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Do internal viscosity models satisfy the fluctuation-dissipation theorem?
Journal of Non-Newtonian Fluid Mechanics, 45 (1992) 47-61 Elsevier Science Publishers B.V., Amsterdam
47
Do internal viscosity models satisfy the fluctuation-dissipation theorem? Jay D. Schieber
’
Department of Chemical Engmeerirg, McGill Utk,ersity, Montrt!al, Qu&ec H3A 2A7 (Canada) (Received
March
3480 UniL~ersity Street,
7, 1992)
Abstract The Brownian forces for a Hookean dumbbell model with internal viscosity are characterized using the fluctuation-dissipation theorem and assuming that the momentum and configuration of the chain fluctuate on separate time scales. A Langevin-type approach in the full phase-space of the dumbbell is used to derive the proper stochastic differential equation. By considering both the stochastic differential equation, or equation of motion, and the equivalent Fokker-Planck equation, or diffusion equation, a contraction is made to the configuration space of the dumbbell. The resulting diffusion equation is compared to the models of previous works employing internal viscosity. It is found that some previous models do indeed satisfy the fluctuation-dissipation theorem (Kuhn and Kuhn, Booij and van Wiechen), whereas others do not. Keywords: Brownian forces; fluctuation-dissipation internal viscosity: Langevin-type approach
theorem;
Hookean
dumbbell
model:
1. Introduction Dilute polymer solutions have been modeled for many years as friction points connected by elastic ‘springs’. The primary motivation behind such models is the fact that polymer chains are composed of such a large number of monomer units that a chain may take effectively an infinite number of different configurations. Thus, if one breaks the chain into many
Correspondence to: J.D. Schieber, Department of Chemical Engineering, 3480 University St., MontrCal. QuCbec H3A 2A7, Canada. ’ Permanent address: Department of Chemical Engineering, University ton, TX 77204-4792, USA. 0377-0257/92/$05.00
0 1992 - Elsevier
Science
Publishers
McGill University, of Houston.
B.V. All rights reserved
Hous-
48
J.D. Schieber / J. Non-Newtonian
Fluid Mech. 45 (1992) 47- 61
subchain segments, one may be able to treat these smaller segments in a statistical mechanical way. Using straightforward statistical mechanical and thermodynamic arguments, one is led to ‘spring’ potentials describing the restoration force between adjacent subchain segments. These spring potentials are conservative forces dependent only on the separation of the segments. One of the very first models for dilute polymer solutions also incorporated internal viscosity [l]. Internal viscosity is the idea that the faster an external force attempts to stretch a polymer chain in solution, the more resistance will be offered by the chain to extension, above that offered by the bead friction alone. This is because extension of the chain requires many internal bond rotations and reconfigurations of the subchain, which will presumably dissipate energy into the solvent. Physically, one can envision the addition of internal viscosity as a ‘dashpot’ in parallel with every ‘spring’ in the bead-spring model. The earliest models used linear Hookean spring potentials and modeled the chain segments as inertialess friction points or ‘beads’. More sophisticated models for dilute polymers employ hydrodynamic interaction [2-41 (perturbation to the solvent flowfield from the motion of the other beads), finitely extensible non-linear springs [3,4,6], inertial beads [7], and complete hydrodynamic drag forces [8]. One crucial assumption in all of these models is the handling of the solvent influences on the polymer chain. The earliest models assumed a simple, linear Stokes-law drag force which depends only on the instantaneous velocity of the chain segments (beads) relative to the solvent, and a random, fluctuating Brownian force. The source of both these forces is the solvent, and one many sense intuitively that the form for these forces may not be independent of one another. The interdependence can be thought of in terms of energy exchange between the solvent and the chain. The solvent essentially ‘heats up’ the polymer segments through the continual bombardment of Brownian forces, while the segments continue to dissipate heat to the solvent through the frictional forces. In fact, a very elegant theorem has been derived using projection-operator methods [9] for a single Brownian particle [lo], and subsequently generalized for interacting Brownian particles [8], explicitly stating the interdependence of the fluctuating and the dissipative forces, called the ‘fluctuation-dissipation theorem’ (FDT; refs. 8-10 are neither the most comprehensive or earliest, but are the most appropriate here). The theorem not only guarantees the satisfaction of the Maxwell-Boltzmann equation at equilibrium, but requires specific time-dependent correlations for the random forces, necessary for the stochastic equations of motion to be consistent with Hamilton’s equations of motion. Thus, one must be very
J.D. Schieber / J. Non-Newtonian Fluid Mech. 45 (1992) 47-61
49
careful in choosing the form for the Brownian forces in such models. Also, if one adds an additional dissipative mechanism to the model, such as internal viscosity, one must correspondingly modify the Brownian forces to satisfy the FDT. Apparently, the first work using internal viscosity was done by Kuhn and Kuhn [l], using a so-called ‘string model’. Actually, the model is treated mathematically as a simple, inertialess, Hookean dumbbell in two dimensions with internal viscosity. The Brownian forces were chosen such that the Maxwell-Boltzmann equation was satisfied. However, satisfaction of the Maxwell-Boltzmann distribution is not sufficient to ensure satisfaction of the FDT. Subsequent work [11,12] used no random, fluctuating forces, but rather included a statistical entropic term that used the gradient of the distribution function; this term is also not sufficient to guarantee satisfaction of the FDT, although such an approach has usually been safe. Since some of these works have derived diffusion equations only in two dimensions, or for steady state, it is difficult to compare them for self-consistency. On the other hand, Phan-Thien et al. [13] have clearly derived a diffusion equation different from that by Manke and Williams [12], for example. Therefore, it is of use to have the correct diffusion equation at hand, by which we can compare all previous works. The goal of this paper is to determine precisely the form of the Brownian forces necessary to satisfy the fluctuation-dissipation theorem for models with internal viscosity. This is accomplished through a Langevin-type approach in the full phase-space of an inertial, Hookean dumbbell model, where employment of the FDT is straightforward. The resulting Fokker-Planck, or diffusion, equation is then contracted into the configuration space. The contraction itself, although straightforward, raises some interesting questions about the neglect of inertia in such models. 2. The proposed model Although such effects may be very important in the dynamics of polymer solutions, we will neglect hydrodynamic interaction, finite-extensibility of the chain segments, and the complete hydrodynamic forces on the beads (Basset forces), so that we may focus strictly on the effects of internal viscosity. A chain of arbitrarily many chain segments is also possible in this formalism, but the essential physics is clear in a simple dumbbell model. Thus, we begin by writing the equation of motion for each bead of the model by summing up all of the forces d rnzijL
d
= -la,
-F,‘”
+ F,” + F,“,
J.D. Schieber / J. Non-Newtonian Fluid Mech. 45 (1992) 47- 61
50
where m is the mass of each bead, 5 := 6n-qsa, and a is the radius of a bead (the symbol ‘ := ’ means ‘is defined as’). F,” is the random, fluctuating Brownian force to be determined by the FDT. F[;,‘”and F,’ are the internal viscosity and spring forces, respectively, for bead i and are given below. 5, represents the velocity of bead i with respect to the solution velocity at the same location, and the solvent flowfield ~7~at point r is given by vg -t K. r. The tensor K := (VU,)+ may be a function of time, but not position. T, is the position of bead i. The notation follows closely that of ref. 14. By writing the equation of motion for each bead in the form of eqn. (I), we are assuming that when the flowfield is switched on, the Brownian forces remain unchanged in the coordinate system that stays concomitant with the macroscopic streaming velocity of the fluid. The validity of this assumption is an important question, but plays no role in satisfying the FDT, since the fluctuation-dissipation theorem (at least in the form considered here) is valid only at equilibrium. The conservative spring forces are described by a linear force law with constant H, so we can write -F;=F;=H(r,-r,).
