Microscopic Stress Tensor Research Articles

Linear Response Theory of Scale-Dependent Viscoelasticity for Overdamped Brownian Particle Systems

We show the linear response theory of spatial-scale-dependent relaxation moduli for overdamped Brownian particle systems. We employ the Irving-Kirkwood stress tensor field as the microscopic stress tensor field. We show that the scale-dependent relaxation modulus tensor, which characterizes the response of the stress tensor field to the applied velocity gradient field, can be expressed by using the correlation function of the Irving-Kirkwood stress tensor field. The spatial Fourier transform of the relaxation modulus tensor gives the wavenumber-dependent relaxation modulus. For isotropic and homogeneous systems, the relaxation modulus tensor has only two independent components. The transverse and longitudinal deformation modes give the wavenumber-dependent shear relaxation modulus and the wavenumber-dependent bulk relaxation modulus. As simple examples, we derive the explicit expressions for the relaxation moduli for two simple models the non-interacting Brownian particles and the harmonic dumbbell model.

Viscoelasticity of a Stretched Semiflexible Polymer Chain with Fixed Ends

Regularities of the decay of time correlation functions of fluctuations of the microscopic stress tensor characterizing viscoelastic properties for statically stretched chains composed of elastic units with bending stiffness are studied by molecular dynamics and analytically. Chains with the persistent length smaller than the contour length at degrees of extension up to break are considered. In the region of high extensions, where distortions of the local structure of a chain appear, the decay of correlation functions can be described by the modified Rouse model consisting of subchains with anisotropic rigidities depending on the equilibrium chain extension. This makes it possible to determine the relaxation times and moduli of strongly stretched chains, which is necessary in studying the strength of polymer networks. For correlation functions, the analytical relation is formulated that takes into account the contribution of transverse bending fluctuations and agrees with the results of computer simulation at low extensions. A change in the correlation functions with the extension of long semiflexible chains is qualitatively similar to that for flexible chains, but quantitatively is appreciably different.

Stresses in non-equilibrium fluids: Exact formulation and coarse-grained theory.

Starting from the stochastic equation for the density operator, we formulate the exact (instantaneous) stress tensor for interacting Brownian particles and show that its average value agrees with expressions derived previously. We analyze the relation between the stress tensor and forces due to external potentials and observe that, out of equilibrium, particle currents give rise to extra forces. Next, we derive the stress tensor for a Landau-Ginzburg theory in generic, non-equilibrium situations, finding an expression analogous to that of the exact microscopic stress tensor, and discuss the computation of out-of-equilibrium (classical) Casimir forces. Subsequently, we give a general form for the stress tensor which is valid for a large variety of energy functionals and which reproduces the two mentioned cases. We then use these relations to study the spatio-temporal correlations of the stress tensor in a Brownian fluid, which we compute to leading order in the interaction potential strength. We observe that, after integration over time, the spatial correlations generally decay as power laws in space. These are expected to be of importance for driven confined systems. We also show that divergence-free parts of the stress tensor do not contribute to the Green-Kubo relation for the viscosity.

Geometric derivation of the microscopic stress: A covariant central force decomposition

We revisit the derivation of the microscopic stress, linking the statistical mechanics of particle systems and continuum mechanics. The starting point in our geometric derivation is the Doyle–Ericksen formula, which states that the Cauchy stress tensor is the derivative of the free-energy with respect to the ambient metric tensor and which follows from a covariance argument. Thus, our approach to define the microscopic stress tensor does not rely on the statement of balance of linear momentum as in the classical Irving–Kirkwood–Noll approach. Nevertheless, the resulting stress tensor satisfies balance of linear and angular momentum. Furthermore, our approach removes the ambiguity in the definition of the microscopic stress in the presence of multibody interactions by naturally suggesting a canonical and physically motivated force decomposition into pairwise terms, a key ingredient in this theory. As a result, our approach provides objective expressions to compute a microscopic stress for a system in equilibrium and for force-fields expanded into multibody interactions of arbitrarily high order. We illustrate the proposed methodology with molecular dynamics simulations of a fibrous protein using a force-field involving up to 5-body interactions.

