Yves Elskens

Etienne Pardoux

DIFFUSION LIMIT FOR MANY PARTICLES IN A PERIODIC STOCHASTIC ACCELERATION FIELD

Keywords: AMS 2000 subject classifications: Primary 34F05, 60H10, 82C05, 82D10; secondary 60J70, 60K40 quasilinear diffusion, weak plasma turbulence, propagation of chaos, wave-particle interaction, stochastic acceleration, Fokker-Planck equation, Hamiltonian chaos Elskens_Pardoux_Nbody_aap.hyper32394.tex

The one-dimensional motion of any number N of particles in the field of many independent waves (with strong spatial correlation) is formulated as a second-order system of stochastic differential equations, driven by two Wiener processes. In the limit of vanishing particle mass m → 0, or equivalently of large noise intensity, we show that the momenta of all N particles converge weakly to N independent Brownian motions, and this convergence holds even if the noise is periodic. This justifies the usual application of the diffusion equation to a family of particles in a unique stochastic force field. The proof rests on the ergodic properties of the relative velocity of two particles in the scaling limit.

1. Introduction. The motion of a particle in the field of many waves [START_REF] Doveil | Diffusion of a test electron beam in a discrete spectrum of waves propagating along a traveling wave tube[END_REF][START_REF] Doveil | Wave-particle interaction and Hamiltonian dynamics investigated in a traveling wave tube[END_REF][START_REF] Tsunoda | Experimental test of quasilinear theory[END_REF] is a fundamental process in classical physics, the understanding of which is a prerequisite to the analysis of many plasma and fluid phenomena. In one space dimension, it can be described by the Hamiltonian model (1.1)

H = p 2 2m + M m=1
A m cos(k m qω m tϕ m )

where the particle with mass m has position q and momentum p, while the force field derives from a potential with time Fourier components A m e iϕm . The wave field comprises M waves, with a smooth dispersion relation associating a wavenumber k m , a pulsation ω m and a phase velocity v m = ω m /k m to each wave -usually determined by fixed properties of the environment, such as the geometry of the domain where waves propagate (then wavenumbers k m and pulsations ω m are discrete). The complex amplitudes A m e iϕm are more easily tuned by the experimenter or affected by simple changes in the environment. The dynamical systems approach to this problem discusses the particle motion after prescribing a single choice for each wave complex amplitude. As it would be quite exceptional to control all waves (though this is e.g. the assumption underlying the standard map, see [START_REF] Bénisti | Nonstandard diffusion properties of the standard map[END_REF] for a discussion), physicists often turn to a probabilistic description of the dynamics, considering an "ensemble" of realizations (A m , ϕ m ). Various arguments are then invoked to reduce the particle evolution equations to a stochastic differential equation, often driven by a "white noise". This results in somewhat tractable models (see e.g. [START_REF] Krommes | Fundamental statistical descriptions of plasma turbulence in magnetic fields[END_REF] about the validity of such derivations).

In this paper we focus on two issues. First, a random field characterized by (A m , ϕ m ) for a given dispersion relation with discrete frequency spectrum may be periodic in time : may the force on the particle be considered as independent over several time periods in a genuine limit ? Second, may one consider several particles subject to the same wavefield as independent in a genuine limit ? The latter issue underlies the frequent application of the Fokker-Planck equation to the evolution of a family of particles in a single turbulent wavefield -though a priori one can only grant that the diffusion equation describes the evolution of the distribution of a single particle for an ensemble of wavefield samples.

From a more general perspective, this work also relates to the issue of "propagation of chaos" in statistical physics [START_REF] Kac | Foundations of kinetic theory[END_REF][START_REF] Kac | Probability and related topics in physical sciences[END_REF], an aspect of Hilbert's 6th problem : how does chaotic dynamics enable a system, in which initial data are independent ("random") but the evolution may generate correlations, to behave as if the evolution regenerated independence ("randomness") or destroyed correlations ? Here, how do two Wiener processes, fully describing a prescribed "turbulent" environment, generate N independent Brownian motions for particles ?

A further motivation for the present work is that physics literature most often focuses on the evolution of particle distribution functions, e.g. by showing that they obey a Fokker-Planck equation, and on instantaneous observables such as p(t 1 ) for given t 1 (pointwise in time). However, the notion of a diffusion process implies rather a measure on the set of trajectories, viz. functions p(•) (globally in time). Here we shall show how our model implies that an arbitrary number of trajectories in a single realization of the dynamics do, jointly, admit the Wiener measure description.

In sections 2 and 3 we motivate the mathematical model more precisely and state our main results, which are proved in the subsequent sections. The crucial Theorem 3.2 is an ergodic theorem for a rescaled process, im- plying that a process V t , describing the relative velocity of one particle with respect to another one (or to its own earlier motion), converges weakly to a Brownian motion. The difficulty in proving the ergodic theorem is that the invariant measure of the diffusion process is infinite (it is the Lebesgue measure dxdy), and we must estimate a continuous additive functional generated by a function (namely f (x, y) = sin 2 x) which is not integrable with respect to this measure (it is only locally integrable). This weak convergence then implies the final many-particle result (Theorem 3.1) by a straightforward application of the Lévy characterization of Brownian motion. Section 8 outlines implications and possible extensions to this work.

2. Physical background. Because the waves have different frequencies and velocities, it is generally unrealistic to assume their phases to be correlated. Their intensities are more easily observed, but both in nature and in the laboratory the accumulation of statistical data on waves often involves only their average power spectra, not the detailed intensity data for each measurement run. We assume here that these complex amplitudes are random data, and investigate the statistics of the particle motion in the resulting time-dependent random field. This dynamics is a "stochastic acceleration problem" for a "passive particle" in weak plasma turbulence [START_REF] Andris | Stochastic particle acceleration and statistical closures[END_REF][START_REF] Doveil | Statistics of charged particles in external random longitudinal electric fields[END_REF][START_REF] Maasjost | Stochastic particle acceleration[END_REF][START_REF] Peter | Stochastic acceleration[END_REF][START_REF] Vanden | Some remarks on the quasilinear treatment of the stochastic acceleration problem[END_REF], and its understanding is a prerequisite to a proper analysis of the case where the particle motion feeds back on the wave evolution [START_REF] Doxas | Numerical observation of turbulence enhanced growth rates[END_REF][START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF].

The Hamiltonian (1.1) generates equations of motion q = p/m , (2.1)

ṗ = m k m A m sin(k m q -ω m t -ϕ m ) . (2.2)
An important observation [START_REF] Chirikov | A universal instability of many-dimensional oscillator systems[END_REF][START_REF] Escande | Stochasticity in classical Hamiltonian systems : universal aspects[END_REF] on the motion of a particle in the field (2.2) is locality in velocity : the evolution of the particle when it has velocity q = v depends only weakly on the waves with a Doppler-shifted frequency ω mk m v much larger than their trapping oscillation frequency k m A m /m. In particular, for a two-wave system the resonance overlap parameter

s 1,2 = 2 A 1 /m + 2 A 2 /m |ω 2 /k 2 -ω 1 /k 1 |
becomes unity when there exists a velocity

u = ω 2 /k 2 -2 A 2 /m = ω 1 /k 1 + 2 A 1 /m (with k 1 > 0, k 2 > 0, ω 2 > ω 1
). For many waves with overlap parameters s ≫ 1, the relevant phase velocity range for waves influencing the particle is a "resonance box", with a width scaling as (A/m) 2/3 . [BeEs97, BeEs98a] A good approximation to typical wave dispersion relations in the strong overlap limit, after a Galileo change of reference frame (see e.g. sec. 6.7 in [START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF]), is

(2.3) k m = k 0 , ω m = 2π(m -M/2)/T ,
for some k 0 , T . Then, in the limit M → ∞, the equations of motion yield for A m e iϕm = A 0 (with real A 0 ) the well-known standard map [START_REF] Bénisti | Nonstandard diffusion properties of the standard map[END_REF].

The case where phases ϕ m are independent random variables uniformly distributed on the circle [0, 2π], while A m = A 0 is given, was investigated notably by Cary, Escande, Verga and Bénisti [BeEs97, CEV90, ElEs03] and occurs in the context of the random phase approximation.

