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- Grand Canonical Monte Carlo Simulation of Confined Hard Sphere Fluids
- supervised by Gerd Schröder-Turk, Klaus Mecke, Roland Roth
- finished 2010-08-18
- [PDF]

Snapshot of a hard sphere liquid (radius is 0.5) confined to one of the domains
given by the Gyroid cubic minimal surface with lattice size *L*=17. The cubic frame is the simulation box which in turn corresponds to a translational unitcell of the surface.

Thermodynamical properties of confined fluids depend on the shape of the confinining cavity. A general hypothesis assumes that this dependence is expressed as a linear combination of four terms, namely the volume of the cavity, its interfacial area and integrated mean curvature and its Euler characteristic [5]. König et al. have shown by means of density functional theory that for hard-sphere fluids and for simple pore shapes this hypothesis is numerically valid [3,4]. Here, this work is extended by grand-canonical Monte Carlo simulation studying the equilibrium fluid density and adsorption in cavities of complex shape. The confining cavities are given by periodic labyrinthine domains bounded by so-called triply-periodic minimal surfaces. The actual solid-pore interface is representated by a triangulated version of these surfaces. Due to previous work of Gerd Schröder-Turk these tringulations are available for all studied structures. However, using the example of the Diamond constant mean curvature family, an alternative method is described for the generation of tringulated triply-periodic geometries. This is done with the help of the surface evolver program [1].

The implementation of the Monte Carlo algorithm as a C++ programis explained in detail. In particular, a fast predicate to testwhether a sphere is inside or outside of the confining domain is described. For bulk simulations, i.e. simulations without confinement, we find good agreement with the Carnahan-Starling theory for hard spheres.

The relative difference (*η*_{0}-η)/*η* as predicted by the morphometric approach
(solid line) compared to both the Monte Carlo simulation and DFT datawhich were obtained by an alternative DFT method (symbols) for simulation box sizes *L*=8, 17, 34. *η* is the packing fraction of the bulk fluid the system is
in contact with, whereas *η*_{0} denotes the confined packing fraction, which is defined as the volume occupied by the hard spheres divided by the accessible volume, i.e. the volume the sphere centers can penetrate.

Results for the confined simulations are compared to 3-dimensional numerical DFT calculations
(carried out by Roland Roth) und to an analytic morphometric approach [2,3,4]. We find
quantitative agreement for all studied surfaces in the range of our simulation input parameters
for both the confined packing fraction and the adsorption. Further, we are able to show, that
the Euler number X of the cavity can be estimated from the equilibrium number of spheres in
the cavity. Finally, we find out that the morphometric approach is working particularly good
for channel widths larger than two times the correlation length *ξ* for the studied triply-periodic
minimal surfaces.

[1] K. Brakke. The surface evolver. Experimental Mathematics, 1(2):141–165, 1992.

[2] Hendrik Hansen-Goos and Roland Roth. Density functional theory for hard-sphere mixtures:
the white bear version mark ii. Journal of Physics: Condensed Matter, 18(37):8413, 2006.

[3] P.-M. König. Influence of Geometry on Thermodynamics. the Structure of Fluids, and Effective Interactions in Key-Lock Systems. PhD thesis, Institut für Theoretische und Angewandte Physik,
Universität Stuttgart, 2005.

[4] P.-M. König, R. Roth, and K.R. Mecke. Morphological thermodynamics of fluids: Shape
dependence of free energies. PRL, 93(16):160601, 2004.

[5] K.R. Mecke. Integralgeometrie in der statistischen Physik. Harri Deutsch (Frankfurt), 1994.