Nucleation
Overview
 
Nucleation is the extremely localized budding of a distinct thermodynamic phase. Some examples of phases that may form by way of nucleation in liquids are gaseous bubble
Liquid bubble
A bubble is a globule of one substance in another, usually gas in a liquid.Due to the Marangoni effect, bubbles may remain intact when they reach the surface of the immersive substance.-Common examples:...

s, crystals or glassy
Glass
Glass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is soda-lime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...

 regions. Creation of liquid droplets in saturated vapor
Vapor
A vapor or vapour is a substance in the gas phase at a temperature lower than its critical point....

 is also characterized by nucleation (see Cloud condensation nuclei
Cloud condensation nuclei
Cloud condensation nuclei or CCNs are small particles typically 0.2 µm, or 1/100 th the size of a cloud droplet ) about which cloud droplets coalesce. Water requires a non-gaseous surface to make the transition from a vapour to a liquid. In the atmosphere, this surface presents itself as tiny...

). Nucleation of crystalline, amorphous and even vacancy clusters solid materials is also important, for example to the semiconductor industry.
Encyclopedia
Nucleation is the extremely localized budding of a distinct thermodynamic phase. Some examples of phases that may form by way of nucleation in liquids are gaseous bubble
Liquid bubble
A bubble is a globule of one substance in another, usually gas in a liquid.Due to the Marangoni effect, bubbles may remain intact when they reach the surface of the immersive substance.-Common examples:...

s, crystals or glassy
Glass
Glass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is soda-lime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...

 regions. Creation of liquid droplets in saturated vapor
Vapor
A vapor or vapour is a substance in the gas phase at a temperature lower than its critical point....

 is also characterized by nucleation (see Cloud condensation nuclei
Cloud condensation nuclei
Cloud condensation nuclei or CCNs are small particles typically 0.2 µm, or 1/100 th the size of a cloud droplet ) about which cloud droplets coalesce. Water requires a non-gaseous surface to make the transition from a vapour to a liquid. In the atmosphere, this surface presents itself as tiny...

). Nucleation of crystalline, amorphous and even vacancy clusters solid materials is also important, for example to the semiconductor industry. Most nucleation processes are physical, rather than chemical, but a few exceptions do exist (e.g. electrochemical nucleation). A good example would be the famous Diet Coke and Mentos eruption.
Nucleation normally occurs at nucleation sites on surfaces contacting the liquid or vapor. Suspended particles or minute bubbles also provide nucleation sites. This is called heterogeneous nucleation. Nucleation without preferential nucleation sites is homogeneous nucleation. Homogeneous nucleation occurs spontaneously and randomly, but it requires superheating
Superheating
In physics, superheating is the phenomenon in which a liquid is heated to a temperature higher than its boiling point, without boiling...

 or supercooling
Supercooling
Supercooling, also known as undercooling, is the process of lowering the temperature of a liquid or a gas below its freezing point without it becoming a solid....

 of the medium. Nucleation is involved in such processes as cloud seeding
Cloud seeding
Cloud seeding, a form of intentional weather modification, is the attempt to change the amount or type of precipitation that falls from clouds, by dispersing substances into the air that serve as cloud condensation or ice nuclei, which alter the microphysical processes within the cloud...

 and in instruments such as the bubble chamber
Bubble chamber
A bubble chamber is a vessel filled with a superheated transparent liquid used to detect electrically charged particles moving through it. It was invented in 1952 by Donald A. Glaser, for which he was awarded the 1960 Nobel Prize in Physics...

 and the cloud chamber
Cloud chamber
The cloud chamber, also known as the Wilson chamber, is a particle detector used for detecting ionizing radiation. In its most basic form, a cloud chamber is a sealed environment containing a supersaturated vapor of water or alcohol. When a charged particle interacts with the mixture, it ionizes it...

.

