M Ismail Abdelrahman
email: m.abdelrahman@fresnel.fr

B Gralak
email: boris.gralak@fresnel.fr

Completeness and divergence-free behavior of the quasi-normal modes using causality principle

A fundamental feature of the quasi-normal modes (QNMs), which describe light interaction with open (leaky) systems like nanoparticles, lies in the question of the completeness of the QNMs representation and in the divergence of their field profile due to their leaky behavior and complex eigenfrequency. In this article, the QNMs expansion is obtained by taking into consideration the frequency dispersion and the causality principle. The derivation based on the complex analysis ensures the completeness of the QNMs expansion and prevents from any divergence of the field profile. The general derivation is tested in the case of a one-dimensional open resonator made of a homogeneous absorptive medium with frequency dispersion given by the Lorentz model. For a harmonic excitation, the result of the QNMs expansion perfectly matches the exact formula for the field distribution outside as well as inside the resonator.

Introduction

Light interaction with nanoparticles enables a wide range of unprecedented applications such as high-resolution spectroscopy, photothermal cancer therapy, optical tweezers, and light steering using metasurfaces [START_REF] Nie | Probing single molecules and single nanoparticles by surface-enhanced raman scattering[END_REF][START_REF] Huang | Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods[END_REF][START_REF] Grigorenko | Nanometric optical tweezers based on nanostructured substrates[END_REF][START_REF] Ni | Broadband light bending with plasmonic nanoantennas[END_REF]. The nanoparticles can be viewed as open resonators that are able to confine light down to subwavelength dimensions. Since the energy can escape towards the surroundings, such open systems are nonconservative and their behavior can be described by quasi-normal modes (QNMs) characterized by complex eigenvalues [START_REF] Leung | Completeness and orthogonality of quasinormal modes in leaky optical cavities[END_REF][START_REF] Leung | Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity[END_REF][START_REF] Ching | Quasinormal-mode expansion for waves in open systems[END_REF].

The QNMs expansion provides direct physical description and interpretation of the lightparticle interaction through the investigation of the resonant modes of the system. In addition, the QNMs expansion may lead to reduced computational efforts since the resonant modes of the system have to be calculated only once and then used to solve the excitation problem by evaluating the contribution of the resonances modes near the excitation frequency. The QNMs approach has shown its relevance in many situations such as the calculation of the effective mode volume for leaky cavities [START_REF] Kristensen | Generalized effective mode volume for leaky optical cavities[END_REF] and of the Purcell factor [START_REF] Sauvan | Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators[END_REF], the modeling of light behavior in plasmonic nanoparticles [START_REF] Ge | Quasinormal mode approach to modelling light-emission and propagation in nanoplasmonics[END_REF][START_REF] Ge | Design of an efficient single photon source from a metallic nanorod dimer: a quasi-normal mode finite-difference time-domain approach[END_REF], the design of angle-independent spectral filters [START_REF] Vial | Resonant metamaterial absorbers for infrared spectral filtering: quasimodal analysis, design, fabrication, and characterization[END_REF], the scattering matrix calculations [START_REF] Alpeggiani | Quasinormal-mode expansion of the scattering matrix[END_REF], the local density of states calculations [START_REF] De Lasson | Semianalytical quasi-normal mode theory for the local density of states in coupled photonic crystal cavity-waveguide structures[END_REF], the modeling of the perturbations of black holes [START_REF] Nollert | Quasinormal modes of schwarzschild black holes: Defined and calculated via laplace transformation[END_REF], and the list goes on. Moreover, it has been recently demonstrated that the peaks of the extinction spectrum of a scattering particle are associated with its QNMs [START_REF] Powell | Interference between the modes of an all-dielectric meta-atom[END_REF], which cannot be elucidated by the multipolar expansion.

The QNMs expansion suffers however from a longstanding limitation: the complex nature of the eigenvalues z q of open systems results in the divergence of the field outside the resonator, i.e. e iz q |r | → ∞ when |r | → ∞. This divergence complicates the normalization of the QNMs since the energy of the modes is unbounded and, more importantly, the QNMs expansion appears to fail in describing the radiation outside the open resonator. Several solutions have been proposed to explicitly mitigate the normalization difficulty for leaky resonators [START_REF] Kristensen | Generalized effective mode volume for leaky optical cavities[END_REF][START_REF] Sauvan | Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators[END_REF][START_REF] Bai | Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure[END_REF]. Other approaches based on the Dyson equation formalism [START_REF] Ge | Quasinormal mode approach to modelling light-emission and propagation in nanoplasmonics[END_REF] lead to the definition of regularized QNMs from the knowledge of standard QNM inside the resonator, which are then propagated in the background using the Green's function of the background [START_REF] Dezfouli | Regularized quasinormal modes for plasmonic resonators and open cavities[END_REF]. The theoretical question of the validity of the QNMs expansion outside the resonator still remains an open problem that doubts the completeness of the QNMs, i.e., the ability of the QNMs to form a complete set of modes to represent the field in the whole space, especially in the case of absorptive and dispersive systems. In Ref. [START_REF] Vial | Quasimodal expansion of electromagnetic fields in open two-dimensional structures[END_REF], it was suggested that the exponential divergence outside the resonator is attributed to a noncausal flaw in the definition of the QNMs. A recent review could be consulted for an exhaustive discussion on the recent advances in the topic of QNMs expansion [START_REF] Lalanne | Light interaction with photonic and plasmonic resonances[END_REF].

