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The physical properties of materials 171 where D is Debye’s maximum frequency. Figure 6.3b shows the atomic heat curves of Figure 6.3a plotted against T/ D ; in most metals for low temperatures T/ D − 1 a T 3 law is obeyed, but at high temper- atures the free electrons make a contribution to the atomic heat which is proportional to T and this causes ariseofC above the classical value. 6.3.3 The specific heat curve and transformations The specific heat of a metal varies smoothly with tem- perature, as shown in Figure 6.3a, provided that no phase change occurs. On the other hand, if the metal undergoes a structural transformation the specific heat curve exhibits a discontinuity, as shown in Figure 6.4. If the phase change occurs at a fixed temperature, the metal undergoes what is known as a first-order trans- formation; for example, the ˛ to , to υ and υ to liq- uid phase changes in iron shown in Figure 6.4a. At the transformation temperature the latent heat is absorbed without a rise in temperature, so that the specific heat dQ/dT at the transformation temperature is infinite. In some cases, known as transformations of the sec- ond order, the phase transition occurs over a range of temperature (e.g. the order–disorder transformation in alloys), and is associated with a specific heat peak of the form shown in Figure 6.4b. Obviously the nar- rower the temperature range T 1 T c ,thesharperis the specific heat peak, and in the limit when the total change occurs at a single temperature, i.e. T 1 D T c ,the specific heat becomes infinite and equal to the latent heat of transformation. A second-order transformation also occurs in iron (see Figure 6.4a), and in this case is due to a change in ferromagnetic properties with temperature. 6.3.4 Free energy of transformation In Section 3.2.3.2 it was shown that any structural changes of a phase could be accounted for in terms of the variation of free energy with temperature. The Figure 6.3 The variation of atomic heat with temperature. Figure 6.4 The effect of solid state transformations on the specific heat–temperature curve. 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, G 0 , 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 C dT the change in free energy of the system dG is dG D dH TdS SdT D C p dT TC p dT/T SdT DSdT so that the free energy of the system at a temperature T is G D G 0 T 0 SdT At the absolute zero of temperature, the free energy G 0 is equal to H 0 ,andthen G D H 0 T 0 SdT which if S is replaced by T 0 C p /TdT becomes G D H 0 T 0 T 0 C p /TdT dT (6.1) Equation (6.1) indicates that the free energy of a given phase decreases more rapidly with rise in tempera- ture 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 H 0 , 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 H 0 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 C p /TdT. From Figure 6.3b it is clear that a large C p /TdT 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 TS 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 temper- atures, can be transformed to fcc at sub-zero temper- atures. It is interesting to note that iron, being bcc (˛-iron) even at low temperatures and fcc (-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 ° C, reappears as the υ-phase above 1400 ° C. This behaviour is attributed to the large electronic spe- cific heat of iron which is a characteristic feature of most transition metals. Thus, the Debye characteristic temperature of -iron is lower than that of ˛-iron and this is mainly responsible for the ˛ to transformation. However, the electronic specific heat of the ˛-phase becomes greater than that of the -phase above about 300 ° C and eventually at higher temperatures becomes sufficient to bring about the return to the bcc structure at 1400 ° C. 6.4 Diffusion 6.4.1 Diffusion laws Some knowledge of diffusion is essential in understanding the behaviour of materials, particularly at elevated temperatures. A few examples include such commercially important processes as annealing, heat-treatment, the age-hardening of alloys, sintering, surface-hardening, oxidation and creep. Apart from the specialized diffusion processes, such as grain boundary diffusion and diffusion down dislocation channels, a distinction is frequently drawn between diffusion in pure metals, homogeneous alloys and inhomogeneous alloys. In a pure material self-diffusion can be observed by using radioactive tracer atoms. In a homogeneous alloy diffusion of each component can also be measured by a tracer method, but in an inhomogeneous alloy, diffusion can be determined by chemical analysis merely from the broadening of the interface between the two metals as a function of time. The physical properties of materials 173 Figure 6.5 Effect of diffusion on the distribution of 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/dt DDdc/dx (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, 1 D. It is usually expressed as units of cm 2 s 1 or m 2 s 1 