Finitely Extendable Nonlinear Elastic Research Articles

Steady-state extensional viscosity of a linear polymer solution using a differential pressure extensional rheometer on a chip

In our earlier theoretical work [Lee and Muller, J. Rheol. 61, 1049–1059 (2017)], we proposed a design for a differential pressure extensional rheometer (DPER) on a chip. Here, we implement the DPER to evaluate the steady-state viscosity of a semidilute poly(ethylene oxide) solution at high and low extension rates. At low extension rates, the extensional viscosity exhibits strain thinning behavior with a power-law exponent of −0.5. At intermediate extension rates, the extensional viscosity exhibits strain thickening. At high extension rates, the extensional viscosity plateau has been estimated, and the corresponding finite extendable nonlinear elastic (FENE) constant is evaluated as 482. A novel method to determine the FENE constant and the extensional relaxation time distribution is presented, which are key parameters for the understanding of the extensional flow of a linear polymer solution.

A multiscale SPH method for simulating transient viscoelastic flows using bead-spring chain model

In this article, a multiscale smoothed particle hydrodynamics (SPH) method is proposed to simulate transient viscoelastic flows by using a bead-spring chain description of polymer molecule. This methodology couples macroscopic conservation equations for mass and momentum with a stochastic differential equation for bead-spring chain dynamics, which can be solved by a three-step semi-implicit algorithm to obtain the polymeric stress. Accordingly, a closed-form constitutive equation is not required. The multiscale SPH method is firstly verified by the plane Couette flow with Hookean and finitely extendable non-linear elastic (FENE) bead-spring chain models with the number of beads M=2, which correspond to Hookean and FENE dumbbell models, respectively. Results for the velocity and shear stress are in excellent agreement with the available numerical and analytical solutions in literature. Then, the numerical method is extended to simulate the plane Couette flow and Poiseuille flow with FENE bead-spring chain model with M >2. In particular, some molecular information of bead-spring chain such as the molecular stretch, the orientation angle and the mean configuration thickness are displayed. The influence of the number of beads M on the evolutions of the molecular information is also analyzed in detail. It is demonstrated that the multiscale SPH method proposed here is a promising computational tool for the multiscale simulations of transient viscoelastic flows.

Advances in the analysis and prediction of turbulent viscoelastic flows

It has been well-known for over six decades that the addition of minute amounts of long polymer chains to organic solvents, or water, can lead to significant turbulent drag reduction. This discovery has had many practical applications such as in pipeline fluid transport, oil well operations, vehicle design and submersible vehicle projectiles, and more recently arteriosclerosis treatment. However, it has only been the last twenty-five years that the full utilization of direct numerical simulation of such turbulent viscoelastic flows has been achieved. The unique characteristics of viscoelastic fluid flow are dictated by the nonlinear differential relationship between the flow strain rate field and the extra-stress induced by the additive polymer. A primary motivation for the analysis of these turbulent fluid flows is the understanding of the effect on the dynamic transfer of energy in the turbulent flow due to the presence of the extra-stress field induced by the presence of the viscoelastic polymer chain. Such analyses now utilize direct numerical simulation data of fully developed channel flow for the FENE-P (Finite Extendable Nonlinear Elastic – Peterlin) fluid model. Such multi-scale dynamics suggests an analysis of the transfer of energy between the various component motions that include the turbulent kinetic energy, and the mean polymeric and elastic potential energies. It is shown that the primary effect of the interaction between the turbulent and polymeric fields is to transfer energy from the turbulence to the polymer.

An Analysis of Peristaltic Flow of Finitely Extendable Nonlinear Elastic- Peterlin Fluid in Two-Dimensional Planar Channel and Axisymmetric Tube

