Hexatic phase and cluster crystals of two-dimensional GEM4 spheres

Structural transition, orientational order, and anomalous specific heat in a two-dimensional dimer crystal of core-softened particles

Systems featuring hard-core-soft-shell repulsive pair potentials can form ordered phases, where particles organize themselves in aggregates with nontrivial geometries. The dimer crystal formed by one such potential, namely, the hard-core plus generalized exponential model of order 4, has been recently investigated, revealing a low-temperature structural phase transition, with the onset of nematic ordering of the dimers. In the present work, we aim to characterize this phase transition via a mean-field theory, by which a detailed analysis of the low-temperature properties of the system is carried out under quadrupole approximation. We determine the transition temperature and identify its order parameter, highlighting the link between the structural transition and the nematic ordering of the system. The first-order character of the transition is established and supported by the Landau expansion of the free energy in powers of the order parameter. The theory is subsequently generalized to take into account lattice vibrations and dimer length fluctuations. Finally, we provide an explanation for the anomalous behavior displayed by the specific heat in the vanishing-temperature limit, which is also supported by Monte Carlo simulations.

Ultrasoft Classical Systems at Zero Temperature

At low temperatures, classical ultrasoft particle systems develop interesting phases via the self-assembly of particle clusters. In this study, we reach analytical expressions for the energy and the density interval of the coexistence regions for general ultrasoft pairwise potentials at zero temperatures. We use an expansion in the inverse of the number of particles per cluster for an accurate determination of the different quantities of interest. Differently from previous works, we study the ground state of such models, in two and three dimensions, considering an integer cluster occupancy number. The resulting expressions were successfully tested in the small and large density regimes for the Generalized Exponential Model α, varying the value of the exponent.

Condensation and Crystal Nucleation in a Lattice Gas with a Realistic Phase Diagram

We reconsider model II of Orban et al. (J. Chem. Phys. 1968, 49, 1778−1783), a two-dimensional lattice-gas system featuring a crystalline phase and two distinct fluid phases (liquid and vapor). In this system, a particle prevents other particles from occupying sites up to third neighbors on the square lattice, while attracting (with decreasing strength) particles sitting at fourth- or fifth-neighbor sites. To make the model more realistic, we assume a finite repulsion at third-neighbor distance, with the result that a second crystalline phase appears at higher pressures. However, the similarity with real-world substances is only partial: Upon closer inspection, the alleged liquid−vapor transition turns out to be a continuous (albeit sharp) crossover, even near the putative triple point. Closer to the standard picture is instead the freezing transition, as we show by computing the free-energy barrier relative to crystal nucleation from the “liquid”.

Multiple 2D crystal structures in bilayered lamellae from direct self-assembly of 3D systems of soft Janus particles

Numerous crystals and Frank-Kasper phases in two-dimensional (2D) systems of soft particles have been presented by theoretical investigations. How to realize 2D crystals or Frank-kasper phases by direct self-assembly of...

Low-temperature ordering of the dimer phase of a two-dimensional model of core-softened particles

Purely pairwise interactions of the core-softened type, i.e., featuring a soft repulsion followed by a hard-core interaction at shorter distance, give rise to nontrivial equilibrium structures entirely different from the standard close packing of spheres. In particular, in a suitable low-temperature region of their phase diagram, such interactions are well known to favor a transition from a fluid to a cluster crystal. The residual mutual interaction between individual clusters can lead to the formation of patterns of their reciprocal orientations. In this work, we investigate two examples of such models in two dimensions, at the density most appropriate to the dimer phase, whereby clusters consist of just two particles, studying them with optimization techniques and Monte Carlo simulations. We focus on the dimer crystal, and unveil a second phase transition at extremely low temperature. This transition leads from a triangular dimer lattice with randomly disordered dimer orientations at high temperature to a reduced-symmetry ground state with nematic orientational order and a slightly distorted structure characterized by a centered-rectangular lattice at low temperature.

Statistical Mechanics and Thermodynamics of Liquids and Crystals

Thermodynamic phases are the most prominent manifestation of emergent behavior [...].

