MODELS AND SIMULATIONS OF INTERFACES

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MODELS AND SIMULATIONS OF INTERFACES

A. Sutton

To cite this version:

A. Sutton. MODELS AND SIMULATIONS OF INTERFACES. Journal de Physique Colloques, 1990,

51 (C1), pp.C1-35-C1-46. �10.1051/jphyscol:1990104�. �jpa-00230000�

COLLOQUE DE PHYSIQUE

Colloque Cl, supplement au nol, Tome 51, janvier 1990

MODELS AND SIMULATIONS OF INTERFACES

A.P. SUTTON

Department Metallurgy and Science of Materials, Oxford University, Parks Road, GB-Oxford, OX1 3PH, Great-Britain

Abstract

-

Recent progress in developing and understanding geometric models such as the structural unit model, quasiperiodicity at irrational interfaces and geometric criteria for low interfacial energy are reviewed. Advances in interatomic forces, notably the implementation of models based on electronic structure rather than interatomic potentials to model grain boundaries, are assessed. The use of temperature dependent interatomic forces to model thermodynamic properties of grain boundaries is described briefly. Clear dependencies of the expansions and cleavage energies on the interplanar spacings of incommensurate boundaries are derived using an analytic model and confirmed by full atomistic relaxation for long period commensurate boundaries.

1 -INTRODUCTION AND REVIEW

The ultimate aim of many models and simulations of interfaces is t o determine the systematics of variations of structure and properties with geometrical and chemical degrees of freedom as well as temperature. I t is only by studying the systematics of structure and property variations that we may hope t o extrapolate the results into unexplored regions of the multi-dimensional parameter space associated with an interface.

Occassionally very detailed information is available from experiment about the the structure of a particular interface and this information can then be used to test models quantitatively. Several reviews of models and simulations have appeared in the last two years and we give only a summary of their main conclusions in this section. The remainder of the paper is devoted to a brief account of an analytic model of grain boundary energies, expansions and ideal cleavage energies. The model probes the significance of periodicity at an interface and the spacing of lattice planes parallel to and on either side of the interface.

Geometric criteria for low interfacial energy were reviewed in [l] using all the available reliable experimental observations of low interfacial energies. It was emphasized in [l] that to evaluate the success of any geometric criterion for low interfacial energy it is crucial that aU interfaces in a particular system that could meet the criterion are identified. Surprisingly it was found that this had not been done in many cases and false claims about the success of various criteria had been made in the literature. I t was concluded that, with one exception, none of the geometric criteria was supported with any statistical si nificance by the body of reliable experimental observations. The exception was the criterion, due to Wolf

h],

of minimizing the area of the planar unit cell in the interface when the interface plane normals in both crystals are fixed.

This is a refined version of the familiar planar coincidence site density criterion of Brandon et al. [3].

However, no convincing explanation for the success of this criterion for metalJmeta.1 and ionic/ionic interfaces or its apparent failure at metal/ionic interfaces has appeared. We shall return to the theme of the relationship between interfacial geometry and energy in section 2.

The structural unit model 41 has been critically reappraised recently [5]. According to this model the structures of longer period

b

oundaries may be described as certain sequences of structural units of shorter period boundaries. Faceting is one example where a boundary decomposes into structural units of other boundaries lying within the same coincidence system. There does not seem to be any difficulty with applying the model here, except that it ignores dislocations at facet junctions that arise from changes in the translation state. However, limitations on the utility of the model were identified when it is used to interpolate between the structures of short period boundaries with differing misorientations. When the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990104

rotation axis becomes of higher index it is found [5] that the Burgers vector of secondary dislocations associated with the minority units may become unrealistically large. By "unrealistically large" we mean that the dislocation is unlikely to be localised within the minority unit unless the minority unit itself is large. If the dislocation is not localised then the assumption of the model, that local misorientation changes occur so as to introduce structural units from other boundaries, breaks down. On the other hand, if large minority structural units are required then the misorientation range between the majority and minority structural unit boundaries is small and the predictive capacity of the model is limited. In practice this means that the model may usefully be applied only to relatively low index rotation axis pure tilt and pure twist boundaries, i.e. <loo>, <110>, < I l l > and possibly <112>. The challenge now is to develop models that apply away from these special cases in the five dimensional parameter space of grain boundaries. At the same time we urgently need some experimental observations of boundary structures in these regions.

The concept of quasiperiodicity at irratzonal grain boundaries was introduced by Rivier [6], Gratias and Thalal [7] and Sutton [g]. This work was reviewed and extended to interphase boundaries by Sutton [g]; see also Thalal and Gratias, this conference. Although it has been recognised [l01 for some time that irrational grain boundaries can have perfectly ordered structures it is only since the introduction of the concept of quasiperiodicity that the precise nature of the order has been elucidated. The simplest way to visualise quasiperiodicity at an irrational grain boundary is to apply the structural unit model to an irrational tilt boundary. Along the tilt axis (which is assumed to be rational) the structure of the boundary is periodic.

