Transient Extensional Flow Research Articles

Viscoelastic fluid flow in microporous media

This paper describes an experimental investigation of the flow of flexible polymer solutions through a distinctive micro-porous structure (typical pore sizes of 500 microns) with the primary focus on the interaction of the viscoelasticity of the working fluid with the micro-porous structure. In particular, we relate the bulk flow properties to measurable rheological parameters, demonstrating that a key parameter in estimating the pressure drop through the porous media (for a variety of polymer types, concentrations, solvents, molecular weights, and states of degradation) is the extensional relaxation time (or, more correctly, a characteristic time for extensional stress growth). A Weissenberg number (Wi) is calculated as a product of this extensional relaxation time and the nominal shear rate in the flow. Results, suitably normalised with Newtonian pressure-drop data, show a critical Wi of roughly 0.01 where all working fluids in two different pore structures reveal the onset of elastic dominance over viscous forces as the flowrate increases. Such a low critical value of Wi is due to the estimate of a nominal shear rate based on pore size, which severely underestimates the maximum shear rates within the complex pore structure. Significant deviation from the universal behaviour is observed for high concentrations of the polyacrylamide, which is thought to be a result of shear-thinning for these systems. Systematic degradation of the polyethylene oxide solutions caused an exponential decay in elasticity, which is reflected in the pressure-drop measurements. These imply that, although the flow through porous media is known to be a complex combination of both shear and extensional flow, the extensional effects are of primary importance in this context (or that transient stretching in the media is well represented by this relaxation time) of viscoelasticity. The findings highlight the significance of the transient extensional flow through the neck regions between the pores of the media, which are regions of contraction/expansion.

Orientation control and nonlinear trajectory tracking of colloidal particles using microfluidics

Suspensions of anisotropic Brownian particles are commonly encountered in a wide array of applications such as drug delivery and manufacturing of fiber-reinforced composites. Technological applications and fundamental studies of small anisotropic particles critically require precise control of particle orientation over defined trajectories and paths. In this work, we demonstrate robust control over the two-dimensional (2D) center-of-mass position and orientation of anisotropic Brownian particles using only fluid flow. We implement a path-following model predictive control scheme to manipulate colloidal particles over defined trajectories in position space, where the speed of movement along the path is a degree of freedom in the controller design. We further explore how the external flow field affects the orientation dynamics of anisotropic particles in steady and transient extensional flow using a combination of experiments and analytical modeling. Overall, this technique offers new avenues for fundamental studies of anisotropic colloidal particles using only fluid flow, without the need for external electric or optical fields.

A constitutive model for entangled polymers incorporating binary entanglement pair dynamics and a configuration dependent friction coefficient

Following recent work [e.g., J. Park et al., J. Rheol. 56, 1057–1082 (2012); T. Yaoita et al., Macromolecules 45, 2773–2782 (2012); and G. Ianniruberto et al., Macromolecules 45, 8058–8066 (2012)], we introduce the idea of a configuration dependent friction coefficient (CDFC) based on the relative orientation of Kuhn bonds of the test and surrounding matrix chains. We incorporate CDFC into the “toy” model of Mead et al. [Macromolecules 31, 7895–7914 (1998)] in a manner akin to Yaoita et al. [Nihon Reoroji Gakkaishi 42, 207–213 (2014)]. Additionally, we incorporate entanglement dynamics (ED) of discrete entanglement pairs into the new Mead–Banerjee–Park (MBP) model in a way similar to Ianniruberto and Marrucci [J. Rheol. 58, 89–102 (2014)]. The MBP model predicts a deformation dependent entanglement microstructure which is physically reflected in a reduced modulus that heals slowly following cessation of deformation. Incorporating ED into the model allows “shear modification” to be qualitatively captured. The MBP model is tested against experimental data in steady and transient extensional and shear flows. The MBP model captures the monotonic thinning of the extensional flow curve of entangled monodisperse polystyrene (PS) melts [A. Bach et al., Macromolecules 36, 5174–5179 (2003)] while simultaneously predicting the extension hardening found in PS semidilute solutions where CDFC is diluted out [P. K. Bhattacharjee et al., Macromolecules 35, 10131–10148 (2002)]. The simulation results also show that the rheological properties in nonlinear extensional flows of PS melts are sensitive to CDFC but not to convective constraint release (CCR) while those for shear flows are influenced more by CCR. The monodisperse MBP toy model is generalized to arbitrary polydispersity.

