Grain Boundary Properties: Energy, Mobility

G.B. properties1Grain Boundary Properties:Energy, Mobility27-765, Spring 2001A.D. RollettG.B. properties2Why learn about grain boundary properties?Many aspects of materials behavior and performance affected by g.b. properties.Examples include:- stress corrosion cracking in Pb battery electrodes, Ni-alloy nuclear fuel containment, steam generator tubes- creep strength in high temp. alloys- weld cracking (under investigation)- electromigration resistance (interconnects)G.B. properties3Properties, phenomena of interest1. Energy (excess free energy wetting, precipitation)2. Mobility (normal motion grain growth, recrystallization)3. Sliding (tangential motion creep)4. Cracking resistance (intergranular fracture)5. Segregation of impurities (embrittlement, formation of second phases)G.B. properties41. Grain Boundary EnergyFirst categorization of boundary type is into low-angle versus high-angle boundaries. Typical value in cubic materials is 15 for the misorientation angle.Read-Shockley model describes the energy variation with angle successfully in many experimental cases, based on a dislocation structure.G.B. properties5LAGB to HAGB TransitionLAGB: steep risewith angle.HAGB: plateau

Dislocation StructureDisordered StructureG.B. properties61.1 Read-Shockley modelStart with a symmetric tilt boundary composed of a wall of infinitely straight, parallel edge dislocations (e.g. based on a 100, 111 or 110 rotation axis with the planes symmetrically disposed).Dislocation density (L-1) given by:

1/D = 2sin(q/2)/b q/b for small angles.G.B. properties71.1 Tilt boundary

bDG.B. properties81.1 Read-Shockley contd.For an infinite array of edge dislocations the long-range stress field depends on the spacing. Therefore given the dislocation density and the core energy of the dislocations, the energy of the wall (boundary) is estimated (r0 sets the core energy of the dislocation): ggb = E0 q(A0 - lnq), whereE0 = b/4(1-n); A0 = 1 + ln(b/2r0)G.B. properties91.1 LAGB experimental resultsExperimental results on copper.

[Gjostein & Rhines, Acta metall. 7, 319 (1959)]G.B. properties101.1 Read-Shockley contd.If the non-linear form for the dislocation spacing is used, we obtain a sine-law variation (Ucore= core energy):

ggb = sin|q| {Ucore/b - b2/4(1-n) ln(sin|q|)}

Note: this form of energy variation may also be applied to CSL-vicinal boundaries. G.B. properties11 vs.

[001][101][111]0.300.260.230.33Low Angle Grain Boundary Energy A. Otsuki, Ph.D.thesis, Kyoto University, Japan (1990)

HighLow[335][323][727][203][205][105][215][117][113][8411]Yang, C.-C., A. D. Rollett, et al. (2001). Measuring relative grain boundary energies and mobilities in an aluminum foil from triple junction geometry. Scripta Materiala: in press.G.B. properties121.2 Energy of High Angle BoundariesNo universal theory exists to describe the energy of HAGBs.Abundant experimental evidence for special boundaries at (a small number) of certain orientations.Each special point (in misorientation space) expected to have a cusp in energy, similar to zero-boundary case but with non-zero energy at the bottom of the cusp.G.B. properties131.2 Exptl. Observations

Hasson, G. C. and C. Goux (1971). Interfacial energies of tilt boundaries in aluminum. Experimental and theoretical determination. Scripta metallurgica 5: 889-894

Tilts

TiltsTwinG.B. properties14Dislocation models of HAGBsBoundaries near CSL points expected to exhibit dislocation networks, which is observed.

twistsHowe, J. M. (1997). Interfaces in Materials. New York, Wiley Interscience.G.B. properties151.2 Atomistic modelingExtensive 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, 1992; also book by Sutton & Balluffi.Grain boundaries in fcc metals: Cu, AuG.B. properties16Atomistic models: resultsResults of atomistic modeling confirm the importance of the more symmetric boundaries.

