63 Supplemental Slides In the slides that follow, various details are given that it was not possible to cover in the lecture

64 Experimental Impact of Energy Wetting by liquids is sensitive to grain boundary energy. Example: copper wets boundaries in iron at high temperatures. Wet versus unwetted condition found to be sensitive to grain boundary energy in Fe+Cu system: Takashima et al., ICOTOM-12 (1999).

65 Wetting comparison High energy (light) boundaries should be wet (“W”). Low energy boundaries (dark) should be dry (“U”). Example of Cu wetting boundaries in Fe with (311) on one side. Takashima, M., P. Wynblatt, and B.L. Adams, Correlation of grain boundary character with wetting behavior. Interface Science, : p ]

66 Low-angle g.b. properties Recently, the properties of low angle grain boundaries have been measured by the MIMP at CMU. The results confirm the Read-Shockley relationship. A variation of energy with misorientation axis was also found: boundaries with misorientation axes had the lowest energies whereas those with axes had the highest. The variation was only over a range of +/- 10%, however.

67  vs. Low Angle Grain Boundary Energy, Yang et al. High Low [335] [323] [727] [203] [205] [105] [215] [117] [113] [8411] [001] [101] [111] "Measuring relative grain boundary energies and mobilities in an aluminum foil from triple junction geometry", C.-C. Yang, W. W. Mullins and A. D. Rollett, Scripta Materiala 44: (2001).

68 Dislocation models of HAGBs Boundaries near CSL points expected to exhibit dislocation networks, which is observed. twists

69 Atomistic modeling Extensive atomistic modeling has been conducted using (mostly) embedded atom potentials and an energy-relaxation method to locate the minimum energy configuration of a (finite) bicrystal. See Wolf & Yip, Materials Interfaces: Atomic-Level Structure & Properties, Chapman & Hall, Grain boundaries in fcc metals: Cu, Au

70 Atomistic models: results Results of atomistic modeling confirm the importance of the more symmetric boundaries. Example of symmetric tilt boundary energy for embedded-atom-method calculations using either Lennard-Jones, copper or gold interatomic potentials. g.b. plane Wolf & Yip

71 Coordination Number Reasonable correlation for energy versus the coordination number for atoms at the boundary: suggests that broken bond model may be applicable, as it is for solid/vapor surfaces. Wolf & Yip

72 Low Angle GB Mobility Huang and Humphreys (2000): coarsening kinetics of subgrain structures in deformed Al single crystals. Dependence of the mobility on misorientation was fitted with a power-law relationship, M*=k  c, with c~5.2 and k= m 4 (Js) -1. Yang, et al.: mobility (and energy) of LAGBs in aluminum: strong dependence of mobility on misorientation; boundaries based on [001] rotation axes had much lower mobilities than either [110] or [111] axes.

73 M vs. [001] [101] [111] [117] [113] [335] [105] [205] [203] [215] [8411] [727] [323] Relative Mobility LAGB Mobility in Al, experimental High Low

74 LAGB: Axis Dependence We can explain the (strong) variation in LAGB mobility from axes to axes, based on the simple tilt model: tilt boundaries have dislocations with Burgers vectors nearly perp. to the plane. boundaries, however, have Burgers vectors near 45° to the plane. Therefore latter require more climb for a given displacement of the boundary.

75 Symmetrical 12.4 o grain boundary=> dislocations are nearly parallel to the boundary normal =>  =  /2 Symmetrical 11.4 o grain boundary=> nearly 45 o alignment of dislocations with respect to the boundary normal =>  = 45 o +  /2

Low Angle GB Mobility, contd. Winning et al. measured mobilities of low angle grain and tilt boundaries under a shear stress driving force. A sharp transition in activation enthalpy from high to low with increasing misorientation (at ~ 13°).

77 Dislocation Models for Low Angle G.B.s Sutton and Balluffi (1995). Interfaces in Crystalline Materials. Clarendon Press, Oxford, UK.

78 Theory: Diffusion Atom flux, J, between the dislocations is: where D L is the atom diffusivity (vacancy mechanism) in the lattice;  is the chemical potential; kT is the thermal energy; and  is an atomic volume.

79 Driving Force A stress  that tends to move dislocations with Burgers vectors perpendicular to the boundary plane, produces a chemical potential gradient between adjacent dislocations associated with the non- perpendicular component of the Burgers vector: where d is the distance between dislocations in the tilt boundary.

80 Atom Flux The atom flux between the dislocations (per length of boundary in direction parallel to the tilt axis) passes through some area of the matrix between the dislocations which is very roughly A≈d/2. The total current of atoms between the two adjacent dislocations (per length of boundary) I is [SB].

81 Dislocation Velocity Assuming that the rate of boundary migration is controlled by how fast the dislocations climb, the boundary velocity can be written as the current of atoms to the dislocations (per length of boundary in the direction parallel to the tilt axis) times the distance advanced per dislocation for each atom that arrives times the unit length of the boundary.

82 Mobility (Lattice Diffusion only) The driving force or pressure on the boundary is the product of the Peach-Koehler force on each dislocation times the number of dislocations per unit length, (since d=b/√2  ). Hence, the boundary mobility is [SB]: See also: Furu and Nes (1995), Subgrain growth in heavily deformed aluminium - experimental investigation and modelling treatment. Acta metall. mater., 43,

83 Theory: Addition of a Pipe Diffusion Model Consider a grain boundary containing two arrays of dislocations, one parallel to the tilt axis and one perpendicular to it. Dislocations parallel to the tilt axis must undergo diffusional climb, while the orthogonal set of dislocations requires no climb. The flux along the dislocation lines is:

84 Lattice+Pipe Diffusion The total current of atoms from one dislocation parallel to the tilt axis to the next (per length of boundary) is where  is the radius of the fast diffusion pipe at the dislocation core and d 1 and d 2 are the spacing between the dislocations that run parallel and perpendicular to the tilt axis, respectively.

