Zener pinning
Encyclopedia
Zener pinning is the influence of a dispersion
of fine particles on the movement of low- and high angle grain boundaries
through a polycrystalline
material. Small particles act to prevent the motion of such boundaries by exerting a pinning pressure
which counteracts the driving force pushing the boundaries. Zener pinning is very important in materials processing as it has a strong influence on recovery
, recrystallization and grain growth
.
. When a boundary passes through an incoherent particle then the portion of boundary that would be inside the particle essentially ceases to exist. In order to move past the particle some new boundary must be created, and this is energetically unfavourable. While the region of boundary near the particle is pinned the rest of the boundary continues trying to move forward under its own driving force. This results in the boundary becoming bowed between those points where it is anchored to the particles.
The figure illustrates a boundary of energy γ per unit area where it intersects with an incoherent particle of radius
r. The pinning force
acts along the line of contact between the boundary and the particle i.e. a circle
of diameter AB = 2πr cosθ. The force per unit length of boundary in contact is γ sinθ. Hence the total force acting on the particle-boundary interface is
The maximum restraining force occurs when θ = 45° and so Fmax = πrγ .
In order to determine the pinning force by a given dispersion of particles, Clarence Zener
made several important assumptions:
For a volume fraction Fv of randomly distributed spherical particles of radius r, the number per unit volume (number density) is given by
From this total number density only those particles that are within one particle radius will be able to interact with the boundary. If the boundary is essentially planar
then this fraction will be given by
Given the assumption that all particles apply the maximum pinning force, Fmax, the total pinning pressure exerted by the particle distribution per unit area of the boundary is
This is referred to as the Zener pinning pressure. It follows that large pinning pressures are produced by:
The Zener pinning pressure is orientation dependent, which means that the exact pinning pressure depends on the amount of coherence at the grain boundaries.
and phase field
simulations have been used in 3D to model the phenomenon. Complex shape of interface can be captured in the computer models. It can provide better approximation for the pinning force.
Colloid
A colloid is a substance microscopically dispersed evenly throughout another substance.A colloidal system consists of two separate phases: a dispersed phase and a continuous phase . A colloidal system may be solid, liquid, or gaseous.Many familiar substances are colloids, as shown in the chart below...
of fine particles on the movement of low- and high angle grain boundaries
Grain boundary
A grain boundary is the interface between two grains, or crystallites, in a polycrystalline material. Grain boundaries are defects in the crystal structure, and tend to decrease the electrical and thermal conductivity of the material...
through a polycrystalline
Polycrystalline
Polycrystalline materials are solids that are composed of many crystallites of varying size and orientation. The variation in direction can be random or directed, possibly due to growth and processing conditions. Fiber texture is an example of the latter.Almost all common metals, and many ceramics...
material. Small particles act to prevent the motion of such boundaries by exerting a pinning pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
which counteracts the driving force pushing the boundaries. Zener pinning is very important in materials processing as it has a strong influence on recovery
Annealing (metallurgy)
Annealing, in metallurgy and materials science, is a heat treatment wherein a material is altered, causing changes in its properties such as strength and hardness. It is a process that produces conditions by heating to above the recrystallization temperature, maintaining a suitable temperature, and...
, recrystallization and grain growth
Grain growth
Grain growth is the increase in size of grains in a material at high temperature. This occurs when recovery and recrystallisation are complete and further reduction in the internal energy can only be achieved by reducing the total area of grain boundary...
.
Origin of the pinning force
A boundary is an imperfection in the crystal structure and as such is associated with a certain quantity of energyEnergy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
. When a boundary passes through an incoherent particle then the portion of boundary that would be inside the particle essentially ceases to exist. In order to move past the particle some new boundary must be created, and this is energetically unfavourable. While the region of boundary near the particle is pinned the rest of the boundary continues trying to move forward under its own driving force. This results in the boundary becoming bowed between those points where it is anchored to the particles.
Mathematical description
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
r. The pinning force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
acts along the line of contact between the boundary and the particle i.e. a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
of diameter AB = 2πr cosθ. The force per unit length of boundary in contact is γ sinθ. Hence the total force acting on the particle-boundary interface is
The maximum restraining force occurs when θ = 45° and so Fmax = πrγ .
In order to determine the pinning force by a given dispersion of particles, Clarence Zener
Clarence Zener
Clarence Melvin Zener was the American physicist who first described the property concerning the breakdown of electrical insulators. These findings were later exploited by Bell Labs in the development of the Zener diode, which was duly named after him...
made several important assumptions:
- The particles are spherical.
- The passage of the boundary does not alter the particle-boundary interaction.
- Each particle exerts the maximum pinning force on the boundary regardless of contact position.
- The contacts between particles and boundaries are completely random.
- The number densityNumber densityIn physics, astronomy, and chemistry, number density is an intensive quantity used to describe the degree of concentration of countable objects in the three-dimensional physical space...
of particles on the boundary is that expected for a random distributionUniform distribution (continuous)In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
of particles.
For a volume fraction Fv of randomly distributed spherical particles of radius r, the number per unit volume (number density) is given by
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
then this fraction will be given by
Given the assumption that all particles apply the maximum pinning force, Fmax, the total pinning pressure exerted by the particle distribution per unit area of the boundary is
This is referred to as the Zener pinning pressure. It follows that large pinning pressures are produced by:
- Increasing the volume fraction of particles
- Reducing the particle size
The Zener pinning pressure is orientation dependent, which means that the exact pinning pressure depends on the amount of coherence at the grain boundaries.
Computer Simulation
Particle pinning has been studied extensively with computer simulations. Monte CarloMonte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
and phase field
Phase field models
A phase field model is a mathematical model for solving interfacial problems. It has mainly been applied to solidification dynamics, but it has also been applied to other situations such as viscous fingering, fracture dynamics, vesicle dynamics, etc....
simulations have been used in 3D to model the phenomenon. Complex shape of interface can be captured in the computer models. It can provide better approximation for the pinning force.