Structural analysis of a binary metallic glass model. I. — The Pd80Si20 alloy

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Structural analysis of a binary metallic glass model. I. - The Pd80Si20 alloy

F. Lançon, L. Billard, J. Laugier, A. Chamberod

To cite this version:

F. Lançon, L. Billard, J. Laugier, A. Chamberod. Structural analysis of a binary metal- lic glass model. I. - The Pd80Si20 alloy. Journal de Physique, 1985, 46 (2), pp.235-241.

�10.1051/jphys:01985004602023500�. �jpa-00209962�

Structural analysis ^{of a} binary ^metallic glass ^model.

I.

The Pd80Si20 alloy (+)

F. Lançon (*), ^L. Billard, ^J. Laugier ^{and A. Chamberod}

Centre d’Etudes Nucléaires de Grenoble, Département de Recherche Fondamentale, Section de Physique ^du Solide, 85X, 38041 Grenoble Cedex, ^France

(Reçu ^le 13 juillet 1984, accepté ^{le 23 octobre} 1984 )

Résumé. 2014 Dans le premier papier ^{de cette} série consacrée à l’analyse structurale d’un alliage métallique amorphe

nous étudions un échantillon de Pd80Si20 obtenu par relaxation numérique. ^Nous discutons la reproductibilité

de la méthode et comparons ^avec les fonctions d’interférence expérimentales. ^{Nous abordons} l’analyse structurale

microscopique ^sous plusieurs angles : ^nous étudions le nombre de voisins de chaque type pour chaque type d’atomes, ensuite, ^nous effectuons une analyse ^{en termes de} polyèdres ^{définis à} partir ^de plans radicaux, enfin, ^{nous montrons} qu’on peut généraliser ^les cinq ^unités géométriques décrites par Bernal de façon à caractériser sans ambiguïté

l’environnement de n’importe quel ^atome de silicium. L’ensemble de ces méthodes révèle en particulier ^{une certaine}

tendance pour les ^atomes de silicium à avoir un environnement prismatique.

Abstract.

In the first paper of this series devoted ^{to a} structural analysis ^{of a} binary ^metallic glass model, ^we study

a Pd80Si20 sample ^obtained by numerical relaxation. We discuss the reproducibility of the method and make a

comparison ^{with the} experimental interference functions. Then we undertake a microscopic structural analysis

from several points ^{of view : we first} study ^{the number of} neighbours ^{of each} type ^{for each} type ^of atoms; secondly,

we analyse ^{the structure} by ^means of the radical plane method; ^{at last, we} show that it is possible ^to generalize

the five fundamental characteristic units introduced by Bernal, ^{so that we can} define the environment of any Si atom without any ambiguity. ^{All these} methods reveal a certain tendency ^{towards a} prismatic environment for the metalloids.

Classification Physics ^Abstracts

61. 40D

1. Introduction.

Numerical relaxation of various random packings ^of spheres ^{has been} widely ^{used to} provide ^{models for} amorphous ^metallic structures. The first attempts ^were naturally ^{done with one} type ^{of atoms and} although only ^metallic glass alloys ^{are stable at room} tempera- ture, this first stage ^{has been} very fruitful towards

achieving ^{a better} understanding ^{of the} amorphous

state. It has yielded sample ^models having ^{both a} high density ^{and a} pair distribution function typical ^of

metallic glass structure, which ^are the two main features to be accounted for in a realistic model. On the other hand, ^a _great ^{deal of} work has been done con-

cerning ^the microscopic analysis of these models [1-5] :

number of neighbours, ^local structural units, ^statistics

(+) ^{The main} part of this work is a part of the thesis of F. Laneon, University ^of Grenoble, ^France (April ^{18, 1984).}

about the Voronoi cells, ^{and the} search for holes or

interstitial sites. It should be noted that these results

depend essentially ^{on the} potential used, ^{but not on the} starting configuration ^{or on the} details of the relaxation

procedure.

