Evidence of a special chiral nematic phase is provided using numerical simulation and Onsager theory for systems of hard helical particles. This phase appears at the high density end of the nematic phase, when helices are well aligned, and is characterized by the C2 symmetry axes of the helices spiraling around the nematic director with periodicity equal to the particle pitch. This coupling between translational and rotational degrees of freedom allows a more efficient packing and hence an increase of translational entropy. Suitable order parameters and correlation functions are introduced to identify this screw-like phase, whose main features are then studied as a function of radius and pitch of the helical particles. Our study highlights the physical mechanism underlying a similar ordering observed in colloidal helical flagella [E. Barry, Z. Hensel, Z. Dogic, M. Shribak, and R. Oldenbourg, Phys. Rev. Lett. 96, 018305 (2006)] and raises the question of whether it could be observed in other helical particle systems, such as DNA, at sufficiently high densities.
High density solutions of helical polynucleotides and polypeptides are known to form liquid–crystal phases that have important physico–biological consequences.1,2 The simplest of these phases is the nematic (N) where the helices, as elongated particles, have their long axes preferentially aligned along a common fixed direction
In this work, using Monte Carlo (MC) simulations complemented by Onsager theory, we provide convincing evidence of the existence of a different chiral nematic phase, originating from the specific helical shape of the particles and stable against other possible phases within a specific range of densities, dependent on the radius r and the pitch p of the constituent helices [Fig. 1(a)].
Similar to the cholesteric, this phase is still nematic in that helices are homogeneously distributed and mobile with their long axis
A transition from the isotropic (I) to the
Consider a pair of helices locally in phase, contrasted with the case where they are in antiphase, as shown in Fig. 1(c). While in the latter case both helices can freely rotate about their
We modeled the helices as a set of 15 partially fused hard spheres of diameter D, our unit of length, rigidly arranged in a helical fashion with a given contour length L = 10D [Fig. 1(a)]. Hence different helix morphologies can be achieved upon changing r and p,8 as in the experiments on flagella.5 We then performed MC isobaric-isothermal (NPT) numerical simulations,9 on systems of N, typically between 900 and 2000, such helices, at many values of pressure P, measured in reduced units P* = PD3/kBT, kB being the Boltzmann constant and T the temperature. Our simulations were organized in cycles, each consisting, on average, of N/2 attempts to translate a randomly selected particle, N/2 trial rotational moves and an attempt to modify shape and volume of the triclinic computational box. Periodic boundary conditions were applied, as these are appropriate in the present case. Initial configurations were taken either as a low density or a highly ordered compact configuration. Typically, 3–4 × 106 equilibration MC cycles were followed by additional 2 × 106 production MC cycles to collect statistics on various quantities.
Fig. 2 shows, in the P*–volume fraction (η = ρv0, with ρ the number density and v0 the helix volume) plane, the MC results for the representative cases with r = 0.2 and r = 0.4 and the same value of p = 8. Points labeled I, N, Sm correspond to the isotropic, ordinary nematic, and smectic phases, respectively, as identified by the usual nematic
The onset of the
The first term in right-hand-side of Eq. (1) represents the entropic cost for the loss of freedom in the azimuthal angle rotation, while the second represents the excess free energy within the second virial approximation characteristic of Onsager theory. Here particle positions and orientations are expressed with respect to the same (laboratory) reference frame, having its origin at the position of the center of the particle 1 and the X axis parallel to the
In short, we have found that systems of hard helical particles undergo a second-order entropy-driven transition from an ordinary nematic N to a screw-like nematic
Our results provide a theoretical explanation of the I–
We are grateful to C. de Michele, M. Dijkstra, Z. Dogic, D. Frenkel, R. Podgornik, R. van Roij, F. Sciortino, and E. Velasco for useful discussions, and to MIUR PRIN-COFIN2010-2011 (Contract No. 2010LKE4CC), Government of Spain via a Ramón y Cajal research fellowship, and Cooperlink Italy-Australia bilateral agreement for financial support.