Perpendicular to the tilt axis in the boundary plane the boundary structure is described as an aperiodic sequence of majority and minorit structural units. As shown in [g] the sequence can be determined by applying the algorithm given in [4{ or by applying the strip and projection construction (e.g. [ll]). The two dimensional lattice from which the sequence of units is obtained in the strip and projection construction is simply the decomposition lattice introduced in [4]. The sequence of structural units displays the property of local isomorphism and this is arguably the most important consequence of quasiperiodicity at an irrational interface found so far. There is no periodicity in the sequence of structural units at an irrational tilt boundary but if we cut out a patch of the boundary and ask whether the sequence within the patch is repeated anywhere else the answer is yes, infinitely many times. Moreover, the distance we have to travel along the boundary before we find a repetition of the patch generally increases as the size of the patch increases. Successive patches are not spaced periodically but there are rigorous bounds on their spacings.

This can be stated precisely 191 using the mathematics of ttalmost periodic functionst1 developed by Bohr [l21 and Besicovitch [13]. Local isomorphism in an irrational interface is thus the analogue of translational periodicity at a rational interface. The main limitation of the analysis of Sutton [8] is that it rests on the validity of the structural unit model. Gratias and Thalal [7] avoided all assumptions about models. They noted that since the diffraction pattern of an irrational bicrystal is a convolution of the diffraction patterns of the separate crystals then it followed that the interface would be locally isomorphic. This is explained in detail in Gratias and Thalal's original paper and also in [g] where the trivial generalization of their analysis to interphase boundaries is carried out. The relationship between Gratias and Thalal's analysis of "hidden symmetries" at irrational interfaces and Pond's general theory of interfacial symmetry [l41 is also discussed in [g].

All simulations ultimately rest on the assumptions they make about interatomic forces. The vast majority of simulations assume some form of potential to describe atomic interactions. The recent introduction of methods for a review see [15]) that take the electronic structure of the solid into account explicitly involves a range o !f new assumptions and approximations, which it is believed are all less restrictive than those of a potential. The hallmark of an electronic structure method is the solution of the Schrodinger equation in one approximate form or another. As the ions are displaced the electrons respond through the Schrodinger equation and influence the force acting on an atom. It is this feature, which is often described as rehybridization, that is not described by a potential. The simplest of all electronic structure methods is the tight binding method and recently it has been embedded within density functional theory [16].

At a grain boundary in a close packed elemental metal the most important consideration as far as rehybridization effects is concerned is the local coordination number, which may be expected to decrease as the boundary expansion increases. However, since both measured and calculated boundary expansions in metals are normally less than 10-'nm the change in the local coordination number is not as great as at a free surface. The simplest way of introducing a dependence of the force acting between two atoms on their local atomic environment is the embedded atom [l71 or Finnis-Sinclair model [18]. In both of these models the total energy is represented by a pairwise repulsion and an attractive N-body potential. The relatively small change in coordination number at a grain boundary enables the N-body potential to be well approximated by an effective pair potential. Thus in a close packed elemental metal we do not expect significant differences between grain boundary structures obtained by the full N-body forms of the embedded atom or Finnis-Sinclair potentials and the effective pair potentials obtained by expanding the N-body potentials about the perfect crystal density. We do expect significant differences at free surfaces and at grain boundaries with large expansions. Neither the Finnis-Sinclair nor the embedded atom method takes into account any degree of directional bonding, which may be significant in b.c.c. elemental metals as

well as intermetallic alloys and a t grain boundaries with segregated impurities.

For group I V elements the extent of rehybridization effects is determined by the extent to which tetrahedral bonding can be preserved. Sutton [l91 has reviewed tight binding, local density and classical interatomic potential calculations of grain boundaries in Si and Ge prior to Easter 1989. Comparison with available experimental measurements of boundary structures and electrical properties was also made in [19]. Although the (relatively few) boundaries that have been modelled in Si and Ge are pure tilt or twist boundaries with no steps, facet junctions or extrinsic dislocations within them it has been found that it is always possible to reconstruct the boundaries without dangling bonds. The extent of rehybridization at the intrinsic (111) stacking fault and the ( I l l ) , (112) and (310 twins in Si was studied in detail using tight binding by Paxton

b

and Sutton [20], who also gave a thoroug review of the experimental measurements of these boundary structures. They found that rehybridization was most significant when dangling bonds were present in the boundary, but the presence of these dangling bonds led to a high energy metastable structure. Kohyama et al. [21] modelled the (112) and 310 twins with a very similar method and obtained almost identical structures and energies as in [20\. Tbe finding that the lowest energy structures contain only four-fold coordinated atoms prompted Sutton [l91 to repeat the calculations for the (112) and (310) boundaries with the Stillinger-Weber potential for Si 221. It was found [l91 that the lowest energy structures were the same as those found by tight binding (20,211. Moreover in the case of the (112) twin the five metastable structures considered in 201 were also found to be metastable with the Stillinger-Weber potential. Although the value of any particu

I

ar boundary energy differed in the two methods by up to a factor of two, the rankings of the five boundary energies were the same. In both [20,21] it was found that there are no midgap states, except when there are dangling bonds present. Kohyama et al. [21] were further able to show that there are no states at all introduced into the minimum gap, i.e. the gap between the valence band maximum and the conduction band minimum. They [21] found there were localized states in the band gap but, owing to the dispersion parallel to the boundary plane, they were buried underneath the band edges in the densities of states. Thus there were no states that might contribute to band tails in these calculations. It has been suggested [l9 that experimentally observed band tail states [23] may be caused by segregated dopant atoms. It is

a

so possible that states may be pulled out of the bands into the minimum gap by strain that exists at boundary imperfections, e.

