# Dynamics of Colloidal Cubes and Cuboids in Cylindrical Nanopores

Alessandro Patti<EMAIL_ADDRESS>Department of Chemical
Engineering and Analytical Science, The University of Manchester, Manchester
M13 9PL, United Kingdom  Alejandro Cuetos  Department of Physical, Chemical
and Natural Systems, Pablo de Olavide University, 41013 Seville, Spain

###### Abstract

Understanding how colloidal suspensions behave in confined environments has a
striking relevance in practical applications. Despite the fact that the
behaviour of colloids in the bulk is key to identify the main elements
affecting their equilibrium and dynamics, it is only by studying their
response under confinement that one can ponder the use of colloids in
formulation technology. In particular, confining fluids of anisotropic
particles in nanopores provides the opportunity to control their phase
behaviour and stabilise a spectrum of morphologies that cannot form in the
bulk. By properly selecting pore geometry, particle architecture and system
packing, it is possible to tune thermodynamic, structural and dynamical
properties for ad hoc applications. In the present contribution, we report
Grand Canonical and Dynamic Monte Carlo simulations of suspensions of
colloidal cubes and cuboids constrained into cylindrical nanopores of
different size. We first study their phase behaviour, calculate the chemical
potential vs density equation of state and characterise the effect of the pore
walls on particle anchoring and layering. In particular, at large enough
concentrations, we observe the formation of concentric nematic-like coronas of
oblate or prolate particles surrounding an isotropic core, whose features
resemble those typically detected in the bulk. We then analyse the main
characteristics of their dynamics and discover that these are dramatically
determined by the ability of particles to diffuse in the longitudinal and
radial direction of the nanopore.

††preprint: AIP/123-QED

##  I INTRODUCTION

The study of the behaviour of simple and complex fluids constrained within
nanoscale environments is referred to as nanofluidics. Fluids confined in such
small media, which for instance include synthetic nanowires as well as natural
nanoporous networks, exhibit thermodynamic, structural and dynamical
properties that deviate from those measured in the bulk. In case of nanopore-
confined colloids, these deviations are especially significant when the size
of the suspended particles or macromolecules is comparable to the size of the
hosting nanopore. Experimental Pieranski (1980); Neser  _et al._ (1997);
Cohen, Mason, and Weitz (2004); Cortes  _et al._ (2016); Martínez-Pedrero  _et
al._ (2021), theoretical Rodríguez-Ponce, Romero-Enrique, and Rull (2001);
Yao, Zhang, and Chen (2018); Mizani  _et al._ (2019); Yao and Chen (2020);
Behzadi, Ghazi, and Aliabadi (2021); Teixeira, Anquetil-Deck, and Cleaver
(2021) and simulation Avendaño, Liddell Watson, and Escobedo (2013); Patti
(2013); Muangnapoh  _et al._ (2014); Workineh and Vanakaras (2016); Avendaño,
Jackson, and Wensink (2018); Luo  _et al._ (2019); Gârlea  _et al._ (2019);
Anquetil-Deck, Cleaver, and Teixeira (2020); Anzivino  _et al._ (2021);
Donaldson, Schall, and Rossi (2021) studies on confined colloidal suspensions
have shown the existence of a rich realm of nanostructures that are not
observed in the bulk. Additionally, depending on the adhesive interactions
established between the fluid and the pore walls and their relevance with
respect to the cohesive interactions between fluid particles, several new
phenomena, such as wetting, capillary condensation and evaporation, layering,
anchoring, are detected Dijkstra, Roij, and Evans (2001); Dijkstra (2004);
Martínez-Ratón (2007); Brumby  _et al._ (2017); Salgado-Blanco, Díaz-Herrera,
and Mendoza (2019); Basurto  _et al._ (2020). If inter-particle and wall-
particle interactions are determined by mere excluded-volume effects, with no
enthalpy-driven forces being involved, the behaviour of the confined fluid is
exclusively determined by geometric details, which include pore and particle
architecture as well as relative particle-to-pore size.

Due to their ability to form nanostructures that are especially relevant in
the design of nanomaterials, anisotropic particles are receiving more and more
attention, especially since new accurate experimental techniques for their
synthesis have become available Sun and Xia (2002); Manoharan, Elsesser, and
Pine (2003); Shankar  _et al._ (2004). In particular, cuboids are especially
interesting particles that can self-assemble into a wide spectrum of liquid
crystal (LC) phases John and Escobedo (2005); John, Juhlin, and Escobedo
(2008); Cuetos  _et al._ (2017); Patti and Cuetos (2018); Yang  _et al._
(2018), including the elusive biaxial nematic phase when some conditions (e.g.
size dispersity, extreme particle anisotropy, application of an external
field) are met Belli  _et al._ (2011); Belli, Dijkstra, and van Roij (2012);
Leferink op Reinink  _et al._ (2014); Dussi  _et al._ (2018); Cuetos  _et al._
(2019); Mirzad Rafael  _et al._ (2021). Under spherical confinement,
experiments and simulations showed that cubes can form face-centered-cubic,
hexagonal close-packed or simple-cubic phases depending on the degree of the
particle sharpness Wang  _et al._ (2018). By applying Onsager second-order
virial theory, Velasco and co-workers studied the effect of constraining
perfectly aligned hard board-like particles (HBPs) within two parallel planar
hard walls and observed suppression of the nematic-to-smectic transition,
usually observed in the bulk, and the occurrence of biaxial phases Varga,
Martínez-Ratón, and Velasco (2010). A similar study was performed by Mizani et
al, who applied Parsons-Lee density functional theory within the restricted-
orientation Zwanzig model to study the orientational ordering and layering of
HBPs in slit pores Mizani  _et al._ (2019). These authors observed that the
hard wall-particle interaction promotes the occurrence of uniaxial nematic
order with strong adsorption at the walls. The behaviour of freely-rotating
hard cubes confined in between parallel hard walls has been studied by Monte
Carlo simulations Khadilkar and Escobedo (2016). In this case, the authors
varied the distance between the walls to accommodate between one and five
layers of particles, in order to assess what phase transitions occur as the
system goes from a quasi-2D geometry to a quasi-3D bulk behavior. Studies of
cubes or cuboids under cylindrical confinement are significantly less common.
Aggregation of silver cubes in cylindrical nanopores of alumina membranes has
been investigated experimentally to establish the effect of the substrate on
the cubes plasmonic sensitivity, which was found to be more than one order of
magnitude larger than that observed on planar solid substrates König  _et al._
(2014). Very recent experiments have assessed the impact of cylindrical
confinement on the ability of cellulose nanocrystals (CNCs), whose rod-like
shape reminds that of the prolate HBPs studied here, of forming cholesteric
(Ch) liquid crystals Prince  _et al._ (2021). The authors observed that
nanoconfined CNCs organise into a core-shell structure, with an isotropic
core, running parallel to the longitudinal axis of the cylindrical capillary,
surrounded by a Ch shell. We will see that similar core-shell morphologies
have also been observed in our simulations.

The purpose of the present article is studying the behaviour of colloidal
cuboids confined in cylindrical nanopores. More specifically, we investigate
equilibrium, structural and dynamical properties of perfect hard cubes as well
as oblate and prolate HBPs constrained in cylindrical nanopores of different
diameters. There are two main reasons why we opted for hard-core potentials.
To the best of our knowledge, there are no available soft potentials that
could be employed to efficiently estimate the minimum distance between pairs
of cubes or cuboids, and the qualitative insight that underpins the physics of
these systems would not be significantly affected by the specific choice of
the repulsive potential. This has been observed by comparing the phase
behaviour of hard spherocylinders Bolhuis and Frenkel (1997) with soft
repulsive oblate Cuetos and Martínez-Haya (2008); Martínez-Haya and Cuetos
(2009) and prolate Cuetos and Martínez-Haya (2015) spherocylinders or with
spherocylindrical particles with Kihara attractive interactions Cuetos  _et
al._ (2003). To determine the system equilibrium properties, we perform Grand
Canonical Monte Carlo (GCMC) simulations, whereas the study of the dynamics is
carried out by Dynamic Monte Carlo (DMC) simulations. We show that the
orientation of particles within the nanopore is not uniform, but changes with
its radial distance from the pore wall, and determines their mobility at both
short and long-time scales. The present article is organised as follows. In
Section II, we describe the model adopted to mimic colloidal cuboids and the
main GCMC and DMC simulation details applied to equilibrate the systems, study
their thermodynamic and structural properties and produce their time
trajectories. Results, which include the chemical potential vs density
equations of state, structural correlation functions and order parameters,
mean square displacements and the self-part of the van Hove correlation
functions are presented and discussed in Section III. Finally, in Section IV,
we draw our conclusions.

