Phase behavior of dipolar hard and soft spheres

Phase separation of a magnetic fluid: Asymptotic states and nonequilibrium kinetics

We study self-assembly in a colloidal suspension of magnetic particles by performing comprehensive molecular dynamics simulations of the Stockmayer (SM) model, which comprises spherical particles decorated by a magnetic moment. The SM potential incorporates dipole-dipole interactions along with the usual Lennard-Jones interaction and exhibits a gas-liquid phase coexistence observed experimentally in magnetic fluids. When this system is quenched from the high-temperature homogeneous phase to the coexistence region, the nonequilibrium evolution to the condensed phase proceeds with the development of spatial as well as magnetic order. We observe density-dependent coarsening mechanisms-a diffusive growth law ℓ(t)∼t^{1/3} in the nucleation regime and hydrodynamics-driven inertial growth law ℓ(t)∼t^{2/3} in the spinodal regimes. [ℓ(t) is the average size of the condensate at time t after the quench.] While the spatial growth is governed by the expected conserved order parameter dynamics, the growth of magnetic order in the spinodal regime exhibits unexpected nonconserved dynamics. The asymptotic morphologies have density-dependent shapes which typically include the isotropic sphere and spherical bubble morphologies in the nucleation region, and the anisotropic cylinder, planar slab, cylindrical bubble morphologies in the spinodal region. The structures are robust and nonvolatile, and exhibit characteristic magnetic properties. For example, the oppositely magnetized hemispheres in the spherical morphology impart the characteristics of a Janus particle to it. The observed structures have versatile applications in catalysis, drug delivery systems, memory devices, and magnetic photonic crystals, to name a few.

Self-Assembly at a Macroscale Using Aerodynamics

Intuitive self-assembly devices are of great significance to the emerging applications of self-assembly theory. In this paper, a novel intuitive device with an aerodynamic system is fabricated for the self-assembly experiment. Table tennis balls were used as the objects to be assembled during the self-assembly process. To understand more about the system, two experiments were designed—the directed assembly experiment was conducted to organize a specific structure and to explore the influences of environmental variables, and the indirect assembly experiment repeated with the “bottom-up” self-organization process and expressed the characteristics of “the optimization” and “the emergence” in the self-organization process. This article expressed a novel self-assembly approach at a macroscale and created a new choice or idea for the structural design and the optimization method.

Electric-Field-Driven Assembly of Dipolar Spheres Asymmetrically Confined between Two Electrodes

Externally applied electric fields have previously been utilized to direct the assembly of colloidal particles confined at a surface into a large variety of colloidal oligomers and nonclose-packed honeycomb lattices (J. Am. Chem. Soc. 2013, 135, 7839-7842). The colloids under such confinement and fields are observed to spontaneously organize into bilayers near the electrode. To extend and better understand how particles can come together to form quasi-two-dimensional materials, we have performed Monte Carlo simulations and complementary experiments of colloids that are strongly confined between two electrodes under an applied alternating current electric field, controlling field strength and particle area fraction. Of particular importance, we control the fraction of particles in the upper vs lower plane, which we describe as asymmetric confinement, and which effectively modulates the coordination number of particles in each plane. We model the particle-particle interactions using a Stockmayer potential to capture the dipolar interactions induced by the electric field. Phase diagrams are then delineated as a function of the control parameters, and a theoretical model is developed in which the energies of several idealized lattices are calculated and compared. We find that the resulting theoretical phase diagrams agree well with simulation. We have not only reproduced the structures observed in experiments using parameters that are close to experimental conditions but also found several previously unobserved phases in the simulations, including a network of rectangular bands, zig zags, and a sigma lattice, which we were then able to confirm in experiment. We further propose a simple way to precisely control the number ratio of particles between different planes, that is, superimposing a direct current electric field with the alternating current electric field, which can be implemented conveniently in experiments. Our work demonstrates that a diverse collection of materials can be assembled from relatively simple ingredients, which can be analyzed effectively through comparison of simulation, theory, and experiment. Our model further explains possible pathways between different phases and provides a platform for examining phases that have yet to be observed in experiments.