G.B. properties17Coordination NumberReasonable 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.

G.B. properties18Experimental Impact of EnergyWetting 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, M., A. D. Rollett, et al. (1999). Correlation of grain boundary character with wetting behavior. ICOTOM-12, Montral, Canada, NRC Research Press, p.1647.G.B. properties19G.B. Energy: Metals: SummaryFor low angle boundaries, use the Read-Shockley model: well established both experimentally and theoretically.For high angle boundaries, use a constant value unless near a CSL structure with high fraction of coincident sites and plane suitable for good atomic fit.G.B. properties20

High Angle BoundariesTransfer of vacancies between two adjacent sets of dislocations by grain boundary diffusion mechanism Low Angle BoundariesTransfer of atoms from the shrinking grain to the growing grain by atomic bulk diffusion mechanism LA->HAGB TransitionG.B. properties212.1 Low Angle G.B. MobilityMobility of low angle boundaries dominated by climb of the dislocations making up the boundary.Even in a symmetrical tilt boundary the dislocations must move non-conservatively in order to maintain the correct spacing as the boundary moves.G.B. properties22Tilt Boundary Motionhboundary displacementdx(Bauer and Lanxner, Proc. JIMIS-4 (1986) 411)Burgers vectors inclined with respect to the boundary plane in proportion to the misorientation angle.glideclimbG.B. properties23Low Angle GB MobilityHuang 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*=kqc, with c~5.2 and k=3.10-6 m4(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.G.B. properties24M vs.

[001][101][111][117][113][335][105][205][203][215][8411][727][323]0.30.10.00040.9Relative Mobility0.030.01

LAGB Mobility in Al, experimentalHighLowG.B. properties25LAGB: Axis DependenceWe 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.G.B. properties26Symmetrical 12.4o grain boundary=> dislocations are nearly parallel to the boundary normal=> = /2

Symmetrical 11.4o grain boundary=> nearly 45o alignment of dislocations with respect to the boundary normal=> = 45o +/2

G.B. properties272.1 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).G.B. properties28Dislocation Modelsfor Low Angle G.B.s

Sutton and Balluffi (1995). Interfaces in Crystalline Materials. Clarendon Press, Oxford, UK. G.B. properties29Theory: DiffusionAtom flux, J, between the dislocations is:

where DL is the atom diffusivity (vacancy mechanism) in the lattice;m is the chemical potential;kT is the thermal energy;and W is an atomic volume.

G.B. properties30Driving ForceA stress t 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.

G.B. properties31Atom FluxThe 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 Ad/2. The total current of atoms between the two adjacent dislocations (per length of boundary) I is [SB].

G.B. properties32Dislocation VelocityAssuming 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.

G.B. properties33Mobility (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/2q). 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, 2209-2232.

G.B. properties34Theory: Addition of a Pipe Diffusion ModelConsider 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:

G.B. properties35Lattice+Pipe DiffusionThe total current of atoms from one dislocation parallel to the tilt axis to the next (per length of boundary) is

where d is the radius of the fast diffusion pipe at the dislocation core and d1 and d2 are the spacing between the dislocations that run parallel and perpendicular to the tilt axis, respectively.

G.B. properties36Boundary VelocityThe boundary velocity is related to the diffusional current as above but with contributions from both lattice and pipe diffusion:

G.B. properties37Mobility (Lattice and Pipe Diffusion)The mobility M=v/(tq) is now simply:

This expression suggests that the mobility increases as the spacing between dislocations perpendicular to the tilt axis decreases.

G.B. properties38Effect of twist angleIf the density of dislocations running perpendicular to the tilt axis is associated with a twist component, then:

where f 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.

G.B. properties39Effect of MisorientationIf the density of the perpendicular dislocations is proportional to the density of parallel ones, then the mobility is:

where a is a proportionality factor. Note the combination of mobility increasing and decreasing with misorientation.

