Nicolas Gast
email: nicolas.gast@inria.fr

Construction of Lyapunov functions via relative entropy with application to caching *

We consider a system of interacting objects that is a generalization of the model of the cache-replacement policy RAND(m) introduced in [6]. We provide a mean-field approximation of this system. We show how to use relative entropy to construct a Lyapunov function for this model. This guarantees that the mean-field model converges to its unique fixed point.

INTRODUCTION

Mean-field approximation is a powerful tools for studying systems of interacting objects. Many papers have established that, under mild condition, a system of interacting objects converges to a deterministic dynamical system ẋ = f (x) over any finite time interval. However, characterizing the asymptotic behavior of a differential equation is often a difficult task. In particular, showing that the differential equation has a unique fixed point does not guarantee the convergence of the stochastic system to this fixed point [START_REF] Benaïm | A class of mean field interaction models for computer and communication systems[END_REF]. A sufficient condition is that all trajectories of this dynamical system converge to a fixed point of the equation. This is often difficult to show. Recently, it has been shown in [START_REF] Budhiraja | Limits of relative entropies associated with weakly interacting particle systems[END_REF] that relative entropy can be used to construct a Lyapunov function that guarantees the convergence of all trajectories to a fixed point. In this paper, we use a similar idea to construct a Lyapunov function for a generalization of the cache-replacement policies RAND(m) introduced in [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF].

This paper makes two contributions. First, we finish the analysis of the mean-field model of [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF] by providing a Lyapunov function for the mean-field model. Second, we illustrate a method for constructing explicitly a Lyapunov function by considering a term of relative entropy plus a corrective term. We believe that this method is generic and can be applied to other mean-field dynamics.

MODEL AND APPROXIMATION

We consider a system of n objects that move between boxes. There are h boxes and the ith box contains mi objects. If the object k is in box i, then at rate λ k,i it is exchanged with an object (taken at random) from box i + 1. A configuration of the system is a vector c = (c1 . . . cn) where c k is the box in which object k is. The set of possible configurations C is the set of vectors c = (c1 . . . cn) such that there are exactly mi objects in box i:

|{k : c k = i}| = mi.
This model is a generalization of the model of the cachereplacement policy RAND(m) introduced in [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF], where the authors study a cache that is decomposed into boxes. The requests for item k occurs at rate λ k . If this item is in box i, it is exchanged with a random item from box i + 1. The model of [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF] can be expressed in our setting by choosing λ k,i := λ k , the request-rate of object k.

It can be verified that this system is reversible and that its stationary measure satisfies, for each c ∈ C:

π(c) = κ n k=1 c k -1 i=1 λ k,i , where κ = ( c∈C n k=1 c k -1 i=1 λ k,i ) -1 is the normalizing constant.
The constant κ can be computed by a dynamic programming algorithm in a time O(n h i=1 (mi + 1)) by using a similar algorithm as the one of [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF]. While this complexity is polynomial in n and m, it is exponential in the number of boxes. This calls for an approximation.

The mean-field approximation

We consider the following set of ODEs (for k ∈ {1 . . . n}, i ∈ {1 . . . h}):

ẋk,i (t) = 1 {i>1} λ k,i x k,i-1 (t) - λ ,i-1 x ,i-1 (t) mi x k,i (t) + 1 {i<h} λ ,i x ,i (t) mi+1 x k,i+1 (t) -λ k,i x k,i (t) (1)
with initial conditions x k,i (0) = 1 {c k (0)=i} , where 1 {A} = 1 if A is true and 0 otherwise. The quantity x k,i is meant to be an approximation of the probability for an object k to be in box i. For object k, this ODE corresponds to the probability measure of a birth-death process of birth rate λ k,i and of death rate µi(x) = n =1 λ ,i-1 x ,i-1 /mi. This birth-death process is represented in Figure 1. The intuition behind this ODE is the following. If object k is in box i, it jumps to box i + 1 at rate λ k,i . It jumps to box i -1 when an object from box i -1 jumps to box i and is exchanged with this object. This occurs at rate n =1 1 {c =i-1} λ ,i-1 . The following result guarantees that x k,i is indeed a good approximation of the stochastic process. It can be proven by adapting the proof of [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF]Theorem 6], that is based on stochastic approximation arguments.

1 2 h . . . . . . λ k,1 λ k,2 λ k,h-1 µ h-1 (x) µ2(x) µ1(x)
(x) = n =1 λ ,i-1 x ,i-1 /mi.
Theorem 1. For any T , there exists a constant C independent of the parameters λ i,k and m i,k such that

E sup t≤T /ε,i,α≥0 k (λ k,i ) α (x k,i (t) -1 {c k (t)=i} ) ∞ ≤ Cε, where ε = max i,k λ i,k + maxi(mi) -1 .
This theorem guarantees that the content of the boxes of the stochastic process converges to the one of the ODE approximation as the values (λ k,i )s go to 0 and the box sizes (mi)s go to infinity. Note that the values of (λ i,k )s can be made arbitrarily small by multiplying all of them by a constant τ < 1. Hence, the main assumption for the convergence to hold is that the box sizes (mi)s go to infinity.

Fixed point analysis

The differential equation ( 1) has the following form:

d dt x k,i (t) = h j=1 x k,j (t)Γ (k) j,i (x(t)),
where Γ (k) (x) is the transition kernel of a birth-death process with a birth rate λ k,i and a death rate that depends on x: µi(x) = n =1 λ ,i-1 x ,i-1 /mi. For a fixed x, the stationary measure of the birth-death process of kernel Γ (k) (x) is π k,. (x), where for i ∈ {1 . . . h}:

π k,i (x) = i-1 j=1 λ k,j /µj(x) h j =1 j -1 j=1 λ k,j /µj(x) (2)
A fixed point of the ODE (1) satisfies x = π(x). By monotonicity arguments similar to the proof of [6, Theorem 7], it can be shown that this equation has a unique fixed point, that we denote x * . This fixed point can be efficiently computed by an iterative procedure (see [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF]Section 5.3]).

LYAPUNOV FUNCTION

In this section, we show that any solution of (1) converges to the unique fixed point x * . We construct explicitly a Lyapunov function, by using relative entropy.

The reason to use relative entropy

As we recall below, relative entropy between a Markov chain and its stationary measure decreases with time. We will see that this implies that it is natural to consider the relative entropy between x and π(x) plus a corrective term.

Recall that for two discrete distributions p and q over a finite set {1 . . . h}, the relative entropy of p with respect to q (also called Kullback-Leibler divergence) is

R(p q) := h i=1 pi log pi qi
If x(t) is the probability distribution at time t of a continuous time Markov process that has kernel Γ and if π is its steady-state distribution, then the relative entropy of x(t) with respect to π decreases with time:

d dt R(x(t) π) = h i=1 d dt xi(t) 1 + log xi(t) πi ≤ 0, (3)
where d dt xi(t)= j x k,j (t)Γj,i. The equality holds only when x(t) = π. A simple proof of this result is in [START_REF] Budhiraja | Limits of relative entropies associated with weakly interacting particle systems[END_REF]Lemma 3.1].

For a given object k, the process x k,. is a Markovian model with kernel Γ (k) (x) that depends on x. The stationary measure of this kernel is π k,. (x), defined in [START_REF] Boudec | The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points[END_REF].

The derivative with respect to time of R(x k,. π k,. (x)) is

d dt R(x k,. (t) π k,. (x(t))) = h i=1 d dt x k,i (t) 1 + log x k,i (t) π k,i (x) - h i=1 x k,i (t) d dt log π k,i (x(t)) ≤ - h i=1 x k,i (t) d dt log π k,i (x(t)) (4)
This inequality holds because of Equation ( 3). If h(x) is such that d dt h(x(t)) equals Equation ( 4), then the function t → R(x(t) π(x(t))) -h(x(t)) decreases with time. When h(x) is upper bounded, this function decreases to some value, which by (3), implies that x(t) converges to a fixed point. Hence, exhibiting such a function h suffices to guarantee the convergence of the ODE to a fixed point.

Main result and proofs

We say that a vector x is admissible if x k,i ≥ 0 and if n k=1 x k,i = mi and h i=1 x k,i = 1. The first condition corresponds to having mi objects per box and the second that the object k is in one of the h boxes.

Theorem 2. For any admissible initial condition x(0), the ODE (1) has a unique solution x. Moreover, for all t, x(t) is admissible and limt→∞ x(t) = x * .

Proof. The existence and uniqueness of the solution is due to the Lipchitz-continuity of the right-hand side of Equation [START_REF] Benaïm | A class of mean field interaction models for computer and communication systems[END_REF]. The fact that x(t) remains admissible when x(0) is admissible can be verified by showing that h i=1 ẋk,i = n k=1 ẋk,i = 0 when x is admissible. We prove in Lemma 1 and 2 that the function ( 5) is a Lyapunov function: it is non increasing and lower bounded. Hence, limt→∞ d dt L(x(t)) = 0. By Lemma 1 and the continuity of π(x), this shows that limt→∞ x(t) = x * .

We define the function L(x) be by:

L(x) = n k=1 R(x k π k (x)) - h i=1 mi log i-1 j=1 µj(x) - n k=1 log h i=1 i-1 j=1 λ k,j /µj(x) (5)
The first term is the relative entropy and the other two are an integral of Equation (4). Lemma 1. Let x be a solution of the ODE (1). Then d dt L(x(t)) ≤ 0 with equality if and only if x(t) = π(x(t)).

Proof. By definition of π(x) in Equation ( 2), we have

log π k,i (x) = log i-1 j=1 λ k,j -log i-1 j=1 µj(x) -log h i=1 i-1 j=1 λ k,j /µj(x) (6)
As d dt log( i-1 j=1 λ k,j ) = 0, this implies that

h i=1 n k=1 x k,i (t) d dt log(π k,i (x(t))) = h i=1 n k=1 x k,i (t) =m i d dt log i-1 j=1 µj(x(t)) + n k=1 h i=1 x k,i (t) =1 d dt log   h j =1 j -1 j=1 λ k,j /µj(x(t))  
This is equal to the the derivative with respect to time of the last two additional terms of Equation [START_REF] Fricker | Incentives and redistribution in homogeneous bike-sharing systems with stations of finite capacity[END_REF]. Applying this and Equation ( 4) implies

d dt L(x(t)) = h i=1 n k=1 d dt x k,i (t) 1 + log x k,i (t) π k,i (x(t))
.

By Equation [START_REF] Budhiraja | Limits of relative entropies associated with weakly interacting particle systems[END_REF], this shows that d dt L(x(t)) ≤ 0 with equality only if x(t) = π(x(t)).

