Finite Shear Rate Research Articles

Stability of uniform shear flow

The stability of idealized shear flow at long wavelengths is studied in detail. A hydrodynamic analysis at the level of the Navier-Stokes equation for small shear rates is given to identify the origin and universality of an instability at any finite shear rate for sufficiently long wavelength perturbations. The analysis is extended to larger shear rates using a low density model kinetic equation. Direct Monte Carlo Simulation of this equation is computed with a hydrodynamic description including non Newtonian rheological effects. The hydrodynamic description of the instability is in good agreement with the direct Monte Carlo simulation for $t < 50t_0$, where $t_0$ is the mean free time. Longer time simulations up to $2000t_0$ are used to identify the asymptotic state as a spatially non-uniform quasi-stationary state. Finally, preliminary results from molecular dynamics simulation showing the instability are presented and discussed.

Multicritical phenomena in flow of viscoelastic liquids. 1. Zaremba-Fromm-De Witt liquid

It has been shown that multicritical phenomena caused by nonlinearity of viscosity and high elasticity, and forced anisotropy at finite shear rates take place during flow of viscoelastic polymer melts which are isotropic in the resting state. The sign of the low-frequency asymptotic values of the dynamic viscosity and elasticity measured during steady flow is a criterion of the appearance of instability. These arguments are illustrated by the solution and analysis of the complex reaction to low-amplitude, periodic shear of a steady-flowing, very simple viscoelastic liquid — ZFD liquid. It was shown that the instability of viscoelastic liquids for a given steady shear rate is due to the effect of perturbations lasting for no less than some limiting value and its manifestations are caused by superposition of different types of instability — multicritical phenomena.

Does the Gaussian thermostat maximize the phase-space compression factor?

A recent hypothesis of D. J. Evans and A. Baranyai according to which the Gaussian thermostat maximizes the average phase-space compression factor λ in nonequilibrium steady states is analyzed for a dilute gas under uniform shear flow. Three routes have been followed: (i) an exact solution of the Bhatnagar-Gross-Krook kinetic equation for arbitrary shear rate, (ii) an exact solution of the Boltzmann equation through super-Burnett order, and (iii) a numerical solution of the Boltzmann equation for finite shear rates. The results show that the above hypothesis does not exactly hold for arbitrary shear rates, although the thermostat that maximizes λ is close to the Gaussian one. In addition, the influence of the thermostat considered on the nonlinear shear viscosity is also analyzed.

Shear flow in the infinite-shear-rate limit.

Computer simulations have revealed the existence of a disorder-order phase transition in simple shearing liquids [J. J. Erpenbeck, Phys. Rev. Lett. 52, 1333 (1984)]. Above a certain shear rate, the motion of the particles becomes ordered. This high-shear-rate structure, the so-called string phase, is studied numerically in order to investigate the possibility of such a transition taking place in real systems. At first we determined the optimal arrangements of spherical particles in the string phase. In the high-density infinite-shear-rate limit, where the random thermal motion of particles is negligible, minimum energy structures can be found as functions of density and interaction. We identify these structures and relate them to equilibrium ones. Utilizing the symmetry of the limit structure, we perform nonequilibrium molecular dynamics simulations at high but finite shear rates in order to study the coexistence conditions of liquid and string phases as a function of periodic boundary symmetry and system size. The stability of the string phase depends significantly on these aspects of the simulations. The larger the system, the less stable the pure string phase is compared to the coexisting liquid-string formation. This calls into question the existence of this phase transition in real shearing fluids. We also study the liquid-string transition in terms of the so-called phase space compressibility factor 〈\ensuremath{\Lambda}〉, since it was found recently [D. J. Evans and A. Baranyai, Phys. Rev. Lett. 67, 2597 (1991)] that this simple phase variable exhibits a local extremum property even far from equilibrium. In the thermodynamic limit, 〈\ensuremath{\Lambda}〉 correctly estimates the shear rate at which the liquid phase becomes unstable in computer simulations.

Shear lift force on spherical bubbles

The shear lift force on a spherical bubble in an unbounded shear flow at low Reynolds number is derived. It is two thirds of that for a solid sphere. An approximate expression for the shear lift force at finite Reynolds number and finite shear rate is obtained by an interpolation using the present result and Auton's result at large Re and small shear rate.

Shear-induced melting of two-dimensional solids.

We have carried out detailed nonequilibrium molecular-dynamics simulation studies of the shear-induced melting transition of a model two-dimensional solid. We find that the shear melting of the two-dimensional soft-disk solid at temperature T=1 and density \ensuremath{\rho}=1.03 occurs in two stages: (1) a transition from elastic to plastic behavior takes place as soon as any finite shear rate is applied; (2) qualitative changes in structural and dynamic behavior occur near a shear rate of \ensuremath{\gamma}=0.07. For \ensuremath{\gamma}\ensuremath{\lesssim}0.07, the system possesses very long-range bond-orientational correlations, and the instantaneous static structure factor exhibits pronounced sixfold anisotropy, with a sixfold pattern that rotates uniformly with time in response to the applied shear. For \ensuremath{\gamma}\ensuremath{\gtrsim}0.07, the system behaves like an ordinary two-dimensional liquid under shear in that the ranges of translational and bond-orientational correlations are comparable and the instantaneous static structure factor does not exhibit persistent sixfold anisotropy. We discuss our results in terms of the two competing possibilities of a Kosterlitz-Thouless-Halperin-Nelson-Young two-stage melting scenario or a single first-order melting transition.

