Phase transitions of systems confined in long cylindrical pores (capillary condensation, wetting, crystallization, etc.) are intrinsically not sharply defined but rounded. The finite size of the cross section causes destruction of long range order along the pore axis by spontaneous nucleation of domain walls. This rounding is analyzed for two models (Ising/lattice gas and Asakura-Oosawa model for colloid-polymer mixtures) by Monte Carlo simulations and interpreted by a phenomenological theory. We show that characteristic differences between the behavior of pores of finite length and infinitely long pores occur. In pores of finite length a rounded transition occurs first, from phase coexistence between two states towards a multi-domain configuration. A second transition to the axially homogeneous phase follows near pore criticality. PACS numbers: 64.75Jk, 64.60.an, 05.70Fh, 02.70Tt Fluids and fluid mixtures in nano-and microporous materials (pore diameters from 1 nm to 150 nm) play important roles in various industries (extracting oil and gas from porous rocks; use as catalysts or for mixture separation in the chemical and pharmaceutical industry; nanofluidic devices, etc.) [1][2][3]. The interplay of finite size and surface effects strongly modifies the phase behavior of such confined fluids [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] in comparison with the bulk. The vapor to liquid transition is shifted ("capillary condensation"), as well as critical points [3,4,9,12]. Effects of wetting [20] on phase coexistence give rise to interesting patterns (plugs versus capsules versus tube structures etc. [7]). However, although various phase diagrams (different from the bulk) have been proposed (e.g. [1,3,7,9,12,13,17]), many aspects hitherto are not well understood. E.g., the "critical point" where adsorption/desorption hysteresis vanishes seems to be systematically lower than the critical temperature where the density difference between the vapor-like and liquid-like states vanishes [12], in contrast to what theories have predicted [14].However, a crucial aspect (stressed only in a few pioneering studies [3,8], and in the context of Ising/lattice gas models [21][22][23]) is the rounding of all transitions, caused by the quasi-one-dimensional character of a fluid in a long cylindrical pore with cross-sectional radius R. With the current progress of producing pores of wellcontrolled diameter varying from the nanoscale (carbon nanotubes [23][24][25]) to arrays of pores in silicon wafers [26], up to 150 nm wide and of well-controlled length, experiments become feasible which are not plagued by effects of random disorder, which occur in porous glasses [1,27]. Thus, it is important to understand the phase transitions in pores more precisely, considering both the radius R and the length L of the pore as variables (the important role of L has so far been largely disregarded). In the present Letter, we elucidate the rounding of vaporliquid type transitions in cylindrical pores, based on Monte Carlo sim...

When a fluid that undergoes a vapor to liquid transition in the bulk is confined to a long cylindrical pore, the phase transition is shifted (mostly due to surface effects at the walls of the pore) and rounded (due to finite size effects). The nature of the phase coexistence at the transition depends on the length of the pore: For very long pores the system is axially homogeneous at low temperatures. At the chemical potential where the transition takes place fluctuations occur between vapor-like and liquid-like states of the cylinder as a whole. At somewhat higher temperatures (but still far below bulk criticality) the system at phase coexistence is in an axially inhomogeneous multi-domain state, where long cylindrical liquid-like and vapor-like domains alternate. Using Monte Carlo simulations for the Ising/lattice gas model and the Asakura-Oosawa model of colloid-polymer mixtures the transition between these two different scenarios is characterized. It is shown that the density distribution changes gradually from a double-peak structure to a triple-peak shape, and the correlation length in axial direction (measuring the equilibrium domain length) becomes much smaller than the cylinder length. The (rounded) transition to the disordered phase of the fluid occurs when the axial correlation length has decreased to a value comparable to the cylinder diameter. It is also suggested that adsorption hysteresis vanishes when the transition from the simple domain state to the multi-domain state of the cylindrical pore occurs. We predict that the difference between the pore critical temperature and the hysteresis critical temperature should increase logarithmically with the length of the pore.

Colloidal systems are often modelled as fluids of hard particles (possibly with an additional soft attraction, e.g. caused by polymers also contained in the suspension). In simulations of such systems, the virial theorem cannot be straightforwardly applied to obtain the components of the pressure tensor. In systems confined by walls, it is hence also not straightforward to extract the excess energy due to the wall (the "wall tension") from the pressure tensor anisotropy. A comparative evaluation of several methods to circumvent this problem is presented, using as examples fluids of hard spheres and the Asakura-Oosawa model of colloid-polymer mixtures with a size ratio q = 0.15 (for which the effect of the polymers can be integrated out to yield an effective attractive potential between the colloids). Factors limiting the accuracy of the various methods are carefully discussed, and controlling these factors very good mutual agreement between the various methods is found.

A model for two-dimensional colloids confined laterally by "structured boundaries" (i.e., ones that impose a periodicity along the slit) is studied by Monte Carlo simulations. When the distance D between the confining walls is reduced at constant particle number from an initial value D(0), for which a crystalline structure commensurate with the imposed periodicity fits, to smaller values, a succession of phase transitions to imperfectly ordered structures occur. These structures have a reduced number of rows parallel to the boundaries (from n to n-1 to n-2, etc.) and are accompanied by an almost periodic strain pattern, due to "soliton staircases" along the boundaries. Since standard simulation studies of such transitions are hampered by huge hysteresis effects, we apply the phase switch Monte Carlo method to estimate the free energy difference between the structures as a function of the misfit between D and D(0), thereby locating where the transitions occur in equilibrium. For comparison, we also obtain this free energy difference from a thermodynamic integration method: The results agree, but the effort required to obtain the same accuracy as provided by phase switch Monte Carlo would be at least three orders of magnitude larger. We also show for a situation where several "candidate structures" exist for a phase, that phase switch Monte Carlo can clearly distinguish the metastable structures from the stable one. Finally, applying the method in the conjugate statistical ensemble (where the normal pressure conjugate to D is taken as an independent control variable), we show that the standard equivalence between the conjugate ensembles of statistical mechanics is violated.

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.