Patrice Bertail

Stéphan Clémençon

Sharp Bounds for the Tails of Functionals of Markov Chains

Keywords: AMS 2000 subject classification: Primary 60E15, 60J27, Secondary 60K05 Probability inequalities, Markov chain, regenerative method

This paper is devoted to establishing sharp bounds for deviation probabilities of partial sums Σ n i=1 f(X i ), where X = (X n ) n✷◆ is a positive recurrent Markov chain and f is a real valued function defined on its state space. Combining the regenerative method to the Esscher transformation, these estimates are shown in particular to generalize probability inequalities proved in the i.i.d. case to the Markovian setting for (not necessarily uniformly) geometrically ergodic chains.

Introduction

Consider a positive recurrent Markov chain X = (X n ) n✷◆ with transition probability Π, state space E with countably generated σ-field ❊ and stationary distribution µ. It is well known that we may then restrict ourselves to the case when the chain X is regenerative, namely when there exists a measurable set A such that µ(A) > 0 and Π(x, .) = Π(y, .) for all (x, y) ✷ A 2 , even if it entails considering a Nummelin extension of the initial chain (see [START_REF] Nummelin | A splitting technique for Harris recurrent chains[END_REF]). By virtue of the strong Markov property, the sequence (τ A (n)) n✷◆ of successive passage times to the regeneration set A forms a (possibly delayed) renewal process in this case and the segments (X 1+τ A (j) , . . . , X τ A (j+1) ), j ✕ 1, obtained by dividing the sample path of the chain according to the regeneration times, are i.i.d. r.v.'s, valued in the torus ❚ = ❬ n✕1 E n . Furthermore, the return time to the atom A has finite expectation and the stationary distribution may be viewed as a Pitman occupation measure:

✽B ✷ ❊, µ(B) = 1 ❊ A [τ A ] ❊ A [ τ A i=1 ■ {X i ✷B} ],
where τ A = inf{n ✕ 1, X n ✷ A}, ❊ A [.] denotes the expectation conditioned on X 0 ✷ A and ■ {❆} the indicator function of any event ❆. One may refer to [START_REF] Nummelin | General Irreducible Markov Chains and Non-negative Operators[END_REF] for an account on renewal theory applied to Markov chain analysis. Let f : (E, ❊) → | be a measurable function.

Asymptotic expansions for the tail probabilities of the partial sums n -1/2 n i=1 f(X i ) have been obtained via the regenerative method (see [START_REF] Smith | Regenerative stochastic processes[END_REF]), which technique consists roughly speaking in applying appropriate results for i.i.d. r.v.'s to the partial sums over regeneration cycles τ A (j+1) i=1+τ A (j) f(X i ), j ✕ 1. Refer in particular to [START_REF] Bolthausen | The Berry-Esseen Theorem for Functionals of Discrete Markov Chains[END_REF], [START_REF] Malinovskii | Limit theorems for Harris Markov chains I[END_REF] and [START_REF] Bertail | Edgeworth expansions for suitably normalized sample mean statistics of atomic Markov chains[END_REF] for such refinements of the CLT theorem. This paper aims at establishing, with the same means, non asymptotic bounds for the probability that the sum n i=1 f(X i ) exceeds a prescribed number x, holding for all x and n, similar to tail bounds proved in the independent framework such as those obtained by W. Hoeffding, G. Bennett or S.V. Nagaev for instance. We point out that the regenerative method is by no means the sole technique for obtaining probability inequalities in the markovian setting. Such non asymptotic results may be established by martingale arguments (see [START_REF] Glynn | Hoeffding's inequality for uniformly ergodic Markov chains[END_REF]), coupling techniques based on decay rate assumptions for mixing coefficients (see [START_REF] Doukhan | Mixing: properties and examples[END_REF] or [START_REF] Rio | Théorie asymptotique des processus aléatoires faiblement dépendants[END_REF] for instance), from results of the quasi-compact operator theory when the spectral gap property is assumed to hold for the transition kernel (see [START_REF] Lézaud | Chernoff-type bound for finite Markov chains[END_REF] and [START_REF] Léon | Optimal Hoeffding Bounds for Discrete Reversible Markov Chains[END_REF]) or by methods based on information inequalities as in [START_REF] Marton | A Measure of Concentration Inequality for Contracting Markov Chains[END_REF] (see also [START_REF] Samson | Concentration of Measure Inequalities for Markov Chains and Φ-Mixing Processes[END_REF]). However, the regenerative method imposes much less restrictions on the ergodicity properties of the chain than most alternative techniques. In particular it may be used without stipulating Doeblin's condition (i.e. uniform ergodicity) to be fulfilled.

The paper is organized as follows. Section 2 first gives an insight into how the regenerative approach may apply for deriving tail bounds in the positive recurrent case, appealing additionally to an exponential change of probability measure as in [START_REF] Talagrand | The missing factor in Hoeffding's inequalities[END_REF] and then states the main results of the paper. Technical proofs are postponed to section 3.

Probability Inequalities for Regenerative Markov Chains

We first introduce further notation. Let ν be some probability distribution on (E, ❊).

Here and throughout, we denote by P ν (respectively, P A ) the probability measure on the underlying space such that X 0 ∼ ν (resp., conditioned on X 0 ✷ A) and by ❊ ν [.] (resp., by ❊ A [.]) the P ν -expectation (resp., the P A -expectation). Furthermore, it is assumed

throughout the paper that ❊ A [τ 2 A ] < ∞ or equivalently that ❊ µ [τ A ] < ∞, since by a standard result in renewal theory P µ (τ A = k) = µ(A)P A (τ A ✕ k) for all k ✕ 1.
In the case when Y 1 , ..., Y n are i.i.d. random variables such that

E[Y 1 ] = 0, ❊[Y 2 1 ] = σ 2
and |Y 1 | ✻ M for some constant M, the following tail estimate has been obtained in [START_REF] Bennett | Probability inequalities for a sum of independent random variables[END_REF] P(

n i=1 Y i ✕ x) ✔ exp -nσ 2 M 2 H( Mx nσ 2 ) , (1)
with

H(x) = (1 + x) log(1 + x) -x.
When the Y i 's are not bounded anymore, this inequality may be extended using truncation arguments, as in [START_REF] Fuk | Probability Inequalities for Sums of Independent Random Variables[END_REF] (see also [START_REF] Nagaev | Large Deviations of Sums of Independent Random Variables[END_REF]) at the price of an additional term related to the tail behavior of the Y i 's on the right-hand side of (1). For arbitrary positive constants x and M, we have the inequality

P( n i=1 Y i ✕ x) ✔ exp x M -( x -nm M M + nσ 2 M M 2 ) log(1 + xM nσ 2 M ) + nP(Y > M) ✔ exp -nσ 2 M M 2 H( Mx nσ 2 M ) + nP(Y > M), (2)
with

m M = ❊[Y 1 ■ {|Y 1 |✔M} ] and σ 2 M = ❊[Y 2 1 ■ {|Y 1 |✔M} ].
This paper is devoted to establishing analogous inequalities under the assumption that

Y i = f(X i ), 1 ✔ i ✔ n, where f : (E, ❊) → | is a µ-integrable function.
Without loss of generality, we assume that µ(f) = ❊ µ [f(X 1 )] = 0 throughout the article.

The regenerative approach

Let l n = n i=1 ■{X i ✷ A} be the number of renewals (i.e. visits to the regeneration set A) up to time n. We denote by α = ❊ A (τ A ) the mean inter-renewal time. For j ✕ 1, define

S j (f) = τ A (j+1)
i=1+τ A (j) f(X i ) and s j = τ A (j + 1)τ A (j). Notice that these are two sequences of i.i.d. random variables with common variance σ 2 f and σ 2 1 respectively, which are assumed to be finite throughout the article. The regenerative method is based on the preliminary observation that the sum i✔n f(X i ) may be decomposed as follows on the event {l n ✕ 2}:

n i=1 f(X i ) = S A (f) + ln -1 j=1 S j (f) + S (n) n (f), (3)
where

S A (f) = τ A i=1 f(X i ) and S (n) n (f) = n i=1+τ A (ln ) f(X i )
with the usual convention regarding empty summation. It is noteworthy that the summands in (3) are generally not independent for fixed n (observe that, when f ✑ 1 for instance, they sum up to n). Hence, when applied for establishing tail bounds, the regenerative method comprises three main steps. The first one consists in partitioning the underlying space Ω according all possible fashions for the chain to regenerate up to time n, so that the summands in (3) be mutually independent on each subset of the partition. The matter is next to establish a tail bound on each of these subsets by relying on the resulting independence structure. And the desired tail estimate is finally computed by summing all the bounds previously obtained in a way that the sum obtained may be identified. Now that an insight into the principle of the method has been given, in the next subsection we turn to developing the argument for proving tail inequalities in a rigorous fashion. Observe first that for all strictly positive real numbers p 1 , p 2 and p 3 such that 1/p 1 + 1/p 2 + 1/p 3 = 1, we have for all x > 0 the immediate bound

P ν ( n i=1 f(X i ) ✕ x) ✔ P ν ( n i=1 f(X i ) ✕ x, l n ✔ 1) + P ν (S A (f) ✕ x/p 1 ) + P ν ( ln -1 j=1 S j (f) ✕ x/p 2 ) + P ν (S (n) n (f) ✕ x/p 3 ), (4)
with the convention that ln -1 j=1 S j (f) = 0 when l n ✔ 1. Whereas bounds for the first two terms on the right hand side of (4), as well as for the last one, may be easily derived from assumptions on the tails of τ A and S A (f) under P ν and P A , special attention must be paid to the tail of ln -1 j=1 S j (f). Beyond the dependence structure among the summands emphasized above, it is noteworthy that the S j (f)'s are generally unbounded (even though one assumes f to be bounded).

Main result

Although the technique we present here for establishing bounds for the tail behavior of ln -1 j=1 S j (f) is very general, we focus on a simple result for the sake of clarity. Precisely, we derive an inequality extending (2), whose right hand side consists of two components: the first one being of the form of an exponential bound involving truncated moments of the (S j (f), s j )'s, while the second one is related to their tail behavior. Other probability bounds can be deduced by a slight adaption of the argument described below. And extensions of moment inequalities of Rosenthal type for i✔n f(X i ) may be established with the same method. We also emphasize that, although the present study is confined to the markovian setting, the method applies in the same way to any (eventually continuous time and not necessarily markovian) stochastic process with a regenerative extension (see [START_REF] Thorisson | Coupling, Stationarity and Regeneration[END_REF]).

Theorem 1 Consider a regenerative positive recurrent Markov chain X = (X n ) n✷◆ with initial and stationary probability distributions ν and µ. Let A be an accessible atom for X and f : (E, ❊) → | some µ-integrable function. Assume further that

• (i) α = ❊ A [τ A ] < ∞ and 0 < σ 2 A = ❊ A [τ 2 A ] < ∞, • (ii) ❊ A [S A (f)] = 0 and 0 < σ 2 f = ❊ A [S A (f) 2 ] < ∞.
Then, for any vector

M = (M 1 , M 2 ) ✷ | ✄2 + of euclidian norm ❦M❦ = (M 2 1 +M 2 2 ) 1/2 , there exists a constant C M depending on σ 2
A and σ 2 f such that , for any n ✕ α,

P ν ( ln -1 j=1 S j (f) ✕ x) ✔ C M exp -n(1 + |ρ|)σ 2 2 ❦M❦ 2 H( ❦M❦ ♣ 2 (1 + |ρ|)σσ f x) + (n -1)P A (|S A (f)| ✕ M 1 ) + (n -1)P A (τ A ✕ M 2 ), (5)
with

σ2 f = Var A (S A (f)■ {|S A (f)|✔M 1 } ), σ2 A = Var A (τ A ■ {τ A ✔M 2 } ), ρ = (σ f σA ) -1 Cov A (S A (f)■ {|S A (f)|✔M 1 } , τ A ■ {τ A ✔M 2 } ), σ = q σ2 f σ2 A /(σ 2 f + σ2 A ),
Var A and Cov A denoting respectively variance and covariance under P A and the constant C M , depending on α, σ2

f and σ2 only, being of order O(❦M❦ 2 ) as ❦M❦ → ∞. Furthermore, we have, for any n ✕ α,

P ν ( ln -1 j=1 S j (f) ✕ x) ✔ C M exp - x 2 σ/σ f 2(n(1 + |ρ|)σσ f + xM ♣ 2 3 ) + (n -1)P A (|S A (f)| ✕ M 1 ) + (n -1)P A (τ A ✕ M 2 ). (6)
Before developping the argument based on the regenerative method, a few remarks are in order at this point.

