Florian Brunier-Coulin
email: florian.brunier-coulin@univ-amu.fr

Nicolas Vandenberghe

Gautier Verhille

Patrice Le Gal

Fluid -Structure Instabilities in the Axial Balancing System of a Turbo-Pump

This article is devoted to the experimental and theoretical modeling of a fluid structure instability that affects the axial balancing system cavities of spatial turbo-pumps. After having explained the design of the laboratory experiment that mimics the real industrial set-up, we describe the observation of the instability that emerges due to the coupling of acoustic modes of a fluid cavity with the deformations of a metallic disc that closes the cavity when a flow passes through the system.

The system bifurcates towards a vibrational state via a Hopf bifurcation that can be super critical or sub critical, depending on the flow rate and inlet aperture. The experimental arrangement is fully described and tested in order to build a theoretical model that predicts very well the observed threshold of the instability.

Therefore, we expect our model to be predictive for real industrial cases.

Introduction

In the aerospace industry, turbo-pumps as illustrated in figure 1 (a) are equipped with very high speed rotating turbines whose rotating frequency can be as high as 100,000 rpm. Their role is to fuel the combustion chamber of the rocket engine with a high flow rate of propellant at optimized pressure. Because of the extreme hydrodynamic regimes encountered in these machines, the axial position of the rotors cannot be solely realized by mechanical components such as ball bearings. Therefore, to operate in this regime of intense mechanical loading, the force balance is ensured by the derivation of a part of the propellant flow into a cavity behind the rotor called an hydrostatic bearing or an Axial Balancing System (ABS) [START_REF] Reddecliff | Hydrostatic bearings for cryogenic rocket engine turbopumps[END_REF][START_REF] Shimura | Stability of an axial thrust self-balancing system[END_REF]. This rotor-stator cavity is delimited radially by an inner and an outer valve positioned respectively at the periphery of the rotor and at its center. The equilibrium of dynamic pressure is responsible for the axial positioning of the rotor. However, in certain specific conditions, the response of the compressible fluid to the rotor displacements and/or deformations can lead to vibrations and instabilities. Verhille and Le Gal [START_REF] Verhille | Couplage d'oscillateurs : l'instabilité d'un système d'équilibrage axial d'une turbo-pompe[END_REF][START_REF] Verhille | Oscillators coupling: The instability of an axial equilibrium system of a turbo-pump[END_REF] considered an ABS with a rigid and axially mobile rotor and showed the appearance of a low frequency instability where the rotor oscillates axially.

In the present study, we consider a second kind of instability involving the deformability of the rotor. So far, two kinds of instability have been observed. The first one is induced by the rotation of the disc and it involves the rotor flexibility and its coupling with the flow generated by the rotation. It has been studied in the context of high speed magnetic hard drives [START_REF] Angelo | Aerodynamically excited vibration and flutter of a thin disk rotating at supercritical speed[END_REF][START_REF] Kim | Prediction Of Aeroelastic Flutter In A Hard Disk Drive[END_REF]. It results in the growth of non axisymmetric deformation modes. The second instability, which is studied here, is due to the coupling between the deformations of the disc and the modulation of the pressure drop across the valve due to these deformations. This instability does not require any rotation of the disc or of the fluid and both axisymmetric and nonaxisymmetric modes can be excited. In the present study we focus on the axisymmetric modes that are relevant in the context of spatial turbo-pumps. These complex interactions between hydrodynamic and solid structures may be critical for turbo-pumps. Thus, a better understanding of the fluid/structure interactions in rotor/stator systems plays a key role to improve the reliability and the performance of spatial turbopumps.

The paper is organised as follows. Section 2 describes the design and the characteristics of the experimental setup that allows us to investigate this instability in well controlled conditions. Then, sections 3 and 4 are devoted respectively to the full characterisation of the setup and to the description of the observed instability. To analyse this fluid/structure instability, we develop a model presented in section 5.

Experimental setup

Design principles

The setup under study in the present work is composed of a disk, clamped at its center, whose upper side near its outer edge is very close to an annular surface (Fig. 1). Above the disk an admission tube is used to inject air at high flow rate. After passing through the thin annular opening, the inner valve, the air flows into a cavity under the disk before going through an exit orifice. Under a given air flow, the system reaches an equilibrium state, characterised by an axisymmetric bending deformation of the disk, thus adjusting the pressure drop through the inner valve. Under some conditions the system presents an instability resulting in self sustained oscillations of the disk. Determining the conditions under which this instability occurs is the goal of the paper. Note that, in our study the disk does not rotate, as it will be shown later, the instability does not require its rotation. In practice, several criteria have to be satisfied to observe the desired instability. To maximise the amplitude of the axisymetric vibration mode of the disc, it is beneficial to tune the shape and the frequency of the acoustic mode in the cavity with the ones of the normal mode of the disk. The eigenfrequencies for the acoustic cavity ω f and for the axisymmetric bending mode of the disc ω d are

ω f = λ f c 0 R (1)
ω d = λ d 1 R 2 D ρ d h (2)
where D = Eh 3 /12(1 -ν 2 ) is the bending modulus of the disc of thickness h, ρ d is its density, E the Young modulus and ν the Poisson coefficient. λ f and λ d are two numbers that characterise the acoustic and the elastic modes. They depend on the boundary conditions in the setup. When these two frequencies are equal the thickness of the disc is given by

h = R λ f λ d c 0 c d 12(1 -ν 2 ), (3)
where c 0 is the speed of sound in the working fluid and

c d = (E/ρ d ) 1/2 .
The geometry of the inner valve is also a key element in the dynamics of the system. The pressure drop across the valve can be written ∆P = ξ(a, Re)ρV 2 0 where ρ is the fluid density, V 0 is the mean velocity proportional to the flow rate Q 0 and ξ(a, Re) the non-dimensional coefficient of the pressure drop that depends on the geometry of the valve and especially on the valve aperture a and on the flow conditions. As we shall see in section 5 the threshold of the instability strongly depends on the stiffness of the valve. Our setup is designed to allow for a change of the initial valve opening a 0 , i.e. the opening at rest, without air flow. From this threshold, one can define a critical flow rate above which the system is unstable. In our facility, this critical flow rate should be lower than 11 L/s (i.e. 40 m 3 /h) constraining the geometry of the valve. For the sake of simplicity of the experimental setup, we decide to work with air at ambient temperature. The disc is made of stainless steel with a density ρ d = 7700 kg.m -3 , a Young modulus E = 203 GPa and a Poisson coefficient ν = 0.3. Gravity is negligible. The radius of the disc is R = 90 mm and its thickness is h = 4 mm. The disk is held by a 300 mm in diameter hub that partially fills the acoustic cavity of height H = 5 mm and radius 95 mm. The walls of the cavity and the upper part of the inlet valve are made of the same steel as the disc to avoid differential thermal dilatation. Pictures of the experimental setup are shown in figure 2.

The valve aperture is controlled by a ring positioned between the bottom of the cavity and the upper part of the inlet valve. We can choose between four different rings with calibrated thicknesses of 9.00, 9.05, 9.10 and 9.15 mm and several intermediate rings made of paper with a thickness of 0.09 mm. The 9.00 mm thickness ring is used to control the initial position of the disc by imposing a 0 = 0 mm. The air flow enters in the cavity by the inner valve and exits to ambiant air at the center of the cavity through an annulus around the hub that is drilled by eight 10 mm diameter holes.

Instrumentation

To characterise the instability several sensors are installed on the setup to measure the different parameters. Upstream of the cavity, a flow meter (Krohn H250-RR) measures the flow rate crossing the cavity within the range 0.35 -35 m 3 /h. As can be seen on the left of Figure 2, an upstream cavity (50 cm long and 170 mm in diameter) is used as a tranquillising chamber upstream the ABS cavity. The clean dry air flow crossing the cavity is provided by a compressor with a maximal pressure of 7 × 10 5 Pa. The flow rate is controlled by several valves which allow both a fine tuning of the desired flow rate and a short transient time to reach the stationary state. A differential pressure sensor (NXP MPX5500) is used to measure the pressure drop at the inlet valve within the range of 0-500 kPa, with a sensibility of 9 mV/kPa and with a response time of 1 ms. Two microphones, one omnidirectional and one unidirectional, provided by PUI Audio, are mounted flush to the cavity floor. They measure the acoustic modes within the range of 0.2-2 kHz. The vibrations of the disc are measured with two laser position sensors (Keyence LK-H052) located at the same radius r = 7 cm but separated by an angle of 90 • or 135 • . The acoustic modes of the cavity are initially characterised with a loud speaker K12.25 provided by Visaton, mounted flush on the bottom of the cavity which correspond to the stator in a rotor-stator cavity (see Figure 2 c). All the data are digitalized simultaneously at 10 kHz with a NI 6220 acquisition card.

Setup characterization

Characterization of bending modes and acoustic modes

Preliminary to the stability study of the axial balancing system, we perform different test to characterize the system. With the disk held in position, we proceed to ping-tests by gently tapping the disk and recording its response with the two laser vibrometers. We describe here the measurements made in the presence of the enclosure, i.e. in the presence of the acoustic cavity underneath the disk and of the upper surface including the inner valve upper surface.

The response of the disk to the ping test is shown in figure 3. The signal is strongly damped. The periodogram exhibits peaks at 485 Hz, 606 Hz, 740 Hz, 1576 Hz. The theoretical values [START_REF] Leissa | Vibration of plates[END_REF] expected for an annular disk clamped at r i = 15 mm and free at R = 90 mm with the material properties given in section 2.1 are 528 Hz for the mode with one Nodal Diameter (1ND), 586 Hz for the axisymmetric mode (0ND), 742 Hz for the mode 2ND and 1523 Hz for the mode 3ND showing a good agreement with the measured values. As will be discussed in details later, a key ingredient in the estimation of the instability threshold will be the damping associated with the vibration of the disk. To estimate the damping coefficient of the axisymmetric mode, we compute the Fourier spectra of the displacement signal for short subsets and then measure the decay in time of the amplitude of the 0ND mode [START_REF] Cawley | A quick metthod for the measurement of structural damping[END_REF] at frequency f 0 = 606 Hz. The results of the analysis are shown in the inset of figure 3. The exponential decay of the amplitude of the 0ND mode at short times is characterized by a decay rate τ -1 d = 78 s -1 , leading a damping coefficient ζ d = τ d /(2πf 0 ) = 0.021. The ping tests have been performed with a clamping torque applied to the upper bolt holding the disk of 70 N.m. Tests have shown that if the clamping torque is lowered the damping is stronger. The value of 70 N.m was chosen to avoid possible wrapping of the disk. We attribute the unusually high value of the damping coefficient to the coupling between the disk vibration and the fluid motion in the thin gap between the disk and the upper surface (i.e. the inner valve). During disk vibration air has to be repetitively sucked in and expelled from the thin interval, and without the upper surface, the damping coefficient was measured lower (by a factor approximately equal to ten).

