Planar Extensional Flow Research Articles

Microfluidic Shape Analysis of Non-spherical Graphite for Li-Ion Batteries via Viscoelastic Particle Focusing.

The size and shape of graphite, which is a popular active anode material for lithium-ion batteries (LIBs), significantly affect the electrochemical performance of LIBs and the rheological properties of the electrode slurries used in battery manufacturing. However, the accurate characterization of its size and shape remains challenging. In this study, the edge plane of graphite in a cross-slot microchannel via viscoelastic particle focusing is characterized. It is reported that the graphite particles are aligned in a direction that shows the edge plane by a planar extensional flow field at the stagnation point of the cross-slot region. Accurate quantification of the edge size and shape for both spheroidized natural and ball-milled graphite is achieved when aligned in this manner. Ball-milled graphite has a smaller circularity and higher aspect ratio than natural graphite, indicating a more plate-like shape. The effects of these differences in graphite shape and size on the rheological properties of the electrode slurry, the structure of the coated electrodes, and electrochemical performance are investigated. This method can contribute to the quality control of graphite for the mass production of LIBs and enhance the electrochemical performance of LIBs.

Dynamics of meniscus-bound particle clusters in extensional flow

Capillary suspensions are three-phase mixtures containing a solid particulate phase, a continuous liquid phase, and a second immiscible liquid forming capillary bridges between particles. Capillary suspensions are encountered in a wide array of applications including 3D printing, porous materials, and food formulations, but despite recent progress, the micromechanics of particle clusters in flow is not fully understood. In this work, we study the dynamics of meniscus-bound particle clusters in planar extensional flow using a Stokes trap, which is an automated flow control technique that allows for precise manipulation of freely suspended particles or particle clusters in flow. Focusing on the case of a two-particle doublet, we use a combination of experiments and analytical modeling to understand how particle clusters rearrange, deform, and ultimately break up in extensional flow. The time required for cluster breakup is quantified as a function of capillary number Ca and meniscus volume V. Importantly, a critical capillary number Cacrit for cluster breakup is determined using a combination of experiments and modeling. Cluster relaxation experiments are also performed by deforming particle clusters in flow, followed by flow cessation prior to breakup and observing cluster relaxation dynamics under zero-flow conditions. In all cases, experiments are complemented by an analytical model that accounts for capillary forces, lubrication forces, hydrodynamic drag forces, and hydrodynamic interactions acting on the particles. Results from the analytical models are found to be in good agreement with experiments. Overall, this work provides a new quantitative understanding of the deformation dynamics of capillary clusters in extensional flow.

Steady-state simulation of the film-casting process for Newtonian and viscoelastic fluids

Thin polymeric films are usually produced in an extensional flow between an extrusion head and a chill roll. The final product is prone to several defects, such as reduced width and non-uniform thickness, and a better understanding of the whole process can improve the final product quality and reduce wastes. This work revisits the film-casting process by exploring the scaling relation between different variables, with particular focus on the width reduction (neck-in). The numerical simulations are carried out for Newtonian, Upper-Convected Maxwell and Giesekus fluids in isothermal and steady-state conditions, for varying film aspect ratio (AR; film width over length), draw ratio (DR) and Deborah number (De). The governing equations are discretized with finite-volumes in their full form (3D model) and using an approximate 2.5D model. However, the latter has been mainly used in this study, as it offered similar accuracy as the 3D model, at a much lower computational cost. The results show that the normalized film shape is independent of the aspect ratio for sufficiently high values of this parameter and that the film thickness scales with DR according to the theoretical prediction for planar and uniaxial extensional flows, in the centre and edge regions, respectively. The neck-in grows logarithmically with the draw ratio for high AR, with a De-dependent growth rate (higher De leads to lower neck-in and lower dependence on DR). There is a reasonable correlation between the neck-in and the ratio between the planar and uniaxial extensional viscosities, as well as between the neck-in and dimensionless variable M = ln(DR)/(De+δ), where δ is a constant which depends on the fluid rheology. However, none of the correlations offer a perfect fit to the neck-in and the search for such a correlation (or master curve) shall be pursued in future works.

