Single class grain growth model in MatCalc 6

Single class grain growth model in MatCalc 6 (MatCalc 6.00.0200) P. Warczok

Grain growth Tendency - to minimize: grain surface area specific grain boundary energy Page 2

Grain growth kinetics General idea: Mobility & Driving force DD = dddd dddd = MMPP DD Page 3 DD PP DD MM tt - Grain size growth rate - Grain boundary mobility - Driving force/pressure for grain growth - Time

Single class grain growth Single quantity: Mean grain size DD 1 DD 2 Page 4

Single class grain growth Single quantity: Mean grain size Page 5

Growth driving pressure Driving pressure dependent on Grain interface energy Grain size PP DD ~ γγ HHHH DD Page 6 γγ HHHH - Grain interface energy DD - Mean grain size (diameter)

Growth driving pressure Driving pressure dependent on Grain interface energy Grain size PP DD = 2kk dd γγ HHHH DD γγ HHHH DD kk dd - Grain interface energy - Mean grain size (diameter) - Scaling factor Page 7

Growth driving pressure Driving pressure dependent on Grain interface energy Grain size PP DD = 2kk dd γγ HHHH DD γγ HHHH DD kk dd Page 8 - Grain interface energy - Mean grain size (diameter) - Scaling factor

Growth driving pressure Driving pressure dependent on Grain interface energy Grain size PP DD = 2kk dd γγ HHHH DD γγ HHHH DD kk dd Page 9 - Grain interface energy - Mean grain size (diameter) - Scaling factor

Mobility general approach General form MM = MM 0 eeeeee QQ/RRRR MM MM 0 QQ RR TT - Grain boundary mobility - Mobility pre-factor - Activation energy - Gas constant - Temperature Page 10

Mobility general approach General form MM = MM 0 eeeeee QQ/RRRR MM MM 0 QQ RR TT - Grain boundary mobility - Mobility pre-factor - Activation energy - Gas constant - Temperature Page 11

No obstacles DD = MM ff PP DD MM ff = ηη ff ωωdd GGGG VV mm bb 2 RRRR PP DD = 2kk dd γγ HHHH DD ωω DD GGGG VV mm bb RR TT ηη ff, kk dd - Grain boundary width (hard coded - 1 nm) - Diffusion coefficient for grain boundary - Molar volume - Burger s vector - Gas constant - Temperature - Scaling factor Page 12 Turnbull D., Trans. AIME, 191 (1951), pp. 661-665

No obstacles DD = MM ff PP DD MM ff = ηη ff ωωdd GGGG VV mm bb 2 RRRR PP DD = 2kk dd γγ HHHH DD ωω - Grain boundary width (hard coded - 1 nm) DD GGGG - Diffusion coefficient for grain boundary VV mm - Molar volume bb - Burger s vector RR - Gas constant TT - Temperature ηη ff, kk dd - Scaling factor Page 13

No obstacles DD = MM ff PP DD MM ff = ηη ff ωωdd GGGG VV mm bb 2 RRRR PP DD = 2kk dd γγ HHHH DD ωω - Grain boundary width (hard coded - 1 nm) DD GGGG - Diffusion coefficient for grain boundary VV mm - Molar volume bb - Burger s vector RR - Gas constant TT - Temperature ηη ff, kk dd - Scaling factor Page 14

No obstacles DD = MM ff PP DD MM ff = ηη ff ωωdd GGGG VV mm bb 2 RRRR PP DD = 2kk dd γγ HHHH DD DD~ tt Page 15

System with precipitates Grain boundary movement Precipitates on the grain boundary Reduction of grain boundary energy https://i.ytimg.com/vi/8eet_t6llwy/maxresdefault.jpg Grain boundary pinning/slowdown Page 16

System with precipitates Growth retarding pressure dependent on Precipitate phase fraction PP ZZ - Pinning force (Zener force) Precipitate radius ff ii,jj - Phase fraction of class j of precipitate i PP ZZ = kk rrγγ HHHH 2 ii αα ii ff ii ββ ii 1 jj ff ii,jj rr ii,jj rr ii,jj kk rr - Mean radius of class j of precipitate i - Scaling factor αα ii - Pinning factor of precipitate i ββ ii - Pinning exponent of precipitate i Page 17

System with precipitates Growth retarding pressure dependent on Precipitate phase fraction Precipitate radius PP ZZ = kk rrγγ HHHH 2 Page 18 ii αα ii ff ii ββ ii 1 jj ff ii,jj rr ii,jj PP ZZ ff ii,jj rr ii,jj kk rr αα ii ββ ii - Pinning force (Zener force) - Phase fraction of class j of precipitate i - Mean radius of class j of precipitate i - Scaling factor - Pinning factor of precipitate i - Pinning exponent of precipitate i

