The following points highlight the three main surface defects in crystals. The types are: 1. Free Surfaces 2. Grain Boundaries 3. Stacking Faults.
Surface defects are defects of two dimensional nature in the mathematical sense (that is, they have two dimensions). These are areas of distortions that lie about a surface, having a thickness of a few atomic diameters (that is, the thickness is almost negligible compared to other two dimensions of the surface). From the thermodynamics point of view, surface defects are not stable-and are present as metastable defects, and under suitable conditions, these defects can be removed.
Type # 1. Free Surfaces:
Free surfaces (external surfaces) of materials are surface defects in the sense that growth of crystal has stopped abruptly at the surface. The atoms in the free surface of a solid (or, even liquid) have no neighbours and have no cohesive-bonds, on one side.
That is why, these atoms have higher free energy as compared to atoms inside, Fig. 4.92. Free surfaces have surface energies that are roughly calculated from the number of ‘broken bonds’ at the surface. For example if the free surface is a close-packed plane of a close-packed crystal structure such as {111} type.
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An atom in free surface has six neighbouring bonds in the surface plane, and three bonds with atoms below this surface but the three bonds above the surface are broken (as coordination number is 12). Thus, three out of total twelve bonds of an atom, that is, one quarter of bonds are missing compared to an atom inside.
The surface energy shall be one-quarter of the cohesive energy per atom, that is, around 1 eV for a metal such as FCC-iron. There are on an average around 1015 atoms cm-2 in a crystal surface. So, the surface energy becomes 1015 eV cm-2 i.e., 1600 ergs cm-2. Table 4.8 shows experimentally measured surface free energy of some metals
Free surfaces can be classified as (a) Singular surface, (b) Vicinal surface as illustrated in Fig. 4.93.
Singular surface has low surface energy and usually have low index planes such as (100), (110), or (111). Vicinal surfaces have orientation near those of singular surfaces and are formed of low index planes separated by mono-atomic or mono-molecular steps or ledges, kinks and double kinks, etc.
In metals, the free (valence) electron bond is unselective and that is why, any two metals with clean surfaces, when brought in close contact adhere well. And, it follows that a liquid metal of low cohesive energy generally spreads freely over a clean surface of a metal of high cohesive energy, because by doing so, a surface of high energy (solid metal) is replaced by a low-energy surface and a low-energy metal-metal interface. The wetting action of solders and brazing alloys is based on this principle.
Type # 2. Grain Boundaries:
Classification of Boundaries:
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Dislocation theory offers good means for broad understanding of the nature and properties of grain boundaries. In a single phase polycrystalline material, a grain boundary is a two dimensional surface, which separates two regions differing in orientation of their crystal axes.
It is a surface defect where orientation of unit cells of one grain changes to other, i.e. at this surface; there is a mismatch of unit cells with unlike orientations. There are many types and gradations of grain boundaries in metals and alloys.
They are classified as:
A. Based on Degree of Difference in Orientation as:
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i. Sub-Boundaries:
These are boundaries supposed to be existing within a grain, when the two regions on both sides of them have as small a difference in orientation as 1°, or less.
ii. Low-Angle Boundaries:
Low-Angle Boundaries are boundaries between two adjacent grains differing in orientation by less than about 10°.
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iii. High Angle Boundaries:
High angle boundaries are common boundaries in materials having large difference in orientation between grains.
B. Based on Type of Difference in Orientation as:
i. Tilt Boundaries:
Tilt boundaries are boundaries between adjacent grains having misorientation in such a way that lattices appear to be tilted with respect to one another.
ii. Twist Boundaries:
Twist boundaries are boundaries between adjacent grains having misorientation in such a way that the lattices appear to be rotated (twisted) with respect to one another.
iii. Coincidence Boundaries:
Coincidence boundaries are boundaries in which atomic positions common to both the adjacent grains form some fraction of the total boundary area.
C. Based on Continuity of Atoms on both Sides of the Boundary:
i. Coherent Boundary:
Coherent boundary is a boundary where there is complete continuity of atoms across it, i.e. there is one to one correspondence of atoms at the boundary/interface.
ii. Semi-Coherent Boundary:
Semi-coherent boundary is a boundary where there is partial continuity of atoms across it, i.e., some planes of atoms are continuous, but some may end at it.
iii. Incoherent Boundary:
Incoherent boundary is a boundary having no coherency of atoms across it at all.
