Julien Bureaux

Nathanaël Enriquez
email: nathanael.enriquez@u-paris10.fr

Nathana Ël Enriquez

On the number of lattice convex chains

Keywords: 2010 Mathematics Subject Classification. 05A16, 11P82, 52A22, 52C05, 60F05 convex polygons, grand canonical ensemble, zeta functions, local limit theorem, polyominoes

An asymptotic formula is presented for the number of planar lattice convex polygonal lines joining the origin to a distant point of the diagonal. The formula involves the non-trivial zeroes of the zeta function and leads to a necessary and sufficient condition for the Riemann Hypothesis to hold.

The paper deals with the enumeration of lattice increasing convex chains starting from the origin and ending at a point (n, n) of the diagonal. Convex chains are defined as finite sequences of points (x i , y i ) 0≤i≤k of Z 2 satisfying:

0 = x 0 < x 1 < x 2 < • • • < x k-1 ≤ x k , 0 = y 0 ≤ y 1 < y 2 < • • • < y k-1 < y k , 0 ≤ y 1 -y 0 x 1 -x 0 < • • • < y k -y k-1 x k -x k-1 ≤ +∞.
The point (x k , y k ) will be called the endpoint of the chain.

(0, 0) (x 2 , y 2 ) (x 3 , y 3 ) (x 4 , y 4 ) (x 5 , y 5 )

x y

(x 1 , y 1 )
In 1979, Arnol'd [START_REF] Arnol | Statistics of integral convex polygons[END_REF] considered the question of the number of equivalence classes of convex lattice polygons having a prescribed area (we say that two polygons having their vertices on Z 2 are equivalent if one is the image of the other by an affine automorphism of Z 2 ). Later, Vershik changed the constraint in this problem and raised the question of the number, and typical shape, of convex lattice polygons included in a large box [-n, n] 2 . The cornerstone in this problem is the estimation of the number p(n) of convex polygonal chains with vertices in (Z ∩ [0, n]) 2 and joining (0, 0) to (n, n) (or more generally (n, cn) for some c > 0). In 1994, a solution to this problem was found independently by Bárány [START_REF] Bárány | The limit shape of convex lattice polygons[END_REF], Vershik [START_REF] Vershik | The limit form of convex integral polygons and related problems[END_REF] and Sinaȋ [START_REF] Ya | A probabilistic approach to the analysis of the statistics of convex polygonal lines[END_REF]. Namely, they showed by different methods that

p(n) = exp 3κ 1/3 n 2/3 (1 + o(1)) ,
where

κ = ζ(3) ζ(2) ,
and that the limit shape of a typical convex polygonal chain is the arc of parabola tangent to the sides of the square, which maximizes the affine perimeter. Note that the approach of Sinaȋ was recently made rigorous and extended by Bogachev and Zarbaliev [4]. This is also this approach that we choose in this paper and we make use of a proposition of this latter paper. Here, we go further and fully exploit Sinaȋ's probabilistic model by giving an exact integral representation of the partition function and by making a precise asymptotic analysis of it.

The appearance of the values of the Riemann zeta function in the above formula suggests the arithmetic aspects of the problem. In this paper, we establish a connection between the combinatorial analysis of the number of convex chains and the zeroes of Riemann's zeta function.

Theorem. The number p(n) of lattice convex chains with endpoint (n, n) satisfies [START_REF] Arnol | Statistics of integral convex polygons[END_REF] p(n) ∼ e -2ζ (-1) (2π) , [START_REF] Bárány | The limit shape of convex lattice polygons[END_REF] and where the function I crit will be defined later by equation [START_REF] Vershik | The limit form of convex integral polygons and related problems[END_REF].

where κ = ζ(3)/ζ
Moreover, under the assumption that the zeroes ρ of the Riemann zeta function inside the critical strip 0 < (ρ) < 1 are simple, this function can be expressed as

(2) I crit (β) = ρ Γ (ρ)ζ(ρ + 1)ζ(ρ -1) ζ (ρ)β ρ ,
where the precise meaning of the series will be given by (10).

A straightforward corollary of this theorem is that, if Riemann's Hypothesis was to hold, the oscillatory term I crit ((κ/n) 1/3 ) inside the exponential would be roughly of order n 1/6 . Both statements are actually equivalent, as we will show in the comments section: the condition

(H) ∀ > 0, log p(n) = 3κ 1 3 n 2 3 + O(n 1 6 +
) holds if and only if Riemann's Hypothesis does.

In [START_REF] Bodini | Asymptotic analysis and random sampling of digitally convex polyominoes[END_REF], Bodini, Duchon, Jacquot and Mutafchiev presented a precise asymptotic analysis of digitally convex polyominoes, based on generating functions and on some version of the circle method. This problem turns out to be strongly related to the enumeration of convex chains. In the last section, we show how a slight modification of Sinaȋ's model makes its treatment possible.

A statistical mechanical model

We start this paper by reminding the correspondence between lattice convex chains and non-negative integer-valued functions on the set of pairs of coprime positive integers.

Let P be the set of primitive vectors, that is to say the set of all vectors (x, y) whose coordinates are coprime positive integers, including the pairs (0, 1) and (1, 0). As already used in Jarník [START_REF] Jarník | Über die Gitterpunkte auf konvexen Kurven[END_REF], the space of lattice increasing convex chains starting from the origin is in one-to-one correspondence with the space Ω of nonnegative integer-valued functions ω : P → Z + with finite support (that is to say, ω(v) 0 only for finitely many v ∈ P ):

• The function ω associated to the convex chain (x i , y i ) 0≤i≤k is defined for all v ∈ P by ω(v) = gcd(x i+1 -x i , y i+1 -y i ) if there is exists i ∈ {0, . . . , k} such that (x i+1x i , y i+1y i ) is proportional to v, and ω(v) = 0 otherwise. • The inverse map is obtained by adding up the vectors ω(v) v by increasing slope order. In particular, the endpoint of the chain is equal to

v∈P ω(v) v.
1.1. Description of Sinaȋ's model and overall strategy. We endow the space Ω with Boltzmann-like probability measures depending on two parameters β = (β 1 , β 2 ) ∈ (0, +∞) 2 and characterized by the fact that the random variables (ω(v)) v∈P are independent and geometrically distributed with parameter e -β•v . In this setting,

v∈P P β [ω(v) 0] < ∞
hence the function ω has almost surely finite support and we can write

P β (ω) = v∈P 1 -e -β•v e -ω(v)β•v = 1 Z(β) exp        -β • v∈P ω(v)v       
where the partition function Z(β) is equal to

(3) Z(β) = v∈P 1 -e -β•v -1 .
Since, as noticed above, X(ω) = v∈P ω(v)v is the endpoint of the chain corresponding to ω, we deduce that the conditional distribution induced by P β on the set of convex chains ending at n = (n 1 , n 2 ) is the uniform distribution. In particular, we obtain the following formula which will be instrumental in the proof:

(4) P β [X = n] = p(n) e -β•n Z(β)
In order to get a formula for p(n), our strategy is to choose the two parameters so that E β [X] = n. This will indeed lead to an asymptotic equality of P β [X = n] due to a local limit result. Together with the analysis of the partition function, this local limit result will constitute the key of the proof.

