Higgins R.A. Engineering Metallurgy: Applied Physical Metallurgy

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concentration and diffusion will ultimately cease when a uniform distri-

bution of solute atoms has been achieved.

The Schottky defect (3.62) is formed by a similar progressive movement

of atoms to that described above, whereby a vacancy is left behind in the

lattice and a corresponding atom is deposited at the surface of the metal.

Such a process takes place due to internal strains within the lattice.

8.25 As long ago as 1855 Adolph Fick proposed rules governing the dif-

fusion of substances in general. Here we are concerned with their application

to diffusion in metallic alloys. Fick's Law states that the flow or movement

(J) of atoms across unit area of a plane (Fig. 8.5) at any given instant is

— at the same instant

but of opposite sign, ie,

Here Z), the diffusion coefficient, is a constant for the system but will

vary for other compositions of alloy. It is in units ^

£—*- and is usually

time

expressed as 10~

/s. The significance of the negative sign preceding D is

that the movement of solute atoms is taking place 'down' the concentration

gradient.

When -j=0, then / =

and this satisfies the requirement that diffusion

ceases when the concentration gradient reaches zero as the solid solution

becomes homogeneous (line CD in Fig. 8.6). In practice coring is never

completely eliminated. It is reduced quite rapidly in the early stages of an

annealing process but traces still remain even after prolonged treatment

—mathematically of course CD (Fig. 8.6) only becomes horizontal after

an infinitely long time interval. Traces of coring are however generally

acceptable in most industrial processing.

Here we have considered diffusion in only one direction—parallel to the

jc-axis in which the concentration gradient is operating. In practice diffusion

will also take place along y and z axes.

Fig.

8.5.

area

Fig.

8.6 AB is the initial concentration gradient and CD the concentration gradient after a

long time during which diffusion has been taking place.

There are similarities in principle between Fick's Law and other Laws

such as Ohm's Law. The concentration gradient operates in a similar way

to potential difference as the driving force and D is independent of the

value of j— in the same way as electrical resistance is independent of

potential difference. Similar principles also govern heat flow through solids

where thermal conductivity is independent of the magnitude of the tem-

perature gradient. All such properties, ie electrical and thermal conduc-

tivities, viscosity and diffusion are known as transport properties of a

substance.

Fick's Law assumes that diffusion is proceeding under steady-state con-

ditions and that the concentration gradient at any given point will not

change with time. That is, we are as it were 'feeding in' solute atoms at

one end of a piece of alloy and extracting an equal number at the other

end. In practice this will not be the case and in a given piece of alloy matter

will be conserved: its overall or 'average' composition remains constant

and only the positions of its atoms vary in terms of x and t.

Consider the situation where, due to coring in an alloy, a certain distri-

bution of concentration is set up and that subsequently, due to heat activa-

tion (eg annealing), the atoms diffuse so as to attempt to reach a uniform

concentration (eliminate coring). Concentrations are therefore changing

with time and atoms must be accumulating in the region between plane

and plane (JCO + dx) or moving from it (Fig. 8.5). Hence the number

crossing area A of the plane, x

, is not equal to that crossing the same area

(JCO

+ dx). The flux entering this section is:

CONCENTRATION

ATOMS

IN SOLID

SOLUTION

DISTANCE X

The flux leaving

the

section

can be

written:

where

Jxo

+ dx = Jxo + h-

)

+ • • • • + (

The

her

terms

x=x

can be neglected)

The rate of movement of atoms from the section Jt

+ dx is equal to the

difference between the two values of AJ and also equal to the volume of

the section, A.dx times the rate of decrease of c:

. , Sc

—. Adx = —. Adx

bx bt

Combining (2) with Fick's Law (1) and eliminating /:

bt bx \ bx)

assuming that D is a constant independent of the concentration, c. This

used to be known as Fick's Second Law of Diffusion but is now more

generally referred to as The diffusion equation. Since c depends upon x

and t it could be written as c

(Xf t)

In this system of three equations (1) is an experimental law linking the

flow of atoms at any point with the concentration gradient there, (2) is the

continuity equation expressing the fact that atoms cannot disappear whilst

(3) combines these two equations. Equation (1) applies only to steady-state

conditions where these conditions do not vary with time. Equation (3)

applies to the general case where the concentration gradient at a certain

plane changes with time. Values of bc/bt and b

c/bx

can be determined

experimentally in order to determine the value of D for a given set of con-

ditions.

