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A DIFFUSE NEUTRON SCATTERING STUDY OF
CLUSTERING KINETICS IN Cu-Ni ALLOYS
J. Vrijen, S. Radelaar, D. Schwahn
To cite this version:
A
DIFFUSE NEUTRON SCATTERING STUDY
OF CLUSTERING KINETICS IN Cu-Ni ALLOYS
J. VRIJEN
Netherlands Energy Research Foundation ECN Petten, The Netherlands
S. RADELAAR
Physics Laboratory, Rijksuniversiteit Utrecht, Utrecht, The Netherlands and
D. SCHWAHN
Institut fiir Fcsthorperforschung. KFA, 517 Jiilich, West-Germany
R b 6 . - Par diffusion des neutrons thermiques on a examine les cinetiques d'agglomeration
dans les alliages Cu-Ni. Pour optimatiser les conditions de l'expkrience on a utilisi: les isotopes 6 5 C ~ et 62Ni. L'tvolution dans le temps de I'intensitk diffuse 1 400 OC a 6te mesurCe pour 8 alliages Cu-Ni de compositions comprises entre 30 et 80 at. % de nickel. La relaxation de la matrice null dans le cas d'un alliage A 56,5 at. % de nickel a ete aussi etudiee A 320, 340, 425 et 450 OC. Ces mesures ont ete analystes .dans le modtle de Cook et des informations en ont ete deduites sur le coefficient de diffu- sion 1 basse temperature et sur les proprietb therrnodynamiques du systeme Cu-Ni. On en conclut finalement que le modtle de Cook n'est pas suffisamment detaille pour pouvoir decrire les premieres etapes de la relaxation.
Abstract. - Diffuse scattering of thermal neutrons was used to investigate the kinetics of clustering in Cu-Ni alloys. In order to optimize the experimental conditions the isotopes 6 S C ~ and 62Ni were alloyed. The time evolution of the diffuse scattered intensity at 400 oC has been measured for eight Cu-Ni alloys, varying in composition between 30 and 80 at. % Ni. The relaxation of the so called null matrix. containing 56.5 at. O/, Ni, has also been investigated at 320, 340, 425 and 450 OC.
Using Cook's model from all these measurements information has been deduced about diffusion at low temperatures and about thermodynamic properties of the Cu-Ni system. It turns out that Cook's model is not sufficiently detailed for an accurate description of the initial stages of these relaxations.
Introduction. - The kinetics of clustering and short range order (s.r.0.) in stable solid solutions were discussed from a theoretical point of view by Cook [I]. His model is based on the lattice diffusion theory for spinodal decomposition as developed by Cook, De Fontaine and Hilliard 123. The model gave good agreement with data on the kinetics of s.r.0. as measured by one of the authors [3,4]. At that time no experimental data on clustering were available. One of the important results of Cook's theory was that he demonstrated the impossibility of describing cluster- ing kinetics, even approximately, by a single relaxation time in contrast t o short-range order kinetics.
Important theoretical contributions have been given by Khachaturyan [5] and Yamauchi [6], but in this paper we will mainly limit ourselves to a comparison with Cook's theory.
For binary alloys he derived the following expres- - -
sion for the- time evolution of the diffuse scattering cross-section
where t is the time, K the scattering vector and O(K) the
relaxation time, which is given by
D is the interdiffusion coefficient, given by
D = [Qc(l
-
C) f " / k , T I [cD,+
(1-
c) D,].
(3)B 2 ( ~ ) is a lattice sum which for polycrystalline fcc materials reads
12
B 2 ( ~ ) = 7 (1
-
sin icrI/icr1).
rlC7-348 J . VRIJEN, S. RADELAAR AND D. SCHWAHN
f" is the second derivative of the Helmholtz free energy per unit volume with respect to the concen- tration c of component 1, K the gradient energy coefficient, Q the atomic volume,
k,
Boltzmann's constant, T the temperature, Dl and D, the tracer diffusion coefficients of component 1 and 2, respec- tively and r , the distance between nearest neighbour- ing atoms. Recently Yamauchi formulated a statistic mechanical approach in the discrete lattice represen- tation. He derived for the time dependence of the diffuse scattering cross-section the following expres- sionA(~)(K, t ) is a complex integral which contains the time evolution of the pair correlations. For t = 00 one
obtains
The superscript (i) denotes the order of approximation of the interatomic interactions. Eq. (5) has been derived for zeroth, first and second order. The result for T(~)(K) is equivalent to that of Cook, given in eq. (2). The most important difference with Cook's model is that eq. (5) contains the time dependent term A(')(K, t). This means that Yamauchi's model does not predict points of zero growth at certain K-values that are constant in time, as Cook's model does, but allows for a gradual shift of these points and, in addition, for small changes in the whole K-region. Both models predict the relaxation behaviour for composition fluctuations of short as well as long wavelengths.
