Sary Drappeau
email: sary-aurelien.drappeau@univ-amu.fr

Daniel Fiorilli
email: daniel.fiorilli@universite-paris-saclay.fr

THE FIRST MOMENT OF PRIMES IN ARITHMETIC PROGRESSIONS: BEYOND THE SIEGEL-WALFISZ RANGE

We investigate the first moment of primes in progressions q≤x/N (q,a)=1 ψ(x; q, a) -x ϕ(q) as x, N → ∞. We show unconditionally that, when a = 1, there is a significant bias towards negative values, uniformly for N ≤ e c √ log x . The proof combines recent results of the authors on the first moment and on the error term in the dispersion method. More generally, for a ∈ Z\{0} we prove estimates that take into account the potential existence (or inexistence) of Landau-Siegel zeros.

Introduction

The distribution of primes in arithmetic progressions is a widely studied topic, in part due to its links with binary additive problems involving primes, see e.g. [START_REF] Iwaniec | Analytic number theory[END_REF]Chapter 19] and [START_REF] Ju | The dispersion method in binary additive problems[END_REF]. For all n ∈ N we let Λ denote the von Mangoldt function, and for a modulus q ∈ N and a residue class a (mod q) we define ψ(x; q, a) := n≤x n≡a (mod q) Λ(n).

In the work [START_REF] Fiorilli | Residue classes containing an unexpected number of primes[END_REF], the second author showed the existence, for certain residue classes a, of an unexpected bias in the distribution of primes in large arithmetic progressions, on average over q. An important ingredient in this result is the dispersion estimates of Fouvry [START_REF] Fouvry | Sur le problème des diviseurs de Titchmarsh[END_REF] and Bombieri-Friedlander-Iwaniec [START_REF] Bombieri | Primes in arithmetic progressions to large moduli[END_REF]; these involve an error term which restricts the range of validity of [START_REF] Fiorilli | Residue classes containing an unexpected number of primes[END_REF]Theorem 1.1]. Recently, this error term was refined by the first author in [START_REF] Drappeau | Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method[END_REF], taking into account the influence of potential Landau-Siegel zeros. This new estimate allows for an extension of the range of validity of [START_REF] Fiorilli | Residue classes containing an unexpected number of primes[END_REF]Theorem 1.1], which is the object of the present paper. In particular, we quantify and study the influence of possible Landau-Siegel zeros, and we show that, in the case a = 1, a bias subsists unconditionally in a large range. Here is our main result.

Theorem 1.1.

There exists an absolute constant δ > 0 such that for any fixed ε > 0 and in the range 1 ≤ N ≤ e δ √ log x , we have the upper bound with an implicit constant depending effectively on ε, and where

(1.1) N x q≤x/N ψ(x; q, 1) - x ϕ(q) ≤ - log N 2 -C 0 + O ε (N -
C 0 := 1 2 log 2π + γ + p log p p(p -1) + 1 .
In other words, there is typically a negative bias towards the class a = 1 in the distribution of primes in arithmetic progressions modulo q. One could ask whether Theorem 1.1 could be turned into an asymptotic estimate. To do so we would need to rule out the existence of Landau-Siegel zeros, because if they do exist, then we find in Theorem 1.3 below that the left hand side of (1.1) is actually much more negative.

In order to explain our more general result, we will need to introduce some notations and make a precise definition of Landau-Siegel zeros. We begin by recalling [START_REF] Fiorilli | Residue classes containing an unexpected number of primes[END_REF]Theorem 1.1]. For N ≥ 1 and a ∈ Z \ {0} we define

M 1 (x, N ; a) = q≤x/N (q,a)=1 ψ * (x; q, a) - x ϕ(q)
,

where 1 ψ * (x; q, a) = 1≤n≤x n≡a mod q n =a Λ(n).
With these notations, [5, Theorem 1.1] states2 that for N ≤ (log x) O (1) (1.2)

M 1 (x, N ; a) φ(|a|) |a| x N = µ(a, N ) + O a,ε,B N -171 448 +ε with (1.3) µ(a, N ) :=        -1 2 log N -C 0 if a = ±1 -1 2 log p if a = ±p e 0 otherwise.
We recall the following classical theorem of Page.

Theorem 1.2 ([9, Theorems 5.26 and 5.28]). There is an absolute constant b > 0 such that for all Q, T ≥ 2, the following holds true. The function s → q≤Q χ (mod q) L(s, χ) has at most one zero s = β satisfying Re(s) > 1-b/ log(QT ) and Im(s) ≤ T . If it exists, the zero β is real and it is the zero of a unique function L(s, χ) for some primitive real character χ.

Given x ≥ 2, we will say that the character χ (mod q) is x-exceptional if the above conditions are met with Q = T = e √ log x . There is at most one such character. By the analytic properties of Dirichlet L-functions, if exceptional zeros exist, their effect can often be quantified in a precise way, and are expected to lead to secondary terms in asymptotic formulas. For instance, it is known [START_REF] Montgomery | Multiplicative number theory I : Classical theory[END_REF]Corollary 11.17] that if the x-exceptional character exists, then there is a distortion in the distribution of primes in the sense that

ψ(x; q, a) = x ϕ(q) -χ(a)1 q|q x β βϕ(q) + O(xe -c √ log x ) (1.4) = x ϕ(q) (1 -η x,a 1 q|q ) + O(xe -c √ log x ) (1.5) with (1.6) η x,a := χ(a) βx 1-β ∈ (-1, 1).
We are now ready to state our more general result. As we will see, the secondary term in (1.4) can potentially yield a large contribution to M 1 (x, N ; a) for N considerably larger than q. For this reason, it is relevant to consider instead the expression

(1.7) M Z 1 (x, N ; a) = q≤x/N (q,a)=1 ψ * (x; q, a) -(1 -1 q|q η x,a ) x ϕ(q) ,
where, by convention, the term involving η x,a is only to be taken into account when the xexceptional character exists.

Our results show that, in the case of the hypothetical two-term approximation (1.7), there is a new bias term, which results from the contribution of the possible x-exceptional character. (i) If there is no x-exceptional character, then

(1.8) M 1 (x, N ; a) φ(|a|) |a| x N = µ(a, N ) + O a,ε N -171 448 +ε .
(ii) If the x-exceptional character χ (mod q) exists, then with C a,q and D a,q as in (2.4) and (2.5) below,

(1.9)

M Z 1 (x, N ; a) ϕ(|a|) |a| x N = µ(a, N ) + N η x,a r≤N (r,a)=1 q|r 1 -( r N ) β ϕ(r) -C a,q log N q + D a,q - 1 β + O a,ε N -171 448 +ε .
(iii) If the x-exceptional character exists and N ≥ q, then the previous formula admits the approximation

(1.10) M Z 1 (x, N ; a) ϕ(|a|) |a| x N = (1-η x,a )µ(a, N )+η x,a μχ (a)+O a,ε N ε (N/q) -171 448 +(log N ) 2 1 -β x (1-β)/2 ,
where μχ (a) = 1 2 1 a=±1 (log q -p|q log p p ).

In (1.9), we have that C a,q a 1/φ(q) and D a,q a 1, hence the secondary term involving η x,a is O a,ε (N (log N )q -1+ε ). Since by Siegel's theorem we have the bound q A (log x) A for any fixed A > 0, we recover [START_REF] Fiorilli | Residue classes containing an unexpected number of primes[END_REF]Theorem 1.1].