Differential coefficients vary with the nature of the solute atoms, the

nature of the solvent structure and with temperature.

8.26 In Fig. 8.1, and the illustrations which follow, both solvent and

solute atoms in substitutional solid solutions have been shown of equal

size,

whilst in interstitial solid solutions solute atoms have been shown as

very small and fitting easily into the interstices of the lattice. In the real

world such is not the case; both solvent and solute atoms have been rep-

resented in this way in order not to complicate the illustration unduly.

Atoms of different elements forming substitutional solid solutions will

never be of exactly the same size and for this reason some distortion of

the lattice structure of the parent metal will be caused by the presence of

atoms—whether larger or smaller—of the solute metal (Fig. 8.7(i)). Simi-

larly solute atoms of a size similar to those of the parent metal will sometimes

dissolve interstitially (Fig. 8.7(ii)). However, Fig. 8.7 does not represent the

situation completely since we are only describing the coplanar distribution

of atoms, ie along the x and y axes. In practice the distribution of solute

atoms and the distortion they produce will also occur along the z axis since

crystals are three-dimensional in nature.

Fig.

8.7 The distortion of lattice structures caused by the presence of (i) substitutional, (ii)

interstitial solute atoms.

Only a coplanar representation can be given here. In practice of course crystals are three-

dimensional.

Intermediate Phases

8.30 In the foregoing section we have been dealing with disordered substi-

tutional solid solutions in which there is no 'ordering' of the atoms of the

solute metal within the lattice of the solvent metal. Such solutions are

formed when the electrochemical properties of the metals are similar.

However, when two metals have widely divergent electrochemical proper-

ties they are likely to associate to form a chemical compound. Thus strongly

electropositive magnesium will combine with weakly electropositive tin to

form the substance Mg

Sn. This is generally described as an intermetallic

compound. Between these two extremes of substitutional solid solution on

the one hand and intermetallic compound on the other, phases are formed

which exhibit a gradation of properties according to the degree of associ-

ation which occurs between the atoms taking part. These phases are collec-

tively termed intermediate phases. At one extreme we have true

intermetallic compounds whilst at the other ordered structures which can

be more accurately classed as secondary solid solutions.

8.31 These intermediate phases can be classified into three main

groups:

1 Intermetallic compounds in which the laws of chemical valence

are apparently obeyed as in Mg

Sn, Mg

Pb, Mg

and Mg

. These

valence compounds are generally formed when one metal (such as mag-

nesium) has chemical properties which are strongly metallic, and the

other metal (such as antimony, tin or bismuth) chemical properties which

are only weakly metallic and, in fact, bordering on those of non-metals.

Frequently such a compound has a melting point which is higher than

that of either of the parent metals. For example, the intermetallic com-

pound Mg

Sn melts at 780

C, whereas the parent metals magnesium and

tin melt at 650 and 232°C respectively. This is an indication of the high

strength of the chemical bond in Mg

Sn.

2 Electron Compounds As was shown earlier (1.70) the chemical

valence of a metal is a function of the number of electrons in the outer

'shell' of the atom, whilst the nature of the metallic bond is such that

wholesale sharing of numbers of electrons takes place in the crystal

structure of a pure metal.

In these 'electron compounds' the normal valence laws are not obeyed,

but in many instances there is a fixed ratio between the total number of

valence bonds of all the atoms involved and the total number of atoms

in the empirical formula of the compound in question.

There are three such ratios, commonly referred to as Hume-Rothery

ratios:

(i) Ratio 3/2 (21/14)—(3 structures, such as CuZn, Cu

Al, Cu

Sn,

Al, etc.

(ii) Ratio 21/13—y structures, such as Cu

, G19AI4, Cu

iSn

, Na

iPb

, etc.

