Simon Villain-Guillot
email: simon.villain-guillot@u-bordeaux1.fr

D Cahn-Hilliard dynamics : coarsening and interrupted coarsening

Many systems exhibit a phase where the order parameter is spatially modulated. These patterns can be the result of a frustration caused by the competition between interaction forces with opposite effects.

In all models with local interactions, these ordered phases disappear in the strong segregation regime (low temperature). It is expected however that these phases should persist in the case of long range interactions, which can't be correctly described by a Ginzburg-Landau type model with only a finite number of spatial derivatives of the order parameter.

An alternative approach is to study the dynamics of the phase transition or pattern formation.

While, in the usual process of Ostwald ripening, succession of doubling of the domain size leads to a total segregation, or macro-segregation, C. Misbah and P. Politi have shown that long-range interactions could cause an interruption of this coalescence process, stabilizing a pattern which then remains in a micro-structured state or super-crystal. We show that this is the case for a modified Cahn-Hilliard dynamics due to Oono which includes a non local term and which is particularly well suited to describe systems with a modulated phase.

I. INTRODUCTION

Many systems exhibit phases where the order parameter is spatially modulated and forms a pattern [START_REF]Domain Shapes and Patterns: The Phenomenology of Modulated Phases" M. Seul and D. Andelman[END_REF]. These phases are the result of a frustration caused by the competition between interaction forces with opposite effects.

For example, in a blend of polymers, the difference of interaction energies between homo and hetero polymers generates locally a repulsion between heteropolymers which leads to a macroscopic segregation. But for diblock co-polymers which are built with two heteropolymers A and B which are attached to each other by a chemical bond, such a macroscopic global phase separation is prohibited. They form a disordered phase at high temperature (when the entropic effects prevail), but below a critical temperature, whereas energetic considerations should lead to segregation, this chemical binding prevents separation between A and B heteropolymers over a long distance : the two components A and B self-organized in patterns or domains of finite size (mainly lamellar or hexagonal) in order to minimize nevertheless contacts between heteropolymers en thus the energy of interaction. The relative density in heteropolymers is thus spatially periodically modulated. This spontaneous microstructuration could be helpfull to design a new generation of solar cells based on organic semi-conductors [START_REF]Semiconducting Block Copolymers for Self-Assembled Photovoltaic Devices" G. Hadziioannou[END_REF].

In all models with local interactions, these ordered phases disappear in the strong segregation regime (low temperature). It is expected, however, that these phases should persist in the case of long-range interactions, which can't be correctly described by a Ginzburg-Landau type model with only a finite number of spatial derivatives of an order parameter (which can be defined in our preceding example from the relative density in the two components A and B).

An alternative approach is to study the dynamics of phase transition. While, in the usual process of Ostwald ripening, succession of coarsening events with doubling of the domain size leads to a total segregation, or macro-segregation, C. Misbah and P. Politi [START_REF] Politi | When does coarsening occur in the dynamics of one-dimensional fronts?[END_REF] have shown that long-range interactions could cause an interruption of this coalescence process, stabilizing a pattern that remains consequently in a micro-structured pattern or super-crystal.

We show here that this is the case for the equation of Oono [START_REF] Oono | Computationally efficient modeling of ordering of quenched phases[END_REF], which is particularly well suited to describe the dynamics of systems with a modulated phase.

II. DYNAMICS OF PHASE TRANSITIONS

A. Time-Dependent Ginzburg Landau equation

Derivation of the model

Different equations can be used to describe the dynamics of a phase transition depending on, for example, if the order parameter is a scalar or a vector, and whether it is conserved by the dynamics or not (for a review see [START_REF] Hohenberg | Theory of dynamical critical phenomena[END_REF][START_REF] Gunton | Phase Transition et Critical Phenomena[END_REF]).

As at equilibrium, this order parameter must minimize a free energy, the dynamics out of equilibrium must then involve deviation from this stable order parameter value or function, just like in a simple mechanical system. The simplest dynamics based on Ginzburg-Landau free energy for a scalar order parameter is the TDGL (Time-Dependent Ginzburg Landau or model A in Hohenberg and Halperin classification [START_REF] Hohenberg | Theory of dynamical critical phenomena[END_REF]) which writes

∂u ∂t (r, t) = - δF GL δu = ∇ 2 u - ε 2 u -2u 3 (1)
In this equation, u (r, t) is a macroscopic order parameter which is a coarse grained of a microscopic order parameter in a small volume around the postition r. And ε is the dimensionless control parameter, usually the reduce temperature ε = T -Tc

Tc

where T c is the critical temperature of the phase transition. This partial differential equation is invariant by the transformations u → -u and x i → -x i + a i . F GL is the Ginzburg-Landau free energy local density or Lyapounov functional in the context of dynamical systems :

F GL = 1 2 (∇u) 2 + ε 2 u 2 + u 4
The non-local term (∇u) 2 prevents discontinuity or roughness of the order parameter and assigns energetic overcost to its variations in proportion with their sharpness. When looking at the temporal evolution of the free energy F GL (r, t)dr :

d dt F GL dr = δF GL δu . ∂u ∂t dr = δF GL δu .(- δF GL δu )dr = -( δF GL δu ) 2 dr < 0
One notices from equation 1 that the dynamics will induce a change of u (r) as long as it hasn't reached a minimum of the free energy density F GL . If one looks for homogeneous states (where the order parameter is independent of the spatial coordinates) to be stationary states of this equation, they will be the extrema of the Landau potential

V (u) = ε 2 u 2 + u 4
FIG. 1: Landau potential as a function of u, the amplitude of the order parameter. We have plotted the profil of this potential above and below the pitchfork bifurcation at ε=0.

