On the sign and critical scaling of the tolman length: a comparison of interfacial and bulk-pressure routes

The Tolman length, , is a fundamental parameter for describing the curvature dependence of surface tension, yet its determination remains controversial due to persistent discrepancies in its sign and magnitude. In this work, we systematically investigate these inconsistencies for the truncated and shifted Lennard-Jones fluid ( ) using molecular dynamics simulations. We evaluate through three distinct methodologies: the geometric difference , the surface tension expansion , and the bulk-pressure route . To address the non-uniqueness of the pressure tensor, we implement a discretized Irving-Kirkwood contour with sampling points, finding that is required for numerical consistency in interfacial-based routes. Our results show that interfacial definitions ( ) are consistently positive, with being approximately twice as large as across all studied conditions. In contrast, the bulk-pressure route yields a small negative Tolman length, at . Both the and frameworks exhibit a remarkable dual self-consistency: they quantitatively recover the planar surface tension for all as curvature vanishes, while capturing the expected qualitative trends for critical point shifts and droplet vapor pressure via the modified Kelvin equation. This dual consistency helps explain the historical difficulty in reaching a consensus on the sign of . However, we suggest that the bulk-pressure route is more robust, as it is found to be independent of the discretization parameter and avoids numerical integration across the anisotropic interfacial region. Finally, we found that all definitions follow a temperature dependence consistent with the universal scaling using the 3D Ising critical exponent . • Comparison of three Tolman length definitions reveals method-dependent sign inconsistencies. • Geometric and surface tension-based Tolman lengths exhibit a systematic 2:1 ratio. • The bulk-pressure route recovers negative Tolman lengths for all T, reconciling simulation and DFT results. • Temperature dependence of all definitions follows universal 3D Ising scaling with v=0.63. • Droplet critical coordinates shift consistently with finite-size scaling laws.