To the extent that the phases and amplitudes of the waves are independent random variables, the physicist usually views the force (2.2) as a mollification of a white noise, with amplitude

σ = E(k 2 m A 2 m ) = |k m | EA 2 m
, where the relevant mode m is the one nearest to the current particle velocity (the mathematical expectation E is called ensemble average with respect to wave amplitudes and phases). 1 This is the core of quasilinear theory [DrPi62, Pe94, RoFi61, VVS62]. With some care, one interprets (2.1)-(2.2) as a stochastic differential equation ; this applies in the case m ∈ Z for dispersion relation (2.3) with Gaussian independent complex amplitudes such that EA 2 m = k -2 0 σ 2 . The particle velocity then has a Brownian evolution, so that for t, t ′ ∈ [0, T ] (2.4)

E(p t -p 0 )(p t ′ -p 0 ) = D QL min(t, t ′ )
with the quasilinear diffusion coefficient D QL = σ 2 T /2. However, the particle evolution for t > T may show a strong correlation to its motion for 0 ≤ t ≤ T because the waves are periodic in time [START_REF] Elskens | Quasilinear limit for particle motion in a prescribed spectrum of random waves[END_REF][START_REF] Elskens | Nonquasilinear evolution of particle velocity in incoherent waves with random amplitudes[END_REF], and the dispersion relation (2.3) may generate a strong spatial correlation between the motions of two particles because all waves acting on a particle at any time have the same wavelength. 2

1 Phases do not appear in σ (nor in s) because the relative phase of two waves ϕm + ωmtϕnωnt varies uniformly over time (hence ϕmϕn can be absorbed in the choice of the time origin).

2 There is a large body of literature on the case of incoherent waves with no dispersion relation. Then the sum m becomes a double sum m,n and one varies wavenumbers kn independently from pulsations ωm. This space-time stochastic environment is more noisy than our model and may also be considered to motivate a quasilinear approximation.

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Set k 0 = 1 and T = 2π by the choice of space and time units. The particle phase space is T × R, where T is the circle modulo 2π, and the particle equation of motion reads, with initial data (q(0), q(0)) = (q 0 , q0 ), (2.5)

q = m -1 µ ′ -1 m=-µ A m sin(q -mt + ϕ m ) = m -1 m A m cos(mt -ϕ m ) sin q -m -1 m A m sin(mt -ϕ m ) cos q
in a galilean frame moving at a velocity inside the spectrum of wave phase velocities (µ + µ ′ = M). We assume µ ≫ 1, µ ′ ≫ 1. For A m = A 0 given and independent random phases (uniform on the circle), in the limit A 0 /m → ∞, with (m/A 0 ) 2/3 M 10, Bénisti and Escande [START_REF] Bénisti | Origin of diffusion in Hamiltonian dynamics[END_REF][START_REF] Bénisti | Finite range of large perturbations in Hamiltonian dynamics[END_REF] have shown that the particle momentum p = m q follows essentially a Brownian motion, with diffusion constant given by the quasilinear estimate

D QL = πA 2 0
as long as the motion does not approach the boundaries of the wave velocity spectrum. Then the particle momenta p for an ensemble of independent realizations of the system will be described by a distribution function verifying the Fokker-Planck equation

(2.6) ∂ t f = D QL 2 ∂ 2 p f
even for t > T . Numerical simulations [START_REF] Elskens | Nonquasilinear evolution of particle velocity in incoherent waves with random amplitudes[END_REF] show similar behaviour for i.i.d. complex Gaussian random variables A m e iϕm . The present work establishes a rigorous version of this result in the frame of stochastic processes.

3. Main results. We first let min(µ, µ ′ ) → ∞ in the model, taking A m cos ϕ m , A m sin ϕ m as i.i.d. Gaussian random variables with zero expectation and E(A m cos ϕ m ) 2 = E(A m sin ϕ m ) 2 = 1/2 for all m ∈ Z. In particular this implies that phases ϕ m are i.i.d. uniformly on T. To follow earlier practice, we now set m = 1/A. Then formally (2.5) becomes the Stratonovich stochastic differential equation for 0 Note that the Stratonovich and Itô integrals define the same solutions for this system, and that the vector fields (sin q)∂ p and (cos q)∂ p commute. For 0 ≤ t ≤ 2π, it is clear that P is a Brownian motion for any value of A ≥ 0 (see the first lines of Sec. 7). However, the model (2.5) defines a dynamical system for all times 0 ≤ t < ∞, and one may wonder how the initial stochastic behaviour over [0, 2π] extends for longer times. Formally, one solves then (3.1)-(3.2) with the periodized field, i.e. with the continuous processes defined by dC t+2kπ = dC t , dS t+2kπ = dS t (3.4) for k ∈ Z. In other words, C t -t 2π C 2π and S t -t 2π S 2π are independent Brownian bridges repeated periodically for t ∈ R, while C 2π and S 2π are independent Gaussian random variables with expectation 0 and variance 2π 2 .

≤ t ≤ 2π dQ t = AP t dt , (3.1) dP t = sin(Q t ) • dC t + cos(Q t ) • dS t , (3.2) with initial data Q 0 = q 0 , P 0 = p 0 = m q0 = q0 /A ; here, from [Ka85], π -1/2 (C, S) is a standard 2-dimensional
For this extended process, the wave field acting on the particle for t ∈ [0, 2π] is not stochastically independent from the wave field acting during [0, 2π]. Therefore one does not expect the particle momentum to proceed as a Brownian motion for all times, and indeed for A small enough the velocity q may remain bounded in a narrow interval for all times. This is easily seen numerically and can be attributed to the existence of Kolmogorov-Arnol'd-Moser invariant tori in the 3-dimensional extended phase space with coordinates (t, q, q). On the other hand, for large A, the dynamics viewpoint [BeEs97, BeEs98a, ElEs03] suggests that the nonlinearity in the equations of motion (due to trigonometric functions of Q) may enable a decorrelation of the force over the period T = 2π, so that the long-time evolution of the velocity would also be close to Brownian. This is what we shall show.

An intimately related issue is the relative motion of several particles, released in the same realization of the wave field. Even though each particle velocity diffuses for t ∈ [0, 2π], their motions are not independent. We shall also show that for large A the motions of any finite family of particles released at initial data (Q

(ν) 0 , P (ν) 0 ), 1 ≤ ν ≤ N ,
approaches a family of N independent processes. This can also be expected from the consideration of the top Lyapunov exponent of the dynamics (3.1)-(3.2) in the limit A → ∞.

Theorem 3.1. For any N > 0, the momentum processes P (ν) defined by with N different initial data (q (ν) 0 , p (ν) 0 ) ∈ T×R, 1 ≤ ν ≤ N , converge as A → ∞ to N independent Wiener processes with variance πt, and convergence is in law in C(R + , R N ).

dQ (ν) t = AP (ν) t dt , Q (ν) 0 = q (ν) 0 , (3.5) dP (ν) t = (sin Q (ν) t )dC t + (cos Q (ν) t )dS t , P (ν) 0 = p (ν) 0 , (3.6)
The key argument in the proof is the following weak convergence theorem, where we write now n = π 1/2 A. Consider the two-dimensional diffusion process indexed by n ≥ 1, solution of the SDE on R 2 (3.7)

     dU n t dt = nV n t , U 0 = u , dV n t = sin(U n t )dW t , V 0 = v , where (u, v) ∈ {(kπ, 0), k ∈ Z} and W is a standard Brownian motion. We prove Theorem 3.2. As n → ∞, V n ⇒ v + 1 √ 2 B ,
where {B t , t ≥ 0} is a standard one-dimensional Brownian motion, and the convergence is in law in C(R + , R).

4.

A change of time scale. Note that for any n ≥ 1, the law of {(U n t , V n t ), t ≥ 0}, the solution of (3.7), is characterized by the statement

       dU n t dt = nV n t , U 0 = u , V n is a martingale, d V n t dt = sin 2 (U n t ) , V n 0 = v .

Now define (like Bénisti and Escande

[BeEs97, BeEs98a]) (4.1) X t = U n n -2/3 t , Y t = n 1/3 V n n -2/3 t . We first note that X 0 = u, Y 0 = n 1/3 v, Y is a martingale, and        dX t dt = n -2/3 dU n dt (n -2/3 t) = n 1/3 V n n -2/3 t = Y t , Y t = n 2/3 V n n -2/3 t , d Y t dt = sin 2 (X t ) .
Using a well-known martingale representation theorem, we can pretend that there exists a standard Brownian motion {B t , t ≥ 0} such that (4.2)

     dX t dt = Y t , X 0 = u , dY t = sin(X t )dB t , Y 0 = n 1/3 v .
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Note that the process {(X t , Y t ), t ≥ 0} still depends upon n, but only through the value of Y 0 . On the other hand, V n t = n -1/3 Y n 2/3 t . Hence

V n t = v + n -1/3 n 2/3 t 0 sin(X s )dB s ,
in other words V n is a martingale such that V n 0 = v and

V n t = n -2/3 n 2/3 t 0 sin 2 (X n s )ds .
Here we recall the fact that the process X depends upon n (through the initial condition of Y ), unless v = 0. Consequently

(4.3) lim n→∞ V n t = t × lim n→∞ 1 n 2/3 t n 2/3 t 0 sin 2 (X n s )ds
and in order to prove Theorem 3.2 it suffices to show that the above limit is t/2.