Wormlike micellar solutions: III. VCM model predictions in steady and transient shearing flows

The two species, scission/reforming Vasquez–Cook–McKinley (VCM) model was formulated to describe the coupling between the viscoelastic fluid rheology and the kinetics of wormlike micellar assembly and deformation-induced rupture. The model self-consistently captures the nonlocal effects of stress-induced diffusion and has been studied in various limits for a number of canonical flow fields including Large Amplitude Oscillatory Shear (LAOS), steady and transient extensional flow as well as steady pressure-driven channel flow. However, a complete study of the spatiotemporal model predictions in shearing flow, both with (and without) inertia, and with (or without) the stress-concentration diffusive coupling, has not yet been reported. In this paper we present a comprehensive investigation of the full VCM model in steady and transient shearing flow including inertial and diffusive (non-local) effects. The consequences of varying the model parameters, the effect of the start-up ramp rate, and the role of geometry on the steady state flow curve are each investigated. As a result of the onset of shear-banding and nonlocal effects in the velocity, stress and concentration profiles, we show that the measured rheological properties in a wormlike micellar solution described by the VCM model can depend on the initial ramp rate as well as specific details of the geometry such as the length scale of the rheometric fixture chosen and its curvature. The complete time evolution of the rheological response at high Deborah numbers is examined, from the initial formation of inertial waves through nonlinear overshoots in the viscoelastic stresses, shear band formation (and elastic recoil in the local velocity), to the long time diffusion-mediated approach to a final steady state.

Application of the MLD “toy” model to extensional flows of broadly polydisperse linear polymers: Part II. Comparison with experimental data

The new diluted stretch tube polydisperse MLD “toy” model developed in Part I is used to simulate experimental data for a “spiked” and “unspiked” polydisperse polystyrene melt with a broad molecular weight distribution in transient uniaxial, planar and bi-axial extensional flows [1–3]. The new model is in good agreement with experimental data sets in transient stretching flows whereas the original “naïve” polydisperse MLD “toy” model without incorporation of the new physics completely fails to capture salient features of stretching flows, such as strain hardening, of these broadly polydisperse systems. Quantitative analysis of the apparent distinctions in material response between different types of transient extensional flow is performed in terms of the new polydisperse MLD diluted stretch tube model. Results of this analysis quantitatively explain why uniaxial extension is the most strain hardening and bi-axial extension is the least strain hardening extensional flow thereby resolving this long standing and previously enigmatic issue.

The effect of pre-shear on the extensional rheology of wormlike micelle solutions

The effect of initial microstructural deformation, alignment, and morphology on the response of wormlike micelle solutions in transient uniaxial extensional flows is investigated using a pre-shear device attached to a filament stretching rheometer. In filament stretching experiments, increasing the strength and the duration of the pre-shear just before stretch is found to delay the onset of strain hardening. In these experiments, the wormlike micelle solution filaments fail through a rupture near the axial midplane. The value of the elastic tensile stress at rupture is found to decrease with increasing pre-shear rate and duration. The most dramatic effects are observed at shear rates for which shear banding has been independently observed. The reduction in the strain hardening suggests that pre-shear before filament stretching might break down the wormlike micelles reducing their size before stretch. Strain hardening is also observed in capillary breakup rheometry experiments; however, the pre-sheared wormlike micelle solutions strain harden faster, achieve larger steady-state extensional viscosities and an increase in the extensional relaxation time with increasing shear rate and duration. The difference between the response of the wormlike micelles in filament stretching and capillary breakup experiments demonstrates the sensitivity of these self-assembling micelle networks to pre-conditioning.

