Engineering Materials Vol II (microstructures_ processing_ design) 2nd ed. - M. Ashby_ D. Jones (1999) WW Part 4 docx

Summary In this chapter we have shown that diffusive transformations can only take place if nuclei of the new phase can form to begin with.. Homogeneous nucleation, in defect-free region

and heterogeneous nucleation If Ω is the volume occupied by one atom in the nucleusthen we can easily see that

If the nucleus wets the catalyst well, with θ = 10°, say, then eqn (7.15) tells us that

r*het= 18.1r*hom In other words, if we arrange our 102 atoms as a spherical cap on a goodcatalyst we get a much bigger crystal radius than if we arrange them as a sphere And,

as Fig 7.4 explains, this means that heterogeneous nucleation always “wins” overhomogeneous nucleation

It is easy to estimate the undercooling that we would need to get heterogeneousnucleation with a 10° contact angle From eqns (7.11) and (7.3) we have

Fig 7.4. Heterogeneous nucleation takes place at higher temperatures because the maximum random fluctuation of 10 2 atoms gives a bigger crystal radius if the atoms are arranged as a spherical cap.

Nucleation in solids

Nucleation in solids is very similar to nucleation in liquids Because solids usuallycontain high-energy defects (like dislocations, grain boundaries and surfaces) newphases usually nucleate heterogeneously; homogeneous nucleation, which occurs indefect-free regions, is rare Figure 7.5 summarises the various ways in which nucleationcan take place in a typical polycrystalline solid; and Problems 7.2 and 7.3 illustratehow nucleation theory can be applied to a solid-state situation

Summary

In this chapter we have shown that diffusive transformations can only take place if

nuclei of the new phase can form to begin with Nuclei form because random atomic

vibrations are continually making tiny crystals of the new phase; and if the ature is low enough these tiny crystals are thermodynamically stable and will grow In

temper-homogeneous nucleation the nuclei form as spheres within the bulk of the material In

Fig 7.5. Nucleation in solids Heterogeneous nucleation can take place at defects like dislocations, grain boundaries, interphase interfaces and free surfaces Homogeneous nucleation, in defect-free regions, is rare.

heterogeneous nucleation the nuclei form as spherical caps on defects like solid surfaces,

grain boundaries or dislocations Heterogeneous nucleation occurs much more easilythan homogeneous nucleation because the defects give the new crystal a good “foothold”.Homogeneous nucleation is rare because materials almost always contain defects

Postscript

Nucleation – of one sort or another – crops up almost everywhere During winterplants die and people get frostbitten because ice nucleates heterogeneously insidecells But many plants have adapted themselves to prevent heterogeneous nucleation;they can survive down to the homogeneous nucleation temperature of −40°C The

“vapour” trails left by jet aircraft consist of tiny droplets of water that have nucleatedand grown from the water vapour produced by combustion Sub-atomic particles can

be tracked during high-energy physics experiments by firing them through heated liquid in a “bubble chamber”: the particles trigger the nucleation of gas bubbleswhich show where the particles have been And the food industry is plagued bynucleation problems Sucrose (sugar) has a big molecule and it is difficult to get it

super-to crystallise from aqueous solutions That is fine if you want super-to make caramel –this clear, brown, tooth-breaking substance is just amorphous sucrose But the sugarrefiners have big problems making granulated sugar, and will go to great lengths toget adequate nucleation in their sugar solutions We give examples of how nucleationapplies specifically to materials in a set of case studies on phase transformations inChapter 9

Further reading

D A Porter and K E Easterling, Phase Transformations in Metals and Alloys, 2nd edition, Chapman

and Hall, 1992.

G J Davies, Solidification and Casting, Applied Science Publishers, 1973.

G A Chadwick, Metallography of Phase Transformations, Butterworth, 1972.

to a temperature at which the equilibrium structure is two-phase (α + β) Duringcooling, small precipitates of the β phase nucleate heterogeneously at α grainboundaries The nuclei are lens-shaped as shown below.

