Moveent of Dislocations Glissile and prisatic dislocation loops Plastic strain through oveent of dislocations Glide and clib Lattice resistance to glide: Peierls stress Kinks and jogs Moveent of dislocations with kinks and jogs Generation of dislocations Frank-Read echanis, cross-slip References: Hull and Bacon, Chs. 3.4-3.9, 10 Allen and Thoas, Ch. 5, pp. 283-294
Review of soe of the dislocation properties Dislocation lines can not end within a crystal they originate on surfaces, grain boundaries, dislocation nodes, etc., or for dislocation loops Dislocation oveent: Edge dislocation oves to shear direction Screw dislocation oves to shear direction Exaple: expansion of a glissile dislocation loop y τ R dr 2 line tension: W 2 Gb T = disl αgb L 2 τ x balance of energy due to the increase of the dislocation line by ΔL and work done by the external stress to increase the slip area by ΔS: τbδs = TΔL ΔL = 2π( R + dr) 2πR = 2πdR τb2πrdr T 2πdR 2 2 ΔS = π( R + dr) πr 2πRdR τ c T br Gb 2R for a dislocation loop of radius R to expand, the external stress should exceed the critical value τ c critical stress decreases as the radius of the loop increases
Glissile and prisatic dislocation loops glissile dislocation loop (b is within the plane of the loop) prisatic loop can be fored by condensation of vacancies in a aterial with high supersaturation of vacancies (e.g., due to rapid quenching or irradiation by energetic particles) b is noral to the plane of the loop of edge dislocation is fored. prisatic loop (b is not within the plane of the loop) at high T and/or presence of vacancy sinks the prisatic loops of vacancy type will shrink at vacancy supersaturation (c > c 0 ) the loops can grows clib defined by balance of F r F r clib and che
Prisatic dislocation loops prisatic loops can be fored fro clusters of vacancies or interstitials t = 0 in t = 213 in disappearance of prisatic dislocation loops fro a thin sheet of Al heated to 102 ºC in TEM t = 793 in t = 1301 in these are partial dislocations and forces due to the stacking faults are contributing (together with the line tension forces) to the clib forces acting on the dislocations Tartour & Washburn, Phil. Mag. 18, 1257, 1968
Plastic strain through oveent of dislocations Plastic deforation is directly related to the otion of obile dislocations dislocations glide under stress as shown by arrows when a dislocation oves distance d across the slip plane, it contributes b to the total displaceent D if N dislocations cross the crystal, plastic deforation would be ε = D / h = Nb / h a dislocation can also oves a distance x i < d and stop due to an obstacle, resulting in incoplete N slip - a contribution to the total displaceent is a fraction x i /d of b. Thus, in general, b ε = D h = b hd N i =1 x i = bρ x where x ρ = 1 N N i = 1 Nl = / x i hld D = d i = 1 - average distance covered by a dislocation - density of obile dislocations x i strain rate: ε & dε / dt = bρ v + bρ& x bρ v where v is the average dislocation velocity
Plastic strain through oveent of dislocations strain rate: ε& = bρ v - this equation is also applicable to screw and ixed dislocations and to the plastic strain due to clib σ xx x i if a dislocation clibs a distance of x i, it contributes bx i /d to the displaceent Δh the total plastic tensile strain is then: h ε ε& = = Δh h = b hd bρ v clib N i=1 x i = bρ x σ xx Δh d ρ order of agnitude estiation of the axiu strain rate at large stresses, ost of the dislocations can be obile: dislocation velocity cannot exceed speed of sound: = 10 14-2 real strain rates, v = 10 3 /s, ε& 10 4 s 1 b = 3 10-10 ρ 10 v ε& = bρ ρ 3000 v = 3 10 /s 7 s 1 14 10 15 2 the length doubles in ~3 10-8 s = 30 ns relatively sall ρ can explain even the fastest plastic deforation
Glide vs. Clib Glide/slip: conservative oveent within the slip plane Clib: non-conservative oveent away fro the slip plane let s consider a dislocation AB that oves to A B in the direction t r r n r r = l t - surface noral b r n r = 0 - conservative otion, glide r b n r n r > 0 - addition of aterial r clib b n r t r < 0 - reoval of aterial l r A A t r l r B B exaple: t r l r r n r r r = l t b n r < 0 - positive clib crystal shrinks in direction parallel to slip plane results fro copressive strain in general, volue change due to clib is r r r r r r ΔV = b l t = b l t