Naturally, ^the experimental situation has called for models with several types ^{of atoms and some models} have been proposed [6-15]. However, the situation is far from clear; ^{on one} hand, this is due to a greater ^com- plexity of the model : necessity ^for introducing ^several types ^of potentials, analysis ^{in terms of} partial pair

correlation functions, difficulties in finding ^{an unambi-}

guous description of local structures, ^{etc. But on} the other hand, this is due to a more fundamental diver- gence among workers on this subject. Schematically,

one can distinguish ^two ways of thinking : ^{either one}

considers that a good ^{choice of} potentials ^{will lead} naturally ^{to a} good representation, ^{or one assumes that}

a good ^{choice of the initial} configuration (even ^{if it}

needs a relaxation) will be the main ingredient ^{in the}

model.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004602023500

236

We propose here ^an approach ^{of the first} type, ^where

the initial configuration is considered to be of lesser

importance. ^After having ^described the construction of the model, ^we discuss the reproducibility ^{of the}

results. Then, ^we analyse ^{the atomic} environments, using several methods. It appears that the local struc- tures are not as disordered as may have been supposed

in such a model : we find strong evidence for local

geometric ^units analogous ^to those which are imposed

« by ^{hand » in other models, but which are here the}

mere result of the relaxation. In a second _paper, we

will study ^the composition dependence and show in

particular ^{that the} density ^{can be well} interpreted

if one makes the approximation ^{that Si atoms are not} neighbours.

2. Construction of the modeL

We first describe the construction of a model for a

Pd80S’20 amorphous alloy. ^{We choose a modified}

Johnson potential 4lj, ^{which has} given good ^{results in}

the monoatomic case [5]. ^{For each} type ^of pair ij (i, j

^{Pd or} Si), ^{we make a} homothetic transformation

on Wj, ^{so that the} position rij ^{and the depth} eij ^{of the} minimum are given by

These values have been taken from [13].

We use periodic boundary conditions, ^{and let the}

size of the ^{periodic box vary according to the minimi-}

zation procedure, ^{in order to} obtain relaxed states under zero pressure.

Two classes of starting configurations ^{are consider-} ed, corresponding ^{to two} different methods :

2.1 THE SUBSTITUTIONAL METHOD. - First we consi- der a monoatomic sample ^{relaxed in a Johnson} poten- tial [5]; it remains stable after a change ^{of the} length

scale with a factor rPdPd/ro, ^where ro is the position ^of

the minimum of 0, with which the sample ^was relaxed, Starting ^{with this} « pure Pd amorphous state », ^we

construct a Pd80S’20 alloy by replacing ²⁰ pct ^of initial atoms by ^Si atoms, ^{at random. In order to test} the reproducibility, ^{two initial} samples ^{are built from}

two monoatomic relaxed samples. Then, they ^are

relaxed in the previously ^defined system ^of pair potentials. ^The energy per atom, E, and the atomic

density po ^are shown on table I, together ^{with the} dispersion between the two states. We see that we

obtain a good reproducibility.

2.2 THE INTERSTITIAL METHOD.

It has been shown that [5] ] the monoatomic amorphous ^{state can be} fully analysed ^{in terms of} interstitial sites which are

deltahedra, ^i.e. polyhedra ^with triangular ^{faces. Start-} ing ^{with the same two} samples ^as above, ^{we introduce}

Si atoms in these interstitial sites, beginning ^{with the}

Table I.

Potential energy per ^atom E (in ^units of 8PdPd) ^{and atomic} density po (number of ^atoms per ^unit

volume, ^{in units} of ^10-2 A- 3). The table gives ^the

means (x) ^{over two} configurations ^{and the} difference (Ax)

between them.

largest of these holes. This is reminiscent of the model of Polk [16], ^{but it must be} emphasized that, ^{in a} relaxed monoatomic model, ^{there are} very few large holes; ⁱⁿ particular, ^{to obtain a 20} pct ^metalloid concentration, ^{some of the Si} atoms must be introduc- ed in smaller (octahedral) ^sites.

Then, ^{these two} « interstitial type » alloys ^are relaxed, ^{and the results are} given ^{in table I. Here} also,

the reproducibility between these two samples ^is very

good.

2.3 THE FINAL METHOD.

However, ^{it can be seen} in table I that the dispersion ^{between the states of the}

substitutional method and those of the interstitial

one is rather large, being ^{of the order of} several per cent in relative value, instead of several _per thousand for the dispersion ^between two states in the same class.