In this article, the QNMs expansion is derived taking into consideration the frequency dispersion and the causality principle. This approach has recently demonstrated the validity of the modal expansion to represent the transient fields produced by a one-dimensional resonator [START_REF] Abdelrahman | Modal analysis of wave propagation in dispersive media[END_REF]. Herein, it is shown that an additional causality-related phase factor cancels the exponential divergence of the "conventional" QNMs of open systems. Arguments are proposed to support the validity of the expansion on the QNMs of the Green's function in the whole space. A simple example is presented for a one-dimensional resonator made of a homogeneous absorptive dispersive medium, where the Green's function of the resonator is expanded using QNMs and then the obtained results are compared to the exact formulation.

Causality of the Green's function

The electromagnetic Green's function G(r, r ; z) of a system of permittivity ε(r, z) is defined as the solution of the Helmholtz equation for a point Dirac current source δ(rr ) located at r :

∇ × ∇ × G(r, r ; z) -z 2 µ 0 ε(r, z) G(r, r ; z) = izµ 0 U δ(r -r ) . (1)
Here the following notations have been adopted: r (and r ) is the position vector of space, ∇× is the curl operator, z is the complex frequency, µ 0 is the vacuum permeability, and U is the unit dyadic tensor. The reference system is defined as the infinite media surrounding the resonator and is characterized by the permittivity ε ref (r, z) and the Green's function G ref (r, r ; z). The difference ∆ε(r, z) = ε(r, z) -ε ref (r, z) corresponding to the resonator must be restricted to a bounded domain. Let G d (r, r ; z) be the "diffracted" (or "difference") Green's function, corresponding to the field diffracted by the resonator, defined as the difference

G d (r, r ; z) = G(r, r ; z) -G ref (r, r ; z) . (2)
For dispersive media, the permittivities ε(r, z) and ε ref (r, z) tend to the vacuum one ε 0 when the frequency z → ∞, which in turn suggests when z → ∞ that

G(r, r ; z) → G ref (r, r ; z) → G 0 (r -r ; z) =⇒ G d (r, r ; z) → 0 , (3)
where G 0 (rr ; z) is the free Green's function in vacuum.

The Fourier transform with respect to the time variable t of the diffracted Green's function can be defined as

G d (r, r ; t) = ∫ Γ η dz e -izt G d (r, r ; z) , (4)
where Γ η is the line parallel to the real axis of complex frequencies z = ω + iη, with real part ω describing R and imaginary part set to the positive number η. It is stressed that the integral ( 4) is well-defined for all t, which results from the following property of the permittivity [START_REF] Gralak | Analytic properties of the electromagnetic green's function[END_REF]:

∫ R dω ε(ω + iη) -ε 0 < ∞.
The resulting time-dependent function ( 4) is the Green's function generated by the spatio-temporal Dirac source δ(rr )δ(t).

The evaluation of the integral expression ( 4) is based on the remark [Eq. ( 3)] stating that the Green's functions G(r, r ; z) and G ref (r, r ; z) tend to the free one G 0 (r, r ; z) when z → ∞, so that the exponential behavior of the Green's functions G(r, r ; z) and G ref (r, r ; z) at high frequencies is governed by e iz |r -r |/c , where c = 1/ √ ε 0 µ 0 is the light velocity in vacuum. Hence it appears reasonable to make this first assumption (I):

G d (r, r ; z) × e -izτ e -iz |r -r |/c -→ z→∞ 0 . ( 5
)
When the complex frequency z has a positive imaginary part, this assumption is always true for τ = 0, while when z has negative imaginary part, it could be necessary to consider the arbitrary small time τ > 0, to ensure the vanishing behavior at the limit z → ∞. A second assumption (II) is based on the bound nature of the resonator and of the difference of permittivities ∆ε(r, z) = ε(r, z) -ε ref (r, z). Under this condition, it can be assumed that the diffracted Green's function G d (r, r ; z), which corresponds to the scattering operator [START_REF] Makialo | Modes and resonances of plasmonic scatterers[END_REF], has only a discrete set of resonances {z q } located in the lower half plane of complex frequencies (arguments based on the analytic Fredholm theorem [START_REF] Reed | Scattering Theory[END_REF] or a boundary integral expression of the scattering operator [START_REF] Makialo | Modes and resonances of plasmonic scatterers[END_REF] can be used to support this assumption). The time-dependent Green's function ( 4) is evaluated considering the two following cases. For t < |rr |/c, the line Γ η is deformed in the upper half plane of complex frequencies, i.e. with positive imaginary part, leading to G d (r, r , t) = 0, since all the Green's functions are analytic in this domain. For t > τ + |rr |/c, the line Γ η is deformed in the lower half plane and the Green's function is given by the contributions of the set of poles {z q }:

G d (r, r , t) = -2iπ {poles z q } e -iz q t G q d (r, r , z q ) , (6)
where G q d (r, r , z q ) is the residue of the function G d (r, r , z) at its pole z q . The application of the residue theorem is justified by the assumption (I) that ensures the vanishing value of the integral (4) on an infinite semi-circle in the lower half plane.

