Ricardo Pérez

Marco Local

Ricardo P Érez-Marco

LOCAL MONODROMY FORMULA OF HADAMARD PRODUCTS

Keywords: 2000 Mathematics Subject Classification. 08A02, 32S05, 30D99, 30F99, 11G55, 35C05, 33C60, 26A33 Hadamard product, Hadamard grade, singularities with monodromy, recurrent monodromy, algebro-geometric singularities, polylogarithm singularities, elliptic integrals, hypergeometric functions, fractional integration

may come

Local monodromy formula of Hadamard products

Introduction.

Given two power series

F (z) = A 0 + A 1 z + A 2 z 2 + . . . = n≥0 A n z n G(z) = B 0 + B 1 z + B 2 z 2 + . . . = n≥0 B n z n
their classical Hadamard product is defined by

F ⊙ G(z) = A 0 B 0 + A 1 B 1 z + A 2 B 2 z 2 + . . . = n≥0 A n B n z n .
In a previous article [START_REF] Érez-Marco | Monodromies of singularities of the Hadamard and eñe product[END_REF] we gave a formula for the monodromies of the singularities of the Hadamard product F ⊙ G of two holomorphic functions F and G with isolated singularities (α), resp. (β), with holomorphic monodromies ∆ α F and ∆ β G. More precisely, ∆ α F = F 1 -F where F 1 is the branch obtained as the local analytic continuation of F around α (following a loop of winding number 1 with respect to α), and the monodromy ∆ α F is holomorphic if it defines a holomorphic germ at the point α. We consider also the operator Σ α = Id +∆ α , thus

F 1 = Σ α F = F + ∆ α F .
The singularity at α is integrable, when we have S-lim z→α (z -α) F (z) → 0 where the limit is a Stolz limit at α, that is, z → α with arg(z -α) bounded. In this case we have Cǫ F (u)du → 0 for a loop C ǫ around α with C ǫ → 0 when ǫ → 0. We prove the following result: Theorem 1.1 (Hadamard iterated monodromy formula for integrable singularities). We consider F and G holomorphic germs at 0 with respective set of singularities (α) and (β) in C. We assume that the singularities are isolated and integrable. For k ≥ 0, we consider the different branches around the singularities

F k = Σ k α F and G k = Σ k β G.
Then the set of singularities of the principal branch of F ⊙ G is contained in the product set (γ) = (αβ) and is composed by isolated singularities, and for N ≥ 1 we have the formula

(1) (Σ N γ -Id)(F ⊙ G)(z) = - 1 2πi α,β αβ=γ N -1 k=0 z/β α ∆ α Σ k α F (u) . ∆ β Σ k β G (z/u) du u
For N = 1 this is the same formula from [START_REF] Érez-Marco | Monodromies of singularities of the Hadamard and eñe product[END_REF] except for the absence of the residual part, and generalized to integrable singularities (it was proved there for holomorphic singularities, which are not necessarily integrable and can produce a residual part).

We use the following notation for the integral convolution1

F ⋆ G = - 1 2πi z/β α F (u) . G(z/u) du u
Then, the main formula can be written as

(Σ N γ -Id)(F ⊙ G)(z) = N -1 k=0 αβ=γ ∆ α Σ k α F ⋆ ∆ β Σ k β G
We have another form of the formula that follows from formula [START_REF] Allouche | Hadamard grade of power series[END_REF] and the formal identity

∆Σ k = Σ k+1 -Σ k = (Σ k+1 -Id) -(Σ k -Id)
Proposition 1.2. We have

∆ γ Σ k γ (F ⊙ G) = αβ=γ ∆ α Σ k α F ⋆ ∆ β Σ k β G
Therefore, the operators (∆ γ Σ k γ ) k∈Z define morphism from the Hadamard algebra to the integral convolution algebra in the associated monodromy spaces.

As Corollary, by induction, we get the formula for the iterated monodromy for a Hadamard product with several factors:

Corollary 1.3. We have (2) (Σ N γ -Id)(F 1 ⊙ F 2 ⊙ . . . ⊙ F n ) = N -1 k=0 α 1 ...αn=γ ∆ α 1 Σ k α 1 F 1 ⋆ . . . ⋆ ∆ α 1 Σ k αn F n 2.
Proof of the iterated monodromy formula.

2.1. Geometric proof. The proof follows the same lines than the proof given in [START_REF] Érez-Marco | Monodromies of singularities of the Hadamard and eñe product[END_REF] by starting with Plancherel-Hadamard convolution formula

(3) F ⊙ G(z) = 1 2πi η F (u)G(z/u) du u
where η is a positively oriented circle centered at 0 of radius r > 0 with |z|/R G < r < R F , where R F and R G are the respective radii of convergence of F and G.

Hadamard Multiplication Theorem is derived from the convolution formula. The singularities of F ⊙ G are located at points γ = αβ where α and β are singularities of F and G respectively. Also from the covolution formula we get that the singularities γ are integrable.

We can consider the case with trivial multiplicity 1, i.e. when there is only one pair of singularities (α, β) that satisfies γ = αβ. The general case is a superposition of this case. We can also assume α = β = γ = 1 to simplify the notation. We follow the analytic continuation of F ⊙ G by moving the point z and deforming the path of integration homotopically into η z so that the point z never crosses it. We consider the situation where z starts at z = z 0 and turns positively N ≥ 1 times around the point z = 1. Figure 2 shows the local homotopy deformation from the integration path η z 0 to ηz 0 near 1 for N = 1. Figure 1 shows the resulting deformed path for N = 2. Using the convolution formula we have

(Σ N 1 -Id)(F ⊙ G)(z 0 ) = 1 2πi ηz 0 -ηz 0 F (u)G(z/u) du u
Thus we are led to compute the last integral in the difference of the two paths that is the multi-loop path shown in Figure 1. As illustrated in Figure 2, we can deform into a train track composed by 4N segments copies of [z 0 , a] and [a, 1] where a ∈ C is an arbitrary point in between z 0 and 1.

The integral is independent of the collapsing point a and we can make a → 1, leaving a train track composed by 2N segments copies of [z 0 , 1]. There are no polar contributions at z = z 0 or z = 1 since we assume that the singularities are integrable.

Considering the 2N paths, in the positive ordering given by the orientation of ηz 0 , and the corresponding signs given by the orientation, we have to integrate on [1, z 0 ] the maps

F 1 G 0 , -F 1 G 1 , F 2 G 1 , -F 2 G 2 , . . . , F N G N -1 , -F N G N , F N -1 G N , -F N -1 G N -1 , . . . , F 0 G 1 , -F 0 G 0 for a total contribution of F 1 (G 0 -G 1 ) + F 2 (G 1 -G 2 ) + . . . + F N (G N -1 -G N ) + F N -1 (G N -G N -1 ) + . . . + F 0 (G 1 -G 0 ) = -F 1 ∆ 1 G 0 -F 2 ∆ 1 G 1 + . . . -F N ∆ 1 G N -1 + F N -1 ∆ 1 G N -1 + . . . + F 0 ∆ 1 G 1 = -∆ 1 F 0 ∆ 1 G 0 -∆ 1 F 1 ∆ 1 G 1 -. . . -∆ 1 F N -1 ∆ 1 G N -1
and finally

(Σ N 1 -Id)(F ⊙ G)(z 0 ) = - 1 2πi z 0 1 N -1 k=0 ∆ 1 F k (u).∆ 1 G k (z/u) du u
which is the main formula for α = β = 1.⋄ 3. Properties of monodromy operators and another derivation.

We develop some general properties of the monodromy operators that provide a second derivation of the iterated monodromy formula from the case N = 1. This section is only used for this purpose and can be skipped in a first reading. The formulas developed here are useful for applications (for example we get a straightforward computation of the monodromy of polylogarithms).

3.1. Properties of monodromy operators for integrable singularities. Proposition 3.1 (Fundamental integro-monodromy formula). Let f be a holomorphic germ with an integrable isolated singularity at z = α. The local germ with an isolated singularity at α

F (z) = z α f (u) du is well defined. We have ∆ α F (z) = z α ∆ α f (u) du or ∆ α z α f (u) du = z α ∆ α f (u) du
In general, for z 0 ∈ C which is path connected to α in a domain where f is holomorphic, we have

∆ α z z 0 f (u) du = z α ∆ α f (u) du
Proof. The general case for z 0 ∈ C follows from the case z 0 = α since

z z 0 f (u) du = α z 0 f (u) du + z α f (u) du
where the integral from z 0 to α is taken on a path where f is holomorphic, and

α z 0 f (u) du is constant on z. Now, we can integrate along the straight segment [α, z], F (z) = [α,z] f (u) du then F 1 (z) = [α,z]∪ηz f (u) du
where η(z) is the circle centered at α passing through z with the positive orientation. Then we have

∆ α F (z) = η(z) f (u) du .
Now, we can collapse homotopically η(z) into the segment [α, z], which gives a double integral along this segment,

∆ α F (z) = z α (f 1 (u) -f (u)) du = z α ∆ α f (u) du .
Example. We recall that the polylogarithms can be defined as Li

1 (z) = -log(1-z) and Li k+1 (z) = z 0 Li k (u) du u .
From the properties of the logarithmic function, we have

∆ 1 Li 1 (z) = -2πi
and by induction using the Proposition 3.1 we prove

∆ 1 Li k (z) = -2πi (log z) k-1 (k -1)!
The one line proof by induction is the computation

∆ 1 Li k+1 (z) = z 1 -2πi (k -1)! (log u) k-1 u du = - 2πi k! (log u) k z 1 = - 2πi k! (log z) k
which gives the result.⋄

We extend Proposition 3.1.

Proposition 3.2. Let u → f (u, z) be a holomorphic germ with an integrable isolated singularity at u = α, depending holomorphically on z in a neighborhood U of α with (u, z) → f (u, z) holomorphic for (u, z) in a neighborhood of z∈U B(α, z) × {z} -{(z, z)}. We assume that u → f (u, z) is uniformly integrable at u = z (i.e. along paths landing at u = z). The local germ with an isolated singularity at α

F (z) = z α f (u, z) du
is well defined. We have

∆ α F (z) = z α ∆ [η(z)] f (u, z) du
where η(z) is the loop starting at z whose support is the circle centered at α passing through z with the positive orientation.

Note that f (u, z) may have an integrable singularity at u = z, as it will be the case in our main application of this formula. The proof follows from the same argument as for Proposition 3.1 since we can integrate along η(z), and then we collapse homotopically η(z) onto the segment [α, z].

We recall the notation for the integral convolution

f ⋆ g = - 1 2πi z γ f (u/β) . g(βz/u) du u .
Corollary 3.3. Consider f , resp. g, having an isolated singularity at α, resp. β, and γ = αβ. We have

∆ γ (f ⋆ g) = ∆ α f ⋆ g + f ⋆ ∆ β g + ∆ α f ⋆ ∆ β g
Proof. We only need to compute the monodromy along η(z) and apply Proposition 3.2. Observe that when the variable u goes along η(z), then u/β winds once around α, and βz/u winds once around β, hence

Σ [η(z)] f (u/β) . g(βz/u) = f 1 (u/β) . g 1 (βz/u) thus ∆ [η(z)] f (u/β) . g(βz/u) = ∆ α f (u/β) . g(βz/u)+f (u/β) . ∆ β g(βz/u)+∆ α f (u/β) . ∆ β g(βz/u)
and the result follows.