System with precipitates Growth retarding pressure dependent on Precipitate phase fraction Precipitate radius PP ZZ = kk rrγγ HHHH 2 Page 19 ii αα ii ff ii ββ ii 1 jj ff ii,jj rr ii,jj PP ZZ ff ii,jj rr ii,jj kk rr αα ii ββ ii - Pinning force (Zener force) - Phase fraction of class j of precipitate i - Mean radius of class j of precipitate i - Scaling factor - Pinning factor of precipitate i - Pinning exponent of precipitate i

System with precipitates MM pp MM pppppppp = PP ZZ MM pp + MM PP ff DD 1 PP ZZ PP DD PP ZZ PP DD PP ZZ < PP DD MM pp = ηη pp MM ff = ηη pp ηη ff ωωdd GGGG VV mm bb 2 RRRR DD = MM pppppppp PP DD Page 20 MM pppppppp MM pp ηη pp - Grain boundary mobility for matrix with precipitates - Grain boundary mobility for pinned interface - Scaling factor

System with precipitates MM pppppppp = PP ZZ MM pp + MM PP ff DD MM pp = ηη pp ηη ff ωωdd GGGG VV mm bb 2 RRRR DD = MM pppppppp PP DD Page 21 MM pp 1 PP ZZ PP DD MM pppppppp MM pp ηη pp PP ZZ PP DD PP ZZ < PP DD - Grain boundary mobility for matrix with precipitates - Grain boundary mobility for pinned interface - Scaling factor

System with precipitates MM pp MM pppppppp,0 = PP ZZ MM pp + MM PP ff DD 1 PP ZZ PP DD PP ZZ PP DD PP ZZ < PP DD MM pp = ηη pp ηη ff ωωdd GGGG VV mm bb 2 RRRR MM ff MM pppppppp DD = MM pppppppp PP DD MM pp PPDD PPZZ Page 22 PP ZZ 0 PP DD

System with precipitates DD = MM pppppppp PP DD MM pp MM pppppppp = PP ZZ MM pp + MM PP ff DD 1 PP ZZ PP DD PP ZZ PP DD PP ZZ < PP DD MM pp = ηη pp ηη ff ωωdd GGGG VV mm bb 2 RRRR PP ZZ = kk rrγγ HHHH 2 ii αα ii ff ii ββ ii 1 jj ff ii,jj rr ii,jj Page 23

Case with solute drag Solutes on the grain boundary Reduction of grain boundary energy http://slideplayer.com/slide/227236/ Grain boundary pinning/slowdown Page 24

Case with solute drag DD = MM eeeeee PP DD 1 MM eeeeee = 1 MM pppppppp + 1 MM ssss MM eeeeee MM ssss - Effective grain boundary mobility - Grain boundary mobility with solute drag MM eeeeee = MM ppppppppmm ssss MM pppppppp + MM ssss Page 25

Cahn impurity drag MM eeeeee - Effective grain boundary mobility MM ssss = ii αα ii cc ii 1 MM ssss αα ii - Grain boundary mobility with solute drag - Inverse mobility cc ii - Solute concentration on αα ii = ωω RRRR 2 EE ii DD CCBB VV mm sssssss EE ii RRRR EE ii RRRR EE ii the grain boundary - Grain boundary/solute interaction energy DD CCBB - Cross boundary diffusion coefficient Page 26

Cahn impurity drag MM eeeeee - Effective grain boundary mobility MM ssss,0 = ii αα ii cc ii 1 MM ssss αα ii - Grain boundary mobility with solute drag - Inverse mobility cc ii - Solute concentration on the grain boundary EE ii - Grain boundary/solute interaction energy αα ii = ωω RRRR 2 EE ii DD CCBB VV mm sssssss EE ii RRRR EE ii RRRR DD CCBB - Cross boundary diffusion coefficient (~tracer diff.) Page 27

Buken-Kozeschnik critical pressure Continuous mobility change dependent on the driving pressure. MM pppppppp MM ff MM ssss PP cccccccc PP cccccccc + PP DD Page 28 PP cccccccc

Buken-Kozeschnik critical pressure Continuous mobility change dependent on the driving pressure. MM pppppppp MM ff MM ssss PP cccccccc PP cccccccc + PP DD Page 29 PP cccccccc

Buken-Kozeschnik critical pressure Continuous mobility change dependent on the driving pressure. MM pppppppp MM ff MM ssss PP cccccccc PP cccccccc + PP DD Page 30 PP cccccccc

Case with solute drag DD = MM eeeeee PP DD Buken H., Kozeschnik E., Metall. Mater. Trans. A, (2016), DOI: 10.1007/s11661-016-3524-5 1 = 1 + 1 MM eeeeee MM pppppppp MM ssss MM eeeeee = MM ppppppppmm ssss MM pppppppp + MM ssss MM eeeeee MM ssss - Effective grain boundary mobility - Grain boundary mobility with solute drag Page 31

Acknowledgments Heinrich Buken Yao Shan Page 32