E x p e r i m e n t a l l y, t w o k i n d s o f o r d e r h a v e b e e n d e t e c t e d

Transcription

1 The Order Disorder Transformation The nature of ordering: - T h e r e are many types of order in s o lid m aterials,as t h e s p atial o rd e r o f the at o ms i n a p ure c r ys t a l.the t e r m o r d e r i n g i n s u b s t i t u t i o n a l s o l i d s o l u t i o n s, h o w e v e r, i s u s u a l l y r e s e r v e d f o r t h e s p e c i a l a t o m i c a r r a n g e m e n t s c h a r a c t e r i z e d b y h i g h e r t h a n r a n d o m p r o b a b i l i t y o f f i n d i n g u n l i k e a t o m s a s n e a r e s t n e i g h b o r s. Concerning on the relationship of the ordering phenomenon to phase diagrams by consider in s o m e d e t a i l t h e c o n c e p t o f o r d e r i n g a n d s o m e p h y s i c a l m o d e l s p r o p o s e d t o a l l o w i t s m a t h e m a t i c a l t r e a t m e n t. T h e q u a s i - c h e m i c a l m a t h e m a t i c a l t r e a t m e n t w h i c h p r e d i c t s t h e p h a s e - d i a g r a m r e l a t i o n s h i p s. E x p e r i m e n t a l l y, t w o k i n d s o f o r d e r h a v e b e e n d e t e c t e d and identif ied in solid solutio ns, s hort-range order and long r ange or der. Both are the result of the tendency of each atom in certain solutions to surround itself with unlike atoms. When this tendency is small, or when at high temperatures the disordering tendency of thermal agitation is high, any specific atom succeeds in surrounding itself with only a slightly greater than random number of unlike atoms and for only very short times. Thermal agitation breaks up these local groups, and new groups form, which are in turn "dissolved,". Nevertheless, if we were to "freeze in" an atomic distribution at any moment and count the number of like and unlike nearest-neighbor pairs we should find the fractional number (the probability) of unlike pairs to

2 be higher than that, for a random distribution. The probabilities of unlike next-nearest pairs, third-nearest pairs, would also be found to deviate from randomness, but the magnitude of the deviation would drop off rapidly with distance, most often falling virtually to zero at a few atomic diameters. Ordering of this type is called short-range ordering. If the ordering tendency is high, or if the temperature is low, the nonrandom correlations between the positions occupied by the two atomic species are higher in magnitude and are effective over greater distances. Below some critical temperature the distance becomes large enough so that time-independent, long-distance correlations, long-range order, appear. In this state it is possible to classify all the lattice sites in the crystal into sublattices, each of which tends to be occupied predominantly by one kind of atom (Fig.1). if, when long-range order exists, we again count pairs of atoms, we Shall now find that not only do nearest-neighbor and next-nearest-neighbor pairs exhibit nonrandom probabilities-of like and unlike pairs but that there are equally prominent and nonvanishing correlations between the states of occupancy of sites that are separated by distances approaching macroscopic dimensions. There is actually a sharper distinction between short-range and longrange order than may at first glance seem to be the case. The presence of long-range order and the magnitude of the critical temperature are clearly defined both theoretically and experimentally. Theoretically, as will be seen later, long-range order is either definitely present or definitely not present, depending on whether the temperature is above or below the critical temperature; theoretically predicted position of the latter is, however, a function of the type of assumptions made in deriving the relationship between temperature and the ordering tendency.

3 Experimentally, the presence of long-range order is delineated primarily by certain characteristic x-ray diffraction effects. This may be understood with reference to Fig.1, which shows the completely ordered arrangement in a 50:50 A B alloy having a bcc space lattice in the disorder state. This ordered structure may be viewed as being made up of two interpenetrating sublattices, on one of which, called the sublattice, the A atoms reside and on the other hand the sublattice, the B atoms reside. Each site is surrounded,by eight equidistant sites and similarly each site by eight equidistant sites. If this ordered arrangement of the atoms extends over thousands of atom diameters, it represents perfect long-range order. It should be noted that in the perfectly ordered condition the space lattice is no longer bcc, since the positions occupied by the A atoms no longer have surroundings equivalent to those occupied by the B atoms. In fact the structure shown in fig.1 is characterized by a simple cubic space lattice with an AB atom pair with each lattice point. Fig.1: the order arrangement in a 50:50 alloy.

4 As may be seen, long-range order lowers the symmetry of the crystal structure; as a result, it is detectable by the appearance in x-ray diffraction patterns of new reflections, called superlattice reflections. The intensities of these reflections are directly related to the degree of longrange order, as to the extent of separation of the atomic species onto the appropriate sublattices. Short-range order alone does not produce superlattices or superlattice reflections but rather affects the intensity of the diffuse (background) scattering in x-ray diffraction patterns, gild is measurable by this effect. The characteristics of short- and long-range order may be further illustrated by noting the following points: 1. Perfect short-range order implies perfect long-range order, and vice versa. 2. Completely random arrangements of the two kinds of atoms are characterized by no short-range and no long-range order. 3. Intermediate arrangements, however, can have quite high degrees of short-range order and little or no long-range order. An extreme case of this is illustrated in Fig.2 for a hypothetical 50:50 A B alloy made up of out-of-step blocks, or antiphase domains. It is seen that in this two-dimensional alloy the and sublattices over long distances are each occupied 50% by A and 50% by B atoms, and therefore there is no long-range order in the alloy. On the other hand, it is also apparent that the arrangement is quite highly ordered on a local scale,the number of unlike nearest-neighbor atom pairs being far greater than for a random arrangement.