.

steps, facet junctions. This has been analysed further in [l91 where it was emphasized that the condition for a state to be localised is dependent not only on the strength of the perturbation to the perfect crystal Hamiltonian but also on the scattering properties of the perfect crystal embodied in its electronic Green function. The challenge now is to place dopant and other impurities, especially oxygen in Si, into the boundaries and also to consider boundary imperfections. There is some experimental evidence [24] that recombination centres occur at facet junctions but whether they are associated with segregated impurities is not clear.

The influence of temperature on the stability and structures of interfaces has been modelled by a number of groups and reviewed critically by Pontikis [25 Molecular dynamics and Monte Carlo techniques have been employed and attention has focussed on Asordering of the boundary region through roughening or premelting. Such disordering processes are nucleated by thermal fluctuations and so molecular dynamics and Monte Carlo techniques are the natural ones to choose. However, many thermal properties of a grain boundary, like those of the perfect crystal, do not depend on thermal fluctuations, but on thermal vibrations averaged over a long period of time, e.g. the thermal expansion coefficient, specific heat and elastic constants. To model these properties Sutton 261 introduced the concept of a temperature dependent interatomic force. The basic idea is to augment t

L

e interatomic potential energy that applies at OK with the vibrational Helmholtz free energy expressed in quasiharmonic theory. By analogy with the Finnis-Sinclair model [l81 the phonon local density of states is expressed in the second moment approximation. The local vibrational free energy is then dependent only on the local stiffness and the temperature. Both the potential energy and the Helmholtz free energy contribute to the force acting on an atom. Not surprisingly, the temperature dependent force varies with the third derivative of the potential energy, reflecting the fact that if the potential energy were harmonic there would be no temperature dependent force. The force is also N-body in nature even when one assumes a pair potential for the potential energy. Using this scheme it is possible to evaluate all the standard thermodynamic properties of a solid with a simple "energy minimization" code. The atomic positions are the thermally averaged positions and since the local stiffness at a grain boundary site may differ from that of the perfect crystal there may be differing temperature dependent contributions. For the same reason that the total phonon density of states of the system may be projected onto local densities of states at each site it is possibld to project the total value of a thermodynamic excess quantity onto individual sites in the boundary. We may think of these projected thermodynamic quantities loosely as the values of the thermodynamic function at each site, although strictly speaking the projected quantity cannot be separated from the local environment. Local Gruneisen constants may also be defined. The X=13 (001) twist boundary in a Lennard-Jones solid was relaxed [26]

with temperature dependent forces and compared with X-ray diffraction measurements [27] of the mean square displacements and thermal expansion coefficents for this boundary in Au. The boundary was relaxed at 0, 600 and 1200K. At each temperature only one stable structure was found with layer group symmetry p42i2'. It was found that the main change in the boundary structure as the temperature was raised was

that the boundary expansion is an increasing function of the bulk equilibrium lattice parameter. This was reflected by the average Gruneisen constants in the boundary region which were also an increasing function of temperature. Good qualitative agreement was obtained with the measurements of [27]. A remarkably strong correlation was found [26] between local thermodynamic quantities and the local state of hydrostatic pressure. The atoms that dominated the boundary's excess specific heat and entropy were in a state of hydrostatic tension. This was also found in an earlier study [28]. An atom that is in a state of hydrostatic tension is loosely bound and has a low mean vibrational frequency so that it can absorb more thermal energy. Atoms that are hydrostatically compressed have a high mean vibrational frequency and may give negative contributions to the boundary's excess entropy, specific heat and thermal expansion.

2

-

AN ANALYTIC MODEL FOR BOUNDARY EXPANSIONS AND CLEAVAGE ENERGIES 2.1 Introduction

We present an analytic model to study the variation of the ideal cleavage energy with the grain boundary plane in the absence of impurity segregation. Thus, the model addresses the questions of which boundaries are thermodynamically and intrinsically more susceptible to intergranular cleavage and why.

Mechanistically, other boundaries may be more intrinsically susceptible to intergranular cleavage owing to variations in the orientation of the relevant slip systems, in either grain, to the direction of the propogating crack in the boundary plane. This effect is not considered here. Our model assumes pairwise atomic interactions and it is worked out in detail for a Lennard-Jones potential. I t is not obvious how the model may be extended t o non-pairwise interactions. However, we believe that the predictions of the model may apply to elemental metals, but not to those systems containing a significant degree of bond bending forces, such as covalent crystals and possibly intermetallic alloys. A more complete account of this work has been submitted for publication elsewhere.

We shall argue that the ideal cleavage energy decreases as the equilibrium boundary expansion increases.

