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\centerline{\bf Coarsening dynamics of the convective Cahn-Hilliard 
equation}
\centerline{\bf and faceted crytal growth}


\bigskip

The coarsening dynamics of a faceted {\em vicinal} crystalline 
surface growing into its melt by {\em attachment kinetics} is 
considered. The convective Cahn-Hilliard equation ($\mathcal{CCH}$) 
is derived as a small amplitude expansion of such surface evolutions
restricted to 1-D morphologies. It takes the form
\begin{align}  \tag{$\mathcal{CCH}$}
q_t - \varepsilon q q_x = \left( \hat{W}^{\,\prime} (q) 
- q_{xx} \right)_{xx},
\end{align}
where the local surface slope $q(x,t)$ serves as the order parameter,
subscripts denote partial derivative with respect to
time $t$ and space $x$ respectively, and $\ ^{\,\prime}$ denotes the
$q$-derivative. 
The {\em effective free energy} $ \hat{W}(q) $ takes the form of a 
symmetric {\em double well} with minima at $ q=  \pm 1 $,
thereby capturing the anisotropy of the crystal surface energy.
Also, the dimensionless small parameter $\varepsilon$
multiplying the convective term $q q_x$ is a 
dimensionless measure of the growth strength.


A {\em sharp interface} theory for $\mathcal{CCH}$ is derived through 
a matched asymptotic analysis.
The theory predicts a nearest neighbor interaction
between two non-symetrically related phase boundaries ({\em kink} and 
{\em anti-kink}),
whose characteristic separation  $\mathcal{L}_{\mathcal{M}}$ grows
as coalescing kink/anti-kinks annihilate one another.
Theoretical predictions on the resulting (skew-symetric) 
coarsening dynamical system 
{$\mathcal{CDS}$} include
\begin{itemize}
\item The  characteristic length  $\mathcal{L}_{\mathcal{M}} \sim t^{1/2} 
$,
provided $\mathcal{L}_{\mathcal{M}}$
is appropriately small with respect to the {\em Peclet} length scale
$\mathcal{L}_{\mathcal{P}}$.
\item Binary coalescence of phase boundaries is impossible
\item Ternary coalescence may only occur through the 
{\em kink-ternary} interaction; two kinks meet an anti-kink resulting in a 
kink.
\end{itemize}
Direct numerical simulations performed on both $\mathcal{CDS}$ and 
{$\mathcal{CCH}$} confirm each of these predictions.
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Last, a linear stability analysis of $\mathcal{CDS}$ identifies a 
{\em pinching} mechanism
as the dominant instability, which in turn leads to
kink-ternaries.
We propose a self-similar period-doubling {\em pinch ansatz}
as a model for the coarsening process, from which an analytical coarsening 
law
for the characteristic length scale $\mathcal{L}_{\mathcal{M}}$ emerges.
It predicts both the scaling constant $c$ of the $t^{1/2}$ 
regime, i.e.,  $ \mathcal{L}_{\mathcal{M}}$ $ = c\  t^{1/2} $ , as 
well as the crossover to logarithmically slow coarsening as
$\mathcal{L}_{\mathcal{M}}$ crosses $\mathcal{L}_{\mathcal{P}} $.
Our analytical coarsening law stands in good qualitative agreement 
with large scale numerical simulations that have been 
performed on $\mathcal{CCH}$. 

\smallskip


\noindent 
In part, joint work with Felix Otto and Stephen H. Davis.

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