For ε > 0, the potential is a convex function and there is only one minimum, u = 0.

For ε < 0, the Landau potential is a concave function around u = 0, which is now a maximum ; two other solutions have now appeared as miminum of the potential, symmetric one each other.

which is plotted in Fig. 1 for the two possible signs of the control parameter. For ε > 0, the only extremum is u = 0, so there is only one homogenous solution, which is stable, being a minimum of the Landau potential (which is a convex function as long as ε > 0). When ε < 0, this potential is now concave in a neighborhood of u = 0, which is now a maximum and thus is now linearly instable. Two other symmetric solutions u = ± √ -ε 2

have now appeared due to this pitchfork bifurcation. They are the new stable homogeneous solutions and correspond to a minimum of the potential V min = -ε 2 /32.

Linear stability analysis

Linear stability analysis consists in computing the growth rate of small fluctuations of a solution. When linearizing equation (1) around u = 0 (i.e. when neglecting the nonlinear

term u 3 ) one gets ∂u ∂t (r, t) = - ε 2 u + ∇ 2 u
Considering this equation in the Fourier space we can decompose u in Fourier series in the case of a finite size problem or Fourier transform in the infinite case :

u(r, t) = q u q e iq•r+σt (2)
where u q is the amplitude of the Fourier mode at t = 0. For example, it can be the thermal fluctuations proportional to T . This mode decomposition enables to compute the qdependence of the amplification factor σ(q) (or growth rate or imaginary part of k = q -iσ) :

σ(q) = -(q 2 + ε 2 ) (3)
σ(q) is negative for ε > 0, and thus the homogeneous solution u = 0 is unstable with respect to fluctuations of the order parameter. The whole band 0 < q < (-ε/2) is linearly unstable as σ(q) > 0 (see Fig. 2)

FIG. 2: Amplification factor σ(q) computed via linear stabily analysis of the time-dependent Ginzburg-Landau equation (TDGL). It is positive (growth of the modulations) for all the modes q < -ε 2 .

Symmetry breaking and conservation law

The linear stability analysis enables to conclude that the most instable mode is for q = 0 :

it is thus a long wave instability, which will give rise to large homogeneous domains and imply spontaneous symmetry breaking. This is the case, for example, in magnetic systems.

But if there is a conservation law, as for example a conservation of mass, such an instantaneous symmetry breaking is prohibited : the matter, or the different species diffuse with a finite characteristic time. Hillert [START_REF] Hillert | A Solid Solution Model for Inhomogeneous Systems[END_REF], Cahn and Hilliard [START_REF] Cahn | Phase Separation by Spinodal Decomposition in Isotropic Systems[END_REF] have proposed a model to describe segregation in a binary mixture. This equation, later on denoted C-H for Cahn-Hilliard, corresponds to model B in the Hohenberg and Halperin classification [START_REF] Hohenberg | Theory of dynamical critical phenomena[END_REF]. Cahn-Hilliard dynamics is the minimal equation describing phase transition for a conserved scalar order parameter. As this conservation law prevents global symmetry breaking, it will generate numerous domains and interfaces separating them. This dynamic governs a whole class of first order phase transition like the Fréedericksz transition in liquid crystals [START_REF] Chevallard | Interface dynamics in Liquid crystals[END_REF], segregation of granular media in a rotating drum [START_REF] Oyama | Mixxing of solids[END_REF], or formation of ripple due to hydrodynamic oscillations [START_REF] Scherer | Sand ripples in an oscillating annular sand-water cell[END_REF][START_REF] Stegner | Dynamical evolution of sand ripples under water[END_REF].

B. Model B or Cahn-Hilliard equation

Derivation of the model

Cahn-Hilliard dynamics is a modified diffusion equation for a scalar order parameter u, which writes :

∂u ∂t (r, t) = ∇ 2 ( ε 2 u + 2u 3 -∇ 2 u) = ∇ 2 ( δF δu ) (4)
In the original work of Cahn and Hilliard, u (r, t) represents the concentration of one of the components of a binary alloy. But it can also be the fluctuation of density of a fluid around its mean value, or concentration of one chemical component of a binary mixture, or the height of a copolymer layer [START_REF] Joly | Early Stage of Spinodal Decomposition in 2D[END_REF]..

As in model A, this equation is invariant by the transformations u → -u and x i → -x i + a i and when looking at the time evolution of the local quantity F (t) , we still have :

dF dt = δF δΦu . ∂u ∂t = δF δu .∇ 2 ( δF δu ) = -(∇ δF δu ) 2 < 0
In order to derive a conservative dynamics, such that Φ(x, t)dx = cste, one can start from a detail balance [START_REF]Theory of spinodal decomposition in alloys[END_REF] , or from a conservation equation for the order parameter Φ.