5. Qualitative properties of the solution of (4.2). We now consider the two-dimensional diffusion process (5.1)

     dX t dt = Y t , X 0 = x , dY t = sin(X t )dB t , Y 0 = y ,
with values in the state-space E = [0, 2π) × R\{(0, 0), (π, 0)}, where 2π is identified with 0. We first prove that the process {(X t , Y t ), t ≥ 0} is a conservative E-valued diffusion. Indeed, Proposition 5.1. Whenever the initial condition (x, y) belongs to E, inf{t > 0, (X t , Y t ) ∈ {(0, 0), (π, 0)}} = +∞ a.s.

Proof: We define the stopping time

τ = inf{t, (X t , Y t ) = (0, 0)} . Let R t = X 2 t + Y 2 t , Z t = log R t , t ≥ 0. A priori, Z t takes its values in [-∞, +∞). Itô calculus on the interval [0, τ ) yields dX 2 t = 2X t Y t dt , dY 2 t = 2Y t sin(X t )dB t + sin 2 (X t )dt , dZ t = dR t R t - d R t 2R 2 t = 2Y t X t + sin 2 (X t ) R t dt -2 Y 2 t sin 2 (X t ) R 2 t dt + 2 Y t sin(X t ) R t dB t .
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Now clearly | sin(x)| ≤ |x|, sin 2 (x) ≤ x 2 , and it follows from the above and standard inequalities that on the time interval [0, τ ),

Z t ≥ Z 0 -2t + t 0 ϕ s dB s ,
where |ϕ s | ≤ 1. Hence the process {Z t , t ≥ 0} is bounded from below on any finite time interval, which implies that τ = +∞ a.s., since τ = inf{t, Z t = -∞}. A similar argument shows that τ ′ = +∞ a.s., where

τ ′ = inf{t, (X t , Y t ) ∈ {(0, 0), (π, 0)}} .
We next prove (here and below B E stands for the σ-algebra of Borel subsets of E) the Proposition 5.2. The transition probabilities {p((x, y); t, A)

:= P x,y ((X t , Y t ) ∈ A), (x, y) ∈ E, t > 0, A ∈ B E }
have smooth densities p((x, y); t, (x ′ , y ′ )) with respect to Lebesgue's measure dx ′ dy ′ on E.

Proof: Consider the Lie algebra of vector fields on E generated by X 1 = sin(x) ∂ ∂y , X 2 = [X 0 , X 1 ] and X 3 = [X 0 , [X 0 , X 1 ]], where X 0 = y ∂ ∂x . This Lie algebra has rank 2 at each point of E. The result is now a standard consequence of the well-known Malliavin calculus, see e.g. Nualart [START_REF] Nualart | The Malliavin calculus and related topics[END_REF].

Proposition 5.3. The E-valued diffusion process {(X t , Y t ), t ≥ 0} is topologically irreducible, in the sense that for all (x, y) ∈ E, t > 0, A ∈ B E with non empty interior,

P x,y ((X t , Y t ) ∈ A) > 0 .
Proof: From Stroock-Varadhan's support theorem, see e.g. Ikeda-Watanabe [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF], the support of the law of (X t , Y t ) starting from (X 0 , Y 0 ) = (x, y) is the closure of the set of points which the following controlled ordinary differential equation can reach at time t by varying the control {u(s), 0 ≤ s ≤ t} It is not hard to show that the set of accessible points at time t > 0 by the solution of (5.2) is dense in E. The result now follows from the fact that the transition probability is absolutely continuous with respect to Lebesgue's measure, see Proposition 5.2.

We next prove the Lemma 5.4.

P (|Y t | → ∞, as t → ∞) = 0 .
Proof: The Lemma follows readily from the fact that

Y t = W t 0 sin 2 (X s )ds ,
where {W (t), t ≥ 0} is a scalar Brownian motion. Then either t 0 sin 2 (X s )ds is bounded and Y t is finite, or the integral diverges and Y t is finite anyway because W is recurrent.

Hence the topologically irreducible E-valued Feller process {(X t , Y t ), t ≥ 0} is recurrent. Its unique (up to a multiplicative constant) invariant measure is the Lebesgue measure on E, so that in particular the process is nullrecurrent. It then follows from (ii) in Theorem 20.21 from Kallenberg [START_REF] Kallenberg | Foundations of modern probability[END_REF] Lemma 5.5. For all M > 0, as t → ∞,

1 t t 0 1 {|Ys|≤M } ds → 0 a.s.
6. A path decomposition of the process {(X t , Y t ), t ≥ 0}. We first define two sequences of stopping times. Let T 0 = 0 and for ℓ odd, Let now τ 0 = T 1 . We next define recursively {τ k , k ≥ 1} as follows. Given τ k-1 , we first define

T ℓ = inf{t > T ℓ-1 , |Y t | ≥ M + 1} , for ℓ even, T ℓ = inf{t > T ℓ-1 , |Y t | ≤ M } .
L k = sup{ℓ ≥ 0, τ k-1 ≥ T 2ℓ+1 } . Now let η k = τ k-1 if τ k-1 < T 2L k +2 , T 2L k +3 if τ k-1 ≥ T 2L k +2 .
We now define

τ k = inf{t > η k , |X t -X η k | = 2π} ∧ inf{t > η k , |Y t -Y η k | > 1} .
It follows from the above definitions that t 0

1 {|Ys|≥M +1} sin 2 (X s )ds ≤ ∞ k=1 τ k ∧t η k ∧t sin 2 (X s )ds ≤ t 0 sin 2 (X s )ds ,
a statement which will be refined in the proof of Proposition 6.3. Define

K 0 = {k ≥ 1, |Y τ k -Y η k | < 1} , K 1 = {k ≥ 1, |Y τ k -Y η k | = 1} , K t = {k ≥ 1, η k < t} , K 0 t = K 0 ∩ K t , K 1 t = K 1 ∩ K t .
We first prove the Lemma 6.1. 1 t

k∈K 1 t (τ k -η k ) → 0 in L 1 (Ω) as M → ∞, uniformly in t > 0.
Proof: We shall use repeatedly the fact that since

|Y η k | ≥ M > 2, |Y η k |-1 ≥ |Y η k |/2. We have that (see the Appendix below), since τ k -η k ≤ 4π/|Y η k |, P(k ∈ K 1 |F η k ) ≤ P sup η k ≤t≤τ k |Y t -Y η k | ≥ 1 F η k ≤ 2 exp(-|Y η k |/(8π)) .
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Consequently, using again the inequality τ kη k ≤ 4π/|Y η k |, we deduce that

E (τ k -η k )1 {k∈K 1 } |F η k ≤ 8π |Y η k | exp(-|Y η k |/(8π)) ≤ 8π |Y η k | exp(-M/(8π)) .
On the other hand, whenever k ∈ K 0 ,

τ k -η k ≥ 2π/(|Y η k | + 1) ≥ π/|Y η k | . Now, provided t ≥ 4π/M , 2t ≥ t + 4π M ≥ E   k∈K 0 t (τ k -η k )   ≥ πE   k∈Kt 1 {k∈K 0 } 1 |Y η k |   ≥ π 2 E   k∈Kt 1 |Y η k |   , since P(k ∈ K 0 |F η k ) = 1 -P(k ∈ K 1 |F η k ) ≥ 1 -2 exp(-M/(8π)) ≥ 1/2 , provided M is large enough. Finally 1 t E   k∈K 1 t (τ k -η k )   ≤ 32 exp(-M/(8π)) E k∈Kt |Y η k | -1 E k∈Kt |Y η k | -1 = 32 exp(-M/(8π)) → 0 , as M → ∞, uniformly in t.
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Now, for any k ∈ K 0 ,

τ k η k sin 2 (X s )ds = τ k -η k 2π 2π 0 sin 2 (x)dx + τ k η k sin 2 (X s ) 1 - Y s (τ k -η k ) 2π
ds , (6.1) and we have

τ k η k sin 2 (X s ) 1 - Y s (τ k -η k ) 2π ds = τ k η k τ k η k sin 2 (X s ) Y r -Y s 2π drds ≤ 1 2π τ k η k τ k η k |Y r -Y s |drds .
Finally we have the Lemma 6.2. Uniformly in t > 0,

k∈K 0 t τ k η k τ k η k |Y r -Y s |drds k∈K 0 t (τ k -η k ) → 0 a.s., as M → ∞. Proof: Since |Y t -Y η k | ≤ 1 for η k ≤ t ≤ τ k , k∈K 0 t τ k η k τ k η k |Y r -Y s |drds k∈K 0 t (τ k -η k ) ≤ 2 sup k∈K 0 t (τ k -η k ) ≤ 8π/M → 0 ,
as M → ∞, uniformly in t.

We are now in a position to prove the following ergodic type theorem, from which Theorem 3.2 will follow :

Proposition 6.3. As t → ∞, 1 t t 0 sin 2 (X s )ds → 1 2
in probability.

