Table of contents

Yaakov Rosenfeld

This special issue on density functional theory is dedicated to the memory of Yaakov (Yasha) Rosenfeld. The Liquid State community lost an exceptional scientist and a special friend on 21 July 2002. Yasha had intended to contribute to this volume. Sadly, he died from lung cancer at the age of 54.

Yasha made outstanding contributions to the statistical mechanics of liquids and dense plasmas. He was the author of more than 100 papers, several of which are regarded as `classics'. His early work was on the theory of freezing in simple fluids and on high pressure equations of state. In 1979, together with N W Ashcroft, he developed the idea of universality of the bridge function for simple fluids. The key publication [1] proved very influential in the theory of bulk liquids and carries >400 citations. At about the same time Yasha began his lifelong studies of the properties of classical plasmas, focussing on the equation of state in the strong-coupling limit. He remains a major figure in the theory of Coulomb fluids to which he successfully applied the ideas of Onsager `charge smearing', providing a lower bound to the electrostatic energy. In the mid-80s Yasha turned his attention to the properties of fluids in the asymptotic high density limit (AHDL), arguing that at sufficiently high densities hard-core liquids and the classical one-component plasma (OCP) share common structural and thermodynamic features and that it is possible to perform a perturbative analysis about the AHDL, i.e. about packing fraction η = 1 for hard spheres and plasma parameter Γ = ∞ for the OCP. Whilst some of the concepts introduced in the AHDL papers appeared esoteric to many researchers, these provided a basis for much of Yasha's later work. An example of his skilful combination of classical and quantum DFT for dense ion-electron plasmas is contained in [2]. This paper provided inspiration for later Thomas-Fermi molecular dynamics simulations developed in France and the USA (Livermore).

In a remarkable paper [3] he developed a new graphical expansion for the pair direct correlation functions c_ij⁽²⁾ of uniform mixturesof hard particles. The expansion involves pair overlap volumes as basis functions and pair excluded volumes as basic variables. Yasha showed that the lowest order version of the theory, the `scaled field particle approximation' yields c_ij⁽²⁾ that are the same as those of Percus Yevick (PY) theory for hard spheres in D (odd) dimensions. The paper, which constitutes a synthesis of scaled particle ideas and PY theory, provided geometric approximations for c_ij⁽²⁾ based on expansions in terms of the fundamental measures of hard particles. It was an important forerunner for Yasha's first contribution [4] to hard-sphere DFT, a Letter which continues to have an enormous influence on the development and on the applications of classical DFT.

We recall some of the history. Independently, Tarazona [5] and Curtin and Ashcroft [6] had constructed weighted-density approaches to DFT, primarily for the hard-sphere fluid. Their strategy was to write the excess free energy functional asF_ex[ρ]=∫ dr ρ(r) Ψ_ex((r)) where the excess free energy per particle Ψ_ex is evaluated at some coarse-grained or weighted density (r). The weight function w(r;), which determines (r), was obtained by requiring the second functional derivative of -β F_ex[ρ] to yield an accurate c⁽²⁾ for the uniform fluid and in practice this meant enforcing the known PY result for hard spheres. Although the approaches proved very successful in a wide variety of applications (including hard-sphere freezing) they are somewhat ad hoc in character and extensions to mixtures are not straightforward. Yasha [4] started from a very different perspective, motivated by his earlier work [3]. His is foremost a theory for hard-sphere mixtures, and it is based upon a deconvolution of the Mayer f functions f_ij(r) in terms of four scalar and two vector weight functions. The functional is written as βF_ex[{ρ_i}]= ∫ dr Φ({n_α(r)}) where Φ is a function of the weighted densities {n_α(r)} with α labelling the six weights. In the uniform fluid the weighted densities reduce to the usual scaled particle variables. Φ is constructed so that F_ex recovers the exact low densitiy limit and satisfies a certain thermodynamic requirement (a brief summary can be found in [7]). The resulting functional generatesc_ij⁽²⁾ which are identical to those of PY theory. Since Yasha's approach is based on the geometrical properties of the spheres he termed it fundamental measure theory (FMT). Unlike the earlier approaches, where the density dependent weight has a range equal to the hard sphere diameter, in FMT the range of the weights is equal to the radius. FMT has now become the theory of choice for most DFT practitioners and the number of applications to inhomogeneous, and, indeed, bulk systems is growing very rapidly-as can be gleaned from several papers in the present volume. In subsequent papers Yasha proposed extensions of his FMT to nonspherical, hard convex bodies.