Brownian dynamics simulations of star-branched polymers in dilute solutions in extensional flow

Using Brownian dynamics (BD) simulations of FENE bead-spring models, the dynamics of star-branched polymers in dilute solutions under extensional flow have been investigated. Studies on star polymers in transient extensional flow reveal that the initial transient stress response at low strains is governed by both the number of arms and the shortest arm. On the other hand, the steady-state behavior of star polymers in elongational flow is limited by the maximum effective “contour” length of the molecules. The distribution of arm extension and birefringence of the star-branched molecule are broader and the mean is shifted to lower values, when compared to equivalent linear systems. As a result, the degree of arm extension at steady-state decreases as the number of arms in the star increases. Both an analysis of individual ensembles in Brownian dynamics simulations and a study of a simple force balance indicate that the constraint imposed on the star arms by the central branch point and contributions from “asymmetric” arm arrangements give rise to overall less extended and oriented star-branched molecules with broader arm extension and birefringence distributions. The results obtained from stress-conformation hysteresis simulation indicate that less-stretched arms exhibit more retarded relaxation, as the number of arms increases in star-branched molecules. The effect of excluded volume (EV) interactions, incorporated through the Lennard–Jones potential, on the dynamics of star polymers in extensional flow appears unimportant.

How dilute are dilute solutions in extensional flows?

We investigate the concentration dependence of the characteristic relaxation time of dilute polymer solutions in transient uniaxial elongational flow. A series of monodisperse polystyrene solutions of five different molecular weights (1.8×106⩽M⩽8.3×106g∕mol) with concentrations spanning five orders of magnitude were dissolved in two solvents of differing solvent quality (diethylphthalate and oligomeric styrene). Optical measurements with a capillary breakup extensional rheometer of the rate of filament thinning and the time to breakup in each fluid are used to determine the characteristic relaxation time. A criterion for a lower sensitivity limit is introduced, in the form of a minimum concentration cmin necessary for experimental resolution of the effects of polymeric viscoelasticity. This criterion is validated by experiment and comparison to numerical calculations with a multimode bead-spring dumbbell model. These calculations also rationalize previous paradoxical observations of extensional thinning in fluid threads of ultradilute polymer solutions in which stress relaxation apparently occurred faster than predicted by the Zimm theory. Above this minimum sensitivity limit we show that the effective relaxation time of moderately dilute solutions (0.01⩽c∕c*⩽1) in transient extensional flow rises substantially above the fitted value of the relaxation time extracted from small amplitude oscillatory shear flow and above the Zimm relaxation time computed from kinetic theory and intrinsic viscosity measurements. This effective relaxation time exhibits a power-law scaling with the reduced concentration (c∕c*) and the magnitude of the exponent varies with the thermodynamic quality of the solvent. The scaling of this “self-concentration” effect appears to be roughly consistent to that predicted when the dynamics of the partially elongated and overlapping polymer chains are described within the framework of blob theories for semi-dilute solutions.

Nonlinear shear and extensional flow dynamics of wormlike surfactant solutions

Nonlinear shear and extensional flow dynamics of a wormlike micellar solution based on erucyl bis(2-hydroxyethyl) methyl ammonium chloride (EHAC) are reported here. The influences of surfactant (EHAC) and salt (NH 4Cl) concentrations on the linear viscoelastic parameters are determined using small amplitude oscillatory shear experiments. The steady and time-dependent shear rheology is determined in a double gap Couette cell, and transient extensional flow measurements are performed in a capillary breakup extensional rheometer (CABER). In the nonlinear shear flow experiments, the micellar fluid samples show strong hysteretic behavior upon increasing and decreasing the imposed shear stress due to the development of shear-banding instabilities. The non-monotone flow curves of stress versus shear rate can be successfully modeled in a macroscopic sense by using the single-mode Giesekus constitutive equation. The temporal evolution of the flow structure of the surfactant solutions in the Couette flow geometry is analyzed by instantaneous shear rate measurements for various values of controlled shear stress, along with FFT analysis. The results indicate that the steady flow bifurcates to a global time-dependent state as soon as the shear-banding/hysteresis regime is reached. Increasing the salt–surfactant ratio or the temperature is found to stabilize the flow, and corresponds to decreasing values of anisotropy factor in the Giesekus model. Finally, we have investigated the dynamics of capillary breakup of the micellar fluid samples in uniaxial extensional flow. The filament thinning behavior of the micellar fluid samples is also accurately predicted by the Giesekus constitutive equation. Indeed quantitative agreement between the experimental and numerical results can be obtained providing that the relaxation time of the wormlike micellar solutions in extensional flows is a factor of three lower than in shear flows.

