Segregation and grain boundary structure

Segregation and grain boundary structure

110 Surface Science 144 (1984) 110-123 North-Holland. Amsterdam SEGREGATION V. VITEK AND GRAIN BOUNDARY and Gui Jin WANG Department of Matenals s...

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Surface Science 144 (1984) 110-123 North-Holland. Amsterdam



and Gui Jin WANG

Department of Matenals sylvania 19104, USA Received


15 November

Science and Engrneering,

1983; accepted


for publication

of Pennsylvania,

30 December




Atomic structure of the ,W?Z = 5(210)/[001] symmetrical tilt boundary in Cu containing different amounts of segregated Bi has been investigated. The interatomic forces were described in terms of empirical pair potentials constructed so as to fit self consistently the enthalpy of mixing of this alloy system. The most important structural effects found in this study are strong anisotropy of the segregation energy for different segregation sites, formation of an ordered alloy structure in the region of the grain boundary and occurrence of structural transformations due to segregation. A saturation level of 1.25 monolayers beyond which no further segregation can occur, was found. Significance of these structural effects for the segregation process and for the influence of segregation upon various grain boundary properties is discussed.

1. Introduction Grain boundaries are the most common “defects” in materials since they are always present if the material is not in a single crystal form. Their presence affects a large variety of material properties ranging from mechanical strength to electronic characteristics (l-31 because many important physical processes, such as self diffusion of alloying elements, decohesion or cavitation, localized electronic these


states etc., occur boundary



at grain boundaries.

it is necessary

To investigate

to understand

the atomic

structure of grain boundaries. The reason is that the long range elastic fields of grain boundaries, unlike e.g. those of dislocations, play only a secondary role and the important physical processes occur in the region of the core of grain boundaries. The structure of grain boundaries has, indeed, been studied extensively (for reviews see refs. [4,5]). The geometrical, crystallographic, description of grain boundaries is now well established and understanding of the general


of the atomic


is emerging.


only very

few direct experimental studies of the atomic structure have been made and the computer simulations of grain boundaries were instrumental in advancing our understanding of their structure. Though the details of the results of these calculations, i.e. exact atomic positions, cannot be usually confirmed by any 0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)


V. Virek, G.J. Wung / Segregation and

groin boundarystructure


direct observations, these results were found to be in good agreement with electron microscopic studies [6,7] as well as with observed trends of various grain boundary phenomena [4,8]. Hence, we can expect general features of grain boundary structure are well reproduced in the atomistic calculations. In particular, these calculations revealed the nature of the basic building blocks of grain boundaries, the compact polyhedra of atoms ]9,10], and led to establishing a structural unit model [ll] which gives a framework for relating the structure of general boundaries to that of some special high coincidence boundaries The structural unit model, deduced on the basis of extensive atomistie studies, provides for a relationship between the structures of various boundaries throughout a n&orientation range for a particular rotation axis and inclination of the boundary plane. To a good approximation all such boundaries can be regarded as composed of mixtures of two structural elements. These structural elements are identified as the units of two relatively short period, low Z, boundaries which delimit the misorientation range. Each delimiting boundary is composed of a contiguous sequence of such units. When this does not contain identifiable units of any other boundary the delimiting boundary is called favoured and its unit is a fundamental structural element of nearby boundary structures. This model is approximate in the sense that the units are always somewhat distorted in the intervening boundaries. A continuous series of boundaries between two delimiting boundaries can then be defined as encompassing those boundaries which are composed of unique sequences of units of delimiting boundaries. The sequence of units in any such boundary is determined by the condition that the minority units are as widely separated as possible. Since any general boundary can .be approximated as closely as required by a high X coincidence boundary, this model establishes a unique relationship between the structures of general boundaries and structures of short period low .X boundaries. This is obviously very important when carrying out atom&tic studies of grain boundary phenomena since such calculations are always iimited to reIativeIy short period boundaries and the model allows us to generalize results of these studies. However, the structure of delimiting boundaries need not be unique [I2-141. In general, several different metastable structures of a given delimiting boundary may exist each composed of different types of units. The intervening boundaries may then contain all these different kinds of units. Thus although a unique sequence of the units of the two delimiting boundaries represents the structure of a grain boundary a large multiplicity of structures may arise when units corresponding to different structures of the delimiting boundaries all appear as structural units. An example is shown in fig. 1 where eight different structures of the B = 73(82t3)/[001] symmetrical tilt boundary are shown. The structure of this bonndary is always composed of one unit of the 2 = 5(210)/[002] and two units of the Z’ = ~~3l~~~~~I] boundaries. Each of these


