Chapter 2: Defects in Materials

"Crystals are like people, it is the defects in them which tend to make them interesting! - Professor Sir Colin Humphreys"

A crystal is at a state of thermodynamic equilibrium when its free energy \( G = H - TS \) is minimized. When are we (humans) at equilibrium?

We have two knobs that we can tune to lower the free energy of the system above and that is entropy and enthalpy. At low temperatures our entropy knob becomes negligible and energy is minimized by minimizing \( H \). This is done via binding and forming the ordered structures that we discussed previously. At the limit of \( T = 0 \) that perfect crystal is the equilibrium configuration. When temperature increases the free energy is minimized by adopting a less well-ordered atomic/molecular arrangement because that increases the entropy. Entropy increases when imperfections exist so when the temperature is not zero the equilibrium state will most likely contain imperfections. We can calculate the equilibrium concentration by looking at the change in free energy \( \Delta G \) and the \( \Delta S \) by adding imperfections and the enthalpic \( \Delta H_f \) cost of forming those defects.

Crystalline Point Defects (0D)

Vacancies

A vacancy exists when a site that is normally occupied in the perfect crystal is unoccupied. This can occur in many ways for example, forcing an atom onto an interstitial site, into a dislocation core, or to a grain boundary or free surface. You can see a couple of examples of a vacancy below

We can actually calculate the equilibrium concentration of vacancies, \( n \)! Consider a reference state of a perfect crystal of \( N \) atoms and a continuous set of vacancies, \( n \). The new state is a crystal containing a mixture of \( N \) atoms and vacancies. The total enthalpy change is then

\[
\Delta H_f = n \, \Delta h_f
\]

where \( \Delta h_f \) is the enthalpy of formation of a single vacancy. We can also get an idea about the change in configurational entropy from creating vacancies if we remember Boltzmann's entropy equation (he had it carved on his tombstone...show off)

\[
S = k \ln \Omega_m
\]

where \( k \) is the Boltzmann constant and \( \Omega_m \) is the number of distinguishable ways you can arrange the system. The configurational entropy \( \Delta S_c \) is then given by

\[
\Delta S_c = S_{mix} - S_{ref} = k \ln\!\bigg(\frac{\Omega_m}{\Omega_{ref}}\bigg)
= k \ln \Omega_m = k \ln \frac{(N+n)!}{N! n!}
\]

where \( \Omega_m \) is the number of distinguishable ways of arranging \( N \) atoms and \( n \) vacancies on \( N+n \) sites and \( \Omega_{ref} \) is the number of configurations in the ideal reference state, which is 1 since there are no defects. So the total Gibbs energy change is

\[
\Delta G = \Delta H - T \Delta S = n(\Delta h_f - T \Delta S_v) - kT \big[ (N+n)\ln(N+n) - N\ln N - n\ln n \big]
\]

where \( \Delta S_v \) is the vibrational entropy of the atoms surrounding the vacancy and we used Stirling's approximation \( \ln X! \approx X \ln X - X \). Now how do we minimize Gibbs? We take the derivative and set it equal to zero:

\[
\frac{\partial \Delta G}{\partial n} = \Delta h_f - T \Delta s_v + kT \ln \frac{n}{n + N} = 0
\]

We can now solve for the equilibrium fraction of vacant sites \( x_v \)

\[
x_v = \frac{n}{n + N} = \exp\!\bigg(\frac{\Delta s_v}{k}\bigg) \exp\!\bigg(-\frac{\Delta h_f}{kT}\bigg)
\]

You should notice in this equation that there is an exponential temperature dependence, known as Arrhenius dependence or Arrhenius's law. This is critical because there are many behaviors in Materials Science which are Arrhenius (project idea???). Let's take a look at how strongly the concentration of vacancies varies with temperature by looking an example Arrhenius plot below:

Now typically we can assume that \( \exp (\frac{\Delta S_v}{k}) \approx 1 \). The enthalpy of formation of vacancies for metals is typically on the order of 15–150 \( kT \) (where \( kT = 0.026 \text{ eV} \)). So we can clearly see that at room temperature the probability of forming vacancies will be pretty low already. Is this supported by the graph?

This method is not the typical way Arrhenius graphs are presented. It is actually better to plot the log of the equilibrium concentration of vacancies as a function of the reciprocal temperature \( \frac{1}{T} \). That way we can extract the activation energy or energy of formation of the vacancy.

Interstitals

An interstitial defect occurs when an atom occupies an interior site which is not normally occupied. If the interstitial is of the same species in a single component crystal it is a self-interstitial and we have the following equation for the equilibrium number of self-interstitials

\[
x_i = \frac{n}{n + N} = \exp\!\bigg(\frac{\Delta s_v}{k}\bigg) \exp\!\bigg(-\frac{\Delta h_f}{kT}\bigg)
\]

however now the \( \Delta s_v \) and \( \Delta h_f \) represent the enthalpy of formation and vibrational entropy for interstitial formation. As you might imagine the energy of formation for interstitials is lower for more open and less densely packed structures. Obviously there are also higher numbers of these 0D defects for nonequilibrium processes like quenching, irradiation, ion implantation, or cold working.

Self interstitials are classified as intrinsic interstitials but the more common and useful type of interstitial for our purposes as material scientists and for determining the mechanical behavior of materials are extrinsic interstitials. These are typically impurity defects of different size than the host or solvent lattice. Imagine C in Fe.

