Diffusion Notes PDF

Diffusion Diffusion Occurs to reduce free energy Reduces concentration differences Occurs away from regions of high concentration to regions of low concentration In systems with miscibility gaps the diffusion is towards the region of high concentration; still reduces free...

Diffusion Diffusion Occurs to reduce free energy Reduces concentration differences Occurs away from regions of high concentration to regions of low concentration In systems with miscibility gaps the diffusion is towards the region of high concentration; still reduces free energy In both cases, diffusion takes place from regions of high chemical potential to regions where it is low Diffusion is faster in cold worked metals It’s all about temperature, vacancies, available spaces, grain boundaries, low atomic packing factors, anisotropy of crystals, etc. Atomic mechanisms Interstitial diffusion – atoms move through the interstices of the matrix by forcing their way between larger atoms Substitutional movement – atoms move by a vacancy mechanism In crystals, atoms usually vibrate about their sites with a thermal energy of 3kT per atom; the energy increases in proportion to absolute temperature. The presence of vacancies allows atoms to “jump” out of position and transfer to an adjacent vacancy Effect of Temperature on Diffusion As the temperature increases: 1) The equilibrium concentration of vacancies increases, producing more space for diffusion 2) The amplitude of vibration (& the energy of vibration) increases creating space for atoms to move into adjacent sites 3) For smaller atoms, interstitial sites are usually not occupied (only a small fraction of sites is occupied) and each interstitial atom will always be surrounded by empty sites 4) Jumps occur when the thermal energy is sufficient to overcome the strain energy barrier Interstitial Diffusion: Activation Interstitial Diffusion Random jump process; always possible due to the large number of empty sites Simple model: B atoms fit into interstitial sites (not the usual case); 6 empty sites surround the initial site of a B atom (octahedral site) Each B atoms jumps ΓB times per second 1|Physical Metallurgy (WSM)  Each jump is in a random direction; 6 possible with equal probability. In a plane with n1 B atoms per m2, the number atoms jumping to another plane / sec is During the same time the number of atoms jumping back from plane 2 to plane 1 is: The net flux of atoms (n1 > n2) from left to right is: Where n1 & n2 are related to the concentration of B in the lattice If (vertical) planes 1 & 2 are separated by a certain distance & α is the concentration of B atoms in plane 1, CB(1) = n1/α atoms m-3; likewise CB(2) = n2/α; therefore n1 –n2 =α[CB(1)- CB(2)]; the quantity in the [] = -α(∂CB/∂x) (the negative of the gradient times α); the flux is now: Substituting DB =1/6 ΓB α2 gives Fick’s Law (1855) Fick’s First Law of Diffusion The partial derivative indicates that the concentration gradient can change with time DB = intrinsic diffusivity or the diffusion coefficient of B, and has units [m2 s-1] Units for J=[quantity m2 s-1]and for the gradient ∂C/∂x =[quantity m-4], where quantity can be atoms, moles, kg, etc as long it is the same for both J and C 2|Physical Metallurgy (WSM) Diffusivity When atomic jumps of B are truly random & with a frequency independent of concentration, DB is: DB is a constant independent of concentration In this case D was derived for a simple cubic lattice. It is applicable to any randomly diffusing atom in any cubic lattice provided the value of α is corrected. This not applicable to non cubic lattices such as hexagonal lattices. Diffusivity & Crystal Lattice In non-cubic lattices the probability of jumps in different crystallographic directions is not equal and D varies with direction. Atoms in hexagonal lattices diffuse at different rates parallel and perpendicular to the basal plane. The condition that atomic jumps occur randomly & independently of concentration is usually not justified in real alloys. Fick’s 1st law is still applicable but D is made to vary with composition. Values of D The value of DC for the diffusion of C in fcc-Fe at T = 1000°C: DC = 2.5 x 10-11 m2 s-1 at 0.15 wt% C DC = 7.7 x 10-11 m2 s-1 at 1.4 wt% C The reason for the increase in DC is that more C (higher concentration) atoms strain the lattice making diffusion easier. Jump Frequency The following data can be used to estimate the jump frequency of a carbon atom in γ-Fe at 1000°C using: Lattice parameter of γ-Fe is ~0.37nm; and jump distance α = 0.37/(2)1/2 = 0.26nm (2.6 Å); for D = 2.5 x 10-11 m2 s-1; Γ = 2 x 109 jumps s-1 If the vibration frequency of atoms is ~1013 then only one jump in 104 results in a jump from one site to another. Random Walk of 1 Atom Consider a single diffusing atom: If each jump is random & independent of the previous one, the result is a random walk. The distance travelled by the atom after n jumps is: r=α n1/2 from its original