Threshold-type shear-thickening in polymeric solutions

Threshold-type shear-thickening has been reported in the literature for relatively complex mixtures and one of the most famous is a solution of poly(vinylalcohol) in water in which a substantial amount of sodium borate is added. As far as we know, relatively few polymer solutions have been shown to be shear-thickening, even less of the threshold type. Two cases of threshold-type shear-thickening are examined. One of them is reversible [HPAM Pusher 700 or partially hydrolyzed poly(acrylamide) aqueous solution], the other one (in vitro-synthesized native dextran aqueous solution) leads to the formation of an irreversible structure provided sufficient strain is applied. In both cases, the occurrence of shear-thickening above a critical value of shear rate is related with a change in macromolecular conformation. This conformational change (reversible) can be followed by the formation of intermolecular bonds (irreversible).Presented at the 57^th Annual Meeting of the Society of Rheology, Ann Arbor, Michigan, Oct. 1985     

Brownian dynamics simulations of shear-thickening in dilute polymer solutions

A versatile model describing the shear thickening behaviour of dilute polymer solutions in high shear flows is presented. The polymer macromolecules are modelled as Hookean elastic dumbbells which deform affinely during flow. In addition, the dumbbells feel a retractive anisotropic hydrodynamic drag and an isotropic Brownian force. Furthermore, it is assumed that high shear rate increases the probability of molecules forming associations and this is described through expressions for the frequencies of association and dissociation, without explicitly accounting for finite extensibility, hydrodynamic interaction or excluded volume effects. Thus, a reversible kinetic process is incorporated into the model, which results in two diffusion equations for the associated and dissociated dumbbells. Numerical simulations predict shear thickening for specific range of parameters, which are physically meaningful and related to molecular characteristics of the polymer. A comparison against experimental data reported in the literature revealed very promising results, thus confirming the ability of this model to predict shear thickening under a wide range of conditions, for various polymer models.Nomenclature A A factor in the frequency of association - B Frequency of dissociation - B ₀ Reference frequency of dissociation - c Concentration of polymer solution - c _i Concentration of singlets (i = 1) and doublets (i = 2) in the solution - c ^* The overlap concentration - D _t Translation coefficient of molecule - F _i (Q) Spring force for a singlet (i = 1) and for a doublet (i = 2) - F Frequency of association - F ₀ Reference frequency of association - H _i Dumbbell spring constant for a singlet (i = 1) and for a doublet (i = 2) - k Boltzman's constant - k _H Huggins constant - MW Molecular weight - MW _c Critical molecular weight for formation of entanglements - n Number density of molecules in the polymer solution - n ₀ Number density of dumbbells at equilibrium - n _i Number density of singlets (i = 1) and doublets (i = 2) - Q Vector defining the size and orientation of a dumbbell - t Time - T Absolute temperature - x Degree of multimerization - W Interaction energy between the two components of a doublet Greek letters a Dimensionless anisotropy parameter - Shear rate - _i Friction coefficient of singlets (i = 1) and doublets (i = 2) - _i Intrinsic viscosity of singlets (i = 1) and doublets (i = 2) - _red Reduced viscosity of solution - _sp Specific viscosity - Viscosity of the polymer solution of concentration c - _s Viscosity of the solvent - (t) White noise - K ^T Velocity gradient tensor - _Hi Time constant of a singlet (i = 1) and a doublet (i = 2) - ₁ Length scale of singlets (standard deviation of singlet lengths at equilibrium) - ₂ Length scale of doublets - T _p Stress tensor - T _xy Shear Stress (xy element of T _p ) - T _pi Contributions to the stress tensor of singlets (i = 1) and doublets (i = 2) - ₀ Equilibrium configuration distribution function of Q - _i Configuration distribution function of singlets (i = 1) and doublets (i = 2)     

A study of the interacting FENE dumbbell model for semi-dilute polymer solutions in extensional flows

The effect of polymer concentration on the conformation of semidilute polymer solutions in extensional flows is studied via the interacting elastic dumbbell model proposed by Hess (1984), here modified to include a nonlinear Warner spring (FENE dumbbell) instead of the linear Hookean spring of the original model. The length of flow-induced conformation changes for the polymer is predicted to be a decreasing function of concentration. In particular, increasing concentration tends to inhibit large extension of the polymer due to polymer-polymer interaction. The specific birefringence is thus proportional to c ^–1 for semi-dilute solutions, in contrast to dilute solutions where it is known to be independent of concentration. However, the correlation between birefringence and the principle eigenvalue of the velocity gradient tensor, also found originally for dilute solutions, is predicted to occur in the semi-dilute regime. All of these predictions agree qualitatively with experimental observations.Some recent exceptions to the neglect of segmental stretch can be found in Marrucci and Grizzuti (1988), Pearson et al. (1991), Mead et al. (1992).     