Efficient calculation of phase coexistence and phase diagrams: application to a binary phase-field-crystal model

We show that one can employ well-established numerical continuation methods to efficiently calculate the phase diagram for thermodynamic systems described by a suitable free energy functional. In particular, this involves the determination of lines of phase coexistence related to first order phase transitions and the continuation of triple points. To illustrate the method we apply it to a binary phase-field-crystal model for the crystallisation of a mixture of two types of particles. The resulting phase diagram is determined for one- and two-dimensional domains. In the former case it is compared to the diagram obtained from a one-mode approximation. The various observed liquid and crystalline phases and their stable and metastable coexistence are discussed as well as the temperature-dependence of the phase diagrams. This includes the (dis)appearance of critical points and triple points. We also relate bifurcation diagrams for finite-size systems to the thermodynamics of phase transitions in the infinite-size limit.

Cluster self-assembly condition for arbitrary interaction potentials

We present a sufficient criterion for the emergence of cluster phases in an ensemble of interacting classical particles with repulsive two-body interactions. Through a zero-temperature analysis in the low density region we determine the relevant characteristics of the interaction potential that make the energy of a two-particle cluster-crystal become smaller than that of a simple triangular lattice in two dimensions. The method leads to a mathematical condition for the emergence of cluster crystals in terms of the sum of Fourier components of a regularized interaction potential, which can be in principle applied to any arbitrary shape of interactions. We apply the formalism to several examples of bounded and unbounded potentials with and without cluster-forming ability. In all cases, the emergence of self-assembled cluster crystals is well captured by the presented analytic criterion and verified with known results from molecular dynamics simulations at vanishingly temperatures. Our work generalises known results for bounded potentials to repulsive potentials of arbitrary shape.

Emergence of an Ising critical regime in the clustering of one-dimensional soft matter revealed through string variables

Soft matter systems are renowned for being able to display complex emerging phenomena such as clustering phases. Recently, a surprising quantum phase transition has been revealed in a one-dimensional (1D) system composed of bosons interacting via a pairwise soft potential in the continuum. It was shown that the spatial coordinates undergoing two-particle clustering could be mapped into quantum spin variables of a 1D transverse Ising model. In this work we investigate the manifestation of an analogous critical phenomenon in 1D classical fluids of soft particles in the continuum. In particular, we study the low-temperature behavior of three different classical models of 1D soft matter, whose interparticle interactions allow for clustering. The same string variables highlight that, at the commensurate density for the two-particle cluster phase, the peculiar pairing of neighboring soft particles can be nontrivially mapped onto a 1D discrete classical Ising model. We also observe a related phenomenon, namely the presence of an anomalous peak in the low-temperature specific heat, thus indicating the emergence of Schottky phenomenology in a nonmagnetic fluid.

Ultracold Bosons on a Regular Spherical Mesh

Here, the zero-temperature phase behavior of bosonic particles living on the nodes of a regular spherical mesh ("Platonic mesh") and interacting through an extended Bose-Hubbard Hamiltonian has been studied. Only the hard-core version of the model for two instances of Platonic mesh is considered here. Using the mean-field decoupling approximation, it is shown that the system may exist in various ground states, which can be regarded as analogs of gas, solid, supersolid, and superfluid. For one mesh, by comparing the theoretical results with the outcome of numerical diagonalization, I manage to uncover the signatures of diagonal and off-diagonal spatial orders in a finite quantum system.

Freezing of a soft-core fluid in a one-dimensional potential: Predictions based on a pressure-balance equation

Using concepts from classical density functional theory, we investigate the freezing of a two-dimensional system of ultrasoft particles in a one-dimensional external potential, a phenomenon often called laser-induced freezing (LIF). In the first part of the paper, we present numerical results from free minimization of a mean-field density functional for a system of particles interacting via the GEM-4 potential. We show that the system does indeed display a LIF transition, although the interaction potential is markedly different from the cases studied before. We also introduce a suitably defined effective density within the potential wells ρ[over ¯]_{eff} as a control parameter of LIF, rather than the amplitude of the external potential as in the common LIF scenario. In the second part, we suggest a theoretical description of the onset of LIF which is based on the pressure-balance equation relating the pressure tensor and the external potential. Evaluating this equation for the modulated liquid phase at effective density ρ[over ¯]_{eff} and combining it with the (known) stability threshold of the corresponding bulk fluid, we can predict the critical effective density or, equivalently, the potential amplitude related to the onset of LIF. Our approach yields very good results for the model at hand and it is transferable, in principle, to other model systems.