Finally we remark that if the x-exceptional character exists and N ≥ q, the associated "secondary bias", that is the difference between the main terms on the right hand side of (1.10) and µ(a, M ), contributes an additional quantity -η x,a (µ(a, N ) -μχ (a)).

The bound q A (log x) A does not exclude the possibility that (1 -β) log x = o(1) in the context of (1.10). Should this happen, we would have that η x,a = χ(a)+o [START_REF] Bombieri | Primes in arithmetic progressions to large moduli[END_REF]. If moreover a = 1 and N ≤ qO(1) , then the main term of (1.10) would becomes asymptotically (1+o(1))μ χ(a), and would not depend on N anymore. In this situation, the additional bias coming from the exceptional character would annihilate the N -dependance of the overall bias.

Remark. The influence of possible Landau-Siegel zeros on the second moment has been investigated by Liu in [START_REF] Liu | Barban-Davenport-Halberstam average sum and exceptional zero of L-functions[END_REF]. As for the first moment, it is closely related to the Titchmarsh divisor problem of estimating, as x → ∞, the quantity 1<n≤x Λ(n)τ (n -1).

After initial works of Titchmarsh [START_REF] Titchmarsh | A divisor problem[END_REF] and Linnik [START_REF] Ju | The dispersion method in binary additive problems[END_REF], Fouvry [START_REF] Fouvry | Sur le problème des diviseurs de Titchmarsh[END_REF] and Bombieri, Friedlaner and Iwaniec [START_REF] Bombieri | Primes in arithmetic progressions to large moduli[END_REF] were able to show a full asymptotic expansion, with an error term O(x/(log x) A ). In the recent work [START_REF] Drappeau | Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method[END_REF], the first author refined this estimate taking into account the influence of possible Landau-Siegel zero, with an error term O(e -c √ log x ).

Proof of Theorem 1.3

2.1. The Bombieri-Vinogradov range. We begin with the following lemma, which follows from the large sieve and the Vinogradov bilinear sums method. Given a Dirichlet character χ mod q, we let

ψ(x, χ) := n≤x χ(n)Λ(n). Lemma 2.1. For 2 ≤ R ≤ Q ≤ √ x, we have the bound q≤Q 1 ϕ(q) χ (mod q) R<cond(χ)≤Q max y≤x |ψ(y, χ)| (log x) O(1) R -1 x + Q √ x + x 5 6 .
Proof. This follows from the third display equation of page 164 of [START_REF] Davenport | Multiplicative number theory[END_REF] with Q 1 = R, after reintegrating imprimitive characters as in [3, page 163].

We deduce the following version of the Bombieri-Vinogradov theorem, with the contribution of exceptional zeros removed. Lemma 2.2. Fix a ∈ Z \ {0}. There exists δ > 0 such that for all x, Q ≥ 1 we have the bound 1) , where η x,a was defined in (1.6).

q≤Q max y≤x max (a,q)=1 ψ(y; q, a) -(1 -η x,a 1 q|q ) y ϕ(q) xe -δ √ log x + Q √ x(log x) O(

Initial transformations, divisor switching.

From now on, we let δ > 0 be a positive parameter that will be chosen later, we fix x ≥ 1 and define χ as the possible x-exceptional character, of conductor q and associated zero β, and recall the notation (1.3). In order to isolate the contribution of the potential Landau-Siegel zero, we define

E(x; q, a) := ψ * (x; q, a) -(1 -η x,a 1 q|q ) x ϕ(q) .
If the x-exceptional character does not exist, then every term involving η x,a can be deleted.

With this notation we have the decomposition

M Z 1 (x, N ; a) = q≤x 1 2 +δ (q,a)=1 E(x; q, a) + x 1 2 +δ <q≤x (q,a)=1 ψ * (x; q, a) - x/N <q≤x (q,a)=1 ψ * (x; q, a) -x x 1 2 +δ <q≤x/N (q,a)=1 1 ϕ(q) + η x,a x x 1 2 +δ <q≤x/N (q,a)=1 q|q 1 ϕ(q) = T 1 + T 2 -T 3 -T 4 -T 5 , (2.1)
say. We discard the first term by using the dispersion estimate [4, Theorem 6.2], which yields that there exists an absolte constant δ > 0 such that in the range |a| ≤ x δ , (2.2)

T 1 = q≤x 1 2 +δ (q,a)=1 E(x; q, a) xe -δ √ log x .
We end this section by applying divisor switching to the sums T 2 and T 3 .

Lemma 2.3. Fix a ∈ Z \ {0} and define T 2 and T 3 as in (2.1). There exists an absolute constant δ > 0 such that in the range N ≤ x

1 2 -δ , |a|N ≤ xe -2δ
√ log x we have the estimate

T 2 -T 3 = x r≤x 1 2 -δ (r,a)=1 1 -( r x 1/2-δ ) ϕ(r) -η x,a x r≤x 1 2 -δ (r,a)=1 q|r 1 -( r x 1/2-δ ) β ϕ(r) -x r≤N (r,a)=1 1 -( r N ) ϕ(r) + η x,a x r≤N (r,a)=1 q|r 1 -( r N ) β ϕ(r) + O(xe -δ √ log x ).
Proof. We rewrite the condition n ≡ a mod q as n = a + qr for r ∈ Z. Summing over r and keeping in mind that |a|N < x, for large enough values of x we obtain the formula

T 3 = 1≤r<N -aN/x (r,a)=1 ψ * (x; r, a) -ψ * (a + rx N ; r, a) .
Recalling that N ≤ x 1 2 -δ , we may apply the Bombieri-Vinogradov theorem in the form of Lemma 2.2. We obtain the estimate (2.3)

T 3 = x r≤N (r,a)=1 1 -( r N ) ϕ(r) -η x,a x r≤N (r,a)=1 q|r 1 -( r N ) β ϕ(r) + O(xe -δ √ log x ).
Replacing N by x 1 2 -δ , we obtain a similar estimate for T 2 , and the result follows.

2.3. Sums of multiplicative functions. In the following sections, we collect the main terms obtained in the previous section and show that they cancel, to some extent, with T 4 and T 5 . We start with the following estimate for the mean value of 1/ϕ(q), which we borrow from [6, Lemma 5.2] with the main terms identified in [8, Lemme 6].

Lemma 2.4. Fix ε > 0. For a ∈ Z \ {0} and q 0 ∈ N such that (a, q 0 ) = 1 and q 0 ≤ Q, we have the estimate

q≤Q (q,a)=1 q 0 |q 1 ϕ(q) = C a,q 0 log Q q 0 + D a,q 0 + O a,ε (q ε 0 Q -1+ε ),
where

(2.4) C a,q 0 := φ(a) aϕ(q 0 ) p aq 0 1 + 1 p(p -1) , (2.5) D a,q 0 := p|a log p p - p aq 0 log p p 2 -p + 1 + γ 0 .
Here, γ 0 is the Euler-Mascheroni constant.

