The Mesoscopic Constitutive Equations for Polymeric Fluids and Some Examples of Viscometric Flows ()
1. Introduction
Describing viscoelastic behavior of the polymer system, one should distinguish the case of highly concentrated (c > 10%) solutions and melts of long polymers—strongly entangled systems (
), where 
is the length (in any units) of a macromolecule and 
is the length of a part of macromolecule between the adjacent entanglements [1-3], and the case of melts of shorter polymers and half-dilute polymer solutions (c ~ 1% - 10%)— weakly entangled systems (
) [4]. The convenient characteristic of a system of entangled linear macromolecules (solutions and melts of polymers) appears to be
, which, for strongly entangled systems, is inversely proportional to number of entanglements for one macromolecule—
. Here 
is the polymer density. The quantity 
can be easily with estimated the value 
of real component of dynamic modulus 
on the typical plateau, according to the formula for weakly and strongly entangled systems, correspondingly
.
In this paper we consider the case of weakly entangled systems and formulate a rheological equation of state (RES) that establishes a relationship between the stress tensor, kinetic characteristics, and internal dynamic parameters. At present, a large number of such equations of various complexity is known for polymeric liquids, but, despite various approaches both phenomenological [5-7] and microstructural [1,2,8-11] ones, the problem how to include specific features of a polymer system into the form of constitutive equations has no complete solution. This is due to both complexity of these systems, which are formed by tangled macromolecules, and mathematical difficulties [11]. The information about the microstructure and micro dynamics of the material ought to be incorporated into the present theory of linear and nonlinear relaxation phenomena in polymer systems. An advantage of the micro structural approach is a possibility of studying the relationship between the micro characteristics of a polymer system (concentration and molecular weight of the polymer) and macroscopically observed quantities (viscosity, shear and normal stresses, etc.). In this connection using microstructural concepts it’s feasible to formulate a sequence of RES that takes into account new molecular effects in each stage. At [4,9,12] obtained and studied a simple rheological model which can be chosen as an initial approximation in formulating such a sequence of RES. In this work, RES [4] is extended to the case of allowance for the additional corrections caused by intrinsic viscosity and the delayed interaction of a macromolecule with its environment. Realization of this approach involves consequent solution of two problems: formulation of the equations of dynamics for a macromolecule and transition from the formulated equations to RES. The resulting equations can be recommended as the first approximation in constructing a sequence of RES. Comparison of the approach with others, Graessley [1,3] and Doi-Edwards [8] approaches, one can find in [9,12].
2. Dynamics of a Macromolecule in Flow
The mesoscopic approach to the description of the dynamics of polymer systems is based on the equations of the macromolecule dynamics which cannot be formulated without additional assumptions, namely:
1) A monomolecular approximation of the system. It means that, instead of the entire set of interacting macromolecules in the volume we consider are non-interacting separate macromolecule, moving in the effective medium formed by the solvent and the other macromolecules.
2) The coarse-grained approximation of a macromolecule. It means that, irrespective of the chemical nature of the polymer, the slow motions of the chosen macromolecule can be described as motions of N centers of friction (beads) connected by elastic entropy forces (springs) in a chain. These assumptions lead to the following equations of the macromolecules dynamics [9] in normal modes:
, (1)
where 
and 
are the i-th components of the normal coordinates and velocity, 
is the mass of a bead. Here 
is the friction coefficient of a bead in a monomer fluid, 
is the velocity-gradient tensor, which is conveniently expressed below as the sum of symmetric 
and anti-symmetric 
parts, and the bracketed expression is the difference between the particle velocity at a given point of space and the velocity of undisturbed flow at this point. The force 
describes the interaction of the polymer chain with the environment via the solvent, while the force 
is an intrinsic resistance force, 
is a random force, and 
is the coefficient of elasticity.
