Coalescence in the 1D Cahn–Hilliard model

Simon Villain-Guillot

► To cite this version:

Simon Villain-Guillot. Coalescence in the 1D Cahn–Hilliard model. Journal of Physics A: Mathematical and General (1975 - 2006), 2004, 37 (27), pp.6929-6941. 10.1088/0305-4470/37/27/005 hal-00000956v2

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Submitted on 2 Apr 2004

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Coalescence in the 1D Cahn-Hilliard model
Processus de coalescence pour l'équation de Cahn-Hilliard
unidimensionnelle

Simon Villain-Guillot

Centre de Physique Moléculaire Optique et Hertzienne,
Université Bordeaux I, 33406 Talence Cedex, France
s.villain@cpmoh.u-bordeaux1.fr

Abstract

We present an approximate analytical solution of the Cahn-Hilliard equation describing the coalescence during a first order phase transition. We have identified all the intermediate profiles, stationary solutions of the noiseless Cahn-Hilliard equation. Using properties of the soliton lattices, periodic solutions of the Ginzburg-Landau equation, we have construct a family of ansatz describing continuously the process of destabilization and period doubling predicted in Langer's self similar scenario [1].

Pacs numbers : 05.45.Yv, 47.20.Ky, 47.54.+r

I. INTRODUCTION

When a homogenous system departs suddenly from equilibrium, the fluctuations around the initial ground state are linearly amplified and the homogenous phase can for example spontaneously separate into two different more stable states. The interfaces which delimit the numerous resulting monophasic domains will interact with each other, either giving rise to formation of a complex pattern, or merging into a single interface when the domains of the same state slowly coalesce, minimizing the overall interfacial energy. It results then in only two well separated domains. This process of first order phase transition arises particularly for binary mixtures [2] alloys [3], or vapor condensation [4].

It can either initiate via a nucleation process, where the homogenous state is put sud-
denly in a metastable configuration, and an energy barrier has to be crossed before the
transition appears. Or via a spinodal decomposition when the system is led into a linearly
unstable configuration. In this latter case, the leading instability selects a modulation of the
order parameter at a well defined length scale, which will grows and, due to non-linearities,
saturates. The resulting pattern is composed of well defined interfaces delimiting domains
containing one of the two stable phases. Remarkably, this intermediate stage conserves the
modulation width, and the resulting stationary pattern is of almost the same length scale as
the one selected initially [5,6]. The dynamics finally ends with a much slower, self-inhibiting
process, dominated by the interactions between the interfaces. The different regions of each
phase coalesce in the so-called Ostwald ripening where the number of domains diminishes
whereas their typical size increases. The asymptotic state is decomposed into two domains,
one for each phase. In this article, by spinodal decomposition, we refer to the first stage of
the dynamics only, while coarsening will denote the second stage. Although this coarsening
dynamics is in fact already present, its influence can be often neglected during the first stage
of the process.

Hillert [3], Cahn and Hilliard [7] have proposed a model equation describing the segregation for a binary mixture. This model, known as the Cahn-Hilliard equation (C-H later on), belongs to the Model B class in Hohenberg and Halperin’s classification [8]. It is a standard model for phase transition with conserved quantities and has applications to phase transition in liquid crystals [9], segregation of granular mixtures in a rotating drum [10], or formation of sand ripples [11,12]. It is a partial differential equation to which a conservative noise is added to account for thermal fluctuations [13].

Figure (1) shows snapshots of a numerical integration of the (C-H) dynamics which rep-
resents the full phase transition process after a quench in temperature. Thermal fluctuations
were present in the initial conditions, but have been omitted in the dynamics. The three
main stages of the spinodal decomposition described above are clearly distinguished: first,
from Fig. (1 (a)) to Fig. (1 (b)), we observe the selection of a typical length scale for the
modulations, then, from Fig. (1 (b)) to Fig. (1 (c)), the non-linear growth and its satu-
ration. We note that the number of peaks has been almost conserved between these two
configurations. On the contrary, during the coarsening dynamics observed between Fig. (1
(c)) and Fig. (1 (d)) the typical length of the pattern is increasing while on the other hand
the amplitude of the modulation slowly growth to reach its asymptotic value.

An important activity has been devoted to the description of the dynamics of phase
transition, using both statistical methods or numerical simulations (for a review see [14]).
The late stage of the spinodal decomposition where the coarsening dynamics dominates
exhibits "dynamical scaling": the dynamics presents a self-similar evolution where time
enters only through a length scale $L(t)$, associated with a typical length of the domains
or the rate of decay of the inhomogeneities. For instance, scaling arguments and stability
criteria give the law $L(t) \sim t^{1/3}$ for spatial dimensions greater than one and a logarithmic
behavior for one dimension in the case of the (C-H) equation [14].

