Journal of Non-CrystallineSolids 75 (1985) 355-360 North-HoUand, Amsterdam
DENSE RANDOM PACKED MODEL FOR AMORPHOUS Ni35Zr65: MIXING*
INFLUENCE OF THE HEAT OF
C. K. SAW and R. B. SCHWARZ Materials Science and Technology Division, Argonne National Laboratory Argonne, IL 60439
The structure of amorphous Ni35Zr65 alloy was simulated by a Dense Random Packed (DRP) model. The model was relaxed by minimizing its elastic energy through the use of a modified Keating potential that included only central forces with interatomic bonds of equal strength. The Bathia-Thornton number concentration (NC) distributions and the Warren chemical short range order parameter, ~w' were deduced from the model and compared with measurements in rapidly quenched Ni35Zr65. The DRP model simulates a truly random alloy for which 0~ = 0. Since 0~ = -0.04 for the actual alloy, the fraction of Ni-Zr nearest neighbors in the alloy exceeds that in the model. The effects of decreasing the strength of any one of the interatomic bonds (Ni-Ni, NI-Zr, or Zr-Zr) were investigated. It was found that the bond relaxations cause a sharpening of some of the peaks in NC distributions but have no effect on the value of ~w" The effect of permuting Ni and Zr atoms, so as to increase the number of Ni-Zr nearest neighbors at the expense of Ni-Ni and Zr-Zr pairs, was also investigated. It was found that these permutations improve the agreement between the calculated and the measured NC distributions and generate an aw value of -0.038.
I. INTRODUCTION Computer models have been used extensively to study the structure of amorphous materials.
Dense Random Packed (DRP) models are the easiest to con-
struct and have provided radial distribution functions quite similar to those measured experimentally [ •
Computer models that involve the packing of small
clusters of atoms instead of individual atoms have also been used to simulate chemical short range order (CSRO) in amorphous structures 2'3.
A limitation of
this deterministic approach is that the atomic arrangement chosen may be only one of the many which develop in the alloy.
Molecular dynamic models require
longer computations but have the advantage that the structure is free to change in response to temperature and external constraints.
of CSRO with dynamic models requires interatomic potentials that are reliable beyond the first nearest neighbors.
Neither DRY nor dynamic models have been
successful in reproducing the CSRO observed in binary amorphous alloys of an early and a late transition metal, which are characterized by large negative heats of mixing, AH << O. *Work supported by the U. S. Department of Energy.
0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
CK. Saw, R.B. Schwa~ / Dense random packed model for amorphous Ni35Zr65
Harris and Lewis 4 used a DRP model of a binary amorphous alloy to investigate the dependence of CSRO on the interatomic bond strengths in Cu33Zr67. For atomic radii corresponding to those of Cu and Zr, an increase in the Cu-Zr bond strength to twice the strength of the Cu-Cu and Zr-Zr bonds resulted in a sharper partial density correlation, Pcc(r).
However, the Warren CSRO
parameter, ~w' only showed a decrease from -0.051 to -0.056.
deviation of these calculations was not reported. In a previous paper 5, a computer code was used to generated DRP models of amorphous binary alloys AxBI_ x as a function of the concentration x and the atomic size ratio RB/R A.
The model was relaxed by using a modified Keating
energy expression which included only central forces and atomic bonds AA, BB, and AB of equal strength.
The Bhatia-Thornton number-concentration (NC) fluctua-
tions GNN(r) , GNC(r) and Gcc(r) , as well as ~w' were deduced from these models.
In all cases ~W was found to be approximately zero (no CSRO), even
though significant structure was observed in Gcc(r) for RB/R A < I. In the first part of the present work, we investigate the influence of the bond strength on CSRO in binary amorphous alloys.
We observe that a factor of
5 decrease in any one of the three bonds has little or no effect on 0~.
conclude that a simple strengthening or softening of one of the bond types in the DRP model cannot reproduce the CSRO that is observed experimentally in amorphous alloys with ~H << 0.
In the second part of this work we assume that
the three bonds have the same strength but we allow for limited atomic motion, as described below. In a superheated molten alloy with ~H << 0, the entropy term T~S opposes the development of significant CSRO.