Chain Formation and Phase Separation in Ferrofluids: The Influence on Viscous Properties

Ferrofluids have attracted considerable interest from researchers and engineers due to their rich set of unique physical properties that are valuable for many industrial and biomedical applications. Many phenomena and features of ferrofluids' behavior are determined by internal structural transformations in the ensembles of particles, which occur due to the magnetic interaction between the particles. An applied magnetic field induces formations, such as linear chains and bulk columns, that become elongated along the field. In turn, these structures dramatically change the rheological and other physical properties of these fluids. A deep and clear understanding of the main features and laws of the transformations is necessary for the understanding and explanation of the macroscopic properties and behavior of ferrofluids. In this paper, we present an overview of experimental and theoretical works on the internal transformations in these systems, as well as on the effect of the internal structures on the rheological effects in the fluids.

Biaxial nematics of hard cuboids in an external field

By computer simulation, we model the phase behaviour of colloidal suspensions of board-like particles under the effect of an external field and assess the still disputed occurrence of the biaxial nematic (NB) liquid crystal phase. The external field promotes the rearrangement of the initial isotropic (I) or uniaxial nematic (NU) phase and the formation of the NB phase. In particular, very weak field strengths are sufficient to spark a direct I-NB or NU-NB phase transition at the self-dual shape, where prolate and oblate particle geometries fuse into one. By contrast, forming the NB phase at any other geometry requires stronger fields and thus reduces the energy efficiency of the phase transformation. Our simulation results show that self-dual shaped board-like particles with moderate anisotropy are able to form NB liquid crystals under the effect of a surprisingly weak external stimulus and suggest a path to exploit low-energy uniaxial-to-biaxial order switching.

Transmutable Colloidal Crystals and Active Phase Separation via Dynamic, Directed Self-Assembly with Toggled External Fields

A diverse set of functional materials can be fabricated by assembling dispersions of colloids and nanoparticles. Two principal engineering challenges prevent efficient production of these materials: first, scalable synthesis of particles with carefully tailored interactions required to generate complex structures, and second, the propensity of such materials to arrest in undesirable metastable states. Active assembly processes, such as dynamic, directed self-assembly in which the interactions among particles are externally controlled and vary over time, offer a promising method to address these challenges. For dispersions of polarizable dielectric or paramagnetic nanoparticles, an effective mode of active assembly can be achieved by toggling an external electric or magnetic field, which induces attractive particle interactions, on and off cyclically over time. Here, we develop computational and theoretical models for such active assembly processes and find that cyclically toggling the external field leads to growth of colloidal crystals at significantly faster rates and with many fewer defects than for assembly in a steady field. The active process stabilizes phases that are only metastable in steady fields, including a dense fluid phase and body-centered orthorhombic crystals. The growth mechanism and terminal structure of the dispersion are easily controlled by the toggling protocol, and the toggle parameters can be used to continuously transmute between crystal structures with different lattice parameters. Finally, we show how results from linear irreversible thermodynamics can be used to predict the dissipative terminal states of the active assembly process in terms of parameters of the toggling protocol.

Magnetic Field-Induced Assembly of Superparamagnetic Cobalt Nanoparticles on Substrates and at Liquid-Air Interface

Superparamagnetic cobalt nanoparticles (Co NPs) are an interesting material for self-assembly processes because of their magnetic properties. We investigated the magnetic field-induced assembly of superparamagnetic cobalt nanoparticles and compared three different approaches, namely, the assembly on solid substrates, at water-air, and ethylene glycol-air interfaces. Oleic acid- and trioctylphosphine oxide-coated Co NPs were synthesized via a thermolysis of cobalt carbonyl and dispersed into either hexane or toluene. The Co NP dispersion was dropped onto different substrates (e.g., transmission electron microscopy (TEM) grid, silicon wafer) and onto liquid surfaces. Transmission electron microscopy (TEM), scanning force microscopy, optical microscopy, as well as scanning electron microscopy showed that superparamagnetic Co NPs assembled into one-dimensional chains in an external magnetic field. By varying the concentration of the Co NP dispersion (1-5 mg/mL) and the strength of the magnetic field (4-54 mT), the morphology of the chains changed. Short, thin, and flexible chain structures were obtained at low NP concentration and low strength of magnetic field, whereas they became long, thick and straight when the NP concentration and the magnetic field strength increased. In comparison, the assembly of Co NPs from hexane dispersion at ethylene glycol-air interface showed the most regular and homogeneous alignment, since a more efficient spreading could be achieved on ethylene glycol than on water and solid substrates.

