Editing Crystallization (section)

==Dynamics==
As mentioned above, a crystal is formed following a well-defined pattern, or structure, dictated by forces acting at the molecular level. As a consequence, during its formation process the [[crystal]] is in an environment where the solute [[concentration]] reaches a certain critical value, before changing status. Solid formation, impossible below the [[solubility]] threshold at the given [[temperature]] and [[pressure]] conditions, may then take place at a concentration higher than the theoretical solubility level. The difference between the actual value of the solute concentration at the crystallization limit and the theoretical (static) solubility threshold is called [[supersaturation]] and is a fundamental factor in crystallization.

===Nucleation===
{{Main|Nucleation}}
Nucleation is the initiation of a phase change in a small region, such as the formation of a solid crystal from a liquid solution. It is a consequence of rapid local fluctuations on a molecular scale in a homogeneous phase that is in a state of metastable equilibrium. Total nucleation is the sum effect of two categories of nucleation – primary and secondary.

====Primary nucleation====
Primary nucleation is the initial formation of a crystal where there are no other crystals present or where, if there are crystals present in the system, they do not have any influence on the process. This can occur in two conditions. The first is homogeneous nucleation, which is nucleation that is not influenced in any way by solids. These solids include the walls of the crystallizer vessel and particles of any foreign substance. The second category, then, is heterogeneous nucleation. This occurs when solid particles of foreign substances cause an increase in the rate of nucleation that would otherwise not be seen without the existence of these foreign particles. Homogeneous nucleation rarely occurs in practice due to the high energy necessary to begin nucleation without a solid surface to catalyze the nucleation.

Primary nucleation (both homogeneous and heterogeneous) has been modeled as follows:<ref name="Tavare">Tavare, N. S. (1995). ''Industrial Crystallization''. Plenum Press, New York.{{page?|date=October 2021}}</ref>

:<math>B = \dfrac{dN}{dt} = k_n(c - c^*)^n,</math>

where
: ''B'' is the number of nuclei formed per unit volume per unit time,
: ''N'' is the number of nuclei per unit volume,
: ''k<sub>n</sub>'' is a rate constant,
: ''c'' is the instantaneous solute concentration,
: ''c''<sup>*</sup> is the solute concentration at saturation,
: (''c'' − ''c''<sup>*</sup>) is also known as supersaturation,
: ''n'' is an empirical exponent that can be as large as 10, but generally ranges between 3 and 4.

====Secondary nucleation====
Secondary nucleation is the formation of nuclei attributable to the influence of the existing microscopic crystals in the magma.<ref name="McCabeSmith">McCabe & Smith (2000). ''Unit Operations of Chemical Engineering''. McGraw-Hill, New York.{{page?|date=October 2021}}</ref> More simply put, secondary nucleation is when crystal growth is initiated with contact of other existing crystals or "seeds".<ref>{{Cite web |url=http://www.reciprocalnet.org/edumodules/crystallization/ |title=Crystallization |website=www.reciprocalnet.org |access-date=2017-01-03 |url-status=live |archive-url=https://web.archive.org/web/20161127173509/http://www.reciprocalnet.org/edumodules/crystallization/ |archive-date=2016-11-27 }}</ref> The first type of known secondary crystallization is attributable to fluid shear, the other due to collisions between already existing crystals with either a solid surface of the crystallizer or with other crystals themselves. Fluid-shear nucleation occurs when liquid travels across a crystal at a high speed, sweeping away nuclei that would otherwise be incorporated into a crystal, causing the swept-away nuclei to become new crystals. Contact nucleation has been found to be the most effective and common method for nucleation. The benefits include the following:<ref name="McCabeSmith" />
* Low kinetic order and rate-proportional to supersaturation, allowing easy control without unstable operation.
* Occurs at low supersaturation, where growth rate is optimal for good quality.
* Low necessary energy at which crystals strike avoids the breaking of existing crystals into new crystals.
* The quantitative fundamentals have already been isolated and are being incorporated into practice.

The following model, although somewhat simplified, is often used to model secondary nucleation:<ref name="Tavare" />
:<math>B = \dfrac{dN}{dt} = k_1 M_T^j(c - c^*)^b,</math>
where
: ''k''<sub>1</sub> is a rate constant,
: ''M<sub>T</sub>'' is the suspension density,
: ''j'' is an empirical exponent that can range up to 1.5, but is generally 1,
: ''b'' is an empirical exponent that can range up to 5, but is generally 2.

[[Image:Crystal growth.PNG|thumb|upright|Crystal growth]]

===Growth===
{{Main|Crystal growth}}
Once the first small crystal, the nucleus, forms it acts as a convergence point (if unstable due to supersaturation) for [[molecules]] of solute touching – or adjacent to – the crystal so that it increases its own dimension in successive layers. The pattern of growth resembles the rings of an onion, as shown in the picture, where each colour indicates the same mass of solute; this mass creates increasingly thin layers due to the increasing surface area of the growing crystal. The supersaturated solute mass the original nucleus may ''capture'' in a time unit is called the ''growth rate'' expressed in kg/(m<sup>2</sup>*h), and is a constant specific to the process. Growth rate is influenced by several physical factors, such as [[surface tension]] of solution, [[pressure]], [[temperature]], relative crystal [[velocity]] in the solution, [[Reynolds number]], and so forth.

The main values to control are therefore:
* Supersaturation value, as an index of the quantity of solute available for the growth of the crystal;
* Total crystal surface in unit fluid mass, as an index of the capability of the solute to fix onto the crystal;
* Retention time, as an index of the probability of a molecule of solute to come into contact with an existing crystal;
* Flow pattern, again as an index of the probability of a molecule of solute to come into contact with an existing crystal (higher in [[laminar flow]], lower in [[turbulent flow]], but the reverse applies to the probability of contact).

The first value is a consequence of the physical characteristics of the solution, while the others define a difference between a well- and poorly designed crystallizer.

===Size distribution===
{{Unreferenced section|date=July 2017}}

The appearance and size range of a crystalline product is extremely important in crystallization. If further processing of the crystals is desired, large crystals with uniform size are important for washing, filtering, transportation, and storage, because large crystals are easier to filter out of a solution than small crystals.<ref>{{Cite journal |last1=Beck |first1=Ralf |last2=Häkkinen |first2=Antti |last3=Malthe-Sørenssen |first3=Didrik |last4=Andreassen |first4=Jens-Petter |date=2009-05-07 |title=The effect of crystallization conditions, crystal morphology and size on pressure filtration of l-glutamic acid and an aromatic amine |url=https://linkinghub.elsevier.com/retrieve/pii/S1383586609000355 |journal=Separation and Purification Technology |volume=66 |issue=3 |pages=549–558 |doi=10.1016/j.seppur.2009.01.018 |issn=1383-5866|url-access=subscription }}</ref> Also, larger crystals have a smaller surface area to volume ratio, leading to a higher purity. This higher purity is due to less retention of [[mother liquor]] which contains impurities, and a smaller loss of yield when the crystals are washed to remove the mother liquor. In special cases, for example during drug manufacturing in the pharmaceutical industry, small crystal sizes are often desired to improve drug dissolution rate and bio-availability. The theoretical crystal size distribution can be estimated as a function of operating conditions with a fairly complicated mathematical process called population balance theory (using [[population balance equation]]s).