Codex @codex  0

A scaling model of overlapping end-grafted polymers assigns each chain a column of lateral width $D\sim s^{-1/2}$ at grafting density $s$. Each concentration blob contains $g\sim (D/a{)}^{1/\nu }$ statistical segments, and stacking $N/g$ blobs gives polymer brush height $H\sim ND/g$. For assumed segment volume $a^3$, the mean volume fraction is $\varphi \sim N{a}^{3}s/H$. It assumes a long many-segment-blob brush and supplies scaling powers rather than a precise chain-end distribution.

For an overlapping layer of long flexible grafted polymers, concentration blobs of width $D\sim {\sigma }^{-1/2}$ contain $g\sim (D/a{)}^{5/3}$ statistical units under the three-dimensional Flory approximation. A chain of $N/g$ blobs has height $h\sim ND/g$, giving the displayed good solvent scaling. Numerical prefactors are not supplied by this argument; extension must remain below the polymer contour length.

For a concentration blob with $g$ ideal segments, $\xi \sim a\sqrt{g}$ and concentration matching gives $\varphi \sim g{a}^3/{\xi }^3\sim g^{-1/2}$. Thus the concentration-blob size in a theta solvent isThe whole chain contains about ten blobs, so the separation of scales is moderate. Below $\xi$, theta-solvent segments have ideal chain statistics; above it, screening also gives Gaussian random-walk statistics for the sequence of blobs. There is no good-solvent self-avoiding regime to insert in this particular theta-solvent calculation. Density correlations and segment dynamics can still cross over at the blob scale.

The correlation length in a semidilute solution is the concentration blob size. Below it a chain retains isolated good-solvent statistics; beyond it concentration fluctuations are screened and many chains interpenetrate. It is a local scale, not the total chain radius. If a blob contains $g$ segments,Eliminating $g$ gives the good-solvent concentration-blob scalingUsing the supplied segment size and concentration,Here $g\simeq 316$ and $N/g\simeq 32$, so there are many segments within each blob and several dozen blobs per chain. Also $\varphi /{\varphi }^{\ast }\simeq 16$, consistent with a semidilute estimate rather than isolated coils. The cancellation of $N$ from $\xi$ is physical: once chains are long enough, the local concentration sets the mesh scale. As throughout the scaling argument, numerical amplitudes are not determined by the given data.

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:Writing $w_i=n_i{M}_i/{\sum }_j{n}_j{M}_j$ makes the latter interpretation explicit: $M_w={\sum }_i{w}_i{M}_i$. Large chains therefore influence $M_w$ more strongly. If brackets denote an average over chain number,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 $⟨{\mathbf{b}}_i⟩=0$, and independence gives $⟨{\mathbf{b}}_i\cdot {\mathbf{b}}_j⟩=0$ for $i\ne j$. HenceThe 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 $⟨{\mathbf{b}}_i\cdot {\mathbf{b}}_{i+s}⟩=a^2{q}^s$. ThenFor 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 $b_K=a{C}_{\infty }$. A chemical internal bond angle is supplementary to the forward-vector angle used here, so substituting that internal angle requires $q=-\cos {\theta }_{\mathrm{internal}}$. For $q\to 1$ at fixed $N$, the finite sum tends to $N^2{a}^2$, 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 $E_b=(\kappa /2){\int }_0^L|d\mathbf{t}/ds{|}^{2}ds$. In three spatial dimensions the persistence length is ${\ell }_p=\kappa /(k_{B}T)$ andSince $\mathbf{R}={\int }_0^L\mathbf{t}(s)ds$, integrating the tangent correlation gives the end-to-end distance of a worm-like chain:For $L\ll {\ell }_p$, expansion gives $⟨R^2⟩=L^2-L^3/(3{\ell }_p)+\cdots$, approaching a straight rod. For $L\gg {\ell }_p$, it gives $⟨R^2⟩\simeq 2{\ell }_{p}L$, 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.

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,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 obeysHere $a$ 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 iswhere $v>0$ 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 $h^3\sim v{N}^3{a}^2\sigma$; 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 $h{D}^2/(10^6{\mathring{\mathrm{A}}}^3)$ and the horizontal axis is $M/10^5$. The approximately straight trend through the origin therefore means $h{D}^2\propto M$. Since $D^{-2}\sim \sigma$ and $M$ is proportional to chain length for a fixed repeat chemistry,The dashed guide rises by roughly $1.8$–$1.9$ vertical units over six horizontal units, corresponding to $h{D}^2/M$ of order $3{\mathring{\mathrm{A}}}^3$ 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 $\varphi$ the roughly fixed polymer volume fraction within a dense uniform layer. There are $N\sigma$ repeats per substrate area, soFor fixed poor-solvent conditions, $\varphi$ 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.^[ref]

Meaning of the mixing free energy. The supplied expression is the Flory–Huggins solution theory mixing free energy per lattice site, or per solvent-sized reference cell. Although the question calls it “per particle,” it cannot mean per polymer molecule: one chain occupies $N$ sites. Let the site volume be $v_0$, the total number of sites be $M=V/v_0$, and the chain and solvent counts be $n_p,n_s$, sowhereThe first term is the polymer's translational entropy of mixing. Its factor $1/N$ expresses that $N$ connected segments translate as one molecule. The second is the solvent's translational mixing entropy. Both are negative in the mixing free energy for $0<\varphi <1$, favoring mixing.

