High-bandwidth one-and two-particle microrheology in solutions of wormlike micelles

We have developed a large-bandwidth two-particle microrheology technique to measure loss and storage moduli of viscoelastic materials from 0.1 Hz to about 100 kHz using laser trapping and interferometry. We found that quantitative agreement between one- and two-particle microrheology exist in entangled solutions of wormlike micelles chosen as a simple model viscoelastic system. These results validate both experimental method and data interpretation. The consistent results also prove that in a simple system, where the solution length scales are much smaller than the micron probe size, one-particle microrheology can accurately measure bulk viscoelastic parameters.This paper was presented at the AERC 2005     

Flow of dilute polyacrylamide solutions through a sudden planar contraction

The flow of partially hydrolyzed polyacrylamide solutions through a 10:1.2 sudden planar contraction was investigated by means of laser-Doppler anemometry. The resultant velocity profiles are compared with those for Newtonian water flow. It is shown that velocity profiles of dilute high molecular weight HPAM solutions of concentrations of 25 ppm and 50 ppm exhibit a velocity maximum upstream of the sudden planar contraction. They first appear near the wall and move towards the channel axis as the flow approaches the contraction. Furthermore, it is found that the centreline velocity profiles of the polymer solution show an earlier response to the downstream flow restriction than water. This is associated with enhanced recirculation regions in front of the sudden channel contraction.Streamlines calculated from the experimentally obtained velocity data reveal all the characteristics of a diverging flow field upstream of the contraction. The experiments reveal that, for the volume flow rate investigated, the flow of dilute polymers remains two-dimensional in the centre throughout the major part of the channel.The addition of small amounts of the divalent salt CaCl₂ reduces the polymer effects to pure Newtonian flow behaviour.     

Flows of dilute polymer solutions through packed porous chromatographic columns

Summary We have found that diffusion of polystyrene into the micropores of packed porous chromatographic columns of a type used for size separations of macromolecules is strongly influenced by the flow rate of the polymer solution through the column and also by the macromolecular size and internal pore size of the column packing. Dilute polymer solutions (0.05 wt.% polystyrene in THF) were pumped continously through the columns (Dupont SEC). Step changes in flow rate were made and the polymer concentration in the column effluent was monitored continuously. When the flow rate was increased, the partition coefficient for polymer between mobile and stationary phase shifted toward more polymer retention in the column, as evidenced by the drop in outlet concentration for several minutes following the step change. Molecular weight, pore size and solvent effects have all been thoroughly investigated. The conclusion is that polymer molecules diffuse into the internal pores of the column packing at a flow ratedependent rate and moreover, they do so to a greater extent the larger the polymer molecule, provided they are not sterically excluded from the pores. This dynamic diffusional partitioning is clearly important in its effect on resolution characteristics of an SEC column and also sheds some light on qualitatively similar observations in polymer flooding for enhanced oil recovery.

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Zusammenfassung Es wurde gefunden, daß die Diffusion von Polystyrol in die Mikroporen von porösen Packungen chromatographischer Säulen, wie sie für die Trennung von Makromolekeln verschiedener Teilchengrößen Verwendung finden, sowohl von der Fließgeschwindigkeit der Polymerlösung durch die Säule als auch von der Größe der Makromolekeln und der inneren Poren erheblich beeinflußt werden. Verdünnte Polymerlösungen (0,05 Gew.-% Polystyrol in THF) wurden kontinuierlich durch die Säulen (Dupont SEC) gepumpt. Dabei wurde die Fließgeschwindigkeit sprunghaft geändert und die Polymerkonzentration im Auslauf kontinuierlich registriert. Wenn die Durch-flußgeschwindigkeit erhöht wurde, verschob sich der Verteilungskoeffizient der Polymeren zwischen bewegter und stationärer Phase zu höheren Polymeranteilen in der Säule hin, was durch einen Konzentrationsabfall in der ausfließenden Lösung während einiger Minuten nach dem Geschwindigkeitssprung angezeigt wurde. Molekulargewicht, Porenweite und Lösungsmitteleinfluß wurden eingehend untersucht. Dabei ergab sich, daß die Polymermolekeln in die inneren Poren der Packung mit einer fließgeschwindigkeitsabhängigen Geschwindigkeit hineindiffundieren und dies in einem um so höheren Maße, je größer die Molekeln sind, vorausgesetzt, daß sie nicht aus sterischen Gründen zurückgehalten werden. Diese dynamische diffusive Verteilung ist natürlich von Bedeutung für die Trenncharakteristik einer GPC-Säule und wirft ebenso einiges Licht auf verschiedene qualitativ ähnliche Beobachtungen beim Fluten mit Polymeren zur Ausbeutesteigerung bei der Ölförderung.

