Behavior Of Dilute Polymer Solutions Research Articles

Measuring the viscoelastic behavior of dilute polymer solutions using high-speed statistical particle microrheology

The viscoelastic behavior of polymer solutions is commonly measured using oscillating shear rheometry, however, the accuracy of such methods is limited by the oscillating frequency of the equipment and since the relaxation time of the dilute polymer solutions is short, this requires measurement at very high frequencies. Microrheology has been proposed to overcome this technical challenge. Yet the equipment for resolving the statistics of particle displacements in microrheology is expensive. In this work, we measured the viscoelastic behavior of Methocel solutions at various concentrations using a conventional epi-fluorescence microscope coupled to a high-speed intensified camera. Statistical Particle Tracking is used in analyzing the mean-squared displacement of the dispersive particles. Relaxation times ranging from 0.76 – 9.00 ms and viscoelastic moduli, G′ between 11.34 and 3.39 are reported for Methocel solutions of concentrations between 0.063 – 0.5%.

The Oldroyd-B fluid in elastic instabilities, turbulence and particle suspensions

The Oldroyd-B fluid has become the starting point for almost all complex flow calculations and analysis involving the behavior of dilute polymer solutions. The reasons for this are clear: based on the kinetic theory of high molecular weight “phantom” chains near equilibrium, the resulting model, in its simplest form, produces a single mode description of the solution constitutive equation in terms of a single symmetric dyad order parameter. The physical connection to the individual chains is therefore unambiguous through the end-to-end configuration tensor. The flow of the fluid resulting from the mathematical solution of the constitutive equation coupled to Cauchy’s equation of motion and continuity has been examined for decades and its strengths and weaknesses in predicting the features of non-linear viscometric and non-viscometric polymer solution flows are well known. Moreover, “simple” extensions (e.g. the FENE-P model) can ameliorate the most serious of these shortcomings. For example, the multi-mode Oldroyd-B model can provide quantitative predictions of the linear viscoelasticity of a wide range of polymer solutions. Remarkably, even with all its shortcomings, the Oldroyd-B fluid has been invaluable in predicting at least qualitatively, features as complex as purely elastic instabilities, turbulent drag modification, and even the effective rheology of particles suspended in polymer solutions. While remaining mindful of the model’s shortcomings, any student of these diverse fields may find the Oldroyd-B fluid an important starting point.

Extrudate swell of Boger fluids

Boger fluids are dilute polymer solutions exhibiting high elasticity at low apparent shear rates, which leads to high extrudate swell. Numerical simulations have been undertaken for the flow of three Boger fluids (including benchmark Fluid M1), obeying an integral constitutive equation of the K-BKZ type, capable of describing the behavior of dilute polymer solutions. Their rheology is well captured by the integral model. The flow simulations are performed for planar and axisymmetric geometries without or with gravity. The results provide the extrudate swell and the excess pressure losses (exit correction), as well as the shape and extent of the free surface. All these quantities increase rapidly and monotonically with increasing elasticity level measured by the stress ratio, S R . It was found that the main reason for the high extrudate swelling is high normal stresses exhibited in shear flow (namely, the first normal-stress difference, N 1). Surprisingly, the elongational parameter of the model or a second normal-stress difference N 2 do not affect the results appreciably. Gravity serves to lower the swelling considerably, and makes the simulations easier and in overall agreement with previous experiments.