In addition, to characterize the acoustic modes of the cavity in the absence of flow, a white noise is generated by the loudspeaker and the variations of pressure in the cavity are measured by the microphone. Two main peaks have been identified in the pressure spectrum. The main one around 2330 Hz corresponds to the first azimutal mode. The second one in the range 575 -830 Hz corresponds to an axisymmetric mode of the cavity and is close to the first axisymmetric bending mode of the disk. In presence of an air flow, acoustic modes of the tranquillising chamber are also excited. To avoid a coupling between these modes and the modes of the disc, a piece of foam in the upper half part of the pipe has been added shifting the chamber modes around 350 Hz minimizing the coupling with the bending modes.

Characterization of the inner valve

A detailed characterization of the inner valve, i.e. the narrow annular section through which the fluid flows into the acoustic cavity, will prove to be crucial in our analysis. In the experimental setup we record the pressure drop ∆P for different flow rates Q 0 and initial apertures a 0 . The pressure drop is measured between two sensors located in the admittance chamber above the disk and in the acoustic cavity at r = 90 mm. As the flow rate Q 0 is increased, the pressure drop increases and the disc bends due to the pressure difference ∆P between its two sides. With the assumption of an elastic response, the transverse displacement field u z (r) is linear with the pressure difference ∆P . The solution for an annulus free at r = R and clamped at r = αR with α < 1 reads [START_REF] Timoshenko | Theoy of Plates and Shells[END_REF]

u z (r) = ∆P R 4 64D r 4 R 4 -8 r 2 R 2 log r R + K 1 r 2 R 2 + K 2 log r R + K 3 , (4)
where the three constants K 1,2,3 depend on the aspect ratio α and the Poisson ratio ν and are determined using the boundary conditions (Appendix A). We assume here that the pressure in the cavity is uniform. The measurement of the displacement u z (r) at r = 72 mm is in good agreement with this theory, as seen in Figure 4.a). The deviation from linear behaviour observed for ∆P > 4000 Pa corresponds to an unstable regime where vibrations are observed. We can use the stationary solution 4 to compute the valve aperture a = a 0 + u z (R).

The pressure drop across the valve can be written ∆P = ρ 0 V 2 0 f (α, Re), with V 0 = Q 0 /πR 2 where the function f depends on the aperture α = a/R and on the Reynolds number Re = aV 0 /µ, with µ the dynamic viscosity. There is no simple way to determine the function f but correlation formula are commonly used [START_REF] Idelc'hik | Handbook of hydraulic resistance[END_REF]. Here we use the formula ∆P where C 1 = 0.1519 and C 2 = 99.85 are obtained by fitting the experimental data. Eq. ( 5) offers a fair description of the experimental data over the range of α and Re explored in the experiment (Fig. 4).

ρV 2 0 = C 1 C 2 α 1/2 + Re α 2 Re , (5)

Description of the instability

Critical flow rate

Figure 5(a) presents a typical evolution of the disc vibration when the flow rate increases slowly up to values above the instability threshold. We observe a rapid increase of the amplitude of vibration when the threshold is crossed (near t = 1 ms). The critical flow rate is defined by the flow rate for which the amplitude of vibration is 4 times larger than the rms of the noise without flow. The evolution of the critical flow rate with the valve aperture has been investigated experimentally. It is represented in Figure 5.b) and compared with the theoretical predictions determined in section 5. This curve will be discussed in more details during the derivation of the analytical model.

As shown in figure 7.c), the frequency of the excited mode is around 610 Hz close to the threshold. This frequency and the analysis of the phase shift between the two vibration sensors evidence that the excited mode is indeed the axisymmetric bending mode.

Bifurcation

A vib = (Q -Q c ) 1/2 for Q Q c .
The oscillations are symmetrical around the equilibrium position of the disc for intermediate flow rates (here between 0.55 and 0.8 L/s). For higher flow rates, vibrations become asymmetrical when the disc comes into contact with the upper part of the cavity. This occurs when the amplitude of vibrations are greater than the mean valve aperture. For larger apertures, the transition becomes subcritical and a small hysteresis is observed, see Fig. 6 (b). The hysteresis is repeatable and each point represents a stationary regime.

For large apertures, for which higher flow rate can be reached before crossing the critical one, we also observe oscillations of small amplitude as shown in Fig. 6 (b). We argue that such small amplitude vibrations are induced by the turbulent flow and not the instability as they do not present the characteristic growth as it is observed in the preceding cases. They do not present the characteristic growth associated with an amplitude. A critical flow rate can still be observed. Above the threshold, the amplitude of vibration is almost six times greater than before the bifurcation and does not seem to depend on the flow rate. At this stage, the disc comes into contact with the upper part of the cavity.

The signature of a supercritical bifurcation is also observed on the growth rate of the most unstable mode. To estimate this timescale, we fit the amplitude of vibration by an exponential law A vib (t) = A 0 exp(-t/τ ), as seen in Fig. 7 (a) when the flow rate varies rapidly from Q = 0 to Q = Q f . The evolution of the growth rate 1/τ is presented in Figure 7 (b). The growth rate increases first linearly with Q f -Q c before reaching a steady value, here τ 26 ms for a 0 = 40 µm. The saturation of the growth rate may either be due to the establishment of the flow inside the cavity or to the complete closing of the valve when the disc collides the upper part of the valve. This evolution of the growth rate is accompanied by a slight modification of the frequency of vibration, as shown on Figure 7.(c). At threshold, the pulsation is close to 610 Hz and reaches frequency as high as ∼ 655 Hz for the largest flow rate.

Theoretical interpretation

Governing equations

To gain insight into the dynamics of the Axial Balancing System we propose here a simplified model coupling a deformable disc, an acoustic cavity and an inner valve whose aperture depends on the disc motion. We build a model taking into account these different ingredients to emphasize the key parameters in this system. The general approach used in the present study of a turbo-pump is similar to the description of wind musical instruments with vibrating reed valves such as clarinets [START_REF] Wilson | Operating modes of the clarinet[END_REF][START_REF] Fletcher | The Physics of Musical Instruments[END_REF]. It consists in the coupling between an acoustic cavity and an elastic deformable element whose motion influences the flow in the cavity. If the general principles are similar in the two systems (wind instrument and turbo-pump) the geometry and the flow conditions are different. We consider a simplified model as illustrated in figure 1 (b). We approximate the Axial Balancing System as a cylindrical cavity enclosing a disc of radius R clamped in r = 0. The thickness of the disc h is assumed to be small enough (h R) so that the elastic response of the plate can be described by the Kirchoff-Love elastic plate theory. Nonlinear terms will not be considered as the deformations remain small. The inner valve is characterized by its aperture, a, between the upper wall of the cavity and the disc. Cylindrical coordinates will be used and the study is restricted to axisymmetric modes.

The description of the dynamics is based on the coupling between the pressure in the lower cavity and the transverse vibrations of the disc. In the lower cavity, the fluid dynamics is described by the classical acoustic wave equation 1 c 0 2

∂ 2 v ∂t 2 -∆ v = 0, ( 6
)
where v is the velocity field and c 0 is the speed of sound. The acoustic modes are coupled with the deformation of the disc through the boundary condition as discussed later.

The vibrations of the disc are described by the bending wave equation with a forcing coming from the differential pressure acting on each face of the disc [START_REF] Landau | Théorie de l'élasticité[END_REF]

m s ∂ 2 u z ∂t 2 + D∇ 4 u z = P in -P c (7)
where u z is the local displacement in the axial direction, P in is the constant and uniform pressure above the disc, P c is the pressure field in the cavity at the boundary of the disk, which depends on the fluid velocity v, see figure 1.b). m s = ρ d h and D = Eh 3 /12(1 -ν 2 ) are respectively the surface density and the bending modulus of the disc where ρ d is the density, E the Young modulus and ν the Poisson coefficient. We note that we do not consider the radial deformation of the disc nor the centrifugal force that may be relevant for a disc rotating at high speeds. They are not relevant in our experimental setup. Moreover, the gravitational force is neglected in this study as it is much smaller than the pressure force (P in -P c )S where S is the surface of the disc. The inner valve controls the pressure drop ∆P = P in -P c (R) according to Eq. ( 5). The pressure drop across the valve thus relates the flow rate and the pressure in the cavity and it depends on the deformation of the disc through the variable a = a 0 + u z (R).

Single mode oscillations of the disc

The general form of the equation for the disc ( 7) must be solved with the space dependent pressure P c . To obtain a simplified system we use an energetic formulation. The kinetic energy of the disc writes

K = 1 2 R 0 m s ∂u z ∂t 2 2πrdr (8)
The potential elastic energy for the bending disc writes

V = 1 2 R 0 D ∂ 2 u z ∂r 2 + 1 r ∂u z ∂r 2 -2(1 -ν) ∂ 2 u z ∂r 2
1 r ∂u z ∂r 2πrdr

(9)

The work of pressure forces writes

W = R 0 (P in -P c )u z 2πrdr (10)
It is customary to obtain an approximate formulation amenable to numerical resolution by introducing an ansatz for the displacement field of the form u z (r, t) = a k (t)φ k (r) where the φ k are a given set of linearly independent approximation functions (see e.g. [START_REF] Leissa | The historical bases of the Rayleigh and Ritz methods[END_REF]). In the present work we use the simplest form of this approach and we focus on a simple form

u z (r, t) = w(t) r R 2 (11)
This form provides a simple approximation for the first mode of the disc (though it does not verify the boundary conditions -free moment and free shear force -at r = R). Using this form and performing the derivative yields an equation

m ẅ + kw = P in -P ef f (12) with m = 2m s /3, k = 16D(1 + ν)/R 4
and an effective pressure

P ef f = 2 πR 2 R 0 P c (r) r R 2 2πrdr ( 13
)
We note here that this approach does not take into account dissipation that will be heuristically incorporated later. We note also that this simplified approach predicts a free vibration eigenfrequency ω app = (D/µR 4 ) 1/2 [24(1 + ν)] 1/2 ≈ 5.58(D/µR 4 ) 1/2 while the exact analytic solution, yields ω ≈ 3.752(D/µR 4 ) 1/2 [START_REF] Leissa | Vibration of plates[END_REF].