Corollary 3.4. We have

∆ γ ∆ α F ⋆ ∆ β G = ∆ α Σ α F ⋆ ∆ β Σ β G -∆ α F ⋆ ∆ β G Proof. We write ∆ α F ⋆ ∆ β G = - 1 2πi z γ ∆ α F (u/β) . ∆ β G(βz/u) du u
and we use Proposition 3.2 applied to the function f (u, z) = ∆ α F (u)∆ β G(z/u), and the singulatity at γ,

∆ γ ∆ α F ⋆ ∆ β G = ∆ 2 α F ⋆ ∆ β G + ∆ α F ⋆ ∆ 2 β G + ∆ 2 α F ⋆ ∆ 2 β G = ∆ α Σ α F ⋆ ∆ β Σ β G -∆ α F ⋆ ∆ β G recalling that Σ = Id +∆.
3.2. Derivation from the case N = 1. We assume Theorem 1.1 for N = 1 and we argue by induction. To simplify we assume that we don't have higher multiplicity for the singularity γ = αβ, i.e. that there is only one pair (α, β) such that γ = αβ. The general case is obtained by linear superposition. We write

Σ N +1 γ (F ⊙ G) = Σ N γ (F ⊙ G) + ∆ γ Σ N γ (F ⊙ G) = Σ N γ (F ⊙ G) + ∆ γ (F ⊙ G) + N -1 k=0 ∆ γ ∆ α Σ k α F ⋆ ∆ β Σ k β G = Σ N γ (F ⊙ G) + ∆ γ (F ⊙ G) + N -1 k=0 ∆ α Σ k+1 α F ⋆ ∆ β Σ k+1 β G -∆ α Σ k α F ⋆ ∆ β Σ k β G = Σ N γ (F ⊙ G) + ∆ γ (F ⊙ G) + ∆ α Σ N α F ⋆ ∆ β Σ N β G -∆ α F ⋆ ∆ β G = Σ N γ (F ⊙ G) + ∆ α Σ N α F ⋆ ∆ β Σ N β G
where we use the induction hypothesis on the second line, we use Corollary 3.4 in the third line, we telescope the sum on the fourth line, and we use ∆ γ (F ⊙ G) = ∆ α F ⋆ ∆ β G (which is the result for N = 1) in the last line. Now, the result follows by using the induction hypothesis in the last equation.⋄ Obviously, if we knew the morphism property for ∆ γ Σ N γ we could shorten the proof using this in the first line. The morphism property is equivalent to the iterated monodromy formula. 4. Applications.

4.1.

Local monodromies of Hadamard products of algebraic functions. We consider holomorphic germs H that are the Hadamard product of a finite number of algebraic functions. The Hadamard grade, defined by Allouche and Mendès-France in [START_REF] Allouche | Hadamard grade of power series[END_REF], is the minimum integer n ≥ 1 such that

H = F 1 ⊙ . . . ⊙ F n
where F 1 , . . . , F n are algebraic functions. The grade is infinite if there is no such decomposition. The Hadamard grade is natural when we study the problem of generating the tower of special functions by Hadamard products à la Liouvile. We observe that for an algebraic function F , for any ramification singularity α, we have that Σ k α F , for k ∈ Z, are the conjugates of F , and (Σ k α F ) k is d-periodic where d is the degree of F (the local minimal period divides d).

We consider in the rest of this section algebraic functions F 1 , . . . F n which have all the singularities integrable. The main iterated monodromy formula puts a heavy restriction on local monodromies of the singularities of the function

H = F 1 ⊙ . . . ⊙ F n . Theorem 4.1. Let H = F 1 ⊙ . . . ⊙ F n where, for 1 ≤ j ≤ n, the F j are algebraic functions of degree d j ≥ 2 with integrable ramification points, then, if d = d 1 . . . d n , we have for N ≥ 1, (Σ N +2d γ -2 Σ N +d γ + Σ N γ H = 0
Proof. We start from formula (2)

(Σ N γ -Id)(F 1 ⊙ F 2 ⊙ . . . ⊙ F n ) = N -1 k=0 α 1 ...αn=γ ∆ α 1 Σ k α 1 F 1 ⋆ . . . ⋆ ∆ α 1 Σ k αn F n
which shows that if N = kd + r the Euclidean division by d, 0 ≤ r < d, we have

(Σ N γ -Id)H = k Σ d γ -Id + Σ r γ -Id H Therefore, we get Σ N +d γ -Σ N γ H = Σ d γ -Id)H and (Σ N +2d γ -2 Σ N +d γ + Σ N γ H = 0
As Corollary from the proof we get, Corollary 4.2. We have the existence of the following limit

lim N →+∞ 1 N (Σ N γ -Id)(F 1 ⊙. . . ⊙F n ) = 1 d 1 . . . d n d-1 k=0 α 1 ...αn=γ ∆ α 1 Σ k α 1 F 1 ⋆ . . . ⋆ ∆ αn Σ k αn F n
This is interesting once we recognize 1 N (Σ N γ -Id) as a Birkhoff sum for ∆ γ associated to the local monodromy dynamics of Σ γ

1 N Σ N γ -Id = 1 N N -1 k=0 (Σ k+1 γ -Σ k γ ) = 1 N N -1 k=0 ∆ γ • Σ k γ
We can be more precise and prove a recurrence result of the local monodromy. The proof is done by induction.

N = K n D n + R n 0 ≤ R n < D n R n = K n-1 D n-1 + R n-1 0 ≤ R n-1 < D n-2 . . . . . . R 2 = K 1 D 1 + R 1 0 ≤ R 1 < D 1 R 1 = K 0
so, the K i for i < n are uniformly bounded on N, 0 ≤ K i < D i+1 , and we have the

(D n )-adic decomposition N = K n D n + K n-1 D n-1 + . . . K 1 D 1 + K 0 .
Then we have

(Σ N γ -Id)(F 1 ⊙ . . . ⊙ F n ) = K n α 1 ...αn=γ Dn-1 k=0 ∆ α 1 Σ k α 1 F 1 ⋆ . . . ⋆ ∆ αn Σ k αn F n + . . . + K n-1 α 1 ...αn=γ d 1 -1 l 1 =0 ∆ α 1 Σ l 1 α 1 F 1 ⋆ D n-1 -1 k=0 ∆ α 2 Σ k α 2 F 2 ⋆ . . . ⋆ ∆ αn Σ k αn F n + . . . + K n-2 α 1 ...αn=γ 0≤l 1 <d 1 0≤l 2 <d 2 ∆ α 1 Σ l 1 α 1 F 1 ⋆ ∆ α 2 Σ l 2 α 2 F 2 ⋆ D n-2 -1 k=0 ∆ α 3 Σ k α 3 F 3 ⋆ . . . ⋆ ∆ αn Σ k αn F n + . . . . . . + K 1 α 1 ...αn=γ 0≤l 1 <d 1 . . . 0≤ln<dn ∆ α 1 Σ l 1 α 1 F 1 ⋆ . . . ⋆ ∆ αn Σ ln αn F n + . . . + α 1 ...αn=γ K 0 k=0 ∆ α 1 Σ k α 1 F 1 ⋆ . . . ⋆ ∆ αn Σ k αn F n 4.2.
Class of recurrent monodromy. We generalize the previous result to functions with recurrent monodromy.

Definition 4.4. An isolated singularity α of a germ F has a recurrent monodromy of order d ≥ 1 if the iterated monodromy Σ d α is a linear combination of monodromies

(Σ k α ) 0≤k≤d-1 , that is, there are constants a k ∈ C, 0 ≤ k ≤ d -1, such that, for all m ≥ 0, Σ m+d α F = d-1 k=0 a k Σ m+k α F .
The recurrent monodromy class is the set of functions with only isolated singularities with recurrent monodromies at all singularities.

A first example are algebraic function which are recurrent of the order equal to their degree and a 0 = 1 and a j = 0 for j = 1, . . . , d. From the result of the previous section, the Hadamard product of algebraic functions with integrable singularities is in the recurrent monodromy class (but it is not necessarily algebraic, see section 4.3). Observe that in the definition the order of the recurrence depends on the singularity. From the main formula we have for k ≥ 0,

Σ m γ F ⊙ G = F ⊙ G + m-1 k=0 αβ=γ ∆ α Σ k α F ⋆ ∆ β Σ k β G
and plugging the recurrence relation by induction we lower the exponents and end-up with

Σ m γ F ⊙ G = F ⊙ G + (k,l) αβ=γ c m,k,l (α, β) ∆ α Σ k α F ⋆ ∆ β Σ l β G where 0 ≤ k ≤ d α , 0 ≤ l ≤ d β and constant c m,k,l (α, β) ∈ C.
If we consider d+1 of these relations, for m = 0, . . . , d, then by linear elimination we find a linear combination that kills all the ∆ α Σ k α F ⋆ ∆ β Σ l β G, and we can express Σ d γ H linearly in function of the lower iterated monodromies Σ m γ H, for m = 0, . . . , d -1.

Observe that if the singularities α and β of F and G have respective orders d α and d β , then if the singularity γ = αβ of F ⊙ G has no multiplicity, then the monodromy at γ is d α d β -recurrent. For m(γ) ≥ 2 we can improve the above proof observing that parts of the sum associated to different pairs (α, β) do not interact. Then this gives that the monodromy at γ is D-recurrent for

D = lcm{d α d b ; (α, β), αβ = γ}
This shows a graduated structure of the Hadamard algebra of functions with recurrent monodromies.

An interesting subclass of the recurrent monodromy class are those functions whose monodromy is K-recurrent for a subfield K ⊂ C, for example when K = Q or K is a number field. Definition 4.6 (K-recurrent monodromy class). Let K ⊂ C be a subfield of C. A recurrent monodromy function has a K-recurrent monodromy if we have a recurrence

Σ m+d α F = d-1 k=0 a k Σ m+k α F . with coefficients in a k ∈ K for all 0 ≤ k ≤ d -1.
Corollary 4.7. Let K ⊂ C be a subfield of C. The K-recurrent monodromy class with integrable singularities is invariant by Hadamard product.

Proof. In the proof of Theorem 4.5 we have all coefficients c m,k (α 1 , . . . , α n ) ∈ K and the linear combination can be done with scalars in K, hence the result.

We can refine the previous results allowing the field K to depend on the singularity, say K α . Then, for the Hadamard product F ⊙ G, the associated field K γ for the singularity is an extension of the fields (K α ) and (K β ) such that αβ = γ.

Examples.

We have some remarkable examples. In all these examples K is a subfield of C.

Example 1. Finite dimensional récurrence. We consider the class of functions F having integrable singularities at places α ∈ C, such that for each singularity α there is a finite K-dimensional space V α of dimension d α ≥ 1, and for F ∈ F and all k = 0, 1, . . . d -1, Σ k α F ∈ V α . Then F is a class of functions with K-recurrent monodromy so they Hadamard products are in the K-recurrent monodromy class.

Given a linear differential equation with rational function coefficients, the set of local solutions at a regular point (out of poles) is a finite dimensional vector space. If we have a bases of solutions that are integrable (which is more precise than fuchsian) then all solutions are integrable. Moreover, since the monodromy operator Σ α commutes with differential operators with rational function coefficients (if the coefficients don't have monodromies at the poles or singularities this also works), then we get that all solutions have recurrent monodromies. Hence their Hadamard products do have recurrent monodromies also.