##  II METHODS

###  II.1 Model

Colloidal cuboids are modelled as HBPs of thickness $T$, length $L$, and width
$W$. For the particular case of hard cubes (HCs), $T=W=L$. We have set the
thickness $T_{p}$ of prolate HPBs as the system unit length. The dimensions of
HCs and HBPs have been set in such a way that the particle volume,
$v_{0}=LWT$, is the same for the three geometries. Details are given in Table
1. All particles interact via a hard-core potential with each other and with
the pore walls. As such, they can freely displace and rotate, with no
restrictions on mutual distances and orientations as long as no overlaps are
produced. The occurrence of overlaps has been assessed via the separating axes
method described by Gottschalk and coworkers Gottschalk, Lin, and Manocha
(1996) and later adapted by John and Escobedo to determine the phase and
aggregation behaviour of tetragonal parallelepipeds John and Escobedo (2005).

Table 1: Reduced length ($L^{\star}\equiv L/T_{p}$), width ($W^{\star}\equiv W/T_{p}$), thickness ($T^{\star}\equiv T/T_{p}$) and volume ($v^{\star}_{0}\equiv v_{0}/T_{p}^{3}$) of the three sets of particles studied in this work. | Perfect Cube | Oblate Cuboid | Prolate Cuboid
---|---|---|---
$L^{\star}$ | $\sqrt[3]{9}$ | 3 | 9
$W^{\star}$ | $\sqrt[3]{9}$ | 3 | 1
$T^{\star}$ | $\sqrt[3]{9}$ | 1 | 1
$v^{\star}_{0}$ | 9 | 9 | 9

###  II.2 Simulation Methodology

Simulations comprised equilibration and production runs. Equilibration runs
consisted of standard MC simulations that were performed in the grand
canonical ensemble at constant pore volume, $V$, and chemical potential,
$\beta\mu$, with $\beta$ the energy unit Frenkel and Smit (2001). We then
computed the excess chemical potential from the ensemble density $\rho=\langle
N\rangle/V$, with $\langle N\rangle$ the average number of particles in the
nanopore, through the equation:

$\beta\mu^{\rm ex}=\beta\mu-\ln\rho.$ (1)

GCMC simulations were also employed to determine the thermodynamic and
structural properties of the confined fluids of HCs and HBPs. By contrast, to
study their dynamics, DMC simulations were performed in the canonical ensemble
at constant $V$ and number of particles, $N$. In all the cases, the particles
have been confined in cylindrical nanopores with periodic boundaries in their
longitudinal direction. Radius and length of the nanopore have been set
depending on the dimensions of the three families of particles studied here.
More specifically, HCs have been arranged in nanopores of radius
$R^{\star}\equiv R/T_{p}=\sqrt[3]{9}\times\\{3,6,9\\}$, while oblate and
prolate HBPs in nanopores of radius $R^{\star}=\\{9,12,15\\}$ and
$R^{\star}=\\{12,15,22.5\\}$, respectively. By contrast, the nanopore length,
$h>>L$, has been set to observe a statistically relevant number of particles
at the smallest chemical potentials. An exemplary configuration of a nanopore
hosting HCs is displayed in Fig. 1 for $\beta\mu^{ex}=5.011$. Specific details
of equilibration and production runs are provided below.

Figure 1: Longitudinal (left) and frontal (right) views of an exemplary
equilibrium configuration obtained at $\beta\mu^{ex}=5.011$ of systems
containing HCs in cylindrical nanopores of radius
$R^{\star}=9\times\sqrt[3]{9}$. Colour gradient indicates different particle
orientations. In particular, particles in blue have their reference unit
vector oriented along the pore longitudinal axis, while particles in red or
green have their reference unit vector oriented perpendicularly to this axis.

####  II.2.1 Equilibration

Initial configurations consisted of randomly positioned, but perfectly aligned
HBPs with their main axis oriented along the longitudinal direction of the
nanopore. In some cases, equilibrium configurations were employed as the
initial configurations of systems at larger chemical potentials. Equilibration
was considered accomplished when $N$ or, equivalently, the packing fraction
$\eta\equiv Nv_{0}/V$ achieved a stationary value within reasonable
statistical fluctuations. We also computed the total nematic order parameters,
which usually take longer to relax as compared to the packing fraction. To
this end, we applied the standard procedure of diagonalising the following
traceless symmetric second-rank tensorEppenga and Frenkel (1984)

${\bf
Q}^{\lambda\lambda}=\frac{1}{2N}\Big{\langle}\sum_{i=1}^{N}\left(3\hat{\lambda}_{i}\cdot\hat{\lambda}_{i}-{\bf
I}\right)\Big{\rangle}$ (2)

where $i$ indicates a generic particle, ${\bf I}$ the second-rank unit tensor
and $\hat{\lambda}_{i}=\hat{T}_{i}$, $\hat{W}_{i}$, $\hat{L}_{i}$ the unit
vector aligned with the particle thickness, width and length, respectively.
The eigenvalues of ${\bf Q^{\lambda\lambda}}$ and their relative eigenvectors
correspond, respectively, to the order parameters ($S_{2,W}$, $S_{2,T}$,
$S_{2,L}$) and nematic directors ($\hat{m}$, $\hat{p}$, $\hat{n}$) coupled to
each particle axis. In the isotropic phase, the order parameters are all close
to zero, whereas if particles tend to align, then their preferential
orientation is identified by the largest positive eigenvalue and its
associated eigenvector. The interested reader is referred to our previous
works for additional details Cuetos  _et al._ (2017); Patti and Cuetos (2018);
Cuetos  _et al._ (2019).

####  II.2.2 Production

Once the systems achieved equilibrium, we investigated their structural and
dynamical properties by performing, respectively, GCMC and DMC simulations.
The initial configurations for the production runs were those equilibrated in
the preliminary equilibration step. In particular, in the GCMC production
simulations the chemical potential was the same as that imposed in the
equilibration run, while in DMC simulations, performed in the canonical
ensemble, we set the number density obtained in the equilibration step. To
study the impact of confinement, the pore was divided in concentric
cylindrical shells. The fluid density in these shells is not homogeneous, but
exhibits significant fluctuations in the vicinity of the pore walls. These
fluctuations depend on the radial distance from the longitudinal axis, as
exemplary shown in Fig. 2, where we report the numerical density along the
nanopore radial direction, normalised with that measured in the bulk at the
same chemical potential, for HCs (left frame) and oblate (middle frame) and
prolate (right frame) HBPs. We observe that, depending on the geometry of
particles and their ability to effectively pack at the pore wall, the density
distribution profiles change. In particular, the packing is especially
effective for cubes that can best adapt to the pore wall’s curvature, but
significantly less effective for rod-like HBPs.

Figure 2: Normalised density, $\rho^{\star}\equiv\rho/\rho_{\rm bulk}$, of HCs
(left), oblate HBPs (middle) and prolate HBPs (right) as a function of the
reduced distance, $r^{\star}\equiv r/T_{p}$, from the nanopore longitudinal
axis. The curves in each of the three frames indicate a different value of the
fugacity $f^{\star}=\exp(\beta\mu)$. Nanopore radius from left to right:
$R^{\star}=9\times\sqrt[3]{9}$ (HCs), 15 (oblate HBPs), and 22.5 (prolate
HBPs).

