Mechanisms of diffusion

6.4.2 Mechanisms of diffusion

The transport of atoms through the lattice may conceiv-

2 0 ably occur in many ways. The term ‘interstitial diffu- sion’ describes the situation when the moving atom

where c 0 is the initial solute concentration in the alloy does not lie on the crystal lattice, but instead occu- and c is the concentration at a time t at a distance

pies an interstitial position. Such a process is likely x from the interface. The integral term is known as

in interstitial alloys where the migrating atom is very the Gauss error function (erf (y)) and as y ! 1,

small (e.g. carbon, nitrogen or hydrogen in iron). In erf ⊲y⊳ ! 1. It will be noted that at the interface where

this case, the diffusion process for the atoms to move x D 0, then c D c 0 / 2 2, and in those regions where the 2 from one interstitial position to the next in a perfect

curvature ∂ c/∂x is positive the concentration rises, lattice is not defect-controlled. A possible variant of in those regions where the curvature is negative the

this type of diffusion has been suggested for substitu- concentration falls, and where the curvature is zero

tional solutions in which the diffusing atoms are only the concentration remains constant.

temporarily interstitial and are in dynamic equilibrium This particular example is important because it can

with others in substitutional positions. However, the

be used to model the depth of diffusion after time energy to form such an interstitial is many times that to t , e.g. in the case-hardening of steel, providing the

produce a vacancy and, consequently, the most likely concentration profile of the carbon after a carburizing

mechanism is that of the continual migration of vacan- time t, or dopant in silicon. Starting with a constant

cies. With vacancy diffusion, the probability that an composition at the surface, the value of x where

atom may jump to the next site will depend on: (1) the the concentration falls to half the initial value, i.e.

probability that the site is vacant (which in turn is pro-

2 , is given by x D ⊲Dt⊳ . Thus knowing portional to the fraction of vacancies in the crystal),

D at a given temperature the time to produce a given and (2) the probability that it has the required activa- depth of diffusion can be estimated.

tion energy to make the transition. For self-diffusion The diffusion equations developed above can also be

where no complications exist, the diffusion coefficient transformed to apply to particular diffusion geometries.

is therefore given by

If the concentration gradient has spherical symmetry about a point, c varies with the radial distance r and,

DD 1 6 2 a exp [⊲S f C S m ⊳/k ] for constant D,

f /kT

m /kT ]

dc d c 2 dc DD 2 C dt (6.8) dr r dr

D D 0 f C E m ⊳/kT ] ⊲ 6.12⊳ When the diffusion field has radial symmetry about a

The factor f appearing in D 0 is known as a correla- cylindrical axis, the equation becomes

tion factor and arises from the fact that any particular diffusion jump is influenced by the direction of the

previous jump. Thus when an atom and a vacancy dt

dc d 2 c 1 dc

D D C (6.9)

dr 2 r dr exchange places in the lattice there is a greater prob- ability of the atom returning to its original site than

and the steady-state condition ⊲dc/dt⊳ D 0 is given by moving to another site, because of the presence there

d 2 c 1 dc of a vacancy; f is 0.80 and 0.78 for fcc and bcc

2 C r D 0 dr (6.10) dr lattices, respectively. Values for E f and E m are dis- cussed in Chapter 4, E f is the energy of formation of which has a solution c D Alnr C B. The constants A

a vacancy, E m the energy of migration, and the sum and B may be found by introducing the appropriate

of the two energies, Q D E f C E m , is the activation

energy for self-diffusion 1 E d . cDc 1 at r D r 1 the solution becomes

boundary conditions and for c D c 0 at r D r 0 and

c 0 /r⊳ C c

1 The entropy factor exp [⊲S f C S m ⊳/k ] is usually taken to be ln⊲r 1 /r 0 ⊳

cD ln⊲r 1 1 ln⊲r/r 0 ⊳

unity.