A better reproducibility ^{is obtained} by ^{« anneal-} ing » by ^{means of a} mechanical cycle, which has been shown to give good ^results in the monoatomic case.

For each sample ^a compression ^{strain is} applied step by step, up ^to 12 % : ^{at each} step, ^the dimensions of the cubic box are decreased by ^one per ^cent and the _energy is minimized with respect ^{to the} atomic coordinates.

Then the strain is gradually ^decreased using ^{the same} procedure. Finally, ^full minimization is made with respect ^to coordinates and density variations. Thus,

each sample ^is subject ^{to 25 new} relaxations. The results are shown in table I, ^{and it is seen} that we have

now a better reproducibility between the two classes.

This is confirmed by ^the comparison ^{between the} pair

correlation functions. Figure ^{1 shows the} partial pair

correlation functions and the partial interference functions obtained in the case of the interstitial ^build- ing after mechanical treatment. We have not repre- sented the curves obtained in the case of the substitu- tional building, ^because they ^are very close ^to the

previous ^ones.

Fig. ^1.

^Partial pair correlation functions and partial

interference functions for a Pd,,Si,o ^model (initial ^intersti-

tial construction followed by ^a relaxation and a mechanical

annealing).

2.4 COMPARISON WITH EXPERIMENTS.

We compare the results given by the simulation to the experimental X-ray diffraction curve. If we make the approximation,

Fig. ^2.

Comparison ^of pair correlation functions with

experiments : a) experimental (dashed line) and simulated

(continuous line) ^{curves for 4} I7po r[gx(r) - 1].

b) ^Neutron experiments ^obtained by [18] using ^several

values of the cut-off Km in the interference functions, ^and corresponding ^curve for the model.

which is shown to be very good ^{in this case, that the} weighting ^{terms of the} partial interference function in the expression ^of Sx(q) ^are q-independent, ^{we have :}

Figure ^{2a shows the curves} 4 nrPo[gx(r) - I], experimental and simulated : in this latter _case, a

rescaling ^of lengths ^{with a} factor 1.012 has been made to adjust ^the phases of oscillations at large ^{r. This}

leads to a value of the density po

^0.0717 (A)- 3 or, for the volumic mass, p.

10.8 g/cm3, ^as compared

with known data of 10.63 g/cm3 [9], 10.3-10.5 g/cm3 [13]

or 10.25 g/cm3 [17].

The curves are also in good agreement : ^{let us} just

notice that the model yields ^{a small} pre-peak, ^which

is due to a contribution of Pd-Si pairs, and this is also obtained in the experiment.

Turning ^{now to neutron} diffraction experiments,

one obtains

which is compared ^{to an} experimental ^{result of} [18]

in figure ^{2b. The} pre-peak ^{is now} clearly ^visible.

As for the partial correlation functions, ^{we have} experimental ^{results of} [19] ^which give ^the partial

Pd-Pd and Pd-Si functions obtained through X-ray

and neutron diffraction experiments ^and the assump- tion that Ss;s,

^1. Combining equations (1) ^and (2)

Fig. ^3.

^Partial pair interference functions obtained by [19] using ^the hypothesis Ssis;

^{1 and} corresponding ^curves

for the model.

238

one obtains Spdpd ^and SPdsi ^{as functions of} Sx, Sn ^and Ss;s;. ^{It will be seen that this} gives pseudo-partial ^func-

tions and

which we have calculated from the model and are

compared ^{with the} experimental ^{results on} figure ^3.

The agreement ^with experimental ^{results can be}

considered as rather satisfactory, taken into account that we have used _very simplified interatomic interac- tions.

3. Structural analysis.

3.1 NUMBERS OF NEIGHBOURS.

One can find in the literature on amorphous alloys different results for these numbers. This _may be due to differences between

experimental situations or between basically ^different models, but it is also clearly ^{due to the lack of a clear}

definition of what is the nearest neighbour distance in a

disordered medium. As for _us, it can be seen on figure ¹

that the partial pair correlation functions _{gPdPd and} gPdSi have a ^clear minimum with zero value, ^situated

at 1.26 times _rPdPd and _rPdSi, respectively, ^{and it is}

rather natural to define the corresponding neighbour- ing pairs ^as those which have a length less than this

limiting ^{value. The} situation is not so clear for Si-Si

pairs, ^as emphasized ^on figure ⁴ which shows the mean

number nij(r) of j-atoms ^{at a distance less than r from}

an i-atom. Thus, ^{we take the same} limiting ^value

1.26 rsisl for ^the Si-Si neighbouring pairs, ^{but it is} quite arbitrary. ^{Table II} shows the results obtained with those assumptions. ^{The most} striking feature is that most of the Si atoms have either 8 or 9 Pd neighbours :

let us recall that this strong ordering ^{has not been} pre-

imposed but is the result of the relaxation.

