= Solution
Write $s$ for the number of grafted chains per unit <area> and $\Sigma=sa^2$ for its dimensionless form; this separates the physical <grafting density> from a dimensionless coverage. A free dilute chain has size $R_0\sim aN^\nu$, with $\nu=1/2$ in a <theta solvent> and approximately $3/5$ in a <good solvent>. An isolated end-grafted chain loses translations and configurations that cross the substrate, but retains the same leading size exponent if the surface is nonadsorbing. When the lateral graft spacing $D\sim s^{-1/2}$ exceeds $R_0$, the layer consists of <polymer mushrooms>. When coils overlap, crowding stretches the chains away from the surface into a <polymer brush>.
The <polymer mushroom-to-brush crossover> is
$$
\boxed{s_*\sim R_0^{-2},\qquad \Sigma_*\sim N^{-2\nu}.}
$$
For the stipulated statistical-segment interpretation of $N$ and $a$, the scaling estimates are
$$
\begin{array}{c|ccc}
&R_0&D_*&s_*\;(\mathrm{nm^{-2}})\\\hline
\theta&14\,\mathrm{nm}&14\,\mathrm{nm}&5.10\times10^{-3}\\
\text{good}&25.5\,\mathrm{nm}&25.5\,\mathrm{nm}&1.54\times10^{-3}
\end{array}
$$
Thus $\Sigma_*\simeq2.50\times10^{-3}$ and $7.54\times10^{-4}$ respectively. The numerical constants are scaling conventions: using a <radius of gyration> rather than an end-to-end size changes order-one amplitudes, not the powers of $N$.
In the <Alexander-de Gennes brush model>, take blobs of lateral size $D$. Within a blob, $D\sim ag^\nu$; beyond it, the $N/g$ blobs are aligned away from the wall. Thus
$$
g\sim(D/a)^{1/\nu},\qquad
\boxed{H\sim ND/g=Na\Sigma^{(1-\nu)/(2\nu)}.}
$$
With coarse-grained segment <volume> $a^3$, conservation gives the <polymer-brush volume fraction> through $\phi H\sim Na^3s$, hence
$$
\boxed{\phi\sim\Sigma^{(3\nu-1)/(2\nu)},\qquad
\begin{array}{c|cc}
&H&\phi\\\hline
\theta&Na\Sigma^{1/2}&\Sigma^{1/2}\\
\text{good}&Na\Sigma^{1/3}&\Sigma^{2/3}
\end{array}}
$$
The <theta-solvent brush height> includes three-segment repulsion even though the effective two-segment interaction vanishes. These are scaling estimates for a roughly uniform brush, not an assertion that every chain end lies at the same height in an actual brush.
The many-segment-blob requirement $g\gg1$ gives $D\gg a$, or $\Sigma\ll1$. Together with overlap, the regime is \b[$N^{-2\nu}\ll\Sigma\ll1$]. At the <dense-brush packing limit>, $\phi\sim1$, $g\sim1$ and $H\sim Na$. Here $s\sim a^{-2}=2.04\,\mathrm{nm^{-2}}$ and $H\sim280\,\mathrm{nm}$. If one formally continues a dense incompressible layer above this boundary, conservation would instead give
$$
\boxed{\phi\simeq1,\qquad H\simeq Na^3s=Na\Sigma.}
$$
This is the requested formal dense-layer expression, but for $\Sigma>1$ it gives $H>Na$, which violates <finite extensibility of a polymer chain>. \b[There is no physically admissible continuation above this limit in the stated segment-volume model.] A <Kuhn length> segment is not necessarily a cube of material: with an independently specified material <volume> $v_0$ per counted segment, the dense-layer law is $H=Nv_0s$, and the packing bound is $s\leq a/v_0$. The data do not determine $v_0$ separately. The simple model therefore establishes its own breakdown, rather than a universal microscopic maximum <grafting density>.
<grafting-to> uses preformed chains whose <lengths> and dispersities can be characterized before attachment. As coverage rises, each incoming coil must penetrate a crowded brush and lose <conformational entropy> to bring its end to the surface; the growing insertion barrier limits achievable coverage. <grafting-from> supplies small <monomers> to surface-bound growing chains and avoids insertion of a whole coil. \b[Choose <grafting-from> for high <grafting density>.] Initiator coverage and reaction control still matter, and inaccessible active sites, termination and unequal growth can broaden the resulting chain-length distribution. Neither route removes the elastic and packing costs of the final brush.
A generic application is <steric stabilization> of synthetic <colloids> in a <good solvent>. Bringing coated particles together compresses and interpenetrates their brushes, increasing <osmotic pressure> and conformational free-energy cost. This repulsion can oppose aggregation. The chains must remain attached and sufficiently cover the surface; poor-solvent collapse, sparse coverage or interparticle bridging can defeat stabilization.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2013/paper-1-brush-scaling.png]
{title=Original brush-height and volume-fraction scaling, showing the segment-packing limit}
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