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Order Parameters
Continuous and Discontinuous Transformations
Free Energy Changes for Conserved and Non-conserved Order Parameters
Free Energy Density and Diffuse Interfaces
Continuous Phase Transformations--Kinetics
The functional gradient is the starting point for the kinetic equations for conserved and non-conserved parameter fields. From an integral over the homogeneous free energy density and a gradient energy term:
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(23-1) |
This is the free energy for a domain
for an arbitrary field
representing an order parameter.
Supposing the the order parameter field is changing (or, flowing) with
velocity field ,
the free energy as a function of time is
and the instantaneous rate of total free energy change:
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(23-2) |
from this equation, it follows that the fastest1decrease
in total free energy if the flow field
is chosen so that it is `parallel' to (minus) the functional gradient
which is defined by:
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(23-3) |
Kinetics of Non-conserved Order Parameters:
The Allen-Cahn Equation
For a non-conserved order parameter ,
Eq.
is the local rate of increase of free energy for a small change
;
therefore
is the driving force to change
.
No long-range diffusion is required (in other words, the order
parameter can change with no flux of order into an element
).
Therefore, assuming kinetics that are linear in the driving force:
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(23-4) |
which is known as the Allen-Cahn equation for describing order-disorder kinetics. It is also called Model A or the non-conserved Ginsberg-Landau equation.
Allen-Cahn: Critical Microstructural Wavelengths
Consider a system where
has two minima at
:
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(23-5) |
where
is the height at the unstable saddle point at
.
Suppose the system is initially uniform at unstable point
(for instance, the system may have been quenched from a higher
temperature, disordered state and
represent two equivalent ordering variants). If the system is
perturbed a small amount by a planar perturbation in the
-direction,
.
Putting this and Eq. 23-5
into Eq. 23-4, and
keeping the lowest order terms in
:
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(23-6) |
so that
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(23-7) |
The perturbations grow if
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(23-8) |
which is approximately equal to the interface width.
Note that the amplification factor is a weakly increasing function of
wavelength (asymptotically approaching
at long wavelengths). This would predict that the longest wavelengths
would dominate the morphology of an order-disorder phase transition.
However, the probability of finding a long wavelength perturbation is a
decreasing function of wavelength and this also has an effect on
morphology.
The Kinetics of Conserved Parameters:
The Cahn-Hilliard Equation
Because
is a (locally) conserved parameter, the flux of
from one volume element to its neighbor will affect the kinetics.
is guaranteed to be conserved if
is the divergence of a flux.
Equation
is the local increase of free energy density due to a local addition
.
The flux is assumed to be linear in the gradient of Eq.
:
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(23-9) |
This is equivalent to the linear assumption in Fick's law. The
proportionality factor
is related to the interdiffusion coefficient. However,
is necessarily positive.
Therefore, the local rate of increase of the composition is given by (minus) the divergence of the flux:
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(23-10) |
if
is constant, then
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(23-11) |
which is known as the Cahn-Hilliard equation describing the kinetics of spinodal decomposition. It is also called Model B or the conserved Ginsberg-Landau equation.
The first term on the right-hand side of Eq. 23-11 looks like the
classical diffusion equation in regions where
can be reasonably approximated by a quadratic function, (for instance
near the minima of
).
The fourth-order term has the effect of stabilizing the shortest
wavelengths when
,
as discussed below.
Cahn-Hilliard: Critical and Kinetic Wavelengths
Consider the following function as an approximation to the regular solution model:
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(23-12) |
which has minima at concentrations
and
and a maximum of height
at
.
Suppose we have an initially uniform solution at
and that we perturb the concentration with a small plane wave:
.
Putting this into Eq. 23-11
and keeping the lowest-order terms in
,
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(23-13) |
Therefore any wavelength
will grow if
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(23-14) |
Taking the derivative of the amplification factor in Eq. 23-13 with respect to
and setting it equal to zero, we find the fastest growing wavelength:
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(23-15) |
It is expected that domain size in the early stage of spinodal
decomposition will be approximately .
Note that this approach for conserved order parameter is analogous to the case of kinetic and thermodynamic stability of a cylinder with axial perturbations:
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Simulations
Below are simulations of Eqs. 23-4 and 23-11 with initial conditions taken as a small perturbation about the unstable (or saddle) point.
Can you determine, by observation, which simulation corresponds to which type of kinetics?
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