Splay-induced order in systems of hard wedges Piotr Kubala Micha l Cie sla and Lech Longa Institute of Theoretical Physics Jagiellonian University in Krak ow Lojasiewicza 11 30-348 Krak ow Poland

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Splay-induced order in systems of hard wedges

Piotr Kubala, Micha l Cie´sla, and Lech Longa

Institute of Theoretical Physics, Jagiellonian University in Krak´ow, Lojasiewicza 11, 30-348 Krak´ow, Poland

The main objective of this work is to clarify the role that wedge-shaped elongated molecules, i.e.,

molecules with one end wider than the other, can play in stabilizing orientational order. The focus

is exclusively on entropy-driven self-organization induced by purely excluded volume interactions.

Drawing an analogy to RM734 (4-[(4-nitrophe-noxy)carbonyl]phenyl2,4-dimethoxybenzoate), which

is known to stabilize ferroelectric nematic (NF) and nematic splay (NS) phases, and assuming that

molecular biaxiality is of secondary importance, we consider monodisperse systems composed of

hard molecules. Each molecule is modeled using six colinear tangent spheres with linearly decreas-

ing diameters. Through hard-particle, constant-pressure Monte Carlo simulations, we study the

emergent phases as functions of the ratio between the smallest and largest diameters of the spheres

(denoted as d) and the packing fraction (η). To analyze global and local molecular orderings, we

examine molecular conﬁgurations in terms of nematic, smectic, and hexatic order parameters. Ad-

ditionally, we investigate the radial pair distribution function, polarization correlation function, and

the histogram of angles between molecular axes. The latter characteristic is utilized to quantify local

splay. The ﬁndings reveal that splay-induced deformations drive unusual long-range orientational

order at relatively high packing fractions (η > 0.5), corresponding to crystalline phases. When

η < 0.5, only short-range order is aﬀected, and in addition to the isotropic liquid, only the standard

nematic and smectic A liquid crystalline phases are stabilized. However, for η > 0.5, apart from

the ordinary non-polar hexagonal crystal, three new frustrated crystalline polar blue phases with

long-range splay modulation are observed: antiferroelectric splay crystal (CrSPA), antiferroelectric

double splay crystal (CrDSPA) and ferroelectric double splay crystal (CrDSPF). Finally, we employ

Onsager-Parsons-Lee Local Density Functional Theory to investigate whether any sterically-induced

(anti-)ferroelectric nematic or smectic-A type of ordering is possible for our system, at least in a

metastable regime.

I. INTRODUCTION

The most spectacular discovery of the last decade in

the ﬁeld of liquid crystal research has been the identiﬁ-

cation of the ferroelectric nematic (NF) [1–4], antiferro-

electric nematic splay (NS) [5], and nematic twist-bend

(NTB) [6–8] phases, all characterized by various forms of

long-range polar order.

Based on current experimental observations, it seems

that the stabilization of these new nematic phases is

linked to the strong softening (i.e., reduction by one to

two orders of magnitude) of one of the Frank elastic

constants, Kii [9], in the parent, uniaxial nematic (N)

phase, near the transition to one of the polar nematics.

Kiis (i= 1,2,3) weight here elementary deformations of

splay (K11), twist (K22), and bend (K33) types of the

undistorted, reference uniaxial nematic state (N) in the

Oseen-Zocher-Frank free energy [9–11]

F=1

2VZVK11[ˆn(∇ · ˆn)]2+K22[ˆn ·(∇ × ˆn)]2+

+K33[ˆn ×(∇ × ˆn)]2; (1)

ˆn denotes the locally preferred orientation of molecules

referred to as the director, and Vis the system’s volume.

While each Frank elastic constant is typically positive

and on the order of 10 pN, the observed softening of

Kii, suggests that the lack of orientational modulation

of Nis no longer energetically favored, and it can pro-

mote the appearance of orientationally modulated phase.

Eﬀectively it means that the softened elastic constant

may become negative. An important observation made

by Meyer many years ago [12] was that this softening

can be attributed to the entropy of packing of molecules

with speciﬁc shape of nonzero steric dipole. In fact, the

dipolar asymmetry of these molecules is expected to in-

duce a ﬂexopolarization eﬀect, which couples to splay and

bend director deformations [13] and eﬀectively reduces

the K11 and/or K33 elastic constants [14]. For instance,

molecules with a bow(banana) shape can reduce the K33

bend elastic constant, leading to the formation of twist-

bent and splay-bent phases [15]. Greco and Ferrarini

were the ﬁrst to explicitly demonstrate this phenomenon

through molecular dynamic simulations on a system of

bow-shaped molecules that interact solely through steric

interactions [16].

