Philippe Langevin
email: langevin@univ-tln.fr

ESTIMATION OF SOME EXPONENTIAL SUM BY MEANS OF q-DEGREE VAL ÉRIE GILLOT AND

Keywords: spectral amplitude, monomial, binomial, Fourier coefficient, exponential sum, Weil bound, Carlitz-Uchiyama bound, Deligne bound

In this paper, we improve results of Gillot, Kumar and Moreno to estimate some exponential sums by means of q-degrees. The method consists in applying suitable elementary transformations to see an exponential sum over a finite field as an exponential sum over a product of subfields in order to apply Deligne bound. In particular, we obtain new results on the spectral amplitude of some monomials.

Introduction

Exponential sums and bounds for them are exploited by coding theorists and communications engineers [START_REF] Paterson | Applications of Exponential Sums in Communications Theory[END_REF]. The minimal distance of dual BCH codes and other cyclic codes can be evaluated in terms of exponential sums. They are also useful in the study of sequences with small correlations, for spread-spectrum and other communications applications. In both case, the estimation of exponential sums is often a key point for the construction of a good code. In this paper, we focus our-self in the estimation of exponential sums over finite fields for some polynomials. First, we begin introducing the tools and known results about spectral amplitude. Then, we generalise results of Kumar and Moreno [START_REF] Kumar | Polyphase sequences with periodic correlation properties better than binary sequences[END_REF] over spectral amplitude of some monomials in odd characteristic. We define the q-degree and the principle of multivariate point of view to be able to apply the results of Deligne [START_REF] Deligne | La conjecture de Weil I[END_REF] for exponential sums over a product of finite fields. Thus, we obtain a bound in terms of q-degree generalising the result of Gillot [START_REF] Gillot | Bounds for exponential sums over finite fields[END_REF].

Spectral Amplitude

All along the paper, L denotes an extension of degree m of a finite field K of order q and characteristic p. The Fourier coefficient of a polynomial f (X) ∈ L[X] at a ∈ L is by definition equal to the value of the exponential sum:

The authors would like to thank Regis Blache, Nicolas Katz and Igor Shparlinsky for theirs useful comments and suggestions while the preparation of the manuscript. where µ L denotes the canonical additive character of L. The maximal value that can take the absolute value of the Fourier coefficients is often called the spectral amplitude of f , we will use the notation R L (f ) = max a∈L | f (a)|. A very difficult question coming from coding theory and cryptography consists in finding polynomials having a small spectral amplitude. If the degree, say d, of f is not divisible by p then the spectral amplitude of f is upper-bounded by the Carlitz-Uchiyama bound :

(2) R L (f ) ≤ (d -1) √ q m .
For each a ∈ L, applying Weil Theorem to the Artin-Schreier curve

y p -y = f (x) + ax of genus g = (p-1)(d-1)

2

, there exists 2g Weil numbers ω i of absolute value √ q m such that Fourier coefficient of f at a in a finite extension L r of degree r of L is given by:

x∈Lr µ Lr f (x) + ax = - 2g i=1 ω r i .
It follows that the bound (2) is optimal in the sense that, when a ∈ L and f (X) ∈ L[X] are fixed, there exists an infinite sequence of finite extensions (L k ) k∈N of the field L of increasing degree r k such that

| x∈L k µ L k f (x) + ax | ∼ R L k (f ) ∼ (d -1) √ q mr k .
On an other side, the Parseval relation :

(

) b∈L | f (b)| 2 = q 2m , implies that √ q m ≤ R L (f ). 3
There exists polynomials of spectral amplitude √ q m , they define generalised bent functions, see for example [START_REF] Kumar | Welch Generalized bent functions and their properties[END_REF].

Remark that only a very small number of bent functions of monomial form are known, see [START_REF] Helleseth | Monomial and quadratic bent functions over finite fields of odd characteristic[END_REF]. Moreover, if p = 2 then for all

f ∈ L[X] is fixed max b∈L × R L (bf ) ≥ 2q m .
This fact proved by Chabaud and Vaudenay [START_REF] Chabaud | Vaudenay Links between differential and linear cryptanalysis[END_REF] is remarked as a consequence of Sidel'nikov bound in [START_REF] Ph | On the Non-linearity of Power Functions[END_REF]. It is not true in odd characteristic as we will see in the next section. The goal of this paper consists in giving an upper-bound on the spectral amplitude of monomial (i.e. polynomial of the form bx d with b ∈ L × ). More precisely, denoting by S(a, b, d) the exponential sum x∈L µ L (bx d + ax), we will present a new upper-bound on

max b∈L × R L (bx d ) = max a∈L max b∈L × S(a, b, d) .
If δ denotes the gcd of d and q m -1 then, by an averaging argument, one can easily prove :

(δ -1)q m ≤ max b∈L × S(0, b, d) ≤ max a∈L max b∈L × S(a, b, d) .
Several authors as Vinogradov, Davenport and Heilbronn, Hardy and Littlewood, Hua and Vandiver, Akulinicev, Karatsuba, Carlitz gave general estimations on the magnitude of trigonometric sums involving binomials [START_REF] Lachaud | Exponential sums as discrete Fourier transform with invariant phase functions[END_REF]. The exponential sum S(0, b, d) is a Gauss sum and

∀b ∈ L × , S(0, b, d) ≤ (δ -1) √ q m .
In the paper [START_REF] Lachaud | Exponential sums as discrete Fourier transform with invariant phase functions[END_REF], Lachaud generalises the inequality of Akulinicev to obtain the bound

∀a ∈ L, ∀b ∈ L × , S(a, b, d) ≤ q m √ δ
It is easy to verify that the fourth power moment method, used by Karatsuba [START_REF] Karatsuba | On estimates of complete trigonometric sums[END_REF] in the case of a prime field, works on the extension fields as well, leading to the estimation:

