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Here $T$ is the absolute temperature and $k_B$ the Boltzmann constant. An adsorption blob contains $g\sim (\xi /b{)}^2$ monomers, so there are $N/g\sim N(b/\xi {)}^2$ such blobs. Confining a blob removes an amount of conformational entropy of order $k_B$; thus the positive confinement contribution to the Helmholtz free energy isThe statement about one blob's entropy is used as an entropy loss relative to the free chain, rather than as an absolute entropy.

In the assumed slab model, contacts occur within a monomer-scale distance $b$ of the substrate. The contact fraction is approximately $b/\xi$, giving $N_c\simeq Nb/\xi$. Multiplying this by the negative contact energy yieldsThe two contributions express the competition between loss of conformational entropy and gain of adsorption energy. The numerical coefficients are the ones stipulated by this blob approximation.

Minimize the Helmholtz free energy at fixed $N,T$. Its derivative isThe physical stationary point for $ϵ>0$ isTo check the minimum directly, put $x=b/\xi$. Then $F/(N{k}_{B}T)=(x-ϵ/2{)}^2-{ϵ}^2/4$, which is globally minimized at $x=ϵ/2$. The binding free energy gain per monomer is consequently $\mu =k_{B}T{ϵ}^2/4>0$.

The weak-adsorption blob picture requires $b\ll {\xi }_{\ast }\ll b\sqrt{N}$, or $2/\sqrt{N}\ll ϵ\ll 2$ in this model. If $ϵ$ is too large, the layer saturates at a monomer-scale thickness; if too small, a many-blob adsorbed layer has not formed.

The stretched segment is a Gaussian chain of $n$ Kuhn length steps, with unperturbed root-mean-square end-to-end distance $l=b\sqrt{n}$. Its end-to-end probability density is proportional to $\exp [-3z^2/(2n{b}^2)]$, so its leading elastic Helmholtz free energy is $3k_{B}T{z}^2/(2n{b}^2)$. Adding the remaining segment's adsorption free energy givesAn additive constant independent of $n,z$ is irrelevant. At the optimal adsorption height this becomesThe original PDF's hint labels the positive quantity proportional to $k_{B}T{R}^2/l^2$ an entropy. It has units of energy and is the stretching free energy. The corresponding change of conformational entropy is $\Delta S=-3k_B{R}^2/(2l^2)$. This dimensional and sign correction is essential to the minimization.

This is the leading extensive Gaussian chain approximation: endpoint normalization and junction corrections give subleading terms. It requires many stretched monomers and $z\ll nb$, rather than extension near the full contour length.

During slow peeling in atomic force microscopy, $n$ adjusts to minimize the Helmholtz free energy at each imposed height. Define the positive binding gain $\mu =k_{B}T[ϵb/\xi -(b/\xi {)}^2]$ for the adsorbed segment. The stationary point satisfiesSince ${\partial }_n^{2}F=3k_{B}T{z}^2/(b^2{n}^3)>0$, this is a minimum while $0<n_{\ast }<N$. The upward pulling force is the height derivative of the minimized free energy. The implicit $n_{\ast }(z)$ term vanishes because ${\partial }_{n}F=0$, givingUsing $\xi =2b/ϵ$,It is independent of $z$: extra pulling height releases more monomers while leaving the fractional extension of the stretched part unchanged. Equivalently $F_{\min }=-N\mu +f_{\mathrm{peel}}z$. This is the equilibrium peeling force of an adsorbed Gaussian chain.

The plateau ends when $n_{\ast }=N$, atThereafter all monomers are detached and the Gaussian chain force is $f=3k_{B}Tz/(N{b}^2)$, agreeing continuously with the plateau at $z_{\det }$. Very near the start and final release, finite-chain and junction effects lie outside the blob approximation.

For the usual interpretation of a real chain in a good solvent, the detachment force is smaller in the weak-adsorption regime. Excluded volume swells the chain, increasing the cost of confinement near the substrate. The adsorbed layer becomes thicker and its binding free energy per monomer decreases. The same contact energy therefore does not mean the same total binding strength.