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Proof: We first note that

[0, t] = B 0 t ∪ B 1 t ∪ C t ,
where

B 0 t = [0, t] ∩ ∪ k∈K 0 t [η k , τ k ] , B 1 t = [0, t] ∩ ∪ k∈K 1 t [η k , τ k ] , C t = [0, t]\(B 0 t ∪ B 1 t ) .
We have

1 t t 0 sin 2 (X s )ds = 1 t t 0 1 B 0 t (s) sin 2 (X s )ds + 1 t t 0 1 B 1 t (s) sin 2 (X s )ds + 1 t t 0 1 Ct (s) sin 2 (X s )ds . Now C t ⊂ {s ∈ [0, t], |Y s | ≤ M + 1}
, so for each fixed M > 0, it follows from Lemma 5.5 that the last term can be made arbitrarily small, by choosing t large enough. The second term goes to zero as M → ∞, uniformly in t, from Lemma 6.1. Finally the first term equals the searched limit, plus an error term which goes to 0 as M → ∞, uniformly in t, see Lemma 6.2 and the following fact, which follows from the combination of Lemma 6.1 and Lemma 5.5 : 1 t

k∈K 0 t (τ k -η k ) → 1 in probability, as n → ∞.
We can finally proceed with the Proof of Theorem 3.2 All we have to show is that (see (4.3))

lim n→∞ 1 n 2/3 t n 2/3 t 0 sin 2 (X n s )ds = 1 2π 2π 0 sin 2 (x ′ )dx ′ = 1 2
in probability. In the case v = 0, the process {(X n t , Y n t )} does not depend upon n, and the result follows precisely from Proposition 6.3. Now suppose that v = 0. In that case, the result can be reformulated equivalently as follows. For some x ∈ R, y = 0, each t > 0, define the process {(X t s , Y Remark 6.4. This proof holds uniformly with respect to initial data (u, v) satisfying |v| ≥ a for any a > 0.

7. Proof of Theorem 3.1. Our Theorem 3.1 now appears as a simple corollary of Theorem 3.2. Proof: We first prove the Theorem for t ∈ [0, 2π]. Then the vector P = (P (1) , . . . , P (N ) ) is a martingale in R N , and to prove our claim it suffices to show that its quadratic variation matrix converges to πt times the identity matrix. The diagonal elements of the matrix are

P (ν) t = t 0 (sin 2 Q (ν) s + cos 2 Q (ν) s )πds = πt
and we only need to compute the cross-variation

P (ν) , P (ν ′ ) t = t 0 (sin Q (ν) s sin Q (ν ′ ) s + cos Q (ν) s cos Q (ν ′ ) s )πds = t 0 cos(Q (ν) s -Q (ν ′ ) s )πds . (7.1) Now, define (with n = π 1/2 A) U n t = 1 2 (Q (ν) t -Q (ν ′ ) t ) , V n t = n -1 dU n t dt , U ′n t = 1 2 (Q (ν) t + Q (ν ′ ) t ) , V ′n t = n -1 dU ′n t dt .
These processes solve the stochastic differential equation

dU n t = nV n t dt , U n 0 = q (ν) 0 -q (ν ′ ) 0 2 , (7.2) V n 0 = p (ν) 0 -p (ν ′ ) 0 2π 1/2 , (7.3) dV n t = 1 2 √ π sin(U ′n t + U n t ) -sin(U ′n t -U n t ) dC t + 1 2 √ π cos(U ′n t + U n t ) -cos(U ′n t -U n t ) dS t = sin U n t dW n t (7.4)
where the process W n is the martingale defined by W n 0 = 0 and

dW n t = π -1/2 (cos U ′n t dC t -sin U ′n t dS t ) .
The quadratic variation of W n is

W n t = π -1 t 0 (cos 2 U ′n s + sin 2 U ′n s ) π ds = t
in view of the quadratic and cross variations (3.3) of (C, S), and this result does not depend on the process U ′n (which follows the center of mass of the two particles ν and ν ′ ). Thus W n is a standard Wiener process, and (U n , V n ) defined by (7.2)-(7.3)-(7.4) satisfies the hypotheses of Theorem 3.2. Hence, with X defined by (4.1),

P (ν) , P (ν ′ ) t = t 0 cos(2U n s )πds = t 0 (1 -2 sin 2 U n s )πds = πt 1 - 2 n 2/3 t n 2/3 t 0 sin 2 X n s ′ ds ′
which converges in probability to 0 for n → ∞ as shown in the proof of the theorem. Now we consider the process over the interval [0, 4π], taking into account that (C, S) over the whole interval is neither a martingale nor Markov. From the given initial data, amplitude A and realization of (C, S), we define a subsidiary set of particles N + 1 ≤ ν ≤ 2N , with initial data

Q (ν) 0 = Q (ν-N ) 2π , P (ν) 0 = P (ν-N ) 2π
, of which a.s. none coincides (modulo 2π for q) with any of the initial data (Q (ν) 0 , P (ν) 0 ). Recalling that at t = 2π the law of any P is Gaussian with variance 2π 2 (so that its probability density is bounded by 1/ √ 2π 2π 2 < 1/(2π)), this ensures that, given ǫ > 0,

P min 1≤ν,ν ′ ≤N |P (ν) 0 -P (ν ′ ) 2π | < ǫ or min 1≤ν<ν ′ ≤N |P (ν) 2π -P (ν ′ ) 2π | < ǫ ≤ 3ǫN 2 √ 2π 2π 2 < ǫN 2 /2 .
For processes such that min ν,ν ′ |P (ν)

0 -P (ν ′ ) 2π | ≥ ǫ and min ν<ν ′ |P (ν) 2π -P (ν ′ )
2π | ≥ ǫ, the above proof holds (uniformly with respect to initial data) for the full set of 2N particles defined here. It follows that a.s. the P (ν)

t -P (ν) 0 , t ∈ [0, 2π],
1 ≤ ν ≤ 2N are mutually independent Wiener processes, each with variance πt. This implies that P (ν) t -P (ν) 0 , t ∈ [0, 4π], 1 ≤ ν ≤ N are also mutually independent Wiener processes.

For any interval [0, 2kπ] with k ∈ N 0 the same argument reduces the N -particle problem to kN particles over [0, 2π], and the proof is complete.

Remark 7.1. Our theorem allows that P

(ν) 0 = P (ν ′ ) 0 for some ν = ν ′ with 1 ≤ ν, ν ′ ≤ N .
8. Perspectives. The implications of Theorem 3.1 are twofold. First, for N = 1, they support the observation [BeEs97, BeEs98a, CEV90, El08] that, in systems with finite M ≫ 1 and A 0 /m ≫ 1 (with appropriate scaling), the long-time behaviour of a single particle in a periodic wave field exhibits statistical properties approaching those of the Brownian motion. To complete the connection with physics literature, one must now discuss how the finite sums in (2.2) approach the right hand side of (3.2), and how this implies that the solutions of the first equation approach the solutions of the latter equation [START_REF] Héctor | On the gap between deterministic and stochastic ordinary differential equations[END_REF]. This will be discussed separately.

Second, and this is conceptually more fundamental, for a single sample of the Wiener wavefield (with M = ∞ formally), with strong spatial correlations (thanks to the single wavevector k 0 in the model), the limit A 0 /m → ∞ leads to independence of the evolutions of all N particles, which can then be collectively described by the diffusion equation. While such an independence is often admitted without proof in physics practice, our work provides an explicit justification to it. We even prove a little more than usual one-or two-time statements, as in our limit p (N ) is independent in law from even the full evolution data {p (ν) (t), 0 ≤ t ≤ T, 1 ≤ ν ≤ N -1}. This second implication is an important issue, as the acceleration of a passive particle is a Hamiltonian process while the diffusion equation is irreversible. While the key to this irreversibility is clearly the fact that the diffusion process only relates to the momentum component of particle evolution, we shall further investigate the interplay of limits N → ∞ and A → ∞, or m → 0, with stochasticity versus Hamiltonian "conservativeness" in (Q, P ) variables in future work.

Finally, we followed general practice in discussing the stochastic acceleration problem in only one space dimension [START_REF] Chirikov | A universal instability of many-dimensional oscillator systems[END_REF][START_REF] Hazeltine | The framework of plasma physics[END_REF]. This is rather classical, and it applies e.g. to particle motion along magnetic field lines in strongly magnetized plasmas ; higher-dimensional motions may call for different elementary models.

