Please enable JavaScript.
Coggle requires JavaScript to display documents.
METALS (Solid State Diffusion (Interstitial vs Substitutional
…
METALS
Solid State Diffusion
Industrial applications
- Doping in the production of electrical components
- Diffusion welding
- Coating processes
- Case hardening
- Vapour deposition processes
- But also some embrittlement phenomena
Factors influencing diffusion
- Diffusing elements
- Host solid
- Temperature
- Microstructure
Diffusion mechanisms
- Interstitial diffusion
- Vacancy diffusion
- Interdiffusion and self-diffusion
- “Shortcuts” (GB, surface, pipe)
Definition
- Brownian motion: individual movement, Specific B molecule will move around right and left, up and down, in a quite disorderedly way
- Diffusion: movement of the group of B molecules
is not disorderly at all; the area occupied by B molecules expands in proportion to √t
- Transfer rate: described by the diffusion coefficient D
Response to a concentration gradient
- A and B miscible: B atoms from B-rich area to A-rich area (and conversaly)
- A and B, miscibility gap “Uphill diffusion”: B atoms from A-rich area to AB-rich area (and conversaly)
- Diffusion stops when Chemical potentials are equal
Interstitial vs Substitutional
- Interstitial diffusion faster than vacancy diffusion
• Bonding of interstitials weaker and more interstitial sites than vacancy sites
- Lattice distortion during jump, energy from thermal energy of atomic vibration (E~ 3kT)
- Concentration of interstitial atoms low > vacant sites available
Substitutional diffusion
- C = (M/(2√(πDT))) exp(-x2/4Dt)
with M= known quantity of Au*
- Effect of vacancies:
- The vacancy conc. on either side of the couple departs from eq.
- Excess of vacancies on one side and a depletion on the other
- Vacancies are destroyed or created at various sinks and sources
- Jump frequency of an indiv. metal atom, Γ is prop. to the vacancy conc.: D=D0.exp(Qsd/RT)
with QSD = ΔHm (vac.migration)+ΔHv(vac.formation)
- Transfer rate of the marker:
Vm=Jv/(cA+cB)=(DB-DA)/(cA+cB).dcB/dz
- Diffusion and flux due to movement of the interphase
- x =0 at the interphase (moving with the interphase), diffusion relative to the lattice
- xbar =0 at other end of the diff. couple (stable axis), flux due to the velocity of the lattice
- Darkens equation (interdiff. coeff.) DAB=DB.xA+DA.xB
- Kirkendall effect: formation of porosity > dissimilar diff. fluxes within a diff. couple !JA!>!JB!
Marker at the diff. interface move in the opposite dir. to the most rapidly moving species ⇒ vacancies can move
Fick's 1st law
- J is the flux of diffusing atoms, (atoms/m2 s-1) or (kg/m2 s-1): J=M/At≅(1/A)=dM/dt
- CB(1)-CB(2)=-α(∂CB/∂x)
- J=-D.dc(x)/dt
- D: Diffusion coefficient (m^2/s)
- Diffusion from higher to lower concentration
Fe-C case
- C atoms strain the austenite lattice => diffusion easier
Steady-State Diffusion
- After time t average atom has moved (radial distance)
from the origin
- r=α√(Γt), r=2.4√(Dt)
- In 1 s each every C atom will move ~0.5 m, but reach only net distance of ~10 μm
- Fraction of successful jumps = exp(- ΔGm/RT)
D=D0.exp(-QID/RT) where QID = Hm
2nd Fick's law and nonsteady state
- The rate of compositional change is equal to the diffusivity times the rate of the change of the concentration gradient
- J varies with time
- dC/dt=D.d2C/dt2
- Concentration increases with time in those parts of the system where concentration profile has a positive curvature
- With boundary conditions:
- Two rods with different c1 and c2 joined at x=0 and are infinitely long: c(x)-c1=(c2-c1)/2.(1+erf(x/(2sqrt(Dt))))
- Homogenisation: c(x)=c+B0.sin(Pi.x/l)
The longer the relaxation time (τ), then the slower decay
Short wavelength shorter time for homogenisation
- Decarburisation: C=C0.erf(x/(2sqrt(Dt))
High Diffusivity Paths
- Effect of GB diffusion combined with volume diff.
- Solute atom differs in size from solvent atoms > strain in the matrix
- Stain energy reduces ear structural discontinuities (dislocations, GB...)