Slot coating of mildly viscoelastic liquids

The region of acceptable quality in the space of operating parameters of a coating process is usually bounded by various coating defects. An important limit of the slot coating process is the low-flow limit. It is the maximum web speed at a given film thickness and distance between coating die and web (coating gap); equivalently, the minimum film thickness at a given web speed and coating gap; again equivalently, the maximum gap at a given film thickness and web speed at which the coating bead remains stable. The condition that defines this limit is a force balance at the downstream meniscus of the coating bead, where the coated layer is carried away by the web translating past the die. Although most of the liquids coated industrially are polymeric solutions, dispersions, or both, that are not Newtonian, most previous investigations of the low-flow limit in slot coating dealt with Newtonian liquids. Recently, the effect of high molecular weight polymer on the low-flow limit of slot coating was examined by visualization experiments and theoretical analysis using an algebraic non-Newtonian model that takes into account the extensional thickening behavior of polymer solutions. Although Generalized Newtonian Models of this class may capture the different ways that dilute polymer molecules behave in extension- and shear-dominated flow zones and the reported predictions display the same trends that are observed experimentally, algebraic models cannot represent viscoelastic stresses. In the present work, the low-flow limit of slot coating of mildly viscoelastic liquids is examined by solving the conservation equations for two-dimensional flow, coupled with either of two differential viscoelastic models that describe the mechanical behavior of dilute polymer solutions (namely the Oldroyd-B and FENE-CR models). The results show how the different rheological properties affect the flow and the critical conditions at the onset of the low-flow limit.

Experimental Investigations on the Influence of the Concentration of Polyacrylamide Solutions on Elongational Flow Behavior

Information about the elongational flow behavior of dilute polymer solutions was obtained, which allows conclusions to be formulated about the extension of polymers and the elastic characteristics of such systems. It is shown that dilute polyacrylamide solutions, which show Newtonian behavior concerning the shear viscosity, can show elastic properties in elongational flow experiments.

Non-Newtonian Viscosity of Dilute Polymer Solutions

The non-Newtonian behavior of dilute polymer solutions is investigated by computer simulation of a bead-and-spring model, using the Brownian dynamics technique to evaluate the shear rate dependence of the intrinsic viscosity. This behavior is a consequence of the combined effects of hydrodynamic interaction, excluded volume, and finite extensibility. The simulations allow the study of the influence of each effect separately. When all the effects are considered, the simulation results can be compared to experimental data of solution viscosities, which are available just in the region of moderately small shear rates. Data from molecular architecture and other solution properties are obtained to parametrize the bead-and-spring model, with emphasis in the description of the chain extensibility. Comparison of simulation and experimental results is presented for both highly flexible vinyl polymers and locally stiff cellulose chains.

Depolarization behaviour of dilute polymer solutions: II. Angular dependence and analyzer absorption

AbstractStudy of the depolarization behaviour of dilute polymer solutions leads to determination of the Cabannes factor, necessary for correcting the Rayleigh ratio used in light‐scattering calculations for deviations due to anisotropy. We present here a method to determine the Cabannes factor for polymers as a simultaneous function of both scattering angle and molar mass, accomplished by coupling size‐exclusion chromatography with depolarization multi‐angle light scattering (SEC/D‐MALS). The depolarization behaviour of brominated polystyrene (PSBr), previously studied at a right‐angle geometry, is seen to possess a non‐trivial angular dependence in addition to being a function of molar mass. We also demonstrate initial attempts at correcting SEC/D‐MALS results for absorption of light by the Polaroid filters that act as analyzers in the optical train of the system. Copyright © 2003 Society of Chemical Industry

Water‐soluble polymers. LXXXII. Shear degradation effects on drag reduction behavior of dilute polymer solutions

AbstractDrag reduction measurements were conducted on extensively characterized poly(ethylene oxide) and poly(acrylamide) utilizing a fully automated rotating disk rheometer equipped with an optical tachometer, torque transducer, and software allowing real‐time data acquisition. The instrument sensitivity allowed the study of concentrations as low as 0.1 ppm. In addition, previously immeasurable concentration‐ and time‐dependent shear degradation effects were readily observed. A power law equation was shown to adequately correlate the percentage of drag reduction and the volume fraction for each polymer. Furthermore, an empirical shift factor was utilized to superimpose data from all the systems that were studied. By conducting measurements in the proper concentration and time domains, it was possible to extract the minimal concentration for the maximum drag reduction efficiency in the absence of shear degradation. The resulting values were significantly higher than those previously reported by our laboratories for poly(ethylene oxide) and poly(acrylamide). © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 82: 1211–1221, 2001

