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Microstructure-Properties: IIPrecipitate Growth

27-302

Lecture 3

Fall, 2002

Prof. A. D. Rollett

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Materials Tetrahedron

Microstructure Properties

ProcessingPerformance

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Objective

• The objective of this lecture is to provide a quantitative description of the kinetics of precipitate growth. Armed with this information, it is possible to construct a simple TTT diagram.

• Subsequent lectures will examine the role of the interface (coherent vs. incoherent) in precipitate morphology and growth, as well as competing transformation mechanisms.

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References• Phase transformations in metals and alloys, D.A.

Porter, & K.E. Easterling, Chapman & Hall.• Materials Principles & Practice, Butterworth

Heinemann, Edited by C. Newey & G. Weaver.

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Growth Rates

• For most precipitates, the growth is diffusion controlled, either through the bulk or along a boundary.– The size is typically proportional to √time (decreasing growth

rate).

• Some transformations are interface controlled, such as recrystallization.– The size is typically proportional to time (constant growth rate).

• Surprisingly, these basic rules hold under most circumstances, from 1D to 3D growth.

• Coarsening is significant at early stages (increasing average size with constant volume fraction).

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Notationv := velocity (of an interface)x := position, or, thicknesst := timeD := diffusion coefficient (be careful of type)C := concentrationL := width of a diffusion zonek := geometrical factor (of order unity)r := radius (of a precipitate tip)r* := critical radius (for nucleation)X := mole fraction; Xe := conc. at interface; X := conc. in a precipitate; X0 := conc. in matrix.u := lateral migration rate of ledgesh := ledge height := ledge spacingn := number of atoms (in Gibbs-Thomson)G := Gibbs free energy (in Gibbs-Thomson)A := Area of grain (in Gibbs-Thomson) := Interface energy (in Gibbs-Thomson)∆µ := chemical potential (in Gibbs-Thomson)

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Growth of Plates (1D)• Assumptions: consider 1D growth of,

say, platelets.• Example: new phase nucleated on a

grain boundary; rapid growth along the boundary, then slower growth away from the boundary (e.g. pro-eutectoid ferrite in austenite; also Sc2O3 in MgO, Chiang fig. 2.16), with incoherent interface.

• Flux of solute B depends on the concentration gradient, which is, in general, a complicated function of time.

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Solutions to Diffusion Equation

• In general terms, what we are dealing with is the diffusion equation, where C is the concentration, and D is the diffusion coefficient (diffusivity in heat flow):

D 2C + f(x,y,z,t) = C/t

• Also, in some cases, we are interested in the behavior of spherical precipitates, which means that spherical coordinates are appropriate:

2C/r2 + (1/r)C/r + (1/r2)2C/2 = 0

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Diffusion Eq. Contd.

• A mathematician would say that we are seeking solutions to the diffusion equation. In the cases that we deal with here, we can ignore source/sink terms (by focusing on the diffusion of solute in the matrix only) and solve only:

D 2C = C/t

• The initial state and the boundary conditions are as important as the differential equation itself, which makes the problem a boundary value problem.

• For precipitates, the initial state is a uniform concentration, C = constant. The boundary conditions are Dirichlet boundary conditions because we specify that the concentration is the alloy composition far away from the precipitate, and is equal to the equilibrium composition at the interface between the matrix and the precipitate.

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1D growth rate

• Velocity of the interface, v = dx/dt• Velocity given by flux divided by number of moles of

B required to form the precipitate:

v = D/(C - Ce) dC/dx

• The concentration gradient (at the interface), however, decreases continuously with time, as the matrix is depleted of solute (B).

• The relevant diffusion coefficient is the interdiffusion coefficient of B diffusing through A. If the solute is an interstitial, then use the appropriate coefficient for interstitial diffusion.

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Linearized gradients

• Clarence Zener* originated the approximation of linearizing the concentration gradient.

• Simplify the concentration gradient by the sawtooth below: conservation of solute dictates that the two shaded areas are equal (x is now the thickness of the precipitate!):

(C - C0)x = L∆C0/2 dC/dx’ = ∆C02/2(C - C0)x

x’

* Zener was in the Physics department of Carnegie Tech. for many years

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Linearized gradients, contd.

