Alejandro
Cuetos
a,
Effran
Mirzad Rafael
b,
Daniel
Corbett
b and
Alessandro
Patti
*b
aDepartment of Physical, Chemical and Natural Systems, Pablo de Olavide University, 41013 Sevilla, Spain
bSchool of Chemical Engineering and Analytical Science, The University of Manchester, Manchester, M13 9PL, UK. E-mail: alessandro.patti@manchester.ac.uk
First published on 13th February 2019
By computer simulation, we model the phase behaviour of colloidal suspensions of board-like particles under the effect of an external field and assess the still disputed occurrence of the biaxial nematic (NB) liquid crystal phase. The external field promotes the rearrangement of the initial isotropic (I) or uniaxial nematic (NU) phase and the formation of the NB phase. In particular, very weak field strengths are sufficient to spark a direct I–NB or NU–NB phase transition at the self-dual shape, where prolate and oblate particle geometries fuse into one. By contrast, forming the NB phase at any other geometry requires stronger fields and thus reduces the energy efficiency of the phase transformation. Our simulation results show that self-dual shaped board-like particles with moderate anisotropy are able to form NB liquid crystals under the effect of a surprisingly weak external stimulus and suggest a path to exploit low-energy uniaxial-to-biaxial order switching.
Such an exclusive particle architecture, referred to as self-dual shape, was predicted to favour the formation of the biaxial nematic (NB) phase in systems of HBPs with rounded3 or square4 corners. Nevertheless, these theoretical predictions were made within the context of the Zwanzig model, which does not allow free rotations of particles and restricts their orientations to only six. Computer simulations that explored the phase behaviour of freely rotating HBPs highlighted the challenge of observing stable NB phases, even when an element of size-dispersity is incorporated.5 The very recent and insightful simulation study by the Dijkstra's group showed that monodisperse HBPs, with a geometry close or equal to the self-dual shape, are able to stabilise the NB phase only if their anisotropy is significantly large, with L* ≥ 23.6 Nevertheless, the highly uniform colloidal board-like particles synthesised by Nie and coworkers were not observed to assemble into the NB phase in a wide spectrum of aspect ratios, between L* = 20 and 180, and very close to the self-dual shape.7 A direct observation of a stable NB phase was reported almost ten years ago in dispersions of purely repulsive and quasi dual-shaped goethite particles,8 whose stability was shown to depend on the extremely large particle size dispersity.4 These computational and experimental findings unambiguously indicate that self-dual shaped HBPs cannot self-assemble into an NB phase, unless their anisotropy and/or size dispersity are especially relevant. Vroege and coworkers investigated the effect of a magnetic field on the phase behaviour of quasi-dual-shaped polydisperse goethite particles in suspension and observed a biaxial-to-uniaxial nematic transition above a certain magnetic field strength.9 To explain the origin of these experimental results, the same authors formulated a mean-field theory to calculate the phase diagram of HBPs in the presence of a magnetic field. Despite the good qualitative agreement between theory and experiments, the former neglects the effect of size polydispersity, being crucial to stabilise the NB phase,8 and is limited by the strong approximations imposed by the Zwanzig model, which was shown to be not especially accurate to describe the phase behaviour of HBPs.2,5
In this work, we perform Monte Carlo (MC) simulations of freely-rotating monodisperse HBPs in the presence of an external field that promotes a phase transition from the isotropic (I) or uniaxial nematic (NU) phase to the NB phase. Because our main interest is gaining an insight into the energetics associated to this process, the field strength is a simulation parameter that assumes values between εf* ≡ εfβ = 0.1 and 3, with β the inverse temperature. At the same time, we propose an alternative route for the formation of the NB phase that does not necessarily imply extreme anisotropies6 or particle size dispersities,8 nor the addition of depletants.10 In the following, we discuss the main aspects of the model and simulation methodology applied. The interested reader is referred to ESI† and our previous works for additional details.2,5
We have simulated fluids of HBPs with L* = 12 and 1 ≤ W* ≤ 12, including , which gives the above mentioned self-dual shape. The particle thickness, T, is our system unit length and is kept constant. The unit vectors, , ŷ, and ẑ, aligned along W, T, and L, respectively, define the particle orientation in space. Interactions are described by a hard-core potential, which only depends on the distance between pairs of particles. Consequently, an MC move is rejected if an overlap occurs and accepted otherwise. The separating axes method, introduced by Gottschalk et al.11 and later modified to study suspensions of tetragonal parallelepipeds,12 has been employed to assess the occurrence of overlaps between HBPs. The interaction between HBPs and external field is described by a potential energy term that favours the alignment of the particle intermediate axis, , with the direction of field, indicated by ê, and reads
(1) |
The external field has been applied to suspensions of HBPs that were either in dense I phases, very close to the border with the nematics, or deeply in the NU− or NU+ phases, at the packing fraction η ≡ Nv/V = 0.34, with v and V the particle and box volume, respectively. At this packing fraction, the nematic phase is stable for the whole range of particle geometries studied here. In particular, NU+ and NU− phases are observed, respectively, at 1 ≤ W* < 3.46 and 3.46 < W* ≤ 12, as shown in the phase diagram available in ref. 2. By contrast, the packing fraction of the systems in the I phase changes with W*, between η = 0.20 and 0.31. All simulations have been run in the canonical ensemble, where number of particles, volume and temperature are kept constant. After equilibrating a number of I and NU phases, the field was switched on and new equilibrium states were achieved.
