The shape of dendritic tips

Scaling laws for two-dimensional dendritic crystal growth in a narrow channel

We investigate analytically and computationally the dynamics of two-dimensional needle crystal growth from the melt in a narrow channel. Our analytical theory predicts that, in the low supersaturation limit, the growth velocity V decreases in time t as a power law V∼t^{-2/3}, which we validate by phase-field and dendritic-needle-network simulations. Simulations further reveal that, above a critical channel width Λ≈5l_{D}, where l_{D} is the diffusion length, needle crystals grow with a constant V<V_{s}, where V_{s} is the free-growth needle crystal velocity, and approaches V_{s} in the limit Λ≫l_{D}.

A Stable Mode of Dendritic Growth in Cases of Conductive and Convective Heat and Mass Transfer

In this paper, we develop a theory of stable dendritic growth in undercooled melts in the presence of conductive and convective heat and mass transfer boundary conditions at the solid/liquid interface of a dendrite. To simplify the matter and construct the analytical theory, conductive and convective mechanisms are considered separately. Namely, the laws for total undercooling and selection criterion defining the stable growth mode (dendrite tip velocity and diameter) are derived for conductive and convective boundary conditions. To describe the case of simultaneous occurrence of these heat and mass transfer mechanisms, we sew together conductive and convective laws using power stitching functions. The generalised selection theory is compared with experimental data for Al24Ge76 and Ti45Al55 undercooled melts.

Dynamics of intracellular clusters of nanoparticles

Background

Nanoparticles play a crucial role in nanodiagnostics, radiation therapy of cancer, and they are now widely used to effectively deliver drugs to specific sites, targeting whole organs and down to single cells, in a controlled manner. Therapeutic efficiency of nanoparticles greatly depends on their clustering distribution inside cells. Our purpose is to find the cluster density using Smoluchowski’s coagulation equation with injections.

Results

We obtain an exact cluster density of nanoparticles as the steady-state solution of Smoluchowski’s equation describing clustering due to the fusion of endosomes. We also analyze the unsteady cluster distribution and compare it with the experimental data for time evolution of gold nanoparticle clusters in living cells.

Conclusions

We show the steady cluster density is in good agreement with experimental data on gold nanoparticle distribution inside endosomes. We find that for clusters containing between 1 and 20 nanoparticles, the exact cluster density provides a better description of the existing experimental data than the well-known approximate asymptotic power-law distribution $$x^{-3/2}$$ x - 3 / 2

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)'.

Microstructure and morphology of Si crystals grown in pure Si and Al-Si melts

Microstructure of Al-40 wt%Si samples solidified in electromagnetic levitation furnace is studied at high melt undercooling. Primary Si with feathery and dendritic structures is observed. As this takes place, single Si crystals either contain secondary dendrite arms or represent faceted structures. Our experiments show that at a certain undercooling, there exists the microstructural transition zone of faceted to non-faceted growth. Also, we analyze the shape of dendritic crystals solidifying from liquid Si as well as from hypereutectic Al-Si melts at high growth undercoolings. The shapes of dendrite tips grown at undercoolings >100 K along the surface of levitated Al-40 wt%Si droplets are compared with pure Si dendrite tips from the literature. The dendrite tips are digitized and superimposed with theoretical shape function recently derived by stitching the Ivantsov and Brener solutions. We show that experimental and theoretical dendrite tips are in good agreement for Si and Al-Si samples.

A review on the theory of stable dendritic growth

This review article summarizes the main outcomes following from recently developed theories of stable dendritic growth in undercooled one-component and binary melts. The nonlinear heat and mass transfer mechanisms that control the crystal growth process are connected with hydrodynamic flows (forced and natural convection), as well as with the non-local diffusion transport of dissolved impurities in the undercooled liquid phase. The main conclusions following from stability analysis, solvability and selection theories are presented. The sharp interface model and stability criteria for various crystallization conditions and crystalline symmetries met in actual practice are formulated and discussed. The review is also focused on the determination of the main process parameters-the tip velocity and diameter of dendritic crystals as functions of the melt undercooling, which define the structural states and transitions in materials science (e.g. monocrystalline-polycrystalline structures). Selection criteria of stable dendritic growth mode for conductive and convective heat and mass fluxes at the crystal surface are stitched together into a single criterion valid for an arbitrary undercooling. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Evaporation kinetics of a polydisperse ensemble of drops