3.2 BERNAL ENVIRONMENTS.

For each Si atom, ^we have searched for a trigonal prismatic environment,

6 Pd atoms making ^two triangles ^linked by ^three

« square » faces : in fact, ^{such a face is a} cycle ^of

order 4, ^the diagonals ^{of which are} greater ^than 1.26 rpdpd (as ^discussed above), the sides of which are

less than 1.26 rPdPd (and ^{so are the sides of the two} triangles). Moreover, the distance from the central Si to each of the six Pd must be less than 1.26 rPds;.

We have found that the proportion ^{of Si atoms in} prisms ^{varies from P}

^{6 ±} 1 per ^{cent to} P

22 ± 2 per ^cent in the substitutional _case, before and after the mechanical treatment, ^{and from P}

³⁶ + _{6 per} ^cent

Table II.

Number of neighbours ^{in the model.}

Fig. ^4.

Mean number niir) ^{of atoms of type j which are}

at a distance less than

from an i atom : neighbouring ^{of Pd}

atoms (left curves) ^{and Si atoms} (right curves).

to P

30 ± 4 per ^cent in the interstitial case. The

dispersion ^{between two} samples ^{in the same class is} roughly of the order of magnitude of the fluctuation which could ^be expected ^{for a set of N Si atoms} (here

N - 300) having ^the probability ^{P of} being ^{in a} prism.

We notice that the mechanical treatment has not been sufficient to reduce the dispersion ^{between the two}

classes of models, but these differences represent very local variations, ^as compared ^{to the} reproducibility

of the density, ^the energy ^or the pair correlation func- tions. Some of these prisms may have one or more

square faces capped ^{with a} supplementary ^Pd atom, and we have found that about 2/3 ^{of the} prisms ^were capped ^three times, ^about 1/3 capped twice, ^and very few capped ^{one time or not} capped ^{at all.}

By ^{the same} method, other Bernal environments

can be searched for : we have found (after ^mechanical treatment) ^about 14 per ^cent Si atoms in Archimedean

antiprisms, 4 per ^cent Si in tetragonal dodecahedra, practically ^{no Si} (1 over 344) ⁱⁿ octahedra, ^{and none}

in tetrahedra.

This describes the environment of about 40 _per cent of the Si atoms and we need a more sophisticated

method to analyse ^{the other ones.}

3. 3 RADICAL PLANE METHOD.

A generalization ^of

the Voronoi polyhedra ^{has been} proposed [20] ^to analyse amorphous alloys, by ^{means of} polyhedra

defined by ^radical planes : ^{these are the} planes ^contain- ing ^{all the} points ^{M for} which the distances from M to the tangency points ^{to two atomics} spheres ^are equal.

Taking ^an atomic radius Rpd

rpdpd/2 ^and Rsi =

rPdSi

Rpd, ^{one can} thus obtain a full decomposition

into polyhedra associated with these radical planes.

Each polyhedron ^can be characterized by ^{the set} (n3l n4, n5...) ^of integers ni, where ni represents ^the number of faces of the polyhedron which have i sides.

As for the polyhedra ^{around Pd} atoms, ^{we observe a}

predominance ^{of the} (0, 2, 8, 4) type (about 11 per ^cent of Pd atoms) ^{which was also the case for the mono-}

atomic amorphous (in ^this case, using ^{the Voronoi} decomposition). ^{But we are} mainly concerned with the

polyhedra ^{around Si atoms.}

The analysis [20] ^of samples ^obtained by ^Gaskell (trigonal prismatic packing and distorted prismatic packing) [13-14], by relaxation under a Lennard-Jones

type potential ^had shown very few topologically- perfect prismatic polyhedra (0, 3, 6). ^In fact, ^other types ^of polyhedra ^{were found, which can be considered}

as resulting ^from slight distortions starting ^{from a} prism, ^{but we have} no measure of the amplitude ^of

such a distortion; ^{for instance a} type ^like (0, 2, 8) corresponds simply ^{to an} Archimedean antiprism

with its two square faces capped by ^{half an} octahedron.