The scaling argument can include both adsorption and stretching. Let $\xi \sim b{g}^{\nu }$, where $\nu =1/2$ for an ideal chain and approximately $3/5$ for a self-avoiding walk in three dimensions. With the same contact-fraction approximation,Minimization gives $\xi /b\sim {ϵ}^{-\nu /(1-\nu )}$ and $\mu /(k_{B}T)\sim {ϵ}^{1/(1-\nu )}$. At detachment, a tensile blob of size ${\xi }_t\sim k_{B}T/f$ costs approximately $k_{B}T$; its monomer count is $g_t\sim ({\xi }_t/b{)}^{1/\nu }$. Equating the stretching gain per monomer $k_{B}T/g_t$ to $\mu$ givesThus $f_{\mathrm{real}}\sim (k_{B}T/b){ϵ}^{3/2}$, compared with $f_{\mathrm{ideal}}\sim (k_{B}T/b)ϵ$, so their ratio scales as ${ϵ}^{1/2}<1$ for weak attraction. These are scaling relations, not exact prefactors. The real-chain adsorption exponents here use the same simple uniform-contact model as the earlier parts; a more detailed surface theory of critical phenomena can refine them. The thicker, more weakly bound layer reflects the higher cost of confining a good solvent chain.^[ref]

“Real” alone does not specify solvent quality: a collapsed chain in a poor solvent, strong adsorption, or substantial bending stiffness requires a different comparison. The smaller-force conclusion concerns the flexible good solvent chain implicit in the excluded-volume correction.

The gap is not a fixed mechanical separation. Thermal fluctuations of the wall compete with drift towards the microtubule under the opposing force. A monomer can be incorporated only after the gap reaches its length. The Smoluchowski equation describes this overdamped drift and diffusion of gap probability density.

For gap potential energy $U(x)=Fx$, the equation and its probability current areThe drift mobility is $D\beta$, the overdamped consequence of the fluctuation-dissipation relation for a Langevin particle. Equivalently, this coefficient is fixed by requiring the zero-current solution to be the Boltzmann distribution $p\propto e^{-\beta U}$. Solve this equation with the appropriate exclusion, incorporation and probability-normalization conditions. The steady flux through the incorporation threshold gives the addition rate $j_0$; the elongation speed is $v=\delta j_0$.

There is a regime distinction in the printed problem. The absorbing boundary imposed in part (c) means instantaneous incorporation once a gap opens, and leads to the diffusion-limited Brownian ratchet with constant load. A genuinely reaction-limited polymerization Brownian ratchet instead assumes fast gap equilibration followed by a finite chemical addition rate, giving an exponential gating factor multiplying that chemical rate. The equations requested in the subsequent parts specify the former model, which is the one solved here.

In this polymerization Brownian ratchet model, a gap of width $\delta$ immediately accommodates one monomer. That event advances the tip by $\delta$ and resets the gap to zero. The threshold is therefore an absorbing boundary condition for a diffusion,All waiting-time probability lies within $0\le x<\delta$, so the steady occupation probability density satisfiesThere is a reinjection flux $j_0$ at $x=0$ equal to the absorption flux at $x=\delta$. The nonpenetration constraint at zero is a reflecting boundary condition for a diffusion during each individual waiting episode. For the stationary reset process it must not be confused with setting the net current $J(0^{+})$ to zero: the source at reset supplies $J(0^{+})=j_0$. This is how a normalized steady density can coexist with a nonzero exit flux.

Each addition starts a new statistically identical waiting episode at zero gap. Thus the addition rate is the reciprocal of a mean first-passage time, not the reciprocal of the mean gap:This can also be derived without the steady density. If $T(x)$ is the mean time to reach $\delta$ starting at gap $x$, its backward Smoluchowski equation isThe first boundary is reflecting and the second absorbing. An integrating factor gives $T^{\prime }(x)=[1-e^{\beta Fx}]/(D\beta F)$, whenceAt weak force, $\tau \simeq {\delta }^2/(2D)$ is the time for a freely diffusing wall to move a monomer length, with reflection at the tip. At strong force, opening the gap requires work $F\delta$, with Boltzmann distribution suppression $e^{-\beta F\delta }$. The drift establishes a near-tip thermal length ${\ell }_F=k_{B}T/F$ and a diffusion time ${\ell }_F^2/D$; consequentlyThe exponentially long wait explains why the growth slows. There is no additional chemical waiting time in the imposed absorbing-threshold model; such a time would be needed to describe the separate reaction-limited polymerization Brownian ratchet regime.

In ideal linear polyethylene, incorporated ethene contributes $-\mathrm{C}{\mathrm{H}}_2-\mathrm{C}{\mathrm{H}}_2-$ monomeric units; the smallest constitutional repeating unit is $-\mathrm{C}{\mathrm{H}}_2-$. See monomer for the counting distinction.

Polymer adsorption binds a polymer near an attractive interface through contacts of some of its monomers. The binding energy competes with the loss of conformational entropy. A chain need not lie flat: noncontacting loops and tails can extend into the solvent. In a weak-adsorption blob model, the layer thickness exceeds the Kuhn length and local chain statistics remain approximately those of the free chain below that thickness.