V. Vitek, G.3. Wang / Segregation





+A *A +

+A *AA





+A *A,

+A+ A*

and grain hounda~ structure



Fig. 1. Eight different metastable structures of the periodic ZZ= 73(~3O)/[~lj symmetrical tilt boundary in Cu. A and + represent atoms in two different (002) planes. The structural units B, B’, C and C’ are marked by solid lines.

delimiting (favoured) boundaries possesses two me&stable structures, marked B and B’ for the (210) and C and C’ for the (310) boundaries, respectively. Thus eight possible combinations, each of which were found to be metastable, may occur in the periodic Z = 73(830) boundary. The different structures of a boundary, in general, possess different energies. Hence, it could be considered that only the lowest energy configuration has a physical significance. This would be true for ideal boundary structures where multiplicity could play a

V. Vitek, G.J. Wang / Segregation and grain boundary structure


role only at very high temperatures. However, if defects such as impurities, extrinsic dislocations, etc. are present in the boundaries, different structures could become favoured in their vicinity in particular if energies of different structures are very similar as it has been found in our calculations [14]. Hence, the multiplicity of structures could play an important role in the physical processes that involve interactions of other defects with grain boundaries. For example, as pointed out earlier [12-141, the ease of absorption or emission of vacancies by grain boundaries may be related to the multiplicity of their structures since different structures transform between each other by removal or insertion of layers of atoms. Similar transformations invoked by segregation will be discussed in this paper. Whilst most of grain boundary phenomena occur in both pure materials and alloys their effects are often strongly enhanced if segregation of one or several alloying elements to the grain boundaries occurs [15]. Thus it is especially important to understand the phenomena of segregation and their influence upon the grain boundary structure and properties controlled by the structure. However, most of the atomistic studies dealt with boundaries in pure metals and did not, therefore, address directly these important questions. The limited calculations which have been performed treated only the cases of very dilute concentration of solutes in grain boundaries [16-201. In this paper we present the main results of an atomistic study of grain boundaries in copper which contain a high concentration of segregated bismuth. The emphasis of this study is upon the structural aspects. In particular, we discuss the anisotropy of segregation and its impact upon the structure of grain boundaries with segregated solutes as well as the structural transformations invoked by segregation.

2. Interatomic forces and method of calculation In the calculation reported here the energy per atom i, E,, in a binary alloy A,_,B,. is written as: N

E; = +


&(rii) + U,(u,),

where N is the total number of atoms, #j is either c$**, +BB or c#J*~,which are the three pair potentials describing A-A, B-B, and A-B interactions, and U, is the density dependent part of the energy which has been written as a function of the volume per atom, u,. Both the potentials +‘j and U, are concentration dependent and have been determined empirically following the scheme developed by Maeda et al. [22]. The potentials +“” and GBB are based on the corresponding potentials for pure elements (see also Vitek and Minonishi [23])


V. Vifek, G.J. Wang / Segregafion and grain boundary sfrwfure

and the #A” potential is constructed so as to fit self consistently the enthalpy of mixing of the alloy. The latter is of primary importance in the studies of segregation. The energy of segregation, Es”8, corresponding to the change of the energy of the system when a solvent atom in the boundary containing already N, solute atoms in the unit cell, is replaced by a solute atom, was given in our previous paper [13] and will not be reproduced here. The relaxation calculations have all been carried out at constant total pressure and the method employed has been described in detail by Sutton and Vitek [21]. It allows for relative rigid body translations of the two grains, relative displacement of any atomic layer parallel to the boundary, as well as individual relaxations of all atoms within the repeat cell of the boundary. Owing to the requirement of the constant pressure an overall expansion or contraction perpendicular to the boundary occurs. This expansion (contraction) is generated in the vicinity of the boundary and atomic layers remote from the boundary experience a rigid body displacement perpendicular to the boundary. Periodic boundary conditions have been used throughout and the repeat cell of a boundary with impurities is always taken as a multiple of the repeat cell of this boundary in the pure metal.

