Diffusion .1 Diffusion laws - The physical properties of materials

The physical properties of materials

6.4 Diffusion .1 Diffusion laws

172 Modern Physical Metallurgy and Materials Engineering relative magnitude of the free energy value governs the stability of any phase, and from Figure 3.9a it can be seen that the free energy G at any temperature is in turn governed by two factors: (1) the value of G at 0 K, Go, and (2) the slope of the G versus T curve, i.e. the temperature-dependence of free energy. Both of these terms are influenced by the vibrational frequency, and consequently the specific heat of the atoms, as can be shown mathematically. For example, if the temperature of the system is raised from T to T + dT the change in free energy of the system dG is

d G = d H - T d S - S d T

= C p d T - T ( C p d T / T ) - S d T

= - S d T

so that the free energy of the system at a temperature T is

f0 T

G = G o - S d T

At the absolute zero of temperature, the free energy Go is equal to Ho, and then

f0 T

G = H o - S d T

which if S is replaced by f o r ( C p / T ) d T becomes

E/o J

G = H o - ( C p / T ) d T d T (6.1)

Equation (6.1) indicates that the free energy of a given phase decreases more rapidly with rise in temperature the larger its specific heat. The intersection of the free energy-temperature curves, shown in Figure 3.9a, therefore takes place because the low-temperature phase has a smaller specific heat than the higher- temperature phase.

At low temperatures the second term in equation (6.1) is relatively unimportant, and the phase that is stable is the one which has the lowest value of H0, i.e. the most close-packed phase which is associated with a strong bonding of the atoms.

However, the more strongly bound the phase, the higher is its elastic constant, the higher the vibrational frequency, and consequently the smaller the specific heat (see Figure 6.3a). Thus, the more weakly bound structure, i.e. the phase with the higher H0 at low temperature, is likely to appear as the stable phase at higher temperatures. This is because the second term in equation (6.1) now becomes important and G decreases more rapidly with increasing temperature, for the phase with the largest value of f ( C l , / T ) d T . From Figure 6.3b it is clear that a large f ( C p / T ) d T is associated with a low characteristic temperature and hence, with a low vibrational frequency such as is displayed by a metal with a more open structure and small elastic strength. In general, therefore, when

phase changes occur the more close-packed structure usually exists at the low temperatures and the more open structures at the high temperatures. From this viewpoint a liquid, which possesses no long-range structure, has a higher entropy than any solid phase so that ultimately all metals must melt at a sufficiently high temperature, i.e. when the T S term outweighs the H term in the free energy equation.

The sequence of phase changes in such metals as titanium, zirconium, etc. is in agreement with this pre- diction and, moreover, the alkali metals, lithium and sodium, which are normally bcc at ordinary temperatures, can be transformed to fcc at sub-zero temperatures. It is interesting to note that iron, being bcc (c~-iron) even at low temperatures and fcc (y-iron) at high temperatures, is an exception to this rule. In this case, the stability of the bcc structure is thought to be associated with its ferromagnetic properties. By hav- ing a bcc structure the interatomic distances are of the correct value for the exchange interaction to allow the electrons to adopt parallel spins (this is a condition for magnetism). While this state is one of low entropy it is also one of minimum internal energy, and in the lower temperature ranges this is the factor which governs the phase stability, so that the bcc structure is preferred.

Iron is also of interest because the bcc structure, which is replaced by the fcc structure at temperatures above 910~ reappears as the 8-phase above 1400~

This behaviour is attributed to the large electronic specific heat of iron which is a characteristic feature of most transition metals. Thus, the Debye characteristic temperature of y-iron is lower than that of a-iron and this is mainly responsible for the c~ to y transformation.

However, the electronic specific heat of the c~-phase becomes greater than that of the y-phase above about 300~ and eventually at higher temperatures becomes sufficient to bring about the return to the bcc structure at 1400~

6.4 Diffusion

The physical properties of materials 173

.. ~.~ ~i~Inltsal dlstriDution

~ \ .... ..~F,n_.al distribution

l__

Distance ---,.- x

F i g u r e 6.5 Effect o f diffusion on the distribution o f solute in an alloy.

Inhomogeneous alloys are common in metallurgical practice (e.g. cored solid solutions) and in such cases diffusion always occurs in such a way as to produce a macroscopic flow of solute atoms down the concentration gradient. Thus, if a bar of an alloy, along which there is a concentration gradient (Figure 6.5) is heated for a few hours at a temperature where atomic migration is fast, i.e. near the melting point, the solute atoms are redistributed until the bar becomes uniform in composition. This occurs even though the individual atomic movements are random, simply because there are more solute atoms to move down the concentration gradient than there are to move up. This fact forms the basis of Fick's law of diffusion, which is

dn / d t -- - D d c / d x (6.2)

Here the number of atoms diffusing in unit time across unit area through a unit concentration gradient is known as the diffusivity or diffusion coefficient, ~ D.

It is usually expressed as units of cm2s -1 or m2s - l and depends on the concentration and temperature of the alloy.

