Dipolar Hard Sphere Fluid Research Articles

Molecular theory of the static dielectric constant of dipolar fluids.

The link between the static dielectric constant and the microscopic intermolecular interactions is the Kirkwood g1 factor, which depends on the orientational structure of the fluid. Over the years, there have been several attempts to provide an accurate description of the orientational structure of dipolar fluids using molecular theories. However, these approaches were either limited to mean-field approximations for the pair correlation function or, more recently, limited to adjusting the orientational dependence to simulation data. Here, we derive a theory for the dielectric constant of dipolar hard-sphere fluids using the augmented modified mean-field approximation. Qualitative agreement is achieved throughout all relevant thermodynamic states, as demonstrated by a comparison with simulation data from the literature. Excellent quantitative agreement can be obtained using a single empirical scaling factor, the physical origin of which is analyzed and accounted for. In order to predict the dielectric constant of the Stockmayer fluid (Lennard-Jones plus dipole potential), we use an adjusted version of the expression for the dipolar hard-sphere fluid. Comparing theoretical predictions with newly generated simulation data, we show that it is possible to obtain excellent agreement with simulation by performing the calculations at a corresponding state using the same scaling factor. Finally, we compare the theoretical orientational structure of the Stockmayer fluid with that obtained from simulations. The simulated structure is calculated following a post-processing methodology that we introduce by deriving an original expression that relates the proposed theory to the histogram of relative dipole angles.

Interparticle correlations in the simple cubic lattice of ferroparticles: Theory and computer simulations

Anisotropic interparticle correlations in the simple cubic lattice of single-domain ferroparticles (SCLF) are studied using both theory and computer simulation. The theory is based on the Helmholtz free energy expansion like classical virial series up to the second virial coefficient. The analytical formula for the Helmholtz free energy is incorporated in a logarithmic form to minimize the effects of series truncation. The new theoretical approach, including discrete summation over lattice nodes coordinates, is compared critically against the classical virial expansion of the Helmholtz free energy for the dipolar hard sphere fluid; the main differences between the Helmholtz free energy of SCLF and dipolar hard sphere fluid are discussed. The theoretical results for the Helmholtz free energy, the magnetization, and the initial magnetic susceptibility of the SCLF are compared against Molecular Dynamic simulation data. In all cases, theoretical predictions using logarithmic form of the Helmholtz free energy are seen to be superior, but they only have an applicability range of the effective dipolar coupling constant λe<1.5. For highest values of λe, the structural transition of the magnetic dipoles in SCLF is observed in Molecular Dynamic simulation. It has been shown that for λe≳2, an antiferromagnetic order appears in the system.

Kinetic Theory and Shear Viscosity of Dense Dipolar Hard Sphere Liquids.

Transport properties of dense fluids are fundamentally challenging, because the powerful approaches of equilibrium statistical physics cannot be applied. Polar fluids compound this problem, because the long-range interactions preclude the use of a simple effective diameter approach based solely on hard spheres. Here, we develop a kinetic theory for dipolar hard-sphere fluids that is valid up to high density. We derive a mathematical approximation for the radial distribution function at contact directly from the equation of state, and use it to obtain the shear viscosity. We also perform molecular-dynamics simulations of this system and extract the shear viscosity numerically. The theoretical results compare favorably to the simulations.

An equation of state for Stockmayer fluids based on a perturbation theory for dipolar hard spheres.

