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The change of an isolated flexible polymer from an expanded coil toward a dense polymer globule as its solvent becomes less favorable. In the Flory approximation, $F/(k_{B}T)\sim R^2/(N{a}^2)+v{N}^2/R^3+u{N}^3/R^6$, where the positive three-segment coefficient $u$ prevents collapse to zero size. Positive $v$ gives swollen-coil scaling, $v=0$ gives the theta temperature condition, and negative $v$ favors a dense state with $R^3\sim uN/|v|$. Finite chain length rounds the change; polymer aggregation is a distinct multi-chain process and must be controlled in an experiment.

The Flory approximation balances the entropic cost of expanding an ideal chain against a mean-field excluded volume interaction. For $N$ segments of length $b$ in $d$ spatial dimensions, minimizing $R^2/(N{b}^2)+v{N}^2/R^d$ gives $R^{d+2}\sim v{N}^3{b}^2$. This yields the scaling exponents $3/5$ in three dimensions and $3/4$ in two dimensions. It is a scaling approximation, not an exact numerical free energy or proof of critical exponents; confinement replaces $R^d$ by the accessible coil volume.

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 long flexible polymer in a good solvent, a tensile blob of size ${\xi }_t\sim k_{B}T/f$ contains $g\sim ({\xi }_t/a{)}^{1/\nu }$ statistical segments. Aligning $N/g$ blobs gives $R\sim (N/g){\xi }_t=Na(fa/k_{B}T{)}^{1/\nu -1}$. The three-dimensional Flory approximation $\nu =3/5$ gives the displayed power. The regime requires $a\ll {\xi }_t\ll a{N}^{\nu }$: below its force range the coil has ordinary linear response, and above it finite link extension matters. Numerical amplitudes depend on solvent quality and segment conventions.^[ref]

The radius of gyration of a dilute linear polystyrene coil in a good solvent depends on its molar mass as $R_g=K{M}^{\nu }$, with a material- and solvent-dependent amplitude. A manufacturer-supplied light-scattering calibration for polystyrene in benzene or toluene near room temperature is the displayed relation, quoted over $M\simeq 2\times 10^5$ to $5\times 10^7\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. The exponent is close to the Flory approximation $3/5$. Values outside the quoted mass range are extrapolations, not extra measured data. Using $c^{\ast }=M/[N_A(4\pi /3)R_g^3]$ then gives the polymer overlap concentration scaling $c^{\ast }\propto M^{1-3\nu }$. ^[ref]

Let $b\simeq 2l_p$ be the Kuhn length, $N=L/b$ the number of statistical segments, and $v$ their effective excluded volume. We first assume a good-solvent, strongly confined self-avoiding walk, rather than an ideal chain. A confinement blob contains $g$ links and has diameter $D\sim b{g}^{3/5}$ in the simple $v\sim b^3$ scaling model. Therefore $g\sim (D/b{)}^{5/3}$. Distant blobs cannot freely pass through one another in a narrow tube, so the $N/g$ blobs form an axial string. The de Gennes confinement scaling is consequentlyThe free energy is of order $k_{B}T$ per blob, not the energy of stretching a straight elastic rod.

The role of solvent and chain thickness can be made explicit by a Flory approximation: an axial extension $X$ costs an entropic term $k_{B}T{X}^2/(N{b}^2)$, while the confined number density gives an interaction term $k_{B}Tv{N}^2/(X{D}^2)$. Minimizing their sum gives $X^3\sim v{N}^3{b}^2/D^2$, henceFor a thin statistical segment of effective diameter $d$, $v\sim b^{2}d$ instead gives $X\sim L(bd/D^2{)}^{1/3}$. Thus the simpler answer implicitly fixes the excluded volume; the DNA thickness and ionic environment can matter quantitatively.

Both blob arguments require $b\ll D\ll R_{\mathrm{free}}$ and $N/g\gg 1$, where $R_{\mathrm{free}}\sim b{N}^{3/5}$ in the simple good-solvent model. The printed comparison with $L$ alone does not ensure confinement relative to the unconfined coil. If $D\gg R_{\mathrm{free}}$, the tube barely perturbs the coil. An ideal chain without excluded volume has axial spread of order $\sqrt{Lb}$ and does not acquire the extensive extension of a nonoverlapping blob string. These are distinct physical regimes, not changes to a numerical prefactor.