The physical properties of materials 175 In alloys, the problem is not so simple and it is

found that the self-diffusion energy is smaller than in pure metals. This observation has led to the sugges- tion that in alloys the vacancies associate preferentially with solute atoms in solution; the binding of vacancies to the impurity atoms increases the effective vacancy concentration near those atoms so that the mean jump rate of the solute atoms is much increased. This asso- ciation helps the solute atom on its way through the lattice, but, conversely, the speed of vacancy migration is reduced because it lingers in the neighbourhood of

Figure 6.8 ˛-brass–copper couple for demonstrating the the solute atoms, as shown in Figure 6.7. The phe-

Kirkendall effect .

nomenon of association is of fundamental importance in all kinetic studies since the mobility of a vacancy through the lattice to a vacancy sink will be governed

some practical importance, especially in the fields of by its ability to escape from the impurity atoms which

metal-to-metal bonding, sintering and creep. trap it. This problem has been mentioned in Chapter 4.

When considering diffusion in alloys it is impor-

6.4.3 Factors affecting diffusion

tant to realize that in a binary solution of A and B The two most important factors affecting the diffu-

sion coefficient D are temperature and composition. equal. This inequality of diffusion was first demon-

the diffusion coefficients D A and D B are generally not

Because of the activation energy term the rate of diffu- strated by Kirkendall using an ˛-brass/copper couple

sion increases with temperature according to equation (Figure 6.8). He noted that if the position of the inter-

(6.12), while each of the quantities D, D 0 and Q faces of the couple were marked (e.g. with fine W or

varies with concentration; for a metal at high temper- Mo wires), during diffusion the markers move towards

atures Q ³ 20RT m ,D 0 is 10 5 to 10 3 m 2 s 1 , and each other, showing that the zinc atoms diffuse out of

D' 10 12 m 2 s 1 . Because of this variation of diffu- the alloy more rapidly than copper atoms diffuse in.

sion coefficient with concentration, the most reliable This being the case, it is not surprising that several

investigations into the effect of other variables neces- workers have shown that porosity develops in such

sarily concern self-diffusion in pure metals. systems on that side of the interface from which there

Diffusion is a structure-sensitive property and, is a net loss of atoms.

therefore, D is expected to increase with increasing The Kirkendall effect is of considerable theoretical

lattice irregularity. In general, this is found experi- importance since it confirms the vacancy mechanism

mentally. In metals quenched from a high temper- of diffusion. This is because the observations cannot

ature the excess vacancy concentration ³10 9 leads easily be accounted for by any other postulated

to enhanced diffusion at low temperatures since D D mechanisms of diffusion, such as direct place-

m /kT⊳ . Grain boundaries and disloca- exchange, i.e. where neighbouring atoms merely

tions are particularly important in this respect and change place with each other. The Kirkendall effect

produce enhanced diffusion. Diffusion is faster in the is readily explained in terms of vacancies since the

cold-worked state than in the annealed state, although lattice defect may interchange places more frequently

recrystallization may take place and tend to mask the with one atom than the other. The effect is also of

effect. The enhanced transport of material along dislo- cation channels has been demonstrated in aluminium where voids connected to a free surface by dislo- cations anneal out at appreciably higher rates than isolated voids. Measurements show that surface and grain boundary forms of diffusion also obey Arrhe- nius equations, with lower activation energies than for volume diffusion, i.e. Q vol ½ 2Q g.b ½ 2Q surface . This behaviour is understandable in view of the progres- sively more open atomic structure found at grain boundaries and external surfaces. It will be remem- bered, however, that the relative importance of the various forms of diffusion does not entirely depend on the relative activation energy or diffusion coefficient values. The amount of material transported by any dif- fusion process is given by Fick’s law and for a given composition gradient also depends on the effective area

Figure 6.7 Solute atom–vacancy association during through which the atoms diffuse. Consequently, since diffusion .

the surface area (or grain boundary area) to volume

176 Modern Physical Metallurgy and Materials Engineering ratio of any polycrystalline solid is usually very small,

it is only in particular phenomena (e.g. sintering, oxi- dation, etc.) that grain boundaries and surfaces become important. It is also apparent that grain boundary diffu- sion becomes more competitive, the finer the grain and the lower the temperature. The lattice feature follows from the lower activation energy which makes it less sensitive to temperature change. As the temperature is lowered, the diffusion rate along grain boundaries (and also surfaces) decreases less rapidly than the dif- fusion rate through the lattice. The importance of grain boundary diffusion and dislocation pipe diffusion is discussed again in Chapter 7 in relation to deformation

Figure 6.9 Anelastic behaviour . at elevated temperatures, and is demonstrated con-

vincingly on the deformation maps (see Figure 7.68), where the creep field is extended to lower temperatures

is often used, where ω 1 and ω 2 are the frequencies on when grain boundary (Coble creep) rather than lattice

the two sides of the resonant frequency ω p 0 at which diffusion (Herring–Nabarro creep) operates.