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Phase field crystal (PFC) theory is extensively used for modeling the phase behavior, structure, thermodynamics, and other related properties of solids. PFC theory can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Here, we carefully identify all of these approximations and explain the consequences of each. One approximation that is made in standard derivations is to neglect a term of form ∇·[n∇Ln], where n is the scaled density profile and L is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and that dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in n. However, the consequences of doing this are (i) the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and (ii) other periodic structures, such as stripes, are erroneously predicted to be thermodynamic equilibrium structures. In general, L consists of a nonlocal convolution involving the pair direct correlation function. A second approximation sometimes made in deriving PFC theory is to replace L with a gradient expansion involving derivatives. We show that this leads to the possibility of the density going to zero, with its logarithm going to -∞ while being balanced by the fourth derivative of the density going to +∞. This subtle singularity leads to solutions failing to exist above a certain value of the average density. We illustrate all of these conclusions with results for a particularly simple model two-dimensional fluid, the generalized exponential model of index 4 (GEM-4), chosen because a DDFT is known to be accurate for this model. The consequences of the subsequent PFC approximations can then be examined. These include the phase diagram being both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal phase. Thus, although PFC models are very successful as phenomenological models of crystallization, we find it impossible to derive the PFC as a theory for the (scaled) density distribution when starting from an accurate DDFT, without introducing spurious artifacts. However, we find that making a simple one-mode approximation for the logarithm of the density distribution lnρ(x) rather than for ρ(x) is surprisingly accurate. This approach gives a tantalizing hint that accurate PFC-type theories may instead be derived as theories for the field lnρ(x), rather than for the density profile itself.

Phase diagram for two-dimensional layer of soft particles

The phase diagram of a monolayer of soft particles described by the Daoud-Cotton model for star polymers is presented. Ground state calculations and grand canonical Monte Carlo simulations are used to determine the phase behavior as a function of the number of arms on the star and the areal coverage of the soft particles. The phase diagram exhibits rich behavior including reentrant melting and freezing and solid-solid transitions with triangular, stripe, honeycomb and kagome phases. These structures in 2D are analogous to the structures observed in 3D. The evolution of the structure factor with density is qualitatively similar to that measured in experiments for polymer grafted nanocrystals [Chen et al., Macromolecules, 2017, 50, 9636].

Melting of a two-dimensional monodisperse cluster crystal to a cluster liquid

Monodisperse ensembles of particles that have cluster crystalline phases at low temperatures can model a number of physical systems, such as vortices in type-1.5 superconductors, colloidal suspensions, and cold atoms. In this work, we study a two-dimensional cluster-forming particle system interacting via an ultrasoft potential. We present a simple mean-field characterization of the cluster-crystal ground state, corroborating with Monte Carlo simulations for a wide range of densities. The efficiency of several Monte Carlo algorithms is compared, and the challenges of thermal equilibrium sampling are identified. We demonstrate that the liquid to cluster-crystal phase transition is of first order and occurs in a single step, and the liquid phase is a cluster liquid.

Formation of cluster crystals in an ultra-soft potential model on a spherical surface

We investigate the formation of cluster crystals with multiply occupied lattice sites on a spherical surface in systems of ultra-soft particles interacting via repulsive, bounded pair potentials. Not all interactions of this kind lead to clustering: we generalize the criterion devised in C. N. Likos et al., Phys. Rev. E, 2001, 63, 031206 to spherical systems in order to distinguish between cluster-forming systems and fluids which display reentrant melting. We use both DFT and Monte Carlo simulations to characterize the behavior of the system, and obtain semi-quantitative agreement between the two. We find that the number of clusters is determined by the ratio between the size σ of the ultra-soft particles and the radius R of the sphere in such a way that each stable configuration spans a certain interval of σ/R. Furthermore, we study the effect of topological frustration on the system due to the sphere curvature by comparing the properties of disclinations, i.e., clusters with fewer than six neighbors, and non-defective clusters. Disclinations are shown to be less stable, contain fewer particles, and be closer to their neighbors than other lattice points: these properties are explained on the basis of geometric and energetic considerations.