A slit of height $D$ leaves two unrestricted directions. Its three-dimensional blobs now form a planar self-avoiding walk, whose size exponent is $3/4$. The polymer coil in slit confinement therefore has in-plane sizeThere is no uniquely preferred axial direction, and the chain is a coil rather than the one-dimensional blob string of a tube. Including a segment excluded volume $v$, the Flory approximation is $\mathcal{F}/(k_{B}T)\sim R_{\parallel }^2/(N{b}^2)+v{N}^2/(R_{\parallel }^{2}D)$. Its minimum satisfies $R_{\parallel }^4\sim v{N}^3{b}^2/D$, agreeing with the displayed scaling when $v\sim b^3$.

If “two-dimensional channel” instead means an intrinsically planar chain confined to a strip of width $D$, a blob is already two-dimensional: $D\sim b{g}^{3/4}$, and the blobs form a one-dimensional string. That different geometry gives $X\sim L(b/D{)}^{1/3}$. Stating which transverse direction is removed is essential. In either geometry the blob description requires many blobs and a width below the free-coil size.

At the theta condition, effective excluded volume approximately cancel and a freely jointed ideal chain has $⟨R^2⟩=N{b}^2$. With the stipulated segment length $b=2a$,In a good solvent repulsive excluded volume swells the coil, giving $R\sim b{N}^{\nu }$ with $\nu \simeq 0.588$ in three dimensions, or the Flory approximation $\nu =3/5$. In a poor solvent effective segment attraction favours a compact globule, whose linear size scales as $b{N}^{1/3}$ at roughly fixed internal density. The latter are scaling laws with model-dependent prefactors. A real chemical monomer need not be a statistical Kuhn length; using $2a$ as the independent segment length is part of this question's idealized model.

With polymer degree $N_A$ and single-site solvent molecules, Flory–Huggins solution theory gives $f/(k_{B}T)=(\varphi /N_A)\ln \varphi +(1-\varphi )\ln (1-\varphi )+\chi \varphi (1-\varphi )$. Expanding the solvent entropy at low concentration gives the dilute-polymer free-energy expansionThe linear term sets a reference chemical potential. The quadratic term controls the leading effective two-segment excluded volume, $v_{\mathrm{eff}}=v_0(1-2\chi )$ for site volume $v_0$.

In a good solvent, $\chi <1/2$ and positive excluded volume swells a dilute chain: the Flory approximation gives $R\sim b{N}_A^{3/5}$ in three dimensions. In a poor solvent, $\chi >1/2$ favors segment contacts and a sufficiently long isolated chain forms a polymer globule, with compact scaling $R\sim b{N}_A^{1/3}$ up to density-dependent factors.

At the theta temperature, the leading two-segment interaction cancels, $\chi =1/2$ in this model, and the long-chain conformation is approximately an ideal chain, $R\sim b\sqrt{N_A}$. The positive cubic term remains. The critical point of a Flory–Huggins polymer solution at finite $N_A$ has ${\chi }_c=\frac{1}{2}(1+N_A^{-1/2}{)}^2$; its limiting value, rather than every finite-chain critical point, is the theta condition.

In ethylbenzene, the good solvent makes effective segment repulsions swell the coil. With the Flory approximation,

Here $R$ denotes a typical chain size, such as an rms end-to-end distance of a polymer chain, rather than the mean end-to-end vector of an isotropic solution, which is zero. The Flory approximation gives $R\sim a{N}^{3/5}$ in a good solvent, an ideal chain has $R\sim a{N}^{1/2}$ in a theta solvent, and a compact polymer globule has size $R\sim a{N}^{1/3}$ in a strongly poor solvent. These are scaling estimates with order-one prefactors omitted, treating the stated $N$ as the segment count associated with the supplied statistical length.

