Adsorption-limited kinetics may be described by phenomenological pseudo-order models. Such models leverage on the general principle that the rate of change of the adsorbed material depends on some power of its concentration, and their solutions provide the quantity of adsorbed molecules per unit mass of the sorbent material as a function of time. The assumptions made about how the solute molecules (adsorbents) are distributed around the sorbent material and whether or not diffusion effects are present are crucial for defining the rate of change. In the first case, the homogeneous uniform distribution of solute molecules and the absence of diffusion effects are well-described by classical modeling (integer-order derivatives). In the second case, fractal modeling arises from a departure from homogeneous uniform distribution, time is apparently contracted, and diffusion effects are still absent. In the third case, deviation from both conditions leads to fractional modeling; unlike fractal modeling, there are memory effects that exert an action on a limited number of process steps. We present briefly solutions for various classical and fractal kinetic models that describe adsorption. For the first time, we present adsorption kinetics under the framework of fractional calculus. In particular, we provide detailed expressions for pseudo-first-order fractional kinetics, while for higher orders, recursive relations amenable to numerical treatment are given. Application of each model is discussed.

Molecular dynamics simulations of a bi-layer membrane made by the same number of 1-palmitoyl-2-oleoyl-glycero-3-phospho-ethanolamine and palmitoyl-oleoyl phosphatidylserine lipids reveal sub-diffusional motion, which presents a crossover between two different power laws. Fractional Brownian motion is the stochastic mechanism that governs the motion in both regimes. The location of the crossover point is justified with simple geometrical arguments and is due to the activation of the mechanism of circumrotation of lipids about each other.

Liquid-cell transmission electron microscopy (LCTEM) is a powerful in situ videography technique that has the potential to allow us to observe solution-phase dynamic processes at the nanoscale, including imaging the diffusion and interaction of nanoparticles. Artefactual effects imposed by the irradiated and confined liquid-cell vessel alter the system from normal “bulk-like” behavior in multiple ways. These artefactual LCTEM effects will leave their fingerprints in the motion behavior of the diffusing objects, which can be revealed through careful analysis of the object-motion trajectories. Improper treatment of the motion data can lead to erroneous descriptions of the LCTEM system’s conditions. Here, we advance our anomalous diffusion object-motion analysis (ADOMA) method to extract a detailed description of the liquid-cell system conditions during any LCTEM experiment by applying a multistep analysis of the data and treating the x/y vectors of motion independently and in correlation with each other and with the object’s orientation/angle.