Field-Directed Self-Assembly of Mutually Polarizable Nanoparticles

Directed assembly of dielectric and paramagnetic nanoparticles can be used to synthesize diverse functional materials that polarize in response to an externally applied electric or magnetic field. However, theories capable of predicting the self-assembled states are lacking. In the proposed work, we develop a complete thermodynamic description of such assemblies for spherical nanoparticles. We show how an important physical feature of these types of particles, mutual polarization, sculpts the free energy landscape and has a remarkably strong influence on the nature of the self-assembled states. Modeling the mutual polarization among nanoparticles requires solving a many-bodied problem for the particle dipole moments. Typically, this computationally expensive task is avoided by neglecting mutual polarization and assuming that each particle in a concentrated dispersion acquires the same dipole moment as a single, isolated particle. Although valid in the limit of small dielectric or permeability contrasts between particles and solvent, this constant dipole assumption leads to qualitatively incorrect predictions for coexisting phases in equilibrium at large dielectric or permeability contrasts. Correctly accounting for mutual polarization enables a thermodynamic theory that describes the equilibrium phase diagram of polarizable dispersions in terms of experimentally controllable variables. Our theoretical predictions agree with the phase behavior we observe in dynamic simulations of these dispersions as well as that in experiments of field-directed structural transitions. In contrast to predictions of a constant dipole model, we find that dispersions of particles with different dielectric constants or magnetic permeabilities exhibit qualitatively different phase behavior. This new model also predicts the existence of a eutectic point at which two crystalline phases and a disordered phase of nanoparticles all simultaneously coexist.

Deswelling behaviour of ionic microgel particles from low to ultra-high densities

The swelling of ionic microgel particles is investigated at a wide range of concentrations using a combination of light, X-ray and neutron scattering techniques. We employ a zero-average contrast approach for small-angle neutron scattering experiments, which enables a direct determination of the form factor at high concentrations. The observed particle size initially decreases strongly with the particle concentration in the dilute regime but approaches a constant value at intermediate concentrations. This is followed by a further deswelling at high concentrations above particle overlap. Theory and experiments point at a pivotal contribution of dangling polymer ends to the strong variation in size of ionic microgels, which presents itself mainly through the hydrodynamics properties of the system.

Dipolar Crystals: The Crucial Role of the Clinohexagonal Prism Phase

We report a new phase called clinohexagonal prism (CHP) that accounts for all the ground states of dipolar hard spheres prepared at any density. This phase merely consists of an oblique prismatic lattice with a hexagonal base. Our calculations show that at intermediate densities, a special close packed body-centered orthorhombic phase coincides with the CHP phase in the ground state for a wide density window. In the high packing regime, i.e., in the vicinity of the density of the hexagonal close packed phase, it is a limiting case of the CHP phase with vanishing obliquity that emerges. These findings provide a unified and clarified view of the solid-solid transitions occurring at zero temperature in dipolar systems and should be relevant in other related molecular or soft matter systems governed by anisotropic (and possibly isotropic) soft potentials.

Colloidal molecules assembled from binary spheres under an AC electric field

Colloidal particles are envisioned as analogues of atoms and molecules, however they often lack the complexities present in their counterparts. In this work, we report the assembly of colloidal molecules from a binary mixture of polystyrene spheres (1, 1.6, 2, and 4 μm) under an alternating current electric field. The rich family of assembled oligomers typically consists of a large sphere that is closely surrounded by a number of smaller petal particles, driven by the dipolar attraction between large and small particles. In deionized water, the number of satellite particles, i.e., the coordination number increases with the increasing size ratio of the constituent particles. For a given size ratio, the coordination number decreases with the increasing frequency of the applied field. These trends have also been correctly captured by computing the electric energy of different oligomers based on induced dipolar and double-layer interactions. By suspending the particles in polyvinylpyrrolidone aqueous solution, we can further tune the bond length of the oligomers independent of their coordination numbers. The addition of polyvinylpyrrolidone also allows us to lock the assembled colloidal molecules so that they remain intact after the electric field is turned off. Our method provides a robust way to produce a family of colloidal molecules with well-defined geometry and high yield.

Self-assembly of three-dimensional ensembles of magnetic particles with laterally shifted dipoles

We consider a model of colloidal spherical particles carrying a permanent dipole moment which is laterally shifted out of the particles' geometrical centres, i.e. the dipole vector is oriented perpendicular to the radius of the particles. Varying the shift δ from the centre, we analyse ground state structures for two, three and four hard spheres, using a simulated annealing procedure. We also compare earlier ground state results. We then consider a bulk system at finite temperatures and different densities. Using molecular dynamics simulations, we examine the equilibrium self-assembly properties for several shifts. Our results show that the shift of the dipole moment has a crucial impact on both the ground state configurations as well as the self-assembled structures at finite temperatures.