Lemma 2. For any admissible x, we have

L(x) ≥ -n log(h) - n k=1 h i=1 log i-1 j=1 λ k,j .
Proof. Expanding the definition of the relative entropy, we have:

L(x) = n k=1 h i=1 x k,i log x k,i - n k=1 h i=1 x k,i log π k,i (x) - h i=1 mi log i-1 j=1 µj(x) - n k=1 log h i=1 i-1 j=1 λ k,j /µj(x)
By using the same transformation as in [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF], we get:

n k=1 h i=1 x k,i log π k,i (x) = n k=1 h i=1 x k,i log i-1 j=1 λ k,j - n k=1 h i=1 x k,i log i-1 j=1 µj(x) - n k=1 h i=1 x k,i log   h j =1 j -1 j=1 λ k,j /µj(x)   This shows that L(x) = n k=1 h i=1 x k,i log x k,i + n k=1 h i=1 x k,i log i-1 j=1 λ k,j ≥ -n log h - h i=1 n k=1 log i-1 j=1 λ k,j ,
where the first part of inequality comes from the entropy the second from the fact that |x i,k (t)| ≤ 1.

CONCLUSION AND DISCUSSION

In this paper, we constructed a Lyapunov function for a generalization of the RAND(m) model of [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF]. This Lyapunov function is composed of a term of relative entropy R(x π(x)) plus an additive correction term that compensates for the variation of π(x). This function guarantees that the ODE approximation converges to its unique fixed point and hence justifies the fixed point analysis.

The construction of the Lyapunov function proposed in this paper is generic. We believe that similar ideas can be applied to many other mean-field systems. The two additive terms in the expression of the Lyapunov function ( 5) form an integral of Equation ( 4). Their simple expression comes form the fact that in the stationary measure π(x), the quantity µi(x) does not depend on the object k while λ k,i is does not depend on time. Note that the Lyapunov function of the model of [START_REF] Fricker | Incentives and redistribution in homogeneous bike-sharing systems with stations of finite capacity[END_REF]Section 3] has the same form.

The stationary measure of the original system is of Gibbs type since it is a product form measure on the set of possible configuration C. However, this measure has not the same form as the Gibbs type measure of Section 4 of the papers [START_REF] Budhiraja | Limits of relative entropies associated with weakly interacting particle systems[END_REF][START_REF] Budhiraja | Local stability of kolmogorov forward equations for finite state nonlinear markov processes[END_REF]. Hence, it seems that their Lyapunov function [START_REF] Budhiraja | Limits of relative entropies associated with weakly interacting particle systems[END_REF]Equation (4.14)] is not directly applicable here although the expressions are close.

Finally, it is shown in [START_REF] Boudec | The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points[END_REF] that, a sequence of reversible process that converges to the solution of an ODE has a sequence of stationary measures that concentrates on the fixed point of the ODE, regardless of the asymptotic behavior of this ODE. A benefit of exhibiting a Lyapunov function is that we are also able to characterize the asymptotic behavior of the ODE. It is not clear if the reversibility has an impact on the particular form of the Lyapunov function L.

Figure 1 :

 1 Figure 1: The birth-death process approximation of the box in which object k is, where µi(x) =

* This work is partially supported by the EU project QUAN-TICOL, 600708.


Jean-Christophe Aval

Frédéric Chapoton

Poset structures on (m + 2)-angulations and polynomial bases of the quotient by G m -quasisymmetric functions

For integers m, n ≥ 1, we describe a bijection sending dissections of the (mn + 2)-regular polygon into (m + 2)-sided polygons to a new basis of the quotient of the polynomial algebra in mn variables by an ideal generated by some kind of higher quasi-symmetric functions. We show that divisibility of the basis elements corresponds to a new partial order on dissections, which is studied in some detail.

Introduction

Let m ≥ 1 be an integer. For every integer n ≥ 1, we define a simple poset structure P m,n on the set of (m + 2)-angulations of a (mn + 2)-gon. This generalizes the construction by Pallo on triangulations [START_REF] Pallo | Right-arm rotation distance between binary trees[END_REF], which is closely related to the Tamari lattice.

Quasisymmetric functions, a generalisation of symmetric functions, were introduced in [START_REF] Gessel | Multipartite P -partitions and inner products of skew Schur functions[END_REF] and are now classical in algebraic combinatorics. Some higher analogues were introduced in [START_REF] Poirier | Cycle type and descent set in wreath products[END_REF] and further studied in [START_REF] Baumann | A Solomon descent theory for the wreath products G S n[END_REF]. We recall some results about quotients of polynomials rings by higher quasisymmetric functions, first obtained for quasisymmetric functions in [START_REF] Aval | Catalan paths and quasi-symmetric functions[END_REF][START_REF] Aval | Ideals of quasi-symmetric functions and super-covariant polynomials for S n[END_REF] and extended to the higher case in [START_REF]Ideals and quotients of B-quasisymmetric polynomials[END_REF].

We then show that the poset P m,n is isomorphic to the divisibility poset of a new particular basis of the quotient of the polynomial ring in m sets of n variables by the ideal generated by G m -quasisymmetric functions without constant term. Our description of a new basis builds upon the basis indexed by m-Dyck paths that was introduced for general m in [START_REF]Ideals and quotients of B-quasisymmetric polynomials[END_REF].

For m = 1, the posets P 1,n were introduced by Pallo in [START_REF] Pallo | Right-arm rotation distance between binary trees[END_REF] and further studied in [START_REF] Csar | On a subposet of the Tamari lattice[END_REF][START_REF] Csar | On a subposet of the Tamari lattice[END_REF], and the basis indexed by triangulations was defined in [START_REF] Chapoton | Une base symétrique de l'algèbre des coinvariants quasisymétriques[END_REF], but the connection between them is new.

The last two sections of the article are devoted to some enumerative results on the posets (enumeration of intervals, rank generating function) and to a recursive description of the intervals as distributive lattices of orders ideals of forests.

A poset structure on M -angulations

Let m ≥ 1 be an integer. For the sake of readability, we will use the expression M -angulation instead of (m + 2)-angulation. An M -angulation of a regular convex polygon is a set of diagonals that cut the polygon into (m + 2)-sided polygonal regions.

We consider the set Q m,n of M -angulations of a (mn+2)-gon. Every element of Q m,n contains n regions, separated by n-1 diagonal edges. The cardinality of Q m,n is given by 1 mn+1 (m+1)n n

, the number of (m + 1)-ary planar rooted trees with n inner vertices (often called a Fuss-Catalan number). A simple bijection between these two classes of objects is given by planar duality. Some elements of Q 2,7 are shown in Figures 1 and2. To define a poset structure on Q m,n , we fix a particular element Q 0 , which is a fan (every diagonal edge involves a fixed vertex, denoted 0 and called the apex, see Figure 2). We consider the following order relation. An element Q of Q m,n is covered in P m,n by the M -angulations obtained by flipping one of its diagonal edges included in Q 0 . Here flipping means removing this diagonal edge and replacing it by another diagonal edge cutting again the (2m + 2)-gon created by the removal into two (m + 2)-gons. Note that any diagonal edge may be flipped in exactly m different ways. As an example the poset P 2,3 is shown on Figure 3. This poset is clearly graded, the rank of an element Q being n -1 minus the number of diagonal edges shared by Q 0 and Q. This is also the number of diagonal edges in Q that are not in Q 0 .

It follows from the description of the coverings that an element of rank r is covered by exactly m(n-1-r) elements. This implies the formula m n-1 (n-1)! for the number of maximal chains in P m,n .

This poset has Q 0 as unique minimum. The fact that Q 0 is smaller than all M -angulations follows from the next lemma by induction on the number of diagonals that are both in Q and Q 0 . This lemma implies that unless Q is Q 0 , there is a cover relation Q Q where Q has one more diagonal in common with Q 0 . Lemma 1. Let Q be distinct from Q 0 . Then there is always at least one diagonal d 0 in Q 0 that cuts exactly one diagonal of Q.

Proof. One can assume without restriction that Q and Q 0 have no common diagonal, otherwise one can find d 0 (by induction) inside one of the parts cut by the common diagonals. In at least one of these parts the restriction of

Q must differ from the restriction of Q 0 , because Q is not Q 0 .
Let us label all the vertices of the regular polygon counter-clockwise by integers, starting from the vertex 0 which is the apex of Q 0 .

Because Q and Q 0 have no common diagonal, there exists a unique region R 0 of Q that contains the vertex 0 in its boundary. Removing R 0 from the ambient polygon, one gets one or more convex polygons, all of them with km + 2 vertices for some k. Let us choose one of these polygons, and let Q be its M -angulation obtained from Q by restriction. Let R be the unique region of Q which is adjacent to R 0 .

Excluding minimal and maximal indices, vertices of Q form a sequence of mk vertices numbered consecutively, in which every residue class modulo m is represented exactly k times.

Because of the tree-like structure of M -angulations, one can build the Mangulation Q by successive additions of M -angles, starting from the region R . Going backwards, one can go from Q to R by removal of M -angles on the boundary.

Using this leaf-removal induction, one can then prove that the boundary of R (excluding minimal and maximal indices) contains exactly one representative of every residue class modulo m.

Note now that in Q 0 , the vertices linked by a diagonal to the apex 0 form a residue class modulo m, when numbered in the same way as the vertices of Q.

It follows that exactly one of the vertices of R (excluding minimal and maximal indices) is the end of a diagonal d 0 in Q 0 . The diagonal d 0 does only cross one diagonal of Q, namely the diagonal separating the regions R 0 and R .

The maximal elements in the poset P m,n are the M -angulations that have no common diagonal with Q 0 . We will call them final M -angulations.

Remark: There are several interesting existing families of posets on objects in bijection with (m + 2)-angulations, including m-Tamari lattices [START_REF] Bergeron | Higher trivariate diagonal harmonics via generalized Tamari posets[END_REF] and m-Cambrian lattices of type A [START_REF] Stump | Cataland: Why the Fuss?[END_REF]. The posets introduced here seem to be new.

m-analogues of B-quasisymmetric functions

In [START_REF] Baumann | A Solomon descent theory for the wreath products G S n[END_REF], Baumann and Hohlweg introduced the ring of B-quasisymmetric functions as the graded dual Hopf algebra of the analog in type B of Solomon's descent algebra. This ring is contained in the polynomial ring in two sets of n variables.

In [START_REF]Ideals and quotients of B-quasisymmetric polynomials[END_REF], a quotient ring of this polynomial ring by an ideal of B-quasisymmetric functions was studied. Moreover, for every integer m ≥ 1, an analog of the ring of B-quasisymmetric functions and an analog of the quotient ring were also defined and studied, involving m sets of n variables.

We refer to [START_REF]Ideals and quotients of B-quasisymmetric polynomials[END_REF] for the original motivations of the study of these rings and for the proof of the results that we will use. Let us now summarize the results of [START_REF]Ideals and quotients of B-quasisymmetric polynomials[END_REF] in their most general form.

Let us denote by M = {x, y, z, . . . , ω} a set of m distinct letters, endowed with a total order x < y < z < • • • < ω. We will mostly illustrate our constructions with the cases m = 2 and m = 3, therefore using only the letters x, y, z.