APPENDIX A

For the convenience of the reader, we prove the following well-known result (see e.g. ex. IV.3.16 in [START_REF] Revuz | Continuous martingales and Brownian motion, 3rd printing of the 3rd edition[END_REF]) Proposition A.1. Let η and τ be two stopping times such that 0 ≤ η ≤ τ ≤ η + T and M t = t 0 ϕ s dB s , where {B t , t ≥ 0} is a standard Brownian motion and {ϕ t , t ≥ 0} is progressively measurable and satisfies We estimate the first term on the right. The second one is bounded by the same quantity. Define for all λ > 0

M λ t =

  Brownian motion. In other words, C and S are martingales, with C 0 = S 0 = 0 and (3.3) C t = S t = πt , C, S t = 0 . imsart-aap ver. 2007/12/10 file: Elskens_Pardoux_Nbody_aap.hyper32394.tex date: November 5, 2008

  imsart-aap ver. 2007/12/10 file: Elskens_Pardoux_Nbody_aap.hyper32394.tex date: November 5, 2008 in the class of piecewise continuous functions : = y(s) , x(0) = x ; dy ds (s) = sin(x(s))u(s) , y(0) = y .

  imsart-aap ver. 2007/12/10 file: Elskens_Pardoux_Nbody_aap.hyper32394.tex date: November 5, 2008

  1 ≤ ν ≤ N ) imsart-aap ver. 2007/12/10 file: Elskens_Pardoux_Nbody_aap.hyper32394.tex date: November 5, 2008

t 0 ϕ 2

 2 s ds ≤ k 2 t for all t ≥ 0. Then for all b > 0,P sup η≤t≤τ |M t -M η | ≥ b ≤ 2 exp -b 2 2k 2 T .Proof: From the optional stopping theorem, it suffices to treat the case η = 0, τ = T . We haveP( sup 0≤t≤T |M t | ≥ b) ≤ P( sup 0≤t≤T M t ≥ b) + P( inf 0≤t≤T M t ≤ -b) .

  The rest of the argument leading to Proposition 6.3 is based upon limits as M → ∞, uniformly with respect to t. It thus remains to check that the fact that Y t 0 now depends upon t does not spoil this uniformity, which is rather obvious.

	We need to show that
	1 t	0	t	sin 2 (X t s )ds →	1 2π	0	2π	sin 2 (x ′ )dx ′
	in probability, as t → ∞. Note that in time t, the process Y t starting from √ ty can come back near the origin.
	It is easily seen, by introducing the Markov time τ t M = inf{s > 0, |Y t s | ≤ M } and exploiting the strong Markov property, that
				1 t	0	t	1 {|Y t s |≤M } ds → 0 a.s.
	follows readily from Lemma 5.5.
	s ≤ t} as the solution of the SDE      dX t s ds = Y t s , X t 0 = x , dY t s = sin(X t s ) dW s , Y t 0 =	√	t y .	t s ), 0 ≤
	imsart-aap ver. 2007/12/10 file: Elskens_Pardoux_Nbody_aap.hyper32394.tex date: November 5, 2008

  exp λM tt ≥ exp(λbλ 2 k 2 T /2) ≤ exp(λ 2 k 2 T /2λb) , imsart-aap ver. 2007/12/10 file: Elskens_Pardoux_Nbody_aap.hyper32394.tex date: November 5, 2008 from Doob's inequality, since {M λ t , t ≥ 0} is a martingale with mean one. Optimizing the value of λ, we deduce that

	P sup 0≤t≤T	M t ≥ b ≤ exp -	b 2 2k 2 T	,
	from which the result follows.
		λ 2 2	0	t	ϕ 2 s ds .
	Then
	P( sup

0≤t≤T M t ≥ b) ≤ P sup 0≤t≤T M λ

imsart-aap ver. 2007/12/10 file: Elskens_Pardoux_Nbody_aap.hyper32394.tex date: November 5, 2008

ACKNOWLEDGEMENTS

This work stems from many stimulating discussions with Dominique Escande, Fabrice Doveil and members of the Turbulence plasma team. Comments from Claude Bardos, Nils Berglund and Cédric Villani were useful. Jonathan Mattingly was helpful in providing the reference for Lemma 5.5. YE was partly supported by CNRS through a delegation position.


domain: spi phys

text:
The onset of pitch-oscillations of a NACA0012 airfoil mounted on a torsion-spring is investigated by the means of a global stability analysis. Experimental studies from literature have shown this configuration undergoing on a flutter behaviour for a close range of Reynolds. This phenomenon is attributed to the laminar character of the boundary layer and to its detachment, transition and subsequent reattachment on the airfoil surface. In this regard, we carried out a linear stability analysis of the mean flow field, issued from a RANS simulation, using a Spalart-Allmaras model. A careful choice of the inlet conditions revealed to be necessary to capture a laminar separation and, as the incidence increases, a laminar separation bubble (LSB).

INTRODUCTION

The first investigation of the aeroelastic behaviour of a NACA0012 airfoil was conducted by [START_REF] Poirel | Self-sustained aeroelastic oscillations of a NACA0012 airfoil at low-to-modelate Reynolds numbers[END_REF], with an experimental set up on a Reynolds number range of 4.5 × 10 4 < Re < 1.35 × 10 5 . The airfoil was free to rotate in the pitch degree of freedom, presenting a self-sustained behaviour around a mean angle of attack of zero degrees and an amplitude around 5.5 • . The primary driver for this behaviour was associated to the laminar boundary layer separation on the second half of the airfoil surface, with a subsequent flow transition to a turbulent state an reattachment, forming a Laminar Separation Bubble (LSB).

Subsequent works have reproduced the same phenomenon, performing large-eddy simulations of forced pitch oscillations [START_REF] Poirel | Aeodynamics of laminar separation flutter at transitional Reynolds number[END_REF] or numerical time-marching fluid-structure-interaction simulations using turbulence modelling for the Navier-Stokes equations [START_REF] Poirel | Computational aeroelastic simulations of self-sustained pitch oscillations of a NACA0012 at transitional Reynolds numbers[END_REF], to name a few. In continuation of these works, [START_REF] Poirel | Experimental small-amplitude self-sustained pitch-heave oscillations at transitional Reynolds numbers[END_REF] performed experiments in the same transitional Reynolds range, but with both pitch and heave degrees of freedom, obtaining both small and large amplitude oscillations.

More recently, an aeroelastic linear stability using a mean flow from 3D DNS simulations was investigated by Negi (2019), showing a static eigenvalue that indicates a divergent behaviour of the airfoil from its equilibrium position at α = 0 • . Further, they associated the subsequent non-linear oscillation to a sub-critical mode, which is non-linearly excited.

The present study aims to understanding of the laminar separation flutter, by highlighting this divergent onset that leads the airfoil to a new equilibrium point where an unstable eigenvalue is then responsible for the non-linear onset dynamics.

MEAN FLOW ANALYSIS

We consider a flow described by the incompressible Reynolds-Averaged Navier-Stokes equations (RANS) with a closure obtained from the Spalart-Allmaras model. This model was intended to be used in a fully turbulent configuration where no laminar separation was able to appear. This turbulent behaviour is commanded by the inlet condition on the eddy viscosity which is recommended to be ν ∞ t = 3ν. Re = 50 000 The key ingredient of this procedure was the use of very small turbulent variable free-stream conditions, ideally at ν ∞ t = 0. Following this idea, we carried out several RANS simulations for different airfoil incidences. The evolution of the mean flow moment coefficient is depicted in figure 1, for two different values of ν ∞ t . It can be observed that the nonlinear behaviour of the flow field at low incidences cannot be captured by a fully turbulent simulation (ν ∞ t = 3ν), whereas a laminar approach was able to capture it, when comparing with the experimental data from [START_REF] Ohtake | Nonlinearity of the Aerodynamic Characteristics of NACA0012 Aerofoil at Low Reynolds Numbers[END_REF]. The comparison between the flow fields is depicted at the left of figure 1, where a recirculation bubble can be observed for the laminar separated case.

SA ν ∞ t = 0 SA ν ∞ t = 3ν Ohtake (2007) ν ∞ t = 3ν: ν ∞ t = 0:

FLUID STRUCTURE LINEAR STABILITY ANALYSIS

In a second time, a global stability analysis was carried out at α = 0 • for the case where ν ∞ t = 0. The FSI stability analysis revealed two additional eigenvalues, associated to the pitching degree of freedom, that are not present when studying a pure hydrodynamic case. This comparison is detailed in figure 2, at left. The FSI stability analysis was carried out with a stiffness, damping and inertia coefficients of K s = 0, D s = 0.0 and I s = 0.00135, respectively. The mode 1 is associated to the divergent linear departure from the equilibrium position. The correspondent spatial structure is depicted at right of figure 2. The second mode, although stable, and can be responsible for the onset of the pitching oscillation, by a non-linear excitation for large enough oscillation amplitudes.

CONCLUSIONS

A global linear stability analysis is performed for a torsion-spring airfoil configuration at the Reynolds number of 50 000. The mean flow field is obtained solving the RANS equations coupled to the Spalart-Allmaras turbulence model, with ν ∞ t = 0. The stability analysis around α = 0 • shows a divergent eigenmode responsible for the departure of the airfoil from this equilibrium position. The subsequent the onset of the pitching oscillation could be explained by a non-linear excitation of a second structure mode.

