Stockmayer Fluid Research Articles

Helmholtz energy models for dipole interactions: Review and comprehensive assessment

Dipolar interactions play an important role for thermodynamic properties of many fluids and accordingly for their modeling. In molecular-based equation of state models, the effect of dipolar interactions is usually described by Helmholtz energy models. There are several Helmholtz energy models for dipolar interactions available in the literature today. In this work, eight dipole contribution models describing the dipole–dipole interactions of fluids were critically assessed by comparing their results with molecular simulation reference data of Stockmayer fluids. Therefore, the dipole contribution models were combined with an accurate Lennard-Jones (LJ) Helmholtz energy model. The following thermodynamic properties were considered in the comparison: vapor pressure, saturated densities, enthalpy of vaporization, surface tension (by using density gradient theory), critical point, second virial coefficient, and thermodynamic properties at homogeneous state points, such as the Helmholtz energy, pressure, chemical potential, internal energy, isochoric heat capacity, isobaric heat capacity, thermal expansion coefficient, isothermal compressibility, thermal pressure coefficient, speed of sound, Joule–Thomson coefficient, and Grüneisen parameter. For the evaluation of the dipole contribution models, molecular simulations for the Stockmayer fluid with the dipole moments of μ2/4πϵ0ɛσ3=0.5,1,2,3,4,5 were carried out. The results indicate, that all considered dipole models exhibit some significant weaknesses. Nevertheless, some dipole contribution models are found to provide a robust description for many properties and state ranges. Overall, the deviations of the dipole contribution models from the Stockmayer simulation data are, in most cases, an order of magnitude higher than the deviations of the LJ EOS from LJ simulation data.

Molecular dynamics simulations of the dielectric constants of salt-free and salt-doped polar solvents.

We develop a Stockmayer fluid model that accounts for the dielectric responses of polar solvents (water, MeOH, EtOH, acetone, 1-propanol, DMSO, and DMF) and NaCl solutions. These solvent molecules are represented by Lennard-Jones (LJ) spheres with permanent dipole moments and the ions by charged LJ spheres. The simulated dielectric constants of these liquids are comparable to experimental values, including the substantial decrease in the dielectric constant of water upon the addition of NaCl. Moreover, the simulations predict an increase in the dielectric constant when considering the influence of ion translations in addition to the orientation of permanent dipoles.

Vapor-liquid equilibria of binary mixtures containing Stockmayer-type model fluids from Monte-Carlo simulations

In this contribution, the vapor-liquid equilibria (VLE) of binary mixtures containing Lennard-Jones (LJ), Stockmayer (LJ + point dipole, ST), and shifted Stockmayer (sST) fluids are investigated with molecular simulations. The LJ and ST fluids are commonly used nonpolar and polar model fluids, respectively. The sST fluid differs from the ST fluid in that the dipole is shifted away from the dispersive center, which results in the sST fluid exhibiting H-bonding-like behavior. By varying the dispersion energy of the LJ fluid and the dipole moments of the ST and sST fluids, and the dipole shift of the sST fluids, different mixtures are generated, which are investigated at several temperatures and concentrations. Differences in the phase behavior between mixtures containing ST and sST fluids are quantified and linked to differences in fluid structure. The reported data include the VLE envelopes, coexisting densities, activity coefficients, and enthalpies for liquid and vapor phases, as well as structural information on the fluids’ intermolecular orientations.

Accelerated inertial regime in the spinodal decomposition of magnetic fluids.

Furukawa predicted that at late times, the domain growth in binary fluids scales as (t) ∼ t2/3, and the growth is driven by fluid inertia. The inertial growth regime has been highly elusive in molecular dynamics (MD) simulations. We perform coarsening studies of the (d = 3) Stockmayer (SM) model comprising of magnetic dipoles that interact via long-range dipolar interactions as well as the usual Lennard-Jones (LJ) potential. This fascinating polar fluid exhibits a gas-liquid phase coexistence, and magnetic order even in the absence of an external field. From comprehensive MD simulations, we observe the inertial scaling [(t) ∼ t2/3] in the SM fluid for an extended time window. Intriguingly, the fluid inertia is overwhelming from the outset - our simulations do not show the early diffusive regime [(t) ∼ t1/3] and the intermediate viscous regime [(t) ∼ t] prevalent in LJ fluids.

