Christian Pedersen

John Niven

Thomas Salez

Kari Dalnoki-Veress

Andreas Carlson
email: acarlson@math.uio.no

Asymptotic regimes in elastohydrodynamic and stochastic leveling on a viscous film

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I. INTRODUCTION

The motion of an elastic sheet supported by a thin layer of viscous fluid is a phenomenon that manifests itself in processes spanning wide ranges of time and length scales, from e.g. magmatic intrusion in the Earth's crust [START_REF] Balmforth | Instability in flow through elastic conduits and volcanic tremor[END_REF][START_REF] Michaut | Dynamics of magmatic intrusions in the upper crust: Theory and applications to laccoliths on earth and the moon[END_REF], to fracturing and crack formation in glaciers [START_REF] Ball | Static and dynamic fluid-driven fracturing of adhered elastica[END_REF], to pumping in the digestive and arterial systems [START_REF] Bilston | Arterial pulsation-driven cerebrospinal fluid flow in the perivascular space: a computational model[END_REF][START_REF] Takagi | Peristaltic pumping of viscous fluid in an elastic tube[END_REF][START_REF] Borcia | Liquid pumping induced by transverse forced vibrations of an elastic beam: A lubrication approach[END_REF], or the construction of 2D crystals for electronic engineering [START_REF] Sanchez | Mechanics of spontaneously formed nanoblisters trapped by transferred 2d crystals[END_REF]. Elastohydrodynamic flows have been studied in model geometries in order to understand their generic features and the inherent coupling between the driving force from the elastic deformations of the material and the viscous friction force resisting motion [START_REF] Hosoi | Peeling, healing, and bursting in a lubricated elastic sheet[END_REF][START_REF] Huang | Wrinkling of a compressed elastic film on a viscous layer[END_REF][START_REF] Al-Housseiny | Two-phase fluid displacement and interfacial instabilities under elastic membranes[END_REF][START_REF] Hewitt | Elastic-plated gravity currents[END_REF][START_REF] Tulchinsky | Transient dynamics of an elastic hele-shaw cell due to external forces with application to impact mitigation[END_REF][START_REF] Elbaz | Axial creeping flow in the gap between a rigid cylinder and a concentric elastic tube[END_REF][START_REF] Arutkin | Elastohydrodynamic wake and wave resistance[END_REF][START_REF] Juel | Instabilities in blistering[END_REF][START_REF] Rivetti | Elastocapillary levelling of thin viscous films on soft substrates[END_REF].

The investigation of an initially flat elastic membrane that is subsequently subjected to an applied deformation has helped disclose how system size, magnitude and direction of elastic deformations and spatial confinement affect the membrane dynamics. Whilst stretching [START_REF] Bernal | Elastic response and wrinkling onset of curved elastic membranes subjected to indentation test[END_REF][START_REF] Box | Impact on floating thin elastic sheets: Dynamic wrinkling is controlled by fluid inertia[END_REF] or compressing [START_REF] Vandeparre | Hierarchical wrinkling patterns[END_REF][START_REF] Kodio | Lubricated wrinkles: Imposed constraints affect the dynamics of wrinkle coarsening[END_REF] of the membrane produces wrinkling patterns [START_REF] Cerda | Geometry and physics of wrinkling[END_REF], it has been shown that slowly deforming the membrane, by means of injecting additional fluid into the supporting layer, leads to a dynamics where the fluid pressure is solely balanced by elastic bending forces [START_REF] Lister | Viscous control of peeling an elastic sheet by bending and pulling[END_REF][START_REF] Berhanu | Uplift of an elastic membrane by a viscous flow[END_REF]. As the out-of-plane deflection increases, a change in the physical mechanism that dictates the dynamics occurs as elastic stretching becomes the dominant driving mechanism until the system reaches a critical size for which gravity starts to dominate the dynamics. For small systems, an analog to such a predicted transition is observed for a thin perturbed elastic plate resting on a nanoscopic fluid layer, where the restoring elastic bending force is opposed by van der Waals forces leading to an elastohydrodynamic touchdown [START_REF] Carlson | Similarity and singularity in adhesive elastohydrodynamic touchdown[END_REF] similar to capillary film dewetting [START_REF] Zhang | Similarity solutions for van der waals rupture of a thin film on a solid substrate[END_REF]. To describe the dynamics theoretically, one can solve the full Navier-Stokes equation in the fluid phase using dynamic boundary conditions at the elastic interface given by the solution of the Föppl-von-Kármán equation [START_REF] Pihler-Puzović | Displacement flows under elastic membranes. part 1. experiments and direct numerical simulations[END_REF], using e.g. the immersed-boundary method [START_REF] Heil | Fluid-structure interaction in internal physiological flows[END_REF]. However, for laminar flows we can describe the dynamics using lubrication theory [START_REF] Batchelor | An introduction to fluid dynamics[END_REF] which allows us to the reduce the number of spatial dimensions by one, thus reducing the computational cost, and opening for additional analytical insight into the problem. Even though the aforementioned studies have had great success in exploring transitions in the dominant driving force for elastohydrodynamic problems, not much has been done to understand elastohydrodynamic flows where the ratio between geometric parameters characterizing the system can undergo large changes without affecting the dominant driving force.

For instance, when an elastic sheet deforms onto a wall pre-wetted by a thin viscous film, the dynamics of the advancing front are dictated by the local curvature of the interface [START_REF] Rivetti | Elastocapillary levelling of thin viscous films on soft substrates[END_REF][START_REF] Lister | Viscous control of peeling an elastic sheet by bending and pulling[END_REF]. This elastohydrodynamic relaxation is reminiscent of the capillary spreading of a viscous drop onto a solid substrate [START_REF] Tanner | The spreading of silicone oil drops on horizontal surfaces[END_REF][START_REF] De Gennes | Wetting: statics and dynamics[END_REF][START_REF] Cormier | Beyond tanner's law: Crossover between spreading regimes of a viscous droplet on an identical film[END_REF][START_REF] Bergemann | Viscous drops on a layer of the same fluid: from sinking, wedging and spreading to their long-time evolution[END_REF]. Similar to capillary flows, elastohydrodynamic relaxation processes are not only limited to very thin pre-wetted films. In fact, an elastic sheet with zero spontaneous curvature but with an initial shape of a bump (Fig. 1) with a height much larger than the pre-wetted viscous film will relax towards a flat equilibrium state. Inevitably, the system must then crossover from a situation where the bump height is larger than the pre-wetted film height, to a situation where instead the pre-wetted Initially, the overall profile presents a localized bump, whose profile is invariant in the y-direction, i.e. quasi two-dimensional. Far away from the perturbation, the viscous film has a constant thickness . In the bump region, the height profiles ĥ(x, t) and h(x, t) = ĥ(x, t) -of the viscous film and the bump, respectively, vary with the horizontal position x and time t, and remain symmetric about x = 0. At x = 0, we define the characteristic height h(x = 0, t) = h0(t) of the bump and its typical radius R(t), with initial values given as h0(t = 0) = hi and R(t = 0) = Ri.

film becomes thicker than the bump. Here we investigate how the elastohydrodynamic leveling changes with the ratio between the bump height and the pre-wetted film thickness. In particular, are there different asymptotic regimes, and how do the system transition from one to another? At the nanoscale, thermal fluctuations are expected to contribute and may even dominate the dynamics [START_REF] Davidovitch | Spreading of viscous fluid drops on a solid substrate assisted by thermal fluctuations[END_REF][START_REF] Démery | Perturbative path-integral study of active-and passive-tracer diffusion in fluctuating fields[END_REF][START_REF] Nesic | Fully nonlinear dynamics of stochastic thin-film dewetting[END_REF][START_REF] Carlson | Fluctuation assisted spreading of a fluid filled elastic blister[END_REF][START_REF] Marbach | Transport and dispersion across wiggling nanopores[END_REF], which we quantify in the leveling dynamics. To answer these questions, we combine numerical solutions of a mathematical model based on the lubrication theory [START_REF] Batchelor | An introduction to fluid dynamics[END_REF] with scaling analysis and experiments.

II. MATHEMATICAL MODEL AND NUMERICAL PROCEDURE

We consider the system depicted in Fig. 1, where we focus on a system where gravity can be neglected, i.e. the bump height is smaller than the elasto-gravity length [START_REF] Hosoi | Peeling, healing, and bursting in a lubricated elastic sheet[END_REF]. Only situations where the bump height h(x, t) is small compared with its horizontal extent and where the film slopes are small, i.e. ∂ ĥ(x, t)/∂x 1, are considered. We describe the viscous flow between the plate and the solid substrate using lubrication theory [START_REF] Batchelor | An introduction to fluid dynamics[END_REF]. When the initial deflection h i of the elastic plate is small compared to its thickness d, we can neglect stretching and the pressure reduces to p(x, t) = B∂ 4ĥ (x, t)/∂x 4 , where B = Ed 3 /[12(1ν 2 )] is the bending rigidity of the plate, E is the Young modulus and ν is the Poisson's ratio [START_REF] Landau | Course of Theoretical Physics[END_REF]. In addition, the system is a spatially unconfined elastic sheet with the two lateral boundaries being free to move relative to the underlying fluid. Thus, the in-plane compression is suppressed, and bending stresses dominate the relaxation process regardless of the ratio d/h i . By assuming incompressible flow and imposing no-slip conditions at the two solid substrates, and considering a one-dimensional geometry as there are no variations along the y-direction, one obtains the governing equation for the evolution of the height profile (see e.g. Ref. [START_REF] Hosoi | Peeling, healing, and bursting in a lubricated elastic sheet[END_REF])

