volume distribution of empty regions in disordered structures

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COMPUTATIONAL MATERIALS SCIENCE

Computational Materials Science 3 (1995) 359-367

Volume distribution of empty regions in disordered structures

Vittorio Rosato, Pier Giuseppe Gabrielli

Dipartimento Innovazione, Settore Nuoci Materiali, ENEA C.R.E. Casaccia, CP2400, 00100 Roma, Italy

Received 28 April 1994; revised 8 August 1994; accepted 19 August 1994

Abstract

We have developed a numerical technique which allows to determine the volume distribution of empty regions and interstitial regions in disordered systems once the coordinates of the point particles are known. The proposed scheme consists of three main tasks: (1) association of a hard-sphere radius to each particle of the system; (2) determination of elemental spherical regions, filling the free volume of the system, with a random search method; (3) percolation of the elemental spherical regions which are connected (or lie very close to each other). The proposed technique has been applied to study disordered structures (Pd and Ni,Al in their liquid and amorphous phases). Results indicates significant differences in the volume distribution of empty regions in the liquid and amorphous structures.

1. Introduction

The technological relevance of the properties of metallic compounds in the amorphous phase (formability, corrosion resistance, magnetic properties etc.) has provided a strong support for the study of these materials. In particular, it has been stressed the strong correlation existing, in amorphous systems, between structure and properties. Diffusion and elastic properties, stress-strain be- havior are almost univocally related to the structural characteristics of disordered media. In this respect, since some decades, much work has been addressed to a more detailed characterization of the amorphous structure from both the theoretical and experimental points of view [l--3].

Since the seminal works of Bernal on the disordered structures [4], most of the theoretical efforts has been devoted to the study of both the microscopic arrangement of the atoms and the “structure” of the empty regions spread over the

whole system. The characteristics of amorphous structures strongly affect the diffusion properties which are, in turn, related to the free volume distribution and kinetics. The study of the topological features of free volume has thus consti- tuted an useful tool to approach this matter [5-81. The study of empty regions and their role in diffusion and growth mechanisms has been exten- sively debated to assess the existence of isolated (and/or clustered) point defects (vacancies) in the amorphous structures [9].

A useful combined theoretical-experimental approach to the study of amorphous structure consists in the determination of the pair correlation function g(r) from diffraction data [lo-111. This function (which contains only an averaged representation of atomic positions) is inserted in the computational framework of the so called “Reverse Monte Carlo” technique (RMC) which allows to associate one (or more> possible atomistic structures compatible with the considered

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360 K Rosuto, P.G. Guhrielli / C’omputationul Matrrials Science 3 (1995) 359-367

data. Once the associated structure is determined able free volume. The volume of the local void is (i.e. the coordinates of all atoms of the system - then identified as the sum of the esr which are or of a representative part of it - are known), it is very close to each other. This “condensation” possible to perform on that structure detailed procedure has been performed in order to ap- microscopic-scale analysis, as the one which is proximate also non canonical empty regions with proposed in the present study. generic geometrical shapes.

Other methods which allow the determination of the atomic coordinates of a given system are Molecular Dynamics and “direct” Monte Carlo (see Ref. [12]) where atomic coordinates are obtained by solving the problem of energy mini- mization of a dynamical system whose particles interact via a given force law.

Giving the coordinates of the particles of the system, the proposed scheme consists of three main tasks:

2. Statement of the problem

The empty space embedding a system of hard- spheres consists of a connected region where larger volumes are separated by narrow passages. An analysis of the topological features of the empty regions in disordered systems has been already attempted by several authors [7] by means of the Delaunay simplices analysis. In the present work we do not attempt a classification of empty region on the basis of their indexes of connectiv- ity defined in the Delaunay geometry. In fact, it has been recognized that the empty regions of disordered systems are complicated configurations consisting of a great number of simplices of non canonical shapes. Moreover, a comparison of data resulting from several works focusing on the volume percentage of Bernal canonical holes in disordered systems has revealed strong discrepan- cies [71 which have been explained as related to the model potentials used to produce the considered atomistic configurations. In this respect, topology does not seem to be useful to describe general features of empty regions in disordered media. We would thus rather focus on a direct evaluation of their volumes. It is, in fact, their associated volume which is the physically relevant quantity used to infer correlations between structure and diffusion properties.