position. This the root mean square displacement after n steps; using time t and Γ jump frequency, r=α (Γt)1/2; using DB = (1/6)ΓB α2 ; or ΓB = 6DB / α2 Gives r = (6DB t)1/2 = 2.4(DB t)1/2 For C in γ-Fe, in 1s, a C atom moves ~0.5m, but r= ~10μm Effect of T- Thermal Activation For a jump to be successful the diffusing atom must successfully lodge itself in between two solvent atoms by moving them apart This requires an activation energy ΔGm (migration) On average the fraction of atoms with a mean energy of ΔG or more is given by exp(-Δ Gm /RT); if the atom is surrounded by z sites, the jump frequency is: And since & 3|Physical Metallurgy (WSM) Diffusion Coefficient – T The expression for DB becomes: This can be simplified into an Arrhenius type equation: where: & Graphical representation of D Grouping terms that are virtually independent of T into a single material constant DO, D or Γ increases exponentially at a rate determined by QID. For interstitial diffusion, Q is only dependent on the activation energy barrier to the movement of interstitial atoms. A convenient graphical representation of D is: The plot of log D vs. (1/T) is a straight line, with slope =(-Q/2.3R) & y-intercept = log DO ; log Do vs. log (1/T) plot Fick’s Second Law of Diffusion Recall: Diffusion Atomic movement Requires activation energy; faster at elevated temperatures Occurs during annealing, heat treatment, age hardening, sintering, surface hardening, oxidation, creep Evidence of Diffusion In pure metals: self-diffusion has been observed using tracer (radioisotope) atoms. In single phase homogeneous alloys: diffusion of each atomic component can be measured by tracer atoms. In non-homogeneous alloys exhibiting segregation (ex. coring): diffusion has been shown to occur with chemical analysis at interfaces; shown by the broadening of interfaces. Special cases: diffusion has been observed along grain boundaries (grain boundary diffusion) and along dislocation channels. Fick’s (First) Law: Steady State 4|Physical Metallurgy (WSM) Unsteady State: Fick’s 2nd Law Fick’s 2nd Law The role of vacancies in diffusion Diffusion through vacancies 5|Physical Metallurgy (WSM) Self diffusion in alloys Kirkendall effect Confirms the vacancy mechanism of diffusion; more vacancies are left behind by the faster diffusing species. Indicates that vacancies change places more frequently with one atomic specie than the other. Important in metal-to-metal bonding, sintering and creep. When the diffusivity of one metal is much higher voiding and weakness of the bond can result in a metal-to- metal bond. More likely to happen in soldering or brazing but not in welding -doc gopez Purple plague- purple phase formed in joints, because too many vacancies during soldering. A.K.A gold aluminium intermetallic Factors affecting diffusion Two most important factors affecting the diffusion coefficient D are: o Temperature o Composition Diffusion is also structure sensitive; D increases with lattice irregularity (decreasing APF) Diffusion is faster in: Metals quenched from high temperature (high vacancy concentration) Grain boundaries Along dislocations; diffusion is faster in cold worked metal Activation Energy Q Values Other information Grain boundary diffusion becomes more important in fine grain metals. Grain boundary diffusion becomes more important at lower temperatures. As the temperature is lowered diffusion in grains decreases faster than in g.b.’s. Pressure has little or no effect on diffusion due to the strong interatomic bonds. Diffusion increases with lower APF. In rhombohedral crystals diffusion is faster in lower (atomic) density directions (ex. Bi 6|Physical Metallurgy (WSM) Equilibrium Concentration of Vacancies: Derivation Equilibrium Vacancy Concentration Vacancies – increase E due to broken bonds but also increases the randomness or Sconfig of the system. The free energy of the alloy will depend on the concentration of vacancies and the equilibrium concentration Xve will be that which gives a minimum free energy. For simplicity, vacancies will be considered as being mixed into an alloy similar to obtaining ΔGmix for A & B atoms when ΔHmix is >0. Vacancies ΔHV The equilibrium concentration of vacancies is small and the interaction between vacancies can be ignored; the increase in enthalpy of the solid is directly proportional to the number of vacancies added: ΔH = ΔHV XV Where XV = mole fraction of vacancies; ΔHV = increase in H per mole of vacancies added; each vacancy increases H by ΔHV/Na With Na = Avogadro’s number Vacancies ΔS Two contributions to entropy of formation of vacancies: ΔSV = a small amount of thermal entropy (due to vibrations) & configurational S, ΔSconfig = -R(XA ln XA +XB ln XB) ΔS = XV ΔSV - R{XV ln XV + (1-XV) ln (1-XV)} The molar free energy of a crystal with XV mol vacancies is: G = GA + ΔG = GA + ΔHV XV - T ΔSVXV +RT {XV ln XV + (1-XV) ln (1-XV)} Equilibrium Concentration of Vacancies XVe To obtain Xve: Differentiating