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 : μ∂ ∂ μ∂c ∂c JL= − = −L = −D ∂x ∂c ∂x ∂x ∂μ DL= ∂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 Carbon indiffusion - T gradient? uC xC JC= − yVa M CVa ∇ μ C T(,) μC uC C u= Vs ∑ x j j∈ S u ⎛ ∂μ ∂μ ⎞ C ⎜ C C ⎟ = − yVa M CVa⎜ ∇uC + ∇T ⎟ ?? Vs ⎝ ∂uC ∂T ⎠ ∂μ Cannot be usedC=depends −S as on the ∂T C reference state! NIST Diffusions workshop 2010March 23-24, 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 : LLkj= jk LLTj= jT NIST Diffusions workshop 2010March 23-24, Industrial Engineering and Management 7 * 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.LLQkT = ∑ kj j In lattice fixed reference framevacancy of mechansim yields uk y Va Lkk = M kVa Vs Lkj = 0 wheni≠ j When no off -coefficien diagonalts : ⎛ ∂ J ⎞ Q* = ⎜ Q ⎟ j ⎜ ⎟ ∂ J j NIST Diffusions workshop ⎝ ⎠∇T 0 = 2010March 23-24, Industrial Engineering and Management 8 Thermal migration – a Kirkendall effect or a cross effect? • Pure component A in lattice-fixed frame: * uA y Va QA J A= − M AVa 2 ∇T Vs T Kirkendall(marker) velocity : vK= −∑ VS k J NIST Diffusions workshop 2010March 23-24, Industrial Engineering and Management 9 Transform to number-fixed frame n ∑ai J i = 0 i=1 uk y Va * LkT = MQkVa k Vs n 2 ui * LT'/()kT =∑ δkiu − ky a i Va M iVa Q/ i T i=1 Vs NIST Diffusions workshop 2010March 23-24, Industrial Engineering and Management 10 e-M-CApplication to F n a∑i iJ= Fe J + J M 0 = i=1 Mi= Va y M iVa 2 1 * * LTFeT'/u=Fe u(() 1 Fe − M ) Fe Q Fe− u Fe u M M/ M Q M T Vs 2 1 * * 2 LTuuMQMT'/ =M() − Fe Fe+( Fe 1 uuMQTLTMMMM − ) /'/= −FeT Vs 2 uC * * * LTuCT'/ Fe M= Fe() Q − Fe u− M M M Q M+ y Va M/ CVa Q C T Vs 2 uC * LTCT'/ ≅yVa M CVa Q/ C T Vs NIST Diffusions workshop 2010March 23-24, 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 −∇⎜ ⋅D − n '∇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 fluxin - Tgradient : u ⎛ ∂μ Q* ∂T ⎞ C ⎜ CC ⎟ JC= − yVa M CVa⎜ + ⎟ Vs ⎝ ∂x T ∂x ⎠ * DC ⎛ ∂uCCQ 1 ∂T ⎞ = − ⎜ + ⎟ Vs ⎝ ∂x T∂μCC/ ∂ u ∂x ⎠ Stationary →t ∞ JC 0 everywhere = * ∂uCC∂μ QC= − ∂ lnT∂ C u For ideal solution : ∂μC * ∂ lnuC =RT/ uC ⇒C Q = R ∂uC (∂ 1T / ) NIST Diffusions workshop 2010March 23-24, Industrial Engineering and Management 14 Okafor et al. 1982 * −1 QC=12 − J 300 mol 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=44 − J 000 mol 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.