Direct calculation of interfacial tensions from computer simulation: Results for freely jointed tangent hard sphere chains

We develop a methodology for the calculation of surface free energies based on the probability distribution of a wandering interface. Using a simple extension of the $NpT$ sampling, we allow the interface area to randomly probe the available space and evaluate the surface free energy from histogram analysis and the corresponding average. The method is suitable for studying systems with either continuous or discontinuous potentials, as it does not require explicit evaluation of the virial. The proposed algorithm is compared with known results for the surface tension of Lennard-Jones and square well fluids, as well as for the interface tension of a bead-spring polymer model, and good agreement is found. We also calculate the interfacial tensions of freely jointed tangent hard sphere chains on athermal walls for a wide range of chain lengths and densities. The results are compared with three different theoretical approaches: scaled particle theory (SPT), Yu-Wu density functional theory, and an analytical approximation based on the latter approach. Whereas SPT only yields qualitative results, the last two approaches are found to yield very good agreement with simulations.

Figure 1

(Color online) Liquid-vapor surface tension of the spherically truncated $(R_c=2.5)$ Lennard-Jones model ${\gamma }^{*}={\gamma }_{lv}{\sigma }^2∕ϵ$ as a function of temperature $T^{*}=k_{B}T∕ϵ$. The squares and circles are obtained from the contertension and the bracketed sampling methods, respectively. The stars are results from [14]. The inset is a plot of the inverse-squared surface area against $1∕\beta \xi$, with slope equal to $\beta \gamma$.

Figure 2

(Color online) Liquid-vapor surface tension of a square well fluid with $\lambda =1.5\sigma$. Results are given in reduced units, with ${\gamma }^{*}={\gamma }_{lv}{\sigma }^2∕ϵ$ plotted as a function of temperature $T^{*}=k_{B}T∕ϵ$. Circles: results from the bracketed sampling method of this work. Squares: anisotropy of the virial tensor [15]. Diamonds: test area method [34]. Triangles: Binder’s method [56].

Figure 3

(Color online) Wall-fluid interface tensions for the LJ-FENE model. The figure shows the spreading coefficient ${\gamma }_{wv}-{\gamma }_{wl}$ as a function of wall strength ${ϵ}_w$. Squares are results obtained in this work from explicit evaluation of ${\gamma }_{wv}$ and ${\gamma }_{wl}$. Circles and lines are results for the spreading coefficients from Ref. 9. The dashed horizontal lines correspond $\pm {\gamma }_{lv}$ and the intersections correspond to wetting and drying transitions. The inset shows the tensions as a function of wall strength. Circles and squares refer to the wall-liquid and wall-vapor tensions, respectively.

Figure 4

(Color online) Surface-area mean-squared displacement as a function of Monte Carlo cycles for chains of length $m=16$. From top to bottom, results refer to monomer densities $\rho =0.10{\sigma }^{-3}$ (solid line), $\rho =0.30{\sigma }^{-3}$ (dashed line, 2 times magnified), and $\rho =0.50{\sigma }^{-3}$ (dash-dotted line, 10 times magnified).

Figure 5

(Color online) Wall-fluid interface tensions for hard sphere chains adsorbed on an athermal substrate as a function of chain length. From top to bottom, results refer to $\rho =0.1{\sigma }^{-3}$, $\rho =0.2{\sigma }^{-3}$, $\rho =0.3{\sigma }^{-3}$, $\rho =0.4{\sigma }^{-3}$, $\rho =0.5{\sigma }^{-3}$, and $\rho =0.6{\sigma }^{-3}$. Symbols show simulation results from this work. Lines are theoretical results: Solid line, TPT1-DFT; dashed line, SPT; dotted lines, DFT-based analytical approximation.

Figure 6

(Color online) Wall-fluid interface tensions for hard sphere chains of length $m=12$ adsorbed on an athermal substrate with the dividing surface located at $z=\frac{1}{2}\sigma$. The circles are simulation results with the volume of the slit pore defined as $V=(L_z-\sigma )$. The solid squares are results obtained by subtracting $\frac{1}{2}p\sigma$ from the simulated data for ${\gamma }_{z=0}$. The lines are theoretical results. Solid line, TPT1-DFT; dashed line, SPT; dotted lines, DFT-based analytical approximation.