G (previous slide) & with XV 10-15° the dislocation spacing is so small that dislocation cores overlap making it difficult to distinguish individual dislocations. At this point the energy of the boundary becomes independent of the orientation. Energy of High Angle Boundaries High angle grain boundaries have a relatively open structure with areas of poor fit. The bonds between atoms are broken or highly distorted. These contribute to the high energy of high angle grain boundaries. Raft Model of High Angle Boundary Grain Boundary Free Energies 17 | P h y s i c a l M e t a l l u r g y ( W S M ) Special High Angle Grain Boundaries Simplest case: boundary between two twins a) Coherent twin: the twin boundary is parallel to the twinning plane; extremely low energy b) Incoherent twin boundary: the twin boundary is at an angle to the twinning plane; high energy Twin Boundaries Equilibrium at Grain Boundaries Forces on grain boundaries can result in a torque but for cases where the boundary energy is independent of orientation the boundary will act like a soap film 18 | P h y s i c a l M e t a l l u r g y ( W S M ) Equilibrium Configuration When boundaries act like soap films metastable equilibrium will result when three boundaries form a junction. The boundary tensions γ1, γ2, and γ3 must balance. This equation applies to any three boundaries, including other solid phases or vapor phase Thermally Activated Migration of GB’s In a polycrystal, grain boundaries are assumed to have the same energy independent of orientation, therefore at the junction of 3 grains, θ1 = θ2 = θ3 = 120°; for 4 grains θ = 109°28’ A curved grain boundary has a net force pulling it in the concave direction. The direction of the concavity of the grain boundary will determine how grain boundaries migrate. Effect of Curvature in GB Migration With θ = 120° grains with six sides will be stable. If there are < 6 grains around the central grain, concavity of the gb will cause the grain to shrink & “dissolve”. When the gbs are concave outward the grain will grow outward and increase in size. Types of Interphase Boundaries Depending on the crystal structures of the two phases and the lattice parameters of both phases, interphase (grain) boundaries can be classified as: (1) Fully Coherent – the two crystals match perfectly at the interface plane so that the two lattices are continuous across the interface. Possible cases: the same crystal structure with nearly identical lattice parameters; crystal structures with identical close-packed planes, i.e., (111)fcc and the (0001)hcp; γcoh = γch due to “wrong” chemical composition across the interface. 19 | P h y s i c a l M e t a l l u r g y ( W S M ) Coherent Interfaces Coherent interfaces are strain-free (no misfit). Strain energy is due to chemical difference, γch Coherent Interfaces with Misfit Coherent interfaces with slight mismatch in lattice parameters will have additional energy due to coherency strains (2) Semicoherent Interfaces – when there are matching crystal structures but a sufficiently large atomic misfit; “periodically” located dislocations can take up the disregistry and create a semicoherent interface. The misfit or disregistry is defined as: where dα & dβ are the unstressed interplanar spacings of matching planes in α and β phases 20 | P h y s i c a l M e t a l l u r g y ( W S M ) Semicoherent Interface When misfit strain of matching lattices becomes large, dislocations can create a semicoherent interface Misfit in Two Directions Misfit in two directions can also be accommodated by dislocations (3) Incoherent Interfaces When the crystal structures are very different and there is no matching across the surface (such as when interatomic distances in the two phases differ by more than 25%), the interface is said to be incoherent. In general incoherent interfaces occur when two randomly oriented crystals of different structures are joined at a surface. It is also possible between two crystals of different structures with an orientation relationship 21 | P h y s i c a l M e t a l l u r g y ( W S M ) Incoherent Interface Randomly oriented crystals with non matching lattice parameters joined at a surface. Complex Semicoherent Interfaces Complex semicoherent interfaces have been observed at boundaries formed by low index planes whose atom patterns and spacings are almost the same. Good fit is restricted to small diamond- shaped areas that only contain ∼8% of the interfacial atoms for both N-W & K-S relationships. Examples of Semicoherent Interfaces Nishiyama- Wasserman (N-W) Kurdjumov-Sachs (K-S) The difference between these two is a rotation of 5.26° in the close packed planes. The figure shows that the matching between a {111}fcc & {110}bcc in N-W is very poor 22 | P h y s i c a l M e t a l l u r g y ( W S M ) Interphase Interfaces in Solids The shape of the grain boundary can indicate whether the interface is semicoherent or incoherent. Particles with Coherent Interfaces When there is a large misfit strain in one direction, the new phase can nucleate on a plane where there is little or no misfit strain; forming a disk. Effect of Aspect Ratio Misfit strain is lowest for disks, highest for spheres and intermediate for needle shaped grains. 