Studies in Rheology: Molecular Simulation and Theory

With an enormous advance in the capability of computers during the last fewdecades, the computer simulation has become an important tool for scientific researches in many areas such as physics, chemistry, biology, and so on. In particular, moleculardynamics (MD) simulations have been proven to be of a great help in understanding the rheology of complex fluids from the fundamental microscopic viewpoint. There are two important standard flows in rheology: shear flow and elongational flow. While there exist suitable nonequilibrium MD (NEMD) algorithms of shear flows, such as the Lees-Edwards purely boundary-driven algorithm and the so-called SLLOD algorithm as a field-driven algorithm, a proper NEMD algorithm for elongational flow has been lacking. The main difficulty of simulating elongational flow lies in the limited simulation time available due to the contraction of one or two dimensions dictated by itskinematics. This problem, however, has been partially resolved by Kraynik and Reinelt’s ingenious discovery of the temporal and spatial periodicity of lattice vectors in planar elongational flow (PEF). Although there have been a few NEMD simulations of PEF using their idea, another serious defect has recently been reported when using the SLLOD algorithm in PEF: for adiabatic systems, the total linear momentum of the system in the contracting direction grows exponentially with time, which eventually leads to an aphysical phase transition.This problem has been completely resolved by using the so-called ‘proper-SLLOD’ or ‘p-SLLOD’ algorithm, whose development has been one of the mainaccomplishments of this study. The fundamental correctness of the p-SLLOD algorithm has been demonstrated quite thoroughly in this work through detailed theoretical analyses together with direct simulation results. Both theoretical and simulation works achieved in this research are expected to play a significant role in advancing the knowledge of rheology, as well as that of NEMD simulation itself for other types of flow in general. Another important achievement in this work is the demonstration of the possibility of predicting a liquid structure in nonequilibrium states by employing a concept of ‘hypothetical’ nonequilibrium potentials. The methodology developed in this work has been shown to have good potential for further developments in this field

Analysis of the configurational temperature of polymeric liquids under shear and elongational flows using nonequilibrium molecular dynamics and Monte Carlo simulations

We present a detailed analysis of the configurational temperature (T(conf)) for its application to polymeric materials using nonequilibrium molecular dynamics (NEMD) and nonequilibrium Monte Carlo (NEMC) methods. Simulations were performed of linear polyethylene liquid C(78)H(158) undergoing shear and elongational flows. At equilibrium, T(conf) is equal to the set point temperature of the simulation. An aphysically large decrease in T(conf) is observed in the NEMD simulations for both flows, especially at strong flow fields. By analyzing separately the individual contributions of the different potential interaction modes to the configurational temperature, it is found that the bonded modes (which constitutes almost 99.5% of the total) dominate the total T(conf) over the nonbonded ones; i.e., bond-stretching (approximate to 86.5%), bond-bending (approximate to 11.8%), bond-torsional (approximate to 1.2%), nonbonded intermolecular (approximate to 0.4%), and intramolecular (approximate to 0.1%) Lennard-Jones. The configurational temperature of the individual modes generally exhibits a nonmonotonic behavior with the flow strength and a dramatic change beyond a critical value of flow strength; this is mainly attributed to the dynamical effect of strong molecular collisions occurring at strong flow fields. In contrast, no such behavior is observed in the NEMC simulations where such dynamical effects are absent. Based on the principal physical concept of the configurational temperature, which represents the large-scale structural characteristics of the system, we propose to exclude the dynamical effects exhibited by the individual interaction modes, in obtaining a physically meaningful T(conf) as the configurational entropy of the system should not be affected by such factors. Since (a) the main difference between equilibrium and nonequilibrium states lies in the change in the overall (global) structure (represented by the bond torsional and nonbonded modes), and (b) the local, very short structure (represented by the bond-stretching and bond-bending modes) is barely changing between equilibrium and nonequilibrium states and its contribution to the total system configurational entropy is negligible compared to the large-scale structural changes, in order to accurately describe the structural changes occurring at nonequilibrium states by use of the configurational temperature, we further propose that only the contributions from the bond-torsional and nonbonded modes to Delta T(conf) between equilibrium and nonequilibrium states should be taken into account to generate a physically meaningful Delta T(conf). Applying the above hypothesis to the analysis of the simulation data, good agreement between the NEMD and NEMC simulations (and between NEMD simulations for different flows) is observed. Furthermore, the configurational temperature obtained in such way is found to match remarkably well with the heat capacity of amorphous polyethylene liquids and the flow-enhanced melting-point elevation reported in experiment.open2

Tension thickening, molecular shape, and flow birefringence of an H-shaped polymer melt in steady shear and planar extension