Another important development of FMT was made in [8]. As the title suggests, this paper encompasses many topics in the theory of liquids; one of us had the (pleasurable!) task of refereeing it-the original version was longer and broader in its scope than the published version of 23 pages in J. Chem. Phys. A key feature is a self-consistent approach for determining the density profiles for general fluid mixtures in external potentials. Crudely speaking, F_ex[{ρ}]is expanded about a uniform reference fluid and terms beyond second order are approximated by a `bridge functional', assumed to be universal. Self-consistency is imposed by the test-particle procedure for the uniform fluid where the method is equivalent to the modified HNC theory. The approach enabled the ideas of FMT to be taken over to a very wide class of liquids, including Coulomb fluids, and has proved very successful for determining the one-body structure of highly inhomogeneous fluids, and bulk fluid pair structure.

The theory of freezing was a favourite topic of Yasha. The original FMT [4] could not account for the freezing transition of the hard sphere fluid. For some time this lead Yasha to ponder whether freezing could/should be described by DFT. (He was not too impressed by results from the earlier DFTs!) However, during a sabbatical year spent in Bristol, Düsseldorf, Lyon, Madrid, München, ... he enquired how the FMT could be modified in order to incorporate freezing. He and his co-workers argued that an accurate DFT should account for dimensional crossover, i.e. situations for which the fluid is confined to effectively lower dimensionality such as in narrow slits or cylindrical pores. In particular they focused on the zero-dimensional limit, which pertains to a cavity that can contain at most one sphere mimicking the situation of a sphere restricted to a site of the crystal lattice, and constructed empirical modifications to the original result for the reduced free energy function Φ({n_α(r)}] that could recover the exact free energy in this limit [9]. The resulting functionals provided a good description of the freezing transition for the hard sphere fluid and found applications in other situations of extreme confinement. In subsequent work with Tarazona [10], Yasha developed a systematic FMT approach based entirely on knowledge of the exact free energy in the zero dimensional limit. Such an approach has provided an excellent account of the properties of the hard-sphere solid and, as discussed by M. Schmidt in this volume, has provided a powerful prescription for generating DFTs for other types of fluids. Moreover it enabled Yasha to make contact with and shed new light on his earlier work on the AHDL for fluids and solids. Among other recent extensions of FMT, Yasha and co-workers made one of the first attempts to include the solvent explicitly in DFT of electric double-layers [11].

Yasha continued working until the end. Bravely he attended the Les Houches Meeting for JPH in April 2002 and his contribution appeared in the special issue [12]. Another paper [13] appeared on July 22, one day after his death.

Of course our brief description does not do justice to Yasha's contributions to physics nor does it reflect properly his inimitable style. Reading his papers (not an easy challenge but one that is ultimately rewarding!) one could not fail to be impressed by Yasha's remarkable insight, ingenuity and technical mastery of the subject. He thought about problems in a unique, sometimes near mystical, way. He made connections where other researchers could not. He was inspirational in discussion-even if one was not really sure where his ideas originated. He enlivened the subject. It was a privilege to work with Yasha and to know him personally. Each of us has his own recollections of times with Yasha. RE remembers walking in the countryside outside of Bristol, discussing details of the AHDL. We got lost and ended up in the grounds of a mental hospital. Fortunately we obtained a ride from a doctor and his patient. RE persuaded Yasha, or was it the other way round, that it was better not to restart the discussion. HL experienced Yasha's ingenuity and `good nose' in fields removed from physics. Yasha persuaded him to buy a particular stock (TEVA) in March 2000 when the high tech stocks were booming (a detailed list of all Yasha's recommendations is available on request). The TEVA stock continues to flourish contrary to the general Nasdaq crash. In February 2000 JPH experienced Yael and Yasha's boundless hospitality when they proudly showed him around many parts of Israel, including the Dead Sea, Masada and Jerusalem. Rapidly science gave way to spirited discussions on archaeology, history and the state of world affairs!

We shall miss a gentleman of science.