(1) attribution of a hard sphere radius to all the particles of the system. This procedure must fulfil the following requirements: (a) the hard-spheres representing system atoms must non overlap; (b) they must fill a significant sample of the total available free volume. In order to allow the simoultaneous fulfilment of points l(a) and (b) as stated above, it will be shown that the hard sphere radii of particles of the same species will be allowed to be some- what different each other, provided that (cl their distribution be sufficiently narrow to pre- serve their identity of equal particles. (2) determination, via a random search, of the maximum possible number of esr with the con- straints that such spheres do not overlap each other and do not overlap with the hard-spheres representing the atoms of the system; (3) percolation of the esr in the case they touch or they lie very close to each other. In the following sections, we describe, in de-

tails, how the three tasks have been accomplished. Furthermore, we give the results obtained by the application of this technique to liquid and amorphous structures in mono and polyatomic systems. Last section will be entirely devoted to the analysis of the resulting data.

3. Evaluation of the hard spheres radii

The proposed method consists in the insertion into the empty regions of the disordered structure of a finite number of elemental spherical regions (esr hereafter) filling most of the avail-

The first issue concerns with the definition of radii associated with the system particles. A generic computer simulation (RMC, Molecular Dynamics or MonteCarlo) results in the set (rJ representing the positions of the center of each particle (the label i runs from 1 to N, the total number of particles in the system). Nothing is directly said about the radii Rj which should be

V. Rosato, P.G. Gabrielli /Computational Materials Science 3 (1995) 359-367 361

associated to each point in a hard-spheres picture of the structure. A reasonable value for Ri could be inferred from the repulsive part of the inter- atomic potential or taken equal to the hard sphere radius of the perfect crystal lattice at the considered temperature. The first suggestion is only qualitative and it cannot be applied on a RMC configuration where there is any potential scheme underlying the resulting atomistic structure. The second suggestion, once applied to a set of point coordinates, always results in the overlap of several spheres. Moreover, if one associates to each atom the same radius, with the constraint to avoid spheres overlap, one must choose a value which satisfies the most pathological structural conditions met in the system, i.e. that associated to the smaller value of Ri. As a consequence of that, the total volume of the system will be not efficiently filled with the hard spheres construction based on a unique radius for all particles (of a given species, in the case of polyatomic systems). The constraint of efficient space filling could be much better fulfilled by associating with each point-particle a radius whose value is related to the local structure of the system. Therefore one expects that particles of the same species will have similar (but not necessarily equal) radii. If the point distribution represents a perfect lattice, the radius distribution will be composed by a &function at the relative hard-sphere radius. In a random points distribution, in turn, one would expect the radii distribution to be a broad one with no small-R cutoff. As we are dealing with a physically grounded point distribution, the radii distribution is expected to be reasonably narrow, centered around an average value which should be very close to the corresponding crystalline hard sphere value and having a clear cutoff at small R values because of the existence of short range repulsion which inhibits infinitesimal ap- proaches among particle centers.

Let us consider the set of atomic coordinates {rJ. For the sake of generality, we will describe the technique on a two components system made up by N, atoms of species 1 and N2 atoms of species 2 such as N, + N2 = N.

Let us consider a generic particle i. For each atom k # i, with k belonging to the first coordi-

nation shell, which lies at a distance d from atom i, we will consider the value of the radius kRi CkRi indicates the radius of the particle i as determined by the presence of the particle k) defined, with the constraint of non-overlap between i and k, as follows:

kRi +‘R, = d, (1)

iRk/ kRi = rk/ri, (2)

where rk and ri are the ionic radii of the atomic species of particles k and i, respectively.

This procedure will define a set (‘R,) of radii values for the particle i. If we will assign to the particle i a value for its hard sphere radius equal to

Ri = min{‘R,} (3)

we will automatically fulfil the non-overlap requirement. In our opinion, if one wishes to satisfy the non-overlap criterion between the hard spheres, this method provides the most efficient filling of the space on the basis of a disordered distribution of points particles.