Remark 1 (On block-moment conditions) Notice that conditions (i) and (ii) do not depend on the atom A chosen. Besides, when dealing with the nonregenerative case, these conditions have to be satisfied by a Nummelin (regenerative) extension of the chain X constructed from a small set S, namely an accessible set to which X returns in a given number of steps with probability uniformly bounded by below (see [START_REF] Nummelin | General Irreducible Markov Chains and Non-negative Operators[END_REF] for an account on the Nummelin splitting technique). A sufficient condition for the latter to hold is

classically that both sup x✷S ❊ x [τ 2 S ] and sup x✷S ❊ x [( τ S
Remark 2 (On constants involved in the exponential term) It is noteworthy that, although the asymptotic distribution of n -1/2 ln -1 j=1 S j (f) as n → ∞ does not depend on the atom A chosen, its distribution for fixed n does (when f ✑ 1 for instance,

ln -1 j=1 S j (f) = τ A (l n ) -τ A
is the time between the first and last visits to the set A before time n). Hence, the constants involved in bound (5) depend on the choice of the atom and ideally the latter should be picked so as to minimize the bound on the right hand side of [START_REF] Borovkov | Some inequalities for sums of multidimensional random variables[END_REF]. We also point out that in the case where the X i 's are i.i.d. and f is bounded, by taking

A = ❳ (so that τ A ✑ 1), M 1 > sup x✷E |f(x)
| and M 2 > 1, the term on the right hand side of (5) (respectively, of ( 6)) boils down to a Bennett's type bound (respectively, to a Bernstein's type bound).

Proof : Treading in the steps of [START_REF] Bertail | Edgeworth expansions for suitably normalized sample mean statistics of atomic Markov chains[END_REF] (see also [START_REF] Bolthausen | The Berry-Esseen Theorem for Functionals of Discrete Markov Chains[END_REF] and [START_REF] Malinovskii | Limit theorems for Harris Markov chains I[END_REF]), the proof is made in four stages as announced in §2.1: 1. truncating the r.v.'s as in [START_REF] Fuk | Probability Inequalities for Sums of Independent Random Variables[END_REF], 2. partitioning the probability space according to all fashions for X to regenerate up to time n, 3. establishing an accurate exponential bound for sums of bounded i.i.d. 1-lattice random vectors and 4. summing the bounds previously obtained over all subsets of the partition.

Step 1: truncation. By the same kind of truncation trick as in [START_REF] Fuk | Probability Inequalities for Sums of Independent Random Variables[END_REF], we shall show that the tail of ln -1 j=1 S j (f) may be bounded by an exponentially decreasing term plus a term related to the tail behavior of the (S j (f), s j )'s. Let M 1 , M 2 be positive thresholds and consider the truncated r.v.'s,

Sj (f) = S j (f)■ {|S j (f)|✔M 1 } and sj = s j ■ {|s j |✔M 2 } , for j ✕ 1.
The Sj (f)'s (respectively, the sj 's) are i.i.d bounded random variables. As l n ✔ n P ν a.s., it follows from the union bound that for all positive x

P ν ( ln -1 j=1 S j (f) ✕ x) ✔ P ν ({ ln -1 j=1 Sj (f) ✕ x} ❭ {s j = sj , 1 ✔ j ✔ l n -1}) + (n -1)P A ({|S A (f)| > M 1 } ❬ {τ A > M 2 }). (7)
Again the last term on the right hand side of (7) may be bounded from assumptions on the speed of return time to A and on the tail of S A (f) under P A . We shall now deal with the first term which we denote by P ν,n (x).

Step 2: partitioning. In order to be brought to the independent framework, we partition the event {l n ✕ 2} according to all possible values for the first and last regeneration times, as well as for the total number of regeneration times up to time n. In this respect, consider the collection of events

U r,l,m = {τ A = r, m j=1 s j = n -r -l, s m+1 > l},
for 1 ✔ r, l ✔ n and 2 ✔ m ✔ n. Combining the formula of total probability to the strong Markov property, we get

P ν,n (x) = n m=2 n r=1 n l=1 P ν (τ A = r) ✂ p r,l,m (x) ✂ P A (τ A > l), (8)
where p r,l,m (x

) = P( 1✔j✔m Sj (f) ✕ x, 1✔j✔m {s j -❊[s j ]} = n-r-l-mα) and α = ❊[s j ].
Since ❊[ Sj (f)] ✔ 0, we have p r,l,m (x) ✔ pr,l,m (x) for all r, l and m, where pr,l,m (x) = P(

1 σf ♣ m m j=1 { Sj (f) -❊[ Sj (f)]} ✕ x σf ♣ m , 1 σA ♣ m m j=1 {s j -❊[s j ]} = λ r,l,m ), (9) with λ r,l,m = (n -r -l -❡ αm)/(σ A ♣ m)
. Hence all boils down to get an accurate bound for p r,l,m (x) using the fact that ( Sj (f), sj ) j✕1 is a sequence of bounded i.i.d. bivariate 1-lattice random vectors. With the notations previously set out, notice that ρ = (σ f σA ) -1 Cov( Sj (f), sj ) for j ✕ 1.

Step 3: exponential bound for 1-lattice random vectors. It is easy to get a crude exponential estimate for (9) by the standard Chernoff method (for instance see [START_REF] Prokhorov | An extension of S.N. Bernstein's inequalities to multidimensional distributions[END_REF] and inequality [START_REF] Samson | Concentration of Measure Inequalities for Markov Chains and Φ-Mixing Processes[END_REF] in §4.1). However such an estimate would not be precise enough to get a summable bound in m for (8) (preventing thus from reproducing the argument in [START_REF] Malinovskii | Limit theorems for Harris Markov chains I[END_REF] or [START_REF] Bertail | Edgeworth expansions for suitably normalized sample mean statistics of atomic Markov chains[END_REF] for instance). Indeed, under adequate assumptions, from limit theorems for k-lattice random vectors with minimum span h (see [START_REF] Dubinskaite | Limit Theorems in | k I[END_REF], [START_REF] Dubinskaite | Limit Theorems in | k II-III[END_REF] or Lemma 6.4 in [START_REF] Bertail | Edgeworth expansions for suitably normalized sample mean statistics of atomic Markov chains[END_REF]), pr,l,m (x) may be shown as asymptotically equivalent to m -1/2 h ∞ t=x φ Σ (t/ ♣ m, λ r,l,m )dt, denoting by φ Σ the density of the bivariate gaussian distribution with mean zero and variancecovariance matrix Σ = Var((σ -1 f Sj (f), σ-1 A sj )). As pointed out in §6.2.2 of [START_REF] Bertail | Edgeworth expansions for suitably normalized sample mean statistics of atomic Markov chains[END_REF], the factor m -1/2 appearing in the latter quantity is of prime importance for obtaining bounds that are summable in m. In order to prove the exponential inequality required, we use the method proposed in [START_REF] Talagrand | The missing factor in Hoeffding's inequalities[END_REF] for improving exponential bounds in the independent setting, which is based on a refined use of the classical argument of the Bahadur-Rao Theorem. An exponential bound for sums of i.i.d. 1-lattice bounded random vectors is stated in the following lemma, of which proof is given in §3.1.

Lemma 2 (Exponential inequalities for 1-lattice bounded random vectors)

Let (S ✄ j , L ✄ j ) 1✔j✔m be i.i.d. centered and square integrable bivariate r.v.'s with covariance matrix Σ = ( 1 ρ ρ 1 ). Assume further that the L ✄ j 's are lattice r.v.'s with minimal span h > 0 and that there exists finite constants B 1 and

B 2 such that |S ✄ j | ✔ B 1 and |L ✄ j | ✔ B 2 for j = 1, ..., m. Set B 2 = B 2 1 + B 2
2 , then there exists a universal constant c such that for all m ✕ 1 and y ✕ 0,

P(m -1 2 m j=1 S ✄ j ✕ y, m -1 2 m j=1 L ✄ j = λ) ✔ e 3 2 ( h 2π ♣ m + 4c B m ) exp - y 2 + λ 2 4B 2 . ( 10
)
Furthermore, we also have the following Bennett/Bernstein type inequalities,

P(m -1 2 m j=1 S ✄ j ✕ y, m -1 2 m j=1 L ✄ j = λ) ✔ e 3 2 ( h 2π ♣ m + 4cB m )e - m(1+|ρ|) B 2 H( B||(y,λ)|| (1+|ρ|) ♣ m ) ✔ e 3 2 ( h 2π ♣ m + 4cB m ) exp    - ||(y, λ)|| 2 2(1 + |ρ| + B||(y,λ)|| 3 ♣ m )    .
Remark 3 An overestimated value of the constant c may be deduced by a careful examination of the proof of limit theorems stated in [START_REF] Dubinskaite | Limit Theorems in | k I[END_REF] and [START_REF] Dubinskaite | Limit Theorems in | k II-III[END_REF] (see §3.1 further). However we do not know at present what the best value for the constant c might be.

Going back to our problem, notice first that, as the L ✄ j 's are lattice with minimal span

h = σ-1 A , lemma 2 applies to the sequence {( Sj (f) -❊[ Sj (f)])/σ f , (s j -❊[s j ])/σ A } j✕1 with ρ = ρ, B 1 = 2M 1 /σ f and B 2 = 2M 2 /σ A . Observing that B = (B 2 1 + B 2 2 ) 1/2 ✔ 2||M||/σ, this yields pr,l,m (x) ✔ e 3 2 ( 1 2πσ A ♣ m +8c ||M|| σm ) exp - m(1 + |ρ|)σ 2 ||M|| 2 H ✥ ||M|| ✁ ||(x/(σ f ♣ m), λ r,l,m )|| (1 + |ρ|)σ ♣ m ✦ . ( 11
)
Recall that, for any a > 0, the function

x ✼ → H(a ♣ x) is concave on | + . Writing ||(x/(σ f ♣ m), λ r,l,m )|| = ✥ 1 2 ( x ♣ 2 σf ♣ m ) 2 + 1 2 (λ r,l,m ♣ 2) 2 ✦ 1/2
, by concavity arguments we get that pr,l,m (x

) ✔ e 3 2 ✒ 1 2πσ A ♣ m + 8c ||M|| σm ✓ e - m(1+| ρ|) σ2 2||M|| 2 H ✏ ||M||x ♣ 2 (1+| ρ|) σf σm ✑ +H ✏ ||M|||λ r,l,m | ♣ 2 (1+| ρ|) σ♣ m ✑ . (12) As the function x ✼ → H(x)
x is increasing on | + , we have for any

m ✔ n exp - m(1 + |ρ|)σ 2 2||M|| 2 H ✥ ||M||x ♣ 2 (1 + |ρ|)σ f σm ✦ ✔ exp - n(1 + |ρ|)σ 2 2||M|| 2 H ✥ ||M||x ♣ 2 (1 + |ρ|)σ f σn ✦ ✔ exp - x 2 σ/σ f 2(n(1 + |ρ|)σσ f + x||M|| ♣ 2 3
) , [START_REF] Malinovskii | Limit theorems for Harris Markov chains I[END_REF] the last bound following from the classical inequality H(x) ✕ x 2 2(1+x/3) , x ✕ 0.