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With 8 figures and 3 tables     

Measurement of dynamic properties of dilute polymer solutions with a Laser-Doppler velocimeter

The complex velocity field of an oscillating Couette flow is measured with a Laser-Doppler velocimeter. Different evaluation methods are used for the determination of material functions such as relaxation times and the dynamic complex viscosity.     

Viscosity enhancement In non-Newtonian flow of dilute polymer solutions through crystallographic porous media

We report results of the flow of dilute mono-disperse solutions of atactic poly(styrene) in di-octyl phthalate through regular crystallographic packed beds of spheres. Pressure drop measurements made as a function of flow rate across simple cubic and body centred cubic arrays of spheres have been used to estimate the specific viscosities of the polymer solutions as a function of the superficial Deborah number. Through both structures the onset Deborah number for the non-Newtonian increase in specific viscosity is found to be low when compared on the basis of well-characterized Zimm relaxation times. Surprisingly it is found that polymer solutions achieve a greater maximum specific viscosity in the simple cubic than in the body centred cubic array, a result contrary to prior expectations due to the absence of trailing stagnation points in the simple cubic structure. It is hypothesised that the trailing stagnation points in the body centred cubic array may be screened from the flow field by strands of oriented polymer and that, as such, the periodic variations in cross-sectional area of the flow (which are more severe in the simple cubic array) may play the most significant role in causing polymer extension and in enhancing the non-Newtonian viscosity.     

Some energetic properties of smooth solutions in rate-type viscoelasticity

A free energy function compatible with the second law of thermodynamics is constructed for the semilinear rate-type viscoelasticity. Under physically acceptable conditions this energy function is a positive and convex function of stress and strain.

It is shown that for a class of one dimensional initial-boundary value problems, the total energy at any time is bounded by the sum of the total energy of the initial data and the energy supplied to the body by nonvanishing body forces. A Maxwell type viscosity approach to non-linear elasticity for an isolated body is also discussed.

The stability in total energy and uniqueness of the smooth solutions of some initial-boundary value problems is discussed for the one dimensional case as well as for the three dimensional case of small deformations.     

Minimum principles in the linear dynamic theory of viscoelasticity

This paper establishes three minimum principles and transformed minimum principles concerning Laplace transformation in the linear dynamic theory of viscoelasticity.     

Viscosity enhancement in non-Newtonian flow of dilute aqueous polymer solutions through crystallographic and random porous media

We report results on the flow of dilute aqueous solutions of hydrolysed poly(acrylamide) (HPAA) through beds of spheres packed in simple cubic and body-centred cubic crystallographic arrays. Pressure drop measurements made across the arrays as a function of the flow rate have been used to estimate the specific viscosities of the HPAA solutions as a function of the superficial strain rate. It is found that greater non-Newtonian increases in the specific viscosity occur in the body-centred cubic array, which is thought to be due to the presence of trailing stagnation points, which are not present in the simple cubic array. Experiments have been performed using HPAA solutions in the presence of mono- and divalent cations at various concentrations and, for validity, have been compared with results obtained from a traditional randomly packed porous medium. In addition, a study of mechanical degradation of the polymer in flow through the crystallographic arrays has been carried out and reveals a greater rate of degradation in the body-centred array and also a significant increase in degradation with salt concentration.     

On the solutions in the linear theory of micropolar viscoelasticity

In the present paper the linear theory of micropolar viscoelasticity is considered. The explicit expression of fundamental solution of the system of equations of steady vibrations is constructed by means of elementary functions and its basic properties are established. The Green's formulas in the considered theory are obtained. The formulas of integral representations of Somigliana-type of regular vector and regular (classical) solution are presented. The representation formulas of Galerkin-type solution of the system of nonhomogeneous equations and of the general solution of the system of homogeneous equations by means of eight metaharmonic functions are presented. The completeness of these solutions is proved.     

