Parallel Hard Walls Research Articles

Structural and dynamical equilibrium properties of hard board-like particles in parallel confinement.

We performed Monte Carlo and dynamic Monte Carlo simulations to model the diffusion of monodispersed suspensions composed of impenetrable cuboidal particles, specifically hard board-like particles (HBPs), in the presence of parallel hard walls. The impact of the walls was investigated by adjusting the size of the simulation box while maintaining constant packing fractions, fixed at η = 0.150, for systems consisting of HBPs with prolate, dual-shaped, and oblate geometries. We observed that increasing the distance between the walls led to the recovery of an isotropic bulk phase, while local particle organization near the walls remained stable. Due to their shape, oblate HBPs exhibit more efficient anchoring at wall surfaces compared to prolate shapes. The formation of nematic-like particle assemblies near the walls, confirmed by theoretical calculations based on density functional theory, significantly influenced local particle dynamics. This effect was particularly pronounced to the extent that a modest portion of cuboids near the walls tended to diffuse exclusively in planes parallel to the confinement, even more efficiently than observed in the bulk regions.

External field-driven property localization in liquids of responsive macromolecules.

We explore theoretically the effects of external potentials on the spatial distribution of particle properties in a liquid of explicitly responsive macromolecules. In particular, we focus on the bistable particle size as a coarse-grained internal degree of freedom (DoF, or "property"), σ, that moves in a bimodal energy landscape, in order to model the response of a state-switching (big-to-small) macromolecular liquid to external stimuli. We employ a mean-field density functional theory (DFT) that provides the full inhomogeneous equilibrium distributions of a one-component model system of responsive colloids (RCs) interacting with a Gaussian pair potential. For systems confined between two parallel hard walls, we observe and rationalize a significant localization of the big particle state close to the walls, with pressures described by an exact RC wall theorem. Application of more complex external potentials, such as linear (gravitational), osmotic, and Hamaker potentials, promotes even stronger particle size segregation, in which macromolecules of different size are localized in different spatial regions. Importantly, we demonstrate how the degree of responsiveness of the particle size and its coupling to the external potential tune the position-dependent size distribution. The DFT predictions are corroborated by Brownian dynamics simulations. Our study highlights the fact that particle responsiveness can be used to localize liquid properties and therefore helps to control the property- and position-dependent function of macromolecules, e.g., in biomedical applications.

Confined Quantum Hard Spheres.

We present computer simulation and theoretical results for a system of N Quantum Hard Spheres (QHS) particles of diameter and mass m at temperature T, confined between parallel hard walls separated by a distance , within the range . Semiclassical Monte Carlo computer simulations were performed adapted to a confined space, considering effects in terms of the density of particles , where V is the accessible volume, the inverse length and the de Broglie’s thermal wavelength , where k and h are the Boltzmann’s and Planck’s constants, respectively. For the case of extreme and maximum confinement, and , respectively, analytical results can be given based on an extension for quantum systems of the Helmholtz free energies for the corresponding classical systems.

Anisotropy-independent packing of confined hard ellipses

We have examined the close-packed structure and ordering properties of hard ellipses between two parallel hard walls by using geometrical conjectures and Monte Carlo simulations. Starting from closely packed hard disk structures, we discover two competing densest packings of ellipses with the stretching of hard disks into different directions. We find that parallel and tilted close-packed structures alternate with widening pore width up to the two-dimensional bulk limit, where both structures produce the same close packing value (η=π/12). Our results highlight that the densest packing does not depend on the aspect ratio of the ellipse. This universal behavior is confirmed for three different aspect ratios (κ=2, 3, and 4) using replica exchange Monte Carlo simulation. Our simulation results show that even the global phase diagram is universal, where layering transitions between parallel structures and tilted-parallel structural changes are present.

Second-Virial Onsager Theory and Its Limitations in the Prediction of the Ordering Transitions of Confined Hard Rods between Two Parallel Hard Walls

We studied the phase behavior of hard rectangular rods with edge lengths L, D, and D. All the calculations have been carried out through Onsager second virial theory within the three-state Zwanzig ...

Dynamical properties of densely packed confined hard-sphere fluids.

Numerical solutions of the mode-coupling theory (MCT) equations for a hard-sphere fluid confined between two parallel hard walls are elaborated. The governing equations feature multiple parallel relaxation channels which significantly complicate their numerical integration. We investigate the intermediate scattering functions and the susceptibility spectra close to structural arrest and compare to an asymptotic analysis of the MCT equations. We corroborate that the data converge in the β-scaling regime to two asymptotic power laws, viz. the critical decay and the von Schweidler law. The numerical results reveal a nonmonotonic dependence of the power-law exponents on the slab width and a nontrivial kink in the low-frequency susceptibility spectra. We also find qualitative agreement of these theoretical results to event-driven molecular dynamics simulations of polydisperse hard-sphere systems. In particular, the nontrivial dependence of the dynamical properties on the slab width is well reproduced.