Self-Assembly of Crystalline Structures of Magnetic Core-Shell Nanoparticles for Fabrication of Nanostructured Materials

A theoretical study is presented of the template-assisted formation of crystalline superstructures of magnetic-dielectric core-shell particles. The templates produce highly localized gradient fields and a corresponding magnetic force that guides the assembly with nanoscale precision in particle placement. The process is studied using two distinct and complementary computational models that predict the dynamics and energy of the particles, respectively. Both mono- and polydisperse colloids are studied, and the analysis demonstrates for the first time that although the particles self-assemble into ordered crystalline superstructures, the particle formation is not unique. There is a Brownian motion-induced degeneracy in the process wherein various distinct, energetically comparable crystalline structures can form for a given template geometry. The models predict the formation of hexagonal close packed (HCP) and face centered cubic (FCC) structures as well as mixed phase structures due to in-plane stacking disorders, which is consistent with experimental observations. The polydisperse particle structures are less uniform than the monodisperse particle structures because of the irregular packing of different-sized particles. A comparison of self-assembly using soft- and hard-magnetic templates is also presented, the former being magnetized in a uniform field. This analysis shows that soft-magnetic templates enable an order-of-magnitude more rapid assembly and much higher spatial resolution in particle placement than their hard-magnetic counterparts. The self-assembly method discussed is versatile and broadly applies to arbitrary template geometries and multilayered and multifunctional mono- and polydisperse core-shell particles that have at least one magnetic component. As such, the method holds potential for the bottom-up fabrication of functional nanostructured materials for a broad range of applications. This work provides unprecedented insight into the assembly process, especially with respect to the viability and potential fundamental limitations of realizing structure-dependent material properties for applications.

Hierarchical self-assembly of colloidal magnetic particles into reconfigurable spherical structures

Colloidal self-assembly has enormous potential as a bottom-up means of structure fabrication. Here we demonstrate hierarchical self-assembly of rationally designed charge-stabilised colloidal magnetic particles into ground state structures that are topologically equivalent to a snub cube and a snub dodecahedron, the only two chiral Archimedean solids, for size-selected clusters. These spherical structures open up in response to an external magnetic field and demonstrate controllable porosity. Such features are critical to their applications as functional materials.

Switching plastic crystals of colloidal rods with electric fields

When a crystal melts into a liquid both long-ranged positional and orientational order are lost, and long-time translational and rotational self-diffusion appear. Sometimes, these properties do not change at once, but in stages, allowing states of matter such as liquid crystals or plastic crystals with unique combinations of properties. Plastic crystals/glasses are characterized by long-ranged positional order/frozen-in-disorder but short-ranged orientational order, which is dynamic. Here we show by quantitative three-dimensional studies that charged rod-like colloidal particles form three-dimensional plastic crystals and glasses if their repulsions extend significantly beyond their length. These plastic phases can be reversibly switched to full crystals by an electric field. These new phases provide insight into the role of rotations in phase behaviour and could be useful for photonic applications.

Nematic ordering of polarizable colloidal rods in an external electric field: theory and experiment

The orientation of dielectric colloidal rods dispersed in a dielectric fluid medium exposed to an external electric field: theory and confocal microscopy measurements.

An experimental and simulation study on the self-assembly of colloidal cubes in external electric fields

When a suspension of colloidal particles is placed in an oscillating electric field, the contrast in dielectric constant between the particles and the solvent induces a dipole moment in each of the colloidal particles. The resulting dipole-dipole interactions can strongly influence the phase behavior of the system. We investigate the phase behavior of cube-shaped colloidal particles in electric fields, using both experiments and Monte Carlo simulations. In addition to a string fluid phase and a body centered tetragonal (BCT) crystal phase, we observe a columnar phase consisting of hexagonally ordered strings of rotationally disordered cubes. By simulating the system for a range of pressures and electric field strengths, we map out the phase diagram, and compare the results to the experimentally observed phases. Additionally, we estimate the accuracy of a point-dipole approximation on the alignment of cubes in string-like clusters.