3. Boundaries with segregated impurities Extensive atomistic studies of grain boundaries with varying concentrations of segregant have been carried out for Cu-Bi, Cu-Ag and Au-Ag substitutional alloys. The following symmetrical tilt boundaries with the [OOl] rotation axis were investigated: 2 = 5(210), Z’= 5(310) and _Z = 17(530). Since the purpose of this work was not to investigate the kinetics of segregation but to study structures of grain boundaries with substitutional impurities we did not investigate equilibrium segregation. Instead, we followed the effect on the boundary structure of a fixed number of impurity atoms (corresponding to various boundary concentrations of solute) positioned at various grain boundary sites at 0 K. Physically, this corresponds to the situation where equilibrium segregation occurs at a high temperature and the corresponding distribution of impurities is frozen in after a rapid cooling of the material. The calculations always started at such low concentrations of segregant that no impurity interacted with another one directly. At this stage only one impurity was present in each repeat cell of the boundary. A large number of possible substitutional segregation sites were always tested. An impurity atom was then placed into a site with low (negative) segregation energy and the concentration increased by placing another atom successively into various possible segregation sites within the repeat cell until the most favoured site for its segregation was again found. This process was then repeated successively, increasing thus gradually the concentration of impurities in the boundary. Since it is reasona-

V. Vitek, G.J. Wang / Segregation

and grain boundary structure


ble to assume that the sites with the lowest segregation energy will always be filled first this procedure is considered to model closely the most probable distributions of solutes in the boundaries at various concentrations. In reality, different levels of impurity concentration will, of course, be achieved as equilibrium distributions of impurities at various temperatures and fluctuations will always occur. The basic structural effects of segregation were found to be of the same nature in all the boundaries studied. Hence, in this paper we concentrate on the 2 = 5(210) boundary and on Cu-Bi system, for which a number of experimental studies have been made. The detailed results of all the other calculations will be presented elsewhere [24]. The repeat cell of the (210) boundary was taken as a[7201 x a[002] (a is the lattice parameter). This unit comprises four basic repeat cells of the boundary in the pure metal. The possible segregation sites in the lowest energy configuration B (cf. fig. 1) of the boundary are shown in fig. 2 where the atoms considered for substitution by impurities are marked by letters. The segregation energies, ES’, for some of these sites, are summarized in table 1. The boundary studied does not possess a mirror symmetry with respect to the boundary plane but the asymmetry arising from the relative displacement of the grains is only small and, therefore, the segregation energies for sites Z, J and K are very similar to those for sites C, E and G, respectively. ES0 is usually large negative for the sites associated with large hydrostatic tension (for more details see ref. [21]) which suggests that the principal driving force for the segregation is in this case the size effect. The variation of Es0 from site to site is very pronounced and thus a strong anisotropy of the segregation can be expected. A similar result was found in

+ A

k +


+ .



Fig. 2. B structure of the B = 5(210)/[001] boundary with segregation sites considered marked by letters.

V. Vitek, GJ.


Wang / Segregation and gruin boundary srructure

Table 1 Segregation energies for different impurity per repeat cell

sites in the I: = 5(210)/[001]


for the case of one












- 1.15



- 1.61





the previous studies of grain boundaries containing dilute impurities [21]. Placing the first Bi atom into the B site the second Bi atom was put successively into the other possible sites. (Sites displaced by a[0011 with respect to those shown in fig. 2 were also considered and are marked by the superscript z, e.g. B’.) The segregation energies, E,‘, for those sites for which Es0 is negative, are summarized in table 2. (All the other sites were also tested but no negative values of E,’ were found.) Increasing the concentration further the second Bi atom was placed into the site B,, which possesses the lowest value of Ei, and the most favourable site for the third Bi atom was then found. Repeating this process the concentration of Bi in the boundary was gradually increased. The segregation energy, EsN~, corresponding always to the most favoured site, is plotted in fig. 3 as a function of N,, the number of Bi atoms per unit cell before the next segregation event occurs. It is seen that E>!B is negative up to Na = 12. For larger values of N, it becomes first positive and than converges to zero. Hence for Nb > 12 any additional segregation is not favoured and a saturation occurs for thirteen Bi atoms per repeat cell, i.e. for the concentration of 1.25 monolayers. The sites filled at this stage are B, B,, B’, B;, E, E,, J, J,, EZ, Ef, JZ, J;, G. The structure corresponding to this case is shown in fig. 4. It is an ordered structure in which every Bi atom is surrounded by Cu atoms and vice versa. This ordering is, of course, a consequence of the crystal like structure of the grain boundary where the sites favoured for segregation form a sublattice. When the first Bi atom was placed into the site E, which is also favoured for segregation, the grain boundary structure transformed to that shown in fig. 5.