∂u ∂t = -∇ • j
where j is a matter current associated with u. This current is related to the gradient of the chemical potential µ via the Hartley-Fick law : j = -∇µ). And this chemical potential is itself related to the functional derivative of the free energy µ = δF δΦ . This phenomenological approach enables to recover the C-H equation(eq 4).

If one looks globally at the quantity u(x, t)dx =< u >, the Cahn-Hilliard gives

d < u > dt = ∂u ∂t (x, t)dx = ∇ 2 ( δF δu (x, t))dx = -(∇ δF δu )
So, apart from boundary terms, the order parameter is indeed a conserved quantity.

Linear stability analysis

∂t (r, t) = ∇ 2 ε 2 u -∇ 4 u (5)
one gets for the amplification factor in the Fourier space σ(q) :

σ(q) = -(q 2 + ε 2 )q 2 (6)
So, as σ(q) is negative for ε > 0, the u = 0 solution is stable with respect to small fluctuations of the order parameter. For negative ε, Fig 3 shows a band of instable Fourier modes, as σ(q) > 0 for 0 < q < (-ε/2). Moreover, linear stability analysis of C-H predicts that the most instable mode is not anymore for q = 0 but for q C-H = √ -ε/2 (for which

σ max = ε 2 16
). This wave number of maximum amplification factor will dominate the first stage of the dynamics which is called the spinodal decomposition; this explains in particular why the homogeneous domains appear at length scales close to L = λ C-H /2 = π/q C-H , half the wave length associated with the instability. For longer times, interfaces separating each domain interact through Ostwald ripening or coarsening, causing < L > to change slowly toward higher values.

III. CAHN-HILLIARD EQUATION

A. On the periodic solutions of Cahn-Hilliard equation When the equation is studied for a constant negative ε, via a rescaling of u (as √ -εu), position r (as r/ √ -ε) and time (as t/|ε| 2 ), we observe that we could restrict the dynamics to the case ε = -1. So later on, we will study the equation

∂u ∂t (r, t) = ∇ 2 (- 1 2 u + 2u 3 -∇ 2 u) (7)
In 1D, a family of stationary solution of this nonlinear dynamics is the so-called interfacelattice solutions (or soliton-lattice), which writes :

U k,ε (x) = k∆Sn( x ξ , k) with ξ = ∆ -1 = 2 (k 2 + 1) (8)
where Sn(x, k) is the Jacobian elliptic function sine-amplitude, or cnoidal mode. This family of solutions is parametrized by the Jacobian modulus k ∈ [0, 1], or "segregation parameter".

These solutions describe periodic patterns of period

λ = 4K(k)ξ, where K(k) = π 2 0 dt 1 -k 2 sin 2 t ( 9
)
is the complete Jacobian elliptic integral of the first kind. K(k) together with k, characterize the segregation, defined as the ratio between the size of the homogeneous domains, L = λ/2, and the width of the interface separating them, 2ξ. The equation ( 9) and the relation ξ = ∆ -1 enable to rewrite this family as :

U k,λ (x) = 4K(k) • k λ Sn( 4K(k) λ x, k). ( 10
)
and using equations ( 8) and ( 9), we find that for a stationary solution, λ, and k have to be related one another through the following implicit equation (or the state equation) :

λ 2 = 2(1 + k 2 ) (4K(k)) 2 . ( 11
)
Using equations [START_REF] Chevallard | Interface dynamics in Liquid crystals[END_REF] we can compute the free energy per unit length

F GL (k, λ) = ( 4K λ ) 2 -ε 4 (1 - E K ) + 1 + 2k 2 6 - E 6K (1 + k 2 ) ( 4K λ ) 2
where E(k)is the complete Jacobian elliptic integral of the second kind. The absolute minimum for F GL (k, λ) is for k = 1 and λ = ∞, i.e. for complete segregation with a single interface.

B. Stationary States of the Cahn-Hilliard Dynamics

The dynamics starts initially with k = 0, for which U(x) describes a sinusoidal modulation of almost vanishing amplitude around the high temperature homogenous stationary solution

u = 0 U k→0,ε (x) = k 1 2 sin( 1 2 x) (12) = k 2π λ C-H sin( 2π λ C-H x) = kq C-H sin(qx)
The spinodal decomposition dynamics will saturate and reach a stationary state which is a periodic pattern with a finite domain length (weak segregation regime) for which λ = λ C-H , and k = k s 0 = 0.687 so as to satisfy [START_REF] Oyama | Mixxing of solids[END_REF], i.e k is solution of the implicit equation :

2(1 + k s2 0 )K(k s 0 ) 2 = - ε 0 λ 2 C-H 16 = π 2 . ( 13
)
The amplitude of the modulation is then k s 0 ∆ s 0 = 0.400 √ -ε 0 , which is different from u b .

Using linear stability analysis, Langer has shown that the stationary profile thus obtained,

u 0 (x) = U k s 0 ,λ C-H (x)
, is destroyed by stochastic thermal fluctuations [START_REF]Theory of spinodal decomposition in alloys[END_REF]. He has identified the most instable mode as an "antiferro" mode, leading to an infinite cascade of period doubling [START_REF] Villain-Guillot | Coalescence in the 1D Cahn-Hilliard model[END_REF]. Disorder of the pattern is also a cause of Ostwald ripening : if the periodicity of the interface-lattice is broken, either when the distance between theses interfaces or when the bulk value in the different domains become non-constant, coarsening is triggered by diffusion of matter between neighboring domains : big domains will then absorb smaller ones [START_REF] Calisto | Bubbles interaction in Canh-Hilliard equation[END_REF].