The relevance of confinement and density fluctuations on particle orientation
has been systematically studied by computing the longitudinal and radial
orientational order parameters, which measure, respectively, the average
alignment of $\hat{T}$, $\hat{W}$ and $\hat{L}$ along the axis of the nanopore
and perpendicularly to it. Both sets of order parameters have been obtained by
diagonalisation of the traceless second-rank symmetric tensor that reads

$\textbf{Q}^{\nu\nu}_{\theta}=\Bigg{\langle}\frac{1}{2N_{\theta}}\sum_{i=1}^{N_{\theta}}\left(3\hat{\nu}_{i}\cdot\hat{\nu}_{i}-\textbf{I}\right)\Bigg{\rangle}_{\theta}$
(3)

where $N_{\theta}$ refers to the instantaneous number of particles contained
in the generic cylindrical shell $\theta$ at distance $r$ from the nanopore
axis, $\hat{\nu}_{i}$ is the vector projection of $\hat{T}_{i}$, $\hat{W}_{i}$
or $\hat{L}_{i}$ on the radial or longitudinal direction of the cylindrical
nanopore, and $\langle...\rangle_{\theta}$ indicates ensemble average within
the cylindrical shell $\theta$. We notice that $\textbf{Q}^{\nu\nu}_{\theta}$
is formally different from $\textbf{Q}^{\lambda\lambda}$, which has been
defined in Eq. (2): the former measures how the particles in a given shell
orient along the nanopore radial and longitudinal directions, whereas the
latter measures the average orientation of all particles in the whole nanopore
in the laboratory frame of reference. In particular, the nematic order
parameter $S^{\theta}_{2,L_{r}}$, which measures the alignment of $\hat{L}$
along the radial ($r$) direction of the nanopore in the cylindrical shell
$\theta$, is the largest eigenvalue of the tensor
$\textbf{Q}^{L_{r}L_{r}}_{\theta}$. Similarly, the nematic order parameter
$S^{\theta}_{2,L_{l}}$, which measures the alignment of $\hat{L}$ along the
longitudinal ($l$) direction of the nanopore in the cylindrical shell $\theta$
is obtained from the largest eigenvalue of tensor
$\textbf{Q}^{L_{l}L_{l}}_{\theta}$. From these definitions, it follows that
if, in a given cylindrical shell $\theta$, prolate HBPs are perfectly aligned
along the nanopore longitudinal axis, then $S^{\theta}_{2,L_{l}}=1$ and
$S^{\theta}_{2,L_{r}}=-0.5$. By contrast, if they are aligned along the
nanopore radial direction, then $S^{\theta}_{2,L_{r}}=1$ and
$S^{\theta}_{2,L_{l}}=-0.5$. Similar expressions to those valid for $\hat{L}$
have been employed to estimate the radial and longitudinal alignment of
$\hat{T}$ and $\hat{W}$.

To study the dynamics, we performed DMC simulations in the canonical ensemble.
The DMC method has been discussed elsewhere Patti and Cuetos (2012); Cuetos
and Patti (2015); Corbett  _et al._ (2018); García Daza, Cuetos, and Patti
(2020); Chiappini, Patti, and Dijkstra (2020). Here we only present its
essential features and refer the interested reader to these works for details.
In a DMC cycle, we attempt $N$ combined displacements and rotations of
randomly selected particles, which are accepted or rejected with probability
$\text{min}[1,\text{exp}(-\beta\Delta E)]$, where $\Delta E$ is the difference
in energy between new and old configuration. Since particles interact via a
hard-core potential, only moves producing a geometrical overlap are rejected,
while all the others are always accepted. In the latter case, the new position
of a particle $i$ is set by summing three elementary displacements along each
space dimension:
$\delta\textbf{r}_{i}=X_{T}\hat{T}_{i}+X_{W}\hat{W}_{i}+X_{L}\hat{L}_{i}$. The
terms $X_{\alpha}$, with $\alpha=T$, $W$, or $L$, are uniform distributions
that depend on the particle translational diffusion coefficients at infinite
dilution. In particular:

$|X_{\alpha}|\leq\sqrt{2D^{\rm tra}_{\alpha,i}\delta t_{\text{MC}}},$ (4)

where $\delta t_{\text{MC}}$ is a DMC time step. Additionally, reorientations
of the particle axes, $\hat{T}_{i}$, $\hat{W}_{i}$, $\hat{L}_{i}$, are
attempted by imposing three consecutive rigid rotations around $\hat{L}_{i}$,
$\hat{W}_{i}$, and $\hat{T}_{i}$, respectively. Similarly to translations,
elementary rotations around are constrained within the limits imposed by the
rotational diffusion coefficients around the three particle axes at infinite
dilution:

$|Y_{\alpha}|\leq\sqrt{2D^{\rm rot}_{\alpha,i}\delta t_{\text{MC}}},$ (5)

Translational and rotational diffusion coefficients, $D^{\rm tra}_{\alpha}$
and $D^{\rm rot}_{\alpha}$, have been obtained by applying the open-source
software HYDRO++, which describes particles of generic shape as an array of
spherical beads of arbitrary size Carrasco and García de la Torre (1999);
García de la Torre, del Rio Echenique, and Ortega (2007). The values of
translational and rotational diffusion coefficients for the geometries
relevant to our study, are reported in Table 2 in units of
$T_{p}^{2}\tau^{-1}$ and $\tau^{-1}$, respectively, with $\tau$ the time unit.

Table 2: Infinite-dilution translational and rotational diffusion coefficients of HCs and HBPs studied in this work as obtained from HYDRO++ software Carrasco and García de la Torre (1999); García de la Torre, del Rio Echenique, and Ortega (2007). | Perfect Cube | Oblate Cuboid | Prolate Cuboid
---|---|---|---
$D^{\rm tra}_{T}\tau/T^{2}_{p}\cdot 10^{-2}$ | 4.237 | 3.352 | 2.628
$D^{\rm tra}_{W}\tau/T^{2}_{p}\cdot 10^{-2}$ | 4.237 | 3.970 | 2.628
$D^{\rm tra}_{L}\tau/T^{2}_{p}\cdot 10^{-2}$ | 4.237 | 3.970 | 3.624
$D^{\rm rot}_{T}\tau\cdot 10^{-2}$ | 1.935 | 0.989 | 0.215
$D^{\rm rot}_{W}\tau\cdot 10^{-2}$ | 1.935 | 1.332 | 0.215
$D^{\rm rot}_{L}\tau\cdot 10^{-2}$ | 1.935 | 1.332 | 2.992

Regardless the DMC time step employed in Eqs. (2) and (3), the actual time
scale of Brownian dynamics, $t_{\text{BD}}$, must be recovered Patti and
Cuetos (2012); Cuetos and Patti (2015); Corbett  _et al._ (2018); García Daza,
Cuetos, and Patti (2020); Chiappini, Patti, and Dijkstra (2020). The rescaling
to the BD timescale is crucial to ensure a consistent comparison between the
dynamics of different families of particles (e.g. HCs and HBPs) and between
the dynamics of the same system under different conditions (e.g. HCs at low
and high density). To this end, the DMC time scale must be rescaled via the
acceptance rate $\mathcal{A}$:

$t_{\rm BD}=\frac{\mathcal{A}}{3}C_{\rm MC}\delta t_{\text{MC}}$ (6)

where $C_{\rm MC}$ is the total number of DMC cycles.

A key aspect that we further stress here is the fluid spatial inhomogeneity
and the existence of density fluctuations as shown in Fig. 2. Because a change
in density affects the probability of accepting a particle move, the
acceptance rate in Eq. (4) is not space invariant, but depends on the radial
coordinate. Consequently, following our recent work on the extension of the
DMC method to heterogeneous systems García Daza, Cuetos, and Patti (2020), we
have discretised the nanopore into $k$ concentric cylindrical shells, within
which $\mathcal{A}$ can be safely assumed constant. From this, it follows that

$3\delta t_{\text{BD}}=\mathcal{A}_{1}\delta
t_{\text{MC},1}=\mathcal{A}_{2}\delta
t_{\text{MC},2}=\dots=\mathcal{A}_{k}\delta t_{\text{MC},k}$ (7)

where $\mathcal{A}_{1},\dots,\mathcal{A}_{k}$ and $\delta
t_{\text{MC},1},\dots,\delta t_{\text{MC},k}$ refer, respectively, to the
acceptance rate and MC time step in the cylindrical shell $1,\dots,k$. It
follows that Eq. 7 establishes a link between the DMC time steps across the
cylindrical shells and guarantees the existence of a unique BD time scale. In
practice, one of the $k$ shells is arbitrarily selected to set a reference DMC
time step, which is kept constant. By contrast, the time steps associated to
all the other shells are obtained by convergence of Eq. 7 over a short
preliminary DMC simulation García Daza, Cuetos, and Patti (2020). In
particular, we chose as a reference the farthest shell from the pore wall and,
in this shell, we set $\delta t_{\text{MC},1}=10^{-3}\tau$. The so-calculated
MC time steps for each shell are then employed in Eqs. 4 and 5 to obtain the
maximum displacements and rotations in each cylindrical shell and consistently
evaluate and characterise the dynamics of HCs and HBPs in the nanopore.
Additional details are available in our recent work García Daza, Cuetos, and
Patti (2020).

