Research issues
My research activity addresses a range of topics in the theory and numerical simulation of the statistical properties of condensed-matter systems. Past and ongoing themes of interest are summarized below:
RESIDUAL MULTIPARTICLE ENTROPY
SURFACE ROUGHENING AND PREROUGHENING
SELF-ASSEMBLY OF A MIXTURE OF DIMERS AND SPHERES
STRIPES IN SYMMETRIC MIXTURES OF HARD SPHERES
ZERO-TEMPERATURE PHASES OF SOFT-CORE BOSONS
PHASES OF BOSONIC PARTICLES IN A BUBBLE TRAP
My current research is focussed on the following problems:
- Phases and disordered motifs of a two-dimensional SALR fluid
- Self-assembly of a binary mixture of hard spheres away from equimolarity
- Phase diagram of a Bose fluid near collapse
I started doing research by investigating the high-density ordering of hard particles embedded in a spherical surface. Although the six-fold triangular order typical of a 2D solid is frustrated by surface curvature, a system of a few thousand particles shows reliable evidence of the formation of a defected solid where 5-fold and 7-fold disclinations cluster together at the vertices of a regular icosahedron [2,3]. This form of cooperative aggregation finds a simple explanation in terms of the maximum-entropy principle. In [4] I carried out an exact analysis of the statistical mechanics of a system of just four calottes, which allowed me to identify an ergodic transition to a high-density "phase" characterized by self-confinement of the particles.
RESIDUAL MULTIPARTICLE ENTROPY
I have long been involved in the testing of a phenomenological ordering criterion for simple fluids, originally devised by Giaquinta and Giunta, based on the evaluation of the so-called residual multi-particle entropy (RMPE), a quantity weighing up the contribution of multi-particle spatial correlations to the configurational entropy of the fluid [12,27]. When approaching the ordered phase from the fluid, the RMPE rapidly increases from negative to positive values, vanishing close to the transition point. Several continuous [1,5,19,22,33,58,67] and lattice systems [13,23] were investigated, both in two and in three dimensions, and in all cases the locus of RMPE zeroes turns out to be in a close correspondence with the phase boundary of the maximally disordered phase in the system. In [82] we have extended the derivation of the correlation expansion also to the entropy of a crystalline system.
SURFACE ROUGHENING AND PREROUGHENING
My initial activity at SISSA has been mainly focussed on the modelling of the thermal behaviour of a crystalline surface approaching the bulk melting point from low temperature. At the interface between crystal and vapour, various phenomena may occur, such as preroughening (PR), roughening, and surface melting (SM), which are all observed in e.g. Ar(111). First, I have devised a solid-on-solid (SOS) model of fcc(111) showing both PR and roughening [6,8]. The crucial ingredient for PR is a strong and sufficiently long-range (elastic) repulsion between parallel surface steps. In the disordered flat (DOF) phase between PR and roughening, the first surface layer is half-occupied. At PR, the solid-vacuum interface substantially broadens, the growth rate becomes a linear function of the disequilibrium, and the diffusion of a tracer particle is critically slowed down. The behaviour is somewhat different in a metal, i.e., in a particle-conserving surface [10]: when the system is prepared in a perfectly flat configuration, but external parameters call for a DOF state, the SOS surface spontaneously separates into two coexisting DOF phases with a step in between (an example is Cu(110)). I have also provided a very accurate variational theory of preroughening [11] which accounts for the switch occurring from integer to half-integer values of the mean surface height in equilibrium and for the broadening of the solid surface at a critical PR. Furthermore, the theory describes the growth mode correctly and shows the reentrant-layering phenomenon in adsorption. The interplay between PR and SM was studied in [9,14,15]. By modelling with a Potts interface the entropy gain from melting of the very first surface layer, it is found that PR, when present, always marks the onset of the SM process. Next, I carried out an extensive study of a terrace-step-kink model of vicinal surfaces where a short-range repulsion is present between parallel kinks [16,21]. Within such a model, a strong kink-kink repulsion induces PR of the vicinal, thus stabilizing an unconventional kind of DOF phase. Finally, I investigated the analog of surface preroughening in a two-dimensional system of unit and double charges on a lattice [23]. Depending on the model parameters, the PR transition of the reference surface may be first-order or continuous. Using the surface temperature as a control parameter, the dual Coulomb-gas system undergoes two consecutive phase transitions through phases whose dielectric behaviour was analyzed by simulation and exact finite-size calculations. At the analog of PR for charges, and provided the transition is continuous, the Coulomb gas behaves like an electric insulator while being metallic both below and above, at least up to the roughening temperature where the conventional metal-insulator transition occurs.
