Planar Extensional Flow Research Articles

Structural response of a prealigned cylindrical block copolymer melt to extensional flow

We report in situ small-angle x-ray scattering studies of a prealigned cylindrically ordered styrene-ethylene butylene-styrene block copolymer melt subjected to extensional flow. Samples are prepared via lubricated planar extensional flow and tested using two initial conditions: “parallel” with cylindrical microdomains oriented along the extensional flow direction and “perpendicular” with domains aligned transverse to the flow direction. The experiments employ a counter-rotating drum extensional flow fixture housed in an oven designed for in situ synchrotron access. The impacts of initial condition, extension rate, and final Hencky strain on the melt structure are analyzed both during and following flow. Stress and flow kinematics are strongly influenced by the initial sample orientation. While parallel samples exhibit uniaxial deformation, perpendicular samples exhibit planar extensional kinematics, attributed to susceptibility to compression along the cylindrical microdomain axis due to a microscopic buckling instability. Scattering data reveal both anisotropic deformation of microdomain spacing and reorientation induced by the flow. Persistence of higher order reflections confirms hexagonal packing throughout the flow process, and strong alignment along the stretching direction is attained for both initial conditions. Flow-induced deformation of microdomain spacing and mechanical stress relax on similar time scales upon flow cessation, while negligible relaxation of orientation is observed.

Analytical solution of steady 2D wall-free extensional flows of UCM fluids

The general analytical solution for the two-dimensional steady planar extensional flow with wall-free stagnation point is obtained for viscoelastic fluids described by the upper convected Maxwell model providing the stress and pressure fields. The two normal stress fields contain terms that are unbounded for |a|De<½, |a|De>½ and even for any |a|De, where De denotes the Deborah number and |a|De denotes the Weissenberg number, but the pressure field is only unbounded for |a|De<½. Properties of the first invariant of the stress tensor impose relations between the various stress and pressure coefficients and also require that they are odd functions of |a|De. The solution is such that no stress singularities exist if the stress boundary conditions are equal to the stress particular solutions. For |a|De<½ the only way for the pressure to be bounded is for the stresses to be constant in the whole extensional flow domain and equal to those particular stresses, in which case the loss of stress smoothness, reported previously in the literature, does not exist. For |a|De>½, however, the pressure remains bounded even in the presence of stress singularities. In all flow cases studied, the stress and pressure fields are contained by the general solution, but may require some coefficients to be null.

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.

The molecular stress function (MSF) model in rheology

The molecular stress function (MSF) model is an integral constitutive equation introduced more than two decades ago. It is based on the time-deformation separability principle. The time contribution encloses the linear viscoelastic information, which can be provided by the phenomenological models or any molecular theory. The deformation contribution is defined in the MSF model as a strain measure describing the orientation and the stretch of the strands of the chain as independent processes. The orientation is described by the second-order tensor of the Doi-Edwards model, considering the independent alignment assumption. The stretch is taken into account by the molecular stress function, the main characteristic being that it is included inside the integral and it is the solution of an evolution equation. Since its proposal, the MSF model has been used to describe quantitatively the non-linear rheology of a broad variety of materials such as rubbers, linear and long-chain branched polymer melts and blends of polydisperse samples relevant to the industry. Nearly, monodisperse systems in solution and melt states have also been studied in samples with different structures like linear, bidisperse blends with linear components, combs and pom-pom molecules. Predictions have been obtained for a variety of deformations like uniaxial, equibiaxial and planar extensional flow as well as for steady, medium and large amplitude oscillatory and exponential shear flow. The quantitative description of polymer melts in transient elongation is crucial for numerical simulations. Therefore, the MSF model has been applied to perform finite element simulations for different processes and freesurface deformations, due to its flexibility, reliability and reduced number of material parameters. The integral constitutive equation and its physical interpretation remains the same since it was first published. The evolution equation of the molecular stress function is material dependent because it considers different molecular mechanisms occurring in different structures. Given its importance to rheology, it is the objective of this contribution to review the antecedents, physical basis and applications of the MSF model.

