Tensorial density functional theory for non-spherical hard-body fluids

Abstract

In a recent publication (Hansen-Goos and Mecke 2009 Phys. Rev. Lett. 102 018302) we constructed a free energy functional for the inhomogeneous hard-body fluid, which reduces to Rosenfeld's fundamental measure theory (Rosenfeld 1989 Phys. Rev. Lett. 63 980) when applied to hard spheres. The new functional is able to yield the isotropic-nematic transition for the hard-spherocylinder fluid in contrast to Rosenfeld's fundamental measure theory for non-spherical particles (Rosenfeld 1994 Phys. Rev. E 50 R3318). The description of inhomogeneous isotropic fluids is also improved when compared with data from Monte Carlo simulations for hard spherocylinders in contact with a planar hard wall. However, the new functional for the inhomogeneous fluid in general does not comply with the exact second order virial expansion. We introduced the ζ correction in order to minimize the deviation from Onsager's exact result in the isotropic bulk fluid. In this article we give a detailed account of the construction of the new functional. An extension of the ζ correction makes the latter better suited for non-isotropic particle distributions. The extended ζ correction is shown to improve the description of the isotropic-nematic bulk phase diagram while it has little effect on the results for the isotropic but inhomogeneous hard-spherocylinder fluid. We argue that the gain from using higher order tensorial weighted densities in the theory is likely to be inferior to the associated increase in complexity.

Fundamental measure theory for non-spherical hard particles: predicting liquid crystal properties from the particle shape

“…This term is missing from the earlier parabolic approximation ( ) ζ S par of Hansen-Goos and Mecke [6]. We further included a term with an exponent 1 < q < 2 that allows the second derivative of ( ) ζ S fit to similarly diverge at the other extremal value S = −1/2; this seems to be a possibility if we consider the numerical data [6]. In the following, we use equation (25) including the polynomial in S of degree 5 Note that the fitted value for q is close to one and, thus, b 2 and a 1 are strongly correlated.…”

“…The parameter ζ in equation (5) is a renormalization factor introduced by Hansen-Goos and Mecke [5,6] to correct for the truncation of this expansion of the Mayer function after the term involving rank-two tensors. Common values for ζ are ζ = 5/4, which is obtained by minimizing the difference between the exact and edFMT excluded volumes [5] and ζ = 1.6, which results from a fit to simulation results for the isotropic-nematic (IN) transition [5].…”

“…Common values for ζ are ζ = 5/4, which is obtained by minimizing the difference between the exact and edFMT excluded volumes [5] and ζ = 1.6, which results from a fit to simulation results for the isotropic-nematic (IN) transition [5]. The edFMT form of the functional as written here has the following drawbacks when considering liquid crystals of elongated particles: (i) The IN transition is not accurately described unless the fitted value ζ = 1.6 is used [5,6]. (ii) The width of the coexistence and the surface tension of the IN interface is too low [33].…”

“…These calculations are very computationally intensive, which is why we proposed an expansion of N 12 in tensors [32] that converges more quickly than the tensor expansion that was originally proposed [6]. Alternatively, one can use Wertheim's expansion of the Mayer bond in spherical harmonics [43], which we will report on elsewhere [34].…”

“…A driving force for these efforts is the possibility that these experimental systems form structures that can be applied as novel materials. While these hard-particle systems were originally studied using computer simulations [2], the application of continuum theories is more natural for some long-wave-length or high-symmetry problems.Density functional theory (DFT) [3,4] is a continuum theory for systems that are inhomogeneous or anisotropic either due to applied external fields [3,5,6] or spontaneous symmetry breaking [7][8][9]. Hard spheres represent a classical and quite tractable system to which density functional theory has been applied in many studies.…”