Direct Measurement of Length Scale Dependence of the Hydrophobic Free Energy of a Single Collapsed Polymer Nanosphere

The physics underlying hydrophobicity at macroscopic and microscopic levels is fundamentally distinct. However, experimentally quantifying the length scale dependence of hydrophobicity is challenging. Here we show that the size-dependent hydrophobic free energy of a collapsed polymer nanosphere can be continuously monitored from its single-molecule force-extension curve using a novel theoretical framework. The hydrophobic free energy shows a change from cubic to square dependence of the radius of the polymer nanosphere at a radius of $\ensuremath{\sim}1\text{ }\text{ }\mathrm{nm}$---this is consistent with Lum-Chandler-Weeks theory and simulations. We can also observe a large variation of the hydrophobic free energy of each polymer nanosphere implying the heterogeneity of the self-assembled structures and/or the fluctuation of the water-polymer interface. We expect that our approach can be used to address many fundamental questions about hydrophobic hydration, which are otherwise inaccessible by ensemble measurements.