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Using classical density functional theory (DFT) in a modified mean-field approximation we investigate the fluid phase behavior of quasi-two dimensional dipolar fluids confined to a plane. The particles carry three-dimensional dipole moments and interact via a combination of hard-sphere, van-der-Waals, and dipolar interactions. The DFT predicts complex phase behavior involving first- and second-order isotropic-to-ferroelectric transitions, where the ferroelectric ordering is characterized by global polarization within the plane. We compare this phase behavior, particularly the onset of ferroelectric ordering and the related tri - critical points, with corresponding three-dimensional systems, slab-like systems (with finite extension into the third direction), and true two-dimensional systems with two-dimensional dipole moments.

Two-dimensional (2D) fluids consisting of particles with classical dipole-dipole interactions such as (para)magnetic nanoparticles at interfaces [1-3], cobalt nanocrystals on solid surfaces [

For theory and computer simulations, exploring the full structural and phase behavior of 2D dipolar systems remains challenging. Apart from the above-mentioned aggregation phenomena, one topic investigated particularly by computer simulations concerns the appearance and characteristics of vapor-liquid transitions [9-14]. Another question touches the structure at high densities close to the range where crystallization is expected to occur. Various Monte Carlo (MC) simulation studies

[9,15] revealed the appearance of ferroelectric (or ferromagnetic, respectively) domains, but overall frustrated (vortex) structures without true long-range orientational ordering. This is consistent with integral equation results [15,16], where predictions on the low-temperature behavior are extracted by analyzing correlation functions. On the other hand, recent Molecular Dynamics (MD) simulations [

Similar to the dense fluid state, the nature of the 2D crystalline structures formed at finite temperatures remains so far unclear [18,19], although ground state calculations indicate ferromagnetism for certain 2D lattice types such as hexagonal lattices [

The diversity of simulation results shows that spatial dimension has a profound influence of the ordering behavior of dipolar fluids. The purpose of the present study is to collect and compare theoretical results on that issue based on a relatively simple, mean-field like approach. Specifically, we employ classical density functional theory (DFT) in the modified mean field approximation [26-28], where the pair correlations are replaced by their low-density limit, i.e., the Boltzmann factor. The application of this approach for three-dimensional (3D) dipolar systems and their mixtures [

Here we apply the approach to a quasi-2D dipolar (Stockmayer) fluid, where the particles are confined to a plane, but carry 3D dipole moments. Evaluating the phase diagram and comparing with corresponding DFT results for 3D systems, slab-like systems, and true 2D systems with 2D dipole moments we can identify, on a mean-field level, the influence of spatial dimension and of the dimension of the order parameter on global ordering in fluid-like dipolar systems. Based on previous experiences one would expect that the mean-field approach for the quasi-2D system will (as it generally does) overestimate the stability of orientationally ordered phases. However, given the importance of meanfield-like approaches in the general context of spin and dipolar systems, and realizing that the mean-field DFT approach is, so far, still the only theory targeting the whole (homogeneous) phase diagram of dipolar systems, we think our results are important for a complete understanding of such systems.

The remainder of the paper is organized as follows. In Section 2 we formulate the quasi-2D model and briefly detail the derivation of the corresponding grand-canonical functional. Numerical results for the phase diagram at a typical dipole moment are presented in Section 3. There we also use Landau expansions to compare the onset of ordering in the quasi-2D system with the cases of 3D, slab-like and true 2D systems. Finally, in Section 4 we summarize our results.

The quasi-2D Stockmayer fluid consists of disk-like particles of diameter at positions in the x-y plane. The orientation of their 3D dipole moments is represented by the Euler angles. The microscopic interactions between the particles stem from anisotropic dipole-dipole and isotropic LJ forces. The resulting pair potential between two particles with coordinates and is given as

where is the connecting vector between the two particles, and. Further,

is the 3D dipole-dipole interaction potential, and

is the LJ potential.

To mimic the fact that the effective range of the repulsion varies with the thermodynamical parameters, we choose in our DFT calculations a temperaturedependent hard core defined via the Barker-Henderson formula [

To analyze the phase behavior we employ classical DFT, where the key quantity is the grand canonical potential as a functional of the singlet density

, with N being the total number of particles [

) and equals for isotropic states

[

, with the coefficients representing orientational order parameters [

. Indeed, transforming the Cartesian components of in terms of spherical harmonics one obtains

[

Within the grand canonical formalism, the system is characterized by its size, the chemical potential and the inverse temperature. In the present study, the dipolar contribution to the excess (interaction) free energy is treated in the modified mean-field approximation, where the pair correlations are approximated by the Boltzmann factor [26-28]. In addition, following our previous study on slab-like systems [

(2)

On the right side of Equation (2), the first line contains ideal gas contributions (involving the thermal wavelength) and the orientational entropy (last term). The second line contains the excess free energy of our reference system, the hard disk fluid [

andwith. Finally, in the last term on the right side of Equation (2),

where

and.