We develop a perturbation theory for the difference between the Helmholtz energy of a Stockmayer fluid, i.e., a fluid interacting by a Lennard-Jones plus point-dipole potential, and a Lennard-Jones fluid. We show that the difference can be approximated by the perturbational Helmholtz energy contribution of a dipolar hard-sphere fluid with a suitably chosen effective hard-sphere diameter, relative to a hard-sphere fluid with the same effective diameter. We analyze both a third and fourth order perturbation theory, both written as Padé approximations. Several recipes for calculating the hard-sphere diameter are investigated; we find that the Weeks-Chandler-Andersen diameter is most suitable. Results of the perturbation theory are shown to be in good agreement with reference data for the Helmholtz energy, internal energy, and isochoric heat capacity as obtained from molecular simulations performed in this work and to vapor-liquid equilibrium data from the literature. Theoretical predictions of the proposed model are compared to results from the perturbation theory of Gubbins and Twu [Chem. Eng. Sci. 33, 863 (1978)], which is a theory based on a Lennard-Jones reference fluid. We find the theories are in good agreement. Our approach can easily be applied to van der Waals potentials, other than Lennard-Jones potentials. If a dipolar Mie fluid is considered, the approach merely requires calculation of the effective hard-sphere diameter for a Mie potential. We further note that the approach has a reduction in the variable space of the underlying correlation integrals, i.e., the correlation functions of a hard-sphere fluid depend on density only, whereas the Lennard-Jones reference correlation functions depend on density and temperature.

The magnetization of dense aggregated dipolar fluids

The magnetization for dipolar fluids is studied treated separately at low and high external magnetic field. Canonical ensemble Monte Carlo simulations have been performed in dipolar hard sphere fluid in order to test these theoretical results. New expressions are introduced at low and high external field and ultimately the synthesized formula is given. The main difference in the structure of dipolar liquids at different external field is the average orientation of the formed chains. In case of infinitesimal external magnetic field there is no a well-specified direction, while at high enough external field actually the chains are parallel to this field. The present theory yields good result in the intermediate region as well where the most conspicuous failures are provided by the former theories.

A molecular Debye-Hückel theory of solvation in polar fluids: An extension of the Born model.

A dielectric response theory of solvation beyond the conventional Born model for polar fluids is presented. The dielectric response of a polar fluid is described by a Born response mode and a linear combination of Debye-Hückel-like response modes that capture the nonlocal response of polar fluids. The Born mode is characterized by a bulk dielectric constant, while a Debye-Hückel mode is characterized by its corresponding Debye screening length. Both the bulk dielectric constant and the Debye screening lengths are determined from the bulk dielectric function of the polar fluid. The linear combination coefficients of the response modes are evaluated in a self-consistent way and can be used to evaluate the electrostatic contribution to the thermodynamic properties of a polar fluid. Our theory is applied to a dipolar hard sphere fluid as well as interaction site models of polar fluids such as water, where the electrostatic contribution to their thermodynamic properties can be obtained accurately.

Fourth and fifth virial coefficients and thermodynamic properties of the dipolar hard sphere fluids in zero external magnetic field

Analytical expressions of the fourth and the fifth virial coefficients for the dipole hard sphere fluid in zero external magnetic field are obtained on the basis of numerical results of Mayer-sampling calculations. The virial coefficients are incorporated in to the so-called logarithmic free energy (LFE) theory [E.A. Elfimova, A.O. Ivanov, P.J. Camp, Phys. Rev. E, 86, 021126 (2012)]. In this theory, the virial expansion of the Helmholtz free energy is re-summed into a logarithmic function. The argument of the logarithm is a polynomial expansion in density with coefficients chosen to give the first few terms in the conventional virial expansion. The theoretical predictions of the thermodynamic functions based on the LFE theory that accounts five virial coefficients are compared critically with published results from Monte Carlo simulations, conventional virial expansion, and known theory based on Padé approximation [G.S. Rushbrooke, G. Stell, J.S. Hoye, Mol. Phys., 26, 1199 (1973)]. It is shown that the LFE theory accurately captures computer simulation results for dipolar coupling constant λ ≤ 4, even at the highest value of the particle volume fraction φ ≲ 0.5, and outperforms thermodynamic theories mentioned above.

Sedimentation equilibria in polydisperse ferrofluids: critical comparisons between experiment, theory, and computer simulation.