Uniaxial extensional flow behavior of immiscible and compatibilized polypropylene/liquid crystalline polymer blends

In this work liquid crystalline polymer (LCP) and thermoplastic (TP) blends with and without compatibilizer were studied with respect to their elongational flow behavior, under uniaxial extensional flow. This knowledge is important because in processes involving dominantly extensional deformations, like the case of the formation of the LCP fibrillation, transient extensional flow properties become more important than transient or steady-shear properties. In systems characterized by disperse phase morphologies (10 and 20 wt%) the LCP acts as a plasticizer, decreasing the viscosity of the system and increasing its durability with respect to that of the matrix. On the other hand, for a system in which a co-continuous morphology is present (40 wt% LCP) fibrils and droplets deformation occurs simultaneously, leading to a much higher strain hardening and durability. Moreover, the addition of compatibilizers to the blends gives rise to an increase of the strain hardening and to a decrease of the durability, which is in accordance with the mechanical properties, namely a higher Young’s modulus and lower elongation at break, in comparison with noncompatibilized systems.

Fingerprinting the processing behavior of polyethylenes from transient extensional flow and peel experiments in the melt state

The flow curves of linear (linear-low and high density) and branched polyethylenes are known to differ significantly. At increasing shear rates, the linear polymers exhibit a surface melt fracture or sharkskin region that is followed by an unstable oscillating or stick-slip flow regime when a constant piston speed capillary rheometer is used. At even higher shear rates, gross melt fracture appears. Unlike their linear counterparts, branched polyethylenes rarely exhibit sharkskin melt fracture and although gross melt fracture appears at high shear rates there is no discontinuity in their flow curve. The various flow regimes of these two types of polyethylenes are examined by performing experiments in the melt state using a unique extensional rheometer (the SER by Xpansion Instruments) that is capable of performing accurate extensional flow and peel experiments at very high rates not previously realized. The peel strength curves of these linear and branched polyethylenes exhibit all of the distinct flow regimes exhibited in their respective flow curves, thereby providing a fingerprint of their melt flow behavior. Moreover, these extensional flow and peel results in the melt state provide insight into the origins and mechanisms by which these melt flow phenomena may occur with regard to rapid tensile stress growth, melt rupture, and adhesive failure at the polymer wall interface.

Extensional properties of monodisperse and bidisperse polystyrene solutions

A filament stretching rheometer is used to study a series of monodisperse and bidisperse polystyrene solutions and to test predictions of the simplified Mead–Larson–Doi (MLD) “toy” model. This model incorporates convective constraint release, diffusive constraint release, and chain stretching into the reptation-based theory of Doi and Edwards but ignores contour length fluctuations (CLFs). The model parameters are the plateau modulus and the two time constants, the Rouse time and the reptation time for each monodisperse component. In order to artificially incorporate CLFs into the toy model, we relate the constants of the toy model to parameters of the Milner–McLeish model obtained using dynamic data. The predictions of this toy MLD model are in good agreement with experimental data for monodisperse solutions in steady shear, startup of shear, and steady state and transient extensional flow. At high strain rates in extensional flow, good agreement is only obtained if the extensibility of the chains is arbitrarily reduced by a factor of 2 below the molecularly derived value, perhaps reflecting the limitations of a single mode model. Without any additional parameters, the model also yields reasonable agreement with experimental data for bidisperse solutions. The study illustrates the advantages of extensional data in justifying parameter estimates and also the sensitivity of extensional measurements to the high molecular weight component of blends.