χ(s; β) := v∈Z 2 + \{0} 1 (β • v) s , (s) > 2. Lemma 1.1. For all β = (β 1 , β 2 ) ∈ (0, +∞) 2 , (5) log Z(β) = 1 2iπ 3+i∞ 3-i∞ Γ (s)ζ(s + 1) ζ(s) χ(s; β) ds.
Proof. We first take logarithms in (3) and then expand in Taylor series,

log Z(β) = - v∈P log 1 -e -β•v = v∈P m≥1 1 m e -mβ•v .
Now, we make use of the Mellin inversion formula

e -z = 1 2iπ c+i∞ c-i∞ Γ (s)z -s ds,
which holds for every real numbers z > 0 and c > 0. For all c > 0, we know, from the exponential decrease of Γ , that its integral on c + iR is absolutely convergent. Taking c > 2 in order to ensure that sup s∈c+iR m≥1

1 m s+1 < ∞ and sup s∈c+iR v∈P 1 (β•v) s < ∞, we can apply Fubini's theorem and write log Z(β) = 1 2iπ c+i∞ c-i∞ v∈P m≥1 1 m Γ (s) (mβ • v) s ds = 1 2iπ c+i∞ c-i∞ ζ(s + 1)Γ (s) v∈P 1 (β • v) s ds.
To conclude the proof of ( 5), notice that the partition Z 2 + \ {0} = d≥1 (dP ) translates into the identity of Dirichlet series

v∈Z 2 + \{0} 1 (β • v) s = d≥1 1 d s v∈P 1 (β • v) s .
Our main contribution is to exploit this integral representation in order to push further the estimates of log Z, leading to a precise estimate of p(n, n). As Sinaȋ does in his paper, we could also get precise asymptotics for p(n, cn), but for the sake of readability of the paper, we limit our scope to p(n, n) which leads to a choice of equal parameters β 1 = β 2 = β. In this case, we observe by elementary manipulations, that for (s) > 2, (6)

χ(s; (β, β)) = v∈Z 2 + \{0} 1 (βv 1 + βv 2 ) s = n≥1 n + 1 β s n s = ζ(s -1) + ζ(s) β s .
Therefore, equation ( 5) becomes

(7) log Z(β, β) = 1 2iπ 3+i∞ 3-i∞ Γ (s)ζ(s + 1)(ζ(s -1) + ζ(s)) ζ(s)β s ds.

Analysis of the partition function

We start by giving the first order estimates of log Z(β) and of its partial derivatives. For this purpose, we make use of the meromorphic continuation of the modified Barnes zeta function χ which is detailed in the appendix.

Lemma 2.1. For all nonnegative integers k 1 , k 2 , for all > 0, and all β = (β 1 , β 2 ) ∈ (0, +∞) 2 , such that <

β 1 β 2 < 1 , ∂ k 1 +k 2 ∂β k 1 1 ∂β k 2 2 log Z(β 1 , β 2 ) ∼ β→0 (-1) k 1 +k 2 ζ(3) ζ(2) k 1 !k 2 ! β k 1 +1 1 β k 2 +1 2 .
Proof. We apply formula [START_REF] Jarník | Über die Gitterpunkte auf konvexen Kurven[END_REF], and shift the line of integration to the left by using the residue theorem. We remind here that, by Corollary A.2 of the Appendix, all the integrated functions can be meromorphically continued, that the only pole in the region (s) ≥ 1 lies at s = 2, and that for all δ ∈ (0, 1),

log Z(β) = ζ(3) ζ(2)
1 β 1 β 2 + 1 2iπ 1+δ+i∞ 1+δ-i∞ Γ (s)ζ(s + 1) ζ(s) χ(s; β) ds.
We can take the iterated derivatives formally in the previous equality, since we control the derivatives of χ(s; β) by Corollary A.2, and since both functions ζ(s + 1) and 1/ζ(s) are bounded on the line 1 + δ + iR.

In the special case β 1 = β 2 = β, we derive a much more precise asymptotic series expansion of log Z(β, β) and of its derivatives in β. The proof is based on formula [START_REF] Titchmarsh | The theory of the Riemann zeta-function, Second[END_REF] and on the residue theorem again. Lemma 2.2. Let A be a positive number such that the contour γ defined below surrounds all the zeroes of ζ inside the critical strip.

The contour γ is defined as the union of the following oriented paths: on the right side the curve γ right (t) = 1 -A log(2+|t|) + it for t going from -∞ to +∞ and on the left side the curve γ left (t) = A log(2+|t|) + it for t going from +∞ to -∞ (see Figure 1). We define the functions I crit and I err respectively by

(8) I crit (β) = 1 2iπ γ Γ (s)ζ(s + 1)(ζ(s -1) + ζ(s)) ζ(s)β s ds.
and

I err (β) = 1 2iπ -1 2 +i∞ -1 2 -i∞ Γ (s)ζ(s + 1)(ζ(s -1) + ζ(s)) ζ(s)β s ds.
(i) For all k ≥ 0, when β goes to 0, the k-th derivative of

I crit (β) is o(β -k-1
).

(ii) When β goes to 0, the function 2)

I err (β) is of order O(β 1/2 ) and I err (β) = o(1/β). (iii) For all β > 0, (9) log Z(β, β) = ζ(3) ζ(
1 β 2 + I crit (β) + 7 6 log 1 β + C + I err (β)
with C = -2ζ (-1) -1 6 log(2π).

