On the derivation of some fundamental expressions for the average stress tensor in systems of interacting particles

The so-called generalized Kramers-Kirkwood expression for the average stress tensor of a system of interacting point particles, derived by Bird and Curtiss on using a phase-space-kinetic formalism has been reconsidered from different points of view. First a derivation based upon volume averaging is discussed, and after that a derivation based upon a virtual work principle. The latter approach offers the possibility of distinguishing reversible (including thermodynamic and Brownian) and dissipative forces and stresses by using a projection operator, associated with the constraints of the system

Microscopic modelling of the flow properties of polymers

The understanding of the flow behaviour of polymeric liquids is of great interest from a practical as well as a theoretical point of view. An important part of the research in this field consists of the development of suitable models, describing the rheological properties of the materials. Depending upon its purpose, such a model may be based upon empirical knowledge of the macroscopic flow behaviour or on information about the microstructure of the materials. Moreover, for a given system, different types of modelling may be possible. In order to provide an overview of the various approaches in this area the basic principles of some important models are discussed: continuum, bead-rod-spring, transient network, reptation and configuration tensor models. Emphasis has been put on a consistent treatment of the fundamentals of the various models and their interrelationship, rather than considering any of them in much detail

About the role of constraints in the linear relaxational behaviour of thermodynamic systems

A formalism is presented by which the linear relaxational behaviour of thermodynamic systems can be described. Instead of using the concept of internal variables of state a set of so-called constraint equations is introduced. These equations represent structural properties of the system and turn out to be related to the shape of the relaxation and retardation spectra of the system. Furthermore some problems concerning the concept of internal variables of state are clarified

The rheological behaviour of suspensions of fat particles in oil interpreted in terms of a transient-network model

The transient-network model for concentrated dispersions, described in a previous paper, is used to describe the rheological behaviour of dispersions of glyceryl tristearate crystals in paraffin oil. The model prediction of the storage modulus of this system is compared with corresponding expressions given in literature. Model calculations are carried out to fit the linear viscoelastic behaviour of the system as well as its stress response in large amplitude shear experiments. Information is thus obtained about the stiffness and strength of the interparticle bonds, and the chance of them breaking in a state of rest or as the result of flow. It is concluded that the probability of interparticle bond fracture strongly depends on the measure of bond stretching. The general findings link up with the Lennard-Jones potential which is assumed to describe the potential energy of the bonds between the particles. Accurate measurements of the temperature dependence of the dynamic moduli by making use of a torsion resonator lead to the conclusion that the energy dissipation at a high frequency originates mainly from the flow of liquid around the particles

Thermodynamic approach to rheological modeling and simulations at the configuration space level of description

The so-called matrix model is a general thermodynamic framework for microrheological modeling. This model has already been proven to be applicable for a wide class of systems, in particular to models formulated at the configuration tensor level of description. For models formulated at the configuration space level of description a matrix formulation is readily obtained, but for the subsequent analysis one still needs an explicit solution of the configuration space distribution functions. In the present paper we describe an approach in which this problem is solved by combining the matrix model with a Lagrangian simulation method in configuration space developed recently by Szeri and Leal. The result is a consistent and unified formulation of stress tensor expressions, including the stress averaging, and the evolution equations. This formulation is also suited for numerical simulations. In this way, the range of applicability of the matrix model is extended substantially. In order to clarify the principles of the method and some aspects of its implementation, a simple example is discussed in some detail

A transient-network model describing the rheological behaviour of concentrated dispersions

Attractive forces acting between particles in dispersions may cause a three-dimensional structure to be built up. A temporary-network model is postulated that describes the rheological behaviour of such systems. Chains of particles are assumed to be created and broken by thermal actions and by applied deformation. The relation between the network structure and the macroscopic stress tensor is deduced. One of the main model features is that no use is made of the common assumption of affinity of the motion of the chain vectors with the gradient of the macroscopic velocity field. Instead, the chain deformations are assumed to depend on the forces acting on them, i.e. their deformations depend on their stiffness and on the applied deformation, whereas fracture of chains may cause stress relaxation in the rest of the network. The chains may behave as highly non-linear springs, whereas the probability that the chains will break in some time interval may be an explicit function of the chain length itself. Integral equations are derived, from which the stress-tensor components can be calculated in any flow experiment, that obeys creeping-flow conditions. Analytical expressions are obtained for the relaxation spectrum of such systems in terms of the microscopic parameters

On the limit of linear viscoelastic response in the flow between eccentric rotating disks

The dependence on frequency of the limiting value of strain, ΨL, for which linear Viscoelastic response occurs in eccentric rotating disks (ERD) flow is studied theoretically and experimentally. The theoretical investigations are based upon the general simple-fluid theory of Coleman and Noll. It is shown that according to this theory ΨL becomes independent of angular velocity, ω, at relatively high frequencies, whereas ΨL becomes inversely proportional to at sufficiently low frequencies. The results of previous investigations, based upon some special rheological models, are discussed. The behavior predicted by the simple-fluid theory is confirmed by experiments on polyisobutylene solutions

A generalized thermodynamic approach to microrheological modelling

A generalized thermodynamic theory is presented which may be applied to microrheological models. The purpose of this theory is to offer a simple framework for many types of modelling at various levels of description. The essential elements of our approach are: a specification of the subsystem in which the reversible storage of energy takes place, the way of coupling of this subsystem to the environment and a proper definition of reversible and irreversible variables. The resulting set of equations, containing a stress tensor expression a microscopic evolution equation and a microscopic equation of state are expressed in a matrix form. Some applications of the theory to well known micro-rheological models are discussed and directions for further developments are indicated