R Evans, J-P Hansen and H Löwen

References

[1] Rosenfeld Y and Ashcroft N W 1979 Theory of simple classical fluids-universality in the short-range structure Phys. Rev. A 20 1208

[2] Ofer D, Nardi E and Rosenfeld Y 1988 Interionic correlations in plasmas- Thomas-Fermi hypernetted-chain density-functional theory Phys. Rev. A 38 5801

[3] Rosenfeld Y 1988 Scaled field particle theory of the structure and the thermodynamics of isotropic hard particle fluids J. Chem. Phys. 89 4272

[4] Rosenfeld Y 1989 Free energy model for the inhomogeneous hard-sphere fluid mixture and DFT of freezing Phys. Rev. Lett. 63 980

[5] Tarazona P 1984 Mol. Phys. 52 81 Tarazona P and Evans R 1984 Mol. Phys. 52 847 Tarazona P 1985 Phys. Rev. A 31 2672

[6] Curtin W A and Ashcroft N W 1985 Phys. Rev. A 32 2909

[7] Roth R, Evans R, Lang A and Kahl G 2002 J. Phys.:Condens. Matter 14 12063

[8] Rosenfeld Y 1993 Free energy model for inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas J. Chem. Phys. 98 8126

[9] Rosenfeld Y, Schmidt M, Löwen H and Tarazona P 1996 Fundamental measure free energy density functional for hard spheres: dimensional crossover and freezing Phys. Rev. E 55 4245

[10] Tarazona P and Rosenfeld Y 1997 From zero-dimensional cavities to free-energy functionals for hard disks and hard spheres Phys. Rev. E 55 R4873

[11] Biben T, Hansen J-P and Rosenfeld Y 1998 Generic density functional for electric double layers in a molecular solvent Phys. Rev. E 57 R 3727

[12] Rosenfeld Y 2002 Structure and effective interactions in multi-component hard-sphere liquids: fundamental-measure density functional approach J. Phys.: Condens. Matter 14 9141

[13] Juranek H, Redmer R and Rosenfeld Y 2002 Fluid variational theory for pressure dissociation in dense hydrogen: multicomponent reference system and non-additivity effects J. Chem. Phys. 117 1768

An introduction is given to this special issue which focuses on recent developments and future perspectives in density functional theory of inhomogeneous classical fluids. Different strategies and applications of classical density functional theory are discussed. The topics briefly treated are different approximation schemes for different interparticle interactions, binary mixtures and polydisperse systems, application to interfaces, wetting, confined geometries and porous media and also dynamical problems treated by density functional theory. Particular emphasis is placed on recent applications of Rosenfeld's fundamental measure approach. Finally a guide is presented to the subsequent articles in this Special Issue.

Density functional theory (DFT) of classical fluids in the canonical ensemble (CE) is studied by means of Legendre transform techniques in an extended variable space. The constraint that arises from having a fixed number of particles N is incorporated into the theory by considering N and its conjugate, the CE chemical potential μ, as additional variables in the DFT analysis of the problem. The method allows for obtaining an Ornstein–Zernike (OZ) equation in the CE. A comparison is made with other approaches to the problem. In particular, μ is identified with the Lagrange multiplier related to the fixed-N constraint and a further Legendre transform leads to an equivalent OZ equation equal to that of the grand canonical ensemble (GCE) in terms of functional derivatives stripped off from their asymptotic behaviour. The CE analogous to the GCE direct correlation function is introduced as an excess (over ideal) quantity in terms of which we obtain a CE compressibility equation.

Fundamental measure theory (FMT) has recently been extended to penetrable spheres and soft spherical interactions (soft-FMT) (Schmidt M 1999 Phys. Rev. E 60 R6291; 2000 J. Phys.: Condens. Matter11 10 163). This paper presents these theories in a unified description for a pure system and also describes a simple procedure that is thought to improve the accuracy of FMT for soft, spherically repulsive interactions. An ultra-soft interaction, which is a model for the interaction of star polymers with arm number about 8 in a good solvent, is investigated and a simple procedure is found to significantly improve the accuracy of bulk thermodynamic and pair-correlation functions generated by soft-FMT when compared to Monte Carlo simulation results. The simple procedure also improves prediction of the bulk pressure–density relationship for a square-shoulder system. Similar gains in accuracy are expected for a wide range of soft interactions.