Flow of dilute to semi-dilute polystyrene solutions through a benchmark 8:1 planar abrupt micro-contraction

We have studied the flow of thermodynamically ideal solutions of a high molecular weight (M_w = 6.9 MDa) atactic polystyrene in the θ solvent dioctyl phthalate (aPS in DOP) through a micro-fabricated 8:1 planar abrupt contraction geometry. The channel is much deeper than most micro-scale geometries, providing an aspect ratio of 16:1 and a good approximation to 2D flow in the narrow channel. The solutions span a range of concentration 0.03 wt.% < c < 0.6 wt.%, encompassing the dilute to semi-dilute regimes and providing a range of fluid viscosities and relaxation times such that we achieve a range of Weissenberg numbers (8.7 < Wi < 1562) and Reynolds numbers (0.07 < Re < 11.2), giving elasticity numbers between 31 < El < 295. We study the flow using a combination of micro-particle image velocimetry (μ-PIV) to characterize the flow field, pressure measurements to evaluate the non-Newtonian viscosity, and birefringence measurements to assess macromolecular strain. Flow field observations reveal three broad flow regimes characterized by Newtonian-like flow, unstable flow and vortex growth in the upstream salient corners. Transitions between the flow regimes scale with Wi, independent of El, indicating the dominance of elastic over inertial effects in all the fluids. Transitions in the flow field are also reflected by transitions in the relative viscosity (determined from the pressure drop) and the macromolecular strain (determined from birefringence measurements). The strain through the 8:1 contraction saturates at a value of ～4–5 at high Wi. The results of these experiments broaden the limited set of literature on flow through micro-fluidic planar contractions and should be of significant value for optimizing lab-on-a-chip design and for comparison with modeling studies with elasticity dominated fluids.     

Flow birefringence of dilute polymer solutions in two-dimensional flows

Summary Flow birefringence measurements have been obtained on three molecular weight samples (2–8 × 10⁶ M _W ,M _W /M _N = 1.14–1.3) of polystyrene in dilute solution (50–100 ppm) in a viscous polychorinated biphenyl solvent. The flows were generated using a four roll mill which could simulate a wide range of two dimensional flows in which the flow type (i.e. the ratio of the rate of rotation to the rate of strain) could be varied independently of the velocity gradient. The normalized birefringence, corrected for concentration, (n/nc), was observed to approach a saturation value at high velocity gradients in purely extensional flow. This saturation value was independent of both the molecular weight and the concentrationc, in agreement with theory. In addition, the magnitude of the saturation value is consistent with nearly fully extended chains and suggests extensions in the range of 20–50 times the rest state size. The data for the birefringence over a wide range of flows was found to be well correlated against the eigenvalue of the velocity gradient tensor in agreement with the results of the strong/weak flow theories ofTanner (1976) andOlbricht et al. (1980).The experiments are compared with a simple dumbbell model which incorporates the effects of a nonlinear spring variable hydrodynamic friction, and internal viscosity. It is shown that this simple model can simulate the experimental results surprisingly well if the effects of molecular weight distribution and finite transit times in the flow device are taken into account.