(iii) Ratio 7/4 (21/12)—e structures, such as CuZn

, Cu

Sn, AgCd

, etc.

Thus,

in the compound CuZn, copper has a valence of 1 and zinc a

valence of 2, giving a total of 3 valences and hence a ratio of 3 valences

to 2 atoms. In the compound Cu

iSn

copper has a valence of 1 and tin

a valence of 4. Therefore 31 valences are donated by the copper atoms

and 32 (ie, 4 X 8) by the tin atoms, making a total of 63 valences. In

all,

39 atoms are present in the empirical formula of Cu

iSn

therefore

the ratio

Total number of valences 63 21

Total number of atoms 39 13

These Hume-Rothery ratios have been valuable in relating structures

which were apparently unrelated. There are, however, many electron

compounds which do not fall into any of the three groups mentioned

above, nor are they valence compounds. Electron compounds are

formed as a result of the nature of the metallic bond and the stability of

such a substance depends upon electron concentration and the lattice

structure involved.

3 Size-factor Compounds These are intermediate phases in which

compositions and crystal structures arrange themselves in such a way as

to allow the constituent atoms to pack themselves closely together.

In the Laves phases compositions are based upon the general formula

eg MgNi

, MgCu

, TiCr

and MnBe

. Their formation depends

upon the fact that the constituent atoms vary in size by about 22.5% but

that they can none the less pack closely together in crystal structures.

A very important group of size-factor compounds are the interstitial

compounds formed between some transition metals and certain small non-

metallic atoms. When the solid solubility of an interstitially dissolved

element is exceeded a compound is precipitated from the solid solution.

In this type of compound the small non-metal atoms still occupy interstitial

positions but the overall crystal structure of the compound is different from

that of the original interstitial solid solution. Compounds of this type have

metallic properties and comprise hydrides, nitrides, borides and carbides

of which TiH

, TiN, Mn

TiB

TaC, W

C, WC, Mo

C and Fe

C are

typical. All of these compounds are extremely hard and the carbides find

application in tool steels and cemented-carbide cutting materials. Fe

C is

of course the phase cementite of ordinary carbon steels. Many of these

carbides are extremely refractory, having melting points well in excess of

300O

8.32 Many intermediate phases are extremely hard and brittle so that

a small ingot of such a substance may often be crushed to a powder by

gentle pressure in the jaws of a vice. This is particularly true of valence

compounds where covalent or ionic bonds tend to replace the metallic

bond, thus making the substance essentially non-metallic in its properties

as indicated by brittleness and a low electrical conductivity. Consequently

such phases are of limited use in engineering alloys and then in relatively

small proportions of the total microstructure.

Small quantities of intermediate phases—usually electron compounds—

are generally present in bearing alloys. Here the hard particles of com-

pound are embedded in a matrix of tough solid solution. The compound

resists wear and has a low coefficient of friction, whilst the solid solution

provides the necessary tough matrix capable of withstanding mechanical

shock and high compressive forces.

Intermediate phases are often easy to identify as such under the micro-

scope. Frequently they are of an unexpected colour but this is explained

by the fact that crystal structures and lattice distances are different from

those of the parent metals. (Lattice distances control the wavelength of

light reflected from a surface.) Thus Cu

iSn

is pale blue whilst Cu

Sb is

bright purple. Since intermediate phases are of ordered structure, coring

is not present and the microstructure appears uniform.

8.33 The good high-temperature properties of many of th£ stable inter-

metallic compounds suggests that they could be used as engineering

materials, particularly in the aerospace industries, but for their low ductility

and extremely poor impact toughness. Nevertheless research is now in

progress with a view to improving ductility and hence 'fabricability' of

some of the more promising compounds.

