F R Éd

Éric Palesi

CONNECTED COMPONENTS OF PGL(2, R)-REPRESENTATION SPACES OF NON-ORIENTABLE SURFACES

The Teichmüller space of a surface naturally embeds as a connected component in the moduli space of representations from the fundamental group of the surface into the group of isometries of the hyperbolic plane. We present invariants that distinguish all the connected components of the space of representations. This allows us to compute the number of connected components of these spaces both in the orientable and in the non-orientable case.

Introduction

Let G be a Lie group and S be a closed surface (orientable or not) with Euler characteristic χ(S) < 0. Its fundamental group is denoted by π = π 1 (S) with respect to a base point. The representation space Hom(π, G) is the set of all homomorphisms φ : π → G. The group G acts by conjugation on the representation space, and the quotient of Hom(π, G) by this action is called the moduli space of representations Hom(π 1 (S), G)/G = X (S, G) Moduli spaces of representations appear naturally in many contexts, depending on the group G. For example we can identify the moduli space as the set of isomorphism classes of flat G-bundles on the surface. These spaces admit a natural action of the mapping class group exhibiting very interesting dynamics ( [START_REF] Goldman | Mapping class group dynamics on surface group representation in "Problems on Mapping Class Groups and Related Topics[END_REF][START_REF] Palesi | Ergodic action of the mapping class group on moduli spaces of representation of non-orientable surfaces Geometriae[END_REF]).

When G is non-compact, the moduli space X (S, G) contains contractible components which correspond to locally homogeneous geometric structures on the surface (see [START_REF] Goldman | Locally Homogeneous Geometric Structures Proceedings of the 2010 International Congress of Mathematicians[END_REF] for a survey on such structures). In particular, when G = Isom(H 2 ) is the group of isometries of the hyperbolic plane, the related geometric structures on the surface will be hyperbolic. Other geometric structures can be recovered from other groups G such as complex projective structures when G = PSL(2, C)(see [START_REF] Dumas | Complex projective structures[END_REF]), or convex projective structures when G = PSL(3, R) (see [START_REF] Choi | The classification of real projective structures on compact surfaces[END_REF]).

The Teichmüller space T (S) is equivalent by the uniformization theorem to the set of equivalence classes of complete hyperbolic structures on S. There is an isomorphism between Isom(H 2 ) and the Lie group PGL(2, R) through the identification of H 2 with the upper-half plane. When the surface S is given a hyperbolic structure, the holonomy map gives a representation of π into PGL(2, R). This representation is only well defined up to conjugation by an element of PGL(2, R) (corresponding to a choice of a base point for π 1 (S)). Hence we have a well defined map hol : T (S) -→ X (S, PGL(2, R)).

This map is a proper embedding, and its image is exactly the set of PGL(2, R)-equivalence classes of discrete and faithful representations.

The subset of such representations is well-understood inside the moduli space as shown in the following theorem (see [START_REF] Goldman | Topological components of spaces of representations[END_REF]):

Theorem 1. Let S be a closed surface (orientable or not). The set of PGL(2, R)-equivalence classes of discrete and faithful representations is a connected component of the moduli space X (S, PGL(2, R)).

Obviously, representation spaces and moduli spaces contain more representations than the discrete and faithful ones, so these spaces have more than one connected component. A natural question is to find some topological invariants that allow one to distinguish the connected components and hence compute the number of components.

For representation φ : π 1 (S) → PGL(2, R), a classical invariant is the so-called Euler class that is associated to the circle bundle over the surface defined by φ. This invariant can be defined in several ways, and the main objective of this note is to give a simple construction valid both in the orientable and the non-orientable case. And we give some insight into how this invariant distinguishes the connected components of the representation space.

In the classical case of orientable surfaces, the holonomy of a hyperbolic structure lies in the group of orientation-preserving isometries Isom + (H 2 ), which is identified with the group PSL(2, R).

In Section 2, we will give some basic properties of the group PSL(2, R) and its universal cover. In particular, we will study the behavior of the map corresponding to the relation of the fundamental group of the surface. This is the key ingredient to define a topological invariant using a simple algorithm.

In Section 3, we first define this invariant for closed orientable surfaces, and then generalize it to orientable surfaces with boundary. This invariant is all that we need to distinguish the connected components of the representation space Hom(π 1 (Σ), PSL(2, R)), and give a characterization for the discrete and faithful representations. In [START_REF] Goldman | Topological components of spaces of representations[END_REF], Goldman proves the following : Theorem 2. Let Σ be a closed orientable surface with χ(Σ) < 0. The representation space Hom(π, PSL(2, R)) has 2|χ(Σ)|+1 connected components, indexed by the Euler class χ(Σ) ≤ e ≤ -χ(Σ).

Moreover, a representation is discrete and faithful if and only if it is in the maximal components satisfying |e| = |χ(Σ)|.

We give only some elements of the proof of this result and refer to Goldman's original paper for a complete proof. This question has also been generalized and solved for a variety of other Lie groups, for example when G = PSL(n, R) (see [START_REF] Hitchin | Lie groups and Teichmüller space[END_REF]), G = Sp(2n, R) (see [START_REF] Burger | Maximal representations of surface groups:Symplectic Anosov structures[END_REF][START_REF] Garcia-Prada | Representations of surface groups in Sp(2n,R)[END_REF]), or when the symmetric space associated to G is Hermitian (see [START_REF] Toledo | Representations of surface groups in complex hyperbolic space[END_REF]).

In Section 4, we extend the construction of the topological invariant defined in Section 3 to the case of a non-orientable surface N . In this case, the invariant is only well defined in Z/2Z and the representation space Hom(π 1 (N ), PSL(2, R)) has only 2 connected components. These components do not contain any holonomy representations for hyperbolic structures (see [START_REF] Palesi | Connected components of representation spaces of non-orientable surfaces Communications in Analysis and Geometry[END_REF]).

In this case, it is natural to consider not only orientation-preserving isometries, but also orientation reversing isometries. Hence, the interesting representation space is Hom(π, PGL(2, R)). For representation into PGL(2, R), we have to define another invariant that takes values in (Z/2Z) k , and then understand the set of representations for each value of this invariant. The main result of [START_REF] Palesi | Non-Teichmüller components for non-orientable surfaces in preparation[END_REF] is the following : Theorem 3. Let N be a closed non-orientable surface with χ(N ) < 0, then the representation space Hom(π, PGL(2, R)) has 2 3-χ(n) -2χ(N )-1 connected components indexed by two obstruction classes.

The group PSL(2, R)

The fundamental group of a surface is a finitely presented group with p generators and q relations. So for any Lie group G the representation space Hom(π, G) can be identified with the subset of G p defined by q equations in words in G, corresponding to the q relations. So to understand the representation space, we study the properties of G and of the map G p -→ G q (g 1 , . . . , g p ) -→ (R 1 (g 1 , . . . , g p ), . . . , R q (g 1 , . . . , g p ))

where the R i are the q relations, each in p G-valued unknowns.

In the following section, we shall note G = SL(2, R) and G = PSL(2, R) = G/{±I}. We see that G is a double cover of G.

2.1. Universal cover. The standard polar decomposition of SL(2, R) allows one to decompose any A ∈ G in a unique way into A = R • P , with R ∈ SO(2) and P ∈ Sym, the group of symmetric positive definite matrices of determinant 1. The group Sym being contractible, we have a natural deformation retraction of G onto the circle SO( 2)

S 1 R/Z. So, the fundamental group of G is π 1 (G) π 1 (S 1 ) Z. Let G be the universal cover of G and p : G → G, the covering map. The center of G, Z( G) = ker(p) is isomorphic to Z and G G/Z. We denote by z a generator of Z( G) so that Z( G) = z .
Two lifts of the same element will differ by an element of Z( G). In other words, if

A, B ∈ G satisfy p(A) = p(B), then there exists n ∈ Z such that A = z n B.
The group G is a connected 2-fold cover of G, and thus G is also the universal cover of G. We have a canonical identification G = G/ z 2 .

Classification of elements.

The orientation-preserving isometries can be classified through their number of fixed points on the boundary. An element f in Isom + (H) 2 \ {Id}, is said to be elliptic (resp. parabolic and hyperbolic) when f has no fixed points on the boundary ∂H 2 (resp. one and two fixed points)

This can be interpreted in terms of the trace of corresponding elements of PSL(2, R). Notice that only the absolute value of the trace is well defined in G. An element A in G \ {I}, is elliptic (resp. parabolic and hyperbolic) when

| tr(A)| < 2 (resp. when | tr(A)| = 2 or | tr(A)| > 2.
Lifts of elliptic (resp. parabolic, hyperbolic) elements in the universal cover G will also be called elliptic (resp. parabolic, hyperbolic) and we denote by E, P and H the corresponding subsets of G. These subsets decompose G into infinitely many disjoint subsets, indexed by integers Z corresponding to different copies of G inside G, according to Figure 1.

We can distinguish these regions using a simple invariant of A ∈ G. The action of G on H 2 extends to an action on the circle at infinity ∂H 2 S 1 . This action lifts to an action of G on S 1 R. Hence for

mA = min{A • x -x | x ∈ R}, mA = max{A • x -x | x ∈ R}.
Using these invariants, we can give the following characterizations [START_REF] Jankins | Homomorphisms of fuchsian groups to PSL(2, R)[END_REF]:

A ∈ E i ⇐⇒ [mA, mA] ⊂]i, i + 1[, A ∈ H i ⇐⇒ i ∈ ]mA, mA[, A ∈ P + i ⇐⇒ mA = i < mA, A ∈ P - i ⇐⇒ mA < i = mA.
There are some direct consequences of these formulas, that are important to understand the case of surfaces with Euler characteristic -1. For example, we have the following proposition (see [START_REF] Goldman | Topological components of spaces of representations[END_REF]):

Proposition 1. If A, B ∈ H 0 and AB ∈ H i then i ∈ {-1, 0, 1}.
We will see in Section 3 how this will be useful to understand the case of the pair-of-pants.