Polymerization converts a monomer, or a mixture of monomers, into a polymer: repeated chemical reactions establish a connected molecular chain or network. The central question is how connections form, not merely whether a material becomes solid. Joining small molecules can produce a thread that can be handled, while the individual molecules remain invisible. See IUPAC, Basic Definitions of Terms Relating to Polymers, definition 2.2.

In step-growth polymerization, complementary reactive groups on monomers and on shorter or longer chains can react with one another. A two-unit chain can join a three-unit chain to make a five-unit chain; growth does not require a specially active end to add only free monomers. An example is the Nylon 6,10 reaction of ${\mathrm{H}}_2\mathrm{N}(\mathrm{C}{\mathrm{H}}_2{)}_6\mathrm{N}{\mathrm{H}}_2$ with $\mathrm{ClC}(=\mathrm{O})(\mathrm{C}{\mathrm{H}}_2{)}_8\mathrm{C}(=\mathrm{O})\mathrm{Cl}$: each amine end can form an amide connection with an acid-chloride end, with formal elimination of $\mathrm{HCl}$. Both reactants have two suitable ends, allowing repeated joining. See Paula Hammond, MIT 10.569 lecture 2: step-growth joining and Hammond, lecture 5, p. 4: acid chloride–amine reaction.

In chain-growth polymerization, a monomer reacts with an active site on a growing chain, and the growth step regenerates an active site. Schematically, $P_j^{\ast }+M⟶P_{j+1}^{\ast }$, where $j$ counts incorporated monomeric units and the star denotes reactivity rather than a particular electric charge. Ethene forming polyethylene is an example: the double chemical bond is converted into single chemical bonds as the carbon backbone grows. Initiation creates the active chains; subsequent reactions can deactivate them or transfer reactivity to another molecule. See IUPAC: chain polymerization and John McMurry, Organic Chemistry, 31.1: chain growth. Whether growth is stepwise or carried by an active site is a different question from whether small byproducts are released: chain-growth polymerization can also release them.

A motivating experiment: drawing nylon from a liquid interface. Feldwinn's UCSB demonstration places $25\mathrm{mL}$ of an aqueous chemical solution of hexamethylenediamine, ${\mathrm{H}}_2\mathrm{N}(\mathrm{C}{\mathrm{H}}_2{)}_6\mathrm{N}{\mathrm{H}}_2$, in a $100\mathrm{mL}$ beaker, then gently adds $25\mathrm{mL}$ of sebacoyl chloride, $\mathrm{ClC}(=\mathrm{O})(\mathrm{C}{\mathrm{H}}_2{)}_8\mathrm{C}(=\mathrm{O})\mathrm{Cl}$, in an organic solvent. The procedure specifies hexane; other parts of that handout list cyclohexane. The layers remain separate. The instructor catches material at their interface with tweezers and winds it slowly onto a glass rod; the handout reports continued spooling of Nylon 6,10. It specifies freshly prepared acid-chloride chemical solution and washing the thread before touching it. These are observations and conditions of a supervised laboratory demonstration, not measurements of chain lengths. See Feldwinn, UCSB Demonstration Library, Demo 044, pp. 1–3.

The mechanistic interpretation is that the monomers encounter one another near the interface, where amide connections produce the film; withdrawing it exposes a region where more material can form. An authored research demonstration independently reports a rapidly formed interfacial film collapsing into a thread on withdrawal, using cyclohexane and an aqueous chemical solution containing potassium hydroxide. See Polymer Day: Outreach Experiments for High School Students, Journal of Chemical Education (2017), nylon demonstration. This is the striking observation: a thread much longer than the beaker can emerge because material is continually supplied to the formation region, rather than because a pre-existing thread was hidden in the liquid.

Useful proposed controls, not results reported in those demonstrations, omit one monomer at a time while retaining the same solvents, liquid amounts, and pulling procedure. The prediction is loss of a renewable drawable Nylon 6,10 film; merely finding cloudiness would not establish polymerization, since other chemical reactions, including acid-chloride hydrolysis, can occur. See McMurry, 21.4: competing acid-chloride hydrolysis. Stopping and restarting withdrawal would probe renewal at the interface. Even successful drawing does not measure the degree of polymerization, establish a chain-length probability distribution, or prove the entire growth mechanism. Chemical identification and a molecular-size measurement would be needed for those stronger conclusions. These controls and limits are inferences from the setup, not invented experimental data.