Figure 3: Excess chemical potential as a function of the number density in
systems of HCs (top), oblate HBPs (middle) and prolate HBPs (bottom). Empty
circles refer to systems in the bulk, while squares, triangles and diamonds to
systems confined in cylindrical nanopores of different radius. Solid lines are
guides for the eye.

##  III RESULTS AND DISCUSSION

In this section, we showcase simulation results that highlight the most
relevant properties of HCs and HBPs in cylindrical nanopores of different
radius. First, we report the impact of confinement on their thermodynamic and
structural properties, and then we discuss the dynamics.

In Fig. 3, plots of the excess chemical potential vs the reduced density are
shown for HCs and oblate and prolate HBPs. Each of these three sets of
equations of state has been obtained in the bulk and in cylindrical nanopores,
as indicated in the figure legends. As a general tendency, the dependence of
$\beta\mu^{\rm ex}$ on $\rho$ observed in confined systems deviates
significantly from that in the bulk, particularly so at large densities and in
narrow tubes, where the effect of walls becomes especially relevant. As an
example of the effect of the walls on the confined particles, in Fig. 4 we
show a sequence of snapshots of prolate HBPs in nanopores of radius
$R^{\star}=22.5$ at increasing value of the excess chemical potential and
volume fraction. In dilute systems, HBPs appear to be randomly oriented,
regardless their position in the pore. At larger densities, we notice a
gradually increasing alignment of the particles located close to the walls,
whereas those in the centre persist in their disordered state. The alignment
along the nanopore longitudinal direction basically sparks the formation of a
nematic-like cylindrical layer of HBPs that separates the walls from the
disordered inner core. We notice that such nematic-like layers develop at
densities that, in the bulk, would be insufficient for the formation of the
nematic phaseCuetos  _et al._ (2017), found at $\rho T_{p}^{3}\approx 0.033$,
being significantly larger than that of the systems studied here (see bottom
frame of Fig. 3).

Figure 4: Prolate HBPs in cylindrical nanopores of radius $R^{\star}=22.5$ at
different normalised densities $\rho^{\star}\equiv\rho/\rho_{\rm bulk}$. The
particles are coloured according to the orientation of their major axis. In
particular, particles in blue are oriented along the pore longitudinal axis,
while particles in red or green are oriented perpendicularly to this axis.
Figure 5: Radial ($S^{\theta}_{2,\lambda_{r}}$, empty symbols) and
longitudinal ($S^{\theta}_{2,\lambda_{l}}$, solid symbols) order parameters of
HCs as a function of the distance from the centre of the cylindrical nanopore
at $f^{\star}=2000$ and $R^{\star}=3\times\sqrt[3]{9}$ (left),
$R^{\star}=6\times\sqrt[3]{9}$ (middle) and $R^{\star}=9\times\sqrt[3]{9}$
(right). Circles, squares and triangles refer, respectively, to the
projections of the particle unit vectors aligned with $\hat{\lambda}=\hat{L}$,
$\hat{W}$ or $\hat{T}$. The insets in each frame show a frontal and side view
of the nanopore, with the particles coloured according to the orientation of
their major axisIn particular, particles in blue are oriented along the pore
longitudinal axis, while particles in red or green are oriented
perpendicularly to this axis. Particles in different snapshots are not in
scale relative to one another. Solid lines are guides for the eye.

The resulting radial and longitudinal nematic order parameters are displayed
in Figs. 5, 6 and 7 for HCs, oblate HBPs and prolate HBPs, respectively, at
three different nanopore radii, but constant fugacity. More specifically, Fig.
5 displays the radial and longitudinal order parameters of confined HCs at
$f^{\star}=2000$, corresponding to $(\beta\mu^{ex},\rho
T^{3}_{p})=(11.043,0.032)$, $(10.954,0.035)$, and $(10.928,0.036)$ at
$R^{\star}=3\times\sqrt[3]{9}$, $6\times\sqrt[3]{9}$ and $9\times\sqrt[3]{9}$
respectively. In all cases, $S^{\theta}_{2,\lambda_{r}}\approx
S^{\theta}_{2,\lambda_{l}}\approx 0$ for $\lambda=$ $\hat{L}$, $\hat{W}$ or
$\hat{T}$, suggesting a random orientation of HCs in both radial and
longitudinal directions and across the concentric cylindrical shells. More
interesting are the results reported in Fig. 6, where the order parameters
$S^{\theta}_{2,\lambda_{r}}$ and $S^{\theta}_{2,\lambda_{l}}$ are shown for
fluids of oblate HBPs. Also in this case, we display results at fugacity
$f^{\star}=2000$, corresponding to values of excess chemical potential and
density that are very similar to those of Fig. 5. In particular,
$(\beta\mu^{ex},\rho T^{3}_{p})=(11.134,0.029)$, $(11.098,0.030)$, and
$(11.078,0.031)$ at $R^{\star}=9$, 12 and 15, respectively. Similar
conclusions can also be drawn at other fugacities and are not discussed here.
The spatial dependence of $S^{\theta}_{2,\lambda_{r}}$ and
$S^{\theta}_{2,\lambda_{l}}$ indicates a significant orientation of oblate
HBPs at the fluid/solid interface, where $\hat{T}$ is strongly aligned with
the nanopore radius (empty triangles), whereas $\hat{W}$ and $\hat{L}$ are
oriented perpendicularly to it (empty squares and circles), but not
necessarily aligned along the nanopore axis (solid circles and squares). Such
a particle anchoring on the wall is observed across the three nanopore radii
studied here and even at relatively dilute systems, where $\rho
T_{p}^{3}\approx 10^{-2}$. At slightly larger distances from the walls, the
order parameter that measures the alignment of $\hat{T}$ with the nanopore
radius decays to $S^{\theta}_{2,T_{r}}\approx-0.25$, suggesting that $\hat{T}$
is almost perpendicular to the nanopore radial direction (empty triangles).
However, since $S^{\theta}_{2,T_{l}}$ is very low, the alignment between
$\hat{T}$ and the nanopore longitudinal direction is basically random (solid
triangle). In other words, at increasing distances from the wall, oblate HBPs
gradually acquire a more and more disordered configuration that, in the
nanopore core, eventually results in a fully isotropic one. Similarly, prolate
HBPs orient at the fluid/solid interface and form nematic-like domains that
are aligned along the longitudinal direction of the cylindrical nanopore. This
is shown in Fig. 7 for $f^{\star}=5000$, a value of fugacity that provides
$(\beta\mu^{ex},\rho T_{p}^{3})=(12.318,0.022)$, $(12.275,0.023)$, and
$(12.219,0.024)$ at $R^{\star}=12$, 15 and 22.5, respectively. In this case,
the particle minor axes, $\hat{T}$ and $\hat{W}$, orient perpendicularly to
the longitudinal direction of the cylindrical nanopore, with
$S^{\theta}_{2,T_{l}}\approx S^{\theta}_{2,W_{l}}\approx-0.5$ (solid squares
and triangles), while the particle main axis, $\hat{L}$, results to be
parallel to it, with $S^{\theta}_{2,L_{l}}\approx 1$ (solid circles). As far
as the radial alignment of the minor axes is concerned, we observe that
$S^{\theta}_{2,T_{r}}\approx S^{\theta}_{2,W_{r}}\approx 0.25$ (empty
triangles and squares), suggesting a very weak orientation at the walls that
becomes completely random in the nanopore core.

