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The properties of ferrogels in homogeneous magnetic fields are studied using a simple microscopic model and Monte Carlo simulations. The main phenomena of interest concern the anisotropy and enhancement of the elastic moduli that result from applying uniform magnetic fields before and after the magnetic grains are locked in to the polymer-gel matrix by cross-linking reactions. The positional organization of the magnetic grains is influenced by the application of a magnetic field during gel formation, leading to a pronounced anisotropy in the mechanical response of the ferrogel to an applied magnetic field. In particular, the elastic moduli can be enhanced to different degrees depending on the mutual orientation of the fields during and after ferrogel formation. The model represents ferrogels by ensembles of dipolar spheres dispersed in elastic matrices. Experimental trends are shown to be reflected accurately in the simulations of the microscopic model. In addition, the simulations yield microscopic insights on the organization of the magnetic grains. Finally, simple relationships between the elastic moduli and the magnetization are proposed. If supplemented by the magnetization curve, these relationships yield the dependencies of the elastic moduli on the applied magnetic field, which are often measured directly in experiments. [ABSTRACT FROM AUTHOR]

The dynamic magnetic response of a ferrofluid to a weak ac magnetic field is studied using statistical mechanical theory and Brownian dynamics simulations, taking account of dipole-dipole interactions between the constituent ferromagnetic colloidal particles, and the presence of a range of particle sizes. The effects of interactions and polydispersity on the frequency dispersion are shown to be substantial: the amplitude of the response can be about twice that of a noninteracting system; the frequency for peak power loss can be reduced by about one half; and polydispersity effects can even change the qualitative appearance of the susceptibility spectrum. [ABSTRACT FROM AUTHOR]

The frequency-dependent magnetic susceptibility of a ferrofluid is calculated under the assumption that the constituent particles undergo Brownian relaxation only. Brownian-dynamics simulations are carried out in order to test the predictions of a recent theory [A. O. Ivanov, V. S. Zverev, and S. S. Kantorovich, Soft Matter 12, 3507 (2016)] that includes the effects of interparticle dipole-dipole interactions. The theory is based on the so-called modified mean-field approach and possesses the following important characteristics: in the low-concentration, noninteracting regime, it gives the correct single-particle Debye-theory results; it yields the exact leading-order results in the zero-frequency limit; it includes particle polydispersity correctly from the outset; and it is based on firm theoretical foundations allowing, in principle, systematic extensions to treat stronger interactions and/or higher concentrations. The theory and simulations are compared in the case of a model monodisperse ferrofluid, where the effects of interactions are predicted to be more pronounced than in a polydisperse ferrofluid. The susceptibility spectra are analyzed in detail in terms of the low-frequency behavior, the position of the peak in the imaginary (out-of-phase) part, and the characteristic decay time of the magnetization autocorrelation function. It is demonstrated that the theory correctly predicts the trends in all of these properties with increasing concentration and dipolar coupling constant, the product of which is proportional to the Langevin susceptibility χL. The theory is in quantitative agreement with the simulation results as long as χL≲1. [ABSTRACT FROM AUTHOR]

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 (B2 and B3) are evaluated as functions of the dipolar coupling constant λ, and the Langevin parameter a. The formula for S3 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 A. and a 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. [ABSTRACT FROM AUTHOR]

The thermodynamic properties of the dipolar hard-sphere fluid are studied using theory and simulation. A new theory is derived using a convenient mathematical approximation for the Helmholtz free energy relative to that for the hard-sphere fluid. The approximation is designed to give the correct low-density virial expansion. New theoretical and numerical results for the fourth virial coefficient are given. Predictions of thermodynamic functions for dipolar coupling constants λ = 1 and 2 show excellent agreement with simulation results, even at the highest value of the particle volume fraction φ. For higher values of λ, there are deviations at high volume fractions, but the correct low-density behavior is retained. The theory is compared critically against the established thermodynamic perturbation theory; it gives significant improvements at low densities and is more convenient in terms of the required numerics. Dipolar hard spheres provide a basic model for ferrofluids, and the theory is accurate for typical experimental parameters λ ≲ 2 and φ ≲ 0.1. This is demonstrated explicitly by fitting osmotic equations of state for real ferrofluids measured recently by analytical centrifugation. [ABSTRACT FROM AUTHOR]