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 c 1 and c 2 are the concentrations of diffusing atoms in these two planes c 1 >c 2 the corresponding number of such atoms in the respective planes is n 1 D c 1 b and n 2 D c 2 b. If the probability that any one jump in the Cx direction is p x ,then the number of jumps per unit time made by one atom is p x ,where 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 x c 1 b and the number making the reverse transition is p x c 2 b so that the net gain of atoms at B is p x bc 1 c 2 D J x 1 The conduction of heat in a still medium also follows the same laws as diffusion. Figure 6.6 Diffusion of atoms down a concentration gradient. with J x the flux of diffusing atoms. Setting c 1 c 2 D bdc/dx this flux becomes J x Dp x v v b 2 dc/dx D 1 2 vb 2 dc/dx DDdc/dx 6.3 In cubic lattices, diffusion is isotropic and hence all six orthogonal directions are equally likely so that p x D 1 6 . For simple cubic structures b D a and thus D x D D y D D z D 1 6 va 2 D D (6.4) whereas in fcc structures b D a/ p 2andD D 1 12 va 2 , and in bcc structures D D 1 24 va 2 . Fick’s first law only applies if a steady state exists in which the concentration at every point is invariant, i.e. dc/dt D 0forallx. 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 dc/dt 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 J x and across the other J x C 1 dJ/dx the difference being dJ/dx. We thus obtain Fick’s second law of diffusion dc dt D dJ x dx D d dx D x dc dx (6.5) When D is independent of concentration this reduces to dc x dt D D x d 2 c dx 2 (6.6) 174 Modern Physical Metallurgy and Materials Engineering and in three dimensions becomes dc dt D d dx D x dc dx C d dy D y dc dy C d dz D z dc dz 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 c D c 0 2 1 2 p x/[2 p Dt] 0 exp y 2 dy (6.7) where c 0 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 !1, erf y ! 1. It will be noted that at the interface where x D 0, then c D c 0 /2, and in those regions where the curvature ∂ 2 c/∂x 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. 1 erfy D 1 2 ,isgivenbyx D p Dt. Thus knowing 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 dt D D d 2 c dr 2 C 2 r dc dr (6.8) When the diffusion field has radial symmetry about a cylindrical axis, the equation becomes dc dt D D d 2 c dr 2 C 1 r dc dr (6.9) and the steady-state condition dc/dt D 0isgivenby d 2 c dr 2 C 1 r dc dr D 0 (6.10) which has a solution c D Alnr CB. The constants A and B may be found by introducing the appropriate boundary conditions and for c D c 0 at r D r 0 and c D c 1 at r D r 1 the solution becomes c D c 0 lnr 1 /r C c 1 lnr/r 0 lnr 1 /r 0 The flux through any shell of radius r is 2rDdc/dr or J D 2D lnr 1 /r 0 c 1 c 0 (6.11) 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 Mechanisms of diffusion The transport of atoms through the lattice may conceiv- ably occur in many ways. The term ‘interstitial diffu- sion’ 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 substitu- tional 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 vacan- cies. 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 activa- tion energy to make the transition. For self-diffusion where no complications exist, the diffusion coefficient is therefore given by D D 1 6 a 2 f exp [S f C S m /k] ð exp [E f /kT]exp[E m /kT] D D 0 exp [E f C E m /kT] 6.12 The factor f appearing in D 0 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 prob- ability 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 E f and E m are dis- cussed in Chapter 4, E f is the energy of formation of a vacancy, E m the energy of migration, and the sum of the two energies, Q D E f C E m , is the activation energy for self-diffusion 1 E d . 1 The entropy factor exp [S f C S m /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 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 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 impor- tant to realize that in a binary solution of A and B the diffusion coefficients D A and D B are generally not equal. This inequality of diffusion was first demon- strated by Kirkendall using an ˛-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 Figure 6.7 Solute atom–vacancy association during diffusion. Figure 6.8 ˛-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 diffu- sion coefficient D are temperature and composition. Because of the activation energy term the rate of diffu- sion increases with temperature according to equation (6.12), while each of the quantities D, D 0 and Q varies with concentration; for a metal at high temper- atures Q ³ 20RT m , D 0 is 10 5 to 10 3 m 2 s 1 ,and D ' 10 12 m 2 s 1 . Because of this variation of diffu- sion 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 temper- ature the excess vacancy concentration ³10 9 leads to enhanced diffusion at low temperatures since D D D 0 c v exp E m /kT. 