= Finite extensibility of a polymer chain
{title2=$R\leq Na$}
The <end-to-end distance of a polymer chain> with $N$ fixed-length links of <length> $a$ cannot exceed its <polymer contour length> $Na$. A <Gaussian chain> <spring> has no such bound, so it eventually becomes an unphysical extrapolation. For a three-dimensional <freely jointed chain> under <force> $f$, the mean extension along the <force> is $Na[\coth x-1/x]$, where $x=fa/(k_BT)$. Its small-$x$ limit is $Nfa^2/(3k_BT)$, and its large-$x$ limit approaches $Na$ from below.
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