Deficiencies of FENE dumbbell models in describing the rapid stretching of dilute polymer solutions

A wide variety of bead-spring kinetic theory models have been proposed to explain the stress growth and hysteretic behavior of dilute polymer solutions in uniaxial extension. We analyze the Kramers chain, a fine-scale model for polymer dynamics, in order to assess the validity of the coarser-grained bead-spring models in these deformations. Whereas the spring force is a simple function of the dumbbell length for the FENE spring, we find that the relationship between the ensemble-averaged end-to-end force and the extension for a Kramers chain depends on the kinematic history to which it has been subjected. In a quiescent fluid, the Kramers chain force–extension relationship is identical to the FENE force law. However, during start up of elongational flow, the ensemble-averaged end-to-end force for a given (end-to-end) length of the molecule increases with strain until steady state is reached. If the extensional flow is suddenly stopped, the Kramers chain force–extension relationship relaxes back to the FEN...

On the hysteretic behaviour of dilute polymer solutions in relaxation following extensional flow

The hysteretic behaviour of dilute polymer solutions in relaxation following extensional flow is studied in the framework of three distinct theoretical models. For ideal kinematics of uniaxial elongation, we show that the kinetic theory of FENE dumbbells and its FENE-L approximation present a hysteresis when plotting polymer stress versus average molecular extension. Similar behaviour is obtained for ideal extensional kinematics using a FENE-P constitutive equation with a spectrum of finite extensibility parameters. Finally, a numerical simulation of the filament-stretching device shows that spatial inhomogeneities of the stress and average conformation fields also lead to hysteretic behaviour with a single-mode FENE-CR constitutive equation. In all three cases, hysteretic behaviour results from the combined effect of dispersity and non-linearity. We also address the validity of the stress-optic law for FENE dumbbells in relaxation following start-up of uniaxial extension. The simulation results show that the stress-optic coefficient remains constant at low strains only. Plots of stress-optic coefficient versus birefringence show hysteresis as well. This rules out a modified stress-optic law for FENE dumbbells wherein the stress-optic coefficient would be a function of the second moment of the configuration distribution function alone. Finally, it is shown in the Appendix that a proper selection of the spectrum of finite extensibilities can be made so that the multi-mode FENE-P model gives essentially the same stress response as the kinetic theory of FENE dumbbells in transient uniaxial extension.

Finitely extensible bead spring chains in time-dependent shear flows

The configurational and rheological properties of bead spring chains in time-dependent shear flows are calculated. The finite extensibility is incorporated through the constraint of constant contour length of the chain. Start-up of shear flow yields a stress overshoot, whereas oscillatory shear flow yields the same frequency dependence of the dynamic moduli as the simple bead spring model. The results show that finite extensibility can lead to non-linear rheological behavior of dilute polymer solutions. The influence of preaveraged hydrodynamic interaction on the obtained results is discussed.

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.

Brownian dynamics of rigid polymer chains with hydrodynamic interactions.

With the idea of a broad investigation of the flow behavior of dilute polymer solutions in mind, the dynamics of polymer chains with rigid constraints and hydrodynamic interaction is formulated in various equivalent ways. Starting from a very general diffusion equation of polymer kinetic theory, equivalent stochastic differential equations of motion both in terms of generalized coordinates and in terms of constraint conditions are derived. Then an efficient Brownian dynamics simulation algorithm is constructed rigorously, and a convenient expression for evaluating stresses in simulations is suggested. Furthermore, a modified simulation algorithm, which is appropriate for infinitely stiff rather than rigid systems, is discussed.