Furthermore, the boundary expansion increases as the average spacing of planes on either side of the boundary decreases. These statements strictly apply only to what Wolf and Phillpot [29] call typical high angle grain boundaries: boundaries in the high angle regime far from any special misorientation. Wolf and coworkers [2,29-311 have repeatedly drawn attention to the systematic variations of the energies of these typical high angle boundaries with the average spacing of planes parallel to and on either side of the boundary. They have also pointed out that the calculated boundary energies increase with the boundary expansion. In this work we confirm their results and extend them to ideal cleavage energies. We have also clearly identified the role that one or two dimensional periodicity in the boundary plane plays in affecting the boundary energy and expansion. We have analysed the reasons for these systematic variations and our explanation is in sharp contrast to that of Wolf and coworkers [2,29-311. Our analytic model is essentially the same as the "rigid body joining step" of Brokman and Balluffi [32] and the "random boundary limit" of Wolf [31,33]. The model is assessed by comparing its predictions with full atomistic relaxations and its qualitative predictions are confirmed.

2.2 The energv of interaction between two ~ a r a l l e l vlanar nets

In order to elucidate the role that one or two dimensional periodicity in the boundary plane plays in affecting grain boundary energy we begin by considering the energy of interaction between two rigid parallel planar nets of atoms. The analysis is similar in spirit to that of Fletcher and Adamson 1341 and McTague and Novaco [35]. Consider the interaction energy between two rigid, parallel, two dimensional, primitive

I I1

lattices L and L

.

Throughout the remainder of this paper it is assumed that the total energy of the atomic assembly is represented by a sum of pair interactions and that there are no other contributions (such as a density dependent energy). The lattices do not have to share the same unit cell or the same orientation. Let

I1 I

t denote an arbitrary relative translation of L relative to L parallel to the interface and let the lattices be

I I I I

separated by z. If a and

b

are primitive, non-parallel, lattice translation vectors in L

,

and c is any vector

I I* I 1 I 1 1 I 1 I 1 1

normal to L

,

then

a =(b

^{x c} )/(a .b ^{x c} ) and bl*=(c

.a

)/(a

.b

^{x c} ) are basis vectors of the layer reciprocal lattice in lattice L I

.

It can be shown that the energy of interaction of the two lattices per unit area is given' by

where is a layer reciprocal lattice vector that is common to both L and L". I A: (A:') is the area of a

I I1

-

primitive unit cell in L (L ). v(q,z) is the two dimensional Fourier transform of the pair potential:

2 2 2

The integral in (2) extends over all two dimensional space and v(z,z) = v(r) where r =

11)

f z . If the two lattices are incommensurate with each other the only entering (1) is G'=Q and the interaction energy is

I I1

independent of the relative translation of the two lattices. At commensurate orientations of L and L the magnitude of the contribution to the interaction energy from each matching vector is a maximum of

I;(@,z)I and it is modulated by the phase factor exp(i@.&). The wavelength of the modulations of the interaction energy as $ is varied is determined by the condition that e x p ( i g . & ) = 1. We see that & may be stated uniquely if it is expressed modulo a reciprocal lattice vector of the (one or two dimensional) lattice of G' vectors. Thus,

t

may be expressed uniquely within the Wigner-Seitz cell formed from the reciprocal lattice vectors of the lattice of vectors and this-Wigner-Seitz cell is precisely the same as the "cell of non-identical displacements" defined in [36].

To gain some insight into the form of the Fourier components of the pair potential in (2), and for use in subsequent sections, we shall take the example of a simple Lennard Jones potential. The Lennard-Jones potential that we shall use is as follows:

This potential has a miniumum at r = ro and the depth of the minimum is E. The parameters E and ro are the natural units of energy and length for this potential. Substituting (3) into (2) we obtain

where K5 and K2 are modified Bessel functions. At q=O this expression reduces to

At large values of qz the asymptotic form of v(q,z) is exponentially damped. The sign of v(q,z) can be positive o r negative depending on both z and q. The magnitude of v(q,z) generally decreases as z or q increases. v(q,z) always diverges to plus infinity as z tends to zero regardless of q owing to the dominance of the repulsive contribution in the Lennard-Jones potential at small separations.

2.3 Surface and grain boundary energies in terms of interlayer interactions

Consider the creation of two surfaces of a crystal by ideal cleavage. We imagine that all atomic interactions across a cleavage plane between two adjacent atomic layers are switched off. Both halves of the crystal may then be separated and in this imaginary process no relaxation is allowed to take place. Thus we create two ideal surfaces with an energy given by

where Ell. is the interaction energy between layers 1 and 1'. Layers parallel to the surfaces have been labelled from -W to +m and the cleavage plane is between layers -1 and +l. There is no layer 0 in order to

keep the symmetry in the layer numbering on either side of the cleavage plane. Using (1) it is straightforeward t o write down an expression for E l k .

The energy of an ideal, unrelaxed grain boundary between crystals I and I1 may also be expressed in terms of layer interactions:

where,

and a similar expression for . E::;''.

-

' E;::.

-

is the interaction energy between layer l in crystal I1 and layer 1.