To identify the long-range order of the phases at the initial and final equilibrium states, we calculated the nematic order parameter and nematic director coupled to each of the three particle axes. In particular, we applied the standard procedure of diagonalising the following traceless symmetric second-rank tensor22
(2) |
In the light of these preliminary considerations, we now discuss our simulation results, which depend on three main factors: (i) the initial state of the system, being either isotropic or uniaxial nematic, (ii) the field strength, εf, and (iii) the particle geometry, which only depends on W*. In Fig. 1 and 2, we show the phases observed, in the new equilibrium state, when a field of strength 0.1 ≤ εf* ≤ 3 is applied to, respectively, I or NU phases of HBPs with 1 ≤ W* ≤ 12. The values of the uniaxial and biaxial nematic order parameters that validate these results are provided in the ESI.† The state points at εf = 0 are included as a reference and identify I phases in Fig. 1, and NU+ (W* < 3.46) or NU− (W* > 3.46) phases in Fig. 2. The criterion adopted in both figures to classify the new equilibrium phases essentially depends on the value of the uniaxial and biaxial order parameters and is summarised in Table 1.
Fig. 1 Equilibrium LCs resulting from the application of the external field given in eqn (1), with strength εf, to an isotropic phase of HBPs of length-to-thickness ratio L* = 12 and width-to-thickness ratio 1 ≤ W* ≤ 12. Packing fractions are in between η = 0.20 and 0.31 (see ESI† for details). Circles, empty and solid squares, empty and solid triangles, and diamonds indicate, respectively, I, NU+, NU−, NB+, NB−, and NB phases. Shaded areas are guides for the eye. The vertical dashed line refers to the self-dual shape, where . |
Fig. 2 Equilibrium LCs resulting from the application of the external field given in eqn (1), with strength εf, to uniaxial nematic phases (η = 0.34) of HBPs of length-to-thickness ratio L* = 12 and width-to-thickness ratio 1 ≤ W* ≤ 12. Circles, empty and solid squares, empty and solid triangles, and diamonds indicate, respectively, I, NU+, NU−, NB+, NB−, and NB phases. Shaded areas are guides for the eye. The vertical dashed line refers to the self-dual shape, where . |
S 2,L | S 2,T | S 2,W | B 2,L or B2,T | Phase | |
---|---|---|---|---|---|
a S 2,T and/or S2,W should be in the range specified. b S 2,L and/or S2,W should be in the range specified. | |||||
0.00–0.20 | 0.00–0.20 | 0.00–0.20 | — | I | |
0.40–1.00 | 0.00–0.35 | 0.00–0.35 | 0.00–0.30 | NU+ | |
0.00–0.35 | 0.40–1.00 | 0.00–0.35 | 0.00–0.30 | NU− | |
0.40–1.00 | 0.35–1.00 | 0.35–1.00 | 0.20–0.35 | NB+ | |
0.35–1.00 | 0.40–1.00 | 0.35–1.00 | 0.20–0.35 | NB− | |
0.40–1.00 | 0.40–1.00 | 0.40–1.00 | 0.35–1.00 | NB |
According to this criterion, two weak biaxial phases, similar to those previously found by the Patras group in suspensions of spheroplatelets,28–30 have been identified. To distinguish weak and strong biaxial phases, we look at the magnitude of the biaxial order parameter that is associated to the largest uniaxial order parameter. If the latter is, for example, S2,L, then the former is B2,L. In a strong biaxial phase, simply labelled as NB, the HBPs are highly orientationally ordered, with their three unit vectors , ŷ and ẑ oriented along three mutually perpendicular directions. This significant degree of orientational order is reflected in the large values of all the uniaxial and biaxial order parameters. By contrast, the weak biaxial phases, labelled as NB+ and NB−, show a modest, but not negligible biaxial order parameter, between 0.20 and 0.35, and at least two pronounced uniaxial order parameters, above 0.40, which give them a prolate or oblate character, respectively.