We now estimate the main terms in Lemma 2.3. For N ∈ N, a ∈ Z \ {0} and q 0 ∈ N we define

J γ (x, N ; q 0 , a) := r≤N (r,a)=1 q 0 |r 1 -( r N ) γ ϕ(r) + q≤x/N (q,a)=1 q 0 |q 1 ϕ(q) .
Lemma 2.5. Fix δ > 0 small enough and a ∈ Z \ {0}. For γ ∈ [ 3 4 , 1], (q 0 , a) = 1 and in the range 1 ≤ N ≤ x 1-δ , 1 ≤ q 0 ≤ x δ , we have the estimate (2.6)

J γ (x, N ; q 0 , a) = Jγ (x; q 0 , a) + γf q 0 ,a;N (1) -f q 0 ,a;N (γ) 1 -γ + O a,ε N x -1+ε + q 171 448 +ε 0 N -1-171 448 +ε ,
where the implied constant does not depend on γ, the value of the second main term at γ = 1 is defined by taking a limit, and Jγ (x; q 0 , a) := C a,q 0 log x q 2 0 + 2D a,q 0 -1 γ ;

(2.7)

f q 0 ,a;N (γ) := (q 0 /N ) γ ϕ(q 0 ) Z(-γ)G q 0 ,a (-γ)ζ(1 -γ)ζ(2 -γ)(1 -γ),
where

Z(s) := p 1 + 1 p s+2 (p -1) - 1 p 2s+3 (p -1)
;

G q 0 ,a (s) := p|aq 0 1 + 1 p s+1 (p -1) -1 p|a 1 - 1 p s+1 .
Proof. Assume that γ < 1. We will obtain error terms that are uniform in γ; this will allow us to take a limit and the result with γ = 1 will follow. We split Mellin inversion and a straightforward calculation gives the identity

J γ (x, N ; q 0 , a) = 1 2πi ˆ(2) 1 q s 0 ϕ(q 0 ) Z(s)G q 0 ,a (s)ζ(s + 1)ζ(s + 2) γN s s + γ + x N s ds s .
Taking Taylor series shows that for R ∈ R ≥1 ,

γR s s + γ + x R s = 2 + s log x - 1 γ + O x,γ,R (|s| 2 )
in a neighborhood of 0. We first shift the contour to the left until (-1 2 ). The residue at s = 0 contributes exactly J(x; q 0 , a). We handle the contribution of the term (x/N ) s similarly as in [7, Lemma 5.12]: a trivial estimation using a truncated Perron's formula shows that it is O a,ε ((x/N ) -1/2+ε ), while shifting back the contour to 2+iR (picking up a residue at s = 0) and applying Mellin inversion, we get

1 2πi ˆ(-1 2 ) 1 q s 0 ϕ(q 0 ) Z(s)G q 0 ,a (s)ζ(s + 1)ζ(s + 2) x N s ds s = q≤x/N (q,a)=1 q 0 |q 1 ϕ(q) -K 1 log x q 0 N -K 2
for some constants K 1 , K 2 depending on q 0 and a. Applying Lemma 2.4, we identify K 1 = C a,q 0 and K 2 = D a,q 0 , and we deduce that the above is O a,ε ((x/N ) -1+ε ). We deduce that

J γ (x, N ; q 0 , a) = J(x; q 0 , a) + O a,ε N x 1-ε + 1 2πi ˆ(- 1
2 )

1 q s 0 ϕ(q 0 ) Z(s)G q 0 ,a (s)ζ(s + 1)ζ(s + 2) γN s s + γ ds s .
Shifting the remaining integral further to the line (-1-ε), we pick up two residues, at s = -1 and at s = -γ. This gives rise to the second term in (2.6). As for the shifted integral, we apply Bourgain's subconvexity estimate for ζ(s) [START_REF] Bourgain | Decoupling, exponential sums and the Riemann zeta function[END_REF]. Note that G q 0 ,a (s)

p|aq 0 1 + 1 p Re(s)+2 p|a 1 + 1 p Re(s)+1 .
As in [START_REF] Fiorilli | Residue classes containing an unexpected number of primes[END_REF]Lemma 5.9], we shift the contour to the line Re(s) = -1 -1/(2 + 4θ), where θ = 13/81 is Bourgain's subconvexity exponent. The shifted integral is a,ε q 1/(2+4θ)+ε 0

N -1-1/(2+4θ)+ε .
The desired estimate follows.

In the next two sections, we will prove approximations for the term (2.8) D γ (q 0 , a; N ) := γf q 0 ,a;N (1) -f q 0 ,a;N (γ) 1 -γ appearing in (2.6).

2.4. The main term for γ = 1. The limit of D γ (q 0 , a; N ) as γ → 1 has a simple expression in terms of derivatives of f q,a,N , namely D 1 (q 0 , a; N ) = f q 0 ,a,N (1) -f q 0 ,a,N [START_REF] Bombieri | Primes in arithmetic progressions to large moduli[END_REF].

Recall that f q 0 ,a,N is given by the Euler product (2.7). A direct computation yields that for q ∈ N, a ∈ Z \ {0} and N ∈ R ≥1 ,

f q,a,N (1) =    0 if a = ±1; -1 2N if a = ±1, f q,a,N (1) =        0 if ω(a) ≥ 2; 1 2N (1 -1 ) log if a = ± ν (ν ∈ N, prime); -1 2N log( q N ) -2C 0 + 1 -p|q log p p if a = ±1.
From these observations, we deduce the following.

Lemma 2.6. We have the exact formula

D 1 (q, a; N ) =        0 if ω(a) ≥ 2; 1 2N (1 -1 ) log if a = ± ν (ν ∈ N, prime); 1 2N log( N q ) + 2C 0 + p|q log p p if a = ±1.
2.5. The main term for γ < 1. Now that we have estimated the main term in Lemma 2.5 for γ = 1, we will do so for γ < 1. Under this restriction, we write

D γ (q 0 , a; N ) -D 1 (q 0 , a; N ) = 1 γ -1 ˆ1 γ ˆ1 δ f q 0 ,a,N (δ )dδ dδ.
By a direct estimation of the Euler product we see that in the range 3

4 ≤ γ ≤ δ ≤ 1, |f q 0 ,a,N (δ )| a q δ 0 |G q 0 ,a (-δ )| φ(q 0 ) (log q 0 N ) 2 N -γ (log q 0 N ) 2 N -γ .
Therefore, when 3 4 ≤ γ ≤ 1, we obtain (2.9)

D γ (q 0 , a; N ) = D 1 (q 0 , a; N ) + O a ((log q 0 N ) 2 (1 -γ)N -γ ).
Along with Lemma 2.6, the above yields the following approximation.

Lemma 2.7. Define R(x, N ) := (1 -β)(log qN ) 2 N β x 1-β .
For (a, q) = 1, ν ∈ N and prime, we have that

D 1 (1, a; N ) -η x,a D β (q, a; N ) =        O a (R(x, N )) if ω(a) ≥ 2; (1 -η χ,a ) 1 2N (1 -1 ) log + O a (R(x, N )) if a = ± ν ; (1 -η x,a ) 1 2N {log N + 2C 0 } + η x,a 1 2N {log q -p|q log p p } + O a (R(x, N )) if a = ±1.
2.6. Cancellation of main terms and proof of Theorem 1.3. In this section we combine the main terms in T 2 , T 3 , T 4 and T 5 and prove our main theorem.