Example 2. Algebro-geometric singularities. We consider the class of functions F having only algebro-geometric singularities, as those considered in [START_REF] Jungen | Sur les séries de Taylor n'ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence[END_REF]. This means that F has only isolated singularities and locally near a singularity α we have

F (z) = (z -α) -aα log(z -α) 2πi nα ϕ α (z)
where a α ∈ C, n α ≥ 0 is a positive integer, and ϕ ∈ O α a local holomorphic germ. All these parameters depend on the singularity α, but we may drop the sub-index α to simplify the notation.

These are the type of singularities appearing in solutions of fuchsian equations. Morevoer, we request that ℜa α > -1 so that the singularities are integrable.

Observe that we have for k ∈ Z,

Σ k α F = Σ k α (z -α) -a Σ k α log(z -α) 2πi n Σ k α ϕ(z) = e -2πika (z -α) -a log(z -α) 2πi + k n ϕ(z)
So, the monodromies are in the finite dimensional space generated by the germs,

E l (z) = (z -α) -a log(z -α) 2πi l ϕ(z)
for 0 ≤ l ≤ n. Using the argument from Example 1 we conclude that we have a recurrent monodromy. We can be more precise. For a finite linear combination we have

k c k Σ k α F = (z -α) -a ϕ(z) k c k e -2πika log(z -α) 2πi + k n = n l=0 n l k c k e -2πika k l log(z -α) 2πi l
Since the (n + 1) × (n + 1) Vandermonde determinant (with convention 0 0 = 1 here)

det[k l ] = k 1 =k 2 (k 1 -k 2 ) = 0 is non-zero, for N = n we can find a non-trivial linear combination (δ l,n is Kronecker symbol) k d k k l = δ l,n
and putting c k = d k e 2πika we prove that the monodromy is recurrent of degree n.

Observe that the monodromy is K-recurrent if and only if e -2πia ∈ K. Thus, if K is a number field then we must have α ∈ Q.

Example 3. Polylogarithm singularities. This is a generalization of the previous example. It gives examples with recurrent monodromy that are not fuchsian.

We assume that for each singularity α we have locally near α that F is a germ of the form

F (z) = (z -α) a L(z)ϕ(z)
where L(z) ∈ C[li 1 (z -α), . . . , li n (z -α)] where the li k are the normalized polylogarithms

li k (z) = - Li k (z) 2πi hence ∆ 1 li k (z) = (log z) k-1 (k -1)! and (4) Σ α li k (z -α) = li k (z -α) + (log z) k-1 (k -1)!
Since L is a polynomial on polylogarithm functions, say L = P (li 1 , . . . , li n ), we denote by d the degree of P , and we call d the polylogarithm degree of L.

Lemma 4.8. We have Σ k α F = e 2πika (z -α) a P li 1 +k, . . . , li n +k (log z) n-1 (n -1)! ϕ(z)

and

P li 1 +k, . . . , li n +k (log z) n-1 (n -1)! = Q(log z, li 1 , . . . , li n ) ∈ C[log z, li 1 (z-α), . . . , li n (z-α)]
and the degree of the polynomial Q is bounded by (n -1)d in the first logarithmic variable, and is the same as the one of P in the other variables.

Proof. It is straightforward from the monodromy formula (4) for polylogarithms.

Corollary 4.9. With the previous notations, a function F with polylogarithm singularities is in the recurrent monodromy class. If P ∈ K[X 1 , . . . , X n ] has coefficients in the field K, then the monodromy of F at the singularity α is K-recurrent.

Elliptic integrals, hypergeometric functions and fractional integration.

In this section we sketch one of the applications of the monodromy formula to elliptic integrals and to hypergeometric functions. We refer to [START_REF] Hancock | Theory of elliptic functions[END_REF] for the classical theory of elliptic integrals, and to [START_REF] Slater | Generalized hypergeometric functions[END_REF] and [START_REF] Andrews | Special functions[END_REF] for the theory of hypergeometric functions.

As is well know, the Hadamard product of algebraic functions is not in general an algebraic function. A notable exception happens in finite characteristic by a Theorem of Deligne [START_REF] Deligne | Intégration sur un cycle évanescent[END_REF]. One simple example from [START_REF] Jungen | Sur les séries de Taylor n'ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence[END_REF] is

F (z) = G(z) = (1 -z) -1/2 = +∞ n=0 1.3 . . . (2n -1) 2.4 . . . (2n) z n
which give by Hadamard product the elliptic integral

F ⊙ G(z) = +∞ n=0 1.3 . . . (2n -1) 2.4 . . . (2n) 2 z n = 2 π 1 0 du (1 -u 2 )(1 -zu 2 )
which is proved by developing the right hand side and observing that

1 0 u 2n √ 1 -u 2 du = π/2 0 (cos x) 2n dx = 1.3 . . . (2n -1) 2.4 . . . (2n) π 2 .
This Hadamard product is the classical modular function in the variable k = z 2 which is not an algebraic function (since it is easy to check that the local monodromies are not of finite order),

K(k) = F ⊙ G(k 2 ) = 2 π 1 0 du (1 -u 2 )(1 -k 2 u 2 )
We can directly compute the monodromy of the modular function by application of the monodromy formula. Both F and G have an integrable singularity at z = 1, and we get the closed form expression

∆ 1 (F ⊙ G)(z) = - 1 2πi z 1 du u(1 -u)(u -z)
For the modular function, using the change of variable formula (Proposition 3.11 from [START_REF] Érez-Marco | Monodromies of singularities of the Hadamard and eñe product[END_REF]), we have

∆ ±1 K(k) = ∆ 1 (F ⊙ G)(k 2 ) = - 1 2πi k 2 1 du u(1 -u)(u -k 2 )
More generally, we can generate some classical and generalized hypergeometric functions from Hadamard products of simple polar functions, for example, we have [START_REF] Erd Éyli | Transformation of hypergeometric integrals by means of fractional integration by parts[END_REF] (

1 -z) -a ⊙ (1 -z) -b = F (a, b, 1; z)
and this generalizes the previous example (where a = b = 1/2 and α = β = 1). Then the classical Euler integral formula

F (a, b, c; z) = Γ(c) Γ(b)Γ(c -b) 1 0 u b-1 (1 -u) c-b-1 (1 -zu) -a du
can be interpreted as a monodromy integral formula. In fact, for c = 1 the formula can be derived from equation [START_REF] Erd Éyli | Transformation of hypergeometric integrals by means of fractional integration by parts[END_REF] observing that

∆ 1 (1 -z) -a = e -2πia -1 (1 -z) -a
and from the knowledge of the monodromy of z → F (a, b, c; z) (see [START_REF] Dlmf | Digital Library of Mathematical Functions[END_REF]

formula 15.2.3 or Theorem 2.3.3 in [2]) ∆ 1 F (a, b, c; z) = Γ(c)Γ(a + b -c) Γ(a)Γ(b) (z -1) c-a-b F (c -a, c -ab, c -a -b + 1; 1 -z)
that can be obtained directly from the hypergeometric differential equation and the Kummer solutions (as in Theorem 2.3.3 in [START_REF] Andrews | Special functions[END_REF]).

The same analysis can be carried out for higher integral formulas of the classical hypergeometric functions as for example

F (a, b, c; z) = Γ(c) Γ(b)Γ(c -b) 1 0 u b-1 (1 -u) c-b-1 F (a, b, c; zu) du or for generalized hypergeometric functions p F q   a 1 , . . . , a p ; z b 1 , . . . , b q   = Γ(b 1 ) Γ(a 1 )Γ(b 1 -a 1 ) 1 0 u a 1 -1 (1-u) b 1 -a 1 -1 p-1 F q-1   a 2 , . . . , a p ; zu b 2 , . . . , q p   du
With this systematic procedure, although computational laborious, we can derive an important number of integral formulas in the theory of hypergeometric functions that appear as particular cases of our monodromy formula. Also this shows a link of the convolution operation in the monodromy formula with the convolutions appearing in Katz's theory of rigid local systems (see in particular the introduction of [START_REF] Katz | Rigid local systems[END_REF] 2 ).

More generally, when we consider a finite dimensional vector space of fuchsian functions, as those arising as solutions of a fuchsian equation, the local monodromies are linear operators, and the monodromy integral formulas provide a multitude of integral relations between these functions. Thus we can see that the integral relations in the theory of generalized hypergeometric functions goes far beyond the hypergeometric class of functions. To develop properly a full theory it is convenient to enlarge the Hadamard product to functions with integrable singularities at 0. One remarkable method of derivation of these integral formulas is trough fractional calculus and fractional integration by parts, as iniciated by Erdélyi in [START_REF] Erd Éyli | Transformation of hypergeometric integrals by means of fractional integration by parts[END_REF] (see also [START_REF] Andrews | Special functions[END_REF] section 2.9). One explanation for the success of this approach is that iterated and fractional integration is another incarnation of the monodromy formula. More precisely, we can compute the iterated integral, for an integer n ≥ 1,

I n (f )(z) =
I α (f )(z) = -2πi Γ(α) - 1 2πi z α u α f (u) z u -1 α-1 du u
Therefore we have (abusing the theory)

I α (f )(z) = - 2πi Γ(α) ∆ α F ⊙ (z -1) α-1
where

F (z) = 1 2πi log(z -α)z α f (z)
so that (we are asuming f regular at α)

∆ α F (z) = z α f (z)
Again, in order to make sense of this formula for ℜα > 1, we need to extend the Hadamard product theory to functions with singularities at 0. Part of such theory can be found in some early articles of S. Mandelbrojt. For example, for ramified germs which are series of the form

F (z) = α a α z α
with the proper convergence conditions, the Plancherel-Hadamard convolution formula makes sense integrating along an infinite circle centered at 0 and our derivation of the monodromy formula carries out with the same arguments. These developments are left for future work.

In conclusion, we would like to remove the local "integrability assumption" in the results. For this we need to find monodromy formulas for the general class of isolated singularities with monodromy. The nature of the formulas is geometric and this is left for the next article on monodromies.

Figure 2 .

 2 Figure 2. Homotopical deformation of the integration path when z0 turns once around 1.

Theorem 4 . 3 (

 43 Recurrence of the local monodromy). For 1 ≤ j ≤ n, let F j be algebraic functions of degree d j ≥ 2 with integrable ramification points. Let D j = d n-j+1 . . . d n and consider the successive euclidean divisions starting with N ≥ 1,

Theorem 4 . 5 (

 45 Invariance of the recurrent monodromy class). Let F, G be functions in the recurrent monodromy class with integrable singularities. Then their Hadamard product F ⊙ G is also in the recurrent monodromy class with integrable singularities.Proof. Let m(γ) ≥ 1 be the multiplicity of γ, i.e. the number of (α, β) such that γ = αβ. Let d = d α d β m(γ) where d α and d β are the degree of the recurrence for the monodromies of F and G at points α and β respectively such that γ = αβ.

f

  (u n ) du n . . . du 1 u) n-1 f (u) duthus the fractional integral is naturally defined for ℜα > 0 asI α (f )(z) = 1 Γ(α) z α (z -u) α-1 f (u) duand we can write this formula as

We use ⋆ instead of ⋆ that we reserve for the eñe product that we consider in[START_REF] Érez-Marco | Monodromies of singularities of the Hadamard and eñe product[END_REF].

I am indebted to J. Fresan for pointing out this reference.


