Poshika
Gandhi
* and
Anja
Kuhnhold
Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany. E-mail: poshika.gandhi@physik.uni-freiburg.de; anja.kuhnhold@physik.uni-freiburg.de
First published on 17th June 2024
We study the phase behavior of twisted convex n-prisms with n = 3 and 4, via Monte Carlo simulations. Biaxial phases, in untwisted prisms, can be induced by choosing specific geometries of the prisms. However, due to the convexity of the twisted particles, a strong twisting disables the formation of biaxial phases and stabilizes uniaxial nematic and smectic phases. Using the increased volume of the twisted convex particles, we define an effective aspect ratio of the twisted prisms and find a homogeneous phase behavior across the geometry of the prisms’ cross-section and even across different shapes of the cross-section. In this representation biaxial phases are found for large aspect ratios, while the low aspect ratio behavior can be compared to the hard cylinder phase diagram. For 3-prisms with a small base angle, we show the influence of the initial configuration; a polar initial configuration results in a (polar) splay nematic phase, whereas a non-polar initial configuration results in a biaxial phase.
The studied prisms are biaxial particles defined by having a long axis and two shorter axes. These particles have competing rod-like and plate-like tendencies which lead to the formation of nematic phases either along the long or a short axis.23 However along Straley's line, where the shape parameters (long axis length u, short axes lengths v and w) fulfill the relation (self-duality),24 the particles are expected to form biaxial phases, as Straley's line is the division between the rod-like and plate-like ordering. This predicted behavior has been observed for a variety of biaxial particle shapes.25–29 Similarly for the n-prisms studied here, the prolate nematic phase is expected for
and the oblate nematic phase for
.
Board-like particles (4-prisms), especially, received attention in theoretical, simulational, and experimental works due to the debates about the stability of their biaxial nematic phases. Cuetos et al. confirmed the division of prolate and oblate nematic phases along Straley's line but could not find a biaxial nematic phase (neither from their theoretical approach nor by simulations).26 They concluded that the formation of a (biaxial) smectic phase preempts the formation of the biaxial nematic. On the other hand, there are experimental realizations of a biaxial nematic phase of self-dual particles that, in addition, are polydisperse.30 Van den Pol et al. also suggested that in this case the polydispersity has a stabilizing effect on the biaxial nematic phase. This effect was theoretically confirmed by Belli et al.31 The same authors also studied the effect of adding depletants to a system of board-like particles, which can result in stabilizing prolate, oblate, or biaxial nematic phases.32 Peroukidis and Vanakaras showed that not only the duality of the particles is important for the formation of biaxial phases but they found a critical aspect ratio (u/w = 9) below which the biaxial phase vanishes even for dual-shaped particles.27 This confirmed the predictions of Shih and Alben, who predicted that the biaxial particle needs to be “sufficiently asymmetrical” to produce biaxial phases23 and also explains previous results, e.g. by Martnez-Ratón et al., who applied fundamental measure theory to a system of hard board-like particles.33
A 3-prism, that has a triangular cross-section, in addition to being biaxial, is also a polar shape, i.e. it does not have the inversion symmetry. From an application point of view, it is interesting to know whether this polarity can be transferred to the macroscopic (phase) behavior. The ferroelectricity of splay nematics is a typical example that has been studied recently.34–36 But also, the effect of the shape of the prism on the behavior of the seemingly well-known isotropic, nematic, and smectic phases found interest.37
Here, we present phase diagrams of 3- and 4-prisms with convex twist and varying shape parameters. Biaxial phases are found for specific sets of parameters and both types of cross-sections, whereas the formation of a polar phase strongly depends on the initial configuration.