Expression (1) is the basis for the description of the dynamics for different polymer systems [12]. The definition of the extra forces 
and 
allows one to specify the polymer system. Different models of these forces correspond to different physical cases. For dilute polymer solutions, in which polymer macromolecules can be considered non-interacting, the extra forces are considered to be equal to zero [9]. In concentrated polymer systems, the macromolecules cannot be considered as not interacting. So, one has to take into account the reaction of the environment and the strengthening of the friction coefficient. The first factor is due to the delayed character of interaction of the macromolecule with its environment, and the second is due to the fact that the chosen bead undergoes resistance not only from the monomer solvent, but also from other macromolecules. Furthermore, one should take into account that, in the flow with nonzero velocity gradients, a macromolecular coil changes its form, and the medium, formed by the coils becomes anisotropic. This mobility anisotropy of beads is called induced and is determined by the shape and orientation of macromolecular coils [4,9,12]. According to all these factors the equation for the force of hydrodynamic entrainment follows:
(2)
Here 
is the relaxation time of the environment, 
is the dimensionless tensor friction coefficient of a bead, 
is the strengthening measure of the friction coefficient
, and 
is a parameter. The bracketed expression on the left side of (2) is a substantial derivative of the vector quantity 
[9]. The presence of this derivative allows one to meet the principle of material objectivity in Equation (2) [6,9]. Numerical parameter 
entering into the definition of the substantial derivative can take different values. For 
= 0, the substantial derivative becomes the Jaumann derivative which has a simpler form while for 
= 1 and –1, it becomes the upper and lower convective derivatives, respectively. The specific value of 
corresponding to one of the above-mentioned derivatives in (2) is determined below.
If macromolecules form a tangled system besides the force of hydrodynamic entrainment, one should take into account the intrinsic viscous force
, the meaning of which is elucidated by Pokrovskii et al. [12]. The specific requirement imposed on the force 
is that this force is vanishing when a macromolecular coil is rotating as a unit. All this allows one to write the equation for this force in similar to (2) manner in the form
, (3)
where: 
is the dimensionless tensor friction coefficient and 
is the strengthening measure of the friction coefficient 
for the intrinsic viscous force
. Intrinsic viscous force 
(since
) has relaxation character and depends on the anisotropic properties of the environment.
We assume that the anisotropy of mobility in considered polymer system is characterized by the second-order symmetric tensor
. Then, for coefficients 
and 
we write [9]
(4)
Thus, (1-4) is the system of equations of dynamics of a macromolecule. The random force 
entering into (1) is the Gaussian random process with a zero average. Its correlation tensor satisfies the corresponding fluctuation-dissipation relation [9,10].
3. Stress Tensor and Rheological Equation of State
Equations (1)-(4) give a microscopic picture of a polymer system flow based on discrete variables. Transition to the continuous case, i.e., to the description of polymer-system flows in terms of continuum mechanics, requires introduction of macroscopic variables—density 
and momentum density
. These variables are introduced in the standard manner [9,11]:
(5)
Here 
and 
are vectors of position and velocity of the bead with number
, 
is the co-ordinate vector of the chosen point in space, and 
is time. Sum is taken over all beads in a unit volume, and averaging is performed over the ensemble of all possible realizations of the random force
.
Differentiating (5) due to time yields an equation of mass conservation, and transformation to generalized coordinates using (1) yields an equation for momentum density. In the latter case, we have an expression for the stress tensor of a polymer system in terms of statistical characteristics of the system (1-4) solutions:
, (6)
where 
is pressure, n is the number of macromolecules in a volume unit, T is the temperature in energy unitsand 
and 
are internal thermodynamic parameters with equilibrium values
,
(7)
In the inertia-free case (m = 0), one can formulate (see Appendix) the relaxation equations for the dimensionless correlation moments 
and 
in the following form
, (8)
, (9)
where: 
is the Jaumann derivative of the tensor quantity 
and 
is the measure of intrinsic viscosity, 
is a set of the Rouse relaxation times. In Equations (8)-(9) symbols 
are used.
(10)
Thermodynamic variables 
entering into (6) characterize the inertial properties of a macromolecular coil and, hence, can be used to determine anisotropy tensor 
in (4). Following [9], we write
.
Therefore it becomes possible to establish the physical meaning of the microanisotropy parameters entering into (4). These parameters take into account dimensions 
and shape 
of a macromolecular coil in the equations for macromolecule dynamics.
The Equations (6), (8), and (9) define a nonlinear, anisotropic, viscoelastic fluid. The behavior of system (6), (8), and (9) is determined by the six dimensionless parameters (
, 
, 
, 
, 
, and
) and two dimensional parameters (
and
). Parameter 
characterizing the ratio of the relaxation time of the environment 
to the maximum relaxation time 
was estimated in [9], where it was shown that 
for sufficiently long polymer chains. As to parameter
, here two cases can be distinguished: 
[13,14-16] and 
[5], which are discussed below. As in [13], it is convenient to consider simpler forms of equations (8) and (9) by using the smallness of the parameters 
and
. We consider in more detail the case
, which corresponds to the dynamics of polymer solutions at a concentration about of 1%.
