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With friction $\gamma (x)={\zeta }_{\ell }x$, inward force $qEx$ gives constant entry speed after capture. With the field off, constant outward entropic force gives $x\dot{x}=-f/{\zeta }_{\ell }$ and a quadratic recoil time. Exterior drag, adsorption and the short final end region are omitted. These ideal transport times alone do not establish steady size separation through a whole uniform channel.

An entropic force is the part of a generalized force arising from the dependence of entropy on a coordinate. At fixed temperature, the system's force is $-\partial A/\partial x=-\partial U/\partial x+T\partial S/\partial x$, where $A=U-TS$ is the Helmholtz free energy. If the internal energy is independent of the coordinate, the force is entirely entropic. For an ideal chain whose configurations become less numerous on extension, it acts toward shorter extension. The external tension holding the chain extended has the opposite sign. Thus the force depends on the statistical number of available configurations, even when the energy of each configuration is unchanged.

In spectrin unfolding by force spectroscopy, the cantilever measures tension through its calibrated deflection, while displacement of the pulling point increases the separation of the protein attachments. The protein extension is obtained after allowing for cantilever compliance and any linker extension; it is not simply the commanded displacement.

A folded spectrin repeat stores much of its backbone in a compact domain. Between unfolding events, the already released chain is stretched: increasingly straight conformations have fewer accessible configurations, and the resisting entropic force rises nonlinearly. A peak is reached when a still-folded domain crosses an unfolding barrier on the timescale of the imposed loading. Its released backbone increases the available contour length abruptly. At nearly the same end-to-end distance the fractional extension $x/L$ is then smaller, so tension drops. Pulling resumes on the next, longer-chain elastic branch. Repeated release of domains produces the saw-tooth sequence.

The record is therefore evidence for sequential mechanical unfolding of modular structure, not repeated stretching of a fixed-length rod. For the nominal eight-domain construct, successive branches correspond to increasing numbers of unfolded repeats. The order and rupture forces can fluctuate because barrier crossing is thermally activated and depends on loading rate, structural heterogeneity and pulling geometry; equivalent repeats need not rupture at identical force.

The horizontal separation between worm-like-chain fits at the same force estimates the added contour length, whereas the vertical force drop at one extension reflects redistribution of tension. These are distinct measurements. A repeat increment near the specified $31.7\mathrm{nm}$ is consistent with repeated similar structural units. The gain is a net release relative to the folded span and attachment geometry; $31.7/106\mathrm{nm}$ is not automatically the universal contour length of a peptide residue.

During unloading, relaxation and possible refolding can restore compact domains. Hysteresis would indicate dissipative or kinetically delayed processes. Neither the peak force alone nor an arbitrary saw-tooth area directly equals an equilibrium unfolding free energy: one must separate chain elasticity, cantilever storage, irreversible loss and pulling-rate effects. The worm-like-chain branches describe released-chain elasticity, not the molecular transition state responsible for a rupture peak.

Configuration count. Choose the end-to-end direction as positive and let $N_R,N_L$ count rightward and leftward links. ThenA configuration is determined by choosing the positions of its leftward links, soThe counts must be nonnegative integers; in particular, allowed signed extensions occur in steps of $2a$. If distance means $|L|$ with neither direction selected, there are two corresponding sets for $L>0$, adding a factor of two. That constant does not affect the tension. There is no excluded-volume penalty in this one-dimensional idealized count.

Entropy from the multiplicity. Put $x=L/(Na)$, so $|x|<1$. Applying the leading Stirling's approximation to all three large factorials givesTerms subleading in $N$, including logarithmic factorial prefactors, have been omitted consistently. Expanding near zero extension givesStretching reduces the number of available configurations and hence the entropy.

Tension and the range of Hooke's law. The model assigns no extension-dependent internal energy. At fixed temperature, the externally applied tension is $\mathcal{F}=(\partial F/\partial L{)}_T=-T(\partial S/\partial L{)}_T$. Differentiating the full leading entropy givesIt is the external force needed to oppose the chain's restoring entropic force. Only when $|L|\ll Na$ can one put $\mathrm{artanh}x\simeq x$, yieldingThe paper's displayed linear law therefore requires this small-extension approximation, which is not explicitly stated in its premise $L<L_0$. It is not the exact force law up to full extension. The entropic spring constant grows with temperature because stronger thermal disorder makes a given reduction in configurational entropy cost more free energy. This is the one-dimensional entropic chain force-extension relation.

A sparse number of backward links. Write $q=N_L\ll N$, so $N_R=N-q$ and $L=a(N-2q)$. Exactly,For fixed $q$ and large $N$, each numerator factor approaches $N$, givingMore precisely, the logarithm of the product is $-q(q-1)/(2N)+O(q^3/N^2)$. A small relative error in the multiplicity requires $q^2/N\ll 1$; merely $q/N\ll 1$ need not make the omitted product close to one. The weaker sparse-link condition does suffice for the leading entropy density and leading high-extension tension. These are sparse backward links near full chain extension.