In our samples, ^the analysis ^has given ^the following

results for the more frequent types ^of environments of the Si atoms :

22 pct : (0, 3, 6)

14 pct : (0, 3, 8)

8 pct : (0, 3, 6, 1)

7 pct : (0, 2, 8,1 )

etc.

Moreover, ^{we can look at what} type ^of neighbour (Pd ^or Si) ^{each face} corresponds ^{and for} instance, ^we

have found that the 22 pct ^of prismatic environments

can be decomposed ^into

16 pct : (OPd ^{+ OSi, 3Pd +} OSi, ^{6Pd +} OSi) ^and

6 pct : (OPd ⁺ OSi, ^2Pd + ISi, ^{6Pd +} OSi).

This corresponds truly ^to prisms ^{of Pd} capped ^with respectively ^{3 or 2} supplementary ^{Pd atoms. Due to a}

different definition, these numbers differ slightly ^from

those obtained in 3.2, ^{but are in rather} good agreement

showing ^the presence of prisms ^and antiprisms.

Nevertheless, ^a further refinement is _necessary if one

whishes a complete description of all Si environments.

3.4 DELTAHEDRA AROUND SI ATOMS. - As previously,

the Pd neighbours ^{of a} given ^{Si atom are defined as}

those Pd atoms which are nearer than 1.26 rPdSi; then,

among these Pd atoms, ^{the « nearest} neighbours »

are defined as those which are nearer than 1.26 rpdpd from each other. We are now interested in the descrip-

tion of such a Pd environment. In fact, considering ^a given environment, ^{we can} find the Si atoms which have such one : that was done, ⁱⁿ 3.2, for the Bernal environments. Can we find all possible environments, i.e., ^{can we} find the Pd environment of each Si atom in a systematic ^{manner without} having defined it at the

beginning ? ^{This seems a} formidable task [1] ^{due to}

the large variety ^of geometric figures ^{which can be} imagined, ^{but we will describe now an easy method}

to succeed in it.

In the case of a monoatomic sample, ^{we have}

shown [5] ^{that the} vertices of the Voronoi polyhedra

were associated to a tetrahedron formed by ^{the four}

atoms which are equidistant ^{from each other.} Here, similarly the vertices of the radical planes polyhedra

are associated to a tetrahedron formed by ^{the four}

atoms which have the same tangent length ^from

them. These tetrahedra form a network which is the

topological inverse of the network of the radical planes polyhedra vertices, ^and provides ^a complete ^and unambiguous description ^{of the structure.}

Now, ^{we can work as} previously ^[5] and cluster

together tetrahedra sharing long pairs, ^i.e. pairs ^of

240

neighbours ^{in the} geometric ^radical plane ^{sense which}

are longer ^{than 1.26} rpdPd ^or 1.26 _rpdS¡’ according

to the nature of the pair. As for Si-Si pairs, being

interested only in Pd environment of Si atoms, ^{we do}

not admit _any Si-Si geometric pair ^{as a short} pair,

i.e. _any Si-Si pair ^is considered ^as long and is thus

a nucleus for our clusterization procedure. The details of the method have been given ^elsewhere [5], ^and

we are finally left with units bounded by triangular faces, ^the edges ^{of which are} near-neighbour pairs

Pd-Pd or Pd-Si only, ^{which do not} overlap ^and completely ^{fill the} space, constituting ^a deltahedron network.

Now, ^let us consider _any Si atom. It is a common vertex to several deltahedra, ^{and thus we have} simply

all the neighbour pairs Si-Pd from the central Si atom.

