Fractal-time stochastic processes and dynamic functions of polymeric systems

Dynamic material functions of polymeric systems are calculated via a defect-diffusion model. The random motion of defects is modelled by a fractaltime stochastic process. It is shown that the dynamic functions of polymeric solutions can be approximated by the defect-diffusion process of the mixed type. The relaxation modulus of Kohlrausch type is obtained for a fractal-time defect-diffusion process, and it is shown that this modulus is capable of portraying the dynamic behavior of typical viscoelastic solutions.The Fourier transforms of the Kohlrausch function are calculated to obtain and. A three-parameter model for and is compared with the previous calculations. Experimental measurements for five polymer solutions are compared with model predictions. D rate of deformation tensor - G(t) mechanical relaxation modulus - H relaxation spectrum - I(t) flux of defects - P _n (s) probability of finding a walker ats aftern-steps - P generating function ofP _n (s) - s(t) fraction of surviving defects - , () gamma function (incomplete) - ₀ zero shear viscosity - ^* () complex viscosity - frequency - t _n n-th moment - F[] Fourier transform - f ^* (u) Laplace transform off(t) - , components of ^* - G _f, _f ^* fractional model - G ₃, ₃ ^* three parameter model - complex conjugate ofz - material time derivative ofD     

Some aspects of dilute suspension theory

A theory proposed by the author as representative of the flow of a general suspension contains three interaction forces, f, S and N. For a quasi-concentrated suspension and for a dilute suspension, N and S, N are omitted, respectively. For the latter special case, we treat diffusion of a fluid through an elastic solid. For a quasi-concentrated suspension, we show that F and S depend on the gradient of the motion gradient. We demonstrate the existence of interesting phenomena: non-simple behavior, dissipative effects, generalized lift and drag forces.Presented at the second conference Recent Developments in Structured Continua, May 23 – 25, 1990, in Sherbrooke, Québec, Canada.     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

Neck propagation as a shock wave

Neck propagation in the stretching of elastic solid filaments having a yield point was analyzed using the space one-dimensional thin filament governing equations developed previously by the authors and other researchers. Constitutive model for the filament was assumed to be expressible as engineering tensile stress(X) (tensile force) given as a function of elongational strain with the(X) curve having a yield point maxima followed by a minima and a breaking point greater than the yield point maxima. Also incorporated into the model is the hysteresis of irreversible plastic deformation. When inertia is taken into consideration, the thin filament equations were found to reduce to the nonlinear wave equation ² (X)/ ² =C ₁ ² X/ ² where is Lagrangean space coordinate, is time, andC ₁ is inertia coefficient. The above nonlinear wave equation yields a solutionX(, ) having a stepwise discontinuity inX which propagates along the axis. The zero speed limit of the step wave solution was found to describe the above neck propagation occurring in solid filaments. Furthermore, it was recognized that the nonlinear wave equation was known for many years to also govern the plastic shock wave which propagates axially within a metal rod subjected to a very strong impact on its end. The one-dimensional atmospheric shock wave also was known to be governed by the nonlinear wave equation upon making certain simplifying assumptions. The above and other evidences lead to the conclusion that neck propagation occurring in the extension of solid filament obeying the above(X) function can be formally described as a shock wave.     

The rheology of powders

This paper studies the slow flow of powders. It is argued that since powders can flow like liquids, there must be equations similar to those of liquids. The phenomenon of a variable density, dilatancy, is described by an analogue of temperature called the compactivity X. Whereas, in thermal physicsT = E/S, powders are controlled byX = V/S. The equations for, v, T of a liquid are replaced by, v, X. An analogy for free energy is described, and the solution to some simple problems of packing and mixing are offered. As an example of rheology, it is shown that the simplest flow equations produce a transition to plug flow in appropriate circumstances.Delivered as a Gold Medal Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Measurements of slip at the wall during flow of high-density polyethylene through a rectangular conduit