Let us start with polynomials in the union of m alphabets of each n variables:

X n = x 1 , . . . , x n , Y n = y 1 , . . . , y n , up to Ω n = ω 1 , . . . , ω n . Denote this polynomial ring by Q[X n , Y n , . . . , Ω n ].
Inside this polynomial ring, one can define a space of G m -quasisymmetric functions, which reduces when m = 1 to the classical quasisymmetric functions.

In [START_REF]Ideals and quotients of B-quasisymmetric polynomials[END_REF], a Gröbner basis for the ideal J m,n generated by constant-termfree G m -quasisymmetric functions was described, and from that was deduced a monomial basis for the quotient R m,n of the polynomial ring Q

[X n , Y n , . . . , Ω n ] by J m,n .
This monomial basis for the quotient R m,n is indexed by m-Dyck paths, which gives the dimension formula dim R m,n = 1 mn+1 (m+1)n n .

Definitions

For these definitions, we follow [START_REF] Baumann | A Solomon descent theory for the wreath products G S n[END_REF], with some minor differences, for the sake of simplicity of the computations we will have to make. The main change is to describe directly the general case for any m ≥ 1 and not the special case m = 2. An m-vector of size n is a vector v = (v 1 , v 2 , . . . , v mn-1 , v mn ) of length mn with entries in N. One must think of m-vectors as the concatenation of n sequences of length m. An m-composition is an m-vector in which there is no sequence of m consecutive zeros.

The integer n is called the size of v. The weight of v is by definition the m-tuple (w 1 , . . . , w m ) where w j = n-1 i=0 v mi+j . We also set |v| = mn i=1 v i . For example (1, 0, 2, 1, 0, 2, 3, 0) is a 2-composition of size 4, and of weight (6, 3).

To make notations lighter, we shall sometimes write m-vectors or m-compositions as words with bars, where the bars separates the word into n blocks of length m. For example, 10|21|02|30 stands for the 2-vector (1, 0, 2, 1, 0, 2, 3, 0) (see also the following definition).

Let us now define the fundamental G m -quasisymmetric polynomials, indexed by m-compositions.

Let c = (c 1 , . . . , c mn ) be an m-composition. One can decompose c as a concatenation of n blocks of m integers. Let us first associate to c a word w c in the alphabet {x, y, z, . . . , ω}, defined as the concatenation, over all blocks b = (b 1 , . . . , b m ) of c, of the word x b1 y b2 . . . ω bm . For example, for the 3-composition 010|201, one obtains w c = yxxz (powers are written as repeated letters).

Then the fundamental G m -quasisymmetric polynomial of index c is

F c = i t∈wc t i(t) ,
where the sum is taken over all maps i from the sequence of letters of the word w c to the set {1, . . . , n} such that i is weakly increasing inside every block of c and strictly increasing between two blocks. Let us give some examples for m = 2:

F 12 = i≤j≤k x i y j y k , (1)
F 02|10 = i≤j<k y i y j x k . (2)
It is clear from the definition that the multidegree (i.e. the m-tuple (degree in x, degree in y, ..., degree in ω)

) of F c in Q[X n , Y n , . . . , Ω n ] is the weight of c. If the size of c is greater than n, we set F c = 0.
The space of G m -quasisymmetric polynomials, denoted by QSym n (G m ) is the vector subspace of the ring Q[X n , Y n , . . . , Ω n ] generated by the F c , for all m-compositions c.

Let us denote by J m,n the ideal QSym n (G m ) + generated by G m -quasisymmetric polynomials with zero constant term.

A monomial basis for R m,n

Let v = (v 1 , v 2 , . . . , v mn-1 , v mn ) be an m-vector of size n. We associate to v a path π(v) in the plane N × N, with steps (0, 1) (up step) or (m, 0) (right step). We start from (0, 0) and for each entry v i (read from left to right) add v i right steps (m, 0) followed by one up step (0, 1). This clearly defines a bijection between m-vectors of size n and such paths of height mn.

As an example, the path associated to the 2-vector (1, 0, 1, 2, 0, 0, 1, 1) is

If the path π(v) associated with an m-vector v always remains above the diagonal x = y, we call this path an m-Dyck path, and say that the corresponding m-vector v is an m-Dyck vector.

Being an m-Dyck vector is equivalent to the condition that, for any 1 ≤ ≤ mn, one has

m(v 1 + v 2 + • • • + v ) < .
For v an m-vector (of length mn), we denote by A v the monomial

A v = (x v1 1 y v2 1 • • • ω vm 1 )(x vm+1 2 y vm+2 2 • • • ω v2m 2 ) • • • (x v m(n-1)+1 n • • • ω vmn n ).
This clearly defines a bijection between m-vectors and all monomials in the polynomial ring

Q[X n , Y n , . . . , Ω n ].
For example, the monomial associated to the 2-vector (1, 0, 1, 2, 0, 0, 1, 1) is x 1 x 2 y 2 2 x 4 y 4 . The following result was proved in [Ava07, Th. 5.1].

Proposition 2. The set B m,n of monomials A v for v varying over m-Dyck vectors of size n is a basis for the space R m,n = Q[X n , Y n , . . . , Ω n ]/J m,n .

A bijection between M -angulations and m-Dyck paths

We assign to each vertex of the mn + 2-gon (except the vertex 0) a letter in M as follows. The vertices are labelled by repeating the sequence x, y, z, . . . , ω in counter-clockwise order around the polygon, in such a way that the final vertex just before the vertex 0 receives the last letter ω of M. It follows that the first vertex just after the vertex 0 also receives the letter ω.

See Figure 4 for an illustration of this labelling when m = 2 with the ordered set of letters x < y. Let us also label the inner diagonals of Q 0 from 1 to n-1 in counter-clockwise order.

Then we consider an M -angulation Q. To any diagonal d of Q, we associate a polynomial m d . If d coincides with a diagonal of Q 0 , we set m d = 1. Otherwise, and reading counter-clockwise, d starts from a vertex labelled by the letter u, intersects consecutive diagonals of Q 0 labelled from i to j and ends at a vertex labelled by the letter v. Then we set m d = v j+1u i . We then associate to Q the polynomial P Q defined as the product of m d over its diagonals.

As an example, Figure 5 shows the polynomials associated to the quadrangulations of Figure 3, in the corresponding positions.

1 (x 3 -y 2 ) (y 3 -x 2 ) (x 2 -y 1 ) (y 2 -x 1 ) (x 3 -y 2 ) (y 3 -x 1 ) (x 3 -y 2 ) (x 3 -y 1 ) (y 3 -x 2 ) (y 3 -x 1 ) (y 3 -x 2 ) (x 2 -y 1 ) (x 2 -y 1 ) (x 3 -y 1 ) (y 2 -x 1 ) (x 3 -y 1 ) (y 2 -x 1 ) (y 3 -x 1 )
Figure 5: The polynomials associated to quadrangulations of Figure 3 To deal with leading terms of polynomials, we will use the lexicographic order induced by the ordering of the variables:

x 1 < y 1 < • • • < ω 1 < x 2 < y 2 < • • • < x n < y n < • • • < ω n .
(3)

The lexicographic order is defined on monomials as follows: A v < lex A w if and only if the last non-zero entry of vw (componentwise) is negative.

Note that the leading monomial of the polynomial P Q attached to an Mangulation is easily described: in every binomial factor written v j+1u i with j + 1 > i, keep only the monomial v j+1 . Proposition 3. The set of leading monomials of the polynomials P Q when Q varies over the set of M -angulations coincides with the monomial basis B m,n .

Proof. We need a bijection Φ between M -angulations Q and m-Dyck paths (or rather m-Dyck vectors v) such that the leading monomial of P Q is equal to

A Φ(Q) .
The idea to define Φ is to compose the leading-monomial application Q → LM (P Q ) with the bijection between monomials and m-Dyck paths described in section 3.2.

Let us instead define the reverse bijection Ψ.

Let v be an m-Dyck vector of size n. We start from the empty set of diagonals on the mn + 2-gon. We shall add iteratively diagonals. Let D denote the current set of diagonals, under construction. We read the m-vector v from left to right. To any non-zero entry c associated to variable z k we add to D a fan with c diagonals as follows. Let t be the letter before z in the cyclic order x < y < • • • < ω < x. The (common) ending point of the added fan is the first vertex labelled z that comes counter-clockwise after the diagonal k -1 of Q 0 , and the starting points are the c last vertices labelled t that are available before the ending point. Here being available means not being separated from the ending point by a diagonal already in D.

See Figure 6 for an example of this construction. The quadrangulation Q associated to the monomial x 5 y 3 5 y 7 with P Q = (x 5y 4 )(y 5x 3 )(y 5x 2 )(y 5x 1 )(y 7x 6 ). Now the key point is that, when adding such a fan, the starting points are always strictly between the apex 0 and the ending point in counter-clockwise order. This is because of the m-Dyck word property of v: for any 1 ≤ ≤ mn, one has

m(v 1 + v 2 + • • • + v ) < .
Indeed, when numbering the vertices of the polygon counter-clockwise from the apex 0 to mn + 1, the index of the first vertex t in counter-clockwise order for the fan added at step is exactly

(1 + ) -m(v 1 + v 2 + • • • + v ) -1.
The leading term of the product of binomials m d attached to the diagonals in the fan just added is therefore the monomial z c k , by definition of the lexicographic order.

At the end of the process, there remains only to add some diagonals of Q 0 to obtain an M -angulation. There is a unique way to do that. This does not change the product of m d over all diagonals.

One has therefore defined a map Ψ from m-Dyck words to M -angulations. This has the property that the leading monomial of P Ψ(v) is equal to A v . Because one can recover the m-Dyck word v from A v , this map is injective.

This proves that Ψ is an injection between sets of equal cardinalities, hence a bijection.

The reverse bijection Φ can be described as first removing from Q all diagonals in Q 0 , then proceeding by successive removals of fans. Proposition 4. The set of polynomials P Q (when Q varies over the set of Mangulations) endowed with the poset structure given by divisibility is isomorphic to P m,n .

Proof. The map Q → P Q is a bijection, with inverse obtained from the factorisation of the polynomial. The flip in Q of a diagonal of Q 0 corresponds to the multiplication of P Q by a linear binomial.

Remark: the fan Q 0 has an obvious involutive symmetry that fixes the apex 0 and flips the ambient polygon. Given the labelling of the polygon by M = {x, y, . . . , ω}, this involution acts on variables by simultaneous substitution of letters ω, x, y, z, . . . ←→ ω, . . . , z, y, x.

and renumbering of indices i ↔ n+1-i. By construction, the set of polynomials P Q is sent to itself (up to signs) by this involution acting on variables.

5 Enumerative aspects

Recursive description

Let T m = n≥1 #Q m,n x n be the generating series for M -angulations according to their size n. Note that the power of x is chosen to correspond to the number of regions.

There is a classical recursive decomposition of M -angulations, which describes them as an M -angle with an M -angulation (or nothing) grafted on all but one sides. This implies that

T m = x(1 + T m ) m+1 . (4)
Recall that the final M -angulations are those which do not contain any diagonal of Q 0 . They are the maximal elements of the posets P m,n .