(2)
The non-conservative internal viscosity forces are presumed to be a linear function of the relative velocity of the two beads projected onto the vector connecting them. Thus, we write
_,,“=,,,=,(r2-r1)‘r2-r1). 2
1
(rl
-r,)’
[i),_i)
+K.(r
1
)]
_r
2
1
’
(3)
where $J is called the ‘internal viscosity parameter’. For convenience we use the generalized coordinates of the system. The ‘connector vector’ pointing from bead 1 to bead 2 is defined by Q := r2 - rl and the relative
(4)
velocity
is
V:=fi,-it,.
(5)
If we subtract the i = 2 version of eqn. (1) from the i = 1 version, and use eqns. (2) through (51, we obtain the following equation of motion for the internal phase-space coordinates of the dumbbell: m&Y=
24
-&‘V-SQQ.[V+~.Q]
-2HQ+F”,
$Q=V+K-Q.
For convenience,
we have introduced
the random
force F” := FT -F,“.
J.D. Schieher / J. Non-Newtonian Flud Me&
45 (1992) 47-61
51
To utilise the FDT, and thereby characterize the Brownian forces, we use harmonic analysis. We first define the Fourier-Laplace transform of a variable f(t) by f[o]
:= kmexp(-iol)f(l)
(7)
dt
where the Fourier-Laplace transform of a function is denoted by ‘f[w]‘. If we denote the initial value of a vector by A, =A( t = O), then for any dynamical variable A(t) to be consistent with Hamilton’s equations of motion, it must satisfy a certain relation at equilibrium. For the problem at hand, the following formulation is appropriate. Namely, if we postulate the ‘equation of motion’ for A(t) to have the form A[@] =n[o]
(A,+F”[o])
(8)
for A(t), where fl[w] is any arbitrary Brownian forces must satisfy [S] (F”[o]E;tl)
= (fi[o]-’
- iwa) * (&A,,)
but given function
of w, then
+ (&A,,),
the
(9)
where the presence of a dot over the variable indicates a derivative with respect to time. The dynamic variables in this case are Q and V, so we consider the six-dimensional vector A (at equilibrium) Now we take the Fourier-Laplace m iwV[w]
-WV,,=
iwQ[ o] - Q, = V[
-lV[w]
w].
transform
-2ffQ[o]
of eqn. (6):
- yyQQ.v[m]
+~“[wl,
~11~ I
Note that we have made a separation of time-scales assumption in finding the first line of eqn. (11) analogous to that made by Chow and Hermans [15] for colloidal particles with hydrodynamic interaction. Namely, we have assumed that the velocity relaxes much more rapidly than does Q for most configurations of the dumbbell, so that Q may be treated as a constant in taking the Fourier-Laplace transform of the third term on the right-hand side. This assumption is crucial to further development of the model, and is made implicitly by all models that satisfy the fluctuation-dissipation theorem and include either hydrodynamic interaction or internal viscosity. The assumption is consistent with the additional assumption made below that inertia is negligible.
J.D. Schieber / J. Non-Newtonian Fluid Mech. 45 (1992) 47-61
52
For our case, using eqns. (S), (lo), and (ll), we find that
(12) All that remains to specify the Brownian forces is to specify (&A,) (k,A,). Using equilibrium statistical mechanics, we find [8]
(A”A,) =
[T”?,:,,].
and
(13)
and
G&J)= [ _2yT62kp].
(14)
Thus, from eqns. (81, (lo), (121, and (14) we find (F”[w]F,“)
(15)
=2++2+.
To invert eqn. (15), we must be consistent with our assumption made above and treat Q as a constant on the time scale of the equation for V. Therefore, inverting eqn. (15) back into the time domain yields (F”(t)F,“)
= 4icra(r)l
gS + 245).
(16)
Equation (16) will be satisfied if we define our random forces as
(17) where w(t) is a vector of three independent Wiener processes, completely defined by their first moments and correlation functions [16]:
&v(t) )=o, &&v(O))=sqq. i
i
(18)
From eqn. (171, one can see how satisfying the fluctuation-dissipation theorem modifies the Brownian forces. When 4 is zero, the usual isotropic
J.D. Schieber / .I. Non-Newtonian Fluid Mech. 45 (1992) 47-61
53
Brownian forces are recovered. However, when the internal viscosity is switched on, the Brownian forces depend upon the instantaneous configuration of the dumbbell. The forces are still delta-correlated in time, because of our assumption that the configuration of the dumbbell does not change on the time scale of relaxation of the velocities. We have now completely characterized the equations of motion for the model, which can be written
miV=-{V-
24
ZQQ.[Yf~.Q]
-2HQ
(19)
+r4kT(iB+Zd3’2.-$V, I -$=V+K-Q.
It is convenient to make all subsequent ducing the following parameters A,:=
-&,
A,:=
;,
l :=-*
equations dimensionless
by intro-
24
(20)
5
The first two parameters are the characteristic time constants for the relaxation of the configuration, and of the velocity of the dumbbell, respectively. The third parameter E describes the magnitude of the internal viscosity friction relative to the hydrodynamic friction of a bead. It is well known that any Markovian stochastic differential equation has a completely equivalent Fokker-Planck or Smoluchowski (sometimes called diffusion) equation for its corresponding probability distribution function [16]. If we define the probability distribution function for the dumbbell as Yr(Q, I’, t) dQ dV:= Prob{The dumbbell has an orientation neighborhood
dQ about Q and velocity in
the neighborhood then the equivalent Fokker-Planck
aur
in the
dV about V at time t},
(21)
equation (FPE) is
a
-=__.
aQ
at
Lv+K*Q]~+:&-
B
2kT +mhB
Equation (22) represents the primary goal of this section: derivation of the Fokker-Planck equation in phase space for a Hookean dumbbell model
J.D. Schieber / J. Non-Newtorziun
54
Fluid Mech. -15 (1992) 47-61
with internal viscosity that satisfies the fluctuation-dissipation theorem. However, all calculations done to date for internal viscosity models have been done in the configuration space, rather than in the full phase space. Thus, we must find some way of contracting the information found in this section to just the configuration space. Two different methods will be considered: contraction of the Fokker-Planck equation, and the limit of zero mass in the stochastic differential equation. The former method leads to unambiguous resuits, whereas the latter, more naive, method involves subtle complications. 3. Configuration 3.1. Contraction
space of the Fokker-Planck
equation
All previous works involving internal viscosity have neglected the inertia of the polymer chain in the equation of motion of the beads. The effects of bead inertia have been shown to be insignificant for simple Rouse chains [7]. If we neglect the inertia of the beads, we are able to consider only the configuration of the dumbbell. Thus, we wish to contract the FPE into a Smoluchowski equation for the distribution function in configuration space. We use a method presented earlier [7]. To perform the contraction, we first define the projection operation indicated by (I. . . JJ I[...]:=
f/(...,Y’
(23)
dV,
where f :=/1v
dV.
