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Continuum modeling using the Lennard-Jones potential has been shown to provide a good estimation for the interaction energy between regular-shaped homogeneous molecules comprising the same type of atoms. However, this method may not be accurate for heterogeneous molecules, which are made up of more than one chemical element. The traditional method to deal with this involves approximating the molecule via multiple surfaces in a piecemeal fashion. While this approach works well for small sized molecules, calculations become intensive for large sized molecules as a large number of sums from multiple surface interactions are involved. To address this issue, we propose a new model that approximates a heterogeneous molecule with a single surface or volume, where attractive and repulsive constants (A and B) in the Lennard-Jones potential are replaced by functions A(r) and B(r), which depend on the parameterization of the surface r. We comment that this technique is suitable for regular-shaped nanostructures where their heterogeneity can be modeled by surface (or volume) parameterization. Validation of the new approach is carried out via two problems, namely, carbon nanotube–methane and carbon nanotube–coronene interactions. For coronene and methane, which are assumed to be radially symmetric, we propose A(r) and B(r) to be sigmoidal functions for which the interaction strength decreases from the inner region of the carbon atoms toward the outer region of the hydrogen atoms. Our results for both cases show that using the sigmoidal profiles for A(r) and B(r) gives rise to interaction energies that are in better agreement with those obtained from molecular dynamics studies compared to results using constant A and B. The new approach provides a significant improvement to the current continuum modeling using the Lennard-Jones potential.

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The Lennard–Jones potential and a continuum approach can be used to successfully model interactions between various regular shaped molecules and nanostructures. For single atomic species molecules, the interaction can be approximated by assuming a uniform distribution of atoms over surfaces or volumes, which gives rise to a constant atomic density either over or throughout the molecule. However, for heterogeneous molecules, which comprise more than one type of atoms, the situation is more complicated. Thus far, two extended modeling approaches have been considered for heterogeneous molecules, namely a multi-surface semi-continuous model and a fully continuous model with average smearing of atomic contribution. In this paper, we propose yet another modeling approach using a single continuous surface, but replacing the atomic density and attractive and repulsive constants in the Lennard–Jones potential with functions, which depend on the heterogeneity across the molecules, and the new model is applied to study the adsorption of coronene onto a graphene sheet. Comparison of results is made between the new model and two other existing approaches as well as molecular dynamics simulations performed using the LAMMPS molecular dynamics simulator. We find that the new approach is superior to the other continuum models and provides excellent agreement with molecular dynamics simulations.

SUMMARYAnisotropic diffusion phenomenon in fluids is simulated using smoothed particle hydrodynamics (SPH). A new SPH approximation for diffusion operator, named anisotropic SPH approximation for anisotropic diffusion (ASPHAD), is derived. Basic idea of the derivation is that anisotropic diffusion operator is first approximated by an integral in a coordinate system in which it is isotropic. The coordinate transformation is a combination of a coordinate rotation and a scaling in accordance with diffusion tensor. Then, inverse coordinate transformation and particle discretization are applied to the integral to achieve ASPHAD. Noting that weight function used in the integral approximation has anisotropic smoothing length, which becomes isotropic under the inverse transformation. ASPHAD is general and unique for both isotropic and anisotropic diffusions with either constant or variable diffusing coefficients. ASPHAD was numerically examined in some cases of isotropic and anisotropic diffusions of a contaminant in fluid, and the simulation results are very consistent with corresponding analytical solutions.

Ceramic membranes are currently favourable in membrane filtration applications due to their excellent mechanical strength, thermal and chemical resistance, backflush capability, and thus a long-service cycle. Coated on top of a mesoporous support, the selective top layer of ultrafiltration ceramic membranes has pore size not exceeding a few tens of nanometers and thickness in the order of O10 μm. In fact, the permeability of an ultrafiltration ceramic membrane can be estimated by the permeability of the top layer due to its smallest pore size. Without impairing the filtration function but still improving the permeability, a gradient conical pore shape is proposed. Two formulae for the filtrate flow rate versus pressure drop relationship through a conical pore exhibiting surface slippage are established here by extending the Hagen-Poiseuille law and an analytical solution for the axisymmetric creeping flow. It is analytically proved that the surface slip length in a conical flow is proportional to a local pore radius by a slip coefficient that is unique for a given pore configuration at a prescribed flow rate. The permeability of a conical-pore membrane is enhanced for radius ratio not exceeding 6.5. The optimum configuration, achieved at a ratio of 2.3, produces an enhancement factor for a membrane permeability of 1.5 for a no-slip surface; this enhancement increases linearly with the slip coefficient if a surface slippage exists.

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