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 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 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 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 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, pres- sure 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 transfor- mation 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 10 6 for different directions in the lattice; in cubic crystals D is isotropic. 6.5 Anelasticity and internal friction For an elastic solid it is generally assumed that stress and strain are directly proportional to one another, but in practice the elastic strain is usually dependent on time as well as stress so that the strain lags behind the stress; this is an anelastic effect. On applying a stress at a level below the conventional elastic limit, a specimen will show an initial elastic strain ε e followed by a gradual increase in strain until it reaches an essentially constant value, ε e C ε an as shown in Figure 6.9. When the stress is removed the strain will decrease, but a small amount remains which decreases slowly with time. At any time t the decreasing anelastic strain is given by the relation ε D ε an exp t/ where is known as the relaxation time, and is the time taken for the anelastic strain to decrease to 1/e ' 36.79% of its initial value. Clearly, if is large, the strain relaxes very slowly, while if small the strain relaxes quickly. In materials under cyclic loading this anelastic effect leads to a decay in amplitude of vibration and therefore a dissipation of energy by internal friction. Internal friction is defined in several different but related ways. Perhaps the most common uses the logarithmic decre- ment υ D lnA n /A nC1 , the natural logarithm of suc- cessive amplitudes of vibration. In a forced vibration experiment near a resonance, the factor ω 2 ω 1 /ω 0 Figure 6.9 Anelastic behaviour. is often used, where ω 1 and ω 2 are the frequencies on the two sides of the resonant frequency ω 0 at which the amplitude of oscillation is 1/ p 2 of the resonant amplitude. Also used is the specific damping capacity E/E,whereE is the energy dissipated per cycle of vibrational energy E, i.e. the area contained in a stress–strain loop. Yet another method uses the phase angle ˛ by which the strain lags behind the stress, and if the damping is small it can be shown that tan ˛ D υ D 1 2 E E D ω 2 ω 1 ω 0 D Q 1 (6.13) By analogy with damping in electrical systems tan ˛ is often written equal to Q 1 . 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 occurs when an elastic stress is applied to a specimen too fast for the specimen to exchange heat with its surroundings and so cools slightly. As the sample warms back to the surrounding temperature it expands thermally, and hence the dilatation strain continues to increase after the stress has become constant. The diffusion of atoms can also give rise to anelastic effects in an analogous way to the diffusion of thermal energy giving thermoelastic effects. A particular example is the stress-induced diffusion of carbon or nitrogen in iron. A carbon atom occupies the interstitial site along one of the cell edges slightly distorting the lattice tetragonally. Thus when iron is stretched by a mechanical stress, the crystal axis oriented in the direction of the stress develops favoured sites for the occupation of the interstitial atoms relative to the other two axes. Then if the stress is oscillated, such that first one axis and then another is stretched, the carbon atoms will want to jump from one favoured site to the other. Mechanical work is therefore done repeatedly, dissipating the vibrational energy and damping out the mechanical oscillations. The maximum energy is dissipated when the time per cycle is of the same order as the time required for the 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 torsion pendulum, shown schematically in Figure 6.10. The specimen can be oscillated at a given frequency 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 E E D 2 E E max ω 1 C ω 2 and has a maximum value when the angular frequency of the pendulum equals the relaxation time of the process; at low temperatures around room temperature this is interstitial diffusion. In practice, it is difficult to vary the angular frequency over a wide range and thus it is easier to keep ω constant and vary the relaxation time. Since the migration of atoms depends strongly on temperature according to an Arrhenius-type equation, the relaxation time 1 D 1/ω 1 and the peak occurs at a temperature T 1 . For a different frequency value ω 2 the peak occurs at a different temperature T 2 ,and so on (see Figure 6.11). It is thus possible to ascribe an activation energy H for the internal process producing the damping by plotting ln versus 1/T, or from the relation H D R lnω 2 /ω 1 1/T 1 1/T 2 In the case of iron the activation energy is found to coincide with that for the diffusion of carbon in iron. Similar studies have been made for other metals. In addition, if the relaxation time is the mean time an atom stays in an interstitial position