Uniaxial Extensional Flow Research Articles

Effect of interfacial slip on the deformation of a viscoelastic drop in uniaxial extensional flow field

The effect of interfacial slip on the deformation of a viscoelastic droplet, suspended in another viscoelastic medium, in the presence of a uniaxial extensional flow, is studied analytically. Using the Oldroyd-B constitutive relation, the Stokes flow problem is solved in the limit of a small capillary number and small Deborah number. Experimentally observed interfacial velocity slip is incorporated using a Navier slip boundary condition. The interfacial slip significantly reduces the magnitude of droplet deformation when the droplet has larger viscosity as compared with the suspending phase. The droplet shape becomes less ellipsoidal in the presence of slip. The effect of slip diminishes for low viscosity droplets. Slip effectively weakens the dependence of the droplet deformation on the droplet to medium viscosity ratio. The viscoelasticity of the suspending phase plays a dominant role on the droplet deformation as compared with the viscoelasticity of the droplet phase when there is velocity slip at the droplet surface. The interfacial slip aids the viscoelasticity of the suspending phase in deforming the droplet, while the effect of viscoelasticity of the droplet phase is suppressed by the interfacial slip.

Influence of Extensional Stress Overshoot on Crystallization of LDPE

Low-density polyethylene (LDPE) shows a stress overshoot in start-up of strong uniaxial extensional flows of constant rate. It is believed that the overshoot is caused by a contraction of the polymer backbone due to alignment of the long chain branches—the consequence being that the molecular strain of the backbone does not increase monotonically with the global strain of the melt. In this study we investigate the semicrystalline morphology of LDPE quenched before, after, and at the overshoot. We do this by combining filament stretching rheometry with ex-situ X-ray scattering. It is found that the overshoot indeed is reflected in the orientation of the crystalline domains of the quenched filaments. In a broader perspective, we show that the final crystalline morphology is determined by the stress at quench—not the strain at quench. With these findings we confirm that the much debated overshoot has a physical origin. More importantly, we conclude that even for complex architectures like branched systems, t...

Extensional Step Strain Rate Experiments on an Entangled Polymer Solution

The dynamics of chain entanglement/disentanglement in concentrated polymer solutions are matters of current debate. We report the results of step strain rate experiments in uniaxial extensional flow on a well-characterized polymer solution containing about 22 entanglements per chain which allows these dynamics to be probed. In these experiments the polymer solution is subjected to homogeneous stretching at a given strain rate up until a predetermined value of strain. After this strain is reached, the strain rate is changed to a new value, and stretching is continued until steady state is acquired in the extensional stress. The strain rates are increased in step-up experiments and decreased in step-down experiments. The strain rates are adjusted so that both the orientation and the stretching dynamics can be probed. Additionally, the predictions of two recent single-mode molecular models are evaluated against the experimental data. These models include the effects of entanglement dynamics in their predicti...

An experimental note on the deformation and breakup of viscoelastic droplets rising in non-Newtonian fluids

The deformation of viscoelastic droplets rising in non-Newtonian fluids is examined experimentally. Similarly to the case of bubbles and Newtonian droplets traveling in viscoelastic fluids, a critical volume exists but there is no jump in velocity nor a drastic change in droplet geometry. Nevertheless, a tail appears as well as a negative wake. For large-enough volumes, the thickness of the tail is much larger than that for the Newtonian droplets; this is also true for the magnitude of the negative wake and for the extension of the tail. The head of the droplet is subjected to a bi-axial elongation which deforms the spherical part of the droplet converting it into an elongated shape. This elongated shape blends into a tail and the droplet is converted into a viscous elongated teardrop. The tail is then subjected to a uniaxial extensional flow under the action of elongational stresses along the length of the tail. This extensional flow is counteracted by the shear stresses acting on the interface between the tail and the outer surrounding fluid. This interaction will determine the tail breakup. The influence of the elastic properties of the surrounding fluid and the extensional viscosity of the droplet will be responsible for the length and thickness of the tail as well as for its breakup mechanism.