Further studies on bow-shaped molecules, focusing on

steric interactions, have provided successful explanations

not only for the formation of the NTB phase [17, 18] but

have also demonstrated the potential for stabilizing vari-

ous intermediate polar states between NTB and NSB [17].

It is well-known that even slight diﬀerences in the ge-

ometry of molecules can have a signiﬁcant impact on the

resulting liquid-crystalline self-assembly. For example, a

smectic A phase is observed in a system consisting of

spherocylinders [19], whereas ellipsoids with a very sim-

ilar shape do not exhibit this behavior [20]. This prin-

ciple also applies to the recently discovered NFand NS

phases. These phases rely on both wedge-like molecular

anisotropy and a strong, nearly longitudinal total molec-

ular dipole moment as crucial molecular characteristics

contributing to their stability. Speciﬁcally, the wedge-like

arXiv:2210.04737v3 [cond-mat.soft] 23 Jun 2023

molecular shapes can result in a negative K11 splay con-

stant [21], thereby contributing to the formation of the

polar splay nematic and related smectic phases [22, 23].

Numerical simulations of systems composed of vedge-

like (e.g., pear-shaped, tapered, etc.) molecules began

in the late 1990s [24, 25]. In these simulations, shape

polarity at the molecular level was induced by a soft in-

teraction potential between molecules, which combined

two rigidly connected centers: an ellipsoidal Gay-Berne

potential[26] and a spherical Lennard-Jones potential.

The reported liquid crystalline phases in these simula-

tions were only ordinary uniaxial nematic and smectic

phases, without any macroscopic polarization, as they

exhibited a preference for antiparallel local arrangement.

Berardi, Ricci, and Zannoni developed a generalized

single-site Gay-Berne potential to model the attractive

and repulsive interactions between elongated tapered

molecules [27]. Through parameter adjustments and

Monte Carlo simulations, they observed stable NFand

ferroelectric smectic liquid crystals. However, while the

introduction of a weak axial dipole did not qualitatively

impact these observations, an increase in dipole strength

resulted in the destruction of long-range ferroelectric or-

dering. It is worth noting that using a standard Gay-

Berne potential with an axial dipole at one end of the

molecule resulted in the formation of a bilayer smec-

tic phase [28–30]. Similar mesophase formation was ob-

served in simulations of single-site hard pears [31].

Purely entropic systems constructed of pear-shaped

molecules also exhibit a cubic gyroid phase [32, 33]. In-

terestingly, the stability of this phase is highly sensitive

to the details of the hard-core interaction. Speciﬁcally,

the cubic gyroid phase is observed when describing the

pear shape using two B´ezier curves with the hard pear

Gaussian overlap model (PHGO). However, it vanishes

when the hard pears of revolution (HPR) model is used.

In the PHGO model, a bilayer smectic phase is also ob-

served, whereas the HPR model exhibits isotropic and

nematic phases [34].

(a)

(b)

FIG. 1. (a) Family of wedges used in the study; they are

built of six co-linear tangent spheres with diameters increas-

ing linearly from dto 1 (d≤1 is a parameter). (b) Wedge-

shaped RM734 molecule known to form a polar nematic

phase. Reprinted from Ref. [4] with the permission of RSC.

In this paper, we seek to explore the fundamental

mechanisms that can lead to the emergence of long-range

splay and potentially polar order in liquid crystal sys-

tems. Speciﬁcally, we focus on the interactions among

hard wedge-shaped molecules in the absence of dipo-

lar electrostatic or dispersion forces. By investigating

whether such long-range order can be stabilized solely

through entropic interactions, we aim to advance our un-

derstanding of the essential features of molecular inter-

actions responsible for stabilizing NFand NSphases.