Improved Taylor analogy model for predicting droplet breakup and large deformation in planar extensional flow

The application of the Taylor analogy has been proven effectively in predicting the breakup of droplet within a spray system. Recently, a model of small deformation at steady and transient states within extensional flow with low Reynolds numbers has been successfully proposed by approaching the Taylor analogy. The objective of this study is to improve the Taylor analogy in laminar flow for the prediction of breakup and large deformation of the droplet in planar extensional flow by theoretical modeling and numerical analysis. The performance of the suggested model is collated using a three-dimensional numerical simulation with vast viscosity ratios and capillary numbers. The available experimental data from the literature is likewise compared with the numerical analysis outcomes for validation purposes. The proposed model could accurately predict the critical breakup condition and large deformation of droplet in extensional flow. These improvements hold significant importance for droplet dynamics studies within microfluidic systems.

Extensional rheometry of mobile fluids. Part II: Comparison between the uniaxial, planar, and biaxial extensional rheology of dilute polymer solutions using numerically optimized stagnation point microfluidic devices

Part I of this paper [Haward et al., J. Rheol. 67, 995–1009 (2023)] presents a three-dimensional microfluidic device (the optimized uniaxial and biaxial extensional rheometer, OUBER) for generating near-homogeneous uniaxial and biaxial elongational flows. Here, in Part II, the OUBER device is employed to examine the uniaxial and biaxial extensional rheology of model dilute polymer solutions, compared with measurements made under planar extension in the optimized-shape cross-slot extensional rheometer [OSCER, Haward et al. Phys. Rev. Lett. 109, 128301 (2012)]. In each case, micro-particle image velocimetry is used to measure the extension rate as a function of the imposed flow conditions, and excess pressure drop measurements enable estimation of the tensile stress difference generated in the fluid via a new analysis based on the macroscopic power balance for flow through each device. Based on this analysis, for the most dilute polymer sample tested, which is “ultradilute”, the extensional viscosity is well described by Peterlin’s finitely extensible nonlinear elastic dumbbell model. In this limit, the biaxial extensional viscosity at high Weissenberg numbers (Wi) is half that of the uniaxial and planar extensional viscosities. At higher polymer concentrations, although the fluids remain dilute, the experimental measurements deviate from the model predictions, which is attributed to the onset of intermolecular interactions as the polymer chains unravel in the extensional flows. Of practical significance (and fundamental interest), elastic instability occurs at a significantly lower Wi in uniaxial extensional flow than in either biaxial or planar extensional flow, thereby limiting the utility of this flow type for extensional viscosity measurement.

Ferrofluid droplets in planar extensional flows: Droplet shape and magnetization reveal novel rheological signatures of ferrofluid emulsions

By subjecting a ferrofluid emulsion to an external magnetic field, the flow at the droplet scale is altered, resulting in significant changes in the emulsion's rheological behavior. We present the first study of the rheological response of a dilute ferrofluid emulsion under extensional flow. Our findings reveal that at the microscopic level, the droplets experience a field-induced internal torque that leads to a complex non-symmetric stress tensor. We identify the remarkable presence of shear stresses within a material experiencing a pure extensional flow.

Dual hydrodynamic trap based on coupled stagnation point flows

Recent advancements in science and engineering have allowed for trapping and manipulation of individual particles and macromolecules within an aqueous medium using a flow-based confinement method. In this work, we demonstrate the feasibility of trapping and manipulating two particles using coupled planar extensional flows. Using Brownian dynamics simulations and a proportional feedback control algorithm, we show that two micro/nanoscale particles can be simultaneously confined and manipulated at the stagnation points of a pair of interconnected planar extensional flows. We specifically studied the effect of strain rate, particle size, and feedback control parameters on particle confinement. We also demonstrate precise control of the interparticle distance by manipulating the strain rates at both junctions and particle position at one of the junctions. We further discuss the advantages and limitations of the dual hydrodynamic trap in comparison to existing colloidal particle confinement methods and outline some potential applications in polymer science and biology. Our results demonstrate the versatility of flow-based confinement and further our understanding of feedback-controlled particle manipulation.

