From atomistic interfaces to dendritic patterns

Exploring discrete space-time models for information transfer: Analogies from mycelial networks to the cosmic web

Fungal mycelium networks are large scale biological networks along which nutrients, metabolites flow. Recently, we discovered a rich spectrum of electrical activity in mycelium networks, including action-potential spikes and trains of spikes. Reasoning by analogy with animals and plants, where travelling patterns of electrical activity perform integrative and communicative mechanisms, we speculated that waves of electrical activity transfer information in mycelium networks. Using a new discrete space-time model with emergent radial spanning-tree topology, hypothetically comparable mycelial morphology and physically comparable information transfer, we provide physical arguments for the use of such a model, and by considering growing mycelium network by analogy with growing network of matter in the cosmic web, we develop mathematical models and theoretical concepts to characterise the parameters of the information transfer.

Analysis of the boundary integral equation for the growth of a parabolic/paraboloidal dendrite with convection

The growth of a parabolic/paraboloidal dendrite streamlined by viscous and potential flows in an undercooled one-component melt is analyzed using the boundary integral equation. The total melt undercooling is found as a function of the Péclet, Reynolds, and Prandtl numbers in two- and three-dimensional cases. The solution obtained coincides with the modified Ivantsov solution known from previous theories of crystal growth. Varying Péclet and Reynolds numbers we show that the melt undercooling practically coincides in cases of viscous and potential flows for a small Prandtl number, which is typical for metals. In cases of water solutions and non-metallic alloys, the Prandtl number is not small enough and the melt undercooling is substantially different for viscous and potential flows. In other words, a simpler potential flow hydrodynamic model can be used instead of a more complicated viscous flow model when studying the solidification of undercooled metals with convection.

Transport phenomena in complex systems (part 2)

This theme issue, in two parts, continues research studies of transport phenomena in complex media published in the first part (Alexandrov & Zubarev 2021 Phil. Trans. R. Soc. A 379, 20200301. (doi:10.1098/rsta.2020.0301)). The issue is concerned with theoretical, numerical and experimental investigations of nonlinear transport phenomena in heterogeneous and metastable materials of different nature, including biological systems. The papers are devoted to the new effects arising in such systems (e.g. pattern and microstructure formation in materials, impacts of external processes on their properties and evolution and so on). State-of-the-art methods of numerical simulations, stochastic analysis, nonlinear physics and experimental studies are presented in the collection of issue papers. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

Nucleation and growth dynamics of ellipsoidal crystals in metastable liquids

When describing the growth of crystal ensembles from metastable solutions or melts, a significant deviation from a spherical shape is often observed. Experimental data show that the shape of growing crystals can often be considered ellipsoidal. The new theoretical models describing the transient nucleation of ellipsoidal particles and their growth with and without fluctuating rates at the intermediate stage of bulk phase transitions in metastable systems are considered. The nonlinear transport (diffusivity) of ellipsoidal crystals in the space of their volumes is taken into account in the Fokker-Planck equation allowing for fluctuating growth rates. The complete analytical solutions of integro-differential models of kinetic and balance equations are found and analysed. Our solutions show that the desupercooling dynamics is several times faster for ellipsoidal crystals as compared to spherical particles. In addition, the crystal-volume distribution function is lower and shifted to larger particle volumes when considering the growth of ellipsoidal crystals. What is more, this function is monotonically increasing to the maximum crystal size in the absence of fluctuations and is a bell-shaped curve when such fluctuations are taken into account. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Ostwald ripening in the presence of simultaneous occurrence of various mass transfer mechanisms: an extension of the Lifshitz-Slyozov theory