Systematic study of vapour–liquid equilibria in binary mixtures of fluids with different polarity from molecular simulations

ABSTRACT For engineering applications, one of the most important characteristics of fluid mixtures is the vapour–liquid equilibrium (VLE). Knowledge of the VLE of mixtures containing simple model components like Lennard-Jones (LJ) and Stockmayer (ST) fluids can offer insight into how different modes of molecular interaction shape the phase behaviour of mixtures. However, studies investigating such mixtures are still scarce in the literature, especially for mixtures with an LJ fluid as the high boiling substance. In this work, the VLE of mixtures consisting of LJ and ST fluids are studied with molecular simulation using the Grand Equilibrium method. The energy parameter and length parameter of the LJ fluids and the dipole moments of the ST fluids are varied systematically to generate different types of mixtures. These mixtures are investigated within a broad temperature range. VLE envelopes and molarities are reported as well as additional data on the bubble and dew line including enthalpies, chemical potentials, and activity coefficients. These data fill a gap in the literature by furthering the understanding of how polarity affects phase behaviour in mixtures and are therefore of critical importance, e.g. for the development of theoretical approaches for the description and prediction of phase behaviour for polar mixtures.

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.

Critical assessment of perturbation theories for the relative permittivity of dipolar model fluids

Different perturbation theories for the relative permittivity of dipolar model fluids are assessed by comparison to a comprehensive set of molecular dynamics (MD) simulation data for Stockmayer fluids. With the exception of dense liquids, all MD data are a univariate function of the so-called dipole strength y. Overall, good agreement is found between some of the theories and the MD data.

Solvation Energy of Ions in a Stockmayer Fluid.

We calculate the solvation energy of monovalent and divalent ions in various liquids with coarse-grained molecular dynamics simulations. Our theory treats the solvent as a Stockmayer fluid, which accounts for the intrinsic dipole moment of molecules and the rotational dynamics of the dipoles. Despite the simplicity of the model, we obtain qualitative agreement between the simulations and experimental data for the free energy and enthalpy of ion solvation, which indicates that the primary contribution to the solvation energy arises mainly from the first and possibly second solvation shells near the ions. Our results suggest that a Stockmayer fluid can serve as a reference model that enables direct comparison between theory and experiment and may be invoked to scale up electrostatic interactions from the atomic to the molecular length scale.

Viscosity and self-diffusion coefficient of dipolar liquids

Equilibrium molecular dynamics simulations were used to calculate the viscosity and self-diffusion coefficient of Stockmayer fluids at different dipole moments from the saturated liquid state to the supercritical fluid state. On the basis of the theory of Longuet-Higgins and Pople (J. Chem. Phys. 25 (1956) 884) a physically based correlation equations was proposed for the correlation of density, temperature and dipole moment dependence of viscosity and self-diffusion coefficient simulation data. As an application, with appropriate Stockmayer parameter sets the viscosity of fluoroalkanes were calculated along the saturated liquid curves, and good agreement has been obtained between the experimental and correlation equation data.

Polar soft-SAFT: theory and comparison with molecular simulations and experimental data of pure polar fluids.

The consideration of polar interactions is of vital importance for the development of predictive and accurate thermodynamic models for polar fluids, as they govern most of their thermodynamic properties, making them highly non-ideal fluids. We present here for the first time the extension of the soft-SAFT equation of state (EoS), named polar soft-SAFT, to explicitly model intermolecular polar interactions (dipolar and quadrupolar), using the approach of Jog and Chapman (P. K. Jog and W. G. Chapman, Mol. Phys., 1999, 97(3), 307-319). The theory is first validated using molecular simulation data for a wide range of polar model systems including Stockmayer fluids, LJ dimers with dipole, and quadrupolar LJ fluids, for a wide range of thermophysical properties such as liquid density, vapour pressure, surface tension and heat capacities. Excellent agreement between polar soft-SAFT and simulation data has been obtained for all examined fluids and properties for systems exhibiting low to intermediate polar strength, while the agreement deteriorates at very high polar strengths. Once validated with simulations, the equation has been applied to calculate vapour-liquid equilibria (VLE), surface tension and second-order derivative properties of systems such as 2-ketone and methane chloride families as showcases for dipolar fluids and the benzene family for quadrupolar fluids, finding very good agreement with experimental data. In order to preserve the robustness of the model, the experimental value of the dipole or quadrupole was used in these calculations, while the additional parameter for the polar fluids was set a priori rather than included in the fitting procedure. The excellent agreement found with simulations and experiments empowers the soft-SAFT equation with new capabilities for the development of robust and accurate molecular models of polar fluids of industrial relevance.