A hot-film probe has been used to measure slip of high-density polyethylene flowing through a conduit with a rectangular cross section. A transition from no slip to a stick-slip condition has been observed and associated with irregular extrudate shape. Appreciable extrudate roughness was initiated at the same flow rate as that at which the relationship between Nusselt number and Péclet number for heat transfer from the probe departed from the behavior expected for a no-slip condition at the conduit wall. A ₁ constant defined by eq. (A3) - C dimensionless group used in eq. (7) - C _p heat capacity - D constant in eq. (13) - f u _s/u - f _lin defined by eq. (A6) - G storage modulus - G loss modulus - k thermal conductivity - L length of hot film in thex-direction - L _eff effective length of large probe found from eq. (A3) - Nu _L Nusselt number, defined for a lengthL by eq. (2) - (Nu _L)₀ value ofNu _L atPe = 0 (eq. (A 1)) - Pe Péclet number,uL/ - Pe ₀ Péclet number in slip flow, eq. (6) - Pe ₁ Péclet number in shear flow, eq. (4) - q _L average heat flux over hot film of lengthL - R _i resistances defined by figure 8 - R _AB correlation coefficient defined by eq. (14) for signalsA andB - T temperature - T _s temperature of probe surface - T ambient temperature - T T _s –T - u average velocity - u _s slip velocity - V _b voltage indicated in figure 8 - W probe dimension (figure 6) - x distance in flow direction (figure 1) - y distance perpendicular to flow direction (figure 1) - thermal diffusivity,k/C _p - wall shear rate - _5% thickness of lubricating layer during probe calibration for introduction of an error no greater than 5%; see Appendix I - ^* complex viscosity - density - time - _c critical shear stress, eq. (13) - _w wall shear stress - frequency characterizing extrudate distortion (figures 12 and 13), or frequency of oscillation during rheometric characterization (figures 18–20) - * quantity obtained from normalized Nusselt number, eq. (A1), or complex viscosity ^* - A actual (small) probe (see Appendix I) - M model (large) probe (see Appendix I)     

Temperature and concentration dependent viscosity and gelation temperature of ABA triblock copolymer solutions

In solutions of ABA-triblock copolymers in a poor solvent for A thermoreversible gelation can occur. A three-dimensional dynamic network may form and, given the polymer and the solvent, its structure will depend on temperature and polymer mass fraction. The zero-shear rate viscosity of solutions of the triblock-copolymer polystyrene-polyisoprene-polystyrene in n-tetradecane was measured as a function of temperature and polymer mass fraction, and analyzed; the polystyrene blocks contained about 100 monomers, the polyisoprene blocks about 2000 monomers. Empirically, in the viscosity at constant mass fraction plotted versus inverse temperature, two contributions could be discerned; one contribution dominating at high and the other one dominating at low temperatures. In a comparison with theory, the contribution dominating at low temperatures was identified with the Lodge transient network viscosity; some questions remain to be answered, however. An earlier proposal for defining the gelation temperature T _gel is specified for the systems considered, and leads to a gelation curve; T _gel as a function of polymer mass fraction.Mathematical symbols {} functional dependence; e.g., f{x} means f is a function of x - p _log logarithm to the base number p; e.g., ¹⁰log is the common logarithm - exp exponential function with base number e - sin trigonometric sine function - lim limit operation - – in integral sign: Cauchy Principal Value of integral, e.g., - derivative to x - partial derivative to x Latin symbols dimensionless constant - b constant with dimension of absolute temperature - constant with dimension of absolute temperature - B dimensionless constant - c mass fraction - dimensionless constant - constant with dimension of absolute temperature - d ^* dimensionless constant - D{0} constant with dimension of absolute temperature - e base number of natural (or Naperian) logarithm - g distribution function of inverse relaxation times - G relaxation strength relaxation function - h distribution function of relaxation times reaction constant enthalpy of a molecule - H Heaviside unit step function - i complex number defined by i ² = –1 - j{0} constant with dimension of viscosity - j index number - k Boltzmann's constant - k _H Huggins' coefficient - m mass of a molecule - n number - N number - p index number - s entropy of a molecule - t time - T absolute temperature Greek symbols as index: type of polymer molecule - as index: type of polymer molecule - shear as index: type of polymer molecule - shear rate - small variation; e.g. T is a small variation in T relative deviation Dirac delta distribution as index: type of polymer molecule - difference; e.g. is a difference in chemical potential - constant with dimension of absolute temperature - (complex) viscosity - constant with dimension of viscosity - [] intrinsic viscosity number - inverse of relaxation time - chemical potential - number pi; circle circumference divided by its diameter - mass per unit volume - relaxation time shear stress - angular frequency     

