scholarly journals Computation of Trajectories and Displacement Fields in a Three-Dimensional Ternary Diffusion Couple: Parabolic Transform Method

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Marek Danielewski ◽  
Henryk Leszczyński
Keyword(s):  
Diffusion Couple ◽  
Three Dimensional ◽  
Transformation Method ◽  
Kirkendall Effect ◽  
Initial Distribution ◽  
Displacement Fields ◽  
Ternary Diffusion ◽  
Transform Method ◽  
Ternary Diffusion Couple ◽  
Diagonalization Method

The problem of Kirkendall’s trajectories in finite, three- and one-dimensional ternary diffusion couples is studied. By means of the parabolic transformation method, we calculate the solute field, the Kirkendall marker velocity, and displacement fields. The velocity field is generally continuous and can be integrated to obtain a displacement field that is continuous everywhere. Special features observed experimentally and reported in the literature are also studied: (i) multiple Kirkendall’s planes where markers placed on an initial compositional discontinuity of the diffusion couple evolve into two locations as a result of the initial distribution, (ii) multiple Kirkendall’s planes where markers placed on an initial compositional discontinuity of the diffusion couple move into two locations due to composition dependent mobilities, and (iii) a Kirkendall plane that coincides with the interphase interface. The details of the deformation (material trajectories) in these special situations are given using both methods and are discussed in terms of the stress-free strain rate associated with the Kirkendall effect. Our nonlinear transform generalizes the diagonalization method by Krishtal, Mokrov, Akimov, and Zakharov, whose transform of diffusivities was linear.

scholarly journals Application of numerical inverse method in calculation of composition-dependent interdiffusion coefficients in finite diffusion couples

10.30544/308 ◽  
2017 ◽  
Vol 23 (3) ◽  
pp. 197-211 ◽  
Author(s):  
Yuanrong Liu ◽  
Weimin Chen ◽  
Jing Zhong ◽  
Ming Chen ◽  
Lijun Zhang
Keyword(s):  
Diffusion Couple ◽  
Inverse Method ◽  
Diffusion Path ◽  
Diffusion Couples ◽  
Ternary Diffusion ◽  
Inverse Approach ◽  
Ternary Diffusion Couple ◽  
Composition Profiles ◽  
Standard Sets ◽  
Interdiffusion Coefficients

The previously developed numerical inverse method was applied to determine the composition-dependent interdiffusion coefficients in single-phase finite diffusion couples. The numerical inverse method was first validated in a fictitious binary finite diffusion couple by pre-assuming four standard sets of interdiffusion coefficients. After that, the numerical inverse method was then adopted in a ternary Al-Cu-Ni finite diffusion couple. Based on the measured composition profiles, the ternary interdiffusion coefficients along the entire diffusion path of the target ternary diffusion couple were obtained by using the numerical inverse approach. The comprehensive comparisons between the computations and the experiments indicate that the numerical inverse method is also applicable to high-throughput determination of the composition-dependent interdiffusion coefficients in finite diffusion couples.

The Effective Interdiffusion Coefficients in Experimental Investigations of Interdiffusion in Ternary Systems

2005 ◽  
Vol 237-240 ◽  
pp. 121-126 ◽  
Author(s):  
Ü. Ugaste
Keyword(s):  
Diffusion Couple ◽  
Ternary Systems ◽  
Diffusion Path ◽  
Experimental Investigations ◽  
Ternary Diffusion ◽  
Ternary Diffusion Couple ◽  
Interdiffusion Process ◽  
Variable Diffusion ◽  
Diffusion Paths ◽  
Interdiffusion Coefficients

The application of the effective interdiffusion coefficients for describing the interdiffusion process in ternary systems is discussed. It is shown that the relative values of effective interdiffusion coefficients, which are directly related to the diffusion path developed in a given diffusion couple, are responsible for deviation of the diffusion paths from linearity. The relationship between effective interdiffusion coefficients and partial (intrinsic) coefficients in ternary systems is analysed. It is shown that Boltzmann’s solution for diffusion equation with variable diffusion coefficient by means of relatively easy calculation procedure gives reliable results for the calculation concentration distributions in a ternary diffusion couple.

Computation of the Kirkendall velocity and displacement fields in a one-dimensional binary diffusion couple with a moving interface

2007 ◽  
Vol 463 (2088) ◽  
pp. 3347-3373 ◽  
Author(s):  
W.J Boettinger ◽  
J.E Guyer ◽  
C.E Campbell ◽  
G.B McFadden
Keyword(s):  
Diffusion Couple ◽  
Kirkendall Effect ◽  
Diffuse Interface ◽  
Interface Problem ◽  
Displacement Fields ◽  
One Dimensional ◽  
Moving Interface ◽  
Interdiffusion Coefficients ◽  
Interface Treatment ◽  
Free Strain

The moving interface problem in a one-dimensional binary α/β diffusion couple is studied using sharp and diffuse interface (Cahn–Hilliard) approaches. With both methods, we calculate the solute field and the Kirkendall marker velocity and displacement fields. In the sharp interface treatment, the velocity field is generally discontinuous at the interphase boundary, but can be integrated to obtain a displacement field that is continuous everywhere. The diffuse interface approach avoids this discontinuity, simplifies the integration and yet gives the same qualitative behaviour. Special features observed experimentally and reported in the literature are also studied with the two methods: (i) multiple Kirkendall planes, where markers placed on the initial compositional discontinuity of the diffusion couple bifurcate into two locations, and (ii) a Kirkendall plane that coincides with the interphase interface. These situations occur with special values of the interdiffusion coefficients and starting couple compositions. The details of the deformation in these special situations are given using both methods and are discussed in terms of the stress-free strain rate associated with the Kirkendall effect.

Numerical Approximations to Solution of Biochemical Reaction Model

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Ahmet Yildirim ◽  
Ahmet Gökdogan ◽  
Mehmet Merdan
Keyword(s):  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Transformation Method ◽  
Reaction Model ◽  
Biochemical Reaction ◽  
Graphical Form ◽  
Kutta Method ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform

In this paper, approximate analytical solution of biochemical reaction model is used by the multi-step differential transform method (MsDTM) based on classical differential transformation method (DTM). Numerical results are compared to those obtained by the fourth-order Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method. Results are given explicit and graphical form.

Age-Related Differences in Throwing Techniques Used by the Catcher in Baseball

1994 ◽  
Vol 6 (3) ◽  
pp. 225-235 ◽  
Author(s):  
Shinji Sakurai ◽  
Bruce Elliott ◽  
J. Robert Grove
Keyword(s):  
High Speed ◽  
High Performance ◽  
Age Groups ◽  
Three Dimensional ◽  
Transformation Method ◽  
High Speed Photography ◽  
Two Phase ◽  
Ball Speed ◽  
Age Related ◽  
Release Speed

Three-dimensional (3-D) high speed photography was used to record the overarm throwing actions of five open-age, four 18-year-old, six 16-year- old, and six 14-year-old high-performance baseball catchers. The direct linear transformation method was used for 3-D space reconstruction from 2-D images of the catchers throwing from home plate to second base recorded using two phase-locked cameras operating at a nominal rate of 200 Hz. Selected physical capacity measures were also recorded and correlated with ball release speed. In general, anthropometric and strength measures significantly increased through the 14-year-old to open-age classifications, while a range of correlation coefficients from .50 to .84 was recorded between these physical capacities and ball speed at release. While many aspects of the kinematic data at release were similar, the key factors of release angle and release speed varied for the different age groups.

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