Measurement of relaxation times in extensional flow of weakly viscoelastic polymer solutions

The characterization of the extensional rheology of polymeric solutions is important in several applications and industrial processes. Filament stretching and capillary breakup rheometers have been developed to characterize the extensional properties of polymeric solutions, mostly for high-viscosity fluids. However, for low concentration polymer solutions, the measurements are difficult using available devices, in terms of the minimum viscosity and relaxation times that can be measured accurately. In addition, when the slow retraction method is used, solvent evaporation can affect the measurements for volatile solvents. In this work, a new setup was tested for filament breakup experiments using the slow retraction method, high-speed imaging techniques, and an immiscible oil bath to reduce solvent evaporation and facilitate particle tracking in the thinning filament. Extensional relaxation times above around 100 μs were measured with the device for dilute and semi-dilute polymer solutions. Particle tracking velocimetry was also used to measure the velocity in the filament and the corresponding elongation rate, and to compare with the values obtained from the measured exponential decay of the filament diameter.     

Simple linear viscoelastic models of dilute solutions of polymer molecules and emulsion droplets

The complex viscosity of microemulsions shows relaxation processes of which the largest relaxation time is about 10^–5 s or less. This time can be attributed to relaxation of stresses in the surface of emulsion droplets pertaining to interfacial tension. Superimposed on a spherical droplet surface shape fluctuations can occur due to thermal energies. Our aim is to show the influence of thermal shape fluctuations on the complex viscosity of emulsions. The method used in the derivation has also been applied to inflexible rods to demonstrate its feasibility by showing the formal rheological equivalence of in length thermally fluctuating rods and Rouse's simple model of polymers. The emulsion results have been applied to a dilution series of a non-ionic microemulsion.     

Dynamics of dilute polymer solutions

Taking volume effects, hydrodynamic no-flow conditions, and internal viscosity into account in the molecular dynamics has made it possible to formulate an equation of flow for a dilute polymer solution which in the region of comparatively slowly varying motions is described by the available experimental facts [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 138–146, January–February, 1988.     

Dissipative stresses in dilute polymer solutions

Planar extensional flows of a dilute polymer solution are investigated using a free-draining bead-rod model. For steady flows, an analytic expression for the probability density of the polymer configuration is available. It is found that part of the associated steady polymer stress is unambiguously viscous at all time scales, in the sense that on cessation of flow it disappears instantaneously, but, except at very high flow rates, the elastic component is larger.A Brownian dynamics simulation of the chain is constructed for start-up flows for which no analytic expression is known. A stress that is apparently viscous is found to develop alongside the elastic stress, having comparable magnitude at moderate flow rates. An interpretation of this result for a system having a wide spectrum of relaxation times is given. This feature is not captured by conventional FENE constitutive equations, and a novel model is developed. The consequences for calculations of complex flows are briefly discussed.     

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.     

Thermal diffusion of dilute polymer solutions

In this paper diffusion of a dilute solution of elastic dumbbell model macromolecules under non-isothermal conditions is studied. Using the center of mass definition for the local polymer concentration, the diffusive flux contains a thermal diffusion dyadic d _T .  To get some idea of thermal diffusion d _T is evaluated for steady state isothermal conditions. Explicit results are presented for some homogeneous flows. It is shown that if the polymeric number density is defined via the beads (of the dumbbell) – termed n _b – then the diffusive flux j contains , where τ ^c is the intramolecular contribution to the bulk stress. Though the form of the diffusion equation for n _b thus differs from the corresponding one for n, it is shown that for essentially unbounded systems differences between n and n _b are small. Since the results involve the translational diffusion coefficient they can readily be taken over for Rouse coils. Received: 23 September 1997 Accepted: 5 June 1998     

Normal stress effect in dilute polymer solutions

Rheologica Acta - The non-linear, steady flow behavior of dilute solutions of poly ethylene oxide at high shear rates is experimentally studied. The shear stress-shear rate relations at low and...     

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.     

Converging radial flow of dilute polymer solutions

A characteristic time of dilute polymer solutions is determined from a converging radial flow experiment. The influence of the intradisk separation and of the polymer concentration on this time is studied. Present results are compared to results obtained with a diaphragm. The main limitations of the experiments are pointed out.     