Anne Julbe
email: anne.julbe@umontpellier.fr

Martin Drobek

André Ayral

About the role of adsorption in inorganic and composite membranes

About the role of adsorption in inorganic and composite membranes

Anne Julbe, Martin Drobek, Andre

Introduction

Membranes are physical barriers allowing selective transport of chemical species such as gases, liquids, or ions. The need for continuous and efficient separation processes promises a very bright future to membranes as a versatile resource for reducing costs and environmental impacts of many industries dealing with both liquid and gas-phase separations. Membrane science is becoming increasingly mature, thanks to the strong efforts realized in both fundamental and technical aspects. The understanding of fluid-solid membrane interfaces is universally recognized as a key to harness new processes and technologies. Indeed, in most membrane applications, interfaces govern membrane performance and adsorption phenomena (either molecular physisorption or dissociative chemisorption) plays a key role. Adsorption can act as either a primary mechanism determining membrane efficiency (e.g. selectivity, permeability, reactivity, stability/durability) or a negatively impacting phenomenon (e.g. fouling) which has to be controlled and possibly avoided to guarantee performance stability.

It should be emphasized that adsorption of species at the surface of membranes always attracted strong attention in the field of membrane research. Huge progress has been made recently for a rational design of membrane surfaces and interfaces, supported by the emergence of a wide diversity of new materials and surface modification methods. Hence, the objective of this minireview is to highlight the crucial role of adsorption in membrane separation applications with a zoom on novel membrane materials and the associated transport mechanisms. The corresponding membrane performance and their fouling behavior will be illustrated, together with emerging surface treatment methods. Finally, the relevance of adsorption-coupled methods for both membrane nanostructure analysis and defects detection will be pointed out.

Adsorption in gas phase separation

Gas permeation mechanisms through (micro)porous membranes might include viscous flow, Knudsen diffusion, surface diffusion, multi-layer diffusion, bulk diffusion (in mixtures), and/or microporous (configurational) diffusion. The relative contributions of these mechanisms depend on process variables such as temperature, pressure, gas composition and also on membrane defects. The milestone theory of Maxwell-Stefan can be advantageously applied to model the transfer mechanisms involved in gas mixture separation [START_REF] Krishna | Describing the diffusion of guest molecules inside porous structures[END_REF]. Gas separation mechanisms in microporous membranes are mainly based on molecular sieving (size exclusion), difference in diffusivity, and/or competitive adsorption (e.g. strongly adsorbing components hinder the permeance of weakly adsorbing ones) (Fig. 1a). Molecular adsorption and subsequent diffusion are two major mechanisms for describing gas transport through porous membranes below 300°C. The adsorption is usually not multilayer and often below a monolayer, so it is well described by the Langmuir adsorption model. Gas mixtures separation can be predicted from data measured for pure gases, provided that accurate sorption isotherms are available to compute the Maxwell-Stefan thermodynamic factors. The most significant theories adopted to model surface diffusion phenomenon are discussed in [START_REF] Lito | Kinetic Modeling of Pure and Multicomponent Gas Permeation Through Microporous Membranes: Diffusion Mechanisms and Influence of Isotherm Type[END_REF]. Gas translation (GT) competing with surface diffusion (SD) can describe light gases permeation through zeolite membranes for both single gas (e.g. H2 and CO2, fig. 1b) and gas mixture diffusion [START_REF] Zito | Discrimination among gas translation, surface and Knudsen diffusion in permeation through zeolite membranes[END_REF].

Zeolites, metal organic frameworks (MOFs) or carbon molecular sieve membranes are investigated for large number of gas separation technologies involving e.g. H2, CO2, N2, CO, CH4, noble gases or heavy hydrocarbons (olefins/paraffins or linear/branched isomers). Due to the strong impact of competitive adsorption, membrane performance is always strongly impacted by fluctuations in the operating parameters and by the presence of impurities/contaminants in the feed gas. Comparison of membrane performance thus requires much care and efficient initial outgassing.

Zeolite membranes

Although molecular sieving and competitive diffusivity are both governing zeolite membrane separation efficiency above 300°C, the presence of strongly adsorbing species such as CO2, steam or other contaminants, drastically modify their performance at lower temperature (competitive adsorption). The separation of gases via zeolite membranes is quite complex, particularly in the presence of water vapour [START_REF] Khulbe | Recent development on the effect of water/moisture on the performance of zeolite membrane and MMMs containing zeolite for gas separation[END_REF]. The high selectivity resulting from competitive adsorption was explored at cryogenic temperature with ultrathin highly permeable MFI (Si-rich) zeolite membrane [START_REF] Ye | Cryogenic Air Separation at Low Pressure Using MFI Membranes[END_REF][START_REF] Ye | Efficient Separation of N2 and He at Low Temperature Using MFI Membranes[END_REF]. Attractive separation factors (SF) were obtained for binary gas mixtures such as O2/N2 (SF=5 @67 K) [START_REF] Ye | Cryogenic Air Separation at Low Pressure Using MFI Membranes[END_REF] and N2/He (SF=62 @124 K) [START_REF] Ye | Efficient Separation of N2 and He at Low Temperature Using MFI Membranes[END_REF]. These membranes also separate CO2/H2 (SF=200 @235 K) [START_REF] Korelskiy | Efficient Ceramic Zeolite Membranes for CO2/H2 Separation[END_REF] or CO2/CO (SF= 21 @258 K) [START_REF] Korelskiy | A Study of CO2/CO Separation by sub-micron b-Oriented MFI Membranes[END_REF]. The modification of MFI zeolite is an option to change the impact of surface adsorption (e.g. by coking, catalytic craking, or ion exchange) but it usually involves a reduction of pore sizes and permeability drops. Smaller pore size zeolites (SAPO-34, and DDR) have been considered for large scale industrial separation of CO2/CH4 mixtures. DDR is pure silica and provides higher steam resistance (hydrophobicity) than lower-silica polar structures [START_REF] Khulbe | Recent development on the effect of water/moisture on the performance of zeolite membrane and MMMs containing zeolite for gas separation[END_REF]. When considering hydrophobisation of membrane surface, the bi-layer (or sandwich) strategy is an attractive option impacting on both competitive adsorption and membrane reactivity. The design of bi-layer MFI membranes (ZSM-5/S-1) was proposed [START_REF] Wang | Highly stable bilayer MFI zeolite membranes for high temperature hydrogen separation[END_REF] for obtaining long term steam stability at 500°C. On the other hand, double-sided zeolite membranes deposited on both sides of the support might provide significantly higher fluxes in comparison with conventional single layer. This is particulary true when considering binary mixtures in which the more strongly adsorbed component preferentially permeates (high loading-not in the Henry regime) [START_REF] Gardner | Enhanced flux through double-sided zeolite membranes[END_REF]. This double-sided strategy, limiting concentration-polarisation effects, was also found to improve the performance of MFI membranes for ethanol/water separation by pervaporation [START_REF]Synthesis of capillary titanosilicalite TS-1 ceramic membranes by MW-assisted hydrothermal heating for pervaporation application[END_REF].

2.1.2.Metal-Organic Framework (MOF) membranes

Significant differences exist between the mineral structure of zeolite and the flexible hybrid structure of MOF membranes. As a direct consequence of MOF flexibility: i) molecules larger than the MOF pore aperture can adsorb and permeate through the membrane at high loading (saturation) and ii) the selectivity of large pores MOF membranes might be increased by increasing the feed pressure (pressure-related structural behaviour). Considering the complexity of MOF systems and the strong impact of operational conditions, large number of screening works was done to define the best MOF membranes for specific applications, e.g. upgrading of natural gas [START_REF] Qiao | High-throughput computational screening of metal-organic framework membranes for upgrading of natural gas[END_REF]. Recent investigations of MOF structure-performance relations [13] revealed that MOFs with narrow pore sizes and low porosities are the best adsorbent materials for separation of CO2 from H2, whereas MOFs with large pore sizes and high porosity should be the best membrane materials for selective separation of H2. Effectively, with ZIF membranes at low temperature, relatively weak H2/CO2 separation factors are usually obtained, caused by preferential adsorption of CO2 blocking H2 diffusion. Working at high temperature (> 200°C) is an option to decrease CO2 adsorption and thus increase H2 permeation flow [START_REF] Li | Zeolitic imidazolate framework ZIF-7 based molecular sieve membrane for hydrogen separation[END_REF]. The design of double-layered ZIF-based membranes (i.e. ZIF-8/ZIF-9 and ZIF-67/ZIF-9) could be another option [START_REF] Cacho-Bailo | On the molecular mechanisms for the H2/CO2 separation performance of zeolite imidazolate framework two-layered membranes[END_REF]. The best and most remarkable performance to date (H2 permeance of up to several thousand GPUs with H2/CO2 selectivity higher than 200) was obtained with ultrathin non-stacked ZIF-7 layered membranes [START_REF] Peng | Membranes. Metal-organic framework nanosheets as building blocks for molecular sieving membranes[END_REF]. Attractively, ZIF-8 membranes can also separate noble gas from air via a combination of molecular sieving, preferential adsorption, and diffusivity differences [START_REF] Wu | Recovery of xenon from air over ZIF-8 membranes[END_REF].

Carbon-based membranes

Carbon molecular sieve (CMS) membranes separate gases (e.g. for CO2/CH4, H2/CH4, CO2/N2) by both molecular sieving and selective surface adsorption/diffusion mechanisms [START_REF] Hägg | Carbon Molecular Sieve Membranes for Gas Separation[END_REF]. Their huge sensitivity to both competitive adsorbing species and contaminants (e.g. steam, oxidizing species) drastically limits their applicability in real gas mixtures and imposes frequent regenerations. The type of thermosetting polymer precursor, carbonization and post-treatment/storage conditions strongly impact on performance. The presence of a "hyperskin" with extremely low permeance was recently detected at the outermost region of carbon hollow fibers [START_REF] Sanyal | Cause and effects of hyperskin features on carbon molecular sieve (CMS) membranes[END_REF]. For high pressure applications (up to 20 MPa) CMS membranes are preferably supported on ceramic tubes [START_REF] Kruse | Carbon membrane gas separation of binary CO2 mixtures at high pressure[END_REF]. Although graphite is hydrophobic, pyrolytic carbon membranes often contain hydrophilic groups on their pore walls when prepared from polymers containing oxygen groups and carbonized below 500°C [START_REF] Llosa Tanco | Recent Advances on Carbon Molecular Sieve Membranes (CMSMs) and Reactors[END_REF]. In such case, water molecules are selectively transported by adsorption-diffusion mechanism, while the permeation of less adsorbable molecules is impeded. Also the hydrophilicity of twodimensional (2D) graphene and graphene oxide (GO) was vively debated. Thick graphene-layers seem to be ideal for hydrophobic applications whether single-layer graphene coatings are rather hydrophilic. The oxygen content and doping in graphene as well as the type of support strongly impact on the membrane hydrophilicity [START_REF] Hong | On the Mechanism of Hydrophilicity of Graphene[END_REF]. Conventional analysis of diffusive transport through graphene layers fails for such atomically thin membranes. The importance of non-covalent interactions towards H2 diffusion through graphene-based membranes has been demonstrated in a theoretical study [23], and revealed the importance of quantum effects such as polarization and electron delocalization as key factors accounting for the energy barrier (in addition to pore size). Gas permeation can proceed via direct gas-phase interaction with the pore, or interaction via the adsorbed phase on the membrane surface (i.e. surface adsorption, surface diffusion, and pore translocation), as shown in figure 1c [24]. The subtle interplay between relative adsorption strengths and mobilities that influence the permeation selectivities is explored in [24] using both Maxwell-Stefan formalism (surface diffusion fluxes) and kinetic Monte Carlo simulations (to quantify the correlation effects in surface diffusion of mixtures). Neglecting slowing-down correlation effects leads to severe underestimations of the membrane permeation selectivities that favour the more strongly adsorbed species.