Examples of nucleation

  • Presence of cloud condensation nuclei
    Cloud condensation nuclei
    Cloud condensation nuclei or CCNs are small particles typically 0.2 µm, or 1/100 th the size of a cloud droplet ) about which cloud droplets coalesce. Water requires a non-gaseous surface to make the transition from a vapour to a liquid. In the atmosphere, this surface presents itself as tiny...

     is important in meteorology
    Meteorology
    Meteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...

     because they are often in short supply in the upper atmosphere
    Atmosphere
    An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...

     (see cloud seeding
    Cloud seeding
    Cloud seeding, a form of intentional weather modification, is the attempt to change the amount or type of precipitation that falls from clouds, by dispersing substances into the air that serve as cloud condensation or ice nuclei, which alter the microphysical processes within the cloud...

    ).
  • All natural and artificial crystallization
    Crystallization
    Crystallization is the process of formation of solid crystals precipitating from a solution, melt or more rarely deposited directly from a gas. Crystallization is also a chemical solid–liquid separation technique, in which mass transfer of a solute from the liquid solution to a pure solid...

     process (of formation of solid crystals from a homogeneous solution) starts with a nucleation event.
  • Bubbles of carbon dioxide
    Carbon dioxide
    Carbon dioxide is a naturally occurring chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom...

     nucleate shortly after the pressure is released from a container of carbonated
    Carbonation
    Carbonation is the process of dissolving carbon dioxide in water. The process usually involves carbon dioxide under high pressure. When the pressure is reduced, the carbon dioxide is released from the solution as small bubbles, which cause the solution to "fizz." This effect is seen in carbonated...

     liquid. Nucleation often occurs more easily at a pre-existing interface
    Interface (chemistry)
    An interface is a surface forming a common boundary among two different phases, such as an insoluble solid and a liquid, two immiscible liquids or a liquid and an insoluble gas. The importance of the interface depends on which type of system is being treated: the bigger the quotient area/volume,...

     (heterogeneous nucleation), as happens on boiling chip
    Boiling chip
    A boiling chip or boiling stone is a small, irregularly shaped piece of material added to liquids to make them boil more smoothly. Boiling chips are frequently employed in distillation and heating. When a liquid becomes superheated, a particle of dust or a stirring rod can cause flash boiling...

    s and string used to make rock candy
    Rock candy
    Rock candy is a type of confectionery mineral composed of relatively large sugar crystals. This candy is formed by allowing a supersaturated solution of sugar and water to crystallize onto a surface suitable for crystal nucleation, such as a string or stick...

    . So-called Diet Coke and Mentos eruptions are a dramatic example.
  • Nucleation in boiling
    Boiling
    Boiling is the rapid vaporization of a liquid, which occurs when a liquid is heated to its boiling point, the temperature at which the vapor pressure of the liquid is equal to the pressure exerted on the liquid by the surrounding environmental pressure. While below the boiling point a liquid...

     can occur in the bulk liquid
    Liquid
    Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...

     if the pressure is reduced so that the liquid becomes superheated
    Superheating
    In physics, superheating is the phenomenon in which a liquid is heated to a temperature higher than its boiling point, without boiling...

     with respect to the pressure-dependent boiling point. More often nucleation occurs on the heating surface, at nucleation sites. Typically, nucleation sites are tiny crevices where free gas-liquid surface is maintained or spots on the heating surface with lower wetting
    Wetting
    Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. The degree of wetting is determined by a force balance between adhesive and cohesive forces.Wetting is important in the bonding or adherence of...

     properties. Substantial superheating of a liquid can be achieved after the liquid is de-gassed and if the heating surfaces are clean, smooth and made of materials well wetted by the liquid.
  • Nucleation is relevant in the process of crystallization of nanometer sized materials
    Nanoparticle
    In nanotechnology, a particle is defined as a small object that behaves as a whole unit in terms of its transport and properties. Particles are further classified according to size : in terms of diameter, coarse particles cover a range between 10,000 and 2,500 nanometers. Fine particles are sized...

