Phase behavior of aligned dipolar hard spheres: integral equations and density functional results

Ferrofluids and bio-ferrofluids: looking back and stepping forward

Ferrofluids investigated along for about five decades are ultrastable colloidal suspensions of magnetic nanoparticles, which manifest simultaneously fluid and magnetic properties. Their magnetically controllable and tunable feature proved to be from the beginning an extremely fertile ground for a wide range of engineering applications. More recently, biocompatible ferrofluids attracted huge interest and produced a considerable increase of the applicative potential in nanomedicine, biotechnology and environmental protection. This paper offers a brief overview of the most relevant early results and a comprehensive description of recent achievements in ferrofluid synthesis, advanced characterization, as well as the governing equations of ferrohydrodynamics, the most important interfacial phenomena and the flow properties. Finally, it provides an overview of recent advances in tunable and adaptive multifunctional materials derived from ferrofluids and a detailed presentation of the recent progress of applications in the field of sensors and actuators, ferrofluid-driven assembly and manipulation, droplet technology, including droplet generation and control, mechanical actuation, liquid computing and robotics.

Self-Assembly of Ionic Microgels Driven by an Alternating Electric Field: Theory, Simulations, and Experiments

The structural properties of a system of ionic microgels under the influence of an alternating electric field are investigated both theoretically and experimentally. This combined investigation aims to shed light on the structural transitions that can be induced by changing either the driving frequency or the strength of the applied field, which range from string-like formation along the field to crystal-like structures across the orthogonal plane. In order to highlight the physical mechanisms responsible for the observed particle self-assembly, we develop a coarse-grained description, in which effective interactions among the charged microgels are induced by both equilibrium ionic distributions and their time-averaged hydrodynamic responses to the applied field. These contributions are modeled by the buildup of an effective dipole moment at the microgels backbones, which is partially screened by their ionic double layer. We show that this description is able to capture the structural properties of this system, allowing for very good agreement with the experimental results. The model coarse-graining parameters are indirectly obtained via the measured pair distribution functions and then further assigned with a clear physical interpretation, allowing us to highlight the main physical mechanisms accounting for the observed self-assembly behavior.

Perturbation Theory versus Thermodynamic Integration. Beyond a Mean-Field Treatment of Pair Correlations in a Nematic Model Liquid Crystal

The phase behavior and structure of a simple square-well bulk fluid with anisotropic interactions is described in detail. The orientation dependence of the intermolecular interactions allows for the formation of a nematic liquid-crystalline phase in addition to the more conventional isotropic gas and liquid phases. A version of classical density functional theory (DFT) is employed to determine the properties of the model, and comparisons are made with the corresponding data from Monte Carlo (MC) computer simulations in both the grand canonical and canonical ensembles, providing a benchmark to assess the adequacy of the DFT results. A novel element of the DFT approach is the assumption that the structure of the fluid is dominated by intermolecular interactions in the isotropic fluid. A so-called augmented modified mean-field (AMMF) approximation is employed accounting for the influence of anisotropic interactions. The AMMF approximation becomes exact in the limit of vanishing density. We discuss advantages and disadvantages of the AMMF approximation with respect to an accurate description of isotropic and nematic branches of the phase diagram, the degree of orientational order, and orientation-dependent pair correlations. The performance of the AMMF approximations is found to be good in comparison with the MC data; the AMMF approximation has clear advantages with respect to an accurate and more detailed description of the fluid structure. Possible strategies to improve the DFT are discussed.

Thermodynamics of ferrofluids in applied magnetic fields

The thermodynamic properties of ferrofluids in applied magnetic fields are examined using theory and computer simulation. The dipolar hard sphere model is used. The second and third virial coefficients (B(2) and B(3)) are evaluated as functions of the dipolar coupling constant λ, and the Langevin parameter α. The formula for B(3) for a system in an applied field is different from that in the zero-field case, and a derivation is presented. The formulas are compared to results from Mayer-sampling calculations, and the trends with increasing λ and α are examined. Very good agreement between theory and computation is demonstrated for the realistic values λ≤2. The analytical formulas for the virial coefficients are incorporated in to various forms of virial expansion, designed to minimize the effects of truncation. The theoretical results for the equation of state are compared against results from Monte Carlo simulations. In all cases, the so-called logarithmic free energy theory is seen to be superior. In this theory, the virial expansion of the Helmholtz free energy is re-summed in to a logarithmic function. Its success is due to the approximate representation of high-order terms in the virial expansion, while retaining the exact low-concentration behavior. The theory also yields the magnetization, and a comparison with simulation results and a competing modified mean-field theory shows excellent agreement. Finally, the putative field-dependent critical parameters for the condensation transition are obtained and compared against existing simulation results for the Stockmayer fluid. Dipolar hard spheres do not undergo the transition, but the presence of isotropic attractions, as in the Stockmayer fluid, gives rise to condensation even in zero field. A comparison of the relative changes in critical parameters with increasing field strength shows excellent agreement between theory and simulation, showing that the theoretical treatment of the dipolar interactions is robust.