∀b ∈ L × , ∀a ∈ L × , S(a, b, d) ≤ (d -1) 1 4 q 3 4 m

q-degree

All the previous bounds do not take in consideration an important parameter that we will call the q-degree. Before giving a definition, let us analyse a paradigm example to introduce this notion. It is the case where p is odd and d = 1 + q r . The function Q b : x → Tr L/K (bx d ) is nothing other than a quadratic form whose the bi-linear associate form φ b is given by

φ b (x, y) = Q b (x + y) -Q b (x) -Q b (y) = Tr L/K b(x + y) d -bx d -by d = Tr L/K bxy q r + bx q r y
The general theory of quadratic forms tells us that the spectral amplitude of Q b takes the form q (m+κ(b))/2 , where κ(b) denotes the dimension of the radical of the K-space L with respect to the bi-linear form φ b (i.e. the space defined by

V b = {x ∈ L | ∀y ∈ L, φ b (x, y) = 0}). In this precise case, φ b (x, y) = Tr L/K (bx + b q r x q 2r )y q r ,
because the bi-linear form (x, y) → Tr L/K (xy) is non-degenerate, the kernel of Q b corresponds to the set of solutions of the linear equation [START_REF] Deligne | La conjecture de Weil I[END_REF] bx + b q r x q 2r = 0.

Assuming b = 0, the non-zero solutions are also solution of the equation x q 2r -1 = -b 1-q r in the multiplicative group of L. By the Euclidean algorithm, we know that gcd(q 2r -1, q m -1) = q ∆ -1 where ∆ = gcd(2r, m). If there exists a pair (x, b)

∈ L × × L × satisfying (4) then κ(b) = ∆ and thus max b∈L × R L (bx d ) = q m+∆ 2 .
The existence of a solution in ( 4) is equivalent to the fact that -1 is in the product of two groups L × q 2r -1 of order q m -1 q ∆ -1 , and L × q r -1 of order q m -1 q ∆ ′ -1 with ∆ ′ = gcd(r, m). This product of cyclic groups has order q m -1 q ∆ -1 × q ∆ -1 q ∆ ′ -1 , which is an odd number if and only if both m ∆ and ∆ ∆ ′ are odd. This case is equivalent to say that -1 is not in the product, and thus, all the quadratic forms Q b are non degenerate, we conclude

R L (bx d ) = q m 2 .
Proposition 3.1. Let q be odd, and let d = 1 + q r . If the dyadic valuation of r is greater or equal to the dyadic valuation of m then

∀b ∈ L × , R L (bx d ) = q m 2 . Otherwise max b∈L × R L (bx d ) = q m+∆ 2
where ∆ = gcd(2r, m).

Proof. Indeed, let us write r = 2 a r ′ and m = 2 b m ′ where m ′ and r ′ are odd integers. So that ∆ = 2 min(b,a+1) gcd(r ′ , m ′ ) and ∆ ′ = 2 min(b,a) gcd(r ′ , m ′ ) whence the dyadic valuation of m ∆ is equal to b -min(b, a + 1) and those of ∆ ∆ ′ is equal to min(b, a + 1) -min(b, a). These valuations are equal to 0 if and only if a ≥ b.

The above proposition generalises the result obtained by Kumar and Moreno in the section II of [START_REF] Kumar | Polyphase sequences with periodic correlation properties better than binary sequences[END_REF]. As there, it could be possible to give a complete description of the Fourier coefficient distribution by means of quadratic Gauss sums.

By definition the q-ary weight of a positive integer d < q m , denoted by wt q (d), is equal to sums of the digits

d 0 + d 1 + d 2 + . . . + d m-1 of the q-ary expansion of d = d 0 + d 1 q 1 + d 2 q 2 + . . . + d m-1 q m-1 .
The q-degree of a polynomial f is defined as the integer :

deg q (f ) = max d∈supp (f ) {wt q (d)}
where supp (f ) = {i | a i = 0} is the support of f (x) = i a i x i . As we will see in next section, the q-degree of f is nothing but the degree of a certain polynomial F in several variables, this is the explanation of the terminology q-degree used in the title.

Multivariate point of view

The principle of the multivariate method detailed in [START_REF] Gillot | Bounds for exponential sums over finite fields[END_REF][START_REF] Kumar | Polyphase sequences with periodic correlation properties better than binary sequences[END_REF], applied to a single variable polynomial f (X) ∈ L[X], consists in transforming the exponential sum:

(5) S(f, L) = x∈L µ L f (x)
in an exponential involving several variables. This is done by choosing an arbitrary basis

{β 1 , β 2 , . . . , β m } of L over K. S(f, L) = x 1 ,x 2 ,...,xm∈K µ L f (x 1 β 1 + • • • + x m β m ) = x 1 ,x 2 ,...,xm∈K µ K F (x 1 , x 2 , . . . , x m ) = S(F, K m )
where µ K is the canonical additive character of K and where F is the multivariate polynomial associate to f . The polynomial F is obtained by reduction modulo the ideal I = (X q 1 -X 1 , . . . , X q m -X m ) of the partial development of the trace operator:

(6) F (x 1 , . . . , x m ) = Tr L/K f (x 1 β 1 + • • • + x m β m ) mod I.
We use the Deligne bound, stated in [START_REF] Deligne | La conjecture de Weil I[END_REF] as follow, to evaluate S(F, K m ).

