The statistics of the ordering of chiral ribbons on a honeycomb lattice

A novel model, devised to describe spontaneous chirality synchronization in complex liquids and liquid crystals, is proposed and studied. Segments of ribbon-like molecular columns with left- or right-handed 180degree twist lie on the bonds of a honeycomb lattice so that three ribbons meet in a vertex of the hexagonal honeycomb. The energy of each vertex is a minimum if the three ribbons have the same chirality, -E, and is +E otherwise, and the ground state is homochiral, i.e. all ribbons have the same hand. The energy levels for two vertices linked by a single ribbon are either -2E, 0 and +2 E in this vertex model. Monte Carlo simulations demonstrate that this model is identical to an Ising spin model on a Kagome lattice, for which the site energy structure is quite different. The equivalence of the ordering of the vertex and Ising spin models is also shown analytically. The energy difference between the disordered and ground states, 4J in the spin model, is related to the transition temperature for the Kagome lattice using the exact result, Tc=2.14J. The ordering energy difference for a single site is 50% higher for the vertex model. The thermodynamic energy for the vertex model is corrected by a factor of 1/3 due to double counting and this makes the specific heat of the vertex model also equal to that of the spin model as expected. Other similar models where there is an unusual relation between the site and thermodynamic energies are discussed briefly.


I Introduction
Chirality is ubiquitous in nature as it is built into the molecules that sustain life, such as proteins, DNA and carbohydrates, as well as in most modern drugs. For example the alpha-helix that form proteins is right handed because natural amino acids are left handed. However, even molecules without intrinsic chirality can form uniform helices.
Most achiral but crystallisable (isotactic, syndiotactic) synthetic polymers crystallize in a helical conformation, with long-range helical order sustained by close packing with neighbouring chains. 1,2 Furthermore, most instantaneous conformations of achiral molecules are chiral but time-averaging in a fluid phase renders them achiral. If the energy barrier between enantiomeric conformations is relatively high, a small amount of a chiral dopant can tip the balance and makes one handedness prevail. This is known as chirality amplification, or the sergeants and soldiers effect. 3 It is promoted by helix formation either through self-assembly in columns in solution forming a gel 4,5 or in non-crystalline but helix-forming polymers with high conformational barriers such as poly(phenyl acetylenes). 6,7 In both cases the barriers for chiral interconversion of the individual molecule is compounded by close packing in a helical column, making the switch in chirality a cooperative process. 8,9 If no chiral impurity or chiral surface are present, such chirogenic molecules can tip either way, sometimes resulting in multidomain samples of random handedness, 10 often referred to as a "dark conglomerate" 11 .
A few years ago, studies of thermotropic (i.e. solvent-free) liquid crystals have revealed that one of the two most common bicontinuous cubic phases, the triplenetwork phase previously thought to have spacegroup 3 ̅ , 12,13 is always chiral and optically active, even if it contains only achiral molecules. 14 Recent re-examination reassigned it to a lower symmetry I23, still retaining the triple network cubic nature but with a modified structure (figure 1b,e). 15 Conversely the other common cubic, the double gyroid 3 ̅ phase (figure 1a), is never chiral. Another LC phase, known as Smectic-Q, has since also been found to belong to the same family of bicontinuous phases but is tetragonal and also shows spontaneous chirality in achiral compounds. 16 The typical molecules forming these phases consist of a rod-like aromatic core with between 1 and 3 flexible chains attached at the ends, typically alkyl or oligo(ethylene oxide). These rod-like molecules lie normal to the segment axis in rafts of 3 or 4 (figure 1c). To explain the chirality of such bicontinuous phases 3 it was proposed that the network segments are ribbon-like and chiral due to the twist in molecular orientation, in all cases by 7-9, in successive rafts along the segment ( Figure 1d). This twist angle is a compromise between the tendency of the aromatic cores of neighbouring molecules to stay parallel and maximize their π-π interaction, while avoiding the steric clash between their bulky molten end-chains. To allow for smooth convergence and close packing, it is proposed that all three or four ribbonlike network segments maintain the same twist sense at a network junction, resulting in macroscopic homochirality. Incidentally, the reason that the double gyroid phase shows no outward chirality is that its two networks have opposite twist sense, cancelling each other's chirality 11 . Such cancellation is not possible in the triple network I23 phase. Furthermore, in the Smectic-Q phase the two networks are of the same hand, hence the phase is always optically active.
Even more surprising than the above finding of spontaneous chirality synchronization in bicontinuous LC phases was the discovery that in some of these compounds macroscopic chirality and strong optical activity are maintained even in the isotropic liquid, which we refer to as Iso * phase, above the isotropization temperature Ti of the LC. 17 Iso * then transforms to the ordinary optically inactive liquid (Iso), typically up to 20 K above Ti through a well defined transition at Tc that appears to be second order. 18 The structure of Iso * is uncertain, but since its optical rotation at Ti is similar to that of the I23 cubic, it is likely that it also contains networks with twisted segments We model the Iso*-Iso transition by considering the statistical mechanics of a system with three-way planar vertices constrained to lie on a two-dimensional honeycomb lattice. In this model we assume that all ribbons connecting two adjoining vertices are twisted by ±180 with chirality ±1 and neglect the possibility of energetically costly helix reversals within the ribbon segment.
The vertex energy is favourable,   , if the three ribbons meeting at a vertex have the same chirality and is unfavourable, with energy   , if the chirality of any one ribbon differs from the other two. This condition is the driving force that can cause a given ribbon to switch chirality. This is an interesting statistical system because there are clearly more unfavourable states of a single vertex, 6, compared with 2 favourable states. This contrasts with the connecting ribbons that have only two states, chirality ±1. A phase transition occurs from a disordered state to a state where one chirality dominates and the material becomes uniformly optically active.
We show that the statistical properties of the vertex model on a honeycomb lattice map exactly on to an Ising model on a kagome lattice with the same transition temperature with one dramatic difference, namely that the average site energy for the vertex model is 50% larger than given for the spin model at high temperatures. We also consider other models where the transition temperature is unrelated to the site energy of ordering as is found here.  Table 1 the energies of the site 0 in the configuration We find the average site energy which is the energy the system will reach in the limit of high temperatures is   .