5 Fig.2: Antiphase domains in a two-dimensional crystal. The broken line,an antiphase domain boundary, separates two perfectly ordered domains which are out of step with each other. The number for a random arrangement is given by Eq. (1) as: (1) In fig.2,z=4 for the 48 interior atoms,2 for the 4 corner atoms, and 3 for the 28 other exterior atoms, and X A =X B =0.5 giving.the actual number of AB pairs in the figure is 134, and the total number of pairs is 142. Thus, it is seen that the fractional number of AB pairs in the alloy would be 0.5 for a random array whereas in the actual array it is 0.95.

6 Physical models of ordering: Because the ordering tendency in alloys is opposed by thermal agitation, there is for each temperature a characteristic equilibrium degree of order at which the two opposing tendencies are just balanced. It is the derivation of the relation between this equilibrium degree of order and temperature which- leads to theoretical order-disorder lines in phase diagrams. Two major steps are involved in this derivation:, (1) visualizing - an appropriate physical model of ordering and (2) devising a mathematical treatment of the model amenable to solution. The most frequently and successfully used physical model of ordering was first proposed by Bethe. This model is the exact analogy of the quasi-chemical model introduced for regular solutions. Since the quasi-chemical model seems more natural and is of more general use, we shall from now on refer to this concept as the quasi-chemical (Q-C) model. According to the model, the ordering is the result of nearest-neighbor atom-pair interaction energies for which the attractive interaction between unlike atoms is greater than the average of that between like atoms. This view has led to important qualitative, and some semi quantitative, agreement between theory and experiment. It should be noted, nevertheless, that it has a number of serious shortcomings. Among these are the neglect of such factors as the interactions between non-nearest-neighbor atoms, the distortion of lattice symmetry and size during the order-disorder transitions, and the changes in vibrational entropy accompanying the transition. In spite of these shortcomings, the Q-C model has been the starting point for more successful treatments of ordering than any other model and thus will be used here as an instructional base. The earliest model leading to a moderately successful mathematical

7 treatment for the stability of superlattices as a function of temperature was devised by Bragg and Williams. Their model proposed that ordering is the result of the long-range influence of all the atoms in a crystal on each individual atom. They derived their order-temperate relation without reference to pair interactions. The Bragg-Williams (B-W) result, however, can be obtained on the basis of the Q-C model, as well, simply by applying to the pair-interaction concept assumptions equivalent to those made in the B-W treatment. *Order parameters:- The degree of order in an alloy may be defined in a number of different ways; the two generally used parameters are those due to Bragg and Williams and to Bethe. Confining themselves to crystals of the type shown in Fig.1, wherein there are only two sublattices, Bragg and Williams defined the degree of long-range order as: Here is the number of A atoms on sublattice sites in a given solution, is the number which would be on sites if the solution contained no long-range order, the number if the solution were perfectly ordered, and, and have the same significance for B atoms and the sublattice. may be seen to vary from zero for no long-range order( ) to unity for perfect order ( The B-W order parameter is called the long-range order parameter because it concerns itself explicitly only with the extent to which the

8 various sublattices are properly occupied over long distances. For any given value of other than 1, however (and even for =1 in alloys where the composition is not at the ideal 50:50 ratio), there may be distinguished many different ways in which the A and B atoms on each sublattice may be arranged. For example, the A. and B atoms may be randomly arranged within each sublattice, or they may have various degrees of local order within each sublattice. Any of these local arrangements could be consistent with a single intermediate value of I S, but each would represent different degrees of short-range order. Recognizing this, Bethe introduced as a convenient measure of the local order a short-range order parameter, which be defined as Here n is the number of unlike nearest-neighbor atom pairs in a given state of a solid solution, n r the number which would be present if the solution were completely random, and n o the number if the solution were perfectly ordered., like, varies from zero to unity; it is zero when the solution is completely random (n = n r ) and unity when the solution is prefectly ordered (n = n ). However, and have the same numerical values only in the completely ordered solutions. In solutions of intermediate order the exact relationship between the two depends on the particular assumptions made with regard to the local arrangements on each long - range sublattice. Before proceeding to the discussion of the mathematical relationship between the equilibrium order parameters and temperature, it seems worthwhile to digress in order to clarify some aspects of the

9 nomenclature attached to ordering about which some confusion seems to exist. In particular, the terms "Bragg-Williams theory of ordering" and "Bethe theory of ordering" have each been rather vaguely and interchangeably applied to the three different concepts: (1) physical model of ordering, (2) order parameter, and (3) mathematical treatment of the degree of order versus temperature. The B-W model of ordering led to the B-W long-range order parameter. Since this parameter can be, and is, used to describe the degree of longrange order in other models as well, to avoid confusion it should properly be called simply the long-range order parameter; we shall adopt this practice here. The B-W mathematical treatment for the equilibrium degree of order versus temperature is based on the B-W model, the long-range order parameter, and the implicit simplifying assumption that the A and B atoms, though tending to separate onto the appropriate sub-lattices, are randomly distributed within each sublattice. Similarly, the Bethe model of ordering led to the Bethe short-range order parameter, which we shall call simply, the short-range order parameter. The Bethe mathematical treatment for the equilibrium degree of order versus temperature, along -with the Q-C treatment, is based on the tendency to form unlike nearest-neighbor pairs and leads to a specific relationship between nd to equations for both The obtained in this case does not, however, have, the same value at a given temperature as the derived by the B-W treatment, since the assumptions in the two treatments with regard to local order are quite different. A relation between for the B-W assumption can also be derived. The appropriate interrelationship between the various B-W, Bethe, and Q-C

10 terms discussed above may be summarized as follows: