Basics of Dislocations

History of Dislocations

• A dislocation is a crystallographic defect, or irregularity, within a crystal structure.
• The theory was originally developed by Vito Volterra in 1905 but the term 'dislocation' was not coined until later by Prof. (Sir) Frederick Charles Frank of the Physics Department at the University of Bristol.
• Until the 1930s, one of the enduring challenges of materials science was to explain plasticity in microscopic terms.
• The shear stress at which neighbouring atomic planes slip over each other in a perfect crystal is typically 20-150 GPa (theoretical shear strength), while shear stresses in the range 0.5 to 10 MPa is observed to produce plastic deformation in experiments.
• Theoretical shear stress was first calculated by Frenkel in 1926.
• In 1934, Egon Orowan, Michael Polanyi and G. I. Taylor, (roughly simultaneously) realized that plastic deformation could be explained by the theory of imperfections (dislocations) in a otherwise perfect lattice

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Dislocations in physiology

Knee dislocation 

Dislocations and theoretical shear strength paradox

• In a perfect crystal, the sliding of one plane past an adjacent plane would have to be a rigid co-operative movement of all the atoms from one position of perfect registry to another.
• Dislocations can move if the atoms from one of the surrounding planes break their bonds and re-bond with the atoms at the terminating edge.
• In effect, a half plane of atoms is moved in response to shear stress by breaking and reforming a line of bonds, one (or a few) at a time.
• The energy required to break a single bond is far less than that required to break all the bonds on an entire plane of atoms at once.

Dislocation motion

Secondary evidences for dislocations

• Crystals in the form of fibers with a diameter of a few μm (‘micro-whiskers’) or even less (‘nanowires’ or ‘nanopillars’) have a very high degree of perfection. When entirely free of dislocations, their strength approaches the theoretical strength.
• Universal acceptance of the existence of dislocations in crystals: Failure of classical theory of crystal growth.
• Perfect crystal having irregular facets growing in a supersaturated vapor: At a low supersaturation, growth occurs by the deposition of atoms on the irregular or imperfect regions of the crystal → formation of more perfect faces consisting of close-packed arrays of atoms.
• Further growth requires the nucleation of a new layer of atoms on a smooth face → more difficult process, and requires approximately 50% supersaturation
• This is contrary to many experimental observations which show that growth occurs readily at a supersaturation of only 1%.
• Presence of dislocations in the crystal during growth could result in the formation of steps on the crystal faces which are not removed by preferential deposition → these steps provide sites for deposition

Dislocation movement

Imaging dislocations

Silicon dislocation: orientation <100>

Silicon dislocation: orientation <111>

Dislocations in motion

Bubble raft experiments

• A bubble raft is an array of bubbles demonstrating materials' microstructural and atomic length-scale behavior by modelling {111} plane of a close-packed crystal.
• Bubble rafts assemble bubbles on a water surface, often with the help of amphiphilic soaps. These assembled bubbles act like atoms, diffusing, slipping, ripening, straining, and otherwise deforming.
• The concept of bubble raft modelling was first presented in 1947 by Sir William Lawrence Bragg and John Nye of Cambridge University's.

Bubble raft experiments

Types of dislocations

• A dislocation moves in response to an applied shear stress in a direction parallel to the Burgers vector.
• Burgers vector was named after Dutch physicist Jan Burgers

Edge Dislocation

In an edge dislocation, localized lattice distortion exists along the end of an extra half-plane of atoms, which also defines the dislocation line.

Screw Dislocation

Screw Dislocation

Mixed dislocations

Burger Vector

• If the same atom-to-atom sequence is made in a crystal containing and the circuit does not close, then the vector required to complete the circuit is the Burgers vector.

Burgers Circuit

• It is essential that the circuit in the perfect crystal passes entirely through 'good' parts of the crystal.
• Burgers circuits taken around other defects, such as vacancies and interstitials, do not lead to closure failures.

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• Two rules are implied by the Burgers circuit construction
• When looking along the dislocation line (positive line sense) the circuit is taken in a clockwise fashion
• Burgers vector is taken to run from the finish to the start point of the reference circuit in the perfect crystal.
• This defines the right-hand/finish-start (RH/FS) convention.

Burgers Circuit of Edge Dislocation

Perfect crystal

Crystal with a dislocation

RHFS: Right Hand Finish to Start convention

Burgers Circuit of Screw Dislocation

Dislocation in high resolution TEM

Burgers Vector & crystal structure

• Burgers vector of a dislocation is a crystal vector, specified by Miller indices, that quantifies the difference between the distorted lattice around the dislocation and the perfect lattice.
• Burgers vector denotes the direction and magnitude of the atomic displacement that occurs when a dislocation moves.
• The periodic force field of a crystal requires that atoms must move from one equilibrium position to another.
• Burgers vector must connect one lattice position to another.

• Burgers vectors are the shortest lattice translation vectors which join two points in the lattice.
• A dislocation whose Burgers vector is a lattice translation vector is known as a perfect or unit dislocation.

Close packed plane & direction

• Close-packed planes and directions define the highest atomic density in crystals, critical for slip and plastic deformation. 
• Atomic Packing Factor (APF): FCC and HCP have the highest APF of 0.74

Exercise: Determine the Burgers vector for SC, BCC, FCC crystals

“Close packed volumes tend to remain close packed,�close packed areas tend to remain close packed &�close packed lines tend to remain close packed”

Burgers vectors in cubic crystals

Crystallography determines the Burgers vector

What are the Burgers vectors?

