Phase transformation and microstructure formation

What is a phase (in materials science)?

• A phase in material science is a chemically uniform, physically distinct, and mechanically separable region of a material with uniform physical properties (e.g., density, crystal structure, composition).
• Phases are not limited to states of matter (solid, liquid, gas) but can include different solid structures within a single mixture.
• A phase diagram is a graphical representation in physical chemistry and materials science that shows the stable phases (solid, liquid, gas) of a substance under varying temperature and pressure (or composition) conditions.
• These diagrams identify equilibrium states, phase boundaries, triple points, and critical points.

Some working definitions and interpretations

• Crystallization: The growth of a crystalline phase(s) which may or may not have the same composition as the original liquid (glass).
• Surface crystallization: Crystal growth begins (i.e. nucleates) from the liquid/glass-atmosphere interface, and usually grows perpendicular to this interface.
• Volume crystallization: Crystal growth begins from 'nucleation sites' within the body of the material.
• Liquid/liquid phase separation: The growth of non-crystalline phases which will have a different composition from the original phase.
• A single-component system (e.g. SiO₂ ) cannot separate in this way.
• Spinodal decomposition: Within a region which separates into two liquid phases, there will be a region where there is no energy barrier to nucleation; phase separation is, therefore, limited by diffusion only.

Importance of crystallization & phase transformation

Glass-ceramics

• Glass-ceramics are polycrystalline materials with crystals dispersed within the glassy phase.

• Functional crystalline phase + glassy phase
• Volume fraction = ppm to almost 100%

Microstructure possibilities

Dendritic structure - Leucite glass ceramics

House of cards structure - Boro silicates glass ceramics

Cabbage structure - Leucite glass ceramics

Needle like structure-

Apatite glass ceramics

Rod like structure-

Mica glass ceramics

Ref.: Wolfram Holand, George H. Beall, Glass Ceramic Technology, Willey, 2012.

Applications

Machinable glass-ceramics

• One of the most important class of glass-ceramics
• Mica-containing glass-ceramics: K₂O–B₂O₃–Al₂O₃–SiO₂–MgO-F
• Unique combination of machinability and good mechanical strength
• Used to fabricate complex shapes by CAD-CAM system
• It can be drilled, milled or turned using standard metal working tool.
• This machinability comes from the unique interlocked micro-structure of mica-crystal phase.
• One of the most popular commercial machinable glass-ceramics is Macor®.

Source: http://www.corning.com/specialtymaterials/advanced_optics/specialty_glass_ceramics/products/macor/

Ref.: Deubener J, Allix M, Davis M, Duran A, Höche T, Honma T, et al. Updated definition of glass-ceramics. Journal of Non-Crystalline Solids. 2018;501:3-10.

Machinable glass ceramics

Crystallization

• Crystallization refers to a combination of two processes: nucleation and crystal growth.
• Crystallization requires the presence of a nucleus (nucleation) on which the crystal will subsequently grow (crystal growth) to a detectable size.
• Nucleus may be either homogeneous (forming spontaneously within the melt), or heterogeneous (forming at a pre-existing surface e.g an impurity, crucible wall etc.)
• When a liquid is cooled below its freezing/melting point, crystallization occurs by the growth of crystals at a finite rate from a finite number of nuclei.
• Glass formation may be attributed to a low rate of crystal growth, a low rate of nuclei formation or a combination of both.
• If no nuclei are present, crystal growth cannot occur and the material will form a glass.
• Even if some nuclei are present, but no growth has occurred, the extremely small size and low volume fraction of the nuclei prevents their detection.

Homogeneous nucleation

• The assumption that no change in nuclei concentration during the quenching or crystal growth stages is always questionable.

Homogeneous nucleation

• Formation of a spherical nucleus: Two barriers
• Thermodynamic barrier: The free energy change in a system when a nucleus is formed (the work required to form a nucleus of critical size, i.e. one which will grow instead of re-dissolve into the melt).
• Kinetic barrier: Requirement that mass be moved or rearranged in space, to allow the growth of an ordered particle (a crystal) from a disordered liquid.
• The stability of a particle of the new phase in homogeneous nucleation will depend on two contributions to the thermodynamic barrier
• One from a difference in free energy between the two phases.
• The other from the interfacial energy.

• Formation of a nucleus involves two changes in the energy of the system i.e. the thermodynamic barrier.
• First, the formation of a crystalline arrangement will lower the volume free energy, since the crystalline state has a lower free energy than the melt.
• This decrease in free energy is countered by an increase in surface energy due to the formation of a new interface between regions of different structures.
• At the melting point, the free energy of a given quantity of a material is the same in the crystalline and in the liquid forms.
• At lower temperatures the crystalline form will have lower free energy and the liquid will crystallize if nuclei are available.

• Since the formation of an embryo involves a positive free energy change, the probability of such an occurrence would be quite small.
• Entropy of the solid-liquid system can be increased by the presence of a number of atom clusters in equilibrium with the atoms of the liquid.