The integrals are over all possible velocities projection of both sides of eqn. (22) to obtain
a_f _=_at
V. We begin by taking the the ‘equation of continuity’:
a;*
(I[vn + K’ Q,.f.
(25)
The next step is to obtain an equation for the projected first moment of V, the ‘equation of motion’ in configuration space, by taking the projection of each side of eqn. (22) multiplied by V:
$nf =- $ .(uw
+ Ke Quvn)f-
& B
wf.
H
(26)
J.D. Schieber
/ J. Non-Newtonian
Fluid Mech. 4.5 (1992) 47-61
55
We multiply this equation by A,, and take the limit A, + 0, while keeping terms only of order ( AuV)” or \lh,V. This operation leads to
(27) These equations are not yet closed, so we find the next equation in the hierarchy by taking the projection of both sides of eqn. (22) after multiplying by the dyadic product W, again keeping terms only of order (h,V)” or &I’, while taking the limit of h, + 0
A,
it
S+E Q2
-[vv]+[IvFg.
i
which is satisfied
pvn
(s+yjyQQ)]= qs+,q,
by the Maxwell-Boltzmann
(28)
relation:
2kT = ms.
(29)
When eqn. (29) is inserted into eqn. (271, which is itself inserted into the equation of continuity eqn. (25), we finally obtain the Smoluchowski equation for the dumbbell:
v-
-= at
Equation (30) represents taking the formal limit of a complete separation of time scales for the two stochastic processes involving the relaxation of the velocity and the relaxation of the configuration of the dumbbell and is the goal of this subsection. As a check on the results, the Maxwell-Boltzmann equation for the distribution function f can be shown to satisfy eqn. (30) when K is set equal to zero. It is interesting to note that the solution of eqn. (30) for steady-state, homogeneous, potential flow is independent of E (see eqn. (13.2-14) of ref. 14). In other words, internal viscosity has no influence on the dynamics of the dumbbell in steady-state, homogeneous, potential flow. 3.2. Naive neglect of inertia in the stochastic differential equation (SDE) Rather than going through such a rigorous formalism to contract to the configuration space, one often makes the naive simplification of setting the acceleration term in the equation of motion, dV/dt in eqn. (61, equal to
J.D. Schieber / J. Non-Newtonian Fluid Mech. 45 (1992) 47- 61
56
zero. If this operation is done, one can solve explicitly obtain a new stochastic differential equation: K.Q-
.
$v(t).
$Q+ H
1ii
4kT
for the velocity
to
l/2
F+-$
(31)
When finding the equivalent Smoluchowski equation, one needs to be very careful, because of the presence of the configuration-dependent prefactor in the random, stochastic term. Whenever the noise in such a stochastic differential equation is multiplicative, one needs to attach an interpretation to the equation. This is because the equation is actually not well defined in the differential form written here, but requires a definition of integration involving the Wiener process. For an excellent discussion of this point, see ref. 16. Because of the way this SDE was found, we have, as yet, no unambiguous interpretation attached to it. Usually, one of two different interpretations is used: Ito or Stratonovich, named after their respective inventors. Phan-Thien et al. [13] used the Stratonovich interpretation in their general approach for dumbbells. However, if one uses this interpretation, a Smoluchowski equation arises which differs from that found in Section 3.1 (the difference arises, as in all such problems, in the exact form of the final, diffusive term in eqn. (30)). The Smoluchowski equation found from taking a Stratonovich interpretation does not satisfy the Maxwell-Boltzmann equation, and, thus, cannot satisfy the fluctuation-dissipation theorem. On the other hand, if one uses an Ito interpretation for eqn. (311, one arrives at yet a third Smoluchowski equation, which also fails to satisfy the Maxwell-Boltzmann equation. Thus, the Ito interpretation cannot be correct in this case. The only proper interpretation, i.e. the interpretation which leads to the unambiguously determined Smoluchowski equation, eqn. (301, is a seldomly-used third interpretation called the ‘kinetic form’ [17]. As mentioned above, the type of interpretation made is determined by the definition of the integration over a Wiener process. For the Ito and Stratonovich interpretations, these integrals are well defined in the Riemann-Lebesgue sense. We define the integration for the third interpretation (indicated by the symbol fl for any arbitrary function g(t) =g[x(t>, l+‘(t), t]:
where
the second
integral
on the right-hand
side of eqn. (321 is defined
as
J.D. Schieber / J. Non-Newtonian Fluid Mech. 45 (1992) 47-61
57
the Ito integral. If one uses this third interpretation for the stochastic differential equation eqn. (311, one then arrives at the correct Smoluchowski equation, eqn. (30). It is curious that only the seldomly used kinetic form is correct when making a naive neglect of inertia for this model. There is no obvious reason a priori to choose this interpretation, and the result suggests that the interpretation may be more physical than those of either Ito or Stratonovich. It is also encouraging to note that if one simply begins with the naive form of the Langevin eqn. (311, and requires that the steady-state, equilibrium distribution be Maxwellian, then one indeed arrives at eqn. (30). 4. Comparison
with previously
used models and conclusions
4.1. Kuhn and Kuhn Apparently, the first model to include internal viscosity was the landmark publication by Kuhn and Kuhn [l], using a so-called string model. However, the physical picture of a thread is used only to derive the Hookean spring constant. For a random walk of N constant step-lengths b, they show that HE.-
3kT
(33)
Nb” *
Otherwise, they consider mathematically a massless dumbbell in two dimensions for a steady shear flow. It is not known how important the neglect of the motion of the dumbbell in the third dimension is. Nevertheless, we can reduce the derived Smoluchowski equation, eqn. (30) into two dimensions to compare with the results of Kuhn and Kuhn. Kuhn and Kuhn use the generalized coordinates Q and 8, which are the end-to-end length of the dumbbell, and the angle that the vector Q makes with the direction of flow. By using the following relations (see Table A.7-2 of ref. 18) for any vector v:
and a -.-
aQ
a aQ
la
= _-
Q aQ
one can rewrite eqn. (30) in steady shear flow. In the above relations, v is any vector with component in the direction of Q ~‘o and the other
58
J.D. Schieher / J. Non-Newtonian
component o0 is in the remaining but straightforward, manipulation,
a’f
dQ2
=
af
1
HQ
---
+apC’kT i
a -“as
4,r.f
-sin20+ i 2kT
i?Q
4kT
Fluid Mech. 45 (1992) 47-61
orthogonal direction. After some tedious, one obtains from eqn. (30): 1
sin 28
+ 2
1 af Q- a0
-;--
a”f z
af ii,
+ ,B ~sin’8
2Hf
+ k~
(36)
