Phenomenology of crystal growth.

by Dr. Leonid Sakharov

The crystal growth has distinct features these can be described here without attracting theoretical concepts of mechanism of the phenomena. Wide variety for different crystals of rates for two its basic processes: nucleation spontaneous appearance of small roots of crystals and their expansion explains great diversity of properties of solid matter in nature and industry.

Crystals are state of the matter with long-range order of atoms in it. It means that in some direction the pattern of atoms locations will periodically repeat themselves. Conventionally this statement will be true in any direction in three dimensional space but for some special cases such as liquid-crystals (yes, it may be the key material in your computer monitor you staring now at) in some directions atoms will be randomly located but in others could have a long-range order.

Solid state of matter not necessary equals to crystal. Glasses are not crystals they are liquids. In the spite you will never observe it in your life the window glass is in the process of crystallization.

But the process is going very slow. So slow that all chances are it will never display itself at room temperature.
But some glasses could display crystal growth at some temperatures. There is some class of materials named glass-ceramics manufactured in the way that glass is partially crystallized containing micro particles of crystals in glass matrix.

If one takes some peace of matter with any chemical content there are some common rules of its behavior when heating and cooling. A diversity of actual manifestations of matter transformation is startling various. But following statements are applied to any of them.

Let's discuss a sample of material in closed space with absolute inert walls where by special means a constant pressure is kept.

To stay close to reality it could be for example an platinum cylinder with perfectly fitting plunger.
For most practical situations on the planet earth just open platinum crucible will make a trick for most minerals or organic compounds.

If heat the described above sample up to high enough temperature and keep it there will be no areas inside with long-range order of atoms. A lowest temperature secures an absence of crystalline phase in sample under very long exposition is called as temperature of liquidus. If very slow cooling down the sample up to temperature called solidus there will be only crystalline state (not necessary crystals of only one structure and content) in it. In same cases when chemical content of sample is exactly the same as chemical content as growing crystal temperatures of solidus and liquidus coincide and called temperature of melting.

Below temperature of liquidus inside noncrystalline areas there will be spontaneous origination of crystalline areas able to grow up. A phenomena is called a nucleation.

Quantitatively of the phenomena can be characterized as number of new isolated crystalline areas appeared per time and in a volume:

I = 1 dN (1)
v dt

where v is a volume of sample, dN/dt - rate of appearance new nucleus.

Rate of nucleation is stochastic value that means that a formation of nuclei is an random event.

A formula (1) presents an average value of crystals these will spontaneous form in the unit of volume during one unit of time measurement. For given temperature and content of sample there could be more then one crystal of different chemical and/or structure born in it. Each of them will has own rate of nucleation.

If the sample cooled down very quickly the time until appearance of the first detectable crystal is named an induction period - τ.
There is direct connection between nucleation rate and induction period:

τ = 1 Tg (2)

where Tg - time of growth of nuclei until it will be detected. This time is depend on condition of experiment including sensitivity of detectors. Experimentally rate of nucleation can be measured by calculation of numbers of crystals in samples as time progresses or with repetition of experiment for defining induction period. Both methods has complications mostly based on fact that sample during experiment is changing its property. In the spite of common notion the structure and property of noncrystalline phases can change in time including being reflected by nucleation in it.

Once crystalline area of big enough size is exist in sample it can grow up to some point. The size of the area is important because there is critical size of the crystal nuclei. Crystals large than critical more probable to grow farther, smaller is more likely will dissolve back into noncrystalline phase. Crystals with overcritical size could grow up to same point.
If chemical composition of crystalline phase is exact the same as sample, all sample could transform into monocrystal. If chemical composition is differ the crystal growth will stop reaching the situation when exactions of some chemical component in feeding noncrystalline surrounding changes its thermodynamic potential to equilibrium and growth will stop.

In the same sample the growth of crystals with different composition and/or structure is possible at the same time. The growth rates of crystal in different directions are depend on its structure and distribution of chemical composition of sample that can vary around its borders. If leaved alone in large volume of feeding material the common rule is that the crystal will grow in form of symmetrical object reflecting the symmetry of its atomic structure. There could be infinite variety of actual shapes of crystals produced at different conditions.
Just take a look at snowflakes to appreciate all range of variants in morphology.

As a rule functions characterized temperature dependence of nucleation and growth rates for one type of crystals have bell-like curve with one maximum. Typically temperature of nucleation maximum is lower than for growth. There are infinite variety of specific manifestations of these parameters for different crystals and depend on environment of feeding matter.

To give more specificity how to use main parameters characterized crystallization to set technological routine let's take a more close look at technology of production of glass-ceramic materials. Raw material for production of glass-ceramic materials has to display following characteristics as shown at picture below.

Areas of maximum nucleation and crystal growth have to be significantly spread so far other in such way that if cooling such material with technologically achievable rate there will be no formation of crystal nucleus in it and pure glass could be able to obtain. On the picture it corresponds to the time-temperature trajectory marked as Glass.
If cool down such material up to temperature where rates of nucleation and growth are overlapped and keep it long enough same number of nucleus will spontaneously and randomly formed and they will grow up until the process is thermodynamically favorable. The time-temperature trajectory for such situation is marked on picture as " Big grains". The main disadvantage of such process is that result output is quite unpredictable as soon location time sequence of formation of overcritical nucleus will vary from one sample to other and from one area in sample to other. Sometimes it could be acceptable if nucleation rate is very fast like it could be for metals but very often such material will have not best mechanical and other properties.

If cool down such material to the temperature of maximum rate of nucleation but close to zero rate of crystal growth and expose until concentration of nucleus reach deliverable level, actually one can set by varying the time of exposure a maximum size of growing space of one grain. Then after increasing temperature to maximum of growth rate nucleus will uniformly spread out forming Fine ceramic material. Size of grains can be controlled up to the point that optically such material will be transparent as pure glass or monocrystal but X-ray pattern will be the same as for fine powder. Specially designed such materials could demonstrate specially desirable properties as optical transparently and high strength and hardness for extended interval of working temperature.

Sep. 24, 2017; 19:48 EST

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