Elastic Behavior of 2D Grain Packing Modeled as Micromorphic Media Based on Granular Micromechanics

AbstractDiscrete simulation and continuum modeling are two competing methods for analyzing the behavior of complex material systems and granular assemblies. In this paper, a two-dimensional (2D) granular micromechanics model for micromorphic media is presented. The granular micromechanics method is enhanced by incorporating nonclassical terms in the kinematics. These include, in addition to the usual macroscale displacement gradient, the fluctuations in displacement gradient and also the second gradient of displacement. Microscopic intergranular force vectors and macroscopic stress tensors conjugate to the kinematic measures are defined, and the macroscopic strain energy density function is set equal to the volume average of grain-pair energies. As a result, continuum stiffness tensors are formulated on the basis of intergranular stiffness coefficients and fabric parameters defining the geometry of grains and their contacts. To demonstrate the applicability of the continuum method, a random granular assem...

On computing stress in polymer systems involving multi-body potentials from molecular dynamics simulation.

Hardy stress definition has been restricted to pair potentials and embedded-atom method potentials due to the basic assumptions in the derivation of a symmetric microscopic stress tensor. Force decomposition required in the Hardy stress expression becomes obscure for multi-body potentials. In this work, we demonstrate the invariance of the Hardy stress expression for a polymer system modeled with multi-body interatomic potentials including up to four atoms interaction, by applying central force decomposition of the atomic force. The balance of momentum has been demonstrated to be valid theoretically and tested under various numerical simulation conditions. The validity of momentum conservation justifies the extension of Hardy stress expression to multi-body potential systems. Computed Hardy stress has been observed to converge to the virial stress of the system with increasing spatial averaging volume. This work provides a feasible and reliable linkage between the atomistic and continuum scales for multi-body potential systems.

Driven colloidal suspensions in confinement and density functional theory: Microstructure and wall-slip

We theoretically investigate general properties of driven (sheared) colloidal suspensions in confinement, based on methods of classical density functional theory. In the absence of an exact closed (Smoluchowski-) equation for the one-particle density under shear, we formulate a set of general conditions for approximations, and show that a simple closure fulfills them. The exact microscopic stress tensor is identified. Exemplifying the situation near a wall (oriented parallel to the direction of shear), we note that the microscopic shear stress is not necessarily homogeneous. Formulating a second equation additional to the Smoluchowski equation, we achieve a homogeneous shear stress, and thereby compute the local flow velocity near the wall. This finally leads to a slip length of the complex fluid at the wall.

Viscoelasticity of a stretched polymer chain with ends exposed to a constant force

The viscoelasticity of a stretched polymer chain with its end particles exposed to oppositely acting forces is studied via collisional molecular dynamics and analytically. A simple model according to which a polymer molecule is a chain of particles linked through freely jointed elastic bonds is adopted. The analytical theory is in good agreement with the results of the computer simulation of time correlation functions in the range of large-scale motions of a polymer molecule. It is found that the decay of correlation functions K αβμν of fluctuations of the microscopic stress tensor of a chain, K zzzz, K zαzα , = K zαzα , (α = x, y; z is the axis along which the forces act), is slowed down, and their value increases relative to the respective correlation functions of a chain with fixed ends. The greater the force, the higher this difference. The correlation functions that are transverse with respect to the z axis do not differ from those for chains with fixed ends. The results show that, in the calculation of time correlation functions of strongly stretched polymer chains, different statistical ensembles are not equivalent; this must be taken into account in the dynamic theories of heavily deformed polymers.

Stress tensor and elastic properties for hard and soft spheres

Properties determined from the microscopic stress tensor are discussed for the related systems of a steeply repulsive pair potential and hard spheres. The microscopic stress tensors determined from the local conservation law for momentum are different in these two cases. Consequences are discussed for both fluid and solid phases.

Dynamical derivation of microscopic stress tensors

The equation of motion technique is used to derive quantum operators corresponding to the microscopic stress tensor and pressure. The technique is demonstrated for a one-dimensional system. Stress tensor and related operators are obtained for a nonrelativistic, many-body, Coulomb system. The stress tensor operators occur as a momentum flux density in continuity equations which are the quantum analog to Newton's second law (F = ma). The derivation also produces operators with obvious classical analogs such as linear and angular momentum densities in a context which makes their physical role evident.