The evolution of linear viscoelasticity during the vulcanization of polyethylene

The evolution of linear viscoelasticity during the vulcanization of polyethylene is studied through the gel point. The material in the vicinity of the gel point is described by two scaling laws: one characterizes the viscoelasticity at the critical point and a second characterizes the evolution of viscoelasticity near the gel point. Time Resolved Mechanical Spectroscopy is used to observe both scaling phenomena. The material at the gel point displays power law relaxation over five decades of time with a power-law relaxation exponent equal to 0.32. This study conforms with previous findings that this exponent is composition-dependent. The longest relaxation time diverges in the vicinity of the gel point as _max |p _c - p| ^–1/, and we find = 0.2. This result conforms with previous reports that this exponent may be independent of composition. The Arrhenius flow activation energy for this material undergoes three-fold changes during crosslinking up to the gel point. A single-adjustable-parameter stretched-exponential-power law relaxation function is an inadequate representation of crosslinked materials over any significant range of extent of the reaction up to the gel point.     

Pulsatile blood flow through arterioles

An analytical study was made to examine the effect of vascular deformability on the pulsatile blood flow in arterioles through the use of a suitable mathematical model. The blood in arterioles is assumed to consist of two layers — both Newtonian but with differing coefficients of viscosity. The flow characteristics of blood as well as the resistance to flow have been determined using the numerical computations of the resulting expressions. The applicability of the model is illustrated using numerical results based on the existing experimental data. r, z coordinate system - u, axial/longitudinal velocity component of blood - p pressure exerted by blood - _b density of blood - µ viscosity of blood - t time - , displacement components of the vessel wall - T _t0,T ₀ known initial stresses - density of the wall material - h thickness of the vessel wall - T _t,T stress components of the vessel - K _l,K _r components of the spring coefficient - C _l,C _r components of the friction coefficient - M _a additional mass of the mechanical model - r ₁ outer radius of the vessel - thickness of the plasma layer - r ₁ inner radius of the vessel - circular frequency of the forced oscillation - k wave number - E ₀,E _t, , _t material parameters for the arterial segment - µ _p viscosity of the plasma layer - Q total flux - Q _p flux across the plasma zone - Q _h flux across the core region - Q mean flow rate - resistance to flow - P pressure difference - l length of the segment of the vessel     

Viscoelastic swirling flow with free surface in cylindrical chambers

The flow of a viscoelastic liquid driven by the steadily rotating bottom cover of a cylindrical cup is investigated. The flow field and the shape of the free surface are determined at the lowest significant orders of the regular domain perturbation in terms of the angular velocity of the bottom cap. The meridional field superposed on a primary azimuthal field shows a structure of multiple cells. The velocity field and the shape of the free surface are strongly effected by the cylinder aspect ratio and the elasticity of the liquid. The use of this flow configuration as a free surface rheometer to determine the first two Rivlin-Ericksen constants is shown to be promising.Nomenclature R, ,Z Coordinates in the physical domain D - , , Coordinates in the rest stateD ₀ - r, ,z Dimensionless coordinates in the rest stateD ₀ - Angular velocity - Zero shear viscosity - Surface tension coefficient - Density - Dimensionless surface tension parameter - ₁, ₂ The first two Rivlin-Ericksen constants - Stream function - Dimensionless second order meridional stream function - ^* Dimensionless second normal stress function - 2 Dimensionless sum of the first and second normal stress functions - N ₁,N ₂ The first and second normal stress functions - n Unit normal vector - D Stretching tensor - A _n nth order Rivlin-Ericksen tensor - S Extra-stress - u Velocity field - U Dimensionless second order meridional velocity field - V Dimensionless first order azimuthal velocity field - p Pressure - Modified pressure field - P Dimensionless second order pressure field - J Mean curvature - a Cylinder radius - d Liquid depth at rest - D Dimensionless liquid depth at rest - h Free surface height - H Dimensionless free surface height at the second order     

Determination of discrete relaxation and retardation time spectra from dynamic mechanical data