The structure of clusters formed by Stockmayer supracolloidal magnetic polymers

.Unlike Stockmayer fluids, that prove to undergo gas-liquid transition on cooling, the system of dipolar hard or soft spheres without any additional central attraction so far has not been shown to have a critical point. Instead, in the latter, one observes diverse self-assembly scenarios. Crosslinking dipolar soft spheres into supracolloidal magnetic polymer-like structures (SMPs) changes the self-assembly behaviour. Moreover, aggregation in systems of SMPs strongly depends on the constituent topology. For Y- and X-shaped SMPs, under the same conditions in which dipolar hard spheres would form chains, the formation of very large loose gel-like clusters was observed (E. Novak et al., J. Mol. Liq. 271, 631 (2018)). In this work, using molecular dynamics simulations, we investigate the self-assembly in suspensions of four topologically different SMPs --chains, rings, X and Y-- whose monomers interact via Stockmayer potential. As expected, compact drop-like clusters are formed by SMPs in all cases if the central isotropic attraction is introduced, however, their shape and internal structure turn out to depend on the SMPs topology.Graphical abstract

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.

Experimental verification of anomalous surface tension temperature dependence at the interface between coexisting liquid-gas phases in magnetic and Stockmayer fluids

Our early experimental investigation has demonstrated the anomalous surface tension temperature dependence σ(T) at the interface between coexisting liquid-gas phases in magnetic fluids that undergo field-induced first-order phase transition. The σ(T) dependence is anomalous because the drops of a liquid phase condensed under the action of the applied magnetic field H at high temperature T2 exhibit larger surface tension σ(T2) > σ(T1) than the drops condensed at low temperature T1 < T2. This study verifies and confirms the results of the previous experimental investigation of σ(T) in magnetic fluids by performing the experiment, which is based on the analysis of the Plateau-Rayleigh instability of a gas-liquid interface in a zero magnetic field. A novel explanation of this phenomenon is given in the framework of the Stockmayer model. The anomalous increase in σ(T) is explained by the increase in particle concentration difference in gas and liquid phases, which can be attributed to the high field intensity H needed to generate the phase transition at high temperature.

Theoretical Prediction of Thermal Polarization.

We present a mean-field theory to explain the thermo-orientation effect in an off-center Stockmayer fluid. This effect is the underlying cause of thermally induced polarization and thermally induced monopoles, which have recently been predicted theoretically. Unlike previous theories that are based either on phenomenological equations or on scaling arguments, our approach does not require any fitting parameters. Given an equation of state and assuming local equilibrium, we construct an effective Hamiltonian for the computation of local Boltzmann averages. This simple theoretical treatment predicts molecular orientations that are in very good agreement with simulation results for the range of dipole strengths investigated. By decomposing the overall alignment into contributions from the temperature and density gradients, we shed further light on how the nonequilibrium result arises from the competition between the two gradients.

Co-Oriented Fluid Functional Equation for Electrostatic interactions (COFFEE)

In chemical engineering often functional groups like OH, CO, or NHx with an asymmetric charge distribution occur. Aside from polar interactions, such groups often show H-bonding interactions. The combination of both types is especially hard to describe with equations of state. This leads to the effect that including polar interactions for clearly polar pure compounds may lead to a worse description of mixtures containing this component. One possible reason for this behavior is that equations of state are usually based on perturbation theory that ignores the change in fluid structure due to interactions. This approximation is good for e.g. dispersive interactions, but is known to be bad for at least the gaseous phase of polar fluids up to moderate densities.The issue is addressed in this article by developing a perturbation theory around the target fluid rather than the reference fluid, as is usually the case. This makes it possible to include ordering effects into the equation of state and to calculate additional structural information in a fraction of the time used by typical methods like molecular simulation or integral equation theory. The perturbation theory is parameterized on the Stockmayer fluid’s structure and vapor-liquid equilibria, calculated from molecular simulation, and agrees quantitatively with them. It is verified on a second simple fluid, the shifted Stockmayer fluid, where the dipole is shifted along its axis away from the Lennard-Jones center. For this fluid also molecular simulations are performed for both structure and vapor-liquid equilibria. Both properties are predicted by the new equation of state and again agree quantitatively with simulation.The new theory is then applied to hydrogen chloride and shows improvements over models for existing theories. Namely, liquid density description is improved over a literature model for PCP-SAFT, where the dipolar nature of hydrogen chloride is considered, and vapor pressure description is improved over a literature model for PC-SAFT, where the weak H-bonding nature is considered. The number of parameters can be kept at three, as for the PCP-SAFT model, by using quantum mechanics results from literature for the dipole shift. The model shows improvements for the predicted vapor density compared to both models from literature. Aside from improving thermodynamic property description and prediction, the new theory allows the fast calculation of a fluid’s mutual orientation structure. This is useful e.g. for assessing availability of two reactant groups to each other, thereby allowing a better description of reaction kinetics. This has not been accomplished by any equation of state before.