Step 4: control of the sum. Now combining the previous bound to ( 8) and ( 12), we deduce that

P n,ν (x) ✔ e 3/2 exp -n(1 + |ρ|)σ 2 2||M|| 2 H ✥ ||M||x ♣ 2 (1 + |ρ|)σ f σn ✦ Γ n , (14)
where

Γ n = n m=2 n r=1 n l=1 P ν (τ A = r) P A (τ A > l) ( c 1 ♣ m + c 2 m )γ r,l,m , with c 1 = (2πσ A ) -1 , c 2 = 8c||M||/σ and γ r,l,m = exp - m(1 + |ρ|)σ 2 2||M|| 2 H ✥ ||M|||λ r,l,m | ♣ 2 (1 + |ρ|)σ ♣ m ✦ ✔ exp    - λ 2 r,l,m 2(1 + |ρ| + |λ r,l,m |||M|| ♣ 2 3σ ♣ m    . (15)
Recall that λ r,l,m = (nrl -αm)/σ A ♣ m and consider the subdivision defined by the points

a n,m = (n -αm)/(σ A ♣ m), for 1 ✔ m ✔ n.
In order to bound this sum, we split the latter into two parts, according to whether r + l is less than σA ♣ m or not. Since γ r,l,m ✔ 1, by repeated use of Markov inequality, we get

n m=2 n r+l>σ A ♣ m P ν (τ A = r)P A (τ A > l)( c 1 ♣ m + c 2 m )γ r,l,m ✔ n m=1 ( c 1 ♣ m + c 2 m ) n l✕σ A ♣ m P A (τ A > l) ✔ ❊ A [τ 2 A ] σ2 A n m=1 ( c 1 m 3/2 + c 2 m 2 ) ✔ ❊ A [τ 2 A ] σ2 A (3c 1 + c 2 π 2 6 ). (16)
From the identity (a-b

) 2 ✕ a 2 /2-b 2 , it follows that λ 2 r,l,m ✕ a 2 n,m /2 -((r+l)/(σ A ♣ m)) 2
and, for r

+ l ✔ σA ♣ m, λ 2 r,l,m ✕ (a 2
n,m /2 -1) + , denoting by a + the positive part of any real number a. The term on the right hand side of (15) being decreasing in λ r,l,m , we have

γ r,l,m ✔ exp      - (a 2 n,m /2 -1) + 2(1 + |ρ| + (a 2 n,m /2-1) 1/2 + ||M|| ♣ 2 3σ ♣ m      . (17)
Consider the set A n = {m ✷ {2, . . . , n} : (a 2 n,m /2 -1)

1/2 + ||M|| ♣ 2/(3σ) ✔ m 1/2 (
✷ A c n , γ r,l,m ✔ exp -3σ ♣ m(a 2 n,m /2 -1) 1/2 + 4||M|| ♣ 2 ✔ exp -9σ 2 (1 + |ρ|) 8||M|| 2 m .
It follows that

n m=1 n r+l✔σ A ♣ m P ν (τ A = r)P A (τ A > l)( c 1 ♣ m + c 2 m )γ r,l,m ✔ α(c 1 + c 2 )(U n + V n ),
where U n = m✷An m -1/2 γ r,l,m and V n = m✷A c n m -1/2 γ r,l,m . As in step 4 of Theorem 5.1's proof in [START_REF] Bertail | Edgeworth expansions for suitably normalized sample mean statistics of atomic Markov chains[END_REF], we shall prove that U n is bounded by a Riemann sum. Notice that for all m ✔ n, a n,ma n,m+1 = ασ -1 A m -1/2 + a n,m+1 ((1 + m -1 ) 1/2 -1) and a n,m ✕ a n,m+1 . If a n,m+1 ✕ 0, then ασ -1

A m -1/2 ✔ a n,ma n,m+1 . On the other hand, if a n,m+1 ✔ 0, we have -a n,m+1 = (α(m + 1)n)σ -1

A m -1/2 ✔ ασ -1 A m 1/2 provided that n ✕ α. Hence, for any m such that a n,m+1 ✔ 0,

ασ -1 A m -1/2 ✔ a n,m -a n,m+1 + 2 -1 ασ -1 A m -1/2 ,
and consequently for any m ✕ 2 and n ✕ α, we have

ασ -1 A m -1/2 ✔ 2(a n,m -a n,m+1 ).
We thus obtain that

U n ✔ 2α -1 σA e c 3 n m=1 (a n,m -a n,m+1 ) exp{-c 3 a 2 n,m /2} ✔ 2α -1 σA e c 3 ∞ x=-∞ exp(-c 3 x 2 /2)dx = 2α -1 σA q 2π/c 3 e c 3 , (18)
with c 3 = 1/(4(1 + |ρ|)) ✷ (1/8, 1/4) (in particular, e c 3 / ♣ c 3 ✔ 2 ♣ 2e 1/4 ). Besides, proceeding in a similar way, we get

V n ✔ m✷A c n m -1/2 exp - 9(1 + |ρ|)σ 2 8||M|| 2 m ✔ ∞ 0 x -1/2 exp - 9(1 + |ρ|)σ 2 8||M|| 2 x dx = 2 ♣ 2||M|| ♣ π 3σ(1 + |ρ|) 1/2 ✔ 2 ♣ 2||M|| ♣ π 3σ . (19)
Now combining ( 16), ( 18) and ( 19), we obtain Γ n ✔ c 4 with

c 4 = σ 2 A + α 2 σ2 A (3c 1 + c 2 π 2 6 ) + 2σ A (c 1 + c 2 )e c 3 α α s 2π c 3 + (c 1 + c 2 ) 2 ♣ 2||M||α ♣ π 3σ . (20)
Using [START_REF] Marton | A Measure of Concentration Inequality for Contracting Markov Chains[END_REF], this yields the tail bound [START_REF] Borovkov | Some inequalities for sums of multidimensional random variables[END_REF] where C M = e 3/2 c 4 with c 4 given by [START_REF] Prokhorov | An extension of S.N. Bernstein's inequalities to multidimensional distributions[END_REF]. Notice that C M = O(||M|| 2 ) as ||M|| → ∞. And using ( 13) we obtain the tail bound ( 6) with the "Bernstein version" for the exponential term. ✷.

3 Technical Details -Proof of Lemma 2

Set ❙ m = m -1/2 i✔n S ✄ j and ▲ m = m -1/2 i✔n L ✄ j for notational convenience. For u = (u 1 , u 2 ) ✷ | + ✂ |, let ψ m (u) = log ❊[exp{< u, (❙ m , ▲ m ) >}]
be the log-Laplace of the random vector (❙ m , ▲ m ) and denote by ψ (1) u,m and ψ

u,m its gradient and its hessian matrix. Consider now the probability measure P u,m defined by the Esscher transformation dP u,m = exp{❤u, (❙ m , ▲ m )✐ψ m (u)}dP. Expectation under P u,m is denoted by ❊ u,m [.] in what follows. By exponential change of probability measure, we get P(❙ m ✕ y, ▲ m = λ) = ❊ u,m [e ψm (u)-❤u,(❙m ,▲m )✐ ■ {❙m ✕y,▲m=λ} ] = e -❤u,(y,λ)✐-ψm (u) ❊ u,m [e -❤u,(❙m -y,▲m -λ)✐ ■ {❙m ✕y,▲m=λ} ].

Now choose u = u ✄ such that ψ (1) m (u ✄ ) = (y, λ), that is u ✄ = arg sup u✷|+ ✂| {< u, (y, λ) > -ψ m (u)}.
And as ❊[e <u,(❙m ,▲m )> ] = e ψm (u) , by differentiating one obtains

❊[e <u ✄ ,(❙m , ▲m)> (❙ m , ▲ m )] = ψ (1) m (u ✄ )e ψm (u ✄ ) = (y, λ)e ψm (u ✄ ) yielding ❊ u ✄ ,m [(❙ m , ▲ m )] = (y, λ) and Var u ✄ ,m [(❙ m , ▲ m )] = ψ (2)
m (u ✄ ) in a similar fashion, denoting by Var u ✄ ,m [.], the variance-covariance matrix under P u ✄ ,m . By integrating by parts combined with straightforward changes of variables, one obtains A m (u) = ❊ u,m [e -❤u,(❙m -y,▲m -λ)✐ ■ {❙m ✕y,▲m=λ} ] = e -❤u,(❙m -y,▲m -λ)✐ ■ {❙m -y✕0,▲m -λ=0} dP u,m

= P u,m (S m ✕ y, L m = λ) -u 1 ∞ y e -u 1 (s-y) P u,m (S m -y ✕ s -y, L m -λ = 0)ds ✔ u 1 ∞ s=0 e -u 1 s P u,m (S m -y ✕ 0, L m -λ = 0) -P u,m (S m -y ✕ s, L m -λ = 0)ds.
Using the results in [START_REF] Dubinskaite | Limit Theorems in | k I[END_REF] and [START_REF] Dubinskaite | Limit Theorems in | k II-III[END_REF] or the local Berry-Esseen Bound proved in [START_REF] Bolthausen | The Berry-Esseen Theorem for Functionals of Discrete Markov Chains[END_REF] (see Theorem 4), we know that there exists a constant C u such that, uniformly in (y, λ),

|P u,m (❙ m -y ✕ s, ▲ m -λ = 0) -h ♣ m ∞ s φ Wu (t, 0)dt| ✔ C u m -1 ,
where W u = Var u,m [(❙ m , ▲ m )] and φ Wu is the density of the bivariate gaussian distribu- tion with covariance matrix W u . From [START_REF] Dubinskaite | Limit Theorems in | k I[END_REF] (see also [START_REF] Dubinskaite | Limit Theorems in | k II-III[END_REF]) a crude bound for C u is given by

ck 3 (P u,1 ) where k 3 (P u,1 ) = ❊ u,1 [|X -❊ u,1 [X]| 3 ]/❊ u,1 [(X -❊ u,m [X]) 2 ] 3/2 and c is a universal constant . Notice that φ Wu (s, 0) = exp{-1 2 s 2 /α 2 u,m } ♣ 2πVar u,m (▲ m ) q 2πα 2 u,m , with α u,m = Var u,m (❙ m ) 1/2 (1 -ρ 2 u,m ) 1/2 ρ 2 u,m = Cov u,m (❙ m , ▲ m ) 2 /Var u,m (▲ m )Var u,m (❙ m ).

This yields for

u = u ✄ , A m (u ✄ ) ✔ u ✄ 1 ∞ 0 e -u ✄ 1 s ( h ♣ m s 0 φ W u ✄ (t, 0)dt + 2C u ✄ m -1 )ds = h ♣ m ∞ 0 e -u ✄ 1 s φ W u ✄ (s, 0)ds + 2C u ✄ m -1 = h ♣ m 1 ♣ 2πVar u ✄ ,m (▲ m ) Φ(u ✄ 1 α u,m ) 2πφ(u ✄ 1 α u,m ) + 2C u ✄ m -1 ,
where φ(x) denotes the density of the standard normal distribution and Φ(x) its survivor function. Recall that for all x > 0 (see [START_REF] Talagrand | The missing factor in Hoeffding's inequalities[END_REF]),

1 ♣ 2π(1 + x) ✔ Φ(x) ♣ 2πφ(x) ✔ 1 ♣ 2π max(x, 1
) , which leads to

A m (u ✄ ) ✔ h ♣ m 1 2πVar u ✄ ,m (▲ m ) 1/2 + 2C u ✄ m -1 . ( 21
)
The matter is now to find an upper bound for the quantity L(u ✄ ) = e -❤u ✄ ,(y,λ)✐-ψm (u ✄ ) = inf u e -(❤u,(y,λ)✐-ψm (u)) .

As ❤u, (❙ m , ▲ m )✐ ✔ 1 + ||u|| 2 B 2 , we have ❊[e ❤u,(❙m,▲m)✐ ] ✔ e exp{||u|| 2 B 2 } for all u, so that L(u ✄ ) ✔ e exp(-(y 2 + λ 2 )/4B 2 ).

(

) 22
It only remains to bound k 3 (P u ✄ ,m ) explicitely. Using [START_REF] Samson | Concentration of Measure Inequalities for Markov Chains and Φ-Mixing Processes[END_REF], observe first that

❊ u ✄ ,1 [(❙ 1 -❊ u ✄ ,1 [❙ 1 ]) 2 ] = ❊ u ✄ ,m [(❙ m -❊ u ✄ ,m [❙ m ]) 2 ] (23)
= (sy) 2 e ❤u ✄ ,(s,l)✐-ψm (u ✄ ) dP(s, l)

✕ e -1 (s -y) 2 e (s 2 +l 2 )/4B 2 dP(s, l)

✕ e -1 (s -y) 2 dP(s, l) = e -1 (1 + y 2 ) ✕ e -1 , and similarly Var u ✄ ,m (▲ m ) ✕ e -1 . Moreover, since we have the bound

❊ u ✄ ,1 [|❙ 1 -❊ u ✄ ,1 [❙ 1 3 ] ✔ 2BVar u ✄ ,1 (❙ 1 ),
and it follows from (23) that k 3 (P u ✄ ,1 ) ✔ 2e 1/2 B.

(

) 25
The desired result follows by combining [START_REF] Rio | Théorie asymptotique des processus aléatoires faiblement dépendants[END_REF] with [START_REF] Samson | Concentration of Measure Inequalities for Markov Chains and Φ-Mixing Processes[END_REF] and the estimate [START_REF] Thorisson | Coupling, Stationarity and Regeneration[END_REF].

