Exploring Crystal Growth: Nucleation, Diffusion, & Energy Minimization

1. Growing Crystalline Materials Jon Price

2. Growth and the construction of defects • How are crystals made? • What types of irregularities are possible? • Why are irregularities so important?

3. States of Matter • Transition between the three states of matter • Gas-solid Condensation • Liquid-solid Precipitation • Crystallization • Solid-solid Transformation

4. The SOLUBILITY is defined as the concentration that is reached in a saturated solution (for T and P). Saturation - the amount of solute going into solution is equal to that which comes out of solution.

5. Oversaturated Crystal grows b/c fewer atoms leave than attach Saturated Crystal unchanged b/c as many atoms leave as attach Undersaturated Crystal dissolves b/c more atoms leave than attach

6. So how does a crystal start growing? Nucleation In a solution, random motions will create crystallite clusters. If undersaturated, the clusters disperse. The solution is oversaturated, clusters hang around, bump into each other, and begin to grow.

7. Nucleation Barrier It takes additional energy to form nuclei. This can limit when and how many crystals form.

8. Growth has started. Next stop: the surface - where all the action is. Unsatisfied bonds Charge distribution upset Incomplete coordination polyhedra Crystalline structure - lowers G Crystalline edges - raises G

9. Controls on external shape What makes a bubble round? Could those same forces work for crystals? What’s the difference between this atom And this one The greater the anisotropy of the structure, the more this is a problem!

10. Which is the more stable configuration of 36 atoms?

11. Hornblende Ca2(Mg, Fe, Al)5 (Al, Si)8O22(OH, F)2 Four crystals growing in melt. Note angular faces typical of the 2/m crystals as seen from the 001 plane. Planes (facets) result from energy minimization along a crystallographic plane - depends on T, P, and X.

12. On this phase diagram, there are two phases, a solid and a liquid. The line represents the conditions where both will be present at equilibrium

13. From Blackburn & Dennen, 1998

15. Crystal growth results from diffusion of components to the crystal surface Crystals can grow in any medium - solids, liquids, gasses, supercritical fluids. Liquids and fluids may be melted rocks, C-O-H or aqueous fluids, or a mixture. All are contingent on component transport.

16. Diffusion In any matter over 0 K, atoms migrate. The rate of movement depends on how well the atoms are bonded. In a gas, atoms or molecules may dance around each other, or switch places.

17. Although diffusion happens everywhere, we can see diffusion in places where atoms are initially separated The atoms will move in random directions. As a consequence, the atoms are no longer in distinct domains. With time, the random movements of the atoms lead to complete random dispersion of the atoms However, if there is a chemical gradient, the diffusion may become directional. This is not to say that the diffusion rate changes.

18. From Blackburn & Dennen, 1998

19. Calcite - step growth LLNL

20. Dendritic growth Corners are higher energy - if diffusion cannot keep up with growth, the corners may grow much more rapidly than the faces. Silver Crystal Potential Face Ice I

21. Several material scientists, like Nikolas Provatas at McMaster are exploring this type of growth numerically This is a really simplified model - but extremely computationally intensive.

23. Grain size General principles The slower the change in conditions, the larger the grain size e.g. - slowly cooled rocks have bigger crystals than ones cooled rapidly Problems: it depends on the material - don’t compare apples and oranges

24. More than one crystal… more than one bubble Image from Smith, 1964

25. In polycrystalline systems, atoms diffuse along the boundaries between crystals. To minimize energy, the chemical potential is to the center of curvature Net result: the boundary moves in the convex direction. Smaller crystals are consumed by larger ones

26. If there is a smaller grain of another insoluble material on the boundary, it resists the movement of the boundary The growing grain exerts a force Fm, the particle exerts a force Fr Equal forces - boundary is pinned Fr<<Fm - particle included Fr<Fm particle is swept

27. The entire system is trying to minimize energy Where three crystals meet, the forces generated by the energy along their boundaries must cancel to reach a minimum value.

28. In three dimensions, grains that minimize their energy have near tetrakeidecahedral shapes

30. An example from an amphibolite Image from Kretz, 1968

31. Myrmekite An intergrowth of quartz and feldspar Likely result of too few nucleation sites Undercooling Viscosity contrasts Rapid diffusion

32. Growth rates for albite crystals in an undercooled silicate melt = 10-6 cm/sec (Fenn, 1977) That’s 10 nm/sec. Compare that to the ionic radii in a SiO2 tetrahedron.

33. Ostwald Ripening Minimizing energy requires that smaller crystals are resolved so that bigger crystals may grow.

34. Reactive growth New minerals may form from recrystallization of reaction of preexisitng grains. Overgrowth, mantling, coronas This is a diffusion driven process. A not so natural example follows Periclase (MgO) + Corundum (Al2O3) reacts to Spinel (MgAl2O4)

35. Prolonged runs at high temperature produce a solid state reaction between MgO and Al2O3, forming a layer of spinel Impetus

36. Growth The width of the spinel layer is linear to the square root of time. Implies a diffusion controlled process.

37. Pressure Growth rate may be parameterized following Tammann (1920) k = (DX2 / 2t) k has the units of diffusivity Apparent Ea= ~410 kJ/mol Apparent Va changes, dependent on T. Growth Constant

38. Stoich. spinel Al enriched EMPA traverses of spinel 1400 oC 4 GPa 89 hr 30 mm Boundary compositions 600 oC 3.2 GPa 16 hr 66 mm 1978 oC 2.5 GPa 0.4 hr 115 mm

39. Ratio of the slopes is always -0.661 (~ -2/3) for all runs Maintains charge balance (Mg 2+ vs. Al 3+) Boundary compositions Formula for the spinel: Mg1-3x, Al2+2x, [_]x, O4

40. Growth requires local oversaturation of the chemical components. Initial crystallization begins with nucleation - energy intensive Post-nucleation growth is controlled by the surface of the phase. Growth is always a trick to reduce the energy of the system Atoms are added to reduce unsatisfied bonds and coordination through diffusion Rapid growth may produce crystals with high surface energy. Adjacent crystals must also minimize their energy