Universal viscometric functions for dilute polymer solutions

The Gaussian approximation has been shown previously to be excellent for the treatment of hydrodynamic effects in dilute polymer solutions. However, the computational time required to find the viscometric functions in simple shear flow is prohibitively long for chains with a large number of beads. Here we introduce a new approximation which retains the accuracy of the Gaussian approximation but is significantly less computationally intensive. Thus the rheological behavior of long chains may be explored. Extrapolation of results obtained numerically for long chains to the infinite chain length limit is shown to lead to predictions independent of model parameters. As a result, within the context of the approximation introduced here, universal viscometric functions for dilute polymer solutions in simple shear flow under theta conditions are obtained.     

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     

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)     

The relaxation of concentrated polymer solutions

The focus of this paper is on the viscoelastic properties of concentrated polymer solutions and polymer melts. Dynamic mechanical measurements were performed on various polystyrene/ethylbenzene solutions with polymer concentrations ranging from 40% up to 100% and temperatures from Tg+30°C up to 70°C (230°C for polymer melts). The basis polymers are two commerical grade polystyrenes (BASF) with M _W = 247 kg/mol and 374 kg/mol, respectively. To avoid solvent loss due to evaporating during the measurements, a special sealing technique was used.A phenomenological model which describes quantitatively the relaxation spectrum of concentrated polymer solutions from the flow regime up to the glass transition regime is developed. The relaxation data of the respective polymer melt and the glass transition temperature of the solution are the only input parameters needed. The temperature dependence is described by a universal, concentration invariant WLF-equation. The relaxation spectra are divided into two parts accounting for the entanglement and the segmental relaxation modes, respectively. The relaxation strength related to the flow and entanglement regime scale with c ^2.3, whereas the segmental relaxation strength does not alter with concentration. All relaxation times change with concentration proportional to c ^3.5. Flow curves can be calculated from these relaxation spectra and thus, our results are useful for engineering applications.Roman Symbols a _T Time temperature superposition shift - factor - a _c Time concentration superposition - shift factor in the flow regime - a _c Time concentration superposition - shift factor in the glassy regime - b _T Modulus temperature superposition - shift factor - b _c Modulus concentration shift factor - in the flow regime - b _c Modulus concentration shift factor - in the glassy regime - B Virial coefficients - c Polymer mass fraction kg/kg - c ₁ WLF-parameter - c2 WLF-parameter K - g Relaxation strength of a relaxation Pa mode - G(t) Relaxation modulus Pa - G Storage modulus Pa - G Loss modulus Pa - GN Plateau modulus of linear flexible Pa polymers - (x) Delta function: (0) = 1, - (x<>0)=0 - h() Damping function - H() Relaxation spectrum Pa - J ₀ ^N Recoverable compliance Pa^–1 - m Mass kg - M _c Critical molecular weight kg/mol - M _e Entanglement molecular weight kg/mol - M _w Weight average molecular weight kg/mol - M Number of datapoints - n Scaling exponent - N Number of discrete relaxation modes - T Temperature °C - T _g Glass transition temperature °C - V Volume 1 Greek Symbols Scaling exponent - _f Thermal expansion coefficient K^–1 - Scaling exponent - Shear deformation - Shear rate s^t–1 - Relaxation time s - _c Characteristic relaxation time of thes Cross model - _e Entanglement relaxation time s - Viscosity Pa s - ₀ Zero shear viscosity Pa s - ₀ First normal stress coefficientPa s2 - Segmental friction coefficient - Frequency rad/s Indices f Flow and entanglement regime - g Glass transition regime - i Count parameter - p Polymer - ref Reference state - s Solvent 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     

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.     

Torsional flow stability of highly dilute polymer solutions

We examine the torsional flow stability, to axisymmetric disturbances, of a variety of multimode and non-linear constitutive models in a bounded parallel plate geometry. The analysis is facilitated by the construction of a regular perturbation scheme in the ratio of polymer to total viscosity. As a model for Boger fluids this corresponds to the assumption that the Boger fluid is highly dilute. The consequent simplification provided by the perturbation scheme allows us to examine the effects of a discrete spectrum of relaxation times, shear thinning, second normal stress difference, and finite extensional viscosity on the torsional instability.