The Ostwald ripening stage of a phase transformation process with allowance for synchronous operation of various mass transfer mechanisms (volume diffusion and diffusion along the block boundaries and dislocations) and the initial condition for the particle-radius distribution function is theoretically studied. The initial condition is taken from the analytical solution describing the intermediate stage of a phase transition process. The present theory focuses on relaxation dynamics from the beginning of the ripening process to its final asymptotic state, which is described by the previously constructed theories (Slezov VV. et al. 1978 J. Phys. Chem. Solids 39, 705-709. (doi:10.1016/0022-3697(78)90002-1) and Alexandrov & Alexandrova 2020 Phil. Trans. R. Soc. A 378, 20190247. (doi:10.1098/rsta.2019.0247)). An evolutionary behaviour of particle growth rates dependent on various mass transfer mechanisms and time is analytically described. The boundaries of the transition layer, which surround the blocking point, are found. The fundamental and relaxation contributions to the particle-radius distribution function are derived for the simultaneous occurrence of various mass transfer mechanisms. The left branch of this function is shifted to smaller particle radii whereas its right branch extends to the right of the blocking point as compared with the asymptotic universal distribution function. The theory under consideration well agrees with experimental data. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

The shape of dendritic tips: a test of theory with computations and experiments

This article is devoted to the study of the tip shape of dendritic crystals grown from a supercooled liquid. The recently developed theory (Alexandrov & Galenko 2020 Phil. Trans. R. Soc. A 378, 20190243. (doi:10.1098/rsta.2019.0243)), which defines the shape function of dendrites, was tested against computational simulations and experimental data. For a detailed comparison, we performed calculations using two computational methods (phase-field and enthalpy-based methods), and also made a comparison with experimental data from various research groups. As a result, it is shown that the recently found shape function describes the tip region of dendritic crystals (at the crystal vertex and some distance from it) well. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Transport phenomena in complex systems (part 1)

The issue, in two parts, is devoted to theoretical, computational and experimental studies of transport phenomena in various complex systems (in porous and composite media; systems with physical and chemical reactions and phase and structural transformations; in biological tissues and materials). Various types of these phenomena (heat and mass transfer; hydrodynamic and rheological effects; electromagnetic field propagation) are considered. Anomalous, relaxation and nonlinear transport, as well as transport induced by the impact of external fields and noise, is the focus of this issue. Modern methods of computational modelling, statistical physics and hydrodynamics, nonlinear dynamics and experimental methods are presented and discussed. Special attention is paid to transport phenomena in biological systems (such as haemodynamics in stenosed and thrombosed blood vessels magneto-induced heat generation and propagation in biological tissues, and anomalous transport in living cells) and to the development of a scientific background for progressive methods in cancer, heart attack and insult therapy (magnetic hyperthermia for cancer therapy, magnetically induced circulation flow in thrombosed blood vessels and non-contact determination of the local rate of blood flow in coronary arteries). The present issue includes works on the phenomenological study of transport processes, the derivation of a macroscopic governing equation on the basis of the analysis of a complicated internal reaction and the microscopic determination of macroscopic characteristics of the studied systems. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Dendritic growth of ice crystals: a test of theory with experiments

Motivated by an important application of dendritic crystals in the form of an elliptical paraboloid, which widely spread in nature (ice crystals), we develop here the selection theory of their stable growth mode. This theory enables us to separately define the tip velocity of dendrites and their tip diameter as functions of the melt undercooling. This, in turn, makes it possible to judge the microstructure of the material obtained as a result of the crystallization process. So, in the first instance, the steady-state analytical solution that describes the growth of such dendrites in undercooled one-component liquids is found. Then a system of equations consisting of the selection criterion and the undercooling balance that describes a stable growth mode of elliptical dendrites is formulated and analyzed. Three parametric solutions of this system are deduced in an explicit form. Our calculations based on these solutions demonstrate that the theoretical predictions are in good agreement with experimental data for ice dendrites growing at small undercoolings in pure water.

Fracto-eutectogels: SDS fractal dendrites via counterion condensation in a deep eutectic solvent