To obtain the Bernstein-Bennett type bound, we replace the crude bound [START_REF] Samson | Concentration of Measure Inequalities for Markov Chains and Φ-Mixing Processes[END_REF] by a more refined one by using standard arguments. We have ❊[e ❤u,(❙m,▲m)✐ ] = m i=1 ❊[exp{

u 1 ♣ m S ✄ i + u 2 ♣ m L ✄ i }].
Since s ✼ → (e s -1s)/s 2 is increasing on | ✄

+ and | u 1 ♣ m S ✄ i + u 2 ♣ m L ✄ i | ✔ m -1/2 ||u||B, we have ❊[exp{ u 1 ♣ m S ✄ i + u 1 ♣ m L ✄ i }] ✔ ❊[1 + u 1 ♣ m S ✄ i + u 1 ♣ m L ✄ i )] + ❊[(u 1 S ✄ i + u 2 L ✄ i ) 2 ] exp{m -1/2 ||u||B) -1 -m -1/2 ||u||B) ||u|| 2 B 2 ✔ 1 + (1 + |ρ|) exp{m -1/2 ||u||B} -1 -m -1/2 ||u||B) B 2 .
The last inequality follows from the trivial inequality

❊[(u 1 S ✄ i + u 2 L ✄ i ) 2 ] = u 2 1 + 2ρu 1 u 2 + u 2 2

i=1 f(X i )) 2 ] are finite. Recall also that these 'block' conditions may be replaced by Foster-Lyapunov drift conditions that generally appear as more tractable in practice (refer for instance to Chapter 11 in[START_REF] Meyn | Markov chains and stochastic stability[END_REF] for further details).

Then, we obtain

) and u colinear to (y, λ), then we get

where H(x) = (1 + x)ln(1 + x)x. The last inequality follows from the classical inequality

, for x ✕ 0.


Ramona Mhanna

Abdel Razzak

Abdel Hamid

Sujeet Dutta

Ronan Lefort

Laurence Noirez

Bernhard Frick

Denis Morineau
email: denis.morineau@univ-rennes1.fr

More room for microphase separation: An extended study on binary liquids confined in SBA-15 cylindrical pores

ABSTRACT

The confinement of liquid mixtures in porous channels provides new insight into fluid ordering at the nanoscale. In this study, we address a phenomenon of microphase separation, which appears as a novel fascinating confinement effect for fully miscible binary liquids. We investigate the structure of tert-butanol-toluene mixtures confined in the straight and mono-dispersed cylindrical nanochannels of SBA-15 mesoporous silicates (D = 8.3 nm). Small angle neutron scattering experiments on samples with carefully designed isotopic compositions are performed to systematically vary the scattering length density of the different compounds and assess the radial concentration profile of the confined phases. The resulting modulation of the Bragg reflections of SBA-15 is compared with the predictions from different core-shell models, highlighting a molecular-scale phase-separated tubular structure with the tert-butanol forming a layer at the pore surface, surrounding a toluene-rich core. The present structural study suggests that the microphase separation phenomenon in confinement, which so far had only been reported for a smaller pore size (D = 3.65 nm) and a unique mixture composition must be considered as a general phenomenon. It also highlights the strength of neutron scattering method with isotopic substitution, which is a unique experimental approach to reveal this phenomenon.

I. INTRODUCTION

In the past decades, the modification of the properties of matter at the nanoscale has been addressed through the impregnation of molecular fluids into the nanochannels of mesoporous structures. These studies have shown that the structural and dynamical properties under spatial confinement could not be accounted in terms of the bulk analogs due to the strong interfacial and finite size effects that dominate on scales comparable to a few molecular diameters. [1][2][3][4][5][6][7][8][9] In particular, the formation of original structures of simple liquids in confined geometry has been intensively investigated. 6,7,[9][10][11][12][13] The influence of the confinement has been extended to the dynamics of binary liquids, providing indirect evidence for a microphase separation of fully miscible mixtures of alcohol aqueous mixtures confined in MCM-41 mesostructured porous silicates. [14][15][16][17][18] In a recent study, we conducted a thorough structural analysis to characterize the nature of microphase-separated systems. 19,20 We determined the concentration profile across the nanochannels by implementing an original application of neutron diffraction properties with isotopic substitution and derived a quantitative model to characterize the binary liquid structure.

We concluded on the existence of a novel surface-induced core-shell structure for fully miscible tert-butanol (TBA)-toluene (TOL) binary mixtures confined in MCM-41 silicates with a pore size D = 3.65 nm. In this system, tert-butanol segregates as a pore surface layer of about one atomic size surrounding a toluene-rich core of two to three atomic layers. The preferential segregation of the alcohol at the surface illustrates the amphiphilic character of TBA, supported by the observation of supramolecular clusters in the bulk TBA-TOL mixtures, 21 and the higher affinity of alcohol to silica, which was demonstrated by binary gas adsorption 22 and by the formation of interfacial H-bonded alcohol clusters in alcohol-alkane mixtures in contact with hydrophilic surfaces. [23][24][25][START_REF] Mizukami | Adsorption and Nanostructures[END_REF][START_REF] Mizukami | [END_REF] The aim of the present study is to question the generalization of this microphase-separation concept. Can we observe such phenomenon for a different pore size and up to which length scale does it extend? Is the phenomenon limited to a specific mixture (i.e. with a volume fraction around 50/50) or can it be observed for a wider range of compositions? To tackle such questions we use SBA-15 silicate hosts which are quite interesting for addressing the question of the pore size effects, because they are rather similar to MCM-41 mesoporous silicates with a 2-D long range triangular order of cylindrical channels but with a pore diameter that is twice as large (D = 8.3 nm). Moreover, changing the mixture composition proved to be an efficient and straightforward way to check the domain of stability of the microphase separated structure and to assess the corresponding core and shell proportions.

II. EXPERIMENTAL DETAILS

Hydrogenated solvents TBA and TOL (>99%) and fully deuterated solvents TBA (C4D10O, 99.8%) and TOL (C7D8, 99.5%) were purchased from Sigma-Aldrich and from Eurisotop respectively, and used directly without further purification. The mesoporous silicates SBA-15 were prepared in our laboratory according to a procedure similar to that described elsewhere 28 with minor modifications in the thermal treatments to optimize the final structure of the product. 29 Nonionic triblock copolymer (Pluronic P123): (EO)20(PO)70(EO)20 was used as template to get a mesostructured triangular array of aligned channels with pore diameter D = 8.3 nm, as confirmed by nitrogen adsorption, transmission electron microscopy and neutron diffraction. The calcined matrix was dried at 120°C under primary vacuum for 12 hours prior to the neutron experiments.

The neutron diffraction experiments were performed on the PAXY small angle neutron scattering diffractometer at the Laboratoire Léon Brillouin (CEA-CNRS Saclay). The chosen sample detector distance was 1.6 m and the neutron wavelength 0.6 nm, allowing to measure the diffracted intensity in a momentum transfer i.e. q-range from 0.3 to 3 nm -1 . The measurements were conducted at room temperature with a thermal bath regulator set to 22°C. The empty SBA-15 was packed in 1 mm thick quartz Hellma cells, and then filled by liquid imbibition with the appropriate amount of TBA-TOL mixtures injected in the sealed cells from a syringe to allow complete loading of the porous volume VP = 1.02 cm 3 g -1 , measured by nitrogen adsorption. The neutron scattering measurements were started about one hour after the liquid injection, and the spectra were acquired periodically (typically one spectrum per sample every 1.5 hour) for a total period of about 15 hours. This allowed monitoring the porous matrix imbibition process in real-time and confirmed the complete filling of the porosity from the time evolution of the spectra. The sample mass was verified after the neutron scattering measurements to confirm the absence of any evaporation from the sealed cell. The intensities were corrected from empty cell contribution by subtraction of the filled and empty sample spectra, divided by their own measured forward transmission. The correction of the detector efficiency and normalization were performed with respect to the measured flat intensity of hydrogenated water filled in the same cell.

III. RESULTS

In this manuscript, we develop a procedure which is adapted from the previous work on MCM-41 materials, and based on the study of two sets of experiments. 19 The first set of results comprises different types of contrast matching liquids (i.e. unary and binary mixtures), the second different mixture compositions, which fulfill selective contrast matching conditions.

A. Contrast matching liquids

The diffraction pattern of the empty SBA-15 host exhibits Bragg reflections at relatively small angle vector, due to the regular large lattice spacing of the pores which can be indexed by the Miller indices (hk), using as base a triangular 2-D lattice (cf. Fig. 1). In the case of a homogeneous pore filling with a constant density profile across the pore, the intensity of the Bragg reflections 𝐼 ℎ𝑘 𝑓𝑖𝑙𝑙𝑒𝑑 of the filled samples are related to the intensity 𝐼 ℎ𝑘 𝑒𝑚𝑝𝑡𝑦 of the empty SBA-15 by a scaling factor defined as:

𝐼 ℎ𝑘 𝑓𝑖𝑙𝑙𝑒𝑑 𝐼 ℎ𝑘 𝑒𝑚𝑝𝑡𝑦 = [(𝑛𝑏 ̅ ) 𝑆𝑖𝑂 2 -(𝑛𝑏 ̅ ) 𝑙𝑖𝑞𝑢𝑖𝑑 ] 2 [(𝑛𝑏 ̅ ) 𝑆𝑖𝑂 2 -0] 2 ( 1
)
where 𝑛 is the number density of the molecules within the pore, 𝑏 ̅ is the sum over the coherent scattering lengths of all atoms within the molecule and (𝑛𝑏 ̅ ) is the molecular scattering length density. 19,20,30 Therefore a complete extinction of the Bragg peaks intensity is expected when

(𝑛𝑏 ̅ ) 𝑆𝑖𝑂 2 = (𝑛𝑏 ̅ ) 𝑙𝑖𝑞𝑢𝑖𝑑 .
This situation is realized when the matrix is filled completely with a contrast matching liquid, i.e. the pore filling liquid has the same average scattering length density as silica, which can be realized by varying the D/H isotopic composition of the liquid. 19 We used 54% D/H volume fraction for TOL and 55% D/H for TBA to obtain compositions that fulfill an almost perfect contrast matching condition with (𝑛𝑏 ̅ ) 𝑆𝑖𝑂 2 = (𝑛𝑏 ̅ ) 𝑙𝑖𝑞𝑢𝑖𝑑 = 34.6 × 10 9 𝑐𝑚 -2 assuming that the matrix consists of amorphous silica (2.2 g.cm -3 ). (b) (10) The scattering intensity of the pure contrast matching TOL and TBA in SBA-15 is shown in Fig. 2 (a) and Fig. 2 (b) as a function of time after injection of the liquid within the sealed cell containing the dry porous matrix. The relative intensity of the Bragg reflections (( 10), ( 11), ( 20) and ( 21)) varies rapidly after injection and finally reaches constant values. Their positions remain constant, which demonstrates that the crystalline structure of the pores is unchanged by liquid confinement.

It is not excluded, and even likely, that the mesoporous material is deformed elastically by the action of capillary forces, but according to the literature the strain of the order of 10 -4 to 10 -3 would be much too small to be observed in our SANS experiment using a wavelength rotating selector with a resolution of about 10%. 31,32 Three important conclusions can be derived from these realtime experiments: (1) using this imbibition method, it appears that a delay of about 10 hours is required to reach an equilibrium state that corresponds to the total imbibition of the porous matrix. This result is consistent with previous studies. 19 (2) The variation of the Bragg peaks, viz. (10) and (21), is non-monotonous, and exhibits maximum values for intermediate times. A possible explanation is the faster adsorption of molecules within the silica microporosity that results in a larger density contrast between the empty porous volume and matrix walls, and so a higher scattered intensity than for the corresponding empty matrix. It is worth pointing out that the dynamics of pore filling, related to a slow and non-monotonic equilibration of the diffraction pattern, possibly depends on many experimental parameters, which are specific to the material (pore size, grain size c.a. 1m, surface chemistry…), but also to the experimental conditions

A first direct evidence for the microphase separation of the TBA-TOL mixture in the SBA-15 nanochannels is obtained with a binary liquid comprising hydrogenated TBA-(H) with deuterated TOL-(D) with a volume composition 37/63, so that it fulfills the contrast matching condition.

