Reentrant and isostructural transitions in a cluster-crystal former

Phase separation dynamics in a symmetric binary mixture of ultrasoft particles

Phase separation plays a key role in determining the self-assembly of biological and soft-matter systems. In biological systems, liquid-liquid phase separation inside a cell leads to the formation of various macromolecular aggregates. The interaction among these aggregates is soft, i.e., they can significantly overlap at a small energy cost. From a computer simulation point of view, these complex macromolecular aggregates are generally modeled by soft particles. The effective interaction between two particles is defined via the generalized exponential model of index n, with n = 4. Here, using molecular dynamics simulations, we study the phase separation dynamics of a size-symmetric binary mixture of ultrasoft particles. We find that when the mixture is quenched to a temperature below the critical temperature, the two components spontaneously start to separate. Domains of the two components form, and the equal-time order parameter reveals that the domain sizes grow with time in a power-law manner with an exponent of 1/3, which is consistent with the Lifshitz-Slyozov law for conserved systems. Furthermore, the static structure factor shows a power-law decay with an exponent of 4, consistent with the Porod law.

Phase behavior of mixtures of hard colloids and soft coarse-grained macromolecules

Effective "soft" interactions between macromolecules such as polymers, amphiphilic dendrimers, and suitably designed DNA based dendritic molecules have been shown to be purely repulsive and non-diverging. We report the structure and phase behavior of a mixture of hard colloids and soft coarse-grained macromolecules. Through the use of Reference Interaction Site Model theory and molecular dynamics simulations we find that hard colloids and soft macromolecules act as depletants toward each other, generating a medium-induced effective attraction. This effective attraction leads to the formation of non-dispersed phases at high densities. At low and high fractions of hard colloids the system macrophase separates into two disparate regions of hard colloids and soft macromolecules. However, this system microphase separates into a hard-rich and soft-rich self-assembled domains at intermediate compositions. The formation of microphase separated structure in this system of isotropic, disconnected, and purely repulsive colloids is surprising and quite novel. This behavior is likely due to a softening of the interface between hard-rich and soft-rich self-assembled domains. Molecular dynamics simulations have revealed that the microphase separated state has an overall disordered bicontinuous morphology. The hard-rich domain forms an ordered FCC structure and the soft-rich domain forms a disordered cluster-fluid, making the structure simultaneously ordered and disordered.

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.

On the yielding of a point-defect-rich model crystal under shear: insights from molecular dynamics simulations

In real crystals and at finite temperatures point defects are inevitable. Under shear their dynamics severely influence the mechanical properties of these crystals, giving rise to non-linear effects, such as ductility. In an effort to elucidate the complex behavior of crystals under plastic deformation it is crucial to explore and to understand the interplay between the timescale related to the equilibrium point-defect diffusion and the shear-induced timescale. Based on extensive non-equilibrium molecular dynamics simulations we present a detailed investigation on the yielding behavior of cluster crystals, an archetypical model for a defect-rich crystal: in such a system clusters of overlapping particles occupy the lattice sites of a regular (FCC) structure. In equilibrium particles diffuse via site-to-site hopping while maintaining the crystalline structure intact. We investigate these cluster crystals at a fixed density and at different temperatures where the system remains in the FCC structure: temperature allows us to vary the diffusion timescale appropriately. We then expose the crystal to shear, thereby choosing shear rates which cover timescales that are both higher and lower than the equilibrium diffusion timescales. We investigate the macroscopic and microscopic response of our cluster crystal to shear and find that the yielding scenario of such a system does not rely on the diffusion of the particles - it is rather related to the plastic deformation of the underlying crystalline structure. The local bond order parameters and the measurement of local angles between neighboring clusters confirm the cooperative movement of the clusters close to the yield point. Performing complementary, related simulations for an FCC crystal formed by harshly repulsive particles reveals similarities in the yielding behavior between both systems. Still we find that the diffusion of particles does influence characteristic features in the cluster crystal, such as a less prominent increase of order parameters close to the yield point. Our simulations provide for the first time an insight into the role of the diffusion of defects in the yielding behavior of a defect-rich crystal under shear. These observations will thus be helpful in the development of theories for the plastic deformation of defect-rich crystals.