Effect of surfactant on motion and deformation of compound droplets in arbitrary unbounded Stokes flows

This study deals with the motion and deformation of a compound drop system, subject to arbitrary but Stokesian far-field flow conditions, in the presence of bulk-insoluble surfactants. We derive solutions for fluid velocities and the resulting surfactant concentrations, assuming the capillary number and surface Péclet number to be small, as compared with unity. We first focus on a concentric drop configuration and apply Lamb’s general solution, assuming the far-field flow to be arbitrary in nature. As representative case studies, we consider two cases: (i) flow dynamics in linear flows and (ii) flow dynamics in a Poiseuille flow, although for the latter case, the concentric configuration does not remain valid in general. We further look into the effective viscosity of a dilute suspension of compound drops, subject to linear ambient flow, and compare our predictions with previously reported experiments. Subsequently, the eccentric drop configuration is addressed by using a bipolar coordinate system where the far-field flow is assumed to be axisymmetric but otherwise arbitrary in nature. As a specific example for eccentric drop dynamics, we focus on Poiseuille flow and study the drop migration velocities. Our analysis shows that the presence of surfactant generally opposes the imposed flows, thereby acting like an effective augmented viscosity. Our analysis reveals that maximizing the effects of surfactant makes the drops behave like solid particles suspended in a medium. However, in uniaxial extensional flow, the presence of surfactants on the inner drop, in conjunction with the drop radius ratio, leads to a host of interesting and non-monotonic behaviours for the interface deformation. For eccentric drops, the effect of eccentricity only becomes noticeable after it surpasses a certain critical value, and becomes most prominent when the two interfaces approach each other. We further depict that surfactant and eccentricity generally tend to suppress each other’s effects on the droplet migration velocities.

Stress relaxation following uniaxial extension of polystyrene melt and oligomer dilutions

The filament stretching rheometer has been used to measure the stress relaxation following the startup of uniaxial extensional flow, on a narrow molar mass distribution (NMMD) polystyrene melt and styrene oligomer dilutions thereof. All samples used here were characterized in molecular weight, mechanical spectroscopy, and constant strain rate uniaxial extension in the work of Huang et al. [Macromolecules 46, 5026–5035 (2013); ACS Macro Lett. 2, 741–744 (2013)]. The stress relaxation following the steady extensional stress was measured on a 285 kg/mole NMMD polystyrene and two 1.92 kg/mole styrene oligomer dilutions thereof {PS-285k, PS-285k/2k-72, and PS-285k/2k-44 in the work of Huang et al. [Macromolecules 46, 5026–5035 (2013)]}. The two dilutions contained 28 and 56 wt. % oligomer, respectively. Further, the stress relaxation on a 545 kg/mole NMMD polystyrene diluted with 48 wt. % 0.972 kg/mole styrene oligomer {PS-545k/1k-52 in the work of Huang et al. [ACS Macro Lett. 2, 741–744 (2013)]} was measured as well. All the terminal relaxations could be predicted by a Doi and Edwards, e.g., pure configurational, type of model. At smaller time scales, agreement with a molecular stress function type of constitutive representation was observed for all measured relaxations.

A new look at extensional rheology of low-density polyethylene

The nonlinear rheology of three selected commercial low-density polyethylenes (LDPE) is measured in uniaxial extensional flow. The measurements are performed using three different devices including an extensional viscosity fixture (EVF), a homemade filament stretching rheometer (DTU-FSR) and a commercial filament stretching rheometer (VADER-1000). We show that the measurements from the EVF are limited by a maximum Hencky strain of 4, while the two filament stretching rheometers are able to probe the nonlinear behavior at larger Hencky strain values where the steady state is reached. With the capability of the filament stretching rheometers, we show that LDPEs with quite different linear viscoelastic properties can have very similar steady extensional viscosity. This points to the potential for independently controlling shear and extensional rheology in certain rate ranges.

Molecular based prediction of the extensional rheology of high molecular weight polystyrene dilute solutions: A hi-fidelity Brownian dynamics approach

A highly efficient Brownian dynamics algorithm for simulation of bead-spring chain micromechanical models that utilizes the Krylov framework and the semi-implicit predictor-corrector scheme is used to study the behavior of dilute solutions of high molecular weight polystyrene in uniaxial extensional flow. The influence of key parameters, namely, appropriate inclusion of hydrodynamic interactions (HI) and excluded volume (EV), the level of fine-graining, and the flow strength on the observed extensional hardening of the dilute solutions over a broad molecular weight range is considered. Specifically, it is demonstrated that the combination of HI and successive fine-graining results in very good predictions of rheological properties of solutions containing 1.95, 3.9, and 10.2 × 106 molecular weight macromolecules. However, for the highest molecular weight system, namely, 20 × 106, some level of discrepancy between experiment and the simulation results is observed. Finally, the incorporation of EV in order to predict material functions that are more consistent with the measurements for the highest molecular weight macromolecule solution is discussed.