Table 2 Segregation energies for different sites in the C= 5(210)/[001] boundary impurities per repeat cell; the first impurity is positioned at the site B

for the case of two


Ed (eV/atom)







- 3.00




- 0.83

- 0.42


V. Vitek, G.J. Wang / Segregation and grain boundary structure

Cu-Bi B(2lOf UNIT: a~l~OJ~2a~OOl~,




8 2tl? 1






+ *




+ +




-4 -

b&/UNIT Fig. 3. Segregation energy of Bi to the 2 = S(210) boundary atoms already present in the repeat cell of the boundary.

This structure is different from any of pure case and it is, therefore, a structure increase of the Bi concentration did not structure. hence,depending on the site

Fig. 4. The structure

of the X = 5(210)/[001]

in Cu as a function

of the number


the structures (B or B’) found in the stabilized by the segregation. Further lead to any significant change in this where the segregation first occurs,


in Cu saturated

by bismuth.


K Vitek, G.J. Wang / Segregation and grain boundary structure

Fig. 5. Structure

of the .Z= 5(210)/[001]


in Cu with the site E occupied

by a Bi atom.

different structures of the boundaries with impurities may arise. As mentioned earlier another structure of the 2 = 5(210) boundary in the pure metal also exists. This structure, marked B’, is shown in fig. 6a. It possesses a somewhat higher energy than the structure B but, as explained in section 1, it may still occur in boundaries with defects and in the multiple structures of more general boundaries. The segregation to the boundary with this structure was, therefore, also studied. The most favoured site is the site D (~5: = - 1.6 eV). However, when a Bi atom is placed into this site the B’ structure starts to transform into the structure B and when Bi atoms are at both D and D, sites (concentration 0.19 monolayer) this transformation is complete. This is shown in fig. 6b where solid lines mark the original B’ units and dashed lines the new B units. It is seen that the grain boundary moved at the same time by one interplanar spacing of (420) layers upwards and the sites D and D, of the B’ structure became sites B and B, of the B structure. Further development of the segregation will be, of course, as described for the structure B. Hence, in this case segregation induced transformation of one possible boundary structure into another and reduced thus the structural multiplicity since the B’ structure is no more metastable when Bi atoms are present in the boundary.

4. Discussion The most remarkable structural effects found in the present study of grain boundaries with substitution~ly segregated impurities are the large variation of

V. Vitek, G..J. Wang / Segregation and grain boundary structure


















+ B

+ A

kB+a+ +A Fig. 6. B’ structure sites D and D,.




of the H = 5(210)/~~1~




s~tllol~ + A *+ +a *+

ip + +A


a _+

in (a) pure Cu (b) Cu with Bi segregated


the segregation energy from one atomic site to another, formation of layers of ordered binary alloy structures at the boundaries as the concentration of the segregant increases, and occurrence of structural transformations induced by the segregation. In the case of Cu-Bi, where the size effect is dominating, segregation is particularly:favoured at the sites associated with high hydrostatic tension. However, a similar anisotropy of segregation sites was also found for other systems [21,24,25] in which the size effect is not so important. Hence, this is a general feature of the segregation to grain boundaries and it arises


