We introduce fractal liquids by generalizing classical liquids of integer dimensions $d = 1, 2, 3$ to a fractal dimension $d_f$. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same non-integer dimension. Realizations of our generic model system include microphase separated binary liquids in porous media, and highly branched liquid droplets confined to a fractal polymer backbone in a gel. Here we study the thermodynamics and pair correlations of fractal liquids by computer simulation and semi-analytical statistical mechanics. Our results are based on a model where fractal hard spheres move on a near-critical percolating lattice cluster. The predictions of the fractal Percus-Yevick liquid integral equation compare well with our simulation results.
The glass transition of mesoscopic charged particles in two-dimensional confinement is studied by mode-coupling theory. We consider two types of effective interactions between the particles, corresponding to two different models for the distribution of surrounding ions that are integrated out in coarse-grained descriptions. In the first model, a planar monolayer of charged particles is immersed in an unbounded isotropic bath of ions, giving rise to an isotropically screened Debye-Huckel- (Yukawa-) type effective interaction. The second, experimentally more relevant system is a monolayer of negatively charged particles that levitate atop a flat horizontal electrode, as frequently encountered in laboratory experiments with complex (dusty) plasmas. A steady plasma current towards the electrode gives rise to an anisotropic effective interaction potential between the particles, with an algebraically long-ranged in-plane decay. In a comprehensive parameter scan that covers the typical range of experimentally accessible plasma conditions, we calculate and compare the mode-coupling predictions for the glass transition in both kinds of systems.
To the present day, the Beenakker-Mazur (BM) method is the most comprehensive statistical physics approach to the calculation of short-time transport properties of colloidal suspensions. A revised version of the BM method with an improved treatment of hydrodynamic interactions is presented and evaluated regarding the rotational short-time self-diffusion coefficient, $D^r$ , of suspensions of charged particles interacting by a hard-sphere plus screened Coulomb (Yukawa) pair potential. To assess the accuracy of the method, elaborate simulations of $D^r$ have been performed, covering a broad range of interaction parameters and particle concentrations. The revised BM method is compared in addition with results by a simplifying pairwise additivity (PA) method in which the hydrodynamic interactions are treated on a two-body level. The static pair correlation functions re- quired as input to both theoretical methods are calculated using the Rogers-Young integral equation scheme. While the revised BM method reproduces the general trends of the simulation results, it systematically and significantly underestimates the rotational diffusion coefficient. The PA method agrees well with the simulation data at lower volume fractions, but at higher concentrations $D^r$ is likewise underestimated. For a fixed value of the pair potential at mean particle distance comparable to the thermal energy, $D^r$ increases strongly with increasing Yukawa potential screening parameter.
Diffusion in bidisperse Brownian hard-sphere suspensions is studied by Stokesian Dynamics (SD) computer simulations and a semi-analytical theoretical scheme for colloidal short-time dynamics, based on Beenakker and Mazurs method [Physica 120A, 388 (1983) & 126A, 349 (1984)]. Two species of hard spheres are suspended in an overdamped viscous solvent that mediates the salient hydrodynamic interactions among all particles. In a comprehensive parameter scan that covers various packing fractions and suspension compositions, we employ numerically accurate SD simulations to compute the initial diffusive relaxation of density modulations at the Brownian time scale, quantified by the partial hydrodynamic functions. A revised version of Beenakker and Mazurs $deltagamma$-scheme for monodisperse suspensions is found to exhibit surprisingly good accuracy, when simple rescaling laws are invoked in its application to mixtures. The so-modified $deltagamma$ scheme predicts hydrodynamic functions in very good agreement with our SD simulation results, for all densities from the very dilute limit up to packing fractions as high as $40%$.
The ionic composition and pair correlations in fluid phases of realistically salt-free charged colloidal sphere suspensions are calculated in the primitive model. We obtain the number densities of all ionic species in suspension, including low-molecular weight microions, and colloidal macroions with acidic surface groups, from a self-consistent solution of a coupled physicochemical set of nonlinear algebraic equations and non-mean-field liquid integral equations. Here, we study suspensions of colloidal spheres with sulfonate or silanol surface groups, suspended in demineralized water that is saturated with carbon dioxide under standard atmosphere. The only input required for our theoretical scheme are the acidic dissociation constants pKa, and effective sphere diameters of all involved ions. Our method allows for an ab initio calculation of colloidal bare and effective charges, at high numerical efficiency.
The pair-correlation functions for fluid ionic mixtures in arbitrary spatial dimensions are computed in hypernetted chain (HNC) approximation. In the primitive model, all ions are approximated as non-overlapping hyperspheres with Coulomb interactions. Our spectral HNC solver is based on a Fourier-Bessel transform introduced by Talman [J. Comput. Phys., 29, 35 (1978)], with logarithmically spaced computational grids. Numeric efficiency for arbitrary spatial dimensions is a commonly exploited virtue of this transform method. Here, we highlight another advantage of logarithmic grids, consisting in efficient sampling of pair-correlation functions for highly asymmetric ionic mixtures. For three-dimensional fluids, ion size- and charge-ratios larger than one thousand can be treated, corresponding to hitherto computationally not accessed micrometer-sized colloidal spheres in 1-1 electrolyte. Effective colloidal charge numbers are extracted from our primitive model results. For moderately large ion size- and charge-asymmetries, we present Molecular Dynamics simulation results that agree well with the approximate HNC pair correlations.
We report on depolarized and non-depolarized dynamic light scattering, static light scattering, and static viscosity measurements on interacting charged gibbsite platelets suspended in dimethyl sulfoxide (DMSO). The average collective and (long-time) translational self-diffusion coefficients, and the rotational diffusion coefficient, have been measured as functions of the platelet volume fraction phi, up to the isotropic-liquid crystal (I/LC) transition. The non-depolarized intensity autocorrelation function, measured at low scattering wavenumbers, consists of a fast and a slowly decaying mode which we interpret as the orientationally averaged collective and translational self-diffusion coefficients, respectively. Both the rotational and the long-time self-diffusion coefficients decrease very strongly, by more than two orders of magnitude, in going from the very dilute limit to the I/LC transition concentration. A similarly strong decrease, with increasing phi, is observed for the inverse zero-strain limiting static shear viscosity. With increasing phi, increasingly strong shear-thinning is observed, accompanied by a shrinking of the low shear-rate Newtonian plateau. The measured diffusion coefficients are interpreted theoretically in terms of a simple model of effective charged spheres interacting by a screened Coulomb potential, with hydrodynamic interactions included. The disk-like particle shape, and the measured particle radius and thickness polydispersities, enter into the model calculations via the scattering amplitudes. The interaction-induced enhancement of the collective diffusion coefficient by more than a factor of 20 at larger volume fractions is well captured in the effective sphere model, whereas the strong declines both of the experimental translational and rotational self-diffusion coefficients are underestimated.