Proof. The existence of a suitable number A is a consequence of [7, Theorem 3.8], and the fact that the set of nontrivial zeroes of ζ is symmetric with respect to the vertical line (s) = 1 2 . In addition, the integrals defining I crit (β) and I err (β) are convergent. Indeed, along the path γ, the function 1/ζ(s) is O(log(| (s)|)) by formula (3.11.8) in [START_REF] Titchmarsh | The theory of the Riemann zeta-function, Second[END_REF]. Moreover, this domination makes it possible to apply Lebesgue's dominated convergence theorem to show that the function I crit (β) is a o(β -1 ) as β → 0. The same domination allows us to differentiate I crit under the integral sign as many times as needed. The dominated convergence theorem implies that the k-th derivative of

I crit (β) is o(β -k-1 ).
Assertion (ii) is easier since the function 1/ζ(s) is bounded on the vertical line (s) = -1 2 . We turn now to the proof of (iii). As in previous lemma, the strategy is to start from the integral representation (7) of log Z. Here, in order to get a sharper asymptotic expansion, we introduce a contour whose left side is the line (s) = -1 2 . Since such a contour crosses the critical strip where the denominator ζ(s) has zeroes, we will use the following result of Valiron [7, Theorem 9.7]: there exists α > 0 and a sequence (T k ) such that for all k ∈ N, k < T k < k + 1 and (s) ≤ 2. Therefore, if one applies the residue theorem with the rectangle [3 ∓ iT k , -1 2 ± iT k ] and let k tends to +∞, the contributions of the horizontal segments tend to 0 and one gets that log Z(β, β) is the sum of I crit (β), I err (β) and of the residues of the integrated function between the lines (s) = -1 2 and (s) = 3 which are outside of the contour γ. See Figure 1 for a landscape of the proof.

3 + iT k 3 -iT k -1 2 + iT k -1 2 -iT k γ right -1 -1 2 2 0 1 2 1 γ left
Since the singularity at s = 1 is cancelled by the presence of ζ(s) in the denominator, these residues come only from the simple pole at s = 2 and from the double pole at s = 0. It is straightforward to observe that the residue at s = 2 equals (ζ(3)/ζ(2))β -2 . A more involved but yet elementary computation yields

Res s=0 Γ (s)ζ(s + 1)(ζ(s -1) + ζ(s)) ζ(s)β s = 7 6 log 1 β -2ζ (-1) - 1 6 log(2π).
The announced formula (9) is thus proven.

The residue theorem applied to the above growing contours also leads to the following alternative expression:

(10) I crit (β) = lim k→∞ | (ρ)|<T k Γ (ρ)ζ(ρ + 1)ζ(ρ -1) ζ (ρ)β ρ ,
where ρ runs through the zeroes of ζ with 0 < (ρ) < 1. Here we have assumed for notational simplicity that these zeroes have multiplicity 1 but this is actually not restrictive.

Remark that in the previous arguments, one could push the left-side of the rectangular contour of integration as far as needed to the left in order to obtain complete asymptotic series expansions.

Proof of the theorem

Let us recall that the endpoint is defined by

X(ω) = (X 1 (ω), X 2 (ω)) = v∈P ω(v) v.
We look for a parameter (

β 1 , β 2 ) such that E β (X 1 ) = E β (X 2 ) = n. If one chooses β 1 = β 2 = β, then E β (X 1 ) = E β (X 2 ) = 1 2 E β (X 1 + X 2 ). It now remains to find β > 0 satisfying E β (X 1 + X 2 ) = 2n. A direct computation shows that E β (X 1 + X 2 ) = -d dβ log Z(β, β)
. By monotonicity of the function β → d dβ log Z(β, β), we can find β > 0 depending on n such that

d dβ log Z(β, β) = -2n.
From now on, β 1 and β 2 are going to be chosen equal to the unique solution β of this equation, depending on n. Note that β tends to 0 as n tends to ∞.

As a consequence of statements (ii) and (iii) of Lemma 2.2, the following asymptotic expansion holds: [START_REF] Bárány | The limit shape of convex lattice polygons[END_REF]. In order to estimate the error made when replacing the argument β of I crit by its first order approximation (κ/n) 1/3 , we use the estimates of Lemma 2.2, statement (i), for I crit and I crit . We obtain therefore (11) 1

2n = - d dβ log Z(β, β) = 2κ β 3 -I crit (β) + 7 6β + o 1 β , where κ = ζ(3)/ζ
β 3 = n κ + 1 2κ I crit κ n 1/3 - 7 12κ n κ 1/3 + o(n 1/3 ).
The covariance matrix of the random vector X = (X 1 , X 2 ) is equal to the Hessian matrix of log Z(β 1 , β 1 ) at (β, β), as seen by a straightforward computation. For the calibrated parameter β, Lemma 2.1 implies that its determinant is asymptotically equal to

2κ β 4 κ β 4 κ β 4 2κ β 4 = 3κ 2 β 8 ∼ 3n 8/3 κ 2/3 .
Therefore, a local limit theorem, which is a simplified version of Theorem 1.3 in [START_REF] Bogachev | Universality of the limit shape of convex lattice polygonal lines[END_REF] (in the case r = 1, with the notations of the paper), gives

(12) P β [X 1 = n, X 2 = n] ∼ κ 1/3 2π √ 3n 4/3 . From formula (4), log p(n) = 2nβ + log Z(β, β) + log P β [X 1 = n, X 2 = n].
Hence, gathering (9), ( 11), ( 12), we are able to state

log p(n) = 2nβ + log Z(β, β) + log κ 1/3 2π √ 3 - 4 3 log n + o(1) = 3κ 1/3 n 2/3 + I crit κ n 1/3 - 17 18 log n + log e -2ζ (-1) (2π) 7/6 √ 3κ 1/18 + o(1).
This concludes the proof of the theorem.

Comments

4.1. Link with Riemann's Hypothesis. As we observed in the proof of statement (i) of Lemma 2.2, the fact that the additional term I crit ((κ/n) 1/3 ) is at most of order o(n 1/3 ) follows from the existence of large zero-free regions of the Riemann zeta function in the critical strip. This estimation can be considerably improved if one assumes Riemann's Hypothesis, namely that all zeroes of the zeta function in the critical strip lie actually on the line (s) = 1 2 . Under this assumption, the oscillating term I crit ((κ/n) 1/3 ) is at most of order O(n 1 6 + ) for all > 0. Actually, Riemann's Hypothesis turns out to be equivalent to the statement

(H) ∀ > 0, log p(n) = 3κ 1 3 n 2 3 + O(n 1 6 + ).
Indeed, hypothesis (H) implies, by our theorem, that for all > 0, I crit ((κ/n)

1 3 )) = O(n 1 6 + ). Since I crit (β) = o(β -2 ) by statement (i) of Lemma 2.2, this implies that for all > 0, I crit (β) = O(β -1 2 -). Statement (iii) of Lemma 2.2 then yields (H') ∀ > 0, log Z(β, β) - κ β 2 = O       1 β 1 2 +       .