Another subject of investigation at SISSA was the structural characterization of ultrathin gold nanowires, which were found experimentally to be chiral, i.e., consisting of long strands of atoms that are wound up helically side by side around a monoatomic chain, to form cylindrically-shaped, two-shell tubes. From the theoretical standpoint, the spontaneous thinning down of gold necks suspended in vacuum between two bulk-like tips is driven by the minimization of the wire Gibbs free energy per unit length, a quantity we termed wire tension. LDA electronic-structure calculations for gold indeed confirmed that the wire tension reaches a local minimum value for a chiral tube, precisely the same as found in the experiment [17,20].
Under extreme conditions, the phase behaviour of many elemental substances markedly deviates from the conventional, Lennard-Jones paradigm. Rather than exceptions, unusual features such as a rich solid polymorphism and reentrant melting (i.e., melting of the crystal under isothermal increase of the pressure) appear to be the rule at very high pressure. Similar oddities are present in water, where various other thermodynamic, structural, and dynamic anomalies occur. In the last few years, a research line brought forward in Messina was aimed at seeking the minimal conditions that would allow a classical two-body isotropic interaction to reproduce the most spectacular water anomalies. The first result was recognizing that the exp-6 potential, a popular interaction for rare gases and metals at very high pressure [31,37,38,41], shares the same phenomenology of isotropic soft-core repulsions [30,32,33,40]. In fact, the full spectrum of three-dimensional melting, from "standard" to "anomalous", can be recovered with a unique shape of potential, of either the modified-inverse-power type [44] or the Yoshida-Kamakura type [39,43,48]. In particular, when the potential is marginally soft, one observes reentrant melting and other anomalies even in the absence of two repulsive length scales, implying that the existence of two such scales is not a necessary ingredient for observing anomalous melting [43]. Another versatile model fluid where unusual phase behaviors occur upon varying just a single parameter is the double-Gaussian fluid. We have considered two distinct attraction strengths, small and large. In the weak-attraction case, we find two distinct lines of reentrant melting and as many as four distinct solid phases [56]. In the strong-attraction case, the phase diagram of the system recalls that of water up to moderate pressures [58]. For Gaussian particles on a line, a thorough analysis revealed that, despite the absence of sharp phase transitions [59,63], a number of waterlike anomalies are present, including a minimum point of the number density at a temperature smaller than the location of the maximum [45], a feature only known for supercooled water confined within Silica nanopores. A similar density minimum has also been found for Gaussian particles embedded in a spherical surface [47] or confined in a narrow tube [51]. Melting is unconventional also in two-dimensional flat space. Here, the triangular crystal melts in two steps, via a bond-ordered hexatic phase [46]. In the reentrant-melting regime, also the hexatic phase happens to be anomalous, in that the system expands when it is cooled at constant pressure. When different solid phases occur in equilibrium, the melting mode is sensitive to the crystal symmetry, being two-stage for a triangular solid while standard one-stage for a square solid [50] or a cluster crystal [60]. Finally, interesting results were found for a double-Gaussian fluid pushed to the thermodynamic-stability threshold and even beyond [65,67,73]. When the strenght of attraction overcomes a critical threshold a double-Gaussian fluid becomes Ruelle-unstable, meaning that it eventually collapses to a finite-size cluster, even in the thermodynamic limit, thus becoming non-extensive. This phenomenon has a counterpart in the irreversible aggregation exhibited by a polymer dispersion when the amount of dissolved salt is increased beyond a certain limit. We find that, on approaching the stability threshold from the stable side, the liquid-vapor coexistence region undergoes an anomalous widening. Above the critical attraction strength the system is unstable, but its collective behavior is far from trivial: two separate regions of the thermodynamic plane can be identified according to the average value of the waiting time for collapse: this is finite and small on one side of the boundary, while apparently infinite in the other region.