Properties of the spindle-to-cusp transition in extensional capsule dynamics

Our earlier letter (Dodson W. R. III and Dimitrakopoulos P., Phys. Rev. Lett., 101 (2008) 208102) revealed that a (strain-hardening) Skalak capsule in a planar extensional Stokes flow develops for stability reasons steady-state shapes whose edges from spindled become cusped with increasing flow rate owing to a transition of the edge tensions from tensile to compressive. A bifurcation in the steady-state shapes was also found (i.e. existence of both spindled and cusped edges for a range of high flow rates) by implementing different transient processes, owing to the different evolution of the membrane tensions. In this paper we show that the bifurcation range is wider at higher viscosity ratio (owing to the lower transient membrane tensions accompanied the slower capsule deformation starting from the quiescent capsule shape), while it contracts and eventually disappears as the viscosity ratio decreases. The spindle-to-cusp transition is shown to represent a self-similar finite-time singularity formation which for real capsules with very small but finite thickness is expected to be an apparent singularity, i.e. formation of very large (but finite) positive and negative edge curvatures.

Effects of membrane hardness and scaling analysis for capsules in planar extensional flows

AbstractIn this paper, we investigate computationally the effects of membrane hardness on the dynamics of strain-hardening capsules in planar extensional Stokes flows. As the flow rate increases, all capsules reach elongated steady-state configurations but the cross-section of the more strain-hardening capsules preserves its elliptical shape while the less strain-hardening capsules become lamellar. The capsule deformation in strong extensional flows is accompanied with very pointed edges, i.e. large edge curvatures and thus small local edge length scales, which makes the current investigation a multi-length interfacial dynamics problem. Our computational results for elongated strain-hardening capsules are accompanied with a scaling analysis which provides physical insight on the extensional capsule dynamics. The two distinct capsule conformations we found, i.e. the slender spindle and lamellar capsules, are shown to represent two different types of steady-state extensional dynamics. The former are stabilized mainly via the membrane’s shearing resistance and the latter via its area-dilatation resistance, associated with the elongation tension normal forces and thus both types differ from bubbles which are stabilized mainly via the lateral surface-tension normal forces. Our steady-state deformation results can be used to identify the elastic properties of a real capsule, i.e. the membrane’s shear and area-dilatation moduli, utilizing a single experimental technique.

Loop subdivision surface boundary integral method simulations of vesicles at low reduced volume ratio in shear and extensional flow

Using an unstructured boundary integral method with curvature determination via Loop subdivision surfaces, we explore a region of moderate reduced volume vesicles in flow that includes prolate, biconcave, and stomatocyte shapes. We validate our Loop subdivision code against previously published spectral method simulations. In shear flow, we report dynamic phase diagrams at reduced volumes ranging from 0.65 to 0.95 and determine the critical viscosity ratio at which the vesicle moves away from tank treading. We examine biconcave shapes and find the elimination of the trembling regime and a tumbling that includes significant stretch in the vorticity direction, as well as a general reduction in shear and normal stresses versus a prolate shape. Finally, we re-examine over a wider range of reduced volume the shape instability originally reported by Zhao and Shaqfeh [“The shape stability of a lipid vesicle in a uniaxial extensional flow,” J. Fluid Mech. 719, 345–361 (2013)] of a vesicle placed in an extensional flow. At sufficiently low reduced volume and high capillary number, we find the steady elongated dumbbell shape is unstable to odd perturbations and the vesicle's dumbbell ends become unequal in size. We also find that the critical capillary number as a function of reduced volume is similar between uniaxial and planar extensional flow.

Numerical Simulation of Orientation Distribution on Fibers Suspensions in Planar Extensional Field

A mathematical model of evolution process is adopted to simulate orientation distribution of fibers suspensions in planar extensional flow, i.e., specific form of Fokker-Plank partial differential equation and Jeffery equation. The analytical solution of differential equation on forecast fiber orientation distribution is deduced.

Fluidic‐Directed Assembly of Aligned Oligopeptides with π‐Conjugated Cores

A microfluidic-based directed assembly strategy is employed to form highly aligned supramolecular structures. Formation of aligned synthetic oligopeptide nanostructures is accomplished using planar extensional flow, which induces alignment of underlying material suprastructures. Fluidic-directed assembly of supramolecular structures allows for unprecedented manipulation at the nano- and mesoscales, which has the potential to provide rapid and efficient control of functional material properties.