Mixed matrix membranes (MMMs)

attract great attention due to their outstanding gas separation performance. They offer huge versatility by the selection of the relevant polymer matrix and the inorganic filler which is often a microporous adsorbent (e.g. zeolites, MOF clays, carbon, graphene or nanotubes…). The filler characteristics (size, morphology, particle aspect ratio, aggregation degree, pore size and size distribution, volume fraction, surface area, surface chemistry, etc.) and its batch adsorption properties (adsorption isotherm, kinetics, and conditions) are crucial for mass transfer mechanisms. The dispersion rate and volume fraction of the filler are also key factors, together with the sorption/solution-diffusion properties of the polymer and the quality of the filler/polymer interface [START_REF] Lin | Metal organic framework based mixed matrix membranes: an overview on filler/polymer interfaces[END_REF]. The transport of molecules through MMM membranes typically involves the following steps: i) surface adsorption and axial diffusion through the membrane thickness, ii) radial diffusion in the polymer matrix to reach the particles surface, iii) intra-particle diffusion and adsorption on active sites of the filler. Considering that defect-free polymer/particle contact is difficult to achieve, non-ideal morphologies are usually obtained, with various types of interfacial defects such as pore blocking, interfacial voids and polymer chain stiffening, which contributions can hardly be defined and controlled [START_REF] Lin | Metal organic framework based mixed matrix membranes: an overview on filler/polymer interfaces[END_REF]. Despite being promised a bright future, the design of efficient separation processes based on MMM membranes still relies to a large part on trial and error strategies. This is mainly because gas transport through such composite materials is a complicated problem in which the interface between the matrix and the dispersed filler is expected to play a key role. The available modeling approaches are often limited to empirical parameters which cannot be derived from molecular adsorption/transport coefficients. A modern computer application (Membrane Optimization Tool) for modeling gas transport processes through MMMs has been recently proposed in [26].

2.1.5.

Dense membranes (polymers, metals or ion-conducting ceramics) offer extremely high gas separation selectivity with strong impact of adsorption phenomena at the surface, as a first step. Due to their low cost, polymer membranes are largely investigated for gas separation, operating by molecular adsorption, solution-diffusion and desorption. Both gas solubility and diffusivity are tunable parameters for designing high-performance membranes [START_REF] Omidvar | Sorption-enhanced membrane materials for gas separation: a road less travelled[END_REF]. Dense inorganic membranes (i.e. both metals and ceramics) are also adsorption-sensitive; although dissociative adsorption (chemisorption) is the involved mechanism for these membranes operating at high temperature. Dense Pd-based membranes are highly efficient for continuous H2 purification thanks to the transport mechanism shown in figure 2a. The competitive adsorption of even small amount of CO, CO2, steam or sulfur compounds (e.g. H2S), strongly reduces the H2 permeating flux [28]. Coating Pd-based membrane surface with a protective film (e.g. H2 selective silica layer prepared by ALD [START_REF] Tsai | Hydrogen selective protective coating[END_REF]) is thus a relevant strategy which could also contribute to reduce the adsorption activation energy of H2. Dense mixed ion-and electron-conductive (MIEC) ceramic membranes are used for either oxygen or hydrogen separation. Adsorption and surface kinetics play key roles in the transport mechanisms (fig. 2b), especially for very thin membranes (i.e. below a critical thickness Lc) [START_REF] Fontaine | Oxygen and Hydrogen Separation Membranes Based on Dense Ceramic Conductors[END_REF]. Surface defects are likely to act as catalytically active centers at which oxygen or hydrogen molecules dissociate. The kinetics of the surface exchange process (which also occurs reversibly at the opposite side of the membrane) are crucial for a membrane in operation and are often rate limiting. There is thus no gain by manufacturing very thin membranes unless steps are taken to improve the flux through the interfacial regions. This can be done by increasing the effective membrane surface area, either by generating irregularities (e.g. chemical etching, sand blasting, ion etching) or deposition of a high specific surface area and ideally catalytically active nanomaterial (e.g. noble metals for spillover effect or oxides) [31, 32].

Adsorption in liquid phase separations

Surface properties and adsorption phenomena are critical to all membrane processes in liquid phase: microfiltration (MF), ultrafiltration (UF), nanofiltration (NF), reverse and forward osmosis (RO, FO), electrodialysis (ED), bioreactors (MBR), pervaporation (PV), vapour permeation (VP) but also membrane distillation (MD) and percrystallisation (PC). Whatever the considered application (e.g. synthesis, concentration or purification of valuable molecules, wastewater treatment, desalination, organic solvents dehydration or separation,...) and concerned species (ions, molecules, colloids, cells,…) adsorption phenomena have to be controlled to optimise fluxes, selectivity and long term stability, while limiting fouling and associated issues. The complexity of interface engineering for water treatment has been recently reviewed [34].

Fouling and options for surface modifications

Fouling is a broad term that includes many types of foulant materials (e.g. organic or mineral particles or dissolved species, colloids, proteins, microbiological organisms). Strategies to combat scaling, biofouling, and non-specific organic fouling are all different. Limitation of fouling phenomena implies higher productivity, less cleaning and longer membrane life, thereby reducing operational and capital costs. The major forces contributing to the attachment of foulants on the membrane surface are dispersion and polar interaction forces [START_REF] Abdelrasoul | Fouling in Membrane Filtration and Remediation Methods[END_REF]. When foulants are adsorbed on the membrane surface via electrostatic, hydrophobic, van der Waal, hydrogen-bonding or other interactions, they might accumulate or aggregate and form cake, gel, oil, biofilm or scaling layers. Accordingly, three different antifouling strategies can be considered [START_REF] Abdelrasoul | Fouling in Membrane Filtration and Remediation Methods[END_REF]: i) fouling resistance (inhibiting interactions via hydration layer and/or steric hindrance effect); 2) fouling release (minimizing interaction intensity via lowsurface-energy effect); 3) fouling attack (cell inactivation and foulant oxidization via active interaction/reaction effects). A conceptual model illustrating membrane fouling and cleaning is shown in Fig. 3a. Hydrophobic attraction results from van der Waals forces between molecules and the membrane material. Hydrophilic pore surfaces with highly negative charge are thus recommended for preventing adsorption of (natural) organic matter & colloids. The recent progress in antifouling membrane surface construction has been highlighted in [36] with a focus on bioinspired adhesion chemistry, supramolecular chemistry, mineralization chemistry, click chemistry and coupling chemistry. Substantial progress was recently achieved in interface engineering by using biomimetic concepts such as mussel-inspired chemistry, mediated by polydopamine or polyglycerol as substrate-independent coatings. This approach enables enormous flexibility in tuning membrane surface properties [START_REF] Yang | Dopamine: Just the Right Medicine for Membranes[END_REF] and more specifically its wettability, almost independently of the native surface properties. Universal super-wetting systems were obtained by combining mussel-inspired adhesion and highly hierarchical lotus-like surface structures [START_REF] Schlaich | Mussel-Inspired Polyglycerol Coatings with Controlled Wettability: From Superhydrophilic to Superhydrophobic, Surface Coatings[END_REF]. Insertion of inorganic nanoparticles (e.g. zwitterionic functionalized MoS2 nanosheets [START_REF] Liang | Zwitterionic functionalized MoS2 nanosheets for a novel composite membrane with effective salt/dye separation performance[END_REF]) can also improve the hydrophilicity of common polymer membranes such as PES. On the other hand, the Grignard reaction was explored as a relevant sustainable approach to mitigate ceramic membrane fouling by surface functionalization [START_REF] Mustafa | New insights into the fouling mechanism of dissolved organic matter applying nanofiltration membranes with a variety of surface chemistries[END_REF]. This strategy, playing with the affinity membrane/solvent/solute, provided strong anti-fouling effect for NF membranes (e.g. TiO2), but also improved the performance of both affinity based-separations and organic solvents nanofiltration (OSN) membranes. Scalable physical and chemical methods are available for the surface modification of existing membranes. For instance the modification of polymer membrane surface energy and functionalisation can be operated at industrial scale by non-equilibrium plasma treatment [START_REF] Al-Jumaili | Plasma Treatment of Polymeric Membranes, Non-Thermal Plasma Technology for Polymeric Materials-Applications in Composites, Nanostructured Materials and Biomedical Fields[END_REF]. Additionnaly, ALD has now to be considered as a highly relevant strategy to tackle the fouling challenges [42, 43]. A recent example of oxide interface engineering by ALD of TiO2 or SnO2 layers (~10 nm thick) for crudeoil-repellent membranes was reported [START_REF] Yang | Crude-Oil-Repellent Membranes by Atomic Layer Deposition: Oxide Interface Engineering[END_REF]. Molecular dynamics simulations revealed that a tightly bound hydration layer play a crucial role in protecting the surface from crude oil adhesion.

New inorganic and hydrid membrane materials for liquid phase applications

The development of new inorganic membrane materials such as MOFs, non-oxide ceramics (SiC, SiNx, BN), graphene, GO and other nanosheet-based membranes has considerably enriched the range of accessible/tunable hydrophilicity, organophilicity and surface charge effects. The versatility of MOF design recently led to membranes with impressive stability and performance in both water and organic media (e.g. water treatment, desalination [START_REF] Kadhom | Metal-organic frameworks (MOFs) in water filtration membranes for desalination and other applications[END_REF], PV and OSN) [46]. Silicon carbide (SiC) offers decisive advantages over conventional oxide ceramic membranes in water filtration processes, namely oily water treatment. The SiC surface is naturally covered with native silicon oxide which attracts water while repelling foulants. The low zeta potential (similar to silica) and high surface hydrophilicity are responsible for high fouling resistance although a slippage effect due to a tighly bond water film on native silica could explain the high measured water fluxes. Boron nitride is another original and promising new membrane material which is slightly polar and has a moderate surface energy (40 mJ/m 2 ), very similar to carbon value (35 mJ/m 2 ). Its high negative surface charge density might dramatically hinders ionic transport [START_REF] Weber | Boron Nitride Nanoporous Membranes with High Surface Charge by Atomic Layer Deposition[END_REF] and amino-functionalized BNlayered membranes revealed to be efficient for solute screening in both water and organic solvents [START_REF] Chen | Functionalized boron nitride membranes with ultrafast solvent transport performance for molecular separation[END_REF]. The hydrophobicity and moderate surface energy of macroporous and mesoporous carbon membranes is well adapted to membrane distillation [START_REF] González | Membrane distillation: perspectives for sustainable and improved desalination[END_REF] and crystallization processes, including the continuous percrystallisation approach [START_REF] Madsen | Fine Control of NaCl Crystal Size and Particle Size in Percrystallisation by Tuning the Morphology of Carbonised Sucrose Membranes[END_REF]. The unique surface properties of graphene, GO and other 2D materials generate high-performance versatile membranes for liquid phase applications [51]. Hydrophilic GO ultrathin membranes offer ultrafast permeation of water but when designed as micrometer-thick laminates, they can also attract small ions (with hydrated radii < 4.5Å) and provide their fast transport [START_REF] Joshi | Precise and ultrafast molecular sieving through graphene oxide membranes[END_REF]. Contrary to water, a non-slip behavior has been evidenced for organic solvents through GO laminates [53]. Both 2D and 3D nanomaterials are original nanofillers that contribute to enhance the development of MMM with extraordinary properties. MMM have huge potential in liquid phase applications, namely for water treatment and desalination [START_REF] Liang | Zwitterionic functionalized MoS2 nanosheets for a novel composite membrane with effective salt/dye separation performance[END_REF], but also PV [START_REF] Roy | Polymeric Nanocomposite Membranes for Next Generation Pervaporation Process: Strategies, Challenges and Future Prospects[END_REF] and OSN [START_REF] Vatanpour | Mixed Matrix Membranes for Nanofiltration Application[END_REF], although there are still huge challenges to optimize MMM design for real industrial applications. Superwetting MOF-based composite membranes for oil/water separations were obtained by integration of ZIF-8 in PAN nanofibers. They display unique wetting properties: superoleophobicity under water and superhydrophobicity under oil (Fig. 3b) and thus offer high separation efficiency (99.95%) for oil-in-water and water-in-oil emulsions [56]. Such membrane directly derives from the concept of adsorptive membranes described in the next paragraph.

