Darken Approximation Research Articles

Multicomponent mutual diffusion in the warm, dense matter regime

We present a study in the warm, dense matter regime of transport properties for a ternary mixture of the isotopes of hydrogen at 5 g/cm3 at temperatures between 50 and 400 eV as well as for a quaternary system with added carbon at 20 g/cm3 and 10 eV. We determine the properties from a series of orbital-free molecular dynamics simulations and calculate the diffusion coefficients in the Maxwell–Stefan (MS) formulation. The Darken approximation, which includes only time correlations between the same atom of the same species, gives reasonable agreement with the full MS results. However, we find that the reasons for this concurrence rest with more complex, and somewhat subtle, details within the MS prescription.

Bootstrapping time correlation functions of molecular dynamics.

Molecular dynamics is often considered as a numerical experiment. The error bars on the results are therefore mandatory, but sometimes difficult to determine and computationally demanding. As a low-cost approach, we describe the application of the bootstrap (BS) method to the quantification of uncertainties pertaining to the time correlation functions. We chose the autocorrelation functions of velocity and interdiffusion current for a binary ionic mixture as a test bed, and we assessed the merit of the Darken approximation relating both of them. The intrinsic errors related to phase space sampling is investigated comparing the BS method with the reference method of replica. We also study how the BS method can assist in addressing the finite-size effects.

Multicomponent mutual diffusion in the warm, dense matter regime.

We present the formulation, simulations, and results for multicomponent mutual diffusion coefficients in the warm, dense matter regime. While binary mixtures have received considerable attention for mass transport, far fewer studies have addressed ternary and more complex systems. We therefore explicitly examine ternary systems utilizing the Maxwell-Stefan formulation that relates diffusion to gradients in the chemical potential. Onsager coefficients then connect the macroscopic diffusion to microscopic particle motions, evinced in trajectories characterized by positions and velocities, through various autocorrelation functions (ACFs). These trajectories are generated by molecular dynamics (MD) simulations either through the Born-Oppenheimer approximation, which treats the ions classically and the electrons quantum-mechanically by an orbital-free density-functional theory, or through a classical MD approach with Yukawa pair-potentials, whose effective ionizations and electron screening length derive from quantal considerations. We employ the reference-mean form of the ACFs and determine the center-of-mass coefficients through a simple reference-frame-dependent similarity transformation. The Onsager terms in turn determine the mutual diffusion coefficients. We examine a representative sample of ternary mixtures as a function of density and temperature from those with only light elements (D-Li-C, D-Li-Al) to those with highly asymmetric mass components (D-Li-Cu, D-Li-Ag, H-C-Ag). We also follow trends in the diffusion as a function of number concentration and evaluated the efficacy of various approximations such as the Darken approximation.

The general form of transport diffusivity of shale gas in organic-rich nano-slits—A molecular simulation study using Darken approximation

As a potential substitute for the gradually depleted conventional oil and gas resources, shale gas is attracting increasing attention worldwide. Because of the ubiquitous nanopores in shale matrix, studying diffusion behavior of shale gas in nano-scale channels is crucial for its development, which also lays the foundation for the optimization of hydraulic fracturing, horizontal well spacing, etc. This paper focuses on shale gas diffusion behavior in organic-rich nano-slits and establishes a calculation method for generalized transport diffusivity using Darken approximation by molecular simulation (MS). The main conclusions are as follows. (1) For the whole gas, compared to its self-diffusivity which decreases monotonically with pressure, there is a break point for its transport diffusivity in each slit. It’s found that the sequence of magnitudes of transport diffusivities in different pores at low pressures and that at high pressures are contrary. (2) For the bulk district (BD) in 2 nm slit, the self- and transport diffusivities don’t differ much at low pressures and their discrepancy enlarges at high pressures, the deviation of which can be 2.34 times at 41 MPa. (3) The transport diffusivity of the molecules in the Knudsen layer (KL) in 2 nm slit is always larger (1.21–3.76 times) than their self-diffusivity and shows fluctuations at large pressures, other than the self-diffusivity in KL which decreases monotonically with pressure. Furthermore, the deficiencies and directions for improvements of the kinetic self-diffusion model and Knudsen diffusion model are analyzed by the data comparison and the theory background on their derivation and expressions. It is expected that this work will give new insights on the flow capacity of shale gas in nano-scale pores and contribute to a deeper understanding for shale gas evaluation and exploitation.