Finally, the harmonic Green's function is retrieved by applying the inverse Laplace transform to the expression [START_REF] Leung | Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity[END_REF]. For an arbitrary small τ and for a frequency z with positive imaginary part, the following function is defined:

G τ,d (r, r ; z) = 1 2π ∫ ∞ τ+|r -r |/c dt e izt G d (r, r , t) . (7)
Notice that the arbitrary small time τ, introduced to ensure the vanishing limit (5), leads to avoid the time τ at the start of the time-dependent Green's function in the Laplace transform [START_REF] Ching | Quasinormal-mode expansion for waves in open systems[END_REF]. It can be shown that the resulting function G τ,d (r, r ; z) tends to the required Green's function G d (r, r ; z) when τ tends to zero since the time-dependent function G d (r, r ; t) is bounded with respect to time t. Hence the diffracted Green's function can be expressed as an expansion of the complex resonances (or QNMs):

G d (r, r ; z) = lim τ→0 {poles z q } G q d (r, r , z q ) z -z q e i(z-z q ) (τ+ |r -r |/c) . (8)
This QNMs expansion is derived using complex analysis while taking into consideration the causality principle that appears in the lower limit of the integral in Eq. [START_REF] Ching | Quasinormal-mode expansion for waves in open systems[END_REF], which ensures that this QNMs expansion cannot lead to an exponential increasing behavior. This can be checked remarking that each residue G q d (r, r , z q ) is damped by the factor e i(z-z q ) |r -r |/c , which allows the representation of the diffracted field in the whole space, inside and outside the resonator. Furthermore, the set of QNMs of the scattering operator forms a complete set for the diffracted field since the Green's function has been expanded using solely these modes and without the continuum associated to the infinite reference system. However, as it will be observed afterwards, the convergence of the QNMs expansion above may appear slow for τ → 0. This may require a special attention to use a reasonable number of QNMs in practice.

Test case: 1D dispersive resonator

Fig. 1. A one-dimensional resonator made of a homogeneous dispersive material is excited by a harmonic source located at x 0 . This interaction results in diffracted fields both in the forward (transmission) and backward (reflection) directions.

To test the validity of the derived QNMs expansion a simple one-dimensional example is shown in Fig. 1 for a homogeneous medium of thickness L centered at the origin x = 0, with a relative permittivity given by the Lorentz model ε(z) = 1 -Ω 2 /(z 2 -ω 2 0 + izγ), where ω 0 is the medium resonance frequency, γ is the absorption parameter, and Ω is related to the electron density. This medium is surrounded by vacuum and hence acts as a Fabry-Pérot resonator.

The resonator is excited by a harmonic source of frequency z located at x 0 which generates the incident electric field E s (x, z) = e iz |x-x 0 |/c e iz(x 0 +L/2)/c , normalized to unity at the input interface x = -L/2 of the resonator. The total electric field in the whole space is given by the exact formulation [START_REF] Siegman | Lasers[END_REF]:

E R (x, z) = E s (x, z) + R(z) e -iz(x+L/2)/c
x ≤ -L/2 ,

E in (x, z) = A(z) e iz √ ε(z)x/c + B(z) e -iz √ ε(z)x/c |x| ≤ L/2 , E T (x, z) = T(z) e iz(x-L/2)/c x ≥ L/2 , (9)
where T(z) and R(z) are the transmission and the reflection coefficients of the dispersive resonator, and A(z) and B(z) can be uniquely determined by matching the fields at the two resonator interfaces. The Green's function of the resonator can hence be analytically obtained for the case presented in Fig. 1. To begin with, the transmission function is given as [START_REF] Siegman | Lasers[END_REF] T

(z) = [1 -r 2 0 (z)] e iz √ ε(z)L/c 1 -r 2 0 (z) e 2iz √ ε(z)L/c , r 0 (z) = 1 -ε(z) 1 + ε(z) . ( 10
)
It has been proven that T(z) is an even function of the square root of the permittivity [START_REF] Abdelrahman | Modal analysis of wave propagation in dispersive media[END_REF], hence it contains no branch-cut. Then, in order to use the QNMs expansion ( 8), the conditions (3) and ( 5) have to be satisfied. For the transmitted part of the field, the general derivation is applied to the function

G T (x, z) = T(z) z e iz(x-L/2)/c x ≥ L/2 . ( 11
)
It can be checked that the expression (11) satisfies Eqs. ( 3) and ( 5) and thus the validity of Eq. ( 8) is guaranteed. The set of poles {0, z q : T -1 (z = z q ) = 0} and the corresponding residues T q (z q ) have been estimated and computed in Ref. [START_REF] Abdelrahman | Modal analysis of wave propagation in dispersive media[END_REF]. The transmitted field expansion is then