Rheological characterization of H-shaped poly(methyl methacrylate)s

Investigation of the extensional flow behavior and development of constitutive equations for well-characterized H-shaped polymers is important both from an industrial and academic viewpoints. For H-shaped polymer systems, little is known about the extensional flow behavior at constant strain rates. We synthesized H-shaped poly(methyl methacrylate)s (PMMAs) with narrow molecular weight distributions and various ratios of the molecular weight of the backbone, Mb, to that of the arm, Ma, by atom transfer radical polymerization. However, 1H NMR spectroscopy showed that the resulting PMMA samples contained substantial amounts of linear and star topologies. In spite of the limited purity of the H-shaped PMMAs obtained, the rheological properties in linear viscoelasticity and uniaxial extensional flow revealed interesting results. The zero-shear viscosity of model PMMAs was well described by an exponential relation of the molecular weight. Qualitative agreement was found between the experimental linear-viscoelastic behavior and predictions of the model of McLeish et al. (Macromolecules 32: 6734–6758, 1999) for H-shaped polymers. The elongational flow behavior was also analyzed by the molecular stress function (MSF) model. Surprisingly, the strain-hardening effect for all investigated model PMMAs was weaker than that for pom-pom polystyrene polymers.

Periodic boundary conditions for the simulation of uniaxial extensional flow of arbitrary duration

It is very common with molecular dynamics and other simulation techniques to apply Lees–Edwards periodic boundary conditions (PBCs) for the simulation of shear flow. However, the behaviour of a complex liquid can be quite different under extensional flow. Simple deformation of a simulation cell and its periodic images only allows for simulations of these flows with short duration. For the simulation of planar extensional flow, it was recognised that the PBCs of Kraynik and Reinelt [Int. J. Multiphase Flow 1992;18:1045] could be used to perform simulations of this flow with arbitrary duration. However, a very common extensional flow in industrial applications and experiment is uniaxial extensional flow. Kraynik and Reinelt found that their method could not be directly generalised to this flow because of the lack of a lattice which reproduces itself during uniaxial extension. PBCs are presented in this article, which avoid this problem by finding a lattice which is compatible with the flow, finding the reduced basis to the lattice at all times and using this basis when calculating the position and separation of particles. Using these new PBCs, we perform nonequilibrium molecular dynamics simulations of a simple liquid and show that the technique gives results which agree with those from simulations using simply deforming PBCs.

Modifying the pom-pom model for extensional viscosity overshoots

We have developed a variant of the pom-pom model that qualitatively describes two surprising features recently observed in filament stretching rheometer experiments of uniaxial extensional flow of industrial branched polymer resins: (i) Overshoots of the transient stress during steady flow and (ii) strongly accelerated stress relaxation upon cessation of the flow beyond the overshoot. Within the context of our model, these overshoots originate from entanglement stripping (ES) during the processes of normal chain retraction and branch point withdrawal. We demonstrate that, for a single mode, the predictions of our overshoot model are qualitatively consistent with experimental data. To provide a quantitative fit, we represent an industrial melt by a superposition of several individual modes. We show that a minimal version of our model, in which ES due to normal chain retraction is omitted, can provide a reasonable, but not perfect, fit to the data. With regard the stress relaxation after (kinematically) steady flow, we demonstrate that the differential version of tube orientation dynamics in the original pom-pom model performs anomalously. We discuss the reasons for this and suggest a suitable alternative.

The effect of strain-hardening on the morphology and mechanical and dielectric properties of multi-layered PP foam/PP film

Polypropylene (PP) foams are important in various industrial applications due to their unique mechanical and electrical properties. However, fabrication of microcellular PP foams is still limited because of polymer's rheological properties. In this work, the influence of rheological properties and the layer confinement effect on the microcellular morphology of foam/film structure and the mechanical and dielectric properties were investigated. Two types of long chain branched polypropylenes (BPPs) were used and named as BPP1 and BPP2, respectively. Talc micro-particles were used as a nucleating agent for crystallization. In uniaxial extensional flow, both BPPs exhibited strong strain-hardening behavior with the Trouton ratios well above 3 at all extensional rates, regardless of the talc content; however, BPP2 exhibited more strain-hardening than BPP1. 16 and 32 layered foam/film structures were then fabricated with BPP2 and 32 layered foam/film was fabricated with BPP1. The lower strain-hardening of BPP1 caused cell collapse and showed very little evidence of film layers separating the foam layers, while the stronger strain-hardening of BPP2 resulted in stable cell layers near the surface of the sheet and a non-coalesced cell structure. Mechanical properties in compression showed large differences in compressive strain due to the difference in cell structures. Dielectric property measurement on biaxially oriented 32-layer BPP2 foam/film, the one that showed the best cell structure and mechanical properties, showed much higher apparent dielectric constant (6–7) and higher release and stored energy densities than those of neat biaxially oriented polypropylene films.

