The nonlocal elastic response function is crucial for understanding many properties of soft solids. This may be obtained by measuring strain-strain autocorrelation functions. We use computer simulations as well as video microscopy data of superparamagnetic colloids to obtain these correlations for two-dimensional triangular solids. Elastic constants and elastic correlation lengths are extracted by analyzing the correlation functions. We show that to explain our observations displacement fluctuations in a soft solid need to contain affine (strain) as well as nonaffine components.Elasticity theory [1,2] is based on the assumption that the displacement field u(r) is smooth and continuous for all po sitions r within a volume V. It is expected that this assump tion breaks down [3] below a limiting length g el . How small is g el in practice, and what are the consequences of the break down? What implications does this have on the mechanical response of the material? While for most conventional engi neering problems such issues are practically irrelevant, they do become important when investigating microelastic and microrheological properties of soft solids [6]. We answer these questions in the context of the nonlocal mechanical response of a soft solid in two dimensions (2D). Nonlocal elastic effects are important whenever strain gradients are large, e.g., near crack tips, grain boundaries, and interfaces [4]. They should also be important if the linear dimension of the solid is not much larger than g el . Both of these criteria are satisfied in colloidal solids under usual experimental condi tions [5].Recently, there has been a lot of interest in obtaining the elastic moduli of soft solids [6,7] from fluctuations of par ticle coordinates [8][9][10][11][12]. These methods have several advan tages. First, no specialized equipment is necessary other than standard video microscopy [7]. Second, no external forces need to be applied which may change the very properties that are being measured. This is especially convenient when try ing to obtain elastic constants near the dislocation unbinding transition [1] where the solid is sensitive to external pertur bations [11]. Lastly, these techniques are general enough so that they may be applied to a wide variety of systems [13]. Here we extend this procedure a step further to study nonlo cal [4] elasticity for solids in two dimensions far from the melting transition.The nonlocal elastic response function, or compliance X ij (r , r') (i = x , y), is defined as the strain e ij (r') produced at position r' due to a stress C ij (r) at r. This is given by X ij = (k B T) −1 G ij , where, for a homogeneous solid, G ij (r') = (e i (0)e ( j r')> is the strain-strain correlation function. The (¯> denote a thermal average (and one over the choice of origin) and k B T is the Boltzmann constant times the tempera ture [1]. The linear combinations of components of the strain tensor e y , x ( u derivatives of the displacement field =1,2,3 i ( i e relevant to a 2D solid are ) defined as ) calcu...

The phase behavior of a 50% binary hard-disk mixture with diameter ratio sigmaB/sigmaA=0.414 , which is exposed to a one-dimensional periodic potential, is examined via Monte Carlo simulations. We find an induced structural crossover in the modulated liquid. At higher densities, depending on the strength of the external potential, the system exhibits a tunable demixing transition, followed by fluid-solid coexistence of an equimolar mixture with the S1(AB) square lattice. We find a decoupled melting of the sublattices of the S1(AB) lattice. The melting of the small-component sublattice perpendicular to the external potential minima leads to fissuring in the large-component sublattice.

Ordering phenomena on surfaces or in monolayers can be successfully studied by model systems as binary hard-disk mixtures, the influence of a substrate being modeled by an external potential. For the field-free case the thermodynamic stability of space-filling lattice structures for binary hard-disk mixtures is studied by Monte Carlo computer simulations. As these structures prove to be thermodynamically stable only in high pressure environments, the phase behavior of an equimolar binary mixture with a diameter ratio of sigma_{B}/sigma_{A}=0.414 exposed to an external, one-dimensional, periodic potential is analyzed in detail. The underlying ordering mechanisms and the resulting order differ considerably, depending on which components of the mixture interact with the external potential. The simulations show that slight deviations in the concentration of large particles x_{A} or the diameter ratio sigma_{B}/sigma_{A} have no impact on the occurrence of the various field-induced phenomena as long as the mixture stays in the relevant regime of the packing fraction eta . Furthermore the importance of the commensurability of the external potential to the S1(AB) square lattice for the occurrence of the induced ordering is discussed.

In soft matter systems the local displacement field can be accessed directly by video microscopy enabling one to compute local strain fields and hence the elastic moduli in these systems using a coarse-graining procedure. Here, we study this process in detail for a simple triangular lattice of particles connected by harmonic springs in two-dimensions. Coarse-graining local strains obtained from particle configurations in a Monte Carlo simulation generates non-trivial, non-local strain correlations (susceptibilities), which may be understood within a generalized, Landau type elastic Hamiltonian containing up to quartic terms in strain gradients (K. Franzrahe et. al., Phys. Rev. E 78, 026106 (2008)). In order to demonstrate the versatility of the analysis of these correlations and to make our calculations directly relevant for experiments on colloidal solids, we systematically study in detail various parameters such as the choice of statistical ensemble, presence of external pressure and boundary conditions. Crucially, we show that special care needs to be taken for an accurate application of our results to actual experiments, where the analyzed area is embedded within a larger system, to which it is mechanically coupled. Apart from the smooth, affine strain fields, the coarse-graining procedure also gives rise to a noise field (χ) made up of non-affine displacements. Several properties of χ may be rationalized for the harmonic solid using a simple "cell model" calculation. Furthermore the scaling behavior of the probability distribution of the noise field (χ) is studied. We find that for any inverse temperature β, spring constant f , density ρ and coarsegraining length Λ the probability distribution can be obtained from a master curve of the scaling variable X = χβf /ρΛ 2 .