2 of the resonant Because of the strong binding between atoms, pres-

the amplitude of oscillation is 1/

amplitude. Also used is the specific damping capacity sure has little or no effect but it is observed that with

E/E , where E is the energy dissipated per cycle extremely high pressure on soft metals (e.g. sodium)

of vibrational energy E, i.e. the area contained in a an increase in Q may result. The rate of diffusion

stress–strain loop. Yet another method uses the phase also increases with decreasing density of atomic pack-

angle ˛ by which the strain lags behind the stress, and ing. For example, self-diffusion is slower in fcc iron

if the damping is small it can be shown that or thallium than in bcc iron or thallium when the

results are compared by extrapolation to the transfor-

ω 2 1 mation temperature. This is further emphasized by the

1 E

tan ˛ D D E (6.13) D ω DQ anisotropic nature of D in metals of open structure.

Bismuth (rhombohedral) is an example of a metal in which D varies by 10 6

By analogy with damping in electrical systems tan ˛ for different directions in the

is often written equal to Q . lattice; in cubic crystals D is isotropic.

There are many causes of internal friction arising from the fact that the migration of atoms, lattice defects and thermal energy are all time-dependent

processes. The latter gives rise to thermoelasticity and For an elastic solid it is generally assumed that stress

6.5 Anelasticity and internal friction

occurs when an elastic stress is applied to a specimen and strain are directly proportional to one another, but

too fast for the specimen to exchange heat with its in practice the elastic strain is usually dependent on

surroundings and so cools slightly. As the sample time as well as stress so that the strain lags behind the

warms back to the surrounding temperature it expands stress; this is an anelastic effect. On applying a stress at

thermally, and hence the dilatation strain continues to

a level below the conventional elastic limit, a specimen increase after the stress has become constant.

The diffusion of atoms can also give rise to gradual increase in strain until it reaches an essentially

will show an initial elastic strain ε e followed by a

anelastic effects in an analogous way to the diffusion constant value, ε e C ε an as shown in Figure 6.9. When

of thermal energy giving thermoelastic effects. A the stress is removed the strain will decrease, but a

particular example is the stress-induced diffusion of small amount remains which decreases slowly with

carbon or nitrogen in iron. A carbon atom occupies time. At any time t the decreasing anelastic strain is

the interstitial site along one of the cell edges slightly given by the relation ε D ε an

distorting the lattice tetragonally. Thus when iron known as the relaxation time, and is the time taken

is stretched by a mechanical stress, the crystal axis for the anelastic strain to decrease to 1/e ' 36.79% of

oriented in the direction of the stress develops favoured sites for the occupation of the interstitial atoms

very slowly, while if small the strain relaxes quickly. relative to the other two axes. Then if the stress is In materials under cyclic loading this anelastic effect

oscillated, such that first one axis and then another is leads to a decay in amplitude of vibration and therefore

stretched, the carbon atoms will want to jump from

a dissipation of energy by internal friction. Internal one favoured site to the other. Mechanical work is friction is defined in several different but related ways.

therefore done repeatedly, dissipating the vibrational Perhaps the most common uses the logarithmic decre-

energy and damping out the mechanical oscillations. ment υ D ln⊲A n /A nC 1 ⊳ , the natural logarithm of suc-

The maximum energy is dissipated when the time per cessive amplitudes of vibration. In a forced vibration

cycle is of the same order as the time required for the experiment near a resonance, the factor ⊲ω 2 1 ⊳/ω 0 diffusional jump of the carbon atom.