∂ ĥ(x, t) ∂t = ∂ ∂x B 12µ ĥ3 (x, t) ∂ 5ĥ (x, t) ∂x 5 + Γ ĥ3/2 (x, t)η(x, t) , (1)
where µ is the fluid's dynamic viscosity. At small scales, thermal fluctuations can also influence the dynamics, which is described by the last term of Eq. ( 1). This term mimics the stress generated by thermal fluctuations, originates from an additional symmetric random stress term in the Navier-Stokes equations and is obtained by an integration in the z-direction (for details see [START_REF] Carlson | Fluctuation assisted spreading of a fluid filled elastic blister[END_REF][START_REF] Grün | Thin-film flow influenced by thermal noise[END_REF][START_REF] Miguel A Durán-Olivencia | Instability, rupture and fluctuations in thin liquid films: Theory and computations[END_REF][START_REF] Nesic | Dynamics of thin fluid films controlled by thermal fluctuations[END_REF]). The noise term η(x, t), is multiplied by a prefactor Γ = k B T A /(6µb) where k B is the Boltzmann constant, T A is the ambient temperature, b is the width of the plate along the y-direction, and η(x, t) is modelled as a spatiotemporal Gaussian white noise such that η(x, t) = 0 and η(x, t)η(x , t ) = δ(xx )δ(tt ), where the symbols indicate average quantities. We non-dimensionalize Eq. ( 1) by: X = x/R i , Ĥ(X, T ) = ĥ(x, t)/h i , T = tBh 3 i /(12µR 6 i ), and Θ(X, T ) = η(x, t) 12µR 7 i /(Bh 3 i ). When Γ = 0, this non-dimensionalization procedure gives us a parameter-free partial differential equation for Ĥ(X, T ). When Γ > 0, the non-dimensional number N = 2k B T A R 3 i /(Bh 2 i b) appears as a pre-factor in front of the stochastic term, and N 2 measures the ratio between thermal and bending energies. For the macroscopic system provided in our experiment and described in detail below, i.e. T A = 300 K, h i = 2.5 µm, R i = 20 µm, µ = 10 4 Pa s and B = 1.3 • 10 -12 Nm we get the noise prefactor Γ = 2.5 • 10 -13 ms -1/2 and the energy ratio N = 1.75 • 10 -6 which is well within the elastic bending dominated regime. However, a transition from a dominant elastohydrodynamic leveling to a dominant stochastic leveling would occur for a system with temperature T A = 300 K, membrane perturbation height h i = 10 nm and radius R i = 5 µm for a bending modulus B in the range of 10 -100 k B T A where k B T A = 4 × 10 -21 Nm which corresponds to N in the range of 0 -8 [START_REF] Carlson | Fluctuation assisted spreading of a fluid filled elastic blister[END_REF].

We solve the dimensionless version of Eq. ( 1) numerically by using a finite element method, and we split it into three coupled equations for: the bump profile H(X, T ) = Ĥ(X, T ) -/h i , the linearized curvature ∂ 2 H(X, T )/∂X 2 , and the bending pressure ∂ 4 H(X, T )/∂X 4 . These fields are discretized with linear elements and solved by using Newton's method from the FEniCS library [START_REF] Logg | Automated solution of differential equations by the finite element method: The FEniCS book[END_REF]. For the deterministic case N = 0, an adaptive time stepping routine has been used with an upper time step limit of ∆T = 0.001 and a discretization in space ∆X ∈ [0.001; 0.01]. For the stochastic case N > 0, we have used a constant time step ∆T = 0.001, together with a discretization in space ∆X = 0.0025. At T = 0 we impose the initial condition: H(X, T = 0) = 1tanh(50X 2 ). We further impose the following boundary conditions at the boundary ∂Ω of the numerical domain:

H X ∈ ∂Ω, T ) = H X ∈ ∂Ω, 0), ∂ 2 H(X ∈ ∂Ω, T )/∂X 2 = 0, and ∂ 4 H(X ∈ ∂Ω, T )/∂X 4 = 0.
The noise Θ(X, T ) is introduced independently at each discrete position and time step using the "random" class with the "randn" Gaussian subclass from the Numpy library [START_REF] Oliphant | NumPy: A guide to NumPy[END_REF], with zero mean and a variance 1/(∆X∆T ). We avoid negative values of Ĥ(X, T ) (that might occur in the stochastic case due to the fluctuations), by imposing that when Ĥ(X, T ) < 10 -6 , it is put back to 10 -6 as in [START_REF] Davidovitch | Spreading of viscous fluid drops on a solid substrate assisted by thermal fluctuations[END_REF][START_REF] Nesic | Fully nonlinear dynamics of stochastic thin-film dewetting[END_REF]. To verify the predictions of Eq.( 1), we construct an experimental setup which is described in the following section.

III. EXPERIMENTAL PROCEDURE

The experimental setup is composed of a fiber of polystyrene (PS) with a glass-transition temperature T g, PS ≈ 100 • C deposited on a film of the same polymer supported on a silicon (Si) substrate. These samples are capped by a thin sheet of polysulfone (PSU) with T g, PSU ≈ 180 • C. Sample preparation is carried out as follows: PS fibers (with number-averaged molecular weight M n = 15.8 kg/mol, and polydispersity index PDI = 1.05, Polymer Source Inc., Canada) are pulled from the melt at 175 • C using a glass rod. Thin PS films are spin cast from a toluene solution onto 10×10 mm 2 Si substrates, leading to a thickness of 25 to 380 nm, measured using ellipsometry (Accurion, EP3). The films are annealed at 110 • C for at least 12 hours in vacuum to remove residual solvent and relax residual stresses. The PS fibers are then transferred onto the PS films and the ensemble is heated briefly above T g, PS . The heating allows the PS to flow, thereby resulting in a bump. Thin PSU films (M n ≈ 22 kg/mol, Sigma-Aldrich) are prepared by spin casting from a cyclohexanone solution onto freshly cleaved mica substrates (Ted Pella, USA). The PSU films have a thickness of ≈ 160 nm, measured using ellipsometry, and are annealed in vacuum at 200 • C for at least 12 hours. The PSU films are floated onto water and then transferred onto a supporting apparatus (described previously [START_REF] Rafael | Liquid dewetting under a thin elastic film[END_REF]), held only by the film edges. These freestanding films can be relaxed to an unstrained state ensuring no in-plane tension. The PSU films are finally transferred onto the PS sample. The part of PSU film at the edges of the Si wafer was then removed using a scalpel blade prior to annealing. This was done to ensure slippage at the boundary between the PSU film and liquid PS layer, thus rendering the relaxation bending dominated as discussed above.

After preparation, the samples were annealed on a hotstage (Linkam, UK) at 130 • C, which is above T g, PS but below T g, PSU . Hence, the PS becomes a viscous liquid while the capping PSU film remains an elastic solid, thus realizing the system illustrated in Fig. 1. The height profile is imaged during annealing using optical microscopy with a red laser line filter (λ = 632.8 nm, Newport, USA), which creates interference fringes in the region of the bump, as shown in Fig. 2(a), due to the light that is reflected from the Si substrate. It is clear from these fringes that the initial fiber and resulting flow are one-dimensional over length scales that are many times the width of the perturbation itself. Each interference fringe corresponds to a change in height of λ/(4n), where n ≈ 1.57 is the average index of refraction of the two polymers that make up the sample (n PS = 1.53 and n PSU =1.61). This allows the bump profile h(x, t) to be reconstructed by fitting a polynomial to the fringe data, as shown in Fig. 2(b). Such profiles can then be used to track the leveling dynamics, and to extract in particular the evolution of the height h 0 (t) of the bump with time for various initial geometries.

x [µm] x [µm] h [µm] h [µm] t [min]

IV. RESULTS

A. Elastohydrodynamic leveling

We first start by investigating the elastohydrodynamic leveling in the absence of thermal fluctuations (N = 0). In Fig. 3 we show the numerical solutions of the dimensionless version of Eq. ( 1) for N = 0 and we can see that the aspect ratio h i / controls both the time scale for leveling and the detailed features of height profile. The smaller h i / , the faster the dimensionless leveling process. Also, the dip created near the advancing front of the perturbation is enhanced both in magnitude and lateral extent for smaller h i / . We remark that for each initial aspect ratio there is a transition period of a few numerical time steps preceding the onset of the leveling process. This part of the data is not included in Fig. 4 as it is considered to depend on the initial condition, but does not influence the later dynamics. For h i / = 43, the numerical height profiles are further compared with our experiments, which are found to be in good agreement. We recall here that the elastic plate is floating on the liquid film and has edges that are free to move. Therefore, the pressure contribution from bending still largely dominates any contribution from stretching and

h i = 0.1 (d)
FIG. 3. (a-b) Bump height profiles for an initial aspect ratio hi/ = 43, from a numerical solution and an experiment at two dimensionless times T as indicated, which correspond respectively to t = 37 min and t = 416 min. The geometrical parameters are hi = 2.9 µm, Ri = 16.5 µm, and = 67 nm. In order to account for the experimental uncertainties in the geometrical parameters, the experimental time t is divided by a free fitting factor α = 0.13. Note that, specifically for these figures, the initial condition for the numerical solution was fixed by a curve fitting of the actual experimental profile at t = 13min. (c-d) Bump height profiles for an initial aspect ratio hi/ = 0.1, from a numerical solution at two dimensionless times T , as indicated, chosen so that the central heights H(X = 0, T ) match the ones in the top row.