G.B. properties40Results: Ni MobilityNickel: QL=2.86 eV, Q=0.6QL, D0L=D0=10-4 m2/s, b=3x10-10 m, W=b3, d=b, a=1, k=8.6171x10-5 eV/K.

T (K)q ()M(10-10 m4/[J s])G.B. properties41Theory: Reduced MobilityProduct of the two quantities M*=Mg 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:

G.B. properties42Results: Ni Reduced Mobility g0=1 J/m2 and q*=25, corresponding to a maximum in the boundary mobility at 9.2.

log10M* (10-11m2/s)q ()T (K)G.B. properties43Results: AluminumMobility vs. T and q

The vertical axis is Log10 M. g0 = 324 mJ/m2, q*= 15, DL(T) 1.76.10-5 exp-{126153 J/mol/RT} m2/s, D(T) 2.8.10-6 exp-{81855 J/mol/RT} m2/s, d=b, b = 0.286 nm, W = 16.5.10-30 m3 = b3/2, a = 1. log10M(m4/s MPa)q ()T (K)G.B. properties44Comparison with Expt.: Mobility vs. Angle at 873K

M. Winning, G. Gottstein & L.S. Shvindlerman, Grain Boundary Dynamics under the Influence of MechanicalStresses, Ris-21 Recrystallization, p.645, 2000.q ()Log10M(m4/s MPa)0

-1

-2

-3

-4

-5Log10M(m4/s MPa)G.B. properties45Comparison with Expt.: Mobility vs. Angle at 473K

q ()Log10M(m4/s MPa)4

3

2

1Log10M(m4/s MPa)G.B. properties46Discussion on LAGB mobilityThe 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.G.B. properties472.1 LAGB mobility; conclusionAgreement between calculated (reduced) mobility and experimental results is remarkably good. Only one (structure sensitive) adjustable parameter (a = 1), 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.G.B. properties48LAGB to HAGB Transitions

Read-Shockley forenergy of low angleboundaries Exponentialfunction for transitionfrom low- to high-angle boundariesG.B. properties49High Angle GB MobilityLarge variations known in HAGB mobility.Classic example is the high mobility of boundaries close to 40 (which is near the S7 CSL type).Note broad maximum.

Gottstein & Shvindlerman: grain boundary migration in metalsG.B. properties50HAGB: Impurity effectsImpurities known to affect g.b. mobility strongly, depending on segregation and mobility.CSL structures with good atomic fit less affected by solutesExample: Pb bicrystals

specialgeneralRutter, J. W. and K. T. Aust (1960). Kinetics of grain boundary migration in high-purity lead containing very small additions of silver and of gold. Transactions of the Metallurgical Society of AIME 218: 682-688.G.B. properties51HAGB mobility: theoryThe standard theory for HAGB mobility is due to Burke & Turnbull, based on thermally activated atomic transfer across the interface.For the low driving forces typical in grian growth, recrystallization etc., it gives a linear relation between force and velocity (as typically assumed).Burke, J. and D. Turnbull (1952). Progress in Metal Physics 3: 220.graduateG.B. properties52Burke-TurnbullGiven a difference in free energy (per unit volume) for an atom attached to one side of the boundary versus the other, P, the rate at which the boundary moves is:

Given similar attack frequencies and activation energies in both directions,graduateG.B. properties53Velocity Linear in Driving ForceThen, for small driving forces compared to the activation energy for migration, Pb3kT, which allows us to linearize the exponential term.