This last stage, as observed in two-dimensional demixion of copolymer [15] and as sug-
gested initially by Langer [1], can be described as a self similar process of synchronous fusion
and evaporation of domains. This observation motivated our work and the aim of this arti-
cle is to present a one dimensional ansatz describing continuously the coalescence process.
This ansatz is in the form of a one parameter family of symmetric profiles which interpo-
lates between two stationary states composed of homogeneous domains of length $\lambda/2$ and
$\lambda$. It allows to realize a self similar sequence of coalescence process in 1D, starting from the
regular micro phase separated states issued from the non-linear saturation of the spinodal
decomposition dynamics and ending with the single interface which characterize the infinite
time, thermodynamic stable state.

The paper is organized as followed: first, we present a brief review on general properties
of phase segregations and on the (C-H) model, mainly to fix the notations. We will repro-
duce briefly the original derivation by Cahn and Hilliard, restricting ourselves to the one
dimensional case. In part III, we present a family of symmetric solutions of the Ginzburg-
Landau equation which is used to study the dynamics of spinodal decomposition and to determine all the symmetric stationary state of the (C-H) dynamics. Then in part IV, the main original part of this work, we introduce a non-symmetric family of solutions of the (G-L) equation which is used to construct a continuous interpolation between two consecutive symmetric stationary states. After a study of the energy landscape associated with this ansatz, we finally discuss the numerical accuracy of our calculations. In the conclusion, we justify the hypothesis we have made and compare the suggested scenario with coalescence in real systems.

## II. THE CAHN-HILLIARD MODEL

The Cahn-Hilliard theory is a modified diffusion equation; it is a continuous conservative model for the scalar order parameter $\Phi$, which reads in its dimensionless form:

$$ \frac{\partial \Phi}{\partial t}(\mathbf{r}, t) = \nabla^2(\frac{\varepsilon}{2}\Phi + 2\Phi^3 - \nabla^2\Phi) + \xi(\mathbf{r}, t). \quad (1) $$

The real order parameter can correspond to the dimensionless magnetization in Ising ferromagnet, to the fluctuation of density of a fluid around its mean value during a phase separation or to the concentration in some region around $\mathbf{r}$ of one of the components of a binary solution. $\varepsilon$ is the dimensionless control parameter of the system; it is often identified to the reduced temperature ($\varepsilon = \frac{T-T_c}{T_c}$ where $T_c$ is the critical temperature of the phase transition). This equation, first derived by Cahn and Hilliard [7], has also been retrieved by Langer [1] from microscopic considerations. As written, the (C-H) equation does account for thermal fluctuations present in the system through a random white noise $\xi(\mathbf{r}, t)$, whose amplitude is proportional to the square root of the temperature of the system.

The homogeneous stationary solutions for the noiseless (C-H) equation are extrema of the effective Ginzburg-Landau potential $V(\Phi) = \frac{\varepsilon}{2}\Phi^2 + \Phi^4$ (G-L later on). For positive $\varepsilon$, there is only one homogenous solution $\Phi = 0$ which is linearly stable; for negative $\varepsilon$, the stationary solution $\Phi = 0$ undergoes a pitchfork bifurcation and three stationary solutions exist. $\Phi = 0$ is still a stationary solution, but it is now linearly unstable; two other symmetric solutions $\Phi = \pm\frac{\sqrt{-\varepsilon}}{2}$ are stable and have the same free energy $F = -\varepsilon^2/32$. Thus, a first order transition can be experienced by quenching the system suddenly from a positive reduced temperature $\varepsilon$ to a negative one. Spinodal decomposition is the resulting dynamics.

The stability of the solution $\Phi = 0$ can be studied by linearizing equation (1) around $\Phi = 0$ (i.e. neglecting the non linear term $\Phi^3$); considering $\Phi$ as a sum of Fourier modes:

$$ \Phi(\mathbf{r}, t) = \sum_{\mathbf{q}} \phi_{\mathbf{q}} e^{i\mathbf{q}\cdot\mathbf{r}+\sigma t} \quad (2) $$

where $\phi_q$ is the Fourier coefficient at $t = 0$, we obtain for the amplification factor $\sigma(q)$:

$$ \sigma(\mathbf{q}) = -(q^2 + \frac{\varepsilon}{2})q^2 \quad (3) $$

It shows immediately that $\Phi = 0$ is linearly stable for $\varepsilon > 0$ while a band of Fourier modes are unstable for negative $\varepsilon$, since $\sigma(\mathbf{q}) > 0$ for $0 < q < \sqrt{-\varepsilon/2}$. Moreover, the most
unstable mode is for $q_{C-H} = \sqrt{-\epsilon}/2$ (with $\sigma_{max} = \frac{\epsilon^2}{16}$). This wave number of maximum amplification factor will dominate the first stage of the dynamics; in particular, it explains why the modulations appear at length scales close to $\lambda_{C-H} = 2\pi/q_{C-H}$, the associated wave length. Later on, interfaces separating each domain interact through coalescence dynamics, causing $\langle \lambda \rangle$ to change slowly toward higher values [17,5]

We will now use known results on non-homogeneous solutions of the (G-L) equation to study both the saturation of the spinodal decomposition and the coalescence.