Note that we have neglected the time dependence and set af/at = 0 in this manipulation. If one converts this equation into the notation of Kuhn and Kuhn (through the substitutions: f-+ u, Q + h, kT/H + hi/2, lj/8kT + a’, and E + ( Dtang/Drad - 11, one should arrive at their Smoluchowski equation, (3, 41) of ref. 1. Except for the presence of f in the first term in the brackets on the right-hand side of eqn. (36), Kuhn and Kuhn’s equation is identical. When comparing their eqns. (3, 40) and (3, 411, however, one sees that the discrepancy (namely, the absence of the f, is because of a typographical error in their eqn. (3, 41). We conclude that, aside from the typographical error, the equation derived by Kuhn and Kuhn for a Hookean dumbbell with internal viscosity in two dimensions does satisfy the fluctuation-dissipation theorem. 4.2. CerJ Peter&, and Bazlia and Williams To account exactly for internal viscosity effects in the motion of the dumbbell, one must project all rotations of the dumbbell out of its motion and consider the changes only in dumbbell lengths, as was done above. However, in order to make the mathematics more tractable, some workers have made use of an approximation utilising a ‘rotation vector’ [19]. The assumption can be expressed mathematically as
QQ d ,*,Q=;Q-WQ,
(37)
where R,,, is the ‘rotation vector’. The rotation vector is used by Cerf [20] and by BazGa and Williams [21] to account for the tumbling of the chain in shear flows. For dumbbells, Cerf assumed a magnitude for R of j/2. However, BazGa and Williams claim that such an assumption leads to a non-symmetric stress tensor. Thus, Bazua and Williams use a prefactor involving the consistently-averaged second moments of the dumbbell:
lOI =
(Q:) (Q;>
+ (Q,‘)”
(38)
J.D. Schieber / J. Non-Newtonian Fluid Me&
45 (1992) 47- 61
59
in order to make the stress tensor symmetric. The moments are averaged in order to make the mathematics tractable. Note that the magnitude of the rotation vector is proportional to the shear rate. Thus, at equilibrium, the rotational motion of the dumbbell is assumed by eqn. (38) to be zero. However, at equilibrium the dumbbell with rotate because of Brownian forces, and the actual rotation of the dumbbell is not removed from the change in Q in accounting for internal viscosity effects. Therefore, the internal viscosity will act in exactly the same manner as the bead friction, and the dumbbell behaves exactly like the usual Hookean dumbbell without internal viscosity, but with the time constant resealed. Strictly speaking then, the dumbbell with the rotational vector does satisfy the fluctuation-dissipation theorem at equilibrium, but only because it, in effect, has no internal viscosity, but rather an increased bead friction. For very small shear rates, the dumbbell is expected to tumble from both Brownian forces and drag forces. However, only that component of rotation arising from the drag forces will be removed from the dumbbell deformation in accounting for internal viscosity effects, whereas the Brownian forces are not modified. Using linear response theory [lo], one could presumably check to see if such an assumption if plausible. However, such a check is outside the scope of this work. 4.3. Booij and van Wiechen, Fuller and Leal, and Manke and Williams Booij and van Wiechen [22] have also derived a Smoluchowski equation for a massless Hookean dumbbell with internal viscosity, which they solved for small values of the relative internal viscosity parameter E using perturbation for some flow fields. Indeed, if one compares their eqn. (14) with eqn. (30) above, one finds that the two equations are identical (the following substitutions apply: p + Q, f + K, and ri -+ A). Thus, we conclude that the model used by Booij and van Wiechen, and the works that follow [11,14,23] do satisfy the fluctuation-dissipation theorem. 4.4. Phan-Thien. Atkinson and Tanner Using strictly a Langevin approach, Phan-Thien et al. [13] write the following stochastic differential equation for a Hookean dumbbell with internal viscosity (their eqn. (16) with the following substitutions: L + K, R -+ Q, 5 + (, D + 2kT/l, m(t) + n(t) -+ dW(t)/dt, 3kT/Na’ + H, and 77(l)+ #J) K.Q-
&Q+
d”“;’
* &t)].
(39)
60
J.D. Schieber / J. Non-Newtonian Fluid Mech. 4.5 (1992) 47-61
To this stochastic differential equation, Phan-Thien et al. give a Stratonovich interpretation. However, none of the three possible interpretations discussed above will lead to the proper Smoluchowski equation. Therefore, we must conclude that the dumbbell model with internal viscosity discussed by these authors violates the fluctuation-dissipation theorem. 4.5. Fixman Fixman has considered chains with internal viscosity using both analytic and numerical means [24-281. For each approach he begins by deriving a Smoluchowski equation for a chain of arbitrary bead number, with hydrodynamic interaction, possibly external forces, general interbead potentials, and a general form for internal viscosity. In each paper, the form of the Brownian forces is assumed to be an entropic force: FB = -kT (a log f/aQ>. When one simplifies Fixman’s final diffusion equation (e.g. in ref. 24, eqns (3.18) and (3.19)) one indeed obtains eqn. (301, and Fixman’s diffusion equation satisfies the fluctuation-dissipation theorem. (The procedure involves the following substitutions and simplifications: G + [l6 + (+/Q2)QQ]-‘, zP + gs, v, + K f Q, P + -2HQ, V - a/aQ, ? +f. The simplifications to Fixman’s results are: no hydrodynamic interaction (i.e. no perturbation to the solvent flow field from bead motion). Hookean springs between adjacent beads, no other bead-bead interactions, only two beads, and the exact form of the internal viscosity (or internal friction) force is assumed to be the same as postulated here.) A question remains, however, as to whether the Brownian forces simulated by Fixman also satisfy the fluctuation-dissipation theorem. As Fixman correctly points out (final two sentences of Section I.A. on p. 1198 of ref. 241, “the actual properties of (the Brownian forces FB> are still to be established from the requirement of equivalence.. This equivalence is tested by comparison of (the Smoluchowski equation (30)) with the corresponding streaming velocity implied by a diffusion equation derived from” the equivalent of eqn. (31). Thus, Fixman uses Brownian forces with non-zero mean, which, in effect, adds a drift term to the deterministic part of his stochastic differential equation. Fixman then uses a forward difference equation, which implies that he is simulating a stochastic differential equation with an Ito interpretation and an addition drift term. Because of the equivalence criterion used by Fixman, the simulated Ito-interpreted stochastic differential equation with drift is completely equivalent to both the stochastic differential equation with a kinetic interpretation and the proper diffusion equation.