Theorem 4.1. Let Q be a polynomial in n variables with degree d over K. Let Q d be the homogeneous part of degree d of Q. Let ψ : K → C * be a non trivial additive character over K. Assume that

(i) d is prime to the characteristic of K; (ii) The homogeneous part Q d defines a smooth hypersurface H 0 in P n-1 (K). Then (7) | x 1 ,...,xn∈K ψ Q(x 1 , . . . , x n ) | ≤ (d -1) n q n/2 .
In most of cases, the homogeneous part of higher degree of F rises from the exponents with greatest q-ary weight in the support of f . Of course, the degree of F is nothing other than the q-degree of the polynomial f . In order to study the singularities of the hypersurface defined by F , we substitute x 1 β q i-1 and noting that the previous transformation is invertible, explicitly

1 + • • • + x m β q i-
x i = λ i y 1 + λ q i y 2 + • • • + λ q m-1
i y m where (λ i ) 1≤i≤m is the trace-dual basis of (β i ) 1≤i≤m (i.e. Tr L/K (β i λ j ) = δ ij , the Kronecker symbol), the study of singularities of F is reduced to those of φ.

In [START_REF] Gillot | Bounds for exponential sums over finite fields[END_REF], a bound for S(f, L) in terms of the q-ary weight for specific cases of degree of f is given, let us state this result with our notations in the following theorem. Theorem 4.2. Let f be a one-variable polynomial defined over L, such that f (x) = bx d + g(x) where d is the only exponent in the support of f with q-ary weight equal to deg q (f ).

Assume that d = 1 + d r q r with (p, d r ) = 1 then |S(f, L)| ≤ wt q (d) -1 m q m/2
Proof. See [START_REF] Gillot | Bounds for exponential sums over finite fields[END_REF].

A new exponential sum bound

In this section, the study of singularities of φ(y 1 , . . . , y m ) lead us to the characterisation of the exponents for which the multivariate method applies, generalising Theorem 4.2.

Lemma 5.1. Let φ the transformed polynomial associated to a monomial f (x) = bx d . If d has more than two digits in its q-ary expansion or if d = d k q k + d l q l with d k = 1 and d l = 1 then φ is singular.

Proof. For a monomial f (x) = bx d of degree d = d 0 + d 1 q + • • • + d m-1 q m-1 , we have F (x 1 , . . . , x m ) = m k=1 b q k-1 (x 1 β q k-1 1 + • • • + x m β q k-1 m ) d φ(y 1 , . . . , y m ) = m k=1 b q k-1 y d 0 k y d 1 k+1 . . . y d m-1 k+m-1 ,
where the indexes are calculated modulo m. In both cases, (1 : 0 . . . : 0) is a singular point of φ.

Lemma 5.2. Let φ be the transformed polynomial of f (x) = bx d . Assume that r and m are co-prime, d = d 0 + d r q r with d 0 = 1 or d r = 1. Then all the components of a singularity of φ are different from zero.

Proof. For d = d 0 +d r q r , from now let d 0 = 1 (the result remain true for the symmetric case d r = 1) to obtain φ(y 1 , . . . , y m ) = m i=1 b q i-1 y i y dr i+r and ∂φ ∂y j (y 1 , . . . , y m ) = b q j-1 y dr j+r + d r b q j-1-r y j-r y dr-1 j Assume that P a singularity of φ with y j = 0. Replacing y j by 0 in the partial derivative ∂φ ∂y j (P ), we obtain y j+r = 0. Now, replacing y j+r by 0 in the partial derivative ∂φ ∂y j+r (P ), we obtain y j+2r = 0. While reiterating the method, we obtain that the components of P are null for the positions {j, j + r, j + 2r, j + 3r, . . . , j + kr}. For (m, r) = 1, the smallest k such that kr = 0 mod m, is m, thus P has m components equal to zero. A contradiction is obtained and a singularity of φ cannot have a component equal to zero. Proof. see [START_REF] Gillot | Bounds for exponential sums over finite fields[END_REF].

Theorem 5.1. Let f (x) = bx d + g(x) ∈ L[x]
be a polynomial such that for any b ∈ L × the q-degree of f only depends on the term bx d that is deg q (f ) = wt q (d) > deg q (g). If (i) The q-ary expansion of d has only two digits d = d 0 + d r q r with d 0 = 1 or d r = 1, where r is any integer co-prime to m.

(ii) d m 0 = (-1) m d m r mod p then |S(f, L)| ≤ wt q (d) -1 m q m/2
Proof. Let F (x 1 , . . . , x m ) be the transformed (6) polynomial of f (x) = bx d + g(x). If d = d 0 + d r q r and (m, r) = 1 then, according to Lemma 5.3, the term Tr L/K (bx d ) is not equal to zero. In the particular case m = 2, the assumption (ii) of the theorem also gives Tr L/K (bx d ) = 0. In both cases, the homogeneous part of higher degree of the transformed polynomial F , say F d , only depends on the term bx d , since the q-degree of f is wt q (d), its degree is wt q (d) = d 0 + d r .

On the other hand, we can associate to F d the polynomial φ as in (8). Lemma 5.1 gives us the restriction on the case where the exponent d has only two digits in its q-ary expansion. Note that d k q k + d r q r = q k (d k + d r q r-k ), after the reduction (9), we just have to study integers of the form d 0 + d r q r , with d 0 = 1 or d r = 1.

The homogeneous polynomial F d satisfies the condition (i) of Deligne theorem since the second assumption implies that wt q (d) is prime to the field characteristic p. Using the simple form of

d = d 0 + d r q r , φ(y 1 , . . . , y m ) = m k=1 b q k-1 y d 0 k y dr k+r
Since the degree of the homogeneous form φ is prime to the characteristic of L, the singularities of φ correspond exactly to the non-zero solutions of partial derivative system [START_REF] Kumar | Polyphase sequences with periodic correlation properties better than binary sequences[END_REF] ∂φ ∂y i (y 1 , . . . , y m ) = 0, ∀i, 1 ≤ i ≤ m.