As the liquid is cooled, and/or during
the early stages of solidification into an amorphous state, CSRO develops and lowers the alloy's free energy.
The simplest atomic motions by which this may
occur are permutations of A-B pairs.
The computer code was modified so as to
perform only those A-B permutations that lower the alloy's free energy. It was found that each iteration of this code throughout the random array increased the degree of CSRO, as evidenced by significant decreases in ~w"
2. MODELING RESULTS 2.1. Influence of the Bond Strength A DRP model of Ni35Zr65 was constructed with atomic radii of 1.10 and 1.58 ~, respectively.
The structure was relaxed by minimizing its elastic
energy through the use of a modified Keating potential which included only central forces.
This potential is given by
C.K. Saw, R.B. Schwarz / Dense random packed model tor amorphous Ni35Zr65
n El(X) = ~ where ~ j
and dij are the strength and the equilibrium length of the bonds
between atoms i and j, and ~. is the position of atom i, I Fig. I are the total reduced pair
The histograms in
correlation functions, G(r), calculated for different relative values of bond strength.
For Fig. l(a), the three
bond strengths are equal.
l(b), ~22 = ~12 = 5~II' corresponding to a softening of the Ni-Ni bonds.
Fig. l(c), ~II = ~22 = 5~12 (softening
of the Ni-Zr bonds), and for Fig. l(d), ~11 = ~12 = 5a22 (softening of the Zr-Zr bonds).
curves in Fig. i are the G(r) ~easured by Wagner and Lee 6.
The first two
peaks in G(r), at r = 2.7 and 3.2 A, correspond to Ni-Zr and Zr-Zr pairs, respectively.
Since Ni has a lower
scattering power for X-rays than Zr,
the Ni-Ni pairs, at a separation of
r (A) FIGURE I Total reduced pair correlation function, G(r), of Ni35Zr65. The smooth curve is the experimental result of Ref. . The histograms are the modeling results obtained with dlfferent relative values of interatomic bond strengths, given in the text.
2.2 A, do not give a measurable peak in either the calculated or the measured G(r). For interatomlc bonds of equal strength (Fig. la), the peak in the calculated G(r) at r = 2.7 A (Ni-Zr
nearest neighbors) is lower than that in the measured G(r).
The softening of
either the Ni-Ni or the Ni-Zr bonds (Figs. Ib and Ic) results in a slight increase in the height of this peak.
The softening of the Zr-Zr bonds
(Fig. Id) produces sharp peaks In G(r). 2.2. Influence of the Palr-wise Permutations The pair-wlse atomic permutations were performed as follows:
total array of approximately 3400 atoms, the computer code selected sub-arrays consisting of a pair of adjacent atoms, A and B, plus all nearest neighbors to both.
The atoms A and B were permuted only if the number of AB bonds (within
the sub-array) increased at the expense of AA and BB bonds.
procedure was applied sequentially to all the AB pairs in the ~ i n
C.K. Saw, R.B. Schwarz / Dense random packed model for amorphous Ni35Zr65
which was then relaxed by using equation (1) with cLij = constant. The smooth curves in Fig. 2 are the total G(r) and the NC correlations of Ni35Zr65 measured by Wagner and Lee 6.
The histograms in Fig. 2(a) are the
total G(r) and the NC correlations deduced from the relaxed DRP model, before applying the SWITCH routine.
The amplitudes of these curves have been nor-
malized to those of the measured ones.
The overall agreement between the
calculated and the measured G(r)'s is quite good. ferences appear in Gcc(r) for 3.5 < r < 6.0 ~.
The most significant dif-
However, these differences are
most likely due to experimental errors 7. The histograms in Figs. 2(b), 2(c), and 2(d) are the total G(r) and the NC correlations of the DRP model after the application of the SWITCH routine once, twice, and three times, respectively.
As seen in Fig. 2a, before the
SWITCH routine is applied, the peak in the calculated GNN(r) at r = 2.7 A is somewhat lower than the corresponding peak in the measured GNN(r ).
significant change caused by the SWITCH routine is an increase in the height and area under this peak, reflecting the increase in the number of Ni-Zr nearest neighbors.