Phase-field Navier–Stokes model for vesicle doublets hydrodynamics in incompressible fluid flow

In this paper, we study the hydrodynamics of a vesicle doublet suspended in an external viscous fluid flow. Vesicles in this study are modeled using the phase-field model. The bending energy and energies associated with enforcing the global volume and area are considered. In addition, the local inextensibility condition is ensured by introducing an additional equation to the system. To prevent the vesicles from overlapping, we deploy an interaction energy definition to maintain a short-range repulsion between the vesicles. The fluid flow is modeled using Navier–Stokes equations under the assumption of incompressible flows, and the vesicle evolution in time is modeled using two advection equations describing the process of advecting each vesicle by the fluid flow. Rather than solving the velocity–pressure saddle point system, we apply the Residual-Based Variational MultiScale (RBVMS) method to Navier–Stokes equations and solve the coupled systems monolithically using isogeometric analysis. We study vesicle doublet hydrodynamics in shear flow, planar extensional flow, and parabolic flow under various configurations and boundary conditions. Based on the fluid flow profile and domain configuration, we have observed various dynamics like tank-threading, locking, doublet separation, sliding, and shape transformation in tubular channels.

Mathematical modelling of coalescence of viscous particles: An overview

AbstractViscous particles of polymer melts and glasses coalesce under the action of surface tension. Resistance is due to viscosity, while inertia is not a contributing factor, with the Ohnesorge number being very high. Russian physicist Yakov Frenkel developed a model for neck growth during the initial stage of the merging process of two spherical particles, assuming uniform biaxial extensional flow. Frenkel's model was extended for prediction of neck size as a function of time to the completion of coalescence, expressed by an ordinary differential equation. The time t is expressed in dimensionless form as (tγ/ηR), where η, γ, and R denote the viscosity, surface tension, and particle radius, respectively. Models were also developed for viscoelastic effects, non‐isothermal conditions, and unequal diameter particles. For the coalescence of infinitely long cylinders, planar extensional flow is assumed. Other investigators presented numerical solutions of the Navier–Stokes equations, which include shear flow components, but the predictions of neck growth are not much different from those of the Frenkel‐based models. Comparisons to experiments are also discussed, involving polymers, glasses, animal tissue cells, and biomacromolecules. The models are also used in additive manufacturing applications for the determination of bonding and pore shrinkage.

Universal flow-induced orientational ordering of colloidal rods in planar shear and extensional flows: Dilute and semidilute concentrations

The design of flow processes to build a macroscopic bulk material from rod-shaped colloidal particles has drawn considerable attention from researchers and engineers. Here, we systematically explore and show that the characteristic strain rate of the flow universally determines the orientational ordering of colloidal rods. We employed the fluctuating lattice Boltzmann method by simulating hydrodynamically interacting Brownian rods in a Newtonian liquid moving under various flow types. By modeling a rigid rod as a chain of nonoverlapping solid spheres with constraint forces and torque, we elucidate rigid rod dynamics with an aspect ratio (L/d) either 4.1 or 8.1 under various rotational Péclet number (Per) conditions. The dynamics of colloidal rods in dilute (nL3=0.05) and semidilute suspensions (nL3=1.1) were simulated for a wide range of Per (0.01<Per<1000) under shear flows including Couette and Poiseuille flows in a planar channel geometry, and an extensional and mixed-kinematics flow in a periodic four-roll mill geometry, where n is the number density, and d and L are the diameter and length of the rod, respectively. By evaluating the degree of orientational alignment of rods along the flows, we observed that there is no significant difference between flow types, and the flow-induced ordering of rods depends on the variation of Per up to moderate Per (Per<100). At a high Per (Per>100), the degree of orientational ordering is prone to diversify depending on the flow type. The spatial inhomogeneity of the strain-rate distribution leads to a substantial decrease in the orientational alignment at high Per.