Lagrangian descriptors of driven chemical reaction manifolds

The persistence of a transition state structure in systems driven by time-dependent environments allows the application of modern reaction rate theories to solution-phase and nonequilibrium chemical reactions. However, identifying this structure is problematic in driven systems and has been limited by theories built on series expansion about a saddle point. Recently, it has been shown that to obtain formally exact rates for reactions in thermal environments, a transition state trajectory must be constructed. Here, using optimized Lagrangian descriptors [G. T. Craven and R. Hernandez, Phys. Rev. Lett. 115, 148301 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.148301], we obtain this so-called distinguished trajectory and the associated moving reaction manifolds on model energy surfaces subject to various driving and dissipative conditions. In particular, we demonstrate that this is exact for harmonic barriers in one dimension and this verification gives impetus to the application of Lagrangian descriptor-based methods in diverse classes of chemical reactions. The development of these objects is paramount in the theory of reaction dynamics as the transition state structure and its underlying network of manifolds directly dictate reactivity and selectivity.

Crystallization of soft matter under confinement at interfaces and in wedges

The surface freezing and surface melting transitions that are exhibited by a model two-dimensional soft matter system are studied. The behaviour when confined within a wedge is also considered. The system consists of particles interacting via a soft purely repulsive pair potential. Density functional theory (DFT) is used to calculate density profiles and thermodynamic quantities. The external potential due to the confining walls is modelled via a hard wall with an additional repulsive Yukawa potential. The surface phase behaviour depends on the range and strength of this repulsion: when the repulsion is weak, the wall promotes freezing at the surface of the wall. The thickness of this frozen layer grows logarithmically as the bulk liquid-solid phase coexistence is approached. Our mean-field DFT predicts that this crystalline layer at the wall must be nucleated (i.e. there is a free energy barrier) and its formation is necessarily a first-order transition, referred to as 'prefreezing', by analogy with the prewetting transition. However, in contrast to the latter, prefreezing cannot terminate in a critical point, since the phase transition involves a change in symmetry. If the wall-fluid interaction is sufficiently long ranged and the repulsion is strong enough, surface melting can occur instead. Then the interface between the wall and the bulk crystalline solid is wetted by the liquid phase as the chemical potential is decreased towards the value at liquid-solid coexistence. It is observed that the finite thickness fluid film at the wall has a broken translational symmetry due to its proximity to the bulk crystal, and so the nucleation of the wetting film can be either first order or continuous. Our mean-field theory predicts that for certain wall potentials there is a premelting critical point analogous to the surface critical point for the prewetting transition. When the fluid is confined within a linear wedge, this can strongly promote freezing when the opening angle of the wedge is commensurate with the crystal lattice.

Probing the existence of phase transitions in one-dimensional fluids of penetrable particles

Phase transitions in one-dimensional classical fluids are usually ruled out by using van Hove's theorem. A way to circumvent the conclusions of the theorem is to consider an interparticle potential that is everywhere bounded. Such is the case of, e.g., the generalized exponential model of index 4 (GEM-4 potential), which in three dimensions gives a reasonable description of the effective repulsion between flexible dendrimers in a solution. An extensive Monte Carlo simulation of the one-dimensional GEM-4 model [S. Prestipino, Phys. Rev. E 90, 042306 (2014)] has recently provided evidence of an infinite sequence of low-temperature cluster phases, however, also suggesting that upon pushing the simulation forward what seemed a true transition may eventually prove to be only a sharp crossover. We hereby investigate this problem theoretically by use of three different and increasingly sophisticated approaches (i.e., a mean-field theory, the transfer matrix of a lattice model of clusters, and the exact treatment of a system of point clusters in the continuum) to conclude that the alleged transitions of the one-dimensional GEM-4 system are likely just crossovers.