Égory Gr

Berhuy

COHOMOLOGICAL INVARIANTS OF CENTRAL SIMPLE ALGEBRAS OF DEGREE 4

In this paper, we prove a result of Rost, which describes the cohomological invariants of central simple algebras of degree 4 with values in H * ( -, µ 2 ) when the base field contains a square root of -1.

Introduction

In [START_REF] Rost | La forme trace d'une algbre simple centrale de degré 4[END_REF], Rost, Serre and Tignol defined a cohomological invariant of central simple algebras of degree 4 with values in H 4 ( -, µ 2 ) when the base field contains a square root of -1. On the other hand, taking the Brauer class of the tensor square of a central simple algebra of degree 4 yields a cohomological invariant with values in H 2 ( -, µ 2 ). In this paper, we prove a result of Rost (unpublished), which asserts that these two invariants are essentially the only ones (see Section 4 for a more precise statement). This paper is organized as follows. After proving somme preliminary results in Section 1, we determine the cohomological invariants of multicyclic algebras in Section 2. Section 3 is devoted to the construction of a generic central simple algebra of degree 4. Finally, in Section 4, we prove that a cohomological invariant which vanishes on cyclic algebras and biquaternion algebras is identically zero (which another result due to Rost). As a corollary, we obtain a complete description of cohomological invariants of central simple algebras of degree 4.

The proofs of all the results of this paper rely heavily on the use of valuations and residue maps. We let the reader refer to [START_REF] Garibaldi | Cohomological invariants in Galois cohomology[END_REF] for the basic definitions and results on these topics.

We are grateful to Alexander Merkurjev for providing us his private lecture notes on Rost's theorem.

Preliminaries

Let F be a field of characteristic different from 2. We will denote by Fields F and Alg F the category of fields extensions of F and the category of commutative F -algebras respectively.

The category of sets will be denoted by Sets, and by Rings the category of of commutative rings.

For any field extension K/F , we will denote by H * (K) the cohomology ring of K with coefficients in µ 2 . We then get a functor H * : Fields F -→ Rings.

If K/F is a field extension, then H * (K) carries a natural structure of a H * (F )module, given by the external law

H * (F ) × H * (K) -→ H * (K) (a, ξ) -→ a•ξ = Res K/F (a) ∪ ξ.
We start with some results on the cohomology of rational extensions.

Let n ≥ 1 be an integer, let t 1 , . . . , t n be algebraically independent indeterminates over k, and let K n = k(t 1 , . . . , t n ). Let us denote by P n the set of subsets of {1, . . . , n}. If I = {i 1 , . . . , i r } ∈ P n , we set

(t I ) = (t i1 ) ∪ • • • ∪ (t ir ).
Notice that this definition does not depend on the numbering of the elements of I, since the cup-product in H * (K) is commutative.

If I = ∅, then (t I ) = 1 (and thus a•(t I ) = Res K/k (a) in this case).

Lemma 1.1. The cohomology classes (t I ), I ∈ P n are linearly independent over H * (F ).

Proof. We proceed by induction on n. Assume that n = 1, and let a 0 , a 1 ∈ H * (F ) such that a 0 •1+a 1 •(t 1 ) = 0. Taking residues with respect to the valuation υ t1 shows that a 1 = 0. Then multiplying by (-t 1 ) and taking residues shows that a 0 = 0.

Assume now that the result is proved for n ≥ 1, and let us prove it for n + 1 indeterminates. Let a I ∈ H * (F ), I ∈ P n+1 such that

I a I •(t I ) = 0. Then we have J a J •(t J ) + J a J∪{n+1} •(t J ) ∪ (t n+1 ) = 0,
where J describes P n . Reasoning as in the case n = 1, we get

J a J •(t J ) = J a J∪{n+1} •(t J ) = 0,
and by induction we have a J = a J∪{n+1} = 0 for all J ∈ P n , that is a I = 0 for all I ∈ P n+1 . This concludes the proof.

Proposition 1.2. Let ξ ∈ H * (K n ) be a cohomology class which is unramified at every discrete k-valuation υ = υ ti , i = 1, . . . , n. Then there exist unique cohomology classes

a I ∈ H * (F ) such that ξ = I a I •(t I ).
Proof. We prove it by induction on n. Assume first that n = 1, and let a 1 ∈ H * (F ) be the residue of ξ with respect to the t 1 -adic valuation. Then ξ -Res K/F (a 1 )∪(t 1 ) is unramified at every discrete F -valuation υ = υ t1 by assumption on ξ and because a 1 is constant. By choice of a 1 , it is also unramified at υ t1 , so ξ -Res K1/F (a 1 ) ∪ (t 1 ) is a constant class, so we may finally write

ξ = Res K1/F (a 0 ) + Res K1/F (a 1 ) ∪ (t 1 ) = a 0 •1 + a 1 •(t 1 ).
Now assume that the result is proved for n ≥ 1, and let ξ ∈ H * (K n+1 ) be a cohomology class which is unramified at every discrete F -valuation υ = υ ti , i = 1, . . . , n+1. In particular, it is unramified at every discrete K n -valuation υ = υ tn+1 .

By the case n = 1, there exist b

0 , b 1 ∈ H * (K n ) such that ξ = b 0 •1 + b 1 •(t n+1 ).
Let υ be a discrete F -valuation on K n , υ = υ ti , i = 1, . . . , n, and let w the Fvaluation on K n+1 extending υ such that w(t n+1 ) = 0. Since ξ is unramified at υ, taking residues shows that Res κ(w)/κ(υ) (r υ (b 1 )) = 0. Now κ(w) = κ(υ)(t n+1 ), so Res κ(w)/κ(υ) is injective and r υ (b 1 ) = 0. Moreover, ξ ∪ (-t n+1 ) is also unramified at υ, and reasoning as before shows that r υ (b 0 ) = 0. Hence b 0 and b 1 are unramified at any discrete F -valuation υ on K n , υ = υ ti , i = 1, . . . , n. Now use the induction hypothesis to conclude that ξ =

I a I •(t I ) for some a I ∈ H * (F ).
The uniqueness of the classes a I comes from the previous lemma. This completes the proof.

Definition 1.3. Let F : Fields F -→ Sets be a covariant functor. A cohomological invariant of F over F is a natural transformation F -→ H * of functors Fields F -→ Sets.
Cohomological invariants of F form an H * (F )-module, that we will denote by Inv(F, H * ).

We will need in the sequel the notion of a classifying pair for a functor F.

Definition 1.4. Let F : Alg F -→ Sets be a covariant functor. Let R be an F -algebra, and let a ∈ F(R). We say that the pair (R, a) is classifying for F, if the following conditions hold:

(1)The ring R is a noetherian domain;

(2)For every field extension L/F , and every a ∈ F(L), there exists a maximal ideal m of R and a morphism of F -algebras f : R -→ L such that ker(f ) ⊃ m and F(f ) maps a onto a .

The Specialization Theorem (cf. [2, Theorem 12.2]) then immediately gives:

Lemma 1.5. Let F : Alg F -→ Sets be a subfunctor of H 1 ( -, G), where G is an algebraic group over F . Assume that (R, a) is classifying for F, and let α, β ∈ Inv(F, H * ). Let K be the quotient field of R. If α(a K ) = β(a K ), then α = β.

Invariants of multicyclic algebras

For all integers r, n ≥ 1, we will denote by CSA n : Alg F -→ Sets the functor of isomorphism classes of Azumaya algebras algebras of rank n.

Let R be a commutative ring. Assume that n char(R) and that R contains a primitive n-th root of 1, that we will denote by ζ n . For all u, v ∈ R × , the R-algebra {u, v} n,R generated by two elements e and f subject to the relations

e n = u, f n = v, f e = ζ n ef
is an Azumaya R-algebra of rank n, called a symbol algebra.

We will denote by MS n,r the subfunctor of CSA n r of isomorphism classes of tensor products of r symbols algebras of degree n.

We will denote by 1 the constant cohomological invariant of CSA n (where CSA n is now viewed as a functor from Fields F to Sets). If n = 2m, then for every field extension K/F and every central simple K-algebra A of exponent dividing n, the class m[A] is killed by 2, and therefore defines a cohomology class of H 2 (K) via the usual isomorphism. This defines a cohomological invariant of CSA n , as well as a cohomological invariant of MS n,r for all r ≥ 1, that we will denote by f m in both cases. In particular, we have

f m ({a 1 , b 1 } n,L ⊗ F • • • ⊗ F {a r , b r } n,L ) = (a 1 ) ∪ (b 1 ) + . . . + (a r ) ∪ (b r )
for every field extension L/F and all a i , b i ∈ L × .

If A is a central simple F -algebra, the trace form of A is the quadratic form

q A : A -→ F a -→ Trd A (a 2 ).
Lemma 2.1. Assume that n char(F ) and that µ n ∈ F . Assume also that n = 2m.

For all u, v ∈ F × , the trace form of {u, v} n,F is Witt-equivalent to n, nu, nv, (-1) m nuv .

Proof. One may check that Trd A (e i f j ) = 0 if (i, j) = (0, 0). It easily follows that the subspaces

F •1, F •e m , F •f m , F •e m f m and F •e i f j ⊕ F •e n-i f m-j ,
where 0 ≤ i ≤ j ≤ m, (i, j) = (0, 0), (0, m), (m, 0), (m, m) are mutually orthogonal. Moreover, the 2m 2 -2 planes above are hyperbolic since e i f j is isotropic. The result then follows from the fact that the reduced traces of 1, e m , f m and e m f m are respectively n, nu, nv and (-1) m nuv. This concludes the proof.

Assume that n char(F ) and that µ n ∈ F . Moreover, assume that -1 ∈ F ×2 (this condition is automatically satisfied if n ≡ 0 [START_REF] Lewis | The Hasse invariant of the trace form of a central simple algebra[END_REF]).

If A = {u 1 , v 1 } n,F ⊗ F • • • ⊗ F {u r , v r } n,F
, it follows from the previous lemma that q A is Witt-equivalent to n r u 1 , v 1 , . . . , u r , v r . Therefore, q A ∈ I 2r F , and we have e 2r (q

A ) = (u 1 ) ∪ (v 1 ) ∪ • • • ∪ (u r ) ∪ (v r ).
This defines an element of Inv(MS n,r , H * ) that we denote by e 2r .

Proposition 2.2. Let F be a field of characteristic different from 2, and let n = 2m.

(1)Assume that n char(F ), and that

ζ n ∈ F, where ζ n is a primitive n th -root of 1. Then Inv(MS n,1 , H * ) is a free H * (F )-module with basis 1, f m . (2)Assume moreover that -1 ∈ F ×2 . Then for all r ≥ 2, Inv(MS n,r , H * ) is a free H * (F )-module with basis 1, f m , e 2r .
Proof. We first prove [START_REF] Albert | Structure of Algebras[END_REF]. Let α ∈ Inv(MS n,1 , H * ). Let t 1 , t 2 be two indeterminates over F , and let

K = F (t 1 , t 2 ). Set A = {t 1 , t 2 } n,K , and let υ = υ ti , i = 1, 2 be a discrete F -valuation. By assumption on υ, t 1 , t 2 ∈ O × υ ,

and we have

A {t 1 , t 2 } n,Oυ ⊗ Oυ K. It follows from [2, Theorem 11.7] that the class α K ({t 1 , t 2 } n,K ) is unramified at υ. By Proposition 1.2, there exist a 0 , a 1 , a 2 , a 3 ∈ H * (F ) such that α K ({t 1 , t 2 } n,K ) = a 0 •1 + a 1 •(t 1 ) + a 2 •(t 2 ) + a 3 •(t 1 ) ∪ (t 2 ).
Now let υ be the (t 1 -1)-adic valuation on F (t 2 ). Applying [START_REF] Garibaldi | Cohomological invariants in Galois cohomology[END_REF]Theorem 11.7] shows by specialization that

α K ({1, t 2 } n,F (t2) ) = a 0 •1 + a 2 •(t 2 ), that is α K (M n (F (t 2 ))) = a 0 •1 + a 2 •(t 2 ). Now M n (F (t 2 )
) is unramified at υ t2 , and therefore, so is α K (M n (F (t 2 ))). Taking residues then yields a 2 = 0. Similar arguments show that a 1 = 0, and thus

α K ({t 1 , t 2 } n,K ) = a 0 •1 + a 3 •(t 1 ) ∪ (t 2 ). Let R = F [t 1 , t -1 1 , t 2 , t -1 2 ]. It is clear that the pair (R, {t 1 , t 2 } n,R
) is classifying for MS n,1 . The previous equality shows that the invariants α and a 0 •1 + a 3 •f m coincide on a classifying pair, hence they are equal by Lemma 1.5.