Let F m be the generating series for final M -angulations according to the number of regions. These objects can be decomposed as one M -angle (the unique region having 0 in its boundary), on which one can graft any M -angulation on all sides that do not contain the vertex 0. One obtains that

F m = x(1 + T m ) m = T m /(1 + T m ). (5)
Let us now describe another simple decomposition of M -angulations, corresponding to the rightmost expression in the equation (5).

To every M -angulation Q, one can associate a list L(Q) of final M -angulations, obtained by cutting Q along its initial diagonals. Let us assume that these pieces are listed in counter-clockwise order.

Proposition 5. Sending Q to L(Q) defines a bijection from M -angulations to lists of final M -angulations.

The inverse bijection is very simple: given a list of final M -angulations, one can glue them back along their sides into one single M -angulation.

This inverse bijection from L(Q) to Q can also be interpreted in the following way, that will be useful later. Given a list of k final M -angulations, one considers the M -angulation Q 0 with k regions. One then replaces the regions of Q 0 by the final M -angulations, in the counter-clockwise order. In the resulting Mangulation Q, the initial Q 0 can be identified with the union of all regions that are adjacent to the vertex 0.

Rank generating function

Let G m be the generating function for the elements of all posets P m,n , according to their size n and their rank, namely

G m = n≥1 x n Q∈Pm,n z rk Q . ( 6
)
The generating function G m satisfies

G m = F m (zx)/z 1 -F m (zx)/z (7)
where F m is the generating series for the final elements. Indeed, any Mangulation can be written uniquely as a list of final M -angulations by cutting along the diagonal edges shared with Q 0 . The rank parameter is multiplicative along this decomposition. And the rank of final M -angulations is just their size minus 1, so that the generating series of final M -angulations according to size and rank is just F m (zx)/z. Using Lagrange inversion followed by a simple summation of binomial coefficients, one can deduce from (4), ( 5) and (7) that the rank generating function of P m,n is given by

n-1 k=0 n -k n mn + k -1 k z k . (8)
For m = 1, this enumeration according to rank was already done in [CSS14, §3].

Decomposition of intervals

Let us now study the intervals in the posets P m,n .

An interval A is a pair of elements (A -, A + ) in P m,n that satisfies A -≤ A + . In every finite poset, the number of intervals is also the dimension of the incidence algebra.

Let us call an interval A initial if its lower bound A -is Q 0 . The set of initial intervals can be identified with the set Q m,n of M -angulations in P m,n . Indeed, Q 0 is smaller than all M -angulations by Lemma 1.

Let I m,n be the set of intervals in P m,n . Let I m = n≥1 #I m,n x n be the generating series for intervals.

One can get a recursive decomposition for intervals, similar to the previous decomposition for M -angulations. For this, one needs the following construction.

Suppose that one has an M -angulation B 0 with k regions, and a list of k final M -angulations B 1 , . . . , B k . From this data, one can build an M -angulation G(B 0 ; B 1 , . . . , B k ) as follows. First build the M -angulation L -1 (B 1 , . . . , B k ). Removing its k -1 initial diagonals creates a region with mk + 2 sides. Place B 0 inside this region to define the M -angulation G(B 0 ; B 1 , . . . , B k ). See Figure 8 for an example.

If B 0 is Q 0 , the construction G is just the inverse of the L map. Given an interval A, one can apply the map L to its bottom element A -. This gives a list of final M -angulations A - 1 , . . . , A - k , such that

A -= G(Q 0 ; A - 1 , . . . , A - k ). (9)
Proof. Let A = [Q 0 , A + ] be an initial interval. Consider the set of diagonals of Q 0 that belong to A + . Then one can cut both Q 0 and A + along these diagonals. Every piece A - i of Q 0 is a smaller Q 0 . Every piece A + i of A + is a final quadrangulation. By definition of the partial order, the same diagonals belong to every element of A. Cutting along them gives an isomorphism with the product of the intervals [A - i , A + i ], which are initial-final intervals. Let us now proceed to a more subtle decomposition. Let Q be a final triangulation. Let R 0 be the unique region in Q with 0 in its boundary. Removing R 0 from Q leaves a certain number k of polygons, that will be called the blocks. Let us call k the width of Q. The width can be 0 only if Q is reduced to R 0 . Otherwise it lies between 1 and m.

Proposition 11. Every initial-final interval A is isomorphic to a product of initial-final intervals of width 1.

Proof. The main idea is that the flips downwards from Q happen completely independently in distinct blocks. Let now us give a detailed argument for this independence.

Let us consider the set D of diagonals that go from vertex 0 to one of the vertices of the region R 0 , except the two vertices that are neighbors of 0. These diagonals are inside the region R 0 and therefore do not belong to Q. The extremities of these diagonals receive exactly once every label from 0 to m -2 in clockwise order, and therefore never get the label m -1. Note that the extremities of the initial diagonal are labeled by m -1.

Let us show by induction downward from Q that for every Q ∈ [Q 0 , Q], the diagonals in D do not belong to Q and do not cross any diagonal of Q . This is clear for Q. The effect of a down flip is to replace a diagonal by an initial diagonal. This initial diagonal is not in D and does not cross any diagonal in Q, because of the labeling of its extremity is m -1.

One can therefore cut along the diagonals in D. Replacing every block but a fixed one by a trivial block gives a map to an initial-final interval of width 1. Taking the product of all these maps gives the desired isomorphism.

Proposition 12. The number of elements covered by the maximum in a initialfinal interval [Q 0 , Q] is the width of Q.

Proof. In the proof of Lemma 1, it was shown that in every polygon of Q minus R 0 , there is exactly one initial diagonal d that cross just one diagonal d of Q. The set of diagonals of Q that can be flipped down in the poset is exactly the collection of these d , and their number is therefore the width of Q.

In particular, initial-final intervals of width 1 have a maximum that covers a unique element.

Corollary 13. The intervals in P m,n can be build iteratively by either adding a maximum to a smaller one or by taking a product of several smaller ones.

Proof. By induction on both the height (difference of ranks) of the intervals and the size of the ambient polygon. Every interval is a product of initial-final ones. Every initial-final interval is a product of initial-final intervals of width 1.

In both cases, if the product is not reduced to one element, the factors live in strictly smaller polygons.

Every initial-final interval of width 1 is obtained by adding a maximum to an interval, which has smaller height.

Let us define a forest poset as a poset where every element is covered by at most one element. These posets can be build iteratively from the empty poset by two operations, namely adding a maximum element and taking the disjoint union. Their Hasse diagrams are forests of rooted trees, where the roots are the maximal elements.

Proposition 14. Every interval in P m,n is a distributive lattice, isomorphic to the lattice of order ideals of a forest poset.

Proof. By the results above, the intervals in P m,n can be obtained from smaller intervals using two operations, namely adding a maximum and taking a product. These two operations correspond to adding a maximum or taking the disjoint union, on the poset of join-irreducible elements. The statement follows by induction.

For m = 1, this property of intervals was already obtained in [CSS14, §2].

Corollary 15. Möbius numbers of intervals in posets P m,n belong to {-1, 0, 1}.

Proof. This property is preserved by adding a maximum or taking a product.

Figure 1 :

 1 Figure 1: A quadrangulation of the 16-gon

Figure 2 :

 2 Figure 2: The fan Q 0 for m = 2 and n = 7

Figure 3 :

 3 Figure 3: The poset P 2,3

Figure 4 :

 4 Figure 4: The labelling of vertices by x and y letters (m = 2)

  Figure6: The quadrangulation Q associated to the monomial x 5 y 3 5 y 7 with P Q = (x 5y 4 )(y 5x 3 )(y 5x 2 )(y 5x 1 )(y 7x 6 ).

Figure 7 :

 7 Figure 7: Illustration of construction L

Figure 8 :

 8 Figure 8: B 0 , B 1 , B 2 and G(B 0 ; B 1 , B 2 )

Proposition 6. There exists a unique M -angulation A + 0 such that

Proof. Uniqueness is clear by the definition of the construction G.

Existence is proved by induction on the difference of initial diagonals in A - and A + . If A + = A -, then the only possible choice is

Because the cover relations flips an initial diagonal d and does not change the other ones, the diagonal d must in fact belong to Q 0 . Therefore one can flip it in

. Keeping the same notations, one also has the following result.

Proposition 7. Every element of the interval [A

Proof. This result follows from the proof of Prop. 6. In fact, more is true: all elements greater than A -can be uniquely written G(Q; A - 1 , . . . , A - k ) for some Q and this bijection identifies the upper ideal of A -with a smaller poset of type P.

Proposition 6 implies the following decomposition.

Proposition 8. The map sending an interval A to the pair (A + 0 , L(A -)) defines a bijection between intervals and pairs (B 0 , (B 1 , . . . , B k )) where B 0 is an Mangulation with k regions and B 1 , . . . , B k are final M -angulations.

The inverse bijection is given by

Corollary 9. The generating series of intervals can be expressed using those of M -angulations and final M -angulations as

The second equality follows from (5).

Isomorphism types of intervals

The aim of this section is to give a description of the intervals as posets, and to prove in particular that they are distributive lattices with Möbius numbers in {-1, 0, 1}. By Proposition 7, every interval is isomorphic to an initial interval. It is therefore enough to study initial intervals.

Let us call an interval initial-final if its minimum is Q 0 and its maximum is a final quadrangulation.

Proposition 10. Every initial interval is isomorphic to a product of initial-final intervals.


Cyril Micheau

Elisabeth Rosenberg

Loïc Barré

Nicolas Pannacci
email: nicolas.pannacci@ifpen.fr

Microfluidic comparative study of foam flow between a classical and a pH sensitive surfactant

Keywords: foam morphology, lamellae, foam stability, microfluidic, pH sensitive surfactant, AKYPO

Comparative study between foams made of a pH sensitive surfactant, nonaoxyethylene oleylether carboxylic acid (AKYPO® RO 90 VG) and a classical foaming surfactant, sodium dodecyl sulphate (SDS) was realised using simple microfluidic devices in the scope of Enhanced Oil Recovery (EOR). The influence of injecting fluid parameters such as gas and foaming liquid pressure was investigated and discussed in terms of foam morphology, and lamellae stability and curvature radii. The increasing stability of AKYPO® foam film lamellae by increasing pH is classically explained by the electrostatic repulsive interaction of the double layer in foam film. In other hand, the addition of dodecanol (DOH) on SDS solution seems to have no effect on foam regime and lamellae stability except for high DOH concentration where the foam film shape becomes double bump. The behaviour difference between AKYPO® and SDS lays on the lamellae shape and stability domain. While SDS lamellae stay unchanged with increasing pressure ratio between the gas and the liquid, AKYPO® lamellae curvature radius increases which could be related to surfactants solubilities and structure. These results pointed out the relevance of solution physicochemical parameters on foam stability and flowing and help the understanding of flowing foam in EOR context.

