Nicolas G. Hadjiconstantinou

Statistical error in particle simulations of hydrodynamic phenomena

We present predictions for the statistical error due to finite sampling in the presence of thermal fluctuations in molecular simulation algorithms. Specifically, we establish how these errors depend on Mach number, Knudsen number, number of particles, etc. Expressions for the common hydrodynamic variables of interest such as flow velocity, temperature, density, pressure, shear stress, and heat flux are derived using equilibrium statistical mechanics. Both volume-averaged and surface-averaged quantities are considered. Comparisons between theory and computations using direct simulation Monte Carlo for dilute gases, and molecular dynamics for dense fluids, show that the use of equilibrium theory provides accurate results.

The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics

This paper reviews basic results and recent developments in the field of small-scale gaseous hydrodynamics which has received significant attention in connection with small-scale science and technology. We focus on the modeling challenges arising from the breakdown of the Navier-Stokes description, observed when characteristic lengthscales become of the order of, or smaller than, the molecular mean free path. We discuss both theoretical results and numerical methods development. Examples of the former include the limit of applicability of the Navier-Stokes constitutive laws, the concept of second-order slip and the appropriate form of such a model, and how to reconcile experimental measurements of slipping flows with theory. We also review a number of recently developed theoretical descriptions of canonical nanoscale flows of engineering interest. On the simulation front, we review recent progress in characterizing the accuracy of the prevalent Boltzmann simulation method known as direct simulation Monte ...

Comment on Cercignani’s second-order slip coefficient

Cercignani’s second-order slip model has been neglected over the years, perhaps due to Sreekanth’s claim that it cannot fit his experimental data. In this paper we show that Sreekanth’s claim was based on an incorrect interpretation of this model. We also show that Cercignani’s second-order slip model, when modified and used appropriately, is in good agreement with solutions of the Boltzmann equation for a hard-sphere gas for a wide range of rarefaction. Given its simplicity, we expect this model to be a valuable tool for describing isothermal micro- and nanoscale flows to the extent that the hard-sphere approximation is appropriate.

Heterogeneous Atomistic-Continuum Representations for Dense Fluid Systems

We present a formulation and numerical solution procedure for heterogeneous atomistic-continuum representations of fluid flows. The ingredients from atomistic and continuum perspectives are non-equilibrium molecular dynamics and spectral element, respectively; the matching is provided by a classical procedure, the Schwarz alternating method with overlapping subdomains. The technique is applied to microscale flow of a dense fluid (supercritical argon) in a complex two-dimensional channel.

Analysis of discretization in the direct simulation Monte Carlo

We propose a continuous-time formulation of the direct simulation Monte Carlo that allows the evaluation of the transport coefficient dependence on the time step through the use of the Green–Kubo theory. Our results indicate that the error exhibits quadratic dependence on the time step, and that for time steps of the order of one mean free time the error is of the order of 5%. Our predictions for the transport coefficients are in good agreement with numerical experiments. The calculation of the cell size dependence, first obtained by Alexander et al. [Phys. Fluids 10, 1540 (1998)], is reviewed and a correction is pointed out.

Mechanisms of Molecular Permeation through Nanoporous Graphene Membranes

We present an investigation of molecular permeation of gases through nanoporous graphene membranes via molecular dynamics simulations; four different gases are investigated, namely helium, hydrogen, nitrogen, and methane. We show that in addition to the direct (gas-kinetic) flux of molecules crossing from the bulk phase on one side of the graphene to the bulk phase on the other side, for gases that adsorb onto the graphene, significant contribution to the flux across the membrane comes from a surface mechanism by which molecules cross after being adsorbed onto the graphene surface. Our results quantify the relative contribution of the bulk and surface mechanisms and show that the direct flux can be described reasonably accurately using kinetic theory, provided the latter is appropriately modified assuming steric molecule-pore interactions, with gas molecules behaving as hard spheres of known kinetic diameters. The surface flux is negligible for gases that do not adsorb onto graphene (e.g., He and H2), while for gases that adsorb (e.g., CH4 and N2) it can be on the order of the direct flux or larger. Our results identify a nanopore geometry that is permeable to hydrogen and helium, is significantly less permeable to nitrogen, and is essentially impermeable to methane, thus validating previous suggestions that nanoporous graphene membranes can be used for gas separation. We also show that molecular permeation is strongly affected by pore functionalization; this observation may be sufficient to explain the large discrepancy between simulated and experimentally measured transport rates through nanoporous graphene membranes.