### III. STATIONARY STATES OF THE CAHN-HILLIARD DYNAMICS

#### A. Symmetric Soliton Lattice Solutions

For $\epsilon < 0$, there exists a stationary solution of the one dimensional (C-H) that relies on the two homogeneous phases $\Phi = \pm \frac{\sqrt{-\epsilon}}{2}$

$$ \Phi(x) = \frac{\sqrt{|\epsilon|}}{2} \tanh\left(\frac{\sqrt{|\epsilon|}x}{2}\right). \qquad (4) $$

Such a monotonic solution describes a continuum interface between the two stable homogeneous phases, and corresponds to the thermodynamically stable solution that ends the phase transition dynamics. But this is a particular member of a one parameter family of stationary solutions of the (G-L) equation

$$ \frac{\epsilon}{2}\Psi + 2\Psi^3 - \nabla^2\Psi = 0 \qquad (5) $$

These solutions, the so-called soliton-lattice solutions, are:

$$ \Psi_{k,\epsilon}(x) = k\Delta\text{Sn}(\frac{x}{\xi}, k) \text{ with } \xi = \Delta^{-1} = \sqrt{2\frac{k^2+1}{-\epsilon}} \qquad (6) $$

where $\text{Sn}(x, k)$ is the Jacobian elliptic function sine-amplitude, or cnoidal mode. This family of solutions is parametrized by $\epsilon^*$ and by the modulus $k \in [0, 1]$, or "segregation parameter". These solutions describe periodic patterns of periods

$$ \lambda = 4K(k)\xi, \text{ where } K(k) = \int_{0}^{\frac{\pi}{2}} \frac{dt}{\sqrt{1 - k^2 \sin^2 t}} \qquad (7) $$

is the complete Jacobian elliptic integral of the first kind. Together with $k$, it characterizes the segregation, defined as the ratio between the size of the homogeneous domains, $0.5 \times \lambda$, and the width of the interface separating them, $2 \times \xi$. The equation (7) and the relation $\xi = \Delta^{-1}$, enable to rewrite this family as:

$$ \Psi_{k,\lambda}(x) = \frac{4K(k) \cdot k}{\lambda} \text{Sn}(\frac{4K(k)}{\lambda}x, k). \qquad (8) $$

This family of profiles (or alternating interfaces) can be obtained exactly as a periodic sum of single solitons and antisolitons [18]

$$ \sum_{n} (-1)^n \tanh(\pi s(x - n)) = \frac{2k(s)K(s)}{\pi s} \text{Sn}(x, k) \text{ with } s = \frac{K(k)}{K(k')} \text{ and } k'^2 = 1 - k^2 \qquad (9) $$
## B. Antsatz for the Spinodal Decomposition Dynamics

The preceding family of profiles can be used to explore the spinodal decomposition dynamics. It can be associated with a micro phase separation, locally limited by the finite diffusion coefficient. For $k = 1$, $Sn(x, 1) = \tanh(x)$, we recover the usual single interface solution (4), of width $2/\sqrt{|\epsilon|}$; it is associated with a one soliton solution and corresponds to a strong, or macroscopic segregation. Note that $K(1)$ diverges; the solution

$$ \Psi_{1,\epsilon}(x) = \frac{\sqrt{|\epsilon|}}{2} \tanh\left(\frac{\sqrt{|\epsilon|}}{2}x\right). \qquad (10) $$

is thus the limit of infinite $s$, when the solitons, entering in relation (9), are far apart one each others. In the opposite limit (weak segregation regime), it describes a sinusoidal modulation

$$ \lim_{k \to 0} \Psi_{k,\epsilon}(x) = k \sqrt{\frac{|\epsilon|}{2}} \sin\left(\sqrt{\frac{|\epsilon|}{2}}x\right) = k \frac{2\pi}{\lambda} \sin\left(\frac{2\pi}{\lambda}x\right) = kq \sin(qx) \qquad (11) $$

It will correspond to the Fourier mode $q = \frac{2\pi}{\lambda}$ of the initial white noise, with an arbitrary small amplitude $\nu = kq$. Since experiences, numerical simulations and linear stability analysis show that $\lambda$, the spatial period of the pattern is constant during the whole spinodal decomposition process, we choose $\lambda$ to coincide with the most instable wave length obtain with the Cahn Hilliard linear approach, $\lambda = \lambda_{C-H} = \frac{4\pi}{\sqrt{-\epsilon_0}}$, where $\epsilon_0$ is the quench temperature. Thus, we obtain a one parameter family of profiles $\Psi^*(x, k) = \Psi_{k,\lambda_{C-H}}(x)$ which describe very well both the linear growth and the saturation. The dynamics is now reduced to the time evolution of the single free parameter: $k(t)$. Using equations (6) and (7), we find that $\lambda, k$ and $\epsilon$ are related to one another through the state equation

$$ \epsilon(k) = -2(1 + k^2) \left( \frac{4K(k)}{\lambda} \right)^2. \qquad (12) $$

So, this implicit equation tells us that if we fix $\lambda = \lambda_{C-H}$, the dynamics can also be reduced to the evolution of $\epsilon(t)$.

