= Solution
In <atomic force microscopy>, one end of the <polyacrylic acid> chain is attached to the substrate and the other to a compliant cantilever tip. Controlled separation displaces the cantilever; calibrated cantilever stiffness converts its deflection into <force>. The molecular extension is the tip–surface separation after correcting for cantilever deflection and instrumental offsets. A quasi-static interpretation requires relaxation to be fast compared with the pulling motion.
At low <force> a coiled <polymer> has many available configurations and extends mainly by changing segment orientations. This is the <entropic spring> regime. As the available contour is aligned, the loss of configurational <entropy> becomes steep. For an inextensible three-dimensional <freely jointed chain>,
$$
\frac{\langle z\rangle}{Nb}=\mathcal L\!\left(\frac{fb}{k_BT}\right),\qquad
\mathcal L(x)=\coth x-\frac1x,
$$
where $\mathcal L$ is the <Langevin function>. The limiting extension is the <polymer contour length> $Nb$; the ideal model approaches it only as $f\to\infty$. Real <polymer backbone> stretching permits extension beyond the fitted rigid-link contour, explaining why the experimental curve lies on the more extensible side at high <force>.
The original trace has a sharp force drop after its peak. Finite contour length explains the steep preceding rise, but does not itself cause that drop: release of an attachment, desorption or mechanical failure ends tensile loading. The graph alone does not distinguish these possibilities. If the intact chain is released from tension, thermal motion restores a coil with characteristic <end-to-end distance of a polymer chain> of order $b\sqrt N$, because this macrostate has many more configurations. Recoil is reversible for an intact equilibrating chain; irreversible <chain scission> or attachment loss can prevent repetition of the same pulling curve.
Back to article page
