Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 5 / b

The unrotated squares have a gap $d>a$, so their facing exclusion layers do not overlap. Rotation can bring them into contact even though their centers remain fixed. The strongest depletion force comes from parallel faces in contact over a long shared length, rather than a corner-to-corner contact.

Let $r=\ell +d$. Rotate both squares through the same angle $\theta$ relative to their original orientation. Their center displacement has a face-normal component $r\cos \theta$ and a tangential component $r\sin \theta$. Face contact requiresThe contacting faces overlap for lengthThese two mirror-related orientations, together with equivalent square rotations, maximize the leading overlap area when $a<d\ll \ell$. A small polymer gap sacrifices overlap width without a compensating leading increase in contact length.

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.