An extended White–Metzner viscoelastic fluid model based on an internal structural parameter

A phenomenological viscoelastic model of differential type is proposed along the lines of the White–Metzner modification where, though, the model constants are functions of the invariants of an internal tensorial structural parameter. This modification arises naturally in the context of the generalized Bracket formulation and leads to a model which does not lose its evolutionarity and satisfies the entropy production inequality. A power‐law dependence of the relaxation time on the trace of the tensorial internal parameter is shown to produce reasonable predictions, including a finite zero shear rate viscosity and nonsingular behavior in extension. A multiple mode version is used in fitting experimental data obtained for polymeric melts and its predictions are compared to those of the Phan‐Thien and Tanner model.

Shear induced order and shear processing of model hard sphere suspensions

Results are presented for light scattering studies of suspensions in a liquid of nearly hard colloidal spheres. Interparticle ordering or microstructure is determined for samples in equilibrium and undergoing both steady and oscillatory shear flows. Four basic structures are found to exist under these conditions: liquid‐like or amorphous, string‐like, sliding or randomly stacked layers, and face centered cubic. The conditions leading to these different microstructures is examined systematically and compared with predictions of microstructure for atomic and molecular or colloidal suspensions undergoing steady shear flow. Generally we find that ordering into layers at a finite shear rate is more pronounced as the volume fraction of particles increases. Equilibrium liquid‐like samples do not evidence a transition into a layer structure with increasing shear rate, and equilibrium crystalline samples do evidence a shear resisting face centered cubic ordering at small shear rates. For oscillatory shear flows we find that equilibrium liquid‐like samples could be induced to have a face centered cubic structure or layer structure depending on the amplitude and frequency of the shear flow. A microstructure phase diagram as a function of volume fraction of particles and oscillatory shear amplitude is constructed and compared with a simple model based on a strained face centered cubic lattice. Studies are presented which demonstrate the effect of shear history in producing desired microstructures.

The Weissenberg effect at finite rod-rotation speeds

According to a known heuristic argument, a given polymeric liquid will be expected to exhibit a positive or negative Weissenberg effect (i.e., will move up or down a rotating vertical rod immersed in it, under conditions where contact angle and centrifugal force effects are negligible) if the value of R:=1+2(σ/N1)dN2/dσ is positive or negative, where σ, N1, and N2 denote the shear stress and first and second normal stress differences in steady shear flow. The value of this heuristic treatment lies in its applicability to finite shear rate ranges for which an exact calculation is not available. The treatment includes (and generalizes) a known exact low-shear-rate limiting condition β0&amp;lt;0.25 for rod climbing, where β0 denotes the low-shear-rate limiting value of β:=−N2/N1. Using published experimental data for N1(σ) and N2(σ) to evaluate R, we find that positive Weissenberg effects are predicted for polyisobutylene and some polystyrene solutions (even when β=0.6), in agreement with observation, but that negative effects are predicted for other polystyrene solutions, in disagreement with observation. We cannot explain this disagreement. We also use the Curtiss–Bird kinetic theory for monodisperse polymers to evaluate R for various values of the ‘‘link tension coefficient’’ ε. Over ranges of shear rate experimentally accessible for molten polymers, we find that R&amp;gt;0 provided that ε&amp;gt;1/8. When ε=0, however, we find that R&amp;lt;0 for most low-to-moderate shear rates; it follows that, according to the present heuristic argument, the Doi–Edwards theory (ε=0) for monodisperse polymers predicts negative Weissenberg effects for a significant range of shear rates, in disagreement with all known measurements for molten polymers. Using the Curtiss–Bird theory for polydisperse polymers having a log-normal distribution with Mw=2Mn, we find R values greater than those found for monodisperse polymers. Inertial forces, neglected in the present treatment, could readily be included if required for application to low-viscosity liquids.

