Examining the Mechanical Equilibrium of Microscopic Stresses in Molecular Simulations

The microscopic stress field provides a unique connection between atomistic simulations and mechanics at the nanoscale. However, its definition remains ambiguous. Rather than a mere theoretical preoccupation, we show that this fact acutely manifests itself in local stress calculations of defective graphene, lipid bilayers, and fibrous proteins. We find that popular definitions of the microscopic stress violate the continuum statements of mechanical equilibrium, and we propose an unambiguous and physically sound definition.

Figure 1

Balance of linear momentum of the microscopic stress in a graphene sheet with a Stone-Wales defect: comparison of the atomic virial (upper row) and the IK (lower row) stress definitions. (a) Trace of the raw stress fields. (b) Spatially averaged trace of the stress field. (c) Norm of the divergence of the spatially averaged microscopic stress. Plots focus on a small region near the defect in the $x-y$ plane. Because the system is quasi-2D, we only consider the in-plane components and express stress in units of surface tension.

Figure 2

Balance of angular momentum of the IK stress in a planar DPPC fluid membrane, and influence of lipid chirality. In-plane components of the stress tensor analyzed with CFD (a) and GLD (b). (c) Visualization of the normal and tangential components of the traction vector along a cylindrical surface perpendicular to the bilayer plane, for both CFD and GLD. The effect of the lipid chirality on the GLD stress for the two DPPC enantiomers, L-DPPC and D-DPPC, is shown in (d), where we consider a lipid membrane composed of monolayers with different chiralities (one with pure L-DPPC and the other with pure D-DPPC, left), and a bilayer with both monolayers having equal numbers of L-DPPC and D-DPPC (right).

Figure 3

The IK stress for force fields beyond four-body interactions. (a) Ribbon representation of a structural coiled-coil protein simulated with the five-body CMAP potential (cross-term energy correction map for adjacent dihedrals used with the CHARMM22 force field). Tractions at the surface of the protein are calculated with different variants of the IK stress: GLD (b), cCFD (c), and nCFD (d, see text).