Self-assembly of binary solutions to complex structures

Self-assembly in natural and synthetic molecular systems can create complex aggregates or materials whose properties and functionalities rise from their internal structure and molecular arrangement. The key microscopic features that control such assemblies remain poorly understood, nevertheless. Using classical density functional theory, we demonstrate how the intrinsic length scales and their interplay in terms of interspecies molecular interactions can be used to tune soft matter self-assembly. We apply our strategy to two different soft binary mixtures to create guidelines for tuning intermolecular interactions that lead to transitions from a fully miscible, liquid-like uniform state to formation of simple and core-shell aggregates and mixed aggregate structures. Furthermore, we demonstrate how the interspecies interactions and system composition can be used to control concentration gradients of component species within these assemblies. The insight generated by this work contributes toward understanding and controlling soft multi-component self-assembly systems. Additionally, our results aid in understanding complex biological assemblies and their function and provide tools to engineer molecular interactions in order to control polymeric and protein-based materials, pharmaceutical formulations, and nanoparticle assemblies.

Statistical Mechanics and Thermodynamics of Liquids and Crystals

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

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.

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.

Slow dynamics coupled with cluster formation in ultrasoft-potential glasses

We numerically investigate the slow dynamics of a binary mixture of ultrasoft particles interacting with the generalized Hertzian potential. If the softness parameter, α, is small, the particles at high densities start penetrating each other, form clusters, and eventually undergo the glass transition. We find multiple cluster-glass phases characterized by a different number of particles per cluster, whose boundary lines are sharply separated by the cluster size. Anomalous logarithmic slow relaxation of the density correlation functions is observed in the vicinity of these glass-glass phase boundaries, which hints the existence of the higher-order dynamical singularities predicted by the mode-coupling theory. Deeply in the cluster glass phases, it is found that the dynamics of a single particle is decoupled from that of the collective fluctuations.

Non-monotonic variations of the nucleation free energy in a glass-forming ultra-soft particles fluid

Using molecular dynamics simulation, we study the impact of the degree of supercooling on the crystal nucleation of ultra-soft particles, modeled with the Gaussian core potential. Focusing on systems with a high number density, our simulations reveal dramatically different behaviors as the degree of supercooling is varied. In the moderate supercooling regime, crystal nucleation proceeds as expected from classical nucleation theory, with a decrease in the free energy of nucleation, as well as in the size of the critical nucleus, as supercooling is increased. On the other hand, in the large supercooling regime, we observe an unusual reversal of behavior with an increase in the free energy of nucleation and in the critical size, as supercooling is increased. This unexpected result is analyzed in terms of the interplay between the glass transition and the crystal nucleation process. Specifically, medium range order crystal-like domains, with structural features different from that of the crystal nucleus, are found to form throughout the system when the supercooling is very large. These, in turn, play a pivotal role in the increase in the free energy of nucleation, as well as in the critical size, as the temperature gets closer to the glass transition.

Cluster Glass Transition of Ultrasoft-Potential Fluids at High Density

Using molecular dynamics simulation, we investigate the slow dynamics of a supercooled binary mixture of soft particles interacting with a generalized Hertzian potential. At low density, it displays typical slow dynamics near its glass transition temperature. At higher densities, particles bond together, forming clusters, and the clusters undergo the glass transition. The number of particles in a cluster increases one by one as the density increases. We demonstrate that there exist multiple cluster-glass phases characterized by a different number of particles per cluster, each of which is separated by distinct minima. Surprisingly, a so-called higher order singularity of the mode-coupling theory signaled by a logarithmic relaxation is observed in the vicinity of the boundaries between monomer and cluster glass phases. The system also exhibits rich and anomalous dynamics in the cluster glass phases, such as the decoupling of the self- and collective dynamics.

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.

