Introduction

Colloidal suspensions have been studied extensively, not only because of their important roles in natural and technological processes, but also as model systems whose collective behavior may reveal deep insights into the microscopic mechanisms of structural phase transitions. Monodisperse colloidal spheres can act like mesoscopic ``atoms,'' sometimes organizing themselves into crystalline arrays or, under other conditions, wandering randomly like the atoms in a fluid. Transformations among ordered and disordered states are phase transitions reminiscent of melting and freezing in conventional materials, particularly in simple metals and inert gases. Colloidal phase transitions conform to such touchstones of conventional phase transitions as the Lindemann melting criterion and the Hansen-Verlet freezing criterion [1]. Intensive thermodynamic properties, such as the molar latent heat of freezing, also are comparable between colloidal and conventional materials [2].

Colloidal suspensions' allure as model systems derives from their experimental accessibility. Unlike atoms in conventional materials, individual spheres in colloidal suspensions can be large enough to track using conventional video microscopy [3]. Sphere trajectories measured from sequences of video snapshots provide detailed information regarding suspensions' microscopic order and dynamics [4]. Such information has been used to measure colloidal particles' electrostatic [5,6,7,8,9] and hydrodynamic [10] interactions. It also has been used to study topological transformations during phase transitions in two [11] and three dimensions [1]. Light scattering and mechanical measurements provide complementary views of colloidal suspensions' macroscopic viscoelastic properties. Few other systems offer such a wealth of information over such a range of length scales simultaneously. The possibility of studying arbitrarily large systems for arbitrarily long periods offers benefits even over computer simulations. The close resemblance between atomic and colloidal phase transitions suggests that quite general principles might be derived from colloidal studies. Understanding limitations of this analogy is an integral part of its usefulness and is one of the goals of the present Article.

The following Sections describe direct imaging studies of the lattice structure and dynamics at the surfaces of colloidal crystals. Analyzing the time-resolved trajectories of individual spheres allows us to relate the microscopic interactions between pairs of spheres to the macroscopic parameters describing their collective behavior. The analytical tools developed in this process should be useful for a range of future studies. The results of the present study shed new light on the ongoing debate regarding the nature of colloidal electrostatic interactions. In particular, we find that the conventional linearized DLVO theory for colloidal electrostatic interactions fails to account for the elastic properties of strongly interacting colloidal crystals.

  

Figure 1: Schematic diagram of the sample cell's cross-section. The microscope's objective lens images a 200 nm thick section of the observation volume through the 150 $\mu$m thick coverslip.

After an overview in Section 2 of the techniques used to create and characterize charge-stabilized colloidal crystals, we describe direct imaging measurements of their structure and dynamics. Section 3 describes particle tracking measurements of the many-body potential of mean force. Most studies of colloidal phase transitions have assumed that the potential of mean force can be constructed by linear superposition of effective pair interactions whose parameters are no worse than simple functions of the sphere concentration. We test this assumption by applying it in Section 4 to interpret the results of Section 3 within the framework of the DLVO theory. The resulting effective pair potential is then used in Section 5 to calculate the crystals' elastic moduli. Comparison with elastic constants measured in Section 6 strongly suggest that linearized theories for colloidal electrostatic interactions fail to describe the elastic properties of charge-stabilized colloidal crystals.