Considering only effects of the first order with respect to 
and
, we note that Equations (6) and (8) do not change, and Equation (9) takes the form
(11)
In the zero-th approximation for 
and
, variable 
and Equations (6) and (8) take the form
, (12)
The parameters of this system are
, 
, and
. Note that when 
= 1, system (12) is the system of equations for a dumbbell model
, (13)
where 
and 
are the initial shear viscosity and the relaxation time, and
. The system (13) under the assumption of isotropic relaxation (
= 0), is followed by the well-known structural phenomenological Vinogradov-Pokrovskii model [13,14-17].
The model (13) is simple and gives high accuracy in describing steady nonlinear effects, though it is only the zero-order approximation model, which does not permit one to predict all features of polymer flow. In case, one needs more details, one can consider the contributions of parameters 
and 
which take into account the relaxation character of the environment and the intrinsic viscosity in the equations for macromolecule dynamics.
4. Linear Effects of the Rheological Model
To obtain an expression for the dynamic shear modulus that corresponds to system (6), (8), and (9), we find a solution to this system in a linear approximation with respect to the velocity gradients. In this case, anisotropy tensor 
is equal to zero, and the terms 
can be omitted. Then (8) and (9) are written as
,
where
; 
The latter equations can be written as
(14)
Solving the first equation of (14) by the method of successive approximations with first-order approximation due to the velocity gradients, we obtain
.
Substitution of this expression into the second equation in (14) yields
In the simple oscillating shear flow 
and the last two expressions together with (6) define the complex shear modulus 
(15)
Next, it is convenient to distinguish the real and imaginary parts in
:
.
If the value of the modulus on the plateau is determined by (15), then
. (16)
This series converges only for p = 0. Thus, the Jaumann derivative in the equations of dynamics of a macromolecule (2) and (3) corresponds to cases a) and b). The definitions of the relaxation times
, 
, and 
are given through parameters 
and
, estimates of which are given in [4,9], where it was shown that, for sufficiently long chains, one can always assume
. As to intrinsic viscosity parameter
, here two alternative cases 
and 
are distinguished.
The curves of 
and 
versus the dimensionless frequency 
calculated by (18) are given in Figures 1-4, from which one can see that the values of 
and 
are mainly determined by parameter
, and the impact of parameter 
(for
) is insignificant. The existence of characteristic plateau is determined by relaxation time
. In the case, when 
= 0, that corresponds to dilute solutions, irrespective of the type of convective derivative, a plateau on 
is absent. For
, which corresponds to the dynamics of melts and strongly concentrated solutions [4], from (16) we have
. The calculation results show that, for
, the modulus on the plateau
.
Therefore, the non-dependence of 
on the molecular weight of a polymer means
. Using the estimate for 
obtained in [11], from the last relation one can obtain
.
The value of the initial shear viscosity
, which can be expressed from (16) as
Figure 1. The contribution of parameter 
to the dimensionless frequency dependence of real component of dynamic modulus
.
Figure 2. The contribution of parameter 
to the dimensionless frequency dependence of imaginary component of dynamic modulus
.
Figure 3. The contribution of parameter 
(internal viscosity) to the dimensionless frequency dependence of real component of dynamic modulus
.
Figure 4. The contribution of parameter 
(internal viscosity) to the dimensionless frequency dependence of imaginary component of dynamic modulus
.
. (17)
To compare the calculation and experimental results, we turn to the data of Menezes and Graessley [16,17], where 
and 
were measured for solutions of polybutadiene with different molecular weights at the same concentration of c = 0.0676 g/cm. The above formulae allow us to find the following estimates of the parameters of the rheological model (6), (8), and (9) were obtained: 
= 0.077; 0.025; 0.011 and 0.005, 
= 0.21; 2.35; 16.27 and 147 sec, and 
= 840.4; 480.2; 321.5 and 206.7 Pa for molecular weights 
, respectively. In all cases, 
= 0.025. The comparison of the results is given in Figures 5 and 6, from which one can see satisfactory agreement between the theoretical and experimental curves of 
and 
for 
< 10 sec–1. The values of 
as a function of 
are given on Figure 7 showing good agreement with (17).
Figure 7 presents parameter 
dependence molecular weight corresponding
.