Determining a Tracer Diffusivity by way of the Darken-Manning Equation for Interdiffusion in Binary Alloy Systems

The self-or tracer diffusivity of one component in a binary alloy is often required when there is knowledge of the other component’s self-or tracer diffusivity and the interdiffusivity (and the thermodynamic factor). In the present paper, this problem is addressed for the random alloy model by applying three possible approximations having different levels of accuracy: Darken (low level of accuracy), Manning (medium level of accuracy) and Moleko, Allnatt and Allnatt (MAA) (high level of accuracy). There are unexpectedly large differences between the results of these approximations that sometimes are reflected in the high sensitivity of the vacancy-wind factor to the level of approximation. Generally, for the application of Manning and the MAA approximations, it is found that there is a difference in the number of self-diffusivity roots depending on whether the tracer diffusivity is available for the faster diffuser or for the slower diffuser and depending on how close the composition is to the forbidden (according to Manning’s description) region. Provided that the interdiffusion coefficient (divided by the thermodynamic factor) is greater than the available self-diffusion coefficient multiplied by its complementary composition, the application of the Darken approximation always results in one self-diffusivity root.

Water in Single-Walled Aluminosilicate Nanotubes: Diffusion and Adsorption Properties

Single-walled aluminosilicate nanotubes are attractive materials for construction of nanofluidic devices. They have a well-defined structure, a hydrophilic interior with periodic wide and narrow regions, precisely tunable length and diameter, and a functionalizable interior for tuning mass transport and adsorption properties. We report a computational and experimental investigation that highlights the unique adsorption and diffusive water transport properties of these nanotubes. Axial self-diffusivities of water molecules (at loadings ranging from near-infinite dilution to near-saturation) are calculated by molecular dynamics (MD) simulations, whereas adsorption properties are computed with grand canonical Monte Carlo (GCMC) simulations and are also compared to experimental data. The transport diffusivities are evaluated through the Darken approximation. Water transport in these nanotubes at room temperature was observed to occur via Fickian diffusion. The self-diffusivity decreases with an increase in wa...

The role of molecular interactions and interfaces in diffusion: Transport diffusivity and evaluation of the Darken approximation

Kinetic Monte Carlo (KMC) simulations are carried out to directly study diffusion of benzene through thin (37-100 nm) NaX zeolite membranes under a gradient in chemical potential. Nonlinearities in adsorbate loading near the membrane boundaries are shown to arise from the difference in adsorbate density between the zeolite and adjacent fluid phase. Direct extraction of the transport diffusivity from gradient KMC simulations enables testing of the Darken approximation. This rigorous approach reveals limitations of the Darken approximation and, for the first time, the potentially complex nonunique functionality and multiplicity of the transport diffusivity for strongly interacting adsorbates. In the companion paper we explore these nonlinear interfacial effects in the context of permeation through both single-crystal and polycrystalline membranes.

The role of molecular interactions and interfaces in diffusion: Permeation through single-crystal and polycrystalline microporous membranes

In this second paper of a two part series, we investigate the implications of the interfacial phenomenon, caused by adsorbate-adsorbate interactions coupled with the difference in adsorbate density between the zeolite and the gas phase, upon benzene permeation through single-crystal and polycrystalline microporous NaX membranes. The high flux predicted for thin single-crystal membranes reveals that substantially enhanced flux should be expected in submicron films. Simulations also indicate that the standard local equilibrium assumption made for larger scale membranes is inapplicable at the submicron scale associated with nanometer size grains of thin and/or polycrystalline membranes. Apparent activation energies predicted for benzene permeation through NaX membranes via kinetic Monte Carlo (KMC) simulations are in good agreement with laboratory experiments. The simulations also uncover temperature-dependent flux pathways leading to non-Arrhenius behavior observed experimentally. The failure of the Darken approximation, especially in the presence of the interfacial phenomenon, leads to a substantial overprediction of the flux. Simulations of polycrystalline membranes suggest that this same interfacial phenomenon leads to resistance that can reduce flux by an order of a magnitude with only moderate polycrystallinity.