The orientation dynamics of rigid rod suspensions under extensional flow

The orientation dynamics of a lyotropic colloidal suspension of sepiolite clay under extensional flow have been explored by combined dichroism and small angle light scattering measurements. Extensional flow was applied using a four-roll mill to a thin film of sepiolite suspension (rigid rods 1 μm long and 0.010 μm in diameter). Analysis of transient extensional flow reversals revealed important characteristics of the orientation dynamics of these suspensions: (i) the existence of a critical volume fraction separating isotropic behavior, where no orientation persists after stretching, from nematic behavior, where permanent orientations persist during the relaxation phase; and (ii) in the nematic domain, a critical strain rate separates two flow regimes corresponding to a stable, so-called strong flow regime above the critical strain rate and an unstable, so-called weak flow regime below it. These experimental observations agree with the theoretical predictions of the model proposed by Marrucci and Maffettone (1989, 1990) who have examined the two-dimensional form of the simple molecular model of Hess (1976) and Doi (1981). What is new in the present case is that the colloidal suspension is a three-dimensional system, whereas previous experimental validations of the model concerned only two-dimensional rodlike polymers systems [Maffettone et al. (1996)] [Maruyama et al. (1998a, 1998b)].

Melt Rheology of Polypropylene Containing Small Amounts of High-Molecular-Weight Chain. 2. Uniaxial and Biaxial Extensional Flow

Uniaxial extensional flow behavior was measured for the high-melt-strength polypropylene using a Meissner-type rheometer under transient extensional flow with constant tensile strain rate. The equibiaxial extensional flow behavior of the high-melt-strength polypropylene was measured via the lubricated squeezing flow method under constant strain rate. The high-melt-strength polypropylene consists of polypropylene (PP) as a main component and high-molecular-weight polyethylene (PE) component as a long relaxation time component (see part 1). This system is generally believed to be an immiscible system, at least under the quiescent state. Nevertheless, in part 1, we have found that the high-melt-strength PP with very high-molecular-weight PE shows distinctly different shear flow behaviors from conventional PP, e.g., high elasticity and two-step viscosity at low shear rates and strong and weak strain-dependent nonlinear damping functions characterizing fast and slow relaxation processes. In this study, the transient uniaxial viscosity of the high-melt-strength PP melts first increased gradually with time, following the linear viscoelastic rule in which the uniaxial extensional viscosity is 3 times the shear viscosity development. Beyond a certain critical strain, the uniaxial extensional viscosity showed rapid increase, which was referred to as strain hardening. Furthermore, the transient biaxial extensional viscosity showed also the strain hardening behavior over a critical strain. These prominent behaviors are unexpected for conventional PP. The nonlinear upturn behavior was discussed from a high-molecular-weight chain stretching point of view via molecular constitutive equations given by Osaki et al. for bimodal polymer blends.

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.

Transient non-uniform extensional flows realized in certain types of rheometers

The previous concept of non-uniform stagnant motions (NUSM) is applied to the analysis of steady and transient extensional flow realized in certain types of rheometers, including, in particular, the spin-line rheometer (SLR), the filament stretching rheometer (FSR) and the rotary clamp extensometer (RCE). It is shown that the corresponding approximate solutions depend on two material functions as well as on the specimen geometry. It also turns out that the transient extensional viscosity function and extensional modulus determined on the basis of quasi-elongational approximations may be overestimated or underestimated as compared with more realistic values resulting from the present considerations.

