Viscoelastic and rheological behavior of concentrated colloidal suspensions

Molecular approaches are discussed to the density (), viscoeleastic (), and rheological () behavior of the viscosity(,,) of concentrated colloidal suspensions with 0.3 < < 0.6, where, is the volume fraction, the applied frequency, and ; the shear rate. These theories are based on the calculation of the pair distribution functionP ₂(r,,), wherer is the relative position of a pair of colloidal particles. The linear viscoelastic behavior(,,=0) follows from an equation forP ₂(r,,) derived from the Smoluchowski equation for small, generalized to large by introducing the spatial ordering and (cage) diffusion typical for concentrated suspensions. The rheological behavior(,,=0) follows from an equation forP ₂(r,) of a dense hard-sphere fluid derived from the Liouville equation. This leads to a hard-sphere viscosity^hs(,) which yields the colloidal one(,) by the scaling relation(,) ₀=^hs(,) _B, where ₀ is the solvent viscosity. _B is the dilute hard-sphere (Boltzmann ) viscosity and the's are appropriately scaled,(,) and(,) agree well with experiment. A unified theore for(,,) is clearly needed and pursued.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994. Boulder, Colorado, U.S.A.     

Shear viscosity of liquid4He and3He-4He mixtures,especially near the superfluid transition

High-resolution measurements of are reported for liquid⁴He and³He-⁴He mixtures at saturated vapor pressures between 1.2 and 4.2 K with particular emphasis on the superfluid transition. Here is the mass density, the shear viscosity, and in the superfluid phase both and are the contributions from the normal component of the fluid ( _n and _n ). The experiments were performed with a torsional oscillator operating at 151 Hz. The mole fraction X of³He in the mixtures ranged from 0.03 to 0.65. New data for the total density and data for _n by various authors led to the calculation of . For⁴He, the results for are compared with published ones, both in the normal and superfluid phases, and also with predictions in the normal phase both over a broad range and close to T. The behavior of and of in mixtures if presented. The sloped/dT near T and its change at the superfluid transition are found to decrease with increasing³He concentration. Measurements at one temperature of versus pressure indicate a decreasing dependence of on molar volume asX(³He) increases. Comparison of at T, the minimum of _n in the superfluid phase and the temperature of this minimum is made with previous measurements. Thermal conductivity measurements in the mixtures, carried out simultaneously with those of , revealed no difference in the recorded superfluid transition, contrary to earlier work. In the appendices, we present data from new measurements of the total density for the same mixtures used in viscosity experiments. Furthermore, we discuss the data for _n determined for⁴He and for³He-⁴He mixtures, and which are used in the analysis of the data.     

Viscosity of biniary mixtures. III. Tri-n-butylamine with alkanes and monoalkylamines at 303.15 and 313.15 K

Viscosities of seven binary systems of n-propylamine, n-butylamine, n-hexylamine, n-octylamine, n-hexane, n-octane, and isoctane (2,2,4-trimethylpentane) with tributylamine have been measured at 303.15 and 313.15 K with an Ubbelohde suspended-level viscometer. Based on Eyring's theory, values of excess Gibbs energy of activation G ^*E of viscous flow have been calculated. Deviations of viscosities from linear dependence on the mole fraction and values of G ^*E are attributed to the H-bonding and to the size of alkylamine and alkane molecules. The free volume theory of Prigogine-Flory-Patterson in combination with work by Bloomfield-Dewan has been used to estimate the excess viscosity ln and the terms corresponding to enthalpy, entropy, and free volume contributions for the present binary mixtures.     

Coupling of order parameter collective modes to spin density in3He-B

We derive a general expression for the dynamic spin susceptibility of³He-B which is valid for all magnetic fields. The coupling of real and imaginary modes by particle-hole asymmetry is taken into account. Then we calculate the contribution of the mode at frequency =2 – 1/4 ( is the effective Larmor frequency) to the transverse susceptibility. The spectral weight of this mode in magnetic resonance absorption is proportional to (/)^1/2 (–)², where and are particle-hole asymmetry parameters. From the experimental coupling strength of the real squashing mode to sound we estimate (–)²10^–4. The dynamic susceptibility satisfies the sum rules of Leggett. Finally we point out the difficulties in calculating the transverse NMR frequency of³He-B. These difficulties arise from theS _z =0 Cooper pairs and from the coupling ofJ _z =±1 modes forJ=1 andJ=2.     