4. Definition of elemental spherical regions and their percolation

Up to now, given the assembly of N atoms defined by the positions {ri}, we have associated to each particles a radius Ri, by fulfilling the two requirements of non-overlap between the spheres. Once the construction of the hard-spheres representation of the atomic particles, centered on {rJ with radii {R,}, is accomplished, the volume will result to be divided into a part occupied by the hard spheres and a series of interconnected empty regions embedding the atomic positions.

The method proposed to evaluate the volume distribution of empty regions can be summarized as a random motion of a small “red balloon” which, put into an empty region, searches its center i.e. the point where it can be inflated up to assuming the maximum allowed spherical volume with the constraint of touching, without overlap- ping, the neighbouring hard-sphere particles.

362 I/: Rosato. P.G. Gabrielli /Computational Materials Science 3 (1995) 359-367

Let us start by choosing a point ri” located in a generic point of the free volume of the sample (i.e. where I r$” - ri I > R, for every i, which irn- plies that the considered point does not belong to the volume occupied by a hard sphere associated to a particle). We then evaluate the minimum distance between this point and the surrounding atoms as being

s~)=min(lr~‘)-r,(-R,) for i= 1, N+k- 1

(4)

(the reason for the range of variation of the index i will be explained in the following). .ri’) will thus define the radius of the biggest sphere which we could inflate with the center in t-f’. We then slightly move the point in a random direction by an amount Sr,, (being sure that the new position still belongs to the empty region) such to define a new point rp’ = ri’) + Sr,,. Now we can evaluate

s~2)=min(lr~z’-r,)-R,) for i=l, N+k- 1

(5)

If sP) > sb’) (i.e. we have displaced the point r-h’) in a way which results in the increase of the volume of the “red balloon”) we accept the move. If not, we will return to ri’) to perform an other trial with a different, randomly chosen, St-,,. This procedure will be iterated up to reach a point t-f’ such that, after several thousands trials, for all the attempts made to generate the subsequent sr+‘), it will result $+l) < ST) (i.e. we have reached a point rp’ which maximizes the minimum distance with the surrounding atoms). We thus introduce in rr) a “virtual particle” of radius R, = sf) representing the esr and we restart the procedure by choosing a new point t-24 ,. To make this further choice, we make sure that the new point does not fall into one of the previously defined esr. This constraint, introduced to avoid double counting of the voids, is contained in the conditions imposed to index-i in Eqs. (4) and (5). In fact, as far as i <N, the associated R, are atomic radii. As i > N, the considered radii (and the corresponding particles) will be those of the esr associated to the explored voids which, once defined, are treated on the same ground as they were real hard-sphere particles.

Using these prescriptions, we introduce a number of esr into the empty regions of the system in a way to considerably reduce the allow- able free volume, defined as the difference between the computer box volume Vr and the volume occupied by the atoms defined as I’&, = Cr=, o, (where oi is the volume of the ith atoms represented as an hard-sphere of radius Ri). We have decreed the end of the filling procedure with esr ‘s when, after the insertion of M of those objects, it happens that

5 I’, = 0.5( VT - V&) 1-I

where ~3;‘s are the volume of the inserted esr. After having recorded the position of the cen-

ters and the relative radii of the inserted esr, we evaluate whether or not two or more of these objects touch or lie very close to each other within a distance of 0.01 A. In this case, we merge them together to form a bigger region. If m esr are merged together, the void distribution, defined by the histogram of the esr, will be changed accordingly. If the voids uI, j = 1, m are connected each other, then we define the cumu- lative volume I/ as

v= Fi;, (7) j=l

the volume distribution will be changed as:

N(V) =N(V) + 1

and

(8)

N(l;) =N(r;) - 1 for j= 1, m. (9)

The final form of the void distributions which will be shown in the following are those obtained after the “condensation” process of the esr. This process considerably reduces the role played by the coefficient at r.h.s. of Eq. (6). We have esti- mated its role by reducing its value from 0.5 down to 0.3. The result has been the definition of bigger number of small size esr which, however, disappears after the condensation process. The dependence of the volume distribution of voids on the number of esr used in the computations will be discussed in the following.

