The influence of matrix viscoelasticity on the rheology of polymer blends

We examine the effects of matrix phase viscoelasticity on the rheological modeling of polymer blends with a droplet morphology. Two contravariant, second-rank tensor variables are adopted along with the translational momentum density of the fluid to account for viscoelasticity of the matrix phase and the ellipsoidal droplet shapes. The first microstructural variable is a conformation tensor describing the average extension and orientation of the molecules in the matrix phase. The other microstructural variable is a configuration tensor to account for the average shape and orientation of constant-volume droplets. A Hamiltonian framework of non-equilibrium thermodynamics is then adopted to derive a set of continuum equations for the system variables. This set of equations accounts for local conformational changes of the matrix molecules due to droplet deformation and vice versa. The model is intended for dilute blends of both oblate and prolate droplets, and droplet breakup and coalescence are not taken into account. Only the matrix phase is considered as viscoelastic; i.e., the droplets are assumed to be Newtonian. The model equations are solved for various types of homogeneous deformations, and microstructure/rheology relationships are discussed for transient and steady-state conditions. A comparison with other constrained-volume rheological models and experimental data is made as well.     

Coalescence suppression in model immiscible polymer blends by nano-sized colloidal particles

The effect of nanometer sized silica particles (R16 nm) on the flow-induced morphology of immiscible polymer blends is studied. Polydimethylsiloxane (PDMS) and polyisobutylene (PIB) are chosen as model components. A stable droplet/matrix microstructure is obtained for blends of 30% PIB in 70% PDMS or vice versa. Rheological measurements are used to show that the silica particles alter the sensitivity of the of dispersed phase/matrix microstructure to shear flow. Coalescence is suppressed or at least slowed down on a practical time scale, especially when PDMS is the matrix phase. The effect of mixing conditions, pre-shear rate and particle concentration on the blend morphology are studied. Cryo-SEM is used to observe the accumulation of the particles at the interface. Blends stabilized by solid particles could provide an interesting alternative to blends compatibilized by block-copolymers.This paper was presented at the first Annual European Rheology Conference (AERC) held in Guimarães, Portugal, September 11-13, 2003.     

Effects of coalescence and breakup on the steady-state morphology of an immiscible polymer blend in shear flow

The steady-state morphology of an immiscible polymer blend in shear flow has been investigated by optical microscopy techniques. The blend is composed by poly-isobutylene (PIB) and poly-dimethylsiloxane (PDMS) of comparable viscosity. Experiments were performed by means of a home-made transparent parallel plate device. The two plates can be independently counterrotated, so that sheared droplets of the dispersed phase can be kept fixed with respect to the microscope point of view, and observed for long times. The distribution of drops and their average size were measured directly during flow at different shear rates and for different blend compositions. It was found that the average drop size in steady-state conditions is a decreasing function of the applied shear rate, and does not depend on blend composition for volume fractions up to 10%. Experiments have proved that, in the shear rate range which could be investigated, the stationary morphology is controlled only by coalescence phenomena, droplet breakup playing no role in determining the size of the dispersed phase. More generally, it has been shown that the steady-state morphology is a function not only of the physical parameters of the blend and of the shear rate, but also of the initial conditions applied to the blend. The steady-state results reported in this paper constitute the first direct experimental confirmation of theoretical models which describe the mechanisms of shear-induced drop coalescence.     

A reversible problem in non-equilibrium thermodynamics: Hamiltonian evolution equations for non-equilibrium molecular dynamics simulations

The reversible contribution to contemporary theories of non-equilibrium thermodynamics is reviewed as a methodology for attacking difficult, conservative problems in complex fluid dynamics. Several examples of past successes are discussed, and a new application is addressed: non-equilibrium molecular dynamics (NEMD) simulations. NEMD simulations of fluids are generally based on either a DOLLS or SLLOD tensor algorithm. The former is always considered to be a Hamiltonian system, but not particularly useful in high strain rate flow simulations, while the latter is considered not to be a Hamiltonian system, but much more practical and accurate in flow simulations. We demonstrate herein using non-canonical transformations of the particle momenta of the system that the SLLOD equations, when written in terms of appropriate non-canonical variables, are completely Hamiltonian, whereas the DOLLS equations are not so. A modified set of DOLLS equations in terms of the non-canonical variables which again is completely Hamiltonian is also derived. Both algorithms then lead to a phase space distribution function which is canonical in both the coordinates and momenta.     

