Conformational Properties of a Back-Folding Wormlike Chain Confined in a Cylindrical Tube

When a semiflexible chain is confined in a narrow cylindrical tube, the formation of a polymer hairpin is a geometrical conformation that accompanies an exponentially large local free energy and, hence, is a relatively rare event. Numerical solutions of the hairpin distribution functions for persistence-length-to-tube-radius ratios over a wide range are obtained in high precision, by using the Green’s function approach for the wormlike-chain model. The crossover region between the narrow and moderately narrow tubes is critically investigated in terms of the hairpin free energy, global persistence length, mean hairpin-tip distance from the tube axis, and hairpin-plane orientational properties. Accurate representations of the solutions by simple interpolation formulae are suggested.

Figure 1

Sectional illustration of a tube of diameter $D$, which contains two hairpins formed by the confined polymer segment. The polymer makes a one-dimensional random walk along the tube axis with an effective, long step length $2g$.

Figure 2

Hairpin free energy $\beta F_0$ and global persistence length $g$ as functions of the reduced tube diameter $D/P$. Common to the two plots are dashed and solid black lines representing the asymptotic behavior $2E_m/\tilde{D}$ with Odijk’s and the corrected values in (4), respectively. In plot (a), squares represent the exact solutions for $\beta F_0$ obtained in this work with the underlying red curve plotted according to (6). The purple curves are (1) with Odijk’s (dashed) and corrected (solid) $E_m$ values stated in (4). In plot (b), open circles represent the exact solutions for $\ln g/P$ obtained in this work with the underlying curve plotted according to (7). The green curves are (3), which are produced with the additional $\alpha$ values listed in (5): dashed for Odijk’s and solid for the corrected. The filled circles are the Monte Carlo data for $g/P$ provided by authors of Ref. [27].

Figure 3

(a),(b) Hairpin-tip distributions as functions of $2\rho /D$ for various $\tilde{D}$ and (c) mean hairpin tip to axis distance $⟨\rho \mathrm{⟩}$ as a function of $\tilde{D}$. The inset in (c) shows a hairpin tip specified by a blue circle located at $\rho$ with a unit vector $\mathbf{u}$ making an angle $\phi$ with $\widehat{\mathbf{\rho }}$.

Figure 4

(a) Parameter measuring the directional ordering of the tangent directions of hairpin tips, $\sigma$, as a function of the hairpin distance from the tube axis, as well as (b)–(d) examples of hairpin configurations where the hairpin tips are labeled by blue circles. Plots (b)–(d) correspond to the parameter regimes $\tilde{D}\gg 1$, $\sim 1$, and $\ll 1$, respectively.