Moreover, if a

0 •1 + a 3 •f m = 0, then applying this equality to {t 1 , t 2 } n,K shows that a 0 •1 + a 3 •(t 1 ) ∪ (t 2 )
. Lemma 1.1 then yields a 0 = a 3 = 0. This proves [START_REF] Albert | Structure of Algebras[END_REF].

Assume now that -1 ∈ F ×2 . The fact that 1, f m , e 2r are linearly independent over H * (F ) if r ≥ 2 easily comes from Lemma 1.1. Let us prove by induction on r that 1, f m and e 2r span Inv(MS n,r , H * ) for all r ≥ 1 and every base field F . For r = 1, this follows from [START_REF] Albert | Structure of Algebras[END_REF]. Assume that the result is proved for r ≥ 1, and let α ∈ Inv(MS n,r+1 , H * ). Let K/F be a field extension, and let A be a symbol algebra over K.

Any field extension L/K yields a field extension L/F . The maps

β L : B -→ MS n,r (L) H * (L) -→ α L (B ⊗ L A L )
fit together into a cohomological invariant of MS n,r . By induction, there exists

α i,K (A) ∈ H * (K) such that β = α 0,K (A)•1 + α 1,K (A)•f m + α 2,K (A)•e 2r , where α 2,K (A) = 0 if r = 1.
By linear independence of 1, f m , e 2r (resp. of 1, f m if r = 1), these classes are unique. It easily follows that α i,K (A) only depends on the isomorphism class of A. Moreover, it follows from the uniqueness of α i,K (A) and the fact that α is a cohomological invariant that the maps α i,K fit together into a cohomological invariant

α i ∈ Inv(MS n,1 , H * ).
By (1), we may write

α i = a i •1 + b i •f m , for some a i , b i ∈ H * (F ) (with a 2 = b 2 = 0 if r = 1). Taking K = F (u 1 , v 1 , . . . , u r+1 , v r+1 ), where u i , v i are indeterminates, B = {u 1 , v 1 } n,K ⊗ K • • • ⊗ K {u r , v r } n,K and A = {u r+1 , v r+1 } n,K , we get α K ({u 1 , v 1 } n,K ⊗ K • • • ⊗ K {u r+1 , v r+1 } n,K ) = a 0 •1 + b 0 •(u r+1 ) ∪ (v r+1 )+ (a 1 •1+b 1 •(u r+1 )∪(v r+1 ))∪ r i=1 (u i )∪(v i )+(a 2 •1+b 2 •(u r+1 )∪(v r+1 ))∪ r i=1 (u i )∪(v i ).
The F -automorphism of K which exchanges u 1 and u r+1 , v 1 and v r+1 , and leaves invariant the other indeterminates induces maps MS n,r+1 (K) -→ MS n,r+1 (K) and H * (K) -→ H * (K). Notice that the first one maps the isomorphism class of

{u 1 , v 1 } n,K ⊗ K • • • ⊗ K {u r+1 , v r+1 } n,K onto itself.
Since α commutes with induced maps, we then get

α K ({u 1 , v 1 } n,K ⊗ K • • • ⊗ K {u r+1 , v r+1 } n,K ) = a 0 •1 + b 0 •(u 1 ) ∪ (v 1 )+ (a 1 •1 + b 1 •(u 1 ) ∪ (v 1 )) ∪ r+1 i=2 (u i ) ∪ (v i ) + (a 2 •1 + b 2 •(u 1 ) ∪ (v 1 )) ∪ r+1 i=2 (u i ) ∪ (v i ).
Comparing with the previous equality and using Lemma 1.1, we get that b 0 = a 1 , and

b 1 = a 2 = 0. Therefore, {u 1 , v 1 } n,K ⊗ K • • • ⊗ K {u r+1 , v r+1 } n,K has same image by α and by a 0 •1 + a 1 •f m + b 2 •e 2(r+1) . Now let R = F [u i , u -1 i , v i , v -1 i , i = 1, . . . , r + 1]. The pair (R, {u 1 , v 1 } n,R ⊗ R • • • ⊗ R {u r+1 , v r+1 } n,R ) is classifying for MS n,r+1
. It follows from the previous considerations that the invariants α and

a 0 •1 + a 1 •f m + b 2 •e are equal.
This finishes the proof by induction.

Central simple algebras of degree 4

We start this section by giving a parametrization of central simple algebras of degree 4.

Let R be a commutative ring such that 2 ∈ R × . If a ∈ R × , we will denote by

R[ √ a] the étale quadratic R-algebra R[X]/(X 2 -a).
Let L be a biquadratic étale R-algebra, that is an R-algebra L generated by two elements α, β subject to the relations

α 2 = d, β 2 = d , αβ = βα,
for some d, d ∈ R × . Such an algebra will be denoted by

F [ √ d, √ d ].
The group G of automorphisms of the R-algebra

L = F [ √ d, √ d ] is isomorphic to Z/2Z × Z/2Z
, generated by the automorphisms σ and τ which are uniquely determined by the following relations:

σ(α) = α, σ(β) = -β, τ (α) = -α, τ (β) = β. Let a ∈ R[ √ d] × , b ∈ R[ √ d ] × , c ∈ R[ √ dd ] × ,

and set

n a = N R[ √ d]/R (a), n b = N R[ √ d ]/R (b), n c = N R[ √ dd ]/R (c). Notice that n a , n b , n c ∈ R × and that we have n a = aτ (a), n b = bσ(b), n c = cσ(c) = cτ (c).
Assume that n c = n a n b , and let 1, e σ , e τ , e στ be the canonical basis of the L-vector space M ap(G, L). We define a product law on Map(G, L) by imposing the relations By definition, the elements e ρ , ρ ∈ G are invertible with respect to this product law. Notice that the missing products e ρ e ρ , ρ, ρ ∈ G may be obtained using the relations above, so that the product law on Map(G, L) is completely determined. For example, we have e τ e στ = e -1 σ e σ e τ e στ = e -1 σ (e στ ) 2 = a -1 e σ c = a -1 σ(c)e σ .

One can check that we obtain an Azumaya R-algebra of rank 4, that we will denote by (a, b, c, L/R). Notice that if we set e = e σ and f = e τ , then we have (a, b, c, L/R) = L ⊕ Le ⊕ Lf ⊕ Lef , and

e 2 = a, f 2 = b, (ef ) 2 = c, eλ = σ(λ)e, f λ = τ (λ)f, for all λ ∈ L.
Notice for later use that we have f e = a -1 σ(cb -1 )ef , since

f e = e -1 (ef ) 2 f -1 = a -1 cb -1 f = a -1 σ(cb -1 )ef.
Lemma 3.1. Let F be a field of characteristic different from 2. Then every central simple F -algebra of degree 4 is isomorphic to some (a, b, c, L/F ). Moreover, one may assume that a, b, c / ∈ F × .

Proof.

Assume first that A is a division F -algebra. By a theorem of Albert (see [1, Theorem 11.9] for example), A contains a biquadratic étale F -algebra L. By Skolem-Noether's theorem, σ and τ extend to inner automorphisms Int(e) and Int(f ), for some e, f ∈ A × . This rewrites eλ = σ(λ)e, f λ = τ (λ)f, for all λ ∈ L.

= f 2 e = f (f e) = f a -1 σ(cb -1 )ef = τ (a -1 σ(cb -1 ))f (ef ) = cτ (a) -1 σ(b) -1 (f e)f = cτ (a) -1 σ(b) -1 a -1 σ(cb -1 )(ef )f = cτ (a) -1 σ(b) -1 a -1 σ(c)σ(b) -1 eb = cτ (a) -1 σ(b) -1 σ(c)a -1 σ(b) -1 σ(b)e = n c n -1 a σ(b) -1 e. Since e is invertible, we get b = n c n -1 a σ(b) -1 , that is n c = n a n b .
The proof of the linear independence of 1, e, f, ef over L is straightforward and left to the reader. Hence A contains a subalgebra isomorphic to (a, b, c, L/F ), and therefore A (a, b, c, L/F ) since they have same dimension over F . Assume now that A is not a division algebra. Then A M 2 (Q), where Q = (d, d ) is a (not necessarily division) quaternion F -algebra.

Let L = F [ √ d, √ d ],
and let us consider the F -algebra (d, 1, d, L/F ), where α 2 = d, β 2 = d and βα = αβ. In particular, we have

e 2 = d, f 2 = 1, (ef ) 2 = d, eα = αe, eβ = -βe, f α = -αf, f β = βf.
Notice that we also have f e = dσ(d -1 )ef = ef .

Set i = e, j = βf, i = f, j = αe. It is straightforward to check that the F -algebra B generated by i and j is isomorphic to Q, and that the F -algebra B generated by i and j is isomorphic to (1, d 2 ) M 2 (F ). Moreover one may check B is contained is the centralizer of B. Since B is a central simple F -algebra, we get

(d, 1, 1, L/F ) B ⊗ F B Q ⊗ F M 2 (F ) M 2 (Q) A.
We now prove the last part. Write A (a, b, c, L/F ), for some biquadratic étale

F -algebra L = F [ √ d, √ d ]. Replacing f by (1 + α + β)f , one may assume that b / ∈ F × . Indeed, if b ∈ F × , we then have [(1 + α + β)f ] 2 = (1 + α + β)(1 -α + β)f 2 = b(1 + d -d + 2β) / ∈ F × .
Write a = a 0 + a 1 α. Notice that, for all s ∈ F × , we have

[(s + α + β)e] 2 = (s 2 + d -d + 2sα)(a 0 + a 1 α) = (a 0 (s 2 + d -d)2sa 1 d) + (a 1 (s 2 + d -d ) + 2a 0 s)α, and
[(s + α + β)ef ] 2 = (s + α + β)(s -α -β)c = c(s 2 -d -d -2sαβ).
Thus, after replacing e by (s + α + β)e for a suitable s ∈ F × , one may also assume that a, c / ∈ F × . This concludes the proof.

Remark 3.2. For any biquadratic F -algebra L, we have M 4 (F ) (1, 1, 1, L/F ).

Indeed, the subalgebra B generated by e and β is isomorphic to (1, d ) M 2 (F ), and its centralizer C is the subalgebra generated by f and αe, which is isomorphic to (1, d 2 ) M 2 (F ). Since B is a central simple F -algebra, we have Proof. Recall from [START_REF] Lewis | The Hasse invariant of the trace form of a central simple algebra[END_REF] that for every central simple F -algebra of degree n, we have det(q A ) = det(q Mn(F ) ), w 2 (q A ) = w 2 (q Mn(F )

(1, 1, 1, L/F ) B ⊗ F C M 2 (F ) ⊗ F M 2 (F ) M 4 (F ).
) + n(n -1) 2 [A].
Since q A and q Mn(F ) have same dimension and determinant, the quadratic form q A ⊥ -q Mn(F ) lies in I 2 (F ), and the second equality may be rewritten as

c(q A ⊥ -q Mn(F ) ) + n(n -1) 2 [A],
where c denotes the Clifford invariant. In particular, if n = 4, we have

c(q A ⊥ -q Mn(F ) ) = 6[A] = 2[A].
Let us introduce some notation. Let E be an F -algebra. For u ∈ E × , we denote by q E,u the quadratic form q E,u :

E -→ F x -→ Tr E/F (ux 2 ).
If u = 1, we denote it by q E .

Let ρ be an F -automorphism of L satisfying ρ 2 = Id E . If u ∈ E × , ρ(u) = u, we denote by q E,ρ,u the quadratic form q E,ρ,u :

E -→ F x -→ Tr E/F (uxρ(x))
If u = 1, we denote it by q E,ρ .

Finally, if α, β ∈ F × , we set α, β = 1, -α ⊗ 1, -β . Now assume that A = (a, b, u, L/F ). It is easy to check that we have

Trd A (x + ye + zf + tef ) = Tr L/F (x)
, for all x, y, z, t, ∈ L.