A rheo-optical study of shear-thickening and structure formation in polymer solutions. Part I: Experimental

Simultaneous measurements of the optical and theological response of solutions of highly fractionated polystyrenes have been made, in-situ, to ascertain the connection between flow-induced structure formation and the phenomenon of shear-thickening. Transient and steady state viscosity, dichroism, birefringence and the associated orientation angles were measured in decalin and bromobenzene in the semi-dilute region using a couette device capable of shear rates up to 8,000 s^–1. A one-to-one correlation has been found between the occurrence of maxima in the dichroism and minima in the viscosity. While the size and shape of the shear-thickening structures could not be directly determined, results suggest they are intermediate in size between a cluster of entangled chains and a completely phase-separated liquid. For solutions exhibiting shear-thinning alone, no maximum in dichroism was observed, the signal instead showed a saturation behavior at high shear rates. Birefringence was found to be insensitive to the structure formation and attributable to that of the dissolved chains or entanglement regions. The kinetics of the structuring process leading to shear-thickening are instantaneous and completely reversible and there is a concentration window, above and below which only shear-thinning occurs.     

Surface tension measurement of aqueous polymer solutions

The surface tension of aqueous polymer solutions of polyacrylamide (PAM), polyacrylic acid (PAA), carboxymethyl cellulose (CMC), and hydroxyethyl cellulose (HEC) was studied over a range of polymer concentrations by using the maximum bubble pressure method at temperatures ranging from 20 to 65°C. The surface tension of water was also measured by the maximum bubble pressure method as well as by the DuNoüy ring method over the same temperature range. The experimental water data are in excellent agreement with the well-established tabulated data in the literature.

For a fixed concentration, all of the polymer solutions exhibited a decrease in surface tension with increasing temperature level. When compared with water at a fixed temperature level, the PAM and CMC solutions showed slightly higher surface tension values, whereas the PAA solutions yielded values equal to those found for water. In the case of the HEC solutions, the measured surface tensions decreased with concentration at a fixed temperature level and were lower than the values found for water. For a concentration of 2000 wppm the surface tension values for the hydroxyethyl cellulose were of the order of 10% lower than those for water at a fixed temperature level.

A comparison of the new measurements with the relatively limited previously published studies showed good agreement.     

A rheo-optical study of shear-thickening and structure formation in polymer solutions. Part II: Light scattering analysis

Light scattering calculations based on Anomalous Diffraction Theory (AD), Rayleigh spheroids, and flexible macromolecules are used to propose a phenomenological explanation for the relationship between shear-thickening and structure formation in polymer solutions. Quantitative comparisons are made to experimental data for the rheo-optical behavior of fractionated polystyrene solutions presented in part I of this paper. Results from the ADA calculations suggest that the viscosity and dichroism behavior can be attributed to the production and growth of micron-size, optically isotropic structures during flow. The saturation dichroism behavior exhibited by the solutions which shear thin can be attributed to the formation of entanglement regions which achieve a fixed size and act as Rayleigh spheroids in their scattering behavior. The magnitude and shear rate dependence of the observed birefringence can be accounted for on the basis of the non-linear, flexible macromolecule model, implying that birefringence is governed by the polymer chains remaining in solution which do not take part in the structure formation. The latter result is consistent with the experimental observation that the birefringence dependence on shear rate is the same whether the solution exhibits shear thickening or shear thinning in its viscosity behavior.     