Thus the compound Ni

Al has a melting point of 1390

C and an ordered

FCC structure. Single crystals of the compound are very ductile, as are most

FCC structures (4.12), but in the normal polycrystalline state it suffers from

extreme grain-boundary brittleness. The addition of as little as 0.05%

boron raises the percentage elongation from zero to 50 as the fracture

mode is changed from being intercrystalline to a mixture of intercrystalline

and transcrystalline. Formability is therefore dramatically improved. This

effect is probably due to the segregation of boron at the crystal boundaries

causing some disorder in the boundary region. Stresses due to dislocation

'pile-ups' there are more evenly distributed and can be relieved by slip

across the boundary rather than by fracture along the boundary only.

Eutectics and Eutectoids

8.40 When two metals which are completely soluble in the liquid state

but completely insoluble in the solid state solidify, they do so by crystallis-

ing out as alternate layers of the two pure metals. Thus a laminated type

of structure is obtained and is termed a eutectic. This eutectic mixture is

always of a fixed composition for the two metals concerned and it melts at

a temperature below the melting points of either of the metals.

Consider the two metals cadmium and bismuth which are completely

soluble in each other as liquids but completely insoluble as solids. If

increasing amounts of cadmium are added to bismuth then the freezing

point of bismuth (TIl

C) is progressively depressed (along AE in Fig. 8.8).

Similarly if increasing amounts of bismuth are added to cadmium then the

freezing point of cadmium (321°C) decreases progressively (along BE).

These depression-of-freezing point lines meet at a minimum in E which is

called the eutectic point. This indicates the composition of the eutectic

mixture (60% wt. Bi/ 40% wt. Cd) and also the temperature (140

C) at

which it freezes. The relative thicknesses of the layers of bismuth and

cadmium will be so adjusted as to give an overall eutectic composition as

indicated by E. Suppose the overall composition of a bismuth-cadmium

alloy were given by X. This indicates that bismuth is present in the molten

alloy in excess of the eutectic composition. Therefore some bismuth will

crystallise out first until by the time that the temperature has fallen to

TEMPERATURE

COMPOSITION (°/owt.)

Fig.

8.8 The

eutectic point

(E) for

cadmium/bismuth alloys.

32I°C

IOOCtf

OBi

140

C the remaining liquid will be of composition E. This liquid will then

crystallise out as eutectic. Similarly if the overall composition of the initial

liquid mixture were given by Y then some cadmium would solidify first until

the remaining liquid were of composition E. Eutectic would then form as

before, ie as alternate layers of pure bismuth and pure cadmium. The

mechanism of this type of crystallisation will be discussed more fully in the

next chapter (9.40).

8.41 In a eutectic of two pure metals there is no question whatsoever

of solution, since the layers of the pure metals forming the eutectic can be

seen quite clearly under the microscope, at magnifications usually between

100 and 500, as definite separate entities. The formation of the eutectic is

therefore a result of insolubility being introduced when the alloy solidifies.

At this point it must be admitted that it is very doubtful whether com-

plete insolubility exists in the solid state in any alloy system. There is nearly

always some solid solubility however slight it may be. However, if the two

metals in question are partially soluble in each other in the solid state, we

may still obtain a eutectic on solidification, but it will be a eutectic

composed of alternate layers of two saturated solid solutions. One layer

will consist of metal A saturated with metal B, and the other a layer of

metal B saturated with metal A. This statement may at first cause some

confusion in the mind of the reader, but it must be understood that the

two layers are not one and the same as far as composition is concerned,

since one layer is rich in metal A and the other is rich in metal B. Consider

as a parallel case in liquid solutions the substances water and ether. If a

few spots of ether are added to a test-tube nearly filled with water and the

tube shaken, a single solution of ether in water will result. If, on the other

hand, a few spots of water are added to a test-tube nearly filled with ether,

and the tube shaken, a single solution of water in ether will result. When,

however, we put into the test-tube equal volumes of ether and water and

again shake, we find that we are left with two layers. The upper layer will be

ether saturated with water, and the lower layer will be water saturated

with ether. The upper layer is a solution containing most of the ether, and

the lower layer a different solution containing most of the water. We have

a similar situation in the case of metallic solid solutions, in that one phase

of a eutectic may be a solid solution rich in metal A and the other phase

a solid solution rich in metal B.