C. Coarsening

When considering the C-H equation 4 as a diffusion equation, P. Politi and C. Misbah have shown that there should be coarsening as long as dν/dλ is positive, where ν is the amplitude of the modulation and λ its [START_REF] Politi | When does coarsening occur in the dynamics of one-dimensional fronts?[END_REF]. As in Cahn-Hilliard dynamics

ν = k∆ = k -ε 2(k 2 + 1) and λ = 4K(k)ξ = 4K(k) 2(k 2 + 1) -ε
are two growing functions of the parameter k, this diffusion coefficient will always remain positive and coarsening will proceed until λ → ∞ (as in Fig. 4

left).

FIG. 4: Left : evolution of the amplitude of the modulation of the stationart states as a function of the period, in the cases of a Cahn-Hilliard dynamics. As dν/dλ is always positive, the pattern will rippen until all the interfaces disappear but one (note that as dν/dλ → 0, there is a slowing down of the coarsening process). Right, a model where dν/dλ changes sign : the coarsening will then be interrupted.

When looking at Figure [START_REF]Domain Shapes and Patterns: The Phenomenology of Modulated Phases" M. Seul and D. Andelman[END_REF], one can see that the bulk energy is decreasing when the amplitude varies from ν = 0 to ν = ± √ -ε 2 , that is, when the segregation increase. Meanwhile, the interfacial energy is proportional to the period, we finally get that the total energy decreases when the period of the stationary solutions gets longer and longer. But for other dynamics (as in Fig. 4 right), dν/dλ can change of sign as we will see in the following : segregation then remains partial. P. Politi et C. Misbah speak then of interrupted coarsening.

IV. OONO'S MODEL A. Derivation of the model

We would like to work out the period of modulated phase systems for which there is a competition between two types of interactions: a short-range interaction which tends to make the system more homogeneous together with a long-range one, or a non-local one, which prefers proliferation of domain walls. This competition results in a microphase separation with a preferred mesoscopic length scale. These systems forming a super-crystal can be studied using a modified Landau-Ginzburg approach, derived from Cahn-Hilliard equation and of practical use for numerical simulations [START_REF] Oono | Computationally efficient modeling of ordering of quenched phases[END_REF]:

∂u ∂t = (∇ 2 δF GL (u) δu ) -β 2 u = ∇ 2 ( -1 2 u + 2u 3 -∇ 2 u) - β 4 2 u. ( 14
)
The -β 2 u term models in the Cahn-Hilliard equation the long-range interactions, which prevents the formation of macroscopic domains and favors the modulation. We will see that the inclusion of such a term, following Oono, enables to describe the behavior of modulated systems at T much lower than T c . If we suppose, for example, that in a 3D problem, the long-range interaction decreases like 1 r , the full free energy density writes

F (u) = F GL + F int (15) = 1 2 (∇u(r)
) 2 + -1 4 u 2 (r) + 1 2 u 4 (r) + u(r ′ )g(r ′ , r)u(r)dr ′ , where g(r ′ , r) = 4π ( β 4 ) 2 |r ′ -r| in D=3, or |x ′ -x| in D=1.
The long-range interaction g(r ′ , r) corresponds to a repulsive interaction when u(r ′ ) and u(r) are of the same sign : thus it favors the formation of interphases. If we want to study the dynamic of this phase separation, we use the Cahn-Hilliard equation :

∂u ∂t = ∇ 2 r δF (u) δu (16) = ∇ 2 r -1 2 u + 2u 3 -∇ 2 u + u(r ′ )g(r ′ , r)dr ′ .
If one recalls that -1 |r ′ -r| is the Green's function associated with the Laplacian operator ∇ 2 r in 3D, the preceding equation then transforms into

∇ 2 r u(r ′ )g(r ′ , r)dr ′ = u(r ′ )∇ 2 r g(r ′ , r)dr ′ (17) = - β 4 2 u(r ′ )δ(r ′ , r)dr ′ = - β 4 2 u(r).
which leads to equation [START_REF] Stegner | Dynamical evolution of sand ripples under water[END_REF]. Note that, even with the new term added by Oono to the usual Cahn-Hilliard dynamics, this equation remains in the class of the conservative models, as it derives from a equation of conservation. Note also that the free energy F int is infinite if u(r) is of the same sign in a macroscopic domain.

B. Linear stability analysis for Oono's model

If we look at the linear stability analysis of the homogenous solution u = 0, we found almost the same results as in the original work of Cahn and Hilliard, except that the amplification factor σ(q) now write:

σ(q) = ( 1 2 -q 2 )q 2 - β 4
2

This shows immediately that u = 0 is linearly instable if β < 1, with a band of unstable Fourier modes 0.5 1 -1β 2 < q < 0.5 1 + 1β 2 (for which σ(q) > 0). The most unstable mode is for q C-H = 0.5 like in the simplest Cahn-Hilliard model(4). Therefore, during the initial stage of the dynamics, the spinodal decomposition the homogeneous domains appear at length scales close to L = 2π, as in the usual Cahn Hilliard dynamics. But one sees that, contrary to the simple Cahn-Hilliard case, the long wave length modulations are now stable as σ(q) < 0 for q < 0.5 1 -1β 2 . This explains qualitatively why, for any finite value of β, the dynamics will end in a micro segregated regime, as it is observed numerically and as we will discuss quantitatively below.

It has been noticed in different models [START_REF] Buzdin | Generalized Ginzburg-Landau theory for nonuniform FFLO superconductors[END_REF] that, if the interaction responsible of the modulation is local (i.e. described in the free energy by local terms only, like -(∇u) 2 in the Swift Hohenberg model), then for low temperature or small β, the macrosegregated regime (one unique interface) will be energetically favored compared to the microphase separation.

However, in this model by Oono, because the interaction is long range (i.e. non-local), no matter how small is β, there will always be a finite region around q = 0 where σ(q) < 0. In-

deed, σ(0) = -β 4
2 . Consequently, a modulated phase should always end the dynamics [START_REF] Andelman | Phase transitions in Langmuir monolayers of polar molecules[END_REF].

C. Direct minimization of the free energy

For D=1, the contribution of the long-range interaction to the free energy per unit length is [START_REF] Liu | Dynamics of phase separation in block copolymer melts[END_REF]

F int = 1 λ λ 0 F int dr = -β 2 2λ λ 2 0 λ 2 0 Ψ(r ′ ) |r ′ -r| Ψ(r)drdr ′ .
When using as ansatz the family of interface-lattice solutions U k,λ (x), we then obtain

F int = -β 2 2λ λ 2 0 λ 2 0 k 2 ( 4K λ ) 2 |r ′ -r| Sn( 4K(k) λ r, k)Sn( 4K(k) λ r ′ , k)drdr ′ = π K -β 2 8 2K 0 2K 0 k 2 |x ′ -x| Sn(x, k)Sn(x ′ , k)dxdx ′ .
Thus, this contribution is independent of λ and the only minimization is with respect to k. Consequently, the minimization with respect to λ concerns only F GL and enables to find λ as a function of k :

λ(k) = 8K 1+k 2 3 + k 2 3(1-E K )
. And the minimization of the free energy F GL (k, λ(k)) + F int (k) is simply with respect to a single variable k, which can be done numerically for different values of the interaction strength β 2 .

D. Stationary microsegregated patterns

The family [START_REF] Chevallard | Interface dynamics in Liquid crystals[END_REF] is not anymore an exact stationary solution of the dynamics ( 14) because of its last term. Nevertheless, it is a good candidate for an approximate solution (especially in the case of small β) and thus can be used as a tool for calculation using a solvability condition or Fredholm's alternative.

Indeed, we can write deviation from a given periodic stationary profile of period λ as u(x, t) = u 0 (φ(x, t)) + εu 1 (φ(x, t)) + ... where ε is a small parameter and u 0 is a periodic function of the phase φ(x, t). For a steady state solution φ(x, t) = qx with q = 2π/λ. In the general case φ(x, t) = q(X, T )x where X = ǫx and T = ǫ 2 t i.e. q = ∂φ ∂x is now a slowly varying function of

∂ ∂x = q∂ φ + ǫ ∂q ∂X ∂ ∂q = q∂ φ + ǫ ∂ 2 Ψ ∂X 2 ∂ q ∂ 2 ∂x 2 = q ∂ ∂φ q∂ φ + ǫ∂ 2 XX Ψ∂ q + ǫ∂ 2 XX ∂ q q∂ φ + ǫ∂ 2 XX Ψ∂ q ∂ 2 ∂x 2 = q 2 ∂ φφ + ǫ∂ 2 XX Ψ∂ φ + 2ǫ∂ 2 XX Ψq∂ q ∂ φ ∂ 2 ∂x 2 = q 2 ∂ φφ + ∂ 2 XX Ψ (1 + 2q∂ q ) ∂ φ
where we have kept only the first order terms in ǫ.

If we consider a stationary profile u 0 which satisfies (zero order equation):

q 2 ∂ 2 ∂φ 2 -1 2 u 0 + 2u 3 0 -q 2 ∂ 2 ∂φ 2 u 0 - β 4 2 u 0 = 0 i.e. ∂ ∂φ -1 2 u 0 + 2u 3 0 -q 2 ∂ 2 ∂φ 2 u 0 = β 4 2 w where ∂ φ w = q -2 u 0 (18)
Oono's equation ( 14) becomes then at order one in ǫ ǫ∂ T Ψ∂ φ u 0 = ǫN 0 (u 1 ) + ǫN 1 (u 0 ) where