A new general form of the multi-Yukawa, multicomponent closure of the Ornstein–Zernike equation for factored interactions is derived. The general solution is given in terms of an M × Mscaling matrix Γ obtained by solving M (equal to the number of Yukawa terms in the closure) equations together with M(M − 1) symmetry conditions where ^{( n)} is of higher order in the density, and all quantities are algebraic functions of Γ.

Explicit formulae for the thermodynamic properties are also provided.

Density functional theory (DFT) is the most successful simple theory for ions near an electrode (the double layer). However, most previous applications of DFT have been for ions that are relatively weakly coupled. Interesting effects have been found in simulations for ions that are strongly coupled. Specifically, drying of the electrode with a resultant large increase in the magnitude of the adsorption is observed. Further, the capacitance decreases with increasing coupling. The DFT formalism requires the direct correlation function of the bulk electrolyte as input. If the bulk electrolyte is treated by means of the mean spherical approximation (MSA), DFT fails to account for these phenomena. However, if the bulk electrolyte is treated by means of a generalized MSA, partial success results. The electrolyte dries the electrode but the lowering of the capacitance is predicted only weakly. Further refinements are necessary for full success.

A new approach to the structure of the inhomogeneous fluid mixture is developed within the framework of density functional theory by expanding the Denton–Ashcroft weighted density-based expression for the first-order direct correlation function as a Taylor series in the weighted density inhomogeneity. The connection of this expansion with the conventional functional Taylor perturbative approach is investigated. A truncation of the proposed expansion coupled with the use of a scaled parameter to mimic the effect of the neglected higher-order contributions, determined self-consistently by forcing the wall theorem to be satisfied, is implemented. The theory is employed to predict the structure of a pure hard-sphere fluid as well as a hard-sphere mixture near a hard wall and the agreement with the available simulation results is found to be very good.

We analyse the structure of the fundamental measure theory for the free energy density functional of hard-sphere mixtures. A comparative study of the different versions of the theory, and other density functional approaches, is carried out in terms of their generic form for the three-point direct correlation function, which shows clearly the main advantages and problems of the different approximations. A recently developed version for the monocomponent case is extended to mixtures of hard spheres with different radii, and a new prescription is presented for obtaining the exact dimensional crossover of those mixtures in the one-dimensional (1D) limit. Numerical results for planar wall–fluid interfaces and for the 1D fluid are presented.

A variational approach based on a functional of the polarization charge density at interfaces between different dielectric bodies is used to calculate the energy of electrostatic interaction between two electric charges embedded in either two spherical dielectric cavities or in one spheroidal cavity. An effective, distance-dependent dielectric function is extracted from these results, which is exact within the macroscopic theory of dielectrics. We show that different dielectric functions must be associated with pairs of equally or oppositely charged ions and with pairs of ions in cavities of the same or differing radii.

Lattice gas based models are usually discussed in terms of spin averages instead of distribution functions. As they are very useful in the study of adsorption phenomena, a density functional (DF) formalism, which would unify the discussion of both the liquid and the adsorbed phases, seems a most useful alternative. Here we present a first step in that direction by deriving the two essential components needed for any DF theory. The first one is a fully developed Ornstein–Zernike (OZ) formalism which we arrive at in two steps. The first one is the definition (through functional differentiation of the grand canonical partition function) of the distribution and correlation functions hierarchies. In the second step we find that the rigid neighbourhood of a lattice gas forces us, if an authentic DF theory is our goal and even in the grand canonical ensemble, to define N-modified distribution and correlation functions much in the same way as we have recently done when discussing DF theory in the canonical ensemble. These N-modified hierarchies of correlation functions are, indeed, linked by a full set of n-body OZ equations. The second ingredient for any DF theory is an expression for the entropy (in terms of the already discussed correlation functions) which we obtain by following previous work by us in fluids. We also generalize the compressibility contribution to the entropy by using the already derived lattice gas formalism in a way immediately translatable to liquids. In summary, we show how a deep and intimate relationship between lattice gases and fluids can be obtained if both are discussed in a DF framework with functional differentiation techniques and, therefore, we think that the beginnings of a DF theory of lattice gases are established.

A fundamental measure functional is used to study the induced freezing and re-entrant melting for the hard-disc fluid in an external periodic potential. The phase diagram obtained shows good agreement with recent experimental studies. We also use the functional to describe the hard-disc fluid density near a hard wall and investigate whether this 2D functional can correctly describe the 0D limit.