Often interstitials will occupy tetrahedral or octahedral sites. Tetrahedral sites are positions in the lattice such that the interstitial atom would fit between 4 lattice atoms that would form a tetrahedron around said atom. The same can be said for the octahedral sites but the only difference being there are 6 atoms.

Schottky and Frenkel Defects

In addition to vacancies and interstitials there are certain defects unique to ionic crystals which are Schottky and Frenkel defects. A Schottky defect is when there is the formation of a pair of vacancies in both the anion and cation of the ionic crystal. A Frenkel defect is formed by the creation of either an anion or cation vacancy and corresponding creation of an anion or cation interstitial.

Dislocation Defects (1D)

Imperfections that are localized along a space curve passing through an ordered medium are line imperfections. A dislocation is one such defect. A dislocation is a linear or one-dimensional line defect which involves translation of one part of a crystal with respect to another part. A disclination involves rotation of one part with respect to another. We will deal with two types of dislocation in this class: edge and screw dislocations.

An edge dislocation is essentially an extra half plane of atoms that is inserted into the perfect crystal and it terminates within the crystal. You can see this below

A screw dislocation is a bit harder to visualize but you can imagine that a screw dislocation can be formed by an applied shear stress, seen below:

You can see that in this example some of the differences between edge and screw dislocations—one being that the screw dislocation motion will be perpendicular to the applied force but more on that in a little bit. There are also wedge and twist disclinations as well.

Associated with each dislocation is a dislocation core which is where the largest displacements of atoms from the ideal sites occur and they are concentrated along a dislocation line. We describe dislocations via a unit tangent vector \( t \), and the Burgers vector \( b \). The tangent vector is a unit vector that is tangential to the dislocation line at any given point. The Burgers vector \( b \) is defined in reference to the Burgers circuit. Now in this class we will use the SF/RH (start to finish, right-hand) convention to define the Burgers vector.

To define the Burgers vector:

1. Choose the positive sense of the unit tangent vector of the dislocation line.
2. By traveling along rows of lattice points make a right-hand circuit in the crystal containing the dislocation. Define a Start Point (S) and Finish Point (F).
3. This circuit would be a closed circuit in the perfect crystal but in this instance the circuit is only closed by the Burgers vector which connects your Start and Finish points.

A couple of other quick notes about some of the properties of dislocations:

- The Burgers vector will be conserved and that for a given dislocation there is one constant Burgers vector, even as the tangent vector changes.
- A dislocation cannot end inside a crystal.
- A pure edge dislocation has \( b \) perpendicular to \( t \) everywhere along the dislocation curve and \( t \times b \) points toward the extra half plane of atoms.
- A pure screw dislocation is either parallel or antiparallel to \( t \) everywhere along the dislocation curve. A right-hand screw is parallel and a left-hand screw is anti-parallel.
- A dislocation that is neither pure edge or pure screw is a mixed dislocation and the vector component that is parallel to \( t \) is the screw component and the component that is perpendicular to \( t \) is called the edge component.
- Reversing the sense of \( t \) also reverses the sense of \( b \).

Edge dislocation:

Right-hand screw dislocation:

Left-hand screw dislocation:

While we are speaking of dislocations one of the key material properties that you will come across is the dislocation density \( \rho \), which has units of \( m^{-2} \). A highly cold worked material may have a dislocation density of \( 10^{12} \text{ cm}^{-2} \) whereas a highly annealed material can be as low as \( 10^{5} \text{ cm}^{-2} \). Dislocations are motile and they can move via slip and climb.

It is often the case that a dislocation will not be either pure edge, left, or right hand screw. Instead a dislocation will be mixed having some component that is edge or screw.

Grain Boundaries, External Surfaces, Phase Boundaries, Twin Boundaries, and Stacking Faults (2D)

Perhaps the most obvious 2D defect is just an external surface where the bonds at the surface are not satisfied which gives rise to an increase in surface energy or surface tension.

Grain Boundaries are interfaces at which crystals of different orientations abut. There are special types of grain boundaries like low-angle tilt and twist boundaries when the angle of misorientation of grains is not too large. In fact they are considered simple periodic arrays of dislocations.

A twin boundary is a special type of grain boundary across which there is a specific mirror lattice symmetry. The region of material between these boundaries is a twin. Twins typically appear due to shear stresses or during annealing heat treatments after deformation. Annealing twins are more common in FCC crystals while mechanical twins are more common in BCC and HCP crystals.

Stacking faults are interfaces in crystals across which one part of the crystal is displaced relative to the other part by a displacement vector that is not a translational symmetry operation for that crystal. Remember that for FCC we had this ABCABC stacking and HCP we had ABABAB. A stacking fault in FCC would be ABCBCABCABC. If the fault is missing a layer that is an intrinsic stacking fault but if there is an extra layer that is an extrinsic stacking fault. Stacking fault energy has severe implications on the strain hardening behavior of materials.

There are also antiphase/interphase boundaries which are separate regions of the crystal by a relative translation that is not a symmetry operation of the crystal. It is a special type of stacking fault, one that connects the crystallographically nonequivalent occupied sites in a perfect crystal. They can be coherent, semi-coherent, or incoherent.

Bulk or Volume Defects (3D)

Finally there are also bulk or volume defects which include pores, cracks, foreign inclusions, and other phases.