Ordering phenomena in colloidal dispersions exposed to external one-dimensional, periodic fields or under confinement are studied systematically by Monte Carlo computer simulations. Such systems are useful models for the study of monolayers on a substrate. We find that the interaction with a substrate potential completely changes the miscibility of a binary, hard disc mixture at low external field amplitudes. The underlying ordering mechanisms leading to this laser-induced de-mixing differ, depending on which components interact with the substrate potential. Generic effects of confinement on crystalline order in two dimensions are studied in a model system of point particles interacting via a potential ∝ r −12. The state of the system (a strip of width D) depends very sensitively on the precise boundary conditions at the two confining walls. Commensurate, corrugated boundary conditions enhance both orientational order and positional order. In contrast, smooth repulsive boundaries enhance only the orientational order and destroy positional (quasi-)long range order. As external fields have a strong impact on the elastic behaviour of colloidal crystals there is a need to analyse the elastic response in such systems for the field-free case first. To this aim we study the strain-strain correlation functions in a two-dimensional crystal formed by super-paramagnetic colloids, as monitored by standard video microscopy.

The nature of the melting transition for a system of hard discs with translational degrees of freedom in two spatial dimensions has been analysed by a combination of computer simulation methods and a finite size scaling technique. The behaviour of the system is consistent with the predictions of the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory.The structural and elastic properties of binary colloidal mixtures in two and three spatial dimensions are discussed as well as those of colloidal systems with quenched point impurities.Hard and soft discs in external periodic (light-) fields show rich phase diagrams including freezing and melting transitions when the density of the system is varied. Monte Carlo simulations for detailed finite size scaling analysis of various thermodynamic quantities like the order parameter, its cumulants, etc, have been used in order to map the phase diagram of the system for various values of the density and the amplitude of the external potential. For hard discs we find clear indication of a reentrant liquid phase over a significant region of the parameter space. The simulations therefore show that the system of hard discs behaves in a fashion similar to charge stabilized colloids which are known to undergo an initial freezing, followed by a remelting transition as the amplitude of the imposed modulating field produced by crossed laser beams is steadily increased. Detailed analysis of the simulation data shows several features consistent with a recent dislocation unbinding theory of laser induced melting. The differences and similarities of systems with soft potentials (DLVO, 1/r 12 , 1/r 6 ) and the relation to experimental data is analysed.

In this work, we focus on low-dimensional colloidal model systems, via simulation studies and also some complementary experiments, in order to elucidate the interplay between phase behavior, geometric structures and transport properties. In particular, we try to investigate the (nonlinear!) response of these very soft colloidal systems to various perturbations: uniform and uniaxial pressure, laser fields, shear due to moving boundaries and randomly quenched disorder. We study ordering phenomena on surfaces or in monolayers by Monte Carlo computer simulations of binary hard-disk mixtures, the influence of a substrate being modeled by an external potential. Weak external fields allow a controlled tuning of the miscibility of the mixture. We discuss the laser induced de-mixing for the three different possible couplings to the external potential. The structural behavior of hard spheres interacting with repulsive screened Coulomb or dipolar interaction in 2D and 3D narrow constrictions is investigated using Brownian dynamics simulations. Due to misfits between multiples of the lattice parameter and the channel widths, a variety of ordered and disordered lattice structures have been observed. The resulting local lattice structures and defect probabilities are studied for various cross sections. The influence of a self-organized order within the system is reflected in the velocity of the particles and their diffusive behavior. Additionally, in an experimental system of dipolar colloidal particles confined by gravity on a solid substrate we investigate the effect of pinning on the dynamics of a two-dimensional colloidal liquid. This work contains sections reviewing previous work by the authors as well as new, unpublished results. Among the latter are detailed studies of the phase boundaries of the de-mixing regime in binary systems in external light fields, configurations for shear induced effects at structured walls, studies on the effect of confinement on the structures and defect densities in three-dimensional systems, the effect of confinement and barriers on two-dimensional flow and diffusion, and the effect of pinning sites on the diffusion.

A brief review focusing on low-dimensional colloidal model systems is given describing both simulation studies and complementary experiments, elucidating the interplay between phase behavior, geometric structures, and transport phenomena. These studies address the response of these very soft colloidal systems to perturbations such as uniform or uniaxial compression, laser fields, randomly quenched disorder, and shear deformation caused by moving boundaries.Binary hard-disk mixtures are studied by Monte Carlo simulation, to investigate ordering on surfaces or in monolayers, modeling the effect of a substrate by an external potential. By weak external laser fields the miscibility of the mixture can be controlled, and the underlying mechanism (laser-induced demixing) is clarified. The stability of various space-filling structures is discussed only for the case where no laser fields are present.Hard spheres interacting with repulsive screened Coulomb or dipolar interaction confined in 2D and 3D narrow constrictions are investigated by Brownian Dynamics simulation. With respect to the structural behavior, it is found that layers or planes throughout the microchannel are formed. The arrangement of the particles is disturbed by diffusion, and can also be modified by an external driving force causing a density gradient along the channel. Then the number of layers or planes gets reduced, adjusting to the density gradient, and this self-organized change of order also shows up in the particle velocities.The experimental work that is reviewed here addresses dipolar colloidal particles confined by gravity on a solid substrate on which a set of pinning sites has been randomly distributed. The dynamics of the system is studied by tracking the trajectories of individual particles, and it is found that the mean square displacements of particles that are nearest neighbors of pinned particles are strongly affected by these