This implies that the subspaces L, Le, Lf and Lef are mutually orthogonal. It follows from the previous observation that we have

q A q L ⊥ q L,σ,a ⊥ q L,τ,b ⊥ q L,στ,c .
Let ρ be an F -automorphism of L, and assume that ρ = Id L , so we may write

L ρ = F [ √ ∆], and L = L ρ [ √ ∆ ]
. Notice that uxρ(x) ∈ L ρ , so that we have Tr L/F (uxρ(x)) = 2Tr L ρ /F (uxρ(x)). Now if x = x 0 + x 1 α where α 2 = ∆ and x i ∈ L ρ , we have xρ(x) = x 2 0 -∆ x 2 1 , and therefore q L,ρ,u 2, -2∆ ⊗ q L ρ ,u .

Write u = u 0 + u 1 √ ∆, u i ∈ F , and set n u = u 2 0 -∆u 2 1 . The representative matrix of the previous quadratic form in the basis 1, √ ∆ is

2u 0 2u 1 ∆ 2u 1 ∆ 2u 0 ∆ .
If u 0 = 0, this 2-dimensional quadratic form is isotropic, hence hyperbolic. If u 0 = 0, this form then represents 2u 0 and has determinant 4n u ∆, so it is isomorphic to 2u 0 ⊗ 1, n u ∆ . We then get

q L,ρ,u u 0 ⊗ -n u ∆, ∆ ,
where this form has to be understood as the hyperbolic form of dimension 4 if u 0 = 0. Taking u = 1, we get q L,ρ -∆, ∆ ,

and thus q L,ρ,u ⊥ -q L,ρ 1, -∆ ⊗ u 0 , u 0 n u ∆, 1, ∆ . Since c(ϕ ⊗ ϕ ) = [(disc(ϕ), disc(ϕ ))] for every even-dimensional forms ϕ, ϕ , we get c(q L,ρ,u ⊥ -q L,ρ ) = [(∆ , n u )].
Remark 3.2 shows that q A ⊥ -q M4(F ) is Witt-equivalent to (q L,σ,a ⊥ -q L,σ ) ⊥ (q L,τ,b ⊥ -q L,τ ) ⊥ (q L,στ,c ⊥ -q L,στ ).

We then finally get

c(q A ⊥ -q M4(F ) ) = [(d , n a )] + [(d, n b )] + [(d, n c )].
Since the quaternion algebras (d, n a ) and (d , n b ) are split, we get

c(q A ⊥ -q M4(F ) ) = [(dd , n a )] + [(dd , n b )] + [(d, n c )] = [(dd , n a n b )] + [(d, n c )] = [(dd , n c )] + [(d, n c )] = [(d , n c )].
This concludes the proof.

We now define a classifying pair for CSA 4 . Let us consider the affine variety V ⊂ A 6 defined by the equation

x 2 -uy 2 -vz 2 + uvt 2 + uv = 0.
This is a rational variety, with coordinate ring

F [V] = F [X, Y, Z, T, U, V ]/(X 2 -U Y 2 -V Z 2 + U V T 2 + U V ).
Let us denote by x, y, z, t, u, v the images of X, Y, Z, T, U, V in F [V] respectively. Notice that u and v are algebraically independent over F . Indeed, we have a surjective F -algebra morphism

F [V] -→ F [V, Z] P (x, y, z, t, u, v) -→ P (0, i, Z, i, V Z 2 , V )
which maps u and v onto V Z 2 and V . Since V Z 2 and V are algebraically independent over F , so are u and v. Now let us consider the open subset U of V defined by the equations

u = 0, v = 0, t = 0, y 2 -vt 2 = 0, z 2 -u = 0.
Then U is also a rational variety, whose coordinate ring

F [U] is the localization of F [x, y, z, t, u, v] at u, v, t, y 2 -vt 2 and z 2 -u. Set R 0 = F [U]
. Then R 0 is a noetherian ring, whose quotient field F 0 is a rational extension of F . Moreover, u, v ∈ R 0 are algebraically independent over F . In particular, there exists a transcendence basis of F 0 /F containing u and v.

Set w = u(y 2 -vt 2 ) ∈ R × 0 and let L 0 = R 0 [ √ u, √ uw]
. We now define three elements a 0 , b 0 , c 0 ∈ L × 0 by

a 0 = z + √ u ∈ R 0 [ √ u] × , b 0 = y t + √ uw tu ∈ R 0 [ √ uw] × , c 0 = x + √ w ∈ R 0 [ √ w] × .
Let us check that n a0 n b0 = n c0 . We have We set

n a0 n b0 = (z 2 -u)(y 2 u 2 -uw) t 2 u 2 = (z 2 -u)(y 2 u -w) t 2 u .
x = γ 0 , y = µ 0 γ 1 λ 1 µ 1 , z = λ 0 , t = γ 1 λ 1 µ 1 , u = λ 2 1 d, v = µ 2 0 -µ 2 1 d .
Using the equality

(λ 2 0 -λ 2 1 d)(µ 2 0 -µ 2 1 d) = (γ 2 0 -γ 2 1 d
), and the fact that λ 1 , µ 1 , γ 1 ∈ F × , one may check that p = (x, y, z, t, u, v) ∈ U(E). Then evaluation at p yields an F -algebra morphism f : R 0 -→ E such that

A 0 ⊗ R0 E A.
Now it suffices to notice that ker(f ) = (x -x, y -y, z -z, t -t, u -u, v -v) is a maximal ideal to conclude.

Cohomological invariants of CSA 4

We now prove the following theorem, due to Rost (unpublished): Theorem 4.1 (Rost). Let F be a field of characteristic different from 2. Assume that -1 ∈ F ×2 . Then the map

Inv(CSA 4 , H * ) -→ Inv(MS 2,2 , H * ) × Inv(MS 4,1 , H * )
is injective. In other words, a cohomological invariant which is zero on biquaternion algebras and cyclic algebras is identically zero.

Proof. Let α ∈ Inv(CSA 4 , H * ). Let (R 0 , A 0 ) be the classifying pair of CSA 4 defined in the previous section, and let F 0 be the quotient field of R 0 . Finally, set

A 0 = A 0 ⊗ R0 F 0 = (z + √ u, y t + √ uw tu , x + √ w, F 0 [ √ u, √ uw]/F 0 ). Let υ be a discrete F -valuation on F 0 . If υ = υ π , with π = u, v, t, y 2 -vt 2 , z 2 -u, then u, v, w, t ∈ O × υ and z + √ u, y t + √ uw tu , x + √ w ∈ O υ [ √ u, √ uw] × .
We then get that

A 0 (z + √ u, y t + √ uw tu , x + √ w, O υ [ √ u, √ uw]/O υ ) ⊗ Oυ F 0 ,
and therefore α F0 (A 0 ) is unramified at υ by [START_REF] Garibaldi | Cohomological invariants in Galois cohomology[END_REF]Theorem 11.7] .

Assume now that υ = υ π , with π = t, z 2 -u or y 2 -tv 2 . Then u, v ∈ O × υ , and thus we have (u) ∪ (v) = j υ ((u) ∪ (v)), where j υ : H 2 (κ(υ)) → H 2 ((F 0 ) υ ) is the canonical injection. Now in all three cases, the quadratic form 1, -u, -v, uv is isotropic over κ(υ). It is clear if π = z 2 -u or y 2 -vt 2 , and follows from the equalities 0 = x 2 -uy 2 -vz 2 + uvt 2 + uv = x 2 -uy 2 -vz 2 + uv mod t, if π = t. Thus (u) ∪ (v) = 0, and therefore (u) ∪ (v) = 0 as well. It follows that the quaternion algebra (u, v) F0 is split over (F 0 ) υ . Hence, by Proposition 3.3, 2[A 0 ] splits over (F 0 ) υ . By a theorem of Albert, A 0 ⊗ F0 (F 0 ) υ is a biquaternion algebra. By assumption, we get

Res (F0)υ/F0 (α F0 (A 0 )) = α (F0)υ (A 0 ⊗ F0 (F 0 ) υ ) = 0.
Consequently, ∂ υ (α F0 (A 0 )) = 0 and α F0 (A 0 ) is once again unramified at υ. Finally, α F0 (A 0 ) is unramified at every discrete F -valuation υ = υ u , υ v . In particular, it is unramified at every F 1 -valuation, where F 1 = F (u, v). Since there exists a transcendence basis of F 0 /F containing u and v, the field extension F 0 /F 1 is rational, and we deduce that α F0 (A 0 ) = Res F0/F1 (β), for some β ∈ H * (F 1 ). Let υ 1 be a discrete F -valuation on F 0 , which different from the u-adic and the v-adic valuations. Let us show that β is unramified at υ 1 .

Let u, v, t 1 , t 2 , t 3 be a transcendence basis of F 0 /F , and let us extend υ 1 to a discrete F -valuation on F 0 by setting

υ F1 = υ 1 , υ(t i ) = 0, i = 1, 2, 3.
Then the corresponding ramification index is 1 and κ(υ 1 ) = κ(υ)(t 1 , t 2 , t 3 ), so that κ(υ)/κ(υ 1 ) is a rational extension. Since α F0 (A 0 ) is unramified at υ, we get

0 = ∂ υ (α F0 (A 0 )) = Res κ(υ)/κ(υ1) (∂ υ1 (β)).
Since κ(υ)/κ(υ 1 ) is rational, the map Res κ(υ)/κ(υ1) is injective, and we get that β is unramified at υ 1 . By Proposition 1.2, we have

β = a 0 •1 + a 1 •(u) + a 2 •(v) + a 3 •(u) ∪ (v), a i ∈ H * (F ). Hence we get α F0 (A 0 ) = a 0 •1 + a 1 •Res F0/F1 ((u)) + a 2 •Res F0/F1 ((v)) + a 3 •Res F0/F1 ((u) ∪ (v)).
Let us consider the (v -1)-adic valuation on F 0 , and let F 2 its residue field. Since A 0 is unramified at this valuation, one may specialize A 0 to a central simple

F 2 - algebra B. Then 2[A 0 ] specializes to 2[B]. But 2[A 0 ] = [(u, v)]
specializes to 0, and therefore B is a biquaternion algebra. Then α F2 (B) = 0 by assumption, and therefore a 0 •1 + a 1 •Res F2/F1 ((u)) = 0. In other words, Res F2/F1 (a 0 •1 + a 1 •(u)) = 0. Since F 1 ⊂ F 2 ⊂ F 0 , and F 0 /F 1 is rational, the map Res F2/F0 is injective and thus

a 0 •1 + a 1 •(u) = 0 in H * (F 1 ).
By Lemma 1.1, we get a 0 = a 1 = 0. Considering the u -1-adic valuation shows that a 2 = 0 in a similar way, so that α F0 (A 0 ) = a 3 •Res F0/F1 ((u) ∪ (v)). By Lemma 1.1, we get a 3 = 0, and thus α = 0. This concludes the proof.

Since Res

We now describe the cohomological invariants of CSA 4 . Let F be a field of characteristic different from 2. Assume that -1 ∈ F ×2 , and let K/F be a field extension. By [START_REF] Rost | La forme trace d'une algbre simple centrale de degré 4[END_REF], for every central simple K-algebra A of degree 4, there exists a 2-fold Pfister form q 2 and a 4-fold Pfister form q 4 such that q A ∼ q 2 + q 4 ∈ W (K).

Taking the cohomology class e 4 (q 4 ) ∈ H 4 (K) yields a cohomological invariant of Inv(CSA 4 , H * ). This invariant restricts to zero on cyclic algebras, and its restriction to MS 2,2 obviously coincide with the invariant e 4 defined in Section 2. Therefore, we still denote this invariant by e 4 .

Corollary 4.2. Let F be a field of characteristic different from 2. Assume that -1 ∈ F ×2 . Then Inv(CSA 4 , H * ) is a free H * (F )-module with basis 1, f 2 , e 4 .

Proof. We first prove that 1, f 2 , e 4 are linearly independent. Assume that we have a 0 •1 + a 1 •f 2 + a 2 •e 4 = 0 for some a i ∈ H * (F ).