The sedimentation equilibrium of dipolar particles in a ferrofluid is studied using experiment, theory, and computer simulation. A theory of the particle-concentration profile in a dipolar hard-sphere fluid is developed, based on the local-density approximation and accurate expressions from a recently introduced logarithmic free energy approach. The theory is tested critically against Monte Carlo simulation results for monodisperse and bidisperse dipolar hard-sphere fluids in homogeneous gravitational fields. In the monodisperse case, the theory is very accurate over broad ranges of gravitational field strength, volume fraction, and dipolar coupling constant. In the bidisperse case, with realistic dipolar coupling constants and compositions, the theory is excellent at low volume fraction, but is slightly inaccurate at high volume fraction in that it does not capture a maximum in the small-particle concentration profile seen in simulations. Possible reasons for this are put forward. Experimental measurements of the magnetic-susceptibility profile in a real ferrofluid are then analysed using the theory. The concentration profile is linked to the susceptibility profile using the second-order modified mean-field theory. It is shown that the experimental results are not consistent with the sample being monodisperse. By introducing polydispersity in the simplest possible way, namely by assuming the system is a binary mixture, almost perfect agreement between theory and experiment is achieved.

Electrostatics on the sphere with applications to Monte Carlo simulations of two dimensional polar fluids

We present two methods for solving the electrostatics of point charges and multipoles on the surface of a sphere, i.e., in the space S2, with applications to numerical simulations of two-dimensional (2D) polar fluids. In the first approach, point charges are associated with uniform neutralizing backgrounds to form neutral pseudo-charges, while in the second, one instead considers bi-charges, i.e., dumbells of antipodal point charges of opposite signs. We establish the expressions of the electric potentials of pseudo- and bi-charges as isotropic solutions of the Laplace-Beltrami equation in S2. A multipolar expansion of pseudo- and bi-charge potentials leads to the electric potentials of mono- and bi-multipoles, respectively. These potentials constitute non-isotropic solutions of the Laplace-Beltrami equation, the general solution of which in spherical coordinates is recast under a new appealing form. We then focus on the case of mono- and bi-dipoles and build the theory of dielectric media in S2. We notably obtain the expression of the static dielectric constant of a uniform isotropic polar fluid living in S2 in terms of the polarization fluctuations of subdomains of S2. We also derive the long range behavior of the equilibrium pair correlation function under the assumption that it is governed by macroscopic electrostatics. These theoretical developments find their application in Monte Carlo simulations of the 2D fluid of dipolar hard spheres. Some preliminary numerical experiments are discussed with a special emphasis on finite size effects, a careful study of the thermodynamic limit, and a check of the theoretical predictions for the asymptotic behavior of the pair correlation function.

Self-consistent generalized Langevin-equation theory for liquids of nonspherically interacting particles.

A self-consistent generalized Langevin-equation theory is proposed to describe the self- and collective dynamics of a liquid of linear Brownian particles. The equations of motion for the spherical harmonics projections of the collective and self-intermediate-scattering functions, F_{lm,lm}(k,t) and F_{lm,lm}^{S}(k,t), are derived as a contraction of the description involving the stochastic equations of the corresponding tensorial one-particle density n_{lm}(k,t) and the translational (α=T) and rotational (α=R) current densities j_{lm}^{α}(k,t). Similar to the spherical case, these dynamic equations require as an external input the equilibrium structural properties of the system contained in the projections of the static structure factor, denoted by S_{lm,lm}(k). Complementing these exact equations with simple (Vineyard-like) approximate relations for the collective and the self-memory functions we propose a closed self-consistent set of equations for the dynamic properties involved. In the long-time asymptotic limit, these equations become the so-called bifurcation equations, whose solutions (the nonergodicity parameters) can be written, extending the spherical case, in terms of one translational and one orientational scalar dynamic order parameter, γ_{T} and γ_{R}, which characterize the possible dynamical arrest transitions of the system. As a concrete illustrative application of this theory we determine the dynamic arrest diagram of the dipolar hard-sphere fluid. In qualitative agreement with mode coupling theory, the present self-consistent equations also predict three different regions in the state space spanned by the macroscopic control parameters η (volume fraction) and T* (scaled temperature): a region of fully ergodic states, a region of mixed states, in which the translational degrees of freedom become arrested while the orientational degrees of freedom remain ergodic, and a region of fully nonergodic states.