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61
Acknowledgments
The author would like to thank Professor John M. Dealy for providing financial support during this work, Prpfessor Charles W. Manke for useful discussions, and Dr. Hans Christian Ottinger and Professor R. Byron Bird for careful readings of the manuscript. References 1 W. Kuhn and H. Kuhn, Helv. Chim. Acta, 28 (1945) 1.533-1579; 29 (1946) 71-94. 2 W. Zylka, J. Chem. Phys., 94 (1991) 4628-4636. 3 H.C. Gttinger, J. Chem. Phys., 90 (1989) 463-473; W. Zylka and H.C. Gttingcr, J. Chcm. Phys., 90 (1989) 474-480. 4 L.E. Wedgewood, J. Non-Newtonian Fluid Mech., 31 (1989) 127-142. 5 L.E. Wedgewood and H.C. Gttinger, J. Non-Newtonian Fluid Mech., 27 (1988) 245-264. 6 J.M. Wiest, Rheol. Acta,,, (1989) 4-12. 7 J.D. Schieber and H.C. Ottinger, J. Chem. Phys., 89 (1988) 6972-6981. 8 J.D. Schiebcr, J. Chem. Phys., 91 (1991) 7526-7533. 9 H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, Springer-Verlag, Berlin, 1982. 10 R. Kubo, M. Toda and H. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, Springer-Verlag, Berlin, 1985, pp. 17-20, 106-108. 11 C.W. Manke and M.C. Williams, J. Rheol., 31 (1987) 495-510. 12 C.W. Manke and M.C. Williams, J. Rheol.. 33 (1989) 949-978. 13 N. Phan-Thien, J.D. Atkinson and R.I. Tanner. J. Non-Newtonian Fluid Mech., 3 (1978) 309-330. 14 R.B. Bird, C.F. Curtiss, R.C. Armstrong and 0. Hassager, Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theory, 2nd edn., Wiley-Interscience, New York, 1987, p. 281. 15 T.S. Chow and J.J. Hermans, J. Chem. Phys., 56 (1972) 3150-3154. 16 C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd edn., Springer-Verlag, Berlin, 1985, pp. 93-101. 17 Yu.L. Klimontovich, Physica A, 163 (1990) 515-532. 18 R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, J. Wiley, New York, 1960, p. 739B. 19 A. Peterlin, J. Polym. Sci. A2, 5 (1967) 179-193. 20 R. Cerf, Fortschritte der Hochpolymeren-Forschung (Advances in Polymer Science), 1 (1959) 382-450. 21 E.R. Bazua and M.C. Williams, J. Polym. Sci. (Phys.), 12 (1974) 825-848. 22 H.C. Booij and P.H. van Wiechen, J. Chem. Phys.. 52 (1970) 5056-5068. 23 G.G. Fuller and L.G. Leal, J. Non-Newtonian Fluid Mech., 8 (1981) 271-310. 24 M. Fixman, Macromolecules, 19 (1986) 1195-1204. 25 M. Fixman, Macromolecules, 19 (1986) 1204-1207. 26 M. Fixman, J. Chem. Phys., 84 (1986) 4085-4090. 27 M. Fixman, Faraday Discuss. Chem. Sot., 83 (1987) 199-211. 28 M. Fixman, J. Chem. Phys., 89 (1988) 2442-2462. 29 See ref. 16, p. 84, eqn. (4.2.10).
47
Do internal viscosity models satisfy the fluctuation-dissipation theorem? Jay D. Schieber
’
Department of Chemical Engmeerirg, McGill Utk,ersity, Montrt!al, Qu&ec H3A 2A7 (Canada) (Received
March
3480 UniL~ersity Street,
7, 1992)
Abstract The Brownian forces for a Hookean dumbbell model with internal viscosity are characterized using the fluctuation-dissipation theorem and assuming that the momentum and configuration of the chain fluctuate on separate time scales. A Langevin-type approach in the full phase-space of the dumbbell is used to derive the proper stochastic differential equation. By considering both the stochastic differential equation, or equation of motion, and the equivalent Fokker-Planck equation, or diffusion equation, a contraction is made to the configuration space of the dumbbell. The resulting diffusion equation is compared to the models of previous works employing internal viscosity. It is found that some previous models do indeed satisfy the fluctuation-dissipation theorem (Kuhn and Kuhn, Booij and van Wiechen), whereas others do not. Keywords: Brownian forces; fluctuation-dissipation internal viscosity: Langevin-type approach
theorem;
Hookean
dumbbell
model:
1. Introduction Dilute polymer solutions have been modeled for many years as friction points connected by elastic ‘springs’. The primary motivation behind such models is the fact that polymer chains are composed of such a large number of monomer units that a chain may take effectively an infinite number of different configurations. Thus, if one breaks the chain into many
Correspondence to: J.D. Schieber, Department of Chemical Engineering, 3480 University St., MontrCal. QuCbec H3A 2A7, Canada. ’ Permanent address: Department of Chemical Engineering, University ton, TX 77204-4792, USA. 0377-0257/92/$05.00
0 1992 - Elsevier
Science
Publishers
McGill University, of Houston.
B.V. All rights reserved
Hous-
48
J.D. Schieber / J. Non-Newtonian
Fluid Mech. 45 (1992) 47- 61
subchain segments, one may be able to treat these smaller segments in a statistical mechanical way. Using straightforward statistical mechanical and thermodynamic arguments, one is led to ‘spring’ potentials describing the restoration force between adjacent subchain segments. These spring potentials are conservative forces dependent only on the separation of the segments. One of the very first models for dilute polymer solutions also incorporated internal viscosity [l]. Internal viscosity is the idea that the faster an external force attempts to stretch a polymer chain in solution, the more resistance will be offered by the chain to extension, above that offered by the bead friction alone. This is because extension of the chain requires many internal bond rotations and reconfigurations of the subchain, which will presumably dissipate energy into the solvent. Physically, one can envision the addition of internal viscosity as a ‘dashpot’ in parallel with every ‘spring’ in the bead-spring model. The earliest models used linear Hookean spring potentials and modeled the chain segments as inertialess friction points or ‘beads’. More sophisticated models for dilute polymers employ hydrodynamic interaction [2-41 (perturbation to the solvent flowfield from the motion of the other beads), finitely extensible non-linear springs [3,4,6], inertial beads [7], and complete hydrodynamic drag forces [8]. One crucial assumption in all of these models is the handling of the solvent influences on the polymer chain. The earliest models assumed a simple, linear Stokes-law drag force which depends only on the instantaneous velocity of the chain segments (beads) relative to the solvent, and a random, fluctuating Brownian force. The source of both these forces is the solvent, and one many sense intuitively that the form for these forces may not be independent of one another. The interdependence can be thought of in terms of energy exchange between the solvent and the chain. The solvent essentially ‘heats up’ the polymer segments through the continual bombardment of Brownian forces, while the segments continue to dissipate heat to the solvent through the frictional forces. In fact, a very elegant theorem has been derived using projection-operator methods [9] for a single Brownian particle [lo], and subsequently generalized for interacting Brownian particles [8], explicitly stating the interdependence of the fluctuating and the dissipative forces, called the ‘fluctuation-dissipation theorem’ (FDT; refs. 8-10 are neither the most comprehensive or earliest, but are the most appropriate here). The theorem not only guarantees the satisfaction of the Maxwell-Boltzmann equation at equilibrium, but requires specific time-dependent correlations for the random forces, necessary for the stochastic equations of motion to be consistent with Hamilton’s equations of motion. Thus, one must be very
J.D. Schieber / J. Non-Newtonian Fluid Mech. 45 (1992) 47-61
49
careful in choosing the form for the Brownian forces in such models. Also, if one adds an additional dissipative mechanism to the model, such as internal viscosity, one must correspondingly modify the Brownian forces to satisfy the FDT. Apparently, the first work using internal viscosity was done by Kuhn and Kuhn [l], using a so-called ‘string model’. Actually, the model is treated mathematically as a simple, inertialess, Hookean dumbbell in two dimensions with internal viscosity. The Brownian forces were chosen such that the Maxwell-Boltzmann equation was satisfied. However, satisfaction of the Maxwell-Boltzmann distribution is not sufficient to ensure satisfaction of the FDT. Subsequent work [11,12] used no random, fluctuating forces, but rather included a statistical entropic term that used the gradient of the distribution function; this term is also not sufficient to guarantee satisfaction of the FDT, although such an approach has usually been safe. Since some of these works have derived diffusion equations only in two dimensions, or for steady state, it is difficult to compare them for self-consistency. On the other hand, Phan-Thien et al. [13] have clearly derived a diffusion equation different from that by Manke and Williams [12], for example. Therefore, it is of use to have the correct diffusion equation at hand, by which we can compare all previous works. The goal of this paper is to determine precisely the form of the Brownian forces necessary to satisfy the fluctuation-dissipation theorem for models with internal viscosity. This is accomplished through a Langevin-type approach in the full phase-space of an inertial, Hookean dumbbell model, where employment of the FDT is straightforward. The resulting Fokker-Planck, or diffusion, equation is then contracted into the configuration space. The contraction itself, although straightforward, raises some interesting questions about the neglect of inertia in such models. 2. The proposed model Although such effects may be very important in the dynamics of polymer solutions, we will neglect hydrodynamic interaction, finite-extensibility of the chain segments, and the complete hydrodynamic forces on the beads (Basset forces), so that we may focus strictly on the effects of internal viscosity. A chain of arbitrarily many chain segments is also possible in this formalism, but the essential physics is clear in a simple dumbbell model. Thus, we begin by writing the equation of motion for each bead of the model by summing up all of the forces d rnzijL
d
= -la,
-F,‘”
+ F,” + F,“,
J.D. Schieber / J. Non-Newtonian Fluid Mech. 45 (1992) 47- 61
50
where m is the mass of each bead, 5 := 6n-qsa, and a is the radius of a bead (the symbol ‘ := ’ means ‘is defined as’). F,” is the random, fluctuating Brownian force to be determined by the FDT. F[;,‘”and F,’ are the internal viscosity and spring forces, respectively, for bead i and are given below. 5, represents the velocity of bead i with respect to the solution velocity at the same location, and the solvent flowfield ~7~at point r is given by vg -t K. r. The tensor K := (VU,)+ may be a function of time, but not position. T, is the position of bead i. The notation follows closely that of ref. 14. By writing the equation of motion for each bead in the form of eqn. (I), we are assuming that when the flowfield is switched on, the Brownian forces remain unchanged in the coordinate system that stays concomitant with the macroscopic streaming velocity of the fluid. The validity of this assumption is an important question, but plays no role in satisfying the FDT, since the fluctuation-dissipation theorem (at least in the form considered here) is valid only at equilibrium. The conservative spring forces are described by a linear force law with constant H, so we can write -F;=F;=H(r,-r,).