MobilitygraduateG.B. properties54HAGB MobilityThe basic Burke-Turnbull theory ignores details of g.b. structure: The terrace-ledge-kink model may be useful; the density of sites for detachment and attachment of atoms can modify the pre-factor.Atomistic modeling is starting to play a role: see work by Upmanyu & Srolovitz [M. Upmanyu, D. Srolovitz and R. Smith, Int. Sci., 6, (1998) 41.].Much room for research!graduateG.B. properties55HAGB Mobility: the U-bicrystalThe 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 forceSimulation and experimentxyvVwDunn, Shvindlerman, Gottstein,...G.B. properties56HAGB M: Boundary velocity

Steady-state migration + initial and final transientsSimulationExperimentG.B. properties57HAGB M: simulation results

Grain Boundary Energy gMisorientation q

Misorientation qMobility MS7S13S19Extract boundary energy from total energy vs. half-loop height (assume constant entropy)M=M*/gG.B. properties58HAGB M: Activation energy

simulationexperimentLattice diffusion between dislocationsspecialboundaryS19S7S13S7Q (e)Q (eV)G.B. properties59HAGB M: Issues; dirtSolutes play a major role in g.b. mobility by reducing absolute mobilities at very low levels.Simulations typically have no impurities included: therefore they model ultra-pure material.G.B. properties60HAGB M: impurity effect on recrystallization

R. Vandermeer and P. Gordon, Proc. Symposium on the Recovery and Recrystallization of Metals, New York, TMS AIME, (1962) p. 211.F. R. Boutin, J. Physique, C4, (1975) C4.355.V (cm.s-1)1/T

decreasing Fe contentincreasing Cu contentG.B. properties61GB Mobility: SummaryThe properties of low angle grain boundaries are dictated by their discrete dislocation structure: energy logarithmic with angle; mobility exponential with angle.The kinetic properties of high angle boundaries are (approx.) plateau dictated by local atomic transfer. Special boundary types have low energy and high/low mobility.

Energy (mJ/m2),T=240oC[001][101][111]

Tilt190170148

Twist200205155

Misorientation Axis [uvw] ;

( = 5o

Chart10.1650.01559914530.01559914530.09549818310.16050.01159022580.01159022580.17107061230.216250.00377491720.00377491720.20020974070.2290.01174734010.01174734010.225437790.260250.00899536920.00899536920.24747479410.272250.01184271930.01184271930.26681598240.25850.01907004630.01907004630.28382312040.3130.01235583530.01235583530.29877216680.278750.00623832240.00623832240.31188061660.3140.01143095210.01143095210.3233243260.333750.01053169820.01053169820.33324843940.34450.01461734130.01461734130.34177479620.01096585610.01096585610.3521664546

Experimental ResultRead-Shockley EquationMisorientation Angle (degrees)Relative Boundary Energy

Sheet1EnergyFoilMobilityMobilityAngleAngles1s2s3s4Mean2 Stan.Dev.Misorientation Angles1s2s3s4MeanStan.Dev.Read-Shoc.11.51.50.01530.00290.0360.020.018550.01368807271.50.1570.1540.1610.1880.1650.01559914530.095498183123.53.50.01590.00003570.010.110.0339839250.05109854093.50.1580.1760.160.1480.16050.01159022580.171070612334.54.50.003570.017770.240.009680.0677550.11497718634.50.2150.2120.2170.2210.216250.00377491720.200209740745.55.50.0690.0005340.010.03850.02950850.03087895115.50.2310.2250.2440.2160.2290.01174734010.2254377956.56.50.01350.003680.020.020570.01443750.00785603116.50.2610.2710.2490.260.260250.00899536920.247474794167.57.50.0640.0008230.02330.0008730.0222490.02977838967.50.2670.2580.2820.2820.272250.01184271930.266815982478.58.50.00180.0010570.00860.00590.004339250.0035501598.50.2540.2380.2840.2580.25850.01907004630.283823120489.59.50.009650.0028390.00980.160.045572250.07635422399.50.3050.3010.3180.3280.3130.01235583530.2987721668910.510.50.0230.0260.0110.19680.06420.088637238210.50.280.2870.2730.2750.278750.00623832240.31188061661011.511.50.0560.0440.260.1970.139250.106330851611.50.320.3240.3140.2980.3140.01143095210.3233243261112.512.50.2340.0820.380.1970.223250.122918333312.50.3360.3220.330.3470.333750.01053169820.33324843941213.513.50.380.10640.10440.2640.21370.133723047113.50.3510.3620.3340.3310.34450.01461734130.34177479621325250.8650.9890.8840.88670.9061750.0560546385150.350.3740.3530.3640.360250.01096585610.352166454616