Proof of Theorem 1.3. Recalling (2.1), we have by (2.2) and Lemmas 2.3 and 2.5 that for some small enough δ > 0,

M Z 1 (x, N ; a) = T 1 + T 2 -T 3 -T 4 + T 5 = x J 1 (x, x 1 2 -δ , 1, a) -J 1 (x, N, 1, a) -η x,a x J β (x, x 1 2 -δ , q, a) -J β (x, N, q, a) + O(xe -δ √ log x ) = -xD 1 (1, a; N ) -η x,a x Jβ (x, q, a) -J β (x, N, q, a) + O a,ε (xN -1-171 448 +ε ).
Here, we used the bound D 1 (q, a, N ) a N -1 (log qN ) along with (2.9). If the x-exceptional character does not exist, then this yields (1.8).

Next, assume that the x-exceptional character does exist, and that N q. Then by definition and since q ≤ e √ log x , Jβ (x, q, a) -J β (x, N, q, a) = C a,q log x q2 + 2D a,q -1 β -r≤N q|r (a,r)=1

1 -(r/N ) β ϕ(r) -q≤x/N q|q (q,a)=1 1 ϕ(q) = C a,q log N q + D a,q -1 β -r≤N q|r (a,r)=1

1 -(r/N ) β ϕ(r)

+ O(x -1 5 ),
where the sum over q was evaluated using Lemma 2.4. Since N ≤ e δ √ log x , this yields (1.9). Assume now that the x-exceptional character exists and that N ≥ q. We use Lemma 2.5 to write Jβ (x; q, a) -J β (x, N ; q, a) = -D β (q, a; N ) + O a,ε (N -1+ε (q/N ) 171 448 ).

Therefore, M Z 1 (x, N ; a) = -x D 1 (1, a; N ) -η x,a D β (q, a; N ) + O a,ε (N -1+ε (q/N ) 171 448 ) .

Our claimed formula (1.10) then follows from Lemma 2.7.

Unconditional bias.

In this last section we prove our unconditional result.

Proof of Theorem 1.1. If the x-character does not exists, then the claimed bound follows from (1.8). We can therefore assume that it does exists. Note that M 1 (x, N ; 1)

x/N = M Z 1 (x, N ; 1)

x/N -N η x,1 q≤x/N q|q 1 ϕ(q) = M Z 1 (x, N ; 1)

x/N -N η x,1 C 1,q log x N q + D 1,q + O(x -1 5 ).

Using our estimate (1.9), and noting that the r-sum is O(log( qN )/ϕ(q)), we obtain that M 1 (x, N ; 1)

x/N = µ(1, N ) + O ε (N -171 448 +ε ) -η x,1 N C 1,q log x q2 + O(log(2 + N/q)) .

Since q, N ≤ e δ √ log x and η x,1 > 0, the last term here contributes a negative quantity for large enough x, and we obtain the claimed inequality.

  171 448 +ε ), Date: March 4, 2020. 2010 Mathematics Subject Classification. 11N13 (Primary); 11N37, 11M20 (Secondary).

1

 1

Theorem 1 . 3 .

 13 Fix an integer a ∈ Z \ {0} and a small enough positive absolute constant δ, and let x ≥ 2 and 2 ≤ N ≤ e δ √ log x .

Note that we have excluded the first term because it has a significant contribution which is trivial to estimate.

The improved exponent is deduced by applying Bourgain's work[START_REF] Bourgain | Decoupling, exponential sums and the Riemann zeta function[END_REF].


Abdal-Azim Al-Terkawi

Fredeŕic Lamaty

Thomas-Xavier Metro

Efficient CO 2 Sequestration by a Solid-Gas Reaction Enabled by Mechanochemistry: The Case of L-Lysine

Keywords: Mechanochemistry, Ball-milling, L-Lysine, Carbamate, CO 2 conversion, CO 2 storage, Regioselectivity, Reversibility

This study describes an efficient and solvent-free method for the regioselective production of L-lysine ammonium εcarbamate by ball-milling L-lysine under a CO 2 atmosphere. The regioselective formation of L-lysine ammonium ε-carbamate via this mechanochemical approach was confirmed by a complete analytical study, including 1 H → 13 C and 1 H → 15 N cross polarization magic-angle spinning NMR measurements as well as liquid NMR analyses, powder X-ray diffraction measurements, thermal analyses, and elemental analyses. The time of milling, rotational speed, milling material, and liquid assistants were screened, while the reversibility of the process was assessed. The milling approach was compared with the synthesis in aqueous solutions and in the solid state without agitation. In addition to being regioselective, the synthesis by solvent-free mechanochemistry was found to be much faster than the other approaches while also producing less waste.

