Question

Explain the role that diffusion plays in both diffusion creep and dislocation creep; also explain how these types of creep can be designed against.

Answer 1

Sol:-   The role that diffusion plays in  Diffusion Creep :

Diffusion creep refers to the deformation of crystalline solids by the diffusion of vacancies through their crystal lattice.Diffusion creep results in plastic deformation rather than brittle failure of the material

Diffusional creep is an important mechanism of plastic deformation in a polycrystalline material at relatively low stress and small grain size. There is evidence that diffusional creep plays an important role in some regions of Earth. At high temperatures, atoms move from their stable positions with some probability due to thermally activated processes. This is referred to as diffusion. The driving force for diffusion is the gradient in chemical potential including the concentration gradient caused by the contact of materials with different chemical compositions or by the stress gradient at grain boundaries created by the applied stress. Consequently, the rate of deformation due to diffusive mass transport is sensitive to diffusion coefficient as well as grain size: the rate of deformation is faster for a smaller grain size. Similar to other processes, diffusional mass transport involves a number of parallel (independent) and sequential (dependent) processes. As a result, the interplay of various diffusing species can be complicated and this also results in a complicated variation in grain-size sensitivity with grain size. Deformation of a polycrystalline material is associated with grain boundary sliding. Large-strain plastic flow involving grain-boundary sliding is sometimes referred to as superplastic flow.

Principle

Crystalline materials are never perfect on a microscale. Some sites of atoms in the crystal lattice can be occupied by point defects, such as "alien" particles or vacancies. Vacancies can actually be thought of as chemical species themselves (or part of a compound species/component) that may then be treated using heterogeneous phase equilibria. The number of vacancies may also be influenced by the number of chemical impurities in the crystal lattice, if such impurities require the formation of vacancies to exist in the lattice.

A vacancy can move through the crystal structure when the neighbouring particle "jumps" in the vacancy, so that the vacancy moves in effect one site in the crystal lattice. Chemical bonds need to be broken and new bonds have to be formed during the process, therefore a certain activation energy is needed. Moving a vacancy through a crystal becomes therefore easier when the temperature is higher.

The most stable state will be when all vacancies are evenly spread through the crystal. This principle follows from Fick's law:

In which J_x stands for the flux ("flow") of vacancies in direction x; D_x is a constant for the material in that direction and is the difference in concentration of vacancies in that direction. The law is valid for all principal directions in (x, y, z)-space, so the x in the formula can be exchanged for y or z. The result will be that they will become evenly distributed over the crystal, which will result in the highest mixing entropy.

When a mechanical stress is applied to the crystal, new vacancies will be created at the sides perpendicular to the direction of the lowest principal stress. The vacancies will start moving in the direction of crystal planes perpendicular to the maximal stress.Current theory holds that the elastic strain in the neighborhood of a defect is smaller toward the axis of greatest differential compression, creating a defect chemical potential gradient (depending upon lattice strain) within the crystal that leads to net accumulation of defects at the faces of maximum compression by diffusion. A flow of vacancies is the same as a flow of particles in the opposite direction. This means a crystalline material can deform under a differential stress, by the flow of vacancies.

Highly mobile chemical components substituting for other species in the lattice can also cause a net differential mass transfer (i.e. segregation) of chemical species inside the crystal itself, often promoting shortening of the rheologically more difficult substance and enhancing deformation.

Types of diffusion creep

Diffusion of vacancies through a crystal can happen in a number of ways. When vacancies move through the crystal (in the material sciences often called a "grain"), this is called Nabarro–Herring creep. Another way in which vacancies can move is along the grain boundaries, a mechanism called Coble creep.

When a crystal deforms by diffusion creep to accommodate space problems from simultaneous grain boundary sliding (the movement of whole grains along grain boundaries) this is called granular or superplastic flow. Diffusion creep can also be simultaneous with pressure solution. Pressure solution is, like Coble creep, a mechanism in which material moves along grain boundaries. While in Coble creep the particles move by "dry" diffusion, in pressure solution they move in solution.

  The role that diffusion plays in  Dislocation Creep

Dislocation creep is a deformation mechanism in crystalline materials. Dislocation creep involves the movement of dislocations through the crystal lattice of the material, in contrast to diffusion creep, in which diffusion (of vacancies) is the dominant creep mechanism. It causes plastic deformation of the individual crystals, and thus the material itself.

Dislocation creep is highly sensitive to the differential stress on the material. At low temperatures, it is the dominant deformation mechanism in most crystalline materials.Some of the mechanisms described below are speculative, and either cannot be or have not been verified by experimental microstructural observation.