Indeed, Fig. 3 (b) shows that the extinction of the structure factor of the pore lattice is not achieved under these circumstances. From the time evolution of the scattered intensity (inset of Fig. 3 (a)), the imbibition process is characterized by the decrease and the eventual total extinction of two Bragg peaks indexed (11) and ( 20). But at the same time the intensity of the (10) peak increases, reaching a final value which is as high as 1.5 times that of the empty SBA-15. This uncommon behavior is also observed for the reflection (21), which was negligible for the empty matrix and increases remarkably after filling. The comparison between the spectra of the filled and empty SBA-15 (Fig. 3 (b)) obtained after the equilibrium period, demonstrates the absence of systematic Bragg peak extinction and reveals that the intensities of the different Bragg orders are not affected in the same manner but are rather modulated. This proves that the approximations supporting Eq.

(1) are no longer fulfilled. Generalizing the observations made for MCM-41, we can conclude without further analysis that the filling of the porous channels of SBA-15 with the contrast matching TBA/TOL mixture is complete but not homogeneous, which means that the concentration of the two types of molecules varies within the pore volume.

B. Twin binary liquids

The second strategy conducted to investigate the nature of the structuring of TBA-TOL mixtures in SBA-15 is the study of "twin binary liquids", which satisfy three specific features:

IV. DATA EVALUATION AND CORE-SHELL MODEL

A. Data evaluation

The diffraction spectra comprise two different contributions: a continuous background and Bragg peaks located at the same qhk positions as the empty SBA-15, with h and k being the Miller indices of the triangular crystalline lattice. The background is mostly related to incoherent scattering possibly due to the presence of hydrogen from surface silanol groups and a low-q diffuse scattering. The structural information related to the confined mixture could be extracted from the intensity of the Bragg peaks peaks after removal of the background IBg(q) interpolated with a pseudo Voigt function. Using the same procedure as ref. 19, the Bragg peaks were fitted by a weighted sum of unitary Gaussian functions 𝐺 ℎ𝑘 (cf. Fig. 5) centered at the Bragg peaks positions 𝑞 ℎ𝑘 , thus reflecting the crystalline order of the triangular lattice and conforming to the Eq. ( 2) :

𝐼 𝑓𝑖𝑡 (𝑞) -𝐼 𝐵𝑔 (𝑞) = 1 𝑞 2 ∑ 𝑀 ℎ𝑘 ℎ𝑘 𝐼 ̃(𝑞 ℎ𝑘 )𝐺 ℎ𝑘 (𝑞 -𝑞 ℎ𝑘 ) (2)
𝑀 ℎ𝑘 is the line multiplicity (𝑀 ℎ𝑘 = 6 for (h0) or (hh) and 12 otherwise), and 𝐼 ̃(𝑞 ℎ𝑘 ) = ∫ 𝐼(𝑞 ℎ𝑘 )𝑞 2 𝑑𝑞, is the integrated Bragg peak intensity.

fluids in mesoporous silicates. [33][34][35][36][37][38][39] The neutron diffraction intensity of the SBA-15 materials, considered as a periodic triangular array of cylindrical pores is given by

𝐼(𝑞) = 𝐾𝑆(𝑞)|𝐹(𝑞)| 2 (3)
where K is an instrumental normalization constant factor, S(q) the powder averaged structure factor of the SBA-15 matrix and F(q) the scattering amplitude of a single pore. S(q) is a sum of Dirac functions , while the form factor F(q) comprises the essential structural information about the binary liquid confined in the porous channel according to Eq. ( 4).

𝐼(𝑞) = 𝐾 𝑞 2 ∑ 𝑀 ℎ𝑘 ℎ𝑘 𝛿(𝑞 -𝑞 ℎ𝑘 )|𝐹(𝑞)| 2 (4)
Now comparing Eqs. ( 2) and ( 4) provides the principle of the data analysis, which consists in the evaluation of |𝐹(𝑞)| 2 for given models and the comparison of the predicted values 𝐾|𝐹(𝑞 ℎ𝑘 )| 2 with the measured Bragg peak integrated intensities 𝐼 ̃(𝑞 ℎ𝑘 ).

B. Core-shell model

Now we detail different models, to describe the structure of the confined phases. It should be mentioned that SBA-15 being a 2D crystal, with a translational invariance along the direction of the channel axis, only the averaged structure of the confined liquids within a plane perpendicular to the channel axis matters. Moreover, considering the cylindrical symmetry of the pores, the structure of the confined liquid could be simply expressed in terms of a radial concentration profile across the pore diameter. 19 A concentric cylindrical core-shell model was chosen as the simplest way to describe the concentration profile in the pore: indeed it assumes only two different values of concentration. The static lattice distortions in the pore position from an ideal triangular lattice were accounted for by a Debye-Waller factor (DWF), e -〈u 2 〉Q 2 2 as shown in Fig. 6 (a). 19,33,35,36 Three levels of the scattering length density 𝜌 = 𝑛𝑏 ̅ are considered, corresponding to the core (𝜌 𝑐𝑜𝑟𝑒 ), shell (𝜌 𝑠ℎ𝑒𝑙𝑙 ), and silica (𝜌 0 ) as shown in Fig. 6

Where 𝜌 𝑆𝑖𝑂 2 is the coherent scattering length density of silica and 𝑅 𝑝𝑜𝑟𝑒 = 4.15 nm is the pore radius determined from nitrogen adsorption based on the improved Kruk-Jaroniec-Sayari method. 40 The function Z is given by Z(x) = 2 J1(x)/x, with J1 being a first-order Bessel function.

Silica Shell

Core

r 0 r shell r core R pore R core (b) (a)
The value of the scattering length density of silica (𝜌 𝑆𝑖𝑂 2 = 34.6 × 10 9 𝑐𝑚 -2 ) was used in agreement with the contrast matching experiments illustrated in the previous part.

In the case of a filled matrix, we have proposed three possible concentration profiles: 1) A homogeneous filling of the pore by the mixture (single concentration profile). The resulting diffraction pattern is simply proportional to that of the empty SBA-15, with a different contrast factor in F(q); (i.e., simple rescaling of the intensity with respect to the empty matrix). 2) A complete microphase separation of the two constituents of the binary mixture into a cylindrical core-shell structure where either TBA (later denoted TBA@surface model) or TOL (later denoted TBA@core model) is segregated at the surface of the pore, and the other component located in the core of the pore. The form factor for this core-shell structure is:

𝑭 𝑪𝒐𝒓𝒆-𝑺𝒉𝒆𝒍𝒍 (𝒒) = 𝟏 𝝆 𝑺𝒊𝑶 𝟐 𝑹 𝒑𝒐𝒓𝒆 𝟐 ∑ (𝝆 𝒊 -𝝆 (𝒊-𝟏) )𝑹 𝒊 𝟐 𝒁(𝒒𝑹 𝒊 ) 𝟐 𝒊=𝟏 (7)
Where 𝜌 0 = 𝜌 𝑆𝑖𝑂 2 , 𝜌 1 = 𝜌 𝑠ℎ𝑒𝑙𝑙 , 𝜌 2 = 𝜌 𝑐𝑜𝑟𝑒 , 𝑅 1 = 𝑅 𝑝𝑜𝑟𝑒 , and 𝑅 2 = 𝑅 𝑐𝑜𝑟𝑒 . It is noteworthy that these models do not require any fitting parameter specific to the confined liquid and thus provide predictive results that could be directly compared to the experimental values. Indeed, the only two unknown values viz. the scaling factor K and the MSD 〈𝑢 2 〉, were extracted from the fits of the empty SBA-15. The core radius Rcore is simply determined from the mixture composition xTBA. while rshell andrcore are the scattering length densities of the pure liquids. In the case where TBA is forming the shell (TBA@surface model), the core size is 𝑅 𝑐𝑜𝑟𝑒 = √(1 -𝑥 𝑇𝐵𝐴 ) 𝑅 𝑝𝑜𝑟𝑒 .

Otherwise, TBA forms the core and corresponds to: 𝑅 𝑐𝑜𝑟𝑒 = √ 𝑥 𝑇𝐵𝐴 𝑅 𝑝𝑜𝑟𝑒 (TBA@core model).

3) Monolayer at surface: this concentration profile is rather a variant of the TBA@surface coreshell model. It represents the case of the segregation of one molecular layer of TBA of thickness emono at the pore surface while the remaining quantity of the TBA (not constituting the interfacial layer) mixes with the TOL inside the pore. In this TBA-layer@surface model, the core

concentration of TBA is 𝑥 𝑐𝑜𝑟𝑒 = 𝑥 𝑇𝐵𝐴 𝛼 2 + (1 - 1 𝛼 2 )
where 𝑥 𝑇𝐵𝐴 is the average composition of the filling liquid and 𝛼 = 1 -

𝑒 𝑚𝑜𝑛𝑜 𝑅 𝑝𝑜𝑟𝑒
with the thickness of the monolayer being set to emono = 0.6 nm (average molecular diameter of one TBA molecule obtained from a structural study on bulk TBA 41 ). The core size is then simply defined as: 𝑅 𝑐𝑜𝑟𝑒 = 𝑅 𝑝𝑜𝑟𝑒 -𝑒 𝑚𝑜𝑛𝑜 .

Some studies have shown that SBA-15 could be described by a variant model where the cylindrical pore structure exhibits corrugated pore walls and/or an interfacial microporous region denoted as a 'corona'. 33,35,36,37,38,39 Therefore, a structural model accounting for a reduced density sponge-like corona around the pores was also evaluated. It is based on the assumption that the cylindrical mesopores that follow a perfect 2-D hexagonal lattice with a circular cross section are surrounded by a microporous corona with constant porosity, while dense silica forms the outer region. Since no significant difference was observed between the DWF and the corona models, we

V. DISCUSSION

The strategy used in this study consists in comparing the scattering patterns displayed by silica nanotubes (SBA-15) filled by isotopically different liquid mixtures with models, without any fitted parameters. It means that all the parameters were first obtained independently from the filled samples : the pore radius was fixed to the value Rpore = 4.15 nm determined from nitrogen adsorption based on the improved KJS method, 40 while the scaling factor K and the mean square displacement (MSD) 〈𝑢 2 〉 parameters were estimated from fitting the form factor of the empty matrix as shown in Fig. 7. The obtained value of the MSD is 〈𝑢 2 〉 = 0.5𝑛𝑚 2 , which corresponds to average displacements in the order of 0.7 nm, i.e. about 7% of the unit cell parameter (

√〈𝑢 2 〉 𝑎 =7%)
, is in a good agreement with previous studies. 19,33 Fig. 7. Experimental value of the integrated intensity of the four Bragg peaks of the empty SBA-15. The solid line is the result from the model fitting the scaling factor K and the mean square displacement 〈𝑢 2 〉. The shading indicates the accuracy limit of the method.

The values of the core radius Rcore and the thickness of the shell eshell are fully determined from the sample composition and pore size. They are given in Table 1 for all the studied compositions and

both TBA@surface and TBA-layer@surface models mathematically equivalent for this particular sample.

Table 1. Geometrical parameters of the two core-shell models and the monolayer model for the studied compositions. Note that these parameters were not fitted, but fixed by the composition of the mixture (Rcore) or retrieved from the study of the empty SBA-15 (Rpore and 〈𝑢 2 〉).

We now test the ability of the computed form factors |F(q)| 2 to reproduce the extracted experimental intensities I ̃hk for different samples and different models. We first consider the TBA-(H)-TOL-(D) sample with the contrast matching composition (37/63). The predictions from the four different models are compared with the experimental values of the Bragg reflection intensity in Fig. 8. The radial profiles of the scattering length density across the pore are also sketched for the different models. Clearly, the model with homogeneous concentration profile fails to reproduce the experimental results, predicting an almost total extinction of the Bragg peaks for this almost perfect contrast matching composition. The predictions of the three other models all provide an explanation of the striking observation of nonvanishing Bragg peak intensities for this contrast-matching composition. They successfully predict a high intensity of the (10) Bragg peak. However, the TBA-layer@surface and TBA@surface models seem to be in better agreement than the TBA@core, especially for predicting the TBA@surface TBAlayer@surface TBA@core Homogeneous Experimental Bragg intensity volume fraction 37/63%. The lines are the predicted results from the model calculations according to Eq. ( 6) for four different cases: full mixing (homogeneous), microphase separated core-shell structure with TBA at the surface (TBA@Surface), microphase separated core-shell structure with TBA monolayer at the surface (TBA-layer@Surface) and microphase separated core-shell structure with TOL at the surface (TBA@Core). Right panel: Sketch of the scattering length density profile.