We have investigated the peristaltic motion of a non-Newtonian fluid characterized by the finitely extendable nonlinear elastic-Peterlin (FENE-P) fluid model. A background for the development of the differential constitutive equation of this model has been provided. The flow analysis is carried out both for two-dimensional planar channel and axisymmetric tube. The governing equations have been simplified under the widely used assumptions of long wavelength and low Reynolds number in a frame of reference that moves with constant wave speed. An exact solution is obtained for the stream function and longitudinal pressure gradient with no slip condition. We have portrayed the effects of Deborah number and extensibility parameter on velocity profile, trapping phenomenon, and normal stress. It is observed that normal stress is an increasing function of Deborah number and extensibility parameter. As far as the velocity at the channel (tube) center is concerned, it decreases (increases) by increasing Deborah number (extensibility parameter). The non-Newtonian rheology also affect the size of trapped bolus in a sense that it decreases (increases) by increasing Deborah number (extensibility parameter). Further, it is observed through numerical integration that both Deborah number and extensibility parameter have opposite effects on pressure rise per wavelength and frictional forces at the wall. Moreover, it is shown that the results for the Newtonian model can be deduced as a special case of the FENE-P model

SPH simulations of 2D transient viscoelastic flows using Brownian configuration fields

Based on a micro–macro approach, the smoothed particle hydrodynamics (SPH) method is developed to simulate 2D transient viscoelastic fluid flows, in which the polymer dynamics is described by the evolution of a number of continuous Brownian configuration fields. This approach combines a macroscopic description of the kinematics with a coarse-grained microscopic description of the evolution of the extra-stress, and accordingly a closed-form constitutive equation is not necessary any longer. In order to validate our approach, both Poiseuille flow and Couette flow using Hookean dumbbell model are first studied. Results for the velocity, the shear stress and the first normal stress difference are in good agreement with the reference data from the literature. The simulations are also extended to the finitely extendable non-linear elastic (FENE) dumbbell model. In particular, the influence of dimensionless finite extensibility parameter b on the evolutions of the velocity and the polymer stress tensor is analyzed in detail. Results for the lid driven cavity flow of FENE dumbbell model are also shown. All numerical results demonstrate that the SPH method is a powerful and flexible tool for meshfree simulations of 2D transient viscoelastic fluid flows using Brownian configuration fields.

Numerical solution of the FENE-CR model in complex flows

A finite difference technique for solving the FENE-CR (Finite Extendable Non-linear Elastic – Chilcott and Rallison) closure constitutive model in complex flows has been developed and tested. The governing equations are solved using a Marker-and-Cell type method on a staggered grid. The momentum equation is integrated employing an implicit method while the FENE-CR constitutive equation is approximated by a second-order Runge–Kutta scheme. To demonstrate that the numerical technique can cope with complex flows governed by the FENE-CR model, three flow problems were analysed: the fully-developed channel flow, the 2D cross-slot flow and the impacting drop problem. The analytic solution for fully-developed channel flow of FENE-CR fluids with a solvent viscosity is also presented for validation purposes. This flow problem is used to verify the numerical method and to quantify its accuracy by comparing numerical results of fully-developed channel flow with the analytic solution. The second flow is employed to assess whether the numerical methodology is capable of capturing the purely-elastic instabilities predicted in the literature for 2D cross-slot confined flows. Additionally, the complex free surface flow corresponding to the filling of a 2D cross geometry has also been investigated. The last problem concerns the flow dynamics of a FENE-CR fluid drop impacting on a rigid surface, which allows the assessment of the capability of the model to deal with free surfaces. The effects of varying the Reynolds number, the Weissenberg number and the finite extensibility of the polymer molecules (L2) on the resulting flow patterns are analysed.

Relaxation Process and Dynamical Heterogeneities in Chemical Gels: Critical Behavior of Self-Overlap and Its Fluctuation

We study the dynamical behavior in chemical gelation, as the gelation threshold is approached from the sol phase. On the basis of the heterogeneous diffusion due to the cluster size distribution, as expected by the percolation theory, we predict the long time decay of the self-overlap as a power law in time t(-3/2). Moreover, under the hypothesis that the cluster diffusion coefficient decreases in size as a power law, s(-x), the fluctuation of the self-overlap, χ(4)(t), exhibits growth at short time as t((3-τ)/x), where τ is the cluster size distribution critical exponent. At longer times, χ(4)(t) decays as t(-3/2) while, at intermediate times, it reaches a maximum at time t*, which scales as s*(x), where s* is the size of the critical cluster. Finally, the value of the maximum χ(4)(t*) scales as the mean cluster size. The theoretical predictions are in agreement with molecular dynamic calculations in a model system, where spherical monomers are bonded by a finite extendable nonlinear elastic (FENE) potential.