Since the restriction of e 4 to MS 4,1 is zero, it follows that a 0 + a 1 •f 2 = 0 ∈ Inv(MS 4,1 , H * ). By Proposition 2.2 (1), we get a 0 = a 1 = 0. Thus, we get a 2 •e 4 = 0 ∈ Inv(CSA 4 , H * ). In particular, a 2 •e 4 = 0 ∈ Inv(MS 2,2 , H * ). By Proposition 2.2 (2), we get a 3 = 0.

We now prove that 1, f 2 , e 4 span Inv(CSA 4 , H * ) as an H * (F )-module. Let α ∈ Inv(CSA 4 , H * ). By Proposition 2.2, we have

α | MS 2,2 = a 0 •1 + a 1 •f 2 + a 2 •e 4 ,
for some a i ∈ H * (F ), and also

α | MS 4,1 = b 0 •1 + b 1 •f 2 ,
for some b i ∈ H * (F ). Since M 4 (F ) {1, 1, } 4,F

(1, 1) ⊗ F (1, 1), we may apply M 4 (F ) to both equalities to get a 0 = b 0 . Now we have M 2 ((u, v)) (1, 1) ⊗ F (u,v) (u, v) {u 2 , v 2 } 4,F (u,v) , and applying M 2 ((u, v)) to both equalities yields

a 1 •(u) ∪ (v) = b 1 •(u) ∪ (v) ∈ H * (F (u, v)).
It follows from Lemma 1.1 that a 1 = b 1 . Since e 4 is zero on MS 4,1 , we conclude that α and a 0 •1 + a 1 •f 2 + a 2 •e 4 coincide on MS 4,1 and MS 2,2 . By the previous theorem, we get α = a 0 •1 + a 1 •f 2 + a 2 •e 4 , and this concludes the proof.

e 2 σ

 2 = a, e 2 τ = b, e 2 στ = c, e σ e τ = e στ , e σ λ = σ(λ)e σ , e τ λ = τ (λ)e τ , for all λ ∈ L, and extending by distributivity.

  Since Int(e) | L = σ, we have Int(e 2 ) | L = (Int(e) | L ) 2 = σ 2 = Id L , so a = e 2 ∈ C A (L) = L (since L is a maximal subfield of A) . Since e ∈ A × , we have a ∈ L × . Similarly, b = f 2 ∈ L × and c = (ef ) 2 ∈ L × . Associativity of the product law of A implies that σ(a) = a, τ (b) = b, στ (c) = c, n c = n a n b . Indeed, we have e 3 = ee 2 = ea = σ(a)e = e 2 e = ae, and since e is invertible, we get σ(a) = a. Similar arguments show that τ (b) = b and στ (c) = c. We also have be

Proposition 3 . 3 .

 33 Let F be a field of characteristic different from 2, and let A = (a, b, c, L/F ). Then 2[A] = [(d , n c )] ∈ Br(F ).

Now y 2 Lemma 3 . 4 .

 234 u-w = uvt 2 , and thus n a0 n b0 = (z 2 -u)v. Now z 2 v-uv = x 2 -uy 2 +uvt 2 = x 2 -w = n c0 , and we are done. Hence we may consider the Azumaya algebraA 0 = (a 0 , b 0 , c 0 , L 0 /R 0 ). The pair (R 0 , A 0 ) is a classifying pair for CSA 4 .Proof. Let E/F be a field extension, and let A be a central simple E-algebra of degree 4. By Lemma 3.1, we have A (a, b, c, L/E), with a, b, c / ∈ E × . Write L = E[ √ d, √ d ], with generators α, β. Write a = λ 0 + λ 1 α, b = µ 0 + µ 1 β, c = γ 0 + γ 1 αβ.

  F0/F1 ((u) ∪ (v)) corresponds to [(u, v)] = 2[A 0 ] in Br 2 (F 0 ) via the usual isomorphism, the previous equality rewrites α F0 (A 0 ) = a 3 •f 2,F0 (A 0 ).By Lemma 1.5, we get α = a 3 •f 2 . By assumption on α, we then have 0 = α F1 ({u, v} 4,F1 ) = a 3 •(u) ∪ (v) ∈ H * (F 1 ).

Email: berhuy@ujf-grenoble.fr Address: Université Grenoble I, UFR de Mathématiques, Institut Fourier 100 rue des maths, BP 74, F-38402 St Martin d'Hères Cedex, France


Régis Guégan
email: regis.guegan@univ-orleans.fr.

Confinement effects on water structure in membrane lyotropic phases

Keywords: nonionic surfactant, lamellar, sponge phase, water structure, Raman scattering

The change of the water structure in aqueous solutions of the tri-ethyleneglycol mono ndecyl ether (C 10 E 3 ) was studied by micro Raman scattering. The results obtained on the O-H stretching band show that the behavior of the hydrogen bonding (H-bonds) water network can be used as a probe to follow the lamellar (L α ) to sponge (L 3 ) phase transition. In the lamellar phase, the stack of the surfactant molecules aggregated into a two-dimensional structure (membrane) acts as a soft confinement system for the H-bond water network of which the regular tetrahedral structure is perturbed. The change of the planar organization of the membranes to a highly disordered and infinite array of bilayers in the sponge phase amplifies the surface of contact between amphiphilic surfactant and water molecules which drives a strong disruption of the regular tetrahedral H-bonding water network.

Introduction

The study of the physico-chemical properties of water in confined geometry at a mesoscopic scale is important to technological and biophysical problems, such as microfluidic system, lab on chip materials, enzymatic activity and functions of membranes [1][2][3][4].

Numerous investigations have been devoted on the understanding of confined water and other liquids into mesoporous materials such as MCM-41, Vycor, clay and porous silicon [5][6][7][8][9][10][11][12][13][14][15][16][17][18].

These studies have revealed effects of confinement which affect both the structure and dynamics of the confined system of which properties differ strictly from these-ones of bulk liquid [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Surfactant systems self assemble into a variety of structures which present particular organizations of the membranes and provide model system to host small or significant amounts of water for mimic confinement in biological objects [1,[START_REF]Micelles, Membranes, Microemulsions and Monolayers[END_REF][START_REF] Mackeben | [END_REF][21][22]. The formed lyotropic liquid phases have been considered as intermediate between complex biological water cavities and rigid wall pores in mesoporous media [21]. Despite numerous studies on nanoconfined water, a little is known about the effect of confinement on water molecules arranged into H-bond networks. The past infra-red and Raman scattering investigations have shown a significant change of the tetrahedral H-bonding water network in micelle systems for a confinement lower than 10 nm [21,22]. In the same range of scale, Marinov et al have highlighted a change of the characteristics of water confined in a lamellar, and reverse micelle phases [22]. However, no studies have focused on the investigation of the change of the H-bond water network confined into several membrane phases or when the system undergoes a membrane reorganization which is of great importance since the water network has been recognized to contribute to the hydration force acting between membranes [23,24].

The bulk water structure is understood as a tetrahedral ice like arrangement which competes to a non tetrahedral arrangement [25][26][27]. The Raman spectroscopy gives vibrational spectra which are sensitive to the local environment of molecule and allow apprehending the disruption of the water network. In particular, the band in the region from 3000 to 3700 cm -1 , corresponding to the O-H stretching can be convoluted into three Gaussian components [22].

The lower frequency component at about 3200 cm -1 refers to collective in phase stretching vibrations of the tetrahedral H-bonding network, while the second component at about 3400 cm -1 is assigned to the not in phase vibrations between the nearest or higher neighbor molecules [25]. The last component at high frequency corresponds to non H-bonds or hydrogen bent structure. The measure of the integral intensities of both components gives information of how the regular tetrahedral H-bonding network of can be disrupted. In this approach, several studies on bulk water and once confined in surfactant systems, have used the ratio of the component at about 3200 cm -1 to the component at about 3400 cm -1 , defined as R OH which represents the fraction of water in the system that takes part in the regular tetrahedral H-bonding network [22,25,[28][29][30].

The tri-ethyleneglycol mono n-decyl ether (C 10 E 3 ) and water system forms internal structures organized by surfactant bilayers such as the lamellar (L α ) and sponge (L 3 ) phases [31,32]. The lamellar phase, L α , is characterized by a stack of surfactant molecules aggregated into two dimensional structures (membranes) and layers of water whose organization is at long range order which displays a smectic order [32,33]. On the other hand, for another range of temperature and concentration, the surfactant self organizes in a disordered and infinitely connected array of bilayers, called sponge phase or 'plumber's nightmare phase'. Both phases are present in neighboring regions in several phase diagrams and the transition from L α to L 3 phase is of first order and is triggered by variations of saddle splay rigidity of the Gaussian curvature K of the bilayer [32][33][34][35][36][37].

Previous investigations focused on the changes in water structure in extended polyatomic surfactant structures (lamellar to reverse micelle structures) by the increase of surfactant concentration [21,22]. The aim of this study is to quantitatively evaluate the extent of perturbation of water structure confined into two lyotropic membrane phases which shows two levels and organization of the surfactant bilayer (1D and 3D confinement for the lamellar and sponge phases respectively) and to follow its change in temperature. For this purpose, the changes in water structure of the lamellar to sponge phase transition in the C 10 E 3 -water system is studied by Raman scattering experiments with an accurate analysis of the O-H stretching mode which allows estimating a contribution of strongly and weakly H-bond water molecules. The intensity and shape of the O-H stretching mode provide information on how the fraction of confined water in the system, that takes part in the regular tetrahedral structure, changes during the reorganization of the surfactant bilayer when the system transits from 1D to 3D organization membrane phase.

Materials and Methods

Materials and experimental setup

The surfactant C 10 E 3 was purchased by Nikko Chemicals, Inc., Tokyo is assumed to have a purity better than 99.8 % and was used without further purification. Distilled and deionized water was used to prepare the aqueous solution of 0, 5, 10, 20, 30, 40, 60, 80 and 100 wt % for C 10 E 3 . The solutions were prepared by weight and were homogenized for at least 24 hours.

Then, the solutions were sandwiched between two plates of cover glasses for microscopy with the use of 10 µm diameter silica spheres as spacers. The samples were sealed by glue to avoid any water evaporation. Special care was taken to prevent nor dust contamination nor contact between glue and aqueous solutions. In this C 10 E 3 -water system, an isotropic sponge phase The Raman Spectra in the 2600-4000 cm -1 region were recorded in the backscattering geometry with a Horiba Jobin-Yvon T64000 micro Raman spectrometer equipped with a symphony CCD detector. A Neodynium Vanadate laser (Verdi Coherent) operating at 514.5

nm with an output power of 100 mW was used as an excitation light source. The spectral resolution was 2 cm -1 whereas the spatial resolution, determined on silicon wafer, was better than 2 µm. The temperature range measured for the lamellar to sponge phase transition for the 20 wt % for C 10 E 3 solution was controlled by a Linkam system. The Raman spectra of various concentration wt % of C 10 E 3 solutions were recorded at room temperature (i.e. 25°C). This implies measurements in three phases, sponge, lamellar and reverse micellar solution phase.

Raman scattering experimental setup for light depolarization

The measured spectra in the lamellar phase cannot be directly comparable to the optically isotropic sponge or micellar phases. In the lamellar phase which shows some anisotropic properties, the light experiences partial or full depolarization. A method has been suggested and successfully applied to prevent this experimental problem of light depolarization [22,29].

The lamellar phase consists to numerous randomly oriented microscopic domains of lyotropic liquid crystals. At each interface between two neighboring monodomains, a light beam is split into ordinary and extraordinary beams with orthogonal polarization orientations.