2.3.

The commutator map. The fundamental group of an orientable surface of genus g is given by :

π 1 (Σ g ) = X 1 , Y 1 , . . . , X g , Y g | g 1 [X i , Y i ] .
Let A 1 , B 1 , . . . , A g , B g be elements of G and choose arbitrary lifts A 1 , . . . , B g in G. The element defined by :

R g (A 1 , . . . , B g ) = [ A 1 , B 1 ] • • • [ A g , B g ],
is independent of the chosen lifts. Indeed two different lifts differ by a central element of Z( G) which will cancel out in R. For example, assume that we have two different lifts A 1 = g A 1 where g ∈ Z( G) then

R g = [ A 1 , B 1 ] • • • [ A g , B g ] = (g A 1 ) • B 1 • ( A 1 -1 g -1 ) • B 1 -1 • • • [ A g , B g ] = gg -1 • [ A 1 , B 1 ] • • • [ A g , B g ] = R g (A 1 , . . . , B g ).
So we can define the map

(1) R g : G 2g -→ G.
When g = 1, the image of the commutator map R 1 can be explicitely described using the following proposition (see [START_REF] Goldman | Topological components of spaces of representations[END_REF]).

Proposition 2. If R 1 (A, B) is in H i then i ∈ {-1, 0, 1}.
We will see in Section 3 how this will be useful to understand the case of the one-holed torus.

2.4. The square map. The fundamental group of a non-orientable surface of genus k is given by

π 1 (N k ) = X 1 , . . . , X k | X 2 1 • • • X 2 k . Let A 1 , . . . , A k be elements of G and choose arbitrary lifts A 1 , . . . , A k in G. The element defined by Q k (A 1 , . . . , A k ) = A 1 2 • • • A k 2 ,
is not independent of the chosen lifts. For example, assume that we have two different lifts A 1 = g A 1 where g ∈ Z( G) then

Q k = A 1 2 • • • A k 2 = g 2 A 1 2 • • • A k 2 = g 2 Q k .
However, we see that this gives a well defined element of G/ < z 2 >= G, hence this defines a continuous map :

(2)

Q k : G -→ G.
Using this, we can define the corresponding lifted square map

Q k : G k -→ G.
When k = 1, the image of Q 1 is easily computed (see [START_REF] Palesi | Connected components of representation spaces of non-orientable surfaces Communications in Analysis and Geometry[END_REF]) to be :

Im(Q 1 ) = J = {K ∈ SL(2, R)| tr(K) > -2} ∪ {I}.
Hence, the image of the lifted square map is :

(3) Im( Q 1 ) = J = G \ k∈Z H 2k+1 ∪ P 2k+1 .
We will see in Section 4 how this will be useful to understand the case of one-holed projective plane.

Components for orientable surfaces

In this section, we give a simple algorithm to compute the Euler class of a representation φ : π → G when the surface is orientable and G = PSL(2, R).

3.1.

A topological invariant. Let π = π 1 (Σ g ) be the fundamental group of the closed orientable surface of genus g and let φ be an element of Hom(π, G). The relation of the fundamental group implies that the element R(φ(X 1 ), . . . , φ(Y g )) defined by (2.3) is an element ker(p) = Z, i.e. there exists n ∈ Z such that

R(φ(X 1 ), . . . , φ(Y g )) = z n .
We define o 2 (φ) = n ∈ Z and call this number the second obstruction class of the representation φ. (We shall see the first obstruction class in the last section). This defines a map :

o 2 : Hom(π, PSL(2, R)) -→ Z.
This map is continuous and takes values in a discrete set, so o 2 is constant on connected components of the representation space.

We can relate this invariant with other classical invariants, such as the characteristic class w 2 (E φ ) of the flat G-bundle E φ → Σ, the Euler class e(φ) of the associated circle bundle over Σ, or also the Toledo invariant T (φ) of the representation. All these invariants coincide when G = PSL(2, R). However, in higher dimension they give different generalizations. For precise definitions and relations between these invariants, we refer to [START_REF] Burger | Higher Teichmüller spaces : From SL(2, R) to other Lie groups[END_REF].

Milnor-Wood inequality and a result of Goldman.

A circle bundle over Σ is simply a fiber bundle E → Σ whose fibers are isomorphic to S 1 . A representation ρ : π(S) → PSL(2, R) defines a circle bundle over Σ through the embedding G → Homeo(S 1 ) defined by the action of G on ∂H 2 .

Consider the free product P = Σ × S 1 and the action of π 1 (Σ) on P given by :

π 1 (Σ) × ( Σ × S 1 ) -→ Σ × S 1 γ • (x, θ) -→ (γ • x, ρ(γ) • θ)
where the action of π 1 (Σ) on Σ is by deck transformations. The quotient of P by this action defines a circle bundle over the surface S : The proof of the theorem goes by induction on smaller surfaces that are necessarily surfaces with boundary. So next, we define the invariant in this case and generalize Theorem 2 for surfaces with boundary.

E ρ = ( Σ × S 1 )/π 1 (Σ),

Surfaces with boundary.

For surfaces with boundary, we need to be a little more precise in the definition of the obstruction class. Let Σ = Σ (g,m) , the genus g surface with m ≥ 1 boundary components denoted ∂ 1 Σ, . . . , ∂ m Σ. Its fundamental group has the following presentation :

π 1 (Σ) = X 1 , Y 1 , . . . , X g , Y g , C 1 , . . . , C m g 1 [X i , Y i ] • C 1 • • • C m .
where, C i corresponds to the boundary circle ∂ i Σ. It is clear that π 1 (Σ) is isomorphic to the free group F N in N = 2g + m -1 generators, because we can express one of the C i as a word in the other generators.

In this case, the representation space Hom(F N , G) is simply the product group G N , which is naturally connected as G is itself connected. Indeed when one tries to apply Milnor's algorithm to the presentation above, we get that the element of π 1 (G) obtained, depends on the choice of the lifts of the image of the m generators C 1 , . . . , C m . So in order to have a non-trivial topological invariant, we need to be able to choose canonical lifts for the image of these generators.

An element g in G will have a canonical lift in G if it has a fixed point on ∂H S 1 . The canonical lift, denoted g 0 , is the unique lift of g in G that also has fixed points on S 1 R. By the classification of hyperbolic isometries, an isometry has a fixed point if and only if it is not elliptic. The canonical lift of a non-elliptic element will belong to {I} ∪ H 0 ∪ P 0 . Now, we can define an invariant on the set

W (Σ) = {φ ∈ Hom(π 1 (Σ), G) | ∀C i ⊂ ∂Σ, φ(C i ) is not elliptic} .
Let φ be an element of W (Σ). Choose arbitrary lifts φ(A 1 ), . . . , φ(B g ) of the images of the generators into G. The relation of the fundamental group implies that the element

R = φ(X 1 ), φ(Y 1 ) . . . φ(X g ), φ(Y g ) • φ(C 1 ) 0 • • • φ(C m ) 0 ,
is a lift of the identity element of G into G that is independent of the chosen lifts. So there exists n ∈ Z such that R = z n . We define o 2 (φ) = n and call it the relative second obstruction class of the representation φ. This defines the map o 2 : W (Σ) -→ Z The following theorem is the generalization of Theorem 2

Theorem 5. Let S be an orientable surface with χ(S) < 0. For all n ∈ Z, if |n| ≤ |χ(S)| then o -1 2 (n) is non-empty and connected. As a consequence, the space W (Σ) has 2|χ(Σ)|+1 connected components. We can see that Theorem 2 is just a particular case of Theorem 5, when Σ has no boundaries.

3.4. Summary of the proof. The proof of this theorem goes by induction. First, we prove the theorem for surface of Euler characteristic -1, namely the pair-of-pants Σ (0,3) and the one-holed torus Σ [START_REF] Bradlow | Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces[END_REF][START_REF] Bradlow | Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces[END_REF] . This is mostly a consequence of Propositions 1 and 2, that can describe the set W (Σ).

Next, we decompose any orientable surface into surfaces of characteristic -1. We want the graph dual to this decomposition to be a tree so that the induction process works. For the surface Σ (g,m) , such a decomposition consists of g one-holed tori and g + m -2 pair-of-pants. One can find a continuous path from any representation to a representation such that the curves defining the decomposition are sent to hyperbolic elements. So it is possible to compute the relative Euler class on each subsurface of the decomposition. Then, using the result on Σ (0,3) and Σ (1,1) , one can show that if two representations have their relative Euler class coincide on each subsurface, then they are in the same connected component.

Then it suffices to show that if two representations have the same Euler class, we can find paths to representations such that the relative Euler classes coincide on each subsurface. First, one can prove this fact for each surface obtained by gluing two surfaces of Euler characteristic -1, namely Σ 2 , Σ (1,2) and Σ (0,4) . Using this, one can prove the result on a general surface by induction along the tree corresponding to the dual graph of the decomposition (see [START_REF] Goldman | Topological components of spaces of representations[END_REF] for a complete proof of these arguments).