Equilibrium Phase Behavior of a Continuous-Space Microphase Former

Periodic microphases universally emerge in systems for which short-range interparticle attraction is frustrated by long-range repulsion. The morphological richness of these phases makes them desirable material targets, but our relatively coarse understanding of even simple models hinders controlling their assembly. We report here the solution of the equilibrium phase behavior of a microscopic microphase former through specialized Monte Carlo simulations. The results for cluster crystal, cylindrical, double gyroid, and lamellar ordering qualitatively agree with a Landau-type free energy description and reveal the nontrivial interplay between cluster, gel, and microphase formation.

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.

Soft-core particles freezing to form a quasicrystal and a crystal-liquid phase

Systems of soft-core particles interacting via a two-scale potential are studied. The potential is responsible for peaks in the structure factor of the liquid state at two different but comparable length scales and a similar bimodal structure is evident in the dispersion relation. Dynamical density functional theory in two dimensions is used to identify two unusual states of this system: a crystal-liquid state, in which the majority of the particles are located on lattice sites but a minority remains free and so behaves like a liquid, and a 12-fold quasicrystalline state. Both are present even for deeply quenched liquids and are found in a regime in which the liquid is unstable with respect to modulations on the smaller scale only. As a result, the system initially evolves towards a small-scale crystal state; this state is not a minimum of the free energy, however, and so the system subsequently attempts to reorganize to generate the lower-energy larger-scale crystals. This dynamical process generates a disordered state with quasicrystalline domains and takes place even when this large scale is linearly stable, i.e., it is a nonlinear process. With controlled initial conditions, a perfect quasicrystal can form. The results are corroborated using Brownian dynamics simulations.

Cluster phases of penetrable rods on a line

Phase transitions are uncommon among homogeneous one-dimensional fluids of classical particles owing to a general nonexistence result due to van Hove. A way to circumvent van Hove's theorem is to consider an interparticle potential that is finite everywhere. Of this type is the generalized exponential model of index 4 (GEM4 potential), a model interaction which in three dimensions provides an accurate description of the effective pair repulsion between dissolved soft macromolecules (e.g., flexible dendrimers). Using specialized free-energy methods, I reconstruct the equilibrium phase diagram of the one-dimensional GEM4 system, showing that, apart from the usual fluid phase at low densities, it consists of an endless sequence of cluster fluid phases of increasing pressure, having a sharp crystal appearance for low temperatures. The coexistence line between successive phases in the sequence invariably terminates at a critical point. Focussing on the first of such transitions, I show that the growth of the two-cluster phase from the metastable ordinary fluid is extremely slow, even for large supersaturations. Finally, I clarify the apparent paradox of the observation of an activation barrier to nucleation in a system where, due to the dimensionality of the hosting space, the critical radius is expected to vanish.

Solidification in soft-core fluids: Disordered solids from fast solidification fronts

Using dynamical density functional theory we calculate the speed of solidification fronts advancing into a quenched two-dimensional model fluid of soft-core particles. We find that solidification fronts can advance via two different mechanisms, depending on the depth of the quench. For shallow quenches, the front propagation is via a nonlinear mechanism. For deep quenches, front propagation is governed by a linear mechanism and in this regime we are able to determine the front speed via a marginal stability analysis. We find that the density modulations generated behind the advancing front have a characteristic scale that differs from the wavelength of the density modulation in thermodynamic equilibrium, i.e., the spacing between the crystal planes in an equilibrium crystal. This leads to the subsequent development of disorder in the solids that are formed. In a one-component fluid, the particles are able to rearrange to form a well-ordered crystal, with few defects. However, solidification fronts in a binary mixture exhibiting crystalline phases with square and hexagonal ordering generate solids that are unable to rearrange after the passage of the solidification front and a significant amount of disorder remains in the system.

Large-area ordered Ge-Si compound quantum dot molecules on dot-patterned Si (001) substrates

We report on the formation of large-area ordered Ge-Si compound quantum dot molecules (CQDMs) in a combination of nanosphere lithography and self-assembly. Truncated-pyramid-like Si dots with {11n} facets are readily formed, which are spatially ordered in a large area with controlled period and size. Each Si dot induces four self-assembled Ge-rich dots at its base edges that can be fourfold symmetric along <110> directions. A model based on surface chemical potential accounts well for these phenomena. Our results disclose the critical effect of surface curvature on the diffusion and the aggregation of Ge adatoms and shed new light on the unique features and the inherent mechanism of self-assembled QDs on patterned substrates. Such a configuration of one Si QD surrounded by fourfold symmetric Ge-rich QDs can be seen as a CQDM with unique features, which will have potential applications in novel devices.