Coupling adsorption and membrane filtration for depollution is an attractive option for waste

water depollution in the presence of both a relevant adsorber (e.g. activated carbon, porous polymers, porous ceramics, MOFs…) and an efficient membrane. The drawback of this approach is related to the required adsorbent regeneration (or replacement). Adsorptive membranes are usually applied for selective metal adsorption/sequestion. They are classically prepared by functionalisation of membranes made from nanofibers such as PAN. The fiber surface can be modified with either thiocompounds for Hg 2+ and Cd 2+ ions adsorption or diazoresin-ethylene-diamine-tetraacetic acid layer for Cu 2+ , Pb 2+ , Hg 2+ [START_REF] Zhang | Advanced Modified Polyacrylonitrile Membrane with Enhanced Adsorption Property for Heavy Metal Ions[END_REF]. Dispersion of water-stable nanoparticles PAN fibers is also an option: Zrbased MOF-808 was used for Cd 2+ and Zn 2+ ions removal [START_REF] Efome | Insight Studies on Metal-Organic Framework Nanofibrous Membrane Adsorption and Activation for Heavy Metal Ions Removal from Aqueous Solution[END_REF], activated biochar for chlortetracycline [START_REF] Taheran | Development of adsorptive membranes by confinement of activated biochar into electrospun nanofibers[END_REF] and functional GO nanosheets for lithium sequestration [START_REF] Sun | Porous nanocomposite membranes based on functional GO with selective function for lithium adsorption[END_REF]. Coupling membrane filtration with a conventional adsorption batch is also a highly relevant strategy for industry. Such adsorption-filtration-membrane (AFM) hybrid process is used for boron removal from water or for removing various micropollutants such as steroids or viruses. A hybrid RO/adsorption desalination plant (with silica gel adsorbent) was recently developed for producing irrigation and drinking water [START_REF] Atab | A hybrid reverse osmosis/adsorption desalination plant for irrigation and drinking water[END_REF].

Membrane bioreactors

Extractive membrane bioreactors are developed for wastewater treatment and combine a membrane separation and the degradation of target contaminants by both an active biofilm and microorganisms in the bioreactor. The physicochemical properties of membrane surfaces on the receiving side facing the bioreactor are critical in controlling the extent and nature of the membrane-attached biofilm. An original "Janus-type" nanofibrous composite membrane with opposing wetting properties (superhydrophobic PVDF/superhydrophilic PDMS) was reported [62] for an efficient extraction of phenol in long-term operation. The fouling-releasing fluoro-polymeric surface was found to attenuate the tendency of microbial attachment and encourage biofilm scouring.

Adsorption for the characterisation of nanopores and defects in membranes

Membrane science requires particular attention to defect detection methods and to the complex characterization of the microstructure of very thin supported nanoporous layers. Coupled methods based on the controlled adsorption of a probe molecule (vapor), appear as highly powerful either for the analysis of accessible porous network in thin membranes at room temperature or for the specific analysis of active pores and defects contributing to fluid transport (Fig. 4).

Relevant physical methods such as quartz crystal microbalance, ellipsometry or X-ray reflectometry, can be used to study the adsorption in the open porosity/free volume of the film versus relative pressure p/po, by measuring the mass up-take, refractive index or electronic density variations, respectively. The film thickness evolution, contraction or swelling during the sorption can also be monitored by the two latter techniques. Kinetic investigation of the penetration rate can be used to estimate both the membrane permeability and diffusivity coefficient of the selected probe [START_REF] Rouessac | Three characterization techniques coupled with adsorption for studying the nanoporosity of supported films and membranes[END_REF]. This could be highly attractive for high throughput-screening of membrane materials. Desorption porosimetry [START_REF] Shkolnikov | Estimation of pore size distribution in MCM-41-type silica using a simple desorption technique[END_REF] and evapororometry [START_REF] Zamani | Evapoporometry adaptation to determine the lumen-side pore-size distribution (PSD) of hollow fiber and tubular membranes[END_REF], both based on the measurement of weight variation during controlled vapor evaporation, are also ranked as adsorption-coupled methods.

Gas permeation measurements coupled with controlled adsorption specifically provide an analysis of active pores and defects contributing to gas transport in membranes. Permporometry is based on the controlled blocking of pores by adsorption/condensation of a vapour (e.g. ethanol, water or cyclohexane) present as a component of a gas mixture and the simultaneous measurement of the flux of a non-condensable dry gas (e.g. He) through the remaining open pores. Size distribution of active pores is derived in the mesoporous range by applying the Kelvin equation. Although this equation is non-valid in micropores, the method was applied to detect non-zeolitic pores in zeolite membranes [START_REF] Kita | Zeolite membranes for pervaporation[END_REF]. Determination of micropore sizes in amorphous membrane materials, even semi-quatitive evaluation, is often a tricky task as far as reference t-curves are available for flat surfaces and does not account for changes in surface chemistry and confinement behaviors in the pores [START_REF] Lépinay | Probing the microporosity of low-k organosilica films: MP and t-plot methods applied to ellipsometric porosimetry data[END_REF]. In addition a swelling of zeolite crystals during the adsorption step (e.g. n-hexane in MFI) might decrease the size of intercrystalline spaces/defects and alter the conclusions about membrane quality [START_REF] Yu | Characterizing Nonzeolitic Pores in MFI Membranes[END_REF]. Based on the permporometry concept, a permeation-mapping (gas sniffer) can also be used [START_REF] Kumakiri | Defect detection by permeation mapping[END_REF] to visualize and locate nano-sized defects in zeolite membranes).

Conclusion and prospects

Adsorption and associated surface phenomena are of paramount importance in governing membrane functionality and overall membrane process efficiency in gas phase and liquid phase applications. Both molecular and dissociative adsorption at the solid-fluid interfaces impact on membrane permeability, selectivity and stability. Huge progress to monitor membrane-fluid interactions has been made recently thanks to the emergence of new membrane materials with unique structure and microstructure, versatile surface modification methods and sophisticated membrane designs in synergy with increasingly powerful modeling investigations. Only few examples of up-to-date developments of inorganic and composite MMMs are illustrated in this mini-review as it is impossible to be exhaustive facing such a large diversity of results. Two-dimensional materials (e.g. graphene, GO, MOFs, MoS2, M-Xene, zeolites, LDHs, COFs) are draining significant attention, aiming at the design of nanosheet membranes with outstanding performance. More sophisticated theoretical models will have important implications for the rational design of the derived complex MMM systems. Non-oxide membranes are also promised to brillant future, thanks to their surface charges and antifouling properties. Finally, very few has been done in the investigation of external stimuli (e.g. ultrasounds, piezo-electricity, magnetism, electro-stimulation) for influencing adsorption phenomena and fouling. This could lead to powerful integrated membrane systems with operando monitoring of both membrane performance and stability.

Figure captions:

Figure 1 -

 1 Figure 1-a) Main gas separation mechanisms in microporous membranes, b) Typical evolution vs. temperature of single gas flux through zeolite membranes-Solid lines : predicted single gas flux of CO2 and H2 through a DD3R zeolite membrane-Dotted lines : surface diffusion (SD) and gas translation (GT) contributions (adapted from [3]), c) Schematic representation of transport mechanisms though a 2D graphene membrane (adapted from [24]).

Figure 2 :

 2 Figure 2: Different steps involved in the selective transport of hydrogen or oxygen species through: a) dense palladium-based membrane for H2 purification (adapted from [33]), or b) dense ion-conducting ceramic membranes for either O2 or H2.

Figure 3 :

 3 Figure 3: a) Conceptual model illustrating membrane fouling and cleaning when considering adsorption of natural organic matter, b) Schematic of the superwetting MOF-based composite membranes for continuous selective separation of oil/water mixtures and emulsions (adapted from [56]).

Figure 4 :

 4 Figure 4: Main membrane characterization methods coupling the controlled sorption of a condensable phase, for analysing either all open (accessible) pores or exclusively active pores and defects contributing to fluid transport.

Figure 1 -

 1 Figure 1-a) Main gas separation mechanisms in microporous membranes, b) Typical evolution vs. temperature of single gas flux through zeolite membranes-Solid lines : predicted single gas flux of CO2 and H2 through a DD3R zeolite membrane-Dotted lines : surface diffusion (SD) and gas translation (GT) contributions (adapted from [3]), c) Schematic representation of transport mechanisms though a 2D graphene membrane (adapted from [24]).

Figure 2 :

 2 Figure 2: Different steps involved in the selective transport of hydrogen or oxygen species through: a) dense palladium-based membrane for H2 purification (adapted from[START_REF] Gallucci | Advances on high temperature Pd-based membranes and membrane reactors for hydrogen purification and production[END_REF]), or b) dense ion-conducting ceramic membranes for either O2 or H2.

Figure 3 :

 3 Figure 3: a) Conceptual model illustrating membrane fouling and cleaning when considering adsorption of natural organic matter, b) Schematic of the superwetting MOF-based composite membranes for continuous selective separation of oil/water mixtures and emulsions (adapted from [56]).

Figure 4 :

 4 Figure 4: Main membrane characterization methods coupling the controlled sorption of a condensable phase, for analysing either all open (accessible) pores or exclusively active pores and defects contributing to fluid transport.

The authors declare no conflict of interest.


Astrid Cordier

Pierre-Alain Breuil

Typhène Michel

Lionel Magna

Hélène Olivier-Bourbigou

Jean Raynaud

Christophe Boisson

Vincent Monteil

Titanium-based

come

Introduction

Short linear alpha-olefins (LAOs) are crucial intermediates for the production of consumer goods such as lubricants, detergents and mostly polyethylene grades (HDPE, LLDPE). To face the soaring demand in LAOs driven by the global population growth, industrial companies invest continuous efforts for process optimization. Selective ethylene oligomerization processes have emerged to meet the specific requirements of the plastic industry. In the field of selective ethylene trimerization, homogeneous chromium catalysis has been extensively studied and is widely utilized for 1-hexene production. [1][2][3] Among alternative metals, titanium has proven its legitimacy in selective 1-hexene production. 4,5 In 2001, Hessen et al. reported the first hemilabile ancillary systems based on monocyclopentadienyl titanium bearing a pendant aryl group (η 5 -C5H4CMe2C6H5)TiCl3/MAO to be able to produce 1-hexene with unprecedented selectivities (> 75 wt %). 6 A decade later, Fujita and co-workers shed light on a second family of titaniumbased trimerization systems comprising a single phenoxy-imine tridentate ligand (SFI). 7 Upon activation with 10 000 equivalents of methylaluminoxane (MAO), tridentate phenoxy-imine complex 1 (Fig. 1) yields 1-hexene (92.3 wt %) with an activity close to the performances of the commercial chromium-based system. 1-hexene selectivity is limited by the formation of higher branched oligomers and polyethylene by-products. Although polymer selectivity seems insignificant (0.4 wt %), its accumulation causes major process issues (e.g. pipe clogging, reactor fouling). Therefore, understanding, controlling and eventually preventing the polymerization is the main challenge for process efficiency improvements and feedstock management.