Introduction

Liquid foams are non-equilibrium materials [START_REF] Cantat | Les Mousses: Structures et dynamique[END_REF] composed of a discontinuous gas phase and a liquid phase. Gas is present in the foam in the form of bubbles, spherical in the case of wet foams and polyhedral in the case of dry foams. Liquid foams are used in many process such as ion separation via flotation, [START_REF] Micheau | Evaluation of ion separation coefficients by foam flotation using a carboxylate surfactant, A collection of papers presented at the 5th European Conference on Foams, Emulsions, and Applications[END_REF] nuclear decontamination [START_REF] Dame | Relations between physicochemical properties and instability of decontamination foams, A collection of papers presented at the 5th European Conference on Foams, Emulsions, and Applications[END_REF] or explosion confinement. [START_REF] Prete | Modeling of aqueous foam blast wave attenuation[END_REF] A recent application [START_REF] Kristiansen | Properties of Flowing Foam in Porous Media Containing Oil[END_REF] proposes using foam in Enhanced Oil Recovery (EOR) process in order to extract oil in natural reservoirs. This specific application requires the generation of foams in porous media with the aim of reaching and displacing the oil in place. In porous media, foam's definition is more ambiguous because it is presented as a succession of parallel films, called lamellae, joined together by wetting films on the pore walls. [START_REF] Huerre | Bubbles and foams in microfluidics[END_REF] Furthermore, when foams are sufficiently confined, 2D-foams are obtained and their bubbles can distort themselves to adopt a spheroid or cylindrical shape. [START_REF] Cantat | Les Mousses: Structures et dynamique[END_REF][START_REF] Huerre | Bubbles and foams in microfluidics[END_REF] Foam's lamellae are stabilized by surfactant molecules which adsorb at the liquid/gas interface that could change the local surface properties such as surface viscosity, surface tension or elastic moduli. [START_REF] Cantat | Les Mousses: Structures et dynamique[END_REF][START_REF] Dollet | Influence of the Elastic Deformation of a Foam on its Mobility in Channels of Linearly Varying Width[END_REF] These properties directly influence the capacity of the foam to transport gas into porous media which is mainly linked to the foam stability during its generation and flowing.

The higher effective viscosity of foam relatively to gas [START_REF] Schramm | Microvisual and Coreflood Studies of Foam Interactions With a Light Crude Oil[END_REF][START_REF] Grassia | Analysis of a model for foam improved oil recovery[END_REF] allows to increase the sweep efficiency in porous media and to reduce viscous fingering and gravity segregation due to the low viscosity and the low density of the gas that makes foam a good gas mobility reducing agent. [START_REF] Schramm | Microvisual and Coreflood Studies of Foam Interactions With a Light Crude Oil[END_REF][START_REF] Simjoo | Novel Insight Into Foam Mobility Control[END_REF][START_REF] Quennouz | Microfluidic Study of Foams Flow for Enhanced Oil Recovery (EOR)[END_REF] Moreover, the capacity of foam to accumulate in high-permeability regions reduces gas by-passing by increasing flowing resistance and diverting gas in low permeability pores that improves the EOR process. [START_REF] Kovscek | Foam Mobility in Heterogeneous Porous Media (I: Scaling Concepts)[END_REF][START_REF] Ettinger | Influence of Texture on Steady Foam Flow in Berea Sandstone[END_REF] The fundamental understanding of such mechanism is of high importance to better adjust surfactant formulation and extraction process.

Previous works have investigated the physical properties of foam in porous media using lab experiments in sand [START_REF] Osterloh | Effects of Gas and Liquid Velocity on Steady-State Foam Flow at High Temperature[END_REF], rocks [START_REF] Simjoo | Novel Insight Into Foam Mobility Control[END_REF] or micro-device [START_REF] Dollet | Influence of the Elastic Deformation of a Foam on its Mobility in Channels of Linearly Varying Width[END_REF][START_REF] Quennouz | Microfluidic Study of Foams Flow for Enhanced Oil Recovery (EOR)[END_REF][START_REF] Jones | Structure-Dependant Mobility of a Dry Aqueous Foam Flowing Along Two Parallel Channels[END_REF]. Some were interested in the influence of fluid injected parameters on the foam morphology [START_REF] Quennouz | Microfluidic Study of Foams Flow for Enhanced Oil Recovery (EOR)[END_REF] or performance in term of mobility reduction factor [START_REF] Osterloh | Effects of Gas and Liquid Velocity on Steady-State Foam Flow at High Temperature[END_REF], and others on the flowing foam behaviour depending on the porous media geometry. [START_REF] Dollet | Influence of the Elastic Deformation of a Foam on its Mobility in Channels of Linearly Varying Width[END_REF][START_REF] Quennouz | Microfluidic Study of Foams Flow for Enhanced Oil Recovery (EOR)[END_REF][START_REF] Jones | Structure-Dependant Mobility of a Dry Aqueous Foam Flowing Along Two Parallel Channels[END_REF]. Less study has been focussed on the influence of physicochemicals parameters on foam generation, aging and flowing. It is well known that, surfactant composition and dynamics, salt nature and ionic strength or pH [16,[START_REF] Sedev | DLVO and non-DLVO surface forces in foam films from amphiphilic block copolymers[END_REF] are of crucial importance on the stability of bulk foams (3D-foams) or films. Moreover, for experiments of foam in porous media, chemical composition of the foaming solution is sometimes modified by salt or dye addition, in order to enhance phase contrast [START_REF] Quennouz | Microfluidic Study of Foams Flow for Enhanced Oil Recovery (EOR)[END_REF][START_REF] Andrianov | Immiscible Foam for Enhancing Oil Recovery: Bulk and Porous Media Experiments[END_REF], or by viscosifying or gelling agent to reduce the foam dynamics (cf. aging).

Microfluidic devices are good supports to investigate the influence of physicochemical parameters on foam generation and flowing at the micro-scale. They allowed direct observation of foam with a microscope without using any chemical additive, and a high control level of reproducibility. Coupling with image analysis, and basic image processing, thresholding, and skeletisation of the foam image gives qualitative and quantitative informations on foam generation and flowing.

This work proposes a comparative study between a classical surfactant used as a model surfactant in EOR context, sodium dodecyl sulphate (SDS), [START_REF] Dollet | Influence of the Elastic Deformation of a Foam on its Mobility in Channels of Linearly Varying Width[END_REF][START_REF] Schramm | Microvisual and Coreflood Studies of Foam Interactions With a Light Crude Oil[END_REF][START_REF] Quennouz | Microfluidic Study of Foams Flow for Enhanced Oil Recovery (EOR)[END_REF][START_REF] Jones | Structure-Dependant Mobility of a Dry Aqueous Foam Flowing Along Two Parallel Channels[END_REF][START_REF] Andrianov | Immiscible Foam for Enhancing Oil Recovery: Bulk and Porous Media Experiments[END_REF][START_REF] Marchalot | 2D Foam Coarsening in a Microfluidic System[END_REF] and a pHsensitive surfactant used in metal separation processes, nonaoxyethylene oleylether carboxylic acid (AKYPO® RO 90 VG), [16,[START_REF] Schwarze | Micellar enhanced ultrafiltration (MEUF) of metal cations with oleylethoxycarboxylate[END_REF][START_REF] Schwarze | Oleylethoxycarboxylate: An efficient surfactant for copper extraction and surfactant recycling via micellar enhanced ultrafiltration[END_REF] using microfluidics. Foam morphology and stability are analysed in regard to the injected fluid pressures (N 2 and surfactant solution) and physicochemical parameters such as pH or dodecanol/SDS concentration ratio. Foam phase diagrams are thus obtained and quantified in terms of bubble size, liquid fraction and topology. Foam flow and stability differences between the two selected surfactants are evidenced regarding lamellae stability domain and curvature radius and discussed putting forward interface properties such as surfactant adsorption and surface viscosity.

Materials and method

Microfluidic device

Standard soft lithography in PDMS (PolyDiMethylSiloxane) was used for the microfabrication, whose principle was previously described by Duffy et al. [START_REF] Duffy | Rapid Prototyping of Microfluidic Systems in Poly(dimethylsiloxane)[END_REF] Quasi 2D device with a constant depth (40 µm) was thereby obtained. Bonding of PDMS on glass with plasma cleaner [START_REF] Ginn | Polymer Surface Modification Using Microwave-Oven-Generated Plasma[END_REF] gives hydrophilic channel surface. Due to the reversibility of this treatment i.e. channel internal surface becomes hydrophobic after few hours, microfluidic device was renewed for each experiment. Channel depth is 40 µm.

Experimental protocol

Entrance pressure above atmospheric pressure was applied for surfactant solution and gas using MCFS TM EZ 7000 mbar (Fluigent). PEEK (polyetheretherketone) flexible tubes of 59.2 cm length and 125 µm inner diameter were used to connect source pressure to gas and liquid inlets. Pressures considered below, named respectively P gas for gas and P water for surfactant solution are upstream applied pressures, above atmospheric pressures. All measurements were carried out when steady flow rate was reached (typically after 30 s). In each experiment, gas pressure drop was kept constant during the flowing (150 mbar or 300 mbar) and the surfactant solution pressure drop varies from to .

Each experiment was performed at least twice. Under such applied pressure and with such aspect ratio of the microfluidic device (1200 µm width and 40 µm depth ), no deformation of the PDMS cell is observed (no defocusing and no loss of sharpness on microchannel images). This is confirmed by the calculation of the deformation criterion , defined by Gervais et al. [START_REF] Gervais | Flow-induced deformation of shallow microfluidic channels[END_REF], where is the pressure difference applied on the upper PDMS wall in the observation chamber (the lower glass wall is supposed not to distort) and is the PDMS Young's modulus ( ≈ 10 6 Pa). In the case of the maximum water pressure difference applied 1200 mbar and considering the hydraulic resistance of the microdevice, the pressure difference applied on the PDMS wall is found to be equal to 3 mbar, leading to a deformation criterion of 8.97 10 -3 smaller than one (see calculation in Supplementary material). Accordingly, no deformation is expected at such aspect ratio and such applied pressure.

Chemicals

Foams were generated by co-injection of nitrogen gas and surfactant solution at around 1.

For SDS foam flow experiments (17.33 mM corresponding to 0.5 % w/w), 1-dodecanol (DOH, Alfa Aeser, 98 %, M W = 186.34 g.mol -1 ) was added to the initial solution to reach different molar ratio SDS/DOH. The solubility of DOH in water is about 1.5 10 -2 mM, [START_REF] Jones | Structure-Dependant Mobility of a Dry Aqueous Foam Flowing Along Two Parallel Channels[END_REF][START_REF] Fang | The dynamic surface tension of SDS-dodecanol mixtures 2. Micellar SDSdodecanol mixtures[END_REF] which means all the SDS/DOH mixtures solution prepared were above the alcohol solubility.