To achieve this goal, we utilize a model consisting of

six co-linear tangent spheres, resulting in a molecule with

C∞v(cone) symmetry [Fig. 1(a)]. The diameters of the

spheres follow an arithmetic sequence, starting from d

and progressing to 1. Speciﬁcally, the diameters are given

by d, (4d+ 1)/5, (3d+ 2)/5, (2d+ 3)/5, (d+ 4)/5 and

1. Here, the parameter drepresents the diameter of the

smallest sphere and serves as a descriptor for the shape

of the molecule. When dequals 1, the molecule reduces

to the linear tangent hard-sphere model (LTHS) as de-

scribed by Vega et al. in Ref. [35].

We have chosen this model to capture essential

features of the eﬀective shape exhibited by the RM734

molecule (4-[(4-nitrophe-noxy)carbonyl]phenyl2,4-

dimethoxybenzoate) [Fig. 1(b)]. As previously men-

tioned, the RM734 mesogen has been shown to stabilize

NFand NSphases [3, 4]. By utilizing a model with

similar shape characteristics, we aim to gain further

insight into the relationship between entropy of packing

and the resulting self-organization in liquid crystal

systems. To this end, we investigate the phase diagram

and properties of stable structures using Monte Carlo

integration. Additionally, we compare our results with

those obtained from previous studies on soft- and

hard-core pear models. Furthermore, we examine how

the gradient of a molecule’s diameter inﬂuences the

presence and extent of the observed phases.

Finally, we recognize that the anisotropic polar shape

of the molecules may give rise to non-trivial dense con-

ﬁgurations. We are particularly interested in exploring

whether these conﬁgurations exhibit crystal-like or glass-

like structures, considering the potential competition be-

tween diﬀerent types of lattices [36–38].

The remainder of this paper is organized as follows: In

Sec. II A we provide a brief description of the Monte

Carlo integrator used. In Sections II B and II C, we

introduce the order parameters and correlation functions

used to monitor properties of equilibrium structures. The

results obtained from Monte Carlo simulations are pre-

sented in Sec. III. In Sec. IV, we formulate the Parsons-

Lee Density Functional Theory to study polar ordering in

the liquid-crystalline regime. We analyze some equilib-

rium and metastable phases in detail. Lastly, we provide

a discussion and outlook in Sec. V. Appendix A con-

tains remarks about non-tilted hexagonal conﬁgurations

of close-packed linear tangent hexamers.

II. METHODS

A. Monte Carlo simulations

We assumed hard-core interactions between molecules.

The equilibrium phases were classiﬁed as a function of

the shape parameter dand the packing fraction η. The

latter one is a natural choice for purely steric repulsion.

System snapshots were obtained numerically using the

Monte Carlo scheme [39] implemented in our RAMPACK

software package (see Sect. Code availability). Integra-

tion was carried out in the NpT ensemble. For hard-core

interactions, only a ratio p/T of pressure and temper-

ature is an independent parameter and can be used to

control the packing fraction η. To allow relaxation of the

full viscous stress tensor (including the sheer part), we

used a triclinic simulation box with periodic boundary

condition. For a system of Nmolecules, a full MC cycle

consisted of Nrototranslation moves, N/10 ﬂip moves,

and a single box move. In a rototranslation move, a sin-

gle shape was chosen at random, translated by a random

vector, and rotated around the random axis by a random

angle (clockwise and anticlockwise rotations were equally

probable to preserve the detailed balance condition). If

the move introduced an overlap, it was always rejected

and accepted otherwise. Flip moves were performed in

a similar way to rototranslation moves; however, instead

of random translation and rotation, the molecule was ro-

tated by 180◦around its geometric center. This type

of move facilitated easier sampling of the phase space of

the system, especially for high η. For a box move, the

three vectors b1,b2,b3that span the box were per-

turbed by small random vectors. The move was rejected

if any overlaps were introduced. Otherwise, it was ac-

cepted according to the Metropolis-Wood criterion with

probability [40, 41]

min 1,exp Nlog V

V0−p∆V

T,(2)

where ∆V= (V−V0) and V0,Vare, respectively,

the volume of the box before and after the move. The

perturbation ranges were adjusted during the thermal-

ization phase to achieve an acceptance probability of

around 0.15. To accelerate simulations in a modern

multi-threaded environment, we used domain decomposi-

tion technique [42] for molecule moves and we parallelized

independent overlap checks for volume moves.

To scan the full phase sequence, from isotropic liquid

to crystal, we used p/T ratios corresponding to pack-

ing fraction covering η∈[0.3,0.58] for d∈[0.4,1.0].

