Igor M. Sokolov

From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion

Einsteins explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.

Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations.

We propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, cannot be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish their relation to the continuous-time random walk theory. We show that the distributed-order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the Wiener process and whose diffusion exponent decreases in time (retarding subdiffusion). This process may lead to superslow diffusion, with the mean square displacement growing logarithmically in time. We also demonstrate that the distributed-order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time (accelerating superdiffusion).

Quantitative Mapping of the Elastic Modulus of Soft Materials with HarmoniX and PeakForce QNM AFM Modes

The modulus of elasticity of soft materials on the nanoscale is of interest when studying thin films, nanocomposites, and biomaterials. Two novel modes of atomic force microscopy (AFM) have been introduced recently: HarmoniX and PeakForce QNM. Both modes produce distribution maps of the elastic modulus over the sample surface. Here we investigate the question of how quantitative these maps are when studying soft materials. Three different polymers with a macroscopic Youngs modulus of 0.6-0.7 GPa (polyurethanes) and 2.7 GPa (polystyrene) are analyzed using these new modes. The moduli obtained are compared to the data measured with the other commonly used techniques, dynamic mechanical analyzer (DMA), regular AFM, and nanoindenter. We show that the elastic modulus is overestimated in both the HarmoniX and PeakForce QNM modes when using regular sharp probes because of excessively overstressed material in the samples. We further demonstrate that both AFM modes can work in the linear stress-strain regime when using a relatively dull indentation probe (starting from ~210 nm). The analysis of the elasticity models to be used shows that the JKR model should be used for the samples considered here instead of the DMT model, which is currently implemented in HarmoniX and PeakForce QNM modes. Using the JKR model and ~240 nm AFM probe in the PeakForce QNM mode, we demonstrate that a quantitative mapping of the elastic modulus of polymeric materials is possible. A spatial resolution of ~50 nm and a minimum 2 to 3 nm indentation depth are achieved.

Enzyme-functionalized mesoporous silica for bioanalytical applications.

AbstractThe unique properties of mesoporous silica materials (MPs) have attracted substantial interest for use as enzyme-immobilization matrices. These features include high surface area, chemical, thermal, and mechanical stability, highly uniform pore distribution and tunable pore size, high adsorption capacity, and an ordered porous network for free diffusion of substrates and reaction products. Research demonstrated that enzymes encapsulated or entrapped in MPs retain their biocatalytic activity and are more stable than enzymes in solution. This review discusses recent advances in the study and use of mesoporous silica for enzyme immobilization and application in biosensor technology. Different types of MPs, their morphological and structural characteristics, and strategies used for their functionalization with enzymes are discussed. Finally, prospective and potential benefits of these materials for bioanalytical applications and biosensor technology are also presented. FigureEnzyme-functionalized mesoporous silica fibers and their integration in a biosensor design. The immobilization process takes place essentially in the silica micropores.

Synthesis of mesoporous silica spheres under quiescent aqueous acidic conditions

A gyroid-to-sphere shape transition has been unveiled in the growth of mesoporous silica morphologies that are synthesized under quiescent acidic aqueous conditions. It can be induced by a decrease of the acidity for a surfactant-based gyroid preparation. As the acidity is gradually lowered from the gyroid domain, the growth process changes from one involving a smooth continuous deposition of silicate–surfactant micellar solute species onto specific regions of an evolving silicate liquid crystal seed, to one in which deposition instead occurs on non-specific regions of the seed. This creates multigranular gyroid morphologies which at lower acidity emerge as sphere shapes. The gyroid-to-sphere metamorphosis appears to correlate with an acidity and/or temperature dependent switch in the mode of formation, from the gyroid involving fast and local polymerization of a growing silicate liquid crystal seed, to the sphere based upon a slower and global polymerization of a silicate liquid crystal droplet. Surface tension will cause such a droplet to adopt a spherical shape, ultimately to be rigidified in the form of a mesoporous silica sphere. Comparative gyroid and sphere information is presented on synthesis-size-shape-channel plan relations, degree of orientational order of the channels, extent of polymerization of the silica, thermal stability and nitrogen adsorption properties. The ability to synthesize 1–10 µm diameter mesoporous silica spheres with a narrow sphere size and pore size distribution portends a myriad of applications in large molecule catalysis, chromatographic separations and nanocomposites.

Human epithelial cells increase their rigidity with ageing in vitro: direct measurements

The decrease in elasticity of epithelial tissues with ageing contributes to many human diseases. This change was previously attributed to increased crosslinking of extracellular matrix proteins. Here we show that individual human epithelial cells also become significantly more rigid during ageing in vitro. Using atomic force microscopy (AFM), we found that the Youngs modulus of viable cells was consistently increased two- to four-fold in older versus younger cells. Direct visualization of the cytoskeleton using a novel method involving the AFM suggested that increased rigidity of ageing cells was due to a higher density of cytoskeletal fibres. Our results identify a unique mechanism that might contribute to the age-related loss of elasticity in epithelial tissues.

Percolation on heterogeneous networks as a model for epidemics

We consider a spatial model related to bond percolation for the spread of a disease that includes variation in the susceptibility to infection. We work on a lattice with random bond strengths and show that with strong heterogeneity, i.e. a wide range of variation of susceptibility, patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the heterogeneity. These results are qualitatively different from those of standard models in epidemiology, but correspond to real effects. We suggest that heterogeneity in the epidemic will affect the phylogenetic distance distribution of the disease-causing organisms. We also investigate small world lattices, and show that the effects mentioned above are even stronger.

Statistical thermodynamics and stochastic theory of nonequilibrium systems

About the history of nonlinear science levels of description and basic concepts Gibbsian distributions for equilibrium and stationary states relaxation of fluctuations and irreversible processes nonlinear dynamics and stochastics of order parameters excitations in molecular systems with nonlinear interactions nucleation phenomena in gases reaction-diffusion systems structures in excitable systems entrophy and information.

Models of anomalous diffusion in crowded environments

A particles motion in crowded environments often exhibits anomalous diffusion, whose nature depends on the situation at hand and is formalized within different physical models. Thus, such environments may contain traps, labyrinthine paths or macromolecular structures, which the particles may be attached to. Physical assumptions are translated into mathematical models which often come with nice mathematical instruments for their description, e.g. fractional diffusion equations. The beauty of the instrument sometimes seduces an investigator to use it without any connection to the physical model. The author hopes that the present discussion will reduce the danger of such inappropriate use.

Reshuffling scale-free networks: From random to assortative

Many social networks exhibit assortative mixing so that the predictions of uncorrelated models might be inadequate. To analyze the role of assortativity we introduce an algorithm which changes correlations in a network and produces assortative mixing to a desired degree. This degree is governed by one parameter p. Changing this parameter one can construct networks ranging from fully random (p = 0) to totally assortative (p = 1). We apply the algorithm to a Barabasi-Albert scale-free network and show that the degree of assortativity is an important parameter governing geometrical and transport properties of networks. Thus, the diameter of the network and the clustering coefficient increase dramatically with the degree of assortativity. Moreover, the concentration dependences of the size of the giant component in the node percolation problem for uncorrelated and assortative networks are strongly different.