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Projection of the Rouse model onto macroscopic equations of motion for polymers under shear
H.-P. Wittmann, G. Fiedrickson
To cite this version:
H.-P. Wittmann, G. Fiedrickson. Projection of the Rouse model onto macroscopic equations of mo- tion for polymers under shear. Journal de Physique I, EDP Sciences, 1994, 4 (12), pp.1791-1812.
�10.1051/jp1:1994221�. �jpa-00247033�
Classification Physics Abstracts
46.608 47.50
Projection of the Rouse mortel onto macroscopic equations of motion for polymers under shear
H.-P. Wittmann(*) and G, H, Fredrickson
Department of Chernical Engineering and Matenals Departrnentj University of Californiaj Santa Barbara, CA 93106, U-S-A
(Received 30 May1994j received m final form 18 August 1994, accepted i September 1994)
Abstract. A projection operator technique is used to show how the Onsager coefàcients
arnong collective polymer variables con be expressed in terras of correlation and response func- tions For a collection of noninteracting Rouse polymers, and with no externat shear imposedj trie matrix of Onsager coefficients is calculated for trie collective variables of concentration and
stress. By mcludmg appropriate convective terms, these Onsager coefàcients are used to system-
atically formulate Langevin equations for concentration and stress variables. Due to memory
elfects, these equations are non-local in both space and time. Moreoverj they can be used to
compute the dynamical response functions and the Green functions in trie presence of shear flow.
The resulting coupled equations for concentration and stress variables tutu out to be appropriate generahzations of those obtained from simple phenomenological constitutive equationsj such as trie upper convected Maxwell model and the second order fluid model.
Introduction.
In recent years a dynarnical theory for concentration fluctuations within sheared polymer so-
lutions has been developed by several authors. Helfand and Fredrickson iii formulated a
phenomenological theory where coupled equations of motion for monomer concentration, the
velocity field of the solvent, and the deviatoric polymeric stress tensor field were presented.
They showed that the coupling between the concentration fluctuations and the stress tensor field is essential in order to explain the anisotropic growth of concentration fluctuations. For this purposej they added a type of elastic stress-induced diffusion term to the equation of mo- tion for the concentration fluctuations and utilized the so-called second order fluid mortel [2] as
a constitutive equation, A key feature leading to the predicted enhancement of concentration
(*)Present address: Institut fùr Physik, WA 339j Johannes-Gutenberg-Universitàt, 55099 Mainzj Gerrnany.
Q Les Editions de Physique 1994
fluctuations was trie assurned concentration dependence of the viscosity and normal stress co- efficients in the constitutive equation. Simultaneously, Onuki [3] presented a theory that coula be used to describe polymer dynamics in solutionj assuming either Rouse or reptation dynam-
ics depending on the polymer concentration In this context, he used Marrucci's constitutive mortel [4] in which a long lived strain variable is used as an alternative to stress in the second
order fluid model. Finally, MiIner
[S~ 6] developed a dynamical theory for entangled polymer
solutions. Included among the basic ingredients of his theory was the "two fluid model" put forward by Brochard and de Gennes [7] and discussed later on by Onuki [8] and by Doi and Onuki il?]. By an appropriate generalization of the two mode picture [7], MiIner was able to calculate a non equilibrium steady state structure factor in excellent agreement with results obtained by Wu et al. [9] via light scattering experiments.
Common to all these theoretical formulations is a set of coupled equations of motion for slow variables including the concentration fluctuations and the fluctuations of the elastic stress
variable, These equations have generally been guessed at, although it was recognized early
on il, Si that~ in principle, they could be projected out from the semi-microscopic Rouse or
reptation models. While such a projection technique has the great advantage of yielding the Onsager coefficients unambiguously [S]~ the procedure was only partially carried out iii Si.
In particular, previous authors have sidestepped the derivation of the equation of motion for
the stress variable~ instead utilizing a strictly phenomenological constitutive equation. This is
unsatisfying in that the concentration fluctuations and the fluctuations of the stress variable are non treated on the same footing. Hence~ the resulting description of polymer solution dynamics
does not result from a systematic and consistent éhain of approximations. A more satisfying approach would be to project the Rouse mortel in one step onto coupled Langevin equations for both concentration and stress, and then show in a second step how various phenomenological
constitutive equations con be obtained from there. This is the approach adopted in the present paper. Moreover, in the simplified approach used herej we neglect hydrodynamic interactions
and treat a polymer solution as a binary polymer blend where the two components have a high
and a very low degree of polymensation. For simplicityj we also neglect interactions between
chainsj other than their effect on establishing the values of the monomeric friction coefficient and statistical segment length, thus allowing us to focus on the dynamics of a single chain of the high molecular weight species. Random phase approximation techniques, which facilitate
corrections for interchain interactionsj are discussed briefly at the end of our analysis.