I I1

in crystal I. G (G ) is a layer reciprocal lattice vector of crystal I (11) in a layer parallel to the grain boundary. It is important to note that the layer reciprocal lattice vectors are not, in general, reciprocal

I I1

lattice vectors of the three dimensional lattices to which layers l and 1' belong.

tl (tl

) is the relative translation of layer l parallel to the grain boundary as determined by the stacking sequence of planes in crystal I (11).

t

is a rigid body translation parallel to the boundary of the whole of crystal I1 relative to crystal 1. is the separation of Layers l and 11 in crystal I and is equal to (l+

1

P l-l)dl where d1 is the spacing of iayirs in crystal I. Setting the density of the unrelaxed grain boundary to be the same as that of

I1 I I1 I I I1

the perfect crystal then results in z l - z l , = M

+

( P

I

d -(d +d )/2.

By comparing (7) with (6) it is seen that the unrelaxed grain boundary energy is equal to

where E: is the surface energy of crystal I on a plane parallel to the boundary plane. By rearranging (8) we obtain an expression for the energy of cleaving the unrelaxed grain boundary:

The advantage of working with the cleavage ener y rather than the grain boundary or surface energy is that the layer interactions on the right hand side of

(8

involve summations over only common reciprocal lattice vectors. For a given boundary plane Es and E:'I are independent of the misorientation about the boundary normal. In that case variations in the unrelaxed boundary energy are determined by the last term on the right of (8). The set {GC) contributing to this term varies in magnitude and direction as the misorientation about the boundary normal varies. For a given pair of planes 1 and 1' the magnitude of the Fourier

I1 I I I1

coefficient v(GC,(kl

+ I

P

I

d - ( d +d )/2)) generally decreases as

l

GC

l

increases. Thus boundaries with the smallest vectors have the potential to display the most significant variations in the boundary energy as a function of the misorientation about the boundary normal. These boundaries correspond to those with the smallest planar unit cell areas in real space. It is stressed that they have only a potential for significant

energy variations because (i) v may be positive or negative for a given depending on the separation of layers l and l and (ii) the phase factor, which affects both the sign and magnitude of v, depends on the relative translation of layers 1 and l t . Thus the stacking sequence of planes on either side of the boundary cannot be ignored.

In the case of an incommensurate interface the set {GC) reduces to @=Q. At a "typical high angle grain boundary" the nearest matching G vectors are long and the Fourier components of the potential to which they correspond are negligible. In that case the unrelaxed boundary energy is independent

- --

of the misorientation about the boundary normal and dependent only on the interplanar spacings d and dll. Thus l

the cleavage energy of an unrelaxed incommensurate grain boundary is simply

2.4 Cleavage energies of typical high anrrle arain boundaries

If (5) is substituted into (10) it is found that almost all unrelaxed incommensurate grain boundaries in f.c.c.

TT

crystals are unstable with respect to cleavage. Indeed as the interplanar spacings d1 and dl1 tend to zero it is found that the unrelaxed grain boundary energy tends to plus infinity. This is obvious from the divergence in (5) as z-0 arising from the repulsive part of the pair potential. In reality the boundary relaxes and the cleavage energy becomes positive. In the case of an incommensurate grain boundary the relaxation consists of an expansion normal to the boundary plane as well as individual atomic relaxation in which each atom optimises its local environment. The divergence in (5) as the interplanar spacing tends to zero may be removed by including the boundary expansion, e, in the model. This observation suggests the following simple model as a first approximation to a relaxed incommensurate grain boundary: the two adjoining crystals are treated as rigid objects that may be displaced normal to the boundary plane. By minimizing the boundary energy (or, equivalently, maximizing the cleavage energy) with respect to e it will be shown that positive cleavage energies may be obtained even in the limit of zero interplanar spacing.

The incommensurate boundary cleavage energy is now expressed as

Substituting (5) in (11) we obtain the following expression

where SZ is the atomic volume and g is a dimensionless function:

and

m m

s n = C

C

( d l / r o ) ( d l l / r o ) (14)

l = l l = l 1 d 1 ' + l s d 1

-

( d l

+

d 1 ' ) / 2 + e

I

^{r o}

Sn may be conveniently expressed as an integral and evaluated. Maximizing the cleavage energy with respect to e yields the following condition, which is effectively a relationship between the equilibrium

Interplanor spacing

Interplanar spacing

Fig.1: (a) The equilibrium expansion as a function of the interplanar spacing for incommensurate pure twist boundaries, in units of ro. The solid line is computed using eqn. (15). The open circles show the calculated expansions for fully relaxed twist boundaries listed in table I. (b) The function g, given by eqn. (13) for the same boundaries as in (a); the cleavage energy is proportional to the negative of g, as seen in eqn.(12).

ezpansion, e, and the interplanar spacings d and dzI: I

Once (15) is solved the equilibrium expansion is substituted into (13) to obtain the maximum cleavage energy in (12). The results are presented graphically in fig.1, where the unit of length is ro. The lattice parameter, af, at OK in an f.c.c. crystal in which the interatomic potential is given by (3) is equal to 1.37353%. Consider first the simpler case where dl=dl', which includes pure twist boundaries. h g . l a shows the equilibrium expansion as a function of the interplanar spacing on either side of the boundary. The equilibrium expansion is zero only when the interplanar spacing is 0.87938ro=0.64023af, which is greater than the maximum spacing (0.57735aF0.79301ro) in f.c.c. crystals; thus the model predicts that there is always an expansion at incommensurate twist boundaries in f.c.c. crystals. The model also predicts that as the interplanar spacing tends to zero the expansion tends to 15-1/6ro=0.46360af and the function g tends to -15*13/24 = -1.54138. The function g is plotted in figlb. It is seen that for all interplanar spacings in f.c.c.

