SIMULATIONDEFECTS IN AMORPHOUS SOLIDS

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SIMULATIONDEFECTS IN AMORPHOUS SOLIDS

P. Chaudhari

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C8, suppllment au n08, Tome 41, aoct 1980, page C8-267

SIMULATION.

DEFECTS I N AMORPHOUS S O L I D S

P. Chaudhari

IBM T.J. ;/atson Research Center, Yorktowz Heights, NY 10598, U.S.A.

Abstract.- We review in a qualitative fashion our current theoretical understanding of the refe- rence states and notion of defects in amorphous solids.

In order to perceive a defect in an object or space, we must know what its defect free state is. It is only by comparison with some reference state that we can say that a defect is present. The comparison can be made in terms of topological properties or in terms of fields. In crystalline solids where the underlying lattice provides a reference, it is readily apparent when a defect is present. In an amorphous solid we have no underlying lattice to which we can refer. The visuali- zation of defects is therefore not so straightforward. Furthermore, the packing of atoms precludes a periodic atomic potential, and it is therefore not clear that defects such as vacancies and dislocations can exist. In this paper we shall review, rather briefly and from a theoretical point of view, what is known or speculated about the existence of defects in amorphous solids such as the metallic alloys.

Point Defects

In a crystalline solid, point defects such as vacancies and interstitials are known to be present and play a significant role in atomic transport and mechanical or electrical properties. If we were to take some suitable model for an amorphous solid and remove an atom from the interior of this solid and place it on the surface, we would have created a vacancy in a manner analogous to that in a crystal. Since our reference state consists of atomic packing where a local and atomically vacant volume approximately equal to that of an atom is never present, the vacancy, as a defect,

can be defined in any structure. In a similar fashion we can define an interstitial defect. _,

Having introduced a vacancy we can inquire if it is a defect that can exist in an amorphous solid. In crystal the energy required to generate

a

vacancy is balanced by the configurational and vibrational entro- py terms. A similar situation exists in amorphous solids. The question then is not can vacancies exist but rather can vacancies associated with a localized volume comparable to that of an atom exist. In a crystalline solid the volume is localized by the periodic ar- rangement of atoms and their interactions. This pre- vents atoms neighboring a vacancy from shuffling in to spread out the volume. There is an activation barrier which a neighboring atom must overcome before it can jump into a vacant site. In contrast, an amorphous solid with a central force potential has no stabilizing influence exerted by the lattice or, more precisely, by the local symmetry. Neighboring atoms can therefore shuffle in and spread out the localized volume until the identity of a vacancy is lost.

In a two-dimensional hard sphere simulation of an amorphous solid, spaepen(') has shown that a local area corresponding to a missing atom disappeared when the atoms were allowed to move by suitable vibrations. Computer simulation studies on three- dimensional models using molecular dynamics or static relaxation procedures and

a

Lennard-Jones potential have been carried out by Bennett et They have

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JOURNAL DE PHYSIQUE

demonstrated that the localized volume associated with the missing atom is delocalized. In contrast to the dense random-packed solids relaxed under a Lennard- Jones Potential, a Vacancy in a continuous random network relaxed under a Keating potential, retained its local volume. This is clearly related to the strongly directional covalent bond which preserves the local symmetry in the amorphous structure.

Line Defects

The absence or presence of dislocations in amorphous solids has been a controversial subject. ~ i l m a n ( ~ ) and ~ i ( ~ ) have invoked dislocations to ex- plain the mechanical properties of metallic glasses. spaepen(') has argued that a dislocation, as a localized defect, cannot exist in a dense random-packed model. Both spaepen(l) and ~ r ~ o n ( ~ ) have explained the

. , n ? e ~ b r s ~ c ~ 1 , ~ ~ 0 ~ r ~ i e ~ _ o f _ e l a s s e s in terms o f models which do not rely on the presence of dislocations.

The melting of solids has been interpreted in terms of dislocations in two- and three-dimensional Suggestive evidence for the presence of dislocations has been pointed out by cotterill(') in computer ~imulation studies of melting. Dislocation theories of melting have been proposed by a number of authors.(8) In these theories the dislocation is defined with reference to the crystalline state, and it is argued that as their density increases, both rotational and translational correlations are lost resulting in a loss of crystalline structure. As most amorphous solids differ from liquids only in detail in their diffraction response, we may assume that the structure of most, but not all, amorphous solids is similar to that of the liquid. If the dislocation theory of melting is valid, the amorphous solid is described by a tangled network of dislocations which arc present with respect to some crystalline reference state. Unlike a crystal these dislocations cannot be annealed away. The ground state

of the amorphous solid requires the presence of dislocations.

Recently, Koizumi and ~ i n o r n i ~ a ( ~ ) have gener- ated amorphous G e by putting in a dislocation density of 4 x 1014 into crystalline Ge and subsequently transforming this to a dense random-packed solid by using a method proposed by Chaudhari er al.(lO) and ~ o n n e l l . ( ' l ) On relaxing their model with a Morse potential they were able to obtain a good fit to the pair distribution function of amorphous nickel.

~ i v i e r ( ' ~ ) has demonstrated that the sum of the odd-membered rings in a continuous random network are even. He has also suggested but not proved that the odd-membered rings form disclination lines that thread the solid. He uses the six-membered ring model of Connell and emk kin('^) as the reference state so that five- and seven-membered rings form disclinations of opposite sign. In contrast to this reference state, which suggests that we should be able to anneal out in pairs the disclinations, Kleman and sadoc(14) propose that the reference state is not three-dimensional euclidean but rather three-dimensional spherical. The disclinations are then required to build a contiguous solid when the polytopes in the spherical space are mapped onto euclidean space. Like the dislocation theory of melting, the line defects present in the amorphous solid are necessary to retain this structure in our local space. As the two reference states from which we start to arrive at the amorphous structure are quite different, we need then quite different densities and character of line defects.