• With this approximation in place, we can now estimate the velocity as follows, where ∆C0= C0-Ce:

v = D/(C - Ce) dC/dx = D/(C - Ce)* ∆C02/ x(C - C0)

• Approximate (C - C0)(C - Ce), and note that (for constant molar volumes) we can replace concentrations by molar fractions and integrate the differential equation; ∆X0 is the supersaturation at the beginning of precipitation, ∆X0= X0-Xe:

x =ΔX0

Xβ −Xe

Dt v =ΔX0

2 Xβ −Xe( )

Dt

• Note the behavior of the gradient with time: as the precipitate drains solute out of the surrounding matrix, so the gradient decreases because the solute has to diffuse in from further away.

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An analogy for diffusion controlled growth

• What can we think of that is analogous to diffusion controlled growth?

• Simple analogy: pumping water out of a flooded basement. As you pump the water out, water flows across the floor, thereby lowering the level of the water near the pump. The height of the water represents the solute level. The removal of water represents the removal of solute by the growing precipitate.

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Effect of temperature• In order to predict growth rate as a function of temperature, one

must convolve the effect of driving force with that kinetics (diffusion). Remember that the diffusion is thermally activated and obeys an Arrhenius law, D D0exp-{Q/RT}. As undercooling increases, so does the supersaturation (driving force) but, conversely, the kinetics slow down (slower diffusion). Therefore there is a peak in the growth rate.

velocity

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More accurate solutions• Martin, Doherty & Cantor provide some discussion of more

accurate solutions, of which the linearized gradients approach is one example.

• Invariant field means that the concentration gradient is calculated at time=zero and then taken to be constant (∂C/∂t=0). Not surprisingly, this works well at short times, but not at long times.

• Invariant size is similar; the precipitate size is constrained to be constant. Similar limitations as above.

• Linearized gradients is similar to the previous exposition - the gradient adjacent to the particle is approximated as linear (dC/dx = constant).

• General form of the solution:x = √(Dt),

where is a supersaturation factor that depends on the geometry of growth and assumptions.

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Flat precipitates (plates)Supersaturation:

k =2 ∆C0/ (C - Ce)

= 2 (C0-Ce) / (C - Ce)

2 (CM-CI) / (CP - CI)in the Martin et al. Notation

Summary: linearized gradientsolution, x = √(Dt), where is given by -k/2 = √πe^erfc( is an excellent approximation!

log(-k)

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Spherical precipitates

Supersaturation, k:

k = -2 ∆C0/ (C - Ce)

= -2 (C0-Ce) / (C - Ce)

2 (CI-CM) / (CP - CI)in the Martin et al. notation

Summary: linearized gradient solution is increasingly accurate with increasing supersaturation.

log(-k)

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Impingement/ overlap• Unless the reaction is associated

with a phase change, precipitation reactions only proceed to a limited volume fraction (<0.1).

• Impingement does not, therefore, occur directly.

• Indirect impingement does occur, however, through overlap in the diffusion fields.

• Figure illustrates the disappearance of solute concentration gradients as the concentration ratio at the interfaces approaches the equilibrium value, and the solute has been fully re-distributed.

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Analogy

• Again, we can use the analogy of pumping water out of a flooded basement. Think of having two pumps in different locations. Both draw down the water level. At first their effect is only local but eventually the water level drops enough that one pump affects the other.

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Grain boundary precipitation

• “4 step program”!– Heterogeneous Nucleation– Diffusion through bulk to the grain boundary– Diffusion along g.b., and along the interphase boundary– Attachment of solute to the precipitate.

• Important effect for substitutional solutes, not interstitial (why?).

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Plate/Needle Growth

• For the (diffusion controlled) lengthening of plates/needles, we have to be concerned with the effect of curvature.

• Why? Because for a plate of constant thickness, the curvature at the growing tip remains constant and small.

• Why this shape? Because the interface is coherent (and difficult to grow) on one face (plates) but incoherent around the edge (tip).

Characteristic diffusion distance, L;dC/dx = ∆C/L; L=kr (k~1)

v = D∆C/{kr(C-Cr)}

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Gibbs-Thomson

• The net effect of the curvature at the tip is to raise the free energy of the precipitate tip relative to a flat interface (in equilibrium).

∆X = ∆X0(1- r*/r)

k is a geometrical coefficient ~ 1. As thetip radius decreases, so thegrowth rate slows down.

v =DΔX0

Xβ −Xe( )

1kr

1−r *r

⎛ ⎝

⎞ ⎠

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Ledge controlled thickening

• Turning now to the problem of how a platelet precipitate can coarsen, i.e. increase in thickness, we consider the effect of ledges on the flat surfaces.

