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## Zener pinning
## Additional recommended knowledge## Origin of the pinning forceA boundary is an imperfection in the crystal structure and as such is associated with a certain quantity of energy. 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
The figure illustrates a boundary of energy γ per unit area where it intersects with an incoherent particle of radius The maximum restraining force occurs when β = 45° and so F In order to determine the pinning force by a given dispersion of particles 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 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 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. ## Notes- According to
*Current issues in recrystallization: a review*, R.D. Doherty et al., Materials Science and Engineering A238 (1997), p 219-274
- For information on zener pinning modeling see:
- "Contribution à l'étude de la dynamique du zener pinning: simulations numériques par éléments finis", Thesis in French (2003). by G. Couturier. |

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Zener_pinning". A list of authors is available in Wikipedia. |