Viscoelasticity of stretched polymer chains: Analytical theory and computer-aided simulation

The viscoelasticity of stretched polymer chains has been studied by the method of collisional dynamics. To this end, time correlation functions of the fluctuations of the microscopic stress tensor are modeled and relaxation moduli are expressed. Before, for stretched polymer networks, correlation functions used to be calculated in terms of an approximate theory that allowed one to estimate the strain dependences of loss modulus. The calculated dependences are shown to agree qualitatively with the results of measurements performed over a wide interval of strains, including prefracture strains. This theory is verified by comparing the time correlation functions of stress tensor fluctuations for a single stretched chain; these functions are found by computer-aided simulation and calculated on the basis of the existing analytical theory. In this case, a simple theory is adopted according to which a polymer molecule represents a chain composed of N atoms connected by freely jointed elastic bonds. The first and Nth atoms of this chain are attached by harmonic springs to immobile points located at a fixed distance. The decay of time correlation functions under study can be resolved into three stages. After a short initial interval provided by local motions, one can observe a region of power-law decay, which is followed by monoexponential decay at long times. The results of computer-aided simulation generally agree with the predictions of analytical theory. Certain discrepancies primarily concern the dependences of the exponent of power-law relaxation on the degree of chain stretching.

Two tube models of rubber elasticity

AbstractPolymer entanglements lead to complicated topological constraints and interactions between neighboring chains in a dense solution or melt. Entanglements can be treated in a mean field approach, within the famous reptation model, since they effectively confine each individual chain in a tube‐like geometry. In polymer networks, due to crosslinks preventing the global reptation and constraint release, entanglements acquire a different topological meaning and have a much stronger effect on the resulting mechanical response. In this article we discuss two different models of rubber elasticity, both utilizing the reptation ideas. First, we apply the classical ideas of reptation statistics to calculate the effective rubber‐elastic free energy of an entangled rubbery network. In the second approach, we examine the classical Rouse dynamics of chains with quenched constraints at their ends by crosslinks, and along the primitive path by entanglements. We then proceed to average a microscopic stress tensor for the network system and present it in a manageable form in the equilibrium t → ∞ limit. Particular attention is paid to the treatment of compressibility and hydrostatic pressure in a sample with open boundaries. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2679–2697, 2006

Equilibrium calculation of transport coefficients for a fluid-particle model

A recently introduced particle-based model for fluid flow, called stochastic rotation dynamics, can be made Galilean invariant by introducing a random shift of the computational grid before collisions. In this paper, it is shown how the Green-Kubo relations derived previously can be resummed to obtain exact expressions for the collisional contributions to the transport coefficients. It is also shown that the collisional contribution to the microscopic stress tensor is not symmetric, and that this leads to an additional viscosity. The resulting identification of the transport coefficients for the hydrodynamic modes is discussed in detail, and it is shown that this does not impose restrictions on the applicability of the model. The collisional contribution to the thermal conductivity, which becomes important for small mean free path and small average particle number per cell, is also derived.

Constrained Rouse model of rubber viscoelasticity

In this work we use a new approach to investigate the equilibrium and linear dynamic-mechanical response of a polymer network. The classical Rouse model is extended to incorporate quenched constraints on its end-boundary conditions; a microscopic stress tensor for the network system is then derived in the affine deformation limit. To test the model we calculate the macroscopic stress in equilibrium, corresponding to the long-time limit of relaxation. Particular attention is paid to the treatment of compressibility and hydrostatic pressure in a sample with open boundaries. Although quite different in general, for small strains the model compares well with the classic equilibrium rubber-elasticity models. The dynamic shear modulus is obtained for a network relaxing after an instantaneous step strain by keeping track of relaxation of consecutive Rouse modes of constrained network strands. The results naturally cover the whole time range--from the dynamic glassy state down to the equilibrium incompressible rubber plateau.

Scalable and portable implementation of the fast multipole method on parallel computers

A scalable and portable Fortran code is developed to calculate Coulomb interaction potentials of charged particles on parallel computers, based on the fast multipole method. The code has a unique feature to calculate microscopic stress tensors due to the Coulomb interactions, which is useful in constant-pressure simulations and local stress analyses. The code is applicable to various boundary conditions, including periodic boundary conditions in two and three dimensions, corresponding to slab and bulk systems, respectively. Numerical accuracy of the code is tested through comparison of its results with those obtained by the Ewald summation method and by direct calculations. Scalability tests show the parallel efficiency of 0.98 for 512 million charged particles on 512 IBM SP3 processors. The timing results on IBM SP3 are also compared with those on IBM SP4.

Mesoscopic model for the viscosities of nematic liquid crystals.