When the number of the domains, which the beam is intersecting, is large enough, the exciting laser beam is completely depolarized and the scattered Raman spectrum is that of randomly oriented molecules illuminated with naturally polarized light. If I || and I ⊥ are the intensities of the scattered light components with polarization parallel and perpendicular to that of the exciting light, respectively, then, in the experimental situation described, the

intensities of the Raman signal components with polarizations parallel and perpendicular to the direction of propagation of the exciting light are proportional to I || + I ⊥ and 2I ⊥ , respectively. As the total Raman signal polarization is scrambled on the way out, the intensities of the observed spectra are proportional to I || + 3I ⊥ . The rational idea of the suggested method is that Raman spectra with the same type of a linear combination of I || and I ⊥ as that for the liquid crystalline phases, namely, I || + 3I ⊥ , are measured for the isotropic micellar and sponge phases. Thus, it is possible without the influence of depolarization effects to compare the Raman spectra of the isotropic solution phase and the anisotropic liquid crystalline phase and accordingly to compare changes in the spectral shape of the bands to be attributed to changes in the molecular structure. Obviously, such a combination of I || and I ⊥ is obtained if Raman scattered light is recorded from the isotropic phase with polarization rotated to 60° with respect to the polarization of the exciting light:

I 60 = [cos²(60°)]I || + [sin²(60°)]I ⊥ = (0.25)I || + (0.75)I ⊥
In order to diminish the error due to the uncertainty in tuning the angle of the polarization rotator at 60°, parallel and perpendicular Raman spectra were measured and their linear combinations were calculated according to eq 2. Moreover, the derivative of the intensity at such angles (0 and 90°) of a polarized light with respect to the polarization rotation angle is zero. An important condition of producing correct results with this experimental setup is that both the exciting light and the Raman scattered light have to experience full depolarization in the liquid crystalline phase. This means that the excitation and Raman photons must cross a sufficiently large number of monodomain borders, or in other words, the size of the domains should be much smaller than the distance at which each photon travels through the sample.

Raman shift (cm -1 ) 2600 2800 3000 3200 3400 3600 3800 4000

Raman Intensity (a. u.)

Processing of the spectra

The Figure 2 represents the Raman spectra of C 10 E 3 -water system in the experimental frequency range from 2600 to 4000 cm -1 . In order to discriminate the O-H stretching band from the C-H stretching band relating to surfactant molecules, we used the fitting procedure of the LabSpec software (Horiba, Jobin-Yvon). The C-H stretching band was decomposed into four Gaussian components, while the O-H stretching was fitted with three main Gaussian components at about 3200, 3400 and 3600 cm -1 , as it is shown in the insert of Figure 2. The calculated I 60 spectra were rescaled so that they correspond to equal surfactant content in the solution. The procedure has consisted to divide each spectrum by the integral of the C-H stretching band obtained by the fitting procedure. The Figure 3 shows the results of the calibrated spectra in the range of the O-H stretching.

Raman shift (cm -1 ) 3000 3200 3400 3600 3800

Relative Intensity (a. u.) Thus, the spectra were corrected from the contribution of the O-H stretching of the surfactant which can be achieved from the spectrum of the neat C 10 E 3 . We realized another fitting procedure on the subtracted corrected spectra for both concentrations on the O-H stretching range. The integrated intensity of the O-H stretching band as a function of the mole ratio of water to surfactant is shown in the Figure 4. The excellent linearity supports the appropriateness and accuracy of the calibration procedure adopted in this study.

Results and Discussion

The Raman spectra of the C 10 E 3 -water in the O-H stretching range from 24 to 43°C for the concentration of 20 wt % are shown on the Figure 5, and the parameter R OH , obtained from the spectra, is presented in the Figure 6 as a function of the temperature when the system transits from lamellar to sponge phase. The Raman spectra in the Figure 5 are normalized to equal area after the correction by surfactant contribution. For each spectrum, the parameter R OH was calculated as the area ratio of the component at about 3200 cm -1 to the component at about 3400 cm -1 (R OH = A 3200 /A 3400 ) using the curve fitting procedure. In this temperature range from 24 to 43°C, the 20 wt % C 10 E 3 -water system transits from lamellar to sponge phase. The two phases coexist on about 8°C as it can be represented in either Figure 1 or 6. The Raman spectra evolution in the Figure 5 displays some interesting features. By increasing temperature, one can note the decrease in intensity of the 3200 cm -1 component whereas the component at about 3400 cm -1 becomes preponderant in the O-H stretching band. One can remark also the increase in the intensity of the component at about 3600 cm -1 , which underlines the rupture of the H-bonds. The temperature evolution of the spectra for the C 10 E 3 -water system indicates that the intermolecular vibrational coupling at long range order in the regular tetrahedral H-bonding network is reduced.

The parameter R OH corresponds to the measure of collectiveness vibrations within the fraction of water molecules that take part in the tetrahedral structure. The Figure 6 shows the temperature evolution of R OH for the lamellar to sponge phase transition of the 20 wt % C 10 E 3water system. The parameter R OH obtained from the spectra of pure water (not presented) in the same experimental conditions does not present any fluctuations and remains at a constant value of 0.7 in the studied temperature range. Previous studies on the bulk water from amorphous ice to super critical water via ambient water highlighted the strong change of water structure with a variation of the R OH ratio from 2 at -180°C to about 0 at 400°C [25].

However, on the ambient temperature range between 20 and 50°C, the fraction of the regular tetrahedral H-bonding water molecules display a R OH parameter close to 0.7 which is coherent to our experimental spectra evolution and reflects the good agreement of the fitting procedure.

The parameter R OH exhibits a different behavior when the system is in the lamellar phase.

The removing of water in the system or the introduction of surfactant at a weight of 20% leads to a decrease in the relative part of the tetrahedrally H-bonding water molecules. In the lamellar phase, the 20 wt % C 10 E 3 -water system exhibits a R OH parameter close to 0.55 and remains at this same value of in the phase domain temperature. The changes in the O-H stretching spectra in the lamellar phase, comparatively to spectra in the bulk water at same temperature, are due to a reduction of bulk water relative to the volume of the water that is in some degree involved in interaction with the surfactant molecules and also a change of the structure of water. Interestingly, the introduction of a C 10 E 3 membrane in water affects the fraction of the regular tetrahedrally H-bonding water molecules at about 20% comparatively to bulk water, which indicates the existence of a correlation between surfactant concentration and R OH parameter. Then, as the temperature reaches the sponge-lamellar coexistence region, an abrupt change in the behavior of the R OH parameter is observed, which decreases linearly and remarkably fast with the temperature. This result indicates a drastic change in the structure of water. Thus, in the sponge phase, R OH remains at a plateau value of 0.46 which underlines that water structure does not evolve.

Previous studies in the structure and properties of water confined at a mesoscopic scale between supramolecular aggregates and in small cavities have highlighted the same behavior [38]. In these studies, the decrease of the intensity of the lower frequency component of the O-H stretching band, resulting to the reduced long range intermolecular vibrational coupling, is interpreted to an effect of finite size on the water domains than to the nature of the surface interstices where the water is confined. Generally, the confinement of water or alcohol between hydrophilic surfaces leads to a distortion of the H-bond network, which can be expressed by a decrease in the value of the parameter R OH [39][40][41]. The O-H stretching Raman spectra of interstitial water in multilamellar lipid liposomes showed a significant deviation from the spectrum of bulk water when the distance between the surfaces was several nanometers. Again, the component at about 3200 cm -1 decreased in intensity compared to the component at about 3400 cm -1 [26].

At such mesoscopic scale, numerous studies have shown that the structure and the physical properties of confined liquids in cavities differ strikingly from the bulk system [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The experiments on the confinement of water or associated liquid such as alcohol in mesoporous materials (aerogels, controlled pore glass, MCM-41...) showed confined liquid presents a density, structure and dynamics different from bulk system. Physical properties such as structure factor or density are also perturbed due to the interaction with the surface which is of first importance in this latter case and also to a finite size effect. In the recent Monte Carlo simulation study of methanol, which is the simplest alcohol system available to create some H-bonds, confined into a silica mesopore, the results obtained have shown that the pore surface significantly affects the spatial arrangement of molecules, leading to the creation of layers which propagate to the entire pore volume [16,17]. The interesting result concerns the interfacial layer molecules which creates more H-bonds with the silanols groups of the surface than between them, leading to a strong disruption of the H-bond network.

The changes in the water structure observed for the present C 10 E 3 -water system in the lamellar phase are in good agreement with the changes of the structure of nanoconfined water observed by other researchers [1,21,22]. In the lamellar phase, the repeat distance d of a bilayer can be calculated for one dimensional swelling system by the eq1. This relation is not exactly valuable because it does not take in account the thermal undulations of the bilayer and also the interbilayer interactions but gives a suitable idea of the repeat distance of the bilayer against dilution with a good accuracy. For a concentration of 20 wt % C 10 E 3 in water, the repeat distance d is equal to 138 Å and implicates that water domains are confined between two flat surfaces in a slit thickness close to 110Å [12,32]. The finite size effect confinement and the presence of a surfactant membrane surface contribute to the disruption of the longrange ordering of the water molecules and lead to the decrease of the R OH parameter. Previous studies on confinement of water in surfactant micelles showed a strong change in water structure for core micelle size below 10 nm which is in good agreement with our observations even if the system geometry differs [21].

The structure of the sponge phase is described in terms of a 3-d isotropic continuous bilayer structure as it represented on the insert of the Figure 1 [33][34][35][36][37]. The previous electric conductivity experience on the sponge phase concluded to the structure of a bicontinuous bilayer cubic phase [37]. The repeat distance for this bilayer is 1.4 wider than this-one of the lamellar phase for the same concentration which means that, in this study, the water domains are confined into a space of about 168 Å if we assume that the thickness of the bilayer remains the same in both membrane phases [32]. However, the elastic curvature, characteristic of the lyotropic membrane phases, differs in the sponge phase. Besides, the shape of the bilayer organization shows a 3D arrangement which drives to an increase of water amount in interaction with the polar headgroup of the surfactant [33,36,37]. Moreover, the high disordered structure in the sponge phase shows defects in the bilayer organization which could favor interaction between confined water with hydrocarbon chains of surfactant and as result disrupt the water structure.

The continuous decreasing of the fraction of the regular tetrahedrally H-bonding water molecules undergoes the transition from the lamellar to the sponge phase. In these two phases, water interacts with the hydrophilic part of the C 10 E 3 surfactant membrane. The fast decreasing of the R OH parameter is due to an increase of water molecules interacting with the polar headgroup of the nonionic surfactant. These results obtained here can be related to the hydration force which results to the ordering from the water molecules between the membranes and its organization [23,24]. This force due to the structure of water with surface changes between the lamellar and sponge phases due to an increase of water molecules interacting with the headgroup of surfactant leading to a strong disruption of the water structure. Finally, these results highlight the importance of the first adsorbed layers of water molecules on surface in the water structure. Indeed, the water structure depends mainly on this particular layer and its morphology [42][43][44]. In the case of the lamellar phase, the planar geometry of the surface leads to a 1D slit confinement and affects only the first layer also called solvation layer, as it has been observed for adsorbed water on various substrate whereas in the sponge phase, the solvation layer thickness should highly increase and perturbs in the same order of magnitude the water structure between the 3D organization of the surfactant bilayer. Even if the size of confinement is wider in the sponge phase, the geometry of the highly disordered structure of the membranes reduces the size of the tetrahedrally H-bonding water clusters and consequently disorganizes the water structure at long range ordering, characteristic of the bulk water.

Conclusion

Water confined in solid pores and cavities of mesoscopic dimensions in surfactant structures exhibit highly unusual properties [1][2][3]. The understanding of the dynamics and the water structure is important to biophysical problems such as functions of membranes and food conservation [1]. However the use of biological systems is rather complex and surfactant structures provide interesting well-controlled aggregates for the understanding of confined water [1]. Effects of confinement have been observed on water structure network confined in micelle structures and lamellar phase by Raman scattering and infra-red spectroscopy investigations [21,22]. However, the investigation of H-bond water structure in the sponge phase of which structure corresponds to a 3D isotropic bilayer organization has not been characterized so far. Moreover, the study on the changes in water structure for the lamellar (L α ) to sponge (L 3 ) phase transition has also not be undertaken which is of great importance since the water network ordering has been recognized to contribute to the hydration force acting between membranes [23,24].

In this work, we have explored using Raman scattering the behavior of the water structure in aqueous solution of tri-ethylene glycol mono n-decyl ether (C 10 E 3 ) for various concentrations in the lamellar and sponge phases. Using a three-Gaussian decomposition of the OH stretching band data, it is clear that the confinement in the lamellar phase disturbs essentially the two lower and slightly the higher energy components, as it has been previously identified. The reorganization of the membrane in one dimensional structure, in the lamellar phase to a three dimensional structure in the sponge phase affects both components, which indicates that the water molecules involved in different supermolecular bonding schemes differ from each other through their respective mean degrees of connectivity. In the sponge phase, even for a wider confinement size, the regular tetrahedral water structure is disrupted quite strongly comparatively to this-one of the lamellar phase. This observation indicates water molecules interacting with the polar head groups of the nonionic surfactant increase considerably when the system self organizes from a 1D to a 3D confinement and reveals the importance of the first layers of adsorbed water molecules in the water structure. The behavior of the water structure can be used as a probe to describe perfectly the lamellar to sponge phase transition and its consequence for the formation of the liquid crystalline phases.