Non-orientable surfaces

In this section, we extend the definition of the obstruction invariant to the representation of the fundamental group of a non-orientable surface into PSL(2, R), and then in PGL(2, R). We denote

G + = PSL(2, R) and G ± = PGL(2, R). So the group G ± is the disjoint union G + G -, where G -is the set of orientation-reversing isometries of H 2 .
4.1. Non-orientable surface group. Recall that a closed non-orientable surface N k is homeomorphic to the connected sum of k projective planes. The number k is called the non-orientable genus of the surface and the Euler characteristic is

χ(N k ) = 2 -k.
The fundamental group is given by the following presentation :

π 1 (N k ) = A 1 , . . . , A k |A 2 1 • • • A 2 k .
In this presentation, the generators are represented by one-sided curves, which are curves whose neighborhoods are Mobiüs strips. In particular, when N k is given an hyperbolic structure on N k , the holonomy of a one-sided curve is sent to an orientation-reversing isometry. As the map Q k is continuous, the map o 2 is also continuous and takes values in a discrete set, so we can infer that o 2 is constant on connected components of the representation space.

We can also generalize the construction to non-orientable surfaces with boundaries, denoted N (k,m) . As in the orientable case, we have to impose the condition that boundary components are sent to non-elliptic elements, so we define

W (N (k,m) ) = φ ∈ Hom(π, G + ) | ∀C ∈ ∂N (k,m) , φ(C) is not elliptic .
The map o 2 : W (N (k,m) ) → Z/2Z is defined as in the case of orientable surfaces. The characteristic class o 2 is sufficient to distinguish the connected components of W (N ), and we can state the following :

Theorem 6. Let N (k,m) be a non-orientable surface with χ(N (k,m) ) < 0. For [n] ∈ Z/2Z, the set o -1 2 ([n]) ⊂ W (N (k,m)
) is non-empty and connected.

This establishes that the representation space Hom(π 1 (N k ), G + ) has exactly 2 connected components.

It may seem surprising that in the non-orientable case, the number of connected components does not depend on the non-orientable genus. In fact, representations in G + do not parametrize hyperbolic structures on the surfaces or oriented circle bundles over the surface. Indeed, if a representation φ : Hom(π 1 (N ), G ± ) defines an oriented circle bundle over N , then necessarily if A ∈ π 1 (N ) is represented by a one-sided curve, then φ(A) ∈ G -. Hence we have to study representations in G ± .

Non-orientable Euler class.

The group G ± is not connected anymore as G ± = G + G -, hence π 0 (G ± ) Z/2Z. When studying the representation into G ± , one has to use another topological invariant for the representation, namely the first obstruction class denoted o 1 and defined as follows.

The obstruction class o 1 (φ) of a representation φ ∈ Hom(π, G ± ) is an element of the set Hom(π, π 0 (G ± )) and is obtained by composition of the representation φ : π → G ± with the epimorphism G ± → π 0 (G ± ). As π is finitely generated by {A 1 , . . . , A k }, the image of the generators by the map o 1 (φ) completely determines o 1 (φ). So we can see Hom(π, π 0 (G ± )) as a subset of (π 0 (G ± )) k (Z/2Z) k . We have the following notation for the first obstruction :

o 1 : Hom(π 1 (N k ), G ± ) → (Z/2Z) k .
For example the representation space Hom(π, G + ) is exactly the set of representations φ ∈ Hom(π, G ± ) such that o 1 (φ) = (0, . . . , 0).

On the other hand, given a hyperbolic structure φ on N , the holonomy along a 1-sided curve is necessarily an orientation-reversing isometry as the curve reverse the orientation on the surface. Hence, the image of the 1-sided curves are all orientation-reversing isometries. The generators in the standard presentation are represented by 1-sided curves, so any holonomy representation of a hyperbolic structure satisfies o 1 (φ) = (1, . . . , 1). 4.4. Square map. To understand the representations inside the nonzero components o -1 1 (ε 1 , . . . , ε k ), we need to understand squares of orientation-reversing isometries. First, we notice that if A ∈ G -, then A 2 ∈ G + . Moreover, all orientation-reversing isometries are glidereflections, i.e. the product of a reflection and a hyperbolic translation along the same axis. Hence, the square A 2 of a glide-reflection is an hyperbolic element or the identity element. Simple matrix calculation (see [START_REF] Palesi | Connected components of representation spaces of non-orientable surfaces Communications in Analysis and Geometry[END_REF]) also shows that the square map :

G --→ H ∪ {I} A -→ A 2 is continuous, surjective with connected fibers. Let ε = (ε 1 , . . . , ε k ) ∈ (Z/2Z
) k and define l as the number of ε i that are zero. The fundamental group π 1 (N (l,k-l) ) can be seen as a subgroup of π 1 (N k ) by cutting the surface along the curve A 2 i whenever ε i = 0. Using this property we can define a map :

T ε : o -1 1 (ε) -→ W (N l,n ) φ -→ T ε (φ),
where T ε (φ)(A i ) = φ(A i ) if ε i = 0 and else T ε (φ)(A i ) = φ(A i ) 2 . This map is continuous, surjective with connected fibers because the square map defined above is. We infer that o -1 1 (ε 1 , . . . , ε k ) and W (N (l,n) ) have the same number of connected components.

For all ε such that l > 0, Theorem 6 asserts that W (N (l,n) ) has two connected components indexed by o 2 ∈ Z/2Z. However, when l = 0, we identify N (0,k) = Σ (0,k) as the n-holed sphere. In this case, Theorem 5 tells us that W (Σ (0,k )) has 2k -3 connected components indexed by the relative obstruction class o 2 ∈ Z.

We can now state the following theorem on the total number of connected component of Hom(π, G ± ). The set of representations such that l = 0 is the fiber o -1 1 (1, . . . , 1). Through the map T ε we can define the obstruction of a representation φ ∈ o -1 1 (1, . . . , 1) as the relative obstruction class of T ε (φ) ∈ W (Σ (0,k) ). It can be shown that this invariant coincides with the Euler class of the oriented circle bundle over N k defined by φ. The components of Hom(π, G ± ) consisting of discrete and faithful representations are exactly the ones with ε = (1, . . . , 1) and maximal Euler class. 4.5. General Formula. In [START_REF] Xia | Components of Hom(π 1 , PGL(2, R))[END_REF], Xia proves that for an orientable surface of genus g, the space Hom(π 1 (Σ g ), G ± ) has 2 2g+1 +4g-5 connected components. When translating the formula in terms of the Euler characteristic of the surface, one obtain the following general result. Theorem 8. Let S be a closed surface orientable or not, with Euler characteristic χ(S) < 0 and G ± = PGL(2, R), then the number N (S) of connected components of the space Hom(π 1 (S), G ± ) is given by : N (S) = 2 3-χ(S) -2χ(S) -1.

Figure 1 .

 1 Figure 1. Domains of G

  since the action on the first factor is free. The Euler class of the representation, denoted e(ρ) ∈ Z is defined to be the evaluation of the characteristic class e(E ρ ) of the circle bundle on the fundamental cycle [Σ]. We have e(ρ) = o 2 (ρ).Euler classes of circles bundles were studied by Milnor and Wood[START_REF] Milnor | On the existence of a connection with zero curvature[END_REF][START_REF] Wood | Bundles with totally disconnected structure group[END_REF] who established the so-called Milnor-Wood inequality, stating that the Euler class of an orientable circle bundle E over Σ is bounded by the Euler characteristic of S, namely |e(E)| ≤ |χ(Σ)|. Hence the Euler class e(ρ) of a representation can only take 2|χ(Σ)|+ 1 values, corresponding to the integers contained in the interval [χ(Σ), -χ(Σ)]. The Theorem 2 establishes the converse result and we can reformulate it as : Theorem 4. Let Σ be an orientable closed surface with χ(Σ) < 0, for all n ∈ Z, if |n| ≤ |χ(Σ)| then e -1 (n) is non-empty and connected.

4. 2 .

 2 Second obstruction class. Let us first understand the connected components of Hom(π, G + ), where π = π 1 (N k ). Let φ ∈ Hom(π, G + ). The relation defining the fundamental group implies that the element Q k (A 1 , . . . , A k ) defined in Section 2 is a lift of the identity in G + to G. This defines a map into the group Z( G) = {±I} Z/2Z. So we have the following map :

o 2 :

 2 Hom(π, G + ) → Z/2Z.

Theorem 7 .

 7 The representation space Hom(π 1 (N k ), PGL(2, R)) has 2 k+1 + 2k -5 connected components.

Acknowledgements

The author would like to thank all the organizers for the very interesting and enriching program. This paper was written while the author was a CIRGET postdoctoral fellow at the Université de Sherbrooke.


Morgan Bertin
email: morgan.bertin@lmt.ens-cachan.fr

François Hild
email: hild@lmt.ens-cachan.fr

Stéphane Roux
email: stephane.roux@lmt.ens-cachan.fr

Florent Mathieu
email: florent.mathieu@lmt.ens-cachan.fr

Hugo Leclerc
email: hugo.leclerc@lmt.ens-cachan.fr

OPTIMIZATION AND VALIDATION OF A CRUCIFORM GEOMETRY FOR THE IDENTIFICATION OF ELASTOPLASTIC MODELS

A method based upon the sensitivity fields is proposed to optimize biaxial experimental procedures. The objective is to improve the identifiability of elastic parameters related to an a priori chosen constitutive law. This is achieved by minimizing the covariance matrix of the material parameters. This approach unifies full field measurements via image correlation, numerical simulations and identification procedure. The latter is performed on a biaxial experiment and aims at identifying elasto-plastic models.