According to Lemma 5.2, we may assume that for all i, y i = 0. Multiplying the i th equation by y i , we obtain a new system [START_REF] Kumar | Welch Generalized bent functions and their properties[END_REF]

y i ∂φ ∂y i (y 1 , . . . , y m ) = d 0 b q i-1 y d 0 i y dr i+r + d r b q i-r-1 y d 0 i-r y dr i = 0, ∀i, 1 ≤ i ≤ m.
Changing, y d 0 i y dr i+r by z i , we obtain [START_REF] Lachaud | Exponential sums as discrete Fourier transform with invariant phase functions[END_REF]

y i ∂φ ∂y i (y 1 , . . . , y m ) = d 0 b q i-1 z i + d r b q i-r-1 z i-r = 0, ∀i, 1 ≤ i ≤ m.
The matrix of this system is

(b q j-1 d i-j ) 1≤i,j≤m with d i-j =      d 0 if i = j, d r if i -j = r, 0 otherwise.
Up to the norm factor m j=1 b q j-1 , the determinant of the previous matrix is

ζ m =1 (d 0 + d r ζ r ) If (-d 0 /d r )
is not a m-th root of unity modulo p, the system (12) has only one solution (0, . . . , 0). According to Lemma 5.2 this solution is not admissible since a singularity of φ cannot have a null component. Thus, under the conditions of the theorem the solution of (10) are trivial and φ is always smooth. Theorem 4.1 can be applied to the polynomial F of degree wt q (d) = d 0 + d r to obtain the bound in terms of the q-degree of f : q max b∈L × R L (bx d ) where L has order q 2 and q ≤ 100.

|S(f, L)| = |S(F, K m )| ≤ d 0 + d r -1 m q m/2 .

Numerical results and final remarks

If we apply Theorem 5.1 in the case of q-degree equal to 2, we refind a theorem of Kumar and Moreno [START_REF] Kumar | Polyphase sequences with periodic correlation properties better than binary sequences[END_REF], but this is also a consequence of Proposition 3.1. In particular, the bound is optimal.

In order to check the interest of the bound given in Theorem 5.1 in the case of q-degree equals to 3, we computed the true spectral amplitude of all the monomials bx d over a quadratic extension for the finite field of odd order q ≤ 100 with degree d = 2 + q. Note that (d, q 2 -1) = 1 or 3 according to q ≡ 2 mod 3 or q ≡ 1 mod 3, in particular S(0, b, d) ≤ 2q. For a such d, the conditions of the main theorem are fulfilled if and only if (p, 3) = 1, and in that case our bound claims

1 q max b∈L × R L (bx d ) ≤ 4. ( 13
)
The values are plotted in the graphic of Fig. 1. The numerical experiments show that the bound seems very good for all q ≡ 2 mod 3 but two times too large in the case q ≡ 2 mod 3. This last point is a probable consequence of cancellations of Weil numbers. It is interesting to notice that these exponential sums can be describe by means of the norm N L/K from L onto K, (

) x∈L µ L (ax + bxN L/K (x)), 14
As it has been pointed to us by Katz, one can use a trick of Deligne [START_REF] Deligne | Cohomologie étale des schémas[END_REF] (4.5 of Sommes trig.) to reduce to the split case in which L is no longer the quadratic extension of K, but rather is the product K × K, with trace function (x, y) → x + y, and the norm (x, y) → N ((x, y)) = xy. In an appropriate extension field E the sum ( 14) becomes Using a classical result on character sum with quadratic argument (see [START_REF] Lidl | Niederreiter Finite Fields[END_REF] Th. 5.33), we can express the inner sum in term of a quadratic Gauss sum G K (ψ, ν) involving the quadratic character of K.

= G K (ν, ψ)ν(-N ) x∈K × ψ(x 3 + ux + N v 2 4x
)ν(x) + qδ 0 (v).

Since the last hybrid sum, is a sum of 3 or 4 Weil's numbers, according to whether v = 0 or not, we get the previous estimation [START_REF] Ph | On the Non-linearity of Power Functions[END_REF]. All the other sums S(a, b, d) are estimated in a similar way. Numerically the case q ≡ 0 mod 3 and q ≡ 1 mod 3 seem very similar, and it will be nice to know when and how to avoid the technical hypothesis (ii) of Theorem 5.1 to obtain a more general bound independent of the characteristic of p. Similar transformations using the works of Adolphson and Sperber [START_REF] Adolphson | Exponential sums and Newton polyhedra: cohomology and estimates[END_REF] is probably a way to get answers but we reserve this approach for future researches.

1

 1

µ

  L f (x) + ax ,

Lemma 5 . 3 .

 53 For an integer d of the form d = d 0 + d r q r and for any b ∈ L × , we have Tr L/K (bx d ) = 0 for any x ∈ L if and only if r = m/2, d 0 = d r and Tr L/F q r (b) = 0.

Figure 1 .

 1 Figure 1. The value of 1q max b∈L × R L (bx d ) where L has order q 2 and q ≤ 100.

  x,y∈E µ E (a(x + y) + b(x + y)xy), and again Deligne's theorem applies as soon as p = 3 to show that the sum depends on four Weil numbers. Using a remark of Blache, one can avoid Deligne result to estimate the above sum, in the case [L : K] = 2, as follow. Let N be an non quadratic residue of K, and let ω ∈ L such that ω 2 = N . Using the basis {1, ω} to decompose a = u + ωv, b = s + ωt, and the elements of L as x + ωy, we haveTr L/K ((u + ωv)(x + ωy)) = 2ux -2N vy Tr L/K ((s + ωt)(x + ωy) q+2 ) = 2(sx -N ty)(x 2 -N y 2 ).In particular, denoting by ψ the composition of the character µ K by the multiplication by 2, S(a, 1, q + 2) = x,y∈K ψ(ux -vN y + x 3 -N xy 2 ) = x∈K ψ(x 3 + ux) y∈K ψ(-vN y -N xy 2 )


Vucko Flavien

Aoufi Asdin

Bosch Cédric

Delafosse David

Mobility and trapping of hydrogen in high-strength steel

Electrochemical permeation and thermo-desorption tests are performed to evaluate hydrogen mobility in high strength steel. Experimental parameters are used in a Krom like phenomenological diffusion model. This model is developed to simulate hydrogen diffusion and trapping in processing zones of specimens subjected to fatigue loadings.