N 0 (u 1 ) = q 2 ∂ 2 ∂φ 2 ( -1 2 u 1 + 6u 2 0 u 1 -q 2 ∂ 2 ∂φ 2 u 1 ) - β 4 2 u 1 = q 2 ∂ 2 ∂φ 2 L(u 1 ) - β 4 2 u 1 and N 1 (u 0 ) = ∂ 2 XX Ψ (1 + 2q∂ q ) ∂ φ -1 2 u 0 + 2u 3 0 -q 2 ∂ 2 ∂φ 2 u 0 -q 2 ∂ 2 ∂φ 2 ∂ 2 XX Ψ (1 + 2q∂ q ) ∂ φ u 0 = β 4 2 ∂ 2 XX Ψ (1 + 2q∂ q ) w -q 2 ∂ 2 ∂φ 2 ∂ 2 XX Ψ (1 + 2q∂ q ) ∂ φ u 0
where we have used ∂ φ w = q -2 u 0 and equation ( 18) to simplify N 1 (u 0 ). So Oono's equation ( 14) writes

ǫ∂ T Ψ∂ φ u 0 - β 4 2 ∂ 2 XX Ψ (1 + 2q∂ q ) w ( 19
)
+q 2 ∂ 2 XX Ψ ∂ 2 ∂φ 2 ((1 + 2q∂ q ) ∂ φ u 0 ) = q 2 ∂ 2 ∂φ 2 L(u 1 ) - β 4 2 u 1

E. Stability of stationary microsegregated patterns

A necessary condition for a solution to exist is that the left-hand side of the system is orthogonal to the kernel of the adjoint operator

N † 0 = q 2 ∂ φφ L -β 4 2 Id † ; if v ∈Ker q 2 ∂ φφ L -β 4
2 Id † then the solvability condition (or Fredholm alternative) writes:

< v|∂ T Ψ∂ φ u 0 -N 1 (u 0 ) >=< v|N 0 (u 1 ) >= 0

As for any v we have

< v|q 2 ∂ 2 ∂φ 2 -1 2 u 1 + 6u 2 0 u 1 -q 2 ∂ 2 ∂φ 2 u 1 - β 4 2 u 1 > =< q 2 ∂ 2 ∂φ 2 v| -1 2 + 6u 2 0 -q 2 ∂ 2 ∂φ 2 )u 1 > - β 4 2 < v|u 1 > =< q 2 -1 2 + 6u 2 0 -q 2 ∂ 2 ∂φ 2 ∂ φφ v|u 1 > - β 4 2 < v|u 1 >
this adjoint operator writes :

N † 0 = q 2 ∂ φφ L - β 4 2 Id † = q 2 -1 2 + 6u 2 0 -q 2 ∂ 2 ∂φ 2 ∂ φφ - β 4 2
If v ∈KerN † 0 , we can define u such that q 2 ∂ φφ v = u and which satisfies

q 2 ∂ 2 ∂φ 2 -1 2 u + 6u 2 0 u -q 2 ∂ 2 ∂φ 2 u = q 2 β 4 2 ∂ φφ v = β 4 2 u. ( 20
)
So u is solution of -1

2 u + 6u 2 0 u -q 2 ∂ 2 ∂φ 2 u = β 4 2 v.
Using equation [START_REF] Calisto | Bubbles interaction in Canh-Hilliard equation[END_REF], we thus find that v defined by q 2 ∂ φφ v(= u)=∂ φ u 0 is an element of Ker q 2 ∂ φφ L -β 4 2 Id † .

As a consequence, the diffusion equation writes

ǫ∂ T Ψ = -q 2 < v| ∂ 2 ∂φ 2 ((1 + 2q∂ q ) ∂ φ u 0 ) > + < v| β 4 2 (1 + 2q∂ q ) w > < v|∂ φ u 0 > ∂ 2 XX Ψ
As q 2 ∂ φ w = u 0 and q 2 ∂ φφ v = ∂ φ u 0 we get the equality v = w.

So < v|∂ φ u 0 >= -< ∂ φ v|u 0 >= -< ∂ φ w|u 0 > = -q -2 < u 0 |u 0 > and consequently equation ( 19) is a diffusion equation

ǫ∂ T Ψ = D ∂ 2 XX Ψ ǫ∂ T Ψ = q 2 ∂ q < q (∂ φ u 0 ) 2 > -β 4 2 ∂ q < q w 2 > < u 2 0 > ∂ 2 XX Ψ V. CONCLUSION
As long as the diffusion coefficient is negative (due to the < ∂ k u 0 | ((1 + 2q∂ q ) ∂ φ u 0 ) >= ∂ q < q (∂ φ u 0 ) 2 > term), the coarsening process goes on, in order to minimize interfacial energy. But, due to its second part in β 2 , the diffusion coefficient will vanishe and thus the coarsening will be interrupted at a finite length scale.

FIG. 3 :-ε 2 .

 32 FIG.3: Amplification factor σ(q) computed from the linear stability analysis of Cahn and Hilliard equation

  ∂u

Figure 5 3 .FIG. 5 :

 535 Figure 5 presents λ (β 2 ) which scales like (β 2 ) 1/3 .

  x and t.

	∂u ∂t	=	∂u ∂φ	∂φ ∂t	=	∂u ∂φ	∂φ ∂T	dT dt	= ǫ 2 ∂φ ∂T	∂u ∂φ
	∂u ∂x	=	∂u ∂φ	∂φ ∂x	=	∂u ∂φ	(q +	∂φ ∂X	dX dx	) = q	∂u ∂φ	+ ǫ	∂u ∂X
	If we denote Ψ(X, T ) = ǫφ(x, t), then the local wave number is q(X, T ) = ∂φ ∂x = ∂Ψ ∂X and
			∂ ∂t	= ǫ∂ T φ

Acknowledgment

The authors would like to thank Dr. Chaouqi Misbah (LIPhy, Grenoble) for fruitful discussions and an invitation in Grenoble where part of this work was done.


