Structure and crystallization of bulk amorphous Pd41Ni10Cu28P21 alloy

Bulk amorphous Pd41Ni10Cu28P21 alloy has been prepared by water quenching method. The system shows excellent glass forming ability (GFA) with a high value of reduced glass transition temperature Trg, 0.714. Structural analyses indicate that the Pd41Ni10Cu28P21 alloy has a dense packing structure closer to "frozen liquid" than that of amorphous Pd40Ni40P20 alloy. Experiments on crystallization reveal that several crystalline phases simultaneously precipitate in the early part of crystallization. Below 710 K, a metastable phase forms, and subsequently disappears at elevated temperatures. In addition, the influence of partial substitute of Cu for Ni on GFA has been discussed with regard to thermodynamics and kinetics.     

Kinetics of glass transition and crystallization in multicomponent bulk amorphous alloys

Differential scanning calorimeter (DSC) is used to investigate apparent activation energy of glass transition and crystallization of Zr-based bulk amorphous alloys by Kissinger equation under non-isothermal condition. It is shown that the glass transition behavior as well as crystallization reaction depends on the heating rate and has a characteristic of kinetic effects. After being isothermally annealed near glass transition temperature, the apparent activation energy of glass transition increases and the apparent activation energy of crystallization reaction decreases. However, the kinetic effects are independent of the pre-annealing.     

Structure and crystallization of bulk amorphous Pd41Ni10Cu28P21 alloy

Bulk amorphous Pd₄₁,Ni₁₀Cu₂₈P₂₁, alloy has been prepared by water quenching method. The system shows excellent glass forming ability (GFA) with a high value of reduced glass transition temperatureT _rg 0.714. Structural analyses indicate that the Pd₄₁,Ni₁₀Cu₂₈,P₂₁, alloy has a dense packing structure closer to “frozen liquid” than that of amorphous Pd₄₀Ni₄₀,P₂₀, alloy. Experiments on crystallization reveal that several crystalline phases simultaneously precipitate in the early part of crystallization. Below 710 K, a metastable phase forms, and subsequently disappears at elevated temperatures. In addition, the influence of partial substitute of Cu for Ni on GFA has been discussed with regard to thermodynamics and kinetics     

Explosive-crystallization waves in thin amorphous layers on heat conducting substrates with thermal contact resistance

A model for explosive crystallization in a thin amorphous layer on a heat conducting substrate is presented. Rate equations are used to describe the kinetics of the homogeneous amorphous-crystalline transition. Heat conduction into the substrate and thermal contact resistance at the interface between layer and substrate are taken into account. The whole process is examined as a wave of invariant shape in a moving frame of reference. A coupled system of an integro-differential equation and ordinary differential equations is obtained and solved numerically. The propagation velocity of the wave is obtained as an eigenvalue of the system of equations. Some representative solutions are shown. Crystallization-wave velocities are compared with experimental values for explosive crystallization in germanium. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formation of nanocrystalline Fe73.5Cu1Nb3Si13.5B9 alloy under high pressure

The formation of nanocrystalline Fe_73.5 Cu₁Nb₃Si_13.5 B₉ alloy by annealing an amorphous Fe_73.5Cu₁Nb₃Si_13.5B₉ alloy at a temperature of 823 K under pressures in the range of 1–5 GPa is investigated by using X-ray diffraction, electron diffraction, and transmission electron microscopy. The high pressure experiments are carried out in belt-type pressure apparatus. Experimental results show that the initial crystalline phase in these annealed alloys is a-Fe solid solution (named a-Fe phase below), and high pressure has a great influence on the crystallization process of the a-Fe phase. The grain size of the a-Fe phase decreases with the increase of pressure (P). The volume fraction of the a-Fe phase increases with increasing the pressure as the pressure is below 2 GPa, and then decreases (Pδ2 GPa). The mechanism for the effects of the high pressure on the crystallization process of amorphous Fe_73.5Cu₁Nb₃Si_13.5B₉ alloy is discussed Project supported by the National Natural Science Foundation of China (Grant No. 19674070) and the Natural Science Foundation of Hebei Province.     

Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows

Summary.  We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation u _t −Δu+ɛ⁻² f(u)=0 arising from phase transition in materials science, where ɛ is a small parameter known as an ``interaction length'. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on ɛ. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u ₀ . In particular, all our error bounds depend on only in some lower polynomial order for small ɛ. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18, 19] and Chen [12] and to establish a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow. Received April 30, 2001 / Revised version received March 20, 2002 / Published online July 18, 2002 Mathematics Subject Classification (1991): 65M60, 65M12, 65M15, 35B25, 35K57, 35Q99, 53A10 Correspondence to: A. Prohl     

Phase transitions of iterated Higman-style well-partial-orderings

We elaborate Weiermann-style phase transitions for well-partial-orderings (wpo) determined by iterated finite sequences under Higman-Friedman style embedding with Gordeev’s symmetric gap condition. For every d-times iterated wpo ${\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}\right)}$ in question, d >? 1, we fix a natural extension of Peano Arithmetic, ${T \supseteq \sf{PA}}$ , that proves the corresponding second-order sentence ${\sf{WPO}\left({\rm S}{\textsc{eq}}^{d}, \trianglelefteq _{d}\right) }$ . Having this we consider the following parametrized first-order slow well-partial-ordering sentence ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}, r\right):}$ $$\left( \forall K > 0 \right) \left( \exists M > 0\right) \left( \forall x_{0},\ldots ,x_{M}\in {\rm S}\text{\textsc{eq}}^{d}\right)$$ $$\left( \left( \forall i\leq M\right) \left( \left| x_{i}\right| < K + r \left\lceil \log _{d} \left( i+1\right) \right\rceil \right)\rightarrow \left( \exists i < j \leq M \right) \left(x_{i} \trianglelefteq _{d} x_{j}\right) \right)$$ for a natural additive Seq ^d -norm |·| and r ranging over EFA-provably computable positive reals, where EFA is an abbreviation for IΔ ₀?+?exp. We show that the following basic phase transition clauses hold with respect to ${T = \Pi_{1}^{0}\sf{CA}_{ < \varphi ^{_{\left( d-1\right) }} \left(0\right) }}$ and the threshold point1.

1. If r <? 1 then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$ is provable in T.

1. If ${r > 1}$ then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$ is not provable in T.

Moreover, by the well-known proof theoretic equivalences we can just as well replace T by PA or ACA ₀ and ${\Delta _{1}^{1}\sf{CA}}$ , if d =? 2 and d =? 3, respectively.In the limit case d → ∞ we replaceEFA-provably computable reals r by EFA-provably computable functions ${f: \mathbb{N} \rightarrow \mathbb{R}_{+}}$ and prove analogous theorems. (In the sequel we denote by ${\mathbb{R}_{+}}$ the set of EFA-provably computable positive reals). In the basic case T?=? PA we strengthen the basic phase transition result by adding the following static threshold clause

1. ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1\right)}$ is still provable in T = PA (actually in EFA).

Furthermore we prove the following dynamic threshold clauses which, loosely speaking are obtained by replacing the static threshold t by slowly growing functions 1 _α given by ${1_{\alpha }\left( i\right)\,{:=}\,1+\frac{1}{H_{\alpha }^{-1}\left(i\right) }, H_{\alpha}}$ being the familiar fast growing Hardy function and ${H_{\alpha }^{-1}\left( i\right)\,{:=}\,\rm min \left\{ j \mid H_{\alpha } \left ( j\right) \geq i \right\}}$ the corresponding slowly growing inversion.

1. If ${\alpha < \varepsilon _{0}}$ , then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1_{\alpha}\right)}$ is provable in T = PA.

1. ${\sf{SWP}\left( {\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2},1_{\varepsilon _{0}}\right)}$ is not provable in T = PA.

We conjecture that this pattern is characteristic for all ${T\supseteq \sf{PA}}$ under consideration and their proof-theoretical ordinals o (T ), instead of ${\varepsilon _{0}}$ .