• Each ledge is a line of incoherent interface, on which solute can be readily attached.

• Velocity = uh/.

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Ledge control, contd.

• Velocity = uh/. • Ledge speed, u

= D∆X0 / k(X - Xe)h

• Combining these equs:v = D∆X0 / k(X - Xe)

• Thickening rate, v, inversely proportional to ledge spacing, provided diffusion fields do not overlap.

• Evidence suggests that actual thickening rates are dependent on nucleation of ledges.

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Particle Coarsening• “For unto every one that hath shall more be given”• Another consequence of the Gibbs-Thomson effect is that, even at

constant volume fraction (all the available precipitate volume fraction has precipitated) there is a driving force for coarsening of the precipitates. The reason for this is that smaller particles have a higher solubility than larger particles.

• Mass conservation means that, if one particle grows then another particle somewhere in the system must shrink.

• The mechanism for coarsening is diffusion, hence the kinetics follow an Arrhenius law.

• The average increases with time because a shrinking small particle will eventually disappear at which point the number of precipitates decreases by one (and the average size increases).

• In most systems, coarsening starts immediately after nucleation.

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Coarsening: thermodynamics

• The smaller particle with size r2 has a higher solubility (Gibbs-Thomson) than the larger particle, r1, and so there is a concentration gradient that causes solute to flow from particle 2 to particle 1.

P&E

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Coarsening, contd.• The rate at which coarsening (also known as Ostwald

Ripening)occurs is clearly proportional to the driving force and the kinetic factors. Thus higher coarsening rates should be associated with higher interfacial energies, higher diffusion coefficients, and higher solubilities.

• The kinetics of coarsening were worked out by Wagner and also by Lifshitz and Slyozov (see ref. 10 for Ch.5). The quasi-steady-state distribution of sizes that emerges from their analysis, while not a particularly good fit to experimentally determined data, is known as the LSW Distribution.

• The particle size varies as (time)1/3, in contrast to grain growth, which follows a t1/2 dependence. The full equation is:

r 3 t( )−r t=03 =

8DγXe 1−Xe( )Vm

9εDarken Xβ −Xe( )2RT

t

εDarken=1+∂ lnγB

∂ lnXBDarken, or thermodynamic factor:

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Coarsening, example of W-Ni-Fe

• r t1/3 verified• See extensive discussion in Martin, Doherty & Cantor, pp 239-298.

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Coarsening - practical impact• The practical impact of particle coarsening is highly significant.• Exposure to high temperatures will coarsen almost any system

and decrease strength (based on resistance to dislocation motion).

• Particle coarsening also reduces resistance to grain growth (particles pin grain boundaries).

• Sintering of powders is the basic consolidation process for ceramics: coarsening limits minimum grain size (especially in liquid-phase sintering because of enhanced diffusion rates).

• Coarsening minimized in systems with low interfacial energies and solubilities (e.g. Ni superalloys).

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Summary

• By making reasonable assumptions about the geometry of the diffusion field, it is possible to describe the rate of precipitate growth with simple equations.

• As undercooling increases, so the driving force increases which initially increases the growth rate.

• As one continues to increase the undercooling, so the diffusion rate slows down and growth rates also decrease.

• Interface structure can drastically affect growth kinetics if solute attachment sites are lacking.

• Coarsening of precipitates is driven by interface curvature.

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Appendix: Gibbs-Thomson relation• The Gibbs-Thomson relationship between pressure and curvature is based

on a differential relationship between rate of change of surface area and the chemical potential.

• The definition of interface energy is through the contribution of the interface to the (free) energy of the system.

G = U + PV + ST + Afrom which dG = dU + PdV + dPV + SdT + dST + dA + dA

• Under isothermal, isobaric conditions with no change in interfacial energy, we can simplify this to read, dG = dU + PdV + dST + dA

• We are interested in characterizing a system that has a constant amount of material and an isolated grain, for example, enclosed within another grain.

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Gibbs-Thomson, contd.• The island grain can expand or contract. Clearly the free energy of the system

decreases as it shrinks so chemical potential associated with the interface is positive.

• Now we equate the change in free energy to the chemical potential multiplied by the number of atoms transferred.

dG = ∆µ dn = dA

but we know that dA= -8πrdr and dn= dV/, where is the atomic volume. Thus,

∆µ = dA/dn = -2 /r

• Similarly, if dG, dS=0 (in a bubble), one obtains P= 2/r

r dr