Introduction

One of the main issues of mechanical engineering is to understand, describe and predict how materials behave and break. The related objectives are to model observed phenomena with the best accuracy. The study focuses on in-plane biaxial tests of cruciform specimens [START_REF] Makinde | Development of an appartus for biaxial testing using cruciform specimens[END_REF]. The first in-plane biaxial loading test system has been proposed by Feron and Makinde [START_REF] Ferron | Design and development of a biaxial strenght testing device[END_REF]. Chen and Matthews (3), and Alan and Peter (4) have discussed various aspects associated with biaxial experimental systems on sheet rolled steel and composite materials. Some of the protocols are able to apply complex loading conditions [START_REF] Boehler | A new direct biaxial testing machine[END_REF]. Two main issues have been raised by the authors cited above: i) how to reveal specific material behaviors? and ii) how to quantify and identify material parameters? These two questions are related to one goal, namely, improving the quality of the identified parameters. The identification quality of constitutive parameters is both related to the experimental procedures and the identification methods. In the sequel, the identification quality is estimated by the covariance matrix of the identified material parameters [C p ]. In the case of a displacement field, it is expressed with the sensitivity field gathered in a matrix [S]. The covariance matrix of the identified parameter [C p ] accounts for different aspects of the problem, namely, the geometry of the studied structure, the chosen constitutive law, the set of parameters, the boundary conditions, the measurement uncertainties, the identification method and to a lesser extent the numerical simulation method.

Theoretical background

Integrated digital image correlation (I-DIC) is a global DIC technique that relies on a mechanical choice for the measured displacement field. It allows to perform the measurement and identification of the sought parameters in only one step. It is based on the global DIC technique (6) that relies on the registration of an image f in the reference configuration and a series of pictures g in the deformed configurations. The problem consists of minimizing the sum of squared differences between the deformed image corrected by the measured displacement u(x) and the reference image (written for each time t independently)

χ 2 f (t) = 1 2γ 2 f N Ω Ω ((g(x + u(x, t), t) -f (x)) 2 (1)
with respect to the sought displacement fields u(x, t), where x is any considered pixel. In this expression Ω denotes the Region of Interest (ROI), N Ω its area in terms of the number of pixels it contains, and γ f the standard deviation (expressed in gray levels) of the white noise assumed to affect each image independently (including the reference one, what is responsible for the factor of 1/2 coming as a multiplicative term in this functional). The load is also in importance for the identification because it increases the number of the measured quantities and as a consequence diminishes the relative uncertainty by increasing the material parameters sensitivities [START_REF] Bertin | Optimization of a biaxial tensile specimen geometry for the identification of constitutive parameters based upon full field measurements[END_REF]. Thus, the same approach is followed with the reaction forces for which χ 2 F is minimized

χ 2 F = 1 N F N t γ 2 F {F m -F c } t {F m -F c } (2)
where {F m } are the measured reaction forces and {F c } are the computed levels with respect to the chosen material parameter set, N F the number of load cells for each actuator (N F = 1), N t the number of steps and γ 2 F the load variance. The identification based upon both observables, i.e., displacement field and reaction force, is achieved by minimizing the global functional χ 2

I χ 2 I = N Ω N Ω + N F χ 2 f + N F N Ω + N F χ 2 F ( 3
)
where the DIC and reaction forces functionals have been introduced in Equations. ( 1) and ( 2) respectively. The choice for the specific weight is issued from a Bayesian foundation for using a weighted quadratic difference including noise covariance [START_REF] Bertin | Optimization of a biaxial tensile specimen geometry for the identification of constitutive parameters based upon full field measurements[END_REF].The covariance matrix of the identified parameters is written as (where • is the mean value of •)

[C I p ] = {δp} ⊗ {δp} (4)

Optimization

The identification involving multiple parameters, it is somewhat subjective to choose a given criterion to express the quality of identification by a single number to be optimized. It is desirable that all parameters be determined accurately, but when increasing the quality of one determination degrades that of another one, expressing a preference is difficult. Because of this freedom, one cannot pretend to provide a universal answer. The choice made herein is to focus on the "worst" determination, namely, the one that has the largest uncertainty due to noise, and optimization is designed to reduce this uncertainty to its minimum. In another word, the maximum level of the minimum eigenvalue of [C I p ] -1 is sought. Furthermore and since prior the experiment the pattern is unknown, it is useful to resort to a "mean-field" assumption. The chosen parameterization is based on (log(E/E 0 ), log(ν/ν 0 )) collectively denoted {q}. Here E 0 and ν 0 are nominal values of the chosen parameters. The choice of using a log scale for the moduli is to favor relative uncertainties. The geometry optimization is applied to a linear elastic law for the sake of simplicity and relies on the four fillet radii size. The material parameters are chosen to be representative of a stainless steel (17-7 PH grade) (E = 200 GPa, ν = 0.3 and σ 0 = 300 MPa). A non-proportional loading path is prescribed. First a displacement along direction e 1 is applied up to a maximum value d 1 = d * while the transverse displacement is kept equal to 0. Then, at fixed displacement d 1 , the displacement d 2 is raised to reach an equibiaxial strain state, d 1 = d 2 = d * and finally both displacement amplitudes are reduced to zero at the same rate so that in the displacement plane (d 1 , d 2 ) the loading path is a triangle. For each filet radius, d * is defined such as the maximum Von Mises equivalent stress is equal to the yield stress.

Biaxial experiment and identification

The previous procedure has been applied to a biaxial experiment with a precipitate hardened stainless steel (17-7 PH grade).

The experimental system with the tested sample is shown in Figure 1(a). The identification rests on three constitutive laws, namely, (A) a linear isotropic elasticity law, (B) an elasto-plastic law with linear kinematic hardening and (C) an elasto-plastic law with exponential kinematic hardening. The latter includes 360 time steps (loads and images) captured at 1 Hz and includes loading and unloading cycles. A 16-bit camera (PCO edge 2.0) with telecentric lens (×0.5) is used. The correlation residual is the natural tool to evaluate the registration quality of DIC approaches [START_REF] Mathieu | Estimation of elastoplastic parameters via weighted femu and integrated-dic[END_REF]. Figure 2 shows the correlation residuals for the DIC solution (black line) and for the three constitutive laws (color lines). The maximum permanent strain reached is equal to = 18.9. ) Table 1 gathers the results. The choice of an elasto-plastic law diminishes the model error with the lowest level's residual. However, the latter is still high with respect to the DIC one equal to χ 2 f = 7.77. This implies that the model doesn't describe efficiently all the mechanical behavior over the loading range. Figure 3 shows the evolution of the parameters along the iterations. The convergence is reached in few steps. However, the Poisson's ratio tends to ν = 0.50 since the kinematic behavior is mostly plastic meaning incompressible. Indeed, since the experimental strain is far to be elastic, the identification of law A is mainly driven by plastic strain. Finally, Figure 3 shows the displacement field residual for component U 1 expressed in pixel (1 pixel = 13.5 µm) at the end of the experiment (F 1 = F 2 = 0 N), a displacement field residual being the difference between the DIC and I-DIC solution.

χ 2 f ( DI C ) χ 2 I ( law C ) χ 2 I ( law B ) χ 2 I ( law A )
The choice of an elasto-plastic law diminishes the model error. In the other hand and for law A, the plastic strain is directly exhibited since no plastic strains are predicted by the model. In conclusion, a complete data procedure has been firstly described, which starts from the acquisition of experimental data (i.e., load values and images) to the extraction of material parameters whereby uncertainties could be tracked all the way down to the identification step. Secondly, the identification of constitute parameters has been performed and a procedure has been set up in order to discriminate constitutive laws with respect to experimental data and to evaluate the best one.

Figure 1 :

 1 Figure 1: (a) Biaxial machine with the chosen sample geometry (b) Eigen value λ I of Hessian [C I p ] -1 with the ({p}) parametrization.

  Figure 1(b) shows the eigenvalues of [C I p ] -1 and the optimal radius r equal to r opt = 1.7 mm.

Figure 2 :

 2 Figure 2: Correlation (DIC) and Integrated Digital Image Correlation (I-DIC) residuals for the three constitutive laws.

Figure 3 :

 3 Figure 3: Evolution of the material parameters respectively for the laws A(a), B(b) and C(c).

Figure 4 :

 4 Figure 4: (a), (b) and (c) difference between DIC and I-DIC of the displacement field component U 1 respectively for laws A, B and C. (d) displacement field component U 1 from DIC.

Table 1 :

 1 Identified parameters and residuals for the three laws (δ p (%)).

	law	χ 2 I	E(GPa)	δ E	ν	δ ν	σ 0 (MPa) δ σ0	C(GPa)	δ C	γ	δ γ
	Ref	×	200	×	0.3	×	300	×	10	×	10	×
	A	225	9	0.05 0.499 0.002	×	×	×	×	×	×
	B	50.7	157	0.20 0.31	0.04	480	0.19	6.7	0.2	×	×
	C	40.2	148	0.25 0.30	0.04	423	0.25	8.6	0.25 10.8 0.04

Acknowledgments

It is a pleasure to acknowledge the support of Région Ile de France (DICCIT and THERMOFLUIDE-RT projects).


Alaedine Kossi
email: alaedine.kossi@gmail.com

Georges Bossis

Jacques Persello

Electro-active elastomer composite based on doped titanium dioxide

Keywords: electrorheological elastomer, electromechanical properties, semiconductorinsulator composites, TiO2 colloidal particles, PDMS cross-linked matrix

Electro-active elastomer composite based on doped titanium dioxide Electro-active elastomer composite based on doped titanium dioxide

Alaedine Kossi, Georges Bossis, Jacques

Introduction

Magnetorheological and electrorheological fluids are suspensions of micro-or submicroparticles dispersed in a liquid matrix whose viscosity can be changed by the application of a magnetic or an electric field, respectively. Their solid counterparts are magnetorheological elastomers (MREs) and electrorheological elastomers (EREs).