Figure1:

  Figure1: Phase diagram of the C 10 E 3 -water system. The phases shown are the lamellar phase (L α ), sponge phase

Figure 2 :

 2 Figure 2: Raman spectra of the C 10 E 3 -water system for a concentration of 20 wt % of surfactant. The bands at about 2900 cm -1 are assigned to the C-H stretching of the aliphatic tails of the surfactant whereas the broad band at higher frequency represents the O-H stretching of both water and surfactant molecules. The insert shows the Gaussian decomposition of the vibration Raman spectra of the O-H stretching mode. The correlation coefficient for all fit r²=0.99.

Figure 3 :Figure 4 :

 34 Figure 3: Raman spectra in the O-H stretching band region for the C 10 E 3 -water system. The spectra are normalized to equal surfactant content.

Figure 5 :Figure 6 :

 56 Figure 5: Temperature evolution of the Raman spectra of aqueous C 10 E 3 solution at a concentration of 20 wt % for the surfactant.


P K Roche
email: philip.roche@univ-cezanne.fr

The landscape disparity index: an ecologically weighted measure of the landscape diversity

Introduction

Typically, the analysis of landscape patterns is based on a quantitative analysis of the composition or/and the spatial arrangements of categorical units mapped from aerial photographs, satellite images or land use maps using so-called landscape metrics [START_REF] Gustafson | Quantifying landscape spatial pattern: what is the state of the art ?[END_REF].

Usually in landscape studies, authors assume that different landscape habitats or elements categorised in the mapping process are well defined, and equally different from each other (Forman 1995 and references therein). But, in most cases the landscape element types are often not really distinct or at least are not equally distinct from each other and the degree of dissimilarity between the mapped landscape elements is not usually considered.

In order to improve the ecological relevance of landscape diversity metrics some ecologically important features should be considered. An index should integrate the habitat number and their spatial arrangements [START_REF] O'neill | Indices of landscape pattern[END_REF]. It should also have a low sensitivity to categorical resolution [START_REF] Frohn | Remote sensing for landscape ecology[END_REF] and/or should be able to handle degrees of thematic resolution (Loehle and Wein 1994). The two last points could be addressed by integrating a measure of the ecological contrast between the thematic categoriesi.e. the disparity.

Methods

I propose a new landscape diversity metric: the Landscape Disparity Index (LDI) that takes into account both the landscape elements adjacencies frequencies and the patches ecological dissimilarities (1)

))) 1 ( /( 2 ( ln ln 2 m m d p = LDI j) (i, j) (i, (1)
The LDI was tested using landscape maps from two sites in southern France. Environmental variables were used to compute the dissimilarity index between and within landscape element types. Highly autocorrelated variables like altitude and slope were eliminated. The remaining variables were tested using Kruskal-Wallis tests for significant differences between landscape elements. Non significant variables were removed. The remaining variables were used to compute the dissimilarity matrices.

The LDI was compared with eight other widely used landscape metrics using the Fragstats 3.1 software (McGarical and Marks, 1994).

Scale issues were analysed by computing the LDI and the eight other landscape metrics on our test area at 4 scales (30, 50, 70 and 90 m). Thematic resolution issues were assessed by computing the LDI and the other landscape metric on the 6 and 11 theme maps. In addition, for the LDI, the sensibility to the dissimilarity measure component using the two variants of the distance matrix for both 6 and 11 theme maps.

As a biological variable of interest we used the species richness at the landscape level using both modelled and field data.

If species richness is considered to increase when landscape heterogeneity increases [START_REF] Duelli | Biodiversity evaluation in agricultural landscapes : an approach at two different scales[END_REF], then landscape metrics measuring landscape heterogeneity indexes should be positively correlated with species richness at the landscape level.

Results and Discussion

The Landscape Disparity Index appeared to belong to the group of landscape diversity metrics with some interesting properties not shared by other diversity metrics: low sensibility to thematic resolution and ecologically significant weighting The Shannon index (SHDI) appears to be the more sensitive to thematic resolution effects with Diff% higher than 40% and four indexes have Diff% values under 10% (PLADJ, LDI, COHESION and AI). Two groups of metrics could be defined. The first one comprised the IJI, RPR, SHDI, LDI that could be defined as diversity metrics. The second group comprised the AI, PLADJ, COHESION, LPI and CONTAG indexes that are landscape compaction-aggregation metrics. According to our results high LDI values are also related to high species richness at the landscape level (Fig1). High LDI values indicate that ecologically contrasted habitats are clustered together on the analysis scale. Traditionally used landscape diversity indexes such as the Shannon Index are not as coherent, because high values can be associated with low species richness. Thus, taking into account the degree of ecological contrast between landscape elements weights the spatial heterogeneity and improves correlation with biological patterns. These preliminary results indicate that the Landscape Disparity Index could be used to identify high diversity spots at the landscape level.

Fig. 1 .

 1 Fig. 1. Biplot of species richness and LDI. Each dot represents a landscape window of 90x90 m. The full line represents a GLM regression using a log link.

  IALE World Congress, Wageningen, NLD, 8-12/07/2007

in : 7th IALE World Congress, Wageningen, NLD, 8-12/07/2007


W Lin

A Delahaye
email: anthony.delahaye@cemagref.fr

L Fournaison

Phase equilibrium and dissociation enthalpy for semi-clathrate hydrate of CO 2 +TBAB

Keywords: Hydrate, CO 2, Tetra-n-butylammonium bromide, Equilibrium, Dissociation enthalpy

The present work investigates equilibrium conditions and dissociation enthalpy of hydrates formed from CO 2 -TBAB(tetra-n-butylammonium bromide)-water mixtures. Differential Thermal Analysis (DTA) was used for Hydrate-Liquid-Vapour (H-L-V) equilibrium condition determination in a TBAB concentration range from 4.43 to 9.01 wt% and in a CO 2 pressure range from 0.3 to 2.5 MPa. The results showed that the presence of TBAB allowed decreasing the formation pressure of CO 2 hydrate by approximately 74 and 87% at 283 and 279 K, respectively. Moreover, pressure reductions were dependent on the TBAB concentration. The dissociation enthalpy and the composition of double hydrate formed from 9.01 wt% TBAB solution were determined by both the DTA and Clapeyron equation. The DTA method resulted in 313.2 kJ per kg of hydrate for the dissociation enthalpy and 2.51CO 2 •TBAB•38H 2 O for composition of the double hydrate. For the use of Clapeyron equation, the volume change was defined by taking into account the gas solubility in the liquid phase. The calculation results showed a discrepancy with the experimental data obtained by DTA, suggesting the limited application of Clapeyron equation in the CO 2 -TBAB-water system.

A c c e p t e d M a n u s c r i p t 2

Introduction

With the enforcement of the Montreal and Kyoto Protocols, the primary refrigerants such as CFCs and HCFCs are phased out gradually from the refrigeration industry, due to their global warming and ozone depletion potentials. As a result, the interests for environment-friendly refrigerants increased considerably. CO 2 is one of the most harmless refrigerants except for water and air. Besides, CO 2 has the advantage of being non-toxic and is easily available [1]. Currently, various researches in the air-conditioning and heat pump field are devoted to the investigation of pure CO 2 as a primary refrigerant [2]. Despite the progress made on these researches, some parallel paths including the use of secondary refrigeration loops deserve to be explored.

Secondary refrigeration is capable of reducing the amount and the risk of leakage of primary refrigerants. Currently, ice slurries are used as secondary refrigerant, but their generators are power-limited because they use mechanical technologies (scraped or brushed-surface heat exchangers). Tetra-n-butylammonium bromide (TBAB) hydrate slurries were also studied in previous work [3][4][5]. Rigorously, TBAB and water form semi-clathrate hydrates. In these hydrates, some cages are broken to encage big guest species such as TBAB, and water molecules of the structure are partially replaced by the atoms of guest species [START_REF] Davidson | Water: A Comprehensive Treatise[END_REF]. TBAB hydrate slurries were studied due to their potential suitability for air-conditioning applications (stable between 273 and 285 K). Nevertheless, the generation of TBAB hydrate slurries could have the same drawbacks as ice slurries related to the use of mechanical generators. Moreover, the dissociation enthalpy of TBAB hydrate (~200 kJ•kg -1 ) [START_REF] Oyama | [END_REF] is lower than that of ice (333 kJ•kg -1 ). CO 2 hydrate slurry is well suited for use as secondary refrigerant, since it can be generated by CO 2 injection in a cold aqueous solution (non-mechanical process) and it has a large thermal storage capacity thanks to the large dissociation enthalpy of CO 2 hydrate. In addition, some hydrate slurries are non-cohesive, so that they can flow steadily through piping systems and heat exchangers without coagulating [4]. The CO 2 hydrate dissociation enthalpy was investigated by many researchers based on experimental or calculation methods [8][9][10]. The results showed that the dissociation enthalpy of CO 2 hydrate (374 kJ•kg -1 ) was higher than that of ice, which has higher thermal storage capacity than most of traditional materials used in refrigeration. However, the CO 2 -hydrate formation pressure is higher than that required in cooling systems such as airconditioning (approx. < 1 MPa). Therefore, another component needs to be added to reduce the A c c e p t e d M a n u s c r i p t (THF) in water could reduce the formation pressure of CO 2 hydrate significantly [11].

In the present work, TBAB was used to reduce the formation pressure of CO 2 hydrate because it could form hydrate at atmospheric pressure, and so was likely to form a double hydrate with CO 2 at a lower pressure than that of single CO 2 hydrate. The equilibrium conditions and the dissociation enthalpy of (CO 2 +TBAB) double hydrates were studied by Differential Thermal Analysis (DTA) and also calculated by the Clapeyron equation. The CO 2 pressure range was from 0.3 to 2.5 MPa. The TBAB concentration range was from 4.43 to 9.01 wt%. Low TBAB concentrations were chosen in order to reduce the amount of TBAB in the case of environmentfriendly refrigeration applications.

One of the objectives of this work is to show that the formation pressures of (CO 2 +TBAB) hydrates are sufficiently low to be suitable for refrigeration application conditions. Moreover, the dissociation enthalpy of (CO 2 +TBAB) hydrates has to be sufficiently high. Finally, (CO 2 +TBAB) hydrates have to be stable at higher temperature than TBAB hydrate in order to be formed by CO 2 injection in a cold TBAB-water solution (non-mechanical process).

Experimental

Apparatus and materials

All the measurements were performed using DTA equipment, which was described in detail in a previous work [9]. The DTA device (Fig. 1) consisted of two symmetric transparent glass cells (inner diameter of 30 mm; volume of 46 cm 3 ) equipped with a magnetic stirrer and immersed in a temperature-controlled bath. The two cells were filled with TBAB aqueous solution and inert solution, respectively. A T-type thermocouple (± 0.3 K) was placed inside each cell, and two pressure gauges (0-4 MPa, ± 0.05% full scale) were used to measure the "in situ" fluid pressure.

Another eight T-type thermocouples were connected in series to measure the temperature difference between the two cells. This temperature difference was then converted into a DTA signal (mV). A syringe pump was equipped for quantifying the amount of injected CO 2 . HPLC grade TBAB (purity > 99 %, Fisher Scientific) and deionized water were used to prepare all the solutions.

Solubility of CO 2 in TBAB solution

The solubility of CO 2 in the TBAB solution was measured as follows: an amount of TBAB solution was introduced into the sample cell at 293.1 K. The sample cell was evacuated to remove the air, and then pressurized by CO 2 injection. The stirrer was used to accelerate the gas dissolution, resulting in a pressure decrease. When the CO 2 dissolution equilibrium was reached, pressure and temperature were recorded, and then temperature was changed to measure another condition. This protocol was applied for various pressures and temperatures. The mole fraction of CO 2 in TBAB solution could be calculated by Eq. ( 1):

L w L TBAB L CO L CO CO 2 2 2 n n n n x    (1)
where n is given by:

ZRT PV n n 2 2 2 CO tot CO L CO   (2)
where tot CO 2 n is the total number of moles of CO 2 in the system, which was determined using the syringe pump.