Using an effective logarithmic–Gaussian pair potential that models the interaction between star polymers, we compare the hypernetted chain (HNC) and random phase approximations (RPA) for calculating the bulk structure (including the Fisher–Widom and Lifshitz lines), thermodynamic functions and phase diagram of a phase-separating binary fluid of star polymers, of two-arm length ratio 2:1. Thereby, the stars considered here are equivalent to linear chains in the mid-point representation of their effective interaction. We find that at densities where the star coronas overlap, the quasi-exact HNC and RPA give very similar results. Using a density functional approach, with a functional which generates the RPA, we calculate properties of the inhomogeneous binary fluid. We determine the surface tension and one-body density profiles at the free fluid–fluid interface. For states well removed from the critical point the profiles exhibit pronounced oscillations. For a purely repulsive planar wall potential that models the effective potential between a star polymer and a hard wall, we find a first-order wetting transition with the associated pre-wetting line.

We consider a model mixture of hard colloidal spheres and nonadsorbing polymer chains in a theta solvent. The polymer component is modelled as a polydisperse mixture of effective spheres, mutually noninteracting but excluded from the colloids, with radii that are free to adjust to allow for colloid-induced compression. We investigate the bulk fluid demixing behaviour of this model system using a geometry-based density functional theory that includes the polymer size polydispersity and configurational free energy, obtained from the exact radius-of-gyration distribution for an ideal (random-walk) chain. Free energies are computed by minimizing the free energy functional with respect to the polymer size distribution. With increasing colloid concentration and polymer-to-colloid size ratio, colloidal confinement is found to increasingly compress the polymers. Correspondingly, the demixing fluid binodal shifts, compared to the incompressible-polymer binodal, to higher polymer densities on the colloid-rich branch, stabilizing the mixed phase.

We develop a density functional for hard-sphere mixtures which keeps the structure of Rosenfeld's fundamental measure theory (FMT) whilst inputting the Mansoori–Carnahan–Starling–Leland bulk equation of state. Density profiles for the pure hard-sphere fluid and for some binary mixtures adsorbed at a planar hard wall obtained from the present functional exhibit some improvement over those from the original FMT. The pair direct correlation function c⁽²⁾ (r) of the pure hard-sphere fluid, obtained from functional differentiation, is also improved. When a tensor weight function is incorporated for the pure system our functional yields a good description of fluid–solid coexistence and of the properties of the solid phase.

We present the extension of Rosenfeld's fundamental measure theory to lattice models by constructing a density functional for d-dimensional mixtures of parallel hard hypercubes on a simple hypercubic lattice. The one-dimensional case is exactly solvable and two cases must be distinguished: all the species with the same length parity (additive mixture), and arbitrary length parity (nonadditive mixture). To the best of our knowledge, this is the first time that the latter case has been considered. Based on the one-dimensional exact functional form, we propose the extension to higher dimensions by generalizing the zero-dimensional cavity method to lattice models. This assures the functional will have correct dimensional crossovers to any lower dimension, including the exact zero-dimensional limit. Some applications of the functional to particular systems are also shown.

We test the accuracy of a recently proposed density functional (DF) for a fluid in contact with a porous matrix. The DF was constructed in the spirit of Rosenfeld's fundamental measure concept and was derived for general mixtures of hard core and ideal particles. The required double average over fluid and matrix configurations is performed explicitly. As an application we consider a model mixture where colloids and matrix particles are represented by hard spheres and polymers by ideal spheres. Integrating over the degrees of freedom of the polymers leads to a binary colloid–matrix system with effective Asakura–Oosawa pair potentials, which we treat with an integral-equation theory. We find that partial pair correlation functions from both theories are in good agreement with our computer simulation results, and that the theoretical results for the demixing binodals compare well, provided the polymer-to-colloid size ratio, and hence the effect of many-body interactions neglected in the effective model, is not too large. Consistently, we find that hard (ideal) matrix–polymer interactions induce capillary condensation (evaporation) of the colloidal liquid phase.

We treat the non-equilibrium process of random sequential adsorption of hard particles onto a solid substrate by means of a geometry-based density functional theory. As a prerequisite we solve the zero-dimensional case exactly and use it to construct density functionals in higher dimensions, permitting the treatment of adsorption onto arbitrary spatially inhomogeneous substrates. As applications we study the influence of a hard boundary of the adsorption region in the one-dimensional car-parking problem and for colloidal deposition on a two-dimensional solid substrate. Comparing to our computer simulation results, we find that the respective density functionals correctly predict the oscillatory density profiles near the boundary, with amplitudes that are considerably smaller than in the corresponding equilibrium models.

