Table of contents

The year 2005 marks the 100th anniversary of Albert Einstein's celebrated paper on Brownian motion, one of his most influential works, which provided further strong evidence for the atomistic hypothesis of matter and became a keystone of a fully probabilistic formulation of statistical mechanics.

Honouring this anniversary, we have edited this special issue on diffusion in liquids, polymers, biophysics and chemical dynamics. In this issue a total of 23 experimental and/or theoretical papers have been contributed from leading scientists, and new developments and perspectives in the field are discussed.

The contributed papers shed light on different facets of this ubiquitous phenomenon and cover a broad range of topics: surface- and bulk-mediated surface diffusion, diffusion-assisted selection of particles, diffusion-controlled spreading of monolayers and droplets, diffusion in polymer films, supercooled liquids, porous media and other confined geometries to name but a few.

We wish to thank all the contributing authors to this special issue.

We present the dynamic propensity distribution as an explicit measure of the degree to which the dynamics in a liquid over the time scale of structural relaxation is determined by the initial configuration. We then examine, for a binary mixture of soft discs in two dimensions, the correlation between the spatial distribution of propensity and that of two local measures of configuration structure: the local composition and local free volume. While the small particles dominate the high propensity population, we find no strong correlation between either the local composition or the local free volume and the propensity. It is argued that this is a generic failure of purely local structural measures to capture the inherently non-local character of collective behaviour.

We use a recently derived reformulation of the diffusion constant (Stillinger and Debenedetti 2005 J. Phys. Chem. B 109 6604) to investigate heterogeneous dynamics and non-Gaussian diffusion in a binary Lennard-Jones mixture. Our work focuses on the joint probability distribution of particles with velocity v₀ at time t = 0 and eventual displacement Δx at time t = Δt. We show that this distribution attains a distinctive shape at the time of maximum non-Gaussian behaviour in the supercooled liquid. By performing a two-Gaussian fit of the displacement data, we obtain, in a non-arbitrary manner, two diffusive length scales inherent to the supercooled liquid and use them to identify spatially separated regions of mobile and immobile particles.

We investigate theoretically and experimentally how the hydrodynamically correlated lateral motion of particles in a suspension confined between two surfaces is affected by the suspension concentration. Despite the long range of the correlations (decaying as 1/r² with the inter-particle distance r), the concentration effect is present only at short inter-particle distances for which the static pair correlation is nonuniform. This is in sharp contrast with the effect of hydrodynamic screening in unconfined suspensions, where increasing the concentration changes the prefactor of the large-distance correlation.

Porous materials, concentrated colloidal suspensions are examples of confining systems developing large specific surface and presenting a rich variety of shapes. Such an interfacial confinement strongly influences the molecular dynamics of embedded fluids and the diffusive motion of entrapped Brownian particles. An individual trajectory near the interface can be described as an alternate succession of adsorption steps and random flights in the bulk. Statistical properties of these random flights in various interfacial confining systems are needed as prerequisites in order to understand the full transport process. Related to first passage processes, these properties play a central role in numerous problems such as the mean first exit time in a bounded domain, heterogeneous catalytic reactivity and nuclear magnetic relaxation in complex and biological fluids. In the present work, we first consider the various possibilities of connecting two points of a smooth interface by a random flight in the bulk. Second, we analyse from the theoretical and experimental points of view a way to probe Brownian flight statistics. From the experimental point of view, we investigate the slow fluid dynamics near some colloidal interfaces by field-cycling NMR relaxometry. This is a way to follow slow dynamical correlations from 1 ns to 10 µs. This spectroscopy appears to be a good choice, considering that the algebraic nature of the probability of the first return to a surface builds a long-time memory. The experimental part confirms that the embedded fluid dynamics is sensitive to possible morphologic crossover and provides information about interface geometry. We also believe that such an approach can be used to probe interfacial dynamics by itself, for example in the case of a colloidal system undergoing a phase transition (dynamical arrest, rotational blockage,...).

We study the influence of the boundary conditions at the solid–liquid interface on diffusion in a confined fluid. Using a hydrodynamic approach, we compute numerical estimates for the diffusion of a particle confined between two planes. Partial slip is shown to significantly influence the diffusion coefficient near a wall. Analytical expressions are derived in the low and high confinement limits, and are in good agreement with numerical results. These calculations indicate that diffusion of tagged particles could be used as a sensitive probe of the solid–liquid boundary conditions.

