__Last time: Particle coarsening.--II__

__Elements of mean-field theory for DCC__

- Growth rate of a particular particle of radius
from supersaturated solution

- The particle-size distribution function and its
associated continuity equation

- Key result (1): Steady-state (normalized) particle-size
distribution function

- Key result (2): Functional dependence of mean particle size on time

__Complications of real systems__

- Nonzero volume fraction; particle-particle interactions

- Coherency stresses

- Applied stresses

__Experimental study of coarsening in semi-solid Pb-Sn alloys
(Hardy and Voorhees 1988__

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__Today: Phase Transformations: Overview__

__Metastability, instability, and mechanisms__

__First-order and second-order transitions__

__Free energy functions; conserved and nonconserved variables__

__Spinodal decomposition--I.__

- Diffusion within the spinodal

__Phase Transformations: Overview__

__Metastability, instability, and mechanisms__

*A phase transformation can occur when a system has an accessible state of lower
free energy. The mechanism of the transformation is critically dependent
on whether the starting state is metastable or unstable.
*

*An unstable system can transform by making changes that are small
in degree but large in extent. Such situations lead to mechanisms that are
called continuous transformations. The main categories of continuous
transformations in materials are spinodal decomposition and
continuous ordering.
*

*
*

*A metastable system can transform by making changes that are large
in degree but small in extent. Such situations require
nucleation of the
new phase. After nucleation takes place, a new particle can grow
until it either
impinges with another particle, or supersaturation of the surrounding material
is depleted.
*

*
*

__First-order and second-order transitions__

*Ehrenfest proposed a useful scheme for classification of phase
transformations based on discontinuities in derivatives of the free
energy function that are characteristic of the transformation.
Simply put, the order or a phase transformation is the lowest
order of the derivative of that shows a discontinuity.
*

*
*

*Examples: melting; ordering in brass
*

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*

**Decomposition into Phases: Conserved Fields**-
**Figure 27-1:**Decomposition requires long-rang diffusion. Such a transformation requires flux of a conserved field, like composition , which has an integral that is conserved for a closed system. **Order-Disorder: Nonconserved Fields**-
**Figure 27-2:**The phase change does not require long-rang diffusion. Such a transformation involves local changes in some field, like the order parameter , which is not conserved for a closed system.

__Order Parameters and Phase Transformations__

*Consider a simple one component phase transformation:
*

*We can express the transformation near the transition as a Landau
expansion
*

(27-1) |

*where might be some measure of a ``hidden parameter'' such as
the diffuseness of a peak in the atomic radial-distribution function.
*

*The equilibrium value of is given by
*

(27-2) |

*We will use functions like to follow evolution towards
equilibrium values
.
*

__Spinodal decomposition__

__The chemical spinodal and "uphill diffusion"__

*Recall that
*

(27-3) |

*Note that since,
, that the
diffusivity has the same sign as the second derivative of the
free energy:
*

(27-4) |

*Consider the following free-energy curve and resulting
phase diagram:
*

*In region III,
, how does the diffusion equation behave
when
? Recall that for initial conditions
the diffusion equation has solution:
*

(27-5) |

*This will be very badly behaved for small wavelengths and give no end
of trouble. It is ill-posed.
*

__Gradient Energy__

*How to fix this problem and calculate a governing equation inside the
spinodal region?
*

*Consider the following profile or variation in field:
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*What kind of penalties can be imposed that ``mimic'' surface energy?
*

*Should the penalty depend on the whether the field is increasing
left-to-right or increasing
right-to-left?
*

*For inhomogenous fields, expand the free energy about its homogenous value:
*

(27-6) |

* is the gradient energy coefficient, it introduces surface energy into the
free energy and will ``regularize'' the diffusion equation within
(and applies outside
as well) the spinodal.
*

*For one-dimensional variations, the free energy density is:
*

__Theory of diffuse interfaces__