A mathematical model of the evaporation of a polydisperse ensemble of drops, with allowance for a nonlinear 'diffusion' term in the kinetic equation for the population density distribution function, is developed. The model describes the interaction of a gas phase with vaporizing drops: it has great potential for application in condensed matter physics, thermophysics and engineering devices (e.g. spray drying, cooling, power engineering). The kinetics of heat transfer between phases is theoretically studied. An analytical solution to the integro-differential equations of the process of droplet evaporation is found in a parametric form. Analytical solutions in the presence and absence of the 'diffusion' term are compared. It is shown that the fluctuations in particle evaporation rates ('diffusion' term in the Fokker-Planck equation) play a decisive role in the evolutionary behaviour of a polydisperse ensemble of vaporizing liquid drops. 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)'.

Dendrite tips as elliptical paraboloids

This review article summarizes current theories of the steady-state growth mode of dendrites in the form of elliptical paraboloids. The shape of dendrite tips is analyzed, temperature and solute concentration distributions are described in its vicinity, and a solution of the hydrodynamic problem of a viscous incompressible fluid flowing against a dendrite tip is developed. A significant difference in analytical solutions describing a dendrite tip as an elliptic paraboloid as compared to an axisymmetric morphology is shown. The system of nonlinear equations for determining the stationary velocity of dendrite growth and the radii of curvature of the dendrite tip along the major and minor axis of the ellipse, respectively, is derived. The developed theory is compared with experimental data on the growth of ice crystals consisting of D₂O or H₂O.

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.

Thin interface limit of the double-sided phase-field model with convection

The thin interface limit of the phase-field model is extended to include transport via melt convection. A double-sided model (equal diffusivity in liquid and solid phases) is considered for the present analysis. For the coupling between phase-field and Navier-Stokes equations, two commonly used schemes are investigated using a matched asymptotic analysis: (i) variable viscosity and (ii) drag force model. While for the variable viscosity model, the existence of a thin interface limit can be shown up to the second order in the expansion parameter, difficulties arise in satisfying no-slip boundary condition at this order for the drag force model. Nevertheless, detailed numerical simulations in two dimensions show practically no difference in dendritic growth profiles in the presence of forced melt flow obtained for the two coupling schemes. This suggests that both approaches can be used for the purpose of numerical simulations. Simulation results are also compared to analytic theory, showing excellent agreement for weak flow. Deviations at higher fluid velocities are discussed in terms of the underlying theoretical assumptions. This article is part of the theme issue 'Patterns in soft and biological matters'.

Patterns in soft and biological matters

The issue is devoted to theoretical, computer and experimental studies of internal heterogeneous patterns, their morphology and evolution in various soft physical systems-organic and inorganic materials (e.g. alloys, polymers, cell cultures, biological tissues as well as metastable and composite materials). The importance of these studies is determined by the significant role of internal structures on the macroscopic properties and behaviour of natural and manufactured tissues and materials. Modern methods of computer modelling, statistical physics, heat and mass transfer, statistical hydrodynamics, nonlinear dynamics and experimental methods are presented and discussed. Non-equilibrium patterns which appear during macroscopic transport and hydrodynamic flow, chemical reactions, external physical fields (magnetic, electrical, thermal and hydrodynamic) and the impact of external noise on pattern evolution are the foci of this issue. Special attention is paid to pattern formation in biological systems (such as drug transport, hydrodynamic patterns in blood and pattern dynamics in protein and insulin crystals) and to the development of a scientific background for progressive methods of cancer and insult therapy (magnetic hyperthermia for cancer therapy; magnetically induced drug delivery in thrombosed blood vessels). The present issue includes works on pattern growth and their evolution in systems with complex internal structures, including stochastic dynamics, and the influence of internal structures on the external static, dynamic magnetic and mechanical properties of these systems. This article is part of the theme issue 'Patterns in soft and biological matters'.