Rheology of Complex Fluids

by H. Pleiner
(external collaboration with H.R. Brand, M. Liu and J.L. Harden)

The flow behavior of complex fluids is rather involved. Viscoelasticity, shear thinning and thickening, normal stress differences are well known for a long time. What is missing is a coherent derivation of the appropriate hydrodynamic-like equations especially in the nonlinear domain. Frequently ad-hoc models are used, which have some success in describing the main features, but lack a solid physical foundation. Our goal is to lay a framework for such nonlinear equations that is in accordance with basic physical principles, like thermodynamics and irreversible thermodynamics.

In order to describe visco-elasticity we start with a decent hydrodynamic description of nonlinear elasticity [B]. This requires the use of a bi-vectorial field, ∇_i a_α, in Eulerian description, where a_α(r,t) is the coordinate of a (translated and rotated) body point measured in the laboratory frame r_i at time t. Since the elasticity of a body must not depend on the orientation and position of the frame, in which its initial state is measured (subcript α), compared to the laboratory frame (subsript i), ∇_i a_α is not a (symmetric) tensor, but contains deformations as well as rotations. The dynamic equation for that field is uniquely determined by general physical laws. From it implicit hydrodynamic equations for strains and rotations can be derived, which can be made explicit in the form of an expansion in the strains.
Allowing for strain relaxation viscoelasticity is obtained [B]. For the Eulerian strain tensor the reversible part of the strain evolution equation resembles that of an ‘lower convected model’, while for other strain tensors different forms are obtained. This is a clear signal that the reversible nonlinearities cannot be postulated by ad-hoc principles like the so-called ‘frame indifference principle’, but have to be derived from first principles. Other non-Newtonian features, like shear thinning or thickening and normal stress differences, are treated at least partially by this approach, but further relaxing fields may also be important.
Meanwhile, we have shown [E] that the hydrodynamically derived model for non-Newtonian fluids in terms of the Eulerian strain tensor contains most of, and is more general than, the standard rheological models (expressed as "constitutive" equations for the stress tensor) as special cases and discards a few of them. The hydrodynamic method allows to discriminate those parts of the dynamics that are due to general principles from the unavoidable phenomenological part. The latter is given in the form of truncated power series in the strain tensor that can systematically be generalized when necessary. For this part we stick to the well-established 'linear irreversible thermodynamics', which, being linear in the generalized forces, nevertheless leads to equations highly nonlinear in the variables like the strain tensor.
A different approach to describe viscoelastic effects in semi-flexible and stiff polymeric systems [C] is based on the use of a nematic-like orientational order parameter tensor (Doi-Edwards). For the true nematic phase we combine the director hydrodynamics in a nematic phase with the relaxing dynamics of the scalar degree of order into an effective dynamic equation for the 2nd rank order parameter tensor (including an external field). The nonlinear convective terms are definitely of the Jaumann type describing the coupling to rotational flow, but there are in addition phenomenological (linear and nonlinear) couplings to elongational flow, which can be of equal importance. Jaumann-type convective nonlinearities are also found in the isotropic case, however, the phenomenological couplings have a very different structure from those in the nematic phase due to the underlying director hydrodynamics. Thus, it is not possible to use the same set of equations for the orientational dynamics in the nematic and the isotropic phase. Furthermore we give the form of the appropriate orientational-elastic stresses in the stress tensor. They are completely fixed by the orientation dynamics and no choices are left. Thus, there are again different expressions for the isotropic and the nematic phase. In particular, a simple stress-optical law is valid for the isotropic phase only -- and only in linear approximation.
As in the case of a relaxing elastic strain tensor, the hydrodynamic model of non-Newtonian fluids described in terms of a transient orientational order can be put approximately into the form of a "constitutive" equation with an effective dynamic equation for the stress tensor [F]. Of the traditional constitutive models some forms of the Johnson-Segalman and the Jeffreys model are compatible with this origin of non-Newtonian behavior, while alternatively, some forms of the Maxwell and Oldroyd models are compatible with a relaxing elasticity [E]. Giesekus-type models can be compatible with both types.

In the two-fluid description [D], each component or phase is treated as a continuum described by local thermodynamic variables (e.g. temperature, density, and relevant order parameters), and dynamical quantities (e.g. velocity or momentum). In general, these variables for the constituents are coupled. For instance, the effective friction between components in a binary fluid mixture leads to a drag force in the macroscopic description that is proportional to the local velocity difference. Such a description is important, e.g. for manifestly heterogeneous systems, in particular for their phase separation dynamics or any other flow-induced structural evolution phenomena. We discuss general 2-fluid hydrodynamic equations for complex fluids, where one kind is a simple Newtonian fluid, while the other is either liquid-crystalline or polymeric/elastomeric, thus being applicable to lyotropic liquid crystals, polymer solutions, and swollen elastomers. The procedure can easily be generalized to other complex fluid solutions. Emphasis is placed on the rigorous derivation of the dynamic equations within the framework of hydrodynamics as contrasted to ad-hoc treatments. The resulting equations are rather general and complicated. They can and have to be simplified for special applications or systems by appropriate and well-defined approximations. One of the advantages of starting from the general theory is the possibility to identify and characterize the approximations made. Of special interest are nonlinearities that originate from the 2-fluid description, like the transport part of the total time derivatives. It is shown that the proper velocities, with which the hydrodynamic quantities are convected, cannot be chosen at will, since there are subtle relations among them. Within allowed combinations the convective velocities are generally material dependent. The so-called stress division problem, i.e. how the nematic or elastic stresses are distributed between the two fluids, is shown to depend partially on the choice of the convected velocities, but is otherwise also material dependent. A set of reasonably simplified equations is given as well as a linearized version of an effective concentration dynamics that may be used for comparison with experiments.

Recent Publications:

Phys. Rev. Lett.

[56 kB pdf-file]

[A]

Slow Dynamics in Complex Systems: 8^th Tohwa University International Conference

[461 kB pdf-file]

[B1]

Phys. Rev. Lett.

[518 kB pdf-file]

Proc. 13th Int. Congr. on Rheology

[106 kB pdf-file]

Phys. Rev. Lett.