A technique for the calculation of mass, energy, and momentum densities at planes in molecular dynamics simulations

A perspective on the microscopic pressure (stress) tensor: History, current understanding, and future challenges

The pressure tensor (equivalent to the negative stress tensor) at both microscopic and macroscopic levels is fundamental to many aspects of engineering and science, including fluid dynamics, solid mechanics, biophysics, and thermodynamics. In this Perspective, we review methods to calculate the microscopic pressure tensor. Connections between different pressure forms for equilibrium and nonequilibrium systems are established. We also point out several challenges in the field, including the historical controversies over the definition of the microscopic pressure tensor; the difficulties with many-body and long-range potentials; the insufficiency of software and computational tools; and the lack of experimental routes to probe the pressure tensor at the nanoscale. Possible future directions are suggested.

Local stress tensor calculation by the method-of-plane in microscopic systems with macroscopic flow: A formulation based on the velocity distribution function

In this work, we developed a calculation method of local stress tensor applicable to non-equilibrium molecular dynamics (NEMD) systems, which evaluates the macroscopic momentum advection and the kinetic term of the stress in the framework of the Method-of-Plane (MoP), in a consistent way to guarantee the mass and momentum conservation. From the relation between the macroscopic velocity distribution function and the microscopic molecular passage across a fixed control plane, we derived a method to calculate the basic properties of the macroscopic momentum conservation law including the density, the velocity, the momentum flux, and the two terms of the stress tensor, i.e., the interaction and the kinetic terms, defined on a surface with a finite area. Any component of the streaming velocity can be obtained on a control surface, which enables the separation of the kinetic momentum flux into the advection and stress terms in the framework of MoP, and this enables strict satisfaction of the mass and momentum conservation for an arbitrary closed control volume (CV) set in NEMD systems. We validated the present method through the extraction of the density, velocity, and stress distributions in a quasi-one-dimensional steady-state Couette flow system and in a quasi-2D steady-state NEMD system with a moving contact line. We showed that with the present MoP, in contrast to the volume average method, the conservation law was satisfied even for a CV set around the moving contact line, which was located in a strongly inhomogeneous region.

Measuring heat flux beyond Fourier's law

We use nonequilibrium molecular dynamics to explore the effect of shear flow on heat flux. By simulating a simple fluid in a channel bounded by tethered atoms, the heat flux is computed for two systems: a temperature driven one with no flow and a wall driven, Couette flow system. The results for the temperature driven system give Fourier's law thermal conductivity, which is shown to agree well with experiments. Through comparison of the two systems, we quantify the additional components of the heat flux parallel and normal to the walls due to shear flow. To compute the heat flux in the flow direction, the Irving-Kirkwood equations are integrated over a volume, giving the so-called volume average form, and they are also manipulated to get expressions for the surface averaged and method of planes forms. The method of planes and volume average forms are shown to give equivalent results for the heat flux when using small volumes. The heat flux in the flow direction is obtained consistently over a range of simulations, and it is shown to vary linearly with strain rate, as predicted by theory. The additional strain rate dependent component of the heat flux normal to the wall is obtained by fitting the strain rate dependence of the heat flux to the expected form. As a result, the additional terms in the thermal conductivity tensor quantified in this work should be experimentally testable.

Control-volume representation of molecular dynamics

A molecular dynamics (MD) parallel to the control volume (CV) formulation of fluid mechanics is developed by integrating the formulas of Irving and Kirkwood [J. Chem. Phys. 18, 817 (1950)] over a finite cubic volume of molecular dimensions. The Lagrangian molecular system is expressed in terms of an Eulerian CV, which yields an equivalent to Reynolds' transport theorem for the discrete system. This approach casts the dynamics of the molecular system into a form that can be readily compared to the continuum equations. The MD equations of motion are reinterpreted in terms of a Lagrangian-to-control-volume (LCV) conversion function ϑ(i) for each molecule i. The LCV function and its spatial derivatives are used to express fluxes and relevant forces across the control surfaces. The relationship between the local pressures computed using the volume average [Lutsko, J. Appl. Phys. 64, 1152 (1988)] techniques and the method of planes [Todd et al., Phys. Rev. E 52, 1627 (1995)] emerges naturally from the treatment. Numerical experiments using the MD CV method are reported for equilibrium and nonequilibrium (start-up Couette flow) model liquids, which demonstrate the advantages of the formulation. The CV formulation of the MD is shown to be exactly conservative and is, therefore, ideally suited to obtain macroscopic properties from a discrete system.