Model

The indication that TBA rather than TOL stands at the pore surface is definitively confirmed by the other binary samples. To illustrate this point, we now consider the predictions for the two twin binary mixtures shown in Fig. 9. Again, the experimental trends are nicely predicted by the TBA@surface and TBA-layer@surface models as they could remarkably illustrate the unexpected difference in the Bragg peak intensities between the two twin samples. Both models have an equally good reproduction of the ( 10 The intensities of the higher orders are also well-reproduced, providing an explanation for the inversion in the relative intensities of the (11) and (20) Bragg peaks between the two twin samples:

On the one hand, the reduced intensity of the (11) Bragg peak in the TBA-(CM) / TOL-(D) (cf. Fig. 10 (a)) can be explained by the proximity of a cancellation point in the form factor. On the other hand, in the TBA-(D) / TOL-(CM) sample, the cancellation point in the form factor in both models is located at the same position as the (20) order peak, leading to its extinction (cf. Fig. 10 (b)). Generally, it is worth mentioning that, although the attained agreement remains semiquantitative, all the experimental systematic trends are successfully reproduced with the two simple models without any adjustment of their parameters. At variance, the TBA@core model fails to reproduce the observed experimental features. TBA@surface TBAlayer@surface TBA@core Experimental Bragg intensity (b) mixed with its contrast matching counterpart. (A): contrast matching TBA mixed with deuterated TOL and (B): deuterated TBA mixed with contrast matching TOL. The solid lines are the predicted results for three different models: microphase separated core-shell structures with TBA at the surface (TBA@surface), with one molecular TBA layer at the surface (TBA-layer@surface) and with TOL at the surface (TBA@core). Right panel: Sketch of the scattering length density profile.

The core-shell microphase separation is original with respect to the current knowledge of confined binary liquids because it concerns a fully miscible mixture. It is interesting to make a comparison with the existing literature on the liquid-liquid phase separation of binary liquids that already exhibit a miscibility gap in the bulk. 42 According to molecular simulation studies of fluids, the thermodynamics and kinetics of demixing are deeply affected by confinement in infinite cylindrical pores. 43,44,45 It was observed that the system does not phase separated completely but form long-lived metastable pluglike microdomains. A dramatic slowdown of the kinetics of phase separation is induced by the collective motions required for the condensation of two domains separated by a plug of the unlike molecules. This situation differs obviously from our observation, because plug-like structures would effectively lead to full extinction of the Bragg intensity for contrast matching mixtures. In the former studies, the formation of tube-like structures was unfavored by the choice of symmetrical interactions between the two types of molecules and the pore. On the contrary, surface-induced phase transition and the formation of an adsorbed film near the pore surface were observed for asymmetric binary systems. 46 In line with our observations, it shows that the strength of the pore wall interaction with one of the two molecules plays on central role in the formation of surface-directed tube-like microstructures that were observed even at temperature above the two-phase coexistence region.

VI. CONCLUSION

There is currently a growing number of observations reporting on microphase separation induced under nanoporous confinement in liquid mixtures, which are under normal bulk conditions fully miscible. 14,15,16,47,48 However direct evidence for such a phenomenon form structural investigations are scarce. 19,20 In this article, we exploit both the unique properties of neutron scattering regarding isotopic labeling and the additional concentration parameter offered for mixtures to carefully design selective contrast conditions which undoubtly demonstrate the spontaneous microphase separation of different TBA-TOL miscible liquid mixtures under confinement in the nanochannels of SBA-15 mesoporous materials (D = 8.3 nm).

These conclusions were obtained from the analysis of the diffraction pattern produced by filled and unfilled SBA-15 pore lattices, which can be directly related to the radial concentration profile of the confined mixture across the channel. In that aspect, we implemented simple core-shell models having the advantage of being fully predictive, with parameters already determined and fixed by the study of the empty matrix. We derived different variant of the model, assuming either the complete phase separation of the two constituents (TBA@surface and TBA@core), or the formation of one molecular layer (TBA-layer@surface). We also accounted for the irregularity of the SBA-15 porous geometry, in line with the existing literature, introducing either an effective Debye-Waller factor or a microporous corona. Regardless of their simplicity, all the core-shell models predictions that assume the preferential segregation of TBA at the pore surface surrounding a TOL-rich core nicely reproduce all the experimental trends. They explain the variation of the Bragg intensity at the semi-quantitative, and most often at the quantitative level, depending on the system.

The obtained concentric core-shell tubular structure corresponds to the segregation of alcohols in an interfacial TBA-rich region of about one to three molecular sizes, surrounding a TOL rich liquid tube. This result extends the observation firstly reported for the same liquid mixture confined in the MCM-41 materials with smaller pores (D = 3.65 nm). 19 In addition, it shows that the microphase separation, which was first exemplified for binary mixtures with a TBA volume x = 0.51 and 0.43, can be observed for a broader range of compositions, at least down to x = 0.3. This indicates that the original nano-demixing phenomenon can be extended to different confining matrices with larger pore sizes and different mixture compositions, answering the questions raised in the introduction part of the article about the general character of the concept of microphase-separation.

However the overall results obtained for SBA-15 and MCM-41 mesoporous materials remain limited to hydrophilic matrices so far. It would be certainly interesting to vary the nature of interfacial interaction and the pore surface, using hydrophobic matrices in order to assess different anticipated core-shell structures. Another approach would be changing the nature of the liquids to test the applicability of this nanostructuring on mixtures with different structural properties (aromaticity, size, hydrophobicity…). It is noteworthy that TBA spontaneously forms supermolecular cyclic clusters in the bulk state, although TBA-TOL mixtures are homogeneous at the macroscopic scale. 21 It would be of great interest to assess how such self-assemblies, which are observed in the bulk mixtures could influence the formation of core-shell structures in the confined state. For that purpose, one should also consider the case of smaller alcohol molecules, such as methanol or ethanol, in order to investigate different chain-like H-bonded structures.

Ongoing structural studies will address these issues making use of the benefits of the crystallinity of the arrangement mesoporous channels and the selective isotopic sensitivity of the neutron interaction in order to achieve a unique level of description.

Finally, our study suggests that microphase separation, as a new type of nanostructure results from the concomitant effects of specific surface interactions and spatial confinement. The modulation of both elements could provide new opportunities to control supermolecular assembling of complex mixtures. tubular structure with the segregation of tert-butanol that forms a layer at the pore surface surrounding a toluene-rich core.

FIG. 1 .FIG. 2 .

 12 FIG. 1. Neutron diffraction patterns of empty SBA-15 (solid black line) and SBA-15 filled with

(FIG. 3 .

 3 FIG. 3. (a) Time evolution of the neutron diffraction patterns of empty SBA-15 (dashed line), and

FIG. 4 .

 4 FIG. 4. Neutron diffraction patterns of SBA-15 filled with twin binary liquids made of an isotopic

FIG. 5 .

 5 FIG. 5. Neutron diffraction intensity of empty SBA-15. The Bragg peaks are labelled by their

FIG. 6 .

 6 FIG. 6. Schematic representation of (a): the static random disorder of the pore positions from the

  𝑭(𝒒) 𝑯𝒐𝒎𝒐 = (𝝆 𝒍𝒊𝒒 -𝝆 𝑺𝒊𝑶 𝟐 ) 𝝆 𝑺𝒊𝑶 𝟐 𝑭(𝒒) 𝑬𝒎𝒑𝒕𝒚 = (𝝆 𝑳𝒊𝒒 -𝝆 𝑺𝒊𝑶 𝟐 )𝑹 𝒑𝒐𝒓𝒆 𝟐 𝒁(𝒒𝑹 𝒑𝒐𝒓𝒆 ) 𝝆 𝑺𝒊𝑶 𝟐 𝑹 𝒑𝒐𝒓𝒆 𝟐(6)

  based our discussion solely on the DWF model, and the results obtained by this model were shown as supplementary material. It suggests that the observed variations of the diffraction patterns with the isotopic composition of the binary liquids are essentially determined by the microphase separation phenomenon with the formation of a core-shell structure and less sensitive to the details of the model used to represent the SBA-15.

  models. Depending on the model and on the mixture composition, these geometrical values span from one to five molecular sizes. It is interesting to note that for the specific composition TBA/TOL 30/70 the shell thickness eshell is equivalent to about one molecular layer. This makes

experimental intensities the second ( 11 )FIG. 8 .

 118 FIG. 8. Experimental value of the integrated intensity of the four Bragg peaks of the SBA-15 filled

  ) Bragg reflection, which is very intense for the TBA-(CM) / TOL-(D) system (cf. Fig. 9 (a)), and weak for the TBA-(D) / TOL-(CM) sample (cf. Fig. 9 (b)).

FIG. 9 .FIG. 10 .

 910 FIG. 9. Experimental value of the integrated intensity of the four Bragg peaks of the SBA-15

ACKNOWLEDGMENTS

The experiments were performed in the frame of the PhD project of R. Mhanna who acknowledges funding by the Institute Laue-Langevin and the Brittany Region (ARED 7784 / NanoBina). Support from Europe (FEDER) and Rennes Metropole is expressly acknowledged.

We thank Cristelle MÉRIADEC (Institut de Physique de Rennes) and Odile MERDRIGNAC (Institut des Sciences Chimiques de Rennes) for their assistance with the X-ray diffraction and Nitrogen isotherms experiments respectively for the characterization of the SBA-15 materials.

composition 43/57% and comprising a fully deuterated component mixed with its contrast matching counterpart. (A): contrast matching TBA mixed with deuterated TOL and (B): deuterated TBA mixed with contrast matching TOL. The solid lines are the predicted results from the model from the model calculations according to Eq. ( 6) for three different microphase separated coreshell structures with TBA at the surface (TBA@surface), TBA monolayer at the surface (TBAlayer@surface), and TOL at the surface (TBA@core). Right panel: Sketch of the scattering length density profile.

We now consider the TBA/TOL samples with a different value of the volume fraction, i.e. 30/70.

In this case, an exceptional quality of the predictions is obtained with both the TBA@surface and TBA-layer@surface models, which are in quantitative agreement with the experimental intensities of the four Bragg peaks for both the TBA-(CM) / TOL-(D) sample (Fig. 10 It is also obvious that the predictions from the TBA@surface and TBA-layer@surface models are virtually undistinguishable. It was expected since these particular samples comprise the TBA volume fraction needed to form one molecular layer at the surface, thus making these two models mathematically equivalent (see Table 1). On the contrary, the opposite hypothesis that TOL would stand preferentially at the surface can be definitively ruled out. Indeed, the TBA@core model fits with none Bragg reflection and systematically fails to reproduce the experimental intensities.

SUPPLEMENTARY MATERIALS

We also implemented the same core-shell model with an additional (corona) layer accounting for the wall structure, which was modeled by corrugated pore walls comprising an interfacial microporous region denoted as a 'corona' with a reduced silica density. The predictions from this variant model are compared with the predictions from the model detailed in the manuscript that uses a Debye Waller factor instead. Despite slight quantitative differences, the conclusions derived from both models are in full agreement, and demonstrate the existence of a micro-phase-separated


Claude Roger

An algebraic background for hierarchies of PDE in dimension (2|1)

Keywords: Schrödinger operators, Superalgebras, Supersymmetry, Hierarchies of PDE

. We shall discuss here the construction of an algebra of pseudodifferential symbols in dimension (2|1); that algebra generalizes the one for d = 1, used in construction of hierarchies from isospectral deformations of stationary Schrödinger type operators.

1 Reminders on the classic d = 1 case This is the famous Korteweg-De Vries equation and hierarchy, together with its very rich related analytical, geometrical and algebraic structures; they are described in an extensive literature, for example the treatise of Dickey [START_REF] Dickey | Soliton equations and Hamiltonian systems[END_REF], which we shall refer to. It begins with stationary Schrödinger type operators in d = 1 with variable x, like L = ∂ 2 + u(x), then one considers its isospectral deformations, by conjugacy in the space of differential operators or pseudo differential symbols, as follows: L → L t = U (t)LU (t) -1 . The latter formula gives infinitesimally a Lax type equation L = [L, M ], where as usual, the dot stands for time derivative, so L = u. For suitable values of operator M , denoted M = M p for integer p, one gets a hierarchy of equations L = [L, M p ]. If moreover, the operators M p are mutually commuting [M p , M q ] = 0, one deduces an infinite family of conserved quantities for those equations, then said to be completely integrable system with an infinite number of degrees of freedom.

