Zener Pinning - Mathematical Description

Mathematical Description

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 F_max = πrγ .

In order to determine the pinning force by a given dispersion of particles, Clarence Zener 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 density of particles on the boundary is that expected for a random distribution of particles.

For a volume fraction F_v 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, F_max, 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.