Figure 6: Radial ($S^{\theta}_{2,\lambda_{r}}$, empty symbols) and
longitudinal ($S^{\theta}_{2,\lambda_{l}}$, solid symbols) order parameters of
oblate HBPs as a function of the distance from the centre of the cylindrical
nanopore at $f^{\star}=2000$ and $R^{\star}=9$ (left), $R^{\star}=12$ (middle)
and $R^{\star}=15$ (right). Circles, squares and triangles refer,
respectively, to the projections of the particle unit vectors aligned with
$\hat{\lambda}=\hat{L}$, $\hat{W}$ or $\hat{T}$. The insets in each frame show
a frontal and side view of the nanopore, with the particles coloured according
to the orientation of their major axis. Particles in different snapshots are
not in scale relative to one another. Solid lines are guides for the eye.
Figure 7: Radial ($S^{\theta}_{2,\lambda_{r}}$, empty symbols) and
longitudinal ($S^{\theta}_{2,\lambda_{l}}$, solid symbols) order parameters of
prolate HBPs as a function of the distance from the centre of the cylindrical
nanopore at $f^{\star}=5000$ and $R^{\star}=12$ (left), $R^{\star}=15$
(middle) and $R^{\star}=22.5$ (right). Circles, squares and triangles refer,
respectively, to the projections of the particle unit vectors aligned with
$\hat{\lambda}=\hat{L}$, $\hat{W}$ or $\hat{T}$. The insets in each frame show
a frontal and side view of the nanopore, with the particles coloured according
to the orientation of their major axis. Particles in different snapshots are
not in scale relative to one another. Solid lines are guides for the eye.

Not only does confinement affect the phase behaviour and structural properties
of HCs and HBPs, but it also determines their ability to diffuse through the
nanopore. In order to estimate the effect of cylindrical confinement on the
dynamics of HBPs, we have calculated their mean square displacement (MSD) in
the direction parallel to the longitudinal axis of the nanopore and in the
radial direction, and compared them with the total MSD. Longitudinal ($l$),
radial ($r$), and total MSDs are defined as follows:

$\langle\Delta
r^{2}(t)\rangle_{l}=\Bigg{\langle}\frac{1}{N}\sum_{j=1}^{N}[r_{j,l}(t)-r_{j,l}(0)]^{2}\Bigg{\rangle},$
(8)

$\langle\Delta
r^{2}(t)\rangle_{r}=\Bigg{\langle}\frac{1}{N}\sum_{j=1}^{N}[r_{j,r}(t)-r_{j,r}(0)]^{2}\Bigg{\rangle},$
(9)

$\langle\Delta
r^{2}(t)\rangle=\Bigg{\langle}\frac{1}{N}\sum_{j=1}^{N}[\textbf{r}_{j}(t)-\textbf{r}_{j}(0)]^{2}\Bigg{\rangle}.$
(10)

with $r_{j,l}(t)$ and $r_{j,r}(t)$ the longitudinal and radial coordinates of
particle $j$ at time $t$, respectively. The MSDs of HCs confined in nanopores
of radius $R^{\star}=3\times\sqrt[3]{9}$ and $R^{\star}=9\times\sqrt[3]{9}$
are reported in Fig. 8 for different values of the excess chemical potential.
Increasing $\mu^{ex}$ reduces the mobility of particles at long-time scales,
where the effect of packing becomes especially relevant. By contrast, at
short-time scales, when the particles are still very close to their original
location, the MSD is practically unaltered by a change in chemical potential
or, equivalently, density. While, at short times, the contribution of the
radial and longitudinal MSDs to the total MSD is essentially
indistinguishable, the effect of confinement appears to be evident at
approximately $t^{\star}>1$, when the radial MSD slows down significantly as
compared to the longitudinal MSD, especially so at large chemical potentials
and narrow nanopore radii. In particular, the radial MSD in narrow nanopores
exhibits a noticeable decrease that indicates the occurrence of a sub-
diffusive dynamics that persists at long times. This is not the case in
nanopores sufficiently large to allow HBPs to enter the diffusive regime.

Figure 8: Total (black symbols), longitudinal (red symbols) and radial (blue
symbols) MSD of HCs confined in cylindrical nanopores of radius
$R^{\star}=3\times\sqrt[3]{9}$ (a) and $R^{\star}=9\times\sqrt[3]{9}$ (b).
Circles, squares and triangles refer, respectively, to $f^{\star}=5$, 50 and
500. The insets report the ratio $\xi(t)$ between the total MSD calculated in
the nanopore and the total MSD calculated in the bulk at $f^{\star}=5$ (solid
line), 50 (dashed line) and 500 (dotted line).

The MSDs of oblate and prolate HBPs, reported respectively in Figs. 9 and 10,
exhibit qualitatively similar tendencies to those of the MSD of HCs, although
some quantitative differences are noticed. At short-time scales, these
differences, which depend exclusively on the diffusion coefficients at
infinite dilution and hence only on particle geometry, are relatively small.
Nevertheless, system packing, degree of confinement and long-range ordering
become more and more relevant as time passes and eventually determine the
ability of HBPs to diffuse in the nanopore. We stress that, although the MSDs
in frames (a) and (b) of Figs. 8-10 have been calculated at the same chemical
potential, the corresponding density changes with the nanopore radius, tending
to the value in the bulk when the nanopore is sufficiently large (see Tables
S1-S3 in the supplementary material). Interestingly, while denser suspensions
would be expected to exhibit slower dynamics, their MSDs unambiguously
indicate that the opposite behaviour is actually observed. This is especially
evident in systems of HCs (Fig. 8), where the total MSD at $t^{\star}\approx
10^{3}$ in nanopores of $R^{\star}=9\times\sqrt[3]{9}$ (frame (b)) is
approximately 4 to 5 times larger than that in nanopores of
$R^{\star}=3\times\sqrt[3]{9}$ (frame (a)), depending on fugacity. When one
separately examines the two contributions to the total MSD, it turns out that
the origin of this unexpected increase of mobility with packing is entirely
due to the radial MSD, the longitudinal MSD being essentially independent on
nanopore radius. We conclude that, in sufficiently narrow nanopores, the
limitations imposed to radial diffusion dramatically determine the particle
global mobility as much as, and even more than, system packing.

Figure 9: Total (black symbols), longitudinal (red symbols) and radial (blue
symbols) MSD of oblate HBPs confined in cylindrical nanopores of radius
$R^{\star}=9$ (a) and $R^{\star}=15$ (b). Circles, squares and triangles
refer, respectively, to $f^{\star}=5$, 50 and 500. The insets report the ratio
$\xi(t)$ between the total MSD calculated in the nanopore and the total MSD
calculated in the bulk at $f^{\star}=5$ (solid line), 50 (dashed line) and 500
(dotted line). Figure 10: Total (black symbols), longitudinal (red symbols)
and radial (blue symbols) MSD of prolate HBPs confined in cylindrical
nanopores of radius $R^{\star}=15$ (a) and $R^{\star}=22.5$ (b). Circles,
squares and triangles refer, respectively, to $f^{\star}=5$, 50 and 500. The
insets report the ratio $\xi(t)$ between the total MSD calculated in the
nanopore and the total MSD calculated in the bulk at $f^{\star}=5$ (solid
line), 50 (dashed line) and 500 (dotted line).

In the insets included in Figs. 8-10, we report the ratio between the total
MSD calculated in the nanopore and the total MSD calculated in the bulk, that
is $\xi(t)\equiv\langle\Delta r^{2}(t)\rangle_{\rm pore}/\langle\Delta
r^{2}(t)\rangle_{\rm bulk}$. At short time scales, when the dynamics is still
insensitive to both packing and degree of confinement, the ratio between the
nanopore and bulk MSDs displays the same general behaviour for all the systems
studied, with $\xi=1$. However, at intermediate-to-long time scales, $\xi$
deviates from 1, suggesting that these factors along with particle geometry
become relevant. The inset in the left frame of Fig. 8, where $\xi<1$ for
$t^{\star}>1$, suggests that HCs diffuse faster in the bulk than in narrow
nanopores. In larger nanopores (right frame of Fig. 8), the behaviour is more
complex, with $\xi$ increasing up to 1.25 at $f^{\star}=500$ and then
decreasing at longer time scales, indicating that, at least over approximately
three time decades, the dynamics under confinement is faster than the dynamics
in the bulk. Beyond our simulation time, we would expect a similar behaviour
to that observed in smaller nanopores, where $\xi<1$ at sufficiently long
times. This view is further supported by the insets of Fig. 9, which display
$\xi$ for suspensions of oblate HBPs. Also in this case, $\xi$ decreases to
values lower than unity when the long-time diffusive regime is achieved. A
similar peak to that observed in the right inset of Fig. 8 (suspensions of HCs
in nanopores of radius $R^{\star}=15$) is also observed in the insets of Fig.
10 (suspensions of prolate HBPs in nanopores of radius $R^{\star}=15$ and
22.5). The occurrence of such a maximum at intermediate time scales, when
particles are still relatively close to their original position and about to
diffuse through the cage formed by their immediate neighbours, must be somehow
connected to the topological differences existing between the cages in the
bulk and those in a confined environment. While in an isotropic system, the
time a particle takes to diffuse through a cage is not dependent on its
direction in space, such a space invariancy does not hold upon constraining
this system in a cylindrical nanopore. Confining the fluid induces relevant
modifications of its structure that are not detected in the bulk, especially
in the vicinity of the wall. The density profiles and uniaxial order
parameters presented above confirm that this is indeed the case, and the
difference between longitudinal and radial MSDs is an unequivocal evidence
that through structure one can control dynamics. Nevertheless, the existence
of a space-dependent cage effect does not necessarily result into a faster
diffusion in a nanopore as compared to that in the bulk. For example, in
suspensions of oblate HBPs, we find $\xi\leq 1$ in both narrow and large
nanopores, from short to long time scales (see insets of Fig. 9). While we
believe that the morphology of the particle cage dramatically influences the
dynamics at these time and length scales, further investigation is needed to
fully understand what other elements contribute to enhance or inhibit the
diffusion of a nanoconfined fluid as compared to its usual behaviour in the
bulk. The presence of clusters, that we have recently detected in very similar
systems of oblate and prolate HBPs, might contribute to answer this question
Tonti, García Daza, and Patti (2021).

