Last time

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:

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

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

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

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

$$f(\eta) = f_s ((1+\eta)(1-\eta))^2$$ (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 $\delta(t)$:

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

so that

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

The perturbations grow if

$$\lambda > \lambda_{crit} = \pi \sqrt{\frac{K_\eta}{f_s}}$$ (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. :

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

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

if $M_c$ is constant, then

$$\frac{\partial c}{\partial t} = M_c [\nabla^2 f'(c(\vec{x})) - K_c \nabla^4 c(\vec{x})]$$ (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:

$$f(c) = \frac{16 f_s}{(c_\beta - c_\alpha)^4} [ (c - c_\alpha)(c - c_\beta) ]^2$$ (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 $\epsilon(t)$,

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

Therefore any wavelength $\lambda$ will grow if

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

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

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?

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.

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.

• About this document ...