Each particle is overlaid with an orthonormal coordinate system to define its orientation in three-dimensional (3D) space as shown in Fig. 1. The vector defines the long axis of the particle (with length vector
=: uû) and the vectors
and
(=: wŵ) together define the cross-sectional n-gon. For the 3-prism, v and w are the base length and height of the triangle, and for the 4-prism, v and w are the width and height of the rectangle. As in the limit of infinite n the n-prisms form a cylinder, we can write the volume of an n-prism with twist θ in terms of an effective cylinder of diameter deff and length u as,
![]() | (1) |
![]() | (2) |
The volume of a twisted n-prism, Vθ, is calculated by considering it as a combination of tetrahedrons. Each face of the n-prism forms the base of the tetrahedron and the center of the prism forms the apex. The center of the prism lies at the halfway point of the line joining the n-gon centers. For a 3-prism, the centroid is considered to be the center and for a 4-prism, the intersection of the diagonals is considered as the center. As all the vertices, including the apex, are known, the volume of the n-prism can be written as,
![]() | (3) |
For the initial configuration the particles are put on a simple cubic lattice and have their long axes aligned. As 3-prisms can form polar phases, we tested both polar (ŵ aligned) and non-polar initial conditions. The polar initial configuration mimics the effect of a temporary aligning field that could be used in applications to bias the formation of polar phases. During the equilibration, the maximum displacement and maximum rotation angle are adjusted to give acceptance rates between 0.45 and 0.55. Translations along the long axis have a larger step size compared to those in other directions to speed up the equilibration and the exploration of configuration space. For all simulations, the parameters u and v have fixed values of 5.0 and 1.0 respectively.
The equilibrium phases are identified using orientational order parameters, positional density data, and visual inspection of the snapshots. The nematic order parameter Sx for x ∈ {û, , ŵ} is defined as the largest eigenvalue of the tensor Qx, whose components are given by
![]() | (4) |
For each molecular axis x ∈ {û, , ŵ}, a biaxial order parameter can be defined using the three tensors (Qu, Qv, Qw) and their eigenvectors. However, it is sufficient to measure the biaxiality with respect to a primary axis only. The eigenvector corresponding to the largest of the three nematic order parameters Sx is chosen as the primary axis
x. The eigenvectors corresponding to the remaining nematic order parameters are labelled as
and
. For example, if the particles are primarily aligned along their long axis û, then the primary axis
is the nematic director regarding û, and the nematic directors w.r.t the molecular axes
and ŵ form the axes
and
. The biaxial order parameter Bp corresponding to this primary axis can be defined as,26,39
![]() | (5) |
![]() | (6) |
A system is labeled ‘equilibrated’ when the order parameters fluctuate around a mean value for at least 105 MC sweeps. For splay nematic phases, where the effect of the initial configuration is found, the order parameters are required to be stable for at least 5 × 105 MC sweeps. The positional heat maps and visual inspections of the snapshots are used to check whether the smectic planes or the splay fault lines are still moving within the simulation box, in which cases the runs are continued even when the order parameters have stabilised.
Depending on the nematic order parameters Sx for x ∈ {û, , ŵ}, the biaxial order parameters Bp, the smectic order parameter Λ, and the heat maps for particle density, we identify the following phases: isotropic (I): random orientation of all particle axes (0.0 < Sx < 0.3); prolate nematic (Nu): alignment of the long û-axes (Su > 0.3, Sw < 0.3, Bu < 0.5); oblate nematic (Nw): alignment of the short ŵ-axes (Sw > 0.3, Su < 0.3, Bw < 0.5); biaxial nematic (Nb): alignment of the long and short axes (Sx > 0.3, Bp > 0.5, Λ < 0.5); smectic A (Smu): alignment of the long axes plus layering along the director
u (Λ > 0.5); biaxial smectic (Smb): alignment of the long and short axes plus layering along the directors
u and
w (Bp > 0.5, Λ > 0.5).
Fig. 2 (top) shows the phase diagram of 3-prisms with β = 1.0 rad (≈57.3°) and . For zero twist, they form the I phase at low volume fractions and the Nu phase for all studied volume fractions above η = 0.30, as expected from previous studies about triangular prisms.37 As θ is increased, the range of the I phase increases. For Fig. 2 (bottom), the zero twist configuration for β = 0.45 rad (≈25.8°), with
shows a small I and Nw region for low volume densities and a Nb phase with
as the primary axis for η ≥ 0.2, which is again in agreement with previous results, where a transition Nw–Nb was found above a threshold of u/w = 11.29 The tendency to align along the long axis becomes apparent as θ is increased, and the Nb phase is lost. For 3-prisms, the Nb phase was also observed for particles with smaller β where
.