A powerful but still easy to use technique is proposed for the processing and analysis of dynamic mechanical data. The experimentally determined dynamic moduli,G() andG(), are converted into a discrete relaxation modulusG(t) and a discrete creep complianceJ(t). The discrete spectra are valid in a time window which corresponds to the frequency window of the input data. A nonlinear regression simultaneously adjust the parametersg _i , _i ,i = 1,2, N, of the discrete spectrum to obtain a best fit ofG, G, and it was found to be essential that bothg _i and _i are freely adjustable. The number of relaxation times,N, adjusts during the iterative calculations depending on the needs for avoiding ill-posedness and for improved fit. The solution is insensitive to the choice of initial valuesg _i,0, _i,0,N ₀. The numerical program was calibrated with the gel equation which gives analytical expressions both in the time and the frequency domain. The sensitivity of the solution was tested with model data which, by definition, are free of experimental error. From the relaxation time spectrum, a corresponding discrete set of parametersJ ₀,, J _d,i and _i of the creep complianceJ(t) can then readily be calculated using the Laplace transform.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Evaluating Herschel-Bulkley fluids with the back extrusion (annular pumping) technique

A mathematical model was developed to describe the behavior of Herschel-Bulkley fluids in a back extrusion (annular pumping) device. A technique was also developed to determine the rheological properties (yield stress, flow behavior index, and consistency coefficient) of these fluids. Mathematical terms were expressed in four dimensionless terms, and graphical aids and tables were prepared to facilitate the handling of the expressions.Nomenclature a radius of the plunger, m - dv/dr shear rate, s^–1 - F force applied to the plunger, N - F _b buoyancy force, N - F _cb force corrected for buoyancy, N - F _T recorded force just before the plunger is stopped, N - F _Te recorded force after the plunger is stopped, N - g acceleration due to gravity, m/s² - H(t) momentary height between plunger and container bottom, m - K a/R, dimensionless - L length of annular region, m - L(t) depth of plunger penetration, m - n flow behavior index, dimensionless - p static pressure, Pa - P _L pressure in excess of hydrostatic pressure at the plunger base, Pa - p ₀ pressure at entrance to annulus, Pa - P pressure drop per unit of length, Pa/m - Q total volumetric flow rate through the annulus, m³/s - r radial coordinate, measured from common axis of cylinder forming annulus, m - R radius of outer cylinder of annulus, m - s reciprocal of n, dimensionless - t time, s - T dimensionless shear stress, defined in Eq. (3) - T ₀ dimensionless yield stress, defined in Eq. (4) - T _w dimensionless shear stress at the plunger wall - _p velocity of plunger, m/s - velocity, m/s - mass density of fluid, kg/m³ - Newtonian viscosity, Pa s - P p ₀ –p _L , Pa - consistency coefficient, Pa sⁿ - value of where shear stress is zero - _–, ₊ limits of the plug flow region (Fig. 1) - r/R - shear stress, Pa - _y yield stress, Pa - _w shear stress at the plunger wall, Pa - dimensionless flow rate defined in Eq. (24) - dimensionless velocity defined by Eq. (5) - _–, ₊ dimensionless velocity outside the plug flow region - _max dimensionless maximum velocity in the plug flow region - _p dimensionless velocity at the plunger wall     

Response of an elastic Bingham fluid to oscillatory shear

The response of an elastic Bingham fluid to oscillatory strain has been modeled and compared with experiments on an oil-in-water emulsion. The newly developed model includes elastic solid deformation below the yield stress (or strain), and Newtonian flow above the yield stress. In sinusoidal oscillatory deformations at low strain amplitudes the stress response is sinusoidal and in phase with the strain. At large strain amplitudes, above the yield stress, the stress response is non-linear and is out of phase with strain because of the storage and release of elastic recoverable strain. In oscillatory deformation between parallel disks the non-uniform strain in the radial direction causes the location of the yield surface to move in-and-out during each oscillation. The radial location of the yield surface is calculated and the resulting torque on the stationary disk is determined. Torque waveforms are calculated for various strains and frequencies and compared to experiments on a model oil-in-water emulsion. Model parameters are evaluated independently: the elastic modulus of the emulsion is determined from data at low strains, the yield strain is determined from the phase shift between torque and strain, and the Bingham viscosity is determined from the frequency dependence of the torque at high strains. Using these parameters the torque waveforms are predicted quantitatively for all strains and frequencies. In accord with the model predictions the phase shift is found to depend on strain but to be independent of frequency.Notation A plate strain amplitude (parallel plates) - A _R plate strain amplitude at disk edge (parallel disks) - G elastic modulus - m torque (parallel disks) - M normalized torque (parallel disks) = 2m/R ³₀ - N ratio of viscous to elastic stresses (parallel plates) =µ A/ ₀ ratio of viscous to elastic stresses (parallel disks) =µ A _R/₀ - r normalized radial position (parallel disks) =r/R - r radial position (parallel disks) - R disk radius (parallel disks) - t normalized time = t — /2 - t time - _E elastic strain - _P plate strain (displacement of top plate or disk divided by distance between plates or disks) - _PR plate strain at disk edge (parallel disks) - ₀ yield strain - _E normalized elastic strain = _E/₀ - _P normalized plate strain = _P/₀ - _PR normalized plate strain at disk edge (parallel disks) = _PR/₀ - ₀ normalized plate strain amplitude (parallel plates) =A/ ₀ — normalized plate strain amplitude at disk edge (parallel disks) =A _R/₀ - phase shift between _P andT (parallel plates) — phase shift between _PR andM (parallel disks) - µ Bingham viscosity - stress - ₀ yield stress - T normalized stress =/ ₀ - frequency     

