236
R. A. FLINN, W. E. WALLACE AND R. S. CRAIG
sites increases by 0.08% if the present data are employed, or decreases by 0.13% if one makes use of Hirabayashi's measurements. Thus there is little if any variation of the fraction of lattice sites which are vacant on going from the ordered to the disordered state. The further increase in Schottky defects on going from 140 to 215" is 0.00%. Inferences about the Energy of Vacancy Formation in MgCd3.-When a lattice site in a pure solid is vacated, both the energy and the entropy of the solid increase. For a small number of vacancies the entropy effect on the free energy is dominant and the solid contains under equilibrium conditions a fraction of vacancies sufficient to minimize its free energy. Experiment has shown that the percentage of vacancies in pure metals is small, being of the order of 0.1% at temperatures approaching their melting points. The existence of 2y0 vacancies in MgCds a t 25" would suggest that, if they originate in the same way as in pure solids, the energy of vacancy formation is unusually small-roughly 2.5 kcal./mole of vacancies. If the energy of vacancy formation were really so small, one would expect approximately a fourfold increase in vacancy concentration on increasing the temperature from 25 to 200". The measurements presented in this paper exclude this as a possibility. It therefore appears that some other mechanism must be responsible for the defects in MgCd3. Elsewhere it has been pointed out that extensive vacancies also occur in certain alkali halide solid solution^.^^*^^ The idea has been advanced13 that (12) W. T. Barrett and W. E. Wallace, J . Am. Chem. Sac., 76, 366 (1954). (13) W. E. Wallace and R. A. Flinn, Nature, 172, 681 (1953).
Vol. 61
in solid solutions wherein particles of unequal size must be arranged on a common lattice, stresses develop and these may be relieved by omitting an occasional particle. For example, when a small amount of KCI is dissolved in solid NaCI, the first neighbors of the potassium ion are so crowded that the energy of the system may actually be lowered by creating vacancies. I n this way one may account for the abundance of vacancies in the KC1:rich NaC1-KC1 system13 and similarly in the cadmmmrich magnesium-cadmium alloys. I n each case a solvent ion is replaced by a larger ion as the solution is formed and considerable stresses must be developed. It is to be noted that these ideas imply that in certain solid solutions vacancies are formed ezothermally. For this reason in such systems vacancies are not expected to become significantly more numerous as the temperature is raised. There is an alternative possibility to the effect that the introduction of the first 1 to 2y0 vacancies is accomplished with a very small expendit,ure of energy per vacancy (due to the simultaneous reIief of strains) after which the energy increases very rapidly. One would expect in this case a large number of Vacancies but a small temperature coefficient. To distinguish between these two possibilities it mould be necessary to study the effect of temperature on the number of defects a t lower temperatures. It is possible that this approach might further clarify the factors responsible for the large number of vacancies, although increasing difficulties in attaining equilibrium conditions a t reduced temperatures might obscure the interpretation of results obtained in this may.
VOLUME-TEMPERATURE RELATIONSHIPS IN MAGNESIUM-CADMIUM ALLOYS. 11. KINETICS OF THE ORDER-DISORDER TRANSFORMATION I N MgCd,',' BY
R. A.
FLINN,'
W. E. WALLACE AND R. s. CRAIG
Department of Chemistry, University of Pittsburgh, Pittsburgh I S , Pa. Received August 7, 1966
The rates of ordering and disordering of MgCds have been determined by a dilatometric method. The transformation appears to occur in two stages both of which are, within the limit of error, first-order processes. Rate constants at a number of temperatures are presented. It has been found that the kinetic behavior of the system is not uniquely determined by temperature but depends on thermal history as well.
Introduction Thermal expansivities of MgCd, a t temperatures covering the order-disorder Curie point have been measured and reported in the preceding papex4 (hereinafter referred to as I). While these mea+ urements were being made, it became apparent that some interesting information pertaining to the (1) This work was supported by the U. S. Atomic Energy Commission. (2) From a thesis submitted by R. A. Flinn t o the Graduate School, University of Pitt,sburgh, June, 1954. (3) Dow Chemical Company Fellow during the aeademio year
1953-1954.
