Figure 6.13 An antiphase domain boundary.
has pointed out that the ease with which interlocking domains can absorb each other to develop a scheme of long-range order will also depend on the number of possible ordered schemes the alloy possesses. Thus, in /~-brass only two different schemes of order are possi- ble, while in fcc lattices such as Cu3Au four different schemes are possible and the approach to complete order is less rapid.
6.6.2 Detection of ordering
The determination of an ordered superlattice is usu- ally done by means of the X-ray powder technique. In a disordered solution every plane of atoms is statisti- cally identical and, as discussed in Chapter 5, there are reflections missing in the powder pattern of the mate- rial. In an ordered lattice, on the other hand, alternate planes become A-rich and B-rich, respectively, so that these 'absent' reflections are no longer missing but appear as extra superlattice lines. This can be seen 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 I F 2 1 = S 2 ( f A - - f B ) 2, from which it can be seen that in the fully-disordered alloy, where S = 0, the superlattice lines must vanish. In some alloys such as copper-gold, the scattering factor difference ( f A - fB) 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 - f a ) is negligible for X-rays and the super-lattice lines are very weak; in copper-zinc, for
Ra~bu ~ s X / 2 ~
t~ ~lane A
~~ ut
' the same~ amplitude
of phase/
Plane A
~--
Figure 6.14 Formation of a weak 100 reflection from an ordered lattice by the inte(erence of diffracted rays of unequal amplitude.
example, the ratio of the intensity of the superlattice lines to that of the main lines is only about 1:3500.
In some cases special X-ray techniques can enhance this intensity ratio; one method is to use an X- ray wavelength near to the absorption edge when an anomalous depression of the f-factor occurs which is greater for one element than for the other.
As a result, the difference between fA and f a is increased. A more general technique, however, is to use neutron diffraction since the scattering factors for neighbouring elements in the Periodic Table can be substantially different. Conversely, as Table 5.4 indicates, neutron diffraction is unable to show the existence of superlattice lines in Cu3Au, because the scattering amplitudes of copper and gold for neutrons are approximately the same, although X-rays show them up quite clearly.
Sharp superlattice lines are observed as long as order persists over lattice regions of about l0 -3 mm, large enough to give coherent X-ray reflections. When long-range order is not complete the superlattice lines become broadened, and an estimate of the domain
1.o 0.8
o 0.6
0
g' o.4 0.2
0
~- ,'0
x degree of order |
o domain size (A) /
looo
i J ' , 0
1 2 log to t 3 4 5
,~o" ~ ,o~oo ,o~0oo
t (rain)
3000
2500
2OOO .e
1500
" O
Figure 6.15 Degree of order ( x ) and domain size (0) during isothermal annealing at 350~ after quenching from 465~ (after Morris, Besag and Smallman, 1974: courtesy of Taylor and Francis).
The physical properties of materials 179 size can be obtained from a measurement of the line breadth, as discussed in Chapter 5. Figure 6.15 shows variation of order S and domain size as determined from the intensity and breadth of powder diffraction lines. The domain sizes determined from the Scherrer line-broadening formula are in very good agreement with those observed by TEM. Short-range order is much more difficult to detect but nowadays direct measuring devices allow weak X-ray intensities to be measured more accurately, and as a result considerable information on the nature of short-range order has been obtained by studying the intensity of the diffuse background between the main lattice lines.
High-resolution transmission microscopy of thin metal foils allows the structure of domains to be exam- ined directly. The alloy CuAu is of particular interest, since it has a face-centred tetragonal structure, often referred to as CuAu 1 below 380~ but between 380~
and the disordering temperature of 410~ it has the CuAu 11 structures shown in Figure 6.16. The (0 0 2) planes are again alternately gold and copper, but half- 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.
6 . 6 . 3 I n f l u e n c e o f o r d e r i n g o n p r o p e r t i e s Specific heat The order-disorder transformation has a marked effect on the specific heat, since energy is necessary to change atoms from one configuration to another. However, because the change in lattice arrangement takes place over a range of temperature, the specific heat versus temperature curve will be of the form shown in Figure 6.4b. In practice the excess spe- cific heat, above that given by Dulong and Petit's law, does not fall sharply to zero at Tc owing to the exis- tence of short-range order, which also requires extra energy to destroy it as the temperature is increased above Tr
_ ~ ] ,. IL . . . L .! ~ e-- R_e__
Z" 9 J.f" 9 ~ " 9 , I " 9 .I ~ 9 ~-~-.o .I.7" oi~ o - o ~ o
Figure 6.16 One unit cell of the orthorhombic superlattice of CuAu, i.e. CuAu 11 (from J. Inst. Metals, 1958-9, courtesy of the Institute of Metals).