(i) Sub-Boundaries:
When a single crystal is bent by cold working, and if one slip system operates (More dislocations are generated on one slip system), then the bent crystal looks as in Fig. 4.94 (a), because excess positive edge dislocations remain in the crystal to accommodate the curved shape of the crystal.
When such a bent crystal is annealed (before the recrystallisation takes place), the curved crystal forms a number of strain-free sub-grains, each preserving the local orientation of the original bent crystal separated by plane sub-boundaries. This process of forming sub-grains and sub-boundaries is called polygonisation, because a smooth curved crystal changes into polygons.
The dislocations have arranged themselves in sub-boundaries (by the processes of slip and climb during annealing), one above the other because in this position, dislocations relieve each others elastic strain fields, Fig. 4.94 (d). The dislocation arrangement of Fig. 4.94 (a) is one of high strain energy, which changes to as in Fig. 4.94 (b) of lower strain energy.
The sub-boundaries between the sub-grains are considered to exist within a grain and thus, the difference of orientations between the two adjacent sub-grains is very small, around 1° or less. Sub-boundaries are small-angle or symmetric tilt-boundaries within the grain having very small difference of orientation between the sub-grains, Fig. 4.94 (e). Sub-boundaries are revealed by etching as dense row of pits under microscope.
(ii) Low-Angle Boundaries or Tilt-Boundaries:
In 1940, both Bragg and Burger, and later Read-Shockley (1949, 1950) proposed the dislocation model of structure of low-angle boundaries with good success. Low angle boundaries are boundaries between adjacent crystals of same structure but with small difference in crystal orientations, and might be considered to have an array of parallel dislocations (along the plane of the boundary).
Two grains in schematic Fig. 4.95 (a) are misoriented or pictured as tilted symmetrically about an axis (horizontal inside the paper), parallel to the direction of edge dislocations. The two lattices can be maintained continuous by periodic addition of an edge dislocation.
The resulting vertical array of parallel dislocations of same sign (here, positial edge dislocations) is just like a sub-boundary but with greater difference of orientation, still less than about 10°, and is in between two adjacent grains. Low-angle boundaries are also observed under microscope as uniformly spaced line of etch pits, Fig. 4.95 (c).
The presence of each dislocation at the surface of a specimen as etch pit is revealed because of the faster rate of chemical attack by the etchant in the elastically strained area of the crystal near the core of the dislocation.
The angle theta is the angle of misorientation. The greater is the angle of misorientation, the greater is the inclination of planes that terminate as dislocations at the boundary, and closer is the spacing of the dislocations in the vertical boundary, i.e., the spacing between the dislocations in the boundary determines the theta, the angular inclination between the lattices. In Fig. 4.95 (b), we find that
where, b is the burgers vector of dislocation in the boundary, and d is the spacing between the two adjacent dislocations. If the angle of misorientation is assumed to be small, then sin θ/2 ≈ θ/2, and
and the results of this relationship have been found to be in good agreement by x-ray diffraction determinations, and confirms the validity of the dislocation model for low-angle boundaries.
This relationship could be used to drive the energy of the boundary as dependent on angle of misorientation. With the increase in misorientation angle, θ, of a symmetric tilt boundary, the value of d decreases, i.e., the number of dislocations increases.
We know by now that total energy of each edge dislocation per unit length is given by:
where, Ec is the energy per unit length of dislocation core. Energy of the dislocation decreases as d becomes small, or θ increases (∵ d = b/θ). If we assume r0 = 2b, the total energy of the dislocation per unit length, 
If ‘d’ is distance between two dislocations, then energy of unit area of the boundary is approximated as
If a graph is plotted between Eb and θ as in Fig. 4.96 and experimental data is superimposed for copper, the curve matches well up to below about 8°. There is no maximum obtained experimentally. The equation (4.77) predicts well that as b and EC approach zero, the energy vanishes.
