Measuring the equation of state of a hard-disc fluid

– We use video microscopy to study a two-dimensional (2D) model fluid of charged colloidal particles suspended in water and compute the pressure from the measured particle configurations. Direct experimental control over the particle density by means of optical tweezers allows the precise measurement of pressure as a function of density. We compare our data with theoretical predictions for the equation of state, the pair-correlation function and the compressibility of a hard-disc fluid and find good agreement, both for the fluid and the solid phase. In particular, the location of the transition point agrees well with results from Monte Carlo simulations. Hard-disc (HD) fluids play a prominent role in liquid-state theories. This is due to the fact that, first, they often serve as reference systems in perturbation theories of two-dimensional (2D) liquids (just as hard-sphere fluids do for liquids in three dimensions), and that, secondly, at high densities the behavior of every 2D fluid is dominated by excluded-volume effects, which in turn depends just on the short-ranged hard-core part of the interparticle potential. Mainly for these two reasons, the HD equation of state (EOS) appears also in many theories on monolayer adsorption on solid surfaces [1,2], an aspect illustrated, for example, in ref. [3], where the HD EOS is used in statistical mechanical theories modeling the binding of peripheral globular proteins on lipid membranes. The important role of the HD system explains the overwhelming number of theoretical studies on the EOS of a HD fluid, starting as early as 1959 with the presentation of results from scaled particle theory [4]. Most of all approaches to the EOS are based on particular resummations of the virial series and the construction of sophisticated Padé approximants [5–7], others use results from integral equation theories [8], or from theories based on overlap volume functions [9]. Not only the EOS, but also the structure of a hard-disc fluid has been explored in detail and is to date well understood [7–10]. Hard discs are popular also as model system to test advanced density functional theories [11, 12], used, for instance, to study discs in cavities [13], or induced freezing and re-entrant melting [14]. Many computer simulation studies of HD systems are available [1, 15, 16], of which the more recent ones have focused mainly on the melting properties of HD solids [16]. All these theoretical efforts contrast with the situation on the experimental side. To our knowledge, the HD EOS has never been tested with experimental data. The present letter aims at closing this gap. We report on experiments with a 2D model liquid of charged colloidal particles suspended in water. Correlation functions computed from real-space images, together with the virial equation, are used to calculate the pressure p of the 2D liquid as a function of the 2D particle density ρ, which in our experiment can be conveniently varied by means of optical tweezers. We have thus realized a 2D colloidal model fluid for which the EOS, i.e. the p(ρ) diagram, can be determined directly. Comparing the experimental to the theoretical Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-413077 Erschienen in: Europhysics Letters (EPL) ; 63 (2003), 6. S. 791-797 https://dx.doi.org/10.1209/epl/i2003-00495-1