■ INTRODUCTION

The immense production of CO 2 , due to humans and industrial activities, and its steady accumulation in the atmosphere are significant causes of global warming, which is expected to reach 1.5 °C between 2030 and 2052, triggering climate-related risks for nature and humans. Therefore, a massive reduction or zero-CO 2 emission is crucial to prevent further climate-related damages. [START_REF] Masson-Delmotte | Global Warming of 1.5°C: An IPCC Special Report[END_REF] In addition, CO 2 is an abundant, economical, and renewable carbon source that could be captured and used for the production of various materials, thereby avoiding its greenhouse effect. [START_REF] Dell'amico | Converting Carbon Dioxide into Carbamato Derivatives[END_REF][START_REF] Sakakura | Transformation of Carbon Dioxide[END_REF][START_REF] Riduan | Recent Developments in Carbon Dioxide Utilization under Mild Conditions[END_REF][START_REF] Aresta | Carbon Dioxide as a Direct Chemical Feedstock[END_REF][START_REF] Liu | Using Carbon Dioxide as a Building Block in Organic Synthesis[END_REF][START_REF] Song | Selective and Sustainable Catalysis of Carbon Dioxide[END_REF] For instance, CO 2 is well-known to react with amines, leading to the formation of carbamic acids or ammonium carbamates depending on, among other factors, the structures of amines and solvents (Scheme 1). [START_REF] Dell'amico | Converting Carbon Dioxide into Carbamato Derivatives[END_REF][START_REF] Masuda | Studies on the Solvent Dependence of the Carbamic Acid Formation from ω-(1-Naphthyl)Alkylamines and Carbon Dioxide[END_REF][START_REF] Rudkevich | Carbon Dioxide and Supramolecular Chemistry[END_REF] This reactivity has been utilized at the industrial scale to capture CO 2 issuing from fossil fuel-burning power plants. [START_REF] Rochelle | Amine Scrubbing for CO 2 Capture[END_REF][START_REF] Macdowell | An Overview of CO 2 Capture Technologies[END_REF][START_REF] Raynal | From MEA to Demixing Solvents and Future Steps, a Roadmap for Lowering the Cost of Post-Combustion Carbon Capture[END_REF][START_REF] Dutcher | Amine-Based CO 2 Capture Technology Development from the Beginning of 2013A Review[END_REF][START_REF] Heldebrant | Water-Lean Solvents for Post-Combustion CO 2 Capture: Fundamentals, Uncertainties, Opportunities, and Outlook[END_REF] Unfortunately, these processes are limited by their cost due to the energy needed at the CO 2 regeneration step, equipment corrosion, volatility, and stability of the amines, along with environmental issues. [START_REF] Macdowell | An Overview of CO 2 Capture Technologies[END_REF][START_REF] Raynal | From MEA to Demixing Solvents and Future Steps, a Roadmap for Lowering the Cost of Post-Combustion Carbon Capture[END_REF][START_REF] Dutcher | Amine-Based CO 2 Capture Technology Development from the Beginning of 2013A Review[END_REF][START_REF] Heldebrant | Water-Lean Solvents for Post-Combustion CO 2 Capture: Fundamentals, Uncertainties, Opportunities, and Outlook[END_REF][START_REF]Health Effects of Amines and Derivatives Associated with CO 2 Capture: Nitrosamines and Nitramines[END_REF][START_REF] Ghayur | Biological and Chemical Treatment Technologies for Waste Amines from CO 2 Capture Plants[END_REF][START_REF]the most widely used amine to capture CO 2 at the industrial scale, is corrosive and highly toxic according to regulation EC[END_REF] While the predominant processes for capturing CO 2 rely on the use of toxic alkanolamines as chemical sorbents, some research groups have studied the reactivity of CO 2 with amino groups of innocuous α-amino acids and their derivatives to form carbamates. [START_REF] Zhang | Supported Absorption of CO 2 by Tetrabutylphosphonium Amino Acid Ionic Liquids[END_REF][START_REF] Stastny | Supramolecular Structures from Lysine Peptides and Carbon Dioxide[END_REF][START_REF] Yamamoto | Kinetic Investigation on Carbamate Formation from the Reaction of Carbon Dioxide with Amino Acids in Homogeneous Aqueous Solution[END_REF][START_REF] Saravanamurugan | Amine-Functionalized Amino Acid-Based Ionic Liquids as Efficient and High-Capacity Absorbents for CO 2[END_REF][START_REF] Shen | Kinetics of CO 2 Absorption into Aqueous Basic Amino Acid Salt: Potassium Salt of Lysine Solution[END_REF][START_REF] Hu | Reaction Kinetics and Mechanism between Histidine and Carbon Dioxide[END_REF] Of note, the vast majority of these reactions are performed in a liquid phase, mostly using water as a solvent. Using a solvent is the standard approach to homogenize reaction mixtures and to facilitate agitation by traditional stirring systems, yet this lowers the theoretical maximum speed of reactions by moving the reactants away from each other. To the best of our knowledge, only one example of the solvent-free reactivity between amino acid derivatives and CO 2 has been described in the literature: the group of Limbach has performed solid-state NMR studies on the formation of carbamates issuing from the exposition of solid poly-L-lysine to atmospheric CO 2 . [START_REF] Schimming | Evidence by 15 N CPMAS and 15 N-13 C REDOR NMR for Fixation of Atmospheric CO 2 by Amino Groups of Biopolymers in the Solid State[END_REF][START_REF] Dos | Acid-Base Interactions and Secondary Structures of Poly-l-Lysine Probed by 15 N and 13 C Solid State NMR and Ab Initio Model Calculations[END_REF] In this case, the reactivity between poly-L-lysine and CO 2 may have been seriously hampered by mass transfer limitations. To avoid such limitations, many research groups have turned to mechanochemistry, predominantly ball-milling, performing an extraordinarily wide array of reactions between solids in the total absence of solvents. [START_REF] Kaupp | Mechanochemistry: The Varied Applications of Mechanical Bond-Breaking[END_REF][START_REF] James | Mechanochemistry: Opportunities for New and Cleaner Synthesis[END_REF][START_REF] Boldyreva | Mechanochemistry of Inorganic and Organic Systems: What Is Similar, What Is Different?[END_REF][START_REF] Frisčǐc | Mechanochemistry for Synthesis[END_REF][START_REF] Beillard | Alternative Technologies That Facilitate Access to Discrete Metal Complexes[END_REF] Additionally, mechanochemistry is capable of modifying the selectivity when compared to solvent-based syntheses and giving access to otherwise inaccessible compounds. [START_REF] Hernańdez | Altering Product Selectivity by Mechanochemistry[END_REF] While mechanochemistry is now commonly used for the chemical transformation of solids sometimes involving liquids, it has been used only scarcely with gaseous reactants [START_REF] Bolm | Mechanochemistry of Gaseous Reactants[END_REF] yet with very high efficiency in specific cases. [START_REF] Beillard | Cu(0), O 2 and Mechanical Forces: A Saving Combination for Efficient Production of Cu-NHC Complexes[END_REF] Examples describing the use of ball-milling to convert CO 2 are even rarer. [START_REF] Mori | Mechanochemical Activation of Catalysts for CO 2 Methanation[END_REF][START_REF] Phung | The Solvent-Free and Catalyst-Free Conversion of an Aziridine to an Oxazolidinone Using Only Carbon Dioxide[END_REF][START_REF] Jeon | Edge-Carboxylated Graphene Nanosheets via Ball Milling[END_REF][START_REF] Leclaire | Metal Extraction from Supramolecular Solids Obtained by Mechanochemistry[END_REF][START_REF] Brekalo | Manometric Real-Time Studies of the Mechanochemical Synthesis of Zeolitic Imidazolate Frameworks[END_REF] In addition, L-lysine is an innocuous and essential α-amino acid, produced at more than a million metric tons per year and is of paramount importance for animal feed, pharmaceutical, and cosmetic industries. [START_REF] Tadepally | Recent Advances in the Industrial Production of L-Lysine by Bacteria[END_REF] This situation prompted us to study the reactivity of L-lysine with gaseous CO 2 under ball-milling. The results of this approach were compared with syntheses in aqueous solutions and in the absence of both solvent and agitation. To the extent of our knowledge, the present work is the first study on carbamates formation by milling an amino acid with CO 2 .

■ RESULTS AND DISCUSSION Reactivity and Selectivity. In a typical milling experiment, 500 mg of commercially available L-lysine was milled in a planetary mill for 5 h under 5 bar of CO 2 gas (a 20 mL ZrO 2 reactor containing 80 ZrO 2 balls of 5 mm diameter was used), leading to the formation of a white powder that was recovered directly from the milling reactor and first analyzed by thermogravimetry measurements. The corresponding thermoanalytical curve depicted a mass loss of -12.4% between 75 and 175 °C, which was attributed to the release of one CO 2 molecule (calc. 12.42%) from an ammonium-carbamate ion pair, thereby confirming the formation of the carbamate bond during milling (see Figure S3b). The mass loss observed up to 75 °C (-5.1%) was assigned to the release of one water molecule (calc. 5.08%), indicating that the ammoniumcarbamate ion pair was obtained as a monohydrate (0.5 H 2 O per one L-lysine molecule). These results were consolidated by the elemental analysis, which was in agreement with the formation of an ammonium carbamate monohydrate (see Figure S3 and Table S2 for details). The formation of a carbamate bond was further confirmed by 1 H → 13 C cross polarization magic-angle spinning (CP MAS) NMR spectrum that depicted a carbon signal at δ = 165 ppm (Figure 1, red spectrum).