At higher angles than 8°, the dislocations are quite closely spaced, and are difficult to be distinguished due to overlap of cores of the dislocations. The structure of the grain boundaries beyond 8° of misorientation appears to acquire more complicated configurations. This model of dislocations appears to be successful for low angle boundaries up to about 8°.
Problem:
Copper shows array of edge dislocation an (110) plane of atoms. Calculate the spacing of dislocations if the angle of tilt is 1°. (Lattice constant of copper 3.62 A).
Solution:
Copper is FCC, and its (110) plane is shown on which dislocations have Burgers vector,
(iii) Degrees of Freedom of Grain Boundary:
All single phase poly-crystalline materials contain a complex network of grain boundaries between adjacent grains that are crystallographically misoriented with respect to one another. The grain boundary described in Fig. 4.95 (a), where the two neighbouring grains appear to be inclined or tilted with each other (by a small angle) about a horizontal axis normal to the paper is a very special boundary, as it has only one degree of freedom, or variable. In fact, the grain boundaries have five degrees of freedom, three to define the rotation, and two to define the boundary plane.
We specify the difference in orientation by the rotations which we would have to give to one crystal to bring it into same orientation as the other crystal. For example, if we rotate the grain 1 in Fig. 4.95 (a) by an angle theta clockwise about a horizontal axis which runs inside normal to the paper, the two grains become in the same orientation.
There are thus, three degrees of freedom available for such rotations, i.e. there are three ways of doing it corresponding to the three perpendicular axes about which rotations can be made. There are two more degrees of freedom for choosing the orientation of the grain boundary itself, that is, the boundary may not be in symmetrical position between the two crystals, but can be rotated about either of the two axes at 90° to each other (Fig. 4.97 d & e).
We can tilt or twist one crystal relative to the other in three ways, and have two ways to align the boundary between the crystals. Fig. 4.97 illustrates schematically these five degrees of freedom of a grain boundary. In Fig. 4.97 (a), the two neighbouring crystals are tilted with respect to each other about a horizontal axis, normal to the plane of paper with grain boundary placed symmetrically.
This boundary has one degree of freedom and is a symmetrical tilt boundary. Fig. 4.97 (b) illustrates also a symmetric tilt boundary but with a vertical tilt axis. In both these cases, the rotating tilt axis lies in the boundary. The boundary has edge dislocations in it. In Fig. 4.97 (c), one crystal has been rotated relative to the other about a horizontal axis perpendicular to the boundary.
This is called a symmetric twist boundary, and normally contains at least, two different arrays of screw dislocations. Fig. (d) and (e) illustrate that the grain boundaries do not have to be in symmetric positions between the two crystals, but may align asymmetrically in two different ways between the crystals—one by rotating the boundary along a horizontal axis, Fig. 4.97 (d) and other along a vertical axis, Fig. 4.97 (e).
Majority of the grain boundaries in single phase polycrystalline materials have a mixed character of these five variables of varying degree, and thus the general or large angle boundaries have a complex structure.
A symmetric low angle boundary, or symmetric tilt boundary, Fig. 4.94 has angle of misorientation equally divided between the two neighbouring grains, i.e., each lattice is displaced by θ/2 from the vertical axis, i.e., has one degree of freedom. But a typical tilt boundary may have two degrees of freedom (or even more), such as asymmetric tilt boundary, Fig. 4.98 (a), where the angle of misorientation is not divided equally between the grain 1 and grain 2.
Let us take a special case, Fig. 4.98 (a), where the angle has been so chosen that grain 1 contributes twice as many edge dislocations of Burgers vector b1→, as by grain 2 of Burgers vector, b2→. Let us assume that 2 dislocations of grain 1 combine with one dislocation of grain 2, by reaction,
2 b1→ + b2→ = b3→
to give a resultant average dislocation of Burgers vector, Although this reaction does not occur, but it can be assumed that these dislocations of Burgers vector, lie along the asymmetric tilt boundary, and is the Burgers vector perpendicular to such an asymmetric boundary.
Depending on the position of the asymmetric tilt boundaries, the angle θ of misorientation may be shared in innumerable ways between the grains to give an average resultant dislocation of different Burgers vectors. Thus, a large number of such asymmetric boundaries can be assumed to exist.