Tension in that limit. In the overlap regime $1\ll q\ll N$, use Stirling also for $q!$:ThusThe full leading entropy would give $(k_{B}T/(2a))\ln [(N-N_L)/N_L]$, which has the same dominant logarithm when $N_L/N\to 0$.

Approaching full extension. The continuum force law diverges logarithmically as $N_L\to 0$, or $L\to Na$. One must not literally apply Stirling to a zero factorial: at the discrete endpoint there is exactly one directed configuration, $\Omega =1$, and its entropy is zero. A continuous fixed-extension derivative is not defined at that endpoint.

An independent force-ensemble calculation makes the limiting physical statement precise. For applied tension $\mathcal{F}$, a link has partition function $z=2\cosh (\beta \mathcal{F}a)$, giving the exact mean extensionIt approaches $Na$ only as $\mathcal{F}\to \infty$ at positive temperature, and inversion reproduces the logarithmic force law for mean extension. In a real chain, bond extension or failure eventually replaces the ideal inextensible-link model.

For a freely jointed Gaussian chain, independent isotropic segment vectors have zero mean and coordinate variance ${\ell }^2/3$. The central limit theorem gives the end-to-end vector densityThe number of conformations in a fixed small vector-volume element is proportional to this density. Taking its logarithm gives the configurational entropyThis is the vector-constrained entropy appropriate to an anchored strand. A density for the scalar length alone also has a $4\pi R^2$ shell factor and is a different constraint. The approximation requires many segments and extensions well below $N\ell$.

If the internal energy has negligible extension dependence, the Helmholtz free energy is $F=F_0+3k_{B}T{R}^2/(2N{\ell }^2)$. Differentiation gives an entropic force with effective spring constant $3k_{B}T/(N{\ell }^2)$. In an affine isotropic network, averaging over initially oriented strands gives rubber elasticityfor an incompressible material; ${\nu }_s$ is the number of elastically active strands per volume. For example a volume-preserving stretch with principal factors $\lambda ,{\lambda }^{-1/2},{\lambda }^{-1/2}$ has strain-energy density $W=G({\lambda }^2+2/\lambda -3)/2$. Its nominal tensile stress is $G(\lambda -{\lambda }^{-2})=3G(\lambda -1)+O((\lambda -1{)}^2)$, proving the stated small-strain Young's modulus.

Chemically, increasing the effective cross-link density, for example by changing vulcanization, generally increases the modulus by shortening strands between cross-links. Dangling ends, loops and changes of network structure mean the chemical junction count need not equal ${\nu }_s$.

The reference area is $A_0=10^{-5}{\mathrm{m}}^2$. Neglect self-weight, thermal expansion and energetic elasticity, keep the same permanent network, and heat quasistatically in the rubbery regime. Then $G$ increases from $20$ to $22\mathrm{MPa}$. With a constant suspended load, the exact affine-network balance isIt gives ${\lambda }_{300}=1.01662$ and ${\lambda }_{330}=1.01509$. ThusLinear elasticity instead gives extensions $16.35$ and $14.86\mathrm{mm}$, a $1.49\mathrm{mm}$ rise, consistent at this small strain. Incompressibility makes the area $A_0/\lambda$, already included by using nominal stress. Heating increases entropic stiffness, so the loaded rubber contracts and performs work on the mass. Strong thermal expansion, changes in cross-linking or leaving the rubbery regime would require additional material data.

Entropic elasticity. A rubber contains flexible polymer strands tied into a network by cross-links. Stretching biases the strands toward relatively extended configurations, reducing their configurational entropy. The restoring entropic force follows from the increase in Helmholtz free energy $F=U-TS$. For a Gaussian strand with $N_s$ statistical steps of length $b$, its deformation-dependent free energy is approximately $3k_{B}T{R}^2/(2N_s{b}^2)$. Its stiffness therefore increases with absolute temperature, even if the underlying bond stiffness changes little.

In the ideal affine-network model of rubber elasticity, the shear modulus iswhere ${\nu }_s$ is the number density of elastically active network strands. Thus heating increases the modulus approximately in proportion to $T$, provided the network density and rubbery state remain approximately unchanged. Thermal expansion, changing relaxation times or damage can modify that simple trend.

Examples are lightly crosslinked natural rubber, based on cis-rich polyisoprene, and crosslinked polydimethylsiloxane. The crosslinks prevent the long-time macroscopic flow of an unconnected chain solution.

Density estimate and the unspecified modulus convention. With room temperature $T\simeq 298\mathrm{K}$, the Boltzmann constant gives $k_{B}T\simeq 4.11\times 10^{-21}\mathrm{J}$. If the stated modulus is $G=2\times 10^5\mathrm{Pa}$, thenThe molar value counts moles of active strands. If “modulus” instead means Young's modulus $E$, a nearly incompressible isotropic rubber has $E\simeq 3G$, and the inferred value isThus the robust order-of-magnitude estimate is $10^{25}{\mathrm{m}}^{-3}$. The cross-link density inferred from rubber elasticity is usually an effective strand density; the actual junction density is ${\nu }_j=2{\nu }_s/f$ for junction functionality $f$ in an ideal network without inactive loops or dangling ends. For four-functional junctions this would be half the strand density. The paper supplies neither modulus type nor junction functionality, so a unique chemical junction count cannot be extracted.