Noteworthy, only one peak was observed in the carbamate region, strongly suggesting the newly formed carbamate bond was selectively installed on one of the two available positions only (α-NH 2 or ε-NH 2 ). This result was confirmed by a 1 H → 15 N CP MAS NMR experiment showing only one peak at δ = 88 ppm, corresponding to a nitrogen atom engaged in a carbamate bond (Figure 2, red spectrum).

On the 1 H → 13 C CP MAS NMR spectrum (Figure 1, red spectrum), the single carbon signal at δ = 178 ppm displaying a similar chemical shift to the signal of the L-lysine carboxylate (δ = 177 ppm, black spectrum) strongly suggested that the carbamate bond was not installed on α-NH 2 but on ε-NH 2 . These results were then compared with the L-lysine ammonium carbamate obtained by bubbling CO 2 for 4 h into an aqueous solution of L-lysine (500 mg of L-lysine dissolved in 5 mL of D 2 O; η = 10 μL/mg) followed by concentrating until dry to remove water. The 1 H → 13 C CP MAS NMR spectrum of the corresponding L-lysine ammonium carbamate (Figure 1, purple spectrum) depicted a peak at δ = 165 ppm and a high shoulder at δ = 161 ppm, which are indicative of the formation of two different carbamates installed at the α-NH 2 and the ε-NH 2 positions, contrary to neat-milling. This spectrum also depicted two signals corresponding to a carboxylate: a peak at δ = 178 ppm (similar to the carboxylate of L-lysine, Figure 1, black spectrum) and a shoulder at 182 ppm, which can be assigned to the carboxylate bearing a carbamate on the α-NH 2 . As this latter peak is not visible on the 1 H → 13 C CP MAS NMR spectrum of the L-lysine ammonium carbamate obtained by neat-milling, this is an additional proof that neat-milling led to the formation of the L-lysine ammonium carbamate on the ε-NH 2 position only (Figure 1, red spectrum). In addition to being a regioselective process, neat-milling provided L-lysine ammonium ε-carbamate in quantitative yields because, for every reaction condition screened (all the experimental conditions are summarized in Table S1), the obtained powders were analytically pure after being recovered directly from the milling reactor without further treatment. Due to the poor spectrum) and spectra of the products obtained by the reaction of L- lysine with CO 2 gas under neat-milling conditions (red spectrum), synthesis in water/methanol (magenta spectrum), water-assisted milling (blue spectrum), bubbling CO 2 gas into an aqueous solution of L-lysine followed by water evaporation (purple spectrum), and storing L-lysine (powder) under a CO 2 gas atmosphere for 15 days (green spectrum). (b) The proposed molecular structures of L-lysine ammonium ε-carbamate (obtained by neat-milling and synthesis in water/methanol methods) and L-lysine ammonium αand εcarbamates (obtained by water-assisted milling and synthesis in water methods). Color codes: the ε-carbamate bond is shaded in red, and the α-carbamate bond is shaded in blue. crystallinity of the obtained powders, ab initio structure determination of the carbamates starting from the powder Xray diffraction (PXRD) data was unfortunately not possible. Efforts for improving the polycrystallinity or the growth of single crystals did not succeed so far. Neat milling was not the only strategy enabling to obtain L-lysine ammonium εcarbamate regioselectively. Indeed, a patent authored by Bru in 1980 described the synthesis of L-lysine ammonium εcarbamate by reacting CO 2 with L-lysine solubilized in a 1:5 water/methanol mixture. [START_REF] Bru | DE2951132A1[END_REF] Although the original paper did not provide any experimental proof of the regioselectivity of the reaction, in our hands, the corresponding protocol produced L- lysine ammonium ε-carbamate regioselectively, as indicated by the 1 H → 13 C CP MAS NMR spectrum (Figure 1, magenta spectrum). However, this strategy requires MeOH (a toxic solvent) and necessitates a greater experimental time than that of the neat-milling procedure. One could explain the selectivity of the reaction under neat-milling by the crystalline structure of L-lysine. Indeed, in the crystal structures reported by Harris and co-workers, [START_REF] Williams | Exploiting Powder X-Ray Diffraction to Complete the Set of Crystal Structures of the 20 Directly Encoded Proteinogenic Amino Acids[END_REF][START_REF] Williams | Understanding the Solid-State Hydration Behavior of a Common Amino Acid: Identification, Structural Characterization, and Hydration/Dehydration Processes of New Hydrate Phases of L-Lysine[END_REF] L-lysine molecules are found as zwitterions, where the α-nitrogen atom is protonated (as α-NH 3 ) while the ε-NH 2 is not (but involved in a hydrogen-bonding network (HBN) with water); this difference could explain the higher reactivity of the ε-nitrogen atom. This selectivity related to the crystal structure of L-lysine is also a strong indication that the carbamate formation during neat-milling is a solidstate process and that the presence of crystallized water in L- lysine hemihydrate is not sufficient to induce the formation of both αand ε-carbamates. In contrast, the protophilic character of water could explain the nonselectivity of the formation of the carbamate when synthesis is performed in water. The selective formation of the ε-carbamate molecule in the water/methanol mixture could therefore be explained by precipitation/crystallization, which is known to be a potentially selective process. [START_REF] Erdemir | Nucleation of Crystals from Solution: Classical and Two-Step Models[END_REF] Speed of Reaction. The influence of the milling time on the formation of L-lysine ammonium ε-carbamate was next investigated (Table S1). To our delight, the conversion of L- lysine was completed after a reduced milling time of only 5 min, indicating that the reaction is very rapid. Comparatively, synthesis in water was much slower, which could be explained by the formation of carbonate/bicarbonate ions as the kinetic products resulting from the reaction of water with CO 2 , as indicated by the typical carbon signals at about δ = 160 ppm (Figure 3, red [START_REF] Dutcher | Amine-Based CO 2 Capture Technology Development from the Beginning of 2013A Review[END_REF] C NMR spectrum). Part of the speed reduction could also be attributed to CO 2 diffusion limitations, which are minimized in industrial CO 2 absorption processes by using dedicated gas-liquid contacting equipment. [START_REF] Macdowell | An Overview of CO 2 Capture Technologies[END_REF][START_REF] Raynal | From MEA to Demixing Solvents and Future Steps, a Roadmap for Lowering the Cost of Post-Combustion Carbon Capture[END_REF][START_REF] Dutcher | Amine-Based CO 2 Capture Technology Development from the Beginning of 2013A Review[END_REF] The carbon signals related to the carbamate bond formation start to appear only after 48 h of reaction (Figure 3, blue spectrum, peaks at δ = 164.8 and 163.7 ppm). Alternatively, a mixture of L-lysine ammonium αand ε-carbamates could be isolated after removing water under reduced pressure, which is yet known to be an energy-intensive process (Figure 3, purple spectrum).

