Analytical perspectives on the Tolman length in van der Waals theory

We derive an ordinary differential equation for the interface profile in van der Waals theory and propose a computational procedure to solve it, enabling analytical determination of the Tolman length. For a planar interface (i.e., infinite interface radius), the Tolman length is slightly negative and varies nonmonotonically, first decreasing and then increasing, with the reduced temperature. For curved interfaces, it exhibits a consistent inverse-linear dependence on the interface radius (positive for droplets and negative for bubbles), changing sign to positive for sufficiently small droplets. The surface tension first increases and then significantly decreases with decreasing droplet radius, but decreases monotonically for bubbles.