Ion transport between two baths of fixed ionic concentrations and applied electrostatic (ES) potential is analysed using a one-dimensional drift-diffusion (Poisson–Nernst–Planck, PNP) transport system designed to model biological ion channels. The ions are described as charged, hard spheres with excess chemical potentials computed from equilibrium density functional theory (DFT). The method of Rosenfeld (Rosenfeld Y 1993 J. Chem. Phys.98 8126) is generalized to calculate the ES excess chemical potential in channels. A numerical algorithm for solving the set of integral–differential PNP/DFT equations is described and used to calculate flux through a calcium-selective ion channel.

It is shown that the density functional theory (DFT) of soft matter is able to capture, in a semi-quantitative way, the essential properties of the phase behaviour in these complex systems. This is illustrated here for the case of the fractionation and segregation of polydisperse fluids of spherical (colloidal) particles and for the case of the self-assembling of monodisperse fluids of non-spherical particles. Both homogeneous and inhomogeneous situations will be considered and described within the same DFT.

Recent simulation and experimental results on molecular alignment at the liquid–vapour interface of dipolar fluids are in disagreement with the predictions of all existing theories. We discuss these problems in the context of our earlier work on association in dipolar fluids and propose new theoretical approaches to tackle them.

This paper summarizes our efforts to develop fast algorithms for density functional theory (DFT) calculations of inhomogeneous fluids. Our goal is to apply DFTs to a variety of problems in nanotechnology and biology. To this end we have developed DFT codes to treat both atomic fluid models and polymeric fluids. We have developed both three-dimensional real space and Fourier space algorithms. The former rely on a matrix-based Newton's method while the latter couple fast Fourier transforms with a matrix-free Newton's method. Efficient computation of phase diagrams and investigation of multiple solutions is facilitated with phase transition tracking algorithms and arclength continuation algorithms. We have explored the performance that can be obtained by application of massively parallel computing, and have begun application of the codes to a variety of two-and three-dimensional systems. In this paper, we summarize our algorithm development work as well as briefly discuss a few applications including adsorption and transport in ion channel proteins, capillary condensation in disordered porous media and confinement effects in a diblock copolymer fluid.

We present a self-consistent field theory (SCFT) for dilute solutions of semiflexible (wormlike) diblock copolymers, each consisting of a rigid and a flexible part. The segments of the polymers are otherwise identical, in particular with regard to their interactions, which are taken to be of an Onsager excluded-volume type. The theory is developed in a general three-dimensional form, as well as in a simpler one-dimensional version. Using the latter, we demonstrate that the theory predicts the formation of a partial-bilayer smectic-A phase in this system, as shown by profiles of the local density and orientational distribution functions. The phase diagram of the system, which includes the isotropic and nematic phases, is obtained in terms of the mean density and rigid-rod fraction of each molecule. The nematic–smectic transition is found to be second order. Since the smectic phase is induced solely by the difference in the rigidities, the onset of smectic ordering is shown to be an entropic effect and therefore does not have to rely on additional Flory–Huggins-type repulsive interactions between unlike chain segments. These findings are compared with other recent SCFT studies of similar copolymer models and with computer simulations of several molecular models.

Glassy dynamics of fluid particles in a supercooled liquid is discussed on the basis of the time-evolution equation obtained through the dynamical density functional theory (DDFT). The advantage, brought about by the coarse-grained nature of the formalism, in treating such strongly correlated motion over other approaches, such as the mode-coupling theories and direct computer simulations, is emphasized. A direction in which the DDFT should prove its worth on examining the phenomena is suggested.

We propose a density functional for anisotropic fluids of hard body particles. It interpolates between the well established geometrically based Rosenfeld functional for hard spheres and the Onsager functional for elongated rods. We test the new approach by calculating the location of the nematic–isotropic transition in systems of hard spherocylinders and hard ellipsoids. The results are compared with existing simulation data. Our functional predicts the location of the transition much more accurately than the Onsager functional, and almost as well as the theory by Parsons and Lee. We argue that it might be suited to study inhomogeneous systems.