In order to deduce macroscopic equations of motion from the Rouse model, it is necessary to have a method available for calculating the matrix of Onsager coefficients in a systematic fash-
ion. This can be achieved by applying the projection operator technique il1-14] in an unusual way, Originally, the projection operator technique was developed in order to deduce equations
of motion for a given set of collective variables, where the microscopic dynamics was assumed to be deterministic Ii.e. satisfying the Liouville equation). The quantities characterizing the
projected Langevin equations are a frequency matrix and a matrix of memory kernels. The latter quantities have the pecuharity that their dynamical evolution is not governed by the Liouville operator L itself, but by the projected Liouville operator Lp = il P)L(1- P),
where P denotes the projection operator. Since Lp is diflicult to handle, it is not easy to derive expirait expressions for the memory kernels. In spite of this traditional perspective on the projection operator techniquej one can view the technique differently. As is well known, [13, 14], the Langevin equations (for the collective variables) that arise from the projection operator technique imply certain equations of motion for the dynamical correlation functions, governed again by the frequency matrix and the matrix of memory kernels. If ail the collective
variables are invariant under time reversal, then the frequency matrix strictly vanishes. In this
case, the equations of motion for the dynamical correlation functions can be used to express
the matrix of memory kernels as the matrix of dynamical linear response functions multiplied by the inverse of the matrix of dynamical correlation functions. Furthermore, there is a close connection between the memory kernels and the Onsager coefficients. Since the projection operator technique is traditionally based on a free energy bilinear in the collective variables, the matrix of Onsager coefficients is given by the matrix of memory kernels times the matrix of static correlation functions. Thus, the projection operator technique is a suitable tool for
calculating Onsager coefficients. Phrased somewhat differently, the projection operator tech- nique can be applied as a method to determine Onsager coefficients, provided that the static and dynamical correlation functions can be calculated by other means. These statements are
strictly valid only if the microscopic dynamics are deterministic and for situations in which the frequency matrix vanishes.
We now discuss the above in the context of the system of interest here, namely a solution of non-interacting polymers~ viewed as a binary polymer blend where trie two components have
a high and very low degree of polymerisation. Assuming at first that trie full microscopic dynamics of all the chemical monomers would be taken into account the dynamics would be
deterministic charactenzed by a Liouville operator. At this level of deterministic description
one can conclude that the frequency matrix must vanish because both collective fields, the concentration field and the stress tensor field are invariant with respect to time reversal. The Onsager coefficients can then be expressed as mentioned above in terms of static and dynamic correlation functions. These quantities~ however, are most conveniently calculated within the
semi-microscopical Rouse model. It should be emphisized that the Rouse model is itself a
coarse-grained model, where each Kuhnian segment represents ll~ to 21~ chemical repeat units.
The dynamics of the Kuhman segments are modeled by a Langevin equation describing an overdamped Brownian particle harmonically coupled to next neighbours along the chain in a viscous background, experiencing stochastical forces. Consequentely the time evolution is no
longer govemed by a Liouville operator but by a Fokker-Planck operator. Although Rouse
originally assumed that the viscous background is provided by solvent molecules the Rouse model can also be used to describe individual components of a polymer blend assuming that the polymers are not entangled. In this case an individual polymer is than embedded in a
viscous background provided by the neighboring polymers. Adopting this point of view the
Onsager coefficients of a component of a polymer blend can be calculated in the framework of the Rouse model. And this will be done for the high molecular weight species of the binary blend~ which is used by us to mimick for the sake of simplicity the non-interacting polymers
in solution Finally it shall be mentioned that Bixon [15] was the first to use the projection operator technique in the context of polymers. Starting from Kirkwood's generahzed diffusion equation~ he explored the theoretical foundation of the Rouse-Zimm model. Zwanzig [16]
subsequently generalized Bixon's approach. In both cases~ the goal was to obtain equations
of motions for semi-microscopic variablesj namely the Rouse modes or the position vectors of Kuhnian segments The approach that will be taken here is somewhat different. Our starting point is the semi-microscopic Rouse model from which smgle-chain correlation and response
functions can be calculated. Projection operator techniques are then used to deduce larger
scale equations for the macroscopic fields of interest. It should be noted that a discussion of the appropnate dynamical operator m the context of the projection operator technique for polymers can be found in Zwanzig's paper [16]. Finallyj we comment that an alternative
description of how to calculate Onsager coefficients in polymenc systems was given by Kawasaki and Sekimoto [18~ 19]. These authors made a local equilibrium approximation and projected
a semi-microscopic Fokker-Planck equation onto a Fokker-Planck equation for the collective density fields. As shall be discussed below, the Kawasaki-Sekimoto approximation recovers a
subset of the results obtained for the Rouse model by our more general approach~ but fails for
certain collective variables such as the deviatoric stress.