Controlled self-assembly of periodic and aperiodic cluster crystals

Soft particles are known to overlap and form stable clusters that self-assemble into periodic crystalline phases with density-independent lattice constants. We use molecular dynamics simulations in two dimensions to demonstrate that, through a judicious design of an isotropic pair potential, one can control the ordering of the clusters and generate a variety of phases, including decagonal and dodecagonal quasicrystals. Our results confirm analytical predictions based on a mean-field approximation, providing insight into the stabilization of quasicrystals in soft macromolecular systems, and suggesting a practical approach for their controlled self-assembly in laboratory realizations using synthesized soft-matter particles.

Phase behavior of a fluid with a double Gaussian potential displaying waterlike features

Pair potentials that are bounded at the origin provide an accurate description of the effective interaction for many systems of dissolved soft macromolecules (e.g., flexible dendrimers). Using numerical free-energy calculations, we reconstruct the equilibrium phase diagram of a system of particles interacting through a potential that brings together a Gaussian repulsion with a much weaker Gaussian attraction, close to the thermodynamic stability threshold. Compared to the purely repulsive model, only the reentrant branch of the melting line survives, since for lower densities solidification is overridden by liquid-vapor separation. As a result, the phase diagram of the system recalls that of water up to moderate (i.e., a few tens of MPa) pressures. Upon superimposing a suitable hard core on the double-Gaussian potential, a further transition to a more compact solid phase is induced at high pressure, which might be regarded as the analog of the ice I-to-ice III transition in water.

Continuous distributions of charges: extensions of the one component plasma

The electrostatic interaction between finite charge distributions, ρ(r), in a neutralizing background is considered as an extension of the one component plasma (OCP) model of point charges. A general form for the interaction potential is obtained which can be applied to molecular theories of many simple charged fluids and mixtures and to the molecular dynamics (MD) simulation of such systems. The formalism is applied to the study of a fluid of Gaussian charges in a neutralizing background by MD simulation and using hypernetted-chain integral equation theory. The treatment of these interactions is extended to a periodic system using a Fourier Transform formulation and, for a rapidly decaying charge distribution, an application of the Ewald method. The contributions of the self-energy and neutralizing background to the system's energy are explicitly included in the formulation. Calculations reveal differences in behavior from the OCP model when the Wigner-Seitz radius is of order and less than the Gaussian charge density decay length. For certain parameter values these systems can exhibit a multiple occupancy crystalline phase at high density which undergoes re-entrant melting at higher density. An exploration of the effects of the various length scales of the system on the equation of state and radial distribution function is made.

Probing the limitations of isotropic pair potentials to produce ground-state structural extremes via inverse statistical mechanics

Inverse statistical-mechanical methods have recently been employed to design optimized short-range radial (isotropic) pair potentials that robustly produce novel targeted classical ground-state many-particle configurations. The target structures considered in those studies were low-coordinated crystals with a high degree of symmetry. In this paper, we further test the fundamental limitations of radial pair potentials by targeting crystal structures with appreciably less symmetry, including those in which the particles have different local structural environments. These challenging target configurations demanded that we modify previous inverse optimization techniques. In particular, we first find local minima of a candidate enthalpy surface and determine the enthalpy difference ΔH between such inherent structures and the target structure. Then we determine the lowest positive eigenvalue λ(0) of the Hessian matrix of the enthalpy surface at the target configuration. Finally, we maximize λ(0)ΔH so that the target structure is both locally stable and globally stable with respect to the inherent structures. Using this modified optimization technique, we have designed short-range radial pair potentials that stabilize the two-dimensional kagome crystal, the rectangular kagome crystal, and rectangular lattices, as well as the three-dimensional structure of the CaF(2) crystal inhabited by a single-particle species. We verify our results by cooling liquid configurations to absolute zero temperature via simulated annealing and ensuring that such states have stable phonon spectra. Except for the rectangular kagome structure, all of the target structures can be stabilized with monotonic repulsive potentials. Our work demonstrates that single-component systems with short-range radial pair potentials can counterintuitively self-assemble into crystal ground states with low symmetry and different local structural environments. Finally, we present general principles that offer guidance in determining whether certain target structures can be achieved as ground states by radial pair potentials.