Show that the free work needed to produce a nucleus is given by

where ∆G is the free work produced when unit volume of β forms from α Youmay assume that mechanical equilibrium at the edge of the lens requires thatγGB = 2γαβcosθ

Hence, show that the critical radius is given by

r* = 2γαβ/∆G.,

7.3 Pure titanium is cooled from a temperature at which the b.c.c phase is stable to atemperature at which the c.p.h phase is stable As a result, lens-shaped nuclei ofthe c.p.h phase form at the grain boundaries Estimate the number of atoms needed

to make a critical-sized nucleus given the following data: ∆H = 3.48 kJ mol–1;atomic weight = 47.90; Te – T = 30 K; T e = 882°C; γ = 0.1 Jm–2; density of the c.p.h.phase = 4.5 Mg m–3; θ = 5°

GB

α

α β

So far we have only looked at transformations which take place by diffusion: the

so-called diffusive transformations But there is one very important class of transformation – the displacive transformation – which can occur without any diffusion at all.

The most important displacive transformation is the one that happens in carbonsteels If you take a piece of 0.8% carbon steel “off the shelf” and measure its mechan-ical properties you will find, roughly, the values of hardness, tensile strength andductility given in Table 8.1 But if you test a piece that has been heated to red heat andthen quenched into cold water, you will find a dramatic increase in hardness (4 times

or more), and a big decrease in ductility (it is practically zero) (Table 8.1)

The two samples have such divergent mechanical properties because they haveradically different structures: the structure of the as-received steel is shaped by adiffusive transformation, but the structure of the quenched steel is shaped by a displacivechange But what are displacive changes? And why do they take place?

In order to answer these questions as directly as possible we begin by looking at

diffusive and displacive transformations in pure iron (once we understand how pure

iron transforms we will have no problem in generalising to iron–carbon alloys) Now,

as we saw in Chapter 2, iron has different crystal structures at different temperatures.Below 914°C the stable structure is b.c.c., but above 914°C it is f.c.c If f.c.c iron iscooled below 914°C the structure becomes thermodynamically unstable, and it tries tochange back to b.c.c This f.c.c → b.c.c transformation usually takes place by a diffu-sive mechanism But in exceptional conditions it can occur by a displacive mechanisminstead To understand how iron can transform displacively we must first look at thedetails of how it transforms by diffusion

Table 8.1 Mechanical properties of 0.8% carbon steel

Fig 8.1. The diffusive f.c.c → b.c.c transformation in iron The vertical axis shows the speed of the b.c.c.– f.c.c interface at different temperatures Note that the transformation can take place extremely rapidly, making

it very difficult to measure the interface speeds The curve is therefore only semi-schematic.

The diffusive f.c.c → b.c.c transformation in pure iron

We saw in Chapter 6 that the speed of a diffusive transformation depends strongly ontemperature (see Fig 6.6) The diffusive f.c.c → b.c.c transformation in iron shows thesame dependence, with a maximum speed at perhaps 700°C (see Fig 8.1) Now wemust be careful not to jump to conclusions about Fig 8.1 This plots the speed of anindividual b.c.c.–f.c.c interface, measured in metres per second If we want to know

the overall rate of the transformation (the volume transformed per second) then we need to know the area of the b.c.c.–f.c.c interface as well.

The total area of b.c.c.–f.c.c interface is obviously related to the number of b.c.c.nuclei As Fig 8.2 shows, fewer nuclei mean a smaller interfacial area and a smallervolume transforming per second Indeed, if there are no nuclei at all, then the rate oftransformation is obviously zero The overall rate of transformation is thus givenapproximately by

We know that the interfacial speed varies with temperature; but would we expect thenumber of nuclei to depend on temperature as well?