Minimization of the functional (2) with respect to and yields the Euler-Lagrange equations for this problem [

In the following we characterize the state of the quasi- 2D Stockmayer fluid by the dimensionless density, temperature, chemical potential

, and dipole moment

. The quantity measures the strength of the dipolar interactions in an antiparallel side-by-side configuration relative to the LJ interactions. We note that the coupling parameters and are equivalently defined in 3D (or slit-pore) dipolar systems [

Following earlier DFT studies on confined Stockmayer fluids [

For small and intermediate densities (or chemical potentials) we find a state where all order parameters are equal to zero, and those with are either zero or negative. Thus, there is no global polarization and neither a global ordering of the dipole axes; we therefore refer to this state as “isotropic fluid” (IF). The negative values of merely indicate that the dipoles tend to avoid to be oriented parallel or antiparallel to the -axis; rather they prefer to lie (with random orientations) in the -plane. This is an expected effect in a dilute, quasi-2D dipolar system (consistent with simulations and other theoretical studies,

see e.g. [

The transition between the IF and the FF phase is discontinuous in and ρ (yet not in) for temperatures below a tricritical temperature (see

). To this end we expand the integral in Equation (2)that is, the orientational entropy, in a Taylor serios around the isotropic state (where

). Collecting those terms in the resulting approximate functional, , that are proportional to, we obtain [

where the first term stems from the orientational entropy, whereas the second term results from the interaction free energy in Equation (2). The second order phase transition is characterized by a change of sign of the factor of in Equation (3). We thus obtain

This is approximatively the equation of a straight line in the density-temperature plane, consistent with what one sees in

In Figures 1(a) and (b) we have included DFT data for tricritical points of Stockmayer fluids in 3D and in slit-pore geometries. Within the latter situation, the particles are confined between two planar, attractive walls separated by a distance [

. A somewhat different behavior emerges in the chemical potential-temperature representation depicted in

The reduction of spatial dimension not only shifts the TCP, it also has a profound influence on the topology of the phase diagram. Indeed, while the 3D Stockmayer fluid with the same dipole moment exhibits, in addition to the IF-FF transition, a condensation transition within the isotropic liquid (IL) phase, such a transition is absent in the quasi-2D system (see

(at least) [13,14,17]. Therefore, the DFT seems to overestimate the stability of the ferroelectric phase, similar as it does in 3D [

We now discuss in more detail the influence of spatial dimension and its interplay with the dipolar coupling strength on the tricritical point, above of which the low-temperature discontinuous transition between the IF and FF states changes into a (line of) second-order transition(s). Specifically, we are interested in the position of the TCPs, in the quasi-2D system and its 3D counterpart, as functions of the parameter. In previous DFT studies [30,31,34] it was already shown that the coupling strength influences the quantity much more than, at least as long as. Therefore, to estimate the dependence of on in the quasi-2D system, we set the density equal to tricritical density at, that is, to

. The resulting function can then be easily determined from Equation (4). The same procedure is used for the 3D case, where the analog of Equation (3) reads [

with being the volume and being the density of the bulk system. Equation (5) yields

. Fixing the density to that of the tricritical point at [

For both systems, the tricritical temperature increases with, as one may expect when the dipolar interactions (which stabilize the FF state) become more and more important as compared to the spherical attractive ones. More interestingly,

Finally, we briefly discuss the influence of the dimen-

sion of the dipole vector (rather than that of the space accessible for the particles) on the appearance of ferroelectric order. Specifically, we consider a “true” 2D system where, in addition to the spatial confinement of the particles within the x-y-plane, the orientations of the dipole vectors are restricted to that plane as well. Indeed, as shown in previous simulations and theoretical studies (see, e.g., [

A direct comparison of Equation (6) with its quasi-2D analog in Equation (3) shows that, at fixed density, the ferroelectric ordering in the true 2D system occurs at a higher temperature. This is a consequence of the decrease of the dipolar fluctuations (and thus, the orientational entropy) due to their restriction to the plane. Moreover, as revealed in

In this work we have calculated the fluid phase diagram of a quasi-2D Stockmayer fluid by means of density functional theory in the modified mean-field approximation. At the dipole moment considered the system exhibits an isotropic fluid phase where the dipole moments are randomly oriented, yet with a preference for in-plane directions, and a ferroelectric fluid phase characterized by global, in-plane polarization. Apart from exploring the quasi-2D phase behavior, another focus of our study was to identify the role of the dimension of accessible space, as well as that of the dimension of the dipole vector. To quantify these effects on a mean-field level, we have considered the location of the tricritical point. Regarding the impact of space dimension, we have found that decreasing the system’s dimension in z-direction from the bulk limit over slab systems towards the 2D limit shifts the TCP towards lower temperatures and densities. Furthermore, the disappearance of the isotropic liquid phase in the quasi-2D system also shows that the confinement enhances the stability of dense ordered phases relative to disordered ones. Clearly, care has to be taken with respect to the predictions of our mean-field-like DFT approach on a quantitative level. Indeed, from computer simulations [13,14,17] it is known that a quasi-2D Stockmayer fluid at does have a stable isotropic liquid phase at densities beyond the isotropic vapor-liquid critical point, which is absent in our study. This discrepancy reflects the well-known tendency of the DFT to overestimate the stability of ordered phases. However, based on previous DFT studies for bulk and confined systems one would expect a recovery of the isotropic liquid state in the quasi-2D case upon further decrease of. A further interesting result of our study concerns the role of the spin dimension. Here we have found that complete restriction of the dipole moments on in-plane directions yields ferroelectric ordering at temperatures not only higher than those in the quasi-2D system, but even higher than those in 3D.

There remains the question whether fluid states with long-range ferroelectric order, as predicted by DFT, exist at all in quasi-2D and true 2D systems. As mentioned in the introduction, computer simulations give conflicting answers, which may also depend on the number of particles considered in the simulation (indeed, the MD study on quasi-2D systems by Ouyang et al. [

Despite these pitfalls, the DFT approach provides a general overview of the phase diagrams and highlights the dimensionality effects by providing the leading order terms in the free energy. From that perspective, it would be interesting to extend the study towards 2D systems in external fields.