180 Modern Physical Metallurgy and Materials Engineering
Figure 6.17 Electron micrographs of (a) CuAu 11 and (b) CuAu 1 (from Pashley and Pres/and, 1958-9; courtesy of the Institute of Metals).
Electrical resistivity
As discussed in Chapter 4, any form of disorder in a metallic structure (e.g. impuri- ties, dislocations or point defects) will make a largecontribution to the electrical resistance. Accordingly, superlattices below Tc have a low electrical resistance, but on raising the temperature the resistivity increases, as shown in Figure 6.18a for ordered Cu3Au. The influence of order on resistivity is further demonstrated by the measurement of resistivity as a function of com- position in the copper-gold alloy system. As shown in Figure 6.18b, at composition near Cu3Au and CuAu, where ordering is most complete, the resistivity is extremely low, while away from these stoichiomet- ric compositions the resistivity increases; the quenched (disordered) alloys given by the dotted curve also have high resistivity values.
Mechanical properties
The mechanical properties are altered when ordering occurs. The change in yield stress is not directly related to the degree of ordering, however, and in fact Cu3Au crystals have a lower yield stress when well-ordered than when only partially- ordered. Experiments show that such effects can be accounted for if the maximum strength as a result of ordering is associated with critical domain size. In the alloy Cu3Au, the maximum yield strength is exhibited by quenched samples after an annealing treatment of 5 min at 350~ which gives a domain size of 6 nm (see Figure 6.15). However, if the alloy is well-ordered and the domain size larger, the hardening is insignificant. In some alloys such as CuAu or CuPt, ordering produces a change of crystal structure and the resultant lattice strains can also lead to hardening. Thermal agitation is the most common means of destroying long-range order, but other methods (e.g. deformation) are equally effective. Figure 6.18c shows that cold work has a negligible effect upon the resistivity of the quenched (disordered) alloy but considerable influence on the well-annealed (ordered) alloy. Irradiation by neutrons or electrons also markedly affects the ordering (see Chapter 4).Magnetic properties
The order-disorder pheno- menon is of considerable importance in the application of magnetic materials. The kind and degree of orderE 15 - Disordered
~' alloy ~s ~'[
.~
Cu3Au
9
?- alloy~
_ 1 I
o 200 400
Temperature--~ ~
(a)
15
I O ~o
x
. - - 5 .>_
._m
O~sordered 1//~ ~alloy
1 I l
25 5O 75 100
Gold : At %
(b)
J Quenched Cu3Au alloy
.- J / Cu3Au alloy
60 v I l [ J
o 25 50 75 100
Reduction in c.s.a. % (c)
Figure 6.18 Effect of (a) temperature, (b) composition, and (c) deformation on the resistiviO' of copper-gold alloys (after Barrett, 1952; courtesy of McGraw-Hill).
The physical properties of materials 181 affects the magnetic hardness, since small ordered
regions in an otherwise disordered lattice induce strains which affect the mobility of magnetic domain boundaries (see Section 6.8.4).
6.7 Electrical properties
6.7.1 Electrical conductivity
One of the most important electronic properties of met- als is the electrical conductivity, to, and the reciprocal of the conductivity (known as the resistivity, p) is defined by the relation R = p l / A , where R is the resis- tance of the specimen, 1 is the length and A is the cross-sectional area.
A characteristic feature of a metal is its high electri- cal conductivity which arises from the ease with which the electrons can migrate through the lattice. The high thermal conduction of metals also has a similar expla- nation, and the Wiedmann-Franz law shows that the ratio of the electrical and thermal conductivities is nearly the same for all metals at the same temperature.
Since conductivity arises from the motion of con- duction electrons through the lattice, resistance must be caused by the scattering of electron waves by any kind of irregularity in the lattice arrangement. Irregularities can arise from any one of several sources, such as tem- perature, alloying, deformation or nuclear irradiation, since all will disturb, to some extent, the periodicity of the lattice. The effect of temperature is particularly important and, as shown in Figure 6.19, the resistance increases linearly with temperature above about 100 K up to the melting point. On melting, the resistance increases markedly because of the exceptional disor- der of the liquid state. However, for some metals such as bismuth, the resistance actually decreases, owing to the fact that the special zone structure which makes