The nucleation rate is, in fact, critically dependent on temperature, as Fig 8.3 shows

To see why, let us look at the heterogeneous nucleation of b.c.c crystals at grain aries We have already looked at grain boundary nucleation in Problems 7.2 and 7.3.Problem 7.2 showed that the critical radius for grain boundary nucleation is given by

Fig 8.2. In a diffusive transformation the volume transforming per second increases linearly with the number of nuclei.

Grain boundary nucleation will not occur in iron unless it is cooled below perhaps910°C At 910°C the critical radius is

As Fig 8.3 shows, grain boundary nuclei will be geometrically similar at all

temper-atures The volume V* of the lens-shaped nucleus will therefore scale as (r*)3, i.e

Now, nucleation at 910°C will only take place if we get a random fluctuation of about

102 atoms (which is the maximum fluctuation that we can expect in practice) Nucleation

at 900°C, however, requires a random fluctuation of only (102/43) atoms.* The chances

of assembling this small number of atoms are obviously far greater than the chances

of assembling 102 atoms, and grain-boundary nucleation is thus much more likely

at 900°C than at 910°C At low temperature, however, the nucleation rate starts todecrease With less thermal energy it becomes increasingly difficult for atoms to dif-fuse together to form a nucleus And at 0 K (where there is no thermal energy at all)the nucleation rate must be zero

The way in which the overall transformation rate varies with temperature can now

be found by multiplying the dependences of Figs 8.1 and 8.3 together This final result

is shown in Fig 8.4 Below about 910°C there is enough undercooling for nuclei to

form at grain boundaries There is also a finite driving force for the growth of nuclei, so

the transformation can begin to take place As the temperature is lowered, the number

of nuclei increases, and so does the rate at which they grow: the transformation rateincreases The rate reaches a maximum at perhaps 700°C Below this temperaturediffusion starts to dominate, and the rate decreases to zero at absolute zero

Fig 8.3. The diffusive f.c.c → b.c.c transformation in iron: how the number of nuclei depends on

temperature (semi-schematic only).

* It is really rather meaningless to talk about a nucleus containing only two or three atoms! To define a b.c.c crystal we would have to assemble at least 20 or 30 atoms But it will still be far easier to fluctuate 30 atoms into position than to fluctuate 100 Our argument is thus valid qualitatively, if not quantitatively.

Fig 8.4. The diffusive f.c.c → b.c.c transformation in iron: overall rate of transformation as a function of temperature (semi-schematic).

The time –temperature –transformation diagram

It is standard practice to plot the rates of diffusive transformations in the form of time–temperature–transformation (TTT) diagrams, or “C-curves” Figure 8.5 shows the TTTdiagram for the diffusive f.c.c → b.c.c transformation in pure iron The general shape

of the C-curves directly reflects the form of Fig 8.4 In order to see why, let us startwith the “1% transformed” curve on the diagram This gives the time required for 1%

of the f.c.c to transform to b.c.c at various temperatures Because the transformation

rate is zero at both 910°C and −273°C (Fig 8.4) the time required to give 1%

transforma-tion must be infinite at these temperatures This is why the 1% curve tends to infinity

as it approaches both 910°C and −273°C And because the transformation rate is amaximum at say 700°C (Fig 8.4) the time for 1% transformation must be a minimum

at 700°C, which is why the 1% curve has a “nose” there The same arguments apply, ofcourse, to the 25%, 50%, 75% and 99% curves

The displacive f.c.c → → b.c.c transformation

In order to get the iron to transform displacively we proceed as follows We start with

f.c.c iron at 914°C which we then cool to room temperature at a rate of about 105°C s−1

As Fig 8.6 shows, we will miss the nose of the 1% curve, and we would expect to end

up with f.c.c iron at room temperature F.c.c iron at room temperature would beundercooled by nearly 900°C, and there would be a huge driving force for the f.c.c →b.c.c transformation Even so, the TTT diagram tells us that we might expect f.c.c iron

to survive for years at room temperature before the diffusive transformation could getunder way

In reality, below 550°C the driving force becomes so large that it cannot be

con-tained; and the iron transforms from f.c.c to b.c.c by the displacive mechanism Small

lens-shaped grains of b.c.c nucleate at f.c.c grain boundaries and move across the

Fig 8.5. The diffusive f.c.c → b.c.c transformation in iron: the time–temperature–transformation (TTT) diagram, or “C-curve” The 1% and 99% curves represent, for all practical purposes, the start and end

of the transformation Semi-schematic only.