Noncanonical Poisson brackets for elastic and micromorphic solids

This paper investigates the Lagrangian-to-Eulerian transformation approach to the construction of noncanonical Poisson brackets for the conservative part of elastic solids and micromorphic elastic solids. The Dirac delta function links Lagrangian canonical variables and Eulerian state variables, producing noncanonical Poisson brackets from the corresponding canonical brackets. Specifying the Hamiltonian functionals generates the evolution equations for these state variables from the Poisson brackets. Different elastic strain tensors, such as the Green deformation tensor, the Cauchy deformation tensor, and the higher-order deformation tensor, are appropriate state variables in Poisson bracket formalism since they are quantities composed of the deformation gradient. This paper also considers deformable directors to comprise the three elastic strain density measures for micromorphic solids. Furthermore, the technique of variable transformation is also discussed when a state variable is not conserved along with the motion of the body.     

Droplet retraction in the presence of nanoparticles with different surface modifications

We studied the influence of nanoparticles with different surface modifications on the interfacial tension and relaxation of model polymer blend after cessation of different strains. The droplet retraction experiments were carried out on a model system composed of polydimethylsiloxane (PDMS) as the suspending fluid and polyisobutylene (PIB) as droplet at room temperature in the presence of hydrophobic and hydrophilic nanosilica. Different weight fractions of particles were dispersed in the PIB droplet before forming a dispersed droplet by using a microsyringe in shear cell. We found that applied strain, nanoparticle concentration and their thermodynamically preferred localization affect both nominal interfacial tension and droplet retraction process. By addition of nanoparticles at a concentration as low as 0.2%wt, the nominal interfacial tension decreases from 3.12?±?0.15 mN/m for neat PIB-PDMS interface depending on the surface characteristics of nanosilica. Hydrophilic nanosilica has the most effect on nominal interfacial tension and decreases it as low as 0.2?±?0.21 mN/m at 1 wt.% loading under a strain of 7. The results show that the retraction process in this system is mainly controlled by interfacial phenomena rather than bulk rheological properties. Additionally, the shape evolution of droplets changes and the retraction rate slows down in the presence of nanoparticles.     

Particle-induced bridging in immiscible polymer blends

Pickering emulsions are emulsions whose drops are stabilized against coalescence by particles adsorbed at their interface. Recent research on oil/water/particle systems shows that particles can sometimes adsorb at two oil/water interfaces. Such “bridging particles” can glue together drops of oil in water or vice versa. We hypothesize that the same effect should apply in immiscible polymer blends with droplet-matrix morphologies, viz., added particles should glue together drops and give rise to particle-bridged drop clusters. We test this hypothesis in PIB-in-PDMS blends [PIB, poly(isobutylene); PDMS, poly(dimethylsiloxane)] with fumed silica particles. Direct visualization shows that the particles can indeed induce clustering of the drops, and the blends appear to show gel-like behavior. Such gel-like behavior is confirmed by dynamic oscillatory experiments. However, we are unable to conclusively attribute the gel-like behavior to droplet clustering: Association of the fumed silica particles in the bulk, which itself causes gel-like behavior, confounds the results and prevents clear analysis of the gluing effect of the particles. We conclude that PIB/PDMS/fumed silica is not a good model system, for studying particle-containing polymer blends. We instead propose that spherical monodisperse silica particles can offer a far more convenient model system, and provide direct visual evidence of gluing of PIB drops in a PDMS matrix.     

Dynamics and rheology of the morphology of immiscible polymer blends – on modeling and simulation

 The material properties of heterogeneous polymer blends are crucially influenced by their morphology, i.e., by the spatial structure of the blend components and by the specific configuration of the interfaces separating the phases. Hence, in order to understand the behavior of experimentally obtained morphologies, one is interested in modeling the relevant dynamics of the morphology subject to external flow. Thus one can study, e.g., through the interfacial stress tensor the rheological properties due to the interfaces. The balance equations used for that purpose are based on a Cahn-Hilliard equation for the local concentration, the continuity equation, and a modified Navier-Stokes equation for the local velocity. The essential material and processing parameters such as surface tension, viscosity and volume fraction of both polymers, and imposed shear rate are taken into consideration as model coefficients. By regarding hydrodynamic interaction, which is proved to be important in case of immiscible blends, the interfacial relaxation is described properly. Simulations in both three and two dimensions agree at least qualitatively with experimental results concerning droplet deformation, droplet coalescence, and interfacial rheological properties of the blend. Received: 25 September 2000 Accepted: 24 April 2001     

Singular Casimir Elements of the Euler Equation and Equilibrium Points

The problem of the nonequivalence of the sets of equilibrium points and energy-Casimir extremal points, which occurs in the noncanonical Hamiltonian formulation of equations describing ideal fluid and plasma dynamics, is addressed in the context of the Euler equation for an incompressible inviscid fluid. The problem is traced to a Casimir deficit, where Casimir elements constitute the center of the Poisson algebra underlying the Hamiltonian formulation, and this leads to a study of singularities of the Poisson operator defining the Poisson bracket. The kernel of the Poisson operator, for this typical example of an infinite-dimensional Hamiltonian system for media in terms of Eulerian variables, is analyzed. For two-dimensional flows, a rigorously solvable system is formulated. The nonlinearity of the Euler equation makes the Poisson operator inhomogeneous on phase space (the function space of the state variable), and it is seen that this creates a singularity where the nullity of the Poisson operator (the “dimension” of the center) changes. The problem is an infinite-dimension generalization of the theory of singular differential equations. Singular Casimir elements stemming from this singularity are unearthed using a generalization of the functional derivative that occurs in the Poisson bracket.     