Introduction

Hydrogen is known to be very detrimental to the mechanical properties of steels [1,2]. The sensitivity to hydrogen embrittlement increases with the strength of the material [3]. Many premature failures of high-strength steels in industrial applications are due to the presence of hydrogen. Hydrogen embrittlement mechanisms are closely linked to the mobility and the segregation of hydrogen in the material. Microstructural defects, as trapping sites, are particularly involved in these mechanisms as reservoirs for diffusible hydrogen (reversible traps) or as promoters of its mobility (mobile traps like mobile dislocations [4]). A phenomenological model was proposed by Krom [5] to simulate hydrogen diffusion and trapping. The model is well established for monotonic mechanical loading, but needs further development in cyclic strain conditions. Electrochemical permeation (EP) and thermal desorption spectroscopy (TDS) are two interesting techniques to study hydrogen mobility and trapping [6][7][8][9]. The first one informs us about the diffusion coefficient and the content of trapped hydrogen, and the second one gives access to the trapping energy. These methods are applied to a quenchedtempered martensitic steel S690QL in order to develop a model of diffusion/trapping during fatigue loading. Additional analyses are used to establish evolution laws for trapping sites. This paper is divided into three parts. The first part is related to the material and experimental procedures. The second part focuses on the diffusion model laws constructed from experimental data. Finite element simulations are described in the third part.

Material and experimental procedures 2.1 Material

The material of the study is a high strength steel S690QL (EN 10137-2). It exhibits a tempered martensitic microstructure consisting of laths of about 200 nm wide. Laths are gathered in blocks and packets which are included in prior austenite grain with a size ranging from 10 to 20 µm. The steel chemical composition is given in table 1. The tempering duration and temperature were two hours at 550°C after austenitization at 920°C. No retained austenite was revealed by X-ray analysis. This kind of microstructure exhibits many heterogeneities which are hydrogen trapping sites (see figure 1) with characteristic energies. Table 1: Chemical composition of steel S690QL (weight %)

Electrochemical permeation

EP tests are performed in Devanathan and Stachurski [8] link device consisting of a charging cell and an oxidation (or detection) cell. Both cells are filled with 0.1 M sodium chloride solution deoxygenated by nitrogen. The steel sample is cylindrical membrane with a thickness of 0.6 mm and a diameter of 40 mm. Mechanical polishing with SiC paper is performed on both faces up to grade 4000. A thin palladium coating (230 nm) is deposited on detection side by an electrochemical process. The membrane is mounted in a holder which reduces the contact surface to a circular area of 1 cm² on both faces. Platinum counter electrode and saturated calomel electrode are used in each cell and the steel sample is the common working electrode. A floating ground galvanostat is plugged to the charging cell and a potentiostat to the detection side. The detection side is continuously polarized at -100mV/ECS. The charging side of the sample is subjected to a cathodic current (-5 to -30 mA/cm²). The recorded anodic current on the detection side is the permeation rate of hydrogen through the membrane.

Thermal desorption spectrometry

TDS experiments are widely used to study hydrogen trapping in metals [7,10,11]. This technique consists in heating a metal sample at a determined rate in vacuum (<10 -5 mbar) and recording the amount of gas desorbed with a mass spectrometer. Hydrogen pre-charging is performed by cathodic polarization in a 30 g.L -1 NaCl solution at E c =-1200 mV/SCE for 64 hours. The solution was de-aerated and buffered at pH 4.5 by bubbling a gas mixture of N 2 -7%CO 2 . Cylindrical samples are machined from tensile or fatigue specimens. As demonstrated in literature [10,12], analytical models based on Kissinger equilibrium [13] are quite accurate for low hydrogen concentration and low trapping site occupancy, i.e. low trapping energy. Choo and al. [7] demonstrated that detrapping energy E TL can be related to the temperature at the maximum desorption peak T p , i.e. the maximum hydrogen desorption rate, with the equation (Eq.

    R E T T a TL p p     1 ln
where a is the heating rate of the thermal desorption test. As a consequence, trapping energies are determined by testing samples at different heating rates. But spectrum deconvolution is necessary to separate each peak.

Additional analyses

Plastic strain controlled low cycle fatigue tests are analysed using Handfield and Dickson's method [14]. The internal stress X, the thermal effective stress Σ* and the athermal effective stress Σ μ are determined for each hysteresis cycle. These stresses are related to microstructural defects evolution. Dislocation density is investigated by X-Ray line profile analysis [15]. Indeed, peak broadening is related to dislocation density. The momentum method [16][17][18] is applied to determine the dislocation density evolution during cyclic loading. Data are recorded using Panalytical MRD device with Co K α radiation (1.790 Å) and optics equipped with a slit of 1/8°. Microstructural investigation by transmission electron microscopy (TEM) on fatigue specimens is performed using a PHILIPS FEI CM 200 device operating at 200 kV. Thin samples with a diameter of 3 mm and a thickness of 100 µm are machined from fatigue specimens by saw-wire cutting. Final thickness (< 200 nm) is obtained by electropolishing in 10%-perchloric acid with ethanol at 15 V for about 40 s.