(4) R. A. Flinn, W. E. Wallace and R. S. Craig, THISJOWRNAL, 81, 234 (1967).
velocity of the order-disorder transformation in MgCd, was emerging. This paper contains an account of studies of the kinetics of ordering and disordering in MgCd, as determined dilatometrically. Experimental Details Measurements were made in the dilatometer described in paper I. Three types of measurements were performed. In the t y e A experiments temperature was decreased in intervals ofabout IOo and the rate of ordering observed by watching the fall in the meniscus of the dilatometric liquid. This gave information covering the rate of attainment of the equilibrium amount of order characteristic of various temeratures in the region below the critical temperature. f n the type B experiments the sample was quenched to
t
Feb., 1057
ICINETICS OF THE
ORDER-DISORDER TRANSFORMATION IN M G C D ~
gree of undercooling. The type C experiments were similar to the type A experiments except that in the former case the temperature was increased a t intervals of about 10" and the kinetics of disordering were determined. The sample of MgCda was the same as that used in paper I.
Experimental Results and Treatment of Data The variation of meniscus height relative to the final position of the meniscus for the three types of rate studies is shown in Figs. 1, 2 and 3, which give data for the A, B and C types of experiment.
500-
400-
-*
3
300-
?ZOO-
10065 3 "c
I
It will be assumed that each of these processes is first order kinetically with rate constants kl and IC2 for the first and second stages, respectively. As indicated above the first stage is faster and hence kl is larger than kz. Let us first consider the meniscus displacement (m.d.) due to the second process. Ah" will represent the m.d. from the final position due to process 2 and is of course a function of time. The m.d. in the time interval dt due to process 2 is dh
dC dt
- ~ z - dt
I
(4)
In such a reaction sequence as shown in (2) when both steps are first order, the concentration of the (5) R. A. Flinn, Ph.D. Thesis, University of Pittsburgh (1954). ( 6 ) C. Sykes and H. Evans, Proc. Roy. SOC.(London), 8157, 213 (1930). (7) C. Sykes and F. W. Jones, J . Inst. M e t d s , 58, 225 (1936). ( 8 ) G. Borelius, L. E. Larsen and H. Selberg, Arkiv. Fysik, 2, 161 (1950).
(9) N. W. Lord, J . Chsm. Phys., 21, 692 (1953).
2 004
',"1
IOO-
i
(3)
where c2 is a positive constant and dC/dt is the rate of formation of C from B. Since the rate of formation of C is first order with respect to the concentration of B, equation 3 may be rewritten as
$ = -czkzB
237
1000
I%
I
0 00
3000
L
64 4
1000
2000TIME
"C
IN HOURS. 36 00
4d00
Fig. 3.-Dilatometrically measured rate of disorderin of MgCdr at successively higher temperatures (type experiments).
8
intermediate substance is given by the expression (5)
