Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / Section D / 25 / d / iii

A depletion force requires mobile nonadsorbing polymers excluded from the gap, rather than grafted chains alone. If their depletion layers overlap when the plate separation is $h<2\delta$, the overlap restores accessible solution volume and increases polymer entropy. For ideal, dilute depletants of bulk osmotic pressure $\Pi$, an idealized interaction free energy per plate area isIts attractive pressure is $-\Pi$ in the overlap interval. The layer thickness is of order the ideal-coil radius of gyration, $\delta \sim R_g=b_0\sqrt{N/6}$, so the range scales as $N^{1/2}$. For the supplied chain parameters, $R_g\simeq 81.6$ nm and an order-of-magnitude two-layer range is 163 nm. Grafted brushes instead oppose compression through stretching and osmotic penalties; they can coexist with depletion attraction if free depletants are present, but the mere existence of a grafted layer does not create a reservoir of depleting chains. The source does not specify that reservoir's concentration, so its attraction magnitude cannot be calculated.