Hydrodynamical changes due to polymer migration in very dilute solutions

The slip hypothesis, based on thermodynamical arguments, has been extended to obtain the flow characteristics of polymer solutions flowing in a nonhomogeneous flow field. An asymptotic analysis, valid for both channel and falling film flows, is presented that predicts the flow enhancement due to polymer migration. Concentration-viscosity coupling is shown to be a critical factor in the hydrodynamic analysis. The analysis, which essentially provides an upper bound on flow enhancement, explicitly accounts for the influence of wall shear stress, initial polymer concentration etc. A comparison with the pertinent experimental data shows reasonable agreement. c concentration - c ₀ concentration in shear-free region - c _i initial concentration - d rate of deformation tensor - g acceleration due to gravity - g ₁ function defined in eq. [13] or [15] - g ₂ function defined in eq. [18] or [20] - H half-channel thickness or film thickness - K gas law constant - L length of the channel or film - q flow rate per unit width - q ^* normalized flow rate - T temperature - v velocity - V mean velocity - y transverse distance - y _c location of solvent layer - _w /µ _s - _w /c ₀ KT - /t convected derivative - dimensionless cenentration,c/c ₀ - _c dimensionless interface concentration - _w dimensionless wall concentration - relaxation time - µ _eff effective viscosity - µ _s solvent viscosity - dimensionless transverse distance,y/H - _c dimensionless interface location - density - stress tensor - _w wall shear stress - c _i KT/ _w - ns no slip NCL-Communication No. 3155     

The crossover between linear and non-linear mechanical behaviour in polymer solutions as detected by Fourier-transform rheology

 The application of oscillatory shear strain leads, in the non-linear regime, to the appearance of higher harmonic contributions in the shear stress response. These contributions can be analyzed as spectra in Fourier space, with respect to different frequencies, amplitudes and phase angles. In this article, we present an application of this new characterization method to a solution of the linear homopolymer polyisobutylene. The degree of non-linear response during oscillatory shear is quantified using the normalized intensity of the third harmonic contribution. We were able to show experimentally on polyisobutylene that there is an immediate onset of the non-linear response even for very small shear strain amplitudes. Received: 21 June 1999/Accepted: 21 August 1999     

Drag reduction by polymer solutions in a riblet-lined pipe

The flow of 3 to 100 wppm aqueous solutions of a polyethyleneoxide polymer,M _w=6.2×;10⁶, was studied in a 10.2 mm i.d. pipe lined with 0.15 mm V-groove riblets, at diametral Reynolds numbers from 300 to 150000. Measurements in the riblet pipe were accompanied by simultaneous measurements in a smooth pipe of the same diameter placed in tandem. The chosen conditions provided turbulent drag reductions from zero to the asymptotic maximum possible. The onset of polymer-induced drag reduction in the riblet pipe occurred at the same wall shear stress, * _w =0.65 N/m², as that in the smooth pipe. After onset, the polymer solutions in the riblet pipe initially exhibited linear segments on Prandtl-Karman coordinates, akin to those seen in the smooth pipe, with specific slope increment . The maximum drag reduction observed in the riblet pipe was independent of polymer concentration and well below the asymptotic maximum drag reduction observed in the smooth pipe. Polymer solution flows in the riblet pipe exhibited three regimes: (i) Hydraulically smooth, in which riblets induced no drag reduction, amid varying, and considerable, polymer-induced drag reduction; this regime extended to non-dimensional riblet heightsh ⁺<5 in solvent andh ⁺<10 in polymer solutions. (ii) Riblet drag reduction, in which riblet-induced flow enhancementR>0; this regime extended from 5<h ⁺<22 in solvent and from 10<h ⁺<30 in the 3 wppm polymer solution, with respective maximaR=0.6 ath ⁺=14 andR=1.6 ath ⁺=21. Riblet drag reduction decreased with increasing polymer concentration and increasing polymer-induced flow enhancement S. (iii) Riblet drag enhancement, whereinR<0; this regime extended for 22<h ⁺<110 in solvent, withR;–2 forh ⁺>70, and was observed in all polymer solutions at highh ⁺, the more so as polymer-induced drag reduction increased, withR<0 for allS>8. The greatest drag enhancement in polymer solutions,R=–7±1 ath ⁺=55 whereS=20, considerably exceeded that in solvent. Three-dimensional representations of riblet- and polymer-induced drag reductions versus turbulent flow parameters revealed a hitherto unknown dome region, 8<h ⁺<31, 0<S<10, 0<R<1.5, containing a broad maximum at (h ⁺,S,R) = (18, 5, 1.5). The existence of a dome was physically interpreted to suggest that riblets and polymers reduce drag by separate mechanisms.     