Let the channel length be ${\ell }_c\gg R$ and assume uniform solution conductivity. Its Ohmic resistance scales as ${\ell }_c/(\sigma R^2)$, whereas access resistance of a reservoir mouth scales as $1/(\sigma R)$. The fraction of voltage in the entrance regions is therefore of order $R/{\ell }_c$. Most of the potential drop lies inside the long channel; the small entrance field region is neglected in the model, not asserted identically zero.

Write $x$ for inserted contour length and choose $qE>0$ to pull the chain inward. The charge already inside feels force $qEx$, givingConfinement removes polymer configurations. In the stiff-chain regime, the confinement deflection length obeys $R^2\sim {\lambda }^3/{\ell }_p$, hence $\lambda \sim (R^2{\ell }_p{)}^{1/3}$. A free-energy cost of order $k_{B}T$ per deflection gives a roughly constant resisting entropic force $f\sim k_{B}T/\lambda$. The insertion barrier of a charged polymer follows from $F(x)\simeq fx-qE{x}^2/2$, with an entry maximum at $x_{\ast }=f/(qE)$ when that point lies within the insertion interval. Its height is $f^2/(2qE)$. The field supplies work against this confinement entropy.

Put ${\zeta }_{\ell }=2\pi \eta /\ln (R/a)$, so $\gamma (x)={\zeta }_{\ell }x$. After capture, neglect the entry barrier, exterior-chain drag and thermal corrections to strong directed motion. ThenThe confined-polymer entry and recoil times have different length dependence; this entry graph is linear through the origin. Charge and drag both grow with inserted length, leaving a constant velocity. The same cancellation explains length-independent mobility in an ideal free-draining electrophoresis model. It does not include a capture waiting time or claim that every free-solution chain has identical hydrodynamic drag.

A measured superlinear entry time can arise when exterior-coil drag, tension propagation, wall adsorption or nonuniform friction adds length dependence absent from this model. For example $\gamma (x)\propto x^p$ with the same $qEx$ drive gives $\tau \propto L_0^p$ for $p>0$. Under the assumptions of uniform wall charge and polymer entry scaling, a homogeneous thin-double-layer electro-osmotic model adds a length-independent fluid velocity $u_{\mathrm{eo}}=-ϵ{\zeta }_{w}E/\eta$. It changes the drift coefficient, not its length exponent. This does not rule out charge heterogeneity, adsorption or near-cancellation followed by diffusion in a real channel; those are additional mechanisms.

With the field off, the confinement entropy drives outward motion. Assuming the constant force $f$ and the same local friction,The small final end region lies outside the long-confined-segment approximation but gives only a small correction. Thus ideal insertion scales as $L_0$, while recoil scales as $L_0^2$.

TODO: user-deferred DNA-specific numerical stopping-voltage estimate and its biological parameter-based inference. No biological quantities are evaluated.

For entropic-trap length separation, a reservoir/channel array can repeat length-dependent capture, confinement and relaxation delays, turning them into different mean residence times. The uniform inside-channel drift speed derived above is itself length independent, so that drift alone cannot separate lengths. Separation must retain length dependence of the barrier or of the capture/recoil dynamics and can be tuned with a forward-biased time-dependent field. For large bulk throughput, pressure-driven advection is preferable to diffusion alone; a pressure gradient can be combined with electric capture control. In the unobstructed continuum limit $Q=\pi R^4\Delta P/(8\eta {\ell }_c)$. Excessive driving erases trapping differences, so throughput and resolution compete; the question supplies no geometry or adsorption law selecting a unique optimal drive.

In a flexible crosslinked polymer network, deformation restricts the configurations of chain segments between cross-links. The resulting decrease of entropy produces a restoring entropic force. For an ideal affine Gaussian network, the shear modulus is $G={\nu }_s{k}_{B}T$, where ${\nu }_s$ is the number density of elastically active strands. A nearly incompressible isotropic rubber has Young's modulus $E\simeq 3G$. At fixed network density within the rubbery regime, both increase approximately linearly with absolute temperature. This statement does not describe crossing the glass transition or chemical degradation.

The exact count is $\Omega =\left(\genfrac{}{}{0ex}{}{N}{q}\right)=N^q{\prod }_{j=0}^{q-1}(1-j/N)/q!$. For fixed $q$ and large $N$, this is asymptotic to $N^q/q!$. Uniform relative accuracy requires $q^2/N\ll 1$, since the logarithm of the product starts at $-q(q-1)/(2N)$. The weaker assumption $q/N\ll 1$ gives the leading sparse-link entropy but need not give a small relative multiplicity error. In the continuum range $1\ll q\ll N$, the entropic force behaves as $(k_{B}T/(2a))\ln (N/q)$. The fixed-tension mean extension approaches $Na$ only as the tension diverges; at the discrete endpoint $q=0$ the exact entropy is zero and a continuous fixed-extension derivative is not defined.