Proof of the identity between the depletion layer thickness and half the average span for an arbitrary polymer chain

The confinement analysis from bulk structure (CABS) approach [Y. Wang et al., J. Chem. Phys. 128, 124904 (2008)] is extended to determine the depletion profiles of dilute polymer solutions confined to a slit or near an inert wall. We show that the entire spatial density distributions of any reference point in the polymer chain (such as the center of mass, middle segment, and end segments) can be computed as a function of the confinement size solely based on a single sampling of the configuration space of a polymer chain in bulk. Through a simple analysis based on the CABS approach in the case of a single wall, we prove rigorously that (i) the depletion layer thickness delta is the same no matter which reference point is used to describe the depletion profile and (ii) the value of delta equals half the average span (the mean projection onto a line) of the macromolecule in free solution. Both results hold not only for ideal polymers, as has been noticed before, but also for polymers regardless of details in molecular architecture and configuration statistics.