Graphene and other two-dimensional materials offer a new approach to controlling mass transport at the nanoscale. These materials can sustain nanoscale pores in their rigid lattices and due to their minimum possible material thickness, high mechanical strength and chemical robustness, they could be used to address persistent challenges in membrane separations. Here we discuss theoretical and experimental developments in the emerging field of nanoporous atomically thin membranes, focusing on the fundamental mechanisms of gas- and liquid-phase transport, membrane fabrication techniques and advances towards practical application. We highlight potential functional characteristics of the membranes and discuss applications where they are expected to offer advantages. Finally, we outline the major scientific questions and technological challenges that need to be addressed to bridge the gap from theoretical simulations and proof-of-concept experiments to real-world applications.

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.

We present a new Monte Carlo method for obtaining solutions of the Boltzmann equation for describing phonon transport in micro and nanoscale devices. The proposed method can resolve arbitrarily small signals (e.g. temperature differences) at small constant cost and thus represents a considerable improvement compared to traditional Monte Carlo methods whose cost increases quadratically with decreasing signal. This is achieved via a control-variate variance reduction formulation in which the stochastic particle description only solves for the deviation from a nearby equilibrium, while the latter is described analytically. We also show that simulating an energy-based Boltzmann equation results in an algorithm that lends itself naturally to exact energy conservation thereby considerably improving the simulation fidelity. Simulations using the proposed method are used to investigate the effect of porosity on the effective thermal conductivity of silicon. We also present simulations of a recently developed thermal conductivity spectroscopy process. The latter simulations demonstrate how the computational gains introduced by the proposed method enable the simulation of otherwise intractable multiscale phenomena.

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.

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.

Starting from the recently proposed energy-based deviational formulation for solving the Boltzmann equation [J.-P. Péraud and N. G. Hadjiconstantinou, Phys. Rev. B 84, 2011], which provides significant computational speedup compared to standard Monte Carlo methods for small deviations from equilibrium, we show that additional computational benefits are possible in the limit that the governing equation can be linearized. The proposed method exploits the observation that under linearized conditions (small temperature differences) the trajectories of individual deviational particles can be decoupled and thus simulated independently; this leads to a particularly simple and efficient algorithm for simulating steady and transient problems in arbitrary three-dimensional geometries, without introducing any additional approximation.In a previous paper 1 , we presented a low variance Monte Carlo method for solving the Boltzmann transport equation (BTE) for phonons in the relaxation-time approximation whereby computational particles simulate only the deviation from an equilibrium distribution. The benefits of such control-variate formulations 2 , which we will refer to as deviational, are twofold: first, in the limit of small temperature differences, deviational methods exhibit substantial computational speedup compared to traditional Monte Carlo methods 1 ; this speedup increases quadratically as the characteristic temperature difference goes to zero. Second, by simulating only the deviation from equilibrium, deviational methods seamlessly and automatically focus the computational effort on regions where it is needed and can thus be used for solving otherwise intractable multiscale problems. In the present article, we show that for problems exhibiting sufficiently small temperature differences such that the BTE can be linearized, deviational computational particles may be treated independently, thus lending themselves to a simulation algorithm that is simpler, does not use any approximation in space or time, and, depending on the application of interest, can be several orders of magnitude faster than the one presented in Ref. 1.The deviational approach can be introduced by writing the governing equation (with no approximation) in the formwhereis the relaxation time (ω, p and T respectively referring to the angular frequency, the polarization and the temperature), f = f (x, ω, p, θ, φ) is the occupation number of phonon states, V g is the phonon-bundle group velocity and f−1 is a Bose-Einstein distribution at the "control" temperature T eq (k b denotes Boltzmann's constant). Since this distribution does not depend on (T loc − T eq ) once normalized, a particle undergoing a scattering event can be drawn from (the normalized form of) (2) without knowledge of T loc ; energy conservation is simply ensured by conserving the particle. Although this formulation was originally introduced 1 as a means of truncating the discretization of a semi-infinite simulation domain (by limiting the region where computational cells were used), ...

We present a hybrid atomistic-continuum computational framework for the treatment of dense fluid problems with emphasis on the coupling of molecular dynamics with continuum (finite element/spectral) methods for problems involving multi-fluid dynamics in the presence of multi-fluid interfaces. The technique is an extension of the single-fluid framework already presented by the author. The well-known moving contact-line problem is used as a validation example. A hybrid solution that employs molecular dynamics close to the walls where molecular effects are important and continuum fluid mechanics in the remainder of the domain (far field region) is obtained. A fully molecular solution of the same problem serves as an exact solution. Various issues related to dense fluid atomistic-continuum techniques are discussed and contrasted to the already existing but less general dilute gas techniques. Numerical considerations are discussed with particular emphasis on efficiency, and a formulation that reduces computational cost is proposed.

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 Carlo. We also present recent variance reduction ideas which address the prohibitive cost associated with the statistical sampling of macroscopic properties in low-speed flows.