-> Segregation of atoms in those area
- Dapp/D1=1+g.Dp/D1
D1: matrix, Dp: pipe, g=area of pipe/unit area of lattice (Grain bondary length/Grain size)
- Apparent diffusion coefficient
- At low T shortcuts important
- At high T lattice diffusion dominant
- Increasing T: Surface > GB > Volume
Uphill diffusion
- Multicomponent mixtures in which the diffusion flux of any species is strongly coupled to that of its partner species
- We observe transient overshoots during equilibration; the equilibration process follows serpentine trajectories in composition space
- Slide 3.88
SUMMARY
- Diffusion rate increases exponentially with temperature
- Diffusion buy interstitial mechanism is usually faster than by vacancy mechanism
- Diffusion coefficients and activation energies are different for every solute-solvent pair
- Diffusion is faster in polycrystalline materials than in single crystals due to grain boundary diffusion
Phase Diagrams
Intermediate phases
- Minimum free energy after mixing
- Crystal structure different than that of pure components
- Laves phases, MgCu2, Fe2Nb, Fe2Mo
- Interstitial compounds
- Peritectic reaction: α + L → β
- Eutectoid reaction: α → β + γ
Heterogenous system
- A and B have different crystal structures → 2 free energy curves
- Compo. of phases determined by the other phases in eq. (common tangent)
- X0<αe → α phase only
- X0>βe → β phase only
- αe<X0<βe then min free energy is Ge → 2 phases
- When 2 phases exist in eq., the activities of the compo. must be equal in the 2 phases:
Miscibility
- A and B complete miscible: 2.slide 17
- Miscibility gap in solid state 2.slide 19
:warning: Most important param. DHmix
- ΔHmix > 0: A and B dislike, ie repulsive
Reduced Tm below Tm of individual compo.
- ΔHmix >> 0: A and B strongly repulsive (+same microstructure)
Miscibility gap extends to
liquid phase → eutectic phase diagram
- ΔHmix < 0: A and B attractive
Melting more difficult + ordered structures at low T
Strong attraction → ordered phases stable up to Tm
Gibbs Phase Rule
- P + F = C + N
- P = number of phases present
- F = degrees of freedom (T, p, compo.)
- C = components (or compounds)
- N = noncompositional variables
- Tells how many phases can coexist within a system at eq.
- *Single phase field: 2 param. needed (T and compo.)
- 2 phases: F = 2 + 1 – 2= 1 (T or composition)
- 3 phases: F = 0 (T and composition fixed)
- Melting point: F=0
- Solidification : over a T range → F = 2-2+1 = 1
Limit of solubility
- Max. conc. of B soluble in A (XeB): μαB = μβB ~ GβB
- For regular solution μαB = GαB + Ω(1-XB)^2+RTlnXB
RTlnXeB - Ω(1-XB)^2 = ΔGB
→ if XeB <<1, XeB = exp(-((GB + Ω)/(RT)))
- XeB = A exp (-Q/RT)
- A= exp(ΔS B/R)
- Q = ΔH B + Ω
Effect of vacancies
- Removal of atoms increases the internal energy due to broken bonds
- Also increase randomness
- Small increase in the thermal entropy :question:
- Larger effect in configurational entropy due to broken bonds
Microstucture
- Depends on its composition
- rate of cooling
Determine
For a given T and compo. of the system
- The number and types of phases present
- The compo. of each phase
- Weight fraction: Wa=(C0-Cl)/(Ca-Cl)
- Volume fraction: Va=(Wa/ρa)/((Wa/ρa)+(Wb/ρb))
Fe-C diagram
Thermal arrest during cooling
- A0: Ferro/paramag. trans. in cementite
- A1: Eutectoid trans. in hypoeutectoid alloys
- A13: Eutectoid trans. in hypereutectoid alloys
- A2: Ferro/paramag. trans. in ferrite
- A3: Ferrite/austenite trans. above A2
- A32: Ferrite/austenite trans. below A2
- A4: Austenite/δ-ferrite trans.
- Acm: Austenite/cementite trans.
- Austenite: FCC
- Ferrite: BCC
- Pearlite: alternating layers of α-ferrite and Fe3C
Interfaces
-
-
Coincident site lattice (CLS)
- Slide 4.34
- Another way to present misorientation across a GB
- Extend the 3D lattice from one crystalline grain across the boundary
- Observe the density of atoms in the adjacent grain that coincide
- Repetitive array called coincident site lattice (CLS)
- Σ = inverse density of CLS
Ex: Σ = 15 => 1 in 15 atoms sites common to the extended crystal lattice of the other grain
- For randomly misoriented grains Σ can exceed 100
Σ = 1 => almost total coincidence of lattice sites
- High density of lattice points in a CLS is expected to have low energy
Uses
- Lifetime of a lead-acid lead lifetime based on intragranular degradation (corrosion=> low sigma extended life)
- Stress corrosion cracking tendency reduced in alloy 690 (in nuclear power plants)
- Intergranular corrosion resistance of Al2124
- Corrosion of Inconel 600, if deviation from Σ 3 orientation => cracking
- Fatigue of nickel base super-alloys
-
Interfacial free energy γ
- G=G0+Aγ
- DG=γdA+Adγ > F=γ+Adγ/dA
Basic Concepts
- Phase transformation diagram: Diagram showing eq. phases as a function of T and composition, used in heat treating, alloy design, to identify phases, etc.
