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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

3.21 Spring 2002: Lecture 23

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:

$\displaystyle F[y(x)] = \int_\Omega ( f(y) + \frac{K}{2} \nabla y \cdot \nabla y) dV$ (23-1)

This is the free energy for a domain $ \Omega$ for an arbitrary field $ y(\vec{x})$ representing an order parameter.

Supposing the the order parameter field is changing (or, flowing) with velocity field $ v(\vec{x})$, the free energy as a function of time is $ F(y + vt)$ and the instantaneous rate of total free energy change:

$\displaystyle \frac{\partial F}{\partial t} \ensuremath{\left.\mbox{\rule{0pt}{16pt}}\right\vert}_{t=0} = \int_\Omega ( f'(y) - K \nabla^2 y ) v dV$ (23-2)

from this equation, it follows that the fastest1decrease in total free energy if the flow field $ v(\vec{x})$ is chosen so that it is `parallel' to (minus) the functional gradient $ -\delta F/\delta y$ which is defined by:

$\displaystyle \frac{\delta F}{\delta y} \equiv f'(y) - K \nabla^2 y$ (23-3)

Kinetics of Non-conserved Order Parameters:
The Allen-Cahn Equation

For a non-conserved order parameter $ \eta(\vec{x})$, Eq. [*] is the local rate of increase of free energy for a small change $ \delta \eta(\vec{x})$; therefore $ -(f'(\eta) - K_\eta \nabla^2 \eta)$ is the driving force to change $ \eta$. No long-range diffusion is required (in other words, the order parameter can change with no flux of order into an element $ dV$). Therefore, assuming kinetics that are linear in the driving force:

$\displaystyle \frac{\partial \eta}{\partial t} = M_\eta [K_\eta \nabla^2 \eta - f'(\eta)]$ (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 $ f(\eta)$ has two minima at $ \eta = \pm 1$:

$\displaystyle f(\eta) = f_s ((1+\eta)(1-\eta))^2$ (23-5)

where $ f_s$ is the height at the unstable saddle point at $ \eta=0$. Suppose the system is initially uniform at unstable point $ \eta=0$ (for instance, the system may have been quenched from a higher temperature, disordered state and $ \eta = \pm 1$ represent two equivalent ordering variants). If the system is perturbed a small amount by a planar perturbation in the $ z$-direction, $ \eta(\vec{x}) = \delta(t) \sin(\omega z)$. Putting this and Eq. 23-5 into Eq. 23-4, and keeping the lowest order terms in $ \delta(t)$:

$\displaystyle \frac{d \delta(t)}{dt} = M_\eta ( 4 f_s - K_\eta \omega^2) \delta(t)$ (23-6)

so that

$\displaystyle \delta(t) = \delta(0) \exp[ M_\eta ( 4 f_s - K_\eta \omega^2 )t]$ (23-7)

The perturbations grow if

$\displaystyle \lambda > \lambda_{crit} = \pi \sqrt{\frac{K_\eta}{f_s}}$ (23-8)

which is approximately equal to the interface width.

Note that the amplification factor is a weakly increasing function of wavelength (asymptotically approaching $ 4 M_\eta f_s$ 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 $ c(x)$ is a (locally) conserved parameter, the flux of $ c$ from one volume element to its neighbor will affect the kinetics. $ c(\vec{x})$ is guaranteed to be conserved if $ \partial c/\partial t$ is the divergence of a flux.

Equation [*] is the local increase of free energy density due to a local addition $ \delta c(\vec{x})$. The flux is assumed to be linear in the gradient of Eq. [*]:

$\displaystyle \vec{J}_c = -M_c \nabla [ f'(c(\vec{x})) - K_c \nabla^2 c]$ (23-9)

This is equivalent to the linear assumption in Fick's law. The proportionality factor $ M_c$ is related to the interdiffusion coefficient. However, $ M_c$ is necessarily positive.

Therefore, the local rate of increase of the composition is given by (minus) the divergence of the flux:

$\displaystyle \frac{\partial c}{\partial t} = \nabla \cdot M_c \nabla [ f'(c(\vec{x})) - K_c \nabla^2 c(\vec{x})]$ (23-10)

if $ M_c$ is constant, then

$\displaystyle \frac{\partial c}{\partial t} = M_c [\nabla^2 f'(c(\vec{x})) - K_c \nabla^4 c(\vec{x})]$ (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 $ f(c)$ can be reasonably approximated by a quadratic function, (for instance near the minima of $ f$). The fourth-order term has the effect of stabilizing the shortest wavelengths when $ f''< 0$, as discussed below.

Cahn-Hilliard: Critical and Kinetic Wavelengths

Consider the following function as an approximation to the regular solution model:

$\displaystyle f(c) = \frac{16 f_s}{(c_\beta - c_\alpha)^4} [ (c - c_\alpha)(c - c_\beta) ]^2$ (23-12)

which has minima at concentrations $ c_\alpha$ and $ c_\beta$ and a maximum of height $ f_s$ at $ c = c_{avg} \equiv (c_\alpha + c_\beta)/2$. Suppose we have an initially uniform solution at $ c = c_{avg}$ and that we perturb the concentration with a small plane wave: $ c(\vec{x}) = c_{avg} + \epsilon(t) \sin \omega z$. Putting this into Eq. 23-11 and keeping the lowest-order terms in $ \epsilon(t)$,

$\displaystyle \frac{d \epsilon}{dt} = \frac{M_c \omega^2}{(c_\beta - c_\alpha)^2} [16 f_s - K_c \omega^2(c_\beta - c_\alpha)^2] \epsilon$ (23-13)

Therefore any wavelength $ \lambda$ will grow if

$\displaystyle \lambda > \lambda_{crit} \equiv \frac{\pi}{2} (c_\beta - c_\alpha) \sqrt{\frac{K_c}{f_s}}$ (23-14)

Taking the derivative of the amplification factor in Eq. 23-13 with respect to $ \omega$ and setting it equal to zero, we find the fastest growing wavelength:

$\displaystyle \lambda_{max} = \sqrt{2} \lambda_{crit} = \frac{\sqrt{2}\pi}{2} (c_\beta - c_\alpha) \sqrt{\frac{K_c}{f_s}}$ (23-15)

It is expected that domain size in the early stage of spinodal decomposition will be approximately $ \lambda_{max}$.

Note that this approach for conserved order parameter is analogous to the case of kinetic and thermodynamic stability of a cylinder with axial perturbations:

Figure 23-1: The amplification factor for spinodal decomposition and order-disorder reactions.
\begin{figure}\resizebox{6in}{!} {\epsfig{file=figures/Spinodal/lambda_crit-lambda_max.eps}} \end{figure}


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?

Figure 23-2: A simulation-can you determine which type of kinetics? If you are viewing in HTML, click on the figure to see the simulation.
\begin{figure}\resizebox{6in}{!} {\epsfig{file=figures/Spinodal/A.eps}} \end{figure}

Figure 23-3: A simulation-can you determine which type of kinetics? If you are viewing in HTML, click on the figure to see the simulation.
\begin{figure}\resizebox{6in}{!} {\epsfig{file=figures/Spinodal/B.eps}} \end{figure}

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W. Craig Carter 2002-04-08