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)

186

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)