A three-dimensional self-avoiding walk has swollen size $R\sim b{N}^{\nu }$, with $\nu \simeq 0.59$; the three-dimensional Flory approximation gives the nearby estimate $3/5$. Isotropic projection gives $⟨R_{\mathrm{proj}}^2⟩=(2/3)⟨R^2⟩$, preserving this exponent. For the mean distance rather than its root mean square, the orientation average instead gives $⟨R_{\mathrm{proj}}⟩=(\pi /4)⟨R⟩$. Neither prefactor changes its position between the ideal $N^{1/2}$ walks and the planar swollen $N^{3/4}$ walk.

The three solvent conditions describe effective interactions between segments of a generic polymer. In a good solvent, solvent contacts favor a swollen, self-avoiding coil with positive excluded volume. At a theta solvent condition, leading pair repulsion and attraction cancel; the chain has approximately ideal random-walk statistics. In a poor solvent, effective attraction contracts the chain, eventually producing a dense polymer globule for a sufficiently long chain.

For $N$ statistical segments of length $a$, the radius of gyration has the scalingThe good-solvent power $3/5$ is the Flory approximation; the asymptotic three-dimensional exponent is closer to $0.588$. In the ideal freely jointed chain, the more precise result is $R_g^2\simeq N{a}^2/6$, whereas the end-to-end mean square is $N{a}^2$. Scaling notation suppresses those geometry-dependent factors. The globule power follows from a volume proportional to $N$, and presumes a dense collapsed state away from the theta crossover. The dilute condition excludes overlap with other chains; it does not eliminate contacts between distant segments of the same chain.

The coil-globule transition is the contraction of an isolated flexible polymer as solvent quality worsens. A swollen chain in a good solvent has size $R\sim a{N}^{3/5}$ in the Flory approximation. At the theta temperature the effective two-segment excluded volume vanishes and $R\sim a{N}^{1/2}$. A poor solvent favors a dense polymer globule, with $R\propto N^{1/3}$ at fixed internal density.

The competition is visible in the scaling free energyFor $v>0$, elastic cost balances two-segment repulsion to give the swollen-coil law. At $v=0$, three-segment repulsion stabilizes ideal-chain scaling; for $v<0$, attraction balanced by the positive three-segment term gives $R^3\sim uN/|v|$. The dense polymer globule also has an interfacial free energy. The scaling expression explains regimes, not all fluctuation corrections or a detailed finite-chain transition curve. A finite chain crosses over smoothly; the sharp theta-point limit refers to indefinitely long chains.

A dilute-solution light-scattering experiment can follow radius of gyration while changing temperature or solvent composition. Measure at several concentrations and extrapolate toward isolated chains; check molecular-weight distribution, reversibility and the absence of growing multi-chain aggregates. dynamic light scattering measures hydrodynamic size, which is related to but distinct from radius of gyration. A falling scattering-derived size accompanied by a rising aggregate population cannot by itself establish single-chain collapse. solvent quality controls conformation through the balance of segment interactions and conformational entropy.

For a long ideal chain of $N$ statistical segments of length $b$, $R_G^2=N{b}^2/6$. At fixed repeat chemistry and statistical segment length, molar mass is proportional to $N$. The molecular-weight scaling of an ideal-chain gyration radius thus givesThe supplied size and ideal statistics are used as stipulated; an independently tabulated coil size would not replace the given baseline.

solvent quality changes the balance of segment–segment and segment–solvent contacts. In a good solvent, excluded-volume repulsion swells a sufficiently long isolated chain, with $R_G\propto N^{\nu }$, $\nu \simeq 0.588$ in three dimensions; the Flory approximation gives $\nu =3/5$. At theta conditions the effective two-segment excluded volume vanishes and the ideal exponent $1/2$ is recovered for long chains. A poor solvent favors a compact polymer globule, whose large-chain size scales approximately as $N^{1/3}$ at fixed internal density. solvent and temperature can also change local persistence length and hence the prefactor. These different conformations need not have the same radius at the same molar mass.