A theory for the frequency-dependent magnetic susceptibility of a ferrofluid in a static uniform magnetic field is developed, including the dipolar interactions between the constituent particles. Interactions are included within the framework of modified mean-field theory. Predictions are given for the linear responses of the magnetization to a probing ac field both parallel and perpendicular to the static field and are tested against results from Brownian dynamics simulations. The effects of the particle concentration and dipolar coupling constant on the field-dependent static susceptibilities and the frequency dispersions are shown to be substantial, which justifies taking proper account of the interactions between particles. The theory is reliable provided that the volume concentration and dipolar coupling constant are not too large and within the range of values for real ferrofluids. [ABSTRACT FROM AUTHOR]

The effects of particle-size polydispersity on the magnetostatic properties of concentrated ferrofluids are studied using theory and computer simulation. The second-order modified mean-field (MMF2) theory of Ivanov and Kuznetsova [Phys. Rev. E 64, 041405 (2001)] has been extended by calculating additional terms of higher order in the dipolar coupling constant in the expansions of the initial magnetic susceptibility and the magnetization curve. The theoretical predictions have been tested rigorously against results from Monte Carlo simulations of model monodisperse, bidisperse, and highly polydisperse ferrofluids. Comparisons have been made between systems with the same Langevin susceptibility and the same saturation magnetization. In all cases, the new theoretical magnetization curve shows better agreement with simulation data than does the MMF2 theory. As for the initial susceptibility, MMF2 theory is most accurate for the monodisperse model, while the new theory works best for polydisperse systems with a significant proportion of large particles. These results are important for the analysis and characterization of recently synthesized polydisperse ferrofluids with record-breaking values of the initial magnetic susceptibility. [ABSTRACT FROM AUTHOR]

The dynamics of the growth and relaxation of the magnetization in ferrofluids are determined using theory based on the Fokker-Planck-Brown equation, and Brownian-dynamics simulations. Magnetization growth starting from an equilibrium nonmagnetized state in zero field, and following an instantaneous application of a uniform field of arbitrary strength, is studied with and without interparticle interactions. Similarly, magnetization relaxation is studied starting from an equilibrium magnetized state in a field of arbitrary strength, and following instantaneous removal of the field. In all cases, the dynamics are studied in terms of the time-dependent magnetization m(t). The field strength is described by the Langevin parameter α, the strength of the interparticle interactions is described by the Langevin susceptibility χ_{L}, and the individual particles undergo Brownian rotation with time τ_{B}. For noninteracting particles, the average growth time decreases with increasing α due to the torque exerted by the field, while the average relaxation time stays constant at τ_{B}; with vanishingly weak fields, the timescales coincide. The same basic picture emerges for interacting particles, but the weak-field timescales are larger due to collective particle motions, and the average relaxation time exhibits a weak, nonmonotonic field dependence. A comparison between theoretical and simulation results is excellent for noninteracting particles. For interacting particles with χ_{L}=1 and 2, theory and simulations are in qualitative agreement, but there are quantitative deviations, particularly in the weak-field regime, for reasons that are connected with the description of interactions using effective fields.

When a ferrofluid is magnetized in a strong magnetic field, and then the field is switched off, the magnetization decays from its saturation value to zero. The dynamics of this process are controlled by the rotations of the constituent magnetic nanoparticles, and for the Brownian mechanism, the respective rotation times are strongly influenced by the particle size and the magnetic dipole-dipole interactions between the particles. In this work, the effects of polydispersity and interactions on the magnetic relaxation are studied using a combination of analytical theory and Brownian dynamics simulations. The theory is based on the Fokker-Planck-Brown equation for Brownian rotation and includes a self-consistent, mean-field treatment of the dipole-dipole interactions. The most interesting predictions from the theory are that, at short times, the relaxation of each particle type is equal to its intrinsic Brownian rotation time, while at long times, each particle type has the same effective relaxation time, which is longer than any of the individual Brownian rotation times. Noninteracting particles, though, always relax at a rate controlled only by the Brownian rotation times. This illustrates the importance of including the effects of polydispersity and interactions when analyzing the results from magnetic relaxometry experiments on real ferrofluids, which are rarely monodisperse.

The dynamic magnetic susceptibility, χ(ω), of a model ferrofluid at a very low concentration (volume fraction, approximately 0.05%), and with a range of dipolar coupling constants (1≤λ≤8), is examined using Brownian dynamics simulations. With increasing λ, the structural motifs in the system change from unclustered particles, through chains, to rings. This gives rise to a nonmonotonic dependence of the static susceptibility χ(0) on λ and qualitative changes to the frequency spectrum. The behavior of χ(0) is already understood, and the simulation results are compared to an existing theory. The single-particle rotational dynamics are characterized by the Brownian time, τ_{B}, which depends on the particle size, carrier-liquid viscosity, and temperature. With λ≤5.5, the imaginary part of the spectrum, χ^{''}(ω), shows a single peak near ω∼τ_{B}^{-1}, characteristic of single particles. With λ≥5.75, the spectrum is dominated by the low-frequency response of chains. With λ≥7, new features appear at high frequency, which correspond to intracluster motions of dipoles within chains and rings. The peak frequency corresponding to these intracluster motions can be computed accurately using a simple theory.