Figure 1 :

 1 Figure 1: At right, the evolution of the mean flow moment coefficient with the incidence. At left, the streawise velocity for Re = 50 000, α = 0 • for ν ∞ t = 3ν and ν ∞ t = 0, showing a detached flow for the second set of parameters.* Corresponding author. E-mail: diogo.ferreirasabino@toulouse-inp.fr.

Figure 2 :

 2 Figure 2: At left, a comparison between the fluid and a fluid structure spectra, the latter presenting a static unstable eigenvalue. At right, the streamwise velocity field for the two FSI modes.


Yixuan Guo

Qian Zhang

Qianqing Hu

Weiliang Tian

Fabrice Leroux

Pinggui Tang

Dianqing Li

Yongjun Feng
email: yjfeng@mail.buct.edu.cn

Size-dependent Effect of MgAl-Layered Double Hydroxides Derived from Mg(OH) 2 on Thermal Stability of Poly(vinyl chloride)

Keywords: Layered double hydroxides, green route, poly(vinyl chloride) (PVC), particle size, thermal stabilizer

Developing green thermal stabilizer for poly(vinyl chloride) (PVC) today is a great challenge for materials as polymer filler. Here, the "salt-oxide/hydroxide" route in mild conditions is used to fabricate a series of Mg 2 Al-CO 3 -LDH samples from Mg(OH) 2 precursors with different average particle sizes from 202 ± 10 nm to 334 ± 13 nm. A linear correlation is observed for the lateral size of the platelets between Mg 2 Al-CO 3 -LDH samples and their associated Mg(OH) 2 precursors. After surface-organo-modification (SOM), organophilic Mg 2 Al-CO 3 -LDH samples are found to be highly dispersed into PVC and investigated as environmentfriendly thermal stabilizers. From the static/dynamic tests, the performances are strongly enhanced and related to the particle size of the LDH stabilizer, with the yellowish color aspect appearing later than for the commercial HT-3/PVC. Among the LDH series, the platelets with an average particle lateral size of about 220 ± 10 nm perform the best for the thermal stability for PVC polymer. Among the series, the corresponding PVC composite film presents comparatively the minimum color value in static/dynamic discoloration test, exhibiting the longer ignition time for proton initial release as well as the longer stability time in dehydrochlorination test. It underlines that the salt-oxide/hydroxide route is an efficient and environmentally friendly process in producing high-performance green LDH stabilizer for PVC.

Introduction

Poly(vinyl chloride) (PVC) is the third-most widely used polymer around the world, and largely applied in various fields such as construction, chemical, and packaging, electric and electronic products, and so forth [START_REF] Jia | Self-plasticization of PVC materials via chemical modification of mannich base of cardanol butyl ether[END_REF][START_REF] Chiellini | Perspectives on alternatives to phthalate plasticized poly(vinyl chloride) in medical devices applications[END_REF][START_REF] Ahmad | Perspectives on polyvinyl chloride and carbon nanofiller composite: a review[END_REF]. In addition, the statistics anticipate a global demand for PVC to reach 46.3 million tons in 2019 and to continuously increase [START_REF] Bahls | Polyvinyl chloride (PVC)[END_REF]. However, PVC resins have a poor thermal stability: this is mostly due to the fact that particularly, its processing is occurring at temperature higher than the dehydrochlorination and thermal degradation temperature thus making it highly fragile. Therefore, a thermal stabilizer is usually required to add during the processing of PVC to enhance its thermal stability as well as to retard its thermal degradation [START_REF] Yu | Thermal degradation of PVC: a review[END_REF]. Hitherto, various thermal stabilizers have been developed and generally used in the market, for instance, lead salts, metal soaps, organotin, and other organic composites [START_REF] Zhang | A novel La-containing additive for the long-term thermal stabilization of poly(vinyl chloride)[END_REF].

However, most of them are toxic towards human being in a certain degree. Indeed, the lead salts have been strictly limited by the European Union's RoHS and REACH directives. Usually, more than one organic stabilizer is needed to stabilize thermally PVC, moreover they are mostly hazardous chemicals. To challenge high-performance and environment-friendly thermal stabilizers for PVC, the layered double hydroxides (LDH) appear promising. Indeed, for the last forty years and since the first patent (US4299759) in the 1980's [START_REF] Miyata | Method for inhibiting the thermal or ultraviolet degradation of thermoplastic resin and thermoplastic resin composition having stability to thermal or ultraviolet degradation[END_REF], LDHtype materials have drawn more and more attention as a potential "green" thermal stabilizer for PVC [START_REF] Zhang | Performance of layered double hydroxides intercalated by a UV stabilizer in accelerated weathering and thermal stabilization of PVC[END_REF][9][START_REF] Wen | Zn-Al-La hydrotalcite-like compounds as heating stabilizer in PVC resin[END_REF][START_REF] Aisawaa | Preparation of dipentaerythritol-combined layered double hydroxide particle and its thermostabilizing effect for polyvinyl chloride[END_REF]. The thermal stability mechanism of LDHs is found to proceed in two ways: one is the anion-exchange reaction between Cl -in HCl from PVC and CO 3 2-into the LDH interlayer region, and the other is the possible neutralization reaction between HCl produced during PVC degradation and the alkaline LDH layers [START_REF] Wen | Green preparation and characterization of a novel heat stabilizer for poly(vinyl chloride)-hydrocalumites[END_REF]. The first is to trap the aggressive Cl -anions, LDH host is acting as a scavenger, while in the second LDH is dissolved under pH, thus buffering the medium. Most interestingly, LDH filler is acting under pH and chloride anions concentration stimuli, and silent otherwise. This aspect of a dormant load until reacting when necessary is reminiscent of LDH filler used in polymer coatings to inhibit corrosion phenomenon occurring onto metal substrate [START_REF] Stimpfling | Amino acid interleaved layered double hydroxides as promising hybrid materials for AA2024 corrosion inhibition[END_REF]. In decades, the effects of different Mg/Al ratios, intralayer cations and interlayer anions substitution of LDHs on the thermal stability property of PVC have been investigated in detail in the literature [9-11,14], but t it is here beyond the scope of the study.

Since the thermal stability performance of the LDH/PVC composite should also depend on the state of dispersion of the LDH particles within the composite, the focus is on their particle size and size distribution [START_REF] Taviot-Guého | Tailoring hybrid layered double hydroxides for the development of innovative applications[END_REF][START_REF] Schrijvers | Ex-ante life cycle assessment of polymer nanocomposites using organo-modified layered double hydroxides for potential application in agricultural films[END_REF][START_REF] Qiu | Synthesis and properties of polypropylene/layered double hydroxide nanocomposites with different LDHs particle sizes[END_REF][START_REF] Nagendra | Polypropylene/layered double hydroxide nanocomposites: influence of LDH intralayer metal constituents on the properties of polypropylene[END_REF][START_REF] Nagendra | Polypropylene/layered double hydroxide (LDH) nanocomposites: influence of LDH particle size on the crystallization behavior of polypropylene[END_REF][START_REF] Liao | Size effect of layered double hydroxide platelets on the crystallization behavior of isotactic polypropylene[END_REF]. Usually, LDH-type materials were prepared by the so-called coprecipitation from the reaction between metal salts in alkaline medium as reported in academy as well as in industry. However, in this route, huge amount of water is produced in the washing steps that makes irrelevant in terms of sustainable development at any large scale-up [START_REF] Theiss | Synthesis of layered double hydroxides containing Mg 2+ , Zn 2+ , Ca 2+ and Al 3+ layer cations by co-precipitation methods-a review[END_REF][START_REF] Okamoto | Factors affecting the crystal size of the MgAl-LDH (layered double hydroxide) prepared by using ammonia-releasing reagents[END_REF][START_REF] Gu | Hierarchical layered double hydroxide nanocomposites: structure, synthesis and applications[END_REF][START_REF] Oh | The effect of synthetic conditions on tailoring the size of hydrotalcite particles[END_REF][START_REF] Intasa-Ard | Mechanochemical methods for the preparation of intercalation compounds, from intercalation to the formation of layered double hydroxides[END_REF].

More recently, the so-called "green" synthesis route has been developed using mixture between metal oxide and hydroxide or salt-oxide and hydroxide to reduce drastically the amount of water used during the washing step [START_REF] Williams | A kinetic and mechanistic study into the formation of the Cu-Cr layered double hydroxide[END_REF][START_REF] Wang | Synthesis of Li/Al LDH using aluminum and LiOH[END_REF][START_REF] Shin | Phase transformation from brucite to highly crystalline layered double hydroxide through a combined dissolution-reprecipitation and substitution mechanism[END_REF][START_REF] Nishimura | Characterization, synthesis and catalysis of hydrotalcite-related materials for highly efficient materials transformations[END_REF][START_REF] Meng | Novel synthesis of layered double hydroxides (LDHs) from zinc hydroxide[END_REF]. However, the former is difficult to yield LDH platelets suitable as thermal stabilizer for transparent PVC due to their sizes too large beyond micrometer [START_REF] Labuschagné | Green synthesis of hydrotalcite from untreated magnesium oxide and aluminum hydroxide[END_REF], while the latter is not reported to fabricate LDH filler as PVC stabilizer so far [START_REF] Liao | Size effect of layered double hydroxide platelets on the crystallization behavior of isotactic polypropylene[END_REF].

