Phase diagram of chiral biopolymer Wigner crystals

We study the statistical mechanics of counterion Wigner crystals associated with hexagonal bundles of chiral biopolymers. We show that, due to spontaneous chiral symmetry breaking induced by frustration, these Wigner crystals would be chiral even if the biopolymers themselves were not chiral. Using a duality transformation of the model onto a “spin-charge” Hamiltonian, we show that melting of the Wigner crystal is due to the unbinding of screw dislocations and that the melting temperature has a singular dependence on the intrinsic chirality of the biopolymers. Finally, we report that, if electrostatic interactions are strongly screened, the counterions can condense in the form of an intermediate achiral Wigner solid phase that melts by the unbinding of fractional topological charges.

Figure 1

Ground state of polyvalent counterions (valency $Z$), shown as small spheres, neutralizing a single cylindrical macroion of diameter ${\sigma }_c$ (a) or two cylindrical macroions in close proximity (b). The counterions are excluded from the macroion interior volume. The charge per unit length of the macroion is $-e{\rho }_0$. In (a), counterions are positioned along a helix with a pitch that depends on the range of the electrostatic interaction. The axial spacing $Z∕{\rho }_0$ between adjacent counterions is determined by the condition of charge neutrality. In (b), counterions are positioned along a column on the line of closest approach between the two macroions. The counterions produce a net short-range attraction between the macroions.

Figure 2

Ground state of polyvalent counterions neutralizing an extended, hexagonal bundle of macroions. (a) Top view of the bundle. Counterions are positioned along columns situated on the sites of a Kagomé lattice. The columns define triangular “plaquettes” as indicated by a circle. (b) side of a prismatic triangular plaquette. Triangular plaquettes are frustrated because geometry prevents perfect pairwise stagger of the three columns with their respective nearest-neighbors, as favored by electrostatic repulsion. (c) Minimum-energy compromise structure. Neighboring columns are shifted vertically by $d∕3$. The counterions form a helical spiral. The sign of the helicity is not determined.

Figure 3

Schematic hase diagram. The horizontal axis $f$ is a measure of the intrinsic chirality of the biopolymers. The Wigner crystal can adopt three different structures. In the “chiral” state, counterions adopt the helical structure of Fig. 2. At zero temperature, all triangular plaquettes have the same helicity sign determined by the sign of $f$. In the “staggered” chiral phase, the pitch of the counterion helix has doubled, while the counterions are arranged in identical layers in the $a$-chiral uniform phase (see Fig. 7). Transitions between the different structures are first order, with spontaneous chiral symmetry breaking along the line $f=0$. The solid lines indicate continuous melting transitions of the Wigner crystal by the unbinding of integer-screw dislocations (see Sec. 4). The phase diagram is periodic under $f\to f+3$.

Figure 4

Helical charge distributions of the polyelectrolyte core (solid line) compared with the charge distribution of the counterions (a). The pitch $p$ of the polyelectrolyte charge distribution in general need not equal the pitch of the counterion helix but “lock-in” will take place (white arrows) if the electrostatic interaction between the counterions and the helical charge distribution is strong. (b) shows the orientation of the terms $A_{ij}$ entering the phase Hamiltonian (2.1).

Figure 5

(a) The Kagomé lattice (solid dots) and nearest-neighbor bonds of the Kagomé lattice (solid lines). The dual sites of the Kagomé lattice are shown as gray dots and the bonds of the dual lattice are shown as dashed gray lines. (b) Current configurations on the bonds of the dual lattice that satisfy the conservation condition are constructed from currents circulating around the hexagonal and triangular plaquettes.

Figure 6

The “external” current that must be included to compute the contribution to the dual theory from the next-nearest-neighbor interactions between ${\varphi }_i$ and ${\varphi }_j$ in Eq. (2.16). This term leads to a coupling between the spin variables $S\left({\mathbf{R}}_t\right)$ located on the two shaded triangular sites (see Appendix ).

