Diffusion in Temperature Gradients
Lars Höglund and John Ågren Dept of Materials Science and Engineering KTH 100 44 Stockholm
Acknowledgement: TCSAB NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 1 Background
• Strong temperature gradients are often technically important in energy applications, i.e. they may occur in heat exchangers, superheater tubes etc. • Often composites or joints between dissimilar materials, e.g. Stainless steels/high strength steels
NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 2 Content
• Background- importance of temperature gradients • Coupling effects and irreversible thermodynamics treatment • Thermal migration in pure metal and the Kirkendall effect • Implementation in DICTRA • Thermomigration of carbon in steel
NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 3 General approach for isothermal diffusion
:Flux ∂ ∂μμ∂c ∂c −= LJ −= L −= D ∂x ∂c ∂x ∂x ∂μ = LD ∂c Kinetic parameters Darken’s thermodynamic factor, from model. e.g. from Calphad analysis.
Base models on a vacancy mechanism! NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 4 diffusion Carbon diffusion Tin - gradient?
uC xC JC −= My CVaVa ∇ C μμC C ),( uuT C = Vs ∑ x j ∈Sj u ⎛ ∂μ ∂μ ⎞ C ⎜ C C ⎟ −= My CVaVa ⎜ uC +∇ ∇T ⎟ ?? Vs ⎝ ∂uC ∂T ⎠ ∂μ as used be Cannot be used as C −= S depends on the ∂T C reference state!
NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 5 Coupling effects
Flux Heat Electric diffusion
Force Temperature Fourier Seebeck Soret gradient Thermo- migration
Voltage Peltier Ohm Electro migration
Chemical Dufour Volta Fick potential (galvanic cell) gradient
NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 6 Irreversible thermodynamics treatment L L J kj μ kT ∇−∇−= T k ∑ T j T 2 L L J Tj μ TT ∇−∇−= T Q ∑ T j T 2 Onsager reciprocity :laws
= LL jkkj
= LL jTTj NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 7 * transport of Heat of transport Q j : L L L ⎛ Q* ⎞ J kj μ kT T kj ⎜ μ j ∇+∇−=∇−∇−= T ⎟ k ∑ j 2 ∑ ⎜ j 2 ⎟ T T T ⎝ T ⎠ * i.e. kT = ∑ QLL jkj of frame fixed lattice In lattice fixed frame of reference vacancy mechansim yields
yu Vak Lkk = M kVa Vs
Lkj = 0 when ≠ ji diagonal-off no When no diagonal-off coefficien :ts ⎛ ∂ J ⎞ Q* = ⎜ Q ⎟ j ⎜ ⎟ ∂ J j NIST Diffusions workshop ⎝ ⎠ T =∇ 0 March 23-24, 2010
Industrial Engineering and Management 8 Thermal migration – a Kirkendall effect or a cross effect? • Pure component A in lattice-fixed frame: * yu VaA QA J A −= M AVa 2 ∇T Vs T Kirkendall (marker) :velocity
K −= ∑ JVv kS
NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 9 Transform to number-fixed frame
n ∑ Ja ii = 0 i=1
yu Vak * LkT = QM kkVa Vs n 2 ui * kT TL ∑ δ −= au ikki )(/' iiVaVa /TQMy i=1 Vs
NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 10 Application to Fe-M-C
n ∑ JJJa MFeii =+= 0 i=1
= MyM iVaVai
2 1 * * FeT /' TL ()−= )1( − MMMFeFeFeFeFe /TQMuuQMuu Vs
2 1 * * 2 MT /' TL ()−= FeFeFeM −+ )1( MMMM −= FeT /'/ TLTQMuuQMuu Vs
2 uC * * * CT /' TL ()−= − + CCVaVaMMMFeFeFe /TQMyQMuQMu Vs
2 uC * CT /' TL ≅ CCVaVa /TQMy Vs NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 11 Implementation in DICTRA
• Temperature field i.e. T(x,t), given as input, i.e. heat flow equation not solved. • The partial differential equations with the extra term solved with the Galerkin method. ∂ u ⎛ n 1 L ' ⎞ k ⎜−⋅−∇= Dn ' u kT ∇−∇ T ⎟ ⎜ ∑ ki i 2 ⎟ ∂ t ⎝ i=1 VS T ⎠
NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 12 Thermomigration of carbon in steel
Okafor et al. 1982
NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 13 Carbon flux Tin - :gradient u ⎛ ∂μ Q* ∂T ⎞ C ⎜ CC ⎟ JC −= My CVaVa ⎜ + ⎟ Vs ⎝ ∂x T ∂x ⎠ * DC ⎛ ∂u QCC 1 ∂T ⎞ −= ⎜ + ⎟ Vs ⎝ ∂x μ / ∂∂ uT CC ∂x ⎠
Stationary Jt C =∞→ 0 everywhere
* ∂u ∂μCC QC −= ∂ ln ∂uT C :solution idealFor idealFor :solution
∂μC * ∂ lnuC / C C =⇒= RQuRT ∂uC ∂ T )/1( NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 14 Okafor et al. 1982 * −1 QC −= 30012 molJ
NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 15 ∂u ∂T Stationary conditions C ∝ ? ∂x ∂x
NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 16 NIST Diffusions workshop March 23-24, 2010
Industrial Engineering and Management 17 t →∞
100 h
Höglund and Ågren 2009 10 h * −1 QC −= 00044 molJ t=0 Δ: exp, Okafor et al 1982
A 200 K T-difference has about the same effect as carbon concentration differece of 1 at%. Steady state was not established during the NIST Diffusions workshop March 23-24, 2010 experiments by Okafor et al.!
Industrial Engineering and Management 18