Prediction of the Rheological Properties of Dilute Polymer Solutions with Brownian Dynamics Simulations

Significant Achievements, Anticipated Outcomes and Future Work

Micro-macro simulations of complex flows:
Free surface flows occur when a layer of liquid meets a gas at an interface. Such flows arise in a variety of commercial applications, such as coating (e.g. slot coating, roll coating etc.), ink-jet printing, fiber spinning, and micropipetting. Frequently, these applications involve liquids that are viscoelastic due to the presence of polymer as final product (e.g. coating) or as rheology modifier (e.g. ink-jet printing). Most of these flows are time dependent and their dynamics is controlled by the elasticity and capillarity of the liquid. Modelling such flows requires computational methods which can describe and predict the molecular conformation of polymers in the flow field while simultaneously capturing accurately the shape of free surfaces. A micro-macro approach based on combining the Brownian configuration fields (BCF) method with an Arbitrary Lagrangian-Eulerian (ALE) Galerkin finite element method, using elliptic mesh generation equations coupled with time - dependent conservation equations, is applied to study slot coating flows of polymer solutions. The polymer molecules are represented by dumbbells with both linear and non-linear springs; hydrodynamic interactions between beads are incorporated. Calculations with infinitely extensible (Hookean) and pre-averaged finitely extensible (FENE-P) dumbbell models are performed and compared with equivalent closed-form macroscopic models in a conformation tensor based formulation. The BCF equation for linear dumbbell models is solved using a fully implicit time integration scheme which is found to be more stable than the explicit Euler scheme used previously to compute complex flows. We find excellent agreement between the results of the BCF based formulation and the macroscopic conformation tensor based formulation. The computations using the BCF approach are stable at much higher Weissenberg numbers compared to the purely macroscopic conformation tensor based approach, which fail beyond a maximum Wi. A novel computational algorithm is introduced to compute complex flows with non-linear microscopic constitutive models (i.e. non-linear FENE dumbbells and dumbbells with hydrodynamic interactions) for which no closed-form constitutive equations exist. This algorithm is fast and computationally efficient when compared to both an explicit scheme and a fully implicit scheme involving the solution of the non-linear equations with Newton’s method for each configuration field. Fast and accurate differential constitutive models for dilute polymer solutions:
Understanding the viscoelastic nature of dilute polymer solutions is crucial to the optimization of applications such as turbulent drag reduction, and for the control of instabilities observed in processes like roll coating, etc. Unfortunately, quantitative prediction of the behaviour of polymer solutions in complex flows cannot at present be achieved using standard computational fluid dynamics. This is so because typical constitutive equations currently in use for relating the stress due to the polymer to the velocity gradient do not fully capture the intricacies of the behaviour of isolated polymer molecules in solution. In this Project, we have developed several new approximate models that address this lacuna. The models account for two key phenomena that dominate the dynamics of dilute polymer solutions in strong flows: the finite extensibility of a molecule, and the existence of fluctuating hydrodynamic interactions between its different parts. These models have shown to lead to differential constitutive equations for the polymer contribution to the total fluid stress in a dilute polymer solution undergoing deformation. The predictions obtained are in excellent agreement with the results of Brownian dynamics simulations for dynamic properties of dilute polymer solutions in strong shear and extensional flows. A highlight of the new constitutive models is that they are able to accurately replicate the existence of the phenomenon of coil-stretch hysteresis, wherein multiple steady-state values are observed for macroscopic properties such as the polymer stress. With these improved models, we have been able to show that the phenomenon of coil-stretch hysteresis plays an important role in the necking and break-up of thin filaments of dilute polymer solutions. Future Work:
1) Thus far, the micro-macro approach has relied extensively on simulations of simple dumbbell models of polymer molecules. We plan to extend the algorithms developed in this Project to more realistic simulations of bead-spring chains in complex flows. It is also possible to use dumbbell models with a tensorial configuration-dependent friction coefficient to mimic the behaviour of a long polymer molecule. Such a model is likely to be useful in exploring the influence of coil-stretch hysteresis on free-surface flows. 2) The differential constitutive models developed in the Project do not account for the influence of excluded volume interactions. Equations for a model simultaneously incorporating finite-extensibility, hydrodynamic and excluded volume interactions have been developed, and will be implemented numerically.

Computational Techniques Used

A polymer molecule in solution can be thought of as a long, slender, flexible string-like object whose motion is driven by the frictional drag and Brownian forces exerted by the surrounding solvent medium. This motion occurs under three important constraints: (1) the chain cannot be stretched beyond its maximum permissible length, (2) the chain cannot cross itself, and (3) the motion of any part of the chain is communicated to all other parts of the chain by the solvent medium. The Brownian dynamics simulations employed in this Project require the numerical integration of multi-dimensional, stochastic differential equations (SDE's). Besides being highly non-linear, these SDE's can be very stiff while simulating strong extensional flows, since the polymer configuration can be highly stretched. To handle the problems of numerical stability that can arise in such situations, a stochastic-adaptive time-step algorithm using a novel combination of predictor-corrector and semi-implicit methods is used to integrate the SDE's. The accurate prediction of the macroscopic properties of the system requires the calculation of averages over large ensembles of polymer chains. Since the solution is assumed to be dilute, these molecules are essentially isolated systems, and can be simulated independently. Closure approximations result in large sets of coupled nonlinear ordinary differential equations. These are solved using efficient algorithms for stiff systems provided in the NAG library. Micro-macro simulations are based on algorithms that build on and extend the efficient finite-element code for handling complex free-surface flows developed in-house by Matteo Pasquali and others at Rice University, Houston.

Publications, Awards and External Funding

External Funding and Awards

1. ARC Discovery: T. Sridhar, R.P. Jagadeeshan, "The flow properties of proteins and other biopolymers"
2. ARC Linkage-International: R.P. Jagadeeshan, T. Sridhar, E.S.G. Shaqfeh, "DNA Dynamics in Shear and Extensional Flows: Simulation and Single Molecule Experiments"
3. ARC Discovery: T. Sridhar, R.P. Jagadeeshan, M. Pasquali, E.S.G. Shaqfeh, "Understanding the Behaviour of SWNTs in Liquids"
4. ARC LEIF: Total of 20 investigators, "Biomedical Engineering Sensing and Imaging Facility"

Publications

1. Prabhakar, R and J. R. Prakash, "Gaussian approximation for finitely extensible bead-spring chains with hydrodynamic interaction", Accepted for publication in the Journal of Rheology.
2. M. Bajaj, P. P. Bhat, J. R. Prakash and M. Pasquali, "Multiscale simulation of viscoelastic free-surface flows", Accepted for publication in the Journal of Non-Newtonian Fluid Mechanics.
3. Prabhakar, R, J. R. Prakash and T. Sridhar, "Effect of configuration-dependent intramolecular hydrodynamic interaction on elasto-capillary thinning and break-up of filaments of dilute polymer solutions", Submitted to the Journal of Rheology.
4. R. Prabhakar and J. R. Prakash, 2005, “Fast and accurate closure approximations for bead-spring models of dilute polymer solutions”, ANZIAM J., 46 (E), pp. C379-C393.