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]

A grafted polymer chain is tethered to a surface at one end. Let its isolated coil size be $R\sim b{N}^{\nu }$, where $N$ is the number of statistical units and $b$ their statistical segment length. For a good solvent, the Flory approximation gives $\nu \simeq 3/5$; an ideal chain has $\nu =1/2$. Let ${\sigma }_g$ be the number of tethers per unit area, distinct from the membrane tension in Question 1. Their typical lateral separation is $D\sim {\sigma }_g^{-1/2}$.

At low coverage, $D\gg R$, the chains hardly overlap. Each forms a polymer mushroom, retaining roughly the isolated-coil conformation apart from the constraint of the solid surface. Raising coverage until $D\sim R$ makes neighboring coils overlap. The crossover criterion isFor $D<R$ in a good solvent, maintaining isolated coil shapes would crowd segments into the same lateral regions, increasing excluded-volume and osmotic free energy. Chains stretch predominantly away from the surface to relieve that crowding, at the cost of their conformational entropy. The resulting dense tethered layer is a polymer brush.

A simple scaling estimate makes the balance explicit. For brush height $H$, the monomer concentration is of order $N{\sigma }_g/H$. The free energy per chain in units of $k_{B}T$ iswhere $v>0$ is the effective segment excluded volume. The first term is Gaussian stretching; the second is the cost of segment crowding in a good solvent. Minimizing with respect to height givesWhen $v\sim b^3$, this is $H\sim Nb({\sigma }_g{b}^2{)}^{1/3}$. Height increases with chain length and grafting density, while the lateral spacing is fixed by the attachment points. The estimate assumes stretching strong compared with an isolated coil but still much smaller than contour length $Nb$; it does not describe fully extended chains. Poor-solvent attraction can instead collapse the layer, so solvent quality must be specified before using the good-solvent formula.

The mushroom-to-brush change is a crowding crossover for finite chains, rather than automatically a sharp thermodynamic phase transition. A swollen brush can provide steric stabilization: bringing two coated surfaces together further compresses and crowds their chains, producing a repulsive free-energy cost. The controlling comparison is tether separation versus isolated coil size, and the brush height follows from competition between chain stretching and segment interactions.

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.

The polymer overlap concentration is reached when there is roughly one chain per coil volume. If $M$ is the chain molar mass, choose the explicit geometric convention $V_{\mathrm{coil}}=4\pi R_g^3/3$, where $R_g$ is the radius of gyration. ThenHere $c^{\ast }$ is mass per solution volume and $N_A$ is the Avogadro constant. A good solvent gives $R_g\propto N^{\nu }$ with $\nu \simeq 0.588$, approximated by $3/5$ in the Flory approximation. With a polystyrene repeat-unit molar mass of about $104.15\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, the chain has approximately $N=960$ repeat units. This fixes the chain length but not the size amplitude: the problem supplies no statistical segment length or measured coil size.

For a numerical estimate, use the good-solvent size calibration for polystyrene,This empirical calibration is quoted for benzene or toluene near room temperature at molar masses above $2\times 10^5\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$; using it here is a modest extrapolation to $M=10^5\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. It gives $R_g\simeq 11.3\mathrm{nm}$. ThusThe unrounded calibration gives about $27\mathrm{g}{\mathrm{L}}^{-1}$. This is an estimate under a stated extra material input, not a unique number determined solely by molecular weight and the phrase good solvent. Choosing a different coil-volume convention changes its prefactor. The robust chain-length dependence is $c^{\ast }\propto M^{1-3\nu }\simeq M^{-4/5}$ for a flexible good-solvent coil.

Ideal and self-avoiding walks. An ideal chain has independent isotropic coarse-grained steps and permits self-intersections in its mathematical model. For $N$ steps of length $b$, the cross terms average to zero, so its root-mean-square end-to-end distance of a polymer chain obeysA self-avoiding walk excludes repeated occupancy or intersection at the segment scale. This represents positive excluded volume in a dilute good solvent. The nonlocal constraint correlates otherwise distant portions of the chain and swells it:The Flory approximation replaces this exponent by $3/5$, by balancing stretching free energy $R^2/(N{b}^2)$ with an interaction contribution $v{N}^2/R^3$. The result is a more extended and less internally dense coil than the ideal walk; self-avoidance changes the statistics, not merely the local step length.