First, to roughly determine the phase boundaries, pre-

liminary simulations were performed by gradually com-

pressing a small system of N= 400 molecules in a cubic

box from a highly diluted simple cubic lattice. For each

p/T , the integration consisted of the thermalization run

with 9.5×106full MC cycles and the production run

with 0.5×106cycles to gather averages. The ﬁnal snap-

shot of a run was used as a starting point for the next

with a slightly higher p/T . Using the results as guidance,

the main simulations were performed on a much larger

system with N > 5000 in a triclinic box. The initial con-

ﬁguration in the whole range of dwas smectic A with

η≈0.45 (see Sec. III for the description of phases) pre-

pared by thermalizing diﬀerent types of slightly diluted

crystals for (1-5)×108cycles. Initial conﬁgurations were

then independently compressed or expanded to all target

densities in parallel. Thermalization runs were performed

for (0.9-4.5)×108cycles, while production runs were per-

formed for (0.1-0.5)×108cycles. Additionally, in order to

estimate maximal packing fractions, the densest conﬁgu-

rations for each dwere compressed under exponentially

increasing pressure for 3×108cycles, reaching p/T = 104

at the end.

B. Order parameters

Phases in the system can be easily classiﬁed using a

carefully chosen set of order parameters, whose values

have jumps on the boundaries of phase transitions. The

nematic order along the director ˆn is detected by the

average value ⟨P2(ˆa ·ˆn)⟩of the second-order Legendre

polynomial, where ˆa is the long axis of the molecule.

Director ˆn can be inferred directly from the system using

the second-rank Qtensor [43], which can be numerically

computed as

Q=1

i=1

2ˆai⊗ˆai−1

3,(3)

where the summation is done over all molecules in a single

snapshot. P2is then the eigenvalue of Qwith the highest

magnitude and ˆn – the corresponding eigenvector. En-

semble averaged ⟨P2⟩is calculated by averaging P2over

non-correlated system snapshots. The nematic order pa-

rameter has a minimal value −0.5, when all molecules

are perpendicular to ˆn, and reaches its maximum 1 for

molecules perfectly aligned with ˆn (please note that ˆn

and −ˆn directions are equivalent). In a disordered sys-

tem ⟨P2⟩= 0.

Density modulation can be quantitatively described by

the smectic order parameter ⟨τ⟩[44]. It is deﬁned as

⟨τ⟩=1

N*

i=1

exp(ik·ri)+,(4)

where kis the modulation wavevector compatible with

PBC and riis the center of the ith molecule. As the

drift of the whole system is a Goldstone mode, the abso-

lute value |···| is taken before the ensemble averaging to

eliminate it. All possible kcan be enumerated using re-

ciprocal box vectors g1,g2,g3[45] and taking linear com-

binations of them with integer coeﬃcients h, k, l (Miller

indices [46]): k=hg1+kg2+lg3. Here, as the initial

conﬁguration is always a smectic with six layers stacked

along the zaxis, hkl = 006. The smectic order ranges

from 0 for a homogeneous system to 1 for a perfectly

layered one.

Another feature of the system that is measured in the

study is the hexatic order appearing for high packing

fractions η, where molecules tend to form hcp-like struc-

tures. The local hexatic order can be measured using the

so-called hexatic bond order parameter ⟨ψ6⟩[47]. For a

two-dimensional system, it is deﬁned as

⟨ψ6⟩=1

N*N

i=1

6

j=1

exp(6iϕij )+,(5)

where ϕij is the angle between an arbitrary axis in the

plane and the vector that joins the center of the ith

molecule with its jth nearest neighbor. It can be gen-

eralized to three-dimensional systems by projecting the

positions of molecules onto the nearest smectic layers and

computing ψ6within the planes deﬁned by them. Ran-

dom points give ⟨ψ6⟩ ≈ 0.37, while a perfect hexatic or-

der yields ⟨ψ6⟩= 1. The local hexatic order can also be

computed for a system without layers by projecting all

centers on a single plane.