Therefore, there is still a great challenge to fabricate high-performance and environment-friendly LDH thermal stabilizer for PVC composites.

In this work, a series of Mg 2 Al-CO 3 -LDH samples with different particle sizes was first prepared by the salt-hydroxide route using Mg(OH) 2 , Al(NO 3 ) 3 •9H 2 O, and light magnesium carbonate (LMC) as the feeding materials and by adjusting the particle size of Mg(OH) 2

precursor. In a second step, the LDH filler was organo-modified onto the platelets, leading to a surface compatibilized hybrid structure so called hereafter surface organo-modified LDH, SOM-LDH, towards polymer. Finally, the PVC/SOM-LDH composites were melt-blended in an open twin-wheel mill, and subsequently investigated for their thermal stability performance as a function of the LDH particle size. Finally, the precipitate was centrifuged and washed with water for 1 cycle. The obtained slurry was separated in two parts: one was for the preparation of the organic-modified Mg 2 Al-CO 3 -LDH; and the other was for further physical characterization. The LDH samples were noted as LDH from a to g, corresponding to Mg(OH) 2 precursor from 0 h to 16 h.

Experimental section

Organic modification of Mg 2 Al-CO 3 -LDH

In order to enhance the compatibility with the PVC, the LDH samples were further organicmodified by stearic acid as described in the literature [START_REF] Kotal | Polymer nanocomposites from modified clays: recent advances and challenges[END_REF][START_REF] Basu | Advances in layered double hydroxide (LDH)-based elastomer composites[END_REF][34]. For example, the LDH-b slurry (31.20 g, solid content = 18.7 %, 12 mmol) was dispersed in 500 mL of water (solid content < 2 %), and then transferred into a 1 L three-neck flask under mechanical stirring. Subsequently, 0.47 g (1.65 mmol) stearic acid was dissolved into 50 mL of hot water, and then placed into the above-mentioned flask when the reaction temperature reached 80 °C. The resulting slurry was reacted at 80 °C for 2 h. Finally, the suspension was centrifuged and dried at 60 °C overnight leading to the surface organic-modified LDH (SOM-LDH) powder.

Preparation of SOM-LDH/PVC composites

A series of SOM-LDH/PVC composites were prepared by melting blending method. Typically, in the static thermal stability test, 100.00 g PVC resin, 2.00 g stabilizers (0.44 g Ca(st) 2 , 0.44 g Zn(st) 2 and 1.12 g SOM-LDHs) and 40.00 g DOP were mixed in a beaker. The above mixtures were plasticized in an open twin-wheel mill (BP-8175-A, Baopin Ltd., China) at 36 rpm and 30

rpm roll speeds at 185 °C for 4 min, and then the gap between the two rolls was controlled at 0.4 mm to prepare transparent SOM-LDH/PVC composite films, respectively. In the dynamic thermal stability test, a blend of 100.00 g PVC resin, 1.00 g stabilizers (0.22 g Ca(st) 2 , 0.22 g Zn(st) 2 and 0.56 g SOM-LDHs) and 25.00 g DOP was pressed by twin-wheel mill at 190 °C to obtain transparent SOM-LDH/PVC film. In both cases, a neat PVC film was fabricated as the blank sample for comparison without a SOM-LDH filler using the same mass of other stabilizer and DOP as for SOM-LDH/PVC composite.

Characterization

Powder X-ray diffraction (PXRD) patterns were recorded on a Shimadzu XRD-6000 X-ray diffractometer (40 kV, 30 mA, Cu Ka radiation, λ= 0.154 nm) in the range from 3 ° to 70 °/2θ with a scan step of 10 °/min. The morphology and particle sizes of the samples were measured using a Zeiss Supra 55 scanning electron microscope (SEM) and the average particle size was calculated from more than 100 particles for each sample. Fourier transform infrared (FT-IR) spectra in the range of 4000 to 400 cm -1 with 2 cm -1 resolution were collected on a Bruker Vector 22 infrared spectrophotometer using the KBr disk method with a ratio of sample/KBr of 1: 100 and the film method for SOM-CO 3 -LDH/PVC films. The thickness of the LDH samples and the dispersion of LDHs in PVC resins were estimated using a Digital Instruments Version 6.12 atomic force microscope (AFM). Differential scanning calorimeter (DSC) measurements were carried out in N 2 gas flow using a differential scanning calorimeter (DSC, METTLER TOLEDO, DSC1 STARe System).

Thermal stability tests of SOM-LDH/PVC composites

Transparent test of SOM-LDH/PVC composites

100.00 g PVC, 8.00 g stabilizers (1.76 g Ca(st) 2 , 1.76 g Zn(st) 2 and 4.48 g LDHs), and 50.00 g DOP were placed into a beaker and mixed uniformly. The mixture was plasticized at 185 °C by twin-wheel mill, and then hot-pressed and moulded with a thickness of 5 mm using a plate vulcanizing machine. A series of Mg 2 Al-CO 3 -LDH samples were synthesized by the salt-hydroxide method using the above-prepared Mg(OH) 2 precursors, Al(NO 3 ) 3 and LMC as the feeding materials. Fig. 1 shows the PXRD patterns of the seven resulting Mg 2 Al-CO 3 -LDH samples. All the samples exhibit the typical PXRD patterns of LDH with the well-defined Bragg reflection peaks, e.g.

Results and discussion

Structural analysis of Mg 2 Al-CO 3 -LDH

(003), (006), ( 009) and (110) as marked in the graph. The basal d-spacing of Mg 2 Al-CO 3 -LDH is 0.78 nm calculated from the first three Bragg reflection peaks, which is in agreement with that of CO 3 -type LDH reported in the literature [START_REF] Guo | Micrometer-sized dihydrogenphosphate-intercalated layered double hydroxides: synthesis, selective infrared absorption properties, and applications as agricultural films[END_REF]. from Kyowa Chemical Industries of Japan. The morphology of the as-made LDH samples and the HT-3 is for all platelet-like structures but with some slight differences: the as-made series exhibit a rather homogeneous hexagonal shape similar to Mg(OH) 2 precursor when lying flat on the substrate with the observation of thin-layer when exposed perpendicular. The reference HT-3 presents circular roundish aspect. Besides, Fig. 3a displays the particle size variation of Mg(OH) 2 and LDH extracted from Figs. S2 (Supporting information) and 2 as a function of the aging time of Mg(OH) 2 . Interestingly, both series present a similar trend with a first rapid growth up to 8 h then to remain constant after. Indeed, Fig. 3b further demonstrates a close linear relationship in terms of particle size between LDHs and Mg(OH) 2 precursors. For example, the LDH platelets with the smaller particle lateral size of about 202 ± 10 nm were prepared from the smaller particles of Mg(OH) 2 presenting a lateral size of 49 ± 5 nm, while the larger particle size of 334 ± 13 nm was from larger particles of Mg(OH) 2 of 176 ± 10 nm. This linear dependence may suggest a topochemical reaction between both materials, reminiscent of previous observation for the transformation from single to double layered hydroxides [START_REF] Stimpfling | In situ platelets formation into aqueous polymer colloids: the topochemical transformation from single to double layered hydroxide (LSH-LDH) uncovered[END_REF]. An in-situ characterization beyond the scope of the study will be necessary to unravel the reason of such lateral size dependence between the reactant and the reaction product, i.e., between the brucite and the LDH phase and why this LDH series mirrors that of the precursors. The above results suggest that it is an available and efficient route to prepare LDHs with different particle sizes by controlling that of Mg(OH) 2 precursor employed in the salt-hydroxide route. The particle size of HT-3 is 219 nm, close to that of LDH when using Mg(OH) 2 -1h. Interestingly, the AFM image and associated height profile in Fig. 3c,d show that the platelets thickness of LDH-b sample is ca. 3.5 nm consisting of ca. 4 layers of LDH nanosheets, which is much thinner than ca. 24 nm measured for HT-3 (Fig. 2h) and as usually obtained by the traditional coprecipitation [START_REF] Yu | Preparation of two dimensional layered double hydroxide nanosheets and their applications[END_REF][START_REF] Wang | Ultrathin and vacancy-rich CoAl-layered double hydroxide/graphite oxide catalysts: promotional effect of cobalt vacancies and oxygen vacancies in alcohol oxidation[END_REF]. Besides, all the samples from LDH-a to -g display a stacking of platelets of similar thickness, that is close to that observed from samples prepared by another green route using MgO, Al 2 O 3 and NaNO 3 [START_REF] Zhang | Mechano-hydrothermal synthesis of Mg 2 Al-NO 3 layered double hydroxides[END_REF].

