183
Journal of Non-Newtonian Fluid Mechanics, 8 (1981) 183-190
Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
Short Communication
DERIVATION OF THE MARRUCCI MODEL FROM
TRANSIENT-NETWORK THEORY
R. J. J.
JONGSCHAAP
Department of Applied Physics, Twente University of Technology, Enschede (The
Netherlands)
(Received July 3, 1980)
1. Introduction
Many theories about the rheological behaviour of melts and concentrated
solutions of high molecular weight polymers are based upon the transient-
network model, originally developed by Green and Tobolsky [l], Lodge [2],
and Yamamoto [ 31. Among these theories a model proposed by Marrucci
and coworkers [4,5] turns out to be rather successful [6-111.
The basic equations of this model, as formulated in ref. [ 51, are:
Gi = GoPi 9
(3)
Ai = xerx;.4 )
(4)
(5)
where (1) represents a spectral decomposition of the stress tensor, Xi and Gi
are the i-th relaxation time and elastic modulus, respectively (the same quan-
tities with zero subscript are the corresponding equilibrium values), 6/St is
the contravariant convective derivative (see eqn. (22) below), D is the rate-of-
strain tensor and where the dimensionless quantities xi (3~~ Q 1) can be
regarded as structural variables which describe how far the existing structure
is away from equilibrium. They are related to the degree of connectivity of
0377-0257/81/0000-0000/$2.50 @ 1981 Elsevier Scientific Publishing Company
184
the macromolecular network. The constant Q is an adjustable parameter.
The model has proven to predict correctly the mechanical response of vari-
ous polymer melts in different stress and deformation histories. From a physi-
cal point of view it is appealing because of the explicit introduction of an
expression (eqn. (5)) that describes the change of structure of the system.
Despite these attractive features, the derivation of the model, as given in refs.
[4] and [5], is not entirely satisfactory. In fact, eqns. (l-6) are introduced
in a semi-empirical way: the form of the constitutive equation (2) is
suggested by a springdashpot model with a variable spring modulus. The con-
travariant convective derivative is chosen in accordance with results of the
network theory for concentrated systems [2,12] and of bead-spring models
for dilute polymer solutions [13]. The dependence (eqn. (3)) of Gi on xi is
suggested by the theory of rubber elasticity. The dependence (eqn. (4)) of Xi
on Xi is chosen in such a way that the zero shear viscosity becomes propor-
tional to c3R, where c is the number of macromolecules per unit volume.
Finally, the kinetic expression (5) is proposed on the basis of certain micro-
scopic arguments. An objection that may be raised against this procedure is
that the constitutive equation (2) and the kinetic equation (5) are not derived
from the transient-network model (or any other consistent microscopic
model), but, instead, are introduced as separate ad hoc assumptions.
In this note it will be shown that it is possible to derive both the constitu-
tive and the kinetic equations of the Marrucci model from a balance law of
the segment-distribution function in the transient-network model. This makes
it possible to see which assumptions about this balance law are implicitly
made in the Marrucci theory and allows a comparison of this theory with
other theories based upon the transient-network model.
2. Distribution functions, balance law
Consider an incompressible fluid with c macromolecules per unit volume.
Let Ni be the number per unit volume of segments, consisting of i freely
jointed rigid links of length I (segments of this type will be called i-segments),
and \ki(q, t) d3q the number of these segments that have their end-to-end vec-
tor q in the volume element d3q in the configuration space. It follows that
s
\ki(q, t) d3q = Ni *
(f-5)
The number of i-segments per molecule will be denoted by ni, so
ni E Ni/C .
For highly entangled systems ni >> 1; then the quantity
(7)
n=
C ni
i
equals the average number of entanglements per molecule. The equilibrium
185
value of ni will be denoted by noi.
Finally we introduce the structural vari-
ables [ 4,5]
Xi E Yti/Tloi ,
(9)
which describe how far the existing structure is away from equilibrium.
Besides the total distribution functions \ki, we also introduce the distribu-
tion functions
9i(q9 t) G tllNi) *it49 t, -
(10)
Assuming Gaussian statistics, in equilibrium we have:
It/p = (bi/x)3’2 eXp(--bifJ2) ,
(11)
where bi = 3/2i12 .
