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In this paper, we analyze the shape of two-dimensional and three-dimensional drops on flat and curved substrates by minimizing free energy with the variational approach. The process initiated from the flat substrate and has proved the drop shape equation. Then, the drop shape equation for these models has been determined by extending the method to curved substrates. The constraints in 2D and 3D models are constant surface area and volume, respectively. The Euler–Lagrange differential equation can be solved analytically in the two-dimensional case and the drop height as a function of x has been driven which is the equation of a circle. In the 3D model, a fundamental partial differential equation (PDE) has been obtained but it cannot be solved analytically and has been solved through a numerical method. This PDE becomes a first-order ordinary differential equation (ODE) after two reduction steps and is solved by the Euler explicit method. Six different systems are simulated and a logical algorithm is introduced to compare the results of the simulation, numerical, and analytical. The simulation results determined there is an iso-density curve or surface that is closest to the analytical result. It has revealed the solution of the three-dimensional case is not a sphere. The spherical cap approximation (SCA) has been widely used in many wettability studies. Comparison of numerical and SCA results have determined this assumption is valid only for contact angles far from 90 degrees. The present results are determined all the drops on each surface can be described by only one equation if the contact angle is measured for the line (in the case of a 2D drop) or the passing plate (in the case of a 3D drop) from the drop edges. The substrate characteristics at the three-phase contact point (TPCP) and the three-phase contact line (TPCL) are the only factors determining the drop shape. The other points and areas away from TPCP and TPCL have not influenced the drop shape.

Nanotubes have extraordinary properties, which have attracted the attention of many researchers from diverse fields. The interaction between carbon nanotube (CNT) or boron nitride nanotube (BNNT) and polymer/copolymer/surfactant has shown potential improvement in properties and performance. This paper reviews the recent studies in this field obtained from molecular dynamics simulation calculations, focusing on the interaction energies between nanostructure and polymer, radial distribution function, diffusion and radius of gyration and some other physical chemistry properties. Recent studies show that the intermolecular interaction in mentioned systems is strongly influenced by the specific monomer structure of polymers. The high values of intermolecular interaction energy of such composites offer that an efficient load transfer exists at the interface between nanotube and polymer, which is of a key role in the composite reinforcement practical applications. Our study reviewed the possibility of wrapping CNT/BNNT/CNT bundles by polymers and also the effects of CNTs/CNT bundles’ length on the conformational behavior of polymer adsorbed on these nanostructures.

In this study, the dispersion mechanism of aggregated carbon nanotubes (CNTs) using Triton X-100 surfactant under various concentrations is investigated with and without water molecules via molecular dynamics simulation. The obtained results showed that because of interaction between water molecules and hydrophilic segments of surfactant, water molecules play a significant role in the manner of adsorption of the surfactant on the CNT surface. In the presence of water molecules, the surfactant molecules are not able to wrap the CNTs, and they are located in the neighborhood of the CNTs. The results suggested that the creation of space between two CNTs in the absence of the surfactant is performed slowly, while, in the presence of the surfactant molecules, the creation of space between two CNTs which leads to the dispersion of the CNTs is remarkably rapid. The surfactant molecules cause to introduce more numbers of water molecules in the vicinity of and between the CNTs, and with the increasing radial distances between two CNTs, the number of water molecules is rapidly increased. The interfacial angle between two CNTs, surfactant gyration radius, and diagrams of radial distribution function between water molecules, the CNTs, and the surfactant molecules were calculated for a better description of dispersion mechanism of the CNTs by the surfactant and water molecules.