Based on the definition of the mesoscopic concept by Blenk et al. [Physica A 174, 119 (1991); J. Noneq. Therm. 16, 67 (1991); Mol. Cryst. Liq. Cryst. 204, 133 (1991)] an approach to calculate the Leslie viscosity coefficients for nematic liquid crystals is presented. The approach rests upon the mesoscopic stress tensor, whose structure is assumed similar to the macroscopic Leslie viscous stress. The proposed form is also the main dissipation part of the mesoscopic Navier-Stokes equation. On the basis of the correspondence between microscopic and mesoscopic scales a mean-field mesoscopic potential is introduced. It allows us to obtain the stress tensor angular velocity of the free rotating molecules with the help of the orientational Fokker-Planck equation. The macroscopic stress tensor is calculated as an average of the mesoscopic counterpart. Appropriate relations among mesoscopic viscosities have been found. The mesoscopic analysis results are shown to be consistent with the diffusional model of Kuzuu-Doi and Osipov-Terentjev with the exception of the shear viscosity alpha(4). In the nematic phase alpha(4) is shown to have two contributions: isotropic and nematic. There exists an indication that the influence of the isotropic part is dominant over the nematic part. The so-called microscopic stress tensor used in the microscopic theories is shown to be the mean-field potential-dependent representation of the mesoscopic stress tensor. In the limiting case of total alignment the Leslie coefficients are estimated for the diffusional and mesoscopic models. They are compared to the results of the affine transformation model of the perfectly ordered systems. This comparison shows disagreement concerning the rotational viscosity, whereas the coefficients characteristic for the symmetric part of the viscous stress tensor remain the same. The difference is caused by the hindered diffusion in the affine model case.

Simulation studies of shear viscosity time correlation functions in liquid CS2

The Green—Kubo time correlation function for the shear viscosity in liquid CS2 has been simulated by molecular dynamics at several thermodynamic state points. The breakdown of this function into its kinetic and potential contributions as well as the cross-term between the two has been performed. Intermolecular interactions were obtained from a three-centre atom—atom (12/6) Lennard-Jones potential model. The time correlation functions for the potential part of the shear viscosity contain component two-, three- and four-body terms that were explicitly evaluated to show that they partially cancel each other at short times but at long times, they exhibit approximately exponential decays with magnitude ratios corresponding to nearly perfect cancellation. In this respect, the correlation functions for CS2 resemble those of liquid argon. In addition, the microscopic stress tensor was separated into the portions arising from the repulsive and attractive branches of the Lennard-Jones model. This split gives rise to positive autocorrelation functions involving the repulsive and the attractive forces plus a negative cross-correlation function between the two that partially cancels the contributions of the autocorrelation functions. It is argued that the breakdown of the potential part of the shear viscosity into its component parts is helpful in elucidating the role of molecular re-orientation in determining the separate short and long time behaviours of this time correlation function for liquids such as CS2.

Shear thinning in concentrated dispersions

A theory is presented to predict the effective shear viscosity of concentrated dispersions in steady state shear flow as a function of the volume fraction, particle size, interparticle interactions, and shear rate. Shear thinning has been qualitatively argued to be a result of shear-induced change in microstructure, which we shall discuss in terms of the distribution function. Analysing the distribution function and the microscopic stress tensor, we have derived a memory function in steady state shear flow. It is used to calculate all the key features of shear thinning quantitatively without any adjusting parameter. The predicted shear-rate-dependent effective shear viscosity is in good agreement with the experiments of sterically stabilized colloidal suspensions. Since shear thinning has been observed in steady state shear flow as well as in oscillatory shear flow, the difference between these two flows will also be discussed in terms of the distribution functions and structural relaxations.

Microscopic Stress Tensor of Nematic Liquid Crystal Binary Mixtures

A microscopic stress tensor for the binary nematic mixture of anisotropic molecules is presented. Its simple additive character is beneficial for hydrodynamics studies. We pay attention to mixtures of disc and rodlike molecules which may exhibit interesting hydrodynamic properties.

Uniqueness of the microscopic stress tensor

Schofield and Henderson have shown that the balance equation for the density of linear momentum does not provide a unique definition for the microscopic stress tensor. It is argued here that additional conditions of symmetry and physical interpretability, which reasonably can be imposed upon this tensor, are sufficient to establish the uniqueness of the familiar Irving–Kirkwood formula.