crystals the cleavage energies of twist boundaries increase monotonically as the interplanar spacing I 11.

increases. For the more general case where d f d it is convenient to work with the average interplanar

I I1 I I1 I I1

spacing <d>=(d +d )/2 and 6=(d -d )/(d +d ). It is found that for a given average interplanar spacing I I1

<d> the boundary with d =d (i.e. 6=0) always has a greater expansion than the boundary with

I I1 I- 11

d =2<d>, d =O (i.e. b=l). The cleavage energy of a boundary with d -d =<d> is always less than that

T TT

of the boundary with d1=2<d>, dll=O. Moreover, the larger the average interplanar spacing, <d>, the greater the difference in expansions and cleavage energies for the two extreme cases. For a given <d>

--

the boundary expansion increases monotonically, and the cleavage energy decreases monotonically, as d varies l1 from 0 to <d>, i.e. as 6 varies from 1 t o 0. Thus the equilibrium expansion of an incommensurate mixed tilt and twist boundary is less than the expansion of an incommensurate twist boundary with the same value of

<d>. The cleavage energy of the former is predicted to be always greater than that of the latter.

2.5 Com~arison with full atomistic relaxations of twist boundaries

The model we have used in the previous section neglects individual atomic relaxation because the boundary energy is minimized by varying the boundary expansion while otherwise keeping all atoms in their ideal crystal positions. The severity of this approximation is examined in this section where comparison is made with expansions and cleavage energies of twist boundaries that are fully relaxed using the potential of (3).

The twist boundaries selected for this study are listed in table I. Ideally we would relax incommensurate twist boundaries but this is not compatible with the use of periodic boundary conditions in the boundary plane. For this reason we have selected commensurate twist boundaries with relatively large unit cells in the boundary plane and misorientations far from the perfect crystal and twin boundary orientations as approximations to incommensurate twist boundaries. Table I gives the smallest non-parallel, non-zero vectors in the boundary planes. In addition to individual atomic relaxation one grain was free to translate relative to the other both parallel and perpendicular to the boundary plane. This relative translation occurred over several planes adjacent to the geometrical boundary plane but the largest contribution always arose between the first layers of either crystal. Thus our simplifying assumption in section 2.4 that the boundary expansion is confined between the first layers of either crystal is well supported by these calculations. Table I summarises the results. The boundary expansion was calculated by examining the normal displacement of layers remote from the boundary. These expansions are plotted as circles in fig. la.

The ideal cleavage ener ies were calculated by subtracting the relaxed twist boundary energy from twice the energy of a relaxed surface parallel to the twist boundary. The g functions corresponding to these cleavage energies are plotted on fig. l b as circles.

It is seen in fig.la that the boundary expansion increases monotonically as the interplanar spacing decreases for the fully relaxed twist boundaries in agreement with our model. However the model overestimates the expansion as the interplanar spacing tends to zero by about a factor of two. The quantitative agreement between the predictions of the model a d the full relaxation for the cleavage energies (fig.lb) is better. It is noted that the energy of the unrelaxed (114) twist boundary in table I, in which no expansion is allowed, is

- -

of the order of 10" ,/I:, the exact value depending on the relative translation $ parallel to the boundary plane. Allowing the boundary to expand, while not allowing any individual atomic relaxation, not only brings the boundary energy down through 6 orders of magnitude to the correct order but the trends in the

boundary expansion to increase and the cleavage energy to decrease as the interplanar spacing decreases are correctly reproduced. This provides strong support for the model of the previous section. Further support for our model is provided by the atomistic calculations of Wolf and Phillpot [29]. Those authors report calculations for the twist boundaries on the two densest planes in f.c.c. and b.c.c. metals and find that the boundary expansion is lower and the cleavage energy is greater for the denser planes.

Table I

Parameters for fully relaxed, long period, twist boundaries. 8 is the twist angle, denotes a common planar reciprocal lattice vector, d is the interplanar spacing, e is the equilibrium expansion, EGB is the grain boundary energy, Es is the energy of a surface parallel to the grain boundary plane and g is the function given by eqn.(l3) and it is proportional to the negative of the cleavage energy.

Boundary

l

^cos0

l

^Smallest

I

d(ro)

I

e(ro) I'~B('/~;)

I

E ~ ( E / ~ ; ) 1g=-5eC1n ²

/

Plane

I I

vectorsx21r/af

I I ^{1 I I}

^{Tero 4}

2.8 Discussion

Using a simple model for the relaxation a t an incommensurate typical high angle grain boundary we have derived the variation of the ideal cleavage energy with the interplanar spacing on either side of the boundary. The model consists of regarding the boundary expansion as the only variational parameter with which to minimise the boundary energy. By comparison with full atomistic relaxations we have shown that the trends predicted by the model are correct but the model overestimates the boundary expansion by as much as a factor of 2. Nevertheless, the predicted cleavage energies are in good agreement with full atomistic calculations. In all cases the model predicts that the cleavage energy decreases as the boundary expansion increases. The model also predicts that for incommensurate boundaries the expansion increases uniformly to a well defined upper limit as the average interplanar spacing tends to zero. Moreover, for a given average interplanar spacing a boundary with more dissimilar interplanar spacings on either side of the boundary plane is predicted to have a lower expansion and greater cleavage energy.