85 Boundary Velocity The boundary velocity is related to the diffusional current as above but with contributions from both lattice and pipe diffusion:

86 Mobility (Lattice and Pipe Diffusion) The mobility M=v/(  ) is now simply: This expression suggests that the mobility increases as the spacing between dislocations perpendicular to the tilt axis decreases.

87 Effect of twist angle If the density of dislocations running perpendicular to the tilt axis is associated with a twist component, then: where  is the twist misorientation. On the other hand, a network of dislocations with line directions running both parallel and perpendicular to the tilt axis may be present even in a pure tilt boundary assuming that dislocation reactions occur.

88 Effect of Misorientation If the density of the perpendicular dislocations is proportional to the density of parallel ones, then the mobility is: where  is a proportionality factor. Note the combination of mobility increasing and decreasing with misorientation.

89 Results: Ni Mobility Nickel: Q L =2.86 eV, Q  =0.6Q L, D 0L =D 0  =10 -4 m 2 /s, b=3x m,  =b 3,  =b,  =1, k=8.6171x10 -5 eV/K. T (˚K)  (˚) M ( m 4 /[J s])

90 Theory: Reduced Mobility Product of the two quantities M*=M  that is typically determined when g.b. energy not measured. Using the Read-Shockley expression for the grain boundary energy, we can write the reduced mobility as:

91 Results: Ni Reduced Mobility  0 =1 J/m 2 and  *=25˚, corresponding to a maximum in the boundary mobility at 9.2˚. log 10 M* ( m 2 /s)  (˚) T (˚K)

92 Results: Aluminum Mobility vs. T and  The vertical axis is Log 10 M.  0 = 324 mJ/m 2,  *= 15°, D L (T) exp-{ J/mol/RT} m 2 /s, D  (T) exp-{81855 J/mol/RT} m 2 /s,  b, b = nm,    m   b 3 /√2,  = 1. log 10 M (µm 4 /s MPa)  (˚) T (K)

93 Comparison with Expt.: Mobility vs. Angle at 873K M. Winning, G. Gottstein & L.S. Shvindlerman, Grain Boundary Dynamics under the Influence of Mechanical Stresses, Risø-21 “Recrystallization”, p.645,  (˚) Log 10 M (µm 4 /s MPa) Log 10 M (µm 4 /s MPa)

94 Comparison with Expt.: Mobility vs. Angle at 473K  (˚) Log 10 M (µm 4 /s MPa) Log 10 M (µm 4 /s MPa)

95 Discussion on LAGB mobility The experimental data shows high and low angle plateaus: the theoretical results are much more continuous. The low T minimum is quite sharp compared with experiment. Simple assumptions about the boundary structure do not capture the real situation.

LAGB mobility; conclusion Agreement between calculated (reduced) mobility and experimental results is remarkably good. Only one (structure sensitive) adjustable parameter (  ), which determines the position of the minimum. Better models of g.b. structure will permit prediction of low angle g.b. mobilities for all crystallographic types.

97 HAGB Mobility: the U-bicrystal The curvature of the end of the interior grain is constant (unless anisotropy causes a change in shape) and the curvature on the sides is zero. Migration of the boundary does not change the driving force Simulation and experiment x y v V w Dunn, Shvindlerman, Gottstein,...

98 HAGB M: Boundary velocity Steady-state migration + initial and final transients SimulationExperiment

99 HAGB M: 2D simulation results Grain Boundary Energy  Misorientation  Mobility M  Extract boundary energy from total energy vs. half- loop height (assume constant entropy) M=M * /  Note: misorientation angle shown in plots is 1/2 of total angle.

100 HAGB M: Activation energy simulation experiment Lattice diffusion between dislocations special boundary   Q (  ) Q (eV) Simulations exhibit much smaller activation energies than experiments, possibly because solutes affect experimental results.

101 3D simulations: reduced mobility (M*) vs. Misorientation  13 77 (m 4 /Js) (deg) Zhang, Upmanyu, Srolovitz

102  13 77 Mobility and Energy vs. Misorientation 77  13 (J/m 2 ) (m 4 /Js) (deg) Zhang, Upmanyu, Srolovitz

103 Mobility vs. Misorientation  13 77 (m 4 /Js) (deg) Zhang, Upmanyu, Srolovitz

104 Reduced Mobility, M* In many experiments on g.b. mobility, only the migration rate can be measured and the boundary curvature. If the energy of the boundary is not known (or must be assumed to be constant) then one can only derive the reduced mobility, M*. Where M and E are the mobility and energy defined in the standard way, M* = M E.

105 High Angle GB Mobility Large variations known in HAGB mobility. Classic example is the high mobility of boundaries close to 40° (which is near the  7 CSL type). Note broad maximum. Gottstein & Shvindlerman: grain boundary migration in metals

106 Mobility of HAGBs with stored energy driving force Huang & Humphreys, The effect of solute elements on grain boundary mobility during recrystallization of single-phase aluminum alloys, Proc. Conf. Rex & Gr.Gr., Aachen, vol (2001). As previously observed, broad peak in mobility observed centered on 40° misorientation with 10° FWHM w.r.t. misorientation angle. Similar decrease with deviation from axis.

107 Simulation Results: Misorientation Axis Dependence tilt misorientations = fastest moving boundaries Dramatic decrease in mobility with deviation in tilt axis T = 0.7T m 001 (45 o ) 110 (54.7 o ) 112 (56.9 o ) 023 (50.7 o ) 111 (54.7 o ) 144 (30.0 o )122 (55.3 o ) Unpublished work by Upmanyu