Figure 7

The three Wigner crystal ground states (lower part) are shown with the corresponding chiral spin configurations (top part). The values of the integer indices $M\left({\mathbf{R}}_h\right)$ are given inside the hexagonal cells and the values of $S\left({\mathbf{R}}_t\right)$ are given inside the triangular cells. The topological charge $Q\left({\mathbf{R}}_h\right)=M\left({\mathbf{R}}_h\right)+\frac{1}{6}{\sum }_{{\mathbf{R}}_t}^{\prime }S\left({\mathbf{R}}_t\right)$ on the hexagonal sites is zero in all three cases. The stability range of the three states is indicated.

Figure 8

The ground-state structure of the chiral as functions of the ratio $V_2∕V_1$ of the strength of next-nearest-neighbor and nearest-neighbor electrostatic interaction between counterion columns and the chiral parameter $f$. Shaded regions correspond to chiral structures.

Figure 9

Composite spin-charge defects. (a) $Q=1$ defect composed of three separate $Q\left({\mathbf{R}}_h\right)=+1∕3$ defects located on the vertices of an equilateral triangle. Inside the triangular domain, the spin variables are $S\left({\mathbf{R}}_t\right)=+1$ (shaded), outside the domain $S\left({\mathbf{R}}_t\right)=-1$ (white). (b) A charge-neutral single flip (shaded triangle) requires three fractionally charged defects.

Figure 10

Chiral droplets. (a) shows a neutral chiral droplet in the form of a paralellogram with the minimum number of fractional defects decorating the four corners. $Q\left({\mathbf{R}}_h\right)=+1∕3$ charges are indicated as solid dots and $Q\left({\mathbf{R}}_h\right)=-1∕3$ as open dots. Note that the fractional charges are separated on length scales of the order of the droplet size. (b) shows a more complex domain shape. The kink-steps of the domain wall are decorated by a distribution of $Q\left({\mathbf{R}}_h\right)=\pm 1∕3$ neutral dipole pairs. Here, $Q\left({\mathbf{R}}_h\right)=+1∕3$ charges are labeled as filled circles and $Q\left({\mathbf{R}}_h\right)=-1∕3$ are labeled as open circles.

Figure 11

Chiral worms. Spin variables with $S\left({\mathbf{R}}_t\right)=+1$ are shown as shaded triangles. Endpoints, branch points and bends of the domains are decorated by fractional-defect dipoles. Chiral worms become more prominent compared with chiral droplets when the chiral field is large. Fractional defect charge is labeled as in Fig. 10.

Figure 12

Phase diagram for $T_{\chi }<T_{1∕3}<T_1$ and small $f$ (schematic). Along $f=0$, broken chiral symmetry is restored at the Ising critical point, $T_{\chi }$. Below $T_{\chi }$, the Wigner crystal is chiral and has a periodicity $d$. Above $T_{\chi }$, the Wigner crystal is achiral and has a reduced periodicity of $d∕3$. At $T=T_{1∕3}$, the achiral Wigner crystal melts by the unbinding of fractionally charged topological defects. Along the dashed lines, the order parameter correlation function $g_{ij}$ associated with the primary counterion density mode at $G=2\pi ∕d$ changes from algebraic decay to exponential decay $g_{ij}$. This change in correlations is associated with the percolation of chiral domain walls that span the system.

Figure 13

External “current” flowing from $i$ to $j$ that enters the computation of the primary correlation function $g_{ij}$ in the dual representation. The hexagonal sites bordering the receive contributions to ${\zeta }^{\prime }\left({\mathbf{R}}_h\right)$ from neighboring gray triangles. An edge-sharing triangle contributes $+2\zeta \left({\mathbf{R}}_t\right)$, while non-edge-sharing triangles contribute $-\zeta \left({\mathbf{R}}_t\right)$ to ${\zeta }^{\prime }\left({\mathbf{R}}_h\right)$ (see text). The resulting distribution of ${\zeta }^{\prime }\left({\mathbf{R}}_h\right)$ used in Eqs. (D4, D6) is shown in the hexagonal cells.