Now, we compute the Mellin transform of log Z(β, β) = v∈P m≥1

1 m e -mβ(v 1 +v 2 ) for (s) > 2 by manipulations similar to the proof of Lemma 1.1:

+∞ 0 log Z(β, β)β s-1 dβ = Γ (s)ζ(s + 1)(ζ(s -1) + ζ(s)) ζ(s) .
We obtain therefore the following identity for (s) > 2:

Γ (s)ζ(s + 1)(ζ(s -1) + ζ(s)) ζ(s) - κ s -2 = 1 0 log Z(β, β) - κ β 2 β s-1 dβ + +∞ 1 log Z(β, β)β s-1 dβ.
The condition (H') implies that the right-hand side function is holomorphic in the region (s) > 1/2, hence defining a holomorphic continuation of the left-hand side function in the same region. Since Γ (s), ζ(s + 1) and ζ(s -1) do not vanish in the region 0 < (s) < 1, this prevents ζ(s) from having zeroes in the region 1/2 < (s) < 1.

4.2. Numerical considerations. In view of numerical computations, one can truncate the series defining I crit in (10) at the two conjugate zeroes of the zeta function with smallest imaginary part, which is approximately 1 2 ± i14.1347. Indeed, the next zeroes have an imaginary part around 21.0220 and the exponential decrease of the Γ function makes this next term around 10 4 times smaller than the first one. Therefore, with a relative precision of 10 -4 , I crit (β) ≈ 6.0240 • 10 -11 cos(14.1347 log β) + 9.5848 • 10 -10 sin(14.1347 log β)

β
for usual values of β = (κ/n) 1/3 . In addition, one can see from this approximation that for a large range of values of n, the importance of the oscillatory term in the exponential part of ( 1) is actually extremely small in comparison with the polynomial prefactor.

4.3. Digitally convex polyominoes. We present here a modification of our model which leads to an asymptotic analysis of digitally convex polyominoes as in [START_REF] Bodini | Asymptotic analysis and random sampling of digitally convex polyominoes[END_REF]. Our probabilistic approach notably differs from the circle method used in this paper. We were not able to follow all the steps of the computations in [START_REF] Bodini | Asymptotic analysis and random sampling of digitally convex polyominoes[END_REF], and we obtain eventually a slightly different result.

Let us first recall that a digitally convex polyomino is the set of all cells of Z 2 included in a bounded convex region of the plane. The contour of a digitally convex polyomino can be decomposed into four specifiable sub-paths through the standard decomposition of polyominoes. In [START_REF] Bodini | Asymptotic analysis and random sampling of digitally convex polyominoes[END_REF], the authors focus on one of these paths, namely the one joining the rightmost lowest point to the highest rightmost one. More precisely, they want to find the asymptotics of the number p(n) of such paths with total length n.

Taking the convex hull of such a path defines a one-to-one correspondence with (increasing) convex chains having no horizontal segment. For this reason, we slightly modify the model by changing the set P into P = P \ {(1, 0)} and the probability measure on configurations which are now integer-valued functions on P :

P β (ω) = v∈ P 1 -e -β•v e -ω(v)β•v = 1 Z(β) exp         -β • v∈ P ω(v)v        
where β = (β, β) and where

Z(β) = v∈ P 1 -e -β•v -1 .
Again, the ending point is equal to

X(ω) = (X 1 (ω), X 2 (ω)) = v∈ P ω(v)v.
Moreover the connection between combinatorics and the probabilistic model is encompassed in the formula

P β [X 1 + X 2 = n] = p(n) Z(β, β) e -βn .
A computation similar to [START_REF] Ya | A probabilistic approach to the analysis of the statistics of convex polygonal lines[END_REF] shows that the integral representation (7) of the logarithmic partition function is changed into

log Z(β, β) = 1 2iπ c+i∞ c-i∞ Γ (s)ζ(s + 1)ζ(s -1) ζ(s)β s ds,
which leads to the following asymptotic expansion as β tends to 0:

log Z(β, β) = κ β 2 + I crit (β) + 1 6 log 1 β + C + o(1),
where the constants κ and C are still ζ(3)/ζ(2) and -2ζ (-1) -1 6 log(2π). In order have the correct calibration E β [X 1 + X 2 ] = n for the length of the path, we choose β > 0 depending on n such that -d dβ log Z(β, β) = n. This leads to the following analogue to (11):

1 β 3 = n 2κ + 1 2κ I crit 2κ n 1/3 - 1 12κ n 2κ 1/3 + o(n 1/3 ).
An statement analogous to (12) asserts the one-dimensional local limit result:

P β [X 1 + X 2 = n] ∼ 1 √ 3π κ 4 1 6 1 n 2 3
.

We have finally obtained

p(n) ∼ κ 4 5 18 e -2ζ (-1) √ 3π 2 3 1 n 11 18 exp       3 κ 4 1 3 n 2 3 + I crit       2κ n 1 3             .
This amends the constant in the prefactor term of [3], Proposition 2.5.

Appendix A. Analytic continuation of the Barnes function

We introduce here the Barnes zeta function defined for all β = (β 1 , β 2 ) ∈ [0, +∞), w > 0 and s ∈ C in the region (s) > 2 by Proof. We need only prove that the equality holds for (s) > 2 since the righthand side of the equation is meromorphic in (s) > 1 with a single pole at s = 2. Let {x} = xx denote the fractional part of x. We apply a first time the Euler-Maclaurin formula to the partial summation defined by F(x) = n 2 ≥0 (w + β 1 x + β 2 n 2 ) -s , leading to

ζ 2 (s, w, β) := v 1 ,v 2 ≥0 1 (w + β 1 v 1 + β 2 v 2 ) s .
n 1 ≥1 F(n 1 ) = ∞ 0 F(x) dx - F(0) 2 + ∞ 0 ({x} - 1 2
)F (x) dx.

We use again the Euler-Maclaurin formula for each of the summations in n 2 .