Despite recent advances in the design of extensional rheometers optimized for strain and stress controlled operation in steady, dynamic, and transient modes, obtaining reliable steady-state elongational data for macromolecular systems is still a formidable task, limiting today's approach to trial-and-error efforts rather than based on a deep understanding of the deformation processes occurring under elongation. Guided, in particular, by the need to understand the special rheology of branched polymers, we studied a model, unentangled H-shaped polyethylene melt using nonequilibrium molecular dynamics simulations based on a recently developed rigorous statistical mechanics algorithm. The melt has been simulated under steady shear and steady planar extension, over a wide range of deformation rates. In shear, the steady-state shear viscosity is observed to decrease monotonically as the shear rate increases; furthermore, the degree of shear thinning of the viscosity and of the first- and second-normal stress coefficients is observed to be similar to that of a linear analog of the same total chain length. By contrast, in planar extension, the primary steady-state elongational viscosity ??1 is observed to exhibit a tension-thickening behavior as the elongation rate ?? increases, which we analyze here in terms of (a) perturbations in the instantaneous intrinsic chain shape and (b) differences in the stress distribution along chain contour. The maximum in the plot of ??1 with ?? occurs when the arm-stretching mode becomes active and is followed by a rather abrupt tension-thinning behavior. In contrast, the second elongational viscosity ??2 shows only a tension-thinning behavior. As an interesting point, the simulations predict the same value for the stress optical coefficient in the two flows, revealing an important rheo-optical characteristic. In agreement with experimental indications on significantly longer systems, our results confirm the importance of chain branching on the unique rheological properties of polymer melts in extension.open5

Structure of a sheared soft-disk fluid from a nonequilibrium potential

The distortion of structure of a simple, inverse-12, soft-disk fluid undergoing two-dimensional plane Couette flow was studied by nonequilibrium molecular dynamics (NEMD) simulation and by equilibrium Monte Carlo (MC) simulation with a nonequilibrium potential, under which the equilibrium structure of the fluid is that of the nonequilibrium fluid. Extension of the iterative predictor-corrector method of [Reatto Phys. Rev. A 33 3451 (1986)] was used to extract the nonequilibrium potential with the structure input from the NEMD simulation. Very good agreement for the structural properties and pressure tensor generated by the NEMD and MC simulation methods was found, thus providing the evidence that nonequilibrium liquid structure can be accurately reproduced via simple equilibrium simulations or theories using a properly chosen nonequilibrium potential. The method developed in the present study and numerical results presented here can be used to guide and test theoretical developments, providing them with the "experimental" results for the nonequilibrium potential.open2

A proper approach for nonequilibrium molecular dynamics simulations of planar elongational flow

We present nonequilibrium molecular dynamics simulations of planar elongational flow (PEF) by an algorithm proposed by Tuckerman et al. [J. Chem. Phys. 106, 5615 (1997)] and theoretically elaborated by Edwards and Dressier [J. Non-Newtonian, Fluid Mech. 96, 163 (2001)], which we shall call the proper-SLLOD algorithm, or p-SLLOD for short. [For background on names of algorithms see W. G. Hoover, D. J. Evans, R. B. Hickman, A. J. C. Ladd, W. T. Ashurst, and B. Moran, Phys. Rev. A 22, 1690 (1980) and D. J. Evans and G. P. Morriss, Phys. Rev. A 30, 1528 (1984).] We show that there are two sources for the exponential growth in PEF of the total linear momentum of the system in the contracting direction, which has been previously observed using the so-called SLLOD algorithm. The first comes from the SLLOD algorithm itself, and the second from the implementation of the Kraynik and Reinelt [Int. J. Multiphase Flow 18, 1045 (1992)] boundary conditions. Using the p-SLLOD algorithm (to eliminate the first source) implemented with our simulation strategy (to eliminate the second) in PEF simulations, we no longer observe the exponential growth. By analyzing the equations of motion, we also demonstrate that both the SLLOD and the DOLLS algorithms are intrinsically unsuitable for representing a nonequilibrium system with elongational flow. However, the p-SLLOD algorithm has a rigorously canonical structure in laboratory phase space, and thus can represent a nonequilibrium system not only for elongational flow but also for a general flow.open313

An examination of the validity of nonequilibrium molecular-dynamics simulation algorithms for arbitrary steady-state flows

Nonlinear-response theory of nonequilibrium molecular-dynamics simulation algorithms is considered under the imposition of an arbitrary steady-state flow field. It is demonstrated that the SLLOD and DOLLS algorithms cannot be used for general flows, although the SLLOD algorithm is rigorous for planar Couette flow. Following the same procedure used to establish SLLOD as the valid algorithm for planar Couette flow [D. J. Evans and E. P. Morriss, Phys. Rev. A 30, 1528 (1984)], it is demonstrated that the p-SLLOD algorithm is valid for arbitrary flows and produces the correct nonlinear response of the viscous pressure tensor. (c) 2005 American Institute of Physics.close161

Structural and Dynamical Characteristics of Short-Chain Branched Ring Polymer Melts at Interface under Shear Flow