Eq. ( 1) is still valid. We now turn to a scaling analysis of Eq. ( 1) for N = 0. When h 0 (t)/ 1, the equation can be linearized and reduces to 12µ∂h/∂t = B 3 ∂ 6 h/∂x 6 and we deduce the long-term scaling for the temporal evolution of the horizontal length of the bump: R(t) ∼ [B 3 t/(12µ)] 1/6 . Since there is area conservation in the (x, z)-plane, we assume R(t)h 0 (t) to be constant, that is evaluated to R i h i at t = 0. By combining these scaling relations we get for h 0 (t)/ 1

h 0 (t) ∼ τ t 1/6 , (2)
where τ = 12µh 6 i R 6 i /(B 9 ) is the characteristic time scale for the bending-driven leveling dynamics. As we operate within the regime where bending dominates over stretching, a similar result is obtained by considering the force balance between the viscous and bending forces [START_REF] Vandeparre | Hierarchical wrinkling patterns[END_REF]. Also if we include isotropic stretching due to clamped boundaries a similar scaling law appear, but now with an additional logarithmic term, R(t) ∼ (t/log(t)) 1/6 [START_REF] Kodio | Lubricated wrinkles: Imposed constraints affect the dynamics of wrinkle coarsening[END_REF]. However, when h 0 (t)/ 1 we must match the curvature of a traveling-wave solution localized near the advancing front with the 10 -30 10 -24 10 -18 10 -12 10 -6 10 0 10 6 10 12 10 18 10 -4 1) with N = 0, and the black circle-shaped markers are scaled experimental data points. The exponents of the two asymptotic regimes of Eqs. ( 2) and (3) are indicated with triangles. The inset provides a zoom in the region containing the experimental data for the three samples, with initial aspect ratios hi/ = 30, 43, 56; corresponding respectively to = 50, 67, 26 nm; hi = 1.52, 2.9, 1.48 µm; and Ri = 9.6, 16.5, 9.9 µm. The uncertainties in all experimental length scales are about 5%. To compensate for those, the characteristic time τ for each sample is multiplied by a free fitting factor α = 0.7, 0.13, and 1.3, respectively. quasi-static solution to obtain the correct scaling [START_REF] Lister | Viscous control of peeling an elastic sheet by bending and pulling[END_REF], i.e. constant pressure in the bump, leading to [START_REF] Carlson | Fluctuation assisted spreading of a fluid filled elastic blister[END_REF]

h 0 (t) ∼ τ t 2/17 . (3)
By balancing the two asymptotic predictions above, we expect the crossover between them to occur around t/τ ≈ 1.

In addition, these asymptotic regimes suggest that h 0 (t)/ is essentially a function of t/τ only, independent of the value of h i / in particular. In order to test our scaling predictions, we compute numerical solutions of the dimensionless version of Eq. ( 1) for N = 0, with h i / ∈ [10 -2 , 10 3 ] and extract h 0 (t)/ as a function of t/τ . These numerical results are plotted in Fig. 4 and compared to the experimental data. For each sample, the experimental data is matched to the numerical data through one fitting parameter α in front of the time scale τ . The values of the fluid viscosity and elastic Young modulus are highly sensitive to the temperature in the experiments, and we estimate them to be µ ≈ 10 4 Pa s [START_REF] Mcgraw | Self-similarity and energy dissipation in stepped polymer films[END_REF][START_REF] Ilton | Capillary leveling of freestanding liquid nanofilms[END_REF] and E ≈ 2.6 GPa [START_REF]Springer Handbook of Condensed Matter and Materials Data[END_REF], respectively. Since all experiments were carried out at the same temperature and with the same polymer, sample-to-sample variations in τ result only from uncertainties in the geometrical parameters h i , R i , d, and . The obtained α values are 0.13, 0.7, and 1.3 for the three samples and each of these values are reasonably close to unity. More importantly, the sample-to-sample variations in α do not exceed a factor of 10, which is well within the expected relative error arising from the high sensitivity of τ to the geometrical parameters. The general agreement between the experimental data and the numerical predictions is good, over about 5 orders of magnitude in t/τ . The systematic early-time tail in the experimental data might be attributed to the initial compressive thermal stresses in the elastic layer, which arise due to the rapid heating of the samples from room temperature to T = 130 • C, which relax prior to leveling and the time needed for the initial shape to enter the asymptotic regime.

The master curve in Fig. 4 confirms that h 0 (t)/ is a function of t/τ . Furthermore, the two scaling regimes predicted above are indeed present, with pre-factors close to unity, and the crossover between the two being located near t/τ ≈ 1 as predicted. Any bump that initially starts in a thin pre-wetted film regime h 0 (t)/ 1 will eventually crossover to a thick-film regime h 0 (t)/ 1, with the corresponding power laws in time. As a final remark, a similar combination (not included here) of numerical simulations and scaling analysis can been performed for an axisymmetric geometry, leading to h 0 (t) ∼ t -2/11 for h i / 1, and h 0 (t) ∼ t -1/3 for h i / 1.

B. Stochastic leveling

We investigate here the leveling process when it is dominated by thermal fluctuations (N > 0). As shown in Fig. 5, the numerical solutions suggest that the aspect ratio h i / is again essential, as it sets the time scale for leveling where the smaller h i / , the faster the dimensionless leveling process. Moreover, by comparison with the deterministic (N = 0) case in Fig. 3, the stochastic (N > 0) profiles exhibit spatiotemporal fluctuations and adopt different average shapes and leveling dynamics.

To go further, we propose a scaling analysis Eq. ( 1), inspired by Ref. [START_REF] Cormier | Beyond tanner's law: Crossover between spreading regimes of a viscous droplet on an identical film[END_REF]. We consider specifically the N 1 limit, for which the thermal fluctuations are the dominant driving contribution to the dynamics and we assume that we can neglect the bending term so that Eq. ( 1) reduces to ∂h/∂t = Γ∂[( + h) 3/2 η]/∂x. We consider the average quantities h 0 (t) and R(t) , where we invoke the ∼ (tx) -1/2 scaling [START_REF] Davidovitch | Spreading of viscous fluid drops on a solid substrate assisted by thermal fluctuations[END_REF] for the root mean square value of the averaged noise over a space interval x and a time interval t. By assuming that the average area conservation in the (x, z)-plan can be expressed as h

0 (t) R(t) ∼ h i R i , we get h 0 (t) 1 + h 0 (t) 3 ∼ τ Γ t (4)
where

τ Γ = 6µh 3 i R 3 i b/(k B T A 4
) is the characteristic time scale for the stochastic leveling dynamics. Interestingly, Eq. ( 4) describes a complete crossover between two asymptotic regimes in the stochastic leveling dynamics: for h 0 (t) / 1, we obtain h 0 (t) / ∼ (τ Γ /t) 1/4 , and thus we recover h 0 (t) ∼ t -1/4 [START_REF] Davidovitch | Spreading of viscous fluid drops on a solid substrate assisted by thermal fluctuations[END_REF]; while for h 0 (t) / 1, we get h 0 (t) / ∼ τ Γ /t. We expect the crossover between the two asymptotic regimes to occur around h 0 (t) / ≈ 1, i.e. around t/τ ≈ 1/8.

In order to test the prediction in Eq. ( 4), we compute the numerical solution of the dimensionless version of Eq. (1) for 5 ≤ N ≤ 8, with h i / ∈ [10 -1 , 10 2 ]. By averaging over a minimum of 30 realizations, we can extract h 0 (t) / as a function of t/τ Γ and the results are plotted in Fig. 6. The data from the numerical solutions is in good agreement with Eq. ( 4) for all h 0 (t) / , with no adjustable parameter. Our results highlight that Eq. ( 4) gives an accurate prediction of the stochastic leveling dynamics and show that the missing pre-factor is close to unity. Finally, in order to further highlight the underlying self-similarity associated with each of the two asymptotic regimes, the insets of Fig. 6 show the corresponding bump height profiles rescaled according to Eq. ( 4). In each asymptotic regime the height profiles collapse onto a universal shape which confirms the overall self-similarity in the leveling dynamics.

V. CONCLUSION

We have described the elastohydrodynamic and stochastic leveling of an elastic plate placed atop a viscous film. By combining numerical solutions, scaling analysis, and experiments, we identified various canonical regimes. Our results highlight the importance of the driving mechanism, either by elastic bending of the plate or thermal fluctuations, and the influence of the aspect ratio of the bump height to the film height. For each of these two driving mechanisms, a crossover between two distinct asymptotic regimes is controlled by the aspect ratio. These findings can be helpful to explain elastohydrodynamic leveling dynamics found in biological, engineering, or geological processes.

  FIG.1. Schematic of the system we are studying. An elastic plate is supported by a viscous film on a solid substrate, surrounded by ambient air. The elastic plate has a thickness d and a width b. Initially, the overall profile presents a localized bump, whose profile is invariant in the y-direction, i.e. quasi two-dimensional. Far away from the perturbation, the viscous film has a constant thickness . In the bump region, the height profiles ĥ(x, t) and h(x, t) = ĥ(x, t) -of the viscous film and the bump, respectively, vary with the horizontal position x and time t, and remain symmetric about x = 0. At x = 0, we define the characteristic height h(x = 0, t) = h0(t) of the bump and its typical radius R(t), with initial values given as h0(t = 0) = hi and R(t = 0) = Ri.

FIG. 2 .

 2 FIG. 2. (a) Typical optical microscopy images showing the temporal evolution of the interference fringes due to the liquid bump capped by the elastic plate (20 µm scale bar). The image at 20 minutes is uncropped, showing the invariance in the y-direction, while the later images are cropped at the red box. (b) Intensity profile (averaged along the y-direction) of the bump at a given time t, and corresponding reconstructed bump profile at 400 minutes. (c) Temporal evolution of the bump profile.

T = 9 . 7 ×

 97 10 -6

FIG. 4 .