Fig 8.6. If we quench f.c.c iron from 914°C to room temperature at a rate of about 10 5 °C s −1 we expect

to prevent the diffusive f.c.c → b.c.c transformation from taking place In reality, below 550°C the iron will transform to b.c.c by a displacive transformation instead.

Fig 8.7. The displacive f.c.c → b.c.c transformation in iron B.c.c lenses nucleate at f.c.c grain boundaries and grow almost instantaneously The lenses stop growing when they hit the next grain boundary Note that, when a new phase in any material is produced by a displacive transformation it is always referred to as

“martensite” Displacive transformations are often called “martensitic” transformations as a result.

Table 8.2 Characteristics of transformations

Displacive (also called diffusionless,

shear, or martensitic)

Atoms move over distances ⱽ interatomic spacing.

Atoms move by making and breaking interatomic

bonds and by minor “shuffling”.

Atoms move one after another in precise sequence

(“military” transformation).

Speed of transformation ≈ velocity of lattice vibrations

through crystal (essentially independent of

temperature); transformation can occur at

temperatures as low as 4 K.

Extent of transformation (volume transformed)

depends on temperature only.

Composition cannot change (because atoms have no

time to diffuse, they stay where they are).

Always specific crystallographic relationship between

martensite and parent lattice.

Speed of transformation depends strongly on temperature; transformation does not occur below 0.3 T m to 0.4 T m

Extent of transformation depends on time as well as temperature.

Diffusion allows compositions of individual phases to change in alloyed systems.

Sometimes have crystallographic relationships between phases.

Details of martensite formation

As Fig 8.8 shows, the martensite lenses are coherent with the parent lattice Figure 8.9shows how the b.c.c lattice is produced by atomic movements of the f.c.c atoms in the

“switch zone” As we have already said, at ≈ 550°C martensite lenses form and growalmost instantaneously As the lenses grow the lattice planes distort (see Fig 8.8) andsome of the driving force for the f.c.c → b.c.c transformation is removed as strainenergy Fewer lenses nucleate and grow, and eventually the transformation stops Inother words, provided we keep the temperature constant, the displacive transforma-tion is self-stabilising (see Fig 8.10) To get more martensite we must cool the irondown to a lower temperature (which gives more driving force) Even at this lowertemperature, the displacive transformation will stop when the extra driving force hasbeen used up in straining the lattice In fact, to get 100% martensite, we have to coolthe iron down to ≈ 350°C (Fig 8.10)

Fig 8.8. Martensites are always coherent with the parent lattice They grow as thin lenses on preferred planes and in preferred directions in order to cause the least distortion of the lattice The crystallographic relationships shown here are for pure iron.

Fig 8.9 (a) The unit cells of f.c.c and b.c.c iron (b) Two adjacent f.c.c cells make a distorted b.c.c cell.

If this is subjected to the “Bain strain” it becomes an undistorted b.c.c cell This atomic “switching” involves the least shuffling of atoms As it stands the new lattice is not coherent with the old one But we can get coherency

by rotating the b.c.c lattice planes as well (Fig 8.8).

Fig 8.10. The displacive f.c.c → b.c.c transformation in iron: the volume of martensite produced is a function of temperature only, and does not depend on time Note that the temperature at which martensite starts to form is labelled M s (martensite start); the temperature at which the martensite transformation finishes

is labelled M (martensite finish).