An external field with strength εf* < 1 is able to reorient prolate and oblate HBPs along a common direction to form NU+ and NU− phases, respectively (see Fig. 1). Interestingly enough, at the self-dual shape, even a very gentle field is sufficient to spark a direct I–NB− transition and then, at εf* = 0.3, the formation of the NB phase, which becomes dominant at larger field strengths across the complete set of particle geometries. Similar tendencies are noticed when the same field is applied to a uniaxial nematic phase. In this case, however, the I–NB phase transition is detected at an almost insignificant field strength, εf* = 0.1, with no intermediate formation of weak biaxial phases (see Fig. 2). Snapshots showing the phase order before and after the application of a relatively weak field to I and NU phases are shown in Fig. 3.
Therefore, regardless of the initial system configuration, the dual-shape is found to be the optimal particle geometry to form the NB phase in terms of energy costs as the strength of the field applied at is at its minimum. In spite of being relatively low, such energy costs are crucial to achieve the desired orientational order with no increase of the translational order, which remains negligible and would otherwise imply the onset of biaxial smectic, rather than nematic, LCs. As a matter of fact, the calculation of the pair distribution functions along the nematic director, g‖(r), excludes the formation of positionally ordered phases (see ESI†). At particle geometries that are different, but still relatively similar, to the self-dual shape, stabilising the NB phase becomes more energetically demanding, whether the initial phase is I or NU. For instance, an I phase incorporating HBPs with W* = 4 requires a field strength of approximately εf* = 1. At larger anisotropies, stronger fields are required, up to εf* = 1.5 at W* = 12. As expected, the energy to reorient HBPs originally in uniaxial nematics into biaxial nematics is sensibly lower. At W* = 4, this is approximately 50% of that needed to align them in the isotropic phase.
In summary, our results suggest that it is indeed possible to observe biaxial nematics in colloidal suspensions of monodisperse HBPs. To this end, one needs to apply an external field whose strength is comparable to the particle thermal energy. In particular, the self-dual shape is the most appropriate particle geometry for energy-efficient I–NB and NU–NB phase transitions. Although these conclusions agree well with past theoretical works,3,4,27,31de facto they indicate that this agreement is the consequence of restricting particle orientations and/or neglecting the occurrence of positionally ordered phases, which artificially enhance the stability of the NB phase. Mere excluded volume effects were shown to be insufficient to observe biaxial nematics in monodisperse systems of HBPs,2 unless extremely long particles are employed.6 Applying an external field to monodisperse freely-rotating HBPs produces a scenario that the above mentioned theories, despite their strong assumptions, had predicted: the NB phase can be stabilised at moderate particle anisotropies and a direct I–NB transition, so far only detected experimentally in systems of significantly size dispersed goethite particles,8 observed. The external field induces the desired alignment of particles with no effect on their positional order an at a relatively low energy cost. While promoting the particle alignment along one direction, this field does not prevent them from rotating freely, especially around the remaining two directions. By gradually increasing the field strength, the particles, regardless their geometry, assume a more and more narrow distribution of orientations, which eventually end up fluctuating around three main orthogonal directions, being the signature of the formation of the NB phase.
Footnote |
† Electronic supplementary information (ESI) available: Simulation details; pair distribution functions of a biaxial nematic phase; uniaxial and biaxial order parameters. See DOI: 10.1039/c8sm02283f |
This journal is © The Royal Society of Chemistry 2019 |