Glide vs. Clib (1) slip: conservative oveent of dislocations perpendicular to b l, i.e., within the slip plane the otion is reversible - if the sign of τ is reversed, the dislocation can ove in the opposite direction and eventually restore the original configuration oveent does not involve point defects an edge or ixed dislocation has only one slip plane defined by b and l for screw dislocation, the nuber of slip planes is defined by its orientation and structure of the crystal (typically 2-4 easy slip planes) otion of a screw dislocation is always conservative (2) clib: non-conservative oveent of dislocations away fro the slip plane the otion cannot be easily reversed - siple change of sign of τ does not reverse the process since additional work has to be done to create point defects oveent is only possible with the help of point defects (vacancies or interstitials) clib is slow (involves diffusion of point defects) and has a strong dependence on T direct contribution of clib to the deforation rate ε& = bρv clib is typically sall (except for ostly screw dislocation with sall edge segents - jogs) clib plays an iportant role in plastic deforation since it enable dislocations circuvent otherwise insurountable obstacles
Lattice resistance to glide: Peierls stress periodicity of lattice translates into the periodic variation of energy as a function of displaceent of the dislocation core along a direction of high syetry - Peierls-Nabarro potential W energy iniu 0 b 2b 3b high energy state WP x energy iniu approxiate evaluation: 2 Gb 2πa W P = exp πk Kb WP 2πx W = W0 cos 2 b Fax 1 dw πw τp = = = b b dx b ax P 2 a τ P = G K exp 2πa Kb K = 1 for screw dislocation K = 1 - ν for edge dislocations b (W and W P are per unit length) assuing a = b, ν = 1/3, K = 2/3 (edge): τ P 1.2 10-4 G << τ 0 G/6 G/30 slip tend to occur in ost widely spaced planes (large a) and for sall b
Lattice resistance to glide: Peierls stress the dependence of τ P on b/a is very strong, e.g. changing b/a fro 1 to 1.5 increases τ P fro 1.2 10-4 G to 2.8 10-3 G, i.e., 23 ties only a sall nuber of slip systes with large a and sall b are norally activated W P and τ P are larger for aterials with angular dependence of interatoic interactions and saller for spherically-syetric long-range interactions: for fcc and hcp τ P 10-6 -10-5 G (dissociation into partials additionally reduce τ P ) for bcc (ixed type of bonding) τ P ~ 10-4 G for covalent aterials τ P ~ 10-2 G edge dislocations tend to be ore obile than screw ones, e.g., a = b, K = 2/3 (edge): τ P 1.2 10-4 G a = b, K = 1 (screw): τ P 1.9 10-3 G Peierls energy landscape defines special low core energy directions in which the dislocation prefers to lie if dislocation is unable to lie in one inia of the Peierls potential, it fors kinks that cross fro one iniu to the next one - these are geoetrically necessary kinks shape/width of a kink is defined by a balance of W disl and W P
Peierls potential and shape of dislocations Low-energy configuration of a dislocation is defined by 3 factors: (1) due to the line energy/tension (W disl ~ L), dislocations tend to be straight (2) due to the lower energy of screw coponents (W disl ~ 1/K), screw segents tend to be longer (3) due to the Peierls energy landscape (W P ), dislocations tend to lie along closely-packed directions for which the core energy is lowest (and τ P is highest) per length of b: W disl ( L = b) 3 Gb 2 b W P = Gb πk 3 exp 2πa Kb 6 10 4 3 Gb (for a = b, K = 1) usually, WP << Wdisl / L Frank-Read source in a Si crystal the dislocation lines tend to lie along <110> directions, where the core has lower energy fro Hull and Bacon what is the direction of the Burgers vector in this iage?
Dislocation otion by kinks Kinks are steps of atoic diension in the dislocation line that are contained in the glide plane of the dislocation τ < τ P The barrier to ove a kink along the line is uch saller (W P2 << W P ) and for etals with predoinantly etallic bonding is negligible lateral otion of kinks can take place at low τ << τ P sall plastic strain (pre-yield icroplasticity) Kinks on a screw dislocation is a short segent with edge character screw dislocation with a kink can slide in only one glide plane (the one that contain the kink). If it glides in a different plane, the kink serves as an anchor point for the screw dislocation. Motion of kinks can be studied in internal friction experients (easureents of energy losses in a vibrating aterial): sall oscillating stresses can generate reversible oveent of kinks - since there is no irreversible deforation, the vibrations are still in the elastic regie. Frequency dependence of the elastic response contains inforation about the kinks.