We state here the results we need on the modified Barnes zeta function χ, which is nothing but a byproduct of the above Barnes function. (i) The function χ(s; β) has a unique pole at s = 2 which is simple and the residue is equal to 1 β 1 β 2 . (ii) For all nonnegative integers k 1 , k 2 , for all δ ∈ (0, 1), for all > 0, there exists a constant C > 0 such that for all β = (β 1 , β 2 ) ∈ (0, +∞) 2 satisfying < β 1 β 2 < 1 , for all s such that 1 + δ ≤ (s) ≤ 3, We then apply Lemma A.1 and observe that the integral terms in the right-hand side of the equation can be differentiated formally.

∂ k 1 +k 2

1. 2 .

 2 Integral representation of the partition function. It turns out that the Mellin inversion formula leas to an exact integral representation of the logarithmic partition function log Z of Sinaȋ's model in terms of the Euler Γ function, the Riemann ζ function, and the modified Barnes zeta function

Figure 1 .

 1 Figure1. Localization of poles (in blue) and contours of integration with which the residue theorem is applied to get the asymptotic expansion of log Z(β, β) as β tends to 0.

Lemma A. 1 .

 1 The meromorphic continuation of ζ 2 (s, w; β) to the half-plane (s) > 1 is given by ζ 2 (s, w; β) β 1 x + β 2 y) s+2 dxdy.

Corollary A. 2 .

 2 For all β = (β 1 , β 2 ) ∈ [0, +∞), the function χ(s;β) = v∈Z 2 + \{0} 1 (β•v) scan be meromorphically continued to the region (s) > 1.

2 +

 2 ; β) ≤ C|s| C |β| k 1 +k 2 + (s) |s -2| .Proof. Elementary manipulations on sums yieldχ(s; β) = ζζ 2 (s, β 1 + β 2 ; β).

Acknowledgements. The authors are pleased to thank Axel Bacher, Philippe Duchon and Jean-Franc ¸ois Marckert for bringing to our knowledge the reference [3] on digitally convex polyominoes.


G Wille
email: g.wille@brgm.fr

C Michel

J Hellal

A Richard

A Burel

P Ollivier

Andl Joly

SEM TECHNIQUES FOR BIO-AND ORGANO-MINERAL INTERFACE CHARACTERIZATION IN THE ENVIRONMENT

Introduction

Bacteria, both as planktonic cells or biofilms (group of microorganisms embedded in an exopolymeric matrix in which cells interactwitheach other) are involved in numerous processesinteracting with geological materials, i.e. rocks, soils, minerals… This has led scientists to focus on the effectsof such interactions on soils, rocks (for example, biomineralizationi.e. synthesis of a mineral by micro-organisms),and their potential applicationsfor industry or environment such as biolixiviationor bioremediation. An example of bio-mineral interactions is biolixiviation. Biolixiviation is a natural process involving the attack of minerals by bacteria that can be exploited in the mining industry as an extraction method of metals from ore. The first example of natural biolixiviation was demonstrated by Arthur R. Colmer andM. E. Hinkle in 1947 [1]. While studying the pollution of the river Ohio (U.S) by drainage water of coal mines, they noted evidence of a massive amount of sulphuric acid. This can be partially explained by the atmospheric oxidation of pyrite (FeS 2 ), a mineral species often present in coal but also by the presence of bacteria such asAcidithiobacillus ferrooxidans orThiobacillus ferrooxidans, capable of oxidizing Iron (Fe 2+ ) or sulphur (S 2-). . Since then, several studies have been conducted on the use of biolixiviation for the extraction of metals of strong economic or technological interest such as gold, cobalt, uranium, zinc or copper from ores. These studies have led to the development and exploitation of biolixiviation in industrial processes [2,3,4,5]. Microbial-mineral interactions are also now strongly involved in bio-remediation processes. Bioremediation is the decontamination of polluted environments by techniques involving biochemical activity or properties of living organisms.As an example, the ability of biofilms to interact with pollutants or to catch them in their spatial structurecan be used for remediation. But in some cases, biofilms can have a negative impact on the bioremediation process (mainly biofouling that inhibitswater purification [6]). In this context, part of this work is the evaluation of the interactions of multi-species biofilms with metals (presently iron nanoparticles) and sand, and its impact on depollution processes. The microbial-mineral interface usually requires micrometric or nanometric scale observations; electron microscopy is thus an observation and analysis technique perfectly suited for that.However electron microscopy applied to living samples is faced by numerous technical challenges includingthe hydrated state of the initial sample, degradation of the sample during drying, freezing... and finally a high sensitivity to electron beam. Thus specific techniques are required for maintaining water or removing it with adapted protocols. The first experiments on STEM in "wet" environmental cells (with water or gas) were reported by M. Von Ardenne at the late 30s [7]. In 1970, the first SEM images of a hydrated sample were reported by W. C. Lane [8], using a specimen stub with a water reservoirwhich ejected a vapor jet to maintain a humid gas layer over the specimen, with no othermodification made to the microscope. Since then the interest of scientists concerned by life science for electron microscopy has grown continuously. We applied conventional, low vacuum and cryo-SEM, STEM-in SEM and TEM techniques for the observation of biofilms involved in bio-remediation processes in interaction with mineral and metallic materials.For this, appropriate procedures and protocols were applied or adapted to address the different problems caused by these specific samples made of a mixture of "soft" hydrated biological samples and "high hardness" mineral particles. These observations enabledto understand the behavior and the internal structure of biofilms in contact with mineral or metallic particles, and the ability of biofilms to fix iron nanoparticles or to bind to sand particles.

I -Materials and methods

The first challenge for good quality SEM/TEM observations of bio-mineral interfaces is sample preparation. For biologicalsamples alone, several preparation and observation methods have been developed and evaluated in the last years [9,10].