Hopping and diffusion of ultrasoft particles in cluster crystals in the explicit presence of a solvent

We have investigated diffusion and hopping processes in a cluster crystal formed from mesoscopic, ultrasoft particles. In contrast to previous contributions we have explicitly included in our investigations the microscopic solvent by using a simulation scheme that takes the induced hydrodynamic interactions into account as faithfully as possible. In our investigations we first focused on the processes of migration of the ultrasoft particles. By evaluating dynamical correlation functions we were able to demonstrate that the presence of the solvent does indeed have an important impact on the diffusion and hopping processes of the particles: this applies in particular to the diffusive behaviour, to the angular orientation of the jump events and to the spatial extents of these events. In a second set-up we have added non-cluster-forming ultrasoft particles to the system, investigating thus the impact of the solvent and that of the mutual interaction of the two species of ultrasoft particles on their respective dynamic behaviours. Our investigations clearly demonstrate, beside the expected significant role that the solvent plays in this set-up, that diffusion and the jump processes show distinct differences for the two particle species.

Microscopically resolved simulations prove the existence of soft cluster crystals

We perform extensive monomer-resolved computer simulations of suitably designed amphiphilic dendritic macromolecules over a broad range of densities, proving the existence and stability of cluster crystals formed in these systems, as predicted previously on the basis of effective pair potentials [B. M. Mladek et al., Phys. Rev. Lett. 96, 045701 (2006)]. Key properties of these crystals, such as the adjustment of their site occupancy with density and the possibility to heal defects by dendrimer migration, are confirmed on the monomer-resolved picture. At the same time, important differences from the predictions of the pair potential picture, stemming from steric crowding, arise as well, and they place an upper limit in the density for which such crystals can exist.

[N]pT ensemble and finite-size-scaling study of the critical isostructural transition in the generalized exponential model of index 4

First-order transitions of system where both lattice site occupancy and lattice spacing fluctuate, such as cluster crystals, cannot be efficiently studied by traditional simulation methods, which necessarily fix one of these two degrees of freedom. The difficulty, however, can be surmounted by the generalized [N]pT ensemble [J. Chem. Phys. 136, 214106 (2012)]. Here we show that histogram reweighting and the [N]pT ensemble can be used to study an isostructural transition between cluster crystals of different occupancy in the generalized exponential model of index 4 (GEM-4). Extending this scheme to finite-size scaling studies also allows us to accurately determine the critical point parameters and to verify that it belongs to the Ising universality class.

Phonon dispersions of cluster crystals

We analyze the ground states and the elementary collective excitations (phonons) of a class of systems, which form cluster crystals in the absence of attractions. Whereas the regime of moderate-to-high temperatures in the phase diagram has been analyzed in detail by means of density functional considerations (Likos et al 2007 J. Chem. Phys. 126 224502), the present approach focuses on the complementary regime of low temperatures. We establish the existence of an infinite cascade of isostructural transitions between crystals with different lattice site occupancies at T = 0 and we quantitatively demonstrate that the thermodynamic instabilities are bracketed by mechanical instabilities arising from long-wavelength acoustical phonons. We further show that all optical modes are degenerate and flat, giving rise to almost perfect realizations of Einstein crystals. We calculate analytically the complete phonon spectrum for the whole class of models as well as the Helmholtz free energy of the systems. On the basis of the latter, we demonstrate that the aforementioned isostructural phase transitions must terminate at an infinity of critical points at low temperatures, brought about by the anharmonic contributions in the Hamiltonian and the hopping events in the crystals.