The dynamics of magnetization relaxation in ferrofluids are studied with statistical-mechanical theory and Brownian dynamics simulations. The particle dipole moments are initially perfectly aligned, and the magnetization is equal to its saturation value. The magnetization is then allowed to decay under zero-field conditions toward its equilibrium value of zero. The time dependence is predicted by solving the Fokker-Planck equation for the one-particle orientational distribution function. Interactions between particles are included by introducing an effective magnetic field acting on a given particle and arising from all of the other particles. Two different approximations are proposed and tested against simulations: a first-order modified mean-field theory and a modified Weiss model. The theory predicts that the short-time decay is characterized by the Brownian rotation time τ_{B}, independent of the interaction strength. At times much longer than τ_{B}, the asymptotic decay time is predicted to grow with increasing interaction strength. These predictions are borne out by the simulations. The modified Weiss model gives the best agreement with simulation, and its range of validity is limited to moderate, but realistic, values of the dipolar coupling constant.

In a recent paper [Phys. Rev. E 75, 011506 (2007)] Hentschke presented new results on the vapor-liquid transition in the Stockmayer fluid. In this Comment, we analyze the data further and demonstrate that at high dipole strength, the isotropic attractive part of the Stockmayer potential is negligible compared to the thermal energy at the vapor-liquid critical point. The results therefore suggest that the transition is largely driven by the dipolar interactions.

The neighbor network in a two-dimensional polydisperse hard-disk fluid with diameter distribution p(sigma) approximately sigma(-4) is examined using constant-pressure Monte Carlo simulations. Graphs are constructed from vertices (disks) with edges (links) connecting each vertex to k neighboring vertices defined by a radical tessellation. At packing fractions in the range 0.24< or =eta< or =0.36, the decay of the network degree distribution is observed to be consistent with the power law k(-gamma) where the exponent lies in the range 5.6< or =gamma< or =6.0 . Comparisons with the predictions of a maximum-entropy theory suggest that this apparent power-law behavior is not the asymptotic one and that p(k) approximately k(-4) in the limit k-->infinity. This is consistent with the simple idea that for large disks, the number of neighbors is proportional to the disk diameter. A power-law decay of the network degree distribution is one of the characteristics of a scale-free network. The assortativity of the network is measured and is found to be positive, meaning that vertices of equal degree are connected more often than in a random network. Finally, the equation of state is determined and compared with the prediction from a scaled-particle theory. Very good agreement between simulation and theory is demonstrated.

Experimental magnetization curves for a polydisperse ferrofluid at various concentrations are examined using analytical theories and computer simulations with the aim of establishing a robust method for obtaining the magnetic-core diameter distribution function p(x). Theoretical expressions are fitted to the experimental data to yield the parameters of p(x). It is shown that the majority of available theories yield results that depend strongly on the ferrofluid concentration, even though the magnetic composition should be fixed. The sole exception is the second-order modified mean-field (MMF2) theory of Ivanov and Kuznetsova [Phys. Rev. E 64, 041405 (2001)] which yields consistent results over the full experimental range of ferrofluid concentration. To check for consistency, extensive molecular dynamics and Monte Carlo simulations are performed on systems with discretized versions of p(x) corresponding as closely as possible to that of the real ferrofluid. Essentially perfect agreement between experiment, theory, and computer simulation is demonstrated. In addition, the MMF2 theory provides excellent predictions for the initial susceptibility measured in simulations.

The kinetics of aggregation in a monolayer of dipolar particles are studied using stochastic dynamics computer simulations. Transient concentrations of end defects (at low density) and Y-shaped defects (at high density) clearly exceed those at equilibrium. Although very large dipole moments are expected to disfavor such defects at equilibrium, it is found that the transient defect concentrations increase with increasing dipole moment. The results suggest that the conditions for defect-driven condensation--as proposed by Tlusty and Safran [T. Tlusty and S. A. Safran, Science 290, 1328 (2000)]--could be met by kinetic trapping, giving rise to a metastable phase transition between isotropic fluid phases.