(2)
The non-conservative internal viscosity forces are presumed to be a linear function of the relative velocity of the two beads projected onto the vector connecting them. Thus, we write
_,,“=,,,=,(r2-r1)‘r2-r1). 2
1
(rl
-r,)’
[i),_i)
+K.(r
1
)]
_r
2
1
’
(3)
where $J is called the ‘internal viscosity parameter’. For convenience we use the generalized coordinates of the system. The ‘connector vector’ pointing from bead 1 to bead 2 is defined by Q := r2 - rl and the relative
(4)
velocity
is
V:=fi,-it,.
(5)
If we subtract the i = 2 version of eqn. (1) from the i = 1 version, and use eqns. (2) through (51, we obtain the following equation of motion for the internal phase-space coordinates of the dumbbell: m&Y=
24
-&‘V-SQQ.[V+~.Q]
-2HQ+F”,
$Q=V+K-Q.
For convenience,
we have introduced
the random
force F” := FT -F,“.
J.D. Schieher / J. Non-Newtonian Flud Me&
45 (1992) 47-61
51
To utilise the FDT, and thereby characterize the Brownian forces, we use harmonic analysis. We first define the Fourier-Laplace transform of a variable f(t) by f[o]
:= kmexp(-iol)f(l)
(7)
dt
where the Fourier-Laplace transform of a function is denoted by ‘f[w]‘. If we denote the initial value of a vector by A, =A( t = O), then for any dynamical variable A(t) to be consistent with Hamilton’s equations of motion, it must satisfy a certain relation at equilibrium. For the problem at hand, the following formulation is appropriate. Namely, if we postulate the ‘equation of motion’ for A(t) to have the form A[@] =n[o]
(A,+F”[o])
(8)
for A(t), where fl[w] is any arbitrary Brownian forces must satisfy [S] (F”[o]E;tl)
= (fi[o]-’
- iwa) * (&A,,)
but given function
of w, then
+ (&A,,),
the
(9)
where the presence of a dot over the variable indicates a derivative with respect to time. The dynamic variables in this case are Q and V, so we consider the six-dimensional vector A (at equilibrium) Now we take the Fourier-Laplace m iwV[w]
-WV,,=
iwQ[ o] - Q, = V[
-lV[w]
w].
transform
-2ffQ[o]
of eqn. (6):
- yyQQ.v[m]
+~“[wl,
~11~ I
Note that we have made a separation of time-scales assumption in finding the first line of eqn. (11) analogous to that made by Chow and Hermans [15] for colloidal particles with hydrodynamic interaction. Namely, we have assumed that the velocity relaxes much more rapidly than does Q for most configurations of the dumbbell, so that Q may be treated as a constant in taking the Fourier-Laplace transform of the third term on the right-hand side. This assumption is crucial to further development of the model, and is made implicitly by all models that satisfy the fluctuation-dissipation theorem and include either hydrodynamic interaction or internal viscosity. The assumption is consistent with the additional assumption made below that inertia is negligible.
J.D. Schieber / J. Non-Newtonian Fluid Mech. 45 (1992) 47-61
52
For our case, using eqns. (S), (lo), and (ll), we find that
(12) All that remains to specify the Brownian forces is to specify (&A,) (k,A,). Using equilibrium statistical mechanics, we find [8]
(A”A,) =
[T”?,:,,].
and
(13)
and
G&J)= [ _2yT62kp].
(14)
Thus, from eqns. (81, (lo), (121, and (14) we find (F”[w]F,“)
(15)
=2++2+.
To invert eqn. (15), we must be consistent with our assumption made above and treat Q as a constant on the time scale of the equation for V. Therefore, inverting eqn. (15) back into the time domain yields (F”(t)F,“)
= 4icra(r)l
gS + 245).
(16)
Equation (16) will be satisfied if we define our random forces as
(17) where w(t) is a vector of three independent Wiener processes, completely defined by their first moments and correlation functions [16]:
&v(t) )=o, &&v(O))=sqq. i
i
(18)
From eqn. (171, one can see how satisfying the fluctuation-dissipation theorem modifies the Brownian forces. When 4 is zero, the usual isotropic
J.D. Schieber / .I. Non-Newtonian Fluid Mech. 45 (1992) 47-61
53
Brownian forces are recovered. However, when the internal viscosity is switched on, the Brownian forces depend upon the instantaneous configuration of the dumbbell. The forces are still delta-correlated in time, because of our assumption that the configuration of the dumbbell does not change on the time scale of relaxation of the velocities. We have now completely characterized the equations of motion for the model, which can be written
miV=-{V-
24
ZQQ.[Yf~.Q]
-2HQ
(19)
+r4kT(iB+Zd3’2.-$V, I -$=V+K-Q.
