Grain Boundary Properties: Energy (L21) - Materials Science and ...

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8 Grain Boundary Energy: Definition • Grain boundary energy is defined as the excess free energy associated with the presence of a grain boundary, with the perfect lattice as the reference point. • A thought experiment provides a means of quantifying GB energy, γ. Take a patch of boundary with area A, and increase its area by dA. The grain boundary energy is the proportionality constant between the increment in total system energy and the increment in area. This we write: γ = dG/dA • The physical reason for the existence of a (positive) GB energy is misfit between atoms across the boundary. The deviation of atom positions from the perfect lattice leads to a higher energy state. Wolf established that GB energy is correlated with excess volume in an interface. There is no simple method, however, for predicting the excess volume.
9 Measurement of GB Energy • We need to be able to measure grain boundary energy. • In general, we do not need to know the absolute value of the energy but only how it varies with boundary type, i.e. with the crystallographic nature of the boundary. • For measurement of the anisotropy of the energy, then, we rely on local equilibrium at junctions between boundaries. This can be thought of as a force balance at the junctions. • For not too extreme anisotropies, the junctions always occur as triple lines.
Page 1 and 2: 1 Carnegie Mellon MRSEC Grain Bound
Page 3 and 4: 3 Outline • Motivation, examples
Page 5 and 6: 5 Properties, phenomena of interest
Page 7: 7 • Grain boundary sliding should
Page 11 and 12: 11 Zero-creep Method • The zero-c
Page 13 and 14: 13 Definition of Dihedral Angle •
Page 15 and 16: 15 Unequal energies • If the inte
Page 17 and 18: 17 Wetting • For a large enough r
Page 19 and 20: 19 Triple Junction Quantities • G
Page 21 and 22: 21 Derivation of Herring Equs. A vi
Page 23 and 24: 23 γ S2 Analysis of Thermal Groove
Page 25 and 26: Bi impurities in Ni have the opposi
Page 27 and 28: 27 Grain boundary energy: current s
Page 29 and 30: 29 Read-Shockley model • Start wi
Page 31 and 32: 31 Read-Shockley contd. • For an
Page 33 and 34: 33 Read-Shockley contd. • If the
Page 35 and 36: 35 Exptl. vs. Computed E gb Tilts
Page 37 and 38: 37 2 i i+2 r ij1 l’ ij j 1 j 3 r
Page 39 and 40: 39 Tilt versus Twist Boundaries Iso
Page 41 and 42: 41 λ(Δg, n) λ(n|5°/[100]) λ(n|
Page 43 and 44: 43 For all grain boundaries in MgO
Page 45 and 46: 45 Symmetric [110] tilt boundaries
Page 47 and 48: 47 Inclination Dependence • Inter
Page 49 and 50: 49 The torque term Change in inclin
Page 51 and 52: 51 Herring’s Relations
Page 53 and 54: 53 Aluminum foil, cross section •
Page 55 and 56: 55 How to Measure Dihedral Angles a
Page 57 and 58: 57 Calculation of G.B. Energy • I
Page 60 and 61:
60 3 ⎪⎪⎧⎧ ∑ ⎨⎨σ ˆ j
Page 62 and 63:
62 Grain Boundary Energy Calculatio
Page 64 and 65:
64 # of total TJs : 8694 # of {111}
Page 66 and 67:
66 Capillarity Vector • The capil
Page 68 and 69:
68 Capillarity vector definition
Page 71 and 72:
71 Computer Simulation of Grain Gro
Page 73 and 74:
73 • Steady state population deve
Page 75 and 76:
75 G.B. Energy: Summary • For low
Page 77 and 78:
77 Supplemental Slides
Page 79:
79 Expanded Young Equations • Pro
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