To explore the possibility of an instability developing in the Axial Balancing System we consider the sum between a stationary state, called "reference state" hereafter, and a random perturbation with a zero mean materialized by a tild over the variable (see e.g. [START_REF] Young | Dynamic stability of rotor-bearing systems subjected to random axial forces[END_REF]).

Reference state

To explore the possibility of an instability developing in the Axial Balancing System, we consider a stationary state we first consider the stationary state associated with a constant flow rate Q 0 = πR 2 V 0 . The flow in the cavity is then obtained from the continuity equation

v r (r) = V R R r = Q 0 2πHr = V 0 R 2 2Hr ( 14
)
with V R = Q 0 /(2πRH). We have assumed that the small deflection of the disc is negligible compared to H. The pressure field in the lower cavity is given by Bernoulli's relation

P 0 (r) -P 0 (R) = 1 2 ρ 0 V 2 R 1 -(R/r) 2 (15)
The pressure field can be used to compute the effective pressure from eq. ( 13)

P ef f = P 0 (R) - 1 2 ρ 0 V 2 R = P 0 (R) - 1 2 ρ 0 V 2 0 R 2H 2 (16)
To close the system we use the characteristic equation of the inner valve (eq. 5)

∆P 0 = P in -P 0 (R) = ρ 0 V 2 0 C 1 C 2 α 1/2 0 + Re 0 α 2 0 Re 0 ( 17
)
where α 0 = (a 0 + w 0 )/R and Re 0 = ρ 0 V 0 (a 0 + w 0 )/µ yielding an equation for the displacement in the reference state w 0

kw 0 = P in -P 0 (R) + 1 2 ρ 0 V 2 R = ρ 0 V 2 0 C 1 C 2 α 1/2 0 + Re 0 α 2 0 Re 0 + 1 2 R 2H 2 (18)
We note that the second term in the brace is negligible compared to the first one.

The value of w 0 can be computed from this equation for a given set of parameters. The result is shown in figure 8, together with experimental measurements. Despite the simplifying hypotheses, the agreement between theory and the measurements is good. We note that for small values of V 0 , an asymptotic solution at leading order in V 0 writes w 0 ∼ C 1 C 2 µV 0 /(ka 0 )(R/a 0 ) 3/2

Stability analysis

The experiment is characterized by the appearance of self sustained oscillations as the flow rate reaches a critical value. To study the threshold we perform here a perturbation analysis of the reference state. A perturbation of the system is added on the stationary state calculated earlier:

w(t) = w 0 + w(t) (19)
Q(t) = πR 2 V 0 + Q(t) (20)
P c (r, t) = P 0 (r) + Pc (r, t) 18) and the dots show experimental measurements (note that the measured displacements are multiplied by (9/7) 2 because measurements were performed at r = 72 mm and not at the outer edge R = 90 mm. See also eq. 11). For the computations we use the geometry and the material properties for the disk given in the text (section 2) and for the fluid ρ 0 = 1.29 kg m -3 , µ = 18.1 × 10 -6 Pa s.

The dynamics of the disk is given by the relation deduced from eq. ( 12)

ẅ + 2ζ d ω d ẇ + ω 2 d w = - 1 m Peff ( 22
)
where ω d = (k/m) 1/2 . The damping term 2ζ d ω d ẇ is not present in eq. ( 12). We introduce it to account for the damping of the vibrations of the disk. The damping coefficient ζ d can be deduced directly from the estimation of the damping of the vibrations of the disk presented in section 3.2 and in Fig. 3.

To allow for a better understanding of the stability of the system, we adopt a simple model for the acoustic cavity. We consider that the cavity behaves like an acoustic resonator near the first eigenfrequency. The derivation of the model is detailed in Appendix B and in this section we use the following form to describe the cavity near its resonance

PR + 2ζ c ω c ṖR + ω 2 c PR = Z 0 1 τ c Q (23)
where PR (t) = Pc (R, t) and Z 0 = ρ 0 c 0 /(πR 2 ) is the acoustic impedance associated with the cavity. The resonator is characterized by its eigenfrequency ω c whose values is typically a constant of order 1 times the characteristic frequency c/R, by a damping coefficient ζ c that depends on the exit condition, and by a time constant τ c that characterizes the stiffness of the cavity. The different parameters depend on the geometry of the cavity and can be determined by using the method of Appendix B. In addition to this relation we assume that in the regime under study, the effective pressure is related to the pressure at r = R by a linear relationship Peff = γ PR . The value of γ is also an outcome of the analysis of Appendix B. Finally the behaviour of the inner valve is linearised. Starting from eq. ( 5) a linear relationship between the pressure perturbation at the exit of the inner valve PR = Pc (R, t), the perturbation flow rate Q and the perturbation of the position w reads Q

= -A PR + B w (24)
where

A = πR 2 ρ 0 V 0 Re 0 α 2 0 C 1 2Re 0 + C 2 α 1/2 0 and B = πRV 0 4Re 0 + 5C 2 α 2 0 2α 0 2Re 0 + C 2 α 1/2 0 (25)
A and B vary with the imposed flow rate Q 0 = πR 2 V 0 . In the limit of small flow rate (Q 0 → 0),

A ≈ πa 5/2 0 R 1/2 /(C 1 C 2 µ), B ≈ 5πR 2 V 0 /(2a 0 ) (26)
Using Eq. (24), the system formed by Eqs. ( 22) and ( 23) has two unknown variables, w(t) and PR (t). To study the stability of the reference state, we look for solutions of the form w(t), PR (t) = ŵ, PR exp(st). The growth rate s is then given by the characteristic equation

s 4 + A 2 s 3 + A 3 s 2 + A 4 s + A 5 = 0 (27)
where

A 2 = 2ζ c ω c + 2ζ d ω d + AZ 0 τ c A 3 = ω 2 c + ω 2 d + 2ζ d ω d 2ζ c ω c + AZ 0 τ c A 4 = 2ζ d ω d ω 2 c + ω 2 d 2ζ c ω c + AZ 0 τ c + B Z 0 γ mτ c A 5 = ω 2 c ω 2 d
All the coefficients A i are positive. When Q 0 = 0, B is zero and the two damped oscillators described by Eqns ( 22) and ( 23) are not coupled. The polynomial has four complex roots, conjugate by pairs with negative real parts. We assume here that the damping coefficients are not too high, a condition fulfilled in the experiment (otherwise real negative eigenvalues may be obtained).

As Q 0 increases, the eigenvalues move (continuously). The solution can become unstable when two complex eigenvalues cross the imaginary axis with a non-zero imaginary part (a single real eigenvalue crossing in zero is not possible because A 5 > 0). The evolution of the real and imaginary parts of the eigenvalues is shown in figure 9 for a set of parameters deduced from the experiment. It is then straightforward to show (Appendix C) that the stability is maintained when

S = A 2 A 3 A 4 -A 2 4 -A 5 A 2 2 < 0 (28)
The different terms that appear in the stability equation evolve when Q 0 changes. Eq. ( 28) presents a complicated variation with the different physical parameters and it is not easy to extract a general relation for the critical flow rate for a given set of parameters. However, using the physical parameters of the experiment, it is possible to determine the value of the critical flow rate for each value of the initial valve aperture a 0 . In figure 10 we present a stability boundary in the plane (a 0 , Q 0 ). The system is unstable above the stability threshold. The theoretical value is compared with the measurements and it shows a good agreement for the measured value of the structural damping ζ d .

Approximate stability boundary

Finally it is worth examining a simplified situation. Using the approximations for small flow rates given in Eq. ( 26), the stability variable S takes a simpler form. This approach emphasizes the importance of the valve stiffness B = ∂Q/∂w that triggers the instability through the term A 2 4 in Eq. ( 28). With the approximations of Eq. (26), A does not change with the flow rate and we define

ζ c = ζ c + AZ 0 2τ c ω c
A 4 is the only variable varying with Q 0 and the stability equation can be written

-A 2 4 + A 2 A 3 A 4 -A 5 A 2 2 = 0 (29)
with an associated solution

A 4 = A 2 2 A 3 + A 2 3 -4A 5 (30)
The other solution (with a "-" in front of the square root) lead to a negative flow rate and is not considered in this analysis. An additional simplification is obtained by writing A 3 ≈ ω 2 c + ω 2 d , i.e. discarding the term which is multiplied by the small factor η c η d . With this approximation, we obtain

A 4 ≈ A 2 2 ω 2 c + ω 2 d + (ω 2 c -ω 2 d ) 2 = A 2 max(ω 2 c , ω 2 d ) (31)
Finally replacing A 4 we obtain a stability condition

if ω c > ω d , 5Q 0 2a 0 Z 0 γ mτ c 2η c ω c (ω 2 c -ω 2 d ) (32) if ω d > ω c , 5Q 0 2a 0 Z 0 γ mτ c 2η d ω d (ω 2 d -ω 2 c ) (33)
This approximate formula emphasizes the different conditions that influences the instability threshold. The instability occurs at lower flow rates for small damping, for matched eigenfrequency between the disk and the cavity and for a small response time of the cavity τ c . Quantitatively the validity of this approximation is limited to small values of Q 0 (see 10) and thus cannot be used in practical design situations.

Conclusion

The instability of a deformable disc in an Axial Balancing System of a turbopump was demonstrated in a well controlled experiment. By matching the eigenfrequencies of an acoustic cavity and a deformable disk and by carefully controlling the inner valve geometry, we were able to trigger self-sustained oscillations. We emphasize here the efficiency of the resonant interactions that lead to significant oscillations of the rigid disk. We provide a complete characterization of the experimental system to allow for comparison with models or numerical simulations.

We propose a model to study the coupling between the axisymmetric modes of the disk and the cavity. The model is used to compute the critical flow rate at which the stationary state of the disk becomes unstable. The prediction of the model is compared with the critical flow rate determined in the experiment and it shows a very good agreement. These results attest the validity of our analytical interpretation that involves a competition between two coupled oscillators, the acoustic cavity and the disc, and emphasizes the role of the stiffness of the inner valve.

The model was validated for a specific range of pressure loss and flow condition. An exact formulation of the pressure loss through the inner valve may be used to extend the stability condition to a specific geometry of the Axial Balancing System. In addition a further improvement to the predictive aspect of the model would be to improve the model to allow for the determination of the damping of the disk.

