https://www.tau.ac.il/%7Eklafter1/258.pdf

The random walk's guide to anomalous diffusion: a fractional dynamics approach

Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

Wetting phenomena are ubiquitous in nature and technology. A solid substrate exposed to the environment is almost invariably covered by a layer of fluid material. In this review, the surface forces that lead to wetting are considered, and the equilibrium surface coverage of a substrate in contact with a drop of liquid. Depending on the nature of the surface forces involved, different scenarios for wetting phase transitions are possible; recent progress allows us to relate the critical exponents directly to the nature of the surface forces which lead to the different wetting scenarios. Thermal fluctuation effects, which can be greatly enhanced for wetting of geometrically or chemically structured substrates, and are much stronger in colloidal suspensions, modify the adsorption singularities. Macroscopic descriptions and microscopic theories have been developed to understand and predict wetting behavior relevant to microfluidics and nanofluidics applications. Then the dynamics of wetting is examined. A drop, placed on a substrate which it wets, spreads out to form a film. Conversely, a nonwetted substrate previously covered by a film dewets upon an appropriate change of system parameters. The hydrodynamics of both wetting and dewetting is influenced by the presence of the three-phase contact line separating "wet" regions from those that are either dry or covered by a microscopic film only. Recent theoretical, experimental, and numerical progress in the description of moving contact line dynamics are reviewed, and its relation to the thermodynamics of wetting is explored. In addition, recent progress on rough surfaces is surveyed. The anchoring of contact lines and contact angle hysteresis are explored resulting from surface inhomogeneities. Further, new ways to mold wetting characteristics according to technological constraints are discussed, for example, the use of patterned surfaces, surfactants, or complex fluids.

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Wetting and Spreading

We review the current state of knowledge of phase separation and phase equilibria in porous materials. Our emphasis is on fundamental studies of simple fluids (composed of small, neutral molecules) and well-characterized materials. While theoretical and molecular simulation studies are stressed, we also survey experimental investigations that are fundamental in nature. Following a brief survey of the most useful theoretical and simulation methods, we describe the nature of gas‐liquid (capillary condensation), layering, liquid‐liquid and freezing/melting transitions. In each case studies for simple pore geometries, and also more complex ones where available, are discussed. While a reasonably good understanding is available for phase equilibria of pure adsorbates in simple pore geometries, there is a need to extend the models to more complex pore geometries that include effects of chemical and geometrical heterogeneity and connectivity. In addition, with the exception of liquid‐liquid equilibria, little work has been done so far on phase separation for mixtures in porous media.

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Phase separation in confined systems

We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical modelling of viscoelastic bodies (Sect. 1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (Sect. 2). In the former analysis fractional calculus leads us to introduce intermediate models of viscoelasticity which generalize the classical spring-dashpot models. The latter analysis induces us to introduce a hydrodynamic model suitable to revisit in Sect. 3 the classical theory of the Brownian motion, which is a relevant topic in statistical mechanics. By the tools of fractional calculus we explain the long tails in the velocity correlation and in the displacement variance. In Sect. 4 we consider the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order $0< \beta <2$. Led by our analysis we express the fundamental solutions (the Green functions) in terms of two interrelated auxiliary functions in the similarity variable, which turn out to be of Wright type (see Appendix), and to distinguish slow-diffusion processes ($0 < \beta < 1$) from intermediate processes ($1 < \beta < 2$).

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Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics

The global phase diagram for wall and surface critical phenomena is analyzed in the space of temperature, surface enhancement, and bulk and surface fields, on the basis of Landau phenomenological theory. Features probably valid for various real systems are the physical unity of prewetting and pure surface criticality, and novel exponents for critical wetting and for a wetting tricritical point, which terminates the region of normal first-order wetting.

Multicriticality of Wetting, Prewetting, and Surface Transitions

Critical behavior in thin films is discussed with attention to the example of phase separation in binary fluid mixtures between parallel plates. The analyses focus on the dependence of the shift in critical temperature ΔTc, critical field, etc., on the film thickness D, and on the nature of the walls as modeled by a surface field or chemical potential h1 which acts near the walls and leads to preferential adsorption of one of the bulk phases. Mean field theory for an Ising/lattice‐gas model is utilized and the resulting asymptotic scaling functions for the shifts ΔTc etc. are computed within Landau theory by analytic and numerical methods. Series analyses for simple cubic lattice Ising model films with h1=0 are used to estimate universal features of three‐dimensional systems: specifically, if ξ(ΔT) is the bulk correlation length, determined, say, via scattering experiments, at ΔT=T−T∞c≳0 then the shift ratio D/ξ(‖ΔTc‖) is about 2.89 for h1=0 but 4.61 for h1→∞, compared with mean field values π and 5.0699....

Critical point shifts in films

On the basis of a scaling hypothesis for the criticality of fluids, or fluid mixtures, held between parallel plates, we analyze the combined effects of finite wall separation and ’’symmetry breaking’’ boundary conditions at the walls (i.e., adsorption, or preferential adsorption of one component, at the surfaces). The resulting bulk (but ’’two‐dimensional’’) critical point is shifted both in temperature and in density or composition (and in the conjugate field variables) by amounts described by scaling. The scaled form of the phase boundary is consistent with results from a low temperature Ising lattice gas model expansion.

Scaling theory for the criticality of fluids between plates

A novel concept of ‘‘surface’’ spinodal decomposition is introduced in the context of wetting and prewetting first order transitions for simple fluids in contact with a bounding wall. Qualitative and semiquantitative discussions are made within the framework of a phenomenological Landau theoretical model for semi‐infinite systems with a (short‐ranged) wall potential. A concept of ‘‘extended wetting’’ is also proposed, whereby even for T<TW (the wetting temperature) a macroscopically thick ‘‘wetting’’ layer can form near the wall upon rapidly quenching the solution into the coexistence region. These ideas are carried over to polymer solutions and blends near their consolute points where the characteristic time scales should be more manageable.

Hisao Nakanishi

Papers

Multicriticality of Wetting, Prewetting, and Surface Transitions

Critical point shifts in films

Scaling theory for the criticality of fluids between plates

Surface spinodals and extended wetting in fluids and polymer solutions

Scaling studies of percolation phenomena in systems of dimensionality two to seven: Cluster numbers