A model of dilute polymer solutions with hydrodynamic interaction and finite extensibility. I. Basic equations and series expansions

A bead-spring chain model is used to describe dilute polymer solutions. Two important effects, the hydrodynamic interaction between the beads of a single chain and the finite extensibility of the springs connecting the beads, are treated in a self-consistently averaged form. For the resulting model, a rheological equation of state, a third-order retarded-motion expansion, and a second-order codeformational memory-integral expansion are derived. Furthermore, the parameters characterizing the model and the limit of infinitely long chains are discussed in great detail.     

A new ALS model for dilute polymer solutions in flows with strong shear components

Ghosh et al. (J Rheol 46:1057–1089, 2002) developed a new model for dilute polymer solutions in flows with strong extensional components. The model based on introducing an adaptive length scale (ALS) as an internal variable was developed to reproduce the fine-scale physics of the Kramers chain. The ALS model describes the polymer molecule as a set of identical segments in which each segment represents a fragment of the polymer that is short enough so that it can sample its entire configuration space on the time scale of an imposed deformation and therefore stretch reversibly. As the molecule unravels, the number of the segments decreases, but the maximum length of each segment increases, so that the constant maximum contour length of the molecule is preserved. Though the single-segment-based ALS model accounts for the orientability of the polymer molecules, it cannot describe the internal motions of the molecules due to the lack of internal nodes. Hence, in this work we consider the more realistic chain (multi-segments) model composed of N _seg springs connected linearly. The model presented in this work is an extension of the ALS model developed by Ghosh et al. (J Rheol 46:1057–1089, 2002). We demonstrate that the ALS varies with the flow strength. Specifically, it is found that as the flow strength increases, the ALS decreases. This implies that as the flow strength increases, the polymer molecule is required to divide into finer and finer segments such that each segment can locally equilibrate with the imposed flow. However, there is a critical number of such subdivisions beyond which further subdivision of the polymer molecule is not required to capture the polymer dynamics for a given flow strength. Both shear viscosity and first normal stress coefficient predictions from ALS model show shear thinning behavior with Weissenberg number. In weak flows, the ALS model and the finitely extensible non-linear elastic model exhibit the same behavior.     

The influence of finite extensibility on the eigenspectrum of dilute polymeric solutions

Linear stability of plane Couette flow of dilute polymeric solutions modeled using the FENE-P constitutive equation has been examined. Specifically, the spurious, discrete and continuous spectra are identified as a function of the ratio of the solvent to the total viscosity β, maximum chain length L, disturbance wavenumber α and the Weissenberg number (We). It is observed that reducing L shifts the entire spectrum to the left (in the eigenspectrum plane) by a factor directly related to the steady-state value of the Peterlin function f. Additionally, decreasing the value of L causes the splitting of the regular continuous spectrum, initially located at −1/We, into two branches. Overall, the plane Couette flow of a FENE-P model is found to be unconditionally stable. Specifically, decreasing L increases the decay rate of the most dangerous disturbance while increasing the number of discrete and spurious modes. However, increasing both β and α decreases the number of discrete and spurious modes whereas increasing We has no influence on the number of spurious modes but results in an increase in the number of discrete modes.     

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     

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.     

A possible alternative to the FENE dumbbell model of dilute polymer solutions

The dumbbell model of dilute polymer solutions is simple and successful, and its FENE version has progressively become a paradigm. In some transient extensional flows however, an increased dissipation was observed which could not be understood within the FENE model. This prompted us to look for a new dumbbell-like model in which the finite extensibility of the polymer is taken more thoroughly into account, i.e. not only through the Warner potential. The main lines of this alternative model are presented.     

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.     

Degradation effects of dilute polymer solutions on turbulent drag reduction in pipe flows

An important practical problem in the application and study of drag reduction by polymer additives is the degradation of the polymer, for instance due to intense shearing, especially in recirculatory flow systems. Such degradation leads to a marked loss of the drag-reducing capability of the polymer.Three different polymer types were tested on degradation effects in a closed pipe flow system. The polymers used were Polyox WSR-301, Separan AP-273 and Superfloc A-110, dissolved in water in concentrations of 20 wppm each. The flow system consisted of a 16.3 mm pipe of 4.25 m length. Two different pumps were used: a centrifugal pump and a disc pump. Different solution-preparation procedures were tried and the experiments were performed at different flow rates.Superfloc A-110 proved to be both the most effective drag reducer and most resistant to degradation. Because of very fast degradation, Polyox WSR-301 was found to be unsuitable for being used as a drag reducer in re-circulatory systems. The disc pump proved to be much better suited for pumping the polymer solutions than the centrifugal pump. The degradation curve of the combination Superfloc/disc pump showed a plateau-like region with reasonable drag reduction, which makes it possible to perform (laser Doppler) measurements under nearly constant circumstances during a sufficient time.     

The charged dumbbell model for dilute polyelectrolyte solutions in strong flows

A charged dumbbell model is used to investigate the behavior of dilute polyelectrolyte solutions in a general linear two-dimensional flow. The model studied has a nonlinear spring, conformation dependent friction and a Coulombic repulsive force due to an effective electrostatic charge on the two beads. The relative importance of the electrostatic charge is reflected by an effective charge density parameter,E. Equilibrium properties such as end-to-end distance and intrinsic viscosity are strongly dependent onE. In strong flows, which produce a dramatic increase in the dumbbell dimensions (a coil-stretch transition), the onset behavior is influenced byE. IncreasingE causes the onset velocity gradient to shift to much lower values. Large values ofE change the qualitative behavior to that of rigid (or slightly extensible) macromolecules or fibers. Results are presented for a charged dumbbell at equilibrium, in steady flows, and in transient flows.     

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     

A constitutive equation for a dilute solution of Hookean dumbbells with approximate hydrodynamic interaction

In a recent series of papers, Öttinger's consistently averaged hydrodynamic interaction has been shown to yield shear-rate dependent viscosity and normal stress coefficients in steady shear flow for dilute solutions of elastic dumbbells and chains. Even more recently, Fan has numerically solved the diffusion equation for the Hookean dumbbell with complete hydrodynamic interaction and he has compared his results with those of the Öttinger model.In this paper, a new approximation¹ for the Oseen—Burgers tensor is proposed where the configuration-dependent terms are replaced by appropriate averages rather than averaging the Oseen—Burgers tensor as a whole as in the Öttinger model. The proposed model leads to a differential constitutive equation which at low shear rates is similar to the Giesekus constitutive equation for a Hookean dumbbell with anistropic drag and anisotropic Brownian motion. The steady shear viscosity and normal stress coefficients for the proposed model are shear-rate dependent and are in close agreement with Fan's numerical calculations. Elongational viscosity for both positive and negative elongation rates are calculated.     

Scale-up criterion of turbulent flows of dilute polymer solutions and a generalized dependence of the friction coefficient

Dilute polymer solutions that reduce turbulent friction are treated as viscoelastic liquids for which, in addition to the Reynolds number, the scale-up criteria include the Deborah or elasticity number. Introduction of a generalized (viscoelastic) Reynolds number makes it possible to reduce the experimental curves for the drag coefficients obtained in pipes of various diameters at different and concentrations to a unified dependence. St. Petersburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 191–196, July–August, 1998.     

Kinetic theory and rheology of dilute polymer solutions

Summary A summary is given of some recent attempts to relate the results of the kinetic theory of rigid and flexible macromolecules to continuum mechanics results.With 1 table     

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.     

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.