The equivalence between volume averaging and method of planes definitions of the pressure tensor at a plane

It is shown analytically that the method of planes (MOP) [Todd, Evans, and Daivis, Phys. Rev. E 52, 1627 (1995)] and volume averaging (VA) [Cormier, Rickman, and Delph, J. Appl. Phys. 89, 99 (2001)] formulas for the local pressure tensor, P(α, y)(y), where α ≡ x, y, or z, are mathematically identical. In the case of VA, the sampling volume is taken to be an infinitely thin parallelepiped, with an infinite lateral extent. This limit is shown to yield the MOP expression. The treatment is extended to include the condition of mechanical equilibrium resulting from an imposed force field. This analytical development is followed by numerical simulations. The equivalence of these two methods is demonstrated in the context of non-equilibrium molecular dynamics (NEMD) simulations of boundary-driven shear flow. A wall of tethered atoms is constrained to impose a normal load and a velocity profile on the entrained central layer. The VA formula can be used to compute all components of P(αβ)(y), which offers an advantage in calculating, for example, P(xx)(y) for nano-scale pressure-driven flows in the x-direction, where deviations from the classical Poiseuille flow solution can occur.

Structural and Dynamic Properties of Concentrated Alkali Halide Solutions: A Molecular Dynamics Simulation Study

The physicochemical properties of alkali halide solutions have long been attributed to the collective interactions between ions and water molecules in the solution, yet the structure of water in these systems and its effect on the equilibrium and dynamic properties of these systems are not clearly understood. Here, we present a systematic view of water structure in concentrated alkali halide solutions using molecular dynamics simulations. The results of the simulations show that the size of univalent ions in the solution has a significant effect on the dynamics of ions and other transport properties such as the viscosity that are correlated with the structural properties of water in aqueous ionic solution. Small cations (e.g., Li+) form electrostatically stabilized hydrophilic hydration shells that are different from the hydration shells of large ions (e.g., Cs+) which behave more like neutral hydrophobic particles, encapsulated by hydrogen-bonded hydration cages. The properties of solutions with different types of ion solvation change in different ways as the ion concentration increases. Examples of this are the diffusion coefficients of the ions and the viscosities of solutions. In this paper we use molecular dynamics (MD) simulations to study the changes in the equilibrium and transport properties of LiCl, RbCl, and CsI solutions at concentrations from 0.22 to 3.97 M.

Local dynamic mechanical properties in model free-standing polymer thin films

High-frequency sinusoidal oscillations of a coarse-grained polymer model are used to calculate the local dynamic mechanical properties (DMPs) of free-standing polymer thin films. The storage modulus G(') and loss modulus G(") are examined as a function of position normal to the free surfaces. It is found that mechanically soft layers arise near the free surfaces of glassy thin films, and that their thickness becomes comparable to the entire film thickness as the temperature approaches the glass transition T(g). As a result, the overall stiffness of glassy thin films decreases with film thickness. It is also shown that two regions coexist in thin films just at the bulk T(g); a melt-like region (G(')<G(")) near the free surfaces and a glass-like region (G(')>G(")) in the middle of the film. Our findings on the existence of a heterogeneous distribution of DMPs in free-standing polymer thin films provide insights into recent experimental measurements of the mechanical properties of glassy polymer thin films.

Pressure tensor and heat flux vector for inhomogeneous nonequilibrium fluids under the influence of three-body forces

We present a derivation of the pressure tensor and heat flux vector for inhomogeneous fluids under the influence of three-body forces. The derivation is based on the method of planes formalism of Todd, Evans, and Daivis [Phys. Rev. E 52, 1627 (1995); 51, 4362 (1995)]. Our derivation is validated against nonequilibrium molecular dynamics simulations of a confined fluid acted upon by a two-body Barker-Fisher-Watts force coupled with the Axilrod-Teller three-body force. Our method of planes calculations agree perfectly with the equivalent mesoscopic route of integrating the momentum and energy continuity equations directly from the simulation data. Our calculations reveal that three-body forces have an important consequence for the isotropic pressure, but have negligible influence on the shear stress (hence viscosity) and heat flux vector (hence thermal conductivity) for a confined simple fluid.