Figure 2. The characteristic length of confinement δ.

(a) A local characteristic length δ_i can be defined at any atom i (with non-vanishing SAS). (b) The contribution to the total potential energy of neighbours along a direction orthogonal to the SAS (line l with local coordinate n) is computed. (c) The thermal energy level provides a criterion to (locally) define the characteristic length δ_i, which typically varies along the whole SAS. (d) A characteristic length of confinement δ for the whole structure can be defined by a weighted average of all δ_i. (e) The length of confinement is reported for a SiO₂ nanopore (δ₁≈0.33 nm) and for a Fe₃O₄ NP (δ₂≈0.50 nm). For SiO₂, only the first water layer (with distance r_w,1 from the pore axis) is located within the volume of influence. For Fe₃O₄, both the first and the second water layers (with distances r_w,1 and r_w,2 from the particle centre, respectively) are found within the volume of influence (more cases are reported in the Supplementary Figs 1 and 2).