The time dependence of the segment-distribution functions is governed by a
balance equation of the following form:
a\ki
- = -a: m (\kii) + ki - hi\ki e
at
(12)
The first two terms are of the usual form of an equation of continuity in
q-space; the terms ki and hi\ki represent the formation and annihilation of
i-segments respectively. Equation (12) was first derived by Yamamoto [3 3
and can be shown to be consistent with the theory of Lodge [2,12]. An exten-
sive discussion of (12) can be found in two papers by Wiegel and de Bats [14],
u51.
The function 4 = $4, t) in eqn. (12) represents the motion of the segment
vectors in the configuration space. If affine deformation of the network is
assumed,
cj=L.q,
(13)
where L is the macroscopic velocity gradient of the fluid flow. In this note we
will assume that eqn. (13) holds; the subsequent discussion, however, also
applies for a special type of non&fine deformation, namely if 4 = L - q,
where z = AL) is some effective velocity gradient of the network. In that case
one simply has to replace L by 1 in all our results. A theory of this type has
been formulated recently by Phan Thien and Tanner [16,17]. For our pur-
pose it is of interest to note that the assumption of non&fine deformation in
this case leads to a constitutive equation which is not of the contravariant
Maxwellian type, the form proposed in the Marrucci model.
The creation function ki and the destruction function hi will depend [3,15]
in general on the variables q and i. Usually certain specific forms of these
functions are assumed. The theory of Lodge [12], for instance, essentiaIly
corresponds to the assumption that
ki =giILiO(q) ,
(14)
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where gi are constants and J/p is given by eqn. (ll), and that the hi are con-
stants. The form (14) of the creation functions is a consequence of the
assumption that the segments are created at a constant rate, and that they
have, at the instant of creation, the same distribution as free chains. The con-
stancy of the hi means that all i-segments have the same constant probability
per unit time of leaving the network
*. In more elaborate theories (see for
instance [16]) the form (14) is often used in combination with the assump-
tion that gi and hi are functions of the mean square extension of the segments,
i.e. gi =gi(<q’>) and hi = hi(<q2>).
We shah make no specific assumptions about the functions k,(q, t) and
h,(q, t) at this stage of the development. For convenience, however, ki will be
written in the form (l4), with gi =gi(q, t).
Substitution of (14) in (12) and integration over all configurations gives
the following differential equations for Ni :
dNi/dt = <gi>O - Ni <hi> 9
(15)
with
and
<hi> = $-
i
Jhi9id3q = JhiJltd3q m
Substitution of (7) and (9) gives the following expression for the rate of
change of the structural variables Xi :
&i
dt = $ <gi>O -xi <hi> s
(17)
(18)
If \ki is written as ‘Pi = N~I,!J~ and substituted in (12), one obtains, using (15):
atii
-=
at
-a:*($iQ)+i(g’$P -<gi>O Jli)-(hi-<h>) $1.
(19)
By multiplying this equation with the diadic qq and integrating over the con-
figuration space, the following expression for the averages
(20)
* In fact, in the theory of Lodge [ 121 an additional parameter, the complexity, on
which the quantities gi and hi may depend, is used. This, however, is not essential for
the present discussion.
187
can be obtained:
it <44>i = ’ NI (<giqq>O - <gi>O <qq>i) - <hiqq> + <hi> <qq>i 3
with the contravariant convected derivative 6/6t defined as
(21)
(22)
for an arbitrary second-order tensor A.
3. Specific assumptions
From now on we assume that gi and hi do not depend explicitly on q but
only on average properties of the state of the fluid. It will be shown that for
deriving the Marrucci model this assumption is sufficient. The specific forms
of the functions gi and hi in that case will be discussed later; at present it is
sufficient to know that gi and hi do not depend on q. The basic equations
(15) and (21) then reduce to
dNi/dt = gi - Nihi
and
(23)
~t<qq>i = ~
i
(<qq>P -<qq>i) P
(24)
respectively .
4. Constitutive equation
The contribution of the i-segments to the stress tensor is given by the usual
expression
Z =Ni<fiq) 9
(25)
in which the entropic forcefi is related to the parameter bi of the equilibrium
distribution (11) in the following way:
fi = 2kTbiq s
(26)
Here k is the Boltzmann constant and T the absolute temperature. Since the
fluid is incompressible, the partial stresses (25) are determined up to an arbi-
trary isotropic pressure. So we may also use partial stresses Ti defined as
Ti = Ti -PiI 9
(27)
where pi are chosen in such a way that Ti = 0 at equilibrium. From (25) and
(26) it follows that
Ti = 2NikTbi(<qq>i -+<q2>fl) 9
(28)