- Gibbs free energy: G=H-TS, in eq. it has the min value
- TTT: Isothermal Time-Temperature-Transformation diagram
CCT: Continuous Cooling Transformation diagram
Shows non-eq. phases, for ex. in steels martensite and
bainite, in aluminium GP-zones, different precipitations (start and stop of the reaction)
- Coherent precipitation: Crystal planes of the precipitate and matrix are equal (continue through the precipitate)
- Incoherent precipitate: separate crystal planes and can have different crystal structure
- Strength change: Coherent precipitation => GP-zones, Partially coherent precipitates θ’, Non coherent θ’’
Thermodynamic background
Gibbs free energy
- G=H-TS, H=E+PV
- E consist of 2 components:
- Kinetic energy of the atoms
- Potential energy of the system atoms
- E = Q + W, dE = TdS-pdV
- pV des. the inter. with the system and its env.
- S = dQ/T is a measure of the system disorder
- Solid state: low Ek or E (low T and small S)
- Liquid state: high E (high T and large S)
- Eq.: dG = 0
- Highest stability: low H and high S
- Heat capacity: C=dH/dT
Phase transformations
- Possible if ΔG=Gf-Gi>0
- 1st order
- Can be seen: change in crystal structure, density, change in outlook,DTA (Differential Thermal Analysis), DSC (Differential Scanning Calorimetry)
- dG/dT=-S, dG/dP=V
- 2nd order
- Can be seen: can not be seen in eq. phase diagrams, change in specific heat
- d2G/dT2=-dS/dTp=-Cp/T,
- When T and P change → dG = -SdT + VdP
- When dP = 0 → 𝜕𝐺/𝜕𝑇 = -S
Solidification
- Driving force: ΔG ~L𝚫T/Tm
- ΔT (undercooling) small => difference in specific heats can be ignored
- ΔG ~ L - TL/Tm→ when ΔT small
Binary system - Definitions
- Single-component system → (T, p) system variables
- Alloys → composition also a variable
- Molar fraction X: XA+XB = 1
- Chemical potential or DG governs how the G changes with respect to the addition/subtraction of atoms
- Solid solution: Solute atoms randomly arranged
- Hume-Rothery rules for substitutional solubility:
- Atomic size difference <±15%
- The same crystal structure
- Similar electronegativity
- Valencies close to each other (complete solubility => the solvent and solute have the same valency)
- Hume-Rothery rules for interstitial solid solutions:
- Solute atoms must be smaller than the interstitial sites in the solvent lattice
- The solute and solvent should have similar electronegativity
- No atomic interactions
Binary system
- Before mixing: G1=XA.GA+XB.GB
- After mixing: G2= G1+ ΔGmix
G2= XAGA+XBGB+RT(XA.lnXA+XB.lnX)
- S = k ln ω (Boltzman equation)
- S = Sth+Sconfig
Rem: ΔHmix = 0 + before mixing S1=k.ln1=0 => ΔSmix = S2
- ω=(NA+NB)!/(NA! NB!) ways to arrange atoms
NA = XANa,NB = XBNb + Stirling’s approximation
- ΔSmix= -R(XA.lnXA+XB.lnXB)
- ΔGmix= RT(XA.lnXA+XB.lnXB)
Ideal solution
- :<3: ΔHmix= 0, ΔGmix= -TΔSmix
- Free energy change is only due to entropy
- A and B have the same crystal structure (no volume change)
- A and B mix forms substitutional solid solution
- Ex: Fe-Mn
Chemical potential
- μ (chemical potential or partial molar free energy) governs the response of the system to adding atoms
- dG’ = μA dnA(+μB dnB)
- Correct proportions: dnA/dnB=XA/XB => G=μA.XA+μB.XB
- μA = GA +RTlnXA, μB = GB +RTlnXB
Regular solutions
- ΔHmix not 0, only due to the bond energies between adjacent atoms
- Bonds: A-A, B-B, A-B with energies εAA, εBB, εAB
- Assumptions:
Volumes of A and B are equal, no change
Due to mixing => interatomic distances and bond energies do not change
- E=PAB.εAB+PAA.εAA+PBB.εBB
- Before mixing only A-A and B-B => DHMIX= PABε
- PAB is the number of A-B bonds and ε= εAB-ε ½(εAA +εBB)
PAB=Na.z.XA.XB
- If ε near 0 => ΔHmix=ΩXAXB
- G= G1+ ΔGmix= XAGA+XBGB+ΩXAXB+RT(XAlnXA+XBlnXB)
- ΔHmix < 0 => Exotermic reaction