The physical properties of materials 177

Figure 6.10 Schematic diagram of a KOe torsion pendulum . The simplest and most convenient way of studying

this form of internal friction is by means of a KOe Figure 6.11 Internal friction as a function of temperature torsion pendulum, shown schematically in Figure 6.10. for Fe with C in solid solution at five different pendulum

frequencies (from Wert and Zener, 1949; by permission of The specimen can be oscillated at a given frequency

the American Institute of Physics) . by adjusting the moment of inertia of the torsion bar. The energy loss per cycle E/E varies smoothly with the frequency according to the relation

discussed above, and hence occurring in different fre- E

E quency and temperature regions. One important source D2 2 E of internal friction is that due to stress relaxation across E

grain boundaries. The occurrence of a strong internal and has a maximum value when the angular frequency

max

friction peak due to grain boundary relaxation was first of the pendulum equals the relaxation time of the

demonstrated on polycrystalline aluminium at 300 ° C process; at low temperatures around room temperature

by Kˆe and has since been found in numerous other this is interstitial diffusion. In practice, it is difficult to

metals. It indicates that grain boundaries behave in vary the angular frequency over a wide range and thus

a somewhat viscous manner at elevated temperatures it is easier to keep ω constant and vary the relaxation

and grain boundary sliding can be detected at very low time. Since the migration of atoms depends strongly on

stresses by internal friction studies. The grain boundary temperature according to an Arrhenius-type equation,

1 D 1/ω 1 and the peak occurs at a temperature T 1 . For a different frequency value

the melting point, assuming the boundary thickness to so on (see Figure 6.11). It is thus possible to ascribe

ω 2 the peak occurs at a different temperature T 2 , and

be d ' 0.5 nm.

an activation energy H for the internal process Movement of low-energy twin boundaries in crys- tals, domain boundaries in ferromagnetic materials and or from the relation

dislocation bowing and unpinning all give rise to inter- ln⊲ω 2 /ω 1 ⊳

nal friction and damping.

H D R 1/T 1 2

In the case of iron the activation energy is found to coincide with that for the diffusion of carbon in iron.

6.6 Ordering in alloys

Similar studies have been made for other metals. In

6.6.1 Long-range and short-range order

, and

an atom stays in an interstitial position is ⊲ 3 An ordered alloy may be regarded as being made up

of two or more interpenetrating sub-lattices, each con-

taining different arrangements of atoms. Moreover, the previously the diffusion coefficient may be calculated

from the relation D D 24 a v for bcc lattices derived

term ‘superlattice’ would imply that such a coher- directly from

ent atomic scheme extends over large distances, i.e.

1 a 2 the crystal possesses long-range order. Such a perfect DD 36 arrangement can exist only at low temperatures, since

the entropy of an ordered structure is much lower than Many other forms of internal friction exist in met-

that of a disordered one, and with increasing tempera- als arising from different relaxation processes to those

ture the degree of long-range order, S, decreases until

178 Modern Physical Metallurgy and Materials Engineering

at a critical temperature T c it becomes zero; the general

form of the curve is shown in Figure 6.12. Partially- ordered structures are achieved by the formation of small regions (domains) of order, each of which are separated from each other by domain or anti-phase domain boundaries, across which the order changes phase (Figure 6.13). However, even when long-range order is destroyed, the tendency for unlike atoms to be neighbours still exists, and short-range order results

above T Figure 6.13 An antiphase domain boundary

c . The transition from complete disorder to . complete order is a nucleation and growth process and

may be likened to the annealing of a cold-worked has pointed out that the ease with which interlocking

domains can absorb each other to develop a scheme are more than the random number of AB atom pairs,

structure. At high temperatures well above T c , there

of long-range order will also depend on the number of and with the lowering of temperature small nuclei

possible ordered schemes the alloy possesses. Thus, in of order continually form and disperse in an other-

ˇ -brass only two different schemes of order are possi- wise disordered matrix. As the temperature, and hence

ble, while in fcc lattices such as Cu 3 Au four different thermal agitation, is lowered these regions of order

schemes are possible and the approach to complete

become more extensive, until at T c they begin to link

order is less rapid.

together and the alloy consists of an interlocking mesh

of small ordered regions. Below T c these domains

absorb each other (cf. grain growth) as a result of

6.6.2 Detection of ordering

antiphase domain boundary mobility until long-range The determination of an ordered superlattice is usu- order is established.