Importance of Chemical Mismatch in Tricritical Polymer Solutions

Using a renormalization group analysis for the tricritical behavior of dilute polymer solutions with many chemically different polymers, we show that the chemical mismatch is indeed important. This is contrary to the irrelevance of chemical mismatch in good solutions. We find the second virial coefficients at the tricritical point to vanish with chain length L as L -1/2 [ln(L/a)] -9/11 , where a is the small cut-off. behavior is in a different universality class from the one for the tricritical solutions of polymers of the same chemical type. The generalization to block copolymers is also addressed

Tricritical theory of dilute polymer solutions with quenched disorder

We have performed a renormalization group analysis of the tricritical behavior of dilute polymer solutions containing quenched impurities in three dimensions. We have treated the quenched disorder using the replica formalism. The logarithmic corrections to the mean field results for the quenched problem are found to be in a different universality class from those for the pure or annealed problem. For example, the leading corrections to the second virial coefficient and the specific heat are proportional to L1/2(ln L/a0)−2/11 and L(ln L/a0)7/11, respectively, where L is the chain length and a0 is the cutoff; the coexistence curve is found to follow the universal equation u∼−𝒞(‖ln 𝒞2a0‖)−9/11 when 𝒞2a0≪1, where 𝒞 is the monomer concentration and u is the two-body excluded volume interaction parameter.

Shear‐thickening behavior of dilute polymer solutions: Kinetic interpretation

An equation of the kinetic of the formation of reversible associations of macromolecules in flow is proposed. The results are compared with experimental data which were derived from viscosity measurements displaying a shear‐thickening behavior of dilute polymer solutions.

The interaction of polymer additives with vorticity (a case study on vortex rings)

Laminar vortex rings were created in a Newtonian fluid and in dilute and semi-dilute polymer solutions, then impinged normally onto a plane, smooth wall. The interaction of the vorticity of these rings with the boundary layer on the wall is studied. The results show the importance of a combined shear and elongation on the elongational behaviour of dilute polymer solutions. They indicate that a combination of shear and elongation can enhance the reflection of vortices at the wall in turbulent, drag-reduced flows.

Scaling behavior of dilute polymer solutions confined between parallel plates

The average size and shape of a polymer coil confined in a slit between two parallel plates depends on the distance L between the plates. On the basis of numerical results, four different regimes can be distingubhed. For large values of L the coil is essentially unconfined. For intermediate values the coil orientates without being squeezed considerably. For still smaller values the coil is squeezed with a corresponding decrease in the value of the radius of gyration. A further reduction in slit width is accompanied by a sharp increase in the value of the radius of gyration, signaling the onset of the transition from three- to two-dimensional behavior. L-Dependent scaling relations are supposed to be valid in the latter regime. Using Monte Carlo results for self-avoiding walks on a cubic lattice, scaling predictions for the parallel component of the radius of gyration and the end-to-end point distance as well as for the osmotic pressure and the number of segments at the wall are presented and verified.

Rheological measurements and light-scattering experiments of dilute solutions of high molecular polystyrene. Light-scattering of flowing polymer solutions

We describe the behavior of dilute polymer solutions by means of light-scattering under shear flow. Solution properties of polystyrene in benzene over a wide range of molecular weight has been studied to determine the coefficientsa andK of the Mark-Houwink relationship and to estimate the rheological conditions with regard to light-scattering experiments of flowing polymer solutions. The investigations were carried out to measure the shear-rate dependence of macromolecules in solution, e.g., to observe an orientation and changing of the mean-square radius of gyration.

A finitely extensible bead-spring chain model for dilute polymer solutions

The FENE-P dumbbell, that is, the finitely extensible non-linear elastic dumbbell with the Peterlin approximation, has been used extensively to describe the rheological behavior of dilute polymer solutions. The model is, however, severely limited, since it cannot describe the broad distribution of relaxation times that real polymer molecules possess. Hence, it is important to extend the model to a chain of finitely extensible springs. The FENE-P chain, however, has not been as widely used, presumably because of the large number of coupled equations that must be solved simultaneously in order to calculate the stress tensor. We present here a new model, the FENE-PM chain, as an alternative to the FENE-P chain. The stress tensor for the FENE-PM chain is calculated from a much smaller set of equations than the FENE-P and at the same time yields nearly the same results for the material properties in shear and elongational flow as the FENE-P. The reduced number of equations greatly expedites calculations for longer chains.