Concentration fluctuations in polymer solutions under extensional flow

In this work, we extend the classical analysis of concentration fluctuations in polymer solutions under shear flow to consider the same phenomenology under extensional flow. Experimental work by van Egmond and Fuller [Macromolecules 26, 7182–7188 (1993)] revealed a four-lobe scattering pattern for a polystyrene solution in a planar extensional flow field. Similar to earlier results found in shear, they find the existence of finite-wavelength peak intensity locations. To investigate this phenomenon, we couple stress and concentration using a two-fluid model with fluctuations driven by thermal noise incorporated through a canonical Langevin approach. The polymer stress is governed by the Rolie–Poly model augmented with finite extensibility to account for large stretching of chains at high Weissenberg numbers. Perturbing the equations about homogeneous planar extensional flow for weak amplitude inhomogeneities, but arbitrary flow strength, we solve for the steady correlations. The resulting structure factor ...

Extensional flow of hyaluronic acid solutions in an optimized microfluidic cross-slot device.

We utilize a recently developed microfluidic device, the Optimized Shape Cross-slot Extensional Rheometer (OSCER), to study the elongational flow behavior and rheological properties of hyaluronic acid (HA) solutions representative of the synovial fluid (SF) found in the knee joint. The OSCER geometry is a stagnation point device that imposes a planar extensional flow with a homogenous extension rate over a significant length of the inlet and outlet channel axes. Due to the compressive nature of the flow generated along the inlet channels, and the planar elongational flow along the outlet channels, the flow field in the OSCER device can also be considered as representative of the flow field that arises between compressing articular cartilage layers of the knee joints during running or jumping movements. Full-field birefringence microscopy measurements demonstrate a high degree of localized macromolecular orientation along streamlines passing close to the stagnation point of the OSCER device, while micro-particle image velocimetry is used to quantify the flow kinematics. The stress-optical rule is used to assess the local extensional viscosity in the elongating fluid elements as a function of the measured deformation rate. The large limiting values of the dimensionless Trouton ratio, Tr ∼ O(50), demonstrate that these fluids are highly extensional-thickening, providing a clear mechanism for the load-dampening properties of SF. The results also indicate the potential for utilizing the OSCER in screening of physiological SF samples, which will lead to improved understanding of, and therapies for, disease progression in arthritis sufferers.

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.

Single Cell Microbioreactor for observing Dynamic Gene Expression and Cell Response to Environment in Bacteria

Biological systems have been shown to implement a low-pass filter in order to distinguish high frequency noise from a lower frequency input signal, which is essential to a cellular system to adapt or sustain fitness in fluctuating environment conditions. Gene expression has been shown to exhibit noise leading to phenotypic heterogeneity of a cell population. Many microfluidic techniques have been developed in order to study single cell inducible gene expression, however rely solely on diffusion timescales to effectively alternate between inducer concentrations. In our approach, we developed a microfluidic platform for single cell analysis that allows for dynamic control of a target cell in oscillating well-defined culture media and inducer concentrations. The hydrodynamic trap enables confinement and manipulation of single cells in free solution using the sole action of fluid flow. Automated feedback control is integrated into the device using an “on-chip” valve, which allows for precise confinement of cells in free solution. During observation, cells are confined at a fluid stagnation point generated by planar extensional flow in a cross-slot microfluidic geometry, thereby enabling non-perturbative trapping of cells for long time scales. Using this platform, we investigated the effect of small molecule inducers on gene expression in the lac operon using fluorescent reporter proteins and cell growth rates as a proxy of cellular fitness. We applied our technique to determine the cut-off frequency associated with periodic stimuli for the lac circuit in E. coli, wherein the cut-off frequency in the case of the low-pass filter is the higher frequency when response begins to attenuate. We observed that single cell gene expression depends on the correlation between growth rate and frequency of exposure to inducer concentrations.

Elastic capsule deformation in general irrotational linear flows

Knowledge of the response of elastic capsules to an imposed fluid flow is necessary for predicting the deformation and motion of biological cells and synthetic capsules in microfluidic devices and in microcirculation. Capsules have been studied in shear, planar extensional and axisymmetric extensional flows. Here, the flow gradient matrix of a general irrotational linear flow is characterized by two parameters, its strain rate, defined as the maximum of the principal strain rates, and a new term, q, the difference of the two lesser principal strain rates, scaled by the maximum principal strain rate; this characterization is valid for ellipsoids in irrotational linear flow, and gives good results for spheres in general linear flows at low capillary numbers. We demonstrate that deformable non-spherical particles align with the principal axes of an imposed irrotational flow. Thus, it is most practical to model deformation of non-spherical particles already aligned with the flow, rather than considering each arbitrary orientation. Capsule deformation was modeled for a sphere, a prolate spheroid and an oblate spheroid, subjected to combinations of uniaxial, biaxial and planar extensional flows; modeling was performed using the immersed boundary method. The time response of each capsule to each flow was found, as were the steady-state deformation factor, mean strain energy and surface area. For a given capillary number, planar flows led to more deformation than uniaxial or biaxial extensional flows. Capsule behavior in all cases was bounded by the response of capsules to uniaxial, biaxial and planar extensional flows.