 4 FIG.4. Non-dimensional bump height as a function of dimensionless time in the bending case, for various initial values of hi/ ∈ [10 -2 ; 10 3 ]. The coloured diamond-shaped markers are rescaled data points from the numerical solutions of the version of Eq. (1) with N = 0, and the black circle-shaped markers are scaled experimental data points. The exponents of the two asymptotic regimes of Eqs. (2) and (3) are indicated with triangles. The inset provides a zoom in the region containing the experimental data for the three samples, with initial aspect ratios hi/ = 30, 43, 56; corresponding respectively to = 50, 67, 26 nm; hi = 1.52, 2.9, 1.48 µm; and Ri = 9.6, 16.5, 9.9 µm. The uncertainties in all experimental length scales are about 5%. To compensate for those, the characteristic time τ for each sample is multiplied by a free fitting factor α = 0.7, 0.13, and 1.3, respectively.

2 FIG. 5 . 2 ( 1 FIG. 6 .

 25216 FIG. 5. Contour plot of the dimensionless bump height profile H(X, T ) as a function of both the dimensionless position X and time T , as obtained from numerical solutions of the dimensionless version (see text) of Eq. (1), with N = 5, and for hi/ = 10 (left) or hi/ = 0.2 (right). The thick solid lines indicate H(X, T ) = 0.03 as an arbitrary reference.

ACKNOWLEDGMENTS

The financial support by the Research Council of Norway (project grant 263056), the Natural Science and Engineering Research Council of Canada and the Joliot Chair of ESPCI Paris is gratefully acknowledged. The authors also thank Tak Shing Chan, Miroslav Kuchta, Corentin Mailliet, René Ledesma-Alonso, Maxence Arutkin, Elie Raphaël, Howard Stone, Pauline Olive, Katelyn Dixon, Paul Fowler, Mark Ilton and David Dean, for interesting discussions and preliminary works in different geometries.


1965

Astérix Among the Stars

In 1965, the cartoon hero Astérix accomplished his fictional tour of Gaul and gave his name to the first French satellite for its inaugural orbit of the earth. The French press was ecstatic.

Was the nation's global power about to be relaunched too? Amid the fanfare surrounding the launch, it was sometimes forgotten that the nation's space autonomy owed much to earlier international cooperation.

On November 26, 1965, France officially joined the elite of space powers. With the launch of Astérix, the first French satellite, from the military base of Hammaguir in Algeria, France became the third country after the Soviet Union (1957) and the United States (1958) to possess its own space agency, the National Center for Space Studies (CNES, created in 1962) and its own access to space beyond the world's atmosphere. This exploit was all the more astonishing in that, prior to the late 1950s, the country had barely engaged in space research at all.

Astérix showed that French technology was capable of tackling a completely new domain, and with a rapidity that beggared belief, by pioneering new methods of organizing scientific research and institutions that would coordinate that research. In purely technical terms, success was sealed by the development of a rocket capable of carrying payload into orbit. The launcher, called Diamant-A, was part of a program known as Pierres Précieuses (Precious Stones); it carried a satellite, A-1, which the public later came to know under the more telling name Astérix. It consisted of a small capsule weighing about ninety pounds whose principal role was to communicate back to Earth a variety of data concerning the launch itself. Even though this capsule soon stopped functioning, it turned France into a serious European competitor in the race for independent access to outer space.

For many years-and this is particularly true of human activity in space-historians have favored a binary interpretative approach that reduced the Cold War to a confrontation between the United States and the USSR. Yet despite the undeniable importance of these two principal world powers, the Astérix satellite and French space activity in general make clear that it would be wrong to discount the European nations in the field of space exploration.

Viewed transnationally, globally, an intense circulation of techniques and individuals had existed for some time, along with extensive sharing and transmission of knowledge about space. The historian Asif Siddiqi, a specialist in space activities, has noted that, just as in the nuclear domain, "every nation involved [in the development of ballistic missiles and space technology] has been a proliferator and has taken advantage of proliferation."

Informal international space-research networks first took shape in Europe and the United States in the 1920s, while the development of ballistic missiles in Germany shortly afterward benefited greatly from exchanges of technology and knowledge with other countries. In the 1950s, German engineers and scientists who had contributed to the construction of wartime V-2 missiles were able to help with the space programs of the two superpowers, with France and the United Kingdom also gaining from their knowledge. Later, China and Japan took full advantage of the new technologies flowing out of the USSR and United States. Thus only ten years after Astérix reached orbit, all these countries had become members of the exclusive club of space powers. They were joined in the 1980s by India and Israel, who also relied on the support offered by Western Europe, the United States, and the Soviet Union.

Even so, the scale of transnational exchange should not blind us to the relevance of the nation itself as a basis for analysis. Technologies developed through space research played a vital role in the construction of a new national identity for postwar France. Indeed, the burgeoning space research sector in the late 1950s contributed to a broader venture: the modernization and reinvention of France as a dynamic, diplomatically and militarily autonomous technological power. There was an overriding military imperative to this impetus, embraced by de Gaulle after his election as president in 1958, as demonstrated by the production, in parallel with the development of space technologies, of new nuclear weapons as a means to project French power. France's participation in both the arms and the space races added political and symbolic clout to the country's advances in the postwar decades.

The Diamant launcher was the product of this uneasy relationship between military goals and civil applications. Although Diamant was a civil project, much of its technology came from the military domain, and more specifically the Pierres Précieuses program that France had put in place in the early 1960s. Persuaded that in the event of a new war it could not depend on the United States or the United Kingdom, de Gaulle's France sought to attain its strategic objectives by developing its own nuclear deterrent and its own independent capability to deliver nuclear warheads. When in 1961 de Gaulle gave the SEREB ballistic missile research consortium the green light to continue developing a launcher, French engineers had already successfully tested powder-propelled missiles using trial vehicles. By combining powder-propelled and liquid-propelled stages, they came up with the Saphir guided missile-which became the blueprint for the Diamant launcher that blasted Astérix into orbit.

Predictably, the French press celebrated Astérix as a feat of huge national importance.

Yet France had been quietly engaged in a variety of cooperative scientific endeavors from the time of her earliest space forays, beginning with a research program known as the International Geophysical Year (1957)(1958). One early French satellite rode on a NASA launcher. Ever since its own creation in 1958, NASA had routinely invited other nations to participate in the conquest of space; France (like several other European nations) had welcomed the opportunity to send engineers and scientists to the United States, where they were able to acquire new knowledge and learn how to orchestrate complex technical projects.

All this contributed to the establishment of a common language for the field of space activities on both sides of the Atlantic. Within Europe, too, thanks to the initiatives of influential scientists like Edoardo All this stemmed from the original Astérix, a modest technological advance that remains to this day the historic symbol of France's entry into the vast area of space politics and activities, to which it has made significant contributions ever since the 1950s.

Sebastian Grevsmühl

  Amaldi and Pierre Auger, in the early 1960s efforts in the domain of space research and exploration began to be shared and coordinated. France took part in the creation of the European Space Research Organization (ESRO), which was founded in 1962 in the spirit of the European Organization for Nuclear Research (CERN) in Geneva, in order to pool the European resources dedicated to the development of scientific experiments and satellites for space. The much more sensitive question of a European launcher, which was explicitly excluded from ESRO, was entrusted to another European consortium set up in March 1962 under the name of the European Launcher Development Organization (ELDO). France pledged to build one of the three stages of this launcher, which was named Europa. Even though ESRO had many difficulties and ELDO was ultimately a failure, this dual approach made it possible for France to take the lead in the domain of European space research, notably by militating in favor of a European space agency in the mid-1970s. This brought the two sectors of innovation and research-launchers and satellites-together under a single umbrella organization.


Mauro Barisione

The transformation of the Radical Right Gender Gap: The case of the 2014 EP Election

. Using the 2014 European Election Studies dataset, this paper revisits the RRGG in a comparative perspective, analyzing gender as a predictor of vote choice for Radical Right Parties in Austria, Belgium, Denmark, France, Netherlands and Sweden. While the gender gap persists once controlled for education, age, religion and occupation, the interaction of gender with political attitudes and leftright ideology significantly affects electoral support for RR parties in the six cases studied. And the RRGG appears tightly entangled with the "traditional gender gap" on the one side and the "modern gender-generation gap" on the other.

Introduction

Since the pioneer insight of Hans Georg [START_REF] Betz | Radical right wing populism in Western Europe[END_REF] a large body of literature has tried to explain the unequal support of men and women for the Populist Radical Rights that have developed in Europe since the 1980s -or "Radical Right Gender Gap" [START_REF] Givens | The Radical Right Gender Gap[END_REF][START_REF] Norris | Radical Right: Voters and Parties in the Electoral Market[END_REF][START_REF] Rippeyoung | When women are Right :the influence of gender, work and values on European far right party support[END_REF][START_REF] Mudde | Populist Radical Right Parties in Europe[END_REF][START_REF] Immerzeel | Explaining the gender gap in radical right voting: a cross national investigation in 12 Western European countries[END_REF][START_REF] De Bruijn | The gender gap in radical right voting: explaining differences in the Netherlands[END_REF][START_REF] Immerzeel | Explaining the gender gap in radical right voting: a cross national investigation in 12 Western European countries[END_REF] 1 . However until recently few studies had attempted a systematic study of the RRGG, controlling for the possible effect of other variables. Those who did show large variations in the gap from one country to another [START_REF] Immerzeel | Explaining the gender gap in radical right voting: a cross national investigation in 12 Western European countries[END_REF]. In addition, even where this gap existed, things might be changing [START_REF] Barisione | Marine Le Pen et les femmes : la fin du Radical Right Gender Gap en France et en Europe? Paper presented at the French Political Science Association Congress[END_REF], at a time when Radical Right Parties increasingly present themselves as women friendly, targeting Islam and Muslims in the name of democratic values and women's rights (Akkerman and Haggelund, 2007;[START_REF] De Lange | Gender and right-wing populism in the Low Countries: ideological variations across parties and time[END_REF]. In France for instance, while gender had a significant impact on the votes for Jean-Marie Le Pen, it had none on the votes for his daughter in the 2012 presidential election, once controlled by age, education, religious practice, occupation and left right position (Mayer, 2013a and2013b). Therefore this paper proposes a re-examination of the impact of gender on voting in the 2014 EP (European Parliament) Election.