A principal outcome of the present study is a set of exphcit expressions for Onsager coeffi- cients in the framework of the Rouse model, where both concentration and stress fluctuations
are included as collective variables. Having these transport coefficients in hand, we then con-
struct linear Langevin equations for the concentration and stress dynamics of non-interacting
polymer assemblies in the presence of external fields conjugate to the collective variables. A
peculiarity of the stress variable, however~ is that there is no single relaxation time, but rather
a set of relaxation times corresponding to the different Rouse modes. As a result, the stress
Langevin equation is not local in time, but contains a memory kernel reflecting the Rouse spectrum. Following this approachj we show explicitly how the semi-microscopic Rouse mortel in the absence of imposed shear can be mapped onto a coupled set of hydrodynamic equations.
All trie transport coefficients entering these equations can be obtained in the framework of the Rouse model. While the static correlation functions can be calculated exactly in closed form, the dynamical correlation and response functions of the Rouse model can only be expressed in
terms of quadratures for arbitrary wavevectors and frequencies. Howeverj for small wavevectors
(1.e. the hydrodynamic regime)~ simple projected equations result for the collective variables.
Our next step is to generalize these equations to incorporate the effects of an externally im-
posed shear flow. As is well known, shear flow convects as well as rotates Kuhnian segments.
By including both effects~ we deduce coupled equations of motion for the concentration and
stress fluctuations of Rouse polymers under shear. This set of equations can be viewed as
a generalization (to arbitrary frequencies and to include spatial inhomogeneities) of certain
phenomenological constitutive equations [2]. Explicitly~ we show how the upper convected Maxwell model [2], Marrucci's model [4], the Lodge equation [21~, 21] equation and the second order fluid model [22] can be deduced on the basis of microscopic calculations.
The present paper is organized as follows: in section we set up the notation and define the collective variables of interest. Section 2 is dedicated to developing the projection operator
technique. Expirait expressions for the bare transport coefficients
can be found in section 3, while alternative derivations of these coefficients and the local equilibrium approximation are
discussed in section 4. Projected hydrodynamic equations in the absence of shear flow can be found in section S. In section 6~ the necessary changes resulting from the imposition of
a flow are introduced. Finally~ the phenomenological constitutive equations are discussed in section 7. For completeness, we have also included two appendices that describe details of our
calculations: Appendix A outlines the calculation of the correlation and response functions in the Rouse mode( while Appendix B summarizes a perturbative calculation for small shear
rates.
l. Rouse model and collective variables.
In the framework of the Rouse model the dynamics of the position r(s, t) of a Kuhnian segment at contour position s is governed by the following Langevin equation:
ôtrjs,t) = -(
~
~~~
+ 9rjs~t) il)
with the effective Hamiltonian H
= ~))/Î /~ds~~l'~~ ~~l'~~ (2)
o 8
and where b, N and ( are the segment size, the number of Kuhnian segments per chain, and the friction coefficient. The stochastic velocity 9~(s,t) is modeled as a Gaussian white noise
with zero mean and characterized by the covariance
< g~(s, i)g~(s',1') >
= 2k~Tç-1 1 à(s s')à(i
1') (3)
In order to go from the microscopic description as given by equations (1) and (2) in a systematic way to a macroscopic description, one has to assemble an appropriate set of collective variables.