This activity has been focussed on the study of a statistical model of closed surface which is meant to represent the interface between a nucleating phase (e.g. solid) and the metastable hosting phase (e.g. liquid). The reference theory for the kinetics of phase transitions is the time-honoured classical nucleation theory (CNT), which describes the nucleus as spherical and in equilibrium with the mother phase. CNT also assumes that the (size dependent, anisotropic) interface tension of a solid embryo is a simple scalar. My new theory [49], worked out in collaboration with Laio and Tosatti at SISSA, goes beyond CNT since it allows for a smooth, rather than sharp, interface and for small deviations of the nucleus away from the spherical shape. The main result is an expression of the interface tension as a function of the nominal radius R of the nucleus, where a universal non-mean-field, R -2lnR correction is also present. For the paradigmatic case of magnetization reversal in the three-dimensional Ising model, we have checked that the nucleation barrier derived from Monte Carlo simulation is well fitted by our formula, which thus provides a way to extract in one shot both the interface tension of the planar interface and the Tolman length. In [52], the problem of surface-tension anisotropy is specifically addressed. We have shown that a strong asphericity of the nucleus is reflected in the sign of the logarithmic prefactor in the free-energy cost of nucleation, which is opposite with respect to the isotropic case. Based on this result, we argue that the preferred shape of a big nucleus could be inferred from the sign of the prefactor, as determined from the optimization of the barrier profile obtained from simulation. In [57] we expressed the free-energy cost of nucleation also in terms of the surface area. Compared to the simpler theory based on a single collective variable, i.e., the cluster volume, the new barrier height is systematically larger. More importantly, depending on the physical situation at hand, the most probable shape of the nucleus may be highly non-spherical, even when the surface tension and stiffness of the model interface are isotropic. Finally, I have carried out a study of ice nucleation in water, using the simple monatomic-water model and three different protocols to identify solid-like particles in the simulation [74]. The chosen protocol and the ensuing reaction coordinate turn out to have a stronger effect on the nucleus size than on the height of the nucleation barrier. As a byproduct of the analysis, I have determined the structure of the nucleation cluster, finding that the relative amount of ice phases in the cluster depends on the method used for classifying solid-like particles. In particular, the phase which is most favored during the earlier stages of crystallization may happen to be different from the stable polymorph. Therefore, the quality of a reaction coordinate cannot be simply assessed from the barrier height computed, and collective variables appropriate to simple fluids may fail to provide a robust method of particle classification for monatomic water.
The decay of a supercooled liquid to equilibrium usually starts with the rapid (i.e., adiabatic) growth of the solid, which heats the system up to the equilibrium freezing temperature (this process, which is signaled by a sudden glowing of the material, is also known as recalescence). Only later will solidification be completed through the exchange of energy with the colder bath. We have used standard thermodynamics to predict what will be the immediate outcome of the recalescence stage, assuming it to be a near-equilibrium solid-liquid coexistence state. Two different systems were investigated: a mean-field fluid [53] and water [54,55], in two different virtual experimental settings, i.e., under isobaric and isochoric conditions, respectively. We compute the final temperature, pressure, and fraction of solid as a function of undercooling and of the amount of foreign gas which may possibly be present in the vessel. The calculation was then repeated under the more realistic hypothesis of a non-zero cost of the interface between solid and liquid. In this case, we identify a condition where the onset of solidification inevitably occurs at the wall in contact with the bath. In further experimental work [61] we reported the existence of a sharp dynamic transition in supercooled water between a fast and a slow decay regime at about 266.6 K.
We have recently demonstrated that thermodynamic integration works correctly on a path running through the liquid-vapor coexistence region [62]. In this region the system is heterogeneous; for a finite system simulated under periodic boundary conditions the shape of the liquid droplet goes through a series of abrupt changes on increasing the density, initially switching from spherical to cylindrical, then turning to slab-like. We have also formulated a simple theory reproducing the gross details of these morphological transitions [64].
SELF-ASSEMBLY OF A MIXTURE OF DIMERS AND SPHERES
We have studied the low-temperature structure of a diluted mixture of asymmetric dimers and spherical particles, a model relevant to the encapsulation of solutes in a colloidal dispersion [68,70,72,79]. Dimers and spheres are hard particles, with an additional short-range attraction between spheres and small monomers. During the simulation we observe the onset of sphere aggregates, held together by dimers, whose shape and typical size depend on the relative concentration of the species. When spheres have the same size of large monomers and the sphere concentration is low, we observe the formation of capsules of dimers wrapped around 1-2 spheres. As the concentration increases, aggregates get bigger and stretched until they form a network percolating throughout the simulation box. At still higher concentrations, we see in equilibrium a bilayer of spheres that are sandwiched between two layers of dimers. The same variety of self-organized structures is found in two dimensions, where bilayers are reduced to worm-like filaments. Occasionally, the two ends of a worm join together, thus giving rise to a two-dimensional vesicle. When the size of a sphere is 2-3 times bigger than the large monomer, the self-assembling scenario changes, getting richer overall with the addition of flexible membrane sheets with crystalline order and monolayer vesicles. In [85] the self-assembled structures of the same model are investigated in the confined geometry provided by a spherical surface. Again, we observe the formation of aggregates of various shapes as a function of the composition of the mixture and of the size of the guest particles. This study is relevant for understanding the behavior of colloidal particles deposited on an oil droplet.