Since the cytoskeletal protein actin is one of the principal building blocks of mammalian cells, it has recently been arousing much interest. Here, we address questions concerning the mechanical and dynamic behaviour of individual actin filaments in confining geometries which mimic the dense cytoskeletal network in eukaryotic cells. Microfluidic devices fabricated by soft photolithography in combination with fluorescence microscopy are used to manipulate, observe and characterize these biopolymers. The polymer statistics is strongly dependent on the characteristics of the surroundings such as the degree of confinement and hydrodynamical flow. Besides this, the intrinsic mechanical properties of the filaments are dominated by the persistence length and the contour length. We analyse the tangent–tangent correlation and the radial distribution function in terms of a confining potential and the contour length of the filaments. In addition, we show that hydrodynamic flow can be successfully used to apply controlled local stress on actin filaments. Our results can be surprisingly well described by a straightforward model which approximates the confining energy of the microchannels using a parabolic potential.

Ultrafast optical Kerr effect spectroscopy has been used to study the temperature-dependent orientational dynamics of a series of nitriles with n-alkyl chains ranging from one to 11 carbons in length. In all cases the orientational diffusion is found to be described by a single-exponential decay. Analysis of the orientational correlation times using the Debye–Stokes–Einstein equation suggests that the molecules adopt extended configurations and reorient as rigid rods. The liquids with shorter alkyl chains undergo an apparent ordering transition as they are cooled.

Solvent evaporation from homopolymer and heteropolymer films along with the interdiffusion of solvent into these films are studied by molecular dynamics simulations. Due to the high viscosity of polymer melts, in many cases polymer films are made by first dissolving the polymer in a low viscosity solvent, spreading the solution on a substrate and subsequently evaporating the solvent. Here we study the last part of this process, namely the evaporation of solvent from a polymer film. As the solvent evaporates, the polymer density at the film/vapour interface is found to increase sharply, creating a polymer density gradient which acts as a barrier for further solvent evaporation. For both homopolymer and heteropolymer films, the rate of solvent evaporation is found to decrease exponentially as a function of time. For multiblock co-polymer films the resulting domain structure is found to be strongly affected by the relative stiffness of the two blocks. The reverse process, namely the interdiffusion of solvent into a polymer film, is also studied. For homopolymer films the weight gain by the film scales as t^1/2, which is expected for Fickian diffusion. The diffusivity D(c) determined from the one-dimensional Fick's diffusion equation agrees well with that calculated from the corrected diffusion constant using the Darken equation. Far above the polymer glass transition temperature, D(c) is nearly independent of concentration. However, as the temperature decreases D(c) is found to depend strongly on the state of the polymer and is related to the shape of the solvent concentration profile. Finally, the swelling of a multiblock copolymer film in which the stiffer block is below its glass transition temperature is also studied. While the solvent swells only the softer block of the copolymer, the weight gain by the film remains Fickian.

An elementary process occurring on surfaces is diffusion. The dynamics is simplest when the concentration of adsorbates is sufficiently small that interaction between adsorbates can be ignored. But even for this tracer diffusion process, much remains to be uncovered. Here, we present the interplay between experimental measurement of tracer diffusion and its theoretical interpretation, which leads to good estimates of the interaction of the adparticle with the surface. We show how the results from three different experimental techniques—field ion microscopy, scanning tunnelling microscopy and quasielastic helium atom scattering—can be interpreted. Using the generalized Langevin equation as a model for the diffusion dynamics, we show how the turnover theory for activated diffusion may be used to describe the measured time evolution of the adparticle distribution on the surface. The different activation energy measured for hopping over single or double lattice lengths is shown to come from the added energy loss to the surface, as the particle moves over the longer path. We discuss some of the issues which are not yet clear; these include quantum effects, such as the quantum suppression of diffusion, vibrationally assisted diffusion, multidimensional effects and diffusion in the presence of external fields.

Particles moving on perfect periodic surfaces under the influence of external forces may move along directions that deviate from that of the force. We briefly recall previous results for transport of particles on surfaces with periodic traps or periodic obstacles driven by a constant external force, and present new results for particles moving in a harmonic periodic potential. The sorting properties are explored as a function of a number of control parameters, specifically the friction, force amplitude and direction, temperature, and lattice constants.