Motivated by the diversity and complexity of two-dimensional (2D) crystals formed by triangular proteins and protein trimers, we have investigated the structures and phase behavior of hard-disk trimers. In order to mimic specific binding interactions, each trimer possesses an "attractive" disk which can interact with similar disks on other trimers via an attractive square-well potential. At low density and low temperature, the fluid phase mainly consists of tetramers, pentamers, or hexamers. Hexamers provide the structural motif for a high-density, low-temperature periodic solid phase, but we also identify a metastable periodic structure based on a tetramer motif. At high density there is a transition between orientationally ordered and disordered solid phases. The connections between simulated structures and those of 2D protein crystals--as seen in electron microscopy--are briefly discussed.

The dynamical properties of a model one-component core-softened fluid with purely repulsive interactions are found to be very complex. At low temperature the fluid structure exhibits cluster motifs including dimers, stripes, and polygons, depending on density. Single-particle diffusion and the velocity, shear-stress, and wave-vector-dependent current correlation functions have all been calculated using molecular dynamics simulations. The results highlight the presence of well-resolved single-particle and collective motions, which is remarkable for what is essentially a "simple" one-component fluid.

The phase behavior of model linear and bent-core molecules has been studied using isothermal-isobaric Monte Carlo computer simulations. The molecular model consists of seven Lennard-Jones spheres rigidly arranged in a "V" shape, with external bond angle, gamma. With gamma=0 degrees (linear molecules), we find isotropic, nematic, untilted smectic A, and two layered phases in which the molecules are tilted with respect to the layer normal. The latter two phases correspond to distinct branches in the equation of state, and possess different types of ordering within and between the layers; these phases are tentatively assigned as being smectic B and crystal. Apart from the possible existence of a tilted smectic B, the phase behavior of this system is broadly in line with earlier simulation studies on related linear molecular models. In the gamma=20 degrees system, isotropic, nematic, and tilted smectic- B phases are observed. Interestingly, the range of stability of the nematic phase is enhanced compared to the gamma=0 degrees system. In simulations of the gamma=40 degrees system, nematic phases are absent, and only isotropic and tilted phases are in evidence. The in-layer structure of the tilted phases shows a very clear change from smectic- B to smectic- A ordering upon increasing the temperature. In all instances of a tilted phase, the degree of molecular tilt is in the region of 30+/-5 degrees, with respect to the smectic layer normal, which corresponds closely to typical experimental observations in real bent-core liquid crystals. In our model, the tilt provides efficient packing of the spheres and favorable attractive interactions between molecules. The relevance of the present simulation model to real bent-core liquid crystals is discussed critically.

The structure and phase behavior of a two-dimensional system with purely repulsive core-softened and long-range interactions are studied using Monte Carlo computer simulations. The pair interactions are of the form, u(r)=4epsilon[(sigma/r)(12)-(sigma/r)(6)]+epsilon(')(sigma/r)(3), with the energy parameter, epsilon(')=(8sqrt[6]/9)epsilon chosen to give a stationary point of inflection in the pair potential at r=6(1/6)sigma. This potential approximates the effective interparticle interactions for a two-dimensional dipolar system in a strong field aligned perpendicular to the plane. The low-temperature portion of the phase diagram is sketched out, and the static properties of the various phases are analyzed in some detail. At low temperatures a variety of interesting states are in evidence, including: fluids with chainlike, striped, and 6-10 sided polygon structural motifs; low-density and high-density triangular crystalline phases; and defective Kagomé lattices. It is shown that clustering is driven by the presence of the repulsive shoulder in the pair potential. Other features, such as the presence of a disordered phase with a network structure, are due to the long-range (1/r(3)) repulsive tail in the potential. The relevance of the simulation results to experimental work, including materials synthesis, is briefly discussed.

The phase diagrams of systems made up of hard spheres interacting with attractive potentials of the form -1/r(3+sigma) are calculated using Monte Carlo simulations, second-order thermodynamic perturbation theory, and an augmented van der Waals theory. In simulations of the systems with sigma=0.1, 1, and 3, fluid-solid coexistence results are obtained using the Gibbs-Duhem integration technique; simulation data for the vapor-liquid coexistence envelopes and critical points are taken from previously published work [P. J. Camp and G. N. Patey, J. Chem. Phys. 114, 399 (2001)]. It is shown that the agreement between the theoretical and simulated phase diagrams improves as the range of the potential is increased, reflecting the decreasing role of short-range correlations in determining the bulk thermodynamics. In the extreme case of sigma=0.1 both theories are in excellent agreement with simulations. Phase diagrams for systems with sigma=4, 5, and 6 are computed using second-order thermodynamic perturbation theory. The results indicate that the vapor-liquid transition becomes metastable with respect to freezing when sigma > or approximately equal to 5, in broad agreement with results for the hard-sphere attractive Yukawa system which is commonly used to model colloidal particles, globular proteins, and nanoparticles.