The results presented in table I are consistent with those of Wolf and coworkers [2,29-311. It is seen in table I that although the twist boundary energies change by more than a factor of 4 between the (111) and (114)

planes the relaxed surface energies change by a much smaller factor. Thus, the reason for the decrease in the ideal cleavage energy as the average interplanar spacing decreases is predominantly the concomitant increase in the boundary energy. The boundary expansion is driven by the short range repulsive part of the potential; it is opposed by the long range attractive part of the potential which is attempting to maximize the atomic density. However, once the boundary has expanded and relieved the atomic overlap it is less clear wh the boundary energy should still depend on the average interplanar spacing. Wolf and coworkers p,29-31ihave offered the following explanation.

Although the boundary expansion increases the average interatomic separation some atoms at the boundary remain closer together than the ideal first neighbour separation. By examining the radial distribution functions in the relaxed boundaries they demonstrate that there are more atoms too close together in boundaries with smaller interplanar spacings. They then argue that the repulsive part of any interaction potential rises steeply at less than the ideal first neighbour separation and therefore the boundary energies are dominated by these short range interactions. Since there are more such short range interactions in boundaries with a smaller interplanar spacing, even after the boundary has expanded, the boundary energy increases as the interplanar spacing decreases.

The larger number of short range interactions in boundaries with smaller interplanar spacings is consistent with the model of section 2.5 where the separation between layers immediately adjacent to the boundary continues to decrease slightly as the interplanar spacing decreases, even after the boundary has been allowed to expand. However, we shall show that this is not the reason for the increase in the boundary energy as the interplanar spacing decreases. We have analysed the sources of the grain boundary energy for each of the boundaries listed in table I as follows. Let Er denote the sum of the pairwise interaction energies in the b bicrystal for interatomic separations less than the ideal first neighbour separation. Let Ea.denote the sum of b the remaining pairwise interaction energies in the bicrystal. Let E: and E: denote the corresponding energies in a piece of perfect crystal containing the same number of atoms as the bicrystal. Clearly E:=O

b b c b

and the boundary energy is equal to Er+Ea-Ea. According to Wolf and coworkers Er should dominate the b . boundary energy and increase as the interplanar spacing decreases. In every case we find that Er is large

b c

and negative. The boundary energy is positive because E,-E, is larger and positive. For example, for the

b b c 2 b

(112) boundary Er=-87.3249, Ea-Ea=91.4282 giving a boundary energy of 4.1033, in units of €/I,. Er is negative -

..

because the Lennard Jones potential is positive only at interatomic separations of less than 2-l/'r,=0.6486at, i.e. at separations of less than 91.7% of the ideal first neighbour separation. We have arrivedat the same conclusion for the Finnis-Sinclair potential for molybdenum [l81 applied to a series of (001)/(221) mixed tilt and twist boundaries relaxed in [37]. The total energy is expressed in terms of a purely repulsive pair potential and a purely attractive N-body potential. We have calculated the contributions to the boundary energy from the pair potential and the N-body potential separately. We find that the contribution to the boundary ener y from the repulsive potential is always negative, and that the boundary energy is positive only because o f t h e larger positive contribution from the N-body term. In our analysis the boundary energy increases as the interplanar spacing decreases because the larger expansion reduces further the attractive interactions across the boundary. This is also the reason why the cleavage energy decreases. However, the reason for the larger expansion at a smaller interplanar spacing is the larger repulsion arising from the short range part of the potential.

A note of caution may be injected here about the increase in the boundary energy with expansion. The analysis of section 2.4 has been applied to incommensurate interfaces. At commensurate interfaces there are Fourier components that contribute to the boundary energy in (7). Those additional terms may affect

-

the expansion and alter the monotonic increase in boundary energy with expansion. For example, we have obtained five metastable structures of the (310) twin in a b.c.c. crystal using a pair potential constructed by Miller [38] for molybdenum. The expansions of these structures are 0.2132, 0.2777, 0.0843, 0.5031, and 0.6483 in units of the crystal lattice parameter and their energies are 1320, 1324, 1486, 2861 and 3317

n

m~/rn' respectively. Although the two highest energy structures have the highest expansions the three lowest energy states do not correlate with the expansion.

Equations similar to (1) have been derived earlier in [34,35]. Both papers drew attention to the matching I I1 c

condition G =G =G as giving rise to local minima in the energy of interaction between two layers as a

function of their lattice structure and orientation. But it is clear that the phase factor e x p ( i ~ ' . t ) in (1) is I I1

equally important since it can reverse the sign of the contribution V ( & , Z ) / ( A ~ A ~ ) t o the interaction energy between two layers. These phase factors appear in both the surface and grain boundary ener ies. Indeed

f

Blandin e t al. [39] showed t h a t the phase factors are almost entirely responsible for t h e stacking ault energy using their reciprocal space formulation of the layer interaction energy. This amounts t o saying that the stacking sequence of planes on either side of the boundary is as important a t a periodic grain boundary as the interplanar spacing and the size of the boundary unit cell i n determining the boundary energy.