Colloidal structures of asymmetric dimers via orientation-dependent interactions

We apply an AC electric field to induce anisotropic interactions among asymmetric colloidal dimers. These anisotropic interactions, being shape-specific and orientation-dependent, can create complex and unique structures that are not possible for spherical particles or symmetric dimers. More specifically, we show a series of novel structures that closely resemble one- and two-dimensional antiferromagnetic lattices, including small clusters, linear chains, square lattices, and frustrated triangular arrays. All of them are uniquely formed by alternating association between dimers with opposite orientations. Our theoretical model attributes those patterns to an exquisite balance between electrostatic (primarily dipolar) and electrohydrodynamic interactions. Although similarly oriented dimers are strongly repulsive, the oppositely oriented dimers possess a concave shoulder in the pair interaction, which favors clustering to minimize the number of overlaps between neighboring particles. By combining the anisotropy in both particle geometry and field-induced interaction, our work suggests a new way to tailor colloidal interactions on anisotropic particles, which is important for both scientific understanding and practical applications.

Effect of external electric fields on the phase behavior of colloidal silica rods

We examine the effect of external electric fields on the behavior of colloidal silica rods. We find that the electric fields can be used to induce para-nematic and para-smectic phases, and to reduce the number of defects in smectic phases. At high field strengths, a new crystal structure was observed that consisted of strings of rods ordered in a hexagonal pattern in which neighboring rods were shifted along their length. We also present a simple model to describe this system, which we used in computer simulations to calculate the phase diagram for rods of L/D = 6, with L the end-to-end length of the rods and D the diameter of the rods. Our theoretical predictions for the phase behavior agree well with the experimental observations.

Size effect and stability of polarized fluid phases

The existence of a ferroelectric fluid phase for systems of 1000-2000 dipolar hard or soft spheres is well established by numerical simulations. Theoretical approaches proposed to determine the stability of such a phase are either in qualitative agreement with the simulation results or disagree with them. Experimental results for systems of molecules or particles with large electric or magnetic dipole moments are also inconclusive. As a contribution to the question of existence and stability of a fluid ferroelectric phase this simulation work considers system sizes of the order of 10 000 particles, thus an order of magnitude larger than those used in previous studies. It shows that although ferroelectricity is not affected by an increase of system size, different spatial arrangements of the dipolar hard spheres in such a phase are possible whose free energies seem to differ only marginally.

Orientation of a dielectric rod near a planar electrode

We present experimental and theoretical results on suspensions of silica rods in DMSO–water, subjected to an applied electric field, in particular on the interaction exhibited between the rods and the electrode used for generating the electric field.

Simulation of a two-dimensional model for colloids in a uniaxial electric field

We perform Monte Carlo simulations of a simplified two-dimensional model for colloidal hard spheres in an external uniaxial ac electric field. Experimentally, the external field induces dipole moments in the colloidal particles, which in turn form chains. We therefore approximate the system as composed of well-formed chains of dipolar hard spheres of a uniform length. The dipolar interaction between colloidal spheres gives rise to an effective interaction between the chains, which we treat as disks in a plane, that includes a short-range attraction and long-range repulsion. Hence, the system favors finite clustering over bulk phase separation, and indeed we observe at low temperature and density that the system does form a cluster phase. As the density increases, percolation is accompanied by a pressure anomaly. The percolated phase, despite being composed of connected, locally crystalline domains, does not bear the typical signatures of a hexatic phase. At very low densities, we find no indication of a "void phase" with a cellular structure seen recently in experiments.

Non-equilibrium magnetic colloidal dispersions at liquid-air interfaces: dynamic patterns, magnetic order and self-assembled swimmers