Ao is the initial concentration of A. Substituting
238
R. A. FLINN, w. E. WALLBCE AND R.
s. CRAIG
Vol. 61
Treatment of the disordering process leads to an equation of the same form as eq. 12 and the evaluation of constants is accomplished in the same way. Examination of the data for the early stages of the transformation shows that in the main log Ah' is linear with time within the limit of experimental .2 1.5 I error. Two representative cases are shown in Fig. 5 5. I n those cases where there is departure from linearity simple explanations can be provided. For M the type X experiment a t 34.8" and the type C ex2 1.0 periment a t 88.5' there is clear evidence for a faster process in the earlier stages but the Ah' values are too small and too few in number to confirm the exponential dependence on time. In the type B experiments a t 61.8 and 48.8' the data suggest the 0.5 possibility of a two stage process, the first being the 0 10 20 30 40 rapid one. However, at these temperatures the Time in hours. Fig. 4.-Dilatometrically measured rate of ordering in entire reaction occurs so rapidly that if there is a MgCds a t 54.6'. first stage, it occurs for the most part in the first eq. 5 into 4 and integrating from infinite time to half hour during which thermal contraction is very important and is mainly responsible for the m.d. time t one obtains The rapid variation of meniscus position seemed to persist for a longer time than one would expect for the attainment of thermal equilibrium but one where = czklAo/(kl - kz). Since lcl > kz, the cannot regard the existence of a fast first process as second term in the parentheses becomes negligible a t established in these cases. The existence of a rapid large values of t and Ah" = The constants first stage and the exponential variation of Ah' P and kz are readily elaluated by plotting log ( h - with time were found for the type A experiments a t hf) against time. For times near the completion of 54.6 and 44.6", the type C experiments a t 64.4 and the reaction h - hf = Ah", the first stage being so 77.0" and after 5 hours for the type A experiment nearly complete as to make a negligible effect on the a t 65.3. The rates during the first 5 hours in this displacement of the meniscus. After p and /cZ latter experiment were abnormally slow. It seems have been obtained, it is then possible to obtain kl, likely that during this period nucleation was either rate controlling or a t least reducing the rate of the rate constant for the more rapid first stage. Let us denote by Aho'' the total m.d. due to the reaction] since this temperature is only slightly second process. Aho" can, of course, be evaluated below the transformation temperature. The exby setting t = 0 in eq. 6. Call hf' the final posi- istence of an induction period during which nuclei tion which the meniscus would have reached if only are being formed and the rate is extremely slow is very noticeable in the type B experiment a t process 1 had occurred. 70.4'. Here the plot of log ( h - hf) versus time hi' = hr f Aho" (7) shows a t first glance no evidence of a rapid procTo obtain the m.d. from hf' due solely to process 1for ess for small values of t. The plot is linear for t < t' it is necessary to correct the observed menis- large values of t but for times less than 50 hr. the cus position h for the displacement due to process 2 log ( h - hf) points fall well below the extension of during the interval from 0 time up to time t. Let us the straight line. Again it seems reasonable to call this corrected meniscus position h'. ascribe the small rate of the reaction in the early h' = h + Aho" - Ah" stages to the slow formation of nuclei. It is possi(8) Now the m.d. associat,ed with the conversion of the ble that here too there is a rapid first stage but that alloy in state A to state B is h' - hf' = h - hf - its presence is masked by the growing number of Ah". Since the rate of the conversion of A into B nuclei and the resulting opportunity for accelerais assumed to obey first-order kinetics tion of the transformation as time elapses. This tendency combined with the dying away of the h' - hr' = C ' e - k l t (9) rapid first stage process may be responsible for the C' is a constant, the initial value of the n1.d. due to rather odd linear variation of h - hf with time from process 1. We thus see that t = 5 hr. to t = 45 hr. (see Fig. 2). h - hf - Ah" = C ' e - k l t The opposing effects of nucleation and a rapid (10) first process are also apparent in the type B experiwhich with eq. 6 gives the relationship ment at 65.2'. In that case they so nearly cancel h - hr - B e - k z t = ( y e - k i l (11) that the log ( h - hr) is almost linear over the enWhere a = C' - pk?/k,. We thus have the com- tire duration of the experiment. The values of the constants a, p, IC1 and ka are plete expression for the variation of h - hf, the poshown in Table I together with the time beyond sition of the meniscus relative to its final position which eq. 12 is applicable. h - hf = C y e - k l t + (12) Discussion of Results a and ICl may be evaluated by plotting log (Ah') against time for times less than t', where Ah' = The data are consistent with the notion that the h - hf ordering or disordering of MgCda occurs in two 2.0
h
I1
\
c
KINETICS OF
Feb., 1957
THE
ORDER-DISORDER TRANSFORMATION IN M G C D ~
TABLE I 0
CONSTANTS FOR EQ.12 IN TEXT
Time in hours for 54.6'. 2 4 6
OC.