Why almost complete reaction matters in step growth. For ideal linear step-growth polymerization, assume exactly two reactive groups per starting molecule, equal numbers of complementary groups, only joining of different chains, no rings, no branching, and no loss of chain material. Let $N_0$ count starting monomer molecules, including both reactant species, and let $p$ be the fraction of either type of reactive group consumed. There are initially $N_0$ groups of each type, so $p{N}_0$ connections reduce the number of chain-bearing molecules to $N=N_0(1-p)$; the count excludes small byproducts but includes unreacted monomers. The number-average degree of polymerization, measured in monomeric units, is thereforeThus $p=0.90$, $0.99$, and $0.999$ give $X_n=10$, $100$, and $1000$. Losing the last fraction of unreacted groups matters enormously. See Hammond, lecture 3: balanced step-growth conversion. This counts residues of both feed species; a Nylon 6,10 interior constitutional repeating unit contains two such monomeric units.

For an $AA+BB$ feed with $A_0\le B_0$ initial reactive-group counts, put $r=A_0/B_0$ and now let $p$ denote the fraction of the limiting $A$ groups consumed. Under the same linear, no-ring assumptions, $N_0=(A_0+B_0)/2$ and $N=N_0-p{A}_0$, givingFor $r=0.98$, even complete consumption of $A$ gives $X_n=(1.98)/(0.02)=99$. Stoichiometry controls chain length as well as yield. See Hammond, lecture 2, pp. 2–3: stoichiometric imbalance. The liquid-interface demonstration is not a homogeneous, balanced reactor, so its thread length cannot be inserted into this formula as a molecular chain length.

A contrasting chain count. In ideal living polymerization, assume no chain termination or irreversible chain transfer, all chains initiated before the counted growth, one chain per effective initiator, a fixed chain count $C$, and every consumed monomer molecule incorporated into those chains. If $M_0$ is the initial number of feed monomer molecules and $q$ their incorporated fraction, conservation gives $X_n=q{M}_0/C=qR$ in monomeric units, excluding initiator fragments. For $R=200$, incorporating half the feed gives $X_n=100$. If initiation consumes one monomer molecule per chain, the fixed-chain interval starts at $q=1/R$, rather than $q=0$. See Hammond, lecture 21, p. 1: ideal living-chain count. The mean does not imply identical chain lengths. Living polymerization does not require reversible loss of monomers; the defining absence of chain termination and irreversible chain transfer is explained in Devon Shipp, IUPAC Polymer Education workshop, Radical Polymerizations II, slide 50. Ordinary chain-growth polymerization with new chains, termination, or transfer need not follow the straight line.

The useful connection is between local chemistry and a macroscopic material: selecting reactive ends and their proportions controls connectivity, while the resulting polymer's conformation is a separate problem described by models such as the freely jointed chain or Gaussian chain. Formation of chemical bonds, the count of monomeric units, and the count of statistical segments must be kept distinct.

A constitutional repeating unit is the smallest chemical unit whose repetition describes a regular chain. Its boundaries mark repetition, not necessarily the origin of the chemical atoms from a single monomer. See IUPAC: constitutional repeating unit.

Chain growth by successive olefin metathesis steps that open cyclic alkenes. Relief of ring strain can offset the loss of monomer translational entropy.

In step-growth polymerization, suitable reactive groups on molecules of different chain lengths can join. The linear count and its assumptions are given in polymerization.

A one-unit monomer, a two-unit chain, and a twenty-unit chain can all be joining partners if their functional groups are compatible and accessible. Schematically, $P_j+P_k⟶P_{j+k}$: the subscripts count monomeric units, and end-group chemistry is suppressed. In chain-growth propagation, by contrast, $P_j^{\ast }+M⟶P_{j+1}^{\ast }$ requires a free monomer and a polymerization-active site. An arbitrary pair of existing chains is not a propagation pair in that mechanism. The practical consequence of step-growth polymerization is that most monomers can have joined something while the average chain is still short.

Whether a joining step releases a small molecule is a separate question. Polycondensation releases small products during the joining chemistry, as in the Nylon 6,10 demonstration in polymerization. Polyaddition joins suitable chains without that elimination: reacting a diol with a molecule bearing two isocyanate groups can build a polyurethane. Thus step-growth polymerization and polycondensation are not synonyms.^[ref]

A tensile blob is the scale below which thermal conformations of a stretched polymer are weakly disturbed by tension $f$. Balancing stretching work $f{\xi }_t$ with $k_{B}T$ gives ${\xi }_t\sim k_{B}T/f$. For free-chain size law $R\sim b{g}^{\nu }$, each blob contains $g_t\sim ({\xi }_t/b{)}^{1/\nu }$ monomers. A stretched string of blobs has extension $z\sim (n/g_t){\xi }_t$ and stretching free energy of order $k_{B}Tn/g_t$. This description requires blobs larger than individual statistical segments and smaller than the unperturbed coil.