Given a periodic function $\Phi$ (obtained either from experimental data or numerical simulation of equation (5)) at time $t$, the ansatz assumes that it corresponds to a soliton lattice of the same period: i.e., there exists $k(t)$ such that $\Psi_{k,\lambda_{C-H}}(x) \sim \Phi(x, t)$ for each time $t$. For this purpose, we have developed three different algorithms, taking advantage of the general properties of the family of solutions $\Psi^*(x, k)$: either, $k$ can be deduced both from the amplitude of the oscillation equals to $4kK(k)/\lambda$, or from the relation $k = 1 - ((\Psi(\lambda/2, k)/\Psi(\lambda/4, k))^2 - 1)^2$; thirdly, a straightforward computation relates $k$ to the ratio of the two first Fourier modes of the soliton lattice $\Psi^*(x, k)$. We have observed that the three methods show in general similar results within an error of one percent.

In this approach, $\epsilon(t)$ can be then interpreted as a fictitious temperature or “local temperature” of the domains: it is the temperature extracted from the profile at a given time, using the correspondence between $\epsilon$ and $k$ of equation (12). For instance, at $t = 0$, the amplitude is small and we find that $k(0) = \frac{\nu\lambda_m}{2\pi} \to 0$ and thus $\epsilon^*(0) = 8\pi^2/\lambda^2$, different a priori from $\epsilon_0$ ($\epsilon^*(0) = \frac{\epsilon_0}{2}$ for $\lambda = \lambda_{C-H}$).
Somehow, the dynamics of (C-H) can be projected at first order onto a dynamics along the sub-family $\Psi^*(x, k) = \Psi_{k,\lambda_{C-H}}(x)$, which can be considered as an attractor of the solutions, i.e. the density profile of the system will evolve with time, staying always close to a function $\Psi^*(x, k)$. And using a solubility condition, it is possible to compute the full non linear part of this dynamics, the saturation of the spinodal decomposition, which leads the system in a well defined stationary state [6].

### C. Saturations of the Spinodal Decomposition Dynamics

According to the previous interpretation of the parameter $\epsilon$, as $\epsilon(t=0) = \frac{\epsilon_0}{2}$, the system is initially out of equilibrium. The dynamics will saturate when this fictitious temperature will reach the real thermodynamic one, i.e. the quench temperature $\epsilon_0$; that is, using equation of state (12) for $\lambda = \lambda_{C-H}$, when $k = k^s$ solution of the implicit equation :

$$2(1 + k^{s2})K(k^s)^2 = -\frac{\epsilon_0\lambda_{C-H}^2}{16} = \pi^2 \quad \text{that is} \quad k^s = 0.687 \quad (13)$$

Note that in this case, the width of the interface, which was initially, just after the quench, proportional to $\frac{2}{\sqrt{-\epsilon_0}}$ has now become proportional to $\frac{\pi}{\sqrt{-\epsilon_0 K(k^s)}} = \frac{\sqrt{2(1+k^{s2})}}{\sqrt{-\epsilon_0}} \approx \frac{1.7}{\sqrt{-\epsilon_0}}$: the segregation has slightly increased. Using linear stability analysis, Langer has shown that the profile thus obtained, $\Psi^*(x, k^s) = \Psi(x, k^s, \lambda_{C-H})$, is destroyed by stochastic thermal fluctuations and he has identified the most instable mode as an "antiferro" mode, leading to a period doubling. The result of this destabilization is another profile of alternate interface, where the length of the domains is now :$\lambda = 2\lambda_{C-H} = \frac{8\pi}{\sqrt{-\epsilon_0}}$. This means that the new stationary profile is given by $\Psi(x, k_1^s, 2\lambda_{C-H})$, where $k_1^s$ is solution of the implicit relation

$$2(1 + k_1^{s2})K(k_1^s)^2 = -\frac{\epsilon_0(2\lambda_{C-H})^2}{16} = 4\pi^2 = 8(1 + k^{s2})K(k^s)^2 \quad \text{that is} \quad k_1^s = 0.985 \quad (14)$$

The interface of this new profile is relatively sharper (the width of the interface is now proportional to $\frac{2\pi}{\sqrt{-\epsilon_0 K(k_2^s)}} = \frac{\sqrt{1+k_1^{s2}}}{\sqrt{2\sqrt{-\epsilon_0}}} \approx \frac{2\times0.99}{\sqrt{-\epsilon_0}}$) compare to the size of the homogeneous domains which has now double, see Fig. (2). Again, this new stationary profile turns out to be linearly instable with respect to an "antiferro" perturbation of period $4\lambda_{C-H}$.