is 3 2 ,and from the relation D D 1 24 a 2 v for bcc lattices derived previously the diffusion coefficient may be calculated directly from D D 1 36 a 2 Many other forms of internal friction exist in met- als arising from different relaxation processes to those Figure 6.11 Internal friction as a function of temperature for Fe with C in solid solution at five different pendulum frequencies (from Wert and Zener, 1949; by permission of the American Institute of Physics). discussed above, and hence occurring in different fre- quency and temperature regions. One important source of internal friction is that due to stress relaxation across grain boundaries. The occurrence of a strong internal friction peak due to grain boundary relaxation was first demonstrated on polycrystalline aluminium at 300 ° C by K ˆ e and has since been found in numerous other metals. It indicates that grain boundaries behave in a somewhat viscous manner at elevated temperatures and grain boundary sliding can be detected at very low stresses by internal friction studies. The grain boundary sliding velocity produced by a shear stress is given by D d/Á and its measurement gives values of the viscosity Á which extrapolate to that of the liquid at the melting point, assuming the boundary thickness to be d ' 0.5nm. Movement of low-energy twin boundaries in crys- tals, domain boundaries in ferromagnetic materials and dislocation bowing and unpinning all give rise to inter- nal friction and damping. 6.6 Ordering in alloys 6.6.1 Long-range and short-range order 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 term ‘superlattice’ would imply that such a coher- ent atomic scheme extends over large distances, i.e. the crystal possesses long-range order. Such a perfect arrangement can exist only at low temperatures, since the entropy of an ordered structure is much lower than that of a disordered one, and with increasing tempera- 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 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 structure. At high temperatures well above T c ,there are more than the random number of AB atom pairs, and with the lowering of temperature small nuclei of order continually form and disperse in an other- wise disordered matrix. As the temperature, and hence thermal agitation, is lowered these regions of order become more extensive, until at T c they begin to link 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 antiphase domain boundary mobility until long-range order is established. Some order–disorder alloys can be retained in a state of disorder by quenching to room temperature while in others (e.g. ˇ-brass) the ordering process occurs almost instantaneously. Clearly, changes in the degree of order will depend on atomic migration, so that the rate of approach to the equilibrium configu- ration will be governed by an exponential factor of the usual form, i.e. Rate D Ae Q/RT . However, Bragg Figure 6.12 Influence of temperature on the degree of order. 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 ˇ-brass only two different schemes of order are possi- ble, while in fcc lattices such as Cu 3 Au 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 jF 2 jDS 2 f A f 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 f 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 f B is negligible for X-rays and the 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 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 f A and f B 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 Cu 3 Au, 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 10 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 Figure 6.15 Degree of order ð and domain size (O) during isothermal annealing at 350 ° C after quenching from 465 ° C (after Morris, Besag and Smallman, 1974; courtesy of Taylor and Francis). 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 ° C, but between 380 ° C and the disordering temperature of 410 ° Cithasthe CuAu 11 structures shown in Figure 6.16. The 002 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 Influence of ordering on properties 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 T c owing to the exis- tence of short-range order, which also requires extra energy to destroy it as the temperature is increased above T c . 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 0.05µ 0.05µ (a) (b) Figure 6.17 Electron micrographs of (a) CuAu 11 and (b) CuAu 1 (from Pashley and Presland, 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 large contribution to the electrical resistance. Accordingly, superlattices below T c have a low electrical resistance, but on raising the temperature the resistivity increases, as shown in Figure 6.18a for ordered Cu 3 Au. 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 Cu 3 Au 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 Cu 3 Au 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 Cu 3 Au, the maximum yield strength is exhibited by quenched samples after an annealing treatment of 5 min at 350 ° C 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 order Figure 6.18 Effect of (a) temperature, (b) composition, and (c) deformation on the resistivity of copper–gold alloys (after Barrett, 1952; courtesy of McGraw-Hill). [...]