A new dipolar potential for numerical simulations of polar fluids on the 4D hypersphere.

We present a new method for Monte Carlo or Molecular Dynamics numerical simulations of three-dimensional polar fluids. The simulation cell is defined to be the surface of the northern hemisphere of a four-dimensional (hyper)sphere. The point dipoles are constrained to remain tangent to the sphere and their interactions are derived from the basic laws of electrostatics in this geometry. The dipole-dipole potential has two singularities which correspond to the following boundary conditions: when a dipole leaves the northern hemisphere at some point of the equator, it reappears at the antipodal point bearing the same dipole moment. We derive all the formal expressions needed to obtain the thermodynamic and structural properties of a polar liquid at thermal equilibrium in actual numerical simulation. We notably establish the expression of the static dielectric constant of the fluid as well as the behavior of the pair correlation at large distances. We report and discuss the results of extensive numerical Monte Carlo simulations for two reference states of a fluid of dipolar hard spheres and compare these results with previous methods with a special emphasis on finite size effects.

Low-temperature magnetic susceptibility of concentrated ferrofluids: The influence of polydispersity

In this paper we address the question of theoretical explanation of extremely high low-temperature initial magnetic susceptibility of concentrated ferrofluids . These laboratory synthesized samples [A.F. Pshenichnikov, A.V. Lebedev, J. Chem. Phys. 121(11) (2004) 5455; Colloid J. 67(2) (2005) 189] demonstrated the record-breaking values χ ~ 120 – 150 at temperatures ~ 230–240 K. The existing models predict such high susceptibility only under the assumption of unreasonably large dipolar coupling constant, which is out of the range of applicability. Here we calculate the second virial contribution to susceptibility for polydisperse ferrofluid, modeled by the dipolar hard sphere fluid. In the resulting expression there exists the parameter, which plays a part of dipolar coupling constant and which is defined in a form of double averaging of high powers of particle sizes over the granulometric distribution. For real particle size distribution this effective parameter at least twice exceeds the commonly defined polydisperse dipolar coupling constant. We show that the low-temperature magnetic susceptibility of the record-breaking ferrofluids could be explained theoretically on the basis of the first terms of the polydisperse second virial contribution in combination with the second-order modified mean field model. • Record-breaking magnetic susceptibility of ferrofluids at low temperatures. • Second virial contribution to magnetic susceptibility of polydisperse ferrofluids. • Interparticle dipole correlations are more pronounced in dense ferrofluids. • Presented model describes the ferrofluid susceptibility χ ~ 120 at temperature 240 K.

Exact second virial coefficient for dipolar hard spheres

The second virial coefficient B2 for a fluid of dipolar hard spheres has been given several approximate forms valid in the limits of weak and strong interactions. They have been formulated as asymptotic expressions in a dimensionless interaction parameter λ. In this paper, B2 is determined exactly for all values of λ in both the three-dimensional case, where spheres may access the whole space and their dipole moments may be oriented in all directions, and in the quasi-two-dimensional case, where spheres have their centres bound to glide on a plane, while their dipole moments are still freely orientable in space.