Thus these families of profiles and instabilities enable to describe the one dimensional coarsening as a cascade of doubling process, leading from a pattern of wave length $\lambda_{C-H}$ composed of domains separated by interfaces to a single $\tanh(\frac{\sqrt{-\epsilon_0}}{2}x)$ interface separating two semi infinite domains. Each of these successive intermediate profiles can be described by an element of the above family of soliton lattice $\Psi(x, k_n^s, 2^n \times \lambda_{C-H})$. We thus have a family of segregation parameter $\{k_n^s\}$, which are determined by the implicit relations

$$2(1 + k_n^{s2})K(k_n^s)^2 = -\frac{\epsilon_0(2^n \lambda_{C-H})^2}{16} = \pi^2 2^{2n}. \quad (15)$$

We have found numerically for the first of them
$$
\begin{array}{l}
\begin{array}{l}
k_0^s = k^s = 0.6869795924 \\
k_1^s = 0.9851675587 \\
k_2^s = 0.99997210165 \\
k_3^s = 0.9999999999027
\end{array}
\qquad
\begin{array}{l}
k_0^s \Delta_0^s = 0.400 \sqrt{-\epsilon_0} \\
k_1^s \Delta_1^s = 0.496250 \sqrt{-\epsilon_0} \\
k_2^s \Delta_2^s = 0.499990 \sqrt{-\epsilon_0} \\
k_3^s \Delta_3^s = 0.49999846 \sqrt{-\epsilon_0}
\end{array}
\qquad (16)
\end{array}
$$

We see that {$k_n^s$} converges toward $k_\infty^s = 1$ (single interface case) ; meanwhile the amplitude of the modulation $k_n^s \Delta_n^s$ goes toward $\sqrt{|\epsilon_0|}/2$, as can be seen in the second column of the table above. For large $n$, we can conclude from the implicit relation (15) that the ratio of the domain size to the interface width characterized by $K(k_n^s)$ behaves as $\pi 2^{n-1}$. Each of the stationary profiles

$$
\Psi_n(x) = \Psi(x, k_n^s, 2^n \lambda_{C-H}) = \frac{\sqrt{-\epsilon_0} K(k_n^s) \cdot k_n^s}{2^n \pi} \mathrm{Sn}\left(\frac{\sqrt{-\epsilon_0} K(k_n^s)}{2^n \pi} x, k_n^s\right), \quad (17)
$$

for which the interface width is proportional to $\frac{2^n \pi}{\sqrt{-\epsilon_0} K(k_n^s)}$ (which tends to $\frac{2}{\sqrt{-\epsilon_0}}$, in agreement with $\tanh(\frac{\sqrt{-\epsilon_0}}{2}x)$), is identically destroyed by the Langer "antiferro" instability.

IV. AN ANSATZ FOR THE 1D COARSENING PROCESS

A. Non-symmetric soliton lattice Profile

In order to describe one step of the coalescence process, i.e. the dynamics that start from
$\Psi_n(x)$ and ends with the profile $\Psi_{n+1}(x)$, we will use another family of equilibrium profiles
[19], solutions of (G-L) equation, which write:

$$
\psi(a, k, x) = \frac{\alpha(a, k) - k/\sqrt{a}\beta(a, k)\mathrm{Sn}(4x\frac{K(k)}{\lambda}, k)}{1 - k/\sqrt{a}\mathrm{Sn}(4x\frac{K(k)}{\lambda}, k)} \quad (18)
$$

$$
\text{where } \alpha(a, k) = \frac{-2k^2/a+1+k^2}{((1+k^2)^2-12k^2+2(a+k^2/a)(1+k^2))^2} \text{ and } \beta(a, k) = \frac{2a-1-k^2}{((1+k^2)^2-12k^2+2(a+k^2/a)(1+k^2))^2}.
$$

It is still a periodic lattice of interfaces, but now, the mean value of the order parameter
is non zero (non symmetric case). It is controlled by the parameter $a \ge 1$: if $a$ goes infinity,
we recover the previous family of periodic profiles.

B. Ansatz for the continuous interpolation between two stationary states

If we choose $a$ to be equal to $1+k'$ (where $k'^2 = 1-k^2$), we can then construct symmetric profiles using the sum of two non-symmetric ones. Indeed, using Gauss' transformation (or descending Landen transformation [20]), which relates the soliton lattice of spatial period $2\lambda$ (and of modulus $k$) to the soliton lattice of period $\lambda$ (and of modulus $\mu = \frac{1-k'}{1+k'}$), we have

$$
1 - \frac{\sqrt{5-k^2}}{2} \left( \psi(k, x - \frac{\lambda}{2}) + \psi(k, x + \frac{\lambda}{2}) \right) = k \mathrm{Sn} \left( 2x \frac{K(k)}{\lambda} \right), \quad (19)
$$

$$
1 - \frac{\sqrt{5-k^2}}{2} \left( \psi(k, x - \lambda) + \psi(k, x + \lambda) \right) = (1-k') \mathrm{Sn} \left( (4x + 2\lambda) \frac{K(\mu)}{\lambda} \right), \quad (20)
$$

where $\psi(k, x) = \psi(1 + k', k, x)$. Thus, we then can show from equation (19) that
$$ K(k) \left[ 1 - \frac{\sqrt{5-k^2}}{2} \left( \psi(k, x - \frac{\lambda}{2}) + \psi(k, x + \frac{\lambda}{2}) \right) \right] = kK(k) \mathrm{Sn}(2x \frac{K(k)}{\lambda}, k). \quad (21) $$