Two ideal-chain regimes. At the theta temperature for a dilute polymer–solvent pair, leading effective segment attractions and repulsions cancel. Large-scale chain statistics are approximately ideal, even though microscopic atoms still have finite size. In a homogeneous polymer melt, surrounding chains screen long-range excluded volume interactions of a chosen chain, again making its equilibrium large-scale size approximately ideal. Local bond-angle correlations must be included in an effective statistical length such as the Kuhn length. Polymer entanglements can slow the dynamics without invalidating this leading equilibrium size law.

Semiflexible DNA. The nucleic acid double helix gives double-stranded DNA a bending stiffness and substantial local orientational persistence. Independent step orientations are therefore a poor description on short scales, while self-avoidance alone does not specify the energy cost of bending. The worm-like chain describes the molecule as a continuously bendable, locally inextensible curve $\mathbf{r}(s)$ parameterized by arc length $s$, with unit tangent $\mathbf{t}(s)=d\mathbf{r}/ds$. Its bending energy iswhere $A$ is bending stiffness. In the free three-dimensional model, the persistence length is ${\ell }_p=A/(k_{B}T)$ andThus ${\ell }_p$ is the contour separation over which tangent alignment falls by the factor $e^{-1}$; it quantifies stiffness relative to thermal bending. The model bridges locally rod-like and globally coil-like conformations. It omits long-range self-avoidance in its simplest form, so sufficiently long dilute DNA can require excluded-volume corrections.

Flexible and rigid limits. Integrating the tangent correlation over the two contour positions gives the end-to-end distance of a worm-like chain:For a very flexible chain on the scale of its total length, $L/{\ell }_p\gg 1$, the exponential is negligible andThis is the ideal chain result with Kuhn length $b_K=2{\ell }_p$ and effective step count $N_K=L/(2{\ell }_p)$. Many persistent sections have decorrelated orientations. The flexible limit means a large ratio $L/{\ell }_p$ within the continuum model's range, not an extrapolation of the persistence length below all molecular length scales.

For a nearly rigid chain, $L/{\ell }_p\ll 1$, the constant and linear terms in the exponential cancel those outside it. The Taylor series givessoThis is the expected rod limit: all tangents are nearly aligned, and the end-to-end distance approaches the arc length.

Helix–coil curves. A helix-coil transition model allows ordered helical runs and disordered coil regions. Use $s$ and $\sigma$ with exactly the free-energy conventions defined in the paper. A negative $\Delta F_{\mathrm{hc}}$ gives $s>1$ and favors the helical state; a positive junction cost gives $0<\sigma <1$ and penalizes starting new domains. The supplied helical fraction is the long-chain form associated with the Zimm-Bragg model.

Its main features follow directly without assuming a sharp transition. Put $Q=(s-1{)}^2+4s\sigma$. For every $\sigma >0$,and differentiating givesThus all curves rise monotonically and cross at $(1,1/2)$. Increasing the boundary free-energy cost reduces $\sigma$, suppresses frequent coil–helix boundaries and makes longer runs favorable. The transition region becomes narrower and steeper: balancing $(s-1{)}^2$ against $4s\sigma$ gives a width of order $\sqrt{\sigma }$ near $s=1$.

For $\sigma =1$, the fraction simplifies to $\boxed{f_h=s/(1+s)}$, the noncooperative two-state result. For $0<\sigma \ll 1$, the steep curve describes cooperative helix formation. In the singular limit $\sigma \downarrow 0$, it approaches zero for $s<1$ and one for $s>1$; its midpoint remains $1/2$ when approached through positive $\sigma$. Setting $\sigma =0$ directly makes the formula indeterminate at $s=1$. For any fixed positive $\sigma$, the given long-chain curve is smooth: it is a cooperative crossover, not a true discontinuity. The plot varies $s$ at fixed $\sigma$; in an actual temperature scan both parameters may change, and a finite chain can further round the crossover.