The first collective variable that comes to mind is the density field, which contains information regarding the spatial distribution of the Kuhnian segments. The density fluctuations for a
polymeric system prove to be influenced by elastic forces, described by an elastic force field
F(x, t). In the framework of the Rouse mortel, this field is given by
~~~'~~ ~~ ~~~ ~~~'~~~ôrÎÎ) ~ ~~~'~~
where the collective field Îi(x~t) denotes the elastic stress tensor
associated with an individual polymer. This collective field contains not only information about the position but also concerning the orientation of the Kuhnian segments. The dynamics of the
density field and the stress tensor field are not independent from one another. In particular, we shall later show that the equation of motion for the density field contains a term proportional
to the divergence of the elastic force field F(x~ t), hence proportional to 5757 Îi (x, t). One can take advantage of this specific sort of couphng by decomposing the Fourier transform Îi(q,t)
of the stress tensor into longitudinal and transverse components, ail(q,t) = #a#paap(q,t) and
a~(q,i)
= (ô~p 4~4p)a~p(q,i), with 4~ = q~/q.
Up to this point only a single polymer was considered for the sake of simplicity. However, for the case of non-interacting polymers in solution or in melts, the single polymer picture is
retained and the collective fields gain a factor of the average polymer concentration:
c(qj i) = ) /~ds e-~~ r~S,~~ coôq,o (5)
o
~~'~~ Î ~~~
~ ~~ ~~~~~~
~Î~ ~~
~~Î~~~ ~) ~~~
Here, c(q~t) is the fluctuation of the monomer concentration about its average value, co, and
Q
co/N.hus~ we retain (c~ Q", Q~) as a set of
ollective variables.
2, Projection operator
technique.
The projection operator technique
be used to formulate the transport efficients in erms rrelation and response unctions
for a melt of Rouse polymers without
shear flow. Our starting point is the
ynamical function f~(q, t) of the Rouse model(~ ) among two variables
A(q~ t), B(q, t) an be nterpreted as a product [23, 14]
q[e+~~°~[Bq) % Cf~(qjt) (7)
(1)
Here
and trie
urned off.
In this context, La is the Fokker-Planck operator associated with the dynamics of the Rouse model [24]. Although the projection operator technique was origmally invented for the case of a deterministic dynamics [11-13] this technique was later on reformulated for the case of
a stochastic dynamics [25-27] governed by a Fokker Planck operator. Such an operator is
not necessarily self-adjoint as it is the case for a Liouville operator. However, as will become obvious belowj it will not be necessary for the present purposes to exphcitly invoke Lo. Our next step is to introduce the projection operator [14]
p_ ~ ~ )~-1jAB(~
q oq q
Aq~Bq
where Aq~Bq E (cqjQÎ, Q~) and with, e.g.~ cq e c(q~t = l~). C~/ denotes in this context the inverse of the matrix of static correlation functions in the Rouse model. From the work of
Zwanzig and Mori~ the Onsager coefficient A(~(q,t) is defined in terms of Q a 1- P and Lo
as [14]
A(~(q,t)
= (iQLoAq[e~Q~°Q~[iQLoBq) (8)
It is also important to recall that the projection operator technique provides not only a Langevin-type equation of motion for the collective variables, but also an equation for the Mon products or correlation functions [13, 14]:
jctBjq,t)
=
irtjciB(q, t) /~ dl' rtC(q, t t'jci~(q, t') (9)
Here and in the following the Einstein summation convention has been adopted. As discussed
in the introduction~ the frequency matrix ilt~( vanishes for the system under consideration due to our particular choice of collective variables, provided that no shear flow is taken into
account. By making use of equation (8)~ the memory kernel r(~ is given by
rt~(qi t)
= t~(qj t)C£/i~~ (la)
To facilitate the following discussionj it proves more convenient to work with the Laplace
transforms of the above relations. We define the Laplace transform of an arbitrary function
f(q~tl bY
f(qj~l = / ~
dl e-~tf(q~ij
o
Since the dynamical linear response function xo~~(qjt) is related to the correlation function
C(~(q~t) by the dissipation fluctuation theorem
~ AB~~ ~~ _i
~ ~AB~~ ~~ j~~~
° ' k~T ° '
where Bill denotes the Heaviside step function due to causahty. A Laplace transform of equation (9) for ml( + 0 yields immediately
rt~(q,~)
= kBTxo~~lq,~)Ci~(qi ~)i~~ (12)
Combining equations (ll~) and (12) allows one to express the Onsager coefficients entirely in
terms of correlation and response functions of the Rouse model:
A(~(q,mi
= kBTxo~~(q~w)Cj~(q~w)[~~C$~ (13)