Finally, it is worth noting that non-axisymmetric mode may also become unstable (at higher flow rates). Our analysis could be adapted to account for such cases but this is out of the scope of the present study.

The authors wish to acknowledge Giuseppe Fiore from the CNES (French space agency) and Martin Seive and David Testa from Ariane Group for scientific discussions during this research collaboration founded by the CNES under contract RT-CT-253-0000-1804-CNES.

with q 0 = i(ω/c 0 ).

We now impose the condition B.5. We consider a harmonic output flow rate Qx exp(-iωt) resulting in v z = Vx exp(-iωt) for r < R x and v z = 0 otherwise, with Vx = Qx /(πR 2

x ) The coefficients A k can be determined by using the relation

R 0 r J 0 (p k r) J 0 (p k r) dr = 1 2 R 2 J 2 0 (p k R)δ(k -k ) (B.11)
that holds because J 1 [p k R] = 0 (see e.g. [START_REF] Sneddon | Special functions of mathematical physics and chemistry[END_REF]) and one obtains for k = 1, 2, ...

A k = Vx 2R x J 1 (p k R x ) p k q k R 2 J 2 0 (p k R) sinh(q k H) (B. 12
)
and

A 0 = Vx R 2 x q 0 R 2 sinh(q 0 H) (B.13)
The pressure field in the cavity that couples with the vibration of the disk can then be obtained through the relation P c = -ρ 0 ∂Ψ/∂t in z = 0. In the following, the non dimensional quantities will be used

ω = ωc 0 R , η = H R , σ = R x R (B.14)
together with the variables r = r/R and z = z/H and the solution for the velocity potential can be written

ψ = Q πR Ã 0 J 0 ( p0 r) + Qx πR Ã0 cosh( q0 z) + Qx πR ∞ k=1 Ãk J 0 ( pk r) cosh( qk z) e -iωt (B.15) p0 = ω, q0 = iη ω, pk = λ k , qk = η λ 2 k -ω2 (B.16)
where λ k is a zero of the J 1 Bessel function and

à 0 = 1 2η p0 J 1 (p 0 )
, Ã0 = η q0 sinh(q 0 ) , Ãk = 2ηJ 1 (p k σ) σ pk qk sinh(q k )J 2 0 (p k ) , (B.17)

The acoustic field depends on the exit flow rate Qx that is not known a priori. We consider that the exit conditions are those of a massless piston, thus assuming a homogeneous exit velocity. Computing the impedance associated with such condition is a classical problem of acoustics (see e.g. [START_REF] Rayleigh | The theory of Sound[END_REF]) and the impedance can be written

Z x = Px Qx = ρ 0 c 0 πR 2 x (a x -ib x ) = Z 0 1 σ 2 (a x -ib x ) (B.18)
with, for R x small compared with the wavelength, Here, the characteristic impedance of the cavity is Z 0 = (ρ 0 c 0 )/(πR 2 ).

a x = 1 2 ωR x c 2 = 1 2 ω σ
To address the dynamics of the system the impedance Z e in r = R should be computed. Since the acoustics has been computed with an exit flow rate Q x one can use the two formal relationship

P x = iZ 0 (α x Q + β x Q x ) (B.19) P R = iZ 0 (α e Q + β e Q x ) (B.20)
In these equations, P x is the (mean) pressure at the exit section (r ≤ R x , z = H) and the coefficients α x and β x can be computed using Eq. (B.10) To perform the stability analysis of the whole system, a simplified form is useful. We note that the dynamics of the cavity near its first resonance is well approximated by a damped resonator of the form

α x = ω Ã 0 2J 1 (p 0 σ) p0
Z e = Z 0 -iω/τ c ω 2 c -ω 2 -2iη c ω c ω (B.26)
where τ c is a typical time scale of the cavity, ω c is the frequency of the first resonance of te cavity and η c is a damping coefficient associated with acoustic radiation through the exit orifice. Finally, it is worth noting that a similar computation can be performed for the effective pressure P ef f (the equivalent pressure acting on the disk)

P ef f = 1 πR 2 R 0 2πrP (r, z = 0, t) r R 2 dr (B.27)
Similar results can be found and in particular, we obtain that in the regime of interest, the effective pressure can be written P ef f = γP R .

To make the simplified more quantitative, we use the different geometrical parameters of the experiment and notably the value R x = 14.1 mm that was determined by using the relation πR 2

x = S x where S x is the total area of the exit orifices. With these values we obtain (see also

Figure 1 :

 1 Figure 1: (a) Example of Liquid Oxygen turbo-pump with Axial Balancing System framed in red [8].( b) Simplified model of the Axial Balancing System. The flow is depicted with the blue arrows. The motion of the disk (sketched in light green) changes the gap a and thus changes the flow condition. This motion may couple with a fluctuating cavity pressure Pc thus triggering an instability.

Figure 2 :

 2 Figure 2: (a) Photograph of the setup with the 50 cm long inlet pipe having a 170 mm inner diameter. (b) Disc fixed above the cavity. The upper part of the cavity has been removed. (c) The bottom part of the cavity cavity with 3 inductive sensors, 2 laser sensors, one pressure port, 2 microphones and one loudspeaker that can be replaced by a visualisation window.

Figure 3 :

 3 Figure 3: (a) Signal measured during a ping test by one of the laser vibrometer. (b) The periodogram of this signal (computed) between t = 0 and t = s. The peak corresponding to the axisymmetric mode at 606 Hz is marked by an arrow. (Inset) Computing periodograms on subset of the signal of lengths 51 ms, the decay rate of the mode at 606 Hz can be determined. The red line corresponds to the exponential fit of the points above noise level. The decay rate is τ -1 d = 78 s -1 .

Figure 4 :

 4 Figure 4: (a) Transverse displacement of the disc measured by the laser sensors at r = 72 mm as a function of the differential pressure across the inner valve for different valve apertures a 0 . The blue curve represents the theoretical solution of equation 4. (b) Pressure loss across the valve ∆P/(ρV 2 0 ) as a function of the non dimensional parameter of Eq.( 5) with C 1 = 0.128 and C 2 = 54.7.

Figure 6

 6 Figure 6 represents two typical evolutions of the maximum and minimum values of the transverse deformation of the disc, measured at r = 72 mm. For small apertures, Fig 6 (a), the behaviour is typical of a supercritical bifurcation. Below the threshold (Q < Q c ), vibrations are negligible. Above the threshold Q > Q c , the amplitude of vibration A vib increases continuously asA vib = (Q -Q c ) 1/2 for Q Q c .The oscillations are symmetrical around the equilibrium position of the disc for intermediate flow rates (here between 0.55 and 0.8 L/s). For higher flow rates, vibrations become asymmetrical when the disc comes into contact with the upper part of the cavity. This occurs when the amplitude of vibrations are greater than the mean valve aperture.

Figure 5 :

 5 Figure 5: (a) Time evolution of the disc displacement uz (top) measured at r = 72 mm and the flow rate (bottom) for an initial aperture a 0 = 170 µm . The amplitude of the vibrations increases when the flow rate crosses the critical flow rate (near t = 1.0 s). (b) Evolution of the critical flow rate as a function of the valve nominal aperture a 0 .

Figure 6 :

 6 Figure 6: Evolution of the amplitude of the displacement of the disc in r = 72 mm as the flow rate changes. The red symbols are for the higher position of the disc (larger aperture), and the blue ones are for the lower position (smaller aperture). (a) Displacement for an initial valve aperture a 0 = 45 µm. A supercritical bifurcation occurs at Q 0 ≈ 0.55 L/s. (b) Displacement for an initial valve aperture a 0 = 170 µm. The bifurcation is subcritical. (c) For larger initial valve aperture a 0 = 185 µm oscillations with limited amplitudes are observed before a sharp transition to vibration of high amplitude. The different graphs share the same scale for the displacement.

Figure 7 :

 7 Figure 7: (a) Evolution of the amplitude of the vibrations as a function of time after a fast flow increment from 0 to 9.72 L/s with a 0 = 170 µm (blue circles). The red curve represents the fit by the exponential function. (b)1/τ (top) and minimal and maximal displacement of the disc in r = 72 mm (bottom) as a function of the flow rate Q f for a 0 = 45 µm. (c) Evolution of the measured frequency with the differential pressure accross the inner valve.

Figure 8 :

 8 Figure8: The displacement of the disc w 0 in the reference state for different values of the initial opening a 0 . The solid lines show the solutions of eq. (18) and the dots show experimental measurements (note that the measured displacements are multiplied by (9/7) 2 because measurements were performed at r = 72 mm and not at the outer edge R = 90 mm. See also eq. 11). For the computations we use the geometry and the material properties for the disk given in the text (section 2) and for the fluid ρ 0 = 1.29 kg m -3 , µ = 18.1 × 10 -6 Pa s.

Figure 9 :

 9 Figure9: Evolution of the real and imaginary parts of the eigenvalues (solutions of Eq. 27). The system is stable for low flow rates with four complex eigenvalues (conjugate by pair) with negative real parts. As the flow rate increases a pair of eigenvalues crosses the imaginary axis. For the figure the geometrical parameters and material properties of section 2 are used with the properties of the cavity obtained in Appendix B and given in figure B.11. The damping coefficient of the disk is η d = 0.025 and the initial aperture is a 0 = 50 µm.

Figure 10 :

 10 Figure 10: Stability threshold for the Axial Balancing System (solid line) obtained from Eq. (28). Above the critical flow rate Q 0 the system is unstable. The boundary is computed for the parameters of section 2 and the properties of the cavity obtained in Appendix B and given in figure B.11. The damping coefficient of the disk is ζ d = 0.021, the dots represent the critical flow rates measured in the experiment and the dashed line shows the approximate solution of Eq. (32).

  σ (B.21)β x = ω Ã0 cosh(q 0 η) + ∞ k=1 Ãk cosh(q k η) 2J 1 (p k σ) pk σ (B.22)P R is the pressure at the entry section (r = R, z = 0), and, using Eq. (B.10),α e = ω Ã 0 J 0 (p 0 ) (B.23)β e = ω Ã0 + ∞ k=1 Ãk J 0 (p k ) (B.24) Combining eqns. (B.19) and (B.20) with eq. (B.18) yieldsZ e = P R Q = Z 0 iα e -β e α x Z x /Z 0 -iβ x (B.25)The different coefficients α and β can be computed by performing the summation numerically for a given set of parameters. The quantity of interest is Z e . It depends on the geometry (i.e. the parameters η and σ) and Z 0 . To make the analysis more concrete, we plot the input impedance for a set of parameters related to the experimental setup in Figure B.11.