Diffusion model

The diffusion equation of the model is given by Krom [5]:

( ) ̇ Eq.2
For cyclic loading, the number of traps N T is a function of both the plastic strain ε p and the cumulative plastic strain p. Evolution laws are determined for each kind of trap.

Five different traps are detected by TDS analyses. Their trapping energies are reported in table 2. Total hydrogen concentration is also determined by katharometry and its concentration in each trap is given by TDS peak area ratios. With the assumption that the steel lattice is filled with hydrogen in our charging conditions, the number of traps can be directly extracted from this data.

Table 2: Trapping sites energies from TDS TDS peak 1 2 3 4 5 Energy (eV) 0.14+/-0.05 0.11+/-0.02 0.23+/-0.05 0.70+/-0.10 0.64+/-0.09 Traps are attributed to possible microstructural defects. Some of them are strongly influenced by plastic strain (peaks 2 and 3) and probably related to dislocation density and cells formation respectively. The model is mainly focused on these defects. Dislocation density is determined by X-Ray diffraction and cell walls developed during fatigue loading are measured by both TEM and X-Ray diffraction. Their evolutions are modeled using equations similar to those used by Kumnick and Johnson [19] to fit experimental data: As shown in figure 2, the global dislocation density decreases with plastic strain accumulation, whereas cell walls density increases until saturation state.

| | Eq.3 Eq.4

Finite elements simulations 4.1 Two traps model

One-dimensional model is developed to compute diffusion and trapping of hydrogen with two different trapping energies. The geometry is constructed to simulate electrochemical permeation test with a main part (Ω L =[0,e L ]) of e L =600 µm and a thin layer, called "coating", (Ω c =[e L ,e L +e c ]) of e c =0.23 µm. Hydrogen diffusion is nonlinear in the main part as proposed by Sofronis [20]. Equation is presented below.

Mechanical loading

Hydrogen diffusion is affected by the elasticity and the plasticity. A two-dimensional model is currently developed to take into account mechanics in the model. The geometry is representative of hydrogen assisted cracking test specimens, with micronotch as presented in figure 4. Hydrogen concentration at the notch-tip is expected during cyclic loading.

Conclusion

Hydrogen mobility and trapping in high-strength steel is studied by electrochemical permeation and thermal desorption spectrometry. Five different trapping energies are detected and related to microstructural defects. An analytical relationship between plastic strain and trap density is used for the mechanical/diffusion coupling. The diffusion coefficient is weakly affected by trapped hydrogen in low trapping energy sites. Finite element analysis is under investigation to fully simulate hydrogen assisted cracking tests.

Figure 1 :

 1 Figure 1: Trapping sites in quench-tempered microstructure

Figure 2 :

 2 Figure 2: Evolution of the dislocation density from equation 3 for low cycle fatigue tests at half strain amplitude of a) 0.5% and b) 0.2%. c) Evolution of the cell walls density from equation 4.

5 C∑. 6 D

 56 T and C L are respectively the trapped and lattice hydrogen concentration. N T is the number of trapping sites. The apparent lattice diffusion coefficient D eff is calculated from Eq.6:EqL is the true lattice diffusion coefficient without any trapping. The diffusion coefficient is constant in the coating and no trapping occurs. Diffusion respects the 2 nd Fick's law. Different boundary conditions are tested at the entrance and at the exit sides to match experimental data. At the interface x=e L , flux continuity with or without concentration continuity is tested. Concentration and diffusion coefficient profiles are presented in figure3. These simulations are computed with two trapping energies 0.11 and 0.23 eV. As shown in figure3b, the diffusion coefficient is weakly affected by trapped hydrogen, and quite similar, except the maximum value. These profiles are strongly changed for higher trapping energies.

Figure 3 :

 3 Figure 3: a) Concentration profiles of lattice hydrogen. b) Evolution of the apparent diffusion coefficient in the steel. c) and d) Concentration profiles of trapped hydrogen.

Figure 4 :

 4 Figure 4: Geometry of notched specimens


Sylvie Renaud

Bogdan Catargi

Jochen Lang

Biosensors in diabetes -how to get the most out of evolution and transpose it into a signal

The increase in diabetes is a major threat to health and to economic development in the 21st century as stated by the UN in the Millennium Development Goals. In 2013 380 million people had diabetes and almost 600 million are expected by 2035. Diabetes is the major cause of non-traumatic amputation and of blindness in developed countries. It is also an expansive disease. In the US one in five dollars for health care is spent on diabetes. Net losses in national income from diabetes amount to hundreds of billions international dollars in China or India. Clearly novel therapies are required.

The burden for the patients, their carers and the national economies drives the search for novel treatments as new technologies may provide crucial human and economic benefits. One recent approach is continuous health monitoring, such as continuous glucose monitoring (CGM). A striking example is the recent proposal by Google: to implant a biosensor into contact lenses to determine tear glucose levels and transmit the data to a pump in the patient's body. The pump will inject the required level of insulin, the lacking hormone. This device circumvents multiple daily blood tests and injections in certain types of diabetes. Is the problem solved by contact lenses? Can all diabetic patients quote Humphrey Bogart's Rick "Here's looking at you, kid" and get their blood values straight? Well, life may be more complex.

Let's have a look what is diabetes to grasp the problem. We distinguish two major forms of the disease: type 1, caused by destruction of insulin-producing beta-cells in the pancreatic islets, and type 2, caused by insufficient insulin release and reduced efficacy of the hormone. Type 2, the major form, is found in elder patients and a panel of life style interventions and drug therapies are available. In contrast, type 1 declares earlier in life and needs hormone replacement therapy by insulin injections. As insulin is extremely potent and can cause brain damage and death by hypoglycemia, glucose levels have to be controlled very precisely, requiring blood samples to be taken, a cumbersome burden for patients. Although fewer patients are concerned by type 1 diabetes, still some 80.000 children develop the disease each year and numbers are on the rise. This situation provides a strong impetus to develop continuous glucose monitoring via sensors that determine the amount of insulin to be injected. A number of different physicochemical methods have been explored for non-invasive approaches: surface-enhanced Raman, thermal or impedance spectroscopy, fluorescence, reverse iontophoresis, photo-acoustic or electromagnetic sensing. In general, these approaches are elegant but suffer from lack in specificity, problems in miniaturization or provoke major local reactions impeding their use.