Conventional SEM Conventional SEM observation of biological samples, especially biofilms, is limited by the requirement of sampledehydration during preparation. As biofilms consist mainly of water, dehydrationmight alter its structure and morphology.Conventional SEM requires a complex and multi-step preparation which involves fixation, dehydration, drying, and coating with metal or carbon. All these steps may incur damage to samples and morphological information can be altered. For instance, hydrated biofilm is subjected to glutaraldehyde fixation and postfixation (osmium tetroxide), followed by ethanol or acetone dehydration, critical-point drying (CPD), and metal coating. The fixation with crosslinking agents (i.e. glutaraldehyde fixation) is used to stabilize the network structure of the biofilm. This step, together with the post-fixation, is designed to prevent breakdown of biological structures. But in some case,swelling or shrinkage at the same time is reported by some authors [9]. The last step of this preparation is coating, which can createdistortion due to the energy dispersed by carbon evaporation or plasma, depending on the type of coating and the performances of the coating system. This step can be deleted by using a low vacuum SEM (LV-SEM), but the small size of bacteria embedded in the biofilm requires high resolution imaging, for which a field emission low vacuum SEM (FE-LV-SEM) is required. On the other hand, the use of low vacuum conditions is known to cause more or less significant reduction in image resolution. Desiccation followed by high/low vacuum SEM can collapse EPS matrices which, in turn,deform biofilm appearance. Alternatively, "wet" Environmental Scanning Electron Microscopy (ESEM) can be performed without biofilm drying.In ESEM, specimens are imaged at a moderate vacuumwithout conductive coating and in a water vaporatmosphere which better preserves the biofilm structure. This approach was not evaluated in this study.

Cryo-SEM Cryo-SEM is widely used for the observation of biological samples. The sample is frozen in a suitable device in order to maintain water in the sample for SEM observations. Due to the high volume of solid crystalline water (ice)compared to liquid water, the sample has to be cooled as fast as possible otherwise the sample can be deformed by ice crystallization. The goal is to obtain amorphous or nano-crystalline water. The sample is frozen by plunging in a convenient medium, such as liquid ethane (or propane) cooled at liquid nitrogen temperature (-210°C) or sub-cooled nitrogen (nitrogen slush). Some other technical solutions can be used, such as cooling between 2 copper plates, projection on a copper plate cooled at liquid nitrogen temperature. The sample is then transferred on the cold stage of the SEM. Fracturation of the sample and/or sputter coating (gold, platinum) can also be performed if necessary in the cryo-SEM preparation chamber. Two different SEM options were used for our study. The first solution for cryo-SEM is the use of a "conventional vacuum" FE-SEM (Hitachi S4500 cold-FEG SEM equipped with a Quorum Technology Polaron LT 7400fig. 1-a). The sample is frozen in nitrogen slush at -210°C, and then transferred in the preparation chamber. Cryo-fracturation (if necessary) is processed in the preparation chamber at -180°C to prevent sublimation of water contained in the sample. Then, SEM observations were performed at -108°C under low voltage (1 kV) using a SE detector. A pre-observation step at -70°C is applied to remove the ice layer formed during the transfer on the surface of the sample. These conditions give good quality SE images, but are problematic to perform BSE images and EDS analysis. The second solution is the use of a Peltier stage (up to -50°C) in a low vacuum SEM (Tescan Mira3XMU equipped with Edax Team EDS systemsee fig. 1-b). The sample is frozen then transferred on the cooled stage under low vacuum conditions. Due to the temperature / pressure stability diagram of water, solid water can be keep at higher temperatures than in the conventional vacuum SEM by using gas pressure in the SEM low vacuum chamber (typical conditions used are -40/-50°C and P = 80 Pa nitrogen). Any high voltage (HV) can be used to perform observations using the SE and BSE detectors of the SEM. EDS analysis is also available (taking into account limitations imposed by low vacuum conditions). However, this results in a loss of resolution due to the level of gas pressure in the SEM chamber. This can be detrimental to the observation, for example in the case of labeling by gold-conjugated aggregates.

STEM-in-SEM and (S)TEM

The STEM concept was described by von Ardenne in thelate 1930s [7].He performed a STEM modeexperiment in 1938, by adding scan coils to a transmissionelectron microscope.Using the transmission mode in a scanning electronmicroscope, both contrast and resolution are improved dueto the lower accelerating voltages, which increase the cross-sectionsand reduce the interaction volume of the incidentelectron beam [11,12]. High contrast results from the use of low voltages. On the other hand, highresolution is available thanks to limited interaction volume and low chromatic aberration [13].STEM implemented in a FE-SEM is an option now offered by mostSEM manufacturers. In addition to the flexibility of SEM for nanometric resolution imaging, ease of use (compared to TEM) allows to extend the usefulness and capabilities of the SEM. However, imaging resolution is limited by thespatial broadening of the electron probeat the exit surface of the sample [12]. Due to a limited need for high resolution and, conversely, a significant need for contrast, STEMin-SEM is well suited to the observation of low-Z samples [14,15,16], which is the case ofbiological samples. For this study, a STEM detector is used in a LV-FE-SEM (Tescan Mira3XMU equipped with Edax Team EDS system) using bright and dark field imaging in combination with SE and BSE detectors. A schematic view of the STEM detector is presented below (fig. 2). EDS punctual analysis is used to identify individualparticles, but mapping is not really possible, due to the non-optimized configuration of EDS systems in SEMs compared to TEM.

II -Application to the study of biofilm / mineral particles interface

Objectives and electron microscopy requirements Managing soil and water contamination problems is faced with new methods for « in-situ remediation ». For this, the high reactivity and the large surface/volume ratio of materials with specific properties (active coal, nanoparticles) may offer benefits compared to existing solutions.

As an example, the high surface/volume ratio of an activated carbon grain is presented on fig.

3. The fixation and development of a biofilm on an activated carbon grain is presented on Fig. 3 (a) and (b). As illustrated by fig. 3 and detailed in Mercier et al. [START_REF] Mercier | Biofilm formation vs PCB adsorption on granular activated carbon in PCBcontaminated aquatic sediment[END_REF], grain surface is gradually covered by a biofilm. This occursin three steps: initial fixation of individual bacteria on the grain surface,increase in biomass and modification of bacterial diversity, and finally, development of the exopolymeric matrix [START_REF] Mercier | Biofilm formation vs PCB adsorption on granular activated carbon in PCBcontaminated aquatic sediment[END_REF]. In this particular case, we concluded that the development of the biofilm on activated carbon grains did not preventPolychlorinated Biphenyls (PCBs) adsorption on activated carbon in polluted aquatic sediments.

Zero-valent Iron nanoparticles (Fe 0 NPs) have demonstrated in-vitro their efficiency in degrading problematic contaminants (nitrates,chlorinated solvents…). But in decontamination conditions, it has been evidenced that the efficiency of Fe 0 NPs is dependent of their reactivity towards the contaminant as well as their accessibility to contaminants. Laboratory studies were conducted in order to observe interactions between Fe 0 NPs and multi-species biofilms in the context of a European project (FP7 Nanorem) evaluating impact of biofilm on NP mobility and reactivity.In this framework,biofilm interactions with NPs were evaluated using various microscopic approaches. Biofilm growth, sampling, labelling and treatment strategies were developed for biofilm analysis for each type of microscopy to access the surface or inside of the biofilm, biofilm composition, and metal location.