Figure B. 11 :

 11 Figure B.11: Non dimensional input impedance Ze/Z 0 and effective impedance Z ef f /Z 0 for R = 90 mm, H = 5 mm, Rx = 14.1 mm. The dots show the real and imaginary parts obtained by computing the sums with 300 modes. The continuous lines show the approximation by a resonator (eq. B.26) with ωc = 0.976 c 0 /R, ηc = 0.0129, τc = 0.0461 R/c 0 and γ = 0.575.

  Fig. B.11) τ c = R c 0 0.0461105, ω c = c0 R 0.976313, η c = 0.0129, γ = 0.5749 (B.28)

Appendix A. Static solution for an annular disk with a uniform pressure drop across its faces

The solution for a disk clamped at αR (with α < 1) and free at R with a uniform pressure difference ∆P applied to its faces reads

with

This form is compared with the measurements in Fig. 4.

Appendix B. Acoustic cavity

In this section, we propose a model to characterise the response of the acoustic cavity. Since the forcing and the viscous terms are neglected in the wave equation 6, the velocity field is irrotational and it is convenient to solve equation ( 6) using the velocity potential ψ defined as v = ∇ψ. The acoustic equation then becomes

We consider that the geometry of the cavity is imposed i.e. we neglect the influence of the small displacement of the disc on the boundary condition at z ≈ 0. The different boundary conditions for an harmonic perturbation write

where H(x) = 1 if x < 0 and 0 otherwise. We assumed that the motion of the disc does not significantly modify the acoustic response of the cavity and therefore we enforce no motion at the contact with the disc. In addition we consider that the entry conditions result in a plug flow with a uniform radial velocity VQ = Q/(2πRH) and that the output of the cavity is also characterized by a plug flow with an harmonic exit flow rate Qx exp(-iωt) localised in r ≤ R x with a velocity Vx = Qx /(πR 2 x ) where R x is the radius of the exit section.

Considering equation (B.1) in cylindrical coordinates and looking for the harmonic response, we can write it under the form of a Bessel equation [START_REF] Watson | Theory of Bessel Functions[END_REF][START_REF] Dattoli | Theory and applications of generalized Bessel functions[END_REF]. Considering only the axisymmetric modes, the solution writes

A k e q k z + B k e -q k z e -iωt (B.6) with p 2 k = q 2 k + (ω/c 0 ) 2 , where q k is the wave number in the axial direction z. The p k 's form a discrete set imposed by the boundary conditions as shown below. J 0 is the zero order Bessel function of the first kind, and the magnitudes A k and B k are to be determined with the boundary conditions.

In the specific radial and axial directions, the velocities v r and v z are obtained from the derivative of ψ in r and z respectively. Equation B.2 yields

To enforce the boundary condition B.3 we write the solution for v r under the form

with A 0 = VQ /[p 0 J 1 (p 0 R)] and p 0 = ω/c 0 . For all other modes, the p k form a discrete set of values verifying J 1 (p k R) = 0 for k = 1, 2, • • • . The equation J 1 (λ) = 0 has an infinity of discrete solutions λ k , the zeros of the Bessel function J 1 . The general solution then writes,

Appendix C. Stability criterion

In the marginally stable state, when a pair of eigenvalues have a zero real part, the characteristic polynomial can be written

When expanding and identifying with the form 27 one obtains

To obtain a solution of this set of equations, a necessary condition is

Therefore, this equation acts as a stability boundary.


Charlotte Devoisin-Lagarde

Bastien Trémolière

Serge Caparos

Elodie Charbonnier

Effects of a cognitive schema account on the stigma of schizophrenia: An exploratory study in a French university student sample

Keywords: Stigma, causal attributions, empathic concern, maladaptive cognitive schema

Background: Amongst people with psychiatric disorders, people experiencing schizophrenia are among those who are victims of high levels of stigmatization.

Aims: The present study explored how educational strategies emphasizing different types of etiological accounts of schizophrenia may help reduce the level of stigma towards the pathology.

Methods: A total of 378 students were randomly assigned to one of four groups. Three experimental groups were presented with a vignette which introduced one out of three different etiological accounts of schizophrenia (Cognitive distortion, Cognitive Schema, or Biogenetics). The fourth group was presented with a vignette unrelated to schizophrenia. The participants completed questionnaires which measured their attitudes, empathic concern and social distance towards people with schizophrenia, before and after reading the vignette.

Results: The "Cognitive Schema" etiological account, which emphasized maladaptive cognitive schemas, was the only one which led to a significant decrease in stigma. The latter decrease was mostly driven by an increase in the level of empathic concern towards people with schizophrenia.

Conclusions: The results of the present study suggest that using cognitive-schema cues may be a promising avenue to reduce the stigmatization of schizophrenia. Future research is warranted which will explore the many factors jointly that explain schizophrenia for use in anti-stigma campaigns.

Introduction

About one in four people will suffer from a psychiatric disorder in their lifetime (World Health Organization, 2001). Not only do individuals with psychiatric disorders have to deal with their symptoms, but they must also cope with the stigmatization that they generally suffer from (Corrigan & Watson, 2002a). Stigmatization is a process of exclusion leading an individual, on the basis of a given characteristic that does not conform to a social norm, to be identified as abnormal or deviant, and reduced to this characteristic [START_REF] Crocker | Social stigma and self-esteem: The self-protective properties of stigma[END_REF][START_REF] Goffman | Stigma: Notes on the management of spoiled identity[END_REF].

Levels of stigmatization differ a lot, and are mainly function of the diagnosis [START_REF] Parle | How does stigma affect people with mental illness?[END_REF]: among all psychiatric disorders, people experiencing schizophrenia are victims of high levels of rejection and stigmatization [START_REF] Angermeyer | Reinforcing stereotypes: focus on forensic cases in news reporting may influence public attitudes towards the mentally ill[END_REF][START_REF] Castillo | Étude des représentations sociales de la schizophrénie dans la population générale et dans une population de patients schizophrènes[END_REF][START_REF] Pescosolido | A disease like any other"? A decade of change in public reactions to schizophrenia, depression, and alcohol dependence[END_REF]. They are commonly perceived by lay people as dangerous, incompetent, and unpredictable [START_REF] Angermeyer | Public beliefs about and attitudes towards people with mental illness: a review of population studies[END_REF]Angermeyer & Matschinger, 2004;[START_REF] Corrigan | Familiarity with and social distance from people who have serious mental illness[END_REF][START_REF] Crisp | Stigmatisation of people with mental illnesses[END_REF][START_REF] Dietrich | The relationship between public causal beliefs and social distance toward mentally ill people[END_REF][START_REF] Graves | Psychophysiological evaluation of stigma towards schizophrenia[END_REF][START_REF] Link | Public conceptions of mental illness: labels, causes, dangerousness, and social distance[END_REF]. Schizophrenia is frequently and intimately associated with the term "violence" and is perceived as the "model of madness" by society [START_REF] Castillo | Étude des représentations sociales de la schizophrénie dans la population générale et dans une population de patients schizophrènes[END_REF]. Consistent with this observation, some of the symptoms of schizophrenia, such as inappropriate affect and the propensity to talk aloud to oneself, are manifest indicators of a psychiatric disorder and may lead to stigmatizing reactions (see [START_REF] Corrigan | Mental health stigma as social attribution: Implications for research methods and attitude change[END_REF].

Stigmatization is likely to have detrimental consequences for people with schizophrenia. The consequences of stigmatization can be observed either within the psychological sphere [START_REF] Berge | Self-esteem and stigma among persons with schizophrenia: implications for mental health[END_REF], the social and professional sphere [START_REF] Daumerie | Discrimination perceived by people with a diagnosis of schizophrenic disorders. INternational study of DIscrimination and stiGma Outcomes (INDIGO): French results[END_REF], the family sphere [START_REF] González-Torres | Stigma and discrimination towards people with schizophrenia and their family members[END_REF], or the sphere of medical care [START_REF] Demyttenaere | Prevalence, severity, and unmet need for treatment of mental disorders in the World Health Organization World Mental Health Surveys[END_REF][START_REF] Fung | Self-stigma of people with schizophrenia as predictor of their adherence to psychosocial treatment[END_REF]. In the specific case of France, recent studies have confirmed that the levels of stigmatization of schizophrenia are high [START_REF] Angermeyer | Attitudes and beliefs of the French public about schizophrenia and major depression: results from a vignette-based population survey[END_REF][START_REF] Durand-Zaleski | A first national survey of knowledge, attitudes and behaviours towards schizophrenia, bipolar disorders and autism in France[END_REF][START_REF] Lampropoulos | Analyzing the presentation and the stigma of schizophrenia in French newspapers[END_REF].

According to attribution theory [START_REF] Corrigan | Mental health stigma as social attribution: Implications for research methods and attitude change[END_REF][START_REF] Weiner | An attributional analysis of reactions to stigmas[END_REF][START_REF] Weiner | Judgments of responsibility : a foundation for a theory of social conduct[END_REF], an important factor that may contribute to the stigmatization of a disorder is the etiological account that people attribute to this disorder. Specifically, level of control (i.e., the extent to which people can actively act on their disease) and stability of the disorder (i.e., the extent to which the disorder is maintained over time) are thought key elements of stigma. Anti-stigma campaigns which have emphasized biogenetic or psychosocial factors to account for the development of schizophrenia, have had mixed effects on measured levels of stigma (see the meta-analyses of Kvaale et al., 2013a;Kvaale et al., 2013b). Endorsing biological factors as the cause of schizophrenia was associated with a greater desire for social distance, i.e., the desire not to interact with them [START_REF] Angermeyer | Causal beliefs and attitudes to people with schizophrenia[END_REF][START_REF] Dietrich | The relationship between public causal beliefs and social distance toward mentally ill people[END_REF]. Biogenetic explanations may reinforce feelings of dangerousness and therefore the desire for social distance [START_REF] Corrigan | At issue: Stop the stigma: call mental illness a brain disease[END_REF][START_REF] Mehta | Is being "sick" really better? Effect of the disease view of mental disorder on stigma[END_REF][START_REF] Phelan | Geneticization of deviant behavior and consequences for stigma: The case of mental illness[END_REF][START_REF] Phelan | Effects of attributing serious mental illnesses to genetic causes on orientations to treatment[END_REF]. Biogenetic causal beliefs are also positively related to prejudice and fear [START_REF] Read | Prejudice and schizophrenia: a review of the "mental illness is an illness like any other" approach[END_REF]. It is possible that biogenetic accounts promote the idea that the person may be considered to have little control over his/her behavior, which may lead to expectations of higher levels of unpredictable behaviors and may increase fear responses in lay people [START_REF] Angermeyer | Public beliefs about schizophrenia and depression: similarities and differences[END_REF].