We now turn our attention to the distribution of the particle displacements
over time. To this end, we calculated the self-part of the van Hove
correlation function (s-VHF) in the longitudinal and radial directions of the
cylindrical nanopore, which respectively read

$G_{s,l}(r_{l},t)=\frac{1}{N}\Bigg{\langle}\sum_{j=1}^{N}\delta(r_{l}-|{\bf{r}}_{j,l}(t)-{\bf{r}}_{j,l}(0)|)\Bigg{\rangle}$
(11)

and

$G_{s,r}(r_{r},t)=\frac{1}{N}\Bigg{\langle}\sum_{j=1}^{N}\delta(r_{r}-|{\bf{r}}_{j,r}(t)-{\bf{r}}_{j,r}(0)|)\Bigg{\rangle}.$
(12)

In particular, the symbol $\delta$ is the Dirac delta and the angular brackets
denote ensemble average over the $N$ particles and at least 100 uncorrelated
DMC trajectories. Longitudinal and radial s-VHFs have been normalised as
follows: $\int_{0}^{\infty}G_{s,l}dr_{l}=\int_{0}^{\infty}G_{s,r}dr_{r}$=1.
The s-VHFs are especially relevant to estimate the probability of observing
particles that are slower or faster than the average. Considering the
tendencies uncovered by radial and longitudinal MSDs, these probabilities are
expected to be different along the nanopore axis and in the radial direction.
This is indeed the case as one can appreciate from Fig. 11, where radial and
longitudinal s-VHFs of the same systems as those considered in Figs. 8-10 are
reported. More specifically, we present the radial (solid lines) and
longitudinal (dashed lines) s-VHFs of HCs and HBPs at $f^{\star}=5$ (top
frames) and 500 (bottom frames) and at time $t^{\star}\approx 10^{3}$. While
the two sets of s-VHFs appear relatively similar, thus suggesting quasi-
isotropic particle displacements in the nanopore, there are distinctive
features that deserve further attention. In particular, the longitudinal
s-VHFs exhibit a peak at small distances that appears slightly more pronounced
at increasing fugacity, where the system density is larger and it is more
difficult for the particles to diffuse. This behaviour is especially evident
in systems of HCs and oblate HBPs, while the s-VHFs of prolate HBPs display a
weaker dependence on fugacity. For the three particle geometries, the
longitudinal s-VHF gradually vanishes with the distance, suggesting that the
probability of finding fast particles, namely particles that are able to
displace longer distances over the same time window, is smaller and smaller
and eventually decays to zero. As far as the radial s-VHFs are concerned, the
very high peak at $r^{\star}\approx 0$ suggests that, even at relatively long
time scales, the mobility of particles across the nanopore concentric shells
is quite limited. Additionally, the analysis of these s-VHFs discloses a few
interesting elements that are strongly related to the density profiles of Fig.
2.

Figure 11: Longitudinal (black dashed line) and radial (solid red line) s-VHFs
obtained at $t^{\star}\approx 10^{3}$ in suspensions of HCs (left), oblate
HBPs (middle) and prolate HBPs (right) confined in cylindrical nanopores of
radius $R^{\star}=9\times\sqrt[3]{9}$, 15 and 22.5, respectively. Top and
bottom frames refer to $f^{\star}=5$ and $f^{\star}=500$, respectively. The
arrows indicate the occurrence of a pick or shoulder observed in the radial
s-VHFs of HCs and oblate HBPs, respectively.

In particular, the radial s-VHFs of confined HCs (left frames in Fig. 11)
exhibit a minor peak between $r^{\star}=3$ and 4. As the left frame of Fig. 2
indicates, the fluid in the cylindrical nanopore does not exhibit a
homogeneous density distribution in the vicinity of the pore wall, but
organises in concentric shells intercalated by domains that are less likely to
be occupied. Correspondingly, particles displacing back and forth between
these contiguous shells are less likely to linger in between them, producing a
probability of displacements that, at sufficiently large densities, results in
the minor peaks observed in the left frames of Fig. 11. Similar considerations
are still valid for the radial s-VHFs of oblate HBPs (middle frames of Fig.
11). However, in this case the minor peaks are much less pronounced than those
observed for HCs, almost degenerating into a smooth shoulder. This is
consistent with the corresponding density profiles of Fig. 2 (middle frames),
which indicate a milder density differences between the two outermost
cylindrical shells. By contrast, no secondary peaks are observed in the radial
s-VHFs of prolate HBPs (right frames of Fig. 11), in agreement with the
particle density distribution of Fig. 2, which exclude the existence of well-
defined shells at the nanopore wall as observed for HCs and oblate HBPs. We
finally notice that the radial s-VHFs gradually decrease to zero at distances
approximately between $r^{\star}=15$ and 20. Such a decrease indicates a non-
vanishing probability to observe especially fast particles that have displaced
significantly longer distances than the average particle. The simultaneous
presence or slow and fast particles is also detected in the longitudinal
s-VHFs (dashed lines in Fig. 11), although in this case their decrease is
monotonic with no evidence of peaks or shoulders.

##  IV CONCLUSIONS

In summary, we have here analysed the behaviour of colloidal hard cubes and
hard cuboids that have been confined in cylindrical nanopores. We have
specifically assessed the effect of packing and particle and pore geometry on
the system equilibrium and dynamical properties, finding that both factors
directly determine the ability of HCs and HBPs to orient at the fluid/solid
interface and their mobility along the nanopore longitudinal and radial axes.
In particular, oblate and prolate HBPs, at sufficiently large density,
organise in ordered nematic-like shells that surround an inner core of
randomly-oriented particles. This is not the case for HCs, which maintain a
random orientation throughout the cylindrical nanopore. Regardless the
occurrence of nematic-like ordering, the radial density profiles, obtained
along the nanopore radial direction, indicate the existence of a layered
arrangement that is especially evident in systems of HCs, but less so in
systems of prolate HBPs. These structural signatures dramatically determine
the dynamical properties of the systems under confinement, which we have
evaluated in terms of MSD and s-VHFs. The computation of the MSD is very
useful to analyse the mobility of HBPs under confinement as compared to that
in the bulk. We have shown that at short-time scales, HBPs diffuse in the
nanopore as they were in the bulk. Then, at intermediate times, the combined
action of layering and alignment in confined suspensions enhances the ability
of particles to diffuse through their surrounding neighbours, leading to a
dynamics that is faster than that in the bulk. However, these tendencies are
only limited to a few time decades. As soon as the system enters the long-time
diffusive regime, the ratio between the MSD in the nanopore and that in the
bulk gradually decreases, eventually approaching values lower than one beyond
our simulation time. Finally, the s-VHFs have been key to distinguish the
distribution of particle displacements along the nanopore longitudinal and
radial directions. Both of them suggest the existence of particles that
displace shorter or longer distances than the average particle. In particular,
radial s-VHFs are especially interesting as they reveal the simultaneous
presence of slow particles that remain close to their original position even
at relatively long times, and fast particles that are able to cover distances
as large as 15 to 20 times that covered by most particles over the same time
scale. A similar tendency, although on a minor scale, is also found along the
longitudinal direction of the nanopore.