The size of the bacteria and EPS filaments in the biofilm is on the order of a few microns at most. These dimensions are difficult or impossible to observe by optical microscopy. Several electron microscopy techniques available in the SEM were applied: cryo-SEM, STEM-in-SEM, TEM. Due to the specificity of the samples and the presence of mineral particles, specific preparation protocols are required. Culture and sampling of biofilms The growth of biofilms was conducted in PVC tubes (fig. 4). The main advantage of such support is the opportunity to easily and quickly obtain a large amount of biological material. In a first step, biofilms were growth during 2-3 weeks without NPs in order to adjust culture and sampling parameters. For this, PVC tubes were inoculated with aquifer water spiked with a nutritive solution to enhance denitrification [START_REF] Hellal | Interaction entre biofilms bactériens et nanoparticules de fer réactives dans le cadre de procédés de nanoremediation7th National Biofilms network (RNB) symposium[END_REF]. In a second timenanoparticles were introduced. Tubes were placed in a temperature-controlled chamber at 20°C during biofilms culture. Biofilm sampling can be achieved by scraping tubes with a sterile scalpel. But in many cases, this was not possible so a protocol based on the use of a gelatin / agarose gel was used [START_REF] Kämper | Methods for in situ detection and characterization of extracellular polymers in biofilms by electron microscopy[END_REF]. This protocol enables to observe the internal face of the biofilm while the "scalpel" protocol is used for observing thebiofilm surface.

Labelling andsample preparation for observation

Depending on the observation objectives, labelling lectins coupled to gold particles (15 nm) was applied to the sample before SEM observations according to Michel et al [START_REF] Michel | Role of exopolymeric substances (EPS) in the stability of the biofilm of Thiomonasarsenivorans grown on a porous mineral support[END_REF].Lectin-Au labelling is applied to locate exopolysaccharides. For this reason, the same lectines as those used for fluorescence microscopy were chosen(PNA and ConA -EY laboratories, USA), a gold aggregate is attached to the lectines instead of a fluorochrome [START_REF] Michel | Analysis of biofilm-nanoparticles interaction using microscopy (fluorescence, MEB, STEM, MET, EDS)[END_REF].Other labelling may be considered, such as FISH probes coupled with gold [START_REF] Gerard | Fluorescence in situ hybridation coupled to ultra smallimmunogold detection to identify prokaryotic cells using transmission and scanning electron microscopyJournal of[END_REF] or heavy elements like Bromide or Iodine. Labelling is performed on the fresh biofilm after sampling. Biofilm is washed once using Phosphate Buffered Saline (PBS). Then the biofilm is covered with a solution of lectines coupled to gold clusters during 90 minutes in the dark. After removing the lectin-Au solution, the biofilm is washed 3 times using PBS buffer..For STEM, labelling was performed before glutaraldehyde fixation.For cryo-SEM observation, labelling was performed just before freezing on nitrogen slush. For STEM-in-SEM and TEM observation, sample is fixed in glutaraldehyde (2.5%) in Cacodylate (Caco) Buffer 0.15 M at pH 7.2, postfixed in Osmium tetroxide (Caco buffer), then included in Agar-Agar and progressively dehydrated with acetone. The sample was finally included in Epon and cut on a Leica UC6 microtome at 80nm. For STEM-in-SEM observation, samples are coated with 2 nm Carbon in a Cressington 208C or 0.5 nm Pt/Pd in a Cressington 208HR. Indeed, non-coated cuts do not support the electron beam energy then a serious degradation, even the formation of a hole, is observed when increasing magnification.

Results and discussion -Cryo-SEM observation Cryo-SEM was applied to observe biofilm development and location of lectins-Au labelling andFE 0 NPs nanoparticles, and also to observe bacterial attachment on sand grains. For this, samples were frozen in nitrogen slush. Observations were performed on a cold-FEG SEM at low HV, and on a LV-FE-SEM at high HV for BSE and EDS. We established that a fresh biofilm frozen in nitrogen slush can be kept in liquid nitrogen for few days before observation. SEM observations were performed on bulk samples with or without fracturation in the preparation chamber. Secondary electrons enable to observe the organic moleculesand thus biofilm on its own, biofilm colonization of sand grainsand their interactions with NPs. In contrast, the location of lectin-Au labelsand iron NPs is more efficient using backscattered detector. Biofilms in contact with iron NPs or lectin-Au labelling observed by cryo-SEM are presented on fig. 5. SE image (fig. 5) clearly shows the organization of the biofilm on the sand grain, and the iron NPs interaction with bacteria. It appears that NPs remain aggregated and do not disperse in the biofilm. This observation indicates that the biofilm / NP interactions are limited due to a low contact area. The main hypothesis is that these forms are bacteria embedded in an exopolysaccharide matrix, cannot really be visible for this reason. Then cutting slices into the biofilm for TEM/STEM observation is the solution to conclude.