STRIPES IN SYMMETRIC MIXTURES OF HARD SPHERES
We have recently introduced [92,94] a model that possibly provides the simplest binary mixture endowed with stripe order (i.e., a one-dimensional periodicity of the composition). The model consists of two species of identical hard spheres with equal concentration, which mutually interact through a square-well potential. By using Monte Carlo simulations and density functional theory, we show that stripes are present in both the liquid and solid phases when the attraction range is rather long. Our theory is accurate in reproducing the phases of the model, at least insofar as the composition inhomogeneities occur on length scales quite larger than the particle size. Then, using Monte Carlo simulations, we prove the existence of solid stripes even when the square well is much thinner than the particle diameter, making our model more similar to a real colloidal mixture. Finally, when the width of the attractive well is equal to the particle diameter, we observe a different and more complex form of compositional order in the solid, where each species of particles forms a regular porous matrix holding in its holes the other species, which witnesses a surprising variety of emergent behaviors for a very basic model of interaction.
ZERO-TEMPERATURE PHASES OF SOFT-CORE BOSONS
The thermodynamic properties of many weakly interacting soft-core bosons at zero temperature are well described by the Gross-Pitaevskii mean-field theory. As the pressure increases, particles undergo a quantum transition from superfluid to cluster supersolid. We have carried out a rigorous study of the various system phases using a variational formulation of the condensate wave function which is amenable to analytic manipulations. Not only is the adopted description quantitatively accurate, but it is also able to exactly predict the order (and sometimes even the location) of the mean-field transition. We have considered a few crystalline structures in two and three dimensions [76]. Depending on the lattice, the transition from fluid to solid may be discontinuous or continuous, a lower coordination entailing a milder transition. The analysis is then repeated for a system of one-dimensional penetrable bosons, which give rise to a quasi-condensate at zero temperature [77], and for a system of soft-core bosons embedded in a spherical surface. In the latter case, for radii up to a few interaction ranges we have examined the stability of a number of crystal-like arrangements having the symmetry of a regular or semi-regular polyhedron. As the radius increases, we find a sequence of transitions between different cluster phases, which, within our mean-field description, are all supersolid [78]. Ultracold bosonic atoms loaded in an optical lattice are instead described by (variants of) the Bose-Hubbard model. In addition to the common insulating and superfluid phases, density waves and supersolids may show up in the presence of a short-range interparticle repulsion and also depending on the geometry of the lattice. We herein explore this possibility, using the graph of a convex polyhedron as a lattice and playing with the coordination of nodes to promote the wanted finite-size ordering. To accomplish this we employ a decoupling approximation. We report zero-temperature results for two Catalan solids, for which a thorough ground-state analysis reveals the existence of insulating phases with polyhedral order and a widely extended supersolid region. The key to this outcome is the unbalance in coordination between inequivalent nodes of the graph. These predictions could be probed in systems of ultracold atoms using holographic optical tweezers [83,84,88].
PHASES OF BOSONIC PARTICLES IN A BUBBLE TRAP
We have considered a system of identical bosons at low temperature under an external field mimicking an isotropic bubble trap, which constrains particles to a portion of space close to a spherical surface. Using path integral Monte Carlo simulations, we examine the spatial structure and superfluid fraction in two cases [95]. For soft-core bosons, we find the existence of supersolid cluster arrangements with polyhedral symmetry (i.e., the same as at zero temperature) and we characterize the temperature behavior of the cluster phases. In the other case, of more immediate experimental interest, of a dipolar condensate on the sphere, we demonstrate how a quasi-one-dimensional supersolid of clusters is formed on a great circle for realistic values of density and interaction parameters. Crucially, this supersolid phase is only slightly disturbed by gravity. The predicted phases can be revealed in magnetic traps with spherical-shell geometry, possibly even in a lab on Earth.
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