= Dense-brush packing limit
{title2=$s\leq a/v_0$}
A dense uniform <polymer brush> with $\phi\simeq1$ must have height $H\simeq Nv_0s$ by <polymer-brush volume fraction>. The contour bound $H\leq Na$ therefore requires $s\leq a/v_0$. If $v_0=a^3$, the limit is $sa^2\leq1$; beyond it the putative incompressible branch demands chains longer than their <polymer contour length>. This is a failure of that microscopic packing model. A <Kuhn length> does not alone determine $v_0$.
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