V. Rosato, P.G. Gabrielli /Computational Materials Science 3 (1995) 359-367 363

5. Computations

We have applied the proposed scheme with the aim of characterizing the volume distribution of empty regions contained in liquid and amorphous structures as they have been provided by Molecular Dynamics simulations of mono (liquid and amorphous Palladium) and polyatomic systems (liquid and amorphous Ni,Al). All these systems, each of them containing N = 500 particles, have been simulated by using empirical n- body potentials derived by a second moment ap- proximation of a tight binding hamiltonian. Po- tential parameters and scheme reliability have been discussed in previous, detailed studies [13,14].

We will shown, in the following, the results obtained in the mentioned cases, i.e. amorphous Pd at T = 300 K, liquid Pd at T = 1750 K, amorphous Ni,Al at T = 300 K and liquid Ni,Al at T = 2100 K. All configurations refer to constant pressure <P,,, = 01, constant temperature simulations. The instantaneous configurations have been taken after quite long equilibration runs (lasting at least lo5 time-steps, for a simulated real time of lo-” s). When several configurations of the same systems have been used to perform configurational averages, they have been taken, from the same simulation run, with a time separation of at

0300 /

0250

0 200

1

.oo 20 .40 60 .a0 1.00 1.40 1.80

r (A)

Fig. 1. Probability distribution of radii in the amorphous Pd

structure (T = 300 K, Pext = 0). Dotted line indicates the value

of the hard sphere radius of crystalline Pd in the same

thermodynamic conditions.

0.300

0250-

0 zoo-

0.150. c z

I! , , , / ~, , / .oo .20 .40 50 .a0 1.00 1.40 1.80

r (A)

Fig. 2. Probability distribution of radii in the liquid Pd struc-

ture (T = 1750 K, Pext = 0). The dotted line indicates the

value of the sphere radius of crystalline Pd in the same

thermodynamic conditions.

least lo4 time steps in order to ensure sufficient independence.

6. Amorphous versus liquid systems

As a first step we have evaluated the hard- sphere radii of the atoms belonging to a given instantaneous configuration. Figs. 1 and 2 show the distributions of hard-spheres radii associated to Pd atoms in amorphous and liquid configurations, respectively (data refer to averages over four independent configurations for each phase). The average values of the Pd radius in the amor, phous and liquid structures are ( r)am = 1.275 A and ( r )liq = 1.238 A which must be compared with the hard-sphere radius of crystalline Pd at the same temperature (r >‘” = 1.374 A (dotted line in Figs. 1 and 2).

Using the radii distributions reported in Figs. 1-2, we have evaluated the volume distribution of voids in those structures, resulting in the reported Figs. 3 and 4. In order to explain the effects on the void distribution of the number of inserted esr and that related to the subsequent condensation process, we report the shape of the voids distribution in the amorphous Pd (a) before the condensation process (Fig. 5) and (b) after the condensation process (Fig. 6). As it is clear from

364

80 T

70

60

50

g40

30

20

13

0 .oo 50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5 00

V (A**3)

V. Rosato, P. G. Gabrielli / Computational Materials Science 3 (1995) 359-367

Fig. 3. Volume distribution of empty regions in amorphous Pd

at T = 300 K and P,,, = 0. It refers to an average over 4

different configurations, each of them filled with - 5000 esr’s,

in order to fulfil Eq. (6).

Fig. 5, the use of a bigger number of esr results in the increase of the small volume peak. This peak almost disappears upon condensation of esr, as it can be noticed from Fig. 6. The distributions reported in Figs. 5 and 6 refer to a single amorphous configuration.