Magnetization of multicomponent ferrofluids

The solution of the mean spherical approximation (MSA) integral equation for isotropic multicomponent dipolar hard sphere fluids without external fields is used to construct a density functional theory (DFT), which includes external fields, in order to obtain an analytical expression for the external field dependence of the magnetization of ferrofluidic mixtures. This DFT is based on a second-order Taylor series expansion of the free energy density functional of the anisotropic system around the corresponding isotropic MSA reference system. The ensuing results for the magnetic properties are in quantitative agreement with our canonical ensemble Monte Carlo simulation data presented here.

Dipolar sticky hard spheres within the Percus-Yevick approximation plus orientational linearization

We consider a strongly idealized model for polar fluids, which consists of spherical particles, having, in addition to a hard-core repulsion, a "surface dipolar" interaction, acting only when particles are exactly at contact. A fully analytic solution of the molecular Orstein-Zernike equation is found for this potential, within the Percus-Yevick approximation complemented by a linearization of the angular dependence on molecular orientations (Percus-Yevick closure with orientational linearization). Numerical results are also presented in a detailed analysis about the local orientational structure. From the pair correlation function g(1,2), we first derive the best orientations of a test particle which explores the space around an arbitrary reference molecule. Then some local and global order parameters, related to the polarization induced by the reference particle, are also calculated. The local structure of this model with only short-ranged anisotropic interactions turns out to be, at least within the chosen approximation, qualitatively different from that of hard spheres with fully long-ranged dipolar potentials.

Phase diagram for stimulus-responsive materials containing dipolar colloidal particles

Dipolar colloidal particles self-assemble into a rich variety of microstructures ranging from co-crystals of unusual symmetry, to open networks (gels) of cross-linked chains of particles. We use molecular dynamics computer simulation to explore the self-assembly, structure, crystallization and/or gelation of systems of colloid particles with permanent dipole moments immersed in a high-dielectric solvent. Particle-particle interactions are modeled with a discontinuous potential. The phase diagram in the temperature-packing fraction plane is calculated. Several types of phases are found in our simulations: ordered phases including face-centered-cubic, hexagonal-close-packed, and body-centered-tetragonal at high packing fractions, and fluid, string-fluid, and gel phases at low packing fractions. The very low volume fraction gel phases and the well-ordered crystal phases are promising for advanced materials applications.

Phase behavior of dipolar hard and soft spheres

We study the phase behavior of hard and soft spheres with a fixed dipole moment using Monte Carlo simulations. The spheres interact via a pair potential that is a sum of a hard-core Yukawa (or screened-Coulomb) repulsion and a dipole-dipole interaction. The system can be used to model colloids in an external electric or magnetic field. Two cases are considered: (i) colloids without charge (or dipolar hard spheres) and (ii) colloids with charge (or dipolar soft spheres). The phase diagram of dipolar hard spheres shows fluid, face-centered-cubic (fcc), hexagonal-close-packed (hcp), and body-centered-tetragonal (bct) phases. The phase diagram of dipolar soft spheres shows, in addition to the above mentioned phases, a body-centered-orthorhombic (bco) phase, and is in agreement with the experimental phase diagram [Nature (London) 421, 513 (2003)]. In both cases, the fluid phase is inhomogeneous but we find no evidence of a gas-liquid phase separation. The validity of the dipole approximation is verified by a multipole moment expansion.

Equilibrium polymerization in the Stockmayer fluid as a model of supermolecular self-organization

A diverse range of molecular self-organization processes arises from a competition between directional and isotropic van der Waals intermolecular interactions. We conduct Monte Carlo simulations of the Stockmayer fluid (SF) with a large dipolar interaction as a minimal self-organization model and focus on basic thermodynamic properties that are needed to characterize the polymerization transition that occurs in this fluid. In particular, we determine the polymerization transition lines from the maximum in the specific heat, C(v), and the inflection point in the extent of polymerization, Phi. We also characterize the geometry (radius of gyration R(g), chain length L, chain topology) of the clusters that form in this associating fluid as a function of temperature, T, and concentration, rho . The pressure, P, and the second virial coefficient, B2, were determined, since these properties contain essential information about the strength of the isotropic (van der Waals) interactions. Our simulations indicate that the locations of the polymerization lines are quantitatively consistent with a model of equilibrium polymerization with the enthalpy of polymerization ("sticking energy") fixed by the minimum in the intermolecular potential. The polymerization transition in the SF is accompanied by a topological transition from predominantly linear to ring polymers upon cooling that is driven by the minimization of the dipolar energy of the clusters. We also find that the basic interaction parameters describing polymerization and phase separation in the SF can be estimated based on the existing theory of equilibrium polymerization, but the theory must be refined to account for ring formation in order to accurately describe the configurational properties of this model self-organizing fluid.