On extensibility effects in the cross-slot flow bifurcation

The flow of finite-extensibility models in a two-dimensional planar cross-slot geometry is studied numerically, using a finite-volume method, with a view to quantifying the influences of the level of extensibility, concentration parameter, and sharpness of corners, on the occurrence of the bifurcated flow pattern that is known to exist above a critical Deborah number. The work reported here extends previous studies, in which the viscoelastic flow of upper-convected Maxwell (UCM) and Oldroyd-B fluids (i.e. infinitely extensionable models) in a cross-slot geometry was shown to go through a supercritical instability at a critical value of the Deborah number, by providing further numerical data with controlled accuracy. We map the effects of the L 2 parameter in two different closures of the finite extendable non-linear elastic (FENE) model (the FENE-CR and FENE-P models), for a channel-intersecting geometry having sharp, “slightly” and “markedly” rounded corners. The results show the phenomenon to be largely controlled by the extensional properties of the constitutive model, with the critical Deborah number for bifurcation tending to be reduced as extensibility increases. In contrast, rounding of the corners exhibits only a marginal influence on the triggering mechanism leading to the pitchfork bifurcation, which seems essentially to be restricted to the central region in the vicinity of the stagnation point.

The role of entanglements on the stability of microphase separated diblock copolymers in shear flow

Various possible orientations of lamellar structures of diblock copolymers under shear are investigated with respect to their stability. A Brownian dynamics model is put forward in which each diblock is modeled as a dumbbell. The blobs in each dumbbell are held together by a finite extendable nonlinear elastic (FENE) potential and interact with all surrounding blobs by soft dissipative particle dynamics (DPD) potentials. In addition to this, the blobs have the possibility to entangle with each other. The corresponding interactions lead to large viscosities, which, however, quickly diminish with increasing shear rate. This fact turns out to be crucial for the stabilization of structures with the lamellae parallel to the velocity-vorticity plane. As a second result it is found that asymmetry in the entanglement interactions stimulates the actual reorientation into this state.

Formation of double helical and filamentous structures in models of physical and chemical gels.

This discusses two recent models, one that captures physical network formation starting from the molecular architecture of its constituents and another that contains the basic features of phase separation in cross-linked polymer gels: A) the Janus chain (multibead bead-spring type) model exhibiting semiflexibility and induced curvature and B) a stretched elastic network of Lennard-Jones particles. The length scales and related structures predicted by the two generic models are different. Model B, a generic soft solid model, exhibits hysteresis and the formation of filamentous structures in two dimensions. The Janus chain model A is able to describe the process of the formation of double helical superstructures, will be operated in three dimensions, and its internal parameters are directly deduced from atomistic simulation. Both models rely on classical ingredients which have been separately studied extensively: i) the Lennard-Jones particle system, ii) the elastic solid, and iii) the FENE-B model for semiflexible, finitely extendable nonlinear elastic (FENE) polymer chains. While model A combines i) and iii), model B combines i) and ii). This aspect of technical simplicity, however, is contrasted by the rich phenomenology observed for these models. The Janus model even resolves structure formation on the molecular scale. Intriguingly, the coarse dynamical models capture a wide range of superstructures known for polymeric networks and therefore clearly serve to understand their underlying physical mechanisms.

Simulation of 2D transient viscoelastic flow using the CONNFFESSIT approach

In this paper, a modified form of the time-dependent DEVSS-G/SUPG scheme associated with the penalty function method was developed to simulate the viscoelastic flow under the high Weissenberg number by using the Calculation of Non-Newtonian Flow: Finite Elements and Stochastic Simulation Technique (CONNFFESSIT) approach. The element condensation method is used to eliminate the velocity gradient term and still maintain good stability. The variance reduction method was also taken for the Brownian configuration field (BCF) equation to reduce the noise level in calculating the polymer stresses. A planar 4:1 abrupt contraction flow was taken as a benchmark case to verify this modified scheme. The stress contours and vortex evolution with the Hookean dumbbell model were compared with the literature results. The simulation was also extended to the finitely extendable non-linear elastic (FENE) dumbbell model. The second case is the planar flow around a confined cylinder at non-zero Reynolds and Weissenberg numbers. Both the inertial and viscoelastic effects were examined in the second case. For the Hookean and FENE dumbbell models, it was found that without the convective term, the streamlines shift upstream behind the cylinder compared with the Newtonian fluid flow, while an opposite result is observed at a high Reynolds number (i.e., Re = 10).