Sheet10000000000000000000000000000000000000000000000000000000000000000000000000

AllHalf1Halh2Half3Half4Read-ShockleyMisOrientation AngleRelative Energy

Sheet200.01559914530.0155991453000.01159022580.0115902258000.00377491720.0037749172000.01174734010.0117473401000.00899536920.0089953692000.01184271930.0118427193000.01907004630.0190700463000.01235583530.0123558353000.00623832240.0062383224000.01143095210.0114309521000.01053169820.0105316982000.01461734130.014617341300.01096585610.01096585610

Experimental DataRead-Shockley EquationMisorientation Angle (degrees)Relative Boundary Energy

Sheet3000000000000000000000000000000000000000000000000000000000000

Misorientation AngleRelative Mobility

00.01368807270.013688072700.05109854090.051098540900.11497718630.114977186300.03087895110.030878951100.00785603110.007856031100.02977838960.029778389600.0035501590.00355015900.07635422390.076354223900.08863723820.088637238200.10633085160.106330851600.12291833330.122918333300.13372304710.1337230471

MeanMisorientation Angle (degrees)Relative MobilityMean Relative Mobility + - 2 Syandard Deviation

Chart10.032390.0240.010.003550.0180.00080.0372870.0380.02390.0290.0050.0290.1460.0850.0057280.0050.0080.009340.0110.0060.01070.0090.350.050330.0270.160.2546980.1440.170.2824480.1390.0420.4040450.2110.211

Misorientation Angle (degrees)Relative Boundary Mobility

Sheet1Energy Data from : ENERGY_2.nbMobility Data from: MOBILITY_1.nbRaw Data from : LB_3_m.xls11.50.032390.03220.0240.0111.50.150.09667628040.0430.052240.003550.00650.0180.000823.50.1560.17381950590.0910.041350.0372870.03540.03834.50.20.20374403250.0490.066460.02390.03470.0290.00545.50.220.229757480.0490.061-3.64276270161.50.02617991670.0952363244570.0290.0280.1460.08556.50.250.25257988220.0640.0473.50.06108647220.170459653680.0057280.00790.0050.00867.50.260.27270646870.0530.0254.50.078539750.1994242386790.009340.00860.0110.00678.50.250.29049900490.0490.0785.50.09599302780.22447774858100.01070.01080.0090.3589.50.30.30623344950.0420.1156.50.11344630560.24634021589110.050330.08680.0270.16910.50.310.32012729740.0630.1227.50.13089958330.265506869810120.2546980.2440.1440.171011.50.320.3323564050.0690.0558.50.14835286110.282339475411130.2824480.27560.1390.0421112.50.3390.34306591660.0550.0249.50.16580613890.297113991112140.4040450.3880.2110.2111213.50.3490.35237767150.0850.10210.50.18325941670.310047911913300.82790.83760.719150.363947427211.50.20071269440.3213170937130.370.0540.052112.50.21816597220.331066680813.50.235619250.339418512414.50.25307252780.3464763238150.26179916670.3495483853A= -0.005~0.05

Sheet100.0240.0100.0180.000800.03800.0290.00500.1460.08500.0050.00800.0110.00600.0090.3500.0270.1600.1440.1700.1390.04200.440.211

Misorientation Angle (degrees)Relative Boundary Mobility

Sheet2000.0430.052000.0910.041000.0490.066000.0490.061000.0640.047000.0530.025000.0490.078000.0420.115000.0630.122000.0690.055000.0550.024000.0850.10200

Experiment DataRead-Shockly EquationMisorientation Angle (degrees)Relative Boundary Energy

Sheet3