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Chapter 4_Diffusional Phase
Transformations in
Solids_Part
The different types of phase transformations can be divided into the following groups: (a) Precipitation Reactions (b) Eutectoid Transformations (c) Ordering Reactions (d) Massive Transformations (e) Polymorphic changes
(a) Precipitation transformation
α` → α + β
α`→ metastable supersaturated solid solution
β → stable or metastable precipitate
α → more stable solid solution with the same crystal structure as α` , but with a composition closer to equilibrium
(b) Eutectoid transformations
γ → α + β
Long-range diffusion is required
(c) Ordering reactions
(d) Massive transformation
the same composition as the parent phase, but different crystal structures.
β→α
n* = the number of stable nuclei (having radii greater than r* )
n* increases as the temperature is lowered.
Nucleation involves the clustering of atoms by short range diffusion during the formation of nuclei. This diffusion effect is related to the frequency (νd) at which atoms from the liquid attach themselves to the solid nucleus.
During the cooling of a liquid, an appreciable nucleation rate will begin only after the temperature has been lowered to below the equilibrium melting temperature (Tm). This phenomenon is termed supercooling (or undercooling). The degree of supercooling for homogeneous nucleation may be significant.
Heterogeneous Nucletion
The activation energy for nucleation (ΔG*) lowered when nuclei form on preexisting surfaces or interfaces, since the surface free energy is reduced. Easier for nucleation to occur at surfaces and interfaces than at other sites.
S(θ) term is a function only of (θ) S(θ) term has a numerical value between zero and unity. For θ =30°, S(θ) = 0.01 and for θ =90°, S(θ) = 0. Critical radius for hetero is same as for homo Activation energy barrier for hetero is smaller than homo by S(θ) term
Much smaller degree of supercooling is required for heterogeneous nucleation.
4.1. Homogeneous Nucletion Solids
The free energy change associated with the nucletion process has three contributions:
- At temperature where the β phase is stable, the creation of a volume V of β will cause a volume free energy reduction of VΔGV.
- Assuming that the α/β interfacial energy is isotropic the creation of an area A of interface will give a free energy increase of Aγ.
- In general, the transformed volume will not fit perfectly into the space originally occupied by the matrix and this gives rise to a misfit strain energy ΔGs per unit volume of β. ΔG = -VΔGV+ Aγ + VΔGs γ can vary widely from very low values for coherent interfaces to high values for incoherent interfaces.
The concentration of critical-sized nuclei C* is C* = C0 exp (-ΔG*/kT)
Supercritical at a rate off f
Nhom = f C f depends on how frequently a critical nucleus can receive an atom from the α matrix.
The driving force for precipitation increases with increasing undercooling (ΔT) below the equilibrium solvus temperature. At small undercoolings, N is negligible because the driving force ΔGv is too small Very high undercoolings N is negligible because diffusion is too slow. Maximum nucleation rate is obtained at intermediate undercoolings. For alloys containing less solute the critical supercooling will not be reached until lower absolute temperatures where diffusion is slower. For alloys containing less solute the critical supercooling will not be reached until lower absolute temperatures where diffusion is slower. The most effective way of minimizing ΔG* is by the formation of nuclei with the smallest total interfacial energy. Incoherent nuclei have such a high value of γ that incoherent homogeneous nucleation is virtually impossible. The nucleus has an orientation relationship with the matrix, and coherent interfaces are formed, ΔG* is greatly reduced and homogeneous nucleation becomes feasible.
4.2. Heterogeneous Nucleation
Creation of a nucleus results in the destruction of a defect, some free energy (ΔGd) will be released by reducing the activation energy barrier.
Nucleation on Grain Boundary
Optimum shape for an incoherent grain-boundary nucleus
Excess free energy associated with the embryo:
V → volume of embryo
Aαβ → area of α/β interface of energy γαβ created
Aαα → area of α/α grain boundary of energy γαα destroyed during the process
S(θ) is a shape factor:
V* and ΔG* can be reduced even further by nucleation on a grain edge or grain corner High-angle grain boundaries are particularly effective nucleation sites for incoherent precipitates with high γαβ. If the matrix and precipitate are sufficiently compatible to allow the formation of lower energy facets then V* and ΔG* can be further reduced.