V. V&k,

G.J. Wang / Segregation

and grain boundary structure

primarily as a consequence of their crystal like structure. At the same time in a given boundary different sites may be favoured in different alloy systems and the anisotropy may have different strength. For example, while it is very strong in the Cu-Bi system it is much weaker in the case of Au-Ag [21,24]. Thus while the phenomenon of the anisotropy is general the details will vary in dependence on the alloy system considered. Since the anisotropy of the segregation energy is very pronounced when considering different sites in one boundary it is reasonable to expect that similar large differences in the propensity to segregation will be found when considering different grain boundaries. A marked anisotropy in the amount of Bi segregated to different grain boundaries in Cu has indeed been observed by a number of authors [25-281. However, at this point it is necessary to distinguish between the following two situations: The variation of the boundary plane while the misorientation of the grains is fixed and variation of the misorientation of the grains while keeping the mean boundary plane fixed [29]. In the latter case the structures of boundaries corresponding to different misorientations throughout a misorientation range are closely related. As explained in section 1 their structures correspond to different mixtures of structural units of boundaries delimiting this range of misorientations [ll]. The propensity to segregation, similarly as other boundary properties, will, therefore, vary smoothly with the misorientation [8,30]. Furthermore, if the segregation propensity is not too different in the two delimiting (favoured) boundaries no large variations for misorientations between them can be expected. Sudden changes may, however, occur when crossing the favoured boundaries [8]. On the other hand, in the former case a much stronger anisotropy is likely since very different atomic configurations occur for different boundary planes. Depending on the strength of the anisotropy this may induce faceting if some of the segregated boundaries are of particularly low energy. An extensive faceting has, indeed, been observed in the Cu-Bi system [31,32] for which the calculations suggest a strong anisotropy of the available segregation sites. The conclusion that smooth and relatively small variation of the propensity to segregation may be expected when varying the misorientation of the grains, could appear to be in conflict with the experimental observations which suggest a significantly lower segregation to some special low 2 boundaries when compared with other boundaries close in misorientation to these special boundaries [33]. However. as pointed out by Donald and Brown [32] it is likely that, particularly in the Cu-Bi system, faceting always occurs in other than low 2 boundaries leading thus to a rapid increase of segregation in these boundaries. The anisotropy of segregation sites and the formation of specific ordered binary alloy structures at grain boundaries are closely related. If we assume that for a given concentration of the segregant the sites occupied by the solute are those possessing the lowest segregation energies then an ordered alloy structure will always be formed during segregation if the boundary structure is

V. Vitek, G.J. Wang / Segregarion and grain boundary structure


not amorphous. This type of ordering in the boundary is not related to any ordering tendency in the bulk although the basic features of this ordered structure, such as the average separation of the solute atoms, will depend on the corresponding atomic interactions. For example, in the structure shown in fig. 4 the Bi-Bi separations closely coincide with the distance at which the Bi-Bi potential has a minimum. Such an ordering is, of course, enabled by wider variability of atom separations in the grain boundary when compared with the perfect lattice. The properties of such layers of ordered alloys may be very different from those of pure metals and therefore the formation of the ordered structures may be primarily responsible for the remarkable changes of grain boundary properties with segregation, such as the marked decrease of the cohesive strength across the grain boundaries. The present calculations show that saturation has been reached for Bi concentration of 1.25 monolayer. While variations from boundary to boundary must be expected, this agrees remarkably well with the segregation levels found at fractured boundaries in the Cu-Bi system [26,34,35]. If a layer of an ordered binary alloy is formed at the boundary during segregation saturation must always occur when ordering is not favoured in the bulk. This is the case in the Cu-Bi system. On the other hand, if the bulk ordering is favoured the ordered layer at the boundary can serve as a nucleus for the growth of the ordered phase in the bulk. No saturation limit would then exist. The phenomena discussed above may all occur while the original structure of the boundary in the pure metal is not drastically changed. However,’ the calculations also suggest that segregation may sometimes involve significant structural transformations. This has also been found in the study of segregation of P to grain boundaries in iron 1301. Two types of such transformations were found in the present work. First a new structure, which is not stable in the pure case, is formed and stabilized by segregation (structure shown in fig. 5). Secondly, when several metastable structures of the boundary in the pure metal exist, segregation may provoke transformations between them (e.g. transformation of the B’ structure into the B structure, fig. 6). As discussed in section I the different metastable structures of the delimiting boundaries may appear in general boundaries leading thus to an extensive multiplicity of their structures. If the latter transformation occurs upon segregation the multiplicity of structures will be reduced. This may be related to the effect of segregation on the ability of boundaries to act as sources and sinks of vacancies and on the grain boundary diffusion since the local transformations between different multiple structures of the type shown in fig. 1 may be involved in these processes [12,14]. Whether a transformation occurs or not depends, however, on the sequence in which the solutes entered the boundary. For example, if the solute first segregates to the site B in the 2 = 5(210) boundary no transformation will occur while if the segregation starts at site E the structure transforms. Thus different structures may develop at different parts of a given boundary during

