1. 27-302 Lecture 10 Fall, 2002 Prof. A. D. Rollett
Microstructure-Properties: II Particle Pinning, Grain Growth, Hard Magnets
27-302
Lecture 10
Fall, 2002
Prof. A. D. Rollett

2. Materials Tetrahedron
Processing
Performance
Microstructure
Properties

3. Objective
The objective of this lecture is to explain how the presence of second phase particles leads to pinning of grain and (magnetic) domain boundaries. This effects sets an upper limit to grain size during grain growth (although not necessarily recrystallization).
Magnetic hardness is also sensitive to the same effect because domain walls are also pinned by particles.

4. References
Phase transformations in metals and alloys, D.A. Porter, & K.E. Easterling, Chapman & Hall, p
Magnetism and Metallurgy of Soft Magnetic Materials, Chen, Dover.
Materials Principles & Practice, Butterworth Heinemann, Edited by C. Newey & G. Weaver.
Bozorth, R. M. (1951). Ferromagnetism. New York, IEEE Press.

5. Notation r particle radius R grain size (or domain size)
Rmax maximum (limiting) grain size (or domain size)
f or vf volume fraction (of particles)
g grain boundary energy (or domain wall energy)
q angle made by boundary at pinning point
Pdrag drag force
Is Saturation magnetization
∆E Energy trap
∆U Change in energy of system
µ0 Permeability of free space
NV Number density of voids
NA Voids per unit area of domain wall
BHmax “BH product” - induction times field
HC Coercivity

6. Basics E Why do particles have a pinning effect?
Answer: once a boundary has intersected with a particle, a certain amount of boundary area is removed from the system. In order for the boundary to move off the particle, the “missing area” must be re-created. This restoration of boundary area requires an energy increase. Through the principle of virtual work, this requires a force. E

7. Analogy to Pinning Effect
Remember how to blow bubbles?
You take a ring on the end of a stick, dip the ring into a soap solution to get a film inside the ring, and then blow bubbles out of the ring.
The soap+water film “sticks” to the ring for the same reason as a grain boundary (or domain boundary) sticks to a particle: it is simply trying to minimize its surface area.

8. Boundary-particle interaction
The drag effect of the particles can be quantified by considering a force balance at the (immovable) particle surface.
Length of boundary attached to a particle = 2πr cos q.
Force per unit length exerted by boundary on particle = g sin q.
Total force = length * force.length-1.
For q=45°, the reaction force on the boundary is at a maximum.
Maximum force = πgr
grain growth pressure

9. Drag pressure
To find the point at which grain growth stagnates, we have to equate the driving force (pressure, really) to the drag pressure.
We cannot equate a per-particle force to a pressure, so we must make an assumption about the fraction of boundary area that intersects with particles.
Zener’s assumption was that the boundaries can be assumed to intersect randomly with the particles (not always true, but a good place to start!).
Stereology (again!): the fraction of particles with radius r and volume fraction f that intersect unit area of a random oriented section plane is 3f/2πr2.
Multiply the maximum force per particle by the number per unit area of boundary to obtain the drag pressure: Pdrag = πgr * 3f/2πr2 = 3fg/2r

10. Stagnation of grain growth
The point at which grain growth will stop is (approximately) determined by a balance between the driving force (pressure) for grain growth, g/D, and the drag force (pressure). 3fg/2rparticle = g/Rstagnation = g/Rmax  Rmax ≤ 2r/3f, or, Dmax ≤ 4r/3f (Zener-Smith Equation)
Note that Underwood gives a more precise analysis and arrives at a larger limiting grain size, Rmax ≤ 4r/3f which illustrates the approximate nature of the derivation given.

11. Grain growth kinetics D2-D02t
The effect of the presence of fine particles is to slow down,and eventually stop grain growth.
D2-D02t

12. Pinning: technological impact
The technological impact of particle pinning is considerable.
Most commercial structural materials, especially for elevated temperature service, rely on fine second phase particles to maintain a fine grain size.
Limitations: particle dispersions are less effective when not stable, or not fully stable.
A common observation: abnormal grain growth frequently occurs in materials annealed close to a solvus, where a particle dispersion is barely stable.

13. Abnormal grain growth
Abnormal grain growth is defined visually by the growth a small fraction of grains to sizes much larger than the matrix structure. More precisely, it is marked by the appearance of a bi-modal grain size distribution (a more severe test!).
See Rollett & Mullins for theory, Scripta metall., (1996).
[Martin, Doherty, Cantor]

14. Abnormal grain growth applications
Abnormal grain growth, although often undesirable, is occasionally crucial for technological materials.
Transformer steels, Fe-3Si, depend on abnormal grain growth, allied with texture, to develop extremely strong Goss texture, {110}//ND, <001>//RD.
Pole Figure Analysis
100 pole
Sample

15. Kinetics of Abnormal gr gr
Particles play an important, if little understood role in this process.
Without particles to pin the matrix, grain growth is normal (self similar).
With particles, the matrix coarsens but only slowly; a small fraction of grains are able to grow much faster than all others and eliminate all others.