A first section revisits the literature about the successive electoral gender gaps through recent decades. A second one presents our research strategy and methodology. A third one tests our hypotheses, using data from the 2014 European Election Studies in Austria, Belgium, Denmark, France, Netherlands, and Sweden.

Theoretical framework: revisiting the electoral gender gap(s)

From traditional to modern gender gap

The term "gender gap" was coined in the US by the National Organization of Women (NOW) at the time of the election of Ronald Reagan. In the 1980 American presidential race, for the first time since they got the right to vote, women gave more support to the Democrats. Since then this gap has persisted, and there is a robust relation between gender and vote, still significant after controlling for age, class, race and religion [START_REF] Manza | The gender gap in U.S. presidential elections : When? Why? Implications ?[END_REF][START_REF] Whitaker | Voting. The gender gap[END_REF][START_REF] Abendschön | The Gender Gap in Voting Revisited: Women's Party Preferences in a European Context[END_REF]. A similar divide has appeared since in several post industrial democracies, and notably in Europe [START_REF] Inglehart | Rising Tide[END_REF][START_REF] Mossuz-Lavau | Genre et politique. Y-a-t-il une exception française?[END_REF][START_REF] Giger | Towards a modern gender gap in Europe? A comparative analysis of voting behaviour in 12 countries[END_REF], women moving more to the left than men whatever the indicator used. It was called the "modern gender gap", in opposition to the previous "traditional" one.

Women in most countries got the right to vote long after men. They first tended to be more conservative, and give more support than men to right wing parties with a traditional view of women's role and family [START_REF] Mossuz-Lavau | Enquête sur les femmes et la politique en France[END_REF]. Their progressive dealignment from the right and realignment towards the left, according to the developmental theories framed by Inglehart and Norris (1999), reflects structural and cultural changes occurring in most post industrial societies after World War 2.

Women's rising level of education and their massive entrance in the labour market on the one hand, the process of secularization, the break-up of traditional family units, and the emancipating influence of post-materialist values and feminist movementsencouraging autonomy and self expression -on the other hand, brought them gradually closer to left-wing parties. This was particularly the case among the younger generations, leading Pippa Norris to coin the term of "generation gender gap" [START_REF] Norris | Mobilizing the Women's Vote: The Gender-Generation Gap in Voting Behaviour[END_REF]. In the long run, generational replacement should even expand the process: "If a generational rather than a life-cycle effect, as seems most likely, this suggests that the process of generational turnover will probably continue to move women leftwards. In the long-term, as younger voters gradually replace older generations, through secular turnover, the modern gender gap should therefore strengthen and consolidate in established democracies. " (Inglehart and Norris, 2000, p. 459).

The emergence of a Radical Right Gender Gap

Since the 1980s though, the electoral boom of radical right anti immigrant parties, especially in Europe, has brought about a third cleavage, women appearing more reluctant to give their votes to these parties. Hans-Georg Betz was the first to outline the phenomenon (Betz, 1994, p.142-148), and the difference seems to persist [START_REF] Givens | The Radical Right Gender Gap[END_REF][START_REF] Norris | Radical Right: Voters and Parties in the Electoral Market[END_REF][START_REF] Immerzeel | Explaining the gender gap in radical right voting: a cross national investigation in 12 Western European countries[END_REF][START_REF] De Bruijn | The gender gap in radical right voting: explaining differences in the Netherlands[END_REF]. Both structural and attitudinal factors have been put forward to account for what has become known as the "Radical Right Gender Gap" [START_REF] Givens | The Radical Right Gender Gap[END_REF].

A first line of argument considers the gendered division of labour [START_REF] Mayer | Ces Français qui votent Le Pen[END_REF][START_REF] Givens | The Radical Right Gender Gap[END_REF][START_REF] Rippeyoung | When women are Right :the influence of gender, work and values on European far right party support[END_REF][START_REF] Studlar | Explaining the gender gap in voting: a cross national analysis[END_REF]. Men are still overrepresented in manual occupations, particularly among industrial blue collars (ouvriers), the "globalization losers" by excellence. In competition with cheap labour in developing countries on the one side and with immigrants inside the country on the other, they are the most likely to support the Radical Right [START_REF] Betz | Radical right wing populism in Western Europe[END_REF][START_REF] Kitschelt | The Radical Right in Western Europe. A Comparative Analysis[END_REF][START_REF] Kriesi | West European Politics in the Age of Globalization[END_REF][START_REF] Oesch | Explaining Voters' support for right-wing populist parties in Western Europe; Evidence from Austria, Belgium, France, Norway and Switzerland[END_REF][START_REF] Bornschier | The populist right, the working class, and the changing face of class politics[END_REF]. Conversely, women are more often employed in non-manual clerical or services jobs and in the public sector, and hold on the whole economically more secure positions and less exposed to immigration. Therefore they should be less inclined to turn to the Radical Right.

A second line of research stresses the greater religiosity of women. Christian Churches all over Europe have repeatedly condemned the anti-immigrant and inegalitarian message of the Radical Rights in the name of the Evangels. Although a process of secularization is taking place in post-industrial European societies, women, especially the elderly, still attend religious services more often than men. They should therefore be more likely to hear the warnings of the Church. Sineau (2004, p.220) has showed that in Catholic France, not only women go to church more often than men, they are also more receptive to the Christian message. At similar level of religious practice than men, they were far less inclined to vote for Jean-Marie Le Pen in the 2002 Presidential election, the gap reaching its peak among elderly women that were catholic and regular church-goers. These remnants of the "traditional gender gap" could explain the RRGG of today.

A third block of research, more in line with the modern gender gap perspective, points on the contrary to the gradual diffusion of feminist ideas at all levels of society -a "rising tide" [START_REF] Inglehart | Rising Tide[END_REF]. This could prevent women, especially the new highly educated generations, from supporting far right parties that defend a traditional ideology, reducing them to their role of spouses and mothers. Conversely, the very spread of feminist ideas, the claims for equality and the growing presence of women in the work force could be seen as a threat for masculine supremacy, breeding insecurity and resentment and feeding an authoritarian anti-feminist vote for the far right [START_REF] Perrineau | Le symptôme Le Pen[END_REF]. This is even more likely in working class and blue collars milieus, where norms of strong manhood and masculinity, based on physical strength, still predominate [START_REF] Frader | Breadwinners and Citizens: Gender in the Making of the French Social Model[END_REF][START_REF] Molinier | Déconstruire la crise de la masculinité[END_REF].

A last line of research stresses the persistence of gender stereotypes from early socialisation and their impact in the political realm [START_REF] Mossuz-Lavau | Genre et politique. La marche vers l'indifférenciation[END_REF][START_REF] Huddy | Gender Stereotypes and the Perception of Male and Female Candidates[END_REF][START_REF] Lahire | Héritages sexués : incorporation des habitudes et des croyances[END_REF]. Girls are still brought up differently than boys, expected not to be violent or aggressive. Women in surveys, for instance, appear systematically less in favour of war and conflict 2 . The image of extremism associated to Radical Right Parties could be a deterrent factor for them. Women are also less trained to assert themselves, to confront disagreement or question social norms [START_REF] Immerzeel | Explaining the gender gap in radical right voting: a cross national investigation in 12 Western European countries[END_REF]. One of the main finding of a study based on three waves of the CSES data in 32 countries shows that men are politically more "assertive" than women, more likely to cast a vote for new or ambiguous parties, whatever their political stands: "the more other respondents indicated to highly dislike the party, the smaller the relative share of female voters" [START_REF] Harteveld | The Gender Gap in Voting: Extremity, Ambiguity and Social Cues[END_REF] p.12; also see [START_REF] Immerzeel | Explaining the gender gap in radical right voting: a cross national investigation in 12 Western European countries[END_REF].

Beyond the "Radical Right Gender Gap" approach

That having been said, one may ask if the different explanations put forward to explain the RRGG are still relevant today, in a context of economic recession and political disaffection. Indeed, new conditions potentially favoring a shift toward the opposite direction, i.e. an erosion of the RRGG, could easily be identified.

Since 2008, the deteriorating economic situation is blurring the bordure between manual and non--manual positions, with both white--collar women in service jobs and blue--collar men in manufacturing jobs being increasingly exposed to part time and low paid insecure positions. As a consequence, women's level of support for the Radical Right should come closer to men's 3 . In addition, religion could work less as a protection against intolerance and feed, on the contrary, anti--Muslim and anti--minorities' resentment in a context of anti--Muslim polarization. A similar backlash trend may concern feminist attitudes. As a belief in equality between men and women, feminism should less clearly deter women from supporting Radical Right Parties, which are gendering the immigration issue and presenting Islam as a threat to women's rights. Finally, the ideologically less extreme and thus more acceptable image given by some of these Radical Rights could weaken the reluctance of female voters to vote for them. Hence, the RRGG might well be in transition, with forces pulling in different directions.

One may even go further, considering that the exclusive focus on the latest research on the "RRGG" and its size are more an obstacle than a help to understand what gender does to electoral behavior. Indeed, the emergence of a RRGG does not necessarily mean that the two previous electoral gaps have disappeared; they need to be articulated together. Moreover, what matters is not just the size of the gap, but the level of support for RR. The fact that there is no radical right gender gap left could have two opposite meanings: that women give the same high support to RR than men or, conversely, that they show the same rejection of these parties. Similarly, the same gap in size can have totally different meanings. Our assumption, in line with Sineau's finding, is that two types of women can be less willing than men to vote for the radical right for opposite reasons. In certain parts of the electorate, this will reflect a modern gender gap, with young educated women being ideologically more progressive and left-wing than men, thus more reluctant to support RR. In other parts, a traditional gender gap may subsist, with elderly, religious women being more prone than men to support conservative mainstream parties rather than the extreme right. This implies that one should explore more closely the specific interactions between gender and the other variables that explain support for the radical right: age, education, occupation, religion, left-right position and social attitudes (especially relative to immigration and feminism/post materialism), and see how gender gaps -in the plural -appear, disappear or overlap.