V. Vitek, G.J. Wang / Segregation and grnrn boundary structure


the segregation cooling. lowest


If a complete energy




will, of course,

the complete

The possibility

this “disorder” may be frozen in by a rapid is reached during segregation only the


of structural



If this is the transformed



will have taken place.

has been discussed



thermodynamics grounds [36-381 and the present study shows the possible microscopic features of such transformations. However, their role in segregation and other grain boundary phenomena is not entirely clear at the present time and further research in this direction

is needed.

Acknowledgement This research has been supported by the Department Basic Energy Sciences, Contract No. DE-A502-79ER-10429.

of Energy,

Office of

References [l] R.W. Balluffi, Ed., Grain Boundary Structure and Kinetics (ASM, Cleveland, OH, 1980). [2] J. Pask and A. Evans, Eds., Surfaces and Interfaces in Ceramic and Ceramic-Metal Systems (Plenum New York, 1981). [3] H.J. Leamy, G.E. Pike and C.H. Seager, Eds., Grain Boundaries in Semiconductors (NorthHolland, New York, 1982). [4] R.W. Balluffi, Met. Trans. 13A (1982) 2069. [5] H. Gleiter, Mater, Sci. Eng. 52 (1982) 91. [6] R.C. Pond and V. Vitek, Proc. Roy Sot. (London) A357 (1977) 453. (71 A. Bourret, C. d’Anterroches and J.M. Penisson, J. Physique 43 (1982) C6-83. [S] A.P. Sutton and V. Vitek, Phil. Trans. Roy. Sot. London A309 (1983) 55. [9] M.F. Ashby, F. Spaepen and S. Williams, Acta Met. 26 (1978) 1647. [lo] R.C. Pond, V. Vitek and D.A. Smith, Acta Cryst. A35 (1979) 689. [ll] A.P. Sutton and V. Vitek, Phil. Trans. Roy. Sot. London A309 (1983) 1. [12] V. Vitek, A.P. Sutton, Gui Jin Wang and D.S. Schwartz, Scripta Met. 17 (1983) 183. [13] V. Vitek and Gui Jin Wang, J. Physique 43 (1982) C6-147. [14] Gui Jin Wang, A.P. Sutton and V. Vitek, Acta Met. 32 (1984), in press. (151 W.C. Johnson and J.M. Blakely, Eds., Interfacial Segregation (ASM, Metals Park, OH, 1977). [16] M.J. Weins and J.J. Weins, J. Physique 36 (1975) C4-81. [17] J.R. Beeler, R.E. Dahl and R.D. Bourquin, J. Physique 36 (1975) C4-97. [18] S. Nichols, Scripta Met. 15 (1981) 423. [19] E.S. Machlin and A. Levi, Scripta Met. 14 (1980) 127. [20] H.K. Chang, J.K. Lee and D.F. Stein, in: Interatomic Potentials and Crystalline Defects, Ed. J.K. Lee (TMS AIME, Warrendale, 1981) p. 373. [21] A.P. Sutton and V. Vitek, Acta Met. 30 (1982) 2011. [22] K. Maeda, V. Vitek and A.P. Sutton, Acta Met. 30 (1982) 2001. [23] V. Vitek and Y. Minonishi, Surface Sci. 144 (1984) 196. [24] Gui Jin Wang, PhD Thesis, University of Pennsylvania (1984). [25] M. Hashimoto, Y. Ishida, R. Yamamoto, M. Doyama and T. Fujiwara, Scripta Met. 16 (1982) 267.

V. Vitek, G.J. Wang / Segregation

[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

and grain boundary structure

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New York,