Surface tension of pure SDS solution and SDS/DOH mixture was obtained from the literature and presented in Table 1.

Image analysis and error estimations

A fast camera (800 fps) with optical microscopy were used for observations. Image analysis with ImageJ was used for the determination of bubble area (see Fig. 5,a), lamella's radius of curvature (radius of circle tangent to the lamella, see Fig. 8,b), bubble Feret's diameter (bubble diameter along flowing direction, see Fig. 5,b), equivalent diameter (diameter of a disc with the same area as the bubble, , see Fig. 5,c), areal bubble number density (number of bubbles per surface unit in bubbles/mm 2 ), topology number (number of bubble rows across the channel width), and areal liquid fraction (ratio between the number of black pixels and the total pixel number inside the microchannel). Each analysis were realised from 50 to more than 1000 bubbles depending on the foam flow regime considered. The image processing was carried out using ImageJ software package. Brightness and contrast of each image were first adjusted and then all the images were automatically thresholded in order to clearly separate the black network of edges or Plateau border and film from white bubbles. A classical skeletonization method was then used when needed to separate bubbles from each other. This image processing is robust and thanks tohigh contrastedpictures no variation of the quantities measured is observed when varying manually the threshold.

Standard errors in measured foam parameters were estimated using the standard deviation method based on the number of analysed bubbles and standard deviation : .

In the bubbly liquid regime, where bubbles are spheroids and well separated from one another, there is a discrepancy between the measured areal liquid fraction and the real volume liquid fraction which vanishes in the very dry foam regime where bubbles are squashed together. This difference between the measured areal liquid fraction and the real volume liquid fraction can be estimated by calculating real bubble volume and equivalent bubble volume and contributes to a systematic error on bubble volume in addition to the standard error . Considering a bubble squeezed between two planes (see Fig. 2), [START_REF] Marchalot | 2D Foam Coarsening in a Microfluidic System[END_REF] it can be divided into two volumes and that represent a cylinder of radius and a piece of torus of minor radius with the height of the microchannel equal to 40 µm. The volume of the cylinder can be calculated as follows:

(1)

The volume of the piece of torus can be calculated using the Pappus-Guldinus theorem [START_REF] Goodman | Generalizations of the Theorems of Pappus[END_REF] which states that the volume of a solid of revolution generated by rotating a plane figure (half disc of radius ) about an external axis z is equal to the product of the area of the plane figure 2 and the distance travelled by its geometric centroid . The volume is then calculated as follows:

(

The real bubble volume of radius is the sum of and and is equal to:

It can then be calculated by replacing by . is determined from the apparent measured radius . When a bubble is illuminated in transmission by a diffuse extended light source, its image is a dark ring with a radius corresponding to the apparent bubble radius that can be calculated as follows: [START_REF] Van Der Net | Simulating and interpretating images of foams with computational ray-tracing techniques, A collection of papers presented at the 5th European Conference on Foams, Emulsions, and Applications[END_REF] (

where is the critical angle of total light reflexion and is equal for water/air interface to 48.8° (see Fig. 2,c). In the case of a directed light source, by contrast, as is usually done for optical microscopy, the apparent radius can be taken to be equal to the real bubble radius .

The equivalent bubble volume used for the calculation of the areal liquid fraction corresponds to the volume of a cylinder with a radius and a height that is obtained from the projection of the dark ring on the whole height of the microchannel and is calculated as follows:

The two volume and can then be calculated from experimental apparent radius and microchannel and used to estimate the systematic volume error using the following expression:

The highest volume error on bubble is obtained in the wet limit regime where bubbles are the smallest ( = 39 µm) and is equal to 24.2 % per bubble. This error decreases with increasing bubble radius to 2.19 % for the biggest bubble ( = 397 µm) at = 1.15. For a given foam of volume and composed of bubbles, the gas fraction is defined as the ratio between the total gas volume (volume occupied by the bubbles) divided by the foam volume . [START_REF] Cantat | Les Mousses: Structures et dynamique[END_REF] The systematic relative error on gas liquid fraction between the measured areal gas fraction and the real volume gas fraction can then be estimated as follows:

The relative error on the gas fraction is thus equal to the systematic bubble volume error . The liquid fraction of a foam is finally defined as: [START_REF] Grassia | Analysis of a model for foam improved oil recovery[END_REF] This leads to an absolute error on the measured areal liquid fraction between around 7 % for the wettest foam ( = 82 %) and 2 % for the driest one ( = 5.23 %).Therefore the measured areal liquid fraction can be considered as a good approximation to estimate the real volume liquid fraction of the foam inside the microchannel.

The error on pressure ratio was estimated using the law of propagation of uncertainty considering an error of ± 10 mbar on water pressure . This explains the increasing error on pressure ratio with decreasing water pressure.

Results and discussion

AKYPO® foam flow.

Foam flow experiments with AKYPO® solution at different pressure ratio and different pH with gas pressure of 150 mbar were first performed. As detailed in Quennouz et al. [START_REF] Quennouz | Microfluidic Study of Foams Flow for Enhanced Oil Recovery (EOR)[END_REF] diphasic flow regime is always obtained due to applied gas pressure exceeding Laplace pressure DP L at the cross junction (DP L = 31 mbar for non-wetting conditions, 60 µm in width and 40 µm in depth channel and for 37.5 mN/m in case of pH 10.1). These experiments allowed to obtain a phase diagram presented in Fig. 3 that shows the morphology of foam for each physicochemical conditions. This phase diagram clearly presents 5 different foam morphology regime already described in the literature: [START_REF] Quennouz | Microfluidic Study of Foams Flow for Enhanced Oil Recovery (EOR)[END_REF][START_REF] Cubaud | Transport of Bubbles in Square Microchannels[END_REF] -Bubbly liquid ( < 0.6). The bubbles flow freely with no interactions between them. A sub-regime can sometimes be observed within this regime, called dripping, in which the bubbles are formed one by one.

-Wet foam (0.6 < < 0.9). Bubbles begin to interact with one another and an overall movement is observed. This regime can be recognized from the previous one by the liquid fraction of the dispersion. The inter-bubble film is thick enough to limit gas diffusion and hence ripening. [START_REF] Marchalot | 2D Foam Coarsening in a Microfluidic System[END_REF] -Polyhedral foam (0.9 < < 1.1). The foam becomes very dry and the bubbles interact strongly with each other. In this regime, staircase (ex: pH = 10. For each experiment, quantitative physicochemical data were obtained by image analysis with ImageJ. The images are processed by a conventional thresholding and segmentation protocol to separate the different bubbles. Using these processed images, it is thus possible to examine several quantitative magnitudes: bubble area (see Fig. 5,a), Feret's (bubble diameter along flowing direction, see Fig. 5,b) and equivalent diameters (diameter of a disc with the same area as the bubble, , see Fig. 5,c), topology number (number of bubble rows across the channel width), liquid fraction (ratio between the number of black pixels and the total pixel number inside the microchannel) and bubble number density (number of bubble per surface unit). However, it must be remembered that liquid is only observed in the equivalent 2D-Plateau borders (junction between three liquid films: two wetting films on channel wall and one perpendicular between two bubbles). The high optical index ratio between the gas and the liquid, and the good configuration used allowed to observe well defined bubble cell boundaries and Plateau border thickness which is favourable for our quantitative analysis. The results for the determination of the bubble area are presented in Fig. 4. Quantitative analysis stops when bubble size becomes higher than the observation window ( > 1.2). This shows that, by increasing the pressure ratio the bubble size increases. The largest bubble size result from the bubble production and not to rupture, related to pinch-off or constriction, occurring during the transport. However, no significant effect of pH on bubble area can be highlighted because all the variations are included in the pressure ratio error bar (see Fig. 5,a). This last observation is verified for all of the different physical quantities determined by image analysis. In the following, only the averaged values, presented on Fig. 5 will be discussed.

The geometrical parameters, , and show the same trend (see Fig. 5, a, b andc). The bubble size increases when the ratio increases as previously observed with SDS foam. [START_REF] Quennouz | Microfluidic Study of Foams Flow for Enhanced Oil Recovery (EOR)[END_REF][START_REF] Cubaud | Transport of Bubbles in Square Microchannels[END_REF] The comparison between the Feret's diameter and the equivalent diameter underlines that, under a pressure ratio of 0.8, bubbles are quite spherical (more precisely cylindrical due to the confinement) and becomes polyhedral after this value which is confirmed by the capture picture on Fig. 3. After this value, bubbles become polyhedral that corresponds to the beginning of the dry foam regime ( ≈ 0.3 on Fig. 5,d). The topology number (see Fig. 5,e) which represents the number of bubble rows is defined from the beginning of bubble interaction (wet foam) to lamellae flow. This corresponds to a pressure ratio of 0.55 with a topology number of around 7. Then, due to the increasing bubble size (see Fig. 5, a, c ande) the number of rows decreases until it reaches 1 that corresponds to lamellae flow. The transition between three rows and two rows foam and two rows and one row foam occurs by T1 transition (bubble neighbour switching) sometimes called T1-wave (see inserts on Fig. 5,d). [START_REF] Raven | Generation, Flow and Manipulation of a Microfoam[END_REF][START_REF] Raven | Periodic Microfluidic Bubbling Oscillator: Insight into the Stability of Two-Phase Microflows[END_REF] After these two transitions, foam lamellae are not stable enough and break down depending on pH. The bubble number density curve (see Fig. 5,f) gives also interesting informations on foam transition when comparing with Fig. 5, a even if it is difficult to state on the phenomenon with confidence due to the large error bars. The bubble number density increases until it reaches a local maximum at around ≈ 0.4 that could correspond to the beginning of bubble area increase. Then bubble number density decreases accompanied with bubbles area growth until no bubbles are formed.