We present a detailed analysis of the interfacial chain structure and dynamics of confined polymer melt systems under shear over a wide range of flow strengths using atomistic nonequilibrium molecular dynamics simulations, paying particular attention to the rheological influence of the closed-loop ring geometry and short-chain branching. We analyzed the interfacial slip, characteristic molecular mechanisms, and deformed chain conformations in response to the applied flow for linear, ring, short-chain branched (SCB) linear, and SCB ring polyethylene melts. The ring topology generally enlarges the interfacial chain dimension along the neutral direction, enhancing the dynamic friction of interfacial chains moving against the wall in the flow direction. This leads to a relatively smaller degree of slip (ds) for the ring-shaped polymers compared with their linear analogues. Furthermore, short-chain branching generally resulted in more compact and less deformed chain structures via the intrinsically fast random motions of the short branches. The short branches tend to be oriented more perpendicular (i.e., aligned in the neutral direction) than parallel to the backbone, which is mostly aligned in the flow direction, thereby enhancing the dynamic wall friction of the moving interfacial chains toward the flow direction. These features afford a relatively lower ds and less variation in ds in the weak-to-intermediate flow regimes. Accordingly, the interfacial SCB ring system displayed the lowest ds among the studied polymer systems throughout these regimes owing to the synergetic effects of ring geometry and short-chain branching. On the contrary, the structural disturbance exerted by the highly mobile short branches promotes the detachment of interfacial chains from the wall at strong flow fields, which results in steeper increasing behavior of the interfacial slip for the SCB polymers in the strong flow regime compared to the pure linear and ring polymers

Quantifying chain reptation in entangled polymer melts: Topological and dynamical mapping of atomistic simulation results onto the tube model

The topological state of entangled polymers has been analyzed recently in terms of primitive paths which allowed obtaining reliable predictions of the static (statistical) properties of the underlying entanglement network for a number of polymer melts. Through a systematic methodology that first maps atomistic molecular dynamics (MD) trajectories onto time trajectories of primitive chains and then documents primitive chain motion in terms of a curvilinear diffusion in a tubelike region around the coarse-grained chain contour, we are extending these static approaches here even further by computing the most fundamental function of the reptation theory, namely, the probability ?? (s,t) that a segment s of the primitive chain remains inside the initial tube after time t, accounting directly for contour length fluctuations and constraint release. The effective diameter of the tube is independently evaluated by observing tube constraints either on atomistic displacements or on the displacement of primitive chain segments orthogonal to the initial primitive path. Having computed the tube diameter, the tube itself around each primitive path is constructed by visiting each entanglement strand along the primitive path one after the other and approximating it by the space of a small cylinder having the same axis as the entanglement strand itself and a diameter equal to the estimated effective tube diameter. Reptation of the primitive chain longitudinally inside the effective constraining tube as well as local transverse fluctuations of the chain driven mainly from constraint release and regeneration mechanisms are evident in the simulation results; the latter causes parts of the chains to venture outside their average tube surface for certain periods of time. The computed ?? (s,t) curves account directly for both of these phenomena, as well as for contour length fluctuations, since all of them are automatically captured in the atomistic simulations. Linear viscoelastic properties such as the zero shear rate viscosity and the spectra of storage and loss moduli obtained on the basis of the obtained ?? (s,t) curves for three different polymer melts (polyethylene, cis-1,4-polybutadiene, and trans-1,4-polybutadiene) are consistent with experimental rheological data and in qualitative agreement with the double reptation and dual constraint models. The new methodology is general and can be routinely applied to analyze primitive path dynamics and chain reptation in atomistic trajectories (accumulated through long MD simulations) of other model polymers or polymeric systems (e.g., bidisperse, branched, grafted, etc.); it is thus believed to be particularly useful in the future in evaluating proposed tube models and developing more accurate theories for entangled systems.open342

Multiscale simulation of polymer melt viscoelasticity: Expanded-ensemble Monte Carlo coupled with atomistic nonequilibrium molecular dynamics

We present a powerful framework for computing the viscoelastic properties of polymer melts based on an efficient coupling of two different atomistic models: the first is represented by the nonequilibrium molecular dynamics method and is considered as the microscale model. The second is represented by a Monte Carlo (MC) method in an expanded statistical ensemble and is free from any long time scale constraints. Guided by recent developments in nonequilibrium thermodynamics, the expanded ensemble incorporates appropriately defined "field" variables driving the corresponding structural variables to beyond equilibrium steady states. The expanded MC is considered as the macroscale solver for the family of all viscoelastic models built on the given structural variable(s). The explicit form of the macroscopic model is not needed; only its structure in the context of the general equation for the nonequilibrium reversible irreversible coupling or generalized bracket formalisms of nonequilibrium thermodynamics is required. We illustrate the method here for the case of unentangled linear polymer melts, for which the appropriate structural variable to consider is the conformation tensor c???. The corresponding Lagrange multiplier is a tensorial field ??. We have been able to compute model-independent values of the tensor ??, which for a wide range of strain rates (covering both the linear and the nonlinear viscoelastic regimes) bring results for the overall polymer conformation from the two models (microscale and macroscale) on top of each other. In a second step, by comparing the computed values of ?? with those suggested by the macroscopic model addressed by the chosen structural variable(s), we can identify shortcomings in the building blocks of the model. How to modify the macroscopic model in order to be consistent with the results of the coupled micro-macro simulations is also discussed. From a theoretical point of view, the present multiscale modeling approach provides a solid framework for the design of improved, more accurate macroscopic models for polymer melts.open151