It is convenient to make all subsequent ducing the following parameters A,:=
-&,
A,:=
;,
l :=-*
equations dimensionless
by intro-
24
(20)
5
The first two parameters are the characteristic time constants for the relaxation of the configuration, and of the velocity of the dumbbell, respectively. The third parameter E describes the magnitude of the internal viscosity friction relative to the hydrodynamic friction of a bead. It is well known that any Markovian stochastic differential equation has a completely equivalent Fokker-Planck or Smoluchowski (sometimes called diffusion) equation for its corresponding probability distribution function [16]. If we define the probability distribution function for the dumbbell as Yr(Q, I’, t) dQ dV:= Prob{The dumbbell has an orientation neighborhood
dQ about Q and velocity in
the neighborhood then the equivalent Fokker-Planck
aur
in the
dV about V at time t},
(21)
equation (FPE) is
a
-=__.
aQ
at
Lv+K*Q]~+:&-
B
2kT +mhB
Equation (22) represents the primary goal of this section: derivation of the Fokker-Planck equation in phase space for a Hookean dumbbell model
J.D. Schieber / J. Non-Newtorziun
54
Fluid Mech. -15 (1992) 47-61
with internal viscosity that satisfies the fluctuation-dissipation theorem. However, all calculations done to date for internal viscosity models have been done in the configuration space, rather than in the full phase space. Thus, we must find some way of contracting the information found in this section to just the configuration space. Two different methods will be considered: contraction of the Fokker-Planck equation, and the limit of zero mass in the stochastic differential equation. The former method leads to unambiguous resuits, whereas the latter, more naive, method involves subtle complications. 3. Configuration 3.1. Contraction
space of the Fokker-Planck
equation
All previous works involving internal viscosity have neglected the inertia of the polymer chain in the equation of motion of the beads. The effects of bead inertia have been shown to be insignificant for simple Rouse chains [7]. If we neglect the inertia of the beads, we are able to consider only the configuration of the dumbbell. Thus, we wish to contract the FPE into a Smoluchowski equation for the distribution function in configuration space. We use a method presented earlier [7]. To perform the contraction, we first define the projection operation indicated by (I. . . JJ I[...]:=
f/(...,Y’
(23)
dV,
where f :=/1v
dV.
The integrals are over all possible velocities projection of both sides of eqn. (22) to obtain
a_f _=_at
V. We begin by taking the the ‘equation of continuity’:
a;*
(I[vn + K’ Q,.f.
(25)
The next step is to obtain an equation for the projected first moment of V, the ‘equation of motion’ in configuration space, by taking the projection of each side of eqn. (22) multiplied by V:
$nf =- $ .(uw
+ Ke Quvn)f-
& B
wf.
H
(26)
J.D. Schieber
/ J. Non-Newtonian
Fluid Mech. 4.5 (1992) 47-61
55
We multiply this equation by A,, and take the limit A, + 0, while keeping terms only of order ( AuV)” or \lh,V. This operation leads to
(27) These equations are not yet closed, so we find the next equation in the hierarchy by taking the projection of both sides of eqn. (22) after multiplying by the dyadic product W, again keeping terms only of order (h,V)” or &I’, while taking the limit of h, + 0
A,
it
S+E Q2
-[vv]+[IvFg.
i
which is satisfied
pvn
(s+yjyQQ)]= qs+,q,
by the Maxwell-Boltzmann
(28)
relation:
2kT = ms.
(29)
When eqn. (29) is inserted into eqn. (271, which is itself inserted into the equation of continuity eqn. (25), we finally obtain the Smoluchowski equation for the dumbbell:
v-
-= at
Equation (30) represents taking the formal limit of a complete separation of time scales for the two stochastic processes involving the relaxation of the velocity and the relaxation of the configuration of the dumbbell and is the goal of this subsection. As a check on the results, the Maxwell-Boltzmann equation for the distribution function f can be shown to satisfy eqn. (30) when K is set equal to zero. It is interesting to note that the solution of eqn. (30) for steady-state, homogeneous, potential flow is independent of E (see eqn. (13.2-14) of ref. 14). In other words, internal viscosity has no influence on the dynamics of the dumbbell in steady-state, homogeneous, potential flow. 3.2. Naive neglect of inertia in the stochastic differential equation (SDE) Rather than going through such a rigorous formalism to contract to the configuration space, one often makes the naive simplification of setting the acceleration term in the equation of motion, dV/dt in eqn. (61, equal to
J.D. Schieber / J. Non-Newtonian Fluid Mech. 45 (1992) 47- 61
56
zero. If this operation is done, one can solve explicitly obtain a new stochastic differential equation: K.Q-
.
$v(t).
$Q+ H
1ii
4kT
for the velocity
to
l/2
F+-$
(31)
When finding the equivalent Smoluchowski equation, one needs to be very careful, because of the presence of the configuration-dependent prefactor in the random, stochastic term. Whenever the noise in such a stochastic differential equation is multiplicative, one needs to attach an interpretation to the equation. This is because the equation is actually not well defined in the differential form written here, but requires a definition of integration involving the Wiener process. For an excellent discussion of this point, see ref. 16. Because of the way this SDE was found, we have, as yet, no unambiguous interpretation attached to it. Usually, one of two different interpretations is used: Ito or Stratonovich, named after their respective inventors. Phan-Thien et al. [13] used the Stratonovich interpretation in their general approach for dumbbells. However, if one uses this interpretation, a Smoluchowski equation arises which differs from that found in Section 3.1 (the difference arises, as in all such problems, in the exact form of the final, diffusive term in eqn. (30)). The Smoluchowski equation found from taking a Stratonovich interpretation does not satisfy the Maxwell-Boltzmann equation, and, thus, cannot satisfy the fluctuation-dissipation theorem. On the other hand, if one uses an Ito interpretation for eqn. (311, one arrives at yet a third Smoluchowski equation, which also fails to satisfy the Maxwell-Boltzmann equation. Thus, the Ito interpretation cannot be correct in this case. The only proper interpretation, i.e. the interpretation which leads to the unambiguously determined Smoluchowski equation, eqn. (301, is a seldomly-used third interpretation called the ‘kinetic form’ [17]. As mentioned above, the type of interpretation made is determined by the definition of the integration over a Wiener process. For the Ito and Stratonovich interpretations, these integrals are well defined in the Riemann-Lebesgue sense. We define the integration for the third interpretation (indicated by the symbol fl for any arbitrary function g(t) =g[x(t>, l+‘(t), t]:
where
the second
integral
on the right-hand
side of eqn. (321 is defined
as
J.D. Schieber / J. Non-Newtonian Fluid Mech. 45 (1992) 47-61
57
the Ito integral. If one uses this third interpretation for the stochastic differential equation eqn. (311, one then arrives at the correct Smoluchowski equation, eqn. (30). It is curious that only the seldomly used kinetic form is correct when making a naive neglect of inertia for this model. There is no obvious reason a priori to choose this interpretation, and the result suggests that the interpretation may be more physical than those of either Ito or Stratonovich. It is also encouraging to note that if one simply begins with the naive form of the Langevin eqn. (311, and requires that the steady-state, equilibrium distribution be Maxwellian, then one indeed arrives at eqn. (30). 4. Comparison
with previously
used models and conclusions
4.1. Kuhn and Kuhn Apparently, the first model to include internal viscosity was the landmark publication by Kuhn and Kuhn [l], using a so-called string model. However, the physical picture of a thread is used only to derive the Hookean spring constant. For a random walk of N constant step-lengths b, they show that HE.-
