Comparison to one-dimensional model

We see that fluctuations play a crucial role in the formation of the chiral pattern because the same initial state can lead to a chiral or nematic state depending on the fluctuations. Moreover, seeding the system with the precursor of a chiral pattern (two parallel, nematically ordered planes whose local directors form an angle of aboutπ/2) does not always lead to a chiral pattern but only increases the probability. Hence, the fluctuations are still of high importance. Moreover, as we will see in Section 4.4.7, a chiral pattern can untwist and form a nematic pattern only by the appropriate fluctuations. In this section we will elucidate the role of the fluctuations by examining a one-dimensional system of non-active interacting spins (simulations performed by R. Selinger, Kent State University, USA, published in Breier et al., 2016) similar to the classical XY-model. TheN_r spins (or rotors) are placed next to each other (see inset of Fig. 4.15) on a line with PBCs. Each spin can rotate around the axis which connects all spins and hence is described by its rotation angleθ_i. The spins interact (just like the SPPs) via the Lebwohl-Lasher potential (Eq. 3.1) which for this model can be rewritten as

V =−

Nr

X

i=1

Jcos[2(θ_i+1−θ_i)] (4.27)

with the coupling constant J. This potential is very similar to the potential of the classical XY-model except for the additional factor of two which leads to nematic

4.4 Spontaneous chiral symmetry breaking

1 2 3 4 5 6 7 8

0 0.1 0.2 0.3 0.4

0.5 |n|=0

|n|=1

|n|=2

|n|=3

number |n| of twists

probability

Figure 4.15: Results of the one-dimensional Lebwohl-Lasher model. Inset: It can evolve to twisted metastable states with ±n twists or the untwisted ground state.

Main panel: Probability of the number of twists for a long chain with N_r = 800 (green O) and for a short chain with Nr = 100 (blue ♦). The lines are guides for the eye. Reprinted figure with permission from Breier et al., Physical Review E, 93(2):022410, 2016. Copyright 2016 by the American Physical Society.

instead of polar symmetry. The system is described by the Hamiltonian H=V +

Nr

X

i=1

1

2Iω_i² , (4.28)

with the moment of inertiaI (which can be set to unity) and the angular velocityω_i of the i-th rotor. The torque of the i-th rotor is given by

τ_i =−I∂V

∂θi

(4.29) such that the rotor is influenced by its two neighbors. The angular acceleration α_i is then proportional to the torque

αi = τ_i

I (4.30)

and needs to be integrated forward in time to obtain the angular velocity ω_i. The temporal integration of the latter then leads to the angular position of the rotor (i.e. the angle θ_i). This integration in time is done using a velocity Verlet algorithm and a nonequilibrium rapid quench (Langevin thermostat) fromT = 10 toT = 10⁻⁷ in 6×10⁵ time steps which successively removes kinetic energy from the system. The initial configuration consists of random directions of the rotors and zero angular ve-locities. For each chain length 200 such annealing trials are carried out independently.

We find that for long chains (N_r ≥ 200) the most likely final state is a chiral state with a twist of±πwhile the nematically ordered state is the most probable for shorter chains (see Fig. 4.15). This behavior can be inferred from the associated energies.

The energy of the nematic ground state is E₀ = −J. Possible chiral states show a rotation angle of nπ along the chain due to the PBCs and the nematic interaction.

The potential energy of such a chiral state is given by

∆E_n=h−Jcos(2∆θ)i −E₀ (4.31)

where the subscript n refers to the number of half-twists. The angle between neigh-boring rotors (for a homogeneous twist) is given by

∆θ = nπ

N_r . (4.32)

For long enough chains, we can make use of the small-angle approximation and find

∆E_n≈^D−J(1−2∆θ²)^E−E₀

= 2J

nπ N_r

2

. (4.33)

4.4 Spontaneous chiral symmetry breaking

This means that there are considerable energy barriers between the different states and if a system is quenched from a high temperature, random state to a low temper-ature one it is possible that it moves (in the energy landscape) into a twisted state instead of the untwisted ground state. Moreover, the height ∆E_nof the energy barrier decreases with increasing chain length. The fugacity of the corresponding equilibrium model exp(−∆E_n/(k_BT)) forms a Gaussian bell-curve as a function of the number of twists n and hence always exhibits a maximum at n = 0. However, if we do not distinguish left- and right-handed helices and study the fugacity as a function of |n|, we find (for long enough chains) a maximum at|n|= 1. This means that the twisted state is more probable than the untwisted ground state if the chain is long enough.

Moreover, if we increase the chain length further, chiral states with more twists be-come more likely. In the limit of an infinitely long chain this destroys the long-range order in the system at any finite temperature and leads to the phase transition at T = 0 (just like in the XY-model). In Fig.4.15 we plot the probability of the states with different numbers|n| of twists and compare systems with different chain length.

We find that in a short chain (N_r = 100) the nematic state is the most probable.

However, for a long chain (N_r = 800) the chiral state with a twist of |n| =π occurs more often than the nematic ground state just as predicted. If we used instead of |n|

the number n of twists (distinguishing between left- and right-handed helices), the curve would always be bell-shaped with a maximum at n = 0. However, for larger chain lengths twisted states are more likely than for shorter chain and the distribu-tion alongn gets broader, so that the combined probability ofn = +1 andn =−1 is larger than the probability of n = 0 for a chain that is long enough (i.e. N_r ≥ 200).

An additional finding of this one-dimensional model is that the mean square number of twists hn²i increases linearly with the number of rotor N_r and hence follows the same statistics as a random walk.

An equivalent effect can also be found in the system of SPPs: The probability of a chiral state is more than ten times larger in a long box of aspect ratio 10 : 1 : 1 (N = 230 ×23× 23, ρ = 1, P = 3.13) than in a cubic box. Moreover, not all chiral patterns in such a box show one half twist inside the simulation box but also the spontaneous formation of a fully twisted chiral (pitch of L_x) can be observed.

However, it is important to note that such a long box requires longer relaxation times until a true steady-state is reached. The fully twisted chiral is hence also expected to be metastable just like the chiral state in general in the cubic simulation domain.

We also find metastable configurations where the local nematic director is oriented perpendicular to the long axis of the simulation box. If one moves along this axis, the orientation of the local nematic director undulates but does not lead to a helix. Such a state is only possible due to the long time it takes until information is propagated along the long axis of the simulation domain.

chiralorderparameterS′ χ

global nematic order parameter0 0.2 0.4 0.6 0.8 S1

−0.2 0 0.2

(a) Example trajectories (time increases from white to black) in nematic and chiral order parameter (N = 66³).

chiralprobabilityPχ

seed with chiral fraction ∆Nχ/N

0.01% 0.1% 1% 10%

0%

10%

20%

(b) Probability of a persistent chiral pattern when the box is seeded with a fraction of Nχ/N particles which follow a chiral pat-tern (N = 50³). For each data point 50 independent simulations are performed.

Figure 4.16: Simulations ofSPPs with P → ∞(ρ= 1, γ = 0.1).