    , and plays an important role in atmospheric processes.
  • Nucleation is a key concept in polymer
    Polymer
    A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...

    , alloy
    Alloy
    An alloy is a mixture or metallic solid solution composed of two or more elements. Complete solid solution alloys give single solid phase microstructure, while partial solutions give two or more phases that may or may not be homogeneous in distribution, depending on thermal history...

     and ceramic
    Ceramic
    A ceramic is an inorganic, nonmetallic solid prepared by the action of heat and subsequent cooling. Ceramic materials may have a crystalline or partly crystalline structure, or may be amorphous...

     systems.
  • In chemistry
    Chemistry
    Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

     and biophysics
    Biophysics
    Biophysics is an interdisciplinary science that uses the methods of physical science to study biological systems. Studies included under the branches of biophysics span all levels of biological organization, from the molecular scale to whole organisms and ecosystems...

    , nucleation can also refer to the phaseless formation of multimers which are intermediates in polymerization processes. This sort of process is believed to be the best model for processes such as crystallization
    Crystallization
    Crystallization is the process of formation of solid crystals precipitating from a solution, melt or more rarely deposited directly from a gas. Crystallization is also a chemical solid–liquid separation technique, in which mass transfer of a solute from the liquid solution to a pure solid...

     and amyloid
    Amyloid
    Amyloids are insoluble fibrous protein aggregates sharing specific structural traits. Abnormal accumulation of amyloid in organs may lead to amyloidosis, and may play a role in various neurodegenerative diseases.-Definition:...

    ogenesis.
  • In molecular biology
    Molecular biology
    Molecular biology is the branch of biology that deals with the molecular basis of biological activity. This field overlaps with other areas of biology and chemistry, particularly genetics and biochemistry...

    , nucleation is the term used to describe the critical stage in the assembly of a polymer
    Polymer
    A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...

    ic structure, such as a microfilament, at which a small cluster of monomer
    Monomer
    A monomer is an atom or a small molecule that may bind chemically to other monomers to form a polymer; the term "monomeric protein" may also be used to describe one of the proteins making up a multiprotein complex...

    s aggregates in the correct arrangement to initiate rapid polymerization. For instance, two actin
    Actin
    Actin is a globular, roughly 42-kDa moonlighting protein found in all eukaryotic cells where it may be present at concentrations of over 100 μM. It is also one of the most highly-conserved proteins, differing by no more than 20% in species as diverse as algae and humans...

     molecule
    Molecule
    A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

    s bind weakly, but addition of a third stabilizes the complex. This trimer then adds additional molecules and forms a nucleation site. The nucleation site serves the slow, or lag phase of the polymerization process.
  • Some champagne stirrers operate by providing many nucleation sites via high surface area and sharp corners, speeding the release of bubbles and removing carbonation from the wine.
  • Sodium acetate heating pads use cavitation voids caused by the deflection of a metal disk as nucleation centres for exothermic crystallization.

Homogeneous nucleation

Nucleation generally occurs with much more difficulty in the interior of a uniform substance, by a process called homogeneous nucleation. The creation of a nucleus implies the formation of an interface at the boundaries of a new phase.

Liquids cooled below the maximum heterogeneous nucleation temperature (melting
Melting
Melting, or fusion, is a physical process that results in the phase change of a substance from a solid to a liquid. The internal energy of a substance is increased, typically by the application of heat or pressure, resulting in a rise of its temperature to the melting point, at which the rigid...

 temperature), but which are above the homogeneous nucleation temperature (pure substance freezing
Freezing
Freezing or solidification is a phase change in which a liquid turns into a solid when its temperature is lowered below its freezing point. The reverse process is melting....

 temperature) are said to be supercooled
Supercooling
Supercooling, also known as undercooling, is the process of lowering the temperature of a liquid or a gas below its freezing point without it becoming a solid....