Likewise, psychosocial accounts of mental disorders (e.g., accounts such as stress, death of a loved one, insults at school) have generally shown mixed effects on the level of stigma (see the systematic review and meta-analysis of [START_REF] Schomerus | Evolution of public attitudes about mental illness: a systematic review and meta-analysis[END_REF]. Stress-related accounts of mental disorders have sometimes been shown to be associated with greater tolerance of the disorders [START_REF] Martin | Of fear and loathing: The role of "disturbing behavior," labels, and causal attributions in shaping public attitudes toward people with mental illness[END_REF][START_REF] Schnittker | An uncertain revolution: Why the rise of a genetic model of mental illness has not increased tolerance[END_REF][START_REF] Van 't Veer | Determinants that shape public attitudes towards the mentally ill[END_REF], but not in all studies [START_REF] Bag | Factors influencing social distance from people with schizophrenia[END_REF][START_REF] Jorm | The public's stigmatizing attitudes towards people with mental disorders: how important are biomedical conceptualizations[END_REF][START_REF] Martin | The construction of fear: americans' preferences for social distance from children and adolescents with mental health problems[END_REF]. Similar mixed results have been observed with early-stressor accounts [START_REF] Dietrich | The relationship between public causal beliefs and social distance toward mentally ill people[END_REF][START_REF] Schnittker | An uncertain revolution: Why the rise of a genetic model of mental illness has not increased tolerance[END_REF][START_REF] Van 't Veer | Determinants that shape public attitudes towards the mentally ill[END_REF]. According to [START_REF] Schomerus | Causal beliefs of the public and social acceptance of persons with mental illness: a comparative analysis of schizophrenia, depression and alcohol dependence[END_REF], lay people may infer from some psychosocial accounts that people experiencing schizophrenia are responsible for their disorder (e.g., with accounts such as the use of psychoactive substances), and this may increase stigma as a result. On the other hand, other psychosocial accounts (e.g., childhood sexual abuse) may reduce the perceived level of personal responsibility and, consequently, improve social acceptance. However, psychosocial explanations, such as childhood sexual abuse, may drive lay people to consider schizophrenia as incurable, and this may increase stigma as a result.

In an attempt to reduce stigma, the use of cognitive accounts of schizophrenia have been largely overlooked. It appears that, so far, such accounts have been mobilized mainly in the context of depression [START_REF] Botha | The influence of emphasizing psychological causes of depression on public stigma[END_REF]. Yet, cognitive factors -for instance cognitive distortion factors and maladaptive cognitive schemas -are known to hold an important role in the development of symptoms in schizophrenia [START_REF] Bortolon | Early maladaptive schemas predict positive symptomatology in schizophrenia: A crosssectional study[END_REF][START_REF] Sundag | Elucidating the role of early maladaptive schemas for psychotic symptomatology[END_REF]. Early maladaptive schemas develop during childhood, via interpersonal interactions, and form a template that guides the interpretation of later experience. They are extremely stable, considered to be more unconditional, automatic, and to increase the risk of developing a mental disorder [START_REF] Schmidt | The schema questionnaire: Investigation of psychometric properties and the hierarchical structure of a measure of maladaptive schemas[END_REF]. Cognitive models of schizophrenia suggest that exposure to negative life events during childhood (e.g., school harassment) promotes the development of maladaptive cognitive schemas, including a threatening worldview, external attribution of negative events and experiences, which may favor the development of schizophrenia, and more specifically, paranoid delusions [START_REF] Bentall | The cognitive and affective structure of paranoid delusions: a transdiagnostic investigation of patients with schizophrenia spectrum disorders and depression[END_REF][START_REF] Garety | A cognitive model of the positive symptoms of psychosis[END_REF][START_REF] Howes | Schizophrenia: an integrated sociodevelopmentalcognitive model[END_REF]. Early maladaptive schemas are higher people experiencing schizophrenia [START_REF] Boyda | Childhood maltreatment and psychotic experiences: Exploring the specificity of early maladaptive schemas[END_REF][START_REF] Giblin | Psychosocial correlates of lateonset psychosis: life experiences, cognitive schemas, and attitudes to ageing[END_REF][START_REF] Khosravani | Early maladaptive schemas in patients with schizophrenia and non-patients with high and low schizotypal traits and their differences based on depression severity[END_REF][START_REF] Sundag | Elucidating the role of early maladaptive schemas for psychotic symptomatology[END_REF]. Negative schemas easy up the development of delusions and hallucinations, including a decrease in performance at reality tests [START_REF] Beck | Cognitive approaches to schizophrenia: theory and therapy[END_REF][START_REF] Birchwood | The power and omnipotence of voices: Subordination and entrapment by voices and significant others[END_REF][START_REF] Garety | A cognitive model of the positive symptoms of psychosis[END_REF].

Cognitive distortions refer to erroneous interpretations of situations (e.g., minimizing positive accomplishments, personalizing neutral events; [START_REF] Beck | Cognitive therapy of depression[END_REF]. Cognitive distortions are important early risk factors for psychosis [START_REF] De Paula | Cognition in at-risk mental states for psychosis[END_REF][START_REF] Gawęda | Self-reported cognitive distortions in the psychosis continuum: A Polish 18-item version of the Davos Assessment of Cognitive Biases Scale (DACOBS-18)[END_REF]. More precisely, an exaggerated tendency to pay attention to threat (Gaweda et al., 2015;Prochwicz & Klosowska, 2017), external attribution bias [START_REF] So | Dimensions of delusions and attribution biases along the continuum of psychosis[END_REF], jumping to conclusions [START_REF] Ross | Jumping to conclusions about the beads task? A meta-analysis of delusional ideation and datagathering[END_REF][START_REF] Mclean | Association of the jumping to conclusions and evidence integration biases with delusions in psychosis: a detailed meta-analysis[END_REF] or bias against disconfirmatory evidence [START_REF] Zawadzki | Cognitive factors associated with subclinical delusional ideation in the general population[END_REF] are important factors driving delusions. In the same way, deficits in recognizing the source of information plays a central role in hallucinations (see the meta-analysis of Waters et al., 2012).

Because cognitive distortions may be perceived by lay people as controllable [START_REF] Cook | Causation beliefs and stigma against depression: Results from a population-based study[END_REF][START_REF] Goldstein | Etiological paradigms of depression: The relationship between perceived causes, empowerment, treatment preferences, and stigma[END_REF], accounts based on such factor may increase public stigma. On the other hand, maladaptive cognitive schemas emerge during childhood, through experiences of harmful events (e.g., critical, abusive or neglectful events which do not meet the needs of the child) [START_REF] Young | Schema therapy : a practitioner's guide[END_REF]. Thus, cognitive schema accounts emphasize the importance of external factors as causes of the disorder, they support the idea that patients have a reduced control over the disorder and, as a result, cannot be responsible for their pathology [START_REF] Schomerus | Causal beliefs of the public and social acceptance of persons with mental illness: a comparative analysis of schizophrenia, depression and alcohol dependence[END_REF]. Cognitive schema accounts may therefore reduce public stigma. Table 1 summarizes

Method

Participants

An a priori power analysis was conducted using Gpower [START_REF] Erdfelder | GPOWER: A general power analysis program[END_REF]. The analysis indicated that a sample size of 332 participants (83 participants per group) would be sufficient to detect a significant interaction between our text conditions and moment of measurement (pre/post text exposure), with a power of .95 and an alpha of .05, based on a small effect size of 0.05 ( p 2 ). Anticipating drop out and exclusion issues, a total of 415 students

Compliance with Ethical Standards

All participants signed a consent form before participation and were informed that their answers would remain anonymous and that their participation was voluntary and could be withdrawn at any time. All the data were stored on a secure university computer. We respected the ethical principles of psychologists and the code of conduct of the American Psychological Association (American Psychological Association, 2002).

Measures

In the absence of validated tools in French language for our variables of interest, all scales used were translated in French via a back-and-forth translation procedure. Firstly, the English version was translated into French by an English native speaker specialized in this research field. Then, a French native speaker translated this document back into English. The two translations (English to French and French to English) were compared in order to ensure that they were both faithful to the original and that the items in the French version were clear.

Familiarity with schizophrenia and mental illness

To assess people's familiarity with schizophrenia, we used a French translation of the Familiarity with mental illness scale [START_REF] Corrigan | An attribution model of public discrimination towards persons with mental illness[END_REF]. This 7-item questionnaire measures participants' familiarity with schizophrenia through everyday encounters with the pathology (e.g., interaction in the workplace, presence of people experiencing the pathology in the entourage or at home). Participants answered using a dichotomous "Yes"/"No" response to each item. Scores ranged from 0 to 7 and higher scores indicated higher familiarity with schizophrenia. We also included two additional questions in order to assess participants' general familiarity with mental illnesses: "Do you have a diagnosis of mental disorder" and "Do you have a member of your family with a diagnosis of mental disorder". Both questions were answered using a dichotomous "Yes"/"No" response.

Attitudes

We used a French translation of the Brief Version of the Attribution Questionnaire (AQ-9) (Corrigan et al., 2002) in order to assess attitudes towards people experiencing schizophrenia. Each item from the attitude questionnaire assesses a reaction towards people with schizophrenia, exploring the domains of blame, anger, pity, help, dangerousness, fear, avoidance, segregation, and coercion (e.g., How angry do persons with schizophrenia make you feel?). Participants gave their answer using 9-point Likert scales (ranging from 1: "Not at all" to 9: "Very much"). Scores ranged from 9 to 81, and higher scores indicated more negative attitudes towards people with schizophrenia. For the sake of comparability with the other measures, we subsequently reversed the questionnaire score and transformed it into a percentage, leading to a percentage score of positive attitude (from 0 to 100). The scale showed acceptable internal consistency (Cronbach's α at T0 = .71 and T1 = .73).