Microstructural Dynamics and Rheology of Worm-like Diblock Copolymer Nanoparticle Dispersions under a Simple Shear and a Planar Extensional Flow.

We investigate the shear and extensional flow behavior of dispersions composed of two types of worm-like nanoparticles (WLNPs) with comparable cross-sectional diameters, similar persistence lengths but differing contour lengths, and thus differing flexibility. By measuring the flow-induced birefringence (FIB) of WLNP dispersions in two contrasting microfluidic devices, we obtain an experimental quantification of the role of shearing and planar extensional flows at aligning a short and stiff WLNP (S-WLNP) and a relatively long and flexible WLNP (L-WLNP). We show that shear and extensional flows induce the alignment of both types of WLNPs. However, extensional deformations are more effective than shear deformations at triggering the onset of alignment of the WLNP. The difference between shear and extensional deformations for WLNP alignment is explained based on the ratio of extensional and shear viscosity of the solvent fluid (Trouton ratio of the solvent) and a structural parameter related to the WLNP extensibility and flexibility. Under shear flow, these WLNP dispersions display shear-thinning behavior, with an exponential reduction in viscosity with increasing alignment. Under extensional flow, the WLNP alignment leads to extensional thinning, making WLNP ideal additives for industrial and biotechnology formulations exposed to extensional dominated flows (e.g., jetting, spraying, and printing processes).

Odd viscosity in chiral passive suspensions

Prior studies have revealed that nonzero odd viscosity is an essential property for chiral active fluids. Here we report that such an odd viscosity also exists in suspensions of non-active or non-externally-driven but chirally-shaped particles. Computational simulations are carried out for monolayers of dense ratchets in simple shear and planar extensional flows. The contact between two ratchets can be either frictionless or infinitely-frictional, depending on their teeth and sliding directions at the contact point. Our results show that the ratchet suspension has the intermediate shear/extensional viscosity as compared with the suspensions of smooth and gear-like particles. Meanwhile, the ratchet suspensions show nonzero even and odd components of the first normal stress coefficient regarding the flow rate, which indicates the mixed feature of conventional complex fluids and chiral viscous fluids.

Breakups of Chitosan microcapsules in extensional flow

The controlled rupture of a core–shell capsule and the timely release of encapsulated materials are essential steps of the efficient design of such carriers. The mechanical and physico-chemical properties of their shells (or membranes) mainly govern the evolution of such systems under stress and notably the link between the dynamics of rupture and the mechanical properties. This issue is addressed considering weakly cohesive shells made by the interfacial complexation of Chitosan and PFacid in a planar extensional flow. Three regimes are observed, thanks to the two observational planes. Whatever the time of reaction in membrane assembly, there is no rupture in deformation as long as the hydrodynamic stress is below a critical value. At low times of complexation (weak shear elastic modulus), the rupture is reminiscent of the breakup of droplets: a dumbell or a waist. Fluorescent labelling of the membrane shows that this process is governed by continuous thinning of the membrane up to the destabilization. It is likely that the membrane shows a transition from a solid to liquid state. At longer times of complexation, the rupture has a feature of solid-like breakup (breakage) with a discontinuity of the membrane. The maximal internal constraint determined numerically marks the initial location of breakup as shown. The pattern becomes more complex as the elongation rate increases with several points of rupture. A phase diagram in the space parameters of the shear elastic modulus and the hydrodynamic stress is established.