Slip-link simulations of entangled polymers in planar extensional flow: Disentanglement modified extensional thinning

In the present work, we have extended a slip-link based model, originally proposed by Masubuchi et al. [J. Chem. Phys. 115, 4387 (2001)], to simulate planar extensional flow. The salient features of the extended model are (i) an unrestricted simulation time for planar extensional flow using Kraynik–Reinelt periodic boundary conditions, (ii) an implicit time stepping scheme to enhance numerical stability, and (iii) a mesh-free Lagrangian method for the evaluation of the osmotic force. With the new code, we have simulated both the static and dynamic responses of monodisperse linear worm-like and inverse Langevin chains and compared the simulations results with the experimental data reported by Bach et al. [J. Rheol. 47, 429 (2003)]. The simulations were run for monodisperse linear chains including up to 50 entanglements. The model is able to reasonably capture the rheological behavior of entangled systems in the nonlinear range. Based on our simulations, we also present an alternative explanation for the dynamics behind extensional thinning based on disentanglement and a physical explanation for the failure of standard models, i.e., Doi-Edwards (DE)/Doi-Edwards-Marrucci-Grizzuti (DEMG) (which predict a thinning exponent of −1) in predicting the extensional thinning exponent of near −0.5 (as seen in various experiments [Bach et al., J. Rheol. 47, 429 (2003); Luap et al., Rheol. Acta 45, 83 (2005)]).

A Microfluidic-based Hydrodynamic Trap for Single Particles

The ability to confine and manipulate single particles in free solution is a key enabling technology for fundamental and applied science. Methods for particle trapping based on optical, magnetic, electrokinetic, and acoustic techniques have led to major advancements in physics and biology ranging from the molecular to cellular level. In this article, we introduce a new microfluidic-based technique for particle trapping and manipulation based solely on hydrodynamic fluid flow. Using this method, we demonstrate trapping of micro- and nano-scale particles in aqueous solutions for long time scales. The hydrodynamic trap consists of an integrated microfluidic device with a cross-slot channel geometry where two opposing laminar streams converge, thereby generating a planar extensional flow with a fluid stagnation point (zero-velocity point). In this device, particles are confined at the trap center by active control of the flow field to maintain particle position at the fluid stagnation point. In this manner, particles are effectively trapped in free solution using a feedback control algorithm implemented with a custom-built LabVIEW code. The control algorithm consists of image acquisition for a particle in the microfluidic device, followed by particle tracking, determination of particle centroid position, and active adjustment of fluid flow by regulating the pressure applied to an on-chip pneumatic valve using a pressure regulator. In this way, the on-chip dynamic metering valve functions to regulate the relative flow rates in the outlet channels, thereby enabling fine-scale control of stagnation point position and particle trapping. The microfluidic-based hydrodynamic trap exhibits several advantages as a method for particle trapping. Hydrodynamic trapping is possible for any arbitrary particle without specific requirements on the physical or chemical properties of the trapped object. In addition, hydrodynamic trapping enables confinement of a "single" target object in concentrated or crowded particle suspensions, which is difficult using alternative force field-based trapping methods. The hydrodynamic trap is user-friendly, straightforward to implement and may be added to existing microfluidic devices to facilitate trapping and long-time analysis of particles. Overall, the hydrodynamic trap is a new platform for confinement, micromanipulation, and observation of particles without surface immobilization and eliminates the need for potentially perturbative optical, magnetic, and electric fields in the free-solution trapping of small particles.