To illustrate, we may consider the flow of atoms in one direction x, by taking two atomic planes A and B of unit area separated by a distance b, as shown in Figure 6.6. If ci and c2 are the concentrations of diffusing atoms in these two planes (c] > c2) the corresponding number of such atoms in the respective planes is n l " - c ] b and n2 = c2b. If the probability that any one jump in the + x direction is Px, then the number of jumps per unit time made by one atom is pxv, where v is the mean frequency with which an atom leaves a site irrespective of directions. The number of diffusing atoms leaving A and arriving at B in unit time is (p~ vcl b) and the number making the reverse transition is (pxvc2b) so that the net gain of atoms at B is

pxvb(cl - c2) -" Jx

~The conduction of heat in a still medium also follows the same laws as diffusion.

! e 3x~

B I

n 1 atoms

diffusing

~n2 atoms

l

tO C

Figure 6.6 Diffusion of atoms down a concentration gradient.

with Jx the flux of diffusing atoms. Setting c~ - c 2 = - b ( d c / d x ) this flux becomes

- l v b 2 ( d c / d x ) Jx -- - P x V v b 2 ( d c / d x ) --

= - D ( d c / d x ) (6.3)

In cubic lattices, diffusion is isotropic and hence all six orthogonal directions are equally likely so that Px -- g. 1 For simple cubic structures b - - a and thus

Dx = Dy -" Dz = -~va 2 -- D 1 (6.4) whereas in fcc structures b = a / V c2 and D - - I r a 2 , and in bcc structures D - - ~ v a 2.

Fick's first law only applies if a steady state exists in which the concentration at every point is invariant, i.e. ( d c / d t ) = 0 for all x. To deal with nonstationary flow in which the concentration at a point changes with time, we take two planes A and B, as before, separated by unit distance and consider the rate of increase of the number of atoms ( d c / d t ) in a unit volume of the specimen; this is equal to the difference between the flux into and that out of the volume element. The flux across one plane is Jx and across the other (Jx + 1 ) d J / d x the difference being - ( d J / d x ) . We thus obtain Fick's second law of diffusion

dt - dx = ~ Ox (6.5)

When D is independent of concentration this reduces to

dcx d2c

- D x ~ (6.6)

d t d x 2

174 Modern Physical Metallurgy and Materials Engineering and in three dimensions becomes

dc d ( ~ _ ~ c ) d ( d ~ ) d ( d c ) d t = ~ Dx +-~y Dy +dzz Dzdzz An illustration of the use of the diffusion equations is the behaviour of a diffusion couple, where there is a sharp interface between pure metal and an alloy.

Figure 6.5 can be used for this example and as the solute moves from alloy to the pure metal the way in which the concentration varies is shown by the dotted lines. The solution to Fick's second law is given by

co[ ]

c = ~ 1 - ~ ,Io exp ( - y 2 ) d y (6.7) where co is the initial solute concentration in the alloy and c is the concentration at a time t at a distance x from the interface. The integral term is known as the Gauss error function (erf (y)) and as y--~ 0r erf (y) --~ 1. It will be noted that at the interface where x - - 0 , then c = c0/2, and in those regions where the c u r v a t u r e OEc/Ox 2 is positive the concentration rises, in those regions where the curvature is negative the concentration falls, and where the curvature is zero the concentration remains constant.

This particular example is important because it can be used to model the depth of diffusion after time t, e.g. in the case-hardening of steel, providing the concentration profile of the carbon after a carburizing time t, or dopant in silicon. Starting with a constant composition at the surface, the value of x where the concentration falls to half the initial value, i.e.

I is given by x = ~ . Thus knowing 1 - eft(y) = ~,

D at a given temperature the time to produce a given depth of diffusion can be estimated.

The diffusion equations developed above can also be transformed to apply to particular diffusion geometries.

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

dc ( d 2 c 2 dc

d t = O \ ~ r 2 + B rdrr

J

^(6.8)

When the diffusion field has radial symmetry about a cylindrical axis, the equation becomes

dc (dEc l d c )

d t = D \ dr 2 + - rdrr (6.9) and the steady-state condition (dc/dt) = 0 is given by

d2c 1 dc

dr 2 + rdrr 0 (6.10)

which has a solution c = Alnr + B. The constants A and B may be found by introducing the appropriate boundary conditions and for c = co at r = r0 and c --- c~ at r -- r~ the solution becomes

r -- co ln(rl/r) + cl ln(r/ro) ln(rl/ro)

The flux through any shell of radius r is -2:rrD(dc/dr)

2rrD

J -- - ~ ( c l - Co) (6.11)

ln(rl/ro)

Diffusion equations are of importance in many diverse problems and in Chapter 4 are applied to the diffusion of vacancies from dislocation loops and the sintering of voids.