Uniaxial extensional flow behavior of comb-shaped poly(methyl methacrylate)

Comb-shaped poly(methyl methacrylate)s (PMMAs) with various mass fractions and various molecular weights of the side chains were prepared by free-radical polymerization followed by atom transfer radical polymerization (ATRP). The rheological properties were characterized in linear-viscoelasticity and in uniaxial extensional flow. Due to branching and thereby slow relaxation of the backbone, significant strain hardening was observed and well analyzed by the molecular stress function (MSF) model. The model parameter fmax2 increased with increasing Mn,bb at almost constant Mn,br and \( \overline{p} \), and with increasing \( \overline{p} \) at almost constant Φn,br, where Mn,bb and Mn,br are the number-average molecular weights of the backbone and the branched chains, and \( \overline{p} \) and Φn,br are the average number per backbone and the number-average mass fraction of the branched chains, respectively. These results provide guidelines for the polymer design with the aim of enhancing strain hardening in elongational flow.

Influence of the potential well on the breakage rate of colloidal aggregates in simple shear and uniaxial extensional flows.

In this work we build on our previous paper (Harshe, Y. M.; Lattuada, M. Langmuir 2012, 28, 283-292) and compute the breakage rate of colloidal aggregates under the effect of shear forces by means of Stokesian dynamics simulations. A library of clusters made of identical spherical particles covering a broad range of masses and fractal dimension values (from 1.8 to 3.0) was generated by means of a combination of several Monte Carlo methods. DLVO theory has been used to describe the interparticle interactions, and contact forces have been introduced by means of the discrete element method. The aggregate breakage process was investigated by exposing them to well-defined shear forces, generated under both simple shear and uniaxial extensional flow conditions, and by recording the time required to reach the first breakage event. It has been found that the breakage rate of clusters was controlled by the potential well between particles as described by DLVO theory. A semiempirical Arrhenius-type exponential equation that relates the potential well to the breakage rate has been used to fit the simulation results. The dependence of the breakage process on the radius of gyration, on the external shear strength, and on the fractal dimension has been obtained, providing a very general relationship for the breakage rate of clusters. It was also found that the fragment mass distribution is insensitive to the presence of electrostatic repulsive interactions. We also clarify the physical reason for the large difference in the breakage rate of clusters between simple shear and the uniaxial extensional flow using a criterion based on the energy dissipation rate. Finally, in order to answer the question of the minimum cluster size that can break under simple shear conditions, a critical rotation number has been introduced, expressing the maximum number of rotations that a cluster exposed to simple shear could sustain before breakage.

Tuning viscoelastic and crystallization properties of polypropylene containing in-situ generated high aspect ratio polyethylene terephthalate fibrils

This study explores how the rheological and crystallization properties of linear polypropylene (PP) are changed by incorporating fibrillated polyethylene terephthalate (PET). PET domains are fibrillated by melt blending with PP followed by drawing. In uniaxial extensional flows, the PP/fibrillated-PET shows strain hardening behavior, not observed for PP/PET blends with spherical PET domains or for the neat PP. At PET fibril contents ≥3 wt%, a plateau appears in the storage modulus at low frequencies, indicating the formation of a percolated physical network of fibrils that responds elastically over long timescales. Avrami analysis of the crystallization of PP/fibrillated-PET at ambient and elevated CO2 pressures reveals that the PET fibrils enhance the kinetics of isothermal crystallization of PP under both pressure conditions. WAXS study shows that when PP/fibrillated-PET is isothermally crystallized at high pressures, γ-phase crystal formation of PP is promoted which is representative of superior mechanical properties.