E T (x, z) = z z q
T q (z q ) e iz q (x-L/2)/c z q (zz q ) e i(z-z q )(x+L/2)/c

+ T 0 e iz(x+L/2)/c , (12)
where T 0 = T(z = 0) = 1. Hence it is found that the series above equals the difference E T (x, z)e iz(x+L/2)/c which is precisely the diffracted field in the half-space x > L/2. It can be noticed that the additional causality-related factor cancels the exponential growing term in the residue expression which ensures the absence of exponential growing: the terms that describe the transmitted field outside the resonator (of x-dependency) appear as a function oscillating at the excitation frequency z only, as in [START_REF] Dezfouli | Regularized quasinormal modes for plasmonic resonators and open cavities[END_REF]:

E T (x, z) -e iz(x+L/2)/c = z z q T q (z q ) e -iz q L/c z q (z -z q ) e iz(x+L/2)/c . ( 13
)
Similarly, the reflection coefficient is given by [25]

R(z) = r 0 (z) 1 -e 2iz √ ε(z) L/c 1 -r 2 0 (z) e 2iz √ ε(z) L/c . (14)
The reflection coefficient R(z) is also an even function of ε(z) and hence its spectrum is restricted to a discrete set of resonances. For the reflected part of the field, the following function is considered

G R (x, z) = R(z) z e -iz (x+L/2)/c e -izτ x ≤ -L/2 . ( 15
)
The infinitesimal time τ > 0 is introduced to ensure that the conditions (3) and ( 5) are satisfied. Therefore, the QNMs expansion ( 8) is valid and the reflected field expansion is

E R (x, z) = E s (x, z) + z z q R q (z q ) z q (z -z q ) e -iz(x+L/2)/c e i(z-z q )τ , (16)
where the poles z q of the function R(z) are the same as T(z), since they share the same denominator, and R q (z q ) are the residues of the function R(z) at its poles. Notice that the pole at z = 0 has no contribution since R(0) = 0. The expression [START_REF] Powell | Interference between the modes of an all-dielectric meta-atom[END_REF] shows no exponential growth for the reflected fields.

Figure 2 shows the transmission function T(z) and the reflection function R(z) both using the exact expression (9) and the QNMs expansion (12) at x = L/2 for T(z), and Eq. ( 16) at x = -L/2 for R(z) while subtracting the incident field term. The given test case is for a Lorentz medium of parameters ω 0 L/c = 10, γL/c = 0.2, and ΩL/c = 20, and for below-resonance excitation with unity amplitude |E s | = 1. The parameter τc/L is set to 0.1 for the presented simulation.

The convergence analysis is also presented while the results of the exact formulation and the QNMs expansions calculations show an excellent agreement, if enough summation terms are included. As mention earlier, this parameter τ ensure the condition (5) at the expense of introducing a small error. It is possible to decrease the value of τ, however, the number of modes needed to ensure the convergence is increased accordingly.

The field distribution, outside as well as inside the resonator, of the given test case is presented in Fig. 3 for a near-resonance excitation at ω s L/c = 8. The results are shown for both the exact formulation [START_REF] Sauvan | Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators[END_REF] and the QNMs expansion [Eqs. [START_REF] Alpeggiani | Quasinormal-mode expansion of the scattering matrix[END_REF][START_REF] Powell | Interference between the modes of an all-dielectric meta-atom[END_REF]]. The field inside the cavity is determined by the field continuity at the two interfaces of the resonator, i.e., the residues A q (z q ) and B q (z q ) are derived from T q (z q ) and R q (z q ) by continuity, provided that each "conventional" QNM-before the regularization factor e i(z-z q ) |x+L/2|/c is introduced by Eq. ( 8)-is a mode of the Helmholtz equation. The results of the QNMs expansion perfectly match the exact formulation, with a less than 2% error using 31 QNMs, and show no divergence outside the resonator and therefore validate the proposed approach. Figure 3 also shows the field distribution of the five nearest QNMs to the excitation frequency where the mode 0 is the central mode (the dominant mode) with the closest real frequency part to ω s .

Figure 4 explicitly identifies the crucial result of this article. The conventional QNMs formulation is compared to the QNMs derived taken into account the causality [Eq. ( 8)]. The field distribution of two QNMs from the previous example is plotted up to a large distance from the resonator. The QNMs derived using causal Green's function show no divergence behavior, as predicted by the calculations, on the contrary to the "conventional" QNMs without the causality-related regularization factor. That finally proves that taking into account the causality principle enables the construction of well-behaved QNMs that can accurately describe the behavior of open systems. Fig. 4. A comparison between the field distribution of QNMs formulation with the additional causality-related factor (solid lines) that shows no divergence outside the resonator, and the conventional QNMs formulation (dashed lines) that exhibits a divergence behavior. The results are shown for the two nearest QNMs to the excitation frequency of the example in Fig. 3. It is worthy to note that the convergence is even slower for the excitation around the resonance frequency ω 0 and above, as it can be observed for the transmission coefficient on Fig. 5. That is due to the infinite number of modes around the resonance ω 0 [START_REF] Abdelrahman | Modal analysis of wave propagation in dispersive media[END_REF] and to the divergence of the residues T q and R q as ε(z)→1. This divergence is then compensated by the 1/z q factor in Eq. ( 12), but in this case it is needed to include many summation terms to ensure convergence.