ally done by means of the X-ray powder technique. In Some order–disorder alloys can be retained in a

a disordered solution every plane of atoms is statisti- state of disorder by quenching to room temperature

cally identical and, as discussed in Chapter 5, there are while in others (e.g. ˇ-brass) the ordering process

reflections missing in the powder pattern of the mate- occurs almost instantaneously. Clearly, changes in the

rial. In an ordered lattice, on the other hand, alternate degree of order will depend on atomic migration, so

planes become A-rich and B-rich, respectively, so that that the rate of approach to the equilibrium configu-

these ‘absent’ reflections are no longer missing but ration will be governed by an exponential factor of

appear as extra superlattice lines. This can be seen the usual form, i.e. Rate D Ae

. However, Bragg

from Figure 6.14: while the diffracted rays from the

A planes are completely out of phase with those from the B planes their intensities are not identical, so that

a weak reflection results.

Application of the structure factor equation indicates that the intensity of the superlattice lines is proportional to jF 2 jD S 2 ⊲f A B ⊳ 2 , from which it can be seen that in the fully-disordered alloy, where S D 0, the superlattice lines must vanish. In some alloys such as copper–gold, the scattering

factor difference ⊲f A B ⊳ is appreciable and the superlattice lines are, therefore, quite intense and easily detectable. In other alloys, however, such as iron–cobalt, nickel –manganese, copper–zinc, the

term ⊲f A B ⊳ is negligible for X-rays and the Figure 6.12 Influence of temperature on the degree of order .

super-lattice lines are very weak; in copper–zinc, for

Figure 6.14 Formation of a weak 100 reflection from an ordered lattice by the interference of diffracted rays of unequal amplitude .

The physical properties of materials 179 example, the ratio of the intensity of the superlattice

size can be obtained from a measurement of the line lines to that of the main lines is only about 1:3500.

breadth, as discussed in Chapter 5. Figure 6.15 shows In some cases special X-ray techniques can enhance

variation of order S and domain size as determined this intensity ratio; one method is to use an X-

from the intensity and breadth of powder diffraction ray wavelength near to the absorption edge when

lines. The domain sizes determined from the Scherrer an anomalous depression of the f-factor occurs

line-broadening formula are in very good agreement which is greater for one element than for the other.

with those observed by TEM. Short-range order is

much more difficult to detect but nowadays direct increased. A more general technique, however, is to

As a result, the difference between f A and f B is

measuring devices allow weak X-ray intensities to be use neutron diffraction since the scattering factors

measured more accurately, and as a result considerable for neighbouring elements in the Periodic Table can

information on the nature of short-range order has

be substantially different. Conversely, as Table 5.4 been obtained by studying the intensity of the diffuse indicates, neutron diffraction is unable to show the

background between the main lattice lines.

High-resolution transmission microscopy of thin scattering amplitudes of copper and gold for neutrons

existence of superlattice lines in Cu 3 Au, because the

metal foils allows the structure of domains to be exam- are approximately the same, although X-rays show

ined directly. The alloy CuAu is of particular interest, them up quite clearly.

since it has a face-centred tetragonal structure, often referred to as CuAu 1 below 380 °

C, but between 380 Sharp superlattice lines are observed as long as ° C

order persists over lattice regions of about 10 mm, and the disordering temperature of 410

C it has the large enough to give coherent X-ray reflections. When

CuAu 11 structures shown in Figure 6.16. The ⊲0 0 2⊳ long-range order is not complete the superlattice lines

planes are again alternately gold and copper, but half- become broadened, and an estimate of the domain

way along the a-axis of the unit cell the copper atoms switch to gold planes and vice versa. The spacing between such periodic anti-phase domain boundaries is 5 unit cells or about 2 nm, so that the domains are easily resolvable in TEM, as seen in Figure 6.17a. The isolated domain boundaries in the simpler superlat- tice structures such as CuAu 1, although not in this case periodic, can also be revealed by electron micro- scope, and an example is shown in Figure 6.17b. Apart from static observations of these superlattice struc- tures, annealing experiments inside the microscope also allow the effect of temperature on the structure to

be examined directly. Such observations have shown that the transition from CuAu 1 to CuAu 11 takes place, as predicted, by the nucleation and growth of anti-phase domains.