Viscosities and Excess Molar Volumes of Binary Mixtures of Alkyl Acetates with Di-, Tri-, and Tetrachloroethane

The excess molar volume V ^E, viscosity deviation , excess viscosity ^E, and excess Gibbs energy of activation G*^E of viscous flow have been investigated from density and viscosity measurements of nine binary mixtures of methyl acetate, ethyl acetate, and amyl acetate with dichloroethane, trichloroethane, and tetrachloroethane at 303.15 K. The results were fitted to polynomials of variable degree. The viscosity data have been correlated with the equations of Grunberg and Nissan, Hind, McLaughlin, and Ubbelohde, Tamura and Kurata, Katti and Chaudhri, McAllister, Heric, and Auslaender. The results have been analyzed in terms of molecular interactions between alkyl acetates and chloroethanes.     

Separable functions in load separation for the ηpl and ηpl CMOD plastic factors estimation

The objective of this paper is to develop the load separation method for evaluating the _pl and _pl ^CMOD plastic factors used in the J estimation approach based on load versus displacement records. Appropriate forms for the geometry and deformation functions have been suggested from the EPRI Handbook solutions to produce the separable form for the load. The obtained functions are applied to evaluate the _pl and _pl ^CMOD plastic factors for center cracked tension specimen. The present load separation method gave results which are somewhat different from the estimated values of _pl given in the literature. For shallow cracks, the _pl and ^CMOD _pl plastic factors show considerable variation with crack size and the strain hardening exponent. For a deeply cracked CCT specimen, the ^CMOD _pl factor tends to the _pl factor and equals approximately unity. Abbreviations: CCT – center cracked specimen; CMOD – Crack Mouth Opening Displacement; EPRI – Electric Power Research Institute; FEM – Finite Element Method; LLD – Load Line Displacement.     

Effect of the elastic factor on the hydrodynamic stability of a structurally viscous medium

The effect of relaxation phenomena on the hydrodynamic stability of the plane gradient flow of a structurally viscous medium is investigated using linear theory.Notation _ij stress tensor deviator - U_i components of the velocity vector - x_i coordinates - t time - P pressure - =₀L/^*V plasticity parameter - _o limiting shear stress - andc dimensionless wave number and the perturbation frequency - Re=VL/* Reynolds number - density - F_ij deformation rate tensor Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 35, No. 5, pp. 868–871, November, 1978.     

The viscosity and some related properties of liquid3He at the melting curve between 1 and 100 mK

The viscosity of liquid ³ He has been measured along the melting curve from 1 to 100 mK by means of a vibrating wire viscometer. In the normal Fermiliquid region we find 1/T² = 1.17–3.10T, where is in P and T in K. At the transition temperature T _A = 2.6 mK a rapid decrease occurs in _n , the viscosity of the normal component. Within 0.3 mK below T _A , _n decreases to about 25% of _A, but then becomes essentially constant. In the B phase _n first decreases to 20% of _A and then seems to increase below 1.4 mK. Data on _n , the density of the normal component, are also presented in the A and B phases. The results show that viscous flow is accompanied by a flow of zero dissipation, thus proving superfluidity in the A and B phases. The viscosity data at magnetic fields up to 0.9T have been related to theoretical calculations of the energy gap of superfluid ³ He near T _A . The splitting of the A transition and the suppression of the B phase in an external field were also measured.     

Shear viscosity measurements of liquid 4He and 3He-4He mixtures near the tricritical point

A torsional oscillator cell is described, by means of which simultaneous precision measurements of () and of the molar volume can be made in liquid ⁴He-⁴He mixtures over the temperature range between 0.5 and 3 K. Here is the mass density, the shear viscosity and in the superfluid phase they become the contributions _n and _n of the normal component. The results of for ⁴He near the superfluid transition are compared with the predictions by Schloms, Pankert and Dohm, and by Ferrell. Measurements of () are reported for mixtures with 0.64X0.74, where X is the ³He mole fraction. Those for X = 0.67 and 0.70 are compared with data by Lai and Kitchens. The viscosity experiments show no evidence of a weak singularity at the tricritical point.     

Shear viscosity measurements of liquid 3He-4He mixtures above 0.5 K

Simultaneous measurements of () and of the molar volume are reported for liquid mixtures of ³He in ⁴He over the temperature range between 0.5 and 2.5 K. Here is the shear viscosity and is the mass density. In the superfluid phase, the product of the normal components, _n and _n , is measured. The mixtures with ³He molefractions 0.30 < X < 0.80 are studied with emphasis on the region near the superfluid transition T and near the phase-separation curve. Along the latter, they are compared with data by Lai and Kitchens. For X > 0.5, the viscosity singularity near T becomes a faint peak, which however fades into the temperature-dependent background viscosity as X tends to the tricritical concentration X _t. Likewise, no singularity in is apparent when T _t is approached along the phase separation branches _– and ₊. Furthermore, viscosity data are reported for ³He and compared with previous work. Finally, for dilute mixtures with 0.01 X 0.05, the results for are compared with previous data and with predictions.     