Constant-Wall-Temperature Nusselt Number in Micro and Nano-Channels

We investigate the constant-wall-temperature convective heat-transfer characteristics of a model gaseous flow in two-dimensional micro and nano-channels under hydrodynamically and thermally fully developed conditions. Our investigation covers both the slip-flow regime 0≤Kn≤0.1, and most of the transition regime 0.1<Kn≤10, where Kn, the Knudsen number, is defined as the ratio between the molecular mean free path and the channel height. We use slip-flow theory in the presence of axial heat conduction to calculate the Nusselt number in the range 0≤Kn≤0.2, and a stochastic molecular simulation technique known as the direct simulation Monte Carlo (DSMC) to calculate the Nusselt number in the range 0.02<Kn<2. Inclusion of the effects of axial heat conduction in the continuum model is necessary since small-scale internal flows are typically characterized by finite Peclet numbers

Variance reduction for Monte Carlo solutions of the Boltzmann equation

We show that by considering only the deviation from equilibrium, significant computational savings can be obtained in Monte Carlo evaluations of the Boltzmann collision integral for flow problems in the small Mach number (Ma) limit. The benefits of this variance reduction approach include a significantly reduced statistical uncertainty when the deviation from equilibrium is small, and a flow-velocity signal-to-noise ratio that remains approximately constant with Ma in the Ma⪡1 limit. This results in stochastic Boltzmann solution methods whose computational cost for a given signal-to-noise ratio is essentially independent of Ma for Ma⪡1; our numerical implementation demonstrates this for Mach numbers as low as 10−5. These features are in sharp contrast to current particle-based simulation techniques in which statistical sampling leads to computational cost that is proportional to Ma−2, making calculations at small Ma very expensive.

A low-variance deviational simulation Monte Carlo for the Boltzmann equation

We present an efficient particle method for solving the Boltzmann equation. The key ingredients of this work are the variance reduction ideas presented in Baker and Hadjiconstantinou L.L. Baker, N.G. Hadjiconstantinou, Variance reduction for Monte Carlo solutions of the Boltzmann Equation, Physics of Fluids, 17 (2005) (art. no, 051703)] and a new collision integral formulation which allows the method to retain the algorithmic structure of direct simulation Monte Carlo (DSMC) and thus enjoy the numerous advantages associated with particle methods, such as a physically intuitive formulation, computational efficiency due to importance sampling, low memory usage (no discretization in velocity space), and the ability to naturally and accurately capture discontinuities in the distribution function. The variance reduction, achieved by simulating only the deviation from equilibrium, results in a significant computational efficiency advantage for low-signal flows (e.g. low flow speed) compared to traditional particle methods such as DSMC. In particular, the resulting method can capture arbitrarily small deviations from equilibrium at a computational cost that is independent of the magnitude of this deviation. The method is validated by comparing its predictions with DSMC solutions for spatially homogeneous and inhomogeneous problems.

Implications of Permeation through Intrinsic Defects in Graphene on the Design of Defect-Tolerant Membranes for Gas Separation

Gas transport through intrinsic defects and tears is a critical yet poorly understood phenomenon in graphene membranes for gas separation. We report that independent stacking of graphene layers on a porous support exponentially decreases flow through defects. On the basis of experimental results, we develop a gas transport model that elucidates the separate contributions of tears and intrinsic defects on gas leakage through these membranes. The model shows that the pore size of the porous support and its permeance critically affect the separation behavior, and reveals the parameter space where gas separation can be achieved regardless of the presence of nonselective defects, even for single-layer membranes. The results provide a framework for understanding gas transport in graphene membranes and guide the design of practical, selectively permeable graphene membranes for gas separation.