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Measure the force and extension of an individual molecular tether under controlled displacement or load. A calibrated cantilever or optical trap converts deflection to force. Synthetic polymer stretching provides a nonbiological application; the observed curve also contains instrument and attachment compliance, and its loading-rate dependence can distinguish equilibrium elasticity from activated events.

Size-exclusion chromatography separates dissolved molecules by their hydrodynamic size in the chosen solvent. A larger species that cannot enter the packing's pores travels with the liquid between particles and elutes earlier. Smaller species explore more internal liquid and elute later. The column is not a sieve that permanently traps the large species: exclusion makes them leave first. “Gel filtration” is common for aqueous separations; “gel permeation chromatography” is common for organic-solvent analysis of polymers. SEC molar mass calibration translates size-dependent elution into a standard-relative estimate of molar mass, rather than a universal absolute measurement.

This makes SEC useful for separating proteins from small dissolved species, comparing protein aggregation, and characterizing dissolved polymer chains. The residual-group monitoring of polyurethane formation experiment already illustrates polymer calibration: its reported molar masses are relative to polystyrene standards. Size-exclusion characterization of chain polymers explains why an elution trace alone cannot count chains or establish their growth mechanism.

Soft matter studies materials whose large-scale structures respond strongly to modest mechanical or thermal disturbances, including polymers, liquid crystals and colloids. Polymer physics connects chain conformations to material response; the coil-stretch transition is one example.

The extent to which a solvent favors polymer–solvent contacts relative to polymer–polymer contacts.

Measurement of time-averaged scattering intensity as a function of angle and concentration. For suitable dilute polymer solutions its small-wave-number behavior gives a radius of gyration; intensity weighting and interparticle correlations require attention.

An effective step length describing a specified chain-count convention: an ideal polymer with $N$ counted units has mean-squared end-to-end distance of a polymer chain $⟨R^2⟩=N{b}^2$. If $N$ counts freely jointed coarse-grained units, $b$ is their Kuhn length. If $N$ counts chemical repeat units, the effective $b$ incorporates local conformational correlations and is not automatically a physical monomer diameter. A repeat-unit volume $v_0$ is independent information; setting $v_0\sim b^3$ is a scaling approximation.

Tacticity describes the stereochemical ordering of configurational units along a polymer chain. It is distinct from the head-to-tail or head-to-head connectivity described by regioselectivity.

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.

Journal of Chemical Physics 10, 51–61. doi.org/10.1063/1.1723621

Primary-paper scan: www.appliedpolymertechnology.org/uploads/1/0/8/8/108867241/jchemphys_10_51.pdf

The lattice model treats connected polymer segments and derives the nonideal combinatorial entropy before adding the energetic mixing contribution.

A solvent under conditions where the leading effective pair excluded volume of a long dilute polymer vanishes. In the simplest Flory–Huggins solution theory this means $\chi =1/2$. The chain is approximately an ideal chain, while three-segment interactions can still affect the osmotic pressure of a concentrated solution. A particular polymer-solvent pair often meets this condition only near its theta temperature.

In the Alexander-de Gennes brush model, a theta solvent gives $D\sim a{g}^{1/2}$. Stacking $N/g$ blobs then gives the displayed height, with mean polymer volume fraction $\varphi \sim (s{a}^2{)}^{1/2}$ for segment volume $a^3$. Cancellation of the two-segment excluded volume does not remove three-segment repulsion or make the crowded layer freely interpenetrable at no cost.

The theta temperature is the temperature at which the leading effective two-segment interaction of a dilute long polymer vanishes, so its large-scale size is approximately that of an ideal chain. In the simplest Flory–Huggins solution theory this is the condition $\chi =1/2$. It is the infinite-chain limit of the upper critical temperature in the usual energetic model, not the critical temperature of every finite chain.

entropy associated with positions of independently translating objects. Connecting segments into one polymer leaves fewer independently translating objects than treating the same segments as separate molecules; this reduces mixing entropy per segment.

Cloudiness measured through attenuation or scattering of light. actin filaments, bundles and aggregates can contribute differently, so the signal is not automatically proportional to polymer mass.

For a planar polymer with $N$ segments of length $\ell$, approximate its coil free energy by $F/(k_{B}T)\sim R^2/(N{\ell }^2)+v_2{N}^2/R^2$. The first term is the entropic stretching cost of an ideal chain, while $v_2$ is an excluded-area coefficient. Minimizing gives $R^4\sim v_2{\ell }^2{N}^3$ and size exponent $\nu =3/4$. For $v_2\sim {\ell }^2$, this becomes $R\sim \ell N^{3/4}$. A parameter multiplying $N^2/R^2$ must carry area units; using an unsquared segment length there would be dimensionally inconsistent.

Suppose a synthetic polymer population contains three chains of molar mass $10000\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$ for every one chain of molar mass $40000\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. Multiplying both chain counts by any common factor leaves the averages unchanged. Substitution into the existing number-average molar mass and weight-average molar mass definitions givesThe larger chains are only one quarter of the chain count but supply $40000/70000=4/7$ of the mass. Thus a weighted mean that emphasizes mass gives a larger result than one that weights each molecule equally. Total mass divided by total chain amount of substance gives $M_n$, while $M_w/N_{\mathrm{A}}$ is not the mean mass of a randomly chosen chain. Inferring chain counts from size-exclusion chromatography still requires the existing SEC detector response and SEC molar mass calibration assumptions.

Western blot response saturation means that increasing target amount no longer increases the recorded signal proportionally. It can arise before imaging, through limited binding or reagent availability, as well as through detector limits. Shortening exposure can address a detector limit without repairing an already unresponsive assay. Test the target and normalization readouts with a dilution series under the actual assay conditions; merely seeing a gray band does not establish a proportional response.

In a 2013 methods experiment, a biological cell extract containing added purified yeast protein was separated on gels, transferred with a Trans-Blot Turbo to polymer membranes, and read with an enzyme-linked secondary antibody. A ChemiDoc MP imager and film recorded the same dilution-series blots. The target and an abundant reference protein had different usable response ranges; transferred total-protein imaging and the reference bands did not track loading equally well. This demonstrates the need for assay-specific response checks, rather than a universal conversion from darkness to amount. ^[ref]

An illustrative response model separates background $b$, target amount $T$, retained fraction $q$, response scale $S$ and characteristic amount $K$:This is an assumed model, not a fitted Western blot calibration. When $qT\ll K$, $I-b\approx (Sq/K)T$. When $qT\gg K$, $I-b$ approaches $S$. For two otherwise identical lanes, doubling $T$ gives the background-corrected ratioAt $qT=K$ the ratio is $4/3$, despite a true twofold amount change. Thus Western blot normalization cannot recover a lost amount difference merely by dividing two saturated readouts.