Shape of a Stretched Polymer

The shape of a polymer plays an important role in its interactions with surrounding molecules. We characterize the shape and the orientational properties of a polymer chain under tension in a good solvent, a physical condition that is often realized both in single-molecule experiments and in vivo. Our findings reveal the existence of hitherto unobserved universal laws encompassing polymers with different rigidities and including the possible presence of excluded-volume effects, showing that both shape and orientation are solely determined by the force contribution to the free energy. In doing so, they also provide a simple way to retrieve these quantities from the knowledge of the force-versus-extension curve.

Figure 1

Schematic illustration of the effect of an external force on a polymer in a good solvent.

Figure 2

Asphericity as a function of $\eta =\mathit{f}·⟨{\mathit{R}}_{\mathit{e}}⟩/k_{B}T$ for different models. The data correspond to WLCs of length $L=1000$ characterized by rigidities $k=1$ (red diamonds), $k=2$ (blue hexagrams), and $k=4$ (orange triangles). The green pentagrams correspond to a self-avoiding walk. Full and empty symbols respectively denote chains with ($E$) and without ($P$) excluded volume. The continuous black line represents the FJC formula reported in Eq. (4). Inset: Enlargement of the plot for small values of $\eta$.

Figure 3

Top panel: normalized eigenvalues ${\lambda }_2/N{b}^2$ (main) and ${\lambda }_3/N{b}^2$ (inset) as a function of $\gamma$, for several values of chain size $N$. The continuous curves are the corresponding fitting functions reported in the main text. Bottom panel: aspect ratio of the $xy$ projection of the chain and of the transverse section of the ellipsoid. The continuous red line shows the aspect ratio of a pure two-dimensional FJC, which we found to be equal to $4.95\pm 0.01$. Inset: sketch of the shape of the transverse section of the chain for different values of $\gamma$. The arrow indicates the direction of increasing $\gamma$. The blue dashed line sketches the typical shape of a two-dimensional FJC.

Figure 4

Collapse of the average values of $\cos {\psi }_1$ (main figure) and $\cos {\psi }_2$ (inset) when plotted as a function of $\eta$ for different models. The continuous curves are the predictions obtained from the electric-dipole analogy [Eqs. (5) and (6), respectively]. The symbols used are the same as in Fig. 2. The sketch in the inset qualitatively describes the dipole analogy.