Colloidal dispersions of interacting particles subjected to an external periodic forcing often develop nontrivial self-assembled patterns and complex collective behavior. A fundamental issue is how collective ordering in such non-equilibrium systems arises from the dynamics of discrete interacting components. In addition, from a practical viewpoint, by working in regimes far from equilibrium new self-organized structures which are generally not available through equilibrium thermodynamics can be created. In this review spontaneous self-assembly phenomena in magnetic colloidal dispersions suspended at liquid-air interfaces and driven out of equilibrium by an alternating magnetic field are presented. Experiments reveal a new type of nontrivially ordered self-assembled structures emerging in such systems in a certain range of excitation parameters. These dynamic structures emerge as a result of the competition between magnetic and hydrodynamic forces and have complex unconventional magnetic ordering. Nontrivial self-induced hydrodynamic fields accompany each out-of-equilibrium pattern. Spontaneous symmetry breaking of the self-induced surface flows leading to a formation of self-propelled microstructures has been discovered. Some features of the self-localized structures can be understood in the framework of the amplitude equation (Ginzburg-Landau type equation) for parametric waves coupled to the conservation law equation describing the evolution of the magnetic particle density and the Navier-Stokes equation for hydrodynamic flows. To understand the fundamental microscopic mechanisms governing self-assembly processes in magnetic colloidal dispersions at liquid-air interfaces a first-principle model for a non-equilibrium self-assembly is presented. The latter model allows us to capture in detail the entire process of out-of-equilibrium self-assembly in the system and reproduces most of the observed phenomenology.

Self-assembly of nanoparticles into heterogeneous structures with gradient material properties

We present a mechanism to form self-assembled functional gradient superlattice structures by subjecting binary nanoparticles in an electric field. The interaction among different dipoles leads to the controllable formation of diverse structures, including particle columns with gradient material properties from inside to outside and various hierarchical layered or three-dimensional particle chain networks. We elucidate how permittivity, volume fraction, particle size, and the frequency of the electric field can be utilized to control the morphology of the induced structures, which would enable designed nanofabrication.

Directed self-assembly of colloidal dumbbells with an electric field

We demonstrate the assembly of colloidal particles with the shape of diatomic molecules ("dumbbells") into crystals that we study with confocal microscopy. The literature on the preparation of nonspherical colloidal particles has grown steadily. Assembly of these particles into regular three-dimensional crystalline lattices, however, is rarely, if ever, achieved and has not yet been studied quantitatively in 3D real space. We find that, by application of an electric field, such particles assemble quite readily. By varying the particle aspect ratio, range of interactions, and electric field strength, we find several different crystal structures of which three have never before been observed. Moreover, the electric field can be used to switch between different structures and manipulate/switch the photonic properties. Moreover, our work sheds light on fundamental questions related to the self-assembly of nonspherical particles.

Low-density ordered phase in brownian dipolar colloidal suspensions

We study the low volume fraction and electric field phase behavior of a Brownian colloidal suspension. On the application of a uniform ac field, we find a novel phase where chains of particles aggregate to form a well defined cellular network, consisting of particle-free "voids" surrounded by a percolating network of particle-rich walls. This cellular structure is stable to very long times, indicative of an equilibrium thermodynamic phase. The cell-cell spacing is not sensitive to the concentration of the sample but scales with sample thickness. Any self-consistent mechanism for the existence of this void phase must consist of long-ranged repulsions and shorter-ranged attractions.

Dipolar structures in magnetite ferrofluids studied with small-angle neutron scattering with and without applied magnetic field

Field-induced structure formation in a ferrofluid with well-defined magnetite nanoparticles with a permanent magnetic dipole moment was studied with small-angle neutron scattering (SANS) as a function of the magnetic interactions. The interactions were tuned by adjusting the size of the well-defined, single-magnetic-domain magnetite (Fe3O4) particles and by applying an external magnetic field. For decreasing particle dipole moments, the data show a progressive distortion of the hexagonal symmetry, resulting from the formation of magnetic sheets. The SANS data show qualitative agreement with recent cryogenic transmission electron microscopy results obtained in 2D [Klokkenburg, Phys. Rev. Lett. 97, 185702 (2006)] on the same ferrofluids.

Simple models of complex aggregation: vesicle formation by soft repulsive spheres with dipolelike interactions

Structural and thermodynamic properties of spherical particles carrying classical spins are investigated by Monte Carlo simulations. The potential energy is the sum of short range, purely repulsive pair contributions, and spin-spin interactions. These last are of the dipole-dipole form, with however, a crucial change of sign. At low density and high temperature the system is a homogeneous fluid of weakly interacting particles and short range spin correlations. With decreasing temperature particles condense into an equilibrium population of free floating vesicles. The comparison with the electrostatic case, giving rise to predominantly one-dimensional aggregates under similar conditions, is discussed. In both cases condensation is a continuous transformation, provided the isotropic part of the interatomic potential is purely repulsive. At low temperature the model allows us to investigate thermal and mechanical properties of membranes. At intermediate temperatures it provides a simple model to investigate equilibrium polymerization in a system giving rise to predominantly two-dimensional aggregates.