Andrey Shobukhov

Mathematical modeling of hysteresis in porous electrodes

We study the mathematical model of the Li+ ions’ intercalation from the electrolyte into the porous graphite surface of the negatively charged electrode and further Li diffusion inside the electrode particle. For proper approximation of experimental data we use the cubic polynomial. We prove the multiplicity of the steady state solutions in a certain range of the electrode potential values. This multiplicity may be explained by the simultaneous existence of several phases at the graphite electrode surface. Numerical investigation allows us to demonstrate the experimentally observed hysteresis. After including the diffusion of Li into the model we compare the charging time for various electrode structures.

Exact steady state solutions in symmetrical Nernst–Planck–Poisson electrodiffusive models

We compare three one-dimensional Nernst–Planck–Poisson systems that describe ion distribution near the electrode surfaces with planar, cylindrical and spherical symmetry respectively. These three models take into account ion diffusion and migration. In particular they describe the diffusive layers formed by Li+ ions in the vicinity of the graphite electrode particles. The three types of symmetry appear due to three different ways of particle ordering inside the electrode. In this paper we construct the exact steady state solutions to these systems and approximate solutions in form of power series. Then we solve the systems numerically and compare the results. We discuss the influence of symmetry in electrode particle ordering on the steady state distribution of ions in the diffusive layer.

Internal phase transition induced by external forces in Finsler geometric model for membranes

In this paper, we numerically study an anisotropic shape transformation of membranes under external forces for two-dimensional triangulated surfaces on the basis of Finsler geometry. The Finsler metric is defined by using a vector field, which is the tangential component of a three-dimensional unit vector σ corresponding to the tilt or some external macromolecules on the surface of disk topology. The sigma model Hamiltonian is assumed for the tangential component of σ with the interaction coefficient λ. For large (small) λ, the surface becomes oblong (collapsed) at relatively small bending rigidity. For the intermediate λ, the surface becomes planar. Conversely, fixing the surface with the boundary of area A or with the two-point boundaries of distance L, we find that the variable σ changes from random to aligned state with increasing of A or L for the intermediate region of λ. This implies that an internal phase transition for σ is triggered not only by the thermal fluctuations, but also by external mechanical forces. We also find that the frame (string) tension shows the expected scaling behavior with respect to A∕N (L∕N) at the intermediate region of A (L) where the σ configuration changes between the disordered and ordered phases. Moreover, we find that the string tension γ at sufficiently large λ is considerably smaller than that at small λ. This phenomenon resembles the so-called soft-elasticity in the liquid crystal elastomer, which is deformed by small external tensile forces.

Monte Carlo simulations of Landau–Ginzburg model for membranes

The Landau–Ginzburg (LG) model for membranes is numerically studied on triangulated spheres in R3. The LG model is in sharp contrast to the model of Helfrich–Polyakov (HP). The reason for this difference is that the curvature energy of the LG (HP) Hamiltonian is defined by means of the tangential (normal) vector of the surface. For this reason, the curvature energy of the LG model includes the in-plane bending or shear energy component, which is not included in the curvature energy of the HP model. From the simulation data, we find that the LG model undergoes a first-order collapse transition. The results of the LG model in the higher-dimensional spaces Rd(d > 3) and on the self-avoiding (SA) surfaces in R3 are presented and discussed. We also study the David–Guitter (DG) model, which is a variant of the LG model, and find that the DG model undergoes a first-order transition. It is also found that the transition can be observed only on the homogeneous surfaces, which are composed of almost uniform triangles according to the condition that the induced metric ∂ar ⋅ ∂br is close to δab.

Surface Tension, Pressure Difference and Laplace Formula for Membranes

The surface tension γ and the pressure difference Δp for spherical membranes are calculated using Monte Carlo simulation technique. We study the so-called tethered and uid surface discrete models that are defined on the fixed-connectivity (tethered) and dynamically triangulated (uid) lattices respectively. Hamiltonians of the models include a self-avoiding potential, which makes the enclosed volume well defined. We find that there is reasonable accuracy in the technique for the calculation of γ using the real area A if the bending rigidity κ or A/N is sufficiently large. We also find that γ becomes constant in the limit of A/N → ∞ both in the tethered and uid surfaces. The property limA/N→∞ γ = const corresponds to certain experimental results in cell biology.

Branched-polymer to inflated transition of self-avoiding fluid surfaces

Abstract We study phase transition of self-avoiding fluid surface model on dynamically triangulated lattices using the Monte Carlo simulation technique. We report the continuous transition between the branched polymer and inflated phases at Δ p = 0 , where Δ p ( = p in − p out ) is the pressure difference between the inner and outer sides of the surface. This transition is characterized by almost discontinuous change of the enclosed volume vs. the variations of the bending rigidity κ and the pressure difference Δ p . No surface fluctuation transition accompanies this transition up to the surface with the number of vertices N = 2562 .

Glass phase in anisotropic surface model for membranes

A Finsler geometric surface model for membranes is studied by using the Monte Carlo simulation technique on connection-fixed triangle lattices with sphere topology. An in-plane three-dimensional unit vector σ is assumed to be the in-plane tilt variable of the surface. The interaction with σ is described by the XY-model Hamiltonian. Since this variable σ is considered as a vector field on the surface, a Finsler metric is defined by using σ. We find that the model has three different phases. They change from the para-magnetic phase to the ferromagnetic and to the glass phases when the strength of the XY interaction increases. Both the para-magnetic and the glass phases are characterized by random configuration of σ; the variable σ randomly fluctuates in the para-magnetic phase while it is randomly frozen in the glass phase. We also find that the surface becomes spherical in both phases. On the contrary, in the ferro-magnetic phase the surface shape becomes tubular or discotic due to the anisotropic bending rigidity and surface tension coefficient, which are dynamically generated by ordered configurations of σ.

Mathematical modeling of aluminum electrolysis over a long interval of time

A new approach is proposed to the modeling of magnetohydrodynamic and chemical processes involved in aluminum electrolysis. In this approach a medium under study is presented as a mixture of reagents in unknown concentrations, making it possible to describe mixing and the chemical interaction of the electrolyte and the metal melt. Therefore, all stages of the electrolysis process can be modeled over a long interval of time.

Rise of source-generated ions in dry air under the action of an electric field

Electrodynamic and plasma chemical tropospheric processes have been considered in order to develop a cloud control technology. A mathematical model has been constructed for ionic flow from a ground-based plasma generator in the electric field of a charged cloud. The problem of the rise of four types of anions and four types of cations and the problem of electric field strength have been solved using a system of transfer equations and the Planck equation. The final distribution of the charged particle concentrations has been analyzed, and it has been demonstrated that a considerable quantity of O3- can rise up to an altitude of 2000 m.

Rise of negative ions from an external source in the lower atmosphere under the action of the earth electric field: a one-dimensional model

A one-dimensional model of the rise of O– negative ions generated by a ground-based external source to altitudes of several kilometers under the action of the Earth electric field is developed. The model takes into account the contributions from the diffusion and drift of the ions, as well as the effect of a rising air flow. The characteristic time of rise of negative ions to an altitude of 2 km is determined, altitude distributions of the concentration of ions are obtained, and the distortion of the Earth electric field due to the presence of an excess of negative ions is demonstrated.