Mohammed Serhir
email: mohammed.serhir@supelec.fr

Dominique Picard Dre

An Efficient Near Field to Near or Far Field Transformation in Time Domain

Keywords: near-field to far-field transformation, time domain, antenna measurement, sampling theorem I

This paper presents a complementary analysis to the computation scheme previously introduced by Hansen and Yaghijan [1] and [2] for the time-domain near-field to near-or far-field transformation technique. The approach presented here aims to reduce the computation time and memory requirements, which represent the main drawbacks that make difficult the application of the technique to practical cases of interest. Computations with electromagnetic simulation of a circular Xband horn antenna illustrate the accuracy of the proposed method and the rapidity of the resulted near-field to near-or farfield transformation, while using the minimum number of measurement samples.

INTRODUCTION

The Ultra Wide Band (UWB) antennas operating over large bandwidth have attracted a lot of interest over wide telecommunication applications. These polyvalent antennas have to be characterized over the operating frequencies in order to complete the validation of the electromagnetic simulation process results. For these antennas, a time domain analysis appears to be very attractive since using the frequency domain approach, a large number of measurements at different frequencies must be carried out in order to cover the whole working band of the antenna under test (AUT).

In the last decades a quantity of work has been dedicated to the characterization of radiating systems by means of near-field to far field transformation techniques [START_REF] Yaghjian | An overview of near-field antenna measurements[END_REF]. These techniques are predominantly developed in the frequency domain and are used to characterize the near field and far field or using the near field data to calculate the radiation pattern at a single frequency.

In addition, the near field time domain technique has many advantages over the conventional frequency domain methods. Time domain gating is an efficient way to filter the multiple reflections and presents a promising tool in the measurement accuracy improvement. Also, using time gating, Hansen has shown in [START_REF] Hansen | Plane-Wave theory of Time-Domain Fields, near-field scanning applications[END_REF] that the scan plane truncation error can be removed. Also, using time domain technique, one can characterize non continuous wave radiating structures (nonsinusoidal signals).

However, the main difficulty related to the practical use of time-domain approach concerns the quantity of measured data and measurement time. The required computation time and memory storage can become exorbitant for large measurement surface and large observation time window.

In this paper, an efficient near field to near-or far-field transformation computational scheme is described and discussed. The proposed analysis is based on the exploitation of the minimum number of measurement points, while providing accurate results. We propose to use the reconstruction formulas to rigorously express the near-field between the measurement points and to efficiently calculate the near-field at different distances using a minimal set of timespace measurements. The computation scheme for near-field to near-or far-field transformation we are proposing is described and validated using electromagnetic simulation software (CST Micro Wave Studio) [START_REF]CST Microwave Studio[END_REF].

II. FORMULATION AND MATHEMATICAL ANALYSIS

We have developed the near-field to near-field transformation algorithm based on the Green's function representation to calculate the antenna radiation pattern at a different distance x1 using the data measurement at the distance x0<x1.

The Matlab routine we have developed uses the direct time domain formula based on the Green's function representation [START_REF] Hansen | Planar near-field scanning in the time-domain. Part 2: Sampling theorems and computation schemes[END_REF]. This direct time formula which expresses the near field to near field transformation is written as

0 2 0 0 0 3 1 ( , ) 1 ( , ) 2 ( , ) E x r t R c cR t E r t R dy dz x E r t R c R                              
The field vector 0 ( , ) E r t is the time domain measured field. The near field to far field transformation is a particular case of the near-field to near-field situation. This time x goes to infinity and Eq. 1 is expressed as:

0 0 0 0 1 ( , , ) ( , . ) 2 r r E E t e x r t e r c dy dz c t                      with r e R R 
, and ( , , ) E t  the far-field in the spatial direction ( , )

.

Using the sampling theorem, ( 1) and ( 2) are transformed to the following double summation  can be determined using a numerical differentiation technique whose accuracy depends on the considered discretization of the measured field (sampling criteria). The measured near field data have to be over-sampled to achieve a consistent result and, as a consequence, the computation time and memory requirement, involving a large number of measured samples in (3) or ( 4), become prohibitive.

0 2 0 0 0 3 1 ( , ) 1 ( , ) 2 ( , ) Ny mn Nz m Ny n Nz mn E x r t R c cR t E r t R y z x E r t R c R                               0 0 0 0 ( , , ) 1 ( , .
To overcome this difficulty we will exploit the fact that the antenna radiated field is band limited. The key point of the method we propose consists of using of the reconstruction formula for the E-field [START_REF] Hansen | Plane-Wave theory of Time-Domain Fields, near-field scanning applications[END_REF].