###### Acknowledgements.

A.P. acknowledges financial support from the Leverhulme Trust Research Project
Grant No. RPG-2018-415 and the use of the Computational Shared Facility at the
University of Manchester. A.C. acknowledges the Spanish Ministerio de Ciencia,
Innovación y Universidades and FEDER for funding (Project No.
PGC2018-097151-B-I00) and C3UPO for the HPC facilities provided.

##  V Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * Pieranski (1980) P. Pieranski, “Two-dimensional interfacial colloidal crystals,” Phys. Rev. Lett. 45, 569–572 (1980).
  * Neser  _et al._ (1997) S. Neser, C. Bechinger, P. Leiderer, and T. Palberg, “Finite-size effects on the closest packing of hard spheres,” Phys. Rev. Lett. 79, 2348–2351 (1997).
  * Cohen, Mason, and Weitz (2004) I. Cohen, T. G. Mason, and D. A. Weitz, “Shear-induced configurations of confined colloidal suspensions,” Phys. Rev. Lett. 93, 046001 (2004).
  * Cortes  _et al._ (2016) L. B. G. Cortes, Y. Gao, R. P. A. Dullens, and D. G. A. L. Aarts, “Colloidal liquid crystals in square confinement: isotropic, nematic and smectic phases,” J. Condens. Matter Phys. 29, 064003 (2016).
  * Martínez-Pedrero  _et al._ (2021) F. Martínez-Pedrero, A. González-Banciella, A. Camino, A. Mateos-Maroto, F. Ortega, R. G. Rubio, I. Pagonabarraga, and C. Calero, “Static and dynamic self-assembly of pearl-like-chains of magnetic colloids confined at fluid interfaces,” Small 17, 2101188 (2021), https://onlinelibrary.wiley.com/doi/pdf/10.1002/smll.202101188 .
  * Rodríguez-Ponce, Romero-Enrique, and Rull (2001) I. Rodríguez-Ponce, J. M. Romero-Enrique, and L. F. Rull, “Orientational transitions in a nematic liquid crystal confined by competing surfaces,” Phys. Rev. E 64, 051704 (2001).
  * Yao, Zhang, and Chen (2018) X. Yao, H. Zhang, and J. Z. Y. Chen, “Topological defects in two-dimensional liquid crystals confined by a box,” Phys. Rev. E 97, 052707 (2018).
  * Mizani  _et al._ (2019) S. Mizani, R. Aliabadi, H. Salehi, and S. Varga, “Orientational ordering and layering of hard plates in narrow slitlike pores,” Phys. Rev. E 100, 032704 (2019).
  * Yao and Chen (2020) X. Yao and J. Z. Y. Chen, “Rodlike molecules in extreme confinement,” Phys. Rev. E 101, 062706 (2020).
  * Behzadi, Ghazi, and Aliabadi (2021) F. Behzadi, S. M. Ghazi, and R. Aliabadi, “From $n$-layer planar ordering to the monolayer homeotropic structure of confined hard rods: The effect of shape anisotropy and wall-to-wall separation,” Phys. Rev. E 103, 022702 (2021).
  * Teixeira, Anquetil-Deck, and Cleaver (2021) P. I. C. Teixeira, C. Anquetil-Deck, and D. J. Cleaver, “Ordering of oblate hard particles between symmetric penetrable walls,” Liquid Crystals 48, 75–87 (2021).
  * Avendaño, Liddell Watson, and Escobedo (2013) C. Avendaño, C. M. Liddell Watson, and F. A. Escobedo, “Directed self-assembly of spherical caps via confinement,” Soft Matter 9, 9153–9166 (2013).
  * Patti (2013) A. Patti, “Modeling the aggregation behavior of amphiphiles in the continuous phase of highly concentrated emulsions,” Colloids and Surfaces A: Physicochemical and Engineering Aspects 437, 90–100 (2013).
  * Muangnapoh  _et al._ (2014) K. Muangnapoh, C. Avendaño, F. A. Escobedo, and C. M. Liddell Watson, “Degenerate crystals from colloidal dimers under confinement,” Soft Matter 10, 9729–9738 (2014).
  * Workineh and Vanakaras (2016) Z. G. Workineh and A. G. Vanakaras, “Self-organisation and alignment properties of homeotropically confined model liquid crystalline dendrimer systems,” Liquid Crystals 43, 944–954 (2016), https://doi.org/10.1080/02678292.2016.1149624 .
  * Avendaño, Jackson, and Wensink (2018) C. Avendaño, G. Jackson, and H. H. Wensink, “Nanorings in planar confinement: the role of repulsive surfaces on the formation of lacuna smectics,” Molecular Physics 116, 2901–2910 (2018).
  * Luo  _et al._ (2019) A. M. Luo, J. Vermant, P. Ilg, Z. Zhang, and L. M. Sagis, “Self-assembly of ellipsoidal particles at fluid-fluid interfaces with an empirical pair potential,” Journal of Colloid and Interface Science 534, 205–214 (2019).
  * Gârlea  _et al._ (2019) I. C. Gârlea, O. Dammone, J. Alvarado, V. Notenboom, Y. Jia, G. H. Koenderink, D. G. A. L. Aarts, M. P. Lettinga, and B. M. Mulder, “Colloidal liquid crystals confined to synthetic tactoids,” Scientific Reports 9, 20391 (2019).
  * Anquetil-Deck, Cleaver, and Teixeira (2020) C. Anquetil-Deck, D. J. Cleaver, and P. I. C. Teixeira, “Ordering of oblate hard particles between hybrid penetrable walls,” The Journal of Physical Chemistry B 124, 7709–7716 (2020).
  * Anzivino  _et al._ (2021) C. Anzivino, G. Soligno, R. van Roij, and M. Dijkstra, “Chains of cubic colloids at fluid–fluid interfaces,” Soft Matter 17, 965–975 (2021).
  * Donaldson, Schall, and Rossi (2021) J. G. Donaldson, P. Schall, and L. Rossi, “Magnetic coupling in colloidal clusters for hierarchical self-assembly,” ACS Nano 15, 4989–4999 (2021), pMID: 33650847, https://doi.org/10.1021/acsnano.0c09952 .
  * Dijkstra, Roij, and Evans (2001) M. Dijkstra, R. v. Roij, and R. Evans, “Wetting and capillary nematization of a hard-rod fluid: A simulation study,” Phys. Rev. E 63, 051703 (2001).
  * Dijkstra (2004) M. Dijkstra, “Capillary freezing or complete wetting of hard spheres in a planar hard slit?” Phys. Rev. Lett. 93, 108303 (2004).
  * Martínez-Ratón (2007) Y. Martínez-Ratón, “Capillary ordering and layering transitions in two-dimensional hard-rod fluids,” Phys. Rev. E 75, 051708 (2007).
  * Brumby  _et al._ (2017) P. E. Brumby, H. H. Wensink, A. J. Haslam, and G. Jackson, “Structure and interfacial tension of a hard-rod fluid in planar confinement,” Langmuir 33, 11754–11770 (2017).
  * Salgado-Blanco, Díaz-Herrera, and Mendoza (2019) D. Salgado-Blanco, E. Díaz-Herrera, and C. I. Mendoza, “Effect of the anchoring strength on the phase behaviour of discotic liquid crystals under face-on confinement,” Journal of Physics: Condensed Matter 31, 105101 (2019).
  * Basurto  _et al._ (2020) E. Basurto, P. Gurin, S. Varga, and G. Odriozola, “Ordering, clustering, and wetting of hard rods in extreme confinement,” Phys. Rev. Research 2, 013356 (2020).
  * Sun and Xia (2002) Y. Sun and Y. Xia, “Shape-controlled synthesis of gold and silver nanoparticles,” Science 298, 2176–2179 (2002).
  * Manoharan, Elsesser, and Pine (2003) V. N. Manoharan, M. T. Elsesser, and D. J. Pine, “Dense packing and symmetry in small clusters of microspheres,” Science 301, 483–487 (2003).
  * Shankar  _et al._ (2004) S. S. Shankar, A. Rai, B. Ankamwar, A. Singh, A. Ahmad, and M. Sastry, “Biological synthesis of triangular gold nanoprisms,” Nature Materials 3, 482–488 (2004).
  * John and Escobedo (2005) B. S. John and F. A. Escobedo, “Phase behavior of colloidal hard tetragonal parallelepipeds (cuboids): A monte carlo simulation study,” The Journal of Physical Chemistry B 109, 23008–23015 (2005).
