= Solution
Minimize the <Helmholtz free energy> at fixed $N,T$. Its <derivative> is
$$
\frac{\partial F}{\partial\xi}=Nk_BT\left(-\frac{2b^2}{\xi^3}+\frac{\epsilon b}{\xi^2}\right).
$$
The physical <stationary point> for $\epsilon>0$ is
$$
\boxed{\xi_* =\frac{2b}{\epsilon},\qquad F_{\mathrm{ads},*}=-\frac14Nk_BT\epsilon^2.}
$$
To check the minimum directly, put $x=b/\xi$. Then $F/(Nk_BT)=(x-\epsilon/2)^2-\epsilon^2/4$, which is globally minimized at $x=\epsilon/2$. The binding <free energy> gain per <monomer> is consequently $\mu=k_BT\epsilon^2/4>0$.
The weak-<adsorption> blob picture requires $b\ll\xi_*\ll b\sqrt N$, or $2/\sqrt N\ll\epsilon\ll2$ in this model. If $\epsilon$ is too large, the layer saturates at a monomer-scale thickness; if too small, a many-blob adsorbed layer has not formed.
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