On the modelling of a PIB/PB Boger fluid in extensional flow

In this paper we study the transient elongational viscosity of a PIB/PB Boger fluid (fluid B) by performing both experiments and model calculations. The experimental results have been obtained by using a filament stretching device was described by van Nieuwkoop and Muller von Czernicki (J. Non-Newtonian Fluid Mech. 67 (1996) 105–123). We have been able to obtain large strains, up to a Hencky strain of 8. A plateau for the Trouton ratio of close to 10 4 is found while approaching a strain of 8. The plateau value appears to be independent of the strain rate history. The predictions using various multi-mode constitutive models (Giesekus, FENE-P, Hinch, FENE) show that the FENE model is the only model which predicts values for the Trouton ratio that are very close to the experiments for the whole range from strain 1 up to 8. The predictions of the models are computed with the actual strain rate ε ̇ t as determined during the experiments in the middle of the filament. During relaxation, both the FENE-P and the FENE model perform well. In order to be able to make predictions without carrying out the relatively expensive FENE calculations, we have developed a new closed form constitutive equation. The model is based on the dumbbell theory in which the connector force used leads to a viscous stress term. The predictions of the model in transient extensional flows are very good and comparable to the FENE model.

Brownian Dynamics Simulation of FENE Dumbbell Model for Dilute Polymer Solutions in Extensional Flow: Effect of Distribution of FENE Parameter and Relaxation Time.

Brownian dynamics (BD) simulations of the steady and transient rheological behaviors of FENE dumbbell model for dilute polymer solutions in uniaxial extensional flow are reported. The simulations include examination of the effects of distributions of maximum allowable extension (FENE parameter b) and of relaxation time parameter α. Stretch-thickening and strain-hardening behaviors, which can be often observed in polymer solutions, are confirmed in the simulations either with or without the distribution of b or α. However, overshoot phenomenon in a start-up extensional flow can never be observed unlike the situation in a start-up shear flow. Curves of transient extensional viscosity vs. strain clearly show that the extensional viscosity is independent of the stretch rate in the regime of αλ0e>1~2 where polymer solutions exhibit significant strain-hardening. Furthermore, it is found that the distribution of b has little influence on extensional flow properties while the distribution of α can affect both the steady and transient extensional flow properties under the distribution condition used in the simulations. BD simulation can precisely predict the extensional viscosity even in the low stretch rate region.

Fracture of DNA in transient extensional flow. A numerical simulation study

Using the Brownian dynamics simulation technique, we studied the fracture process of DNA chains subjected to transient extensional flow, letting the solution with DNA molecules pass through a very small orifice (radius = 0.0065 cm), thus experiencing extensional flow of the convergent (sink) type. The DNA molecules were modeled as FENE bead-spring chains with the springs obeying a modified Warner force law, as proposed by Reese and Zimm. The fracture yield was strongly dependent on flow rate and molecular weight, reaching, in our setup, a level of 100% at a flow rate of around 0.001 cm3/s for DNA with molecular weight 26 × 106 (T7 DNA). There was found to exist a critical flow rate (Qcrit) below which fracture did not occur, in accordance with what was observed in studies on polystyrene in transient extensional flow. We found that for DNA, the critical flow rate depended on the molecular weight as Qcrit ∼ M−14 when the hydrodynamic interaction effect (HI) was not included in the simulations. When HI was accounted for, the relation was found to be Qcrit ∼ M−1.1, close to the theoretical prediction for fracture of partly extended chains in transient extensional flow. © 1996 John Wiley & Sons, Inc.

Fracture of flexible polymer chains in dilute solution under transient extensional flow

Applying the technique of Brownian dynamics simulation, we have studied the fracture process of flexible polymer chains when they encounter an extensional flow field of transient character. For this purpose, a mathematical model was made of an experimental device used earlier by other authors to study fracture of polystyrene in dilute solution. The polymer/solvent system studied was a very dilute solution in theta conditions. The polymer molecule was modeled as a FENE bead-spring chain, including a modification to allow for chain fracture. The simulation results showed that the fracture yield depended strongly on flow rate and on molecular weight. We have characterized the molecularweight dependence of the critical flow rate which is necessary for fracture to occur. The distribution of the resulting fragments is interpreted in terms of the conformation of the chains prior to fracture.