Dislocation otion by double kink foration Double kinks can for spontaneously due to theral fluctuations. Nucleation of a double kink corresponds to an energy increase: 4 2 αgb ΔWdk = 2W kink τb l 2π(1 υ) l ΔW l dk l=l 0 = 0 work done by τ αg l 0 = b >> b 2π(1 υ) τ work against attraction of the kinks τ < τ P ax 3 αgτ ΔWdk = ΔWdk ( l = l0) = 2Wkink 2b 2W 2π(1 υ) W kink, b, W P ev ev Si 2.2 0.45 Ge 1.5 0.23 Bi 0.31 0.01 Fe 0.2 0.004 Cu 0.1 0.001 kink Ag 0.085 0.0007 The effect of lattice resistance to τ is only significant at low T Al 0.09 0.0008 rate of nucleation: 2W R ω 0 exp kbt 12 1 ω 10 0 s kink Double kink nucleation plays a role at low τ. At higher τ dislocations can ove without help of double kink foration
Clib of ostly screw dislocation with sall edge segents Let s consider a dislocation in a siple cubic lattice that consists of long screw segents of length l s >> a and short edge segents of length l e ~ a z the screw segents have 2 slip planes, (001) and (100) the edge segents are siply one (l e = a) or two (l e = 2a) rows of atos located above the edge segents along z- axis these segents can only glide in [010] direction in (001) plane x l s ϕ y otion in [001] direction would require clib of all edge segents by a. Such clib would allow the whole dislocation to ove up by a the effective clib velocity can be large if the rate of vacancy absorption per unit length is n&, the effective clib velocity is this clib velocity can be uch larger than the clib velocity of an edge dislocation: jogs v eff clib 2 ls edge = a n& = vclib ctg ( ϕ) l e v eff = clib a v edge = clib 2 a l n& l 2 s e n& vacancies active clib can result in substantial increase in the concentration of point defects, typically vacancies (c > c 0 ) (the exaple in this page is for absorption, rather than eission of vacancies)
Kinks and Jogs Real dislocations are not straight - they always contain kinks and jogs Kinks and jogs are steps of atoic diension in the dislocation line Kinks are contained in the glide plane of the dislocation Jogs are not contained in the glide plane of the dislocation z kink jog y x kinks in edge and screw dislocations Jogs always for during clib. Clib proceeds by oveent of jogs through eission or absorption of point defects. clib by eission of interstitials jogs in edge and screw dislocations
Effect of kinks and jogs on dislocation otion Kinks: Kinks do not ipede glide of the dislocation in the plane of the kink, on the contrary, double kink foration can help dislocation to ove at τ < τ P A screw dislocation with a kink can glide in a specific glide plane (the glide plane of the kink) in other planes the kink serves as an anchor point for the screw dislocation Jogs: Jogs of screw dislocations have edge character and can only glide along the line oveent in other directions involves clib jogs ipede glide and results in the generation of point defects (ostly vacancies since E vf < E if ) Generation of kinks and jogs: Geoetrical kinks, therally activated generation of double kinks Generation of jogs by absorption or eission of point defects in response to F che (super-/under-saturation of point defects) Intersection of dislocations 2 when two dislocations intersect, each acquires a jog equal in direction and length to b of the other dislocation 1 1 2
Motion of dislocations with super-jogs When a point defect is created, the jogs oves forward one atoic spacing, ~b, resulting in work done by τ being τb 2 l s If the point defect foration energy is E f, the critical stress to ove the dislocation can be obtained fro the energy balance: τ = E 2 0 f / b l s for a segent of length l s the axiu stress is τ c 2 Gb = R Gb l sall (atoic) jog: τ c > τ 0 screw drags the jog along, creating a trail of vacancies (or, less likely, interstitials) l s s longer jog: n point defects have to be generated for each step forward, τ c < τ 0 = ne f /b 2 l s a dipole of edge dislocations of opposite sign is fored. even longer jog: the interaction between the dislocations in the dipole is weaker and the two edge segents can pass each other two parts of the dislocation ove independently fro each other.
Jogs and prisatic loops Prisatic loops can for (1) by condensation of point defects, e.g., the ones generated by the plastic deforation (interstitials can diffuse and self-organize into loops at lower T as copared to vacancies) (2) pinch-off of dislocation dipoles fored during propagation of screw dislocations with jogs or interaction between edge/ixed dislocations (3) foration of loops due to ultiple cross-slip (e.g., interaction of dislocations with obstacles) II III II III II II I I I I
Interaction of dislocations with obstacles Orowan echanis shear loop is fored fro Hull and Bacon Hirsch echanis cross-slip occurs 2-3 ties, leading to foration of prisatic loops
Molecular dynaics siulations of dislocation - precipitate interactions in Al(2139) alloy AlCu 5 Mg 0.5 Ag 0.4 (Mn) CuAl 2 Ω phase has hexagonal plate-like shape with broad face aligned along the Al (111) planes Thickness < 6 n pinning of dislocation and release via cutting: cobination of Orowan and cutting echaniss Shear stress / MPa 350 300 250 200 150 100 50 strain rate 10 8 0 D.W. Brenner, L. Sun, and M.A. Zikry, NCSU -50 0.000 0.005 0.010 0.015 0.020 0.025 Shear Strain
Generation and ultiplication of dislocations Although there are no equilibriu dislocations at any T, dislocations are always present in crystals. They are introduced during crystal growth (e.g., due to internal stresses generated by ipurity particles, T, condensation of point defects into prisatic loops, at interfaces). It is difficult to reduce dislocation density below ~10 10-2 Plastic deforation results in a rapid increase in dislocation density up to ~10 14-10 15-2 Mechaniss of dislocation ultiplication include: Frank-Read source, ultiple crossslip, eission of dislocation fro grain boundaries, etc. Siulation of Frank-Read source using Dislocation Dynaics ethod http://zig.onera.fr/disgallery/index.htl τ > αgb/r in