We give a method for the exact computation of the mobility of a particle diffusing in a lattice gas with defects. We describe the method by considering its application to the computation of the mobility of a particle diffusing in a lattice with defects in the form of dead-ends. The mobility initially increases linearly with field intensity, then crosses over to a nonlinear behaviour and reaches a maximum. Beyond this point the mobility decreases with increasing field intensity, showing an anomalous behaviour which may be attributed to partial confinement imposed by the field. The proposed method is general and may be applied to the computation of arbitrary moments of the probability function of a single particle diffusing in a lattice gas with defects of arbitrary characteristics.

We analyse the dynamics of adsorbed molecules within the bulk-mediated surface diffusion framework. We consider that the particle's desorption mechanism is characterized by a non-Markovian process, while the particle's adsorption and its motion in the bulk are governed by Markovian dynamics, and include the effect of an external field in the form of a bias in the normal motion to the surface. We study this system for the diffusion of particles in a semi-infinite lattice, analysing the conditional probability to find the system on the reference absorptive plane as well as the surface dispersion as functions of time. The agreement between numerical and analytical asymptotic results is discussed.

The use of ultra-thin, i.e. monolayer, films plays an important role in the emerging field of nano-fluidics. Since the dynamics of such films is governed by the interplay between substrate–fluid and fluid–fluid interactions, the transport of matter in nanoscale devices may eventually be efficiently controlled by substrate engineering. For such films, the dynamics is expected to be captured by two-dimensional lattice-gas models with interacting particles. Using a lattice-gas model and the non-linear diffusion equation derived from the microscopic dynamics in the continuum limit, we study two problems of relevance in the context of nano-fluidics. The first one is the case in which along the spreading direction of a monolayer a mesoscopic-sized obstacle is present, with a particular focus on the relaxation of the fluid density profile upon encountering and passing the obstacle. The second one is the mixing of two monolayers of different particle species which spread side by side following the merger of two chemical lanes, here defined as domains of high affinity for fluid adsorption surrounded by domains of low affinity for fluid adsorption.

The structure of a polymer nanodroplet adsorbed on a flat lyophobic substrate chemically decorated with a lyophilic stripe of width 2R_D is studied by molecular dynamics simulation of a coarse-grained bead–spring model of short macromolecules (containing N = 20 effective monomers). Varying the stripe width, the strength of the monomer–wall attraction and the temperature, the equilibrium morphology of the resulting droplets is studied and discussed in terms of current phenomenological theories.

In the second part, the behaviour of a liquid bridge connecting two such lyophilic stripes a distance L apart is analysed. It is shown that for large enough L such free-standing films are unstable and rupture, after a hole has nucleated and grown into an elongated cavity in the lateral direction leaving only a cylindrical bridge between the stripes immediately before break-up. The forces acting between the two walls of the slit pore in equilibrium due to the thin film forming the liquid bridge are also estimated as a function of distance between the walls.

The evaporation of wetting drops deposited on a substrate at thermal equilibrium under normal atmosphere is discussed. The evaporation rate appears to be controlled by the stationary diffusion of vapour molecules in the gas phase. Experiments with alkanes and water drops are fairly well accounted for by an isothermal model, taking into account the specific properties of thin films.

Recently, photo-driven directional motion of glycerol droplets on the surface of a liquid crystal doped with photochromic azobenzene derivatives has been reported. We present a theoretical model for this phenomenon. The motion of droplets is induced by a gradient in surface tension, which is produced by the combined effect of photo-isomerization and diffusion of surface azobenzenes. The theoretical relation between the surface tension and the surface concentration of cis isomers is proposed. The experimentally observed depletion zone of droplets can be reasonably well explained in terms of diffusion of droplets in the presence of the gradient in the surface tension.

Over the past few decades, spontaneous pattern formation in reaction–diffusion systems has been actively studied both theoretically and experimentally. In this article, we report novel phenomena that are generated beyond the framework of usual reaction–diffusion systems: (1) chemical wave propagation under the effect of a boundary, (2) convection in a reaction–diffusion system, and (3) dynamical phase separation under nonequilibrium conditions caused by a focused laser. All of these phenomena are generated as a result of cross-hierarchical interaction between macroscopic and mesoscopic scales.