-STEM-in-SEM and TEM STEM or TEM observation requires a specific sample preparation. But samples are very sensitive to preparation and conservation conditions before fixation. As an example, two images are presented on fig. 7. On fig. 7-a On fig. 7-a, bacteria present an abnormal aspect, with distortions, degraded membranes, compared to bacteria on fig. 7-b, where the expected aspect is observed. These observations illustrate the need to pre-fix the biofilm with glutaraldehyde immediately after sampling. Samples were fixed with glutaraldehyde after lectin-gold labelling but without previously separating them from their support (PVC tube) although. It appeared that the biofilms then came off their support during the fixation process. However, this appeared to be an advantage for observations, since no manipulation had to be performed on fresh biofilm and therefore its structure has not been changed, and it was easier to make cross-sections on the fixed biofilm after its detachment from the growth support. STEM observations were performed at various voltages. However, it appears that such biological samples are very sensitive to contamination at low voltage (10 kV or less), compared to higher voltage (20 -30 kV) that were thus chosen for STEM imaging. Moreover, STEM resolution is strongly affected by HV value. Due to low mean Z of the biological sample (mainly composed of carbon, hydrogen, nitrogen, oxygen), the most of the electrons are transmitted with little or no energy loss at any HV, but the dispersion of the electrons trajectories increase strongly when lowering HV (fig. 8). On fig. 9, the same area is simultaneously imaged using STEM BF / DF, BSE and SE detectors. Some bright particles are clearly visible on BF or DF images (with reverse contrast on the other). Due to the origin of contrast in TEM images, these bright particles cannot be associated to a difference in composition. BSE images of these particles can be used to discriminate between strong diffraction contrast and high Z. On this image, bright particles observed on the BSE image together with DF images were analyzed by EDS and appeared to be composed of calcium and phosphorus (i.e. aggregates of hydroxyl-apatite, the major mineral in the human body, present for example in the composition of bone and dental tissues). This composition was confirmed by EDS mapping by TEM (fig. 10).This kind of map is very useful for the understanding of the internal structure of cells. Biofilms put in contact with Fe 0 NPs were prepared for (S)TEM imaging. A lectin-gold labelling was performed on the fresh biofilm to locate EPS matrix. A microtome section of the biofilm preparation is presented on fig. 12. The STEM-in-SEM enables to observe the biofilm section from the millimetric to the nanometric scale. STEM DF imaging shows that Fe 0 NPs were present as aggregated particles in the biofilm, but no NPs were visible at more than 10-15 µm from the surface and NP penetration in the biofilm was limited to the surface area. Observations at higher magnifications were used to precisely describe the connection between bacteria and NPs. Lectin-gold labelling was applied to the biofilms to highlight EPS in the biofilm. Some images are presented on fig. 13.  Gold labelling appeared everywhere in the biofilm. No labelling was in contact with the membrane of the bacteria, but surrounded the cells at a distance of approximatively 0.2-0.5 µm. This suggested that the targeted EPS were capsular type EPS.

 Fe 0 NPs were only present on or close to the surface of the biofilm. Like gold labelling, no NPs were in direct contact with bacteria, which suggested that EPS formed a protective layer around them.  Using STEM-in-SEM imaging only, it was not possible to concludeor excludeto a co-location of Fe 0 NPs and gold labellingi.e. a connectionand an interactionbetween Fe 0 NPs and EPS

The third point can be evidenced using EDS analysis. In the SEM, EDS punctual analysis is possible using a long counting time (fig. 14). It also suggests that iron NPs were present in the biofilm as aggregates in contact with EPS, but not directly with bacteria. It also confirms that no direct interaction between iron NPs and microorganisms can be considered, because of the presence of a thick layer of EPS surrounding the bacteria.

Conclusion

We applied cryo-SEM and STEM-in-SEM to the observation of the bio-mineral interface. Biofilms were grown on the surface of PVC tubes with a nutritive solution to enhance denitrification, or on sand, and then put in contact with zero-valent iron nanoparticles. Biofilms were studied for understanding the behavior of NPs in the presence of biofilms, using electron microscopy techniques cryo-SEM and STEM-in-SEM. Specific protocols were developed or adapted for these observations. Cryo-SEM samples exhibited a low number of bacteria visible in the biofilm but showed rather a network with forms whose shape and size are similar to that of bacteria. We suggest that these forms may be bacteria embedded in an exopolysaccharide network. This hypothesis was confirmed by STEM observations with lectin-gold labelling for EPS detection, which showed that the bacteria were encapsulated in a matrix of EPS. STEM analyses also showed Fe 0 NPs aggregates on or near the surface (up to 10µm) of the biofilm, but not in direct contact with bacteria. Complementary TEM/EDS analysis also revealed that Fe 0 NPs areconnected to the EPS matrix, as confirmed by EDS mapping of gold and iron which exhibit co-location of lectin-gold (EPS markers) with iron aggregates. Then, the combination of cryo-SEM and STEM-in-SEM, supplemented by TEM/EDS (or EELS) analysis appeared to be a performant tool for understanding the biofilm / mineral interface and interaction.

of Orleans (France). TEM and STEM preparation were carried out by MRic TEM-Biosit electron microscopy platform at the university of Rennes (France). TEM /EDS/EELS (Jeol ARM200F) analysis were performed at the Centre Raymond Castaing (Toulouse -France) and at the demonstration center of JEOL Ltd (Tokyo -Japan)

Fig. 1 :

 1 Fig. 1: (a) Cryo-chamber on a Hitachi S4500 (CME -Univ. Orleans) (b) Peltier stage on a Tescan Mira3XMU (BRGM Orleans) (c) Water phase diagram and area of use of the Peltier stage under LV conditions

Fig. 2 :

 2 Fig. 2 : Schematic view of the STEM detector (doc. Tescan)

Fig. 3 :

 3 Fig. 3 : Cryo-SEM observation of an active coal grain porosity and development of bacteria on activated coal grains after (a) 2 days and (b) 30 days (bacteria location is indicated by white arrows) ([17])

Fig. 4 :

 4 Fig. 4: Biofilm culture in a PVC tube

Fig. 5 :

 5 Fig. 5: Biofilm growth on a sand in presence of iron nanoparticles (SE --50°C / 80 Pa) -Dotted arrow: bacteria / Dashed arrow: NPs

  Fig. 7: STEM-in-SEM images of biofilms a/ sampling of the biofilm few days before fixation; b/ sampling of the biofilm followed by fixation

Fig. 8 :

 8 Fig. 8: Beam dispersion at 5, 15 and 25kV and BF/DF corresponding images

Fig. 9 :

 9 Fig. 9: Observation of a TEM preparation of a biofilm. Top-left: STEM bright field / Topright: STEM dark field Bottomleft: BSE detector / Bottomright: SE detector

Fig. 10 :

 10 Fig. 10: EDS analysis and mapping on a bacteria (TEM)

Fig. 12 :

 12 Fig. 12: Observation of the repartition of Iron nanoparticles in a biofilm (a) view field 148 µm (a complete grid hole) (b) 67.8 µm (c) 12.6µm

Fig. 13 :

 13 Fig. 13: Distribution of Iron nanoparticles and EPS (gold labelling) in a biofilm

Fig. 14 :

 14 Fig.14: EDS analysis (STEM-in-SEM) of a Fe 0 NPs aggregate on EPS matrix.

Fig. 15 :

 15 Fig. 15: STEM (TEM) BF / DF and EDS mapping on Fe and Au

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Acknowledgments

This work is in the project BIOMEB supported by BRGM research direction in collaboration with projects BIOSORP (PDR13D3E80) and NANOREM (FP7 2007-2013 Grand agreement No. 309517). Cryo-SEM and TEM (CM20) observations were performed at the Electron microscopy center (CME) at the university