This is the solution of the G-L equation of period $2\lambda$. Moreover, if we use the fact that

$$ K(k) = \frac{2}{1+k'} K(\mu) \text{ or } K(\mu) = \frac{1}{1+\mu} K(k) \quad (22) $$

we can write

$$ (1-k')K(k)\mathrm{Sn}((4x+2\lambda)\frac{K(\mu)}{\lambda}, \mu) = 2\mu K(\mu)\mathrm{Sn}((2x+\lambda)\frac{2K(\mu)}{\lambda}, \mu). \quad (23) $$

Then, using relation (20), the solution of the (G-L) equation of period $\lambda$ can be expressed as

$$ K(k) \left[ 1 - \frac{\sqrt{5-k^2}}{2} \left( \psi(k, x - \lambda) + \psi(k, x + \lambda) \right) \right] = 2\mu K(\mu) \mathrm{Sn}((2x+\lambda)\frac{2K(\mu)}{\lambda}, \mu). \quad (24) $$

So, we see that both the initial state $\Psi^*(x, k_{n-1}^s, 2^{n-1}\lambda_{C-H})$ and the final state $\Psi^*(x, k_n^s, 2^n\lambda_{C-H})$ of a step of the coalescence process can be described, modulo a phase shift, by the same function :

$$ \Phi(x, k, \phi) = \frac{4K(k)}{\lambda} \left[ 1 - \frac{\sqrt{5-k^2}}{2} \left( \psi(k, x - (1-\phi/2)\lambda) + \psi(k, x + (1-\phi/2)\lambda) \right) \right] \quad (25) $$

with $k = k_n^s$ and $\lambda = 2^n \lambda_{C-H}$. Therefore we can describe the coalescence by a transformation at constant segregation parameter $k$, while the degree of freedom $\phi$, associated with the relative phase between the two profiles, evolves in time from 0 to 1 according to the C-H dynamics.

This non-symmetric lattice of interfaces can be interpreted as a periodic sum of alternating single interfaces (kinks and antikinks). In the same spirit as relation (9), if one forget in the infinite sum every two out of four interfaces, one gets :

$$ \psi(x) \sim \sum_{p} \left[ \tanh(\pi s(x - 4 * p)) - \tanh(\pi s(x - 4 * p + 1)) \right]. \quad (26) $$

Then (see Fig.(3)) adding $\psi(x+2)$ to $\psi(x)$ enables to recover relation (9), while, after a translation, adding $\psi(x+1)$ and $\psi(x)$ gives the soliton lattice of double period, because of the cancelation of half of the interfaces (annihilation of kinks and antikinks).

If we look at the time evolution of the profile $\Phi(x, k, \phi)$, starting from the region $\phi = 0$, we can transform the (C-H) equation into a phase field equation, replacing $\frac{\partial}{\partial t} \Phi(x, k, \phi)$ by $\frac{\partial}{\partial \phi} \Phi(x, k, \phi(t)) \cdot \frac{d\phi}{dt}$. The dynamics will be similar to a spinodal decomposition, with $\phi$ growing and saturating exponentially. $\frac{\partial}{\partial \phi} \Phi(x, k, \phi)$ is the most unstable mode founded in Langer's linear stability analysis, characterized by the alternate growth and decrease of domains ("antiferro" mode). Note that when Langer was studying the most unstable perturbation, he was looking at the linearized version of C-H equation around $\Psi^*(k, x) = \Psi(x, k, \lambda_{C-H})$:

$$ \mathcal{L}\varphi = \left(\frac{\varepsilon}{2} + 6\Psi^{*2} - \nabla^2\right)\varphi = \left(\frac{\varepsilon}{2} + n \times (n+1)\Psi^{*2} - \nabla^2\right)\varphi. \quad (27) $$
$\mathcal{L}\varphi = E\varphi$ is the Lamé equation, for $n = 2$ (here $\varepsilon_0 = 1$). This equation doesn't have simple (algebraic) exact eigenfunction of period $2\lambda_{C-H}$ [21]. $\frac{\partial}{\partial\phi}\Phi(x, k, \phi)$ for $\phi = 0$ is not an exact eigenfunction either [22]. Nevertheless, it happens to be a good approximation for the eigenfunction of lowest eigenvalue. Due to the concavity of $\mathcal{F}(\phi)$ around $\phi = 0$ (see below Figure (5)), this eigenvalue will be negative, triggering a linear destabilization and an exponential amplification of the perturbation, i.e. an exponential growth of the translation $\phi$ with time.