The best overall fit betwen the modeled and measured
NC correlations is that in Fig. 2(b), obtained after a single application of the SWITCH routine. Column 2 in Table I gives the calculated aw as a function of the
TABLE I. Warren's CSRO parameter, ~W' and number of permutations for each application of the SWITCH routine
number of successive applications of the SWITCH routine.
mental value of ~w Is6 -0.04, which is in good agreement with the results shown in Fig. 2(b) for one application of the SWITCH routine.
For two or more
# of SWITCH iterations 0 1 2 3
0.002 -0.038 -0.058 -0.055
0 551 659 601
applications of this routine, 0~ reaches a steady value.
Nevertheless, the SWITCH routine continues t o make a
significant number of permutations~ as seen in column 3 of Table [.
not necessarily increase the number of AB pairs, as the majority of these permutations may Just nullify one another owing to overlapping sub-arrays. Because entropy effects are neglected here, successive applications of this routine would cause the model to develop the CSRO expected at 0 K. effects can, however, be easily incorporated in the routine.
have recently shown 8 that a code which increases the number of Ni-Zr pairs and, at the same time, decreases the number of Ni-Ni pairs not only modifies the Ni-Zr distribution and makes aw negative, but also changes the Ni-Ni distribution near next nearest neighbor distances.
These changes could
C.K. Saw, R.B. Schwarz / Dense random packed model/br amorphous Ni35Zr65
S mzz<: ~~>-~ I
~ - - - -
- - - - T ~ - - ~
r(A) FIGURE 2 Total reduced pair correlation function, G(r), and Bathia-Thornton numberconcentration fluctuations, GNN(r), Gcc(r) , and GNC(r), of Ni35Zr65. The smooth curves are the experimental results of Ref. . The histograms are the modeling results (a) before and (b-d) after permuting Ni-Zr nearest neighbors, The SWITCH routine was applied (b) once, (c) twice, and (d) three times.
C.K. Saw, R.B. Schwarz / Dense random packed model for amorphous Ni35Zr65
explain the so-called prepeak in the Interference Functions,
I(K), observed in
several Ni-based amorphous alloys 9.
3. CONCLUSIONS A softening by a factor of 5 in the strength of any of the three bonds of a binary amorphous alloy produces small changes in the Bathla-Thornton NC fluctuations but has no significant effect on CSRO. The CSRO that develops in rapidly quenched binary amorphous alloys with a large negative heat of mixing can be simulated by a DRP model in which permutations are made to increase the number of AB nearest neighbors at the expense of AA and BB pairs.
For amorphous Ni35Zr65 , a single sequential
application of such permutations throughout the DRP model decreases 0~ from 0 (for the initial random model) to -0.038, which is in good agreement with the experimental value of -0.04.
REFERENCES I) G. S. Cargill III, Dense Random Packing Models for Amorphous Rare Earth Metal-Transition Metal Alloys, in: Amorphous Materials, The Metallurgical Society of AIME Conference Proceedings, St. Louis, Missouri, Oct 25-26, 1982 (TMS-AIME, Warrendale, PA, 1983), p. 15. 2) P. H. Gaskell, J. Non-Cryst.
Solids 32 (1979) 207.
3) D. S. Boudreaux and J. M. Gregor, J. Appl. Phys. 48 (1977) 5057. 4) R. Harris and L. J. Lewis, Phys. Rev. B 25 (1982) 4997 and J. Phys. F: Met. Phys. 13 (1983) 1359. 5) C. K. Saw and J. Faber, Jr., these Proceedings. 6) C. N. J. Wagner and D. Lee, J. Phys. (Paris) 41 (1980) C8-242. 7) After the completion of this work, we became aware of a more recent set of experimental extracted NC fluctuations from Prof. C. N. J. Wagner, obtained by using the Ridge Least Square p r o c e d u r e . Our results and concluslons in the present paper remain unchanged. 8) C. K. Saw and R. B. Schwarz, unpublished results (1985). 9) N. Hayashi, T. Fukynaga, N. Watanabe and K. Suzuki, KENS Report V, National Laboratory for High Energy Physics, Ibaraki-ken, Japan, 1984.