Ice homogeneous nucleation rate for a molecular model of water

Phenomenological theory: Kinetic barrier

• The rate of nucleus formation is very sensitive to variations in temperature.
• For water at - 40°C, the nucleation rate increases by a factor of about 9 for every 1^oC fall of temperature

Nucleation rate

• Probability of a nucleus reaching such a large size is extremely low: the melt will remain effectively free of nuclei, even though the temperature is below T_m.

Metastable zone of undercooling

• As the kinetic barrier increases, the nucleation rate begin to decrease and eventually fall to zero.
• The conflicting changes in the nucleation rate due to changes in the thermodynamic and kinetic barriers, result in a maximum in the temperature-dependence of the nucleation rate.

Limitations of classic theory

• The nucleation theory discussed is often called the classical theory, because of several assumptions made during the derivation of the nucleation equation.
• It is assumed that embryos can be treated as bulk material, free energy being independent of the size of the embryo.
• For very small embryos, the surface energy will also depend on the size of the embryo.
• It is assumed that each embryo has a sharp boundary with a well-defined surface energy.
• In fact there will be a gradient, both in degree of ordering and in chemical composition (in a system of more than one component) as one passes from the crystal phase to the liquid.
• The use of the Boltzmann function to describe the distribution of embryos is not strictly correct, although this requires only a slight adjustment to the classical theory.

Heterogeneous nucleation

Heterogeneous nucleation�

• In heterogeneous nucleation, the nucleus develops on the surface of a foreign solid (substrate).
• The substrate may be the container wall or it may be a solid dispersed throughout the liquid.

• The mechanical equilibrium of the solid growing on the crystal is expressed by the Young–Dupree equation

Heterogeneous nucleation

What should be the criteria for a suitable substrate?

• Lithium disilicate is an orthorhombic crystal with a = 5.80, b = 14.66 A, and c = 4.806 A
• Platinum is a cubic crystal with a= b =c = 3.92A
• (111) plane of platinum matches the (002) plane of lithium disilicate with ~6% difference
• Disregistry: ~15% maximum for metal systems
• Glass-ceramics: a disregistry of 8% would seem to be the maximum
• After only six lattice spacing there will be complete mismatch

• Substrate needs to be thoroughly wetted by the liquid (θ should be small)
• Epitaxial growth or oriented overgrowth: If we put into a undercooled liquid a 'seed' crystal which has a low-index plane in which the atomic spacing and arrangement are similar to those of one of the low-index planes in the crystal that 'wishes' to form, then the liquid will start to crystallize on that foreign nucleus.
• If there is no near match, the liquid will not crystallize.

Importance of heterogeneous nucleation

• Production of glass-ceramics: Crystals of about one micrometre size.
• It is necessary to produce 10¹² of nuclei per cm³ of glass.
• Two broad classes of nucleating agents in glass:
• Group A: Substances capable of forming minute crystals of low solubility in the glass.
• They have a high tendency to reduce from the ionic form to the neutral state in the melt, e.g. Pt, Au, Ag, Cu.

• Group B: Substances such as TiO₂, ZrO₂, P₂O₅ etc. which are soluble in silicate glasses, and large amounts (1-20 wt %) can be added before nucleation occurs.
• These substances rarely crystallize out as the independent oxides at the start of crystallization, but precipitate as a complex compound, e.g. MgTiO₃ , Li₂TiO₃

Rate of crystal growth

• A simple general model can be derived using arguments similar to those used for the nucleation rate.
• A large number of expressions describing crystal growth can be found in the literature: Many of these equations deal with specific models for different crystal growth mechanisms.
• The rate at which a crystal nucleus grows will depend on the rate at which atoms arrive and remain at the surface of the nucleus.

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• Initially, where the nucleus is microscopic (but greater than r*), the growth rate will also be affected by the curvature of the nucleus-liquid interface, since as the nucleus grows there will be an increase in the interfacial surface energy.
• As the nucleus becomes macroscopic in size, i.e. when it can be considered a crystal, this increase in surface energy can be neglected in comparison with the decrease in free energy. Thus the crystal can be treated as having a planar interface.

• The exponential term represents the probability of finding an atom with sufficient thermal energy to leave the liquid and join the crystal.

Nucleation vs. crystal growth

• The temperature-dependence of the crystal growth rate is very similar to that for the nucleation rate.
• Near T_m the growth rate is controlled by the thermodynamic driving force for crystallization.
• As T_g is approached, mobility in the liquid limits the rate at which the crystal can grow. 
• Lack of a metastable zone: Crystal growth can occur at any temperature below T_m if a nucleus is available: Detectable growth rates can occur at any temperature ≤T_m.
• The nuclei involved need not even have the same composition as that of the growing crystal, which is frequently the case for heterogeneous nucleation, particularly at surfaces.

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• As in the nucleation process, the temperature corresponding to the maximum growth rate rises as the diffusion activation energy, ΔG_a increases.
• Maximum in growth rate occurs at much smaller under-cooling than is the case for the maximum nucleation rate.
• Diffusion activation energies for growth and nucleation may not necessarily be equal, since the atomic movements involved may be quite different for the two processes.

• Experimentally measured crystal growth rates for supercooled tris(naphthylbenzene).