Numerical Simulation of the Distribution Function and Free Energy of a Single Wormlike Polymer Confined between Hard Walls

We focus on the distribution and free energy of a wormlike polymer confined between two parallel hard walls. The variation in the distribution and free energy of the wormlike chain as the spacing between the walls decreases (or as the total contour length of the wormlike chain increases or as the persistence length of the chain increases) is simulated. The main reason for these changes is a degradation of the long wormlike chain into a Gaussian long chain under weak confinement.

Validation study of using the free volume approximation to confined thermotropic and lyotropic liquid-crystalline fluids.

We examined the accuracy of the free volume approximation (FVA) to calculate the isotropic-nematic (IN) transition properties of thermotropic and lyotropic rods between two parallel hard walls. This approximation has been proposed to ease the calculation of the confined systems. It approximates the free energy of the confined particles with a bulk free energy. It predicts a special point for these two types of liquid crystals where the first-order IN transition changes to the second one by decreasing either the temperature, the density or the pore width. This prediction is in contradiction (in spite of some qualitative agreement) with those of the other publications where the authors note that the discontinuous transition terminates at the critical point when the walls are completely impenetrable.

Ordering transitions of weakly anisotropic hard rods in narrow slitlike pores.

The effect of strong confinement on the positional and orientational ordering is examined in a system of hard rectangular rods with length L and diameter D (L>D) using the Parsons-Lee modification of the second virial density-functional theory. The rods are nonmesogenic (L/D<3) and confined between two parallel hard walls, where the width of the pore (H) is chosen in such a way that both planar (particle's long axis parallel to the walls) and homeotropic (particle's long axis perpendicular to the walls) orderings are possible and a maximum of two layers is allowed to form in the pore. In the extreme confinement limit of H≤2D, where only one-layer structures appear, we observe a structural transition from a planar to a homeotropic fluid layer with increasing density, which becomes sharper as L→H. In wider pores (2D<H<3D) planar order with two layers, homeotropic order, and even combined bilayer structures (one layer is homeotropic, while the other is planar) can be stabilized at high densities. Moreover, first-order phase transitions can be seen between different structures. One of them emerges between a monolayer and a bilayer with planar orders at relatively low packing fractions.

Ordering of hard rectangles in strong confinement.

Using transfer operator and fundamental measure theories, we examine the structural and thermodynamic properties of hard rectangles confined between two parallel hard walls. The side lengths of the rectangle (L and D, L>D) and the pore width (H) are chosen such that a maximum of two layers is allowed to form when the long sides of the rectangles are parallel to the wall, while only one layer is possible in case the rectangles are perpendicular to the wall. We observe three different structures: (i) at low density, the rectangles align mainly parallel to the wall, (ii) at intermediate or high density, two fluid layers form in which the rectangles are parallel to the wall, and (iii) a dense single fluid layer with rectangles aligned mainly perpendicular to the wall. The transition between these structures is smooth without any non-analytic behaviour in the thermodynamic quantities; however, the fraction of particles perpendicular (or parallel) to the wall can exhibit a relatively sudden change if L is close to H. In this case, interestingly, even three different structures can be observed with increasing density.

Phase behavior of polyhedral nanoparticles in parallel plate confinement.

Monte Carlo simulations are used to investigate the phase behavior of hard cubes, truncated cubes, cuboctahedra and truncated octahedra when confined between two parallel hard walls. The walls are separated by a distance H* which is varied to accommodate a different number of layers, from a monolayer up to approximately 5 layers, hence allowing us to probe the transitional phase behavior as the system goes from a quasi-2D geometry to a quasi-3D bulk behavior. While our results do reveal some phases whose structures resemble those that have been observed before for such systems in 2D and 3D spaces, other phases are also detected, including buckled phases, rotator plastic phases, and solids with significant translational disorder. Ordered phases formed for H* values that are a little too narrow to accommodate an additional particle layer are particularly interesting as they tend to have complex structures. The maximum density for such frustrated phases is low compared to that of non-frustrated ones for the same system at different H*. As the asphericity in the shapes is reduced, the simulated phases show structural features that approach those of the phases that have been reported for hard spheres under similar confinement.

