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S. Wagner, G. Kahl, A. Baumketner:
"Entropic differences between lattices formed by hard ellipses";
Poster: 11th Liquid Matter Conference 2021, Prague; 19.07.2021.

Hard ellipses show in a two-dimensional setup and at high packing fractions an intricate vari-
ety of possible lattice conformation [1]; these structures all emerge from a hexagonal arrangement
of discs via suitable deformations with a chosen rotation (characterized by the parameters ω and
τ, which specify orientational and positional order of the particles). Thus, for a given packing
fraction there is an infinite number of possibilities to arrange ellipses in a lattice structure: they
all differ - for a given aspect ratio κ - in the orientational and positional order of the particles
(see Fig.1a).
In this contribution we show that for ellipses with aspect ratio κ = 2 only two relevant lattice
types survive within the plethora of possible ordered configurations. Our simulation-based inves-
tigations rely on new type of Monte Carlo moves that directly probes the underlying parameter
space of configurations, spanned by ω and τ. By investigating the entropic landscape we find out
that ordered lattice types characterized either by ω = 0 ◦ (corresponding to a diagonal lattice state)
or ω = 30 ◦ (representing a parallel lattice state) are the most stable ones (see Fig.1b and Fig.1c).
Further we provide evidence that for each of these two scenarios there is only one optimal value
of τ.
In an effort to find the true ground state (global entropy-maximum or free energy-minimum),
the exact entropic difference between the parallel and diagonal state must be computed.
This is achieved with the help of a variety of complementary computational methods, such as a
Potential Mean Force Method in combination with the Reweighting Histogram Technique [2] as
well as the Einstein Crystal Method [3]. Particular challenges in this enterprise originate from
the strong dependence of this entropic difference between the two states on system size and the
resulting challenges in terms of estimating the thermodynamic limit. These issues will also be
discussed in detail in this contribution.
The thermodynamics of this two-state system will be discussed in terms of the contribution of
translational and rotational degrees of freedom to the entropy as well as the importance of the
coupling of the two degrees of freedom. The studying of the trade-off between translational and
rotational degrees of freedom in this particular system will provide the basis to understand related
features in other two dimensional, shape-anisotropic systems encountered in nature.
Background. Our investigations should be seen within the following context: While some sci-
entific contributions have focused on the liquid and nematic phases of the hard ellipses [4, 5],
others have investigated also the ordered solid phases of this system [6, 7, 8]; interestingly the
latter ones observed in their investigations only either type of the above scenarios: a solid in the
parallel state was reported in [6] while a solid in diagonal state was reported in [7].
References
[1] J. Viellard-Baron, J. Chem. Phys. 56, 4729 (1972).
[2] G.M. Torrie and J.P. Valleau, J. Comput. Phys. 23, 187 (1977).
[3] C. Vega and E. Noya, J. Chem. Phys. 127, 154113 (2007).
[4] S. Varga and I. Szalai, Mol. Phys. 95, 515 (1998).
[5] X. Wen-Sheng et.al., J. Chem. Phys. 139, 024501 (2013).
[6] J.A. Cuesta and D. Frenkel, Phys. Rev. A 42, 2126-2136 (1990).
[7] G. Bautista-Carbajal and G. Odriozola, J. Chem. Phys. 140, 204502 (2014).
[8] S. Davatolhagh and S. Foroozan, Phys. Rev. E 85, 061707 (2012).

ellipse lattices

https://publik.tuwien.ac.at/files/publik_302045.pdf