Densities,Viscosities, and Refractive Indices of Binary Mixtures of Benzene with Isomeric Butanols at 30°C

The densities, , viscosities, , and refractive indices, n, of binary mixtures of benzene with 1-butanol, 2-methyl-1-propanol, 2-butanol, and 2-methyl-2-propanol, including those of the pure liquids, were measured over the complete composition range at 30°C. The dependence of , , and n on composition was checked by using an empirical relation. The experimental data were used to calculate excess molar volumes, V^E, deviations in viscosity, , excess free energies of activation of viscous flow, G^*E, deviations in refractive index, n, apparent molar volumes, V_,1 and V_{, 2}, and partial molar volumes, , of benzene in alcohols and alcohols in benzene, respectively, at infinite dilution. The variations of these parameters with composition and the effect of branching in alcohols were discussed from the point of view of intermolecular interactions in these mixtures.     

The quantitative application of Fermi-Dirac functions to two- and three-dimensional systems

Expressions for the various physical parameters of the ideal Fermi-Dirac gas in two dimensions are derived and compared to the corresponding three-dimensional expressions. These derivations show that the Fermi-Dirac functions most applicable to the two-dimensional problem are F ₀(), F ₁(), and F₀(). Analogous to the work of McDougall and Stoner in three dimensions, these functions and parameters derived from them are tabulated over the range of the argument, –420. The physical applications of the tabulated 2D and 3D functions to ³He monolayer and bulk liquid ³He nuclear magnetic susceptibilities, respectively, are considered. Calculational procedures of fitting data to theoretical parameters and criteria for judging the quality of fit of data to both two- and three-dimensional Fermi-Dirac values are discussed.This work was performed under the auspicious of the U.S. Department of energy, Office of Solar Application. Work also supported in part by the National Science Foundation and a Navy Equipment Lone Contract.     

Prediction of material damping of laminated polymer matrix composites

In this study the material damping of laminated composites is derived analytically. The derivation is based on the classical lamination theory in which there are eighteen material constants in the constitutive equations of laminated composites. Six of them are the extensional stiffnesses designated by [A] six of them are the coupling stiffnesses designated by [B] and the remaining six are the flexural stiffnesses designated by [D]. The derivation of damping of [A], [B] and [D] is achieved by first expressing [A], [B] and [D] in terms of the stiffness matrix [Q]^(k) andh _k of each lamina and then using the relations ofQ _ij ^(k) in terms of the four basic engineering constantsE _L,E _T, G_LT andv _LT. Next we apply elastic and viscoelastic correspondence principle by replacingE _L,E _T...by the corresponding complex modulusE _L ^*,E _T ^*,..., and [A] by [A]^*, [B] by [B]^* and [D] by [D]^* and then equate the real parts and the imaginary parts respectively. Thus we have expressedA _ij ,A _y ,B _ij ,B _ij , andD _ij in terms of the material damping _L ^(k) and _T ^(k)...of each lamina. The damping _L ^(k), _T ^(k)...have been derived analytically by the authors in their earlier publications. Numerical results of extensional damping l _ij =A _ij /A _ij coupling dampingc _ij =B _ij /B _ij and flexural damping F _ij =D _ij /D _ij are presented as functions of a number of parameters such as fibre aspect ratiol/d, fibre orientation , and stacking sequence of the laminate.     

Viscosities of nonelectrolyte liquid mixtures. I. binary mixtures containing p-dioxane

Viscosity measurements are reported for p-dioxans with cyclohexane, n-hexane, benzene, toluene, carbon tetrachloride, tetrachloroethane, chloroform, pentachloroethane, and ethyl acetate at 303.15 K. Excess Gibbs energies of activation G ^*E of viscous flow have been calculated with Eyring's theory of absolute reaction rates. The deviations of the viscosities from a linear dependence on the mole fraction and values of G ^*E for binary mixtures have been explained in terms of molecular interactions between unlike pairs. The Prigogine-Flory-Patterson theory has been used to estimate the excess viscosity, ln , and corresponding enthalpy ln _H, entropy ln _S, and free volume ln _v terms for binary mixtures of p-dioxane with cyclohexane, n-hexane, benzene, toluene, carbon tetrachloride, and chloroform. Estimates of excess viscosities from this theory for p-dioxane with benzene, toluene, and carbon tetrachloride are good, while for the other three mixtures they are poor. The local-composition thermodynamic model of Wei and Rowley estimates the excess viscosity quite well even for p-dioxane mixtures with cyclohexane and n-hexane.     