. This is useful for making amorphous solid
Amorphous solid
In condensed matter physics, an amorphous or non-crystalline solid is a solid that lacks the long-range order characteristic of a crystal....

s and other metastable
Metastability
Metastability describes the extended duration of certain equilibria acquired by complex systems when leaving their most stable state after an external action....

 structures, but can delay the progress of industrial chemical processes
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

 or produce undesirable effects in the context of casting
Casting
In metalworking, casting involves pouring liquid metal into a mold, which contains a hollow cavity of the desired shape, and then allowing it to cool and solidify. The solidified part is also known as a casting, which is ejected or broken out of the mold to complete the process...

.
Supercooling brings about supersaturation, the driving force for nucleation. Supersaturation occurs when the pressure in the newly formed solid is less than the liquid pressure, and brings about a change in free energy
Gibbs free energy
In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...

 per unit volume, , between the liquid and newly created solid phase. This change in free energy is balanced by the energy gain of creating a new volume, and the energy cost due to creation of a new interface. When the overall change in free energy, is negative, nucleation is favored.

Some energy is consumed to form an interface, based on the surface energy
Surface energy
Surface energy quantifies the disruption of intermolecular bonds that occur when a surface is created. In the physics of solids, surfaces must be intrinsically less energetically favorable than the bulk of a material, otherwise there would be a driving force for surfaces to be created, removing...

 of each phase. If a hypothetical nucleus is too small (known as an unstable nucleus or "embryo"), the energy that would be released by forming its volume is not enough to create its surface, and nucleation does not proceed. The critical nucleus size can be denoted by its radius, and it is when r=r* (or r critical) that the nucleation proceeds.

For example in the classic case of a spherical cluster that liberates -Gv Joules per cubic centimeter during formation (here Gv is a negative quantity), but which must pay the positive cost of σ Joules per square centimeter of surface interfacing with the surrounding, the free energy needed to form a spherical cluster of radius r is,

Where the first term shows the energy gain of creating a new volume and the second term shows the energy loss due to surface tension () of the new interface.

It costs free energy to add molecules to this cluster (because ) until the radius reaches,


where,

Addition of new molecules to clusters larger than this critical radius
Critical radius
Critical radius is the minimum size that must be formed by atoms or molecules clustering together before a new-phase inclusion is stable and begins to grow. Formation of such stable "nuclei" is called nucleation. In precipitation models this is generally a prelude to models of the growth...

 releases, rather than costs, available work. In other words at that point growth of the cluster is no longer limited by nucleation, but perhaps by diffusion (i.e. the supply of molecules) or by reaction kinetics instead.

The free energy needed to form this critical radius can be found by

which occurs at the maximum where

The term can be related to the equilibrium temperature, enthalpy of fusion (), and the degree of undercooling () as follows,
By evaluating this equation at the equilibrium point () at the melting temperature we achieve,
Substitution of into the first equation leads to,
Which by using common denominators and the definition of provides,
As the phase transformation becomes more and more favorable, the formation of a given volume of nucleus frees enough energy to form an increasingly large surface, allowing progressively smaller nuclei to become viable. Eventually, thermal activation will provide enough energy to form stable nuclei. These can then grow until thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

 is restored.

A greater degree of supercooling favors phase transformation, and we can relate to supercooling and find r* and as a function of by the substitution of ,

and


The greater the supercooling, the smaller the critical radius and the less energy needed to form it.

The spontaneous nucleation rate in, say, water changes very rapidly with temperature, so the spontaneous nucleation temperature can be quite well defined. 'Film boiling' on very hot surfaces and the Leidenfrost effect
Leidenfrost effect
The Leidenfrost effect is a phenomenon in which a liquid, in near contact with a mass significantly hotter than the liquid's boiling point, produces an insulating vapor layer which keeps that liquid from boiling rapidly...

 are both believed to be stabilized by spontaneous nucleation phenomena.

