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 C_{2} 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

^{3}The intrinsically chiral character of the helical constituents may translate into a chiral organization, the cholesteric phase,

^{3}in which

^{4}

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

_{2}) symmetry axes of the helices, that become long-range correlated and preferentially oriented along a second common director,

*screw-like*and denote it by

_{2}symmetry axis, respectively. In the

A transition from the isotropic (I) to the

*et al.*,

^{5}using polarizing and differential interference contrast microscopy, combined with experiments on single-particle dynamics. A striped birefringent pattern was observed consistent with a picture where the local tangent to each helix, tilted with respect to

^{6,7}this phase was denoted as conical, although the physical underlying mechanism and detailed structure is different.

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* = 10*D* [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*^{*} = *PD*^{3}/*k*_{B}*T*, *k*_{B} 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 × 10^{6} equilibration MC cycles were followed by additional 2 × 10^{6} production MC cycles to collect statistics on various quantities.

Fig. 2 shows, in the *P*^{*}–volume fraction (η = ρ*v*_{0}, with ρ the number density and *v*_{0} 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

_{1}= |⟨exp (

*i*2π

*Z*/

*d*)⟩| (

*d*being the layer spacing, and

*Z*being the position along the

^{3}as well as appropriate correlation functions. Points identified by C correspond to compact phases. Note that, unlike the case of spherocylinders,

^{10}the high-density phase diagram of helices is not known, and constitutes an interesting property on its own right. We determined the maximum packing configuration by adapting a methodology proposed in Ref. 11, that hinges on an annealing reorganization scheme for a unit cell toward the most compact configuration, in the approximation where we consider a single layer of helices. The results of this analysis are summarized in the color map of Fig. 3 displaying the largest obtained volume fraction as a function of

*r*and

*p*of the helices. Interestingly, while there exists a large variation of the maximal packing, depending on the helix morphology, similar values can be achieved for different

*r*,

*p*pairs. The high density state points in the phase diagram of Fig. 2 were obtained using these maximal packing configurations as initial conditions, upon applying the appropriate pressure until equilibration. Points denoted by C in Fig. 2 are then associated with highly ordered configurations compatible with solid-like ordering. Finally, points identified by

_{2}symmetry axis of helix

*i*(Fig. 1(b)), and the subscript

*i*and

*j*with a specific

*R*

_{‖}, the projection of the interparticle separation

**R**

_{ij}along the

_{2}symmetry axes of two helices as a function of their distance projected along the main director. Fig. 4 shows this correlation function calculated for helices with

*r*= 0.2 (a) and 0.4 (b) and

*p*= 3 and 6, at different values of η. For both radii, a sinusoidal structure with a periodicity equal to

*p*is clearly visible. It persists with a constant amplitude at long interparticle distances. This behavior reflects the helical correlation of the

*r*= 0.4 the N phase is either absent altogether or surviving in a very narrow range of η. Additional insights on the onset of the

*P*

_{1, c}⟩ distinguishes the

*P*

_{2}⟩. Figs. 5(a) and 5(b) show both ⟨

*P*

_{2}⟩ and ⟨

*P*

_{1, c}⟩ as a function of η, for helices with

*r*= 0.2 and increasing value of the

*p*= 3, 6, and 8. As pitch increases, the location of the I-N phase transition moves to lower η, as indicated by the ⟨

*P*

_{2}⟩ behavior, in agreement with results reported earlier.

^{8}This can be understood in terms of an increase of the effective aspect ratio that tends to stabilize the N phase. The location of the N–

*P*

_{2}⟩. This can be ascribed to the fact that the nematic order has first to set in and reach a very high degree before the C

_{2}symmetry axes start twisting around

*P*

_{1,c}⟩ in the neighborhood of the N–

*r*= 0.4, depicted in Figs. 5(c) and 5(d). While there is a similar trend, albeit much less pronounced, of the I-N phase transition approximatively shifting toward larger η for decreasing

*p*, the seemingly simultaneous rise of both ⟨

*P*

_{2}⟩ and ⟨

*P*

_{1,c}⟩ is suggestive of a very narrow ordinary N phase or even of a direct transition from the I to the

The onset of the

^{13}Since numerical simulations indicate that the

*P*

_{2}⟩, we here assume perfectly parallel helices (⟨

*P*

_{2}⟩ = 1). The single-particle density can be then expressed as a function of

*f*(α

^{′}) is the local orientational distribution function. In the N phase the latter is a constant,

*f*= 1/2π, with the normalization condition

*Z*the position of a particle along

^{′}+ 2π

*Z*/

*p*(Fig. 1(b)),

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

**R**

_{12}= (

*X*

_{12},

*Y*

_{12},

*Z*

_{12}) defines the position of particle 2 in this frame, α

_{i}being the angle between the unit vector

*i*and the

*X*axis. The factor (4 − 3η)/(4(1 − η)

^{2}) is a correction introduced to account for higher virial contributions

^{14}and

*a*

_{excl}(

*Z*

_{12}, α

_{1}, α

_{2}) = −∫

*dX*

_{12}∫

*dY*

_{12}

*e*

_{12}(

**R**

_{12}, α

_{1}, α

_{2}) (with

*e*

_{12}the Mayer function

^{15}) is the section of the volume excluded to particle 2 by particle 1 cut by a plane normal to

*Z*=

*Z*

_{12}.

^{16}The inset of Fig. 5 shows the result from Onsager theory for the dependence on η of the order parameter

*r*=0.4 and

*p*= 3 and 6. While a quantitative comparison is clearly not possible, Onsager theory qualitatively agrees with simulation results and clarifies a number of additional issues. As shown in the inset of Fig. 5, the second-order N–

*p*. As helices with

*p*= 3 tend to interpenetrate less than those with

*p*= 6, the entropic gain driving the formation of

*r*= 0.4. The small pitch dependence of the I–N–

*r*= 0.4 observed in Figs. 5(c) and 5(d) can then be interpreted as a result of two competing effects. On the one hand, the shorter effective aspect ratio of the helix associated with smaller

*p*tends to push the formation of the N phase at higher densities. On the other hand, this is balanced by a larger tendency to develop screw-like ordering. As a result, an almost direct transition to a

*p*.

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

_{2}symmetry axes. By obtaining the full phase diagram of hard helices up to the most compact phases, the exact boundaries of the

Our results provide a theoretical explanation of the I–

^{5}This raises the expectation that the same transition could also be observed in other similar systems, including chiral colloidal particles (e.g., bacteria and viruses), helical (bio)polymers, and concentrated DNA solutions, although the much smaller length scales and the specificity of the interactions involved in this latter case may constitute a formidable experimental challenge.

^{12}Our work highlights the generality of the entropic mechanism driving the formation of the

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.