Theoretical expectations

In this paper, due not only to limitations of space but also in the number of variables available in the 2014 EES dataset, we focus on a specific set of expectations arising from our theoretical framework, which combines -as suggested abovepresence/absence of the RRGG with high/low levels of support for the radical right.

In addition, we are interested in disentangling "traditional" and "modern" forms of radical right gender gap.

The framework presented below precisely combines levels of gender gap (yes=significant; no= absent or statistically non significant) with levels of support for radical right parties (yes=highest; no=lowest). This four-cell table enables us to differentiate between cases of inter-gender "equal rejection" (no Gender Gap and no Radical Right support) and "equal support" (high RR score with no GG) on the one side; between cases of "traditional" gap in radical right support (high RR score, especially among men voters) and "modern" patterns of rejection of the RR (women scoring particularly low) on the other side. Within this bi-dimensional framework for analysis of the radical right gender gap, the conventional concern about the magnitude of this gap is only a starting point -a broad picture that necessitates zooming in further into the complexities of this topic.

As for this "broad picture", we expect -given the contradictory trends outlined in the previous section -to find only a mild gender gap in our overall measure of support for radical right parties in the six countries considered. We also anticipate that this gap should be reduced further, if not entirely suppressed, when introducing controls for the main sociological variables -especially education and religiosity -due to compositional gender differences along these societal lines. Finally, more general ideological orientations (such as left-right self-placement) and social attitudes (e.g. toward immigrants) should result as the fundamental "mediators" of support for radical right parties, i.e. the radical right gender gap should disappear when holding these factors equal across gender.

After investigating this general pattern, however, we will move on to consider more specific interactions of gender with a set of theoretically relevant variables, which might act as "moderators" of the radical right gender gap 4 . Indeed, we argue that the issue of gender heterogeneity -i.e. women and men voters being considered not as homogenous blocs but in their intra-group differentiations -is too often neglected in the study of political gender gaps in general, and of the radical right gender gap in particular. Yet we know from scholarly literature that two gender-based mechanisms might coexist and drive the overall RRGG: the first one implies that a category of women present, both in social and ideological terms, all the typical features of RR voters but actually tend to vote less for these parties than men, possibly due to their lower political "assertiveness". This category of women would thereby reflect the idea of a "traditional gender gap", with higher support for mainstream right-wing parties and lower support for radical right parties. We expect to find evidence of this mechanism possibly among elder and more religious women, and even more clearly among those with conservative ideology and less tolerant attitudes toward social outgroups.

The second mechanism points, on the contrary, to the importance of a category of women with social and political profiles completely averse to the radical right, and which we would expect to reject RR parties even more strongly than men. This expectation is based on the notion of a "modern gender gap" resting on new generations of highly educated women characterized by culturally more liberal and politically more progressive attitudes. Alongside a "modern" pattern of greater women' rejection of the radical right, we might also find evidence of "equal rejection" across similar categories of women and men voters. Finally, social economic insecurity aggravated by the economic crisis exposes women in service jobs as well as blue--collar men to unemployment, part time jobs and low paid insecure jobs. This should bring women, especially those already excluded from labour force, close to men in their level of support for the Radical Right. Hence, our set of theoretical expectations transcends -as suggested above -the mere issue of the overall magnitude of the radical right gender gap and aims to attain a more fine-grained comprehension of the phenomenon in its multiple dimensions.

Theoretical framework: Gender Gap (GG) and Radical Right support (RR)

Although our main interest lies in radical right parties as a relatively homogenous ideological family, we are not assuming cross-country homogeneity in patterns of Radical Right Gender Gap. Not only do general levels of support of RR parties differ across countries, but also the gender gap itself could differ across countries, especially depending on whether or not the RR party has pursued a strategy of normalization in the national political arena. To the extent that a normalization strategy -such as that pursued by National Front, Party of Freedom, FPÖ, and Danish People's Partymakes the RR more socially acceptable and less perceived as an ideological "outsider", women might come to present roughly the same support as men for RR parties. Of course, results concerning this last hypothesis will be suggestive at best, given the very limited number of national cases selected for this study.

Data and Method

A special attention will be given to methodological issues. Indeed, very few studies attempt a systematic analysis of the RRGG. Secondly, we test the measure of association between the two variables (vote choice and propensities to vote for a RR party) using an ANOVA. Both their overall correlation (eta= .737) and effect size (eta squared = .542) prove very strong.

Therefore, voting probabilities as collected in an immediate post-election context provide an excellent piece of information about the behavioral propensities of each respondent.

As shown in Appendix figure 1, most respondents will give the radical right party the score '0' -almost 60% overall, which corresponds to the portion of those who do not vote, and do not consider to do it, for a radical right party. But the presence of a continuous scale prevents from losing important information about: (1) radical right voters who did not turn out at a "second-order election" such as the EP election [START_REF] Reif | Nine Second order National Elections-a Conceptual Framework for the Analysis of European Election Results[END_REF][START_REF] Hix | Second-order effects plus pan-European political swings: An analysis of European Parliament elections across time[END_REF]; (2) actual or would-be radical right voters who are not declaring their vote choice due to a "social desirability" bias [START_REF] Holbrook | Social desirability bias in voter turnout reports tests using the item count technique[END_REF].

Using a stacked data matrix, we thus create a new "PTV" variable, which measures propensities to vote for the six parties altogether. To account for the country-level differences, we use a mixed effect multilevel method in which individual respondents represent level 1 and their country level 2. Our multilevel models include random (or varying) intercepts and slopes, because we assume not only that average levels of support for radical right parties vary significantly across countries, but also that gender differently predicts radical right support in different countries.

Results

The broad picture: multilevel regression models predicting the radical right gender gap

Our first hypothesis concerns the presence, magnitude and direction of a gender gap in the support of radical right parties at the 2014 EP election. As anticipated, we expect to find only a mild gender gap referencing higher male support for the European radical right, and this gender gap to possibly disappear after controlling for a set of key 'sociological' variables -namely education and religiosity.

Table 1 shows patterns of RRGG (Radical Right Gender Gap) in the six countries altogether, across different multilevel models that include varying sets of control variables: model 0 (or "Basic", i.e. with only gender as an independent variable and no other controls); model 1 ("Societal", with age, education and religiosity as controls); model 3 ("Political", which includes the respondents' participation in the 2014 EP election, their degree of political interest and their left-right self-placement); model 4 ("Policy-related", which adds two more variables tapping respondents' orientations toward two key issues in radical right parties' platforms and discourses, namely immigration and same-sex marriage). 6

As the coefficient for gender in model 0 shows, there is a fully significant gender gap in favor of higher male support for the six European radical right parties altogether.

Since the independent variable is dichotomous (male vs. female), interpretation of the coefficient is straightforward: the average PTV score (our measure of support for radical right parties on a 0-10 scale) is, on average, 0.46 points lower among women.

While this gender gap is statistically fully significant, however, in substantive terms it also looks relatively mild: if translated in percentage terms, it amounts to a gap of less than 5 percentage points between men and women voters.

The second important result comes from model 1, which reveals that the radical right gender gap is virtually unaffected by potential compositional gender differences in terms of education and religiosity. Of course, both control variables do exert an impact on radical right support, as well as age does. It is among younger, loweducated voters who do not often attend religious services that the propensities to vote for radical right parties are relatively higher. But even taking this into account, women continue to appear more reluctant than men to support these parties. As a result, our starting hypothesis is only partially confirmed: the RRGP is substantively mild, but it fails to decrease when controls by education and religiosity are introduced.

*** Table 1 here It is only when the policy-related dimension of radical right votingagreement/disagreement with restrictive migration policies and with same-sex marriage (model 3) -are entered in addition to left-right position that the gender gap loses any residual significance. This implies that men are more prone to vote for the radical right partly because they are, on average, in 2014, more right wing-oriented than women, but also slightly more restrictive than women on migration policies, as well as clearly more conservative on gender-related issues such as same-sex marriages. Hence, while there is no evidence of a compositional basis of the RRGG in societal terms -i.e. it is not because they are, on average, more religious that women vote less for the radical right (and education levels do not differ, overall) -we do find evidence of a compositional effect in terms of ideology and policy orientations across However, the broad picture is fundamentally the same across countries: a mild radical right gender gap, which is not affected by social/compositional control factors but tends to disappear when introducing left-right ideology and attitudes toward social outgroup-related policies. Hence, the hypothesis that the degree of normalization and gender friendliness of these parties matters does not hold, as the gender gap is not higher in Flanders where such a normalization strategy is not much developed, while it persists in Denmark where the People's Party has definitely made an effort to gender the immigration issue.

Going further: interaction effects of gender in the support for the European radical right

If women still are, on average, slightly less prone to support European radical right parties than men, is this gap amplified or, conversely, suppressed within certain categories of women by socio-occupational and ideological lines? Our second set of hypotheses posits the existence of significant interactions between gender and other variables that are, as shown in section 1, crucial to explaining electoral support for radical right parties. Moreover, we expect not only the magnitude, but also the meaning of the radical right gender gap to vary across categories of voters, given that no gender gap, for instance, can imply men and women equally supporting or, on the contrary, equally rejecting radical right parties.