Fig. 3 shows the phase diagrams of 4-prisms in the η − θ representation. For 4-prisms with w = 1.0 and 0.5, as , the Nu phase is expected. This is confirmed for w = 1.0 in Fig. 3 (left). The result is comparable to the phase behavior of hard colloidal tetragonal parallelepipeds studied by John and Escobedo.28 However, they conclude that the prolate nematic phase only becomes stable for u/v > 5 while we already find it for u/v = 5. For volume fractions larger than the ones we studied, they find a smectic phase; this appears in our volume fraction range for non-zero twist. As θ is increased, the Nu phase region shrinks with a direct I − Smu transition appearing for θ = 0.6. For w = 0.5 in Fig. 3 (center), the Smu phase is observed for θ = 0 and η ≥ 0.45. As seen before, the increasing θ shrinks the Nu phase to allude to a direct I − Smu for θs larger than considered for this paper. For 4-prisms with w = 0.2 (Fig. 3 (right)), the board-like particles at zero twist show I, Nw, Nb, and Smb phases. As θ increases the uniaxial Smu phase appears between the Nb and the Smb phase. On further increase of θ, the Nw and lower-η Nb phases turn into Nu, and Smb disappears from the parameter space explored. The Nb phase was obtained for the zero twist dual shape with u/w = 25 and v/w = 5, which is in agreement with the predictions that u/w > 23 is needed for the Nb phase.29 However, ref. 29 also predicts an intervening Smu phase for u/w < 28, whereas we observe a direct Nb − Smb transition for u/w = 25, as shown in Fig. 3 (right). Thin board-like particles showed a stable Smw phase in ref. 26 however we did not find traces of such a phase.
For both 3- and 4-prisms, a direct I − Nb transition was not found in agreement with several studies,26,29 but in contrast to the thin board-like spheroplatelet particles that showed a I − Nb − Smu transition along the dual shape.27,41 In ref. 42 a direct I − Smu transition was found for 4-prisms with a square cross-section and not a Nu phase. This is in contrast to our simulations (Fig. 3 (left)).
Twisting distorts the sides of the n-prism in order to accommodate the rotation, making the shape no longer conducive for stacking. The increase in the particle volume and decrease in the aspect ratio with θ further alludes to the fact that the particle shape is no longer capable of sustaining oblate and biaxial nematic phases and shows more rod-like tendencies than plate-like (in terms of the distinction using Straley's line). The shift of the phase boundaries towards larger η is, hence, a consequence of the decreasing aspect ratio of the particle.
The n-prisms with a concave twist were reported to form several chiral nematic phases with periodic boundary conditions in ref. 22. In general, it is not trivial to simulate chiral nematic phases because the cholesteric pitch can be much larger than the simulation box length, and periodic boundaries suppress the formation of a chiral nematic phase with equilibrium pitch. Several approaches have been introduced to tackle this problem, e.g. twisted or self-determined boundary conditions or using hard walls.43–45 The convex twist has however, for the simulated parameters and volume fractions, been incapable of producing chiral nematic phases, both for periodic boundaries and hard walls. As compared to the concave twisting, nematic phases also appear at much larger volume fractions. This can be due to the fact that convex twisting increases the volume of the particle significantly, thus lowering its aspect ratio. As the defined aspect ratio of an n-prism is seemingly independent of the shape of the particle, we plotted the results of all the 3-prisms together in Fig. 4a, and the results of all the 4-prisms together in Fig. 4b. The grey lines in both the graphs correspond to θ = 0 for the particle shape shown above it. The sizes of the 3- and 4-prisms are characteristic of their actual sizes in the simulations in relation to other 3- and 4-prisms. As θ increases, the aspect ratio αθ decreases and the data points lie to the left of the grey line. Therefore, overlaps in the data between twisted n-prisms and the untwisted n-prisms with a different cross-section occur frequently. These graphs show strong correlations between the stable phases at a particular volume fraction with the aspect ratio without much dependency on the particle shape. The regions of transition between the phases follow consistent trends, even when the exact transition lines seem to depend on the particle shape. Next, in Fig. 5 the data for both 3- and 4-prisms is plotted together, and the phase diagrams coincide surprisingly well. This implies that the volume occupied by a particle is a bigger determining factor in the entropic stability of the resulting phase as compared to the particle's shape. The observed stable phases around αθ = 10, however, vary between the I, Nu phases found for n-prisms with or large θ, and the I, Nw, Nb phases found for n-prisms with