Nucleation on Dislocations
Main effect of dislocations is to reduce the ΔGs contribution to ΔG* by reducing the total strain energy of the embryo. A coherent nucleus with a negative misfit, smaller volume than the matrix, can reduce its ΔG* by forming in the region of compressive strain above an edge
For grain boundary nucleation:
δ → boundary thickness
D → grain size
The type of site which give the highest volume nucleation rate will depend on the driving force ( Gv). At very small driving forces, when activation energy barriers for nucleation are high, the highest nucleation rates will be produced by grain corner nucleation. •As the driving force increases , grain edges and then boundaries will dominate the transformation. •At very high driving forces it may be possible for the (C1/C0) term to dominate and then homogeneous nucleation provides the highest nucleation rates.
4.3. Precipitate Growth
Successful critical nuclei are those with the smallest nucleation barrier. In the absence of strain energy effects, the precipitate shape satisfying this criterion is that which minimizes the total interfacial free energy. Incoherent interfaces on the other hand are highly mobile. If there are problems in maintaining a constant supply of ledges, the incoherent interfaces will be able to advance faster than the semicoherent interface
4.3.1. Growth behind Planar Incoherent Interfaces
Imagine that such a slab of solute-rich precipitate has grown from zero thickness and that the instantaneous growth rate is v. •Since the concentration of solute in the precipitate (C ) is higher than in the bulk (C0) the matrix next to the precipitate will be depleted of solute. Since the interface is incoherent, diffusioncontrolled growth and local equilibrium at the interface can be assumed The growth rate (v) will depend on the concentration gradient at the interface dC/dx
An ever-increasing volume of matrix so that dC/dx decreases with time.
Important Points:
- x √(Dt) precipitate thickening obeys a parabolic growth law.
- v X0, i.e. for a given time the growth rate is proportional to the supersaturation.
- v √(D/t).
Growth rates are low at small undercoolings due to small supersaturation X but are also low at large undercoolings due to slow diffusion.
Examples
Chapter 4_ Diffusional Phase
Transformations in Solids_ Part
4.4. Overall Transformation Kinetics & TTT
Diagrams
The progress of an isothermal phase transformation can be represented by plotting the fraction transformation (f) The factors that determine the fraction transformed at a given temperature and time are: the nucleation rate the growth rate the density and distribution of nucleation sites the overlap of diffusion fields from adjacent transformed volumes the impingement of adjacent transformed volumes.
Possible Scenario # 1
Nuclei form throughout the transformation so that a wide range of particle sizes exists at any time.
Fraction of volume transformed in time t:
The equation for randomly distributed nuclei:
4.5. Precipitation in Age-Hardening Alloys
Requirements:
- an appreciable maximum solubility of one component in the other, on the order of several percent,
- a solubility limit that rapidly decreases in concentration of the major component with temperature reduction,
- the composition of a precipitation-hardenable alloy must be less than the maximum solubility.
4.5.1. Precipitation in Aluminum- Copper Alloys
Al-4 wt% Cu is heated to a temperature of about 540°C. If the alloy is now aged by holding for a period of time at room temperature or some other temperature below about 180°C, it is found that the first precipitate to nucleate is not θ but coherent Cu-rich GP zones. Why? GP zones are fully coherent with the matrix and therefore have a very low interfacial energy, whereas the θ phase has a complex tetragonal crystal structure which can only form with high-energy incoherent interfaces. Driving force for precipitation of GP zones (ΔGv - ΔGs) is less than for the equilibrium phase, the barrier to nucleation (ΔG*) is still less, and the GP zones nucleate most rapidly.
Transition Phases (θ” and θ’)
Total Precipitation Sequence
α0 → original supersaturated solid solution, α1 → comp. of the matrix in equilib. with GP zones, α2 → composition in equilibrium θ”
GP zones and the matrix have the same crystal structure they lie on the same free energy curve. θ” and θ’ less stable than s θ and have higher free energies. Transition phases have a lower activation energy barrier for nucleation than equilibrium phase. Free energy of the ally decrease more rapidly via the transition phases than by direct transformation to the equilibrium phase
The full sequence of GP zones and transition precipitates is only possible when the alloy is aged at a temperature below the GP zones solvus.
Also the maximum volume fraction of θ” is reduced
The volume fraction of θ” increases with time causing the hardness to increase
Both of these factors contribute to a lower peak hardness on ageing at the higher temperature.
Diffusion rates are faster at higher temperatures and peak hardness is therefore achieved after shorter ageing times.