Measurement of extensional viscosity of polymer solutions

A filament stretching technique for measuring the extensional viscosity of polymer solutions at constant stretch rate is presented. The liquid sample is held between two coaxial discs and stretched by moving the bottom disc downwards with a speed that increases exponentially with time. This is illustrated using a constant viscosity, elastic fluid consisting of 0.185% polyisobutylene in a solvent of kerosene and polybutene. For the case of this particular fluid, two distinct stretch rate regions are found to arise. The stretch rate in the first region is much higher than in the second, which is, in most cases, close to the overall stretch rate imposed on the sample. Nonetheless, all the results of any given run can be represented using an average extensional rate. The extensional stress growth data, plotted as the Trouton ratio against time, show an initial linear viscoelastic region where T_R rises to a value of 3, independent of extensional rate. Beyond this region, T_R depends on the stretch rate and rises dramatically to values in excess of 10³; the higher the extensional rate, the faster is the increase in T_R. These data do not seem to reach a steady state and appear to be similar to polymer melt data obtained by others in the past. The reproducibility of the results is very good and all this suggests that it is now possible to obtain unambiguous constant-stretch-rate stress-growth data for polymer solutions stretched from a state of rest.     

Bead-spring chain model for the dynamics of dilute polymer solutions : Part 2. Comparisons with experimental data

Comparisons are made between experimental rheological data and theoretical predictions obtained from a recently developed algorithm which incorporates three major molecular concepts in a theory for dilute polymer solutions (hydrodynamic interaction, excluded volume and nonlinear springs). These predictions include the radius expansion factor, the apparent chain expansion factor, the molecular weight dependence of the intrinsic viscosity, the frequency dependence of oscillatory flow birefringence, and the shear rate dependence of the intrinsic viscosity. This paper shows that a bead-spring chain model quantitatively predicts these quantities when the relevant molecular concepts are incorporated, suggesting that the rheological properties of dilute polymer solutions can be explained and predicted in terms of these molecular parameters.     

Influence of polymer conformation on the entrance length in capillary flow

An analysis of the flow of polymer solutions through capillaries by means of pressure profiles was performed in this work. The experiments were carried out with aqueous solutions of flexible and semirigid macromolecules, Polyox and Xanthan respectively. The influence of the macromolecules' conformation on the flow field development is reflected on the different magnitudes of the entry length, being of the order of 60 diameters for the solutions of rigid macromolecules, and of the order of 10 diameters for flexible ones. In addition, the calculated end effects were larger for the rigid macromolecules than for the flexible ones. From these results, it is found that an increase in the entrance length is observed when the pseudoplastic behavior of the polymer solutions is increased.Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

Structure and linear viscoelasticity of flexible polymer solutions: comparison of polyelectrolyte and neutral polymer solutions

The current state of understanding for solution conformations of flexible polymers and their linear viscoelastic response is reviewed. Correlation length, tube diameter, and chain size of neutral polymers in good solvent, neutral polymers in θ-solvent, and polyelectrolyte solutions with no added salt are compared as these are the three universality classes for flexible polymers in solution. The 1956 Zimm model is used to describe the linear viscoelasticity of dilute solutions and of semidilute solutions inside their correlation volumes. The 1953 Rouse model is used for linear viscoelasticity of semidilute unentangled solutions and for entangled solutions on the scale of the entanglement strand. The 1971 de Gennes reptation model is used to describe linear viscoelastic response of entangled solutions. In each type of solution, the terminal dynamics, reflected in the terminal modulus, chain relaxation time, specific viscosity, and diffusion coefficient are reviewed with experiment and theory compared. Overall, the agreement between theory and experiment is remarkable, with a few unsettled issues remaining.     

Incorporation of polymer diffusivity and migration into constitutive equations

Given a general one-particle constitutive equation for the stress tensor, we discuss how to incorporate the additional effects of polymer diffusivity and migration into that constitutive equation within the framework of continuum mechanics. For the example of an upper-convected Maxwell model representing the polymer contribution to the stress tensor of a dilute polymer solution, we describe i) how to modify the constitutive equation for the stress tensor to include diffusion and migration effects, ii) how to formulate a balance equation for the polymer mass density in order to describe the nonhomogeneous composition of the polymer solution resulting from migration, and iii) how to close the extended set of coupled equations by means of further constitutive equations for the migration velocity and the diffusion tensor. In order to guarantee the material objectivity for all equations, we formulate them in the body tensor formulation of continuum mechanics (and then translate them into Cartesian space). The proposed equations are compared to results of a recent kinetic-theory approach.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.     