Magnetorheological elastomers can exhibit a change of their Young's modulus of several hundred kPa under the application of a magnetic field of 0.2-0.5 Tesla and have numerous applications as actuators, vibration isolators or variable capacitors. [START_REF] Behrooz | Performance of a new magnetorheological elastomer isolation system[END_REF][START_REF] Bica | Magnetic field intensity effect on plane electric capacitor characteristics and viscoelasticity of magnetorheological elastomer[END_REF] Electrorheological elastomers (EREs) are made by suspending semiconducting particles in a cross-linkable insulating polymer. The semiconducting particles are aligned under an applied electric field to form columns in the insulating medium and then the structured dispersion is cross-linked to maintain the anisotropic structure in the polymeric network. In response to an applied electric field ERE composites manifest a rapid and reversible change of modulus. The main cause inducing the ER effect is the mismatch in dielectric constant and conductivity between semiconducting particles and insulating medium that generate dipolar interactions in the presence of an AC or DC electric field. Theoretical models based on linear electrostatics have been developed to understand the physical mechanisms involved in the ER effect, regarding the role of the permittivity and conductivity in the interaction forces between particles. These studies showed that the attraction force in ER fluids increases with the applied electric field according to a quadratic law 𝐹 𝑒𝑙𝑒𝑐 ∝ 𝐸 0 2 . [START_REF] Tanaka | Electrically Induced Microstructures in Micro and Nano-Suspensions and Related Physical Properties[END_REF][START_REF] Anderson | Electrostatic forces in an ideal spherical-particle electrorheological fluid[END_REF][START_REF] Parthasarathy | Electrorheology: mechanisms and models[END_REF] However, lower exponents (1≤n<2) in the dependence of Felec on E0 were observed experimentally at a high applied electric field. [START_REF] See | Relationship between electric current and matrix modulus in electrorheological elastomers[END_REF][START_REF] Liu | Electrorheology of filled silicone[END_REF] This deviation from the quadratic law was interpreted theoretically by considering, in addition to the solid particle conductivity, the dependence of the matrix conductivity on the electric field which gives rise to a saturation of the local electric field. 8,-16 The possibility to vary the electromechanical characteristics of EREs and their structural stability make them attractive for developing new smart materials with potential application in the automotive industry, robotics and prosthetic limbs, vibration isolators, building applications and electro-active actuators. [START_REF] Shiga | Electroviscoelastic effect of polymer blends consisting of silicone elastomer and semiconducting polymer particles[END_REF][START_REF] Shiga | Electroviscoelastic effect of polymeric composites consisting of polyelectrolyte particles and polymer gel[END_REF][START_REF] Zhu | Experimental research about the application of ER Elastomer in the shock absorber[END_REF][START_REF] Cao | Tunable stiffness of electrorheological elastomers by designing mesostructures[END_REF][START_REF] Li | Research and applications of MR Elastomers[END_REF][START_REF] Byung | Modulus-switching viscoelasticity of electrorheological networks[END_REF][START_REF] Kakinuma | Application of the electrorheological gel to fixture devices for micro milling process[END_REF][START_REF] Martin | Electrostriction in field-structured composites: Basis for a fast artificial muscle?[END_REF][START_REF] Wei | Vibration characteristics of electrorheological elastomer sandwich beams[END_REF] For this purpose, ERE composites containing different inclusions and dispersed in different matrices, PANI/PDMS, [START_REF] Byung | Modulus-switching viscoelasticity of electrorheological networks[END_REF] PbTiO3/AR7, [START_REF] Tangboriboon | Electrorheology and characterization of acrylic rubber and lead titanate composite materials[END_REF] Starch/Silicone oil/Silicone rubber, [START_REF] Hao | A novel electric elastomer based on starch/transformer oil drop/silicone rubber hybrid[END_REF][START_REF] Hao | mechanical behaviour of starch/silicone oil/silicone rubber hybrid electric elastomer[END_REF] polythiophene/polyisoprene, [START_REF] Puvanatvattana | Electric field induced stress moduli in polythiophene/polyisoprene elastomer blends[END_REF] Pb(Zr0.5,Ti0.5)O3/Acrylic rubber, [START_REF] Tangboriboon | Electrorheological properties of novel piezoelectric lead zirconate titanate Pb(Zr0.5,Ti0.5)O3-acrylic rubber composites[END_REF] polydiphenylamine/poly(styrene-block-isoprene-block-styrene, [START_REF] Thongsak | Electroactive polydiphenylamine/poly(styrene-block-isoprene-blockstyrene) (SIS) blends: Effects of particle concentration and electric field[END_REF] poly(p-phenylene)/Acrylic elastomer, [START_REF] Kunanuruksapong | Poly(p-phenylene) and acrylic elastomer blends for electroactive application[END_REF] or cellulose/BMIMCl gel, [START_REF] Kunchornsup | Physically cross-linked cellulosic gel via 1-butyl-3methelimidazolium chloride ionic liquid and its electromechanical responses[END_REF] have been investigated. These systems exhibited an increase of their electromechanical properties, and, in particular, of their storage modulus sensitivity {(G'(E)-G'(0))/G'(0)}. However their relative modulus change {G'(E)-G'(0)} was still low and rarely exceeded 100 kPa (compared for instance to 1 MPa for the magnetorheological elastomers (MREs)). Furthermore the matrices used for these EREs were too soft (zero field modulus around 100 kPa: cf. table 2)

for most of the applications. Other type of ER materials based on a particular morphology of semiconducting particles emerged and showed an interesting ER effect when used as electrorheological fluids (ERFs). [START_REF] Lee | Enhanced electrorheological performance of a graphene oxide-wrapped silica rod with a high aspect ratio[END_REF][START_REF] Kim | Pickering emulsion-fabricated polystyrene-graphene oxide microspheres and their electrorheology[END_REF][START_REF] Wu | Preeparation of rod-like calcium titanyl oxalate with enhanced electrorheological activity and their morphological effect[END_REF][START_REF] Dong | Preparation and enhanced electro-responsive characteristic of graphene/layered double-hydroxide composite dielectric nanoplates[END_REF] It has been reported that the use of mesoporous Cedoped TiO2 enhanced the ER response of Ce-doped TiO2/silicone oil ER fluid and its yield stress was 20 times higher than that of pure dense TiO2/PDMS fluid. [START_REF] Jian | Preparation and Electrorheological Activity of Mesoporous Rare-Earth-Doped TiO2[END_REF][START_REF] Zhao | Preparation and Electrorheological Characteristics of Rare-Earth-Doped TiO2 Suspensions[END_REF] Other workers reported that ERFs based on nano-sized particles showed a so called giant electrorheological effect (GER) when the nanoparticles were coated by molecules having high dipole moments. [START_REF] Wen | The giant electrorheological effect in suspensions of nanoparticles[END_REF][START_REF] Ying | Electrorheological fluids: smart soft matter and characteristics[END_REF] In this work, we investigated the response of an electrorheological elastomer (ERE) based on doped titanium dioxide. The specificity of this material lies in three main points: (i) the modification of the intrinsic properties of pure TiO2 by adding an aluminium cation (Al 3+ ) as substitution impurities of (Ti 4+ ) in the TiO2 lattices during their growth, (ii) the synthesis of TiO2 aggregated nanoparticle with a high specific surface area and (iii) the adsorption of Acetylacetone dipolar molecules (Acac) as a doping agent on the TiO2 surface. The aim underlying these choices was the improvement of relative modulus change (ΔG') developed by conventional electrorheological elastomers. Indeed, by proceeding so, it was expected that the dielectric properties of the semiconducting particles were improved, the attraction force between particles (when subjected to an external electric field) was amplified, and therefore the electro-rheological response was enhanced. In a previous study we have shown that by doping TiO2 nanoparticles with Acac we could obtain a good ER fluid with a large increase of shear modulus. [START_REF] Kossi | Acetylacetone stimulus effect on electrorheological properties of TiO2 aggregated nanoparticles[END_REF] The aim of this work was to obtain such a high field induced modulation, not in the fluid phase but in an elastomeric matrix having a zero field modulus high enough, like PDMS, to be used for active damping applications. This result is not at all granted because the cross-linking pre-polymers are about one hundred times more viscous than the silicone oil used for ER fluid and also because the chain like structure induced by the field can be partly destroyed by the cross-linking process. Here we investigated the electrorheological response of 20 v. % Acac-doped TiO2 semiconducting particles that were embedded and aligned in PDMS cross-linkable matrix as an ERE. In the first section we present the process used for synthesizing the TiO2 particles, their morphological characterization, then the preparation of the TiO2-Acac/PDMS ERE and the experimental conditions for measuring the shear moduli (elastic modulus G' and loss modulus G"). In the second section we first study the kinetics of polymerization by following the viscoelastic properties of samples under an oscillatory shear with and without field during curing. Then we present the electrorheological response of the cross-linked material in terms of the shear modulus increase and of induced current versus electric field amplitude and frequency; here the obtained results were discussed qualitatively in relation to the non linear model of conductivity. Finally we demonstrate the variation in the viscoelastic properties of the TiO2-Acac/PDMS ERE versus the magnitude of the shear strain, for an AC and DC applied electric field.