Heterogeneous nucleation

Heterogeneous nucleation occurs much more often than homogeneous nucleation. It forms at preferential sites such as phase boundaries or impurities like dust and requires less energy than homogeneous nucleation. At such preferential sites, the effective surface energy is lower, thus diminishing the free energy barrier and facilitating nucleation. Surfaces promote nucleation because of wetting
Wetting
Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. The degree of wetting is determined by a force balance between adhesive and cohesive forces.Wetting is important in the bonding or adherence of...

 – contact angles greater than zero between phases encourage particles to nucleate.
The free energy needed for heterogeneous nucleation is equal to the product of homogeneous nucleation and a function of the contact angle :


where

The barrier energy needed for heterogeneous nucleation is reduced, and less supercooling is needed. The wetting angle determines the ease of nucleation by reducing the energy needed. The critical radius remains unchanged: however, the volume can be significantly less for heterogeneous nucleation due to the wetting angle affecting the shape of the cluster.

In the case of heterogeneous nucleation, some energy is released by the partial destruction of the previous interface. For example, if a carbon dioxide bubble forms between water and the inside surface of a bottle, the energy inherent in the water-bottle interface is released wherever a layer of gas intervenes, and this energy goes toward the formation of bubble-water and bubble-bottle interfaces. The same effect can cause precipitate
Precipitation (chemistry)
Precipitation is the formation of a solid in a solution or inside anothersolid during a chemical reaction or by diffusion in a solid. When the reaction occurs in a liquid, the solid formed is called the precipitate, or when compacted by a centrifuge, a pellet. The liquid remaining above the solid...

 particles to form at the grain boundaries
Crystallite
Crystallites are small, often microscopic crystals that, held together through highly defective boundaries, constitute a polycrystalline solid. Metallurgists often refer to crystallites as grains.- Details :...

 of a solid. This can interfere with precipitation strengthening
Precipitation strengthening
Precipitation hardening, also called age hardening, is a heat treatment technique used to increase the yield strength of malleable materials, including most structural alloys of aluminium, magnesium, nickel and titanium, and some stainless steels...

, which relies on homogeneous nucleation to produce a uniform distribution of precipitate particles.

Rate of nucleation

The nucleation rate, I, depends on the average number of critical clusters, n* and the diffusion of molecules to the cluster, .


where the average population of critical nuclei is


where:
  • ΔG* is critical free energy needed corresponding to that of the critical radius.
  • N is the number of potential nucleation sites per unit volume
  • kB is the Boltzmann constant


The number of clusters of a certain size formed is a function of the total number of atoms in the system, the free energy to create a cluster (of that size), and the temperature. The number of clusters increases with increasing temperature.

Plugging in for , we get

The rate of addition of one extra atom to the critical nucleus as estimated by the Volmer-Weber theory is


where A is a term that incorporates the shape factor of the atoms and the area to which atoms can join, and the vibrational frequency of the particles.
And Q is the activation energy for atomic migration.

This term gives us the diffusion of the atoms to the site of nucleation. However, a problem with the Volmer Weber theory is that it ignores formation of particles of r>r* and assumes that size distribution is maintained (fluctuations are occurring fast)

The rate of nucleation can be expressed as:


Where:
  • γ is the surface tension
    Surface tension
    Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...

    .
  • ΔHs is the enthalpy
    Enthalpy
    Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

     per unit volume.
  • Tm is the melting temperature.
  • Θ is the wetting angle.


At very low temperatures, rate of diffusion is low. As temperature increases, the rate of diffusion increases; molecules are able to get to the site of nucleation at a fast enough rate to promote growth of the nucleus. At temperatures significantly below melting temperature, fluctuation of molecules is very low; the molecules are in a low energy state and do not have enough energy to move around and nucleate. Nucleation rate is dominated by diffusion. However, as temperature increases, molecular fluctuations increase and molecules tend to escape from the nucleus, causing a decreased rate of nucleation.

The time required for steady state nucleation is known as the time-lag and can be found by

Where:
  • a is the average particle size.

