Self-Avoiding Wormlike Chain Confined in a Cylindrical Tube: Scaling Behavior

Within a confining tube section, the multithreads of a strongly confined, backfolding polymer exert the excluded-volume repulsions on each other and produce physical properties that are very different from those of a confined ideal chain. The conformational properties of a such confined wormlike chain are of fundamental interest and are also practically useful in understanding the DNA confinement problems. Here, the excluded-volume effects are added to the standard wormlike-chain model by a self-consistent field theory. The numerical solutions are examined in light of their scaling properties.

Figure 1

Length scales in the wormlike-chain confinement problem. Plots (a)–(d) illustrate the physical scales of the extended de Gennes (EdG) regime, Odijk regime, almost unfolded regime, and unfolded regime. Plot (e) quantitatively defines the Odijk regime according to the SCFT solution of the free energy. Plot (f) summarizes main findings of the present work, which identifies the scaling behavior in the white and green areas, in terms of reduced excluded-volume diameter $d/P$ and confinement tube diameter $D/P$. The recent experimental results on DNA confinement are conducted for systems having parameters in the gray area.

Figure 2

(a) Cubic $C_0(\tilde{D})$ and (b) ${\chi }_0(\tilde{D})$ as functions of $\tilde{D}$ for cylindrical-tube geometry. Circles represent the numerical solutions to the SCFT. The two asymptotic limits, shown by black and blue lines, are those in Eqs. (5) and (6). The meanings of symbol colors are the same as in Fig. 1.

Figure 3

(a)–(b) Polymer extension $Z$, (c)–(d) free energy $F$, and (e)–(f) mean-square extension deviation as functions of $\tilde{d}$ and $\tilde{D}$ (left panels) as well as $\tilde{d}$ and $\xi$ (right panels). The original SCFT solutions are displayed in (a), (c), and (e) where deep-dark red and deep gray circles represent systems in the unfolded state, dark red and gray in the almost-unfolded state, red in the Odijk regime, pink, light yellow, light blue in the crossover region, and yellow and blue in the EdG regime. The 10 curves, from right to left in (a), correspond to $\tilde{d}=4^0,4^{-1},\dots ,4^{-9}$. The scaled versions in (b), (d), and (f) contain all original data, after $\xi$ is calculated from $[\tilde{d},\tilde{D}]$ according to Eq. (4) in which ${\chi }_0$ and $C_0$ are presented in Fig. 2. The green straight lines in (b) and (d) represent Eqs. (3) and (7). The insets show the nonlinearity where solid black curves represent $\zeta (0,\xi )$, $\mathrm{\Phi }(0,\xi )$, and $\gamma (0,\xi )$, for which the empirical expressions can be found in the Supplemental Material [50].