The theories mentioned above are based on the linearization of the diffusion equation.
Below the critical temperature T, non-linear effects due to decomposition greatly complicate an analytical treatment. The linearized models are only valid for the very initial stages of spinodal decomposition, which makes it very difficult to test them. In the homo- geneous phase above T, non-linearities probably d o not occur. Hence this region of the phase diagram is ideal for verifying these kinetic theories.
We have chosen the Cu-Ni system for a thorough investigation of clustering kinetics. The use of the isotopes 6 5 C ~ and 62Ni makes a diffuse scattering study very attractive, since such alloys have very large cross-sections for the diffuse scattering of thermal neutrons
(b,,-6, = 1.11 x 10-'2crn and
bNi-6, =
-
0.87 x lo-', cm).
Results and discussion. - Preparation of the samples, heat treatments, spectrometer arrangements
and data handling will be discussed in detail else- where [7]. Therefore, only some general remarks will be made. Eight Cu-Ni compositions varying bet- ween 30 and 80 at
%
Ni were prepared. For all speci- mens the time evolution of the diffuse scattering cross section for thermal neutrons at 400 0C has been measured. For the sample with 56.5 at%
Ni, the so called null matrix, this time evolution also has been measured for 320, 340,425 and 450 OC. All specimens were quenched from 700 OC in water. Subsequently they were annealed at the desired temperature for periods ranging from 5 minutes to several hours and even to 500 hours at 340 O C . Between these steps thechange in da/dl2 was measured at room temperature. For all samples the K-range between 0.1 and 3.4
A-'
has been investigated with 2.58
A
neutrons with one of the neutron spectrometers at the reactor HFR at Petten.For the relaxation of the null matrix at 450 OC also the K-range between 0.01 and 0.2
A-'
has been investigated with 7.3A
neutrons at the small angle neutron spectrometer of the KFA at Julich.The annealing times were corrected for the effect of excess vacancies, frozen in during the quench from 700 OC. For every annealing step the relaxation function T(K) has been derived with eq. (1) from the experimental data. Eq. (2) has been fitted to these T(K).
The parameters for the relaxations at 400 O C are given
in table 1 and those for the relaxations of the null matrix in table 11. For K > 0.8 A - I the relative error
in A(da/dQ) is of the order of 30
%.
This and the fact that the first annealing step (300 s) is longer than the shortest relaxation time (- 50 s) makes the data in this K-range less reliable.TABLE I. Relaxation parameters for the various Cu-Ni compositions; 3
-
400°c. - - - at. perc. Ni 30.0 40.0 47.5 56.5 60.0 65.0 70.0 80.0 24 n cNi c S u D C u f"lrf k B T 12.0 7.7 3.8 4.4 2.2 2.1 1.6 2.8 2 4 2 n c N i ~ $ U D C y K / r : k B T 8.2 8.5 19.0 12.0 24.0 16.0 9.4 0.1 (*lo-*)TABLE 11. Relaxation parameters for the null matrix at various rcmperatures.
temperature ("c) 320 340 400 425 450
For this reason and because the values of da/dQ (t = a) are known, only the K-values ranging from 0.2 to 0.8
A
-'
were used to fit eq. (2). The extrapolation of T(K) obtained in this way to K-values < 0.2A-'
F I G . I. -The relaxation function ?(K) for 6 5 C u 0 , , , , 6 2 N i 0 . 5 6 ,
at 450 O C versus K. The dots are the experimental points, measured at Petten and at Jiilich. The drawn line is a two parameter least squares fit of eq. (2) to the experimental points with K between 0.2 and 0.8 A - ' . The parameters of the fit are given in table 11.
F I G . 2. -The various stages of the relaxation of the null matrix
at 450 OC, as measured at Petten and at Jiilich, together with the
corresponding predictions of the model of Cook with the two parameters for T(K) that are given in table 11.
matrix. The curves correspond to the theoretical values calculated from eqs. (I) and (2) for the various stages, using the parameters for 450 OC shown in table 11.
For the relaxations performed at 320 and 340 OC the relative error in A(do/dQ) is roughly 20
%
for the range K > 0.8A - l .
Moreover, the first annealingperiod is here of the order of the relaxation time. Nevertheless the relaxation times T(K) in this region
remain systematically larger than the predicted values. Moreover, there appears to be no fixed cross-over point at K N 1.15
%L-
l , but instead a gradual shift tosmaller K-values is observed.
We will now examine whether the two parameters that are used in eq. (2) yield physically acceptable values of D, K and f
".
Obviously additional information concerning at least one of these parameters is necessary. Tracer diffusion measurements of Ni and Cu in Cu-Ni alloys by Monma et al. [8] show that in general DNi 4 Dcu so that the value of D in eq. (3) is essentially determined by D,,. Using these data for the 400 OC relaxations results in the values for f" and K given in table 111. These seem reasonable, except perhaps for the gradual increase o f f " with increasing Ni content.