The principal weakness of t h e present model is the assumption that all the cohesion of the solid is provided only by pairwise interactions. Another weakness of the present treatment is that no account has been taken of entropic contributions t o the boundary free energy. As the temperature is raised the vibrational free energy promotes increased boundary expansions, as was found in a full atomistic calculation [26]. The increased boundary expansion will reduce the Fourier components v@,z) and tend t o smooth out variations i n the boundary energy as a function of misorientation.

REFERENCES / l / Sutton A.P. and Balluffi R.W. 1987, Acta Metall., 35, 2177.

/2/ Wolf D. 1985, J. Physique, 46, C4-197.

/3/ Brandon D.G., Ralph B., Ranganathan S. and Wald M.S. 1964, Acta Metall., 12, 813.

/4/ Sutton A.P. and Vitek V. 1983, Phil. Trans. R. Soc.,

309,

1, 37, 55.

/5/ Sutton A.P. 1989, Phil. Mag. Lett., 59, 53.

/6/ Rivier N. 1986, J. Physique, 47, C3-299.

/7/ Gratias D. and Thalal A. 1988, Phil. Mag. Letts., 57, 63.

/8/ Sutton A.P. 1988, Acta Metall., 36, 1291.

/g/ Sutton A.P. 1989, Phase Transitions,

16/17,

563.

/10/ Sutton A.P. 1981, PhD thesis, University of Pennsylvania.

/11/ Katz A. and Duneau M. 1986, J. Physique, 47, C3-103.

1121 Bohr H. 1924, Acta Math.,

45,

29; W., 46, 101; ibid., 47, 237.

1131 Besicovitch A.S. 1932, Almost periodic functions, London: Cambridge University Press.

/14/ Pond R.C. 1985, in Dislocations and properties of real materials, ed. M.H. Loretto, London: Inst. of Metals, p.71.

1151 Sutton A.P. 1985, J. Physique, 46, C4-347.

/16/ Sutton A.P., Finnis M.W., Pettifor D.G. and Ohta Y. 1988, J. Phys. C: solid state phys., 2 , 35.

1171 Daw M.S. and Baskes M.I. 1984, Phys. Rev. B, 29, 6443.

/18/ Finnis M.W. and Sinclair J.E. 1984, Phil. Mag. A, 50, 45.

/19/ Sutton A.P. 1989, Proc. 6th Int. Symp. on Structure and Properties of Dislocations in Semiconductors, Institute of Physics, Bristol-Adam Hilger, t o appear.

1201 Paxton A.T. and Sutton A.P. 1989, Acta Metall., 37, 1693.

1211 Kohyama M., Yamamoto R., Ebata Y. and Kinoshita M. 1988, J. Phys. C: solid state phys., 2 , 3205;

Koh ama M., Yamamoto R., Watanabe Y., Ebata Y. and Kinoshita M. 1988,

m.,

^{2, L695.}

/22]~tillin~er F.H. and Weber T.A. 1985, P h y s Rev. B, 31, 5262.

1231 Werner J.H. 1989, i n Polycn~stalline Semiconductors, eds. H.J. Moller, H.P. Strunk and J.H. Werner, p. 345, Springer-Verlag: Berlin.

1241 Maurice J.-L. and Colliex C. 1989, i n PolycrystaUine Semicondz~ctors, eds. H.J. Moller, H.P. Strunk and J.H. Werner, p. 83, Springer-Verlag: Berlin.

1251 Pontikis V., 1988, J. Physique, 49, C5-327.

/26/ Sutton A.P. 1989, Phil. Mag. A, to appear.

1271 Fitzsimmons M.R., Burkel E. and Sass S.L. 1988, Phys. Rev. Letts., 61, 2237.

/28/ Hashimoto M., Ishida Y., Yamamoto R. and Doyama M. 1981, Acta Metall., 29, 617.

1291 Wolf D. and Phillpot S. 1989, Mat. Sci. Eng.,

m,

3.

1301 Wolf D., Lutsko J. and Kluge M. 1989, Proc. Symp. Atomistic modelling i n materials - beyond pair potentials, eds. D.J. Srolovitz and V. Vitek, Plenum Press: New York, t o appear.

1311 Wolf D. 1989, Acta Metall., 37, 1983.

/32/ Brokman A. and Balluffi R.W. 1981, Acta Metall., 2, 1703.

/33/ Wolf D. 1984, Acta Metall., 32, 245.

1341 Fletcher N.H. and Adamson P.L. 1966, Phil. Mag.,

14,

_99.

/35/ McTague J.P. and Novaco A.D. 1979, Phys. Rev. B,

B,

5299.

1361 Vitek V., Sutton A.P., Smith D.A. and Pond R.C. 1980, i n Grain boundary structure and kinetics, ed.

R. W. Balluffi, American Society for Metals, Metals Park Ohio, p. 115.

1371 Sutton A.P. 1988, Mat. Res. Soc. Symp. Proc., m , 8 1 . 1381 Miller K.M. 1981, J. Phys. F ,

11,

1175.

1391 Blandin A., Friedel J., and Saada G. 1966, J. Physique, C3-128.