Those equations are hamiltonian for the Poisson structure of the dual of Virasoro algebra (cf. [START_REF] Guieu | L'algèbre et le groupe de Virasoro[END_REF]) of vector fields u(x)∂, with the well known bracket [u(x)∂, v(x)∂] = (u(x)v (x) -v(x)u (x))∂, and central term; from that Lie algebra one constructs the associative algebra OD of differential operators of the form u 0 + u 1 ∂ + ..... + u n ∂ n , and finally, by formally inverting ∂ into ∂ -1 , one gets the division algebra of pseudo differential symbols ΨD. A symbol D ∈ ΨD has the form

D = n=N n=-∞ a n ∂ n .
Associative algebra ΨD encodes all computations necessary for construction of operators M p , and also the right Poisson structures for those hamiltonian equations, the dual of Virasoro algebra and its higher order generalizations known as W-algebras. Let's recall that the key formula for algebraic calculations in ΨD is the following:

∂ -1 a = ∞ n=0 (-1) n ∂ n (a)∂ -1-n .
That algebra allows computation of square root of L = ∂ 2 +u(x), and further its successive half-integer powers, giving hierarchies and conserved quantities. As well,Miura's transform factorizes

∂ 2 + u(x) = (∂ + v(x)) • (∂ -v(x)
and turns out to be a rather useful tool.

Non stationary Schrödinger operators

Now, we are in the d = 2 case with variables (x, t), let's set for short ∂ ∂x = ∂ ∂ ∂t = ∂ t ; we shall try to generalize the constructions described in part 1 above to non stationary operators of Schrödinger type with potential: Schr = ∂ 2 -∂ t + u(x, t).

Remarks:

1/One may think of the space parametrized by (x, t) as spacetime with space dimension 1, the variable t being the physical time, but it is not necessarily the case.

2/Those operators are, strictly speaking, heat operators, but Wick rotation t → it transforms them into the actual Schrödinger operator.

3/ Hierarchies of PDE in dimension 2 and more have been on the agenda since some time, with various point of view, cf. for example [START_REF] Sergyeyev | Central extensions of cotangent universal hierarchy: (2 + 1)-dimensional bi-Hamiltonian systems[END_REF].

In the present paper, we shall use a supersymmetric trick: we enlarge the space with an odd dimension and work in the superspace of dimension (2|1), parametrized with even variables (x, t) and one odd variable θ. The even variables can be polynomial, analytic or differentiable, we can have as well x ∈ R or x ∈ S 1 , and the same for t. We shall not make things more precise for the moment, and all necessary technical preliminaries on superalgebra will be developed in the next part.

We shall now consider the odd differential operator D θ = θ∂ t + ∂ θ , which satisfies D 2 θ = ∂ t ; so D θ represents the promised "square root of time", and we can write non stationary Schrödinger operator as a difference of two squares Schr = ∂ 2 -D 2 θ +u(x, t).

Some techniques of graded algebra

We shall recall in this part the most basic definitions and formulas for graded algebra, a detailed introduction can be found in [START_REF] Deligne | Notes on supersymmetry (following Joseph Bernstein)[END_REF]. A graded (or super-) algebra is an associative algebra which admits a graduation, following A = ⊕ p∈Z A p ; an element a ∈ A p is said to be of degree p, denoted by |a| = p. The associative multiplication is graded, i.e.

|ab| = |a| + |b|,
and supercommutative (or graded commutative) which means:

ab = (-1) |a||b| ba.
So, a and b anticommute iff they are both of odd degree, otherwise they commute.

The typical examples are the free algebras on even generators x i and odd generators θ α , written as

A = k[x i ] ⊗ Λ(θ α ),
where as usual the symbol Λ denotes exterior algebra, and k[x i ] the polynomial algebra in indeterminates x i , with coefficients in a field of vanishing characteristic k; in most examples k = C. One can also give a more global description of those superalgebras : consider two finite dimensional vector spaces E 0 and E 1 and assume that elements of E 0 (resp.E 1 ) are even (resp. odd), then the free supercommutative algebra on

E 0 ⊕ E 1 will be A = S * (E 0 ) ⊕ Λ * (E 1 ).
In most cases only modulo 2 degree is taken in account, being the only relevant part for sign formulas.

A map f between graded spaces has a degree So one has odd and even derivations. Now we can consider the graded commutator of derivations following the formula:

[δ 1 , δ 2 ] = δ 1 • δ 2 -(-1) |δ 1 ||δ 2 | δ 2 • δ 1 .
This bracket defines a Lie superalgebra structure on the space of derivations (see [START_REF] Deligne | Notes on supersymmetry (following Joseph Bernstein)[END_REF] for precise formalism about Lie superalgebras); the geometric interpretation of derivations as tangent vector fields allows to consider the space of derivations Der(A) as the Lie superalgebra of tangent vector fields on the underlying super manifold whose space of functions is A [START_REF] Roger | Algebras generated by odd derivations[END_REF]. Finally, we shall make use for any graded algebra, of the following involution: a → ā = (-1) |a| a.

The division algebra of pseudodifferential symbols in d = (2|1)

Our differential and pseudo differential operators will act on the superalgebra A of functions on superspace of dimension (2|1). We shall use as generators the following two operators: D = ∂ + iD θ and D = ∂ -iD θ , where i is present mainly for technical reasons. Computations are a bit delicate, since

D(ab) = D(a)b + aD(b) if a is even, but D(ab) = D(a)b + a D(b)
if a is odd. In terms of composition of operators, an element a ∈ A being viewed as an operator of order zero, one has

D • a = a • D + D(a)
if |a| = 0 (mod.2), and

D • a = a • D + D(a) if |a| = 1 (mod.
2). This is why we cannot consider operators in powers of D only, we must add the conjugate D in order to get a closed algebra.

Those elements generate the associative graded algebra of differential operators on A, denoted as OD(D, D). Its generic element has the following form:

f inite k≥0,l≥0 a k,l D k Dl .
We can change the generators from (D, D) to (∂, D θ ), and so OD(D, D) = OD(∂, D θ ). Straightforward computations give easily:

D 2 = ∂ 2 -∂ t + 2i∂D θ D2 = ∂ 2 -∂ t -2i∂D θ so D 2 + D2 2 = ∂ 2 -∂ t = Schr 0 ,
the non stationary Schrödinger operator with zero potential. This is why algebra OD(D, D) is relevant for our problem ! Moreover, one has D D = DD = ∂ 2 + ∂ t , and omitting the i coefficient, we could have obtained ∂ 2 -∂ t , like in [START_REF] Roger | Algebras generated by odd derivations[END_REF], where that formula was used for generalization of Miura transform.

We are now ready to construct the algebra of pseudo differential symbols ΨD(D, D) by localization of OD(D, D); let's invert formally D and D in D -1 and D-1 respectively, so a generic pseudodifferential symbol will have the form:

f inite k=-∞,l=-∞ a k,l D k Dl .
As in the d = 1 case, the difficulty is to establish a coherent formula for the composition D -1 • a; we shall prove by recurrence on the degree that :

D -1 • a = +∞ i,j=0;i+j>0 a i,j D -i D-j .
(Here we omitted symbol • when obvious). As in the d = 1 case the formulas will be obtained from D • D -1 • a = a; we shall use decomposition of any element a ∈ A according parity, a = a 0 + a 1 . We shall proceed by recurrence, from:

a = D( +∞ i,j=0;i+j>0 a i,j D -i D-j ), so: D(a i,j D -i D-j ) = D(a i,j )D -i D-j + a 0 i,j D -i+1 D-j + a 1 i,j D -i D-j+1
. One then readily deduce from above the recurrence formula

D(a i,j ) = -a 0 i+1,j -a 1 i,j+1
The recurrence is easily introduced: if a = a 0 + a 1 , then a 1,0 = a 0 1,0 = a 0 and a 0,1 = a 1 0,1 = a 1 ; so one gets a unique solution to the equation above but the general formulas for a i,j are not obvious to be made explicit; for example one has a i+1,0 = (-1) i ∂ i a 0 , and also a 0,j+1 = (-1) j ∂ j a 1 ; we shall give also some samples in low degree: a 1,1 = -iD θ a, a 1,2 = i∂D θ a -∂ t a 1 and a 2,1 = i∂D θ a -∂ t a 0 . Computations are exactly parallel when one computes D-1 • a.

Remark: In [START_REF] Roger | A Hamiltonian action of the Schrödinger-Virasoro algebra on a space of periodic time-dependent Schrödinger operators in (1 + 1)-dimensions[END_REF] we considered algebras of differential operators and pseudodifferential symbols in dimension(1|1) with variables (x, θ), variable t being a loop space coordinate; in [START_REF] Voronov | Differential operators on the superline, berezinians, and darboux transformations[END_REF] the authors give a detailed study of the algebra of differential operators in dimension (1|1), they award its paternity to Manin and Radul [START_REF] Yu | A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy[END_REF] in their work about SuSy (supersymmetric) extension of KP hierarchy. From

Schrödinger type operators in dimension (2|1)

We shall consider operators of the type D 2 +U, where potential U = u+iθφ has an odd and even part. Moreover, D 2 = ∂ 2 -∂ t + 2i∂D θ = Schr 0 + 2i∂D θ . We get an unwanted supplementary term, it can be considered as the "price to pay" for generalization to d = 2, like in the case of Miura transform [START_REF] Roger | A matrix Miura transform[END_REF].

The wave functions must be looked for among super functions as F = f + iθα and spectral values have odd and even part too: Λ = λ + iθξ. One has D 2 (F ) = Schr(f ) + iθSchr(α) + 2i∂D θ (F ); then one readily computes D θ (F ) = θ∂ t f + iα. Then:

(D 2 + U)(F ) = Schr(f ) + uf -2∂α + iθ(Schr(α) + uα + 2∂∂ t f + φf ) Finally, equation (D 2 + U)(F ) = ΛF
induces the following system:

Schr(f ) + uf -2∂α = λf Schr(α) + uα + 2∂∂ t f + φf ) = λα + ξf.
In the particular case when the wave function F is purely even, so F = f , one obtains a coupled system in f :

Schr(f ) + uf = λf 2∂∂ t f + φf = ξf.
The second equation is of Klein-Gordon type, so the system couples hyperbolic and parabolic system.

6 KdV like equations in d = 2

We shall obtain evolution equations w.r.t. a real parameter s on a pair of functions (u, φ) in variables (x, t, s) appearing as components of a super function U = u + iθφ as above. We now develop a formalism parallel to the classical d = 1 case [START_REF] Dickey | Soliton equations and Hamiltonian systems[END_REF], and thus consider the equation:

4 U s = D 3 + 6 U D(U),
One deduces from it the following system of evolution PDE:

u s = ∂ 3 u -3∂∂ t u + 6u∂u -6uφ -3∂ 2 φ + ∂ t φ φ s = ∂ 3 φ -3∂∂ t φ + 3∂ 2 ∂ t u -∂ 2
t u + 6φ∂u + 6u∂φ -6φ 2 + 6u∂ t u Let's now consider the case when the potential U is purely even, when φ = 0. The second equation yields :

3∂ 2 ∂ t u -∂ 2 t u + 6u∂ t u = 0, so:

∂ t (3∂ 2 u -∂ t u + 3u 2 ) = 0.

One deduces immediately:

3∂ 2 u -∂ t u + 3u 2 = f (x)( * ),
where f is an arbitrary function independent of t. The first equation gives :

u s = ∂ 3 u -3∂∂ t u + 6u∂u
. Now by deriviation of equation ( * ), one obtains 3∂ 3 u -∂∂ t u + 6u∂u = ∂f (x)

and we can cancel the non linear term, getting the linear dispersive equation with sources:

u s = -2∂ 3 u -2∂∂ t u + ∂f (x) ( * * )

  |f | naturally defined by |f (a)| = |f | + |a|, and we shall consider derivations of graded algebras. A map δ : A → A is a derivation if for any a, b ∈ A, one has: δ(ab) = δ(a)b + (-1) |a||δ| aδ(b).

D 2 θ

 2 = ∂ t , one can consider OD(D, D) = OD(∂, D θ ) as a quadratic extension of OD(∂, ∂ 1 ) which is simply the algebra of differential operators in d = 2.; but those algebras are neither commutative nor anticommutative. Let's stress also the fact that OD(D, D) = OD(∂, D θ ) doesn't imply that ΨD(D, D) = ΨD(∂, D θ ), since different choice of generators to be inverted change the global algebraic structure.

Laboratoire associé au CNRS UMR 5208

Another option is to cancel the term with ∂ t between (*) ans the first equation, and one gets an equation in (x, s) without t as follows:

after change of scale s → -s and x → 2x, one obtains finally:

which is exactly Korteweg De Vries equation with source. Now, if U is purely odd, i.e. u = 0, the above system reduces to:

Finally, cancellation of terms in ∂ t will give the following equation in (x, s):

Any solution of that equation in (x, s) can be prolongated to a general solution in (x, s, t), simply by using a heat kernel.