(iv) Twist Boundary:
A small-angle pure twist boundary is obtained, if one of the crystals appears to be rotated or twisted relative to its neighbour about an axis perpendicular to the plane of the boundary, Fig. 4.99.
In Fig. 4.99 (c), we see shape of cross section at the slip plane of a square crystal lattice deformed by a set of vertical uniformly spaced screw dislocations. The two surfaces on either side of the slip plane have been displaced by an angle, θ, but such a set of vertical dislocations have not accomplished the desired rotation needed to get a twist boundary of Fig. 4.99 (a).
If only a horizontal set of screw dislocations are introduced, then the result is illustrated in Fig. 4.99 (d), which also does not match with Fig. 4.99 (a). If simultaneous introduction of two sets of screw dislocations is done, we get rotation, or twist of two crystals relative to each other to get a picture illustrated in Fig. 4.99 (e), & 4.100.
A twist boundary, lying along a crystallographic plane has rotational displacements of atoms symmetric with respect to the boundary. This type of boundary requires crossed grid of screw dislocations to permit continuity from grain 1 to grain 2. If in a unit area of boundary, each set of screw dislocations contain n dislocations, then the twist angle, θ, is
If the twist boundary does not lie along a crystallographic plane, then, it is asymmetric with respect to rows and columns of atoms on either side of it. At least, then, three sets of screw dislocations with different Burgers vector are needed to produce an asymmetric twist boundary.
An asymmetric twist boundary is analogous to asymmetric tilt boundary. The actual small angle boundary of arbitrary orientation may have varying five degrees of freedom that is, varying degree of tilt and twist of crystallographic axis. If the boundary is asymmetric, then, it may, contain edge, screw and mixed dislocations.
(v) Coincidence Boundary:
If we make duplicate square lattice of holes in a black paper, and then pin the two sheets together, say at X in Fig. 4.101 (a) and hold them against light and rotate into positions in which various other points come into coincidence. One such seen is as in Fig. 4.101 (a). Fig. 4.101 (b) illustrates a square lattice.
A rotation of 37° about a lattice point X, makes atom Y to coincide with Y’, and Z to Z’ in the un-rotated lattice. The Fig. 4.101 (c) illustrates two neighbouring grains in which grain 2 has been shown after rotation of 37° relative to grain 1. Dark dots are coincidence points.
It is clear that coincidence sites are continuous in both the grains in spite of grain boundary between them. These coincidence sites appear to form a lattice of themselves (joined by fine lines), but of larger cell dimensions. We see that one atom out of five atoms is a coincidence site, so that the term 1/5 coincidence boundary is applied to this arrangement of points.
Table 4.9 illustrates some coincidence lattices in BCC and FCC metals. Coincidence lattices occur in all of the common crystal structures, and have site densities varying from 1/3 to 1/19 and even lower. Coincidence boundary normally is incoherent boundary like an ordinary grain boundary; that is, most of the atoms at the boundary of one crystal do not match the lattice positions in the other crystal, but mismatch is less, and so have less energy than ordinary boundaries.
As coincidence sites are low energy sites, the grain boundary between the grains may not follow a relatively straight path as in Fig. 4.101 (c) but a zig-zag path of Fig. 4.102 (a), so that the grain boundary has high density of coincidence sites. Most coincidence lattices do not have high densities of coincidence sites, but in certain cases, the coincidence sites lie in moderately close-packed planes as in Fig. 4.102 (b), leading to the formation of coherent boundary.
Coherent, Semi-Coherent and Incoherent Boundaries:
A coherent boundary has all the atom sites in it common to both crystals (regions) joined by it. There is, thus, continuity of crystal structure from one grain to the other, or as we say, there is one to one correspondence of atoms at the boundary. As the atoms in the boundaries fit together well, coherent boundaries have low energies, and that is why all boundaries tend to become as coherent as possible.
Twin boundary is a coherent boundary of Fig. 4.103 (a). Other examples of such boundaries are in widmanstatten structures, martensite, and in precipitates from supersaturated solid solutions. The important requirement for the formation of coherent boundary is that the atoms in the adjoining crystals must, at the common boundary (coherent), have closely similar patterns and spacings.