Specific rheology – morphology relationships for some blends containing LCPs

For the blend melts of isotropic polysulfone (PSF) and LC polyester (PES), differing in viscosity more than 10 times, the flow curves with maxima were observed in cone and plate geometry. The low shear rate branch is located near the PSF flow curve, and the high shear rate branch is close to the PES flow curve. At high strains, the formation of the ring-like morphology of the blend sample, accompanied by appearance of maximum on flow curve, was registered. The scaling analysis of the reasons for the ring morphology formation was based on stretching of the large, low-viscous LC droplet, embedded to the high-viscous polymer matrix, in a homogeneous shear field. It was shown that, if the critical Taylor radius is not exceeded, the droplet may form the closed torus. Under strong flows, the PSF melt manifests the “spurt effect”, consisting of a drastic increase of the shear rate when the critical value of the shear stress is reached. The pattern of the blend flow curves with maxima may be explained by a vanishing PSF input to the total shear stress, inherent for blends, while the PES melt continues to be in a liquid state and, consequently, is responsible for the blend viscosity at the high shear rates. The presence of regular heterogeneities in the blend in the form of LC rings may initiate the rupture of the entanglements network of the matrix PSF (close to LC rings) under strong shear flows. The appearance of the low-viscous “cracks” at the critical shear stress will diminish the contribution of the PSF to the blends rheological response. Received: 20 April 1999 Accepted: 28 January 2000     

Flows of constant stretch history for polymeric materials with power-law distributions of relaxation times

It is herein shown that for separable integral constitutive equations with power-law distributions of relaxation times, the streamlines in creeping flow are independent of flow rate.For planar flows of constant stretch history, the stress tensor is the sum of three terms, one proportional to the rate-of-deformation tensor, one to the square of this tensor, and the other to the Jaumann derivative of the rate-of-deformation tensor. The three tensors are the same as occur in the Criminale-Ericksen-Filbey Equation, but the coefficients of these tensors depend not only on the second invariant of the strain rate, but also on another invariant which is a measure of flow strength. With the power-law distribution of relaxation times, each coefficient is equal to the second invariant of the strain rate tensor raised to a power, times a function that depends only on strength of the flow. Axisymmetric flows of constant stretch history are more complicated than the planar flows, because three instead of two nonzero normal components appear in the velocity gradient tensor. For homogeneous axisymmetric flows of constant stretch history, the stress tensor is given by the sum of the same three terms. The coefficients of these terms again depend on the flow strength parameter, but in general the dependences are not the same as in planar flow.     

An efficient preconditioning scheme for real-fluid mixtures using primitive pressure–temperature variables

An improved preconditioning scheme incorporating a unified treatment of general fluid thermodynamics is developed for treating fluid flows over the entire regime of fluid thermodynamic states at all speeds. All of the thermodynamic and numerical properties (such as eigenvalues and Jacobian matrices) are derived directly from fundamental thermodynamics theories, rendering a self-consistent and robust algorithm. Further efficiency is obtained by employing temperature instead of enthalpy as the primary dependent variable in the preconditioned energy equation. No iterative solution of a real-fluid equation of state is required. This approach, combined with the use of explicit treatments of temporal and spatial derivatives, results in a scheme for which load balance is much easier to achieve in a distributed computing environment. A numerical stability analysis is performed to assess the effectiveness of the scheme at various fluid thermodynamic states. Sample calculations are also carried out. These include injection and mixing of cryogenic fluids and flame dynamics of coaxial jets of liquid oxygen and methane under supercritical conditions. The robustness and efficiency of the present work are demonstrated over a wide range of thermodynamic and flow conditions.     

Evolution Equations for Non-Simple Viscoelastic Solids

Objectivity and compatibility with thermodynamics of evolution equations are examined in connection with the modelling of viscoelastic solids. The purpose of the paper is to show that the evolution equation for the stress is eventually obtained by means of a tensorial internal variable within the framework of the reference configuration. The non-simple character is realized by gradients of the internal variable. The thermodynamic analysis is developed by investigating the entropy inequality in the reference configuration and allowing for a non-zero extra-entropy flux. It follows that the evolution for the Cauchy stress tensor involves the Oldroyd derivative, irrespective of the form of the non-local terms.     