8.42 Not all eutectics, whether of pure metals or solid solutions, appear

under the microscope in the well-defined laminated form mentioned above.

Usually layers of one of the phases are embedded in a matrix of the other

phase, and quite often these layers are broken or disjointed. Fig. 17.2 illus-

trates the aluminium-silicon eutectic structure in which the thinner silicon

layers (the eutectic mixture contains only 11.7% silicon) are broken into

plate form and are embedded in an aluminium matrix. This is indeed fortu-

nate since silicon is brittle and if present as continuous layers would seriously

impair the properties of this very useful alloy.

Again, surface tension may influence crystallisation and the final eutectic

may consist of globules of one phase embedded in a matrix of the other.

The effect of surface tension will be accentuated by slow rates of cooling

which allow layers of one phase to break up; first, into smaller, thicker

plates,

and ultimately, into rounded globules (Fig. 11.8).

8.43 Sometimes a solid solution which has already formed in an alloy

transforms at a lower temperature to a eutectic type of structure. When

this happens, the eutectic type of structure produced is called a eutectoid,

since it was not formed from a liquid solution like a eutectic, but from a

solid solution. The transformation of the solid solution, austenite, to the

eutectoid, pearlite (7.55), is an example of this type of change. Cementite

nuclei form at random at the crystal boundaries of the austenite, and these

nuclei initiate the growth of cementite plates in the direction in which the

concentration of carbon in the austenite is highest. As a result of the

extraction of the carbon from the surrounding austenite to form cementite,

ferrite will nucleate alongside the cementite. In this way cementite and

ferrite plates will develop alongside each other (Fig. 8.9)

Fig.

8.9 The mechanism of the austenite -» pearlite transformation.

If an alloy is cooled very rapidly from a temperature above that at which

the eutectoid begins to form, the solid solution (or possibly a modified

form of it such as martensite in steels) is often retained. This demonstrates

that the eutectoid has not formed from the liquid but had been produced

at a later stage.

8.50 Many useful alloys consist structurally of a single uniform solid

solution in which increased strength is developed as suggested in the follow-

ing section (8.60). Such alloys are generally also ductile and corrosion

austenite

cementite

ferrite

resistant (21.71). Yet other useful alloys are of duplex structure, that is,

they contain two different phases in the microstructure. These two different

phases may be solid solutions in a eutectic type of structure or they may

be a solid solution along with some form of intermediate phase. Since

many intermediate phases are weak and brittle it is usually only those

which are hard and fairly strong which find use in metallurgical alloys, and

then only in relatively small quantities. In addition to their use in bearing

alloys as already mentioned, such phases may be employed to increase

strength by some form of precipitation hardening process (9.92) or disper-

sion hardening mechanism (8.62).

It is the function of the metallurgist, by control of his alloy compositions

along with suitable mechanical and thermal treatments, to produce alloys

which will provide a set of physical and mechanical properties which fulfil

the requirements of the engineer.

Strengthening Mechanisms in Alloys

8.60 As mentioned earlier in this chapter, the main reason for alloying

is to increase the yield strength of a metal. This involves impeding the

movement of dislocations by making alterations to the structure on

approximately the atomic scale.

8.61 Solid-solution Hardening In a cold-worked metal the presence

of a dislocation causes distortion of that part of the lattice structure near

to it. This distortion, and the energy associated with it, can be reduced by

the presence of solute atoms. Large substitutional solute atoms will reduce

distortion if they take up positions where the lattice structure is being

stretched (Fig. 8.10(i)) due to the presence of a dislocation; whilst small

substitutional solute atoms will have a similar effect if they replace solvent

atoms in regions where the lattice is being compressed (Fig. 8.10(ii)). When

these so-called 'atmospheres' of solute atoms are produced, the movement

of dislocations will be impeded and a greater stress must be applied to

move them. That is to say, the yield point has been raised.

In interstitial solid solutions the relatively small solute atoms will tend

to occupy positions where the lattice is being extended (Fig. 8.10(Hi)) since

Fig.

8.10 The 'pegging' of dislocations by the presence of solute atoms.