  * John, Juhlin, and Escobedo (2008) B. S. John, C. Juhlin, and F. A. Escobedo, “Phase behavior of colloidal hard perfect tetragonal parallelepipeds,” The Journal of Chemical Physics 128, 044909 (2008).
  * Cuetos  _et al._ (2017) A. Cuetos, M. Dennison, A. Masters, and A. Patti, “Phase behaviour of hard board-like particles,” Soft Matter 13, 4720–4732 (2017).
  * Patti and Cuetos (2018) A. Patti and A. Cuetos, “Monte carlo simulation of binary mixtures of hard colloidal cuboids,” Molecular Simulation 44, 516–522 (2018).
  * Yang  _et al._ (2018) Y. Yang, G. Chen, S. Thanneeru, J. He, K. Liu, and Z. Nie, “Synthesis and assembly of colloidal cuboids with tunable shape biaxiality,” Nature Communications 9, 4513 (2018).
  * Belli  _et al._ (2011) S. Belli, A. Patti, M. Dijkstra, and R. van Roij, “Polydispersity stabilizes biaxial nematic liquid crystals,” Phys. Rev. Lett. 107, 148303 (2011).
  * Belli, Dijkstra, and van Roij (2012) S. Belli, M. Dijkstra, and R. van Roij, “Depletion-induced biaxial nematic states of boardlike particles,” J. Phys.: Condens. Matter 24, 284128 (2012).
  * Leferink op Reinink  _et al._ (2014) A. B. G. M. Leferink op Reinink, S. Belli, R. van Roij, M. Dijkstra, A. V. Petukhov, and G. J. Vroege, “Tuning biaxiality of nematic phases of board-like colloids by an external magnetic field,” Soft Matter 10, 446–456 (2014).
  * Dussi  _et al._ (2018) S. Dussi, N. Tasios, T. Drwenski, R. van Roij, and M. Dijkstra, “Hard competition: Stabilizing the elusive biaxial nematic phase in suspensions of colloidal particles with extreme lengths,” Phys. Rev. Lett. 120, 177801 (2018).
  * Cuetos  _et al._ (2019) A. Cuetos, E. Mirzad Rafael, D. Corbett, and A. Patti, “Biaxial nematics of hard cuboids in an external field,” Soft Matter 15, 1922–1926 (2019).
  * Mirzad Rafael  _et al._ (2021) E. Mirzad Rafael, L. Tonti, D. Corbett, A. Cuetos, and A. Patti, “Dynamics of uniaxial-to-biaxial nematics switching in suspensions of hard cuboids,” Physics of Fluids 33, 067115 (2021).
  * Wang  _et al._ (2018) D. Wang, M. Hermes, R. Kotni, Y. Wu, N. Tasios, Y. Liu, B. de Nijs, E. B. van der Wee, C. B. Murray, M. Dijkstra, and A. van Blaaderen, “Interplay between spherical confinement and particle shape on the self-assembly of rounded cubes,” Nature Communications 9, 2228 (2018).
  * Varga, Martínez-Ratón, and Velasco (2010) S. Varga, Y. Martínez-Ratón, and E. Velasco, “Competition between capillarity, layering and biaxiality in a confined liquid crystal,” The European Physical Journal E 32, 89–101 (2010).
  * Khadilkar and Escobedo (2016) M. R. Khadilkar and F. A. Escobedo, “Phase behavior of polyhedral nanoparticles in parallel plate confinement,” Soft Matter 12, 1506–1516 (2016).
  * König  _et al._ (2014) T. König, R. Kodiyath, Z. A. Combs, M. A. Mahmoud, M. A. El-Sayed, and V. V. Tsukruk, “Silver nanocube aggregates in cylindrical pores for higher refractive index plasmonic sensing,” Particle & Particle Systems Characterization 31, 274–283 (2014).
  * Prince  _et al._ (2021) E. Prince, Y. Wang, I. I. Smalyukh, and E. Kumacheva, “Cylindrical confinement of nanocolloidal cholesteric liquid crystal,” The Journal of Physical Chemistry B 125, 8243–8250 (2021), pMID: 34259528, https://doi.org/10.1021/acs.jpcb.1c04387 .
  * Bolhuis and Frenkel (1997) P. Bolhuis and D. Frenkel, “Tracing the phase boundaries of hard spherocylinders,” The Journal of Chemical Physics 106, 666–687 (1997), https://doi.org/10.1063/1.473404 .
  * Cuetos and Martínez-Haya (2008) A. Cuetos and B. Martínez-Haya, “Columnar phases of discotic spherocylinders,” The Journal of Chemical Physics 129, 214706 (2008), https://doi.org/10.1063/1.3028539 .
  * Martínez-Haya and Cuetos (2009) B. Martínez-Haya and A. Cuetos, “Columnar phases of discotics with orientation-dependent interactions,” The Journal of Chemical Physics 131, 074901 (2009), https://doi.org/10.1063/1.3207284 .
  * Cuetos and Martínez-Haya (2015) A. Cuetos and B. Martínez-Haya, “Liquid crystal phase diagram of soft repulsive rods and its mapping on the hard repulsive reference fluid,” Molecular Physics 113, 1137–1144 (2015), https://doi.org/10.1080/00268976.2014.996191 .
  * Cuetos  _et al._ (2003) A. Cuetos, B. Martínez-Haya, S. Lago, and L. F. Rull, “Liquid crystal behavior of the kihara fluid,” Phys. Rev. E 68, 011704 (2003).
  * Gottschalk, Lin, and Manocha (1996) S. Gottschalk, M. Lin, and D. Manocha, “Obbtree: A hierarchical structure for rapid interference detection,” (New Orleans, LA, USA, 1996) pp. 171–180.
  * Frenkel and Smit (2001) D. Frenkel and B. Smit, “Understanding molecular simulation. from algorithms to applications,” (Elsevier Science, San Diego, USA, 2001).
  * Eppenga and Frenkel (1984) R. Eppenga and D. Frenkel, “Monte carlo study of the isotropic and nematic phases of infinitely thin hard platelets,” Molecular Physics 52, 1303–1334 (1984).
  * Patti and Cuetos (2012) A. Patti and A. Cuetos, “Brownian dynamics and dynamic monte carlo simulations of isotropic and liquid crystal phases of anisotropic colloidal particles: A comparative study,” Phys. Rev. E 86, 011403 (2012).
  * Cuetos and Patti (2015) A. Cuetos and A. Patti, “Equivalence of brownian dynamics and dynamic monte carlo simulations in multicomponent colloidal suspensions,” Phys. Rev. E 92, 022302 (2015).
  * Corbett  _et al._ (2018) D. Corbett, A. Cuetos, M. Dennison, and A. Patti, “Dynamic monte carlo algorithm for out-of-equilibrium processes in colloidal dispersions,” Phys. Chem. Chem. Phys. 20, 15118–15127 (2018).
  * García Daza, Cuetos, and Patti (2020) F. A. García Daza, A. Cuetos, and A. Patti, “Dynamic monte carlo simulations of inhomogeneous colloidal suspensions,” Phys. Rev. E 102, 013302 (2020).
  * Chiappini, Patti, and Dijkstra (2020) M. Chiappini, A. Patti, and M. Dijkstra, “Helicoidal dynamics of biaxial curved rods in twist-bend nematic phases unveiled by unsupervised machine learning techniques,” Phys. Rev. E 102, 040601 (2020).
  * Carrasco and García de la Torre (1999) B. Carrasco and J. García de la Torre, “Hydrodynamic properties of rigid particles: Comparison of different modeling and computational procedures,” Biophysical Journal 76, 3044–3057 (1999).
  * García de la Torre, del Rio Echenique, and Ortega (2007) J. García de la Torre, G. del Rio Echenique, and A. Ortega, “Improved calculation of rotational diffusion and intrinsic viscosity of bead models for macromolecules and nanoparticles,” The Journal of Physical Chemistry B 111, 955–961 (2007).
  * Tonti, García Daza, and Patti (2021) L. Tonti, F. A. García Daza, and A. Patti, “Diffusion of globular macromolecules in liquid crystals of colloidal cuboids,” Journal of Molecular Liquids 338, 116640 (2021).