Orientational ordering of confined hard rods: the effect of shape anisotropy on surface ordering and capillary nematization.

We examine the ordering properties of rectangular hard rods with length L and diameter D at a single planar wall and between two parallel hard walls using the second virial density-functional theory. The theory is implemented in the three-state Zwanzig approximation, where only three mutually perpendicular directions are allowed for the orientations of hard rods. The effect of varying shape anisotropy is examined at L/D=10,15,and20. In contact with a single hard wall, the density profiles show planar ordering, damped oscillatory behavior, and a wall-induced surface ordering transition below the coexisting isotropic density of a bulk isotropic-nematic (I-N) phase transition. Upon approaching the coexisting isotropic density, the thickness of the nematic film diverges logarithmically, i.e., the nematic wetting is complete for any shape anisotropy. In the case of confinement between two parallel hard walls, it is found that the continuous surface ordering transition depends strongly on the distance between confining walls H for H<L, while it depends weakly on H for H>L. The minimal density at which a surface ordering transition can be realized is located at around H∼2D for all studied shape anisotropies due to the strong interference effect between the two hard walls. The first-order I-N phase transition of the bulk system becomes a surface ordered isotropic I_{B} to capillary nematic N_{B} phase transition in the slit pore. This first-order I_{B}-N_{B} transition weakens with decreasing pore width and terminates in a critical point for all studied shape anisotropies.

Phase diagram and critical properties of Yukawa bilayers.

We study the ground-state Wigner bilayers of pointlike particles with Yukawa pairwise interactions, confined to the surface of two parallel hard walls at dimensionless distance η. The model involves as limiting cases the unscreened Coulomb potential and hard spheres. The phase diagram of Yukawa particles, studied numerically by Messina and Löwen [Phys. Rev. Lett. 91, 146101 (2003)], exhibits five different staggered phases as η varies from 0 to intermediate values. We present a lattice summation method using the generalized Misra functions which permits us to calculate the energy per particle of the phases with a precision much higher than usual in computer simulations. This allows us to address some tiny details of the phase diagram. Going from the hexagonal phase I to phase II is shown to occur at η=0. All second-order phase transitions are proved to be of mean-field type. We also derive the asymptotic shape of critical lines close to the Coulomb and hard-spheres limits. In and close to the hard-spheres limit, the dependence of the internal parameters of the present phases on η is determined exactly.

Quantum calculations for the photodetachment cross sections of [formula omitted] located between two walls

We re-investigate the photodetachment cross sections of H− in a quantum well consisting of two parallel hard walls using a quantum approach. The formulas for the cross sections are explicitly derived and compared with those derived by using closed-orbit theory (G.C. Yang, et al., Physica B 404 (2009) 1576 [15]). The present quantum results confirm the staircase pattern of the cross sections obtained earlier when the polarization of photons is parallel to the normal direction of the walls. However, we find that when the polarization is perpendicular to the normal direction of the walls, oscillations in the cross sections in the present quantum calculations are still present in contrast to the predictions of closed-orbit theory. The differences in the two results are large enough to be observable.

The rate of convergence of the virial series in confined systems

We examine cluster-series methods for the behaviour of simple fluids under confinement, applying the Mayer-sampling Monte Carlo method to evaluate the necessary cluster integrals. We examine two series formulations, one based on the density of a bulk phase in equilibrium with the confined phase (effectively specifying the chemical potential), and another based on the density of the confined phase. The theory and methods are used to calculate the structure and properties of hard-sphere fluids confined between parallel planar hard walls. We present the effects of width of the confined region on the inhomogeneous virial coefficients, and through them, on the thermodynamics of the confined system.

Tunable long range forces mediated by self-propelled colloidal hard spheres.

Using Brownian dynamics simulations, we systematically study the effective interaction between two parallel hard walls in a 2D suspension of self-propelled (active) colloidal hard spheres, and we find that the effective force between two hard walls can be tuned from a long range repulsion into a long range attraction by changing the density of active particles. At relatively high densities, the active hard spheres can form a dynamic crystalline bridge, which induces a strong oscillating long range dynamic wetting repulsion between the walls. With decreasing density, the dynamic bridge gradually breaks, and an intriguing long range dynamic depletion attraction arises. A similar effect occurs in a quasi-2D suspension of self-propelled colloidal hard spheres by changing the height of the confinement. Our results open up new possibilities to manipulate the motion and assembly of microscopic objects by using active matter.

