 Solution

Each face excludes polymer centers out to distance $a/2$. At face contact, the combined overlap strip has width $a$ and length $L$. Ignoring end corrections of order $a^2$, the two-dimensional Asakura-Oosawa model therefore givesThe final expression retains only the leading term in $d/\ell$. Its dimensions are correct because $\rho$ is a number per area. The zero of this effective potential energy is the well-separated, nonoverlapping-exclusion-layer configuration.