Social Distance

In order to evaluate social distance, we used a French translation of the Social Distance Items [START_REF] Link | The social rejection of former mental patients: Understanding why labels matter[END_REF], which is the most used tool for assessing social stigma of schizophrenia [START_REF] Lampropoulos | La stigmatisation sociale des personnes vivant avec la schizophrénie : une revue systématique de la littérature[END_REF]. It includes seven items displaying social situations (e.g., renting a room) and participants are asked to indicate the extent to which they would be willing to accept to share these situations with people experiencing schizophrenia, using a 4-point Likert scale ranging from 1 ("Definitely willing") to 4 (" Definitely unwilling"). Scores ranged from 7 to 28, and higher scores indicated a preference for greater social distance.

Scores were then reversed and transformed into a percentage, leading to a percentage score of social proximity (from 0 to 100). The scale showed very good internal consistency (Cronbach's α at T0 = .86 and T1 = .89).

Empathic concern

In order to measure empathic concern, we used a French translation of the emotional response scale [START_REF] Batson | Prosocial motivation: Is it ever truly altruistic[END_REF][START_REF] Batson | The altruism question: Toward a social-psychological answer[END_REF]. Participants were instructed to indicate how much they felt each of six emotions (sympathy, soft-hearted, warm, compassionate, tender and moved) towards individuals experiencing schizophrenia, using a 7-point Likert scale ranging from 1 ("Not at all") to 7 ("Very much"). Scores ranged from 6 to 42, and higher scores indicated greater empathic concern. Scores were then transformed into a percentage score of empathic concern (from 0 to 100). The scale showed very good internal consistency (Cronbach's α at T0 = .87 and T1 = .90).

Causal explanations of schizophrenia

Four texts were developed in French language and presented to the participants. Three A fourth text, unrelated to schizophrenia and presenting a topic related to quantum physics, was used in the control condition.

Procedure

After giving their consent, participants provided their basic sociodemographic data (age, sex) and indicated their familiarity with schizophrenia. Next, all participants read a short description of the daily life of an individual experiencing schizophrenia (the description is displayed in the Appendix), in order to prime a correct and realistic conception of the disorder (see [START_REF] Corrigan | Changing public stigma with continuum beliefs[END_REF]. Then, participants completed the questionnaires about attitudes (Corrigan et al., 2002), empathic concern [START_REF] Batson | Prosocial motivation: Is it ever truly altruistic[END_REF][START_REF] Batson | The altruism question: Toward a social-psychological answer[END_REF], and social distance [START_REF] Link | The social rejection of former mental patients: Understanding why labels matter[END_REF] Following the experiment, participants were informed about the purpose of the study.

We took caution to emphasize that each scenario they were presented with displayed only one etiological account (Biogenetics, Cognitive Distortion, Cognitive Schema or Control). Then, they received exhaustive information about causal attributions of schizophrenia and the multiple etiological accounts of the disorder.

Statistical analysis

We conducted a mixed-design analysis of variance using Measure (positive attitude, social proximity, and empathic concern; within-subject), Text condition (Cognitive Schema, Cognitive Distortion, Biogenetics, Control; between-subject), and Moment of measurement (pre-text measure vs. post-text measure; within-subject) as the independent variables, and percentage score (for attitude, social proximity, and empathic concern) as the dependent measure. In addition, we added familiarity, self-reported psychiatric disorder, and selfreported psychiatric disorder in the participant's family as control variables in the analysis.

Data were analyzed using the SPSS software (version 24).

Results

Figure 1 displays the detailed results distinguishing the effect of text condition for each of the measures and moment of measurement (see also Table 2 for the full descriptive A1, in appendix, for full descriptive statistics).

Insert Table 2 and Figure 1 here

Discussion

Amongst people with psychiatric disorders, people experiencing schizophrenia are among those who are victims of high levels of stigmatization [START_REF] Angermeyer | Reinforcing stereotypes: focus on forensic cases in news reporting may influence public attitudes towards the mentally ill[END_REF][START_REF] Castillo | Étude des représentations sociales de la schizophrénie dans la population générale et dans une population de patients schizophrènes[END_REF]Corrigan & Watson, 2002a;[START_REF] Durand-Zaleski | A first national survey of knowledge, attitudes and behaviours towards schizophrenia, bipolar disorders and autism in France[END_REF][START_REF] Graves | Psychophysiological evaluation of stigma towards schizophrenia[END_REF][START_REF] Schomerus | Causal beliefs of the public and social acceptance of persons with mental illness: a comparative analysis of schizophrenia, depression and alcohol dependence[END_REF]. A research effort is necessary to identify new strategies that may help reduce the public stigma of schizophrenia, and the associated multiple negative consequences for people experiencing this mental disorder. To date, the few researches that exist on this topic have highlighted mixed effects. Educational interventions using biogenetics or psychosocial accounts of the development of schizophrenia do not always lead to a decrease in stigma, and may even sometimes lead to an increase in stigma [START_REF] Bag | Factors influencing social distance from people with schizophrenia[END_REF][START_REF] Jorm | The public's stigmatizing attitudes towards people with mental disorders: how important are biomedical conceptualizations[END_REF]Kvaale et al., 2013a;Kvaale et al., 2013b;[START_REF] Schomerus | Evolution of public attitudes about mental illness: a systematic review and meta-analysis[END_REF]. In the context of schizophrenia, educational strategies using cognitive accounts have never been evaluated, yet studies performed in the context of depression have shown these strategies to be promising [START_REF] Botha | The influence of emphasizing psychological causes of depression on public stigma[END_REF]. The present study addressed this issue and compared the effect on public stigma of three different educational strategies, based on cognitive distortion, maladaptive cognitive schema, and biogenetics. In line with our expectations, the information that emphasized the role of maladaptive cognitive schemas in the development of schizophrenia was the only one that led to a significant decrease in stigma. In contrast, our results showed that none of the other two accounts of schizophrenia impacted stigma, and neither did stigma vary in the control condition.

Why should a Cognitive Schema account of schizophrenia be more efficient in reducing stigma? In light of attribution theories [START_REF] Corrigan | Mental health stigma as social attribution: Implications for research methods and attitude change[END_REF][START_REF] Weiner | An attributional analysis of reactions to stigmas[END_REF][START_REF] Weiner | Judgments of responsibility : a foundation for a theory of social conduct[END_REF], level of stigma may depend on etiological account, specifically on control and stability criteria.

Previous studies have shown strong positive relationships between patients' control or responsibility in the development of their particular disorder and public stigma towards the disorder in question [START_REF] Dietrich | The relationship between public causal beliefs and social distance toward mentally ill people[END_REF][START_REF] Lincoln | Can antistigma campaigns be improved? A test of the impact of biogenetic Vs psychosocial causal explanations on implicit and explicit attitudes to schizophrenia[END_REF][START_REF] Schomerus | Continuum beliefs and stigmatizing attitudes towards persons with schizophrenia, depression and alcohol dependence[END_REF]. In addition, studies have highlighted a positive link between stability of the disorder (i.e., the impossibility to treat or reduce it) and public stigma [START_REF] Phelan | Geneticization of deviant behavior and consequences for stigma: The case of mental illness[END_REF][START_REF] Phelan | Effects of attributing serious mental illnesses to genetic causes on orientations to treatment[END_REF].

Because Cognitive Schema accounts of schizophrenia emphasize the importance of external factors (e.g., childhood experiences, such as humiliations at school) as causes of the disorder, they highlight its low controllability. In addition, Cognitive Schema accounts draw attention to the fact that the joint experience of similar situations experienced in childhood (during which cognitive schema is built) and adulthood (during which cognitive schema is activated) is partly responsible for schizophrenia symptoms, putting light on the unstable nature of the disease over time.

The Cognitive Schema exposure effect was driven mainly by an increase in the level of empathic concern towards people experiencing schizophrenia. According to [START_REF] Batson | An additional antecedent of empathic concern: Valuing the welfare of the person in need[END_REF], level of empathy for another person in need depends on three dimensions: perceiving that the other is in need, valuing the other's welfare, and adopting the other's perspective. In the Cognitive Schema condition, the example given in the vignette was related to a child being humiliated at school, which is a common situation that the participants may have themselves experienced during childhood. This may have increased their propensity to adopt the other's perspective, and, as a result, empathic concern. This result is all the more important given that much research has evidenced that a high empathic concern is an important factor for prosocial motivation and helping behaviors [START_REF] Batson | An additional antecedent of empathic concern: Valuing the welfare of the person in need[END_REF][START_REF] Eisenberg | The relation of empathy to prosocial and related behaviors[END_REF]. According to this idea, emotional changes may induce cognitive changes (i.e., decrease in negative attitudes) and behavioral changes (decrease in desire for social distance).

We believe that this first step of research may open new avenues for future research on the prevention of stigmatization of schizophrenia. First, the present results shed light on the necessity to raise public awareness of schizophrenia. Educational strategies based on etiological accounts of schizophrenia may be used, and they should emphasize both the low stability and low controllability in the disease. We see a great importance in the necessity to balance these two components in the explanation of schizophrenia, which is consistent with the multidimensional and balanced approach proposed by [START_REF] Corrigan | At issue: Stop the stigma: call mental illness a brain disease[END_REF]. If biogenetic-related explanations (associated with low control but high stability) are used, they may be accompanied by information about the efficacy of treatments, in order to reduce the feelings of dangerousness and bad prognosis induced by this kind of account. Second, our results call for the need to evaluate the effectiveness of the prevention methods used, the non-standardization of the different explanations being an important factor which possibly explains their mixed effects on stigma [START_REF] Lamboy | Réduire la stigmatisation des personnes souffrant de troubles psychiques par une campagne de communication ? Une synthèse de la littérature[END_REF].