Computer simulation of liquid-vapor coexistence of confined quantum fluids

The liquid-vapor coexistence (LV) of bulk and confined quantum fluids has been studied by Monte Carlo computer simulation for particles interacting via a semiclassical effective pair potential Veff(r) = VLJ + VQ, where VLJ is the Lennard-Jones 12-6 potential (LJ) and VQ is the first-order Wigner-Kirkwood (WK-1) quantum potential, that depends on β = 1∕kT and de Boer's quantumness parameter Λ=h/σ√mε, where k and h are the Boltzmann's and Planck's constants, respectively, m is the particle's mass, T is the temperature of the system, and σ and ε are the LJ potential parameters. The non-conformal properties of the system of particles interacting via the effective pair potential Veff(r) are due to Λ, since the LV phase diagram is modified by varying Λ. We found that the WK-1 system gives an accurate description of the LV coexistence for bulk phases of several quantum fluids, obtained by the Gibbs Ensemble Monte Carlo method (GEMC). Confinement effects were introduced using the Canonical Ensemble (NVT) to simulate quantum fluids contained within parallel hard walls separated by a distance Lp, within the range 2σ ≤ Lp ≤ 6σ. The critical temperature of the system is reduced by decreasing Lp and increasing Λ, and the liquid-vapor transition is not longer observed for Lp∕σ < 2, in contrast to what has been observed for the classical system.

Directed self-assembly of spherical caps via confinement

In this work we use Monte Carlo simulations to study the phase behavior of spherical caps confined between two parallel hard walls separated by a distance H. The particle model consists of a hard sphere of diameter σ cut off by a plane at a height χ, and it is loosely based on mushroom cap-shaped particles whose phase behavior was recently studied experimentally [E. K. Riley and C. M. Liddell, Langmuir, 26, 2010, 11648]. The geometry of the particles is characterized by the reduced height χ* = χ/σ, such that the model extrapolates between hard spheres for χ* → 1 and infinitely thin hard spherical caps for χ* → 0. Three different particle shapes are investigated: (a) three-quarter height spherical caps (χ* = ¾), (b) one-half height spherical caps or hemispheres (χ* = ½), and (c) one-quarter height spherical caps (χ* = ¼). These three models are used to rationalize the effect of particle shape on the entropy-driven self-assembly of the particles under strong confinements; i.e., for 1 < H/χ < 2.5. As H is varied, a sequence of crystal structures is observed, including some having similar symmetry as that of the structures observed in confined hard spheres on account of the remaining spherical surface in the particles, but with additional features on account of the particle shapes having intrinsic anisotropy and orientational degrees of freedom. The χ* = ¾ system is found to exhibit a phase diagram that is most similar to the one obtained experimentally for the confined mushroom cap-shaped colloidal particles under confinement; however, simulations also reveal six other phases that occur on approaching the close-packing conditions for a given H. A qualitative global phase diagram is constructed that helps reveal the interrelations among different phases for all the particle shapes and confinements studied.

Weighted correlation approach: An extended version with applications to the hard-sphere fluid

The purpose of this study is to extend the weighted correlation approach (WCA) for inhomogeneous fluids. It now introduces a generic expression to evaluate the single-particle direct correlation function in terms of a series of pair direct correlation functions weighted by different correlation-weight functions and adjustable weight factors. When applied for practical use, however, approximations of the pair direct correlation functions have to be made, together with appropriate definitions of the weighted densities and the choices of the correlation-weight functions. The WCA approach would, then, not only help us to connect and compare different strategies and their underlying assumptions in the density functional approaches, but also enable us to propose and apply density functional theory methods to predict the density profile of, e.g., the hard-sphere fluid confined between a pair of parallel planar hard walls. Numerical results of the extended WCA approach, against the Monte Carlo (MC) simulations in a range of surface separations and bulk densities, suggest that it is capable of representing the fine features of the hard-sphere density distributions. The WCA results also agree well with the calculations from the fundamental measure theory. In addition, the thermodynamic self-consistency of the WCA approach is confirmed by its fairly good agreement with the MC fitted data for the surface tension of a hard-sphere fluid at a planar hard wall. All these tests show that a pure WCA approach can be constructed to investigate the states of ionic hard-sphere fluids.

Glass Transition in Confined Geometry

Extending mode-coupling theory, we elaborate a microscopic theory for the glass transition of liquids confined between two parallel flat hard walls. The theory contains the standard mode-coupling theory equations in bulk and in two dimensions as limiting cases and requires as input solely the equilibrium density profile and the structure factors of the fluid in confinement. We evaluate the phase diagram as a function of the distance of the plates for the case of a hard sphere fluid and obtain an oscillatory behavior of the glass transition line as a result of the structural changes related to layering.