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Zusammenfassung Es werden Doppelbrechungsmessungen an verdünnten Lösungen (50–100 ppm) von drei Polystyrolproben mit verschiedenem Molgewicht (M _W = 2–8 10⁶,M _W /M _N = 1,14–1,3) in einem polychlorierten Diphenyl-Lösungsmittel (Pyralene 4000) durchgeführt. Die Strömung wird in einer Vier-Walzen-Apparatur erzeugt, die zweidimensionale Strömungen in einem weiten Typenbereich (d. h. mit verschiedenen Verhältnissen von Rotations- zu Deformationsgeschwindigkeit) unabhängig vom Geschwindigkeitsgradienten herzustellen gestattet. Die bezüglich der Konzentration korrigierte normierte Doppelbrechung (n/nc) strebt bei reiner Dehnströmung mit hohem Geschwindigkeitsgradienten gegen einen Sättigungswert. Dieser ist in Übereinstimmung mit der Theorie sowohl vom Molgewicht als auch von der Konzentrationc unabhängig. Weiterhin entspricht die Größe dieses Sättigungswerts der Annahme nahezu vollständig gestreckter Ketten mit Ausdehnungen vom 20–50fachen der Ausdehnung im Ruhezustand. Die Doppelbrechungswerte sind in einem weiten Strömungsbereich gut mit dem Eigenwert des Tensors des Geschwindigkeitsgradienten korreliert, was mit den Ergebnissen der Theorien starker und schwacher Strömungen vonTanner (1976), sowieOlbricht und Mitarbeitern (1980) übereinstimmt.Die Experimente werden mit den Voraussagen der Theorie des einfachen Hantelmodells verglichen, allerdings unter Einschluß eines nichtlinearen Federgesetzes, einer variablen hydrodynamischen Reibung und einer inneren Viskosität. Man findet, daß dieses einfache Modell die experimentellen Ergebnisse überraschend genau wiederzugeben vermag, wenn die Einflüsse der Molgewichtsverteilung sowie einer endlichen Einstellzeit in der Strömungsanordnung mitberücksichtigt werden.

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With 17 figures and 1 table     

Heat transfer of dilute viscoelastic solutions in helical exchangers

The study examined heat transfer of two dilute viscoelastic solutions in helical exchangers of circular cross-section. Ten helical coil heat exchangers with diameter ratios ranging from 4 to 50 were constructed. Results showed doubling the concentrations of polymer increased heat transfer performance by 12 %. The results were expressed in forms of some existing equations and were found to be in fair agreement to previous results.     

Optical and hydrodynamical regimes in dilute solutions of polyelectrolytes

A careful analysis of the equations governing dynamical disturbances in dilute solutions of polyelectrolytes (described on the basis of a conformational model with electromechanical interactions) allows one to place in evidence two extreme regimes, one called the optical regime, which justifies the use of optical methods for measuring the main instantaneous characteristics of the solutions, and the other that corresponds to electrohydrodynamical flows where a hierarchy can be exhibited for the various effects contributing to the flow equations.     

Elastic stresses in capillary jets of dilute polymer solutions

The elastic longitudinal stresses associated with the flow of jets of dilute polymer solutions from a short nozzle and their effect on the stability of the free jet are investigated theoretically and experimentally. The results obtained make it possible to take a fresh look at the ways in which a polymer additive affects the stability of high-velocity capillary jets.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–9, Marc–April, 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)     

Fiber spinning of very dilute solutions of polyacrylamide in water

The extensional viscometer developed earlier by the authors was refined and used to extend very dilute (50 ppm) solutions of polyacrylamide in distilled water. A slender liquid filament was stretched by the use of a suction device, and this resulted in the spinning of the fiber. By varying the volumetric flow rate and the filament length, stretch rates in the 100–1000 s⁻¹ range were easily obtained. The corresponding tensile stresses were very large, and these gave apparent extensional viscosities of the order of 200 P (20 Pa s). In contrast to this, the material functions in shear were difficult to measure, except for the shear viscosity which showed pronounced shear thinning. It was found that all the measurements, in shear as well as extension, could be explained based on the four constant Johnson-Segalman constitutive equation.     

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.     

On the mechanism of drag reduction in dilute polymer solutions

Experimental evidence is given that drag reducing polymer molecules are preferentially collected by strained vortices. This can explain why extremely small amounts of additives can be so effective. They become concentrated in areas of a turbulent flow where they are most efficient.     

Thickening behaviour of dilute polymer solutions in non-inertial elongational flows

A molecular interpretation is proposed to interpret the thickening behaviour of dilute solutions of high molecular weight flexible polymers in non-intertial flows having an elongational character. A set of new results has been gathered showing that the onset of very high end-pressure losses at high deformation rates in capillary flow can be explained by a flow-induced coil-stretch transition of macromolecules in solution. When a high degree of elongation is achieved a marked increase in viscous dissipation occurs in elongational flows.