Akihiko Kitada

The effect of curvature on the instability of a solid/liquid interface

An essential factor causing the instability (stability) of a solid/liquid interface during solidification is explored. By examining the qualitative properties of the classical solution of a nonlinear evolution equation, the bifurcation, the coalescence and the growth rate of the interface are discussed. These discussions lead to the relation between the instability (stability) and the curvature of the interface.

A rapid proof for a flattening property of a classical solution of the Mullins equation

Modifying previous proof based on the maximum principles [J. Math. Phys. 27, 1391 (1986); 28, 536, 2982 (1987)], a rapid proof is proposed for a flattening property of a classical solution of the Cauchy problem associated with the Mullins equation ut=uxx/(1+ux2), (x,t)∈R1×(0,∞), u(x,0)=α(x), x∈R1 which describes the development of material surface due to evaporation condensation.

A consideration of the morphological stability of an interface

The evolution equation of an interface based on a mass transfer and that based on a heat transfer are unified into a nonlinear evolution equation for two spatial variables, and the properties of the solution of this equation are discussed without employing any linear approximations. Then, through these discussions, the intrinsic conditions which characterize the morphological stability are exhibited.

Topologization of Polycrystal : Existence of a Compact Domain with a Self-Similarity

We topologize a polycrystal composed of single crystals defined in a finite-dimensional affine space, to be a complete metric space. In the topological space “polycrystal”, we can find a compact domain with a self-similarity which is one of the geometrical properties characteristic of the complete metric space.

An estimate of the Hausdorff dimension of a weak self-similar set

Abstract We propose an estimate to quantitatively evaluate the Hausdorff dimension of a self-similar set based on a system of weak contractions each of whose contraction coefficient is not a constant but a function of a parameter. Using the estimate, we investigate the topological structures specific to this weak self-similar set.

Addendum to “A Consideration of the Morphological Stability of an Interface”

In our previous paper, we discuss the flattening properties of a generalized interface through the harmony between phenomenal facts and a mathematical structure, and we propose an original principle of interface morphology: ‘‘The interface tends to become flat if the driving force decreases with the increase of the curvature’’. However, we see the incompleteness of its universality; that is, a question ‘‘Is the principle applied even if the contribution neglecting the curvature effect AI or AII or the curvature effect fI or fII depends not only on the height of the interface u but also on the time t or the spatial variable x1 or x2 caused by concentration distribution, impurity distribution, temperature distribution and so on?’’ naturally comes to mind. Then, letting AI, AII, fI, and fII be a function of t, x1, x2, and u, we verify the principle in the same way as that in our previous study. First, with the aid of the implicit function theorem and the mean value theorem, the notion of a weak flattening property is acquired as follows. 1) Even if AI, AII, fI, and fII actually depend on t, x, y, and u, the velocity of the peak top (the valley bottom) of the interface is lower (greater) than the function of t; x; y, and u, which corresponds to the velocity without the curvature effect. Then, with the aid of the lemmas in our previous paper on a nonlinear evolution equation and an ordinary differential equation, the notion of the hierarchy of flattening properties is acquired as follows. 2) If AI and AII are independent of x1 (or x2), the velocity of the peak top (the valley bottom) is lower (greater) than the function of t, x2 (or x1), and u, which corresponds to the velocity without the curvature effect. 3) If AI, AII, fI, and fII are independent of x1 (or x2), the velocity of the peak top (the valley bottom) is lower (greater) than the function of t, x2 (or x1), and u, which corresponds to the velocity without the curvature effect. Then, the bifurcation never takes place. 4) If AI and AII are independent of x1, x2 and u and fI and fII are independent of x1 (or x2), the velocity of the peak top (the valley bottom) is lower (greater) than the function of only t, which corresponds to the velocity of a plane interface. In particular, if AI and AII are constant, the function is constant. Then, the bifurcation never takes place. 5) If AI and AII depend only on u [that is, AIðt; x; y; uÞ 1⁄4 ~ AIðuÞ;AIIðt; x; y; uÞ 1⁄4 ~ AIIðuÞ] under the condition 9 s.t. ~ AIð Þ 1⁄4 ~ AIIð Þ, d1⁄2 ~ AIðuÞ ~ AIIðuÞ = duju1⁄4 < 0, which reinforces the flattening property, and fI and fII are independent of x1 (or x2), the interface exponentially approaches a flat plane u 1⁄4 without the bifurcation. Consequently, the difference between the role of the contributions neglecting the curvature effects (AI and AII) and that of the curvature effects ( fI and fII) for the flattening properties is revealed, and the above principle is more universally exhibited.

A note on a property of a weak self-similar perfect set

Abstract Let S be a compact, weak self-similar perfect set based on a system of weak contractions f j , j =1,…, m each of which is characterized by a variable contraction coefficient α j ( l ) as d(f j (x),f j (y)) ≦ α j (l)d(x,y) , d ( x , y ) l , l >0. If the relation ∑ m j =1 α j ( l 0 ) l 0 , then every nonempty compact metric space is a continuous image of the set S .

Time-Dependent Nucleation Rate

This paper deals with a method of describing the asymptotic behaviour of the Fokker-Planck equation. The nucleation rate, J ( t ), approaches a steady state as J ( t )= J s [1-exp (- t / t l a g )] with 1/(6.3 a ( l c ) Z 2 ) ≥ t l a g ≥1/(12.0 a ( l c ) Z 2 ), where J s is the steady state nucleation rate, Z is the Zeldvitch factor and a ( l c ) the rate at which monomers are absorbed by a cluster with critical size l c . This result agrees with the numerical calculation by Kanne-Dannetschek and Stauffe and the prevoius theory.

Analysis of Charge Compensation Mechanisms in Pr1-xAxCoO3-δ (A = Ca, Sr) by X-ray Absorption Near-Edge Structure

The charge compensation mechanisms in polycrystalline Pr1-xCaxCoO3-δ and Pr1-xSrxCoO3-δ here synthesized by the solid-state reaction method have been investigated by analyzing the Pr-L3 and Co-K X-ray absorption near-edge structure (XANES) with the aid of first-principles calculations. The valence states of Pr ions in these materials were determined to be trivalent (Pr3+) and independent of the concentration of alkaline-earth ions observed in the Pr-L3 XANES profile. The Co-K XANES spectra shifted to the higher-energy side with an increase in alkaline-earth concentration and were examined in detail by first-principles calculations. Using calculated XANES results, we successfully determined that the valence state of Co ions is an intermediate value between 3+ and 4+ and increases with an increase in the concentration of alkaline-earth ions in Pr1-xAxCoO3-δ (A = Ca, Sr) accompanying oxygen vacancy.

Existence of a polycrystal filled with an arbitrary finite number of self-similar crystals

We discuss here a self-similar structure of an aggregate of crystals or noncrystalline grains. In particular, the sufficient condition for a polycrystal to be filled with an arbitrary finite number of self-similar crystals is investigated using a topological concept.