Langer's phenomenon of "antiferro" instability appears due to the existence of two possible directions for displacement of the interfaces "tanh" (or of the non-symmetric lattice of interfaces $\psi$), one with a positive velocity $(+\frac{d\phi}{dt})$ and one with a negative one $(-\frac{d\phi}{dt})$. The four different kinds of interfaces present in a cell of length $2\lambda_{C-H}$ have alternately a positive or a negative velocity. This can be seen as the existence of two antisymmetric patterns [23], or building blocks for the leading instability around an intermediate state $\Psi^*(x, k_n^s, 2^{n-1}\lambda_{C-H})$ (see Figure (4)). $\pm\frac{d}{dx}\psi(x)$, these two building blocks, are associated with the two pairs of interfaces, $\psi(x)$ and $\psi(x+2)$ which have been used to construct our ansatz.

Note that in Langer's analysis, the breaking of symmetry for the choice of the antiferro cell, corresponds here to the freedom we have when choosing the range of variation of $\phi$: we could have chosen to go from 0 to $-1$, ending after a step of coarsening with the symmetric pattern, or equivalently, a pattern translated of half a period.

### C. Energy Landscape

In order to prove the usefulness of this ansatz, we have plot the energy averaged over the final period, $\mathcal{F}(\phi) = \int F(\Phi(x, k, \phi))dx$, as a function of the parameter $\phi$, keeping $k$ constant. We see for example in Figure (5) that the value $\phi = 0$ correspond to a local maximum of energy, while $\phi = 1$ (or $-1$) is a minimum. Note that there is no energy barrier in this particular energy landscape, in agreement with linear stability analysis.

### D. Approximation for adiabatic evolution with constant $k$

Relation (15), enables to write an implicit relation between $k_n$ and $k_{n+1}$:

$$ \sqrt{1 + k_{n+1}^{s2}} K(k_{n+1}^s) = 2\sqrt{1 + k_n^{s2}} K(k_n^s) \quad (28) $$

It is in fact different from the relation (22) obtained by the Landen transformation. Nevertheless in the region $k \ge k_0^s = 0.687$, the two transformation almost coincide, as can be seen on Figure (6). As this is especially true close to $k = 1$, and as this region is rapidly reach after the second or third iteration, we pretend that the process of coarsening can be described with a reasonable accuracy by our ansatz at constant $k$. A slight change of segregation parameters during the $n^{th}$ doubling process, from $k = \text{Landen}(k_n^s)$ when $\phi = 0$ to $k = k_{n+1}^s$ when $\phi = 1$, is present in the dynamics, as seen in the following table

$$ \begin{array}{l|l} k_0^s = 0.6869795924 & \text{Landen}^{-1}(k_1^s) = \text{Gauss}(k_1^s) = 0.7070743852 \\ \text{Landen}(k_0^s) = 0.9826346738 & k_1^s = 0.9851675587 \\ \text{Landen}(k_1^s) = 0.9999720868 & k_2^s = 0.99997210165 \\ \text{Landen}(k_2^s) = 0.9999999998 & k_3^s = 0.999999999902745 \end{array} \quad (29) $$
But it has only a minor effect in terms of profile shape $\Phi(x, k, \phi)$. This means that the dynamics of the parameter $k$, which affect the value of the modulus by 0.25% during the first step of coarsening, becomes negligible as $k$ goes closer to 1. So we can conclude that this "k-dynamics" is irrelevant; this parameter can be considered as constant during the evolution of $\phi$.

V. DISCUSSION ON THE HYPOTHESIS

Our analytic method rely on the assumption that at each step of the dynamics, the system can be characterized by a specific spatial period : we need therefore to discuss how this approach is relevant to the general case where noise is present. We have noted numerically that, for the spinodal decomposition, the average size of the modulation is $\lambda_{C-H}$, with a deviation of less than one percent from the value predicted by the linear theory. It does not mean that, in a real system, each domain has a length scale of $\lambda_{C-H}$, but that the distribution of the domains' length will be centered around $\lambda_{C-H}$ [6]. The coalescence events, due to initial fluctuations in the periodicity of the pattern selected just after the quench, can be neglected during the initial growth of the amplitude of the modulation. Only after this initial growth has saturated (as can be seen from Figure (1)), starts the coalescence process, which will then dominate the dynamics : the typical length scale of the structures increases slowly with time.

As suggested by [1,15], we can suppose that during the ideal coalescence process, each lattice of interfaces will experience an antiferro instability. By ideal coalescence, we mean a process which breaks as few symmetries as possible. But in a real system, this instability will concern a region of finite size, where it choose a certain sublattice, or a range for $\phi$ (for example, $\phi$ varies from 0 to 1), while it is the opposite choice in the neighboring region ($\phi$ varies from 0 to -1). Thus on the overall, the global symmetry is recover. During each step of the process, the width of the domains will locally double. But due to non synchronization between regions, for the system as a whole, the average length scale will vary continuously.

