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__Diffuse Interface Approximation__

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__Functional Gradient__

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__Kinetics for Non-conserved Order Parameters__

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__Kinetics for Conserved Order Parameters__

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__Nucleation__

The transformation of a metastable material by the
growth of initially small fluctuations (*e.g.,* composition fluctuations in
the case of spinodal decomposition) was treated as a continuous transformation.

Some metastable systems are stable with respect to infinitesimal fluctuations, but are unstable to a perturbation in the form of a finite fluctuation. In such systems, the transformation does not proceed until a finite fluctuation occurs. If the necessary fluctuation is so large that its probability of it occurring in any observable time is effectively zero, then the system is kinetically `frozen' in its metastable state. Many, if not most, engineering materials are in such non-equilibrium states.

Each of these finite fluctuations is called a *nucleus*;
the metastable medium is called the *matrix.*
The minimum size nucleus required for continued spontaneous growth
is called a *critical nucleus* and all the fluctuations
smaller than the critical nucleus are called *sub-critical nuclei.*
The process by which the sub-critical nuclei change their size
by the (energetically favorable) loss of its atoms to the matrix and
the improbable attachment of new matrix atoms onto the sub-critical
nucleus is called *nucleation.*

Nucleation is commonly observed in carbonated beverages. When a beverage container is capped, the amount of dissolved CO that is in equilibrium at capped pressure is greater than the amount of dissolved CO in equilibrium at atmospheric pressure. The bubbles of CO that form in an opened beverage and float to the top are the result of nucleation process.

Common experience also demonstrates that there are two different categories
of nucleation.
The nucleation of critical nuclei at defects such as surface imperfections is
called *heterogeneous nucleation*;
nucleation that occurs randomly throughout the volume of the metastable matrix
is *homogeneous nucleation.*

__Nucleation Regimes__

Nucleation is what precedes growth by phase transformation.
Consider the case of a solid solution of phase at a
composition that is metastable with respect to the creation
of an -phase.^{1}

A prototypical case can be illustrated with a phase diagram and free energy curve.

A useful theory of nucleation should predict at least two quantities:

**Incubation Time**- The expected delay , after quenching, before critical nuclei form in sufficient quantities for experimental observation.
**Nucleation Rate**- The rate, , at which critical nuclei form per unit volume after incubation.

These quantities are related to the so-called nucleation regimes:

__Homogeneous Nucleation: Simple Theory__

It is possible to formulate a very simple theory for the size of a critical nucleus from macroscopic thermodynamic quantities: interfacial surface tension (energy/area), and free energy change per unit volume nucleus (energy/volume). This simple theory works remarkably well despite the fact that, at the small sizes that critical nuclei are formed (10-50Å), extrapolation of and is questionable. Another advantage is that the model provides a useful understanding of the physical quantities that control nucleation.

An -phase nucleus forming in a -matrix has
only two contributions to its energy: the decrease in energy
due to the volumetric driving force, , and an increase in
energy due to surface tension,
.^{2}

For a spherical isotropic nucleus of radius , the total free energy is:

Suppose nuclei are formed by random fluctuations and a distribution of
nuclei sizes develops.
Any one of the nuclei having a radius such that
can decrease its free energy continously by growing (i.e., increasing
its radius); all other nuclei cannot grow
continuously, but may grow by improbable events such as random attachment of
add-atoms.
Thus, the size of the critical nucleus must satisfy
;
it is the radius that maximizes
^{3}

Ignoring differences in heat capacities between the two phases and any temperature dependence of surface tension, so that the critical size decreases as and the nucleation barrier decreases as .

The growth of subcritical nuclei increases the nucleus free energy with each increment of nucleus size. The process will be treated as a sequence of activated states similar to the activation process for a vacancy exchange as described in Chapter 6 of KPIM. Assuming that the nucleation occurs at constant pressure and temperature, the nuclei sizes will be distributed with a probability proportional to where is the total free energy of a nucleus containing atoms. Equation 24-2 can be converted using to

__A Model for the Steady-State Nucleation Rate__

A subcritical nucleus has a driving force which tends to make it shrink and dissolve back into the matrix. The development and incremental growth of subcritical nuclei is assumed to be a thermally activated process. The steady-state nucleation regime subsists by the development of a steady-state size distribution of subcritical clusters. At steady-state, the source of the material to produce subcritical clusters is sufficient to replenish the material that is lost as a critical cluster becomes a stable precipitate and grows. The rate at which subcritical clusters reach their critical size must be the steady-state nucleation rate, and can be experimentally determined from the slope in Fig. 24-2.

The steady-state nucleation rate depends on the distribution of subcritical nuclei.

The rate of creation of critical nuclei can be extended to subcritical nuclei as follows. The rate at which subcritical nucleus of size (per unit volume) are created must must be related to the concentration of nuclei of size that grow by one unit minus the rate at which those of size lose one unit:

where is the rate of a successful jump across the nucleus interface and is the number of atoms adjacent to the interface.

We assume that there is a steady-state distribution for which the ``flux'' must vanish, so^{4}

where is the steady-state concentration of nuclei of size . It is reasonable that the kinetic factors do not depend on equilibrium, so Eq. 24-5 can be solved and one of the kinetic factors in Eq. 24-4 can be removed, and an expression for a system away from steady-state can be obtained:

(24-6) |

and this can be approximated by a derivative of a continuous function:

(24-7) |

Therefore the rate of change of the concentration of nuclei of size must be related to the divergence of this flux:

which is called the Zeldovich equation and is very similar to the diffusion equation except that the spatial variable is replaced with a nucleus size .

At steady-state, can be integrated with respect to . The integration constant must be and not a function of :

(24-9) |

which can be integrated again from one particle (where it may be assumed that ) to an infinite number of particles where as .

Several approximations must be made to replace the first integral in Eq. 24-10 with simple forms. The integrand over is large only when , which is small near --so the limits of integration can be extended to without addition of significant error. The thermally activated concentration where is the number of possible homogeneous nucleation sites per volume and the nucleation barrier can be expanded around the critical size:

(24-11) |

where, if the simple approximation in Eq. 24-3 is used

(24-12) |

With these approximations and the integration can be carried out and

(24-13) |

where is the Zeldovich factor given by the result of approximating the integral in Eq. 24-10

(24-14) |

For the simple model in Eq. 24-3:

(24-15) |

Typical experimental values of .

__Model for the Nucleation Incubation Time__

Considerations of thermally activated subcritical nuclei resulted in a diffusion equation for the rate of change of the density of nuclei of size in Eq. 24-8 with an effective diffusivity given by .

The incubation time is approximately the amount of time before particles begin reaching the critical size. The nucleation and growth process throughout all regimes implies that the particle size distribution is changing over time:

This diffusion equation could be solved for an initial distribution of clusters with a fixed concentration at small sizes () and zero flux at to yield a characteristic time when appreciable concentrations appear at .

However, the incubation time can be approximated with a random walk model as follows. Near , is nearly flat and the rate of the number of particles crossing can be approximated by the root-mean-square `displacement' relation for a one-dimensional random walk: where is the `distance' on either side of the maximum that can be considered to be flat. Approximating the by the value at which

,

(24-16) |

so that the characteristic incubation time goes like .