This means that, by increasing gas pressure the bubble number density increases until bubbles become close enough to be in contact with each other and then they become to grow through coalescence. In order to compare the structures and the topology transitions observed to previous works, the dimensionless channel width can be considered as suggested by

Claussen et al. [START_REF] Claussen | Packings of monodisperse emulsions in flat microfluidic channels[END_REF]. Such a ratio, used by Claussen et al. to discuss emulsions topology can help to quantify phase diagram for a global view in different geometries and for different systems [START_REF] Raven | Generation, Flow and Manipulation of a Microfoam[END_REF][START_REF] Raven | Dry microfoams: formation and flow in a confined channel[END_REF]. w is plotted as function of areal gas fraction (see Fig. 6). The dimensionless channel width decreases linearly with increasing areal gas fraction related to the increase of bubble area. Similar results were already obtained for foam made with dishwashing detergent (mainly SDS) [START_REF] Raven | Generation, Flow and Manipulation of a Microfoam[END_REF][START_REF] Raven | Dry microfoams: formation and flow in a confined channel[END_REF] in a narrower channel. Fig. 6, b is divided in four region that represent the four main bubble packing observed in the microdevice which correspond to the different morphology regime (see Fig. 6, a): bubbly liquid ( , sometimes referred to as bubbly flow or loose assembly), wet foam ( , sometimes referred to as alternate foam), dry foam ( , sometimes referred to as staircase packing) and bamboo foam ( , sometimes referred to as bamboo packing). [START_REF] Raven | Generation, Flow and Manipulation of a Microfoam[END_REF][START_REF] Raven | Periodic Microfluidic Bubbling Oscillator: Insight into the Stability of Two-Phase Microflows[END_REF][START_REF] Claussen | Packings of monodisperse emulsions in flat microfluidic channels[END_REF] A good correlation is obtained between the topology transitions observed in Fig. 6, b and the packing diagram of monodisperse emulsion droplets in a rectangular channel geometry based on a droplet surface energy model described by Claussen et al. [START_REF] Claussen | Packings of monodisperse emulsions in flat microfluidic channels[END_REF], and notably in the bamboo foam region at high areal gas fraction and low dimensional width. However, no zigzag packing described by Claussen et al. [START_REF] Claussen | Packings of monodisperse emulsions in flat microfluidic channels[END_REF] nor bidisperse bubbles flow described by Raven et al. [START_REF] Raven | Generation, Flow and Manipulation of a Microfoam[END_REF][START_REF] Raven | Dry microfoams: formation and flow in a confined channel[END_REF] were observed in the present geometry because they correspond to lower dimensionless width value (narrower channel). These results confirm the assumption that foam packing mainly depends on injection parameters and obviously to the channel geometry and bubble formation.

This first set of results reveals that, foam morphology depends only on the fluid injection parameters (pressure ratio) and channel geometry and is comparatively independent of pH. This last parameter plays a significant role on lamella stability in the case of a bamboo foam. By increasing pH, lamellae can be generated at high pressure ratio whereas in acidic conditions no lamellae are formed (see Fig. 3). It can be explained by the pH sensitivity of AKYPO® surfactant that bears a carboxylic acid function. Previous studies have pointed out the crucial impact of pH on AKYPO® micelle and foam. [16,[START_REF] Micheau | Tensioactif carboxylique polyéthoxylé pour la flottation ionique, Étude fondamentale de la solution à la mousse[END_REF] In one hand, Small angle neutron scattering (SANS) on steady state AKYPO® foam at 0.5 mM had demonstrated that, by increasing pH foam film thickness decreased from more than 1000 Å at pH 2 to 380 Å at pH 8. Furthermore surface tension and zeta potential measurements on AKYPO® solution had showed that by decreasing pH, surface charge and critical micelle concentration decrease, favouring liquid-air coverage, by increasing surfactant surface excess at the liquid-air interface, and surfactant-surfactant interactions. These observations led to the conclusion that, foam film thickness is related to the cohesive energy of the surfactant layer adsorbed at the liquid-air interface. [16] By decreasing pH, cohesive energy increases leading to thicker film: the increasing in cohesive energy leads to a higher resistance to liquid flow inside the foam film, i.e. drainage, which induces a Poiseuille like flow. In another hand, SANS on AKYPO® micelles had also demonstrated that, by increasing pH micelle volume decreases from 354.2 nm 3 at pH 2 to 121.5 nm 3 at pH 8 for a surfactant concentration of 0.5 mM. [START_REF] Micheau | Tensioactif carboxylique polyéthoxylé pour la flottation ionique, Étude fondamentale de la solution à la mousse[END_REF] In our case of flowing lamellae, the stability of foam seems to be due to the presence of the carboxylate form of the surfactant. This preponderant form at pH 12 induces electrostatic repulsion that is known to stabilize foam thanks to the DLVO theory [START_REF] Sedev | DLVO and non-DLVO surface forces in foam films from amphiphilic block copolymers[END_REF] by preventing the rupture of the film (transverse interactions) even if films are thicker than in acidic conditions, and seems to confer to the film a sufficient resistance to flow by giving the appropriate cohesive forces between surfactants at the liquid-air interface (lateral interactions). The information on micelle size suggests that lamella stability is not linked to a micelle stratification phenomenon [START_REF] Bergeron | Thin-Film Forces in Foam Films Containing Anionic Polyelectrolyte and Charged Surfactants[END_REF] but one can imagine that smaller micelles could migrate more rapidly to the liquid-air interface during the film generation and counterbalance the surfacetension gradient more efficiently than the bigger ones which can be seen as a more efficient interface healing during the formation (cf. Marangoni effect). Lastly, another hypothesis could be that, surfactant in carboxylate form leads to a low value of the surface expansion modulus which facilitates the surface stretching during lamellae generation whereas in its acidic form the surface provides a solid behaviour (high-surface coverage and high expansion modulus) which makes it easy to break. [START_REF] Cantat | Les Mousses: Structures et dynamique[END_REF] The effect of pressure amplitude on 2D-foam generation and morphology was investigated by changing the gas pressure to 300 mbar. The corresponding foam phase diagram is presented in Fig. 7. It clearly presents the 5 different foam morphology regimes previously described at 150 mbar gas pressure. However the second transition regime frontier (vertical dashed line) is shifted to lower pressure ratio values and the foam lamellae domain is reduced and begin earlier partly due to an increase in bubble size with gas pressure increase that can be seen by comparing Fig. 3 and Fig. 7. Moreover, the effect of pH on lamella stability is still observed. A wetting film (or equivalent 2D Plateau border) thinning in the corners of the microchannelis also observed for high pressure ratio and pH. As pressure ratio increases, the remaining liquid left decreases on the small wall of the microchannel (40 µm) as it can be seen on Fig. 7, and by comparing with Fig. 3. This phenomenon cannot be observed on the large walls (1200 µm) of the microchannel with this experimental setup but the same behaviour is expected for various reasons (drainage, convection, relatively less water phase volume is injected, etc.). These observations are in agreement with Bretherton's work.

[36] This shift in lamellae formation domain may be induced by the increase in gas pressure value. Previous work on flow-focusing geometry with commercial dishwashing detergent (Dreft) and nitrogen had demonstrated the influence of gas pressure on bubble flowrate and foam morphology. [START_REF] Raven | Dry microfoams: formation and flow in a confined channel[END_REF] By increasing the gas pressure, bubble flow rate increase and a rearrangement of bubbles from two bubble rows to one is observed associated with a gas pressure drop. This phenomenon is highlighted by the comparison between Fig. 3 and Fig. 7 which clearly shows this transition morphology induced by increasing gas pressure.

In addition, the absence of lamellae at high pressure ratio, even at high pH (12.0), can be explained by the generation dynamics. The lamella formation (expansion) is too quick compared to the adsorption dynamic of the surfactant which leads to an unstable lamellae and thus to a reduction of their domain. [START_REF] Cantat | Les Mousses: Structures et dynamique[END_REF] Further dedicated study is require in order to better understand this phenomenon considering AKYPO® diffusion coefficients (around 2.4x10 -9 m 2 /s) [START_REF] Jones | Structure-Dependant Mobility of a Dry Aqueous Foam Flowing Along Two Parallel Channels[END_REF] and the bubble formation time at high pressure ratio.

Finally, these two different experiments have shown that, the foam morphology in the microfluidic device is mainly controlled by the pressure ratio and gas pressure value whereas the stability of the foam lamellae depends on surfactant properties that can be tuned for example by pH.

Comparison between AKYPO® and SDS 2D-foams.

Foam flow experiment with SDS at 17.33 mM was performed with a gas pressure of 150 mbar. Dodecanol (DOH) was used to change the interface behaviour of SDS. DOH is known to form mixed monolayers with SDS at the air-water interface increasing the surface viscosity. [START_REF] Pitois | Liquid drainage through aqueous foam: study of the flow on the bubble scale[END_REF][START_REF] Durand | Physicochemical approach to the theory of foam drainage[END_REF] The concentration ratio vary from 0 to 1 and for high dodecanol concentration precipitates are formed (see Fig. 8,b) due to low dodecanol solubility and the solution becomes opalescent (the effect of precipitates on foam stability was no studied here).

[39] The corresponding phase diagram is presented in Fig. 8,a. As previously observed with AKYPO®, the phase diagram presents the 5 different foam morphology regimes. Nevertheless, dodecanol seems to have a low impact on lamellae stability because it does not alter the lamellae preponderant domain when its concentration increases (lamellae always formed at = 3). Furthermore, for high dodecanol concentration, lamellae shape is altered to a double bump shape. This can be explain by the low water solubility of dodecanol, around 0.02 mM (≈ 4 mg.L -1 ) [START_REF] Fang | The dynamic surface tension of SDS-dodecanol mixtures 2. Micellar SDSdodecanol mixtures[END_REF] and the formation of solid needle-shape precipitates observed in the flow-focusing channel (see Fig. 8,b).

Precipitates adsorb to the interface and are transport by the lamella into the Plateau border on channel wall. This centered double bump shape is mainly due to the geometry of the microchannel device, the generation process and the pinning of foam films due to the deposited precipitates. They accumulate in the injecting channel of 2700 µm length (see Fig. 8, b) at the entrance and are driven by freshly formed lamella which undergoes a kind of lamellar division process [START_REF] Haugen | Experimental Study of Foam Flow in Fractured Oil-Wet Limestone for Enhanced Oil Recovery[END_REF] when entering in the foam observation chamber. An interesting aspect of the comparison between SDS and AKYPO® foam is the difference in the lamellae shape/radius of curvature with the increasing pressure ratio.

Lamellae pictures on Fig. 9, a clearly highlights the different behaviour of foam film under varying pressure ratio for the two surfactants. In the case of SDS bamboo foam, for low pressure ratio, lamellae are curved backward. By increasing the pressure ratio, lamellae become flat (between 1 and 1.07) and then are curved forward until they breakdown for high pressure ratio. As it can be seen, foam lamella curvature radius is only slightly affected by the increasing pressure ratio and fluctuates around 2500 µm. In the case of AKYPO® foam, the lamellae also provide a curvature direction change at pressure ratio around 1.07. Nevertheless, by increasing the pressure ratio, foam lamellae are strongly deformed that indicates a pressure rising on the left side of the foam film.

To point out these differences, algebraic curvatures of SDS and AKYPO® foam measured with the Three Point Circular ROI plugin of ImageJ are presented in Fig. 9,b.