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enough along, has to be relaxed using some relaxation procedure. It has not been established how essential this relaxation is in obtaining an adequate fit to the data. If it is essential, the dislocation or disclination models might be no more than a method for introducing perturbations, say in the crystal, which the computer then relaxes into an amorphous dense random- packed solid. That such possibilities exist is indicated by computer simulation studies where a continuous random network was transformed into a more dense random-packed solid by merely changing the range of atomic interaction from nearest to second-nearest neighbor.(2) If we are to accept a particular reference state as a valid description from which we can obtain an amorphous structure by adding a prescribed set of defects, the precise role of relaxation must be under- stood. Given that the short-range topological properties of amorphous structures have not been established, can we say anything about the presence or absence of dislocations in these solids? The answer is in the affirmative if we concern ourselves with the elastic fields rather than the topology. The point here is that an edge or a screw dislocation has a characteristic elastic field whose signature is unique. If we can establish that such a field exists, we can say with cer- tainty that such and such line defect is present. The simplest way of doing this is to introduce a defect in a model, relax it, and then analyze the elastic response of the solid.

Edge and screw dislocations have been introduced in the Finney model.(l5) From an analysis of the Airy stress function using a Lennard-Jones pair potential with an interaction distance cut-off at approximately twice the atom diameter, it was concluded that the edge dislocation could not be stabilized. On the other hand, a screw dislocation had a stress field characteristic of such a defect.(l6,17) One advantage

of introducing dislocations in a model is that we can use the undislocatcd model as a reference state. With this approach properties such as the Burgers vector or, in the case of the screw dislocation, the Eshelby twist can be determined.(16,17)

The stability of the isolated screw but not the edge dislocation suggests that the mechanical properties of amorphous solids are quite different from those of their crystalline counterparts. A number of experi- mental observations related to mechanical properties such as locolized slip band formation, change in density, absence of work hardening and high strength can be qualitatively related to the presencc of a screw dislocation alone. Dislocation generation cannot occur by the operation of Frank-Read-type sources. The yield stress of amorphous solids cannot be related to the existence of a network of dislocations if the edge di~slocation cannot exist. Dislocation tangles which lead to work hardening in crystalline solids are also n o t present. The yield stress is to be associated with screw dislocations.

At temperatures low rclative to the glass transition temperature, the deformation is, we believe, pri-, rnarily by the motion of dislocations. Unlike a cryst,al t.he passage of a dislocation results in a s u b s t a ~ . ~ t i ~ l perturbation of the dense packing of a solid. Cc,mput-

e r simulation studies(17) show that there is a decrease in density confined to the vicinity of the surface of slip. As the temperature of deformation approaches the glass transition temperature, the deJformation be- havior is increasingly dominated by fre:e volume availa- bility. The external applied stress comples with screw dislocations. These move, at low Stresses, with the help of the free volume leaving-intact the packing of the solid.

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C8-270 JOURNAL DE PHYSIQUE

temperature. We believe the glass transition temperature can be similarly described by a binding-unbinding transition. The Burgers vector of a screw dislocation in a Lennard-Jones amorphous solid is approximately 0.82 of the atomic diameter, and the shear modulus is approximately seventy per cent of the crystalline value('7) leading t o an energy of the dislocation which is approximately forty to fifty per cent of the crystalline value. This is approximately also the value of the glass-transition temperature to the melting temperature of the corresponding crystalline solid.

Planar Defects

In our computer simulation studies of vacancies in a Lennard-Jones solid, we had noted that static relaxation did not remove the entire localized volume.(2) It is conceivable that at relatively low temperatures an array of such localized volumes

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much like a frozen in free volume

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could be present. Such a situation could conceivably arise during quench freezing. It is highly unlikely that the solid-liquid interface is planar. As the melt freezes, the surface of contact between neighboring solidified portions of the material cannot be in a dense packed solid arrange- ment, and vacant volumes less than a t atomic volume must be present. If the solid is quenched faster than the armealing out of these localized volumes, the solid contains, an array of defects distributed along a surface forming what might be termed planar defects. Now a continuous a n d bounded array of centers of dilatation distributed along a planar surface generates an edge dislocation loop. We might therefore suppose that in a quenched solid we can observe stress fields which are like those of arr edge dislocation. This edge dislocation is differeht from the one we had deliberately introduced in the amorphous solid by removing a row of atoms equal to i3n atom diameter or by gliding one surface relative to the other by a fixed amount. In this

latter case, we had removed a continuous array of atoms whereas in the quenched-in case the localized vacant volume is not distributed continuously but is statistically distributed. We are currently in the process of computer-simulating this process.

It is, of course, obvious that if screw dislocations are stable, then arrays of screw dislocations can also be present. Such arrays can lead to rotation boundaries.

Intrinsic Structure

In the framework of crystalline solids an atom at an equilibrium site is generally associated with no stress field. In an amorphous solid the association is less obvious, and as the packing and local density of the structure varies from volume to volume element, we might suppose that there might be a corresponding internal stress field which fluctuates in sign and magni- tude. Computer simulation studies by Egami ef a1.(18)

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