Calculation of local pressure tensors in systems with many-body interactions

Local pressures are important in the calculation of interface tensions and in analyzing micromechanical behavior. The calculation of local pressures in computer simulations has been limited to systems with pairwise interactions between the particles, which is not sufficient for chemically detailed systems with many-body potentials such as angles and torsions. We introduce a method to calculate local pressures in systems with $n$-body interactions $(n=2,3,4,\dots )$ based on a micromechanical definition of the pressure tensor. The local pressure consists of a kinetic contribution from the linear momentum of the particles and an internal contribution from dissected many-body interactions by infinitesimal areas. To define dissection by a small area, respective $n$-body interactions are divided into two geometric centers, effectively reducing them to two-body interactions. Consistency with hydrodynamics-derived formulas for systems with two-body interactions [J. H. Irving and J. G. Kirkwood, J. Chem. Phys. 18, 817 (1950)], for average cross-sectional pressures [B. D. Todd, D. J. Evans, and P. J. Daivis, Phys. Rev. E 52, 1627 (1995)], and for volume averaged pressures (virial formula) is shown. As a simple numerical example, we discuss liquid propane in a cubic box. Local, cross-sectional, and volume-averaged pressures as well as relative contributions from two-body and three-body forces are analyzed with the proposed method, showing full numerical equivalence with the existing approaches. The method allows computing local pressures in the presence of many-body interactions in atomistic simulations of complex materials and biological systems.

Figure 1

Model of a small cube to illustrate the definition of local pressures. The three shaded faces $A_{\alpha }$ share a point of intersection in the geometric center of the cube. Some particles and molecules are also shown.

Figure 2

Approximation of the local pressure tensor $p_{\beta \alpha }(x,y,z)$ as a function of the three coordinates. We imagine a movable grid along each of the Cartesian axes $\alpha$. For a point function pressure tensor, the microscopic cubes $\Delta V={\left(2\Delta \alpha \right)}^3$ approach a zero volume.

Figure 3

The definition of geometric centers on both sides of the area $A_{\alpha }$, for a three-body interaction and for a four-body interaction. The connecting vector between the geometric centers of the four-body interaction is intersected by $A_{\alpha }$, leading to a contribution to the internal pressure. The three-body interaction does not contribute to the internal pressure across $A_{\alpha }$. If we assume bonds, angles, torsions, and pairwise van der Waals interactions (excluding 1,2 and 1,3 van der Waals interactions) between the particles, we find contributions from two bonds, one angle, one torsion, and eight nonbond interactions to the internal pressure across $A_{\alpha }$.

Figure 4

Average cross-sectional pressures as a function of the coordinate $\alpha$. The calculation of the internal part of the average pressure component ${\overline{p}}_{zz}^{\mathrm{int}}\left(z\right)$ is illustrated schematically.

Figure 5

Snapshot of 382 propane molecules (black) inside a $4\times 4\times 4{\mathrm{nm}}^3$ cubic box bounded by wall atoms (gray).

Figure 6

Contributions to the net atomic forces and their relative significance.

Figure 7

Partition of the cross-sectional area of the box into arrays.

Figure 8

Pressure profile of the $p_{zz}$ component along the $z$ axis for the arrays in Fig. 7 including kinetic, two-body, and three-body contributions. For reference, the density profile for the entire cross section is also shown.