These results indebted to Christophe Josserand for many fruitful discussions and comments together with numerical help.
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[22] $\frac{\partial}{\partial\phi} \Phi(x, k, \phi = 0)$ corresponds to a local maximum of the free energy averaged over one period $\mathcal{F}(\phi) = \int F(\Phi(x, k, \phi))dx$. The (G-L) or stationary (C-H) equation, i.e. the first functional derivative $\frac{\delta F}{\delta\Phi} = 0$, admits $\Phi(x, k, \phi = 0) = \Psi^*(x)$ as solution. But Lamé equation, obtained when linearizing (G-L) equation around $\Psi^*(x)$, is related with the second functional derivative of $F$; therefore one doesn't expect $\frac{\partial}{\partial\phi} \Phi(x, k, \phi = 0)$ to be an exact solution or eigenfunction of $\mathcal{L}$.

[23] P. Coullet, R.E. Goldstein and J.H. Gunaratne, Phys. Rev. Lett J. 63, 1954 (1989).
FIGURES

FIG. 1. Time evolution of the order parameter $\Phi(x, t)$ for $\varepsilon = -1$, $dx = 0.1227$. (a) initial conditions at $t = 0$ are taken randomly with a very low amplitude $(5 \cdot 10^{-4})$; (b) at time $t = 15$, the amplitude of the small scale spatial modulations has been damped by the (C-H) dynamics, while only long wavelength contributions are still present. ; (c) at $t = 225$, the spatial modulation has almost reached its final amplitude, keeping roughly the same number of peaks as before ; (d) at $t = 1800$, we observe that the number of domains has decreased from the coarsening dynamics.

FIG. 2. Profiles of the two first metastable solutions of the (C-H) dynamics, with $k_1^s = 0.687$ and $k_2^s = 0.985$, corresponding to the first coarsening process.

FIG. 3. Construction of the two first steady solutions of the (C-H) dynamics, with $k_1^s = 0.687$ and $k_2^s = 0.985$, using a superposition of the non-symmetric profile $\psi(k, x)$, itself stationary solution of the (C-H) equation. By changing the phase shift between the two profiles entering into the linear combination, one obtains two different symmetric profiles, of periods $\lambda$ and $2\lambda$.

FIG. 4. Langer's most instable perturbation mode of destabilization of the soliton lattice is identified with $\frac{\partial}{\partial\phi}\Phi(x, k, \phi)$ at $\phi = 0$. It is composed of two antisymmetric patterns, plotted in dotted (plain) line, evolving toward right (left) at velocity $+\frac{d\phi}{dt}$ ($-\frac{d\phi}{dt}$), causing an "antiferro" instability leading to a period doubling of the pattern. They are the spatial derivative of the initial non symmetric profile $\psi(x)$ which has been used to construct our ansatz in Figure 3.

FIG. 5. Profile of the free energy landscape during a coarsening process, $F(\phi)$. It starts at $\phi = 0$ for a configuration characterized by the segregation ratio $k_1^s = 0.687$ for which the energy per unit length is $F(\phi = 0) \approx -0.135$; one sees that in this region, the free energy is a concave function of $\phi$ and thus, the associated pattern is linearly instable. The elementary step of the coarsening process ends for $\phi = 1$ associated with a pattern characterized by the segregation ratio $k_2^s = 0.985$ for which the energy per unit length is $F(\phi = 1) \approx -0.45$. In the region $\phi = 1$, the free energy is a convex function of $\phi$.
FIG. 6. Comparison between the Landen transformation (upper solid line) and the implicit relation between consecutive stationary steady states of (C-H) equation $(1 + k_{n+1}^2)^{1/2}K(k_{n+1}) = 2(1 + k_n^2)^{1/2}K(k_n)$ (lower dash line) in the region between $k = k_1^s = 0.68$ and $k = 1$, corresponding to the region of interest for the coarsening process. The Landen transformation relates the segregation parameter of a soliton lattice of period $\lambda$ with the segregation parameter associated with a lattice of period $2\lambda$. It is the generalization for the cnoidal function of the usual relation $\sin 2\theta = 2 \sin \theta \cos \theta$. The circles mark (from left to right) $k_0^s$, $\text{Landen}^{-1}(k_1^s)$, $k_1^s$, and $\text{Landen}^{-1}(k_2^s)$. If the doubling process associated with the coarsening were only a phase shift of $\phi$ between 0 and 1, the two curves would have coincided, i.e. the implicit relation (28) would be equivalent to the Landen transformation. As it is not the case, there is a $k$ change during the doubling process, but only of a few percent, as can be seen on the figure above. Moreover, one sees that, as the coarsening process takes place, $k$ reaches values closer and closer to 1 for which the disagreement becomes negligible : the $k$ change becomes smaller and smaller.