Glyceline, a deep eutectic solvent comprising glycerol and choline chloride, is a green nonaqueous solvent with potential industrial applications. Molecular mechanisms of surfactant self-assembly in deep eutectic solvents are expected to differ from those in their constituent polar components and are not well understood. Here we report the observation of self-assembled SDS fractal dendrites with dimensions up to ∼mm in glyceline at SDS concentrations as low as cSDS ∼ 0.1 wt%. The prevalence of these dendritic fractal aggregates led to the formation of a gel phase at SDS concentrations above ≥1.9 wt% (the critical gelation concentration cCGC). The gel microscopic structure was visualised using polarised light microscopy (PLM); rheology measurements confirmed the formation of a colloidal gel, where the first normal stress difference was negative and the elastic modulus was dominant. Detailed nano-structural characterisation by small-angle neutron scattering (SANS) further confirmed the presence of fractal aggregates. Such SDS aggregation or gelation has not been observed in water at such low surfactant concentrations, whereas SDS has been reported to form lamellar aggregates in glycerol (a component of glyceline). We attribute the formation of the SDS fractal dendrites to the condensation of counterions (i.e. the choline ions) around the SDS aggregates - a diffusion-controlled process, leading to the aggregate morphology observed. These unprecedented results shed light on the molecular mechanisms of surfactant self-assembly in deep eutectic solvents, important to their application in industrial formulation.

Analytical solution of a binary melt solidification model in the presence of a quasi-equilibrium mushy region for the case of the non-linear phase diagram

The process of directional crystallization with a quasi-equilibrium mushy region is considered in the steady-state conditions with allowance for the non-linear phase diagram. A complete analytical solution of model equations is found in a parametric form by introducing a new variable-the solid phase fraction in the mushy layer. The temperature and solute concentration, as well as the solid fraction in the mushy layer, its boundaries, and the crystallization velocity, are determined analytically. It is shown that a non-linear phase diagram substantially influences the behavior of solutions obtained.

The effect of density changes on crystallization with a mushy layer

A nonlinear problem with two moving boundaries of the phase transition, which describes the process of directional crystallization in the presence of a quasi-equilibrium two-phase layer, is solved analytically for the steady-state process. The exact analytical solution in a two-phase layer is found in a parametric form (the solid phase fraction plays the role of this parameter) with allowance for possible changes in the density of the liquid phase accordingly to a linearized equation of state and arbitrary value of the solid fraction at the boundary between the two-phase and solid layers. Namely, the solute concentration, temperature, solid fraction in the mushy layer, liquid and solid phases, mushy layer thickness and its velocity are found analytically. The theory under consideration is in good agreement with experimental data. The obtained solutions have great potential applications in analysing similar processes with a two-phase layer met in materials science, geophysics, biophysics and medical physics, where the directional crystallization processes with a quasi-equilibrium mushy layer can occur. This article is part of the theme issue 'Patterns in soft and biological matters'.

From nucleation and coarsening to coalescence in metastable liquids

The transition of a metastable liquid (supersaturated solution or supercooled melt) occurring from the intermediate stage (where the crystals nucleate and grow) to the concluding stage (where the larger particles evolve at the expense of the dissolution of smaller particles) is theoretically described, with allowance for various mass transfer mechanisms (reaction on the interface surface, volume diffusion, grain-boundary diffusion, diffusion along the dislocations) arising at the stage of Ostwald ripening (coalescence). The initial distribution function (its 'tail') for the concluding stage (forming as a result of the evolution of a particulate assemblage during the intermediate stage) is taken into account to determine the particle-size distribution function at the stage of Ostwald ripening. This modified distribution function essentially differs from the universal Lifshitz-Slyozov (LS) solutions for several mass transfer mechanisms. Namely, its maximum lies below and is shifted to the left in comparison with the LS asymptotic distribution function. In addition, the right branch of the particle-size distribution lies above and is shifted to the right of the LS blocking point. It is shown that the initial 'tail' of the particle-size distribution function completely determines its behaviour at the concluding stage of Ostwald ripening. The present theory agrees well with experimental data. This article is part of the theme issue 'Patterns in soft and biological matters'.

Dissolution of polydisperse ensembles of crystals in channels with a forced flow

A non-stationary integro-differential model describing the dissolution of polydisperse ensembles of crystals in channels filled with flowing liquid is analysed. The particle-size distribution function, the particle flux through an arbitrary cross-section of the channel, the particle concentration profile, as well as the disappearance intensity of particles are found analytically. It is shown that a nonlinear behaviour of solutions is completely defined by the source term of particles introduced into the channel. In particular, the model approximately describes the processes of dissolution and transport of drug microcrystals to the target sites in a living organism, taking into account complex dissolution kinetics of drug particles. This article is part of the theme issue 'Patterns in soft and biological matters'.