Conclusion

The quasi-normal modes (QNMs) expansion of the Green's function is derived using complex analysis and taking into consideration the frequency dispersion, the causality principle and thus the finite velocity of electromagnetic waves. The resulting formulation of the QNMs expansion shows no divergence of the fields when |r | → ∞, leading to the construction of well-behaved modes of the open system that are able to accurately describe the fields distribution inside and outside the open resonator. In this approach, the regularization process (the additional factor) is intrinsic and appears when the QNMs are redefined using the causal Green's function formulation. A simple one-dimensional homogeneous absorptive medium with a Lorentz frequency dispersion is considered to validate the general derivation and the arguments on which it is based. The fields evaluated using the QNMs expansion, inside and outside the resonator, perfectly match the exact expressions if enough summation terms are considered. These results bring new elements to show that the QNMs can form a complete set to express in the whole space the electromagnetic fields diffracted by dispersive and absorptive materials. The general derivation is applicable to systems with a discrete set of resonances like bounded scatterers in two and three dimensions.

Fig. 2 .

 2 Fig. 2. A comparison between the results of the exact formulas (solid lines) of T(z) and R(z) and their corresponding QNMs expansion (dashed lines) for below-resonance excitation ω s < ω 0 . The results are shown for different numbers of summation terms of the expansion.

Fig. 3 .

 3 Fig. 3. Top) The field distribution of a Lorentz-dispersive resonator of parameters ω 0 L/c = 10, γL/c = 0.2, ΩL/c = 20, and for a near-resonance harmonic excitation at frequency ω s L/c = 8. The result of the QNMs expansion using 31 modes perfectly matches the exact formula. Bottom) The field distribution of the five nearest QNMs to the excitation frequency.

Fig. 5 .

 5 Fig. 5. The exact formula for T(z) and its corresponding QNMs expansion for different excitation regions. The results are shown for different numbers of summation terms in dashed lines.

Vol. 1, No. 2 | 15 Oct 2018 | OSA CONTINUUM 342

Vol. 1, No. 2 | 15 Oct 2018 | OSA CONTINUUM 346

Acknowledgments

We express our gratitude to all the participants in the project "Resonance" and to Prof. Aladin Hassan Kamel (Ain-Shams University, Egypt) for the valuable discussions.

Funding

French National Agency for Research (ANR), project "Resonance" (ANR-16-CE24-0013).


V Meiffren

P Lenormand

F Ansart

S Manov

Sol-gel routes to replace chromate based treatments for protection against zinc corrosion

Regarding the environmental consciousness and requirements relative to industrial processes, researchers and end users have to move to more green surface treatments. That is why, the use of toxic compounds such as chromates must be strictly prohibited. So, in order to replace this element preserving an efficient protection against corrosion of metals, a new route using sol-gel process clearly appears as a promising alternative method. In this paper, we investigated three different sol-gel systems in various media (alcohol and/or water) and compared their efficiency in terms of protection against corrosion and environmentally friendliness. Thus, industrial normalized corrosion test and electrochemical analyses such as polarisation curves and EIS measurements were carried out in order to both evaluate and discuss coatings behaviour in corrosive environment.

Introduction

Protection against corrosion is well known as one of the most important issue in many fields. Regarding metals, the improvement of the protection leads to an increase of the lifetime and allows metal using in more applications. Relative to zinc metal, anticorrosive treatments are needed to slow down oxidation of the surface when pollutants, present in the atmosphere react with the metal. In fact, under chloride or sulfur exposure, zinc reacts by forming white rust (1). Instead of the metal degradation, the visual aspect is strongly damaged, which is a main drawback in the building field. Usually, zinc protection was obtained by forming conversion coatings on the metal, and like many other metals, chromates were the most efficient corrosion inhibitor (2). But, nowadays, industrial processes have to turn to more "green" routes and chromate compounds will be strictly forbidden in the next years. So, in this study, our goal is to propose alternative routes to protect zinc avoiding the use of toxic elements as chromates. Therefore, the amount of VOC contained in the solutions has to be studied. In order to reach both objectives: good performances and environmental aspects, sol-gel route has been chosen in association with dip-coating process because no toxic compounds are used, it is easy-to-use, low cost process and layers can form efficient physical barriers (3)(4)(5)(6)(7)(8)(9)(10)(11). Based on our previous works [START_REF] Cousture | Proceeding of "Matériaux 2006[END_REF][START_REF] Meiffren | proceeding of "Eurocorr 2009[END_REF][START_REF] Meiffren | [END_REF], three different coatings have been studied to define the best compromise between protection efficiency and environmentally friendliness. The first one was made from TEOS (tetraethoxysilane) and MAP (γ-methacryloxypropyl trimethoxysilane) which leads to very high protection against corrosion [START_REF] Cousture | Proceeding of "Matériaux 2006[END_REF] but with a too high amount of alcohol. And two other processes correspond to very low alcohol content in the solution or alcohol free solution, respectively called: 1) GPTMS (3-glycidoxypropyltrimethoxysilane) and ASB (Aluminum-tri-sec butoxide) [START_REF] Meiffren | [END_REF]. 2) Bisamino (Bis [3-(trimethoxysilyl)-propyl] amine) [START_REF] Meiffren | proceeding of "Eurocorr 2009[END_REF]. Developed formulas of these precursors are indicated in the figure 1.