... electrons into the higher band and also create vacancies in the valency band, the material is a semiconductor In general, the lowest energy band which is not completely filled with electrons is called a conduction band, and the band containing the valency electrons the valency band For a conductor the valency band is also the conduction band The electronic state of a selection of materials of different... electrons The extra electrons go into empty zones, and as a Figure 6.22 Schematic diagram of an intrinsic semiconductor showing the relative positions of the conduction and valency bands 184 Modern Physical Metallurgy and Materials Engineering result silicon becomes an n-type semiconductor, since conduction occurs by negative carriers On the other hand, the addition of elements of lower valency than... increased, Sm2 Co, Fe, Cu, Zr 17 alloys also rely on the pinning of magnetic domains by fine precipitates Clear correlation exists between mechanical hardness and intrinsic coercivity SmCo5 magnets depend on the very high magnetocrystalline anisotropy of this 192 Modern Physical Metallurgy and Materials Engineering compound and the individual grains are single-domain particles The big advantage of these... solid, liquid or gaseous materials and ceramics, glasses and semiconductors In all cases, electrons of the laser material are excited into a higher energy state by some suitable stimulus (see Figure 6.38) In a device this is produced by the photons from a flash tube, to give an intense Figure 6. 37 Optical guidance in a multimode fibre 196 Modern Physical Metallurgy and Materials Engineering Thus, translucent... because of these properties and low material costs 6.9 Dielectric materials 6.9.1 Polarization Dielectric materials, usually those which are covalent or ionic, possess a large energy gap between the valence band and the conduction band These materials exhibit high electrical resistivity and have important applications as insulators, which prevent the transfer of electrical charge, and capacitors which store... pass through the inner polar sheet and enter the eye A sudden flash of light will activate photodiodes in the goggles, reduce the impressed voltage and cause rapid darkening of the goggles Further reading Anderson, J C., Leaver, K D., Rawlins, R D and Alexander, J M (1990) Materials Science Chapman and Hall, London Braithwaite, N and Weaver, G (Eds) (1990) Open University Materials in Action Series Butterworths,... these latter machines sudden yielding will show as merely an extension under constant load 198 Modern Physical Metallurgy and Materials Engineering Figure 7. 1 Stress–elongation curves for (a) impure iron, (b) copper, (c) ductile–brittle transition in mild steel (after Churchman, Mogford and Cottrell, 19 57) to the elastic portion of the stress–strain curve from the point of 0.1% strain For control purposes... junction rectification with (a) forward bias and (b) reverse bias The physical properties of materials 185 with high-voltage equipment and can protect it from transient voltage ‘spikes’ or overload 6 .7. 3 Superconductivity Figure 6.25 Schematic diagram of a p–n–p transistor and collector respectively, as shown in Figure 6.25, and the base is made slightly negative and the collector more negative relative... electronic, (b) ionic and (c) molecular mechanisms 194 Modern Physical Metallurgy and Materials Engineering usually increases when one of the contributions to polarization is prevented This behaviour is common in microwave heating of polymer adhesives; preferential heating in the adhesive due to dielectric losses starts the thermosetting reaction For moderate increases, raising the voltage and temperature... mass of the isotope Since both the frequency of atomic vibrations and the velocity of elastic waves also varies as M 1/2 , the interaction between electrons and lattice vibrations Figure 6.26 Variation of critical field Hc as a function of temperature for several pure metal superconductors 186 Modern Physical Metallurgy and Materials Engineering (i.e electron–phonon interaction) must be at least one . of Metals). 180 Modern Physical Metallurgy and Materials Engineering 0.05µ 0.05µ (a) (b) Figure 6. 17 Electron micrographs of (a) CuAu 11 and (b) CuAu 1 (from Pashley and Presland, 1958–9; courtesy of. zones, and as a Figure 6.22 Schematic diagram of an intrinsic semiconductor showing the relative positions of the conduction and valency bands. 184 Modern Physical Metallurgy and Materials Engineering result. than that of a disordered one, and with increasing tempera- ture the degree of long-range order, S, decreases until 178 Modern Physical Metallurgy and Materials Engineering at a critical temperature
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