Effective Free Energy for Solidification of Superfluid 4He Under Pressure

Effective free energy for the solidification of the superfluid ⁴ He under pressure is presented. Since this free energy is a function of two parameters, the density change and the translational symmetry breaking parameter , it is expected to describe over both the superfluid phase and the solid phase.The experimentally established values below and equal to the melting pressure P_m are successfully reproducible by the free energy. For P>P_m, where the superfluid is metastable, the free energy predicts that the instability for should occur first and then accompanies. That is, -nucleation is essential for the solid nucleation in the superfluid ⁴ He.     

Effect of the temperature dependence of thermal properties on the thermal shock tests of ceramics

Thermal stress generated during a thermal shock is closely related to the fracture of ceramics. An attempt has been made to obtain thermal stress in a specimen by numerical calculation. The temperature dependence of thermal conductivity and diffusivity were introduced to realize the practical thermal conditions. The maximum thermal stress, _max ^* , was recognized at the Fourier number, but differed from the temperature dependence. Correlative equations of _max ^* and _max ^* with the Biot number, _i, under cooling or heating tests, have been proposed. These equations resulted in the exact _max ^* and _max ^* compared with the previous equations, in which temperature dependence was ignored. The thermal shock resistance parameter was expressed by the correlative equations of _max ^* in order to suggest adequate experimental conditions and specimen size. A comparison of the measured and calculated time to failure of the specimen led to confirmation of the fracture criterion. The measured time disagreed with the calculated one, if the fracture by thermal shocking was not predominant. The correlative equations were also useful to select the kind of ceramics subjected to thermal shocking.     

Correlation of high-pressure thermal conductivity,viscosity, and diffusion coefficients for n-alkanes

Thermal conductivity, viscosity, and self-diffusion coefficient data for liquid n-alkanes are satisfactorily correlated simultaneously by a method based on the hard-sphere theory of transport properties. Universal curves are developed for the reduced transport properties ^*, ^*, and D ^* as a function of the reduced volume. A consistent set of equations is derived for the characteristic volume and for the parameters R , R , and R _D, introduced to account for the nonsphericity and roughness of the molecules. The temperature range of the above scheme extends from 110 to 370 K, and the pressure range up to 650 MPa.     

The behavior of the4He viscosity near the superfluid transition

Measurements of the shear viscosity at saturated vapor pressure through the lambda transition indicate a singular behavior of the form |1 – (/)|=A ^x , (where =|1–(T/T )|, with equal values for the critical exponent on both sides of the transition.Work sponsored by Consiglio Nazionale delle Ricerche, Rome (Italy).     

A finite volume technique to simulate the flow of a viscoelastic fluid

Hydrodynamically developing flow of Oldroyd B fluid in the planar die entrance region has been investigated numerically using SIMPLER algorithm in a non-uniform staggered grid system. It has been shown that for constant values of the Reynolds number, the entrance length increases as the Weissenberg number increases. For small Reynolds number flows the center line velocity distribution exhibit overshoot near the inlet, which seems to be related to the occurrence of numerical breakdown at small values of the limiting Weissenberg number than those for large Reynolds number flows. The distributions of the first normal stress difference display clearly the development of the flow characteristics from extensional flow to shear flow.List of symbols D rate of strain tensor - L slit halfheight - P pressure, indeterminate part of the Cauchy stress tensor - R the Reynolds number - t time - U average velocity in the slit - u velocity vector - u,v velocity components - W the Weissenberg number based on the difference between stress relaxation time and retardation time - W ₁ the Weissenberg number based on stress relaxation time - x,y rectangular Cartesian coordinates - ratio of retardation time to stress relaxation time - zero-shear-rate viscosity, ₁ + ₂ - ₁ non-Newtonian contribution to - ₂ Newtonian contribution to - ₁ stress relaxation time - ₂ retardation time - density - (, , ) xx, yy and xy components of ₁, respectively - determinate part of the Cauchy stress tensor - ₁ non-Newtonian contribution to - ₂ Newtonian contribution to      

Shear viscosity of the B phase of superfluid3He

The shear viscosity (T) in the Balian-Werthamer (BW) state of superfluid ³ He is calculated variationally throughout the region 0t 1(t=T/T _c) from the transport equation for Bogoliubov quasiparticles. Coherence factors are treated exactly in the calculation of the collision integral. The numerical result for =_s= _s(T)/_n(T_c) agree very well with experiment in the range 0.8t1.0. Analytic expressions = 0.577 (1–1.0008t) and =1–(23/64) [=(T)/k _B T] are obtained in the low-temperature region and in the vicinity ofT _c, respectively. From the numerical analysis it is shown that the latter equation is valid only in the temperature range 0.9997t1.0.Supported by the Research Institute for Fundamental Physics, Kyoto University.