Experimental section

2-a Particles synthesis

Nano-sized TiO2 powder was synthesized via a sol-gel method, [START_REF] Kossi | Acetylacetone stimulus effect on electrorheological properties of TiO2 aggregated nanoparticles[END_REF] using titanium tetraisopropoxide (Sigma-Aldrich, 98 %, solution), 2-Propanol (Sigma-Aldrich, ACS reagent, ≥99.5 %), aluminium isopropoxide (Aldrich, 99.99 %), ammonium hydroxide (Sigma-Aldrich, 28 % NH3 in H2O), acetylacetone (Sigma-Aldrich, ReagentPlus®, ≥99 %), nitric acid (Sigma-Aldrich, ACS reagent, 70 %), acetic acid (Sigma-Aldrich, ACS reagent, ≥99.7 %) and ultrapure water (≥18 M.cm) as starting materials. In a typical procedure, titanium (IV) tetraisopropoxide (0.388 mol), was rapidly added to ultrapure water (640 ml) and then stirred for 30 min. A white precipitate formed immediately upon addition of the titanium (IV) isopropoxide. The resultant colloid was recovered by centrifugation (10000 g for 30 min). The centrifugation cake was added into a jacketed three-necked flask reactor equipped with a mechanical stirring containing 750 ml of an aqueous solution, of molar composition ; 0.36 M nitric acid, 1 M acetic acid and 0.05 M aluminium isopropoxide. Al 3+ was used here in small quantity as impurity, in order to create ionic substitution defects in the TiO2 lattice during growth and to improve the dielectric properties (permittivity and conductivity) of the synthesized TiO2 (pure TiO2 had weak electrorheological response). [START_REF] Jian | Preparation and Electrorheological Activity of Mesoporous Rare-Earth-Doped TiO2[END_REF][START_REF] Tang | Giant electrorheological effects of aluminium-doped TiO2 nanoparticles[END_REF] The pH of the colloidal solution after addition of the cake was measured to be between 1 and 2. Peptization occurred after heating the product at 80 °C for 1 h under medium stirring, whereupon the slurry became a stable sol. Then the sol was cooled down to room temperature and a 17 M ammonium hydroxide solution was added drop by drop into the TiO2 sol under low stirring in order to form a white gel. The pH of the resultant colloidal suspension was measured to be close to 7. The gel was then collected by centrifugation (10000 g for 30 min) and dried at 110 °C overnight. The powder was crushed and ground into fine powder using a mortar and pestle and was further calcined in air at 500 °C for 4 h.

2-b Particles characterization

Morphological characterization of the obtained TiO2 was shown in Figure 1. TEM pictures performed on the obtained TiO2 particles in aqueous medium at pH≈ 7 (Figure 1 (a) and (a')) showed spherical particles with a diameter between 12 and 20 nm that were aggregated between them in a form of large clusters having a weak density. After heat treating TiO2 powder at 500 °C for 4h in air, the SEM picture (Figure 1 (b)) showed relatively dense aggregates formed from the TiO2 nanoparticles. The density of the calcined TiO2 was measured using a 5 cm 3 standard flask pycnometer and was found to be about 3.84 g cm -3 . The type of porosity and specific surface area of the calcined TiO2 powder were (20 v. %) giving, respectively, the scattered intensity Iq (cm -1 ) versus wave vector q (A°-1 ) and the corresponding Porod representation q 4 Iq (cm -5 ) as a function of wave vector q (A°-1 ).

determined by a N2 adsorption/desorption technique. Figure 1 (c) shows a hysteresis in the N2 adsorption/desorption isotherm for relative pressures 0.55 < P/P0 < 0.95 which is a typical characteristic of mesoporous or nanoporous aggregates. The BET plot of N2 adsorption/desorption (Figure 1 (d)) showed that the value of the specific surface area of TiO2 powder (Abet) was about 295 m 2 g -1 . Small angle neutron scattering (SANS) was performed on calcined TiO2 particles that were dispersed in PDMS oil (20 v. % TiO2/PDMS).

The SANS spectra giving the scattered intensity Iq (cm -1 ) versus wave vector q (A°-1 ) (Figure

1 (e))
showed a relative maximum of the scattered intensity for a value of wave vector q about 0.356 nm -1 giving an average diameter of TiO2 nanoparticles close to 18 nm which was in agreement with that estimated from TEM observations. The Porod plot of the scattered intensity q 4 Iq (cm -5 ) as a function of the scattering wave vector q (A°-1 ) (Figure 1 (f)) showed two Porod's regimes: the first at low wave vector which characterizes the interface of the envelope of micrometric grains and the second at high q values which was due to the scattering inside the nanoparticles of 18 nm of diameter. This behaviour is characteristic of porous systems formed from nanoparticles assembling where two length scales were largely separated; nanoparticles on one hand and micrometric grains on the other. A in a mortar until obtaining a homogeneous dispersion, then the mixture was degassed under vacuum for 30 min using a vacuum pump for removing any air bubbles that may be traped in the mixture. At this stage no crosslinking reaction was started. The Rhodorsil RTV141 B component was then added to the mixture. After homogenizing the whole by mechanical stirring, the dispersion was degassed again for 5 min and then rapidly poured onto the rheometer plateau to monitor the crosslinking reaction (controlled by temperature and time)

2-c Preparation of the TiO2 ERE

with or without an applied electric field under shear.

2-d Electrorheological measurements

The viscoelastic properties were measured by using an Anton Paar Physica MCR301

Results and discussion

3-a Cross-linking kinetics with and without an applied electric field:

The polymerization kinetics of the TiO2-Acac particles-filled PDMS (20 v. % TiO2-Acac filler in reacting PDMS) was monitored by measuring the storage and loss moduli during cross-linking reaction at constant strain magnitude γ=0.1 % and ω=100 rad/s. the crosslinking reaction is carried out at a constant temperature of 80 °C. The experiments were carried out under three different conditions. In one, pure PDMS pre-polymers were cross-linked without However, before reaching the equilibrium, the increase on G' and G" under the applied electric field was less pronounced than that without pre-alignment or in other words we did not have the addition of the part of modulus induced by the electric field (at t = 0) and the one given by the polymerization of the PDMS network. This behaviour can be easily seen in In subsequent experiments, the viscoelastic properties were measured at a fixed temperature of 20 °C, for different values of DC electric field strength, electric field frequency and oscillatory shear strain magnitude. The dielectric properties ( conductivity and relative permittivity) of both TiO2 particles and crosslinked matrix were given at the same temperature of 20 °C and are shown in Table 1. enhancement G'elec at a small strain magnitude became proportional to Edc. [START_REF] Gonon | Particle-particle interactions in electrorheological fluids based on surface conducting particles[END_REF][START_REF] Wu | A modified conduction model for the electrorheological effect[END_REF][START_REF] Foulc | Macroscopic model of interaction between particles in an electrorheological fluid[END_REF] In our case, from the experimental results, we found that ΔG'∝ 𝐸 𝑎 1.10 (Figure 5 4) for conductivity ratios Γσ =1.4 10 6 and Γσ = 10 9 respectively, dashdotted curve corresponds to ΔG'∝ E dc 1.1 , (d) variation of the current as a function of the applied electric field; diamond: experimental values, red line: calculated current using equation (3) for Γσ =1.4 10 6 (inset the variation of the current as a function of application time of a constant electric field for the different applied Edc).

3-b

(c)). A similar exponent

of attraction force dependence on the electric field was predicted by Gonon et al. (1999) at a high applied field where the saturation regime was reached and they found that Felec∝ 𝐸 𝑑𝑐 1.12 . 11

However Figure 5 (a) shows a difference in the measured ΔG' between its value just after the application of the field at t=t0 (that we call ΔG'min) and after t=270 s of electric field induction (that we call ΔG'max); in particular when the DC applied electric field strength became strong.

This evolution during the time of application of the field could not be attributed to the current variation because, as shown in the inset (d) of Figure 5, the current was practically unchanged during this time even at high applied electric field strength. The more likely explanation is the existence of a slow rearrangement of the particles which get closer from each other, then increasing the moduli. Theoretical estimation of the variation of the current and the attraction force as a function of the applied electric field has been given in the literature using the conduction model. [START_REF] Foulc | Macroscopic model of interaction between particles in an electrorheological fluid[END_REF][START_REF] Atten | The role of conduction in electrorheological fluids: from interaction between particles to structuration of suspensions[END_REF] In this model the authors considered, for the case where the ratio of the conductivities Γσ=σP/σM >>1, that the electrical conduction in the interparticle gap can be separated into two zones for nearly touching particles (see Figure 6).

The first one, dominated by the conductivity σP of the particles; had a radius δ. The second one was the outer zone (x> δ) where authors considered that the sphere surfaces are equipotential and that, at a high applied electric field, the conductivity of the matrix became field-dependent. The radius δ was determined by equating Cp=CM where Cp and CM are respectively the conductance associated with these two zones: x< δ and x> δ. In this model the radius δ was practically proportional to 𝐸 𝑑𝑐 0.5 but decreased with the conductivity ratio as 𝛿 ∝ (𝑙𝑛𝛤 𝜎 ) -1 .

The attraction force between two nearest particles was expressed as:

𝐹 𝑒𝑙𝑒𝑐 = 4𝜋𝑎 2 𝜀 𝑀 𝐸 𝑑𝑐 2 ( 𝑎 𝛿 ⁄ ) 2 (1)
In equation ( 1), the attraction force increases by decreasing δ and therefore by increasing

Γσ.

Using the conduction model, the dependence between the applied electric field Edc and the current IP passing through a single chain of aligned particles was derived in the case of near touching particles and was expressed as: [START_REF] Atten | The role of conduction in electrorheological fluids: from interaction between particles to structuration of suspensions[END_REF]16

𝐼 𝑃 = 4𝜋𝑎𝜎 𝑃 (2𝑎𝐸 𝑑𝑐 ) 2 √ 𝜀 𝑀 𝐹 𝑒𝑙𝑒𝑐 ⁄ (2)
The total current I passing through all the chains was expressed as:

𝐼 = 𝑁 𝐶ℎ𝑎𝑖𝑛𝑠 𝐼 𝑃 = 3 2 𝜙 ( 𝑅 2𝑎 ) 2 𝐼 𝑃 ( 3
)
where ϕ is the particle volume fraction and R is the radius of the plate geometry used on the rheometer (R=12.5 10 -3 m).