On the similarities of the sPTT and FENE-P models for polymeric fluids

<h2>Abstract</h2> For many commonly used single mode viscoelastic constitutive equations of differential type, it is well known that they share many features. For example, in certain parameter limits the models due to Giesekus, Phan-Thien Tanner and FENE-type models approach the Oldroyd-B model. In this talk, I'll compare the response of the linear form of the simplified Phan-Thien Tanner model [due to Phan-Thien and Tanner, Journal of Non-Newtonian Fluid Mechanics, 2 (1977) 353–365] (the "sPTT") and the Finitely Extensible Nonlinear Elastic model that follows the Peterlin approximation [due to Bird et al., Journal of Non-Newtonian Fluid Mechanics, 7 (1980) 213–235] (the "FENE-P"). I'll show that for steady homogeneous flows such as steady simple shear flow or pure extension the response of both models is identical under certain conditions. For more general flows we show analytically that the results from the two models only formally approach each other when both the polymer concentration and Weissenberg number is small. We then use a numerical approach to investigate the response of the two models when the flow is "complex" under two different definitions: firstly, when the applied deformation field is homogenous in space but transient in time (so called "start-up" shear and planar extensional flow) and then for "complex" flows (through a range of geometries) which, although Eulerian steady, are unsteady in a Lagrangian sense. Under the limit that the flows remain Eulerian steady (so the Weissenberg number is typically small), we see once again a very close agreement between the FENE-P and sPTT models.

On the similarities between the simplified Phan-Thien–Tanner model and the finitely extensible nonlinear elastic dumbbell (Peterlin closure) model in simple and complex flows

For many commonly used viscoelastic constitutive equations, it is well known that the limiting behavior is that of the Oldroyd-B model. Here, we compare the response of the simplified linear form of the Phan-Thien–Tanner model (“sPTT”) [Phan-Thien and Tanner, “A new constitutive equation derived from network theory,” J. Non-Newtonian Fluid Mech. 2, 353–365 (1977)] and the finitely extensible nonlinear elastic (“FENE”) dumbbell model that follows the Peterlin approximation (“FENE-P”) [Bird et al., “Polymer solution rheology based on a finitely extensible bead—Spring chain model,” J. Non-Newtonian Fluid Mech. 7, 213–235 (1980)]. We show that for steady homogeneous flows such as steady simple shear flow or pure extension, the response of both models is identical under precise conditions (ε=1/L2). The similarity of the “spring” functions between the two models is shown to help understand this equivalence despite a different molecular origin of the two models. We then use a numerical approach to investigate the response of the two models when the flow is “complex” in a number of different definitions: first, when the applied deformation field is homogeneous in space but transient in time (so-called “start-up” shear and planar extensional flow), then, as an intermediate step, the start-up of the planar channel flow; and finally, “complex” flows (through a range of geometries), which, although being Eulerian steady, are unsteady in a Lagrangian sense. Although there can be significant differences in transient conditions, especially if the extensibility parameter is small L2&amp;gt;100,ε&amp;lt;0.01, under the limit that the flows remain Eulerian steady, we once again observe very close agreement between the FENE-P dumbbell and sPTT models in complex geometries.

Full linear Phan-Thien–Tanner fluid model: Exact analytical solutions for steady, startup, and cessation regimes of shear and extensional flows

Exact, fully explicit, purely real analytical expressions for the material functions describing steady, startup, and cessation regimes of shear flows and of planar, uniaxial, and biaxial extensional flows of full linear Phan-Thien–Tanner fluids are obtained. These expressions, which have no analogs in the literature, are formulated in compact, beautiful forms, partially due to the unique scaling procedure reducing the number of the model parameters from four to one. The properties of the material functions are investigated in detail. For steady extensional flows, the possible shapes of the extensional viscosity curves are described and the conditions for these shapes to occur are determined. For startup flows, it is found when exactly the stress dynamics is oscillatory, and, in this case, a detailed characterization of oscillations is given, which includes expressions for the position and magnitude of stress overshoots and undershoots.