We have evaluated the distribution of atomic radii for Ni and Al in the cases of amorphous and liquid Ni,AI (Figs. 7 and 8, respectively). Figs. 9 and 10 represent the volume distributions of empty regions found in these systems. All these

.oo .50 1.00 :.50 2 00 2.50 3.00 3 50 4.00 4.50 5.00 V (A**3)

Fig. 4. Same that Fig. 3 for four different Pd liquid configura-

tions The dotted line represents our best fit of data using Eq.

(10).

4oom 5000

0 0.5 1

Volume (A3)

1.5

Fig. 5. Distribution of esr volumes (before condensation) for a

Pd amorphous configuration. The number (2500,3500,5000)

refer to the total number of esr inserted in the configuration.

pictures have been evaluated by averaging over four different configurations, each of them filled with around 5000 esr’s to fulfil Eq. (6).

A common feature of all figures has to be noticed. It concerns the unphysical drop to zero of the volume distribution at small values of volume. Two different effects contribute to this feature. First of all, the finitness of our sampling does not allow to explore the smallest empty

140 ,,,,,,,,,,,,,,,,,,, ,,,,

H woo -

0 0.5 1 1.5 2 2.5

Volume (A3)

Fig. 6. As for Fig. 5 after the condensation process.

I/ Rosato, P.G. Gabrielli /Computational Materials Science 3 (1995) 359-367 365

0.16

0.12 i T Tir

o.oe-

0.04-

0.00 / I I / I I’

- (Al) 3 r(Ni)

L_l .oo .20 .40 .60 .I30 1.00 1.40 1.80

r (A)

Fig. 7. Radii distribution of Ni and Al atoms in the amorphous Ni,At (T = 300 K and Pext = 0) averaged over four independent configurations.

regions. However, empty interstitial regions of volumes smaller than that pertaining to a tetrahedral interstice in the fee parent structure are unlikely to occur. Very small holes are not, therefore, self-existing entities but they must appear in the vicinity of a bigger one. In this respect, we can deduce that our volume cut-off should be of the order of the fee tetrahedral interstitice. Re- gions with smaller volumes, if any, would have been certainly merged into some larger confining region.

0.14.

0.12.

0.10

0.08 *;I‘ z

0.06

0.04

0.02

0 004 .oo .20 .40 50 .BO 1.00 1.40 1.80

6)

- w 3 r(Ni)

Fig. 8. Radii distribution of Ni and Al atoms in the liquid Fig. 10. Volume distribution of voids in the liquid structure of

model of Ni,Al (T = 2100 K and Pext = 0). Ni,AI.

1 150-

z =100-

50-

or', .oo so 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

V(A**3)

Fig. 9. Volume distribution of voids in the amorphous structure of Ni,AI (averaged over four independent configurations)

The volume distribution of empty regions in the liquid phase, disregarding the cut-off at small volumes, can be satisfactorily represented by a Turnbull-Cohen distribution 1153 of the type

N(v) =&(7/T/,) exp( -~v/b/,) (10)

Our best fit of fig. 4 (liquid Pd data) with Eq. (10) yields y/l/, = 0.86 A-” and No = 24.56. The average free volume per atoms will result to be (V) = V,/y = 1.16 A3. In our calculations it amounts to around 14% of the average volume of a Pd atom (as deduced from Fig. 2). It must be also considered that the distributions of Figs. 4

14

12

10

68 z

6

4

2

0 .50 1.00 2.00 3.00 4.00 5

V (A**3)

366 V. Rosato, P.G. Gabrielli / Computational Materials Science 3 (1995) 359-367

and 10 have a very slow decay as they are still not vanishing at 10 A3 < I’< 15 A3. This implies the presence in the liquid structures of quite large empty regions whose volumes amount to the same as a “vacant particle” (having a volume of N 7.95 A3 in liquid Pd, as deduced from Fig. 2). Despite Bernal’s liquid picture as “. . an assemblages of atoms containing no crystalline regions or holes large enough to admit another atom. . . ” [ 161, we have shown large empty regions having sizes representing a noticeable fraction of the total available free volume.