Volume averaging for the analysis of turbulent spray flows

Spray flow calculations are usually based upon equations that have been developed by averaging droplet properties locally throughout the flow field. Presently, standard procedure for LES (large-eddy simulations) is to average these averaged equations once again to filter the short-length-scale fluctuations. In this paper, the theoretical foundations for the averaged spray equations are examined; then the volume-averaging process for LES and the volume-averaging process for two-phase flows are unified for the analysis of turbulent, two-phase flows. Comments are provided on the relationship between the averaging volume and the computational-cell volume. This paper provides generality to the weighting-function choice in the averaging process and precision to the definition of the volume over which the averaging is performed. New flux terms that result from the averaging process and appear in the governing averaged partial differential equations are identified and their modelling is discussed. Situations are identified where sufficient stratification of properties on the scale smaller than the averaging volume leads to the significance of these quantities. Evolution equations for averaged entropy and averaged vorticity are developed. The relationship amongst the curl of the average gas-phase velocity, the average of the gas-phase-velocity curl, and the rotation of the discrete droplets or particles is established. The needs and challenges for sub-grid modelling to account for small-vortex/droplet interactions are presented. Applications to spray combustion are discussed.     

On thermodynamic potentials in thermoelasticity under small strain and finite thermal perturbation assumptions

Expressions for thermodynamic potentials (internal energy, Helmholtz energy, Gibbs energy and enthalpy) of a thermoelastic material are developed under the assumption of small strains and finite changes in the thermal variable (temperature or entropy). The literature provides expressions for the Helmholtz energy in terms of strain and temperature, most often as expansions to the second order in strain and to a higher order in temperature changes, which ensures an affine stress–strain relation and a certain temperature dependence of the moduli of the material. Expressions are here developed for the four potentials in terms of all four possible pairs of independent variables. First, an expression is obtained for each potential as a quadratic function of its natural mechanical variable with coefficients depending on its natural thermal variable that are identified in terms of the moduli of the material. The form of the coefficients’ dependence on the thermal variable is not specified beforehand so as to obtain the most general expressions compatible with an affine stress–strain relation. Then, from each potential expressed in terms of its natural variables, expressions are derived for the other three potentials in terms of these same variables using the Gibbs–Helmholtz equations. The paper provides a thermodynamic framework for the constitutive modeling of thermoelastic materials undergoing small strains but finite changes in the thermal variables, the properties of which are liable to depend on the thermal variables.     

On tensorial forms of thermodynamic potentials in mixtures theory

Four thermodynamic tensorial quantities, equivalents of the thermodynamic potentials, namely the chemical potential tensor, the tensor of enthalpy, the tensor of free energy and the tensor of internal energy are presented. The last three of them are proposed in this paper and connections between all of these tensors are derived. The tensorial forms of the thermodynamic potentials are expressed in terms of four possible pairs of independent state variables. A set of sixteen alternative expressions, four for each tensorial form of the thermodynamic potential are derived and their importance discussed.     

An energy-based damage model of geomaterials—I. Formulation and numerical results

In this paper, an anisotropic damage model is established in strain space to describe the behaviour of geomaterials under compression-dominated stress fields. The research work focuses on rate-independent and small-deformation behaviour during isothermal processes. It is emphasized that the damage variables should be defined microstructurally rather than phenomenologically for geomaterials, and a second-order fabric tensor is chosen as the damage variable. Starting from it, a one-parameter damage-dependent elasticity tensor is deduced based on tensorial algebra and thermodynamic requirements ; a fourth-order damage characteristic tensor, which determines anisotropic damaging, is deduced within the framework of Rice, 1971 normality structure in Part II of this paper. An equivalent state is developed to exclude the macroscopic stress⧹strain explicitly from the relevant constitutive equations. Finally, some numerical results are worked out to illustrate the mechanical behaviour of this model.     

Semi-analytical techniques for investigating thermal non-equilibrium effects in wet steam turbines

This paper describes a semi-analytical solution of the polydispersed wet stream equations, valid in regions where the nucleation rate is negligible. The solution can be used in conjunction with any conventional turbomachinery calculation procedure to obtain estimates of the magnitude of departures from thermal equilibrium. For example, from an initial estimate of the pressure distribution, it is a simple matter to calculate the distribution of supercooling and wetness fraction, together with the thermodynamic losses incurred by the flow.The method differs from the usual numerical approach by providing general results which give considerable physical insight. Computational time and effort is also dramatically reduced. The controlling parameters emerge naturally from the analysis, and information concerning the fundamental fluid mechanics of wet steam is revealed. In particular, the analysis demonstrates the role played by the thermal relaxation time and the rate of expansion in controlling the deviation from equilibrium.The versatility and usefulness of the technique in furnishing results for the turbine designer are demonstrated by a number of applications including one-dimensional nozzle flows and two-dimensional blade-to-blade and hub-to-tip flows. In each case it is shown how the droplet size and expansion rate influence the thermodynamic losses and other flow variables of interest.