Irreversible aggregation processes involving reactive and frozen clusters are investigated using the rate equation approach. In aggregation events, two clusters join irreversibly to form a larger cluster; additionally, reactive clusters may spontaneously freeze. Frozen clusters do not participate in merger events. Generally, freezing controls the nature of the aggregation process, as demonstrated by the final distribution of frozen clusters. The cluster mass distribution has a power-law tail, F_k∼k^−γ, when the freezing process is sufficiently slow. Different exponents, γ = 1 and 3, are found for the constant and the product aggregation rates, respectively. For the latter case, the standard polymerization model, either no gels, or a single gel, or even multiple gels, may be produced.

Individual-entity-based models of chemical or biological dynamics usually consider individual entities diffusing in space and performing a birth–death-type dynamics. In this work we study the properties of a model in this class where the birth dynamics is mediated by the local, within a given distance, density of particles. Groups of individuals are formed in the system and in this paper we concentrate on the study of the properties of these clusters (lifetime, size, and collective diffusion). In particular, in the limit of the interaction distance approaching the system size, a unique cluster appears which helps us to understand and characterize the clustering dynamics of the model.

It is often necessary, in scientific or everyday life problems, to find a randomly hidden target. What is then the optimal strategy to reach it as rapidly as possible? In this article, we develop a stochastic theory for intermittent search behaviours, which are often observed: the searcher alternates phases of intensive search and slow motion with fast displacements. The first results of this theory have already been announced recently. Here we provide a detailed presentation of the theory, as well as the full derivation of the results. Furthermore, we explicitly discuss the minimization of the time needed to find the target.

Continuous time random walk (CTRW) models are widely used to model diffusion in condensed matter. There are two classes of such models, distinguished by the convergence or divergence of the mean waiting time. Systems with finite average sojourn time are ergodic and thus Boltzmann–Gibbs statistics can be applied. We investigate the statistical properties of CTRW models with infinite average sojourn time; in particular, the occupation time probability density function is obtained. It is shown that in the non-ergodic phase the distribution of the occupation time of the particle on a given lattice point exhibits bimodal U or trimodal W shape, related to the arcsine law. The key points are as follows. (a) In a CTRW with finite or infinite mean waiting time, the distribution of the number of visits on a lattice point is determined by the probability that a member of an ensemble of particles in equilibrium occupies the lattice point. (b) The asymmetry parameter of the probability distribution function of occupation times is related to the Boltzmann probability and to the partition function. (c) The ensemble average is given by Boltzmann–Gibbs statistics for either finite or infinite mean sojourn time, when detailed balance conditions hold. (d) A non-ergodic generalization of the Boltzmann–Gibbs statistical mechanics for systems with infinite mean sojourn time is found.

We use a master equation approach based on the Poland–Scheraga free energy for DNA denaturation to investigate the (un)zipping dynamics of a denaturation wedge in a stretch of DNA that is clamped at one end. In particular, we quantify the blinking dynamics of a fluorophore–quencher pair mounted within the denaturation wedge. We also study the behavioural changes in the presence of proteins that selectively bind to single-stranded DNA. We show that such a set-up could be well suited as an easy-to-implement nanodevice for sensing environmental conditions in small volumes.

We use long atomistic molecular dynamics simulations to address certain fundamental issues regarding water dynamics in the hydration layer of a 38 base long (GCCGCGAGGTGTCAGGGATTGCAGCCAGCATCTCGTCG) negatively charged hydrated DNA duplex. The rotational time correlation function of surface water dipoles is found to be markedly non-exponential, with a slow component at long time, whose magnitude depends on the initial (t = 0) residence of the water in the major or minor groove of the DNA. The surface water molecules are also found to exhibit anisotropic diffusion in both the major and minor grooves: diffusion in the direction parallel to the DNA surface exhibits a crossover from higher to lower than that in the direction normal to the surface at short-to-intermediate times. In the same time window, translational motion of water molecules in the minor groove is sub-diffusive, with mean square displacement (MSD) growing as t^α with . In general, water molecules in the major group exhibit faster dynamics than those in the minor groove, in agreement with earlier results (Bonvin et al 1998 J. Mol. Biol.282 859–73). We compare these results with dynamics of water molecules at the surface of an anionic micelle, cesium perfluorooctanoate (CsPFO). Water molecules on the surface of CsPFO also exhibit slow translation and non-exponential orientational dynamics.