Following the framework presented in our "four-cell table", we estimate five different interaction models based on mixed-effect multilevel regression model 1 (societal), as

shown in equations 1-5 (see Appendix A). For each model, we will not present all parameter estimates for every interaction term and control variable included, but just the specific marginal effects of gender resulting from the interaction model, i.e. the statistically most meaningful results [START_REF] Brambor | Understanding interaction models: Improving empirical analyses[END_REF].

Figure 2 presents marginal effects of low vs. high education levels on the RRGG. We find no significant gender gap at low education levels but a clear gender gap (0.8, i.e.

significantly higher than the average value of 0.45 drawn from Table 1) at high education levels. These two gaps conceal however different average levels of RR support. Among the less educated respondents, PTV scores are significantly higher than average (3.4 for men, 3.0 for women, 2.3 the overall average); among the more educated, while scores reflect the average value among men (2.2), they reach their lowest level among women (1.4). In other words, what we find here is a pattern of "modern rejection", with educated women presenting -as in the more general case of the "modern gender gap" -more progressive, less conservative political attitudes.

***Figures 2 and3 here*** While levels of religiosity do not appear to significantly affect gender patterns of radical right support (in both cases, the PTV scores decrease at highest levels of religiosity -figure not reported), social economic insecurity and exclusion from labour force does seem to alter the gender balance in this respect. In the 2014 EES questionnaires, our best proxies are such occupational conditions as "manual worker"

(whether skilled or unskilled) and "non active" out of work respondents (i.e. homemaker, unemployed or temporarily not working). We thus isolate these two employment statuses from the rest of the sample ("Else").

Figure 3 shows that the gender gap is neutralized, and tends even to be reversed, among non-actives (value above the zero line indicates women supporting the RR more than men, although the gap is not statistically significant here), whereas it is reinforced among manual workers (1.0). In both cases, however, women's support for the radical right is higher than their average score (2.1) (the score among non-active women is 2.7, among manual works 2.8). But it is among non-actives that women's position appears most noteworthy, firstly because they even tend to score higher then men, secondly because they account for two thirds of this occupational category.

Among manual workers, on the contrary, men are not only more numerous (68%), but also much more strongly supporting the radical right (3.8). As mentioned in section 1, this could be associated with norms of masculinity permeating blue collars milieus and reflecting an "angry white men" basis for RR support.

Overall, non-active workers provide an excellent example of "equal support" across gender, i.e. RR support with no gender gap. Manual workers, on the contrary, fit the cell of "traditional support", or traditional gender gap in support for radical right parties.

***Figure 4 here*** If the societal dimension matters in this respect, intra-gender variations of the RRGG also depend on ideological and policy-related factors. Table 1 has already shown that the latter are strongly associated with voting for radical right parties, but also that they partly mediate the radical right gender gap, which disappears when gender compositional differences in terms of left-right ideology, attitudes toward migration and homosexuality are all simultaneously accounted for. We now test possible interactions of gender and ideology first, gender and policy-related attitudes secondly, on patterns of support for the radical right. This follows our theoretical expectationoutlined in the "four-cell" table -whereby political attitudes should differently affect different categories of women and men voters in their levels of RR support, consistent with the idea that both a "modern" and a "traditional" gender gap contribute to explaining gender-based attitudes toward radical right parties.

*** Figures 5a and5b here *** First, is ideology differently moderating RR support across gender? A clearly significant interaction coefficient (p=0.008, table not reported) suggests that this is the case. Figure 4 presents the estimated marginal effects of left-right self-placement on RR support conditional on gender, as drawn from interaction model 3 (see Appendix A). As it turns out, levels of support for RR parties differ across gender at the right but not at the left of the ideological scale. The average PTV score for RR parties is, for instance, 0.69 points lower among women than men voters positioning themselves at the extreme right of the scale (position 10), but it does not significantly differ across gender within positions 0 to 4 (center-left). Therefore, the effect of gender on RR support is moderated by ideology, in that women placing themselves at the radical right appear to be less assertive than men in their propensities to vote for radical right parties. In our theoretical framework (four-cell table), this stands for a "traditional" pattern of RR support, with right-wing women less disposed to support radical right parties and more favorable to established conservative parties.

In addition, histograms in Figure 4 We finally test interactions of gender with the two policy-related attitudes that proved most strongly associated with RR support, namely migration and same-sex marriage.

Both figures 5a and 5b show very similar patterns of gender-policies interactions, which tend to replicate what we have already observed about left-right ideology. The RRGG is highest among voters who are more hostile to policies favoring immigrants and homosexuals, whereas it disappears among those with the most tolerant stances (positions 7 to 10 for migration, positions 0 to 1 for same-sex marriage) on these issues. Once again, not only are men slightly more radical in their positioning with respect to these issues (see histograms for positions 0 to 3 in figure 5a, for position 10 -especially when compared with position 0 -in figure 5b), they are clearly more supportive of radical right parties when they take these "tougher" stances. Conversely, women who most clearly affirm their tolerance vis-à-vis social outgroups do not differ from men in their firm rejection of the radical right (PTV scores close to 0 for the migration issue, around 1.6 for the moral issue). Hence, we obtain once again a twofold picture in relation to the radical right gender gap: on the one side, disappearance of the gender gap in the presence of left-wing and more tolerant political positioning ("equal rejection"); on the other side, permanence of the radical right gender gap as a result of "traditional support" based on a mix of higher male ideological 'toughness' and lower women's political assertiveness.

Conclusion

A broad picture results from this analysis of voters' support for six European radical right parties. It suggests that, consistent with previous findings in recent decades [START_REF] Betz | Radical right wing populism in Western Europe[END_REF][START_REF] Givens | The Radical Right Gender Gap[END_REF][START_REF] Immerzeel | Explaining the gender gap in radical right voting: a cross national investigation in 12 Western European countries[END_REF], a Radical Right Gender Gap (RRGG) exists, with women significantly less likely than men to support RR parties.

Yet, contrary to conventional wisdom, this gap appears relatively mild (if translated in electoral terms, roughly 4-5 percentage points at the most). And given that women account for some 53 percent of the electorate in general, one should keep in mind that in spite of the RRGG, there are as many if not more women than men in RR electorates.

Above and beyond the broad picture, however, this paper has first investigated the compositional bases of the RRGG and found that this gap does not depend on education-and religion-related gender differences between men and women voters;

secondly, it has identified the fundamental mediators of the RRGP, i.e. ideological and political attitudes that logically precede electoral support. In particular, left-right self-placement and attitudes toward immigrants and homosexuality have emerged as the main factors that mediate between gender and vote, with the gender gap losing all significance when they are simultaneously held constant.

Thirdly, and perhaps more importantly, our paper has demonstrated that at least two patterns lie behind the general idea of "radical right gender gap". Indeed, when the RRGG is there, it can have two opposite meanings across different intra-gender categories of voters. On the one hand, it implies relatively high levels of support for radical right parties, but even higher support among men. On the other hand, it involves clear rejection of the radical right, but even clearer rejection among women.

In the first case ("Traditional support", according to the theoretical framework outlined in the four-cell table), the explanation is compositional in ideological terms (i.e. women are less often on the right of the ideological spectrum and tend to be more tolerant than men vis-à-vis social outgroups), but rests also on a different -i.e. lower -women's propensity to turn right-wing attitudes into explicit support for the radical right. In other words, women tend to "agree" less -although, admittedly, only slightly less -than men with the ideological and political stances of radical right parties, but even when they do agree, they are less willing to support these parties. Hence, and consistent with what other researchers have found in relation to the 2009 EP election and across a more heterogeneous group of populist radical right parties (Harteveld et al., 2015), one may also conclude that political attitudes -namely, political attitudes that are less assertive and less "tough" among women -both mediate and moderate the relationship between gender and RR voting.

Conversely, in the second pattern ("Modern rejection") women appear to be at least as categorical as men in refusing the radical right. The explanation for this apparent paradox -co-presence of "modern" and "traditional" patterns of gender gap -is in truth very simple, and rests on the idea of intra-gender heterogeneity, with women appearing particularly split along societal and ideological lines [START_REF] Dimaggio | Have Americans' social attitudes become more polarized?[END_REF][START_REF] Campbell | Gender and the Vote in Britain: beyond the Gender Gap? Colchester[END_REF][START_REF] Gidengil | Beyond the gender Gap[END_REF][START_REF] Barisione | Debunking the myth of a "Traditional" gender gap in the electoral support for Silvio Berlusconi in Italy (1994-2013[END_REF]. If we consider the similar cases of "modern rejection" and "equal rejection" together, what we are confronted with is the ideal type of a younger highly-educated left-wing woman who clearly does not lack assertiveness in contrasting the radical right, and who is very far from the profile of the woman on which the traditional RRGG rests -i.e. older, less-educated, more religious and politically conservative. In addition, a third type of woman, characterized essentially by socioeconomic insecurity and exclusion from the labour force, tends to assimilate men's inclination to vote for a radical right party ("Equal support").

Although we have not found cross-country clues for the hypothesis that the erosion of the RRGG is associated with a normalization strategy pursued by some RR parties, the comparative perspective was admittedly marginal in our research design, whereby ensuring inter-party homogeneity was more important than enlarging the number of cases. The longitudinal dimension was also missing in this paper, and should be incorporated in an enlarged research design including all of the waves of the European Election Studies at least since 1989.

As for the dependent variable, we maintain that the PTV (propensities to vote) score is particularly suitable to study the structure of attitudes and patterns of behaviors toward radical right parties in Europe, given both the often too small number of observations referencing actual votes for these parties and the overall reliability, as shown in section 2, of these scores as measures of electoral support. A further step would be to test our findings on actual voting for the RR, drawing from national electoral surveys.