Material and methods

Pre-treatment of the substrate

Rolled and mechanically brushed zinc strip (0.7mm thick) was used as substrate. Samples (25x80mm) were cut and cleaned with ethanol, and then immersed in a commercial alkaline solution (Novaclean®) during 5 min at 80°C in order to remove the top oxide layer and to improve the wettability. By this way, the cleaned zinc has an average roughness (Ra) of 0.6µm.

Preparation of sols

Generally, sol-gel precursors for low temperature applications are metallic alkoxides MOR n type, with more often silicon atom as metal. However, many studies have shown good anticorrosive properties using hybrid layers obtained by adding an organic part to the network (6,11,(15)(16)(17). Therefore, precursors containing an organic group linked to the metal are frequently used such as (R' m MOR n ) (5).

Relative to the sol preparation, it consists of mixing precursor(s) with solvent(s) (water and/or alcohol). Two series of reactions started between water and precursor (hydrolysis [1]) and then with the hydrolyzed precursor and another precursor, hydrolyzed or not (homo and hetero-condensation respectively [2] and [3]) to form M-O-M bonds. These bonds are then the elementary units of the future inorganic network obtained after the drying step.

Hydrolysis:

≡ M-OR + H 2 O ≡ M-OH + R-OH [1] Condensation: ≡ M-OH + ≡ M-OH ≡ M-O-M ≡ + H 2 O [2] or ≡ M-OH + ≡ M-OR ≡ M-O-M ≡ + R-OH [3]
In this study, three sols consisting of different precursors have been prepared: Sol A: TEOS (tetraethoxy silane) and MAP (γ-methacryloxypropyl trimethoxysilane) were mixed in a molecular ratio of 8/1 with [TEOS]=1mol/L, then added to the solvent, ethanol and water in molecular ratio of 1. In all sols, 0.01mol/L of cerium nitrate has been introduced with the aqueous phase because of this concentration corresponds to the best compromise for the corrosion inhibitive properties (18,19).

Hybrid coatings processing

Before deposition, sols A and B were aged during 24h and only during 10min for the sol C, depending on the hydrolysis rate and so the reactions kinetics. Coatings were then performed by dip-coating process using a controlled withdrawal rate of 20cm/min followed by different thermal treatments. For coating A (obtained from the sol A), samples were dried at 60°C during 15min. For the coatings B, drying conditions were 20h at 50°C followed by 16h at 110°C. And for the coatings C, thermal treatment corresponded to 72h at 80°C.

Characterisation techniques

Microstructure of sol-gel coatings has been studied using scanning electronic microscopy (surfaces and cross sections).

In parallel, corrosion resistance of coated zinc was evaluated through a cyclic corrosion test which is a more conventional industrial test. Test duration was 10 cycles, 24h each. One cycle consists in different steps alternating wet and dry atmospheres at different temperatures. During the humidity stage, the chamber was maintained at near 100% relative humidity. In order to estimate the white rust formation, the samples were weighted before and after all corrosion cycles.

Anticorrosion properties were also evaluated thanks to electrochemical measurements. On the first hand, polarization curves have been performed in aerated and unstirred 0.5mol/L of NaCl aqueous solution. Saturated calomel electrode, connected to the working solution through a capillary, was used as reference, and a Pt electrode as auxiliary electrode. 1 cm² of the working electrode was exposed to the corrosive solution. Polarization curves were plotted using a PGP 201 Potentiostat/Galvanostat device. Polarization curves were recorded for all samples in a potential range from free open circuit potential to -0.850V, in order to study the anodic behavior of treated zinc. On the Relative to the coating densities, an increase of the network compactness is clearly shown: for the sol A, that is attributed to the addition of MAP because the methacrylate functions can crosslink the inorganic network (20). In the same way, GPTMS, in the sol B, contains an epoxy ring bonded to the organic chain. This ring can be opened during the thermal treatment [START_REF] Meiffren | [END_REF] and can induce an increase of the network reticulation. second hand, EIS measurements were carried out on 5 cm2 covered area in a 0.5mol/L Na 2 SO 4 electrolyte. Spectra were recorded over a frequency range between 65kHz and 100mHz with an amplitude of 10mV. Both Bode and Nyquist plots were used to determine sample behavior. All electrochemical measurements were performed after stabilization of the potential during 1h.

