Lecture 26: Crystallization

PHYS 430/603 material

Laszlo Takacs

UMBC Department of Physics

Nucleation

Heterogeneous nucleation: Nuclei form at pre-existing surfaces, so that little extra surface is created, the energy barrier is small. The most typical places for heterogeneous nucleation are the wall of the container and high-melting particles present in the melt. Temperature independent.

Homogeneous nucleation: Nuclei form due to the random motion of atoms in the melt. Increases with lowering temperature, dominates far below the melting point.

In order to achieve large undercooling, heterogeneous nucleation has to be avoided. The best is a small droplet with “no room” for a seed particle, levitated freely in a rf magnetic field. In industrial settings only a few degrees of undercooling take place, but ~15% of Tm is possible in the laboratory.

Nucleation and crystallization can be avoided with fast cooling between Tm, where the formation of nuclei can start, to T0, where diffusion becomes negligible.

The nucleation rate

Crystallization - can be avoided, by fast enough cooling to avoid the nucleation line. This is how metallic glasses are made.

Finish

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A deep eutectic point often makes the formation of a glassy phase possible. Ni-P is an interesting system because it can also be made amorphous by mechanical alloying and Ni-P coatings deposited by electrochemical methods can also be amorphous.

STM images of a crystalline Zr and a glassy Zr-Ni-Al-Cu alloy

Mechanical property comparison for a bulk metallic glass

Properties Vitreloy 1* Aluminum Alloys

Titanium Alloys

Steel Alloys

Density (g/cm 3) 6.0 2.6-2.9 4.3-5.1 7.8

Yield Strength (GPa)

1.90 0.10-0.63 0.18-1.32 0.50-1.60

Elastic Strain Limit

2% ~ 0.5% ~ 0.5% ~ 0.5%

Fracture Toughness (MPa

m1/2) 20-140 23-45 55-115 50-154

Specific Strength

(GPa/g/cm 3) 0.32 < 0.24 < 0.31 < 0.21

* Composition: Zr: 41.2 Be: 22.5 Ti: 13.8 Cu: 12.5 Ni: 10

Notice the very competitive properties and the uniquely high elastic limit.

• Stability and phase transformation - always of interest in the case of metastable

materials.

Crystallization of Fe(80)B(20) glass Fe + Fe4B Fe + Fe3B Fe + Fe2B

It takes place in several steps, with the formation of simpler (thus easier-to-nucleate)

but still metastable intermediate phases.

• Magnetism - promising for soft magnetic material: no crystal structure, no

magnetocrystalline anisotropy; stress sensitivity (anisotropy due to magnetostriction)

can be minimized by varying the composition

E.g.: (Fe1-xCox)75Si15B10 shows zero magnetostriction at about x = 0.9

• Even if rapid quenching from the melt does not result in a glassy phase, the first phase

to form is not the most stable one but the one that nucleates the most easily. Quite

often metastable crystalline compounds form. An interesting case is quasicrystals,

alloys that have no translational periodicity but possess five-fold rotational symmetry.

• Nucleation is also an important component of solid-solid phase transformations e.g.

during recrystallization.

“Ordinary” crystallization at moderate cooling rates:The role of heat flow during solidification

Heat flow is an important component of solidification. Heat has to be conducted away to lower the temperature to below the melting point. Solidification is an exothermic process, the latent heat has to be take away also. Heat balance of dx:

heat flow into crystal - heat flow from liquid = latent heat

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λCdT

dx

⎛

⎝ ⎜

⎞

⎠ ⎟C

− λ LdT

dx

⎛

⎝ ⎜

⎞

⎠ ⎟L

= hS v

Heat flows toward the (colder) solid: stable solidification front

Heat flows toward the liquid (colder due to undercooling and latent heat): instability

Undercooling and the warming from solidification can lead to inverse temperature gradient even if the melt is solidifying in a cold container. The resulting instability leads to the formation of dendrites - a very common phenomenon, not a rare occurrence.

The mechanism behind crystal habit

These Wulff diagrams show the direction dependence of the surface energy and the resultant external shape of the crystal. The lowest energy faces grow the fastest during crystallization. This is the reason behind crystal habit, the most obvious external feature of crystals. Historically, crystallography developed from the study of habit way before the existence of atoms had been proven.

The crystallization of alloys:

1. Fast diffusion in both S & L; system is always in equilibrium.2. Fast diffusion in L, little in S; coring3. Slow diffusion in L & S; constitutional supercooling, dendrites.Solidification results in concentration differences.

Initial Sn concentration is 23 at.%. On cooling:1. Pb-Sn(12%) crystallizes first.2. The (uniform) Sn content of the liquid increases. The concentration of the solid

also shifts, the (Pb) phase develops coring.3. The liquid reaches the eutectic point, the solid is Pb-Sn(29%).4. Simultaneous crystallization of Pb-Sn(29%) and Sn-Pb(1.4%) usually in a

lamellar structure.

Zone melting

Suppose we have a PbSn(23 at.%) rod, melt a short section at the left end and move the molten region (the heater) to the right. The Sn content of the left end will be only 12%; the Sn will move to the right.Repeating the process several times purifies the left end and concentrates the Sn (or any other impurity) on the right.This is one of the most important methods of material purification (electro-refining is another.)

Solidification in a mold

Heat flow, cooling rate, variation of impurity concentration determines the micro-structure of cast metals:

1. Chill zone, fast cooling fast nucleation, many small grains.2. Columnar growth in the direction of the heat flow. Only grains with low-

energy face in the right direction grow.3. Impurities are swept toward the middle, more random nucleation and the

formation of equiaxed grains can take place.4. Volume decrease results in a shrinkage pipe.

Nucleation in the solid state

Most transformations in the solid state - such as precipitation - begin with nucleation also.Interface energy is the smallest for coherent boundaries, larger for semi-coherent boundaries, the largest for incoherent phase boundaries. A phase with low interface energy can form, even if it is not the phase with the lowest free energy.Other factors: Direction dependence of the interface energy.

Volume change and related elastic energy.

Spinodal decomposition

Consider the free energy of a two-component alloy system that shows phase separation in equilibrium.

If it is cooled very quickly from the melt (quenched) solid solution may be obtained with a concentration outside the equilibrium solubility range.

At a concentration where the G(C) curve is convex from below, e.g c1, decomposing the solid solution to two regions with slightly different concentrations increases G, it will not happen. (Notice that the equilibrium state at c1 is a two-phase state.)

But at a concentration where the G(C) curve is concave from below, e.g c3, decomposing the solid solution to two regions with slightly different concentrations decreases G. There is a driving force for phase separation by gradual change of concentration distribution. Usually a lamellar microstructure develops - Gunier-Preston zones.

For a regular solution with HAA = HBB, the Gibbs free energy as a function of concentration is:

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G =1

2Nz HAA +2c(1− c)H0[ ] + NkT c lnc+(1− c)ln(1− c)[ ]

Spinodal decomposition is possible between the inflection points - the zeros of the second derivative:

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cw (1− cw ) =kT

2zH0

Decomposition is energetically favorable anywhere between the two end-points of the common tangent. But outside the spinodal range, it can only start with nucleation.

Nucleation versus spinodal decomposition