An arbitrary Lagrangian-Eulerian method for simulating interfacial dynamics between a hydrogel and a fluid

Hydrogels are crosslinked polymer networks swollen with an aqueous solvent, and play central roles in biomicrofluidic devices. In such applications, the gel is often in contact with a flowing fluid, thus setting up a fluid-hydrogel two-phase system. Using a recently proposed model (Young et al. [41] 2019), we treat the hydrogel as a poroelastic material consisting of a Saint Venant-Kirchhoff polymer network and a Newtonian viscous solvent, and develop a finite-element method for computing flows involving a fluid-hydrogel interface. The interface is tracked by using a fixed-mesh arbitrary Lagrangian-Eulerian method that maps the interface to a reference configuration. The interfacial deformation is coupled with the fluid and solid governing equations into a monolithic algorithm using the finite-element library deal.II. The code is validated against available analytical solutions in several non-trivial flow problems: one-dimensional compression of a gel layer by a uniform flow, two-layer shear flow, and the deformation of a Darcy gel particle in a planar extensional flow. In all cases, the numerical solutions are in excellent agreement with the analytical solutions. Numerical tests show second-order convergence with respect to mesh refinement, and first-order convergence with respect to time-step refinement.

Dynamics and rheology of ring-linear blend semidilute solutions in extensional flow: Single molecule experiments

Ring polymers exhibit unique flow properties due to their closed chain topology. Despite recent progress, we have not yet achieved a full understanding of the nonequilibrium flow behavior of rings in nondilute solutions where intermolecular interactions greatly influence chain dynamics. In this work, we directly observe the dynamics of DNA rings in semidilute ring-linear polymer blends using single molecule techniques. We systematically investigate ring polymer relaxation dynamics from high extension and transient and steady-state stretching dynamics in a planar extensional flow for a series of ring-linear blends with varying ring fraction. Our results show multiple molecular subpopulations for ring relaxation in ring-linear blends, as well as large conformational fluctuations for rings in a steady extensional flow, even long after the initial transient stretching process has subsided. We further quantify the magnitude and characteristic time scales of ring conformational fluctuations as a function of blend composition. Interestingly, we find that the magnitude of ring conformational fluctuations follows a nonmonotonic response with increasing ring fraction, first increasing at low ring fraction and then substantially decreasing at large ring fraction in ring-linear blends. A unique set of ring polymer conformations are observed during the transient stretching process, which highlights the prevalence of molecular individualism and supports the notion of complex intermolecular interactions in ring-linear polymer blends. In particular, our results suggest that transient intermolecular structures form in ring-linear blends due to a combination of direct forces due to linear chains threading through open rings and indirect forces due to hydrodynamic interactions; these combined effects lead to large conformational fluctuations of rings over distributed time scales. Taken together, our results provide a new molecular understanding of ring polymer dynamics in ring-linear blends in the nonequilibrium flow.

Predictions of microstructure and stress in planar extensional flows of a dense viscous suspension

Abstract

Droplet shape relaxation in a four-channel microfluidic hydrodynamic trap

Here, we trap and control the position of droplets to study their dynamics using hydrodynamic forces alone without an external field. The hydrodynamic trap is adapted from a previously implemented Stokes trap by incorporating a drop-on-demand system to generate droplets at a T-junction geometry on the same microfluidic chip. We then study confined droplet dynamics in response to perturbation by applying a millisecond-pressure pulse to deform trapped droplets. Droplet shape relaxation after cessation of the pressure pulse follows an exponential decay. The characteristic droplet shape relaxation time is obtained from the shape decay curves for aqueous glycerol droplets of varying viscosities in the dispersed phase with light and heavy mineral oils in the continuous phase. Systems were chosen to provide similar equilibrium interfacial tensions (5-10 mN/m) with wide variations of viscosity ratios. It is found that the droplet shape relaxation shows a strong dependence on droplet radius, and a weak dependence on the ratio of dispersed to continuous phase viscosity. The relaxation time is smaller for the highest viscosity ratios, potentially indicating that the dominant viscosity controls the droplet shape relaxation time in addition to the interfacial tension and droplet size. Droplet shape relaxation time can be used inform the response of droplets in an emulsion when subjected to transient flows in various processing conditions. Finally, an application of this platform for directly visualizing individual droplet coalescence in a planar extensional flow is presented. The microfluidic four-channel hydrodynamic trap can thus be applied for studying fundamental physics of droplet deformation and droplet-droplet interactions on the micro-scale to provide an enhanced understanding of emulsion behavior on an individual droplet level.