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6. Crystal growth

Before a crystal grows, it has to be formed. This is called nucleation.

Fig. 6-1. The different types of nucleation.

Depending on the supersaturation, the growth rate of a crystal varies. This is explained bythree different types of growth rate laws: The BCF- or spiral growth rate law, forming crystals with polyhedral morphology,the 2DNG- or KSV growth rate law, forming crystals with skeletal or hopper morphologyand continuous growth, with dendritic crystals.Be aware that each facet of a single crystal might grow following a different growth ratelaw in the extreme case.

Fig. 6-2. Growth rate laws as function of supersaturation.

Hopper crystals manly attach new adatoms at the edges of the crystal, explained by anincrease of concentration in this zone.

nucleation

primary secondary

homogeneous heterogeneous(spontaneous) (induced by foreign particles)

(induced by crystals)(solution without any crystalline matter)

vgrowth

σσ´ σ´´

1 2 31

2

3

BCF, polyhedral2DNG (KSV), hopper or skeletalcontinuous growth, dendritic rough interface

smooth interface}

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Fig. 6-3. Growth at the edges of the crystal results in a hopper morphology, caused by an increased supersaturation.

Using the Burton-Cabrera-Frank (BCF), or spiral growth theory, the growth rate R isdefined as

Very low supersaturation allows an attachment of adatoms on the crystal surface, becausea spiral dislocation behaves as a self-perputating step. For the 2-dimensional nulceation growth mechanism, or Kossel-Stranski-Vollmer mecha-nism, again the growth rate correlates non-linear with the supersaturation

Here, a supersaturation is needed. Imagine a flat crystal facets without any dislocation.Where should a adatom attach? If the adatom does not find any hole, kink, step or some-thing else, there won´t be anything it could stick on. Supersaturation needs to be higher tohold that adatom on the flat surface.The growth rate above the roughening transition (dendritic growth) is a linear relationship.

Each adatom apporaching the surface will immediately incorporated. It even has no timefor a walkabout, to check for nice accretion sites (steps, etc.). Above, I mentioned the supersaturation. That is a solution with a concentration aboveequilibrium, such as in your cup of tea. At 60°C you added sugar until first crystals didn´tdissolve. Now you have a equilibrium concentration in the cup. Heat it up till the crystalon the ground dissolves, then let it cool down on your desk. What´s going on in there?Concentration was in equilibrium at 60°C. It became slightly undersaturated while heatingup, and supersaturated as it cools down IF no sugar crystals start to grow. A measruementof the supersaturation is defined as

Berg effect

x-tal

concentration

R AB

R

R

=

∝∝

σσ

σσ

2

2

tanh

at low supersaturation

at high supersaturation

R n∝ σ

R ∝ σ

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with

σ

called the relative supersaturation.The morphology of a crystal might change during growth. This happens when the growthrate law changes.

Fig.6-4. During cooling a single crystal might change its growth morphology from den-dritic to polyhedral. That corresponds with a movement from right to left in the R-

σ

diagram.

The growth rate can be measured as

σ = − = −Sc

cequil

1 1.

polygonal growth bands

lineage structure (dendritic growth)

R

σ´ σ´´ σ

RA

dm

dtG

dL

dt

dr

dtvcrystal crystal crystal crystal

R kgm s

G ms

v ms

m L

A L

= = = = =

− −−

−−

−−

−

− =

=

1 3 3 6 6

2 1

1

1

3

2

61

αβ

ρ αβ

ρ αβ

ρ αβ

ρ

α αρ

β β

α

β

[ ]

[ ]

[ ]

mass deposition rate

overall linear growth rate

mean linear velocity

volume shape factor, defined as = crystal

surface shape factor, defined as

for spheres and cubes, = 0.816 for octahedra

A - area of the crystal

L - size of the crystal

m - mass of crystal

r -- radius of the crystal corresponding to the equivalent sphere

- crystal densityρ

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Exercise

Grow crystals under the microscope in transmitted-light, using supersaturated solutions ofKCl, Na2SO4, NaNO3, CuSO4, KAl(SO4)2x12H2O. Put a droplet of heated solution(50°C) on a glass plate. You can use a hairdryer to increase supersaturation in the droplet.What type of morphology do you see? Do you see fluid inclusion bands in a single crystal? How are they arranged, and why?Do you find examples where the morphology changes in a single crystal, e.g. from Hopperto polyhedral?

Fig. 6-5. Growth experiment using Na2SO4. Nice fluid inclusion bands

Fig. 6-6. Dendritic growth of alum at high supersaturation (left), hopper/polyhedral of sylvine on the right hand side.