Rheological models for gypsum plaster pastes

Shear stress and shear rate data obtained for gypsum plaster pastes were correlated by means of different rheological models. The pastes were prepared from a commercial calcium sulfate hemihydrate at various water/plaster ratios ranging from 100/150 to 100/190. The tests were performed at 25°C using a rotating coaxial cylinder viscosimeter. The measurements were accomplished by applying a step-wise decreasing shear rate sequence. Discrimination among the models was made: (1) on the basis of the fitting goodness; (2) by checking the physical meaning of the calculated parameters; (3) on the basis of the stability of the parameters and of their prediction capacity beyond the limits of the experimental data. In the light of above, the Casson model seemed to be most effective for application to gypsum plaster pastes. K Consistency - n Power-law index - N Number of experimental data - P Number of parameters - Shear rate (s^–1) - ₀ Viscosity (Pa · s) - _d Dispersing medium viscosity (Pa · s) - _p Plastic viscosity (Pa · s) - Viscosity at zero shear rate (Pa · s) - Viscosity at infinite shear rate (Pa · s) - [] Intrinsic viscosity - ² Variance - Shear stress (Pa) - ₀ Yield stress (Pa) - Solid volume fraction - _m Maximum solid volume fraction     

On some nearly viscometric flows of viscoelastic liquids

Superposition of oscillatory shear imposed from the boundary and through pressure gradient oscillations and simple shear is investigated. The integral fluid with fading memory shows flow enhancement effects due to the nonlinear structure. Closed-form expressions for the change in the mass transport rate are given at the lowest significant order in the perturbation algorithm. The elasticity of the liquid plays as important a role in determining the enhancement as does the shear dependent viscosity. Coupling of shear thinning and elasticity may produce sharp increases in the flow rate. The interaction of oscillatory shear components may generate a steady flow, either longitudinal or orthogonal, resulting in increases in flow rates akin to resonance, and due to frequency cancellation, even in the absence of a mean gradient. An algorithm to determine the constitutive functions of the integral fluid of order three is outlined.Nomenclature A _n Rivlin-Ericksen tensor of order . - A _k Non-oscillatory component of the first order linear viscoelastic oscillatory velocity field induced by the kth wave in the pressure gradient - d Half the gap between the plates - e _x, e _z Unit vectors in the longitudinal and orthogonal directions, respectively - G(s) Relaxation modulus - G History of the deformation - Stress response functional - I() Enhancement defined as the ratio of the frequency dependent part of the discharge to the frequencyindependent part of it at the third order - I ^*() Enhancement defined as the ratio of the increase in discharge due to oscillations to the total discharge without the oscillations - k Power index in the relaxation modulus G(s) - k _i ^–1 Relaxation times in the Maxwell representation of the quadratic shear relaxation modulus (s ₁, s ₂) - m _i ^–1, n _i ^–1 Relaxation times in the Maxwell representations of the constitutive functions ₁(s ₁,s ₂,s ₃) and ₄ (s ₁, s ₂,s ₃), respectively - P Constant longitudinal pressure gradient - p Pressure field - _mx ,⁽³⁾ _nz ,⁽³⁾ Mean volume transport rates at the third order in the longitudinal and orthogonal directions, respectively - ₀,⁽³⁾, ₁,⁽³⁾ Frequency independent and dependent volume transport rates, respectively, at the third order - s = t- Difference between present and past times t and      