Our present results, although promising, need to be taken with caution, for several reasons which we list here. First, it is worth noting that our effect is a small one. The significant pairwise difference that we observed for the Cognitive Schema condition (between T0 and T1)

showed only a small effect size (Cohen's d = .18). We acknowledge that many other factors (and their interaction) contribute to explain stigma. Second, our research follows an experimental paradigm, and aims to untangle the effects of distinct educational strategies presenting different types of etiological accounts of schizophrenia. As a result, each scenario was designed to emphasize only one cause for schizophrenia. It stands to reason that the etiology of schizophrenia is that maladaptive cognitive schema cannot be considered the only explanation for schizophrenia. Future research is warranted which will explore the many factors jointly that explain schizophrenia. Third, because cognitive schemas develop during childhood [START_REF] Young | Schema therapy : a practitioner's guide[END_REF], one may worry about the possibility that

Cognitive Schema accounts of schizophrenia may drive the public to blame the social environment, and especially the parents, for the development of schizophrenia in a child. This would be an undesirable side-effect of an anti-stigma campaign. Fourth, although the present results show a promising effect of Cognitive Schema accounts in the short term, they need to be replicated in the long term. Because the three measures capturing stigma were obtained immediately after the anti-stigma treatment, one must refrain from overgeneralizing these findings as well as to consider that the positive effects would persist over time. Finally, our sample consisted of predominantly French female university students who are certainly sensitive to the issue of mental health. Consequently, replications need to be conducted with a more representative population before one may be able to generalize such findings. "Schizophrenia is largely the result of genetic factors." This is the first conclusion of an article recently published in the "Journal of Molecular Biology and Genetics". The author of this article is Dr Morel Edouard of Paris-Sorbonne University, who is renowned worldwide for his expertise in schizophrenia. Dr Morel and his collaborators have investigated the factors that may be involved in the onset of symptoms of schizophrenia. They concluded: "Taking into account all the data gathered over the past 30 years, we can conclude that genetic factors, linked to an excessive amount of neurotransmitters in the brain, a reduction of the volume of the grey matter, and abnormalities in some cortical circuits (all genetically predisposed), are the main causes of schizophrenia. For example, an individual with a parent experiencing schizophrenia has 30% risk of developing schizophrenia symptoms in their life." These results have been praised by many researchers working on schizophrenia, who now base their work on the results obtained by Dr Morel and his collaborators.

Cognitive distortion condition

Please carefully read the article below. Later, we will ask you a few questions regarding the content of the article. This article was published in the August-28 edition of this year's "Journal of Behavioral and Cognitive Therapy". The "Journal of Behavioral and Cognitive Therapy" is a journal that covers the latest news about psychopathology and cognitive and behavioral therapy. It is highly respected in its academic field.

"Schizophrenia is largely the result of cognitive factors." This is the first conclusion of an article recently published in the "Journal of Behavioral and Cognitive Therapy". The author of this article is Dr Morel Edouard of Paris-Sorbonne University, who is renowned worldwide for its expertise in schizophrenia. Dr Morel and collaborators have investigated factors that may be involved in the onset of symptoms of schizophrenia. They concluded: "Considering all the data gathered over the past 30 years, we can conclude that reasoning errors largely influence the onset of schizophrenia symptoms. One example of reasoning error may be that if someone blames you, it means that everyone blames you, which is called over-generalization. When this type of thoughts develops over time and multiplies, the symptoms of schizophrenia may appear." These results have been praised by many researchers working on schizophrenia, who now base their work on the results obtained by Dr Morel and his collaborators.

Cognitive Schema Condition

Please carefully read the article below. Later, we will ask you a few questions regarding the content of the article. This article was published in the August-28 edition of this year's "Journal of Behavioral and Cognitive Therapy". The "Journal of Behavioral and Cognitive Therapy " is a journal that covers the latest news about psychopathology and cognitive and behavioral therapy. It is highly respected in its academic field.

"Schizophrenia is largely the result of cognitive factors." This is the first conclusion of an article recently published in the "Journal of Behavioral and Cognitive Therapy". The author of this article is Dr Morel Edouard of Paris-Sorbonne University, who is renowned worldwide for its expertise in schizophrenia. Dr Morel and collaborators have investigated factors that may be involved in the onset of symptoms of schizophrenia. They concluded: "Considering all the data gathered over the past 30 years, we can conclude that dysfunctional cognitive schemas, in other words errors of interpretation, which have their origin in events experienced during childhood, are the primary factors influencing the symptoms of schizophrenia. We can take the example of a child who is humiliated by his schoolmates. When this type of behavior is repeated in adulthood, the dysfunctional schema of mistrust built during childhood can then activate and generate very strong negative thoughts and emotions favoring the emergence of erroneous ideas that take the form of delusions. These results have been praised by many researchers working on schizophrenia, who now base their work on the results obtained by Dr Morel and his collaborators.

  the characteristics of the Biogenetic, Cognitive Distortion, and Cognitive schema accounts.Insert Table 1 hereThe present research explored how lay people's beliefs regarding the causes of schizophrenia shape the stigmatization of people experiencing this disorder. To this end, we exposed a sample of French university students to three different etiological accounts of the pathology (Cognitive Distortion, Cognitive Schema, or Biogenetic explanation), and we measured the effect of this exposition on their level of expressed stigma (stigmatizing attitude, desire for social distance, and empathic concern), compared to a control group. We expected the largest decrease in stigma in participants exposed to the Cognitive schema account, a lower or no decrease in participants exposed to the Cognitive Distortion and Genetic accounts, and no decrease in the control group.

  were recruited from an undergraduate introductory psychology course from [omitted for masked review] University, and randomly assigned to one of the four conditions of our experimental manipulation (Cognitive Distortion, Cognitive Schema, Biogenetic, or Control condition). The participants completed the questionnaire in the lecture hall before lecture started. Thirty-seven participants did not complete the questionnaire and were excluded from the analyses, leaving a final sample of 378 participants (263 females, 2 undefined) aged between 16 and 58 (Mage = 19.90 years; SDage = 4.68 years), assigned to the following conditions: Cognitive Distortion condition (N = 99), Cognitive Schema condition (N = 106), Biogenetic condition (N = 107), or Control condition (N = 66).

  of the texts detailed a specific factor identified as the main cause for the development of schizophrenia: biogenetics, cognitive distortion, or cognitive schema. As it was the case in the vignettes designed by[START_REF] Botha | The influence of emphasizing psychological causes of depression on public stigma[END_REF], these three texts were presented in the form of a (fictitious) scientific article, recently published in the "Journal of Behavioral and Cognitive Therapy" (for the Cognitive Schema and Cognitive Distortion conditions) or the "Journal of Molecular Biology and Genetics" (for the Biogenetics condition) by a fictitious researcher, known worldwide and specialized in the field of schizophrenia. In the article, the researcher emphasized that all the data collected over the past thirty years indicate that schizophrenia is caused by factors related to biogenetics/cognitive distortion/cognitive schema (depending on the condition). All three texts had identical structures and similar contents, except for the target sentences regarding the cause of schizophrenia, that differed according to each condition (see Appendix A1 for the full version of the texts).

  towards people experiencing schizophrenia. In the next step, they were randomly assigned to one of the four conditions: Biogenetics, Cognitive Distortion, Cognitive Schema or Control, and read the vignette relevant to their condition. Finally, they completed for a second time the questionnaires about attitudes, empathic concern and social distance towards people experiencing schizophrenia.

  statistics). A main effect of Measure was observed, F(1.70, 608.56) = 4.83, p = .012,  𝑝 2 = .013. Most importantly, with regards to our current purpose, the Moment of measurement x Text condition interaction was significant, F(3, 358) = 4.42, p = .005,  𝑝 2 = .04. Before exploring the origin of this interaction with pairwise comparisons, and because Measure did not significantly interact with any of our independent variables, we computed a composite score of acceptance (by averaging the percentage scores of positive attitude, social proximity, and empathic concern). Pairwise comparisons using paired sample t-tests showed that only the Cognitive Schema condition yielded an increase in acceptance of schizophrenia between T0 (Macceptance = 57.52; SDacceptance = 15.02) and T1 (Macceptance = 60.22; SDacceptance = 15.75), t(105) = 4.41, p < .001, Cohen's d = 0.18. No significant difference was observed between T0 and T1 in the Cognitive Distortion, Biogenetics, or Control condition (all ts < .99, all ps > .32; see Table

Figure 1 .

 1 Figure 1. Differential scores (T1-T0) of positive attitude, empathic concern and social

Table 1 .

 1 Characteristics emphasized in the biogenetic, cognitive distortion, and cognitive schema accounts of Schizophrenia in terms of control of etiology and stability, and the resulting predictions formulated in the present study.

	Biogenetic	Cognitive	Cognitive
	account	Distortion	Schema
		account	account

Table 2 .

 2 Mean (and SD) of positive attitude, empathic concern and social proximity regarding schizophrenia at T0 and T1, as a function of text condition.

					Difference
	Text Condition	Measure	T0	T1
					(T1-T0)
	Cognitive Schema Empathic			3.93
			54.45 (20.68)	58.38 (21.43)
		Concern
		Positive Attitude	71.72 (14.44)	73.55 (13.87)	1.83

Appendix

A1. Short description of the daily life of an individual with schizophrenia

Please read carefully the text below. Later, we will ask you a few questions regarding the content of this text.

"Some people may be affected by a serious mental illness called schizophrenia. These people can go to university, work in the field of justice and live in a self-contained apartment. They do not always present symptoms, but when they do, they can sometimes hear voices, or think that the CIA is watching them. Aside from these specific symptoms of schizophrenia, they can also feel very anxious, which can cause them to miss work on some days. People with schizophrenia are often hospitalized and take medication for their illness."

Control condition

Please carefully read the article below. Later, we will ask you a few questions regarding the content of the article. This article was published in the August-28 edition of this year's "Journal of the American Academy of Quantum Physics". The "Journal of the American Academy of Quantum Physics" is a journal that covers the latest news about quantum physics. It is highly respected in its academic field.

"Fission is largely the result of electrostatic factors." This is the first conclusion of an article recently published in the Journal of the American Academy of Quantum Physics. The author of this article is Dr Morel Edouard from Paris-Sorbonne University, who is renowned worldwide for his expertise in quantum physics. Dr Morel and his collaborators have investigated the factors that may be involved in the appearance of antineutrons in the fission process. They concluded: "Taking into account all the data gathered over the past 30 years, we can conclude that fission is the phenomenon by which the nucleus of a heavy atom (nucleus that contains many nucleons, such as uranium and plutonium nuclei) is divided into several lighter nuclides, usually two nuclides. This reaction results in the emission of neutrons and antineutrons (usually two or three), and a very important release of energy." These results have been praised by many researchers working in quantum physics, more specifically on fission, who now base their work on the results obtained by Dr Morel and his collaborators."

Biogenetic condition

Please carefully read the article below. Later, we will ask you a few questions regarding the content of the article. This article was published in the August-28 edition of this year's "Journal of Molecular Biology and Genetics". The "Journal of Molecular Biology and Genetics" is a journal that covers the latest news about genetics and the improvement of the well-being of the individual. It is highly respected in its academic field.