Branching points in the low-temperature dipolar hard sphere fluid

In this contribution, we investigate the low-temperature, low-density behaviour of dipolar hard-sphere (DHS) particles, i.e., hard spheres with dipoles embedded in their centre. We aim at describing the DHS fluid in terms of a network of chains and rings (the fundamental clusters) held together by branching points (defects) of different nature. We first introduce a systematic way of classifying inter-cluster connections according to their topology, and then employ this classification to analyse the geometric and thermodynamic properties of each class of defects, as extracted from state-of-the-art equilibrium Monte Carlo simulations. By computing the average density and energetic cost of each defect class, we find that the relevant contribution to inter-cluster interactions is indeed provided by (rare) three-way junctions and by four-way junctions arising from parallel or anti-parallel locally linear aggregates. All other (numerous) defects are either intra-cluster or associated to low cluster-cluster interaction energies, suggesting that these defects do not play a significant part in the thermodynamic description of the self-assembly processes of dipolar hard spheres.

Nonmonotonic magnetic susceptibility of dipolar hard-spheres at low temperature and density.

We investigate, via numerical simulations, mean field, and density functional theories, the magnetic response of a dipolar hard sphere fluid at low temperatures and densities, in the region of strong association. The proposed parameter-free theory is able to capture both the density and temperature dependence of the ring-chain equilibrium and the contribution to the susceptibility of a chain of generic length. The theory predicts a nonmonotonic temperature dependence of the initial (zero field) magnetic susceptibility, arising from the competition between magnetically inert particle rings and magnetically active chains. Monte Carlo simulation results closely agree with the theoretical findings.

Effective isotropic potential for dipolar hard spheres

A new effective isotropic potential is proposed for the dipolar hard-sphere fluid, on the basis of recent results by others for its angle-averaged radial distribution function. The new effective potential is shown to exhibit oscillations even for moderately high densities and moderately strong dipole moments, which are absent from earlier effective isotropic potentials. The validity and significance of this result are briefly discussed.

Thermodynamics of dipolar hard spheres with low-to-intermediate coupling constants

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.

Liquid-gas phase behavior of polydisperse dipolar hard-sphere fluid: Extended thermodynamic perturbation theory for central force associating potential

The liquid-gas phase diagram for polydisperse dipolar hard-sphere fluid with polydispersity in the hard-sphere size and dipolar moment is calculated using extension of the recently proposed thermodynamic perturbation theory for central force (TPT-CF) associating potential. To establish the connection with the phase behavior of ferrocolloidal dispersions it is assumed that the dipole moment is proportional to the cube of the hard-sphere diameter. We present and discuss the full phase diagram, which includes cloud and shadow curves, binodals and distribution functions of the coexisting daughter phases at different degrees of the system polydispersity. In all cases studied polydispersity increases the region of the phase instability and shifts the critical point to the higher values of the temperature and density. The larger size particles always fractionate to the liquid phase and the smaller size particles tend to move to the gas phase. At relatively high values of the system polydispersity three-phase coexistence is observed.

Structural properties of the dipolar hard-sphere fluid at low temperatures and densities

Through extensive state-of-the-art numerical simulations, we study the behavior of the dipolar hard sphere model at low temperatures and low densities, shedding light on a region of the phase diagram where a topological phase transition has long been thought to occur. We show that the system exhibits remarkable and unusual behaviors, like a very low density percolation locus and a stabilization of rings over chain structures. This unexpected abundance of rings comes from a delicate balance between the lower ring energy and the end-to-end chain entropy, and hints at a possible mechanism for the suppression of the gas–liquid phase separation. Our results open the possibility for refined theoretical approaches which, in addition to the previously encompassed chain and branched geometries, must also include the significant contribution arising from ring formation.

No Evidence of Gas-Liquid Coexistence in Dipolar Hard Spheres

We report accurate calculations of the particle density of states in the dipolar hard-sphere fluid. Implementing efficient and tailored Monte Carlo algorithms, we are able to explore, in equilibrium, the low temperature region where a phase separation between a dilute gas of chain ends and a high-density liquid of chain junctions has been predicted to occur. Our data clearly show that the density of states remains always single peaked, definitively excluding the possibility of critical phenomena in the investigated temperature and density region. The analysis of the low temperature configurations shows that at low densities particles preferentially self-assemble into closed rings, strongly suppressing the chain ends concentration.