The structure of the volume distribution of voids in amorphous structures is, in turn, quite different. The analysis of Figs. 3 and 9 reveals the presence of a strong peak at values of volume of the order of 0.5 A3 and a broader region which ends up at values of the order of 2.5 A3. This fact could reflect the particular morphology of amorphous structures where a certain number of sta- ble, crystalline nuclei have formed. The empty regions characterized by small volumes ( < 0.5 A”) might be related to the holes (interstitials) within the crystalline-like regions which are present, as the volume value of few tenths of A3 is of the same order of magnitude of the volume of interstitial sites in Pd crystalline lattice (volume of the octahedral site = 0.78 A3, volume of tetrahedral site = 0.125 A3”>. The broad part of the distribution (0.5 A3 < V< 2.5 A31 could be related to the clustering of small voids in between the crystalline grains, if we adopt a picture of the amorphous structure as a “liquid” of compact crystalline seeds randomly distributed in size and orientation. In this frame, in fact, it could happen that two or more interstitial regions outside adja- cent grains lie close to each other because of the different orientation of the grains.

Further insights on the voids structure can be obtained by the analysis of the void-void pair correlation function g,(r) (Fig. 11 for the case of amorphous Pd). This function contains two peaks at r = 0.68 A and r = 0.96 A which account for the small-scale clusterirrg of small voids. At larger scale, a peak at r E 3 A is hardly detectable and could be related to the presence of larger voids. The overall behaviour of g,(r) is liquid-like.

The achievement of this structural information

2.50-

LOO-

~l.SO-

l.OO-

oso-

0.00 'I I I I I I I / I .oo so 1.00 2.00 3.00 4.00 5.00

r (A)

Fig. 11. Voids pair correlation function for amorphous Pd.

The function has been evaluated over the elemental spherical

holes before the reduction process.

can have a revelant impact in determining the Medium Range Order features of a given disordered structure, in particular for which concerns with recent interpretations of the origins of the first sharp diffraction peaks in the structure of covalent glasses and liquids [17].

In conclusion we have proposed a numerical method which allows to evaluate the volume distribution of voids in disordered system, once the atomistic configuration is known (from MD, MC or RMC simulation).

The actual version of the realized code to compute the esr for a given configuration is, however, quite time-consuming, being the number of esr necessary to fulfil the requirement of Eq. (6) of the order of several thousands. This explains the poor statistics which has been used to evaluate configurational averages. Works are in progress to reduce the complexity of the algo- rithm and to port the code on a parallel architec- ture.

The application of the proposed method to the study of liquid and amorphous structures has revealed that:

(11 the volume distribution of voids in a liquid obeys, with good accuracy, to a Turnbull-Cohen distribution. For the case treated (liquid Pd) the results of the fitting procedure confirm a slow exponential decay of the volume of voids in in-

V Rosato, P.G. Gabrielli/Computational Materials Science 3 (1995) 359-367 367

creasing volume with significant contribution for sizes of the order of one or more “particle volumes”;

(2) the volume distribution of voids in amorphous structure has a much faster decay at a large volume values than that pertaining to liquid. Morover there is a strong deviation from an exponential behaviour in the region around l-2 ii” which could originate from the voids at the inter- faces between the small crystalline grains which start nucleating within the amorphous phase.

Acknowledgements

The authors acknowledges a profitable interac- tion with Dr. Jilt Sietsma who has raised their attention to the problem and contributed with illuminating suggestions.

References

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[2] G.S. Cargill, in: Solid State Physics, eds. F. Seitz, H.

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[4] J.D. Bernal, in: Liquids: structure, properties, solid inter- actions”, ed. T.J. Hughel (Elsevier, Amsterdam, 1965).

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Italy) Course XCVII (19851, Molecular Dynamics Simula- tions of Statistical Mechanical Systems, eds. G. Ciccotti and W.G. Hoover, (North-Holland, Amsterdam, 1986). See, also: Simulation of Liquids and Solids, eds. G. Ciccotti, D. Frenkel and I.R. MacDonald (North-Hol- land, Amsterdam, 1990).

[13] V. Rosato, M. Guillope and B. Legrand, Phil. Mag. A 59 (1989) 321.

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volume distribution of empty regions in disordered structures

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