Most of the reported literature on titanium-based trimerization systems focused on improving and rationalizing the oligomerization reaction, paying little attention to polymerization. [8][9][10][11][12][13][14][15][16][17][18][19][20] On the one hand, applying the ligandoriented catalyst design strategy, Fujita and coworkers reported the strong dependence between ligand structure and the reactivity of trimerization phenoxy-imine systems. 9 Indeed, 1hexene selectivity and activity can be tuned by subtle changes of substituents on specific positions of the phenyl rings. Ishii et al. highlighted that the activity and selectivity was modulated by altering the type and position of alkyl substituents. 9 Nevertheless, any modification of the global framework of the ligand generates polymerization active species. 8,11,14 On the other hand, efforts were invested on the mechanism and active species involved in the trimerization process. Even though several experimental and calculation studies showed evidences for a metallacycle mechanism, the activation process remains unclear. 8,11,12 DFT studies reported by Ishii et al. support the reduction of cationic Ti IV via a β-H transfer followed by reductive elimination to afford cationic Ti II active species. 9 Please do not adjust margins Please do not adjust margins Despite these rigorous studies, no clear identification of polymerization active species is reported. Duchateau and coworkers proposed the formation of a "TiRx" polymerization species, resulting from the abstraction of the ligand by trimethylaluminum. 13 Interestingly, they highlighted a decrease in 1-hexene selectivity from 93 wt % at 28 °C to 64 wt % at 58 °C, with a promoted polyethylene selectivity from 1.6 to 35 wt %.

We report herein an original polymer-to-catalyst approach, which consists in gathering information about the polymer to have an insight into the nature of the polymerization active species. For the first time, the influence of temperature on the activity and selectivity was systematically investigated between 30 and 80 °C. From this study, polyethylene properties were analyzed according to reaction temperature, and combined with kinetic studies to propose hypotheses on polymerization active species.

Results and discussion

Effect of the temperature on the reactivity

Selective trimerization of ethylene was studied with the titanium-based tridentate phenoxy-imine complex 2 (Fig. 1) This SFI complex was selected for its straightforward synthesis and yet similar catalytic performances compared to complex 1/MAO. 9,10 Indeed, similar steric hindrance is provided by adamantyl and tert-butyl groups at the R position.

Although the temperature-sensitivity of such system is mentioned in the literature, there are only few examples of reactions performed at temperature higher than 30 °C. 13 To have a clear view of the influence of temperature on the activity and selectivity of the system, a series of 7 experiments were performed from 26 °C to 80 °C (Table 1). All catalytic tests were carried out in a semi-batch mode under 10 bar of ethylene pressure for 30 minutes.

A lower activity reveals the deactivation of the system while increasing temperature. The maximal activities of the system are achieved between 26 and 40 °C (entries 1-3), which is consistent with the observations made in the literature and patents. 7,21 In fact, Fujita and coworkers reported a high activity at 30 °C for the original complex 1 of 260 kgethylene gTi -1 h -1 , calculated after extrapolation to 10 bar of ethylene. 7 Interestingly, the activity drops above 50 °C to reach a minimum at 80 °C (entry 7). This deactivation with the increase of temperature is counter-intuitive regarding pure kinetic considerations and reveals a pronounced sensitivity of the system to temperature. To the best of our knowledge, this is the first study providing a precise description of the activity decay of a phenoxy-imine titanium-based trimerization systems at higher reaction temperature. Along with the activity, the selectivity of this system is temperature-dependent. In the case of low temperature and high activity (Table 1, entries 1-3) 1-hexene is the main product of reaction with more than 85 wt % selectivity (Fig. 2). Low temperatures are therefore necessary to optimize 1-hexene production, which amounts to 520 kg1-hexene gTi -1 h -1 at 42 °C. In terms of secondary products, about 10 wt % of branched C10H20 oligomers are also formed by co-trimerization of 1-hexene and two ethylene molecules. However, no 1-hexene isomerization is observed. Eventually, ethylene polymerization occurs as minor side reaction. Although this polyethylene production seems insignificant compared to trimerization, it reveals the presence of a polymerization species even at low reaction temperature. Whilst reaction temperature increases, polyethylene production is enhanced but trimerization is highly disfavored. Above 60 °C, the catalytic system mainly polymerizes ethylene (80 wt % selectivity at 80 °C) with a moderate activity (Table 1, entry 7). Please do not adjust margins

Please do not adjust margins

The combination of a poorer activity with a higher polymer formation at elevated reaction temperature reveals a new behavior of such systems. Moreover, the omnipresence of polyethylene questions the nature of the active species. As mentioned earlier, the production of polymer increases despite the deactivation of the overall catalytic system above 60 °C. It is still unclear whether the prominence of polyethylene formation at elevated temperature reflects an increase in activity of polymerization catalyst formed during activation and/or the generation of additional active species from trimerization catalyst alteration.

Further investigations support that polymerization catalyst(s) is(are) generated after activation with ethylene rather than from a side reactions during the preactivation step. Indeed, a preheating of the precatalyst/cocatalyst mixture at 80 °C for 30 minutes before running the catalytic test at 30 °C in presence of ethylene has no significant impact on the activity and the selectivity.

Polymer characterization

The investigation of polymer properties is a crucial step for polymerization active species identification. By definition, every catalyst is unique and leaves its fingerprint in the polymer they produce. 22 Generally speaking, polyethylenes displaying a large molar mass distribution (MMD) are obtained in the case of multi-site catalysis. In contrast, polymers highly homogeneous in size (Ɖ ~ 2) can be produced by single-site metallocenes and post-metallocenes. The amount of short chain branching (SCB) and average molar masses depend on the ability of the species to perform LAO insertion and transfer reactions. These reactions are governed by steric and electronic environments around the metal center. Thus, after the analysis of the polymers obtained in the series, their properties are compared with the ones reported for several classes of catalysts. This strategy allows to have more information about the polymeric side product and also to categorize the unknown active species within a specific family of catalyst.

PE-32 to PE-80 (Table 1, entries 2-7) were characterized in terms of molar mass distribution (MMD) and chemical composition employing high temperature SEC, high temperature 13 C NMR and DSC techniques. HT-SEC analyses of molar mass distributions reveal a clear evolution of polymer properties upon increase of reaction temperature. All polymers display a high weight average molar mass (Mw) with a decreasing Mw from 15x10 5 to 2x10 5 g mol -1 for PE-32 and PE-80 respectively (Fig. 3, Table 2). It is well known that high molar masses are synonym of disfavored transfer reactions. Moreover, temperature enhances the rate of transfer reactions leading to the decrease of Mw. Under conditions of favored 1hexene productivity, narrow molar mass distributions are obtained with a low dispersity of about 2. A dispersity around 1.7-1.8 is uncommon in catalytic polymerization except for peculiar pseudo-living bis(phenoxy-imine) systems. 23 Such low values may emerge from a cut-off of the HT-SEC device for ultrahigh molar mass hydrocarbon chains (> 10 6 g mol -1 ). It is worth pointing out that the higher the temperature, the broader the MMD since a dispersity of 4.6 was measured for PE-80.

Chemical composition distribution reflects the ability of the catalyst to incorporate comonomers, 1-hexene in this case. By coupling differential scanning calorimetry and hightemperature 13 C NMR, the length and content of chain branching were evaluated. Only C4 branching is identified from the incorporation of 1-hexene in the polyethylene backbone. For all polymers produced from 32 °C to 80 °C, 1-hexene content is substantially low (< 1 mol %) and even decreases at higher reaction temperature (Table 2). This trend is explained by the lower amount of 1-hexene produced at T > 42 °C (Fig. 2). Noteworthy, high 1-hexene concentration in the reaction media at 32 °C (Table S1) has little impact on the comonomer content in the polymer. Consequently, the polymerization species displays a poor incorporation ability toward short linear alphaolefins.

Polymer analyses provide valuable information about the polymerization species in the SFI system. First of all, PE-32 to PE-68 are likely produced by one active species as low dispersities are commonly ascribed to single-site catalysis. [START_REF] Severn | Tailor-Made Polymers[END_REF][START_REF] Bochmann | [END_REF][26] However, several active sites may be generated at 80 °C given the larger span of PE chain size. To the best of our knowledge, it is the first study reporting the analysis of the polymer produced with such titanium-based phenoxy-imine trimerization systems. Polymers with high molar masses and a low dispersity were also mentioned by Ye et al. with a hemimetallocene (η 5 -C5H4CMe2C6H5)TiCl3/MAO. 27 In addition, the poor ability for LAOs copolymerization (see a comparison with a metallocene in Fig. S3) and limited transfer reaction are common features for bulky post-metallocenes or ligand-free molecular titanium-based Ziegler-Natta species. On the one hand, the ligands provide a significant steric hindrance close to the metal center, which hampers β-H elimination and LAO coordination. On the other hand, it is typical from ZN species that the PE macromolecules of high molar masses exhibit a low LAO content. 28

Kinetic studies

To clarify the relationship between trimerization and polymerization species, kinetic studies were performed at 40, 60 and 80 °C using a medium-throughput automated Chemspeed platform. Catalytic tests were conducted during 5, 10, 20 and 30 minutes in three identical autoclaves.

The kinetic studies reveal that the evolution of ethylene consumption is strongly dependent on the reaction temperature. Fig. 4 shows a clear difference in the reaction rate profile between low and high reaction temperature for experiments carried out for 30 minutes. For all temperatures, activity increases and progressively decreases. The maximum reaction rate of 540 kgethylene gTi -1 h -1 at 40 °C is 1.5 and 3-fold of the activity at 60 °C and 80 °C, respectively. Moreover, the system deactivates quicker at higher reaction temperature. The decay in activity is initiated within the first 10 minutes of reaction at 60 and 80 °C while the system starts deactivating after 18 minutes at 40 °C. Average activities for the three experiments (441 kgethylene gTi -1 h -1 at 40 °C, 161 kgethylene gTi -1 h -1 at 60 °C and 106 kgethylene gTi -1 h -1 at 80 °C) are in the same order of magnitude than for the temperature study (Table 1, entries 3, 5 and 7).