Adsorption and Diffusion of N2 and O2 in LiLSX Studied by Neutron Scattering Techniques

The vibrational frequencies and diffusivities of N2 and O2 adsorbed in LiLSX have been studied by neutron scattering. Experiments were performed with the single components at 260 K, at various loadings. The stretching frequency of N2 in interaction with Li cations, as observed by inelastic neutron scattering, is compared with quantum chemical calculations. The transport diffusivities of N2 and O2 are derived from quasi-elastic neutron scattering, O2 diffusing faster than N2. The Darken approximation is found to be inexact, the corrected diffusivities of O2 and N2 decreasing with increasing loading.

Direct Tests of the Darken Approximation for Molecular Diffusion in Zeolites Using Equilibrium Molecular Dynamics

We have used equilibrium molecular dynamics and grand canonical Monte Carlo simulations to compute the transport diffusivities of CH4 and CF4 adsorbed in a fully atomistic model of silicalite over a wide range of temperatures and pore loadings. Our results provide a direct test of the accuracy of the Darken approximation, an expression frequently used to relate transport and self-diffusivities in zeolites. The Darken approximation is accurate for CH4 in silicalite at 298 K, but is less accurate at higher temperatures. In contrast, CF4 deviates strongly from the Darken approximation at all temperatures we have examined. This observation suggests that substantial deviations from the Darken approximation may be a common feature of molecular diffusion in zeolites.

Collective surface diffusion on a triangular lattice in presence of ordered phases

Diffusion is studied in a two-dimensional gas of triangular symmetry with repulsive interactions. The collective diffusion coefficient, D c, is calculated as the ratio of the average jump rate and the zero-wave-vector static structure factor (Darken approximation). The dependence of D c on coverage, θ, is studied in both the disordered and ordered regions of the phase diagram. It is shown that, around the coverages of the perfect ordered phases ( θ=1/3 and 2/3), D c changes of orders of magnitude in tiny θ ranges.

Lattice-Gas Theory of Collective Diffusion in Adsorbed Layers

A general theory for collective diffusion in interacting lattice-gas models is presented. The theory is based on the description of the kinetics in the lattice gas by a master equation. A formal solution of the master equation is obtained using the projection-operator technique, which gives an expression for the relevant correlation functions in terms of continued fractions. In particular, an expression for the collective dynamic structure factor S c is derived. The collective diffusion coefficient D c is obtained from S c by the Kubo hydrodynamic limit. If memory effects are neglected (Darken approximation), it turns out that D c can be expressed as the ratio of the average jump rate <w> and of the zero-wavevector static structure factor S(0). The latter is directly proportional to the isothermal compressibility of the system, whereas <w> is expressed in terms of the multisite static correlation functions gn. The theory is applied to two-dimensional lattice systems as models of adsorbates on crystal surfaces. Three examples are considered. First, the case of nearest-neighbour interactions on a square lattice (both repulsive and attractive). Here, the theoretical results for D c are compared to those of Monte Carlo simulations. Second, a model with repulsive interactions on the triangular lattice. This model is applied to NH 3 adsorbed on Re(0001) and the calculations are compared to experimental data. Third, a model for oxygen on W(110). In this case, the complete dynamic structure factor is calculated and the width of the quasi-elastic peak is studied. In the third example the gn are calculated by means of the discretized version of a classical equation for the structure of liquids (the Crossover Integral Equation), whereas in the other examples they are computed using the Cluster Variation Method.

Lattice-gas model of diffusion of NH 3 on Re(0001)

A lattice-gas model for the chemical diffusion of NH 3 on Re(0001) is presented. The system is modelled as a triangular lattice gas with nearest-neighbour repulsive lateral interactions. The average jump rate and the static structure factor are calculated within the cluster variation method framework by using a cluster of seven sites. Then the chemical diffusion coefficient is evaluated in the Darken approximation. This results in the jump rate and diffusion coefficient strongly increasing with coverage in agreement with experimental measurements.