The interactions between charged colloidal particles in an electrolyte may be described by usual Debye-Hückel theory provided the source of the electric field is suitably renormalized. For spherical colloids, we reconsider and simplify the treatment of the popular proposal put forward by Alexander et al. [J. Chem. Phys. 80, 5776 (1984)], which has proven efficient in predicting renormalized quantities (charge and salt content). We give explicit formulae for the effective charge and describe the most efficient way to apply Alexander's et al. renormalization prescription in practice. Particular attention is paid to the definition of the relevant screening length, an issue that appears confuse in the literature. PACS numbers:

Abstract. -We experimentally determine effective interparticle potentials in a two-dimensional (2D) colloidal system of charge-stabilized polystyrene particles at different particle densities ρ. Density variation is achieved by means of a scanned optical laser tweezer which serves to create a boundary box for the system. By changing the size of this boundary box, ρ can be systematically varied without having to prepare a new system. From the measured radial distribution functions we can then obtain the effective pair potentials of the particles. While for low particle densities perfect agreement with Yukawa-like potentials is observed, considerable deviations from this form are found at higher densities. We interpret this result as a many-body effect produced by macroion screening which is expected to become more pronounced as the density is increased.There are several analogies between a charge-stabilized colloidal suspension and a fluid metal. A metal essentially consists of highly charged core ions plus the screening charge cloud of the electrons, which the ions are embedded in [1]. Colloidal suspensions, on the other hand, are composed of large and highly charged particles, i.e. macroions which are suspended in a structureless medium and surrounded by a screening atmosphere of microions. Common to both systems is that there are two classes of ingredients (ions/electrons and macroions/microions) which move and respond on totally separated length and time scales. Accordingly, similar theoretical concepts can be applied in both the theory of metals and that of colloidal suspensions. One of the major theoretical tasks in the description of metals is to reduce the many-body electron-ion Hamiltonian to an effective ionic Hamiltonian which is expressed as a sum of volume-, pair-, triple-, and multi-ion interactions. This can be achieved by eliminating the electronic degrees of freedom. In a similar manner, one of the fundamental questions in colloidal systems is, how to integrate out the microionic degrees of freedom and to calculate effective macroion/macroion forces, consisting of the direct Coulomb repulsions and an indirect interaction mediated by the small ions of the electrolyte. In contrast to fluid metal, however, a suspension of charged colloidal particles can directly be observed and studied under an optical microscope. Accordingly, colloidal suspensions provide an ideal testing ground for the various concepts to attack the interesting many-body problem. In the following, we will in particular focus on the widely used concept of "effective" pair interactions [2,3].

Using positional data from videomicroscopy and applying the equipartition theorem for harmonic Hamiltonians, we determine the wave-vector-dependent normal mode spring constants of a two-dimensional colloidal model crystal and compare the measured band structure to predictions of the harmonic lattice theory. We find good agreement for both the transversal and the longitudinal modes. For q-->0, the measured spring constants are consistent with the elastic moduli of the crystal.

Using videomicroscopy data of a two-dimensional colloidal system the bond-order correlation function G{6} is calculated and used to determine both the orientational correlation length xi{6} in the liquid phase and the modulus of orientational stiffness, Frank's constant F{A}, in the hexatic phase. The latter is an anisotropic fluid phase between the crystalline and the isotropic liquid phase. F{A} is found to be finite within the hexatic phase, takes the value 72/pi at the hexatic<-->isotropic liquid phase transition, and diverges at the hexatic<-->crystal transition as predicted by the Kosterlitz-Thouless-Halperin-Nelson-Young theory. This is a quantitative test of the mechanism of breaking the orientational symmetry by disclination unbinding.

Using positional data from video-microscopy we determine the elastic moduli of two-dimensional colloidal crystals as a function of temperature. The moduli are extracted from the wave-vectordependent normal mode spring constants in the limit q → 0 and are compared to the renormalized Young's modulus of the KTHNY theory. An essential element of this theory is the universal prediction that Young's modulus must approach 16π at the melting temperature. This is indeed observed in our experiment.PACS numbers: 64.70. Dv,61.72.Lk,82.70.Dd In the early 70th Kosterlitz and Thouless [1] developed a theory of melting for two-dimensional systems. In their model the phase transition from a system with quasi-long-range order [2] is mediated by the unbinding of topological defects like vortices or dislocation pairs in the case of 2D crystals. They showed that the phase with higher symmetry has short-range translational order. Halperin and Nelson [3,4] pointed out that this phase still exhibits quasi-long-range orientational order and proposed a second phase transition now mediated by the unbinding of disclinations to an isotropic liquid. The intermediate phase is called hexatic phase. This theory, being based also on the work of Young [5], is known as KTHNY theory (Kosterlitz, Thouless, Halperin, Nelson and Young); it describes the temperature-dependent behavior of the elastic constants, the correlation lengths, the specific heat and the structure factor (for a review see [6,7]). Experiments with electrons on helium [8,9] and with 2D interfacial colloidal systems [10,11,12,13,14] as well as computer simulations [15,16,17] have been performed to test the essential elements of this theory. But research has mainly focused on the behavior of the correlation functions (an illustrative example is the work of Murray and van Winkle [11]). Only a few works can be found that deal with the elastic constants, especially the shear modulus, [9,18,19,20] even though the Lamé coefficients and their renormalization near melting take a central place in the KTHNY theory.A very strong prediction of the KTHNY theory has never been verified experimentally. It states that the renormalized Young's Modulus K R (T ), being related just to the renormalized Lamé coefficients µ R and λ R , must approach the value 16π at the melting temperature [4],which is obviously an universal property of 2D systems at the melting transition. This Letter presents experimental data for elastic moduli of a two-dimensional colloidal model system, ranging from deep in the crystalline phase via the hexatic to the fluid phase. These data, indeed, confirm the theoretical prediction expressed by eq. (1).The experimental setup is the same as already described in [23]. The system is known to be an almost perfect 2D system; it has been successfully tested, and explored in great detail, in a number of studies [14,21,22,23]. Therefore we only briefly summarize the essentials here: Spherical colloids (diameter d = 4.5 µm) are confined by gravity to a water/air interface formed by a ...