Results and discussion

Correlation between sols characteristics and coating microstructure Sols are limpid and homogenous excepted for sol C where some particles appeared when water was mixed to the precursor. This formation is probably due to the high hydrolysis rate. Alcohol is generally added to the solution to reduce the hydrolysis kinetics but, Bisamino compound hydrolysis is extremely quick.

Sol viscosities were quite similar, between 2mPa.s and 3mPa.s respectively for the sol TEOS/MAP and sol Bisamino, and around 9mPa.s for sol GPTMS/ASB. Viscosity differences are only due to precursor concentrations. After deposition by dip-coating and drying as described before, samples were weighted and the weight gain attributable to the layer formation was noticed. Layer thicknesses were then evaluated on polished cross sections observed by electronic microscopy (figure 2) and coating densities have been calculated. All these values are reported in the table 1. But, looking at sample photographs (figure 4.) after four and ten cycles, it can be noted that no rust is present on the surface for A & B coated samples. The slight increase of the weight gain can just be attributed to the white rust forming on edges. This oxidation is due to an accumulation of sol before the drying step, which induces layer brittleness on edges (21). However, for coatings C, general corrosion is observed from the fourth cycle, but the weight gain cannot be seen because it remains under the error bar (around 1mg/dm²).

Micrographs of cross sections highlight the very good leveling of sol-gel layers on the substrate for each coated samples. Coatings are conformal and homogenous, and a good adhesion has been reported by industrial tests (ISO 2409 and 6272).

Durability of coatings under corrosive media

To evaluate corrosion protection efficiency, several analyses were investigated. Firstly, a typical industrial accelerated corrosion test has been carried out to highlight white rust formation under alternation of wet and dry atmospheres. This test is frequently used to predict corrosion resistance in real exposure conditions. Secondly, electrochemical analyses as polarization curves and EIS measurements have been performed to underline protection mechanisms of the systems immersed in an electrolyte. Accelerated corrosion test. Three coated samples for each system (A, B, C) were placed in the exposure chamber. For all systems, the average weight gain due to the white rust formation was recorded after each cycle. Plots of sample weight gains, compiled in the figure 3, underline the strong reduction of the oxidation for all coated samples in comparison of the uncoated zinc. Before the fourth cycle, no oxidation is noted for coated samples. Besides, after the seventh cycle, plots show a slight increase of the weight gain. Electrochemical analyses. In order to determine the global electrochemical behavior of the system "layer + substrate", polarization curves have been plotted (figure 5.). It can be noted on these graphs, a shift of the open circuit potential for all coated samples that is correlated to an increase of the metal nobility; surface reactivity has decreased. Furthermore, anodic parts highlight a reduction of the current density at higher potentials that means a similar decrease of the oxidation reaction kinetics. This decrease is linked to the barrier effect of the sol-gel coating, which is a physical barrier to the ionic migration of the electrolyte ions to the metal surface. Indeed, there is a higher reduction of the current density (around three orders of magnitude) for B coated samples that could be attributed to a thicker and denser layer. In addition, for EIS measurements, Nyquist representation underlines the very high resistance supplied by the coating B, with one well defined capacitive loop and a resistance around 1.1×10 6 Ω.cm 2 . For coatings A and C lower polarization resistances are recorded, but around 5 times over than for uncoated zinc resistance (2000-2500Ω.cm 2 ).

On Bode representation, the high resistance of coating B is underlined by an impedance modulus at low frequencies around 10 6 Ω.cm 2 . For coating A, a slight improvement of the impedance modulus is observed and no significant influence is noted for coating C.

Studying the phase angle at high frequencies, similar explanations can be done: for coating C a phase angle of around 90° means a pure capacitance behavior which is a typical signature of the barrier effect (22)(23)(24)(25). On the contrary, the light increase of the phase angle for coatings A and C show the poor barrier effect of these layers. However, for coating A, this weak capacitive behavior detected at high frequencies is balanced by a light capacitive behavior noticed at low frequencies. Therefore, electrolyte can easily enter the coating, but oxidation reaction is yet decreased. In conclusion, coating A does not really protect zinc as a physical barrier, but it contributes to slow down zinc corrosion. Corrosion test and electrochemical analyses highlight the protection efficiency of coatings A and B. However, in this paper, the environmentally friendliness is an important issue, and as shown before, and alcohol content in the sol A is too high with around 60% vol. . So, it has been chosen to focus on protection by coating B. Nevertheless, coating B acts as a physical barrier, and the thickness can be a key parameter of its good protection. Similarly, for coating C, poor protective efficiency can be attributed to the low thickness. So, in order to compare coating B and C avoiding the thickness influence, shaping process has been modified to obtain similar thicknesses in both cases. Concerning coating B, sol concentration was reduced and for coating C, multilayers were carried out with 15min at 80°C of drying between every dip-coating to increase the thickness.