. This can be due to the prediction23 that the particles need to be “sufficiently asymmetrical” to produce biaxial phases. In addition, the regions of the I, Nu, and Smu phases agree with the ones of hard cylinders.16
![]() | ||
Fig. 5 Phase diagrams of both 3-prisms and 4-prisms plotted together as a function of their aspect ratio with a sketch of the size comparison between the particle shapes. |
Just like the chiral nematic phases, the splay nematic phase22 was also not obtained in our simulations. All simulations for 3-prisms were setup with non-polar initial configurations, however introducing polarity at the setup leads to the emergence of the splay nematic phase (NS), see Fig. 6 (bottom). This phase is characterized by a nematic order of the long axes and a splay deformation in the director field of the ŵ-axes. Fig. 6 shows the effect of a polar initial configuration on the phase diagram, and Fig. 7 shows a snapshot of the phase along with the characteristic director profiles. For non-polar initial configurations, the region with β < 0.6 rad (≈34.3°) and η ≥ 0.20 mainly forms the biaxial nematic phase (Nb). However, for polar initial configurations, the splay nematic phase dominates. In addition, the Nu phase emerges at a lower β for a polar initial configuration. The NS phase stays stable till at least θ = 0.2 rad (≈11.4°), beyond which the simulations at suitable volume fractions could not be performed. We did not compute the free energy of the different states to be able to judge which of the modulated nematic phases is the stable equilibrium phase, but it is likely that the barrier between them is very large, and thus a transition very unlikely, which in turn is a relevant finding for possible applications of polar phases. There are further hints that a double splay phase might have a lower free energy than the single splay phase, which has been reported more commonly in both experiments and simulation studies.46 For the rest of the phase diagram, the simulated parameters produced phases independent of the initial configuration.
For the dual shapes, the Nb phase was found for 3-prisms with u/w > 20 and for 4-prisms with u/w = 25, and was stable across a large range of volume fractions. No direct I − Nb transition was found, rather a small region of Nw phase preceded the Nb. For thin board-like 4-prisms, a direct Nb − Smb phase transition was observed without an intervening Smu phase and the existence of a Smw phase could not be confirmed in our simulations. In addition, for all simulated shapes no direct I − Smu was found for untwisted shapes.
Remarkably, we found an almost uniform phase behavior by introducing an effective aspect ratio of twisted n-prisms. Convex twisting leads to an increasing volume of the particles, which in turn reduces their effective aspect ratio. The phases in the volume fraction – effective aspect ratio diagram overlap independent of the geometry (shape and side lengths) of the prisms' cross-section.
For particle shapes with inherent polarity, the initial configuration of the setup played a significant role in the phase they formed. A polar splay nematic phase was formed when the starting configuration was also polar. The NS was stable over large ranges of η and β, and opens opportunities to form polar phases by just using an initial polar configuration.
For larger volume fractions than studied in this paper, one could compare the phase behavior to the one of the corresponding two-dimensional system (hard triangles for n = 3 and hard rectangles for n = 4),47–50 and determine the effect of the third dimension and of the convex (or concave) twist.
The overlap check starts with the face normals, as the process with them is quicker for objects that are further away from each other. If all face normals fail as separating axis then the edge–edge overlap check is performed. All the directors (face normals and cross products) are unit vectors which are placed on an arbitrary vertex of the face or the edge in question. For edge–edge overlap, this arbitrary placement of the director requires a high floating-point precision in calculating the projections. To reduce the necessary precision, we modified the placement of the director and placed it at the point of the shortest distance between the edges. In this way, the required precision is lowered because instead of comparing two floating-point numbers, we only check if the distance between the projection of the vertices and the point of shortest distance is more or less than zero and it outweighs the additional task of finding the shortest distance between the edges. As shown in Fig. 8 (bottom), the measurement process goes from calculating the distance of the blue projection from both the red projections, to just calculating the distance from the black projection. Apart from this modification, the rest of the algorithm given in ref. 38 has been used as-is.
Footnote |
† Note that this cannot be a full list of all related references, as they are too numerous. |
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