= Solution
<Polydispersity> means that a sample contains chains of different <molar masses> or lengths. Let $n_i$ count chains with <molar mass> $M_i$. The <number-average molar mass> gives every chain equal weight, while the <weight-average molar mass> weights a chain population by its contribution to total <mass>:
$$
\boxed{M_n=\frac{\sum_i n_iM_i}{\sum_i n_i},\qquad
M_w=\frac{\sum_i n_iM_i^2}{\sum_i n_iM_i}.}
$$
Writing $w_i=n_iM_i/\sum_jn_jM_j$ makes the latter interpretation explicit: $M_w=\sum_iw_iM_i$. Large chains therefore influence $M_w$ more strongly. If brackets denote an average over chain number,
$$
\frac{M_w}{M_n}=\frac{\langle M^2\rangle}{\langle M\rangle^2}
=1+\frac{\operatorname{Var}(M)}{\langle M\rangle^2}\geq1.
$$
The ratio is <molar-mass dispersity>, conventionally written Đ. It equals one only when every chain has the same <molar mass>. Different distributions can share the same ratio, so it does not specify the full distribution.
For a <freely jointed chain>, write the end-to-end vector as the sum of $N$ independently oriented steps $\mathbf b_i$, each of fixed length $a$. <Isotropy> gives $\langle\mathbf b_i\rangle=\mathbf0$, and <independence> gives $\langle\mathbf b_i\cdot\mathbf b_j\rangle=0$ for $i\ne j$. Hence
$$
\langle R^2\rangle=
\sum_{i,j=1}^N\langle\mathbf b_i\cdot\mathbf b_j\rangle
=\sum_{i=1}^N a^2,
\qquad \boxed{\langle R^2\rangle=Na^2.}
$$
The root-mean-square <end-to-end distance of a polymer chain> is $a\sqrt N$, although the mean vector is zero. Its <polymer contour length> is $L=Na$, an upper bound on extension; it is not the typical unforced end-to-end length.
Stiffness correlates neighboring bond directions, so the discarded cross terms need not vanish. A <freely rotating chain> fixes the angle $\theta$ between successive forward bond vectors but leaves torsion uniformly distributed. Set $q=\cos\theta$, so $\langle\mathbf b_i\cdot\mathbf b_{i+s}\rangle=a^2q^s$. Then
$$
\langle R^2\rangle=Na^2+2a^2\sum_{s=1}^{N-1}(N-s)q^s
=Na^2\left[\frac{1+q}{1-q}
-\frac{2q(1-q^N)}{N(1-q)^2}\right],\qquad |q|<1.
$$
For a chain long compared with its orientational correlation range, the <characteristic ratio of a freely rotating chain> approaches $C_\infty=(1+q)/(1-q)$. Positive forward persistence increases the mean-square size and gives the <Kuhn length of a freely rotating chain> $b_K=aC_\infty$. A chemical internal bond angle is supplementary to the forward-vector angle used here, so substituting that internal angle requires $q=-\cos\theta_{\rm internal}$. For $q\to1$ at fixed $N$, the finite sum tends to $N^2a^2$, the rigid-rod result; taking the long-chain limit first would incorrectly predict an unbounded size. Real torsional energy preferences can be represented by a <rotational isomeric state model>, with weighted torsional states and correlations beyond independent free rotations.
A <worm-like chain> describes continuously varying local orientation. For unit tangent $\mathbf t(s)$ and bending stiffness $\kappa$, its elastic <energy> is $E_b=(\kappa/2)\int_0^L|d\mathbf t/ds|^2ds$. In three spatial dimensions the <persistence length> is $\ell_p=\kappa/(k_BT)$ and
$$
\langle\mathbf t(s)\cdot\mathbf t(s')\rangle=e^{-|s-s'|/\ell_p}.
$$
Since $\mathbf R=\int_0^L\mathbf t(s)\,ds$, integrating the tangent correlation gives the <end-to-end distance of a worm-like chain>:
$$
\boxed{\langle R^2\rangle=
2\ell_pL-2\ell_p^2(1-e^{-L/\ell_p}).}
$$
For $L\ll\ell_p$, expansion gives $\langle R^2\rangle=L^2-L^3/(3\ell_p)+\cdots$, approaching a straight rod. For $L\gg\ell_p$, it gives $\langle R^2\rangle\simeq2\ell_pL$, the long-chain <ideal chain> result with <Kuhn length> $2\ell_p$. These models isolate stiffness; long-range <excluded volume> can additionally swell an isolated chain in a <good solvent>. When coarse-graining, a chemical repeat count and a count of independent statistical steps must not be silently interchanged.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2016/paper-1-chain-and-brush-scaling.png]
{title=Stiff-chain crossover and distinct <polymer brush> density scalings}
The first panel shows the exact <worm-like chain> crossover between a rod and a long flexible coil. The second compares original normalized scaling curves for a uniform <polymer brush>; their prefactors have been chosen for illustration and do not fit the exam's experimental points.