The [START_REF] Dutcher | Amine-Based CO 2 Capture Technology Development from the Beginning of 2013A Review[END_REF] C NMR analysis of a pure sample of L-lysine ammonium ε-carbamate also displays a small peak corresponding to the formation of carbonate/bicarbonate ions (δ = 161 ppm) almost immediately after dissolution in D 2 O, confirming that the formation of ammonium carbamates of L-lysine is a reversible process in water. [START_REF] Ciftja | Carbamate Formation in Aqueous -Diamine -CO 2 Systems[END_REF][START_REF] Lv | An Efficient Absorbent of Amine-Based Amino Acid-Functionalized Ionic Liquids for CO 2 Capture: High Capacity and Regeneration Ability[END_REF] Structural Analysis by Liquid NMR. The rapid formation of carbonate/bicarbonate ions after the solubilization of L- lysine ammonium carbamates in D 2 O did not prevent homoand heteronuclear 2D liquid NMR measurements to give relevant information on the structure of L-lysine ammonium carbamates in D peaks of the proton signal at δ = 3.8 (labeled Hα*) with the carbamyl carbon signal at δ = 164 ppm (labeled CN-α*) and the carboxylate carbon at δ = 181 ppm (CO*) indicate the formation of the α-carbamate ion. Furthermore, the cross-peak between the proton Hα (δ = 3.4 ppm) and the carboxylate carbon CO (δ = 177 ppm) is attributed to the formation of an α-ammonium ion. On the other hand, the cross-peak between the proton signal at δ = 2.85 ppm (Hε) and the carbamyl signal at δ = 165 ppm (CN-ε) indicates the formation of an εcarbamate ion. These measurements suggest that, in water, both the αand ε-carbamates are individually formed in two different lysine molecules, as depicted in Figure 1b. The proton-carbon cross-peak at δ = 3.5 ppm (H α +)/176 ppm (CO+) is attributable to the protons exchange between αamino and carboxylate groups ( + H 3 N-C α HR-COO -↔ H 2 N-C α HR-COOH) indicated by the minimal chemical shift between both carboxylate groups (Δδ = 1 ppm).

Influence of Liquid Additives. Next, we investigated the influence of liquid additives on the formation of L-lysine ammonium ε-carbamate by ball-milling. Indeed, previous studies on solution-based reactions showed that the nature of the solvent could influence the selectivity outcome of the reaction between amino-functionalized samples and CO 2 in solution. [START_REF] Masuda | Studies on the Solvent Dependence of the Carbamic Acid Formation from ω-(1-Naphthyl)Alkylamines and Carbon Dioxide[END_REF][START_REF] Stastny | Supramolecular Structures from Lysine Peptides and Carbon Dioxide[END_REF] In mechanochemistry, the use of a liquid additive during milling, known as liquid-assisted grinding (LAG), is capable of sharply accelerating the reactions, improving the crystallinity of the resulting powder products and modifying the polymorphic outcome. [START_REF] Frisčǐc | The Role of Solvent in Mechanochemical and Sonochemical Cocrystal Formation: A Solubility-Based Approach for Predicting Cocrystallisation Outcome[END_REF][START_REF] Frisčǐc | Mechanochemical Conversion of a Metal Oxide into Coordination Polymers and Porous Frameworks Using Liquid-Assisted Grinding (LAG)[END_REF][START_REF] Frisčǐc | Ion-and Liquid-Assisted Grinding: Improved Mechanochemical Synthesis of Metal-Organic Frameworks Reveals Salt Inclusion and Anion Templating[END_REF][START_REF] Bowmaker | Solvent-Assisted Mechanochemistry[END_REF][START_REF] Hasa | Mechanochemical Synthesis of Multicomponent Crystals: One Liquid for One Polymorph? A Myth to Dispel[END_REF][START_REF] Al-Terkawi | Hydrated and Dehydrated Ca-Coordination Polymers Based on Benzene-Dicarboxylates: Mechanochemical Synthesis, Structures Refinement, and Spectroscopic Characterization[END_REF][START_REF] Riss-Yaw | Association of Liquid-Assisted Grinding with Aging Accelerates the Inherently Slow Slipping-on of a Dibenzo-24-Crown-8 over the N -Hydroxysuccinimide Ester of an Ammonium-Containing Thread[END_REF] In order to facilitate the comparison with other LAG processes, the η parameter was utilized, which is defined by the ratio of the volume of the liquids used (μL) to the total mass of solids (mg). [START_REF] Frisčǐc | The Role of Solvent in Mechanochemical and Sonochemical Cocrystal Formation: A Solubility-Based Approach for Predicting Cocrystallisation Outcome[END_REF] In our case, adding a small amount of water (50 μL of H 2 O/500 mg of L-lysine; η = 0.1 μL/mg) led to the formation of the carbamate bond on the two positions, as indicated by the peak at δ = 165 ppm and the shoulder at δ = 161 ppm in 1 H → [START_REF] Dutcher | Amine-Based CO 2 Capture Technology Development from the Beginning of 2013A Review[END_REF] C CP MAS measurements (Figure 1, blue spectrum). In addition, 1 H → 15 N CP MAS measurement displayed two peaks at δ = 88 and 74 ppm in the carbamate region (Figure 2, blue spectrum). The formation of the carbamates at both the αand ε-NH 2 positions even when minimal amounts of water are present during the synthesis is a clear indication that the total absence of "free" water during milling (as opposed to crystallized water present in the starting material) is necessary to have a selective formation of the carbamate at the ε-NH 2 position. Contrary to neat-milling, grinding with protophilic solvents such as water or DMSO as liquid assistants provided L-lysine ammonium carbamates with a higher crystallinity (Figure S5, red and blue patterns) than the products obtained by neat-milling (Figure S5, black pattern), as indicated by powder X-ray diffraction measurements. However, milling with nonprotophilic CH 2 Cl 2 as a liquid assistant did not improve the crystallinity of the final powder (Figure S5, cyan pattern).

Reactivity in the Solid State without Agitation. To understand the role of agitation during the milling process, gaseous CO 2 was flushed over powders of L-lysine for 4 h without any agitation, followed by storing the powders under a CO 2 atmosphere at ambient pressure for 15 days. 1 H → 13 C CP MAS NMR measurements of the resulting powder indicated the incomplete conversion of L-lysine to the corresponding ammonium carbamate, as indicated by the tiny peak at δ = 165 ppm (Figure 1, green spectrum). This latter result indicates that the reactivity between L-lysine and CO 2 leads to a "negative feedback", as theorized by Boldyreva, [START_REF] Boldyreva | Mechanochemistry of Inorganic and Organic Systems: What Is Similar, What Is Different?[END_REF] confirming that the grinding process is necessary for the rapid conversion of L-lysine to the corresponding ammonium carbamate.