Accepted medical technology for continuous glucose monitoring uses the protein glucose oxidase that generates an oxidation current upon reaction with glucose. Grafting the enzyme on an electrode allows to determine the current and hence glucose levels. First clinical devices were employed in hospitals in the 1970s leading to the concept of the artificial pancreas: an implantable device measures interstitial blood glucose and drives an insulin pump that delivers the hormone via a subcutaneous needle. The first portable continuous glucose monitors (CGM) to display real-time values became available in 2006. Recent data suggest that these devices not only lift an awesome burden from the patients but also improve their long-term health, an important argument for reimbursement by health insurances. They still show a number of drawbacks such as considerable lag-time and are less accurate in hypoglycemia, a situation that may be life-threatening. Moreover, they work fine in between meals, but are not able to cope automatically with the rapid increase in glucose after a meal. As they tend to respond late and overdose, this may lead to a down in blood glucose at later time points. A number of smart algorithms have been developed to improve CGM and insulin therapy. However, current devices work essentially as an open loop because of the difficulty to find the adequate pattern to master the complexity of life.

Can the biosensor at the base of the artificial pancreas be improved? Current approaches rely on a small building block of cells, a protein. Can we respond better to complexity if we step up one level in the "bio" part and move to cells? Indeed, the relevant cells or the micro-organ in our body has been shaped to its current function during half a billion of years by evolution. Thus part of the work is already done by nature and we may perform the next step, the bio engineering, in somewhat less time. With this concept in mind we started out recently to develop a novel type of a hybrid bio-electronic sensor in a multidisciplinary approach between dialectologists, biologists and microelectronics [START_REF] Bornat | Detection of Electrical Activity of Pancreatic Beta-cells Using Micro-electrode Arrays[END_REF].

How do pancreatic islet cells work (see Fig. 1)? The insulin-producing beta-cells in islets completely metabolize glucose, in a way they count each carbon of the molecule, and produce a novel molecule, ATP that is in precise correlation to the glucose levels outside the cell. ATP in turn will regulate a particular ion channel that depolarizes the cell, the K ATP , and allows calcium influx through calcium channels. Calcium influx will trigger the release of the precise amount of insulin, adapted to glucose levels. However, food and nutrient homeostasis is much more than sugar. The beta-cells will not only respond to and metabolize glucose, but also lipids and some amino acids and count their carbons. So different from current sensors, our -cells release insulin whether we pinch a bit of our kid's cotton candy or have a foie gras. More importantly, the cell integrates also a number of other signals, such as different hormones, that reflect the state and activity of the body. Jogging or trying to catch the train? You need your glucose and should not secrete much insulin. To this end the adrenal glands release epinephrine, that inhibits ion channels thus reducing ion fluxes. Therefore the electrical behavior of the cells, their field potentials, is the first signal that integrates clues from different nutrients and takes the state of the whole body into account. Interestingly the islets have two different major cell populations: beta-cells that secrete insulin and bring glucose down, and -cells, that secrete glucagon, a hormone that liberates glucose from the liver. -cells behave exactly inverse of -cells in electrical terms and recording both will considerably increase precision. If we manage to keep these cells alive for days to weeks and capture these very tiny electrical potentials, we may obtain a highly suitable signal to determine the demand for insulin.

We were able to record these signals in islets obtained from mice or most recently from human donors and found two distinct types of signals: short-lived action potentials, just as in nerve cells, lasting some tens of milliseconds, and slower undulations that we termed "slow waves". Whereas the action potentials represent the activity of a single cell, the slow waves represent the further effect of the cell-cell interactions as found in the islet micro-organ. The signals are finely tuned by glucose within the physiological range and fully regulated by the physiological levels of hormones [START_REF] Nguyen | A novel bioelectronic glucose sensor to process distinct electrical activities of pancreatic beta-cells[END_REF]. Thus, cells have followed long-ago the trend that generally comes up in the development of sensors: the implementation of multi-parametric sensors to adapt to diverse lifestyle and the multitude of daily events, and to improve accuracy and safety, a major concern for the use in humans.

Recording quality is a challenge: it must be high enough to discriminate slight differences resulting from slow changes in the physiology. We used multi-electrode arrays (MEA) that contain multiple plates/electrodes to connect the cells to electrical circuitry (Fig. 2A-B) [START_REF] Raoux | Non-invasive long-term and real-time analysis of endocrine cells on micro-electrode arrays[END_REF]. MEAs have typically an arrangement of 60 electrodes with a diameter between 10 and 20 µm. Cell preparations are cultured directly on the MEA. We designed custom integrated circuits able to process the 60 signals online and in parallel in analog then in digital mode. The circuits were embedded in an electronic board (Fig. 2C) providing on board data storage on SD cards and interfacing with standard peripherals using UART and VGA protocols. Data processing is of 2 types: first, conditioning of raw recorded data to extract significant signal markers (action potential and slow waves); second, evaluation of these markers to indicate changes in islets physiology. Thanks to the use of configurable circuits, evaluation can be adapted to the context and provides the user with quantitative information or go/no-go flags. Electronics were designed in low power submicron technology, sized to ensure continuous and real-time processing on up to 60 channels. A computer-interface is available to configure the system, launch recordings and store raw or processed data in user-specific formats.