Differential thermal analysis

The system was cooled slowly (about 0.14 K•min -1 ) with agitation to 268.1 K in order to crystallize the sample. A marked pressure drop (accompanied by a temperature increase) indicated the exothermic formation of hydrates. Once all the liquid was converted into solids and the pressure of the system reached a steady state, the bath temperature was increased to melt the solids with a heating rate of 0.07 K•min -1 . The melting points were determined following two methods. For congruent melting, such as at a eutectic point, the phase-change pressure and temperature corresponded to the extrapolated onset of the DTA curve. For incongruent melting, e.g. of the (CO 2 +TBAB) double hydrate, progressive endothermic peak appeared during the

Equilibrium results and discussion

DTA experiments for the TBAB-water system

Before studying the CO 2 -TBAB-water system, the melting points of solids formed in the TBAB-water system in a TBAB concentration range from 4.43 to 9.01 wt% were measured and compared with literature data. Fig. 2 data related to the CO 2 solubility in pure water [12,13]. The CO 2 solubility in TBAB solution was slightly lower than that in pure water. This difference became larger at higher pressure. The solubility points also showed that the TBAB concentration had no significant effect on the CO 2 solubility in the TBAB concentration range from 4.43 to 10 wt% and in the pressure range from 0.5 to 1.6 MPa. Kamata et al. [14] reported that the CO 2 solubility in 10 wt% TBAB solution at atmospheric pressure was almost equal to that in pure water.

A c c e p t e d M a n u s c r i p t 6

Solid components in the CO 2 -TBAB-water system

Fig. 4 shows DTA and pressure curves in the heating process for the solids formed in the CO 2 -TBAB-water system. The initial conditions were 9.01 wt% for TBAB concentration, 2 MPa for pressure and 293.15 K for temperature. The first endothermic peak was due to the dissociation of the eutectic mixture. The eutectic mixture comprised type A TBAB hydrate (or double hydrate)

and ice (or CO 2 hydrate), since type B hydrate has no eutectic point [START_REF] Oyama | [END_REF]. During the eutectic melting, the pressure decreased firstly as consequence of gas volume increase as well as gas dissolution in the released liquid phase, and then it remained constant. The constant pressure suggested that only CO 2 -free solids (type A TBAB hydrate and ice) were contained in the eutectic mixture. Moreover, if CO 2 -containing hydrate existed in the eutectic mixture, an increase in initial pressure should lead to an increase in the congruent melting temperature. On the contrary, the experimental result showed that congruent melting temperature decreased from 272.8 to 271.8 K when the initial pressure increased from 0.5 to 2 MPa. This phenomenon was ascribed to the fact that the increase of the CO 2 amount in the liquid phase reduced the freezing point of water.

In Fig. 4, the following two peaks were considered to correspond to type A and B (CO 2 +TBAB) double hydrates, as CO 2 was released, resulting in a pressure increase. According to the phase diagram of TBAB hydrates, we deduced that type B (CO 2 +TBAB) double hydrates are more stable than type A in a TBAB concentration range from eutectic point to 19 wt%.

Therefore, the second and third endothermic peaks were due to type A and type B double hydrate, respectively. The pressure curve exhibited an inflection point, after which the pressure increased linearly with time. The pressure and temperature corresponding to the inflection point were the equilibrium conditions of the double hydrate. It was noted that all the equilibrium P-T data determined in this work were for type B double hydrate.

Equilibrium condition for CO 2 +TBAB double hydrate

Three-phase (H-L-V) equilibrium conditions for the CO 2 -TBAB-water system were not reported by other researchers except for Duc et al. [15]. Fig. 5 shows the experimental P-T data for the CO 2 -TBAB-water system determined by both Duc et al. [15] and in this work. It was observed that part of the data of Duc et al. [15] were consistent with our data. For example, at 279.4 K, the measured formation pressure at a TBAB concentration of 4.43 wt% (present work)

A c c e p t e d M a n u s c r i p t 7 was quite similar to that at 4.96 wt% obtained by Duc et al. [15]. In addition, two points from Duc et al. [15] for a TBAB concentration of 9.91 wt% were close to our curve at 9.01 wt%.

However, an important feature of the study of Duc et al. [15] was related to their claim that the TBAB concentration did not have a significant influence on the hydrate formation pressure in the TBAB concentration range from 4.96 to 65.04 wt%. The reason was that the equilibrium pressures of (CO 2 +TBAB) hydrate at 284 K in the two cases of TBAB concentrations of 4.96 and 65.04 wt% were very close, i.e. 0.84 and 0.815 MPa, respectively. Furthermore, for the points at 4.96, 9.91 and 65.04 wt%, the pressure just increased with temperature no matter what the concentration was. On the contrary, our results showed the following characteristics: the formation pressure of double hydrate was dependent on the concentration of TBAB between 4.43 and 9.01 wt%. Moreover, our work showed that the higher the TBAB concentration was, the lower the hydrate formation pressure was. Based on our results, we had a suspicion against remarks and results of Duc et al. [15]. Complementary analyses based on literature data are specified in the subsequent section.

Jager et al. [16] found that in the CH 4 +1,4-dioxane+water system, the equilibrium pressure of the hydrate showed a minimum at a concentration of 6 mol%, around the stoichiometric ratio of the soluble hydrate former and water (ratio of large sII cages to water). Based on their observation, we believed that, at a given temperature, the equilibrium pressure of (CO 2 +TBAB) double hydrates was lowest when TBAB concentration was stoichiometric, i.e. 40 and 32 wt% for type A and B hydrates, respectively, as shown in Fig. 2 [START_REF] Oyama | [END_REF]. This could be responsible for the closeness of equilibrium pressure for TBAB concentration at 4.96 and 65.04 wt% at 284 K in the work of Duc et al. [15].

On the other hand, for each concentration, only two or three points were measured by Duc et al. [15], which was not sufficient to illustrate clearly the influence of TBAB concentration on the equilibrium pressure. In addition, the type of double hydrate in the experimental results of Duc et al. [15] was not specified. In our opinions, the points for TBAB concentration of 4.96 and 9.91 wt% should be for type B double hydrate, while the points at 40.11 wt% should be for type A double hydrate, because type A hydrate was more stable than type B between 19 to 45 wt%.

The experimental data of three-phase (H-L-V) equilibria for type B double hydrate are presented in Table 2 and plotted in Fig. 6. This figure also shows the equilibrium data of Fig. 6 also showed that the reduction of CO 2 -hydrate formation pressure caused by TBAB addition was close to that caused by THF addition. This pressure reduction was approximately 80 %. However, the formation pressure of (CO 2 +TBAB) double hydrate showed a stronger dependency on temperature than that of (CO 2 +THF) hydrate. For example, the formation pressure of (CO 2 +TBAB) (9.01 wt %) hydrate increased from 0.391 to 2.274 MPa between 282.79 and 288.09 K, whereas the formation pressure for (CO 2 +THF) (10.16 wt%) hydrate increased from 0.507 to 1.655 MPa. Although the addition of a small amount of TBAB into water expanded the hydrate stability region, TBAB could have an inhibiting effect at certain composition or temperature conditions. Imai et al. [START_REF] Imai | [END_REF] reported that in the (HFC-32+TBAB(43 wt%)+water) system, TBAB changed from thermodynamic promoter to inhibitor for hydrate formation at 289.4 K.

Dissociation enthalpy results and discussion

Dissociation enthalpies of gas hydrates are usually obtained by two methods: direct calorimetric determination and indirect calculation using the Clapeyron or Clausius-Clapeyron equation and equilibrium P-T data. In this study, the dissociation enthalpy of (CO 2 +TBAB) double hydrate was determined by DTA, and the results were compared to those calculated by the Clapeyron equation. According to Davidson [START_REF] Davidson | Water: A Comprehensive Treatise[END_REF], TBAB semi-clathrate hydrate is stoichiometric, in contrast to gas hydrates. Moreover, no structural transition occurred when CO 2 was encaged in the semi-clathrate [19]. Consequently, the compositions of TBAB and water in the double hydrate were the same as those in the single TBAB hydrate, whereas the composition of CO 2 was unknown. Here, the formula of type B double hydrate was described by

aCO 2 •TBAB•38H 2 O.
Hereinafter, for simplicity, double hydrate referred to type B (CO 2 +TBAB) hydrate unless otherwise stated. V is given by:

Calorimetric method

DH DH TBAB w, L w w L TBAB TBAB tot CO 2 v n n M n M V V      (6)
where V tot is the total volume of the system; DH v is the molar volume of the double hydrate.

Obata et al. [4]

Composition of double hydrate

Eq. ( 4) and Eq. ( 5) allowed us to calculate the composition of CO 2 in the double hydrate, noted a. Table 3 lists the results of dissociation enthalpy and composition of the double hydrate.

The initial conditions of the sample were 9.01 wt% for TBAB concentration, 2 MPa for pressure and 293.15 K for temperature. The dissociation enthalpy of double hydrate was expressed as both per mass of hydrate and per mole of CO 2 . From Table 3, the value of H d of double hydrate expressed in kJ per mass of hydrate was lower than that of CO 2 hydrate but was close to that of ice. On the other hand, the value of H d of double hydrate expressed in kJ per mole of CO 2 was higher than that of CO 2 hydrate. The composition of double hydrate was evaluated to be 2.51CO 2 •TBAB•38H 2 O. We can note that the maximal value of a is 3 [20], corresponding to a fractional occupancy of dodecahedral cages with CO 2 molecule of 84%.

Clapeyron equation

Previous work pointed out that the Clapeyron equation could be used to calculate the dissociation enthalpy of gas hydrate as long as the system was univariant, as in the case of simple hydrates [21]. In the present work, the CO 2 -TBAB-water system is not monovariant. However, other authors used Clausius-Clapeyron equation for systems which were not monovariant such as the CH 4 -C 2 H 6 -water system [START_REF] Sloan | Clathrate hydrates of natural gases, 2nd[END_REF]22]. Considering that DTA results have to be compared to results obtained by another method, the Clapeyron equation is used as in previous work to calculate in a first approach the dissociation enthalpy of the double hydrate.

Theory analysis

When (CO 2 +TBAB) double hydrate is formed from the TBAB solution at stoichiometric concentration, all the water is converted into hydrate. Based on the formula of double hydrate evaluated in paragraph 4.1.3, the double-hydrate formation process can be written as:

O(h) 38H TBAB 2.51CO O(l) 38H TBAB (g) 2.51CO 2 2 2 2      (7)
When the mass fraction of TBAB is lower than the stoichiometric concentration (32 wt%), the water content exceeds the quantity needed to form double hydrate, suggesting that excess water is A c c e p t e d M a n u s c r i p t 12 also in equilibrium with double hydrate. In this case, the formation of double hydrate from gaseous CO 2 and TBAB solution can be expressed by Eq. ( 8):

O(l) 39)H (1/ O(h) 38H TBAB 2.51CO O(l) 1)H (1/ TBAB (g) 2.51CO 2 0 2 2 2 0 2         C C (8)
where C 0 is the initial molar fraction of TBAB in the liquid phase. However, Eq. ( 8) corresponds to an ideal state. In fact, the real state is that part of CO 2 in the vapour phase is dissolved in the liquid phase. Eq. ( 9) describes the real formation process for double hydrate, where "diss" relates to CO 2 dissolved in the liquid phase at equilibrium pressure and temperature.

O(l) 39)H (1/ (diss) CO O(h) 38H TBAB 2.51CO O(l) 1)H (1/ TBAB (diss) CO (g) CO 2 0 2 2 2 2 0 2 2           C e C d b (9)

Determination of H d using Clapeyron equation

In the presence of excess water, according to Eq. ( 9), the dissociation enthalpy of double hydrate is calculated by the Clapeyron equation in the following form:

V T H dT dP    f (10)
The total volume change V related to Eqs. ( 9) and ( 10) was calculated based on Anderson's work [8], as follows:

2 2 2 CO TBAB, w, V CO CO w, DH V bv V V V      (11)
where V CO 2 v denotes the molar volume of CO 2 in the vapour phase at T and P;

V

are the volume of the water and aqueous TBAB saturated with CO 2 , respectively; the coefficient b is given by the following equation based on the mass balance of CO 2 in Eq. ( 9):

d e b    51 . 2 (12)
where d and e are the number of moles of CO 2 dissolved at T and P in the TBAB solution and pure water, respectively. These values could be determined using the measured CO 2 solubility in TBAB solution as well as in water (literature data).