G5.3 54.6 44.6 34.8
a,
om.
6.64 0.570 .319 .16
70.4 65.2 61.8 48.8
... ...
G4.4 77.0 88.5
-0.583 -1.022
...
...
...
0, om.
ki,
hr.-1
Type A experiments 1.G60 0.271 0.621 .401 .470 .3GO ,328 .80
62,
hr.3
0.0370 ,0675 .0368 .0400
8 2.5
Appli-
Temp.,
239
cable beyond hr.
5 1 1 1
Type B experiments 22.1 ... 6.84 ... 2.580 ... 0.728 ...
0.0703 1.142 1.508 0.835
0.55 2.5 1.25 1
Type C experiments -0 562 0.744 - .382 ,753 - .977 ...
0.113 ,0801 .724
0.9 0.8 1.25
2
0
1.5
O
M
2.0 %
8
i
w
8 2
h
2
h
' rs
I
1.0
1.5&
3
0
2
M
bo
3
M
0.5
stages, a fast first stage followed by a slower second stage, although in some cases the first stage is masked by other effects-attainment of thermal equilibrium or nucleation. Like most studies of the kinetics of order-disorder transformations the present study provides no information as to the nature of the two stages. It has been suggested6Jg9that the first stage may be Dhe growth of domains from existing nuclei until impingement occurs and the second stage consists in the coalescence of antiphase domains. Another notionlois that the ordering of the first stage leaves the alloy in a strained condition and the order parameter which is adopted is characteristic of the strained alloy. The second st,age is then ascribed to the disappearance of strains and subsequent relaxation of the order parameter to the value characteristic of the alloy in its equilibrium state. Neither of these suggested explanations is of such a nature as to be amenable to illumination by dilatometric work. The general features of the data suggest that the transformation in MgCd, occurs by a nucleation and growth mechanism. The two stage process under discussion above, of course, applies only to the growth stage. The existence of something in the nature of an induction period in the ordering process which was a t temperatures just below the transformation point is of course characteristic of processes initiated by nucleation. I n this connection the difference between the kinetic behavior in the type A experiment a t 65.3" and the type B ex(lo) G. Borelius, C. H. Johansson and J. 0. Linde, A n n . Phgsik, 86, 231 (1928).
1.0 5 10 15 Time in hours for 65.3'. Fig. 5.-Dilatometrically measured rates of ordering in MgCds: 0, a4 54.6';. 0 , a t 65.3". Dotted lines show effect of fO.O1 ern. in meniscus position. 0
periment at 65.2" are significant. The over-all rate of the former was very much less than that of the latter. The thermal histories of the sample differed markedly in the two cases. In the type A experiment the sample was held a t 207" for 15 hr. and subsequently reduced to temperature over a period of several days while expansivities between 207 and 65.3" mere being measured. I n the other experiment the sample was held a t 120" for 10 hr., cooled to 84", held there an hour and quenched to 65.2". Avrami" has advanced the notion that there are precursors to the actual nuclei. He has called these precursors "germ nuclei." He has further maintained that the number of "germ nuclei" are reduced progressively with the amount and length of superheating above the transition point. Thus, he expects the number of nuclei formed on subsequent cooling, and hence the transformation rate, to be sensitive to the thermal history of the sample. Perhaps Avrami's germ nuclei are lattice defects of some sort which are conducive to nucleation and which are annealed out a t elevated temperature. Whatever the cause, the effect of thermal history is clearly apparent in the two experiments under discussion. This sensitivity to thermal history does not, of course, establish that the ordering in MgCda requires nucleation but it can be most easily rationalized in terms of a nucleation and growth mechanism. (11) 111. iivraini, J. Chem. Phvs., 7 , 1103 (1339).