Microchannel flow of a macromolecular suspension

In the delivery of DNA molecules by microfluidic devices, the channel width is very often in the same order as the size of the DNA molecules and the applicability of continuum mechanics at this level may be questioned. In this paper we use finitely extendable nonlinear elastic (FENE) chains to model the DNA molecules and employ the dissipative particle dynamics (DPD) method to simulate their behavior in the flow. Simple DPD fluids are found to behave just like a Newtonian fluid in Poiseuille flow. However, the velocity profiles of FENE chain suspensions can be fitted with power-law curves, especially for dilute suspensions. Some results on the conformation and migration of FENE chains are also reported.

Structure and Dynamics of Dilute Polymer Solutions under Shear Flow via Nonequilibrium Molecular Dynamics

We present and discuss results obtained by an extensive nonequilibrium molecular dynamics computer simulation study of polymer solutions under shear, where the chain consists of N beads connected by a finitely extendable nonlinear elastic (FENE) spring force and the solvent is explicitly taken into account. Various scaling laws are extracted from the data which allow one to predict the qualitativeand to certain extent also quantitativestructural and rheological behavior of polymer solutions under good solvent conditions. For most quantities, the results drawn from simulation are compared with experimental data and theoretical predictions which are based on similar models (e.g., harmonic bond potentials or Brownian dynamics methods). Specifically, and in contrast to common theoretical approaches, the simulation yield information about a set of different but characteristic relaxation times, which determine the rheological and structural behavior (for example, flow birefringence, structure factor, rotational dynamics) separately, the difference either resulting from the underlying static or dynamic nature or from relaxation processes which act on different length scales.

Finite-Element simulation of the start-up problem for a viscoelastic fluid in an eccentric rotating cylinder geometry using a third-order upwind scheme

We numerically simulate the flow field of a dilute polymeric solution using a finitely extendable nonlinear elastic (FENE) dumbbell model. A third-order accurate finite element upwind scheme is used to discretize the convection term in the FENE dumbbell equations for the configuration tensor. The numerical scheme also avoids unphysical negative values for diagonal components of the configuration tensor. The FENE dumbbell equations are solved along with the momentum and continuity equations at small Reynolds numbers with an accuracy of second order in time. In this work we apply this numerical technique to the motion of a viscoelastic fluid in an eccentric rotating cylinder geometry. We obtain the velocity and the polymer contribution to the stress fields as a function of time, and also examine the steady solutions. A particular focus is the influence of coupling between changes in polymer conformation and changes in the flow that occurs as the polymer concentration is increased to a level where the polymer contribution to the zero-shear viscosity of the solution is equal to that of the solvent.

Circular Couette flow of concentrated polymer solutions: Encapsulated finitely extendable nonlinear elastic (FENE) dumbbell model results

The diffusion equation based on the encapsulated FENE model concept is applied to circular Couette flow. In all cases studied, a nonuniform concentration distribution is predicted, decreasing with increasing distance from the inner cylinder. For highly dilute solutions the degree of nonuniformity increases with increasing shear rate and increasing flexibility. For concentrated solutions the data reported in the literature are used. In contrast to earlier tube flow predictions, all the systems studied behave qualitatively in the same way. Up to the limit of vortex formation the theory presented should be valid. The implications of nonuniform concentration profiles for normal stress measurements are outlined.

Conformational changes of macromolecules in transient elongational flow

AbstractCalculations are reported which describe the elongation of flexible macromolecules in a pulse‐like elongational How. The polymer molecules are modeled as FENE (finitely extendable nonlinear elastic) dumbbells. The results are in qualitative agreement with experimental results on dilute solutions of polyisobutylene in decalin (10) which form a two parameter family of extension versus position curves. The two parameters‐in the FENE model that are used to fit the data are λH, a time constant and b, which is related to the maximum extension that can be achieved by the molecules. Quantitative comparisons are frustrated by difficulty in estimating the model parameter, b. It is suggested, based on this work, that internal self entanglements in the polymers must be considered in the determination of b.