TABLE 111. f" and K for the various compositions. using the tracer d i f f u - sion data for Cu of Honma et al. 18 1.
at. perc. Ni 3 0 . 0 4 0 . 0 4 7 . 5 5 6 . 5 6 0 . 0 6 5 . 0 7 0 . 0 8 0 . 0 Dcu crn2/s) 5 0 0 . 0 2 0 0 . 0 1 0 0 . 0 5 4 . 0 3 0 . 0 1 5 . 0 9 . 0 5 . 3 f" ( l o 8 erg/cm3) 0 . 3 5 5 . 7 6 . 1 16.0 16.0 3 7 . 0 6 0 . 0 3 5 0 . 0 K ( l o - ' e r g / c m ) 0 . 0 6 1 . 6 7 . 9 1 1 . 0 4 6 . 0 7 4 . 0 9 3 . 0 2 . 7
Much less satisfactory are the results for the relaxa- tions of the null matrix, shown in table IVa, where both f " and K decrease strongly with increasing tem- perature. This means that the values used for Dcu vary too much with temperature. Keeping K constant still results in a physically unacceptable behaviour of
f"
(table IVb). Calculating f " with
TABLE IV. f" and K for the null matrix for the various temperatures and calculated with a) tracer diffusion data for Cu of k n m o et a l .
16 I, b) K constant. c) a. ( 6 1 , using 'I1- - 1 0 . 6 meV and V2
-
r 4 . 0 rneV, and d) eq. ( 7 ) .temperature (OC) 320 340 400 425 450
a . D C u ( 1 0 - 2 1 c r n ~ s ) 0 . 1 2 0 . 6 5 5 4 . 0 270.0 1200.0
f " ( l o 9 erg/cm3> 160.0 7 4 . 0 16.0 8 . 3 4.2
C7-350 J. VRIJEN, S. RADELAAR AND D. SCHWAHN
given by Paulson [9], with for the interaction poten- tials V , for the first and V , for the second nearest neighbours the values - 10.6 and
+
4.0 meV, res- pectively [lo], does not give satisfactory results either (table IVc). Physically reasonable results for f",
Kand D,, could be obtained (table IVd) by a calculation o f f " with the following empirical expression :
This means that for T, nearly the same value is predicted ( - 320 OC), but that the slope of
f"
with temperature is about half the theoretical value. This might be caused by elastic contributions and most probably by effects due to many particle interactions, which are present in Cu-Ni [7].Conclusions. - It has been shown that clustering kinetics involves diffusion over all possible distances in the system, each with its own relaxation time. It has also been found that the relaxation process is expo- nential in time, which could be measured over three
decades. A comparison with the model of Cook was satisfactory for the later stages and for the smaller K-values. For the larger K-values the relaxation behaviour is in agreement with qualitative predictions o f the model of Yamauchi. Thermodynamic and diffusion data have been derived for the whole compo- sition range at temperatufes where the usual mqthods fail due to the low atomic mobility. For the null matrix the best results were obtained using an empi- rical expression for f ", where most probably many- particle interactions influence the temperature depen- dence.
Acknowledgments. - We wish to thank
Prof. W. Schmatz for his hospitality at the KFA, Jiilich, Mr. H. J. Bron of the Laboratorium voor Fysische Metaalkunde, Rijksuniversiteit Groningen, for his cooperation in preparing the specimen and Mr. W. Van der Gaauw for his technical assistance in the experiments at Petten. The FOM (Stichting Fundamenteel Onderzoek der Materie) sponsored this work with a grant for the isotopes.
References
[l] COOK, H. E., J. Phys. Chem. Solids 30 (1969) 2427; Acra Met. 18 (1970) 297.
[2] COOK. H. E., DE FONTAINE, D. and HILLIARD, J. E., Acta Mer. 17 (1969) 765.
[3] RADELAAR, S., Phys. Status Solidi 2 l (1968) K63.
[4] RADELAAR, S. and R r r z e ~ , J. M. J., Phys. Status Solidi 31 (1969) 277.
[5] KHACHATURYAN, A. G.. SOY. Phys.-Solid State 9 (1968) 2040 ; 11 (1970) 2959.
[6] YAMAUCHI, H., Thesis Northwestern University, 1973. [7] VRIJEN, J. and RADELAAR, S., to be published.
181 MONMA, K., SUTO, H. and OIKAWA, H., J. Japan. Inst. Metals
28 (1964) 192.
191 PAULSON, W., Thesis Northwestern University, 1972. [lo] VRIJEN, J., VAN DIJK, C. and RADELAAR, S., Proc. Neutron