Intuitively, thinner LDH platelets should improve the dispersion state of the LDH particles into PVC, thus to be more exposed to the polymer chains and more efficient to capture HCl when PVC degrades [START_REF] Yin | Ultrathin two-dimensional layered metal hydroxides: an emerging platform for advanced catalysis, energy conversion and storage[END_REF][START_REF] Wang | Recent advances in the synthesis and application of layered double hydroxide (LDH) nanosheets[END_REF][START_REF] Zhang | An aqueous miscible organic (AMO) process for layered double hydroxides (LDHs) for the enhanced properties of polypropylene-LDH composites[END_REF][START_REF] Yang | Influence of anionic surface modifiers on the thermal stability and mechanical properties of layered double hydroxide/polypropylene nanocomposites[END_REF]. Evidently, the lateral size may play a role as a barrier effect, the LDH platelets acting as a sacrificial reservoir in improving PVC stability. Quantitatively an increase from 202 ± 10 nm to 334 ± 13 nm shifts the time of a "yellow" appearance from 50 min to 30 min, however, they are all black after 80 min. In addition, Fig. 5b further demonstrates the color values of ΔE as a function of thermal degradation time. For example, SOM-LDH-b/PVC composite presents the minimum color value before turning black after 80 min, showing the best thermal stability among the series. In the dynamic test at 190 °C, Fig. 5c andd show that both SOM-LDH-a and -b/PVC exhibit the same thermal stability than that with HT-3/PVC, but in the first 30 min. The performance in the series after 45 min of test is directly ranked by the lateral size variation. One may note that the particle lateral size of the asmade SOM-stabilizers plays a key role in improving the thermal stability of PVC. As intuitively thought, an appropriate particle size of the filler favours its dispersion state into the composite, then being more exposed, it is more efficient to protect PVC from its degradation effect. To sum up, a small thickness and particle lateral size of LDH platelets have both significant influence on the thermal stability of LDH/PVC composites. The former is controlled during the "green" synthesis, the later can be tuned by reaction conditions. During the thermal degradation process, a certain amount of HCl is released from PVC composites. Fig. 6a shows the concentration of protons as a function of the degradation time up to 240 min. In comparison, all the prepared SOM-LDH samples exhibit much higher capacity to inhibit the HCl release from the PVC degradation than neat PVC or HT-3/PVC. The detailed results are scrutinized in terms of the ignition time (t 1 ) and the stability time (t 2 ) for deeper comparison (Fig. 6b). As reported in Ref. 45, t 1 is the time for the protons (H + ) to be released in solution and t 2 represents the time to reach a concentration of 2*10 -3 mol/L. For example, t 1 is rather short of 9 min for neat PVC, and 30 min for HT-3/PVC, while much longer times are observed for the as-made SOM-LDH/PVC. Comparatively, t 1 ranks as follows: neat PVC < HT- In the series SOM-LDH/PVC composites, SOM-LDH-b/PVC exhibits the higher thermal stability associated to the longer ignition time t 1 of 114 min, and the longer stability time t 2 of 234 min. These above results demonstrate that the SOM-LDH is highly efficient to stabilize PVC in temperature, and to endow the polymer with the performance of interest for application.

Structural analysis of SOM-LDH/PVC films

Thermal degradation behaviour of SOM-LDH/PVC composites

Besides, transparency property is another important parameter for PVC product.

Conclusions

In this work, the "salt-oxide/hydroxide" route in mild conditions is adopted to fabricate a series of Mg 2 Al-CO 3 -LDH samples with different particle sizes from 202 ± 10 nm to 334 ± 13 nm from Mg(OH) 2 precursor that is prepared by the SNAS method. The prepared Mg 2 Al-CO 3 -LDH samples are surface-organo-modified to render them organophilic and easily dispersible into PVC. As thermal stabilizer, the as-made series exhibits higher performance to inhibit the PVC thermal degradation when compared to commercially available LDH standard. Using static/dynamic tests, it is demonstrated that enhanced performances are strongly related to the particle size of the LDH stabilizer. Among the investigated LDH samples in our work, the better candidate comparatively into the series presents an average particle lateral size of about 220 ± 10 nm, close to the commercial standard. Computational modeling should be carried out to better understand such lateral size dependence against performance to further optimize PVC stability.

In summary, the salt-oxide/hydroxide green route appears as transferable and environmentally friendly process to produce high-performance green LDH stabilizer for PVC.

2 where

 2 Static thermal stability of SOM-LDH/PVC composites were carried out by discoloration test and dehydrochlorination test. Discoloration test: The discoloration test of PVC sheets was performed according to ISO 305: 1990-4 standard. The PVC film was approximately cut into 1×1 cm sheets and heated in temperature-controlled oven at 195 °C in air. Strips were removed from the oven every 10 min. The thermal stability of SOM-LDH/PVC composites was appraised by the color differences using a Konica Minolta CR-400 type automatic color difference meter. The △L*, △A* and △ B* values were determined by this color meter. The CIE-L*A*B* values could be used to formulate a value for total change as △E by the equation: △E = (△ ) △L*, △A* and △B* values represented the white (+) and black (-), red (+) and green (-), yellow (+) and blue (-), respectively. The △E value was chosen to describe the discoloration of PVC sheets. A low △E value corresponds to a low color difference.Dehydrochlorination test: The release rate of HCl gas produced by the PVC decomposition was estimated by a dehydrochlorination test based on ISO 182-2: 1990-12[START_REF] Labuschagne | Heat stabilising flexible PVC with layered double hydroxide derivatives[END_REF][START_REF] Dong | Design and synthesis of polyol ester-based zinc metal alkoxides as a bi-functional thermal stabilizer for poly(vinyl chloride)[END_REF]. 1.50 g of transparent PVC film were cut into small pieces by 1×1 mm. Fig.S1shows the schematic diagram of dehydrochlorination test equipment. The PVC pieces were placed in a 50 mL threeflask and immersed in an oil bath at 180 °C. The HCl gas released from PVC pieces was recovered into a 100 mL deionized water at room temperature by N 2 gas. The pH value of the water was measured by a Mettler Toledo FE28 type pH meter (resolution, 0.01) and the concentration of H + in water was calculated.Dynamic thermal stability of SOM-LDH/PVC composites was performed by discoloration test under the plastication process with twin-wheel mill at 190 °C. A piece of PVC film was taken out of the twin-wheel mill every 5 min and their color changes were measured by the abovementioned automatic color difference meter.

Fig. 1 .

 1 Fig. 1. PXRD patterns of Mg 2 Al-CO 3 -LDH samples (a-g) by the salt-hydroxide method from the

Fig. 2 .

 2 Fig. 2. SEM images of Mg 2 Al-CO 3 -LDH samples from the corresponding Mg(OH) 2 precursors

Fig. 2

 2 Fig. 2 displays SEM images of the seven Mg 2 Al-CO 3 -LDH samples obtained by the salt-

Fig. 3 .

 3 Fig. 3. (a) The particle size distribution of Mg 2 Al-CO 3 -LDH samples (blue curve) from the

Fig. 4a displays

  Fig. 4a displays the XRD patterns of LDH-b, SOM-LDH-b and SOM-LDH-b/PVC film (using

Fig. 4 .

 4 Fig. 4. (a) Powder X-ray diffraction patterns of LDH-b, SOM-LDH-b and SOM-LDH-b/PVC

Fig. 5

 5 Fig. 5 shows the color change of SOM-LDH/PVC composite sheets in the static (a and b) and

Fig. 5 .

 5 Fig. 5. Results of discoloration (a, c) and color values (b, d) of transparent PVC sheets at 195 °C

3 /

 3 PVC < SOM-LDH-a/PVC < SOM-LDH-e/PVC < SOM-LDH-d/PVC < SOM-LDH-c/PVC < SOM-LDH-b/PVC, and for t 2 neat PVC < HT-3/PVC < SOM-LDH-a/PVC <SOM-LDH-e/PVC < SOM-LDH-c/PVC < SOM-LDH-d/PVC <SOM-LDH-b/PVC. In addition, Fig. S6 displays first derivative curve of the HCl release rate as a function of time for the PVC samples according to data from Fig. 6a, showing that the ignition of HCl release for neat PVC is much faster than for PVC samples with filler as well as comparing the release rate from the relative intensity of the derivative curves. The observed trend agrees well with the discoloration test results in Fig. 5.

  Fig. 6d displays the DSC curves of LDH/PVC composite films. The glass transition temperature

Fig, 6 .

 6 Fig, 6. (a) HCl release curves for the PVC samples heated at 180 °C; (b) The induction time and

Acknowledgments

This work is supported by Natural Science Foundation of China (No. 21627813, 21761029, and U1707603), First Division Alar Science and Technology Plan Project in Xinjiang Corps (2019GJJ04), Program for Changjiang Scholars and Innovative Research Team in University (No. IRT1205) and Distinguished professor of Kunlun scholars in Tarim University.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.