Dislocations in Ionic crystals

• In ionic crystals, if there is an extra half-plane of atoms contained only atoms of one type, then the charge neutrality condition would be violated ⇒ unstable condition
• Burgers vector has to be a full lattice translation
• CsCl → b = <100> Cannot be ½<111>
• NaCl → b = ½ <110> Cannot be ½<100>

CsCl

• This makes Burgers vector large in ionic crystals
• Cu → |b| = 2.55 Å;
• NaCl → |b| = 3.95 Å

Dislocations in Ionic crystals

Diamond cubic (DC)

Source of dislocations

• Thermo-dynamically stable density of dislocations in a stress-free crystal is zero.
• Apart from (a) crystal whiskers and nanowires, and (b) isolated examples in carefully prepared crystals of silicon, dislocations occur in all crystals.

• The dislocation density in well-annealed crystals (crystals which have been heated for a long time close to their melting point) is usually about 10⁴ mm^-2 (10¹⁰ m^-2).
• A similar density of dislocations is present in crystals grown from the melt or produced by strain anneal techniques.
• When annealed crystals are deformed, there is a rapid multiplication of dislocations and a progressive increase in dislocation density with increasing strain.
• After large amounts of plastic deformation, the dislocation density is typically in the range 10¹⁴ to 10¹⁵ m^-2.

Dislocation multiplication: Frank-Reed sources

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• Important source of the new dislocations is existing dislocations, which multiply.

• Dislocations with the same line sense but opposite Burgers vectors (or opposite line senses and the same Burgers vector) are physical opposites.
• If one is a positive edge, the other is a negative edge.
• If one is a right-handed screw, the other is left-handed screw.

Positive edge

Negative edge

Right handed screw

Left handed screw

Dislocation interactions

Dislocation interactions

• The strain fields surrounding dislocations in close proximity interact such that forces are imposed on each dislocation by the combined interactions of all its neighboring dislocations.
• Consider two edge dislocations with the same sign and identical slip plane: compressive and tensile strain fields for both lie on the same side of slip plane
• The strain field interaction is such that there exists, between these two isolated dislocations, a mutual repulsive force that tends to move them apart.

• Two dislocations of opposite sign and having the same slip plane will be attracted to one another, and dislocation annihilation will occur.
• The two extra half-planes of atoms will align and become a complete plane.
• Dislocation interactions are possible between edge, screw, and/or mixed dislocations, and for a variety of orientations.
• The strain fields and associated forces are important in the strengthening mechanisms for metals.

Peierls-Nabbaro (P-N) Stress

• P-N stress is the force needed to move a dislocation within a plane of atoms in the unit cell.
• P-N stress first calculated by Rudolph Peierls and modified by Frank Nabarro
• Peierls was a solid state physicist who had only marginal interest in dislocations: mostly motivated from discussions and interactions with Egon Orowan.
• The PN model assumes that the misfit region of inelastic displacement is restricted to a single plane where the dislocation will glide.
• Linear elasticity applies far from it.
• The aim of the PN model is, within a continuum approach, to describe the extension of the core of a dislocation.

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• The Peierls stress is often described as the critical resolved shear stress (CRSS) at 0K.

Dependence of P-N stress

• The magnitude of P-N stress varies periodically as the dislocation moves within the plane.

Dislocations and slip

• Dislocation is a boundary between the slipped and the un-slipped parts of the crystal lying over a slip plane

Dislocation Motion

Motion of dislocation

Conservative (Glide)

Non-conservative (Climb)

Motion of dislocations

on the slip plane

Motion of dislocation

⊥ to the slip plane

• Glide or conservative motion occurs when the dislocation moves on the plane/surface which contains both line and Burgers vector
• A dislocation able to move by glide is “Glissile”, one which cannot is “Sessile”.
• Climb or non-conservative motion occurs when the dislocation moves out of the glide plane, normal to the Burgers vector.

Dislocation motion

• Dislocation glide under a shear stress must act on the slip plane in the direction of the Burgers vector, irrespective of the direction of the dislocation line

• The slip direction is necessarily always parallel to the Burgers vector of the dislocation responsible for slip

What happens when dislocations move?

Edge Dislocations exiting crystal via glide to form surface steps

The caterpillar or rug-moving analogy

Bubble raft experiment

In-situ TEM experiment

Dislocations of opposite sign glide in opposite directions under the same stress

Motion of Screw Dislocations

• Displacement of atoms (hence the slip step) associated with the movement of a screw dislocation is parallel to the dislocation line, for that is the direction of its Burgers vector.

• A right-handed screw glides towards the front in order to extend the surface step in the required manner whereas a left-handed screw glides towards the back

Formation of steps from Screw and Edge dislocations

• Both screw and edge motion create same steps

• Dislocations operate on only one type of slip system
• Straight slip bands on a single crystal of 3.25% silicon iron

Planar Slip

Single crystal of cadmium

Wavy slip

Climb of dislocations

• At low temperatures where diffusion is difficult, the movement of dislocations is restricted almost entirely to glide.
• At higher temperatures, an edge dislocation can move out of its slip plane by a process called climb.
• The climb processes involve the diffusion of vacancies either towards or away from the dislocation.

• The mass transport involved occurs by diffusion and therefore climb requires thermal activation.

• Pure screw dislocations have no extra half-plane and in principle cannot climb.

• Climb involves addition or subtraction of a row of atoms below the half plane
• Positive climb = climb up → removal of a plane of atoms
• Negative climb = climb down → addition of a plane of atoms
• Positive climb can occur by either diffusion of vacancies or formation of self-interstitials.
• Negative climb can occur either by interstitial diffusion or formation of vacancy.

Cross Slip of screw Dislocation

Geometric properties of dislocations

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