Curvatures were measured in the center of the lamellae (see picture in Fig. 9,b) excepted for difficult cases, for example at the curvature reversal, where the curvature was considered equal to zero. It is clearly seen that, SDS curvature is weakly modified with pressure ratio whereas AKYPO® curvature increase continuously until it reaches a plateau at around 2.6x10 -4 µm -1 . As a first approximation, the pressure gradient can be estimated using the Young-Laplace equation even if the lamellae are non-equilibrated: [START_REF] Chan | Theory of non-equilibrium force measurements involving deformable drops and bubbles[END_REF] (10)

where is the pressure difference of both side of an interface, is the surface tension and and are two principal curvatures. Applying twice the Young-Laplace equation to a foam film (two liquid/gas interfaces), excluding Plateau borders on the upper and lower plates of the microfluidic device, the following equation is obtained: [START_REF] Quennouz | Microfluidic Study of Foams Flow for Enhanced Oil Recovery (EOR)[END_REF] where is the pressure difference on both sides of the foam lamella. The first principal radius of curvature corresponds to the radius along the confining plates, whereas the second principal radius of curvature may vary between the cell depth (20 µm) and infinity. Such curvatures correspond to pressure drops respectively of 18 mbar and 0 mbar for a 36.08 mN.m -1 (see below). This second curvature could not be measured in the present experimental configuration, but the first one is partially characterized quantified through the radius of curvature along the confining plates (view from above the top plate, see Fig. 9). This curvature radius can be related to the pressure difference of both side of the foam film and a viscous drag force through the viscous froth model as follows [START_REF] Chan | Theory of non-equilibrium force measurements involving deformable drops and bubbles[END_REF][START_REF] Grassia | A foam film propagating in a confined geometry: Analysis via the viscous froth model[END_REF]: [START_REF] Kovscek | Foam Mobility in Heterogeneous Porous Media (I: Scaling Concepts)[END_REF] where is a viscous drag force and is the film velocity component along the film normal.

The left-hand side of equation 12 can then be calculated by measuring the foaming solution surface tension and the curvature radius along the confining plates . Surface tension of SDS solution at 17.33 mM (5 g.L -1 ) is nearly 36.08 mN.m -1 . [START_REF] Jones | Structure-Dependant Mobility of a Dry Aqueous Foam Flowing Along Two Parallel Channels[END_REF] In a previous work surface tension of AKYPO® solutions at different pH [16] was measured (see Table 1). From these experiments surface tension of AKYPO® solution at pH 12 can be estimated at 38 mN.m -1 .The pressure difference on both side of the foam film minus the viscous drag force can then be estimated for the two surfactants. In the case of SDS foam varies between 0.05 and 0.42 mbar for 14.2 mm and 1.7 mm curvature radius respectively, whereas for AKYPO® it is between 0.11 and 2.02 mbar for 6.5 mm and 374 µm curvature radius respectively (see Fig. 9,b). We could not measure film velocity here, but these results show that, considering a similar film velocity between the two surfactants due to same injection conditions, AKYPO® bamboo foam sustains a pressure gradient 5 times higher than SDS foam. As experimental conditions (pressure ratio, concentration, microfluidic device) and surface tension are quite similar for each system, the behaviour difference between the two may be attributed to the surfactant structure itself. AKYPO® surfactant is composed of an alkyl chain of 18 carbon and a polyethoxylated chain containing 9 oxygen atoms and a carboxylic function. These latest are highly hydrophilic which allowed AKYPO® to form foam stabilized by Gibbs monolayers or soluble monolayers (surfactant adsorption is reversible). The surfactant is highly soluble and when the foam or lamellae breakdown a monophasic liquid residue is obtained. In other hand, SDS is composed of a 12 carbon alkyl chain and a sulphate polar head. This structure makes SDS less hydrophilic than AKYPO® and allowed to form Langmuir monolayers or insoluble monolayers in hard water (irreversible adsorption leading to a rigid interface, usually for molecules with solubility lower than 10 -6 -10 -7 M). Furthermore, foam fractionation experiment on 17.33 mM SDS solution (results not yet published) and precipitating SDS foams [START_REF] Zhang | Precipitating Sodium Dodecyl Sulfate to Create Ultrastable and Stimulable Foams[END_REF] have shown the formation of solid SDS residue on top of the foam and bubble surface. This solid monolayer behaviour could explain the deformation difference between the two surfactants: by adsorbing at the interface, SDS gives rise to a solid monolayer that could impact the surface viscoelasticity by increasing the elastic modulus which could lead to a lower flow (or pressure gradient) resistance and then to lamellae breakdown. Moreover, the area per head group for AKYPO® at pH 12 is 96.9

Å 2 [16] and is higher than the area per head group of SDS which is approximately equal to 60 Å 2 . [START_REF] Gelbart | Micelles, Membranes, Microemulsions and Monolayers[END_REF][START_REF] Jurasin | Role of the alkyl chain number and head groups location on surfactants self-assembly in aqueous solutions, A collection of papers presented at the 5th European Conference on Foams, Emulsions, and Applications[END_REF] This means that, the surface excess of AKYPO® is lesser than SDS which induces a lesser surface viscosity [START_REF] Huerre | Bubbles and foams in microfluidics[END_REF][START_REF] Durand | Physicochemical approach to the theory of foam drainage[END_REF] and then flow resistance.

Finally, AKYPO® offers the possibility to tune the lamellae stability by varying pH and to generate foam lamellae in basic condition at higher pressure ratio than SDS.

Conclusion

This comparative study between SDS and AKYPO® RO 90 VG at 17.33 mM using microfluidic has demonstrated that, foam generation and morphology depend strongly on the injected fluid parameters such as gas and liquid pressure and are quite independent of physicochemical parameters such as pH and dodecanol concentration. The simple used micro-device has highlighted the five principal foam flow morphologies previously described for SDS surfactant but newly observed for AKYPO®. The impact of the physicochemical parameters was highlighted in the lamella regime at high pressure ratio, underlining the fact that, AKYPO® foam becomes more stable in basic conditions due to deprotonation of the surfactant head group which increases the disjoining pressure by electrostatic repulsion in the lamella. In other hand, the addition of dodecanol in SDS solution, known to increase the interface viscosity, has only a slightly impact on the flowing lamellae except at high concentration where lamella's shape becomes double bump due to the accumulation of surfactant (and co-surfactant) in the front of the bubble.

The comparison between these two surfactants has pointed out some differences in the lamellae shape and stability domain. AKYPO® at pH 12 allows to form lamellae even at very high pressure ratio whereas SDS lamellae break down. Furthermore the AKYPO® lamella curvature radius increases when pressure ratio increases whereas that one for SDS keeps almost constant. This difference is attributed to the surfactants solubilities and structure.

These particular and tuneable properties of AKYPO® make it a potential candidate for oil extraction in EOR context [START_REF] Micheau | Tensioactif carboxylique polyéthoxylé pour la flottation ionique, Étude fondamentale de la solution à la mousse[END_REF] as diverting agent or gas mobility control agent due to its capacity to form lamellae at high pressure ratio in large micro-channel (1.2 mm width).

This study reveals the crucial relevance of physicochemical parameters on flowing foam in porous media that are sometimes ignored on EOR experimental studies [START_REF] Simjoo | Novel Insight Into Foam Mobility Control[END_REF] or modified by the native rock oil reservoir. Perspectives of this work will be studies on other physicochemical parameters such as ionic strength, nature of salt or temperature that has been

  Microfluidic devices used for foam flow experiments are composed of two entrances for gas and surfactant aqueous phase, a cross junction for bubble formation, a chamber for the foam observation of 1200 µm width and 3000 µm length, and a straight channel (60 µm width) connected to the outside (atmospheric pressure) (see Fig 1). The depth of the microchannel is constant and equal to 40 µm.

Fig. 1 .

 1 Fig. 1. Schematic representation of the microfluidic device used to study AKYPO® and SDS foam flow.

Fig. 2 .

 2 Fig. 2. Bubble shape compressed between two plates in the bubbly liquid regime. (a) 3D representation of spherical compressed bubble. (b) 2D bubble cross section. and are the cylinder and piece of torus volume respectively, is the cylinder radius, the microchannel height, the real bubble diameter and the distance between the half circle gravity center G and the z rotation axis. The centroid-to-cylinder distance is obtained by using a surface integral in polar coordinates on the half circle. (c) Scheme of a half-compressed bubble illuminated in transmission by a diffuse extended light source where is the apparent bubble radius and is the critical angle of total light reflexion (i.e. 48.8°).

3 ,= 1 .

 31 05) or double-staircase (ex: pH = 4.30, = 0.95) structure are sometimes observed.[START_REF] Dollet | Influence of the Elastic Deformation of a Foam on its Mobility in Channels of Linearly Varying Width[END_REF][START_REF] Jones | Structure-Dependant Mobility of a Dry Aqueous Foam Flowing Along Two Parallel Channels[END_REF] -Bamboo foam (1.15 <). The bubble size becomes close to the channel width and the foam flows in the form of lamellae. The gas can diffuse easily between the bubbles without necessarily producing coalescence and coarsening.[START_REF] Chambers | Micromodel Foam Flow Study[END_REF] -Corner-flow or stratified regime (1.15 < ). The liquid flows mainly into the corners of the channel and to the wall as wetting films.

Fig. 3 .

 3 Fig. 3. Foam phase diagram of AKYPO® solution (17.33 mM) as function of pressure ratio and pH for a gas pressure 150 mbar.

Fig. 4 .

 4 Fig. 4. Bubble area determined by image analysis as function of fluid pressure ratio at different pH: 1.99 (r), 2.89 (s), 4.30 (£), 6.13 (v), 6.73 (w), 10.3 (™) and 12.0 (¯).

Fig. 5 .

 5 Fig. 5. Average physical parameters of AKYPO® foam at 17.33 mM with 150 mbar gas pressure as function of pressure ratio: (a) Bubble area , (b) Feret's diameter , (c) equivalent diameter , (d) liquid fraction (only the standard error is represented and not the systematic error ), (e) topology and (f) bubble number density . Black lines are guide to the eye.

Fig. 6 .

 6 Fig. 6. a) Four main bubble packing observed in the microdevice channel. From left to right: bubbly liquid ( ), wet foam ( ), dry foam ( ) and bamboo foam ( ). b) Dimensionless channel width as function of areal gas fraction . The (blue) straight line represents the linear regression with a slope of -18.412, an intercept of 18.801 and a correlation coefficient equal to 0.9877.,

Fig. 7 .

 7 Fig. 7. Foam phase diagram of AKYPO® solution (17,33 mM) as function of pressure ratio and pH for a gas pressure 300 mbar.

Fig. 8 .

 8 Fig. 8. (a) Foam phase diagram of SDS solution (17,33 mM) as function of pressure ratio and concentration ratio for a gas pressure 150 mbar. (b) Picture of the cross-junction showing dodecanol precipitates for = 1.

Fig. 9 .

 9 Fig. 9. Comparison of lamellae shape between pure SDS solution (n) and AKYPO® solution (l) at pH 12, for a 17.33 mM surfactant concentration and 150 mbar gas pressure. (a) lamellae pictures as function of pressure ratio/water pressure and (b) lamellae curvature

Table 1 .

 1 Surface tension of AKYPO® solution in function of pH, determined from[16], and surface tension of

	pure SDS and mixed SDS/DOH solution, determined from [15,25], with	.the concentration of dodecanol in
	millimolar.
		AKYPO®		SDS
	pH	(mN/m)	(mM)		(mN/M)
	2.2	31.5	0	40.5[25]/36.08[15]
	4	32.4	> 0.015		21[25]
	6	34.7
	8.2	36.8
	10.1	37.5