Inserting ( 1) and ( 2 We assumed that σP= σeff/ Φv. The frequency dependence of σP as well as the conductivity of the suspension before and after polymerization is represented in Figure 7. We noticed in particular that, as expected, the conductivity was strongly decreased in the elastomer but that the frequency dependency is quite similar in the liquid phase and in the elastomer. The experimental conductivity ratio at a low frequency was then equal to Γσ =2.5 10 6 which is not so far from the one 1.4 10 6 deduced from the fit of the model, taking into account the uncertainty due to the approximation σP= σeff/ Φv. The relative change in the storage modulus can be obtained as follow: 13

∆𝐺 ′ = 3 2 𝜙 1 𝜋𝑎 2 𝐹 𝑒𝑙𝑒𝑐 (1+𝛾 2 ) (4)
The shear strain  appearing in equation ( 4) can be neglected in our case. Experimental results of ΔG' versus the DC applied electric field are compared with those calculated by inserting equation ( 1) into (4). Figure 5 (c) shows that both experimental and computed results had practically the same dependence on the electric field where ΔG'∝ 𝐸 𝑑𝑐 1.1 . However theoretical estimation of ΔG' is lower by about two orders of magnitude compared to our experimental results even if the conductivity ratio Γσ was increased to 10 9 (Figure 5(c)). This high value of ΔG' observed on nanoparticles coated with molecules of high dipole moment could come from the variation of the force at very short separation. In the approach leading to Equation (4) the change of force with the separation distance dF/ds is not considered since only the projection of the constant force near contact on the shear direction is taken into account. The short range force should also include the Van der Waals interaction and likely interactions between dipoles adsorbed on the surface of the particles. This is beyond the scope of this paper but certainly worth being explored. 2). For example, for a DC applied electric field of 2 kV/mm, oscillatory shear strain γ=0.1 % and ω=100 rad/s, TiO2-Acac/PDMS ERE response reached the ΔG' value up to 0.5 MPa. This high performance of TiO2-Acac/PDMS ERE was attributed to the particular morphology of the synthesized TiO2 particles (mesoporous aggregates with a high specific area of 295 m 2 /g formed from nanosized particles) and to the good affinity between TiO2 surface and acetylacetone allowing the adsorption of 1 molecule per nm 2 of Acac dipolar molecules on the TiO2 surface with practically no Acac remaining free in the insulating matrix. [START_REF] Kossi | Acetylacetone stimulus effect on electrorheological properties of TiO2 aggregated nanoparticles[END_REF] If we considered the relative change of modulus, defined as ΔG'/G'(0)(%)=100(G'(E)-G'(0))/G'(0), its value remained modest (33% as shown in Table 2 for 2 kV/mm, γ=0.1%) but it was just due to the high intrinsic modulus of the PDMS matrix obtained using the RTV141 reacting pre-polymers with a molar stoichiometry of groups, Si-H /Si-vinyl close to 1. The intrinsic elasticity of the PDMS matrix can be easily decreased by adding an adequate molecular agent (trimethyl (vinyl) silane for instance) to neutralize some of the Si-H groups and to reduce Si-H/Si-Vinyl molar stoichiometry. In this case the elastic modulus of the matrix will decrease whereas the effect of the field will remain constant, so the storage modulus sensitivity (ΔG'/G'(0)) can be strongly modulated depending on the needs of the applications. In our experiment, the relative change ΔG' of TiO2-Acac/PDMS at its fluid phase (before starting cross-linking) was about 0.70 MPa by the application of electric field strength of 2 kV/mm and after the end of the cross-linking reaction this value was decreased to 0.5 MPa at the same applied electric field.

As shown in

If the matrix is softer, then we expect that ΔG' of TiO2-Acac/PDMS in its elastomeric phase will increase to be closer to that of TiO2-Acac/PDMS in its phase.

3-c Electric field frequency dependency of the electrorheological response of TiO2-Acac/PDMS ERE:

Experiments were carried out to measure G' and G" of TiO2-Acac/PDMS ERE, under shear strain magnitude γ=0.1 % and ω=100 rad/s, depending on the electric field frequency. In The dependence of mechanical properties on electric field frequency can be determined by giving the variation of the attractive force as a function of frequency. In the case nearly touching particles and Γσ=σP/σM > Γε=εP/εM, the attraction force can be expressed as: [START_REF] Wu | Dielectric and conduction effects in ohmic electrorheological fluids[END_REF][START_REF] Wu | A modified conduction model for the electrorheological effect[END_REF] 𝐹 𝑒𝑙𝑒𝑐 = 12𝜋𝑎

Therefore, the relative change in storage modulus can be calculated by inserting the expression of Felec given in equation ( 5) into equation (4). Using these expressions, the authors found that the attractive force and then the elastic modulus had a higher value at DC and very low frequencies, but became frequency dependent and decreased when electric field frequency was increased beyond a critical value. They demonstrated that the parameters determining this critical frequency limit were the conductivity ratio Γσ and the dielectric permittivity ratio Γε. They also found that the calculated current was independent of the electric field frequency when the frequency was smaller than a critical value, but became a linear function of the frequency beyond the critical value. For Γσ= 10 6 >> Γε=5 the authors found that the critical value of the frequency was very low and was less than 6.7 10 -4 Hz. In In summary we found that the electrorheological behaviour of TiO2-Acac/PDMS ERE was governed by the conductivity mismatch; with, for the TiO2-Acac/PDMS ERE, a higher ER response by applying a DC electric field than an AC one.

3-d Strain dependency of the electrorheological response of the TiO2-Acac/PDMS ERE:

The linear viscoelastic regime of the TiO2-Acac/PDMS ERE was investigated by measuring ΔG' and ΔG" as a function of oscillatory shear strain magnitude (γ) at constante shear frequency ωγ=100 rad/s. We carried out measurements without an electric field, with a 2.5 kV/mm DC and 2 kV/mm AC electric field (nE=10 Hz). In Figure 9 we can see that with or without an applied electric field, the TiO2-Acac /PDMS ERE had a small linear regim for γ≤0.1 %, which was independent of DC or AC modes but was dependent on the electric field strength. This small linear regime could be attributed to the infinitesimal deformation of columns of aligned particles locked in the cross-linked matrix that influenced the viscoelastic properties. [START_REF] Byung | Modulus-switching viscoelasticity of electrorheological networks[END_REF] We noticed also (in Figure 9) that, increasing the electric field strength from 2 to 2.5 kV/mm, the linear regime was disappeared. This evolution of nonlinearity by increasing the electric field strength is likely related to the increase of conductivity: the size of the zone r<δ (cf fig.( 6)) extends with the electric field and makes the modulus sensitive to very small strain.

Conclusion

ER elastomers have been prepared by suspending TiO2-doped Acac particles in crosslinkable PDMS oligomers. The particles were aligned by an AC electric field and maintained in chain like structure by an in situ polymerization to form anisotropic network. The change of the shear modulus was monitored during the polymerization process and its decrease was less than 30% indicating that the field induced structure was only slightly disturbed by the polymerization process. Electrorheological properties of TiO2-Acac/PDMS ERE were investigated experimentally under a DC electric field, AC frequency and shear strain magnitude. The results showed that the relative change in elastic modulus (ΔG') increases with the DC applied electric field according to a power law ≈E 1.10 . this deviation from the quadratic law is due to the non linearity in the conductivity of the matrix that affects the field distribution and the attraction force between particles. However TiO2-Acac/PDMS ERE showed a decrease of ΔG' with the increasing AC electric field frequency. This viscoelastic behaviour, depending on the electric field frequency, was attributed to the high conductivity ratio Γσ=σP/σM compared to the permittivity ratio Γε=εP/εM, between TiO2 semiconducting particles and PDMS matrix. TiO2-Acac/PDMS ERE exhibited a high value of relative change in the elastic modulus ΔG' when compared with those of other ER Elastomers that have been investigated before; ΔG' reached 500 kPa under 2 kV/mm of a DC applied electric field at a strain γ=0.1 %. Whereas the conductivity behavior can be quite well predicted by the field dependence on the conductivity this was not at all the case for the increase of the shear modulus where the theory predicted a value which was too small by 2 orders of magnitude; a better consideration of short range forces is needed to improve the model. The high ER performance of TiO2-doped Acac as an ERE makes it a promising candidate for applications by designing an elastomeric blend with controllable matrix hardness.

Supporting information. Electronic supplementary information (ESI) available: The experimental setup for morphological characterization (SEM, TEM, N2 adsorption/desorption isotherm, SANS and FT-IR) and dielectric characterization by impedance spectroscopy. See DOI: 10.1039/c4tc02535k on www.rsc.org/MaterialsC website.

Figure 1 :

 1 Figure 1: Morphological characterization of synthesized TiO2 particles: (a) TEM picture of the dilute TiO2 suspension as obtained at pH 7; inset (a') TEM picture of the same TiO2 with a magnified scale, (b) SEM picture of TiO2 after heat-treatment at 500 °C, (c) and (d) are the N2 adsorption/desorption isotherm and the BET plot of N2 adsorption performed on the calcined TiO2 powder, respectively, (e) and (f) the SANS spectra of TiO2 powder (calcined at 500°C/4h) dispersed in PDMS silicone oil(20 v. %) giving, respectively, the scattered intensity Iq (cm -1 ) versus wave vector q (A°-1 ) and the corresponding Porod representation q 4 Iq (cm -5 ) as a function of wave vector q (A°-1 ).

Figure 2 :

 2 Figure 2: The chemical structure of PDMS crosslinking pre-polymers.

  rheometer which was equipped with an electrorheological temperature device (Anton Paar) consisting of a parallel plate geometry of 25 mm diameter (PP25/E) and a Peltier temperature device (P-PTD200/E) allowing rheological measurements under applied electric field with thermally controlled environment. A regulated DC power supply (XANTREX-Model HPD 30-10SX) and AC voltage (Hameg function generator HM8030-5) combined with a high voltage amplifier (Trek Model 609E-5) were used to apply DC or AC electric field in the range of 0 to 4 kV/mm across a 1 mm gap between plate geometries. Multimeter (TENMA 72-7735) was used to measure the DC or AC electric current passing through the gap during crosslinking under applied electric field and rheological monitoring.