The spinodal region

Phase transition processes can also be explained in terms of spinodal decomposition
Spinodal decomposition
Spinodal decomposition is a mechanism by which a solution of two or more components can separate into distinct regions with distinctly different chemical compositions and physical properties...

 where phase separation is delayed until the system enters the unstable region where a small perturbation in composition leads to a decrease in energy and thus spontaneous growth of the perturbation. This region of a phase diagram is known as the spinodal region and the phase separation process is known as spinodal decomposition and may be governed by the Cahn–Hilliard equation
Cahn–Hilliard equation
The Cahn–Hilliard equation is an equation of mathematical physics which describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component...

.

Classical nucleation

Due to the many unjustified assumptions made by the classical nucleation theory (CNT), it has limited applicability to the solution of practical problems when compared with experimental data. The limitations arise mainly because the CNT assumes that macroscopic properties of molecules can be applied to microscopic activities. This can be a major drawback when dealing with characteristics such as density, surface tension, and saturated vapour pressure of clusters consisting of only several tens of molecules. The classical nucleation theory also does not take into consideration the interaction of particles around the nuclei, which leads to thermodynamics.

Modifications

In the last five decades, due to the ever- increasing accuracy of experimental results, many scientists and engineers have worked to improve and develop more accurate nucleation models. One of the versions of the classical theory which has gained acceptance is the self-consistent theory (SCT) of nucleation. According to this theory,


Where:
  • ΔG is critical free energy needed.
  • kB is the Boltzmann constant
  • S=pv/po is the supersaturation
    Supersaturation
    The term supersaturation refers to a solution that contains more of the dissolved material than could be dissolved by the solvent under normal circumstances...

  • s is the surface area
    Surface area
    Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...

     of a monomer


Under this theory, the nucleation rate becomes


Where I is the rate calculated from the classical nucleation theory. The exponential coefficient in the equation takes into account the surface energy of the monomer.

Another popular modern nucleation theory is the Dillmann-Meier theory where the change in free energy is expressed as


Where:
  • τ,kn and qo are adjustable coefficients of the model
  • V is the volume of the system


The coefficient kn reflects the difference between the surface energies of the cluster and the macroscopic liquid droplet and is represented in the first term of the equation. The second and third term takes into account the translational, vibrational and rotational degrees of freedom to the free energy of the droplet. Lastly the fourth term takes into account relaxation of the metastable state. Many scientists believe this equation takes into account all significant contributions to the energy of cluster formation.

Although these modern modifications to the classical nucleation theory improve the applicability of the model, further exploration is still going on to either improve or develop a better model that can be applied to a wider range of conditions.

Modern technology

Nucleation is a topic of wide interest in many scientific studies and technological processes. It is used heavily in the chemical industry for cases such as in the preparation of metallic ultradispersed powders that can serve as catalysts. For example, platinum deposited onto TiO2 nanoparticles catalyses the liberation of hydrogen from water. It is also an important factor in the semiconductor industry as the gap width in semiconductors is influenced by the size of metal nanoclusters.

Experimental

It is sometimes difficult to experimentally measure nucleation rate. The supercooling necessary to get a measurable rate may be at a temperature too low to allow for measurable growth rates. Gustav Tammann developed a method, known as the Tammann or “development” method.

In this method, crystals are nucleated at a low temperature Tn and then grown at a higher temperature Tg. For validity of this method, the nucleation rate, I , has to be greater at the nucleation temperature Tn than at the growth temperature Tg; I(Tn)>> I(Tg), and the growth rate U must be greater at the growth temperature than at the nucleation temperature Tn ; U(Tg) >> U(Tn). Since the clusters are heated to a larger temperature with a larger critical radius requirement, clusters no longer meet the critical radius requirement and remelt. A method to heat the particles carefully must be used.

Koster proposed a method for nucleation of metallic glasses.
This method considers the sizes of different crystals and attempts to determine when they were formed using data of their growth rates. It can be used for both homogeneous and heterogeneous nucleation.
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