Order-Disorder Transitions in Alloys

Just as the body-centered cubic structure can be considered as made of two interpenetrating simple cubic lattices, the face-centered cubic structure can be made of four simple cubic lattices. There are some interesting cases of ordered alloys with this crystal structure and ratios of approximately one to three of the two components. An example is found in the copper-gold system, where such a phase is found in the neighbor- 

We shall now introduce a parameter w, which we shall call the degree of order. We shall define it so that w = 1 corresponds to having all the atoms a on one of the simple cubic lattices (which we may call the lattice a), all the atoms b on the other (which we call P). w = 0 will correspond to having equal numbers of atoms a and b on each lattice w = — 1 will correspond to having all the atoms b on lattice a, all the atoms a on lattice p. Thus w = 1 will correspond to perfect order, w = 0 to complete disorder. Lot us now define w more completely, in terms of the number of atoms a and b on lattices a and p. Let there be N atoms, N/2 of each sort, and N lattice points, N/2 on each of the simple cubic lattices. Then we assume that 

Clearly the assumptions (1.1) reduce to the proper values in the cases w = 1, 0, and furthermore they give w as a linear function of the various numbers of atoms. 

To find the energy, we must find the number of pairs of neighbors of types aa, ab, bb. The number of pairs of type aa equals the number of a s on lattice a, times 8/(N/2) times the number of a s on lattice p. This is on the assumption that the distribution of atoms on lattice P surrounding an atom a on lattice a is the same proportionally that it is in the whole lattice p, an assumption which is not really justified but which we make 

To find the internal energy at the absolute zero, we now proceed as in Sec. 2 of Chap. XVII, multiplying the number of pairs aa by etc. Then we obtain 

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ORDER-DISORDER THEORY AND APPLICATIONS. Phase transitions in binary liquid solutions, gas condensations, order-disorder transitions in alloys, ferromagnetism, antiferromagnetism, ferroelectncity, anti-ferroelectricity, localized absorptions, helix-coil transitions in biological polymers and the one-dimensional growth of linear colloidal aggregates are all examples of transitions between an ordered and a disordered state. 

With w 7 0 the problem is not simple. It can in fact be shown to be essentially equivalent to certain problems in the theory of the order-disorder transition in alloys, regular solutions, ferromagnetism, etc. 

Tanner, L.E, and Leamy, H.J. (1974) The microstructure of order-disorder transitions, in Order-Disorder Transformations in Alloys, ed. Warlimont, H. (Springer, Berlin) p. 180. 

That part of the entropy of a substance that is due to a disordered arrangement of the particles as opposed to a similar but ordered arrangement. The most clear-cut example is the order-disorder transition in binary alloys, in which virtually the whole entropy change is of this kind. The entropy change on fusion of a solid is largely due to entropy of disorder. 

Fusion, as an order-disorder transition, is the concept that fusion of a crystalline solid is essentially a change from the almost perfectly ordered solid state to a disordered liquid slate. The vacant spaces in the crystal lattice correspond lo the other component in the binary alloys, which undergo order-disorder transition in the pure form. Evidence from x-ray diffraction measurements indicates that short-range order is retained during fusion but long-range order is lost. 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

The third type of phase change of the second order is fortunately easy to treat theoretically, at least to an approximation, and it, is the one which will be discussed in the present chapter. This is what is known as an order-disorder transition in an alloy, and can be better understood in terms of specific examples, which we shall mention in the next section. 

J. Banhart, W. Pfeiler, and J.Voitldnder, Order-disorder transition in CuPt-alloys, investigated by... 

Cu-Zn phase diagram. Note the order-disorder transition in the fJ phase at 460°C. Also note the extent of the a phase to 38.27 At% Zn and the (3 phase to 48.2 At% indicated by the numbers on the diagram. (Reprinted from Massalski, T.B., Handbook of Binary Alloy Phase Diagrams, ASM, 1990. Reprinted with permission of ASM International. All rights reserved.)... 