23 | P h y s i c a l M e t a l l u r g y ( W S M ) Mobility of Grain Boundaries High energy g.b.’s are very mobile. A tendency to lower energy will be favoured. Low energy g.b.’s are stable and will tend not to move or migrate. Curved g.b.’s will be pulled inward due to surface energy. Three g.b.’s meeting at a node will form a stable configuration. This forms a hexagonal grain. Grains with less than six sides will have g.b.’s that are concave inward. The pulling force will tend to “collapse” the grain. Grains with more than six sides will have g.b.’s that are concave outward. The pulling force will tend to make the grain grow Growth of Solidifying Metals Growth of a pure solid Single phase and with less tendency to form many grains the growth of a pure solid has an: 1) atomically rough solid-liquid interface for metallic systems; 2) atomically flat or sharply-defined interface associated with non-metals. Atomically rough interfaces migrate by a continuous growth process while flat interfaces migrate by a lateral growth process involving ledges. Continuous Growth Lateral Growth (smooth interfaces) Lateral growth is associated with materials that a high entropy of melting. It is more complex than continuous growth and requires the nucleation of the solid in three modes: 1) Surface nucleation of a disk of solid on a smooth interface of the solidifying material. This requires large degrees of undercooling to occur. 2) Spiral nucleation about a screw dislocation which is perpendicular to the interface. Less undercooling is required for this mechanism than surface nucleation. 3) Growth from twin intersections can occur when two crystals of different orientation solidify in contact with each other usually with a twinning orientation. Lateral growth Ledge creation by surface nucleation Spiral growth – about a screw dislocation 24 | P h y s i c a l M e t a l l u r g y ( W S M ) Growth with local variations in heat flow In pure metals when there are variations in the heat flow at the metal-liquid interface, supercooling can take place ahead of the interface. This leads to dendritic growth. In alloys the initial interface is planar associated with a rough interface. When there are variations in the heat flow at the solid-liquid interface constitutional supercooling can take place. For moderate gradients at the interface protrusions will form which are parallel to each other and cellular growth will follow. When the temperature gradient exceeds a certain value and becomes steeper the protrusions will further apart and local variations in heat transfer will allow lateral growth from the protrusions resulting in dendritic growth. Dendritic growth Formation of dendrites Cellular growth Eutectic Solidification With sufficient undercooling eutectics will form in a similar fashion to cellular solidification but with a localized compositional variation. If the undercooling is large (steep gradient) segregation will take place without equilibrium being attained with insufficient time for diffusion of atomic species. Coring will take place, in some cases there will be microsegregation. Solidification in ingots and castings Due to chilling at the cold mold walls a layer of equiaxed grains will be formed. Then when the heat transfer goes into a steady state a columnar zone will form with the grains growing in from the equiaxed layer toward the center. For pure metals these columnar grains will meet each other at the center of the mold. While for alloys at some point constitutional supercooling will take place and equiaxed grains will form at the center of the mold. Ingot Cross Section Alloys cast as ingots in permanent molds form a layer of fine equiaxed grains due to chilling at the mold wall. Steady state heat transfer causes columnar grains to form. Then constitutional super cooling causes equiaxed grains to form at the center. Pure metals do not have equiaxed grains at the center of the mold since they do not undergo constitutional supercooling. 25 | P h y s i c a l M e t a l l u r g y ( W S M ) Orowan’s Equation The Basis for Strengthening Orowan’s Equation The higher gamma becomes, the more deformed the metal is and the more weaker. Lower gamma means stronger metal Orowan’s equation: derivation 26 | P h y s i c a l M e t a l l u r g y ( W S M ) Strengthening Annealing weakens the metal; reduces rho, but increases g much more thus lowering strength Cold working strengthens; reduce the glide distance Strengthening: Summary Cottrell atmosphere- jeepney na example, big atom replacing below and small one replacing above. Strengthening Cold Working & Recrystallization EAF DC- 2 electrodes AC- 3 electrodes 27 | P h y s i c a l M e t a l l u r g y ( W S M )