Small-angle boundary is a semi-coherent boundary. If the dislocations get introduced in a coherent boundary, it results in a semi-coherent boundary. There is continuity of atoms in some layers in both the adjacent grains, but ends up in a dislocation in other layers at the boundary.
By suitably and steadily increasing the density of dislocations in a boundary, (to represent increasing rotations of the crystals, or the boundary, or both) from orientation that produces coherent boundaries, a gradual change from coherent to semi-coherent to in-coherent boundaries can be obtained, Fig. 4.104 illustrate this change. In the last extreme limit, the dislocations are too densely packed to overlap cores of neighbouring dislocations, to result in incoherent, or so called normal grain boundaries.
In multi-phase materials, when second phase forms in alloy crystals, then to get a coherent interface between the phases, the crystal structures of both phases have to be matched. If the size of precipitate is very small, it may adopt a crystal structure that allows it to be coherent with the matrix crystal, even if for matching the atomic bonds of two crystal structures at the boundary produce considerable elastic distortions in it, and its immediate surroundings.
The lattice constant of one phase increases and the other decreases. The thin disc shaped particles have least strain energy, and still be coherent. These thin tiny particles, which are coherent, can be thought of as solute-rich clusters in the parent matrix. In quenched and aged aluminium with 4 percent copper alloy, the tiny particles are copper rich discs about 100 A in diameter and 4 A thick on {100} planes of aluminium matrix with coherent interfaces, Fig. 4.105 (a).
When such a coherent particle grows larger by obtaining atoms from matrix and crystal structure changes, it may break away (may be partly in the beginning) from its coherent attachments to the matrix in order to reduce the increasing elastic coherency strains, and adopts a more stable structure by introducing dislocations in its interface to make the interface semi-coherent, which on further change of the precipitate forms an incoherent interface.
(vi) Common Grain Boundaries:
These boundaries are formed when two neighbouring grains, solidifying from liquid metal, or recrystallising, impinge on each other. The resulting boundary can be quite arbitrary because there are five degrees of freedom of such a boundary. The dislocation model of low-angle grain boundary illustrates that with the increase in angle of misorientation between the grains, the dislocations lie closer together.
At sufficiently high angles of misorientation, the individual dislocations lose their identity, because the cores of the dislocations overlap. The structure and the energy of such boundaries, called the large angle boundaries or common boundaries are difficult to be predicted. However, it is clear that grains on both sides of them are truly crystalline up to the boundary, which is a narrow transition region of imperfect order.
The grain boundaries are not necessarily smooth curvilinear surfaces, but with ledges and facets of macro as well as microscopic dimensions, particularly if the grains are held at-sufficiently high temperature to minimise the local elastic strains, Fig. 4.106.
The energies of large-angle boundaries are usually estimated by thermal-etching. When a polycrystalline material is heated at high temperatures, thermal grooves form on the free surface, wherever a grain boundary intersects the surface perpendicularly.
This is an example of general effect of balancing the forces of surface tension at an interfacial triple junction, Fig. 4.107 (a), consisting of one grain boundary and free surfaces to the right and to the left of the point X.
If ϒs is the free surface tension and ϒb is surface energy of the boundary, then the two surface tensions of the free surface balance each other, but the vertically down component due to grain boundary remains unbalanced.
In order to balance this grain boundary energy, a groove must form as illustrated, Fig. 4.107 (b) with a dihedral angle, θ between the two surface energies of free surfaces to satisfy the equation, Fig. 4.107 (c):
Diffusion and evaporation processes at high temperatures enable the necessary boundary movements to take place. It becomes possible to estimate ϒb if θ and ϒs are known. Actually, this equation gives free energy of a boundary, because the total energy is larger than this, due to extra contribution associated mainly with the vibrational entropy of atoms in the boundary. Table 4.7 provides free energies of some large-angle boundaries. We find, ϒb ≈ 1/3 ϒs.
Grain boundaries have dislocations and vacancies in them. Infact, grain boundaries are source and sink of them. Soluble impurities strongly segregate to the grain boundaries to reduce their elastic misfit energies In fact, first few atomic percent of impurity as well as alloy atoms are invariably present at the grain boundaries as these can be accommodated better in imperfectly ordered grain boundaries.