Rheology and Structure of Molten, Olefin Multiblock Copolymers

Several samples of a recently developed olefin multiblock copolymer were studied by means of rheology, differential scanning calorimetry (DSC) and small-angle X-ray scattering (SAXS). The synthesis involves a chain shuttling agent (CSA) that switches the growing chain between two catalysts, one that favors the incorporation of an α-olefin comonomer and one that suppresses incorporation. The data were used to determine the effect of octene comonomer content and CSA level on rheological behavior and the occurrence of mesophase separation transition (MST) in the melt. To distinguish between crystallization and MST, we made calorimetry scans and measured the density and rheological properties over a range of temperatures. Small angle X-ray scattering analysis of a sample that had undergone planar extensional flow revealed strong alignment of lamellar mesodomains, which maintained their alignment after annealing. This result confirmed the hypothesis based on rheological evidence that a lamellar mesophase is pr...

Extensional rheology of active suspensions

A simple model is presented for the effective extensional rheology of a dilute suspension of active particles, such as self-propelled microswimmers, extending previous classical studies on suspensions of passive rodlike particles. Neglecting particle-particle hydrodynamic interactions, we characterize the configuration of the suspension by an orientation distribution, which satisfies a Fokker-Planck equation including the effects of an external flow field and of rotary diffusion. Knowledge of this orientation distribution then allows the determination of the particle extra stress as a configurational average of the force dipoles exerted by the particles on the fluid, which involve contributions from the imposed flow, rotary diffusion, and the permanent dipoles resulting from activity. Analytical expressions are obtained for the stress tensor in uniaxial extensional and compressional flows, as well as in planar extensional flow. In all types of flows, the effective viscosity is found to increase as a result of activity in suspensions of head-actuated swimmers (pullers) and to decrease in suspensions of tail-actuated swimmers (pushers). In the latter case, a negative particle viscosity is found to occur in weak flows. In planar extensional flow, we also characterize normal stresses, which are enhanced by activity in suspensions of pullers but reduced in suspensions of pushers. Finally, an energetic interpretation of the seemingly unphysical decrease in viscosity predicted in suspensions of pushers is proposed, where the decrease is explained as a consequence of the active power input generated by the swimming particles and is shown not to be directly related to viscous dissipative processes.

Numerical study of the rheology of rigid fillers suspended in long-chain branched polymer under planar extensional flow

We report a detailed numerical study of the rheology of two-dimensional rigid fillers suspended in branched polymer melt under planar extensional flow. The polymer melt is modelled using the pom-pom constitutive equation. The numerical method uses a finite element solution of the flow in a unit cell within the self-replicating lattice for planar extensional flow identified by Kraynik and Reinelt [Int. J. Multiphase flow 18 (1992) 1045]. This is implemented using a quotient space representation that maps all points in space onto points within the unit cell. We show that the Kraynik and Reinelt cell allows simulations of suspensions under planar extensional flow to be conducted to large strains in a truly periodic cell. The influence of both the pom-pom parameters and the particle area fraction on the rheology of the suspension are investigated. We find a reduction in the degree of extension-rate thickening with the increase of both particles concentration and Weissenberg numbers in agreement with published experimental and numerical findings on other viscoelastic models. This reduction is found to be due to flow disturbance created by the particles which disrupts the alignment of backbone tube segments with the extensional axis.

Single-molecule sequence detection via microfluidic planar extensional flow at a stagnation point.

We demonstrate the use of a microfluidic stagnation point flow to trap and extend single molecules of double-stranded (ds) genomic DNA for detection of target sequences along the DNA backbone. Mutant EcoRI-based fluorescent markers are bound sequence-specifically to fluorescently labeled ds lambda-DNA. The marker-DNA complexes are introduced into a microfluidic cross slot consisting of flow channels that intersect at ninety degrees. Buffered solution containing the marker-DNA complexes flows in one channel of the cross slot, pure buffer flows in the opposing channel at the same flow rate, and fluid exits the two channels at ninety degrees from the inlet channels. This creates a stagnation point at the center of a planar extensional flow, where marker-DNA complexes may be trapped and elongated along the outflow axis. The degree of elongation can be controlled using the flow strength (i.e., a non-dimensional flow rate) in the device. Both the DNA backbone and the markers bound along the stretched DNA are observed directly using fluorescence microscopy, and the location of the markers along the DNA backbone is measured. We find that our method permits detection of each of the five expected target site positions to within 1.5 kb with standard deviations of <1.5 kb. We compare the method's precision and accuracy at molecular extensions of 68% and 88% of the contour length to binding distributions from similar data obtained via molecular combing. We also provide evidence that increased mixing of the sample during binding of the marker to the DNA improves binding to internal target sequences of dsDNA, presumably by extending the DNA and making the internal binding sites more accessible.