Large-Scale and Electroswitchable Polarized Emission from Semiconductor Nanorods Aligned in Polymeric Nanofibers

We demonstrate the fabrication of a large-scale device allowing for electrical switching of polarized light with a polarization ratio of 0.45 over a 1.5 cm2 area. To achieve this, silica-coated semiconductor nanorods were embedded in polymeric nanofibers by electrospinning. The uniaxial extensional flow experienced by the particles induces alignment of the nanorods within the nanofibers. A subsequent parallel alignment of the nanofibers themselves results in a large and flexible film of massively aligned nanorods that can conveniently be processed further. We demonstrate this by integrating the aligned nanofibers in a liquid crystal cell with a polarizer where applying a voltage allows switching the emitted light on and off.

Response of an entangled polymer solution to uniaxial and planar deformation

Experimental data on homogeneous planar extension on well-characterized polymeric fluids are rare. As a result, the relationship between planar and uniaxial extensional viscosity is also unclear. We present measurements of the steady–state planar extensional viscosity of a well-characterized polymer solution with about 30 entanglements per chain. The measurements are made by modifying the filament stretching rheometer to enable the stretching of the sample in the shape of a cylindrical tube. When a constant strain rate is imposed on the sample, a well-defined steady-state in the planar extensional viscosity is observed in the experiments. Our experiments show that, within the range of strain rates used, the response of the entangled polymer solution in planar extensional flow is identical to that observed in uniaxial extensional flow. Such a finding, if found to be generally valid, facilitates the comparison between simulations and experimental data, since planar flows are easier to simulate than uniaxial flows. We also compare the predictions of a toy-version of the tube model that describes the flow properties of entangled polymer solutions with the experimental results. We find that the predictions of the model are in qualitative agreement with the experimental observations within the range of strain rates used in this study.

In Situ SAXS Studies of Structural Relaxation of an Ordered Block Copolymer Melt Following Cessation of Uniaxial Extensional Flow

A hexagonally ordered styrene–ethylene-co-butylene–styrene triblock copolymer melt is studied during and immediately following uniaxial extensional flow using small-angle X-ray scattering. Strips of polymer melt are stretched between counter-rotating drums inside an oven designed for in situ synchrotron studies. Previous study of this sample demonstrated deformation and reorientation of PS microdomains and identified a critical Hencky strain ∼0.2 at which the structure is dramatically disrupted. To further probe microstructural dynamics, we study relaxation behavior of structure and stress in samples stretched in the melt to various Hencky strains. Upon cessation of stretching below the critical strain, structural relaxation is strongly retarded; stretching to higher strain leads to more rapid relaxation. At all strains, microdomain orientation relaxation following flow is slower than relaxation of flow-induced deformation and mechanical stress. The extension rate dependence of the sample’s response is also explored; higher rates result in faster relaxation of both stress and structure.

Extensional rheometry of magnetic dispersions

This work presents a technique and develops an apparatus that allows the application of homogeneous external magnetic fields (parallel or perpendicular to the deformation axis) to a fluid sample undergoing extensional flow kinematics while measuring the filament thinning using the commercial version of the capillary breakup extensional rheometer (Haake™ CaBER™ 1, Thermo Scientific). We also present innovative rheological measurements of several commercial ferrofluids (FFs) and one magnetorheological fluid (MRF) under uniaxial extensional flow. The experimental results demonstrate that FFs exhibit a Newtonian-like behavior in the absence of magnetic fields. When a magnetic field is applied perpendicular to the extensional flow, no significant effects are observed similar to shear experiments. However, when the external magnetic field is aligned with the extensional flow, the filament takes longer to break up but otherwise behaves as a Newtonian fluid. In the case of the MRF, due to the higher concentration of particles and larger particle size, the differences in the extensional behaviors are much more dramatic regardless of the orientation of the magnetic field compared to the case when no magnetic field is applied.

Rheological behavior and structure development in thermoplastic polyurethanes under uniaxial extensional flow

In this work, the rheological and structural changes induced by uniaxial extensional flows are studied for two aromatic TPUs: an elastomeric (soft) material composed both of hard and soft segments and an amorphous glass (hard) one composed almost exclusively by hard segments. The uniaxial extensional viscosity was measured on a Sentmanat fixture (SER) at single temperature of 175°C, showing that at high strain rates both materials initially strain-soften before eventually strain-hardening at higher Hencky strains. This effect is smoother and occurs at lower strain rates for the soft TPU. Optical microscopy, atomic force microscopy and X-ray scattering show this behavior to be related with structural changes induced by the flow. Strain-softening is attributed to the orientation of hard domains in the flow direction. In the soft TPU strain-hardening is caused by the stretching of entangled soft segments but in the hard one it appears to be associated with microcracks and with the development of a hitherto unreported hierarchical structure in the molten material.