Apparent thickening behavior of dilute polystyrene solutions in extensional flows

An experimental investigation was undertaken to study the apparent thickening behavior of dilute polystyrene solutions in extensional flow. Among the parameters investigated were molecular weight, molecular weight distribution, concentration, thermodynamic solvent quality, and solvent viscosity. Apparent relative viscosity was measured as a function of wall shear rate for solutions flowing from a reservoir through a 0.1 mm I.D. tube. As increased, slight shear thinning behavior was observed up until a critical wall shear rate was exceeded, whereupon either a large increase in or small-scale thickening was observed depending on the particular solution under study. As molecular weight or concentration increased, decreased and, the jump in above , increased. increased as thermodynamic solvent quality improved. These results have been interpreted in terms of the polymer chains undergoing a coil-stretch transition at . The observation of a drop-off in at high (above ) was shown to be associated with inertial effects and not with chain fracture due to high extensional rates.     

A simplified cooperative model of stress relaxation and other consolidation processes in solids

This paper explores the properties of a relaxation function derived from a differential equation mimicking the distribution mechanism of Bose-Einstein statistics in the time domain. Within a significant portion of the process, the relaxation quantity n decreases linearly with log time. The relation between dn/dt and n is an exponential one. In this respect, the present approach produces results largely equivalent to those obtained using the hypothesis of stress-dependent thermal activation or a box-like spectrum of relaxation times, . The spectrum of the model proposed here is discrete, with integer valued fractions of a characteristic -centering the equations.     

The steady shear viscosity of filled polymeric liquids described by a linear superposition of two relaxation mechanisms

Filled polymeric liquids often exhibit apparent yielding and shear thinning in steady shear flow. Yielding results from non-hydrodynamic particle—particle interactions, while shear thinning results from the non-Newtonian behavior of the polymer melt. A simple equation, based on the linear superposition of two relaxation mechanisms, is proposed to describe the viscosity of filled polymer melts over a wide range of shear rates and filler volume fraction.The viscosity is written as the sum of two generalized Newtonian liquid models. The resulting equation can describe a wide range of shear-thinning viscosity curves, and a hierarchy of equations is obtained by simplifying the general case. Some of the parameters in the equation can be related to the properties of the unfilled liquid and the solid volume fraction. One adjustable parameter, a yield stress, is necessary to describe the viscosity at low rates where non-hydrodynamic particle—particle interaction dominate. At high shear rates, where particle—particle interactions are dominated by interparticle hydrodynamics, no adjustable parameters are necessary. A single equation describes both the high and low shear rate regimes. Predictions of the equation closely fit published viscosity data of filled polymer melts. n power-law index - n ₁,n ₂ power-law index of first (second) term - shear rate - steady shear viscosity - ₀ zero-shear rate viscosity - _{0, 1}, _{0, 2} zero-shear rate viscosity of first (second) term - time constant - ₁, ₂ time constant of first (second) term - µ _r relative viscosity of filled Newtonian liquid - ₀ yield stress - ø solid volume fraction - ø _m maximum solid volume fraction     

Pure liquids described as concentrated cluster dispersions

The Fulcher-Tammann-Hesse-Vogel equation, ln = A + B/(T – T ₀ ), is shown to be equivalent to the general viscosity-composition relationship, ln _r =k f /(1 – f ), for binary mixtures. The Cailletet-Mathias law of the Rectilinear Diameter is rearranged to represent a density mixture formula for two components. Temperature-independent viscosities and densities can then be calculated for dense, solid cluster fractions, dispersed in a low-density, low-viscosity non-clustered continuous phase. The cluster fraction decreases with temperature. The value ofT ₀ is shown to be related to the liquid- or solid-like behavior of the clusters. For liquids with a vapor pressure < 1 mm Hg at the melting point, the calculated cluster volume fraction suggests close packing of clusters, ranging in shape from monodisperse spheres to polydisperse non-spherical particles. Examples are given for molecular liquids, molten metals, and molten salts. The size of the clusters is estimated from the heat of evaporation.