Long-range correction for dipolar fluids at planar interfaces

A slab-based long-range correction for dipolar interactions in molecular dynamics simulation of systems with a planar geometry is presented and applied to simulate vapour–liquid interfaces. The present approach is validated with respect to the saturated liquid density and the surface tension of the Stockmayer fluid and a molecular model for ethylene oxide. The simulation results exhibit no dependence on the cut-off radius for radii down to 1 nm, proving that the long-range correction accurately captures the influence of the dipole moment on the intermolecular interaction energies and forces as well as the virial and the surface tension.

Direct summation of dipole-dipole interactions using the Wolf formalism.

We present an expanded Wolf formalism for direct summation of long-range dipole-dipole interactions and rule-of-thumbs how to choose optimal spherical cutoff (Rc) and damping parameter (α). This is done by comparing liquid radial distribution functions, dipole-dipole orientation correlations, particle energies, and dielectric constants, with Ewald sums and the Reaction field method. The resulting rule states that ασ < 1 and αRc > 3 for reduced densities around ρ(∗) = 1 where σ is the particle size. Being a pair potential, the presented approach scales linearly with system size and is applicable to simulations involving point dipoles such as the Stockmayer fluid and polarizable water models.

Thermodynamics of the Stockmayer fluid in an applied field

The thermodynamic properties of the Stockmayer fluid in an applied field are studied using theory and computer simulation. Theoretical expressions for the second and third virial coefficients are obtained in terms of the dipolar coupling constant (λ, measuring the strength of dipolar interactions as compared to thermal energy) and dipole–field interaction energy (α, being proportional to the applied field strength). These expressions are tested against numerical results obtained by Mayer sampling calculations. The expression for the second virial coefficient contains terms up to λ4, and is found to be accurate over realistic ranges of dipole moment and temperature, and over the entire range of the applied field strength (from zero to infinity). The corresponding expression for the third virial coefficient is truncated at λ3, and is not very accurate: higher order terms are very difficult to calculate. The virial coefficients are incorporated in to a thermodynamic theory based on a logarithmic representation of the Helmholtz free energy. This theory is designed to retain the input virial coefficients, and account for some higher order terms in the sense of a resummation. The compressibility factor is obtained from the theory and compared to results from molecular dynamics simulations with a typical value λ = 1. Despite the mathematical approximations of the virial coefficients, the theory captures the effects of the applied field very well. Finally, the vapour–liquid critical parameters are determined from the theory, and compared to published simulation results; the agreement between the theory and simulations is good.

Dielectric and phase behavior of dipolar spheroids.

The Stockmayer fluid, composed of dipolar spheres, has a well-known isotropic-ferroelectric phase transition at high dipole densities. However, there has been little investigation of the ferroelectric transition in nearly spherical fluids at dipole densities corresponding to those found in many polar solvents and in guest-host organic electro-optic materials. In this work, we examine the transition to ordered phases of low-aspect-ratio spheroids under both unperturbed and poled conditions, characterizing both the static dielectric response and thermodynamic properties of spheroidal systems. Spontaneous ferroelectric ordering was confined to a small region of aspect ratios about unity, indicating that subtle changes in sterics can have substantial influence on the behavior of coarse-grained liquid models. Our results demonstrate the importance of molecular shape in obtaining even qualitatively correct dielectric responses and provide an explanation for the success of the Onsager model as a phenomenological representation for the dielectric behavior of polar organic liquids.

Liquid-vapor interface of the Stockmayer fluid in a uniform external field.

The effect of a uniform (nonspatially varying) external field on the liquid-vapor interface of the Stockmayer fluid (Lennard-Jones particles embedded with a point dipole) has been investigated by molecular-dynamics simulations. The long-ranged parts of both the dipole and Lennard-Jones interactions are treated using an Ewald summation, which removes the effects of the cutoff. The direction of the field shifts the critical point and interfacial properties in different directions. For an external field parallel to the interface, the critical temperature increases, while for a field applied perpendicular to the interface, it decreases. The effects of the field on surface tension and interfacial width are also investigated. For zero field, dipoles near the liquid-vapor interface show a weak orientation parallel to the interface. For fields parallel to the interface, ordering in the liquid phase is greater than the vapor, while for fields perpendicular to the interface, the opposite is true.