= 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 gives
$$
\boxed{U_{\rm min}\simeq-\rho k_{\rm B}T\,a
\left[\ell-\sqrt{(\ell+d)^2-\ell^2}\right]
\simeq-\rho k_{\rm B}T\,a\ell.}
$$
The 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.
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