%0 Book Section %1 statphys23_0847 %A Masters, A.J. %A Vlasov, A.Y.U. %A Sokolova, E.P. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K attractions fundamental measure star statphys23 topic-6 virial %T An exact diagrammatic basis for fundamental measure theory %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=847 %X Fundamental Measure Theory (FMT) is a powerful and popular means of predicting the properties of inhomogeneous hard body fluids 1. Its scope is wide-ranging and it has been successfully used to predict the properties of anisotropic phases, such as solids and nematic and smectic liquid crystals, as well as the properties of fluids in external fields. The basic assumption of the theory is that the Helmholtz energy may be written as a functional of a small number of weighted densities. The weights have geometrical interpretations involving the mean radius, surface area and volume of the particles. The detailed forms of the weighting functions are obtained from an analysis of the Mayer f-function. For the case of parallel cubes, a very clear discussion is given in 2. A problem with this type of derivation, however, is that it is not clear what approximations have been made and thus it is not easy to make systematic improvements. Furthermore the derivations require considerable algebra, leading to a lack of physical insight into why the FMT functional takes the form it does. Finally, to our knowledge, FMT is not easily applied to anything other than hard particles, meaning that one cannot treat the important case of attractive forces. To help cast light on this situation, we note that the FMT equation of state for an isotropic fluid of parallel cubes is identical to that obtained by summing the complete star Ree-Hoover diagrams of the virial expansion 3,4. We have now demonstrated that this result also holds for the general inhomogeneous case. Indeed the same thing happens for mixtures of hard parallelipipeds with restricted orientations - FMT is equivalent to summing the complete stars. This equivalence gives an exact diagrammatic foundation for FMT and thereby allows one to make systematic corrections. We present results showing how one might make such corrections and the effects they have. We next show how it is possible to sum the complete star diagrams for sticky parallelipipeds, leading to an FMT-like functional which incorporates attractive forces. We present some predictions of this theory, relating, for example, to gas-liquid co-existence. While parallipipeds provide an interesting system which exhibits a rich phase behaviour, it would be nice to extend our theoretical work, both on systematic improvements to FMT and the incorporation of sticky attractions, to deal with other shaped particles, such as spheres and freely-rotating plates and rods. We offer some speculations as to how this might be achieved. 1) Y. Rosenfeld, Phys. Rev. A 43, 5424 (1991)\\ 2) J. A. Cuesta and Y. Martinez-Raton, J. Chem. Phys. 107, 6397 (1997)\\ 3) S. K. Mitra and A. R. Allnatt, J. Phys. C (Solid State), 12, 2261 (1979)\\ 4) E. P. Sokolova and N. P. Tumanyan, Liq. Cryst. 27, 813 (2000)

@incollection{statphys23_0847, abstract = {Fundamental Measure Theory (FMT) is a powerful and popular means of predicting the properties of inhomogeneous hard body fluids [1]. Its scope is wide-ranging and it has been successfully used to predict the properties of anisotropic phases, such as solids and nematic and smectic liquid crystals, as well as the properties of fluids in external fields. The basic assumption of the theory is that the Helmholtz energy may be written as a functional of a small number of weighted densities. The weights have geometrical interpretations involving the mean radius, surface area and volume of the particles. The detailed forms of the weighting functions are obtained from an analysis of the Mayer f-function. For the case of parallel cubes, a very clear discussion is given in [2]. A problem with this type of derivation, however, is that it is not clear what approximations have been made and thus it is not easy to make systematic improvements. Furthermore the derivations require considerable algebra, leading to a lack of physical insight into why the FMT functional takes the form it does. Finally, to our knowledge, FMT is not easily applied to anything other than hard particles, meaning that one cannot treat the important case of attractive forces. To help cast light on this situation, we note that the FMT equation of state for an isotropic fluid of parallel cubes is identical to that obtained by summing the complete star Ree-Hoover diagrams of the virial expansion [3,4]. We have now demonstrated that this result also holds for the general inhomogeneous case. Indeed the same thing happens for mixtures of hard parallelipipeds with restricted orientations - FMT is equivalent to summing the complete stars. This equivalence gives an exact diagrammatic foundation for FMT and thereby allows one to make systematic corrections. We present results showing how one might make such corrections and the effects they have. We next show how it is possible to sum the complete star diagrams for sticky parallelipipeds, leading to an FMT-like functional which incorporates attractive forces. We present some predictions of this theory, relating, for example, to gas-liquid co-existence. While parallipipeds provide an interesting system which exhibits a rich phase behaviour, it would be nice to extend our theoretical work, both on systematic improvements to FMT and the incorporation of sticky attractions, to deal with other shaped particles, such as spheres and freely-rotating plates and rods. We offer some speculations as to how this might be achieved. 1) Y. Rosenfeld, Phys. Rev. A 43, 5424 (1991)\\ 2) J. A. Cuesta and Y. Martinez-Raton, J. Chem. Phys. 107, 6397 (1997)\\ 3) S. K. Mitra and A. R. Allnatt, J. Phys. C (Solid State), 12, 2261 (1979)\\ 4) E. P. Sokolova and N. P. Tumanyan, Liq. Cryst. 27, 813 (2000)}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Masters, A.J. and Vlasov, A.Y.U. and Sokolova, E.P.}, biburl = {https://www.bibsonomy.org/bibtex/2f107084eb4725eab8983bd9030762f80/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {daf74673f19377062fed2d9af886ba14}, intrahash = {f107084eb4725eab8983bd9030762f80}, keywords = {attractions fundamental measure star statphys23 topic-6 virial}, month = {9-13 July}, timestamp = {2007-06-20T10:16:30.000+0200}, title = {An exact diagrammatic basis for fundamental measure theory}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=847}, year = 2007 }