Figure 1

The effect of gender on the propensities to vote (PTVs) for RR parties by country Note: each point indicates the relative gender gap (men's score minus women's score) on the 0-10 PTV scale, and corresponds to the estimated coefficient (slope) drawn from four mixedeffects (random intercept and slopes) multilevel models (basic, societal, political and policyrelated).

Figures 2 and 3

Marginal effects of Education and Employment on RR support conditional on gender (women voters) (with 95% confidence intervals) (see interaction models 1 and 2) national sovereignty to public spending, from crime to environmental issues. For the sake of parsimony, we decided to focus only on the two issues that emerged from the multivariate analysis as the most strongly associated with voting for radical parties. Flemish Interest

  further in testing the socio-political bases of the radical right gender gap. To do so, it investigates whether the RRGG depends on different gender compositions in terms of actual turnout at the 2014 EP election, of political interest and of distribution on the left-right ideological axis. While participation in the European election makes a barely significant difference on the support of the radical right (which slightly tends to decrease among actual voters in 2014), a more important role is played by political interest (the higher the interest in politics, the lower the chances to vote for a radical right party) and, much expectedly, by left-right selfplacement: as the latter gets closer to the far-right pole by one unit, the average PTV score increases by more than 0.4 points, all else being held constant. But even in this case -including these political control variables -the radical right gender gap remains fully significant, even though substantively reduced (from 0.46 to 0.33 average points).

  Figure1shows the same results (magnitude of RRGG by each model) across countries. Patterns of gender gap do change, of course, across countries. In the basic model, the gap is of about 0.6 points in France and Denmark, but it dos not reach 0.4 points in Sweden and Belgium. Moreover, decrease of the gender gap in the support of the National Front is linear as we proceed from model 0 to model 3, whereas the pattern appears to be "flatter" for other countries.

  provide the actual distribution of left-right selfplacement across gender. This reminds us another fundamental element: in 2014, European women lean, on average, more to the left than to the right or, alternativelyand consistently with persisting lower assertiveness in politics -prefer to position themselves at the center of the ideological spectrum, not choosing between left and right. Hence, to correctly understand the relationship between gender, ideology and voting, one should keep these three elements in mind: women voters are (1) less often on the right than men (modern gender gap); (2) they are more often on the center (lower political assertiveness); (3) when positioned on the right, they are less prone to express support for the RR parties (traditional radical right gender gap). While the first two elements are compositional in nature, the third points to a moderating effect of ideology in predicting gender-based patterns of RR support, with men's greater propensity to vote for these parties having a clearer radical right connotation.

Figure 4

 4 Figure 4Marginal effects of left-right self-placement on RR support conditional on gender (women voters) (with 95% confidence intervals) (see interaction model 3)

  Therefore our choice went to the European Election Studies, and for the most recent election, 2014, all the more interesting to study as the Radical Rights did very good scores (over one quarter for the Danish People's Party, the French FN, the British UKIP). We chose to limit ourselves to six well-established "radical right populist parties" 5 in Europe that are clearly positioned on the right side of a 0-10 left-right scale by public opinions in their countries. We excluded other potentially relevant parties that are less congruent with a radical right ideology, such as the Northern League and True Finns. We excluded Eastern Europe countries as well, because of the specificity of the post-communist context. The selection of cases presents contrasted features, in that the scores of these parties in the EP election range between 26.6 For this reason, our methodological strategy consists of constructing a single measure of support for radical right parties across the six countries, i.e. a unique dependent variable resulting from the aggregation of voting probabilities for each national party. , to properly analyze the RR gender gap and its relationship with other social and political factors it is safer to turn to a continuous measure of party support such as the propensity to vote (PTV) score. This variable indicates the self-assessed

	Thereforedegree of probability (on a 0-10 scale) that each respondent "will ever vote" for a RR
	party, where '0' means "not at all probable" and '10' means "very probable". Using
	this variable, the number of respondents for each RR party increases from a minimum
	of 580 (Vlaams Belang) to a maximum of 1114 cases (Sweden Democrats) (see
	Appendix table 3).
	Immerzeel, Coffé, and van der Lippe, 2013; Norris, 2005; Rippeyoung, 2007), it is
	usually on the basis of large cross-national surveys (European Values Study,
	European Social Survey, World Values Survey) that are not fit to such a purpose and
	result into a distorted image of actual votes for such parties. The reason for this is the Lastly, we opted for voting probabilities rather than for actual votes, since even in
	non-electoral nature of these surveys, which collect voting intentions or past votes in specifically comparative electoral surveys like the EES (European Election Studies),
	the next or last national election, but outside the context of a real election (Banducci the number of self-reported radical right voters is often too small to permit any

Some just acknowledge the gap based on cross tabulations comparing the level of electoral support of these parties by gender and by country, sometimes only at one point in time, not controlling for the possible effect of other variables. When they do attempt systematic controls (for instance

[START_REF] Banducci | Surveys in Context: How Timing in the Electoral Cycle Influences Response Propensity and Satisficing[END_REF]

. Moreover, the country subsamples are usually small, resulting into very small numbers of self-reported far-right voters, a tendency that is further aggravated by moral reprobation being still attached to support for many of these parties.

percent for the Danish People's Party to 4.3 for the Flemish Interest (see Appendix table

1

). Some of these parties are also perceived as more extremist than others. If one compares their average score on an 11-point left-right scale (0-10), the French FN comes ahead with a peak score of 8.8, while the Danish People's Party gets only 7.4

(see Appendix table 2).

However, in spite of their relative differences, these six parties undoubtedly belong to the same line of the "Radical Right" in contemporary Europe (2015, Kallis) and are suitable, as such, for an aggregate study of the radical right gender gap. Hence, this study aims to test and explain the European radical right gender gap altogether -i.e. with a 'pooled' approach -rather than to seize 'in-country' specificities. statistical analysis. In this 2014 EES survey, for instance, in spite of the high score of the National Front at the EP election (25%), the number of self-reported NF voters amounts to 58 (16%, see Appendix table 1).

Not only does this variable present the clear advantage of a much larger number of cases, it also proves to be a very good proxy for real vote choices. We draw this conclusion from two statistical tests. First, appendix table 3 presents the average PTV score by vote choice at the 2014 EP election (RR party voters vs. other voters). In all cases, propensities to vote for RR parties always exceed the average score '8' on the 0-10 scale among RR party voters, with a clear homogeneity across countries.

Table 1 .

 1 Parameter estimates (coefficients with standard errors in parentheses) of four different mixed-effects multilevel regression models predicting the propensities to vote for radical Right parties

		Model 0	Model 1	Model 2	Model 3
		(Basic)	(Societal)	(Political)	(Policy)
	Gender (Man/woman)	-0.455***	-0.453***	-0.333***	-0.146
		(0.12)	(0.10)	(0.09)	(0.11)
	Age		-0.016***	-0.017***	-0.025***
			(0.00)	(0.00)	(0.00)
	Education (1=lower; 3=higher)		-0.741***	-0.598***	-0.425***
			(0.06)	(0.06)	(0.06)
	Religion (Church attendance		-0.062**	-0.103***	-0.120***
	0=never; 7=weekly)
			(0.02)	(0.02)	(0.02)
	Voted at EP 2014 (No/yes)			-0.215*	-0.121
				(0.10)	(0.09)
	Political Interest (1=low; 4=high)			-0.231***	-0.116**
				(0.05)	(0.04)
	Left-Right ideology (0-10)			0.428***	0.307***
				(0.02)	(0.02)
	Immigration (against restrictive				-0.318***
	policy) (0-10)
					(0.01)
	Oppose same-sex marriage (0-				0.106***
	10)
					(0.01)
	Intercept	2.939***	5.704***	3.949***	5.103***
		(0.42)	(0.48)	(0.46)	(0.47)
	Random-effect parameters:
	Gender (std. dev.)	0.188	0.145	0.080	0.178
	Gender (std. err.)	0.116	0.124	0.156	0.103
	Intercept (std. dev.)	0.983	0.979	0.910	0.949
	Intercept (std. err.)	0.309	0.318	0.302	0.301
	N	5853	5853	5853	5853
	p	0.00	0.00	0.00	0.00
	Log Likelihood	-15221.393	-15138.88	-14793.73	-14485.809
	aic	30452.787	30293.761	29609.460	28997.618
	Significance levels: *p<0.05; **p<0.01; ***p<0.001

For a close yet slightly different definition of the « moderating » and « mediating » effects explaining the RRGG also see[START_REF] Harteveld | The gender gap in populist radical-right voting: examining the demand side in Western and Eastern Europe[END_REF].

We refer in this paper to the definition of the populist radical right parties given by Cas Mudde, based on nativism, authoritarianism and populism(Mudde, 2007, p.19).

In the EES 2014 dataset a set of other relevant policy-related variables are available -from

APPENDIX Appendix A

Interaction model 1

All models, which have the PTV (propensities to vote) score for RR party as the dependent variable, are mixed-effects multilevel models with varying intercepts and slopes (for gender) for group indicator j (country).

Appendix Table 1

Vote choice for radical right parties in the 2014 EES sample and at 2014 EP election. Appendix Figure 1 Percentage distribution of 0-10 PTV scores by each RR party and gender

1 For a good introduction to the debate see the special issue of Patterns of prejudice, on Gender and populist radical rights politics, especially the opening paper (Spierings, Zaslove, Mügge & de Lange, 2015 and in the same issue Mudde, Kaltwasser).

2 For an interesting approach of the contrasted reaction of women to the use of torture in the post 9/11 context, because of the opposed impact of feminism and motherhood, see Wemlinger, 2013.

3 See for instance the comparative analysis conducted at OFCE [START_REF] Eydoux | Special Issue of Revue de l'OFCE « European labour Markets in Times of Crisis. A Gender Perspective[END_REF] showing how the recession first had an impact on men and then on women, from "hecession to she-austerity".