Moreover, ^{we have the} connectivity between these Pd atoms, ^{because we} know the short pairs linking them,

as being ^the edges of the deltahedra. For instance, ^{let us}

suppose that : (i) ^{the Si atom is a common vertex of}

deltahedra Dl, D2, ... ^etc..., (ii) ^{this Si atom is linked}

to Pd atoms X1, X2, X3 ^and X4 of the deltahedron Dj, (iii) ^{there are links} (short pairs being edges ^of D1) only ^between Xl X 2, X2 X3, X3 X4, X4 X1. ^{If we}

« rub out » the PdSi pairs from the central Si, ^{we see} it as the «centre» of an environment having ^a

« square » face ^Xi X 2 X3 X4. ^To characterize the latter, ^we imagine ^a fictitious vertex Y linked to X1, X 2, X3 ^and X4. ^thus constituting ^a cap ^{as a} half- octahedron : the _square face X1 X 2 X3 X4 ^{is now} replaced by four fictitious triangular ^faces Y XIX 2’

YX 2 X3, YX 3 X 4, YX4 X1. ^{In this manner, we have}

defined for each Si atom an environment which is a

deltahedron (limited only by triangular faces) : ^{it can}

be characterized by ^{the comer numbers} (n4’ ns ...) where ni ^means the numbers of comers (vertices)

common to i (triangular faces). ^The description ^{will be}

complete ^{when one} says what ^are the fictitious _caps, and we can give simply their numbers (m4, m5, ...).

For example, ^a trigonal prism ^{with two} square faces

capped ^is obtained, starting ^{from a} trigonal prism

with its three _square faces capped and indexed by

p

(3, 6) by removing ^one cap, which is a vertex common to 4 triangular ^{faces so that the} missing caps indices are m

(1, 0).

Table III gives ^{the more} frequent fictitious del- tahedra environments for the 344 Si atoms in the model. Naturally, among them, ^one finds the Bernal environments discussed in 3.2, and it has been verified that the two methods of search coincide in these cases. However, ^some Bernal environments can

be parts ^{of more} complex environments and that

happened ^{in a few cases.}

4. Conclusion

Starting ^{with a} relaxed monoatomic model for an

amorphous metal, ^we have constructed binary samples, using ^two classes of methods : one corresponds ^to introducing ^{Si atoms} in substitutional positions, ^the

other to introducing ^{Si atoms in} interstitial positions.

After relaxation, ^both samples remain different but a

cycle ^of compression greatly reduces the disparities :

in fact, ^{one obtains a} very good reproducibility ^{as far}

as macroscopic values of _energy, density ^or pair

correlation functions are concerned. Moreover, ^the agreement ^with experimental ^{data is} encouraging, taking ^{into account the} necessarily ^crude approxima-

tions which must be made in the choice of the inter- action potentials.

However, ^and unlike the monoatomic _case, the mechanical treatment is not completely sufficient to obtain a state which is well reproducible ^{at a micro-} scopic ^{scale. In} fact, ^{we have made a detailed} analysis

Table III.

Local deltahedra around Si atoms.

of the local structure : it _appears that the PdPd and PdSi pairs show distributions which are similar to those of monoatomic pairs. ^{On the} contrary, ^{the Si}

atoms have practically ^no correlation between them- selves, ^{even at} short distances. The microscopic study

can be developed ^{from several} points ^{of view : one can}

define the neighbour pairs ^as those which correspond

to the first peak ^{of the} correlation function _up to its first (well defined) ^{minimum, or one can use} the radical

planes method for providing ^{a full} analysis ^{of the}

structure. The most frequent ^local structures around the Si atoms have been found to be the trigonal prism

and the Archimedean antiprism, ⁱⁿ respective pro-

portions ^{of 30} pct ^{and 14} pct ^{in the first} type ^{of ana-} lysis, ^{or 22} pct ^{and 14} pct ^{in the second} type. Finally, combining ^the criterium for neighbour pairs ^{with the} decomposition through the radical planes, ^{we have}

been able to describe the local Pd environment of _any Si atom in terms of fictitious deltahedra, ^{and the} predominance ^{of the two} local units just quoted ^has

been confirmed

This type ^of analysis ^{can be} pursued ⁱⁿ any binary glass model, ^{and in} particular, ^{it can be} developed ⁱⁿ Pd, -.,,Si., alloys ^to understand the composition depen-

dence of these properties : this is the field of a second paper.

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