16. Abnormal grain growth microstructure
Abnormal grain growth is dramatically obvious in Fe-Si steels.
The {110}<001> grains are much larger than the matrix grains.
Pinning of the matrix grains can be achieved with either MnS or AlN particles
[Chen]

17. Textured Fe-Si, magnetic properties
The texture of an iron crystal has a strong effect on its magnetic properties.
Goss texture
polycrystal
Cube texture
[Chen]

18. Magnetic Domain Wall Pinning
In at least one important respect, particle pinning of boundaries is universal in application.
The hardness of a magnetic material depends on microstructural features to prevent motion of domain walls. Particles fill this requirement.
[Bozorth]

19. Obstacles to domain wall motion
Anything that interacts with a domain wall will make moving it more difficult. For example, a second phase particle will require some extra driving force in order to pull the domain wall past it.
Domain wall motion
particle

20. Domain Wall obstacles
A more detailed look at what is going on near particles reveals that magnetostatic energy plays a role in forcing a special domain structure to exist next to a [non-magnetic] particle.
Domain wall motion
[Electronic Materials]

21. Voids and domain walls Domain wall motion S S S S S S N N r N N N N
When a void intersects a domain wall, the free magnetic poles on the surface are more nearly balanced, with a consequent reduction in energy.
Domain wall motion
S S S S
S S N N r
N N N N
N N S S
(a)
(b)

22. Pinning of domain walls
The pinning effect can be estimated. Following Chikazumi, pp , we use voids to make this estimate, based on the magnetostatic energy at the surfaces of the voids (and the reduction from having a domain wall intersect a void).
Note that Chikazumi’s stereology is incorrect!
The magnetostatic energy associated with the magnetic dipole in fig (a) above, is given as: U = 0.5 {Is2/3µ0} {4πr3/3} where Is is the saturation magnetization (e.g. =2.15 Tesla for Fe).

23. Pinning, contd.
Once a domain wall intersects with the void, the magnetostatic energy is approximately halved, so ∆U = 0.25 {Is2/3µ0} {4πr3/3}
Next we multiply the energy per void by the number of voids per unit area of domain wall to obtain the magnitude of the energy trap, ∆E. In stereological terms, this is notated as NA, where NA = 2r NV where NV is the number of particles per unit volume.

24. Pinning, contd.
The number per unit volume is simply related to the volume fraction (of voids): f = 4πNVr3/3
Thus we can estimate the depth of the energy trap: ∆E = ∆U*NA = 0.25 {Is2/3µ0} {4πr3/3} 2r NV = 0.25 {Is2/3µ0} {4πr3/3} 2r 3f/4πr3 = 0.25 {Is2/3µ0} 2fr
Thus, larger volume fractions and size give more effective pinning.

25. Voids  Domain walls
The presence of voids (or particle) can also lead to the existence of domain walls, simply in order to minimize internal magnetostatic energy.
If the energy per unit area of the “energy trap” is greater than the energy per unit area of the domain wall, then a domain wall is energetically favored to exist.
Chikazumi gives an example for Fe with voids of diameter 10 µm, where the energy of a 180° domain wall aligned on {100} is 1.6 mJ.m-2: ∆U = 0.25*2.152*/3/4π.10-7*4*π/3*(10-5)3= J So, if ∆E = ∆U*NA > g, then domain walls are stabilized.

26. Voids  Domain walls. contd.
Thus if, {Is2/3µ0} 2fr > g, then domain walls are stabilized, or, 0.25 * */3/4π.10-7 * * f > So, if f > , or, l ≈ /3/ = 53 mm, then voids stabilize domain walls, which is not a large void fraction (and particles would be approximately as effective).

27. Alnico alloys The Alnico alloys are a classic hard magnetic alloy.
Mishima’s invention (1932) of a strong magnet “steel” with Al, Ni and Co.
Homogeneous solid solution > 1300°C.
Phase separation occurs below 900°C by spinodal decomposition.
Precipitates are Fe + Co rich and ferromagnetic
Matrix is Al + Ni rich and only weakly ferromagnetic.
Annealing in a magnetic field aligns the particles.
Magnetic interaction between the particles is important.

28. Permanent magnets - BH product
Remanence := point (3) Coercivity := HC
[Bozorth]

29. Steels- magnetic properties
The magnetic properties of steels follows the same pattern as for mechanical properties.
Annealing removes carbon from solution which increases permeability.
Increasing carbon content decreases permeability and increases coercivity because of increasing volume fraction of carbide particles.
[Bozorth]

30. Magnetic props. of Fe-C, contd.
Note how all the parameters relevant to permanent magnets become “harder” as the carbon content increases.
The quenched condition (with martensite) is the hardest microstructural state.
Martensite has a highly strained structure, which is very effective in pining domain walls (not discussed here).
[Bozorth]

31. Direct magnetic-mechanical hardness comparison
Direct comparison of mechanical hardness with the coercive force (single crystals of Fe) shows high degree of similarity.
In this case, the hardening defect is the dislocation population. [Chen]

32. Hard magnets
In recent years, a variety of structures based on Sm and Nd have been developed with improved magnetic hardness.
The most critical parameter for most applications is the energy product, BHmax.
[Graham: in Microstructure & Properties of Materials – I, ed. J.C.M. Li]

33. Summary
Particle pinning is critical to grain size control, domain walls, etc.
The effect of particles can be quantified with simple (stereological) methods.
The standard equation for predicting grain size controlled by particle pinning: Dmax ≤ 4r/3f (Zener-Smith Equation)