3kT
(33)
Nb” *
Otherwise, they consider mathematically a massless dumbbell in two dimensions for a steady shear flow. It is not known how important the neglect of the motion of the dumbbell in the third dimension is. Nevertheless, we can reduce the derived Smoluchowski equation, eqn. (30) into two dimensions to compare with the results of Kuhn and Kuhn. Kuhn and Kuhn use the generalized coordinates Q and 8, which are the end-to-end length of the dumbbell, and the angle that the vector Q makes with the direction of flow. By using the following relations (see Table A.7-2 of ref. 18) for any vector v:
and a -.-
aQ
a aQ
la
= _-
Q aQ
one can rewrite eqn. (30) in steady shear flow. In the above relations, v is any vector with component in the direction of Q ~‘o and the other
58
J.D. Schieher / J. Non-Newtonian
component o0 is in the remaining but straightforward, manipulation,
a’f
dQ2
=
af
1
HQ
---
+apC’kT i
a -“as
4,r.f
-sin20+ i 2kT
i?Q
4kT
Fluid Mech. 45 (1992) 47-61
orthogonal direction. After some tedious, one obtains from eqn. (30): 1
sin 28
+ 2
1 af Q- a0
-;--
a”f z
af ii,
+ ,B ~sin’8
2Hf
+ k~
(36)
Note that we have neglected the time dependence and set af/at = 0 in this manipulation. If one converts this equation into the notation of Kuhn and Kuhn (through the substitutions: f-+ u, Q + h, kT/H + hi/2, lj/8kT + a’, and E + ( Dtang/Drad - 11, one should arrive at their Smoluchowski equation, (3, 41) of ref. 1. Except for the presence of f in the first term in the brackets on the right-hand side of eqn. (36), Kuhn and Kuhn’s equation is identical. When comparing their eqns. (3, 40) and (3, 411, however, one sees that the discrepancy (namely, the absence of the f, is because of a typographical error in their eqn. (3, 41). We conclude that, aside from the typographical error, the equation derived by Kuhn and Kuhn for a Hookean dumbbell with internal viscosity in two dimensions does satisfy the fluctuation-dissipation theorem. 4.2. CerJ Peter&, and Bazlia and Williams To account exactly for internal viscosity effects in the motion of the dumbbell, one must project all rotations of the dumbbell out of its motion and consider the changes only in dumbbell lengths, as was done above. However, in order to make the mathematics more tractable, some workers have made use of an approximation utilising a ‘rotation vector’ [19]. The assumption can be expressed mathematically as
QQ d ,*,Q=;Q-WQ,
(37)
where R,,, is the ‘rotation vector’. The rotation vector is used by Cerf [20] and by BazGa and Williams [21] to account for the tumbling of the chain in shear flows. For dumbbells, Cerf assumed a magnitude for R of j/2. However, BazGa and Williams claim that such an assumption leads to a non-symmetric stress tensor. Thus, Bazua and Williams use a prefactor involving the consistently-averaged second moments of the dumbbell:
lOI =
(Q:) (Q;>
+ (Q,‘)”
(38)
J.D. Schieber / J. Non-Newtonian Fluid Me&
45 (1992) 47- 61
59
in order to make the stress tensor symmetric. The moments are averaged in order to make the mathematics tractable. Note that the magnitude of the rotation vector is proportional to the shear rate. Thus, at equilibrium, the rotational motion of the dumbbell is assumed by eqn. (38) to be zero. However, at equilibrium the dumbbell with rotate because of Brownian forces, and the actual rotation of the dumbbell is not removed from the change in Q in accounting for internal viscosity effects. Therefore, the internal viscosity will act in exactly the same manner as the bead friction, and the dumbbell behaves exactly like the usual Hookean dumbbell without internal viscosity, but with the time constant resealed. Strictly speaking then, the dumbbell with the rotational vector does satisfy the fluctuation-dissipation theorem at equilibrium, but only because it, in effect, has no internal viscosity, but rather an increased bead friction. For very small shear rates, the dumbbell is expected to tumble from both Brownian forces and drag forces. However, only that component of rotation arising from the drag forces will be removed from the dumbbell deformation in accounting for internal viscosity effects, whereas the Brownian forces are not modified. Using linear response theory [lo], one could presumably check to see if such an assumption if plausible. However, such a check is outside the scope of this work. 4.3. Booij and van Wiechen, Fuller and Leal, and Manke and Williams Booij and van Wiechen [22] have also derived a Smoluchowski equation for a massless Hookean dumbbell with internal viscosity, which they solved for small values of the relative internal viscosity parameter E using perturbation for some flow fields. Indeed, if one compares their eqn. (14) with eqn. (30) above, one finds that the two equations are identical (the following substitutions apply: p + Q, f + K, and ri -+ A). Thus, we conclude that the model used by Booij and van Wiechen, and the works that follow [11,14,23] do satisfy the fluctuation-dissipation theorem. 4.4. Phan-Thien. Atkinson and Tanner Using strictly a Langevin approach, Phan-Thien et al. [13] write the following stochastic differential equation for a Hookean dumbbell with internal viscosity (their eqn. (16) with the following substitutions: L + K, R -+ Q, 5 + (, D + 2kT/l, m(t) + n(t) -+ dW(t)/dt, 3kT/Na’ + H, and 77(l)+ #J) K.Q-
&Q+
d”“;’
* &t)].
(39)
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To this stochastic differential equation, Phan-Thien et al. give a Stratonovich interpretation. However, none of the three possible interpretations discussed above will lead to the proper Smoluchowski equation. Therefore, we must conclude that the dumbbell model with internal viscosity discussed by these authors violates the fluctuation-dissipation theorem. 4.5. Fixman Fixman has considered chains with internal viscosity using both analytic and numerical means [24-281. For each approach he begins by deriving a Smoluchowski equation for a chain of arbitrary bead number, with hydrodynamic interaction, possibly external forces, general interbead potentials, and a general form for internal viscosity. In each paper, the form of the Brownian forces is assumed to be an entropic force: FB = -kT (a log f/aQ>. When one simplifies Fixman’s final diffusion equation (e.g. in ref. 24, eqns (3.18) and (3.19)) one indeed obtains eqn. (301, and Fixman’s diffusion equation satisfies the fluctuation-dissipation theorem. (The procedure involves the following substitutions and simplifications: G + [l6 + (+/Q2)QQ]-‘, zP + gs, v, + K f Q, P + -2HQ, V - a/aQ, ? +f. The simplifications to Fixman’s results are: no hydrodynamic interaction (i.e. no perturbation to the solvent flow field from bead motion). Hookean springs between adjacent beads, no other bead-bead interactions, only two beads, and the exact form of the internal viscosity (or internal friction) force is assumed to be the same as postulated here.) A question remains, however, as to whether the Brownian forces simulated by Fixman also satisfy the fluctuation-dissipation theorem. As Fixman correctly points out (final two sentences of Section I.A. on p. 1198 of ref. 241, “the actual properties of (the Brownian forces FB> are still to be established from the requirement of equivalence.. This equivalence is tested by comparison of (the Smoluchowski equation (30)) with the corresponding streaming velocity implied by a diffusion equation derived from” the equivalent of eqn. (31). Thus, Fixman uses Brownian forces with non-zero mean, which, in effect, adds a drift term to the deterministic part of his stochastic differential equation. Fixman then uses a forward difference equation, which implies that he is simulating a stochastic differential equation with an Ito interpretation and an addition drift term. Because of the equivalence criterion used by Fixman, the simulated Ito-interpreted stochastic differential equation with drift is completely equivalent to both the stochastic differential equation with a kinetic interpretation and the proper diffusion equation.
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Acknowledgments
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