In zone-refined metals, grain boundaries migrate readily at around 10-5m sec-1 at temperatures close to the melting point, and some movement has been also detected at 0.25 Tm.p.. But soluble impurities at the grain boundaries decrease the movement of grain boundaries. The insoluble impurities have an even stronger effect in reducing grain boundary motion (also thus grain growth).
Problem:
A thermal groove was observed on the surface of iron where the grain boundary intercepted it perpendicularly making a dihedral angle of 147°. The surface energy of iron is 1.49 Jm-2. What is the grain boundary energy of iron?
Solution:
We see in diagram:
ϒb = 2 ϒs cos θ/2
= 2 x 1.49 x cos 73.5°
= 0.85 Jm-2
Type # 3. Stacking-Faults:
Stacking-faults are planar surface defects produced by a fault (mistake) in the stacking sequence of atomic planes in crystals. We discussed briefly stacking fault in section 4.5.12 and 4.5.13. The stacking sequence of planes of atoms on {111} planes in FCC crystal structure has been designated as ABCABC …, whereas in HCP crystal structure on (0001) plane as ABABAB.…
A fault in stacking sequence of say FCC crystal structure may be produced by plastic deformation, or by collapse of a large number of vacancies on one plane, or by agglomeration of a large number of interstitialies on a plane, etc.
Suppose a large number of vacancies collapse on a (111) plane, so that one plane of atoms is missing, say plane C is missing indicated by arrow:
then the stacking sequence becomes:
… ABC AB ABC … (Fig. 4.109 b)
There is now a fault in stacking, and this type of fault is called intrinsic stacking-fault as also seen in the formation of Frank loop by the internal collapse of vacancies. If we compare this sequence (underlined stacking sequence) with an HCP stacking sequence without any fault, that is, … AB AB AB… Fig. 4.109 (d), we find that stacking-fault contains four layers of an HCP stacking sequence.
Thus, formation of an intrinsic stacking-fault in an FCC metal is equivalent to the formation of a thin HCP region. This thin region is a surface defect called stacking-fault. The creation of a region with HCP stacking ABAB introduces a region with higher free energy than the FCC structure.
The coordination number in the faulted region is though twelve as in the unfaulted or perfect region, but the second-nearest-neighbour bonds in the faulted region are not of correct type for FCC perfect lattice. Thus, energy is associated for the creation of the stacking-fault, and is called stacking fault energy, which is normally of the order of 0.01 to 0.05 Jm-2. (Table 4.4)
If some interstitialies agglomerate on a plane of atoms, addition of a plane of atom (at least in some area of it) takes place, It is because of this area of extra atoms as a part of plane that the stacking sequence is faulted, and such a stacking fault is called an extrinsic stacking fault. For example, HCP crystal having a stacking sequence of
is made to have an extra plane of atoms added, say, at the place of the arrow, then the atoms of this extra plane occupy a position different than positions of A or B, but say at C. At positions either A or B, the structure shall be highly distorted as then we shall have either position B on B or A on A. Now, at C positions, the stacking sequence becomes
…ABABCAB … (Fig. 1.109 e)
that is, an HCP crystal has a stacking-fault which contains three layers of FCC stacking sequence, and thus it shall require extra energy. Because of the additional lattice discontinuity associated with an extrinsic fault, it has higher energy than an intrinsic fault.
The introduction of an extrinsic fault in FCC crystal structure changes the stacking sequence from:
to ABCACBCABC … (Fig. 4.109 (c))
By adding an extra plane of atoms at C positions in between A and B as indicated by arrow. The three layers ACB constitute a twin (of BCA). Thus, an extrinsic stacking-fault in FCC crystal is also called twin stacking fault.
This stacking fault in FCC metals can also be considered sub-microscopic twin of approximate atomic thickness. The stacking-fault energies of different FCC metals is responsible for their differential deformation behaviour. Metals in general with low-stacking-fault energies strain harden more rapidly, and twin easily on annealing.
Stacking-faults are easily detected after etching under an optical microscope, because of difference in the etching characteristics of faulted and parent structure.