Principle

Dislocations in crystals :

Dislocation creep takes place due to the movement of dislocations through a crystal lattice. Each time a dislocation moves through a crystal, part of the crystal shifts by one lattice point along a plane, relative to the rest of the crystal. The plane that separates the shifted and unshifted regions along which the movement takes place is the slip plane. To allow for this movement, all ionic bonds along the plane must be broken. If all bonds were broken at once, this would require so much energy that dislocation creep would only be possible in theory. When it is assumed that the movement takes place step by step, the breaking of bonds is immediately followed by the creation of new ones and the energy required is much lower. Calculations of molecular dynamics and analysis of deformed materials have shown that deformation creep can be an important factor in deformation processes.

By moving a dislocation step by step through a crystal lattice, a linear lattice defect is created between parts of the crystal lattice. Two types of dislocations exist: edge and screw dislocations. Edge dislocations form the edge of an extra layer of atoms inside the crystal lattice. Screw dislocations form a line along which the crystal lattice jumps one lattice point. In both cases the dislocation line forms a linear defect through the crystal lattice, but the crystal can still be perfect on all sides of the line.

The length of the displacement in the crystal caused by the movement of the dislocation is called the Burgers vector. It equals the distance between two atoms or ions in the crystal lattice. Therefore, each material has its own characteristic Burgers vectors for each glide plane.

Glide planes in crystals :

Both edge and screw dislocations move (slip) in directions parallel to their Burgers vector. Edge dislocations move in directions perpendicular to their dislocation lines and screw dislocations move in directions parallel to their dislocation lines. This causes a part of the crystal to shift relative to its other parts. Meanwhile, the dislocation itself moves further on along a glide plane. The crystal system of the material (mineral or metal) determines how many glide planes are possible, and in which orientations. The orientation of the differential stress determines which glide planes are active and which are not. The Von Mises criterion states that to deform a material, movement along at least five different glide planes is required. A dislocation will not always be a straight line and can thus move along more than one glide plane. Where the orientation of the dislocation line changes, a screw dislocation can continue as an edge dislocation and vice versa.

Origin of dislocations :

When a crystalline material is put under differential stress, dislocations form at the grain boundaries and begin moving through the crystal.

New dislocations can also form from Frank–Read sources. These form when a dislocation is stopped in two places. The part of the dislocation in between will move forward, causing the dislocation line to curve. This curving can continue until the dislocation curves over itself to form a circle. In the centre of the circle, the source will produce a new dislocation, and this process will produce a sequence of concentric dislocations on top of each other. Frank–Read sources are also created when screw dislocations double cross-slip (change slip planes twice), as the jogs in the dislocation line pin the dislocation in the 3rd plane

Design of material in Diffusion Creep

The constitutive equation for diffusional creep was derived from a simple cubic single crystal. In the diffusional creep of polycrystals, atoms are transported from the grain boundaries subjected to compressive stresses to those subjected to tensile stresses. This leads to a change in shape of creeping grains. This change in grain shape must be accommodated by grain boundary sliding (GBS). If GBS does not occur, voids would form on the grain boundaries subjected to compressive stresses.

The role of GBS in diffusional creep differs principally from that in dislocation creep. In the former, GBS is an indispensable requisite for diffusional creep, while in the latter-GBS does not need to occur at all if a sufficient number of slip systems operates, that is, every grain in a polycrystalline material can deform freely without any grain boundary void formation.

Raj and Ashby analyzed the relation between GBS and diffusional creep. They considered an idealized polycrystal in which GBS can occur in two systems of grain boundaries perpendicular to each other, as shown in Fig. 9.2. The simplified case is shown in Fig. 9.3. When the waved boundary slides under the action of the shear stress from the solid line to the dashed line, diffusion of matter must occur along the arrow direction, otherwise void and/or overlap of matter will be produced as shown in Fig. 9.3.

9.2. Grain boundary sliding in two grain boundary systems in an ideslized polycrystal^[4].

9.3. The model of waved boundary along which the GBS occur

For an idealized polycrystal consisting of hexagonal grains (Fig. 9.2), the profile of any grain boundary system can be described by the Fourier series.