II Energetics of the vertex model compared with the spin model
Average energy In spin models the difference between the average energy, always zero, and the ground state energy divided by the difference between the highest and lowest energy is equal to one half. The value for the vertex model is ¾. This may be compared with an n-state Potts model where the value is 1 n n  so the vertex model might have some similarity to the 4-state Potts which is known to have a second order transition 24,25 .
The distribution of energies and their multiplicities of the central site are shown in Figure 3.

III Monte Carlo simulation of the vertex model
The simulations are run using a Metropolis algorithm and Glauber dynamics over honeycomb lattices of varying size using the energetics given in Table I Table II for both models. The transitions are shown in Figure 3 as red for the larger and blue for the smaller energy.
If the energy change is zero the spin is flipped in the Monte Carlo code with probability ½ . This was chosen rather than the standard procedure of accepting the move when the energy change was zero because it improved the convergence of the average site energy at high temperatures for the vertex model. It is shown in Table II that a move of zero energy can occur for an initial vertex state of energy 2 or zero;whereas zero energy moves occur for the spin model only when the initial energy state is zero. We believe that this difference was responsible for the slow convergence at high temperatures that we observed for the vertex model and not the spin model.
The results in Table II show that the energy of configuration 1 minus that for configuration 2 are identical for the same change in orientation of the central spin for vertex and the spin models provided we choose 2J   . This means that the results for the magnetisation, () mT    , the susceptibility, and Binder cumulant will be identical for these two models. 9 The susceptibility is given by, [1] For  The most accurate way to determine the transition temperature is to use a Binder cumulant 27 that is defined for a lattice of size L by, UL is independent of L at a second order transition. The approximation for the transition temperature is found from the intersections of UL as a function of temperature, for different L. The transition temperature for the vertex model is identical to that of the spin model, 2.1433 17 , within the error as illustrated in Figure 5 bounding it above and below by 2.146 and 2.143 with an error of 0.003 respectively. it is clear the high temperature limits are different. This apparent discrepancy is discussed in more detail in section V.

V A unified formulation for the vertex and spin model
The site energy for the vertex model may be rewritten in terms of pseudo-spins,  1 using the relations that The values of the parameters, A and C, are found by requiring that the maximum and minimum energies are given by The energies and degeneracies of this model are shown in figure 8. The maximum and minimum energies are independent of ; however the energy of the fully disordered state is at + for the vertex model and at zero for the spin model.

(a)
(b) 13 The thermodynamic energy for this model is given by, All the thermodynamic properties of these models are independent of .

VI Other similar models with an unusual relation between the site energy and the thermodynamic energy
The site energy for the vertex model given in equation [4] contained terms that did not depend on the central site. These terms contributed to the ordering energy but not to the value of Tc. We consider what other models would show this feature.
14 A four-way vertex model would map on to a triangular lattice as shown in figure 9.  We chose the vertex energy in this case to be The contributions to the ordering energy at T=0 are 12 from the terms involving spin 0  and an equal contribution from the additional terms. In this case the elimination of (a) (b) double counting will require that the relationship between the thermodynamic energy and the site energies is given by 4vertex Another way of introducing extra terms is to consider multiple spin interactions 28 as in the site energy below for a square lattice: The ordering energy for this model is (4 ) J   but Monte Carlo simulations will predict that the transition is independent of  The negative fourth order term would tend to drive the transition to first order; this occurs for within mean field theory applied to this two dimensional lattice.
The models described here have simple solutions as they are not frustrated as occurs when antiferromagnetic interactions are involved. Such frustrated models have been discussed extensively and include the ice-model and the 8-vertex model which may also map on to solvable models 29 . The theory developed here allows us to estimate the energy difference between vertices in Iso * for which all three connecting ribbons are and are not identical, 2  .

VII Conclusion
Using the Iso * -Iso transition at 465K and RTc =2.1433J for the Kagome lattice we find that J =0.430 kcal/mol and since  =2J we find  =0.86 kcal/mol. This is likely to be an overestimate for  because it was obtained from the combination of a phase transition temperature for a three dimensional system with a two dimensional model. 16 The observed transition is indeed second order 18 and the energy of the transition may be estimated from an integral over the specific heat peak. This experimental result gives an estimate of the ordering energy, U = 1.1kcal/mol which is higher than the value of  =0.86 kcal/mol that would be deduced using this model. We conclude that the two dimensional model of the three dimensional Iso * -Iso transition is qualitatively correct although the detailed numbers are subject to error. However this vertex model does have interesting statistical properties in its own right.