Constitutive equation for dilute polymer solutions in strong elongational flow

A constitutive equation is proposed, which is constructed using both phenomenological and structural ideas. In this formulation, the kinematics of the fluid is characterized by the deformation rate and a structural vector. The vector follows an evolutionary law which is inspired by known molecular models. The expression for the stress is given by introducing a dissipative term related to the strong hydrodynamic interaction of the distorted molecules and a deformation term for the molecules, as well as by using the second law of thermodynamics.A study of the general properties of the evolutionary equation and its response in a homogeneous two-dimensional flow provides evidence of the performance of the proposed model.     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

---

Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

---

a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

Extensional flow of polymer solutions through porous media

The flow behaviour of various polymer solutions of non-hydrolyzed polyacrylamide, hydrolyzed polyacrylamide, polyox and Xanthan was investigated in a plexiglass column having a succession of enlargements and constrictions, and compared with the flow behaviour and mechanical degradation of a solution of non-hydrolyzed polyacrylamide in a packed column of non-consolidated sand. The flow behaviour of this solution was found to be very similar in both the sand pack and plexiglass pore.Apart from the Xanthan solution, all other polymer solutions showed a viscoelastic behaviour in the plexiglass pore. The onset of viscoelastic behaviour, which has previously been defined using the shear rate ( ), stretch rate ( _s ) and Ellis number (E ₁), could be more precisely evaluated using a modified stretch rate (S _G). The pressure losses across the plexiglass pore for different polymer solutions of the same type were found to follow a unique curve provided the suggested group (S _G) was used, a situation which was not achieved with the other rheological parameters.The multipass mechanical degradation of the non-hydrolized polyacrylamide was tested through the sand pack against the suggested group (S _G) and Maerker's group (M _a). It was found that the loss of the solution viscoelasticity due to multipass mechanical degradation was better represented usingS _G thanM _a. A cross-sectional area (cm²) - C ^* critical concentration of polymer (ppm) - d plexiglass pore enlargement diameter - D average sand grain diameter (cm) - e equivalent width for the plexiglass pore - E ₁ Ellis number (a Deborah number) - F _R resistance factor - F _Ri resistance factor at the first pass - h height of the flow path of the plexiglass pore - K power-law constant - K _h,K _w effective permeability to hydrocarbon and water, respectively (10^–8 cm²) - M _a Maerker's group for a given porosity (s^–1) - M _ai value ofM _a at the first pass - N _D Deborah number - n power-law index - Q flow rate (cm³/s) - R capillary radius (cm) - R _g radius of gyration - S _G suggested group of rheological parameters representing a modified maximum stretch rate (s^–1) - S _Gi value ofS _G at the first pass - T _R,t characteristic time for the fluid (s) - t _s residence time (s) - V ₀ superficial velocity (cm/s) - V mean velocity of flow through a porous medium (cm/s) - average axial velocity in the enlargement section of the plexiglass pore (cm/s) - V ₁,V ₂ maximum velocity at a plexiglass enlargement neck and centre - [] intrincis viscosity - viscosity (mPa s) - _r relative viscosity (ratio of the viscosity of the polymer solution to that of the solvent) - shear rate (s^–1) - _s stretch rate (s^–1) - characteristic time for the polymer solution (s)     

Anomalous flow at the inlet of pitot tubes in dilute polymer solutions part I

Summary Experimental results about pressure defect of Pitot tubes placed into a laminar jet of dilute polymer solutions are presented.A fast response measuring system of pressure and visualization techniques made evident that the anomalous Pitot tube response is associated with an anomalous flow pattern, in the vicinity of the inlet of the tube and inside it.The flow of effective macromolecular solutions penetrates into the Pitot tubes to a deeper extent than newtonian fluids do; entering the probe, the stremlines break up into vortices, while the pressure signal strongly fluctuates around its mean value. When the recovery of pressure becomes full, these effects disappear.

---

Sommario Vengono presentati alcuni risultati sperimentali sul mancato recupero di pressione di tubi di Pitot immersi in getti di soluzioni acquose diluite di macromolecole.L'uso di un sistema rapido di misura della pressione di arresto ed il ricorso a tecniche di visualizzazione hanno reso evidente che il mancato recupero è associato ad un carattere anomalo del flusso all'ingresso del tubo e nel suo interno.Il flusso di quelle soluzioni macromolecolari che risultano efficaci agli effetti del fenomeno in studio, penetra all'interno del tubo di Pitot più profondamente di quanto non facciano i fluidi newtoniani: entrando nel tubo le linee di corrente si rompono in vortici ed il segnale di pressione fluttua energicamente attorno al suo valore medio.Entrambi questi aspetti svaniscono quando il recupero di pressione torna al suo valore teorico di arresto.