For a <good solvent>, consider long flexible <grafted polymer chains> with <grafting density> $\sigma$ and lateral spacing $D\sim\sigma^{-1/2}$. At high coverage, $D$ is smaller than the isolated coil size, and adjacent chains cannot retain their unperturbed conformations. A scaling model divides each chain into <concentration blobs> of diameter $D$. Inside one blob the chain retains good-solvent statistics,
$$
D\sim ag^{3/5},\qquad g\sim(D/a)^{5/3},
$$
using the <Flory approximation> exponent $3/5$. The $N/g$ blobs per chain form a stretched sequence approximately normal to the surface. The <good-solvent brush height> therefore obeys
$$
h\sim\frac NgD\sim Na^{5/3}D^{-2/3},\qquad
\boxed{h\sim Na(\sigma a^2)^{1/3}.}
$$
Here $a$ and $N$ refer to consistently chosen effective segments, whose microscopic properties are held fixed as chain length changes. The result is a scaling law with an undetermined order-one prefactor, not a fully specified height in nanometres.
A complementary free-energy balance per chain is
$$
\frac{F}{k_BT}\sim\frac{h^2}{Na^2}+\frac{vN^2\sigma}{h},
$$
where $v>0$ is effective segment <excluded volume>. The first term is the chain <entropy> cost of stretching and the second is the cost of crowding segments into a layer of volume $h/\sigma$ per chain. Minimization gives $h^3\sim vN^3a^2\sigma$; with $v\sim a^3$ it recovers the same height. The blob argument incorporates good-solvent correlations more directly. Both descriptions require overlapping chains, many segments per chain, and heights below the <polymer contour length>; the power law cannot be extrapolated to infinite grafting density.
In the printed experimental graph, the vertical axis is $hD^2/(10^6\,\mathrm{\mathring A^3})$ and the horizontal axis is $M/10^5$. The approximately straight trend through the origin therefore means \b[$hD^2\propto M$]. Since $D^{-2}\sim\sigma$ and $M$ is proportional to chain length for a fixed repeat chemistry,
$$
\boxed{h\propto M\sigma\propto N\sigma\qquad\text{(poor solvent).}}
$$
The dashed guide rises by roughly $1.8$–$1.9$ vertical units over six horizontal units, corresponding to $hD^2/M$ of order $3\,\mathrm{\mathring A^3}$ per unit of the plotted $M$. This is an approximate graphical slope, not a fitted material constant; the graph does not specify a numerical monomer count or a repeat-unit volume from which an absolute brush height could independently be calculated.
The <poor-solvent brush height> has a simple volume argument. Let $v_0$ be the volume per repeat and $\phi$ the roughly fixed polymer <volume fraction> within a dense uniform layer. There are $N\sigma$ repeats per substrate <area>, so
$$
\phi h=N\sigma v_0,\qquad
\boxed{h\simeq\frac{Nv_0\sigma}{\phi}.}
$$
For fixed poor-solvent conditions, $\phi$ is approximately independent of $N$ and $\sigma$, giving the observed linear grafting-density dependence. Polymer–polymer contacts are favored and the layer becomes relatively compact; in a <good solvent>, favorable solvent contact and segment repulsion instead produce a swollen layer whose stretching and osmotic costs balance at the weaker $\sigma^{1/3}$ dependence. Sparse isolated <polymer globules> or lateral segregation fall outside the uniform-brush model.
The cited experimental paper's abstract independently reports these two scaling laws, but identifies polydimethylsiloxane chains on porous <silicon dioxide>, rather than the <polystyrene> system named in the exam. The solution uses the printed axes to infer the exponents and does not treat that material identification as verified experimental provenance.<Characterization of the brush regime for grafted polymer layers at the solid-liquid interface by P. Auroy and others>[Physical Review Letters 66 (1991), 719–722, abstract: neutron-scattering measurement, chemical system, and good- versus poor-solvent thickness scaling]{ref}
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