() ( ) sinc ( ) t t N t t l t t l t f t f t l t t                    
The reconstruction formula is used to express the measured field. The E-field is expressed as a function of time as

0 1 0 0 0 1 0 1 0 0 1 sin ( , ) ( , ) ( ).
( , )

t t N mn mn l N mn l tt l t E r t E r t l t tt l t h t E r t l t                                         with 0 ( ) sin tt h t c l t                   0 0 . t t t t N t      
Consequently, the time derivative of the E-field is rigorously written

1 0 0 0 1 () ( , ) ( , ) t N mn mn l E h t r t E r t l t t t             with, 0 0 1 ( ) cos ( ) (
)

tt h t l h t t t t l t t                         
As it can be seen, from (7), (8), ( 9) and (10) one can determine the E-field and its time derivative rigorously between time-samples provided the sampling Nyquist-Shannon criterion is respected. As a matter of fact, the criteria for timeaxis sampling 12 max t freq  and for spatial-sampling

y z c t      can be sufficient to accurately calculate the near field to near or far field transformation for antenna characterization.

III. RESULTS

In order to show the viability of the proposed analysis, we consider the time-domain near-field issued from the simulated circular horn antenna. The antenna under test we have used is a circular X-band horn antenna operating at the frequency range of 8-12 GHz. The excitation signal applied to the antenna's port is a Gaussian pulse with a carrier at 10GHz. The time-domain near field data is collected by measuring the time waveform at discrete grid points on a finite scan plane. The Finite Integration Technique simulation software can provide the near E-field at arbitrary positions and times [START_REF]CST Microwave Studio[END_REF]. Thus, we consider the E-field in Ny x Nz points in a fictitious plane as actual near-field signal over the measurement plane x=x0. We are interested at transforming the actual field to determine the field at the distance x1>x0 using the minimum number of measurement points (Fig. 1). When x1 goes to infinity the transformation is defined as near-field to far-field transformation. In Fig. 3 we present the antenna near-field issued from the simulation software along the z-axis( x=12cm and y=0).

We consider a parameter α to control the over sampling rate of the near-filed collected over the rectangular grid . Respecting each α value we collect over the measurement surface P1 the tangential components of the E-field. Then we calculate the electric field at the planar surface P2 situated at x=24cm using the near-field to near-field transform as shown in Fig. 2. The resulted field is compared to the time-domain near-field over P2 issued directly from the simulation software (Finite Integration technique).

From Fig. 4, we note that the antenna radiated fields over P2 (x=24cm) issued from the simulation software present a good agreement with the fields calculated using the near-field to near-field transformation (from the scan planes P1 to P2) using different sampling parameters (α=1, 1.2, 1.5, and 2). In addition, for α=1, which corresponds to the Nyquist-Shannon criterion, the size of the near field minimum matrix is Ny=Nz=47 and Nt=109. Using α=1, the computation time for each near-field point is 2.66s.

In Fig. 4, we give also the near field matrix size (Nt, Ny and Nz) associated to the sampling parameter α and the corresponding computation time for each calculated near-field point. As it can be seen, considering an over sampled near field data necessitate long calculation duration.

The accuracy of the calculation of the time domain near or far field using over-sampled measured field is not questioned here, nevertheless the efficiency of the field transformation proposed here helps to overcome the main drawback of the near field time domain techniques using the minimum number of measured data.

IV. CONCLUSION

A complementary analysis to the computation scheme previously introduced by [1] [START_REF] Hansen | Planar near-field scanning in the time-domain. Part 2: Sampling theorems and computation schemes[END_REF] for the time-domain near-field to near-field transformation technique has been presented and numerically validated. The approach we have presented aims to optimize the computation time and memory requirements by considering the minimum number of measurement points. The rigorous calculation of the time derivative of the E-field using of the minimum number of near field samples (respecting Nyquist-Shannon criterion) tend to overcome the drawback related to the practical use of time-domain approach. It results that the use of both an optimal representation of the field over the measurement surface and a convenient computation strategy allow the use of a lower number of samples and a faster execution of the near field to near-or far-field transformation in time domain.

Fig. 1 .

 1 Fig. 1. The radiating element near field measurement at the distance x0. The tangential components are regularly recorded over a square surface in the zy plane, d=56cm,x0=12cm, x1=24cm.

Fig. 2 .Fig. 3 .

 23 Fig. 2. The circular X-band horn antenna used to generate the near-field data over the planar surface x=12cm (P1). Time-domain waveforms are sampled over a grid depending on the sampling parameters α≥1.

  the measured field sampling points in the scan plane. The integers Ny and Nz result from the considered scan plane size and sampling criteria. Here we suppose that the scan plane is located outside the reactive zone of the AUT. The measured field can be sampled with a maximum uniform sample spacing defined for band limited functions described by Nyquist-Shannon of approximately 0 near-field to far-field transformation formula (2) only the time derivative of the tangential E-field is involved. In Eq. 1 the calculation of the near field involves the tangential Efield components and their time derivative at the measurement distance. Moreover, for near-field to near-or far-field transformation we have to calculate the measured electric field between time samples at the instants t interpolation effort. The accuracy of the results depends on the chosen interpolation method.

For

Fig. 4 .

 4 Fig. 4. Time domain near field waveform of the horn antenna at the distance of 24cm from the aperture issued from the simulation software (Finite Integral Technique). Comparison are carried out for different near-field sampling criteria α=1 (a), α=1.2 (b), α=1.5 (c), α=2 (d)