N Pinchon

M Ibn-Khedher

O Cassignol

A Nicolas

F Bernardin

P Leduc

J-P Tarel

R Brémond

E Bercier

G Julien

All-weather vision for automotive safety: which spectral band?

Keywords: vision, visibility, bad weather, fog, infrared

The AWARE (All Weather All Roads Enhanced vision) French public funded project is aiming at the development of a low cost sensor fitting to automotive and aviation requirements, and enabling a vision in all poor visibility conditions, such as night, fog, rain and snow. In order to identify the technologies providing the best all-weather vision, we evaluated the relevance of four different spectral bands: Visible RGB, Near-Infrared (NIR), Short-Wave Infrared (SWIR) and Long-Wave Infrared (LWIR). Two test campaigns have been realised in outdoor natural conditions and in artificial fog tunnel, with four cameras recording simultaneously. This paper presents the detailed results of this comparative study, focusing on pedestrians, vehicles, traffic signs and lanes detection.

Introduction

In the automotive industry, New Car Assessment Programs are increasingly pushing car manufacturers to improve performances of Advanced Driver Assistance Systems (ADAS), and especially autonomous emergency braking on vulnerable road users (VRU). For instance the 2018 Euro Ncap roadmap is moving towards pedestrians and pedal cyclists detection in day and night conditions. This trend matches accidentology figures, like those provided by French Road Safety Observatory [ In the longer term, after automated parking and highway driving, all weather and city driving will be the main technical challenges in the automated driving roadmap.

Current ADAS sensors as visible cameras or Lidars are fitting functional requirements of VRU and obstacle detection in normal conditions (day or night). However, these technologies show limited performances in adverse weather conditions such as fog or rain.

Automotive industry is thus facing this new challenge of detecting vehicle environment in all conditions, and especially in poor visibility conditions, such as night, fog, rain and snow. This topic has been addressed in the framework of the AWARE French public funded project, aiming at the development of a sensor enabling a vision in all poor visibility conditions. This paper presents an experimental comparative study of four different spectral bands: Visible RGB, Near-Infrared (NIR), Short-Wave Infrared (SWIR) and Long-Wave Infrared (LWIR). Sensors and field tests are described in section 2 and 3. Experimental results are detailed in section 4, focusing on pedestrians, vehicles, traffic signs and lanes detection.

Sensors

In this project, we only focus on cameras technologies, and not on distance measurement systems like LIDARs or RADARs. But it is wellknown that both technologies are complementary and necessary to bring redundancy and complementary characteristics for improving the detection system's reliability and accuracy [2]. Four cameras were tested during the project. The Table 2 shows their characteristics. The visible RGB CMOS camera is used here as a reference for the test. Extended SWIR camera is based on InGaAs III-V material and extends from a wavelength of 0.6 µm, red to human eye, to 1.7 µm in the SWIR infrared band. SWIR spectral band is typically used for active (reflective) vision in very dark condition with a good contrast as SWIR light is generally more reflective than visible light. LWIR sensor is an array of microbolometers. It detects the thermal radiation in the spectral band extending from 8 µm to 14 µm. Any object emits radiations which depend on its temperature. For a human or an animal at ambient temperature, the maximum of emission corresponds to a wavelength close to 10µm. LWIR is used for the detection of a temperature contrast and do not require an illuminator.

Mid-Wave Infrared (MWIR) has not been added in the study for reasons of cost and compacity, due to the cooling system required for the detectors.

Field tests

Outdoor test campaign

The four cameras were installed during one month in 2015 along the French motorway A75/E11 at the site La Fageole. A weather station located near the cameras was equipped with a diffusometer (meteorological visibility), a rain gauge, a luxmeter (ambient light) and a temperature and humidity sensor (ambient air). The tests were carried in the CEREMA fog tunnel (30m length, 5.5m width and 2.5m height). Artificial fog and rain are reproduced and controlled: fog and rain drop size, meteorological visibility of fog and rain intensity. Two fog classes are available: unimodal drop size distribution (DSP) centred around 1 micron and bimodal DSP centred around 1.5 and 10 microns. 74 scenarios were performed by varying the type of adverse weather conditions (fog and rain), scene illumination (night, day, automotive lighting, aircraft lighting) and glare (front vehicle).

The scene in front of the cameras contained road pavement, road marking, pedestrians and lights (see Figure 2).

Experimental results

In this section, we describe the detection and recognition range performances that were measured in this study. It is important to keep in mind that these results reflect not only the intrinsic characteristics of the spectral bands but also the capability of the chosen cameras. The cameras were selected to be representative of the typical current state-of-the-art.

In order to prevent any detection algorithm artefact, a visual analysis has been performed by two different human observers. As expected, exact detection range values differed from one observer to the other, but the relative values were consistent. In all cases, brightness and contrast were carefully tuned in order to optimize ranges. With respect of each camera's channels created by the four spectral bands, a video database has been created to remotely record videos of relevant scenes for each listed scenario. The Figure 3 provides a sample of the video database (outdoor campaign):  The LWIR camera is the only one that allows pedestrian detection in full darkness.

 The LWIR camera also proved more resilient to glare caused by facing headlamps in the fog.

Other cameras sometimes missed a pedestrian because she or he was hidden by the glare. In VIS, NIR or SWIR, detection was performed using the vehicle headlamps. The case of a vehicle driving with headlamps off while in adverse conditions was not encountered in this study. Should it happen, however, detection ranges in VIS, NIR and SWIR would be on the order of the recognition ranges. In LWIR, detection relied on the observation of hot vehicle parts: the wheels, the motor or the exhaust system.

In some foggy instances, recognition proved difficult or even impossible in VIS, NIR or SWIR because the vehicle remained entirely hidden by the glare of its own headlamps. This was particularly frequent in the SWIR band for facing vehicles. When this happened, the vehicles were not taken into account in the average ranges given in The visual observation of the videos also confirmed the well-known fact that the exploitation of movement by the human visual system greatly increases detection capabilities: the success of the detection task is much higher when performed while watching a film than when performed by observing still images taken from the very same film. This is due to the human visual cortex implementing advanced spatiotemporal denoising algorithms and should inspire detection software developers.

In some recordings, wild animals are visible on the side of the road in the LWIR band. These animals are visible in none of the other spectral bands.

The conclusions are the following:

 For vehicles with headlights on in adverse conditions, detection is better in SWIR. NIR comes next and then VIS and LWIR.

 For the recognition task in the same conditions, the conclusions are opposed.

 The SWIR band is extremely sensitive to glare in foggy conditions, making the recognition task impossible in many cases.

 For both tasks, the LWIR camera gives much more reproducible results than all the others. In particular, its performance is independent of the vehicle's headlights being on or off.

 The LWIR camera was also the only one allowing the detection of hot blooded animals on the side of the road in adverse conditions.

Road marking detection

Road markings ranges were evaluated in the outdoor test campaign by using the road lines of the highway. The same calibration as for vehicle detection was used to measure the distances.

The following table gives the average maximal detection distance for the four scenarios of interest:

Conclusions of the analysis of all the scenarios are the following:

 The SWIR sensor does not allow the identification of the traffic sign, even during daylight. This is due to the fact that the difference of reflectivity between the letters and the background is very low in the SWIR band. Traffic signs are indeed designed for visible spectrum and not for the other bands.

 The NIR camera provides a good vision and a better SNR than the RGB visible camera, especially in adverse weather conditions. This is due to the same reasons as for road markings detection.

 As expected, the traffic sign is never identifiable in the LWIR band. Letters and background have indeed the same temperature and emissivity.

Conclusion

As they are complementary to other distance measurement systems like LIDARs or RADARs, the AWARE project only focused its experiments on cameras technologies that are necessary to bring redundancy and complementary characteristics for improving ADAS detection system's reliability and accuracy.

In order to detect pedestrian, vehicle, road marking or recognize traffic signs, the relevance of four different spectral bands has been evaluated under adverse weather conditions: Visible RGB, Near-Infrared (NIR), Short-Wave Infrared (SWIR) and Long-Wave Infrared (LWIR).

The Table 13 and -In addition to the visible spectral band, only the LWIR spectral band provides outstanding interests.

Targets can be detected with or without additional light. LWIR detection is not sensitive to any dazzle, -NIR and SWIR provide equivalent performances, mainly due to the fact that experiments used extended visible to NIR and visible to SWIR cameras, -Visible RGB extended to NIR (or Red-Clear sensors) combined with LWIR provide the best spectral bands combination to improve ADAS performances of detection such as vehicle, pedestrian, bicycle, animals or road marking, and recognition such as traffic signs.

Caution shall be considered while using LED headlight technology to provide additional light. LED

Figure 1 :

 1 Figure 1: Outdoor test campaign

Figure 2 :

 2 Figure 2: Tunnel test campaign

Figure 3 :

 3 Figure 3: Example of video records by LWIR (top left), Visible (top right), SWIR (bottom left) and NIR (bottom right) cameras

Figure 4 :

 4 Figure 4: Pedestrian detection test setup into Cerema fog tunnel, LWIR (top left), Visible (top right), SWIR (bottom left) and NIR (bottom right) cameras

Figure 5 :Figure 6 :

 56 Figure 5: example of images recorded in the fog tunnel with the four different cameras

Figure 7 :

 7 Figure 7: Reference of range based on site map analysis and T1 lanes type of road marking

Figure 8 :

 8 Figure 8: Vehicle detection ranges

  Figure 10: Images showing the glare effect in the VIS, NIR and SWIR spectral bands.

Figure 12 :

 12 Figure 12: Images of traffic sign acquired in different spectral bands. First line conditions is a sunny day and second line is from scenario 7 (fog class 3 with snow)

Table 1 :

 1 2014 French accidentology data in adverse conditions

			1]:
		Injury casualties	Fatalities
	Night	32%	41%
	Wet road	20%	20%
	Adverse weather	21%	23%

Table 2 :

 2 Camera characteristics Extended NIR camera uses monochrome CMOS photodiodes with a cut-off wavelength close to 1 µm. It detects the reflective visible and NIR light from the scene. It thus requires an illumination by sun, moon or night glow or an illuminator positioned on the vehicle.

  Table3shows the most interesting scenarios, with a class 3 fog (visibility distance < 100 m):

	Scenario	Weather	Visibility distance	Ambient light	Tempe-rature
	7	Day, heavy fog, light snow	75 m	5032 lux	-1°C
	11	Day, heavy fog and snow	69 m	1096 lux	-1°C
		Day low light,
	22	heavy fog and	75 m	104 lux	-0.8°C
		snow
	26	Night, heavy fog	99 m	0 lux	+1.5°C

Table 3 :

 3

Most interesting outdoor scenarios 3.2 Tunnel test campaign

Table 13 :

 13 Comparison of performances of ADAS functions using Visible, NIR, SWIR or LWIR camera technologies, Night fog, headlights

ADAS FUNCTION Camera Spectral band Day Fog, using headlights VIS NIR SWIR LWIR

	Pedestrian,
	Bicycles, Animals	+	++	++	++++
	detection
	Vehicle shape recognition	+	++	++	++
	Traffic signs recognition	+	++	-	--
	Road marking detection	+	++	++	-

Table 14 :

 14 Comparison of performances of ADAS functions using Visible, NIR, SWIR or LWIR camera technologies, Day fog, headlights Results of experiment clearly state that:

Scenario

Visibility [m]

Ambient light

[Lux]  Observation in the LWIR spectral band is only depending on road marking thermal emissivity. In 15 of the 21 scenarios recorded in LWIR, road markings are not visible. The visibility depends on the weather: rain is cleaning the lines while sun exposure enhances them (see Figure 11). Thus road marking observation in the LWIR spectral band is not relevant.

 Visible, NIR and SWIR all required additional lighting to detect road marking at night. Detection in NIR and SWIR are equivalent, and slightly better than VIS. This is due in part to: a broader overall spectral band, monochromaticity (no RGB filter), larger pixels and a higher bit depth. pulsed technology could reduce detection reliability of system based on Visible, NIR and SWIR cameras.

Acknowledgement

The authors acknowledge the contribution of their colleagues to this work: P. Morange, J-L. Bicard and all the pedestrians from CEREMA, A. Picard from Sagem and B. Yahiaoui from Nexyad.

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