A first order dependence of the reaction rate with ethylene concentration was observed (Fig. S6) and a reliable fitting with experimental data was achieved by applying Kissin's kinetic model. 29 The model has been applied to oligomerization reaction considering ethylene insertion in the metallacycle as the propagation step. This specific Kissin's model involves a process with initiation (ki), propagation (kp) and deactivation (kd) steps whose reaction rate is described by the following equation:

Rp = kp[C2][Ti] [exp(-kit) -exp(-kdt)] ki/(kd -ki)
This model accurately corroborates with experimental data especially in conditions of enhanced polymer productivity, i.e 60 °C and 80 °C (Fig. 4 and Table 1, entries 5 and 7). Applying Kissin's model at 40 °C is relevant although the fast initial activation could not be fitted. Details regarding data fitting are provided in the supplementary information (Table S5) As a result, the rate of reaction is governed by an elementary step involving one ethylene molecule in our conditions. Thus, a first order reaction towards ethylene concentration excludes the oxidative coupling as rate determining step of the metallacycle process. DFT calculations reported by Ishii et al., indicate that the insertion of ethylene in the pentacycle is kinetically limiting as it exhibits the highest free energy (122.3 kJ mol -1 ). 9 The accurate reproducibility of Chemspeed reactors in polymerization reaction allows to confidently analyze the products formation over time. For each product, the relative yield is calculated by dividing the amount of a product by its final yield, i.e mass of product obtained after 30 minutes of reaction. From Fig. 5, it is clear that the processes of oligomer and polymer production are different at 40 °C and 80 °C. In conditions of favored trimerization, all products are continuously generated and their relative yields follow the same trend. Indeed, for each compound, about 40 wt % of the overall yield is obtained within the first 5 minutes of reaction (Fig. 5, top). Despite a similar evolution of product formation, the mass of 1-hexene obtained (4.49 g) is significantly greater than the one of polyethylene (0.08 g). In contrast, trimers and polymer production are independent at 80 °C. 1-hexene formation only occurs within the first 5 minutes (0.37 g for 1-hexene) while the quantity of polymer evolves from 1.99 g after 5 minutes to 3.05 g after 30 minutes. Note that the increase of C10H20 oligomers corresponds to the evolution from 0.010 to 0.013 g, which is almost negligible. The kinetics and product studies revealed that the dependence between trimerization and polymerization active species is related to temperature. At 40 °C, the presence of trimers and polymer combined with their continuous formation implies that two catalysts emerged at the beginning of the reaction from the same precatalyst source. While at 80 °C, trimerization catalyst deactivates within the first 5 minutes of reactions and the polymerization catalyst displays an enhanced activity compared to lower temperature conditions. In this case, the polymerization catalyst could be formed either from a degradation of the latter or independently.

The incorporation of 1-hexene in the polymer over time is a reliable indicator of a potential evolution of active species. At 40 °C and 60 °C, 1-hexene content in the polymer (Fig. 6, top) increases owing to the 1-hexene enrichment in the reaction medium (Fig. 5, top). Homopolyethylene is produced at 80 °C due to a negligible amount of 1-hexene produced (Fig. 5, bottom). The relationship between 1-hexene content in the PE and in the reaction media follows the same correlation at 40 °C and 60 °C (Fig. 6, bottom). This similar 1-hexene sensitivity is the characteristic of an identical polymerization catalyst.

These kinetic studies clearly show that the same and unique polymerization catalyst is present between 40 and 60 °C. This species is formed at the early stage of reaction, along with the trimerization catalyst. Besides, as several active species are suspected to polymerize ethylene at 80 °C, trimerization species may be converted into polymerization species. Based on these conclusions, potential active species hypothesized in the literature exhibit a similar behavior for the production of polymer with such features, i.e a high molar-mass polyethylenes with a limited amount of SCB. In one case, a "TiRx" species proposed by Duchateau et al., would result from a ligand abstraction from Ti to Al, probably induced by the TMA contained in MAO. Such homogeneous molecular Ziegler-Natta species would indeed poorly copolymerize LAOs with ethylene. 30,31 A second assumption results from the identification of Ti III species by Sattler and Talsi et al. during catalytic tests with (FI)TiMe3/B(C6F5)3 and complex 1/MAO, respectively. 12,18 Even though Ti III formation has been correlated to the deactivation of the system, one cannot exclude this possibility since few homogeneous complexes of Ti III are reported to polymerize ethylene. 32,33 Eventually, it is also conceivable that a rearrangement or structure alteration of complex 2 would lead to a putative cationic Ti IV species that successively coordinates and inserts ethylene via a Cossee-Arlman mechanism. In this case, one could assume that such species bears a bulky phenoxy-imine ligand providing sufficient steric hindrance to disfavor β-H transfer reaction and 1-hexene insertion in the polymer chain.

Conclusion

In this work, the phenoxy-imine titanium-based system for ethylene trimerization has been investigated according to reaction temperature from 26 °C to 80 °C. Unlike most of Please do not adjust margins Please do not adjust margins previous studies, our focus was oriented towards rationalizing the formation of polymer along with 1-hexene. This study reveals and quantifies a switch of selectivity from 1-hexene to polymer along with a decrease of activity between 26 to 80 °C. An original polymer-to-catalyst strategy revealed that the polymerization catalyst yields polyethylene with high molar masses and poorly incorporates 1-hexene. A joint formation of trimerization and polymerization species at the early stage of reaction is supported by a kinetic study. Further work is in progress to identify the species responsible for the production of polyethylene.

Experimental

General

All air-and moisture-sensitive reactions were performed under an inert argon atmosphere using standard Schlenck and glovebox techniques. Toluene, pentane and dichloromethane were dried from a MBRAUN solvent purification system with activated alumina and copper catalyst columns. Deuterated solvents were stored with molecular sieves in the glovebox. Chemicals were purchased from Sigma-Aldrich, Strem Chemicals, Acros Organics or Tokyo Chemical Industry Co., Ltd. and were used without further purification. Methylaluminoxane 30 wt % in toluene (13.6 wt % of aluminum, 5.24 wt % TMA, 26.2 wt % MAO) was purchased from Albemarle Corporation and stored at -30 °C under inert atmosphere. Ethylene was purified by passing through three columns containing activated molecular sieves, alumina and BASF copper oxide catalyst.

Complex synthesis

Complex 2 was synthesized according to literature procedure. 10 In a 100 mL Schlenk, a solution of titanium tetrachloride 1.0 N in toluene (1.1 mL, 1.1 mmol) was added to a solution of the phenoxy-imine-ether ligand (340.1 g, 0.91 mmol) in 10 mL of toluene at -78 °C. The solution warmed up to room temperature and stirred overnight. 50 mL of pentane was added to precipitate the catalyst. The brown-red particles were washed with 3x10 mL of pentane and dried under vacuum at 40 °C. Yield: 0.38 g, 0.72 mmol, 79 %.

Catalytic tests

Ethylene oligo/polymerizations were performed in a 1L doublejacketed reactor. A diluted TEA solution in heptane (15 mol L -1 ) was used to scavenge the reactor at 80 °C prior to catalytic test. A solution of MAO 30 wt % in 290 mL toluene (1.6x10 -2 mol L -1 ) was introduced in the reactor and heated at desired temperature. 10 mL of precatalyst solution (2.9 mg, 3 µmol) was then injected and ethylene was continuously fed to keep the pressure of 10 bar constant for 30 minutes. Reaction was quenched by 10 mL methanol and the reactor was cooled to 5 °C before depressurization. The liquid phase was treated with a sulfuric acid solution and analyzed by GC using dodecane as internal standard. The polymer was washed with acidified methanol and methanol. Recovered polymers were dried under vacuum at 100 °C for 2 hours.

The kinetic studies were performed on a fully automated Chemspeed homogeneous catalysis platform located at Axel'One Campus, Villeurbanne, France. This unit is included in a Glovebox to guarantee a controlled atmosphere. Catalytic tests were performed in three independent 270 mL-reactors. Toluene (120 mL) and MAO 30 wt % in toluene solution (0.6 mL, 1 500 mmol) were introduced in each reactor. Once the desired temperature was reached, reactors were pressurized to 10 bar of ethylene. Using SWILE© technology, a batch solution of complex 2 (3.2 mg, 6.1 µmol) in toluene (10 mL) from which 3 mL were injected via a high pressure pump in each reactors. The reaction was run in a semi-batch mode for the desired reaction time. Ethanol was introduced under pressure to quench the reaction. Reactors were cooled to 5 °C before depressurization. Further treatments of the solid and liquid phase were performed as described previously.

Polymer characterization

High Temperature Size Exclusion Chromatography (HT-SEC)

HT-SEC analyses were performed using a Viscotek system (Malvern Instruments) equipped with three columns (PLgel Olexis 300 mm × 7 mm from Agilent Technologies). Samples volume of 200 μL with concentrations between 1-2 mg mL -1 were eluted in 1,2,4-trichlorobenzene using a flow rate of 1 mL min -1 at 150 °C. The mobile phase was stabilized with 2,6-di(tert-butyl)-4-methylphenol (200 mg L -1 ). Online detection was performed with a differential refractive index detector, a dual light scattering detector (RALS LALS) and a viscometer detector for absolute molar mass measurement.

Differential Scanning Calorimetry (DSC)

Polyethylene melting temperature, crystallization temperature and crystallinity were determined using a Mettler Toledo DSC1. An average of 7 mg of PE was analyzed in a 40 µL aluminum crucible. After a first heating from 25 °C to 180 °C at 20 °C min - 1 , samples were cooled and heated within the same range of temperature at 5 °C min -1 . Only the second heat was considered for melting temperature measurement.

Fig. 1

 1 Fig. 1 Titanium-based phenoxy-imine complex for selective trimerization

Fig. 2 2 :

 22 Fig. 2 Evolution of product formation with temperature

Fig. 3

 3 Fig. 3 Molar masses distributions for polymers obtained between 26 and 80 °C.

4 |

 4 J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xx Please do not adjust margins Please do not adjust margins

Fig. 4

 4 Fig.4 Evolution of reaction rate from experimental data (solid line) and fitting using Kissin's kinetic model (dashed lines)

5 PleaseFig. 5

 55 Fig. 5 Evolution of relative yield for oligomers and polymers over time

Fig. 6

 6 Fig.6 Evolution of 1-hexene content in the polymer over time (up) and according to the composition of the reaction medium (down)

6 |

 6 J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xx

Table 1 . Catalytic performance of complex 2/MAO Test T (°C) t (min) nTi (µmol) Al/Ti Activity Global a Polym. b

 1

	1	26	29	2.77	1 640	419	1.3
	2	32	35	3.46	1 310	607	5.7
	3	42	32	3.78	1 200	537	17.5
	4	49	31	3.4	1 330	160	72.3
	5	58	31	4.59	1 000	172	84.3
	6	68	28	3.19	1 422	97	68
	7	80	31	3.85	1 180	78	62.4
	Conditions: complex 2, MAO 30 % in toluene (1 mL), 300 mL toluene, 10 bar of
	ethylene
	a in kg ethylene gTi -1 h -1
	b in kg polyethylene gTi -1 h -1

Table 2 .

 2 Average molar masses and chemical compositions for polymers obtained between 26 and 80 °C.

Acknowledgements

This project was funded by Region Auvergne-Rhône Alpes and IFP Energies nouvelles. We are thankful to the NMR Polymer Center of Institut Chimie de Lyon (FR5223), Franck Collas (Mettler Toledo) and Olivier Boyron (CNRS) for their support in NMR, DSC and SEC analyses. Chemspeed technologies is acknowledged for the homogeneous catalysis platform and Axel'One Campus for facilities. We want to express our gratitude to Jaroslav Padevet (Chemspeed) for his technical support.

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High Temperature Nuclear Magnetic Resonance (HT-NMR) 13 C NMR analysis were performed with a Bruker Avance II 400 spectrometer operating at 100.6 MHz. Spectra were recorded at 393 K using a PSEX 10 mm probe for 13 C NMR. A mixture of odichlorobenzene/o-dichlorobenzene-d4 (1/10 v/v) was used as solvent. Samples were prepared as a solution of 100 mg of polymer in 6 mL of solvent after several 150 °C-25 °C cycles.