To evaluate coating thicknesses, polished cross-sections were observed by electronic microscopy. Thicknesses were measured and reported in the table 2. Similar thickness was performed with a sol concentrated at 75% for coating B and for a multilayer for coating C with around 1.5µm. All coatings were conformal and provided good leveling of the surface. The protection efficiency has been investigated by EIS measurements, and Bode representations of initial and 1.5µm coatings were plotted for both B and C coatings in the figure 8. It can be seen that concerning coating C, the increase of thickness leads to a slight improvement of the impedance modulus (nearby one order of magnitude) and a slight Figure 9.a Bode representation of coatings A after 1h (■), 4h (▲), 24h ( ) and one week (•). decrease of the phase angle at high and medium frequencies. Protective efficiency of coating C is not clearly enhanced.

Furthermore, thickness decrease of coating B contributes to a reduction of the impedance modulus. In addition, a shift of the time constant to higher frequencies and lower phase angle is shown. That is due to the reduction of the physical barrier effect. This experiment underlines that, protection properties for coating B remains higher than for coating C at similar thicknesses, particularly for the barrier effect that is clearly better. So, coating B efficiency is not only due to the high thickness, but also to the high compactness that reduces the electrolyte insertion in the network.

Behavior under immersion in Na 2 SO 4 0.5mol/L. As shown before, with the accelerated corrosion test, two coatings provide promising anticorrosive properties; coatings A and B. But electrochemical responses are very different. In order to study and compare the behavior of these coatings, samples were maintained into Na 2 SO 4 0.5mol/L solution during one week and EIS measurements have been carried out after 1h, 4h, 24h and one week. All Bode representations are compiled in figures 9.a and 9.b.

For coating A behavior during immersion, it can be noted that all Bode representations are similar whatever the immersion time, except at low frequencies. Under 10Hz, a drop in phase angle can be observed during the immersion, from 40° to 20°. This is probably due to a lost of oxidation reduction ability probably correlated to the inhibitor consumption. Regarding coating B, plots show that the immersion in the electrolyte damages the barrier effect of the layer. In fact, a shift of the time constant is simultaneously observed with a significant drop in impedance modulus of one order of magnitude during the first day of immersion due to the electrolyte insertion in the coating. In addition, we can say that after one week, the barrier effect disappears and a second time constant linked to the corrosion products arises. After this, no more protection is offered by coating B.

Conclusions

By sol-gel route, three different precursor families have been studied to replace chromate based treatments for zinc protection against corrosion. Two of them have exhibited good protective properties: coating A and B; one performed from an alcoholic solution (coating A) and a second one with only 3% v of alcohol contained in the sol (coating B).

Both systems have shown very good protection in accelerated corrosion test: no white rust was observed on samples after ten cycles in cyclic humidity test. In addition, polarization curves underline an increase of nobility for A and B coated zinc samples, with a reduction of current densities and oxidation kinetics.

Moreover, EIS analyses have highlighted the high barrier effect of coating B with a strong capacitive effect at high frequencies. Whereas low barrier effect was found for coating A, a light capacitive behavior has been noticed at low frequencies. EIS

Figure 1 .

 1 Figure 1. Developed formulas of precursors

Figure 2 .

 2 Figure 2. Cross-sections of a) coating A, b) coating B and c) coating C observed by electronic microscopy.

Figure 3 .

 3 Figure 3. Weight gain of uncoated zinc ( ), coating A (□), coating B (▲) and coating C (■) during 10 cycles.

Figure 4 .

 4 Figure 4. Photographs of all samples after 4 and 10 cycles exposed to the cyclic humidity test.

Figure 5 .

 5 Figure 5. Polarization curves of uncoated zinc (--), coating A (---), coating B (---) and coating C (--) after 1h of immersion in NaCl 0.5mol/L.

Figure 6 .

 6 Figure 6. Nyquist representation of uncoated zinc ( ), coating A (□), coating B (▲) and coating C (×) after 1h of immersion in Na 2 SO 4 0.5mol/L.

Figure 7 .

 7 Figure 7. Bode representation of uncoated zinc ( ), coating A (■), coating B (▲) and coating C (×) after 1h of immersion in Na 2 SO 4 0.5mol/L.

Figure 8 .

 8 Figure 8. Bode representation of coating B 3µm thick (▲), coating B 1.5µm thick (Δ), coating C 0.8µm thick (×) and coating C 1.5µm thick (×)

Figure 9 .

 9 Figure 9.b Bode representation of coatings B after 1h (■), 4h (▲), 24h ( ) and one week (•).

TABLE 1 .

 1 Several values of sols and coatings.

	Sol reference	% v of alcohol	[precursors]	Weight gain	Thickness	Density
	A	60%	1.1mol/L	1.1g/m²	1.0µm	1.1
	B	3%	2.0mol/L	5.0g/m²	3.0µm	1.7
	C	0%	0.6mol/L	0.7g/m²	0.8µm	0.9

TABLE 2 .

 2 Several values of sols and coatings.

	Sol reference	Sol concentration	Number of dip-coatings	Thickness
	B	B 0	1	3.0µm
	B	0.875B 0	1	2.3µm
	B	0.750B 0	1	1.5µm
	C	C 0	1	0.8µm
	C	C 0	2	1.2µm
	C	C 0	3	1.5µm