Atomic interpretation of the interface transfer coefficients for interdiffusion in AB binary phase separating system

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Abstract

During intermixing in phase separating A/B diffusion couple the actual atomic fractions of A atoms (cα and cβ) on the left as well as on the right hand side of the α/β interface gradually will decrease (from unity) as well as increase (from 0) until the equilibrium cα and cβ will be reached (α and β refer to the A- and B-rich phases, separated by the interface). At the same time the interface will also be shifted. Assuming that the atomic flux across the interface, JI, is proportional to the deviations from the equilibrium (Δα=cα-cα and Δβ=cβ-cβ) the JI=(1/Ω)[KIαβΔα+KIβαΔβ) relation is obtained, where Ω=ΩAB is the atomic volume. It is shown that the KIαβ and KIβα interface transfer coefficients are positive, independent of the interface velocity, equal to each other for symmetric miscibility gap, and can be given as KIβα=KIαβ=K=zvIαβcαξ. Here ΓIαβ is the jump frequency from α to β phase across the interface, a is the lattice spacing, zv is the vertical coordination number.ξ=[1+exp(ZV(cα-cβ)/kT)], where Z is the coordination number, V is the well-known solid solution parameter, proportional to the heat of mixing, and kT has its usual meaning. The above expression justifies the conjecture, frequently used in the literature, that only one interface transfer coefficient is enough for the description of the mass transfer across an interface. For short diffusion times the finite value of ΓIαβ will restrict the flux, leading to finite permeability with linear kinetics of interface shift. It is also shown that for an order of magnitude estimation of the critical interface shift (giving the transition from interface to diffusion control) the xc=aΓ(cβ)/ΓIαβ, relation can be used, where Γ(cβ) is the jump frequency in the β phase at cβ. Thus xc is practically independent of the value of V and only the composition dependence of the jump frequencies is important. For composition independent jump frequencies xc≅a, (i.e. it cannot be detected), while for the case when the diffusivity changes by seven orders of magnitude from pure A to B,xc is about 150 nm.

Original languageEnglish
Pages (from-to)203-209
Number of pages7
JournalInternational Journal of Heat and Mass Transfer
Volume113
DOIs
Publication statusPublished - Jan 1 2017

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coefficients
Fluxes
Chemical analysis
Solid solutions
Mass transfer
Solubility
coordination number
Atoms
Kinetics
miscibility gap
shift
mass transfer
diffusivity
unity
permeability
solid solutions
spacing
deviation
heat
kinetics

Keywords

  • Interdiffusion
  • Interface
  • Interface transfer coefficient
  • Phase separation
  • Second phase growth

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanical Engineering
  • Fluid Flow and Transfer Processes

Cite this

@article{9c66763a8a0e43329b42d2c9640b67f4,
title = "Atomic interpretation of the interface transfer coefficients for interdiffusion in AB binary phase separating system",
abstract = "During intermixing in phase separating A/B diffusion couple the actual atomic fractions of A atoms (cα and cβ) on the left as well as on the right hand side of the α/β interface gradually will decrease (from unity) as well as increase (from 0) until the equilibrium cα and cβ will be reached (α and β refer to the A- and B-rich phases, separated by the interface). At the same time the interface will also be shifted. Assuming that the atomic flux across the interface, JI, is proportional to the deviations from the equilibrium (Δα=cα-cα and Δβ=cβ-cβ) the JI=(1/Ω)[KIαβΔα+KIβαΔβ) relation is obtained, where Ω=ΩA=ΩB is the atomic volume. It is shown that the KIαβ and KIβα interface transfer coefficients are positive, independent of the interface velocity, equal to each other for symmetric miscibility gap, and can be given as KIβα=KIαβ=K=zvaΓIαβcαξ. Here ΓIαβ is the jump frequency from α to β phase across the interface, a is the lattice spacing, zv is the vertical coordination number.ξ=[1+exp(ZV(cα-cβ)/kT)], where Z is the coordination number, V is the well-known solid solution parameter, proportional to the heat of mixing, and kT has its usual meaning. The above expression justifies the conjecture, frequently used in the literature, that only one interface transfer coefficient is enough for the description of the mass transfer across an interface. For short diffusion times the finite value of ΓIαβ will restrict the flux, leading to finite permeability with linear kinetics of interface shift. It is also shown that for an order of magnitude estimation of the critical interface shift (giving the transition from interface to diffusion control) the xc=aΓ(cβ)/ΓIαβ, relation can be used, where Γ(cβ) is the jump frequency in the β phase at cβ. Thus xc is practically independent of the value of V and only the composition dependence of the jump frequencies is important. For composition independent jump frequencies xc≅a, (i.e. it cannot be detected), while for the case when the diffusivity changes by seven orders of magnitude from pure A to B,xc is about 150 nm.",
keywords = "Interdiffusion, Interface, Interface transfer coefficient, Phase separation, Second phase growth",
author = "D. Beke",
year = "2017",
month = "1",
day = "1",
doi = "10.1016/j.ijheatmasstransfer.2017.05.074",
language = "English",
volume = "113",
pages = "203--209",
journal = "International Journal of Heat and Mass Transfer",
issn = "0017-9310",
publisher = "Elsevier Limited",