6.4.2 M e c h a n i s m s of diffusion

The transport of atoms through the lattice may conceiv- ably occur in many ways. The term 'interstitial diffusion' describes the situation when the moving atom does not lie on the crystal lattice, but instead occu- pies an interstitial position. Such a process is likely in interstitial alloys where the migrating atom is very small (e.g. carbon, nitrogen or hydrogen in iron). In this case, the diffusion process for the atoms to move from one interstitial position to the next in a perfect lattice is not defect-controlled. A possible variant of this type of diffusion has been suggested for substitutional solutions in which the diffusing atoms are only temporarily interstitial and are in dynamic equilibrium with others in substitutional positions. However, the energy to form such an interstitial is many times that to produce a vacancy and, consequently, the most likely mechanism is that of the continual migration of vacancies. With vacancy diffusion, the probability that an atom may jump to the next site will depend on: (1) the probability that the site is vacant (which in turn is pro- portional to the fraction of vacancies in the crystal), and (2) the probability that it has the required activation energy to make the transition. For self-diffusion where no complications exist, the diffusion coefficient is therefore given by

1 2

D = ga f v e x p [(Sf + Sm)/k]

x exp [-Ef/kT] exp [-Em/kT]

= D0 exp [ - ( E f + Em)/kT] (6.12) The factor f appearing in Do is known as a correla- 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 exchange places in the lattice there is a greater probability of the atom returning to its original site than moving to another site, because of the presence there of a vacancy" f is 0.80 and 0.78 for fcc and bcc lattices, respectively. Values for Ef and Em are discussed in Chapter 4, Ef is the energy of formation of a vacancy, Em the energy of migration, and the sum of the two energies, Q = Ef a t- Em, is the activation energy for self-diffusion I Ed.

IThe entropy factor exp [(Sf + Sm)/k] is usually taken to be 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 association 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 the solute atoms, as shown in Figure 6.7. The phe- 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 by its ability to escape from the impurity atoms which trap it. This problem has been mentioned in Chapter 4.

When considering diffusion in alloys it is important to realize that in a binary solution of A and B the diffusion coefficients DA and DB are generally not equal. This inequality of diffusion was first demonstrated by Kirkendall using an or-brass/copper couple (Figure 6.8). He noted that if the position of the inter- faces of the couple were marked (e.g. with fine W or Mo wires), during diffusion the markers move towards each other, showing that the zinc atoms diffuse out of the alloy more rapidly than copper atoms diffuse in.

This being the case, it is not surprising that several workers have shown that porosity develops in such systems on that side of the interface from which there is a net loss of atoms.

The Kirkendall effect is of considerable theoretical importance since it confirms the vacancy mechanism of diffusion. This is because the observations cannot easily be accounted for by any other postulated mechanisms of diffusion, such as direct place- exchange, i.e. where neighbouring atoms merely change place with each other. The Kirkendall effect is readily explained in terms of vacancies since the lattice defect may interchange places more frequently with one atom than the other. The effect is also of

Solvent . Solute

(a) (b) (c)

(d) (e) (f)

Figure

6.7 Solute atom-vacancy association during diffusion.

9 i 9 9 9

"- ~rt>derum

/ wire markers

Figure

6.8 t~-brass-copper couple for demonstrating the Kirkendall effect.

some practical importance, especially in the fields of metal-to-metal bonding, sintering and creep.

6.4.3 Factors affecting diffusion

The two most important factors affecting the diffusion coefficient D are temperature and composition.

Because of the activation energy term the rate of diffusion increases with temperature according to equation (6.12), while each of the quantities D, Do and Q varies with concentration; for a metal at high temperatures Q-~ 20RTm, Do is 10 -5 to 10 -3 m 2 s - l , and D "~ 10 -12 m 2 s -l. Because of this variation of diffusion coefficient with concentration, the most reliable investigations into the effect of other variables neces- sarily concern self-diffusion in pure metals.

Diffusion is a structure-sensitive property and, therefore, D is expected to increase with increasing lattice irregularity. In general, this is found experi- mentally. In metals quenched from a high temperature the excess vacancy concentration ~109 leads to enhanced diffusion at low temperatures since D = D o c v e x p ( - E m / k T ) . Grain boundaries and disloca- tions are particularly important in this respect and produce enhanced diffusion. Diffusion is faster in the cold-worked state than in the annealed state, although recrystallization may take place and tend to mask the effect. The enhanced transport of material along dislocation 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. Qvol >_ 2Qg.b >__ 2Qs,rf~ce. 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 diffusion process is given by Fick's law and for a given composition gradient also depends on the effective area through which the atoms diffuse. Consequently, since 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 diffusion 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 diffusion rate through the lattice. The importance of grain boundary diffusion and dislocation pipe diffusion is discussed again in Chapter 7 in relation to deformation 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 when grain boundary (Coble creep) rather than lattice diffusion (Herring-Nabarro creep) operates.

Because of the strong binding between atoms, pressure has little or no effect but it is observed that with extremely high pressure on soft metals (e.g. sodium) an increase in Q may result. The rate of diffusion also increases with decreasing density of atomic pack- ing. For example, self-diffusion is slower in fcc iron or thallium than in bcc iron or thallium when the results are compared by extrapolation to the transformation temperature. This is further emphasized by the anisotropic nature of D in metals of open structure.

Bismuth (rhombohedral) is an example of a metal in which D varies by 106 for different directions in the lattice; in cubic crystals D is isotropic.

Dalam dokumen Modern Physical Metallurgy and Materials Engineering (Halaman 183-187)