Crystal structures of two-dimensional magnetic colloids in tilted external magnetic fields

The stability of different crystal lattices of two-dimensional superparamagnetic suspensions that are confined to a planar liquid-gas interface and exposed to a tilted external magnetic field is studied theoretically by lattice sum minimizations. The magnetic field induces magnetic dipoles onto the colloidal particles along its direction, whose strength can be controlled by the amplitude of the external field. The mutual interaction between the colloids is governed by dipole-dipole forces and a short-ranged repulsion having its physical origin at the presence of the colloidal cores. If the direction of the magnetic field is perpendicular to the liquid-gas interface, there is a purely repulsive interaction leading to stable triangular crystals. By tilting the external field, the interaction becomes anisotropic and a mutual attraction appears upon a threshold tilt angle. We have calculated the full phase diagram at zero temperature varying the tilt angle, the colloidal density, and the strength of the magnetic field. Apart from the triangular lattice we find a variety of stable crystal lattices including rectangular, oblique, chainlike oblique, and rhombic structures. We also present the accurate derivation of the Hamiltonian of two polarizable particles of finite arbitrary geometries in external magnetic and electric fields.

Spontaneous ferromagnetic ordering in magnetic fluids

This paper is devoted to the theoretical justification of spontaneous orientational order in magnetic fluids. We study the self-consistent solutions of the Bogoliubov-Born-Green-Kirkwood-Yvon equation connecting the one-particle distribution function with the pair correlation function. This self-consistent approach is used in the specific density functional method and proves to be equivalent to the mean field theory. On the basis of the second-order perturbation method over the intensity of dipole-dipole interparticle interaction the following effect is discovered: the self-consistent density functional approach leads to the spontaneous "ferrimagnetic" state of the magnetic fluid induced by the dipole-dipole interaction. This strange result seems to be physically meaningless and prejudices the validity of the density functional methods and mean field theories applied to orientational microstructure in ferrofluids.

Density functional theory of solvation in a polar solvent: extracting the functional from homogeneous solvent simulations

In the density functional theory formulation of molecular solvents, the solvation free energy of a solute can be obtained directly by minimization of a functional, instead of the thermodynamic integration scheme necessary when using atomistic simulations. In the homogeneous reference fluid approximation, the expression of the free-energy functional relies on the direct correlation function of the pure solvent. To obtain that function as exactly as possible for a given atomistic solvent model, we propose the following approach: first to perform molecular simulations of the homogeneous solvent and compute the position and angle-dependent two-body distribution functions, and then to invert the Ornstein-Zernike relation using a finite rotational invariant basis set to get the corresponding direct correlation function. This rather natural scheme is proved, for the first time to our knowledge, to be valuable for a dipolar solvent involving long range interactions. The resulting solvent free-energy functional can then be minimized on a three-dimensional grid around a solute to get the solvent particle and polarization density profiles and solvation free energies. The viability of this approach is proven in a comparison with "exact" molecular dynamics calculations for the simple test case of spherical ions in a dipolar solvent.

Theory of structural transformations in ferrofluids: chains and "gas-liquid" phase transitions

We consider a ferrofluid consisting of identical spherical particles with a permanent magnetic moment. Under the assumption that linear flexible chains can appear in the ferrofluid, we estimate the distribution function of the number of particles inside the chain. The analysis is done and simple expressions for the size distribution function are obtained in asymptotics of a strong magnetic interaction between the particles inside one chain. We studied the influence of the linear chains on conditions and scenarios of bulk "gas-liquid" phase transition in the ensemble of the particles under an infinitely strong magnetic field. In order to study the influence of the chains on bulk "gas-liquid" phase transition in the ensemble of the particles, their chemical potential mu is calculated in the model of separate interacting particles as well as in the model with chains, taking into account the interaction between them. When the temperature is low enough, van der Waals loops appear on the plots of mu versus volume concentration phi of the particles in the first model; function mu(phi) increases monotonically in the second model for all examined temperatures. This means that the condensation "gas-liquid" phase transition can take place in the model of individual particles; however, formation of the chains in real ferrofluids prevents the appearance of this transition.