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TY - JOUR

T1 - Atomic interpretation of the interface transfer coefficients for interdiffusion in AB binary phase separating system

AU - Beke, D.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - During intermixing in phase separating A/B diffusion couple the actual atomic fractions of A atoms (cα and cβ) on the left as well as on the right hand side of the α/β interface gradually will decrease (from unity) as well as increase (from 0) until the equilibrium cα and cβ will be reached (α and β refer to the A- and B-rich phases, separated by the interface). At the same time the interface will also be shifted. Assuming that the atomic flux across the interface, JI, is proportional to the deviations from the equilibrium (Δα=cα-cα and Δβ=cβ-cβ) the JI=(1/Ω)[KIαβΔα+KIβαΔβ) relation is obtained, where Ω=ΩA=ΩB is the atomic volume. It is shown that the KIαβ and KIβα interface transfer coefficients are positive, independent of the interface velocity, equal to each other for symmetric miscibility gap, and can be given as KIβα=KIαβ=K=zvaΓIαβcαξ. Here ΓIαβ is the jump frequency from α to β phase across the interface, a is the lattice spacing, zv is the vertical coordination number.ξ=[1+exp(ZV(cα-cβ)/kT)], where Z is the coordination number, V is the well-known solid solution parameter, proportional to the heat of mixing, and kT has its usual meaning. The above expression justifies the conjecture, frequently used in the literature, that only one interface transfer coefficient is enough for the description of the mass transfer across an interface. For short diffusion times the finite value of ΓIαβ will restrict the flux, leading to finite permeability with linear kinetics of interface shift. It is also shown that for an order of magnitude estimation of the critical interface shift (giving the transition from interface to diffusion control) the xc=aΓ(cβ)/ΓIαβ, relation can be used, where Γ(cβ) is the jump frequency in the β phase at cβ. Thus xc is practically independent of the value of V and only the composition dependence of the jump frequencies is important. For composition independent jump frequencies xc≅a, (i.e. it cannot be detected), while for the case when the diffusivity changes by seven orders of magnitude from pure A to B,xc is about 150 nm.

AB - During intermixing in phase separating A/B diffusion couple the actual atomic fractions of A atoms (cα and cβ) on the left as well as on the right hand side of the α/β interface gradually will decrease (from unity) as well as increase (from 0) until the equilibrium cα and cβ will be reached (α and β refer to the A- and B-rich phases, separated by the interface). At the same time the interface will also be shifted. Assuming that the atomic flux across the interface, JI, is proportional to the deviations from the equilibrium (Δα=cα-cα and Δβ=cβ-cβ) the JI=(1/Ω)[KIαβΔα+KIβαΔβ) relation is obtained, where Ω=ΩA=ΩB is the atomic volume. It is shown that the KIαβ and KIβα interface transfer coefficients are positive, independent of the interface velocity, equal to each other for symmetric miscibility gap, and can be given as KIβα=KIαβ=K=zvaΓIαβcαξ. Here ΓIαβ is the jump frequency from α to β phase across the interface, a is the lattice spacing, zv is the vertical coordination number.ξ=[1+exp(ZV(cα-cβ)/kT)], where Z is the coordination number, V is the well-known solid solution parameter, proportional to the heat of mixing, and kT has its usual meaning. The above expression justifies the conjecture, frequently used in the literature, that only one interface transfer coefficient is enough for the description of the mass transfer across an interface. For short diffusion times the finite value of ΓIαβ will restrict the flux, leading to finite permeability with linear kinetics of interface shift. It is also shown that for an order of magnitude estimation of the critical interface shift (giving the transition from interface to diffusion control) the xc=aΓ(cβ)/ΓIαβ, relation can be used, where Γ(cβ) is the jump frequency in the β phase at cβ. Thus xc is practically independent of the value of V and only the composition dependence of the jump frequencies is important. For composition independent jump frequencies xc≅a, (i.e. it cannot be detected), while for the case when the diffusivity changes by seven orders of magnitude from pure A to B,xc is about 150 nm.

KW - Interdiffusion

KW - Interface

KW - Interface transfer coefficient

KW - Phase separation

KW - Second phase growth

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U2 - 10.1016/j.ijheatmasstransfer.2017.05.074

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M3 - Article

AN - SCOPUS:85019887695

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