Figure 3 :

 3 Figure 3: TiO2 ERE cross-linking kinetics, (a) G' of pure PDMS and G' of 20 v. % TiO2/PDMS without pre-alignement (E=0kV/mm) and with pre-alignement at E=1.5 kV/mm AC electric field strength (νac=10 Hz), (b) G" of pure PDMS and G" of 20 v. % TiO2/PDMS with and without pre-alignement, (c) ΔG'=G'E=1.5kV/mm-G'E=0 and ΔG"= G"E=1.5kV/mm-G"E=0 during crosslinking of 20 v.% TiO2/PDMS, (d) variation of G' and G" of the TiO2-Acac/PDMS ERE after the end of cross-linking (at E= 0 kV/mm) versus time as the sample was cooled down (linear ramp of temperature from 80 to 20°C) and maintained at constante T= 20°C. Shear moduli are measured at strain amplitude γ= 0.1 % and ω=100 rad/s and for fig. (a), (b) and (c) the temperature was fixed at T=80°C.

Figure 3 (

 3 Figure 3 (c) where the relative change during time, ΔG'=G'E=1.5kV/mm-G'E=0 and ΔG"=

Figure 4 :

 4 Figure 4: Evolution of the current during cross-linking at constant T=80 °C under rheological monitoring of the TiO2-Acac/PDMS ER with pre-alignment under an applied AC amplitude E=1.5 kV/mm and νac=10 Hz.

  The DC Electric field-dependency of electrorheological response of the TiO2-Acac /PDMS ERE: Shear moduli (G' and G") were measured under shear strain magnitude, γ=0.1 % and ω=100 rad/s, as a function of the applied DC electric field strength. The experiments consisted of applying suddenly a DC electric field and maintaining it during 270 s before turning off the field; the change of the moduli was recorded during this field pulse. The effect of the field on moduli under various DC electric field strengths was reported in Figure5. We observed in parts (a) and (b) an instantaneous increase of G' and G" when the DC electric field was applied. As shown in Figure5 (c), where the dash-dotted curve corresponds to a linear behaviour, the relative change in elastic modulus [ΔG'=G'(E)-G'(0)] was nearly proportional to the applied DC electric field Edc and not to the quadratic law (Edc) 2 as it should be the case for linear electrostatics. J. N.Foulc et al. (1994) attributed this deviation from linear electrostatics to the fact that, under DC voltage, the conductivity of the matrix became field-dependent and was dramatically increased when a certain local field strength Em was exceeded; then the attraction force Felec between particles and the storage modulus

Figure 5 :

 5 Figure 5: the DC electric field dependence of the storage modulus and loss modulus by applying the switching electric fields of different values; (a) variation of G' (b) variation of G" (c) the relative change in elastic modulus: diamond: experimental values, blue and red lines: calculated ΔG' using eq. (4) for conductivity ratios Γσ =1.4 10 6 and Γσ = 10 9 respectively, dashdotted curve corresponds to ΔG'∝ E dc 1.1 , (d) variation of the current as a function of the applied electric field; diamond: experimental values, red line: calculated current using equation (3) for Γσ =1.4 10 6 (inset the variation of the current as a function of application time of a constant electric field for the different applied Edc).

Figure 6 :

 6 Figure 6: Schematic view showing the electrical conduction zones and notations used in the conduction model.

Figure 7 :

 7 Figure 7: Experimental measurement of the conductivity; circle: σP= σpellet/ Φv (Φv=0.44),

  ) into equation(3) we found that 𝐼 ∝ .[START_REF] Atten | The role of conduction in electrorheological fluids: from interaction between particles to structuration of suspensions[END_REF]16 Figure5 (d)shows a comparison between the experimental and calculated results using equation (3) for the current passing through the sample. The best fit was obtained for a conductivity ratio Γσ= 1.4 10 6 giving σP=1.4 10 -7 S.m -1 . The conductivity of TiO2/Acac was deduced from the measurement of the conductivity σeff of a pellet of TiO2 having a volume fraction Φv=0.44.

Figure 5 (

 5 a) or 5 (c), the relative change in storage modulus (ΔG'=G'(E)-G'(0)) of TiO2-Acac/PDMS ERE exhibited high values when compared with those of other ER Elastomers that have been investigated before (Table

Figure 8

 8 Figure8we show the increase of modulus obtained by applying an electric field strength E=2

Figure 8 :

 8 Figure 8: the applied electric field frequency dependence of the storage modulus and loss modulus after switching on the electric field; (a) the time dependance of G' for different electric field frequencies, (b) the time dependance of G" for different electric field (c) the relative change in the elastic modulus versus the applied electric field frequency (triangles: experimental results; solid line: theory with s=2.5 10 6 ; dashed theoretical results multiplied by 10 2 , (d) the corresponding electric field current versus electric field frequency; inset: the variation of the current during field application for the different electric field frequencies.

8 (

 8 our case Γσ~2.5 10 6 >> Γε~ 44, we observed the decay of G' and ΔG' (Figure8(a) and (c)) by increasing the frequency and a linear behaviour of the current I versus frequency (Figured)). We noticed that the decrease of ΔG' versus frequency around Hz was well reproduced by the model as shown in Figure8 (c), but, as in the DC case, it underestimated ( in the tested electric field frequency range) our experimental results by two orders of Because of the limit of our instrument sensitivity at very low frequencies and because of the high conduction of the sample at high frequencies we were unable to produce much more experimental measurement points to investigate the transition regimes that were observed theoretically and therefore testing the validity of the model for simulating our experimental measurements over a wide range of frequencies.

Figure 9 :

 9 Figure 9: Strain dependence of ΔG': (closed diamond) Edc=2.5 kV/mm, (closed circle) Eac=2 kV/mm (n=10 Hz), and ΔG": (stars) Edc=2.5 kV/mm, (plus symbols) Eac=2 kV/mm (n=10 Hz).

Table 1 :

 1 dielectric properties measured at 20 °C

		PDMS matrix	TiO2 Particles
	εr	2.7 *	120
	σ (S.m -1 )	1 10 -13 *	2.50 10 -7

* Data from provider.

Table 2 :

 2 Relevant recently published studies on the viscoelastic properties of the ERE under a DC applied electric field.

	ERE sample	Edc	ϒ(%); ω(rad/s) G'(0)(KPa) ΔG' (kPa)	Γσ	Γε	T°(K)	Ref.
	TiO2-Acac/PDMS 50 wt.%	2 kV/mm	0.1; 100	~1500 ~500 ~2.50 10 6	~44	293 This work
	PANI/PDMS 30 wt.%	2 kV/mm	0.1; 100	~30	~110	-	-	-	22
	PbTiO3/AR71 21.30 wt. %	2 kV/mm	0.1; 100	~45	75-85 ~4.39 10 5 * ~3.80 10 3 * 300	26
	Starch/Transformer oil/ silicone	0.6 kV/mm 0.05; 6.280	~20	81.7	-	-	293	27
	rubber (5/47.5/47.5 wt. %)
	Starch/silicone oil/silicone	0.6 kV/mm 0.05; 6.280 ~37.21 90.8	-	-	293	28
	rubber (10/45/45 wt. %)
	Polythiophene/polyisoprene 20	2 kV/mm	1; 1	~19.45 ~21.40	~3.67	-	300	29
	wt. %
	Pb(Zr0.5,Ti0.5)O3/acrylic rubber	2 kV/mm	0.1; 1	~ 18.80 ~11.02 ~1.92 10 5 * ~1.6510 3 * 300 30
	Polydiphenylamine/poly(styrene	2 kV/mm	0.1; 1	~ 88.77 ~123.32 ~2.26 10 13	-	300	31
	-block-isoprene-block-styrene)
	20 v. %
	Poly(p-phenylene)/ acrylic	2 kV/mm	0.1; 1	110.99 ~107.80 ~2.25 10 6	-	300	32
	elastomer 30 v. %
	Cellulose/BMIMCl 13 wt. % gel 1 kV/mm	0.1; 1	172.22	248	-	-	303	33
	* The conductivity and relative permittivity ratios are measured at 1 kHz.
	-Data are not provided by the authors.

  | 2 and |𝑅𝑒(𝛽 * )| depend on the dielectric properties of the particles and the matrix and vary with frequency as:

		2 𝜀 𝑀 |𝛽 * | 2 exp (2|𝑅𝑒(𝛽 * )| 32 )𝐸 0 2	(5)
	|𝛽 * | 2 =	(𝜎 𝑝 -𝜎 𝑀 ) 2 2 +(𝜀 𝑟 𝑃 -𝜀 𝑟 𝑀 ) 2 𝜔 2 𝜀 0 (𝜎 𝑝 +2𝜎 𝑀 ) 2 𝜔 2 𝜀 0 2 +(𝜀 𝑟 𝑃 +2𝜀 𝑟 𝑀 ) 2	(6)
	and |𝑅𝑒(𝛽 * )| =	𝜔 2 𝜀 0 2 (𝜀 𝑟 𝑃 -𝜀 𝑟 𝑀 )(𝜀 𝑟 𝑃 +2𝜀 𝑟 𝑀 )+(𝜎 𝑝 -𝜎 𝑀 )(𝜎 𝑝 +2𝜎 𝑀 ) 𝜔 2 𝜀 0 2 (𝜀 𝑟 𝑃 +2𝜀 𝑟 𝑀 ) 2 +(𝜎 𝑝 +2𝜎 𝑀 ) 2

where |𝛽 *

Acknowledgement. One of the authors gratefully acknowledges the financial support from the national center of scientific reshearch (CNRS). We thank Dr Alice Mija for FT-IR measurments.