Currently we are developing the technology for two distinct "lab-on-chip" applications: Diachip and Diasensor. Diachip will perform multi-parametric testing of islets, to be used outside the human body for a number of defined applications. For example, one treatment of unstable forms of type diabetes is transplantation of islets. Currently there is no quality control of donor islets that have to be transplanted within 6h. We have started a project with clinicians to compare transplantation outcome and islet quality. Another application concerns the differentiation of pluripotent stem cells into -cells. If this differentiation becomes possible and the whole process can be mastered reliably, diabetic patients, either of type 1 or of type 2, can be definitively cured by transplantation of these cells. Current procedures require the treatment of stem cells with very complex mixtures of biological factors, and their composition often changes daily. Diachip will be able to monitor their development on a non-invasive natural base and may also provide a useful element in quality control.

Our long-term goal, Diasensor, is even more ambitious and of high risk: the development of a novel and physiological sensor for an artificial pancreas and to free type 1 diabetic patients from cumbersome programming and complications inherent to glucose sensing only. Whereas islet transplantation needs several donor million islets, which are difficult to obtain, the sensor requires only a few dozen. Diasensor is essentially the core of a closed-loop controlling insulin delivery depending on the patient's state and nutrient levels. Closed-loop control is very common in machinery: cruise control in cars, temperature control at home (fig. 3A). Artificial pancreas based solely on CGM is a step towards closed-loop control for diabetes but still misses essential points: glucose-only dependence, open-loop via the CGM (fig. 3B). We expect Diasensor to act as a real closed-loop with few, ideally noneuser inputs. It will therefore necessitate an extremely robust feedback control, able to handle both fast and slow changes in the patient's physiology. The whole loop signal processing (conditioning, evaluation and feedback control) has to be finely designed and to be flexible enough to suit most patients (Fig. 3C). Diasensor will require highly performing microelectronic technology, as the implanted device should be of minimum size and weight and ultra-low power. Diasensor implies the use of cells that are foreign to the patient's body with the risk of immune reactions. Recently devices have been engineered that retrieve plasma from blood vessels and are positioned outside the patient's body. Considerable progress has also been achieved in material sciences to provide adequate support that ensures long survival of the cells. This field is a very active area and considerable further progress can be expected in the near future.

It took almost fifty years to develop glucose sensors adapted to therapy and they have revolutionized diabetes care. It is now time to go beyond the use of few constituents of cells, such as proteins, already marvels of evolution, and to exploit nature's more complex developments as given by a micro-organ such as the islets. Expertise gained in this field may also help neuroscience to come up with operational biosensors and artificial organ parts to provide relieve to a number of inborn or traumatic pathologies. In any case, the diabetes field, with its longstanding and well organized scientific, medical and patient communities, should be able to benefit from interdisciplinary work in biomedical engineering. In engineering, in a closed-loop control system, a sensor monitors the physical system output and feeds the data to a controller which adjusts the control as necessary to maintain the desired system output, i.e. the « reference ». A closed-loop system is generally modelled by the mathematical description of its dynamics. In blue, the example of the closed-loop temperature regulation when using a steam air heater? B. The artificial pancreas is a system that measures blood glucose levels using a continuous glucose monitoring (CGM) device, and transmits this information (wired or RF) to an controlled insulin pump (in grey) that calculates and releases the required amount of insulin into the body. C. Diaßsensor is a dynamic closed-loop system able to regulate insulin release based on the physiological response of healthy islets to nutrients and hormones from a diabetic patient. All loop items (biosensor, controller, insulin pumpgrey area) are to be integrated in low power implanted devices. External control via a PC or smartphone application will ensure inocuity and will control calibration phases.

Figure 1 .

 1 Figure 1. Sensing the demand in insulin: Biological sensors versus electrochemical sensor. Left part indicates the sensing of the demand in insulin by pancreatic beta-cells in response to nutrients and hormones, whereas current glucose-sensing electrodes respond only to glucose. Details see text.

Figure 2

 2 Figure 2 Two young members of the lab looking at a MEA. Note screen display of recorded activities.

Figure 3

 3 Figure3Open and closed loops. A. In engineering, in a closed-loop control system, a sensor monitors the physical system output and feeds the data to a controller which adjusts the control as necessary to maintain the desired system output, i.e. the « reference ». A closed-loop system is generally modelled by the mathematical description of its dynamics. In blue, the example of the closed-loop temperature regulation when using a steam air heater? B. The artificial pancreas is a system that measures blood glucose levels using a continuous glucose monitoring (CGM) device, and transmits this information (wired or RF) to an controlled insulin pump (in grey) that calculates and releases the required amount of insulin into the body. C. Diaßsensor is a dynamic closed-loop system able to regulate insulin release based on the physiological response of healthy islets to nutrients and hormones from a diabetic patient. All loop items (biosensor, controller, insulin pumpgrey area) are to be integrated in low power implanted devices. External control via a PC or smartphone application will ensure inocuity and will control calibration phases.

Figure 4 :

 4 Figure4: Current prototype. Photographs of (A) the center of a MEA with an array of 60 TiN electrodes (10µmx10µmm, 100µm spacing); wires connect the electrodes to external pads for electronic connection; cultured islets cells are visible on the array. (B) a MEA device (same as in (A). The circular reservoir for the culture is stacked at the center of expoxy laminated sheet. A ring of 60 pads allows connections to external circuits.(C) The electronics board for configurable recording an real time processing of islet signals. It can be used as a stand-alone device without any peripherals. Pre-and post-processed data are stored on SD cards on board. Configuration is possible using the on-board buttons and switches or using a dedicated software on an eternal PC. Serial and parallel outputs are avaiable for signals display or post-processing.

FIGURE 1 FIGURE 2 FIGURE 3 FIGURE 4

 1234 FIGURE 1