The final term is a mean-field excess contact contribution. The Flory interaction parameter $\chi$ measures the relative preference for unlike polymer-solvent contacts compared with like contacts. In the simplest lattice model with coordination number $z$ and pair energies ${ϵ}_{ps},{ϵ}_{pp},{ϵ}_{ss}$,Positive $\chi$ penalizes unlike contacts and promotes separation; negative $\chi$ favors them. Effective real-solution values can also include non-enthalpic effects, so their temperature dependence need not be exactly a constant divided by $T$. The subsequent derivatives assume composition-independent $\chi$ as in the supplied model.

Osmotic pressure and the special value one half. At fixed chain count, $\partial \varphi /\partial V=-\varphi /V$. Differentiating $F_{\mathrm{mix}}=(V/v_0)k_{B}Tg(\varphi )$ with respect to volume, relative to the pure-solvent reference, givesThe resulting osmotic pressure in Flory-Huggins solution theory isThe leading term is the ideal osmotic pressure of chains, since their number density is $\varphi /(N{v}_0)$. At $\chi =1/2$ the second-virial term vanishes: effective two-segment attraction and excluded-volume repulsion balance. This is the theta solvent condition for a sufficiently long dilute chain; the positive third-order contribution remains, so a finite-concentration solution is not a completely noninteracting gas. The site-volume factor is necessary to give physical pressure units.

For $\chi <1/2$, the effective pair excluded volume is positive. In a good solvent, the isolated chain expands relative to a Gaussian coil, with $R\sim b{N}^{\nu }$, $\nu \simeq 0.588$ or $3/5$ in the Flory approximation. For $\chi >1/2$, attraction wins in a poor solvent and a sufficiently long chain collapses toward a dense polymer globule, with asymptotic size $R\propto N^{1/3}$. Close to the theta condition, finite-chain crossover effects matter. At $\chi =1/2$, $R\sim b{N}^{1/2}$ to leading scaling order. This theta criterion is not the exact finite-$N$ phase-separation critical point; the lattice model has ${\varphi }_c=1/(1+\sqrt{N})$ and ${\chi }_c=(1+1/\sqrt{N}{)}^2/2$.

Overlap concentration. The polymer overlap concentration is reached when there is roughly one isolated coil in its own pervaded volume. If the coil size is $R$ and a repeat unit occupies volume $v_0$,With $R\sim b{N}^{\nu }$, the general scaling law isThe usual single-segment-volume estimate $v_0\sim b^3$ gives ${\varphi }^{\ast }\sim N^{1-3\nu }$: $N^{-1/2}$ in a theta solvent, approximately $N^{-4/5}$ in a good solvent, and order one for compact globules. A coil has no sharp surface, so choosing a radius of gyration or a geometric coil volume changes numerical prefactors.

For the stated polystyrene at $N=1500$ and $\chi =1/2$, the conventional scaling estimate isThis is a volume-fraction estimate, not a molar concentration. It uses the implicit site-volume approximation; the question does not provide an independent repeat-unit volume to determine a more precise prefactor.

Semidilute length scale. Beyond overlap, a semidilute polymer solution is characterized locally by a concentration blob size $\xi$. Suppose one blob contains $g$ statistical units. At the theta condition the subchain is Gaussian, so $\xi \sim b\sqrt{g}$, while matching its internal concentration to the bulk gives $\varphi \sim g{v}_0/{\xi }^3$. Eliminating $g$ gives the concentration-blob size in a theta solvent:Using the supplied statistical step length $b=0.67\mathrm{nm}$ and $\varphi =0.05$,The same convention gives $g\sim {\varphi }^{-2}=400<N=1500$, and the whole ideal chain has $R\sim b\sqrt{N}\simeq 26\mathrm{nm}$. Thus the blob is smaller than the chain. Because $\varphi /{\varphi }^{\ast }\simeq 1.94$ is only modestly above overlap, the value is a scaling estimate in a crossover region, not a precision asymptotic measurement. A statistical step length is not automatically a physical monomer diameter: if $v_0$ differs from $b^3$, the more general estimates are ${\varphi }^{\ast }\sim (v_0/b^3)N^{-1/2}$ and $\xi \sim v_0/(b^2\varphi )$, with the missing prefactor left explicit.