C. Correlation functions

Additional insight into both global properties and

supramolecular structures is given by correlation func-

tions. The ﬁrst is a standard radial distribution function

[48], which can be deﬁned in a computationally friendly

way as

ρ(r) =  dN(r, r + dr)

4πr2dr·(N/V )N,(6)

where dN(r, r + dr) is the number of molecules whose

distance from a selected single molecule lies in the range

(r, r + dr), dris the numerical size of the bin, Nis the

total number of molecules and Vis the volume of the

system. It is then averaged over all molecules ⟨···⟩N

and over independent snapshots ⟨. . .⟩. It is normalized

in such a way that, for a disordered isotropic system,

it approaches 1 for r→ ∞. In systems with long-range

translational order, ρ(r) has a series of numerous minima

and maxima.

In a layered system, one can also measure the layer-

wise radial distribution function in the direction orthog-

onal to k

ρ⊥(r⊥) = 1

nL*nL

i=1 dNi(r⊥, r⊥+ dr⊥)

2πr⊥dr⊥·(N/S)Ni+,(7)

where nLis a number of layers, dNi(r⊥, r + dr⊥) is the

number of molecules in ith layer whose distance from a

selected single molecule calculated along layer’s plane lies

in the range (r⊥, r⊥+dr⊥), and Sis total surface area of

all layers [49]. In the end, it is averaged over all molecules

in the layer ⟨···⟩Ni, all layers (1/nL)PnL

i=1 ··· and uncor-

related system snapshots ⟨···⟩.

As the results will show, the system develops a nontriv-

ial polar metastructure. To quantify it, we use the layer-

wise radial polarization correlation [48], deﬁned alike

ρ⊥(r⊥):

S110

⊥(r⊥) = 1

nL*nL

i=1 ˆaij·ˆaikijik+,(8)

where ⟨···⟩ijikdenotes the average over all molecules in

the ith layer, whose centers’ distance along this layer lies

in the (r⊥, r + dr⊥) range.

The range of splay correlations can be quantiﬁed

by the conditional probability P(θ|r⊥) of ﬁnding two

molecules with angle θbetween their molecular axes ˆai

at a transversal distance r⊥, normalized as

Z90◦

0◦

P(θ|r⊥) dθ= 1,∀r⊥.(9)

To be consistent with the polarization correlation func-

tion, this quantity will also be calculated layer-wise. As

splay corresponds to the radial spread of director ﬁeld

lines, the most probable angle θshould grow with r⊥in

systems with non-zero splay deformation mode.

III. RESULTS

Using the method described in Section II A, we were

able to recognize all phases for d∈[0.4,1.0] and η∈

[0.3,0.58]. There are three liquid phases: isotropic liq-

uid (Iso), nematic (N), smectic A (SmA) and four crys-

talline phases: hexagonal crystal (Crhex), antiferroelec-

tric splay crystal (CrSPA), antiferroelectric double splay

crystal (CrDSPA) and ferroelectric double splay crystal

(CrDSPF). The phase diagram is presented in Fig. 2, the

order parameters are shown in Fig. 3, while Figs. 4,5,6

contain correlation functions. Moreover, representa-

tive equilibrium snapshots of all phases can be seen in

Figs. 8,9. The phases for all sampled pairs (d, η) were

manually classiﬁed using order parameters and visual in-

spection of system snapshots. They are thoroughly ana-

lyzed in the following sections.

A. Liquid phases

For the lowest packing densities η, the system forms

an isotropic liquid phase without any long-range transla-

tional or orientational ordering. An example snapshot of

this phase is shown in Fig. 8(a). In this phase, the values

of the nematic order parameter ⟨P2⟩and smectic order

parameter ⟨τ⟩are close to zero, indicating the absence

of alignment or layering. The hexatic order parameter

⟨ψ6⟩also has a minimal value (see Fig. 3), indicating a

lack of hexagonal arrangement. The radial distribution

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Splay-inducedorderinsystemsofhardwedgesPiotrKubala,MichalCie´sla,andLechLongaInstituteofTheoreticalPhysics,JagiellonianUniversityinKrak´ow,Lojasiewicza11,30-348Krak´ow,PolandThemainobjectiveofthisworkistoclarifytherolethatwedge-shapedelongatedmolecules,i.e.,moleculeswithoneendwiderthantheother,canpl...

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Splay-induced order in systems of hard wedges Piotr Kubala Micha l Cie sla and Lech Longa Institute of Theoretical Physics Jagiellonian University in Krak ow Lojasiewicza 11 30-348 Krak ow Poland

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