For the case of a waved boundary, the profile of the grain boundary can be simplified (by taking first term of the Fourier series) as

x=h/2cos(2Π/λ)y   (9.17)

The applied shear stress τ induces a normal stress σ_n acting on the boundary, which is described by

σn=−2τλ/Πhsin(2Π/λ)y (9.18)

where h₀ is the amplitude of the wave. Since only the grain boundaries act as the source and sink of vacancies and in steady state the chemical potential gradient is time independent, the chemical potential of vacancy µ(x, y) in any grain satisfies the equation,

∇²µ(x,y)=0 (9.19)

The chemical potential is related to stress. Then equation (9.19) is solved for proper boundary conditions and the following equation for GBS rate is obtained

u=8/Π(τΩ/kT)λ/h²Deff (9.20)

where h= d/2, λ= d and D_eff is the effective diffusion coefficient given by

Deff=D(1+(Πδ_b/λ)(D_B/D)) (9.21)

The shear strain rate by GBS is then

γ˙=2u˙d   (9.22)

When the two grain boundary systems contribute to the shear strain, the strain rate is expressed as

γ˙=Bγ(1/d²)τΩ/kTDeff (9.23)

Equation (9.23) is completely equivalent to Eq. (9.13), which implies that grain boundary sliding accommodates diffusional creep. Diffusional creep can be seen as a process of diffusion accommodated by grain boundary sliding, or, adversely, as a process of grain boundary sliding accommodated by diffusion.

Design of material in Dislocation Creep

Kinetics:

In general, the power law for stage 2 creep is:

where is the stress exponent and is the creep activation energy, is the ideal gas constant, is temperature, and is a mechanism-dependent constant.

The exponent describes the degree of stress-dependence the creep mechanism exhibits. Diffusional creep exhibits an of 1 to 2, climb-controlled creep an of 3 to 5, and glide-controlled creep an of 5 to 7

Similarly, the backward rate:

The total creep rate is as follows:

Thus, the rate of creep due to dislocation glide is:

At low temperatures, this expression becomes:

The energy supplied to the dislocation is:

where is the applied stress, is the Burgers vector, and is the area of the slip plane.

Thus, the overall expression for the rate of dislocation glide can be rewritten as:

Thus, the numerator is the energy coming from the stress and the denominator is the thermal energy.This expression is derived from a model from which plastic strain does not devolve from atomic diffusion

The creep rate is defined by the intrinsic activation energy ( ) and the ratio of stress-assisted energy () to thermal energy . The creep rate increases as this ratio increases, or as stress-assisted energy increases more than thermal energy. All creep rate expressions have similar terms, but the strength of the dependency (i.e. the exponent) on internal activation energy or stress-assisted energy varies with the creep mechanism.

Dislocation creep (giving power-law creep)

The stress required to make a crystalline material deform plastically is that needed to make the dislocations in it move. Their movement is resisted by (a) the intrinsic lattice resistance, and (b) the obstructing effect of obstacles (e.g., dissolved solute atoms, precipitates formed with undissolved solute atoms, or other dislocations). Diffusion of atoms can “unlock” dislocations from obstacles in their path, and the movement of these unlocked dislocations under the applied stress is what leads to dislocation creep.

Figure 23.1 shows a dislocation that cannot glide because a precipitate blocks its path. The glide force τb per unit length is balanced by the reaction f₀ from the precipitate. But unless the dislocation hits the precipitate at its mid-plane (an unlikely event) there is a component of force left over. It is the component τb tan θ, which tries to push the dislocation out of its slip plane.

Figure 23.1. The climb force on a dislocation.

The dislocation cannot glide upward by the shearing of atom planes—the atomic geometry is wrong—but the dislocation can move upward if atoms at the bottom of the half-plane are able to diffuse away (Figure 23.2). We have come across Fick's law in which diffusion is driven by differences in concentration. A mechanical force can do exactly the same thing, and this is what leads to the diffusion of atoms away from the “loaded” dislocation, eating away its extra half-plane of atoms until it can clear the precipitate. The process is called “climb,” and since it requires diffusion, it can occur only when the temperature is above 0.3T_M or so. At the lower end of the creep régime (0.3–0.5T_M) core diffusion tends to be the dominant mechanism; at the higher end (0.5T_M–0.99T_M) it is bulk diffusion (Figure 23.2).

Figure 23.2. How diffusion leads to climb.

Climb unlocks dislocations from the precipitates that pin them and further slip (or “glide”) can then take place (Figure 23.3). Similar behavior takes place for pinning by solute, and by other dislocations. After a little glide, of course, the unlocked dislocations bump into the next obstacles, and the whole cycle repeats itself. This explains the progressive, continuous nature of creep. The role of diffusion, with diffusion coefficient

Figure 23.3. How the climb-glide sequence leads to creep.

D=D_oe^-Q/RT

explains the dependence of creep rate on temperature, with

ɛɛ_ss = Aσⁿe^-Q/RT (23.1)

The dependence of creep rate on applied stress σ is due to the climb force: the higher σ, the higher the climb force τb tan θ, the more dislocations become unlocked per second, the more dislocations glide per second, and the higher is the strain rate.