Glassy Crystals Low‐frequency and Low‐temperature Properties a

Glasses are amorphous like liquids, but are rigid like crystals. To a chemist, this may seem natural: silicon dioxide, when covalently bonded into a well-relaxed random network, should resist deformations as an elastic solid just as it would in a crystalline network. To a theoretical physicist, this is more mysterious. What broken symmetry’ causes the rigidity‘! Of course, the chemist may in the end be right-the observed rigidity may be due purely to short-range order. I f so, glasses must flow like a (very viscous) liquid on sufficiently long time scales. Even though it is not favorable for any individual bond to break, rearrangements of sufficiently large clusters of atoms will allow the energy to decrease or remain the same. (If the order is short range with a range R, a cluster of size larger than R will typically allow such an internal rearrangement.) At any finite temperature, these rearrangements will proceed by thermal activation at a slow but nonzero rate. Any external stress applied to the system will produce a bias in these rearrangements, producing in the end the finite shear rate characteristic of a liquid. If the glass is truly solid, it must have some sort of long-range order to produce infinite barriers to rearrangements. The study of the glass transition is the study of the phase boundary between the fluid liquid and the rigid glass. This subject is controversial, and has recently seen several interesting developments. In this paper we will not discuss the glass transition. We will deal partly with the universal low-temperature2 properties of glasses, which are not as controversial; indeed, in many ways they are quite well understood. The tunneling center t h e ~ r y ~ , ~ explained why glasses have specific heats proportional to temperature and thermal conductivities proportional to T 2 , in contrast to insulating crystals where both properties are proportional to T 3 for low temperatures T. I t predicted correctly that glasses would have a saturable ultrasonic attenuation and a “time-dependent” specific heat; the experimental specific heat varies logarithmically with the measuring time. We will also deal with the low-frequency response of glassy materials. In the experimental phenomenology, there are two kinds of low-frequency response in glasses, with quite different signatures, which often (or always) coexist. There has been much study of the a-relaxation processes because of their relationship to the glass transition. The a-relaxations typically are observed in the supercooled liquid, and have a

The Shear Viscosity of Polyvinylacetate-3-Heptanone Mixture near the Critical Point

The shear viscosity η of a solution of polyvinylacetate (MW=6.7×104, Mw/Mn=2.2) in 3-heptanone was measured at the critical concentration 20.325g/100cm3 with varying temperature T in the vicinity of the critical solution temperature Tc(=284.52K). The measurements were made at various low shear rates by using a modified Ubbelohde viscometer with variable inclination angle. The critical anomaly in viscosity was observed. The temperature dependence of the anomalous component of viscosity Δη was expressed in the form, Δη/η=A ln[1-(Tc/T)]+B, with A=-0.034 at finite shear rates and A =-0.069 at zero shear rate. Both of these A values, if they are analysed by Oxtoby's and Kawasaki's theories, respectively, yield ν=0.64 for the critical exponent ν of the correlation length. This result is in good agreement with the result of light scattering measurements due to Kuwahara and indicates that the viscosity analysis is useful for the study of the critical behavior of polymer solutions.

Time Dependent Viscoelastic Properties of Concentrated Polymer Solutions

Measurements of stress growth and stress relaxation after onset and cessation of steady simple shear flow in concentrated polymer solutions were carried out with a Weissenberg rheogoniometer R-17 having a gap servo system. By using monodisperse polymers and their blends, the effect of molecular weight distribution on those transient phenomena is discussed. The so-called stress-overshoot was observed in both experiments of shear and normal stress growths. Ratio of the time of the maximum normal stress difference and that of the maximum shear stress is close to 2 at the limit of low shear rate for both monodisperse and polydisperse polymers. In the range of finite shear rate, the ratio is remarkably dependent on shear rate for polydisperse samples, whereas it is almost independent of shear rate for monodisperse polymers. Shear rate dependence of the ratio of the apparent relaxation time of normal stress difference and that of shear stress in stress relaxation experiments is also found to be remarkably affected by molecular weight distribution. From a comparison between these experiments and theories so far published, it is concluded that these transient phenomena may be explained by assuming that the relaxation spectrum is a function of shear rate at least if the shear rates are not too high. The comparison between theory and experiments is carried out without assuming explicit forms for relaxation spectrum.

Steady Flow Properties of Monodisperse Polymer Solutions. Molecular Weight and Polymer Concentration Dependences of Steady Shear Compliance at Zero and Finite Shear Rates

Steady Flow Properties of Monodisperse Polymer Solutions. Molecular Weight and Polymer Concentration Dependences of Steady Shear Compliance at Zero and Finite Shear Rates

Viscoelastic behavior of an amorphous polymer under oscillations of large amplitude

AbstractThe influence of periodic shear deformation and steady flow on a typical amorphous polymer is discussed. Forced sinusoidal vibrations were applied and the complex viscosity was determined. The action of a vibration of finite amplitude is equivalent to steady flow with a definite finite shear rate. Both processes cause truncation of the long‐time part of the relaxation specturm. It may be accepted to a first approximation that the long‐time boundary of the remaining part of the relaxation spectrum conforms to the long‐time part of the initial spectrum, even if the plateau region of the spectrum is truncated. The concept of limiting truncation of the short‐time part of the spectrum is introduced, this corresponding to the minimum absolute value of the complex viscosity versus reduced frequency and the lowest values of the dynamic and apparent viscosities. With an approximate representation of the relaxation spectrum, calculations were made of the maximum values of the viscosity and the coefficient relating the first difference of normal stresses to the square of the shear rate, and also of the apparent viscosity and normal stresses as functions of the shear rate. The calculated values are compared with experimental measurements, and it is shown that the correlation of the apparent viscosity and the absolute value of the complex viscosity is distributed at high frequencies, being superseded by a correlation between the apparent and dynamic viscosities.