V

can be calculated by Eq. ( 13) and ( 14), respectively:

w 0 L CO CO w, ) 39 / 1 ( 2 2 v C v e V      (13) TBAB w, w 0 TBAB L CO CO TBAB, w, ) 1 / 1 ( 2 2  M C M v d V      (14)
where L CO 2 v is the partial molar volume of CO 2 in liquid phase, which is assumed to be the partial molar volume of CO 2 in pure water, i.e. 3.84e -5 m 3 •mol -1 [8]. w v is the molar volume of water.

The results for the dissociation enthalpy of (CO 2 +TBAB) double hydrate are given in Table 4.

In addition, dissociation enthalpies related to other CO 2 containing mixtures as CO 2 -THF-water and CO 2 -N 2 -THF-water systems are listed in Table 4.

Comparison between calorimetric and Clapeyron equation results

In this study, no experimental data obtained by DTA is available at a TBAB concentration of 4.43 wt%, because the endothermic peaks of eutectic and double hydrate are overlapping seriously. For 9.01 wt% TBAB solution, H d obtained from the Clapeyron equation at 282.8 K was found to be about 45 % higher than the value measured by DTA (Table 4). This discrepancy can be explained by the following arguments:

(1) The TBAB concentration has an evident effect on the dissociation enthalpy of the double hydrate between 4.43 and 9.01 wt%, as shown by the results from the Clapeyron equation in Table 4. This behaviour was previously observed with THF-containing hydrates [23]. However, in the case of DTA experiments, the TBAB concentration varied from the eutectic point to the initial state (9.01 wt%) during the melting of double hydrate. This can partly explain the discrepancy between DTA results and results based on the Clapeyron equation at constant TBAB concentration.

(2) Another argument could be related to the accuracy of equilibrium curve. This possibility might lead to uncertainties of the dissociation enthalpy obtained from the Clapeyron equation.

For instance, if the equilibrium temperature corresponding to 0.391 MPa is shifted from 282.8 to 282.5 K, the dissociation enthalpy of hydrate would decrease by about 8%.

A c c e p t e d M a n u s c r i p t 14

(3) It is proposed that 80% of the hydrate dissociation enthalpy is due to the strength of water hydrogen bonds in the hydrate structure [22]. The hydrate dissociation enthalpy is thus strongly related to the hydration number. Consequently, the ratio between the hydration numbers of CO 2 hydrate (7.23) and (CO 2 +TBAB) hydrates (38/2.51=15.1) should be close to the ratio between the dissociation enthalpies of CO 2 and (CO 2 +TBAB) hydrates (Table 4). This assumption is confirmed in the case of calorimetric data but not in the case of the Clapeyron equation results.

(4) Finally, the analysis based on the use of Clapeyron equation may be not completely appropriate to our system. This analysis includes the assumption related to monovariant systems (introduction of part 4.2) and the definition of volume change.

Conclusions

The three phase (H-L-V) equilibrium conditions of hydrates formed in the CO 2 -TBAB-water system were investigated in this work by using a DTA device in a TBAB concentration range from 4.43 to 9.01wt% and in a CO 2 pressure range from 0.3 to 2.5 MPa. The results showed that the addition of TBAB reduced the formation pressures of CO 2 hydrate by approximately 80%.

This leads to values below 1 MPa in the temperature range of 279 -285 K, which fulfilled the requirement of air-conditioning system. Moreover, a higher concentration of TBAB led to higher pressure reduction of the double hydrate.

Based on the solids formed from a 9 wt% TBAB solution at 2 MPa CO 2 pressure, the dissociation enthalpy of CO 2 +TBAB double hydrate measured by DTA was 313.2 kJ per kg double hydrate, lower than that of CO 2 hydrate but close to that of ice. In addition, DTA results

showed that the composition of double hydrate was 2.51CO 9.91 wt% [15] 40.11 wt% [15] 65.04 wt% [15] 4.43 wt% Present work

A c c e p t e d M a n u s c r i p t 5 heating

 5 process. The phase-change pressure and temperature corresponded to the end of the last DTA peak. The measurements were performed at several initial pressures and then a hydrateliquid-vapour equilibrium curve was obtained. At least 2 parallel runs were conducted to guarantee the accuracy of the experimental data.

Fig. 3

 3 Fig. 3 shows the CO 2 solubility (in mole fraction) in TBAB solutions for various TBAB concentrations (range from 4.43 to 10 wt%) as a function of pressure. Fig. 3 also shows literature

A c c e p t e d M a n u s c r i p t 8 CO 2

 82 hydrate and (CO 2 +THF) hydrate for comparison[11,[START_REF] Sloan | Clathrate hydrates of natural gases, 2nd[END_REF]. From Fig.6, a substantial decrease in CO 2 -hydrate formation pressure was observed in the presence of TBAB. The pressure was reduced from 3.495 to 0.763 MPa at 281.46 K when 4.43 wt% of TBAB was added in the aqueous solution.

A c c e p t e d M a n u s c r i p t 9 InM a n u s c r i p t 10 TBABm 2 n 2 n

 91022 order to determine the dissociation enthalpy of double hydrate, the quantity of double hydrate and the heat absorbed during its dissociation had to be accurately measured. As mentioned previously, the third endothermic peak in Fig.4represented the dissociation of double hydrate; the amount of absorbed heat could be obtained by integrating the DTA signal (calibrated with ice) with time.4.1.1. Quantity of double hydrate n DHAfter the entire sample crystallized, the solids obtained were the eutectic mixture (type A TBAB hydrate and ice), type A double hydrate and type B double hydrate. From Fig.4, it was noticed that the peak area of type A double hydrate was considerably smaller than that of type B double hydrate and eutectic, implying the quantity of type A double hydrate in the solids can be neglected. If one assumed that before melting, the solids were only type B double hydrate and eutectic mixture (type A TBAB hydrate and ice), the lever principle could allow us to calculate the masses of TBAB contained in type B double hydrate and type A TBAB hydrate, and also the mass of ice in the eutectic mixture. Then, various mass fractions could be calculated by the following equation set: fraction expressing the mass of TBAB in type B double hydrate divided by the total mass of TBAB and water in the system; A TBAB w is the mass fraction expressing the mass of TBAB in type A TBAB hydrate divided by the total mass of TBAB and water in the system; ICE w is the mass fraction expressing the mass of ice in the eutectic mixture divided by the total mass of TBAB and water in the system; A S w and B S w are the stoichiometric mass fractions of A c c e p t e d in type A and B TBAB hydrate, respectively; E w is the eutectic mass fraction of TBAB inthe TBAB-water system; 0 w is the total TBAB mass fraction in the system at initial state.From Equation set (3), the number of moles of double hydrate DH n , which is the same as that of TBAB in the double hydrate DH TBAB n , could be calculated as follows: are the total mass of TBAB and water in the system, respectively; M TBAB is molecular weight of TBAB. 4.1.2. Quantity of CO 2 in double hydrate DH CO After the eutectic mixture melted, the system was composed of TBAB liquid solution (at eutectic concentration) saturated with CO 2 , double hydrate and vapour. The CO 2 quantity in the double hydrate DH CO could be calculated from PVT data and the CO 2 solubility in the TBAB solution, as shown in Eq. (P and T are the pressure and temperature corresponding to the onset of the DTA peak of the double hydrate; L TBAB n and L w n are the number of moles of TBAB and water in the liquid phase, respectively. They could be obtained from Eq. (3). The gas volume 2 CO

A c c e p t e d M a n u s c r i p t 11 4. 1 .

 111

A c c e p t e d M a n u s c r i p t 13 The

 13

A c c e p t e d M a n u s c r i p t 15 argumentsA c c e p t e d M a n u s c r i p t 18 Caption

 1518 against the application of Clapeyron equation in our case such as non-monovariant system and differentiation accuracy. In order to check the enthalpy results, the differential scanning calorimetry (DSC) method will be used to measure the dissociation enthalpy of CO 2 +TBAB double hydrate in a further work. Moreover, the definitions related to the use of Clapeyron equation in our system could be improved. Nomenclature a molar ratio of CO 2 to TBAB in type B double hydrate C 0 initial molar fraction of TBAB in the solution H-L-V hydrate-liquid-vapour M molecular weight (kg•mol -1 ) of TBAB in the liquid phase at initial state E w mass fraction of TBAB in the liquid phase at eutectic state ICE w mass of ice in eutectic mixture divided by the total mass of TBAB and water in the system A S w stoichiometric mass fraction of TBAB in type A TBAB hydrate B S w stoichiometric mass fraction of TBAB in type B TBAB hydrate A TBAB w mass of TBAB in eutectic mixture divided by total mass of TBAB and water in the system B TBAB w mass of TBAB in type B double hydrate divided by total mass of TBAB and water in the system Z gas compressibility factor Greek symbol  density (kg•m -3 ) H d dissociation enthalpy (kJ•mol -1 ) or (kJ•kg -1 ) A c c e p t e d M a n u s c r i p t 16 H f formation enthalpy (kJ•mol -1 ) or (kJ•kg -1 ) , CO 2 TBAB solution saturated with CO 2 of tables and figures

Fig. 2 .

 2 Fig.2. Phase diagram of TBAB hydrates at atmospheric pressure[START_REF] Oyama | [END_REF].

Fig. 3 .Fig. 4 .

 34 Fig. 3. Solubility of CO 2 in TBAB aqueous at 283.1 K. Fig. 4. DTA and pressure profiles in heating for the solids formed from CO 2 -TBAB-water system (initial TBAB concentration: 9.01 wt%, initial CO 2 pressure: 2 MPa, initial temperature: 293.15 K).

Fig. 5 .

 5 Fig. 5. Equilibrium conditions of hydrates in CO 2 -TBAB-water system.

Fig. 6 .

 6 Fig. 6. Equilibrium conditions of CO 2 hydrate in various systems.

A c c e p t e d M a n u s c r i p t 20

 20

  Fig. 1

Fig. 3

 3 Fig.3

M a n u s c r i p t 26 -

 26 Fig.4

Fig. 5

 5 Fig.5

  2

Table 1

 1 Phase-change temperatures for the solids in TBAB-water system Table2Three-phase (H-L-V) equilibrium conditions for CO 2 +TBAB double hydrate at different TBAB concentrations

Table 3

 3 Comparison of dissociation enthalpies for ice and hydrates of aCO 2 •TBAB•38H 2 O, CO 2 •nb h H 2 O

Table 4

 4 Measured and reported dissociation enthalpies of different hydrates obtained by calorimetric method and

	Clapeyron / Clausius-Clapeyron equation
	Fig. 1. Experimental DTA device: (1) sample cell; (2) reference cell; (3) temperature-controlled bath;
	(4) cooling/heating unit; (5) syringe pump; (6) CO 2 bottle; (7) acquisition interface; (8) thermocouples and
	differential thermal analysis; (9) pressure gauges.

Table 2 Mass

 2 T=273.65 K, for THF+CO 2 +N 2 gas composition is 17 mol%CO 2 +83 mol%N 2 .

			fraction, TBAB (wt%)
	9.01		7.02		4.43
	T (K)	P (MPa)	T (K)	P (MPa)	T (K)	P (MPa)
	282.79	0.391	282.59	0.379	279.40	0.344
	283.34	0.393	282.72	0.379	280.19	0.515
	284.43	0.583	283.40	0.578	280.56	0.513
	285.07	0.859	283.58	0.575	281.46	0.763
	285.33	0.866	284.58	0.842	282.91	1.172
	286.38	1.333	285.38	1.216	283.16	1.179
	286.53	1.339	286.14	1.667	284.66	1.715
	287.6	1.803	286.96	2.199
	288.09	2.274
	Average absolute deviation for pressure results (AADP%): 5%

a : Current work, T=282.8 K. b : Ref. [23], c : Ref. [8], T=283.1 K. d : Ref. [11], T=280 K, calculated with Clausius-Clapeyron equation.