A(x) ~-, x/-Z'/-x, and for x << 1, A ( x ) N constant and explicitly given by
A(x=O)=
0
o-}) 74
0
(5.19)
.
In Fig. 1, we have shown, as an example, the behavior of the real part of dxx(X). Thus the diffusion coefficient for o) >> o90 is the same as in the equilibrium case Eq. (4.8). For o) << w0, the diffusion equation will be reduced to
O P(r,t) = {Dxx ( ff-~2-t- ff-~2) + Dyyff-~2} P(r, t)
(5.20)
with
Dxx- 6zrarl 1 - -~ Dyy
-
6~ar/
1- ~
.
(5.21)
This agrees with the result given by Ryskin [23]. He pointed out that, using the fact that Eq. (5.21) differs from the Stokes-Einstein relation, the constitutive equation is not necessarily invariant under the rigid body motion of the whole system.
54
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
6. Conclusions In this article, we have generalized the Green-Kubo formula and the Stokes-Einstein relation for the diffusion coefficient to the system under arbitrary homogeneous flow. The diffusion equation which was derived in Section 2 has quite different form from that in the equilibrium case. The irreversible part of the equation is no more given in the form of the time convolution but a time dependent position vector had to be introduced. The diffusion coefficient is given by the generalized Green-Kubo formula Eq. (3.18) in which the coefficient is given in terms of the autocorrelation function of the velocity in the comoving frame moving along with the macroscopic flow, or the generalized Stokes-Einstein relation Eq. (3.17) which is written in terms of the time evolution operator H ( t ) and the effective m o b i l i t y / t ( t ) + 6 t t ( t ) . It was pointed out that the autocorrelation function of the velocity in the comoving frame does not satisfy the time translational invariance in general. Due to this property, the diffusion coefficient in the comoving frame is different from that in the laboratory frame. The explicit dependence of the diffusion coefficient on both the frequency and the velocity gradient of the macroscopic flow has been examined for the simple shear case and the pure rotational case. For both cases, the asymptotic behavior for small velocity gradients and for small frequencies exhibits the nonanalytic dependence as Eqs. (1.1) and (1.2), though the numerical factors differ. Especially, for pure rotational flow, the diffusion coefficient is found to be given in terms of the mobility in the corotating frame of reference which rotates with the same angular velocity as the macroscopic flow. It is worthwhile to note the fact that, while in the corotating frame the fluid looks as if it is at rest, the diffusion coefficient is not given by the Stokes-Einstein relation with Stokes value 67ra~7 but there is a deviation from it which is proportional to the square root of coo. This was discussed first by Ryskin [23]. He pointed out that this is a counterexample of the principle of material frame invariance (MFI) which asserts the invariance of the constitutive equation under the rigid body motion of the whole system. In paper III [22], one of the authors discussed the mobility (or the friction coefficient) and showed that the expression for the mobility varies depending not only on the rigid body motion of the system but also on the choice of the frame of reference in which the phenomenon is observed. The mobility in the laboratory frame is related to that in the comoving frame by Eq. (5.4). As is discussed in this paper, the diffusion coefficient, though it is dependent on coo, does not depend on the choice of the frame of reference and is therefore always given by the coo dependent mobility in the comoving frame.
Acknowledgements This work is partially supported by the Centennial Memorial Foundation of the Tokyo Institute of Technology for the stay of one of the authors (K. M.) in the Gorlaeus Laboratory. He is grateful to the kind hospitality of the members of the Laboratory. We are indebted to Prof. K. Kitahara for a fruitful discussion.
K. Miyazaki, D. Bedeaux/Physica A 219 (1995) 39-55
55
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