For most of the intermetallic compounds, deviation from the stoichiometric composition is accommodated by antisite atoms CuZn, CusAu, NiTi, NisAl. Paired antisite defects are also formed thermally in high concentration for those alloys which present an order-disorder transition in the solid state (CU3AU, CuZn) their concentration is directly related to the LRO parameter value. The absence of structural vacancies has been checked by positron annihilation in the case of CuZn (Kim and Buyers, 1980), CujAu (Doyama et fl/.,1985a,b), FejAl (Schaefer et al., 1990), and TiAl (Shirai and Yamaguchi, 1992). In all cases, thermal vacancies form at high temperature. 

Some typical phenomena which have been described as second-order transitions are order-disorder transitions in metal alloys, onset of ferromagnetism, onset of ferroelectricity and onset of superconductivity. ... 

Figure 6.10. Dilatotneuic record of a sample of a Ni-AI-Fe alloy in the neighbourhood of an order-disorder transition temperature (Cahn ei al. 1987).

Using the so-called "block copolymers (a block of Na A-monomers at one end is covalently bonded to a block of Nb B-monomers) one can also realize the analogy of order-disorder phenomena in metallic alloys with polymers one observes transitions from the disordered melt to mesophases with various types of long range order (lamellar, hexagonal, cubic, etc ). We shall not consider these phenomena here further, however... 

The order-disorder transition of a binary alloy (e.g. CuZn) provides another instructive example. The body-centred lattice of this material may be described as two interpenetrating lattices, A and B. In the disordered high-temperature phase each of the sub-lattices is equally populated by Zn and Cu atoms, in that each lattice point is equally likely to be occupied by either a Zn or a Cu atom. At zero temperature each of the sub-lattices is entirely occupied by either Zn or Cu atoms. In terms of fractional occupation numbers for A sites, an appropriate order parameter may be defined as... 

Dilatometric methods. This can be a sensitive method and relies on the different phases taking part in the phase transformation having different coefficients of thermal expansion. The expansion/contraction of a sample is then measured by a dilatometer. Cahn et al. (1987) used dilatometry to examine the order-disorder transformation in a number of alloys in the Ni-Al-Fe system. Figure 4.9 shows an expansion vs temperature plot for a (Ni79.9Al2o.i)o.s7Feo.i3 alloy where a transition from an ordered LI2 compound (7 ) to a two-phase mixture of 7 and a Ni-rich f c.c. Al phase (7) occurs. The method was then used to determine the 7 /(7 + 7O phase boundary as a function of Fe content, at a constant Ni/Al ratio, and the results are shown in Fig. 4.10. The technique has been used on numerous other occasions,... 

Fig.8. Variation with temperature of the average segregant concentration at the Ll2(100) surface (solid lines) and at the first underlayer (dotted lines) in AB3 model alloy calculated in the FCEM approximation for different segregation/order factors r (marked near the plots). Arrows indicate order-disorder transition temperatures (for r =3.5, Ts=Tb).

In addition to the segregation/order factor, and depending on its magnitude, the crystal structure and surface orientation can strongly affect the surface composition in ordered alloys. For example, unlike the case of the equiatomic bulk truncated composition of Ll2(100), LRO tends to maintain the Ll2(lll) surface with nominal bulk concentration (0.25). Therefore, the two ordered surfaces are expected to exhibit quite different segregation characteristics for the same r value (Fig. 10). Moreover, SRO causes pronounced changes of surface sublattice and average compositions associated with a considerable reduction of the order-disorder transition temperature (especially in fee alloys). 

The surfaces of CU3AU alloy (bulk structure LI2) was studied thoroughly by various techniques and theoretical approaches, especially in relation to the order-disorder transition [52-63]. Recently, medium-energy ion-scattering spectroscopy (MEIS) measurements confirmed the stabilization of bulk-truncated equiatomic termination for this surface at low temperatures. Starting at about 500 K, the Au atoms in the surface layer begin to move to the second... 

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