Stability and Reversibility. Next, the thermal stability of L-lysine ammonium ε-carbamate and the reversibility of the formation of this carbamate were assessed. After being stored under ambient conditions for several months, L-lysine ammonium ε-carbamate showed no analytical differences with the freshly prepared compounds, indicating its good stability under ambient conditions. The thermal treatment of L- lysine ammonium ε-carbamate at 175 °C for 1 h or the addition of a few drops of aqueous DCl cleaved the carbamate bond. The release of gaseous CO 2 was indicated by the continuous formation of bubbles while adding aqueous DCl on the aqueous solutions of L-lysine ammonium ε-carbamate, in addition to the disappearance of the carbamate signals in the [START_REF] Dutcher | Amine-Based CO 2 Capture Technology Development from the Beginning of 2013A Review[END_REF] C NMR spectra (Figure 5, red spectrum) (by thermal treatment at 175 °C, Figure 5, green spectrum). The comparisons of [START_REF] Dutcher | Amine-Based CO 2 Capture Technology Development from the Beginning of 2013A Review[END_REF] C NMR spectra in Figure S6 indicate the presence of small quantities of nonidentified products after the DCl treatments, while these products are present in smaller quantities after the thermal treatment. The latter spectrum is similar to the 13 C NMR spectrum of L-lysine, which indicates that the carbamate formation by milling L-lysine with CO 2 is a reversible process upon heating at 175 °C. Of note, this temperature is superior to the one used in classical absorbers that reversibly produce CO 2 at temperatures between 100 to 120 °C. [START_REF] Rochelle | Amine Scrubbing for CO 2 Capture[END_REF] Mechanism. In the literature, two mechanisms were mainly proposed to describe the solution-based reactions between alkylamines and alkanolamines with CO 2 : the termolecular mechanism and zwitterion mechanism. [START_REF] Danckwerts | The Reaction of CO 2 with Ethanolamines[END_REF][START_REF] Donaldson | Carbon Dioxide Reaction Kinetics and Transport in Aqueous Amine Membranes[END_REF][START_REF] Crooks | Kinetics and Mechanism of the Reaction between Carbon Dioxide and Amines in Aqueous Solution[END_REF] The latter mechanism is commonly used to explain the CO 2 reaction with amino acids. [START_REF] Yamamoto | Kinetic Investigation on Carbamate Formation from the Reaction of Carbon Dioxide with Amino Acids in Homogeneous Aqueous Solution[END_REF][START_REF] Shen | Kinetics of CO 2 Absorption into Aqueous Basic Amino Acid Salt: Potassium Salt of Lysine Solution[END_REF][START_REF] Hu | Reaction Kinetics and Mechanism between Histidine and Carbon Dioxide[END_REF] In this proposed reaction pathway (Scheme 2), the carbamate formation is explained by the nucleophilic attack of the nitrogen on the electrophilic carbon of the carbon dioxide molecule, forming a zwitterion (Scheme 2, eq 1). Second, another amino-functionalized molecule acts as a weak base for the formation of an ammonium carbamate ion pair (Scheme 2, eq 2). Such a mechanism could also explain the formation of the carbamates obtained by milling in the absence or quasi-absence of added liquids.

In solution-based reactions, the nature of the solvent can influence the selectivity outcome of the reaction between amino-functionalized samples and CO 2 in solution. [START_REF] Masuda | Studies on the Solvent Dependence of the Carbamic Acid Formation from ω-(1-Naphthyl)Alkylamines and Carbon Dioxide[END_REF][START_REF] Stastny | Supramolecular Structures from Lysine Peptides and Carbon Dioxide[END_REF] It has been reported that carbamic acids of amines and lysine peptides are selectively generated in polar aprotic solvents. [START_REF] Stastny | Supramolecular Structures from Lysine Peptides and Carbon Dioxide[END_REF][START_REF] Yamamoto | Kinetic Investigation on Carbamate Formation from the Reaction of Carbon Dioxide with Amino Acids in Homogeneous Aqueous Solution[END_REF] For our experiments, using LAG conditions of such solvents (dimethyl sulfoxide (DMSO) and dichloromethane (DCM)) did not lead to carbamic acids, only to the formation of ammonium carbamates of L-lysine, as indicated by [START_REF] Dutcher | Amine-Based CO 2 Capture Technology Development from the Beginning of 2013A Review[END_REF] C NMR spectra (Figure S7). The latter result can be rationalized by the poor solubility of L-lysine in the organic solvents, taking into account that the previously reported experiments were performed on α-amino acids completely solubilized in organic solvents.

Although the presence of an organic liquid additive does not seem to influence the outcome of the reaction, water has a critical influence on the course of the reaction. While neatmilling regioselectively produces the carbamate at the ε-NH 2 position, adding a small amount of water during milling (η = 0.1 μL/mg) led to the formation of the carbamate bond on both the αand the ε-NH 2 positions. These carbamates products are confirmed by [START_REF] Dutcher | Amine-Based CO 2 Capture Technology Development from the Beginning of 2013A Review[END_REF] C NMR analyses, where the partial conversion to carbonate/bicarbonate anions is also observed after solubilizing the powder sample in water (Scheme 3, eq 3, and Figure 5, blue spectrum). In contrast, bubbling gaseous CO 2 into a solution of L-lysine in water leads first to the formation of carbonate/bicarbonate anions (Scheme 3, eq 4) and later to the formation of carbamate compounds (2-6 days). Overall, it is strongly suggested that the absence of the added water during neat-milling prevents the competitive formation of the carbonate/bicarbonate anions, which would explain the much faster formation of the carbamate compounds by milling in comparison to during synthesis in water. Comparatively, the rapid appearance (few minutes) of a precipitate of L-lysine ammonium ε-carbamate after bubbling CO 2 in a solution of L-lysine in a 1:5 mixture of water/methanol might indicate that the precipitation mechanism prevails over the formation of the carbonate/bicarbonate anions.

Environmental Impact. Finally, the production of waste by the three protocols was compared (neat-milling, synthesis in water, and synthesis in water/MeOH). The complete E factor (cEF) was calculated for each of these three approaches, of which the values are reported in Table 1. [START_REF] Roschangar | Overcoming Barriers to Green Chemistry in the Pharmaceutical Industrythe Green Aspiration Level TM Concept[END_REF][START_REF] Sheldon | The E Factor 25 Years on: The Rise of Green Chemistry and Sustainability[END_REF] The cEF is defined as the amount of waste divided by the amount of final product. Contrary to the simple E factor (sEF), the cEF takes into account water and all organic solvents, assuming no recycling. Thus, the neat-milling process displays a cEF of 0.4, while the two solvent-based processes have a much higher cEF of 22.5 and 15.0, for synthesis in water and in water/MeOH, respectively.

However, these figures were obtained while the CO 2 quantities were not optimized in any of the three processes, possibly leading to overvalued cEF values. Therefore, to Scheme 2. Proposed Zwitterion Mechanism for the Formation of Ammonium Carbamate by Milling L-lysine (Introduced as R-NH 2 ) under a CO 2 Atmosphere a a Nucleophilic attack of the nitrogen atom (R-NH 2 ) on the electrophilic carbon of CO 2 , forming a zwitterion (eq 1). Another nitrogen atom (R-NH 2 ) acts as a weak base, leading to the formation of an ammonium carbamate ion pair (eq 2).

Scheme 3. Reactions for the Conversion of Carbamate to Carbonate/Bicarbonate (eq 3), the Formation of Ammonium-Bicarbonates by Dissolving Gaseous CO 2 in Aqueous Amine-Containing Substrate (eq 4), and the Formation of Bicarbonate by Dissolving CO 2 in Water (eq 5) facilitate a comparison between the three described protocols, a second set of E factors was calculated, where the CO 2 quantity used for the calculation was the exact quantity of CO 2 ending in the lysine ammonium carbamate (considering excess CO 2 could be 100% recycled). Additionally, an extra assumption was applied, considering that the solvents used could be recycled with a 10% loss. In these conditions, the cEF value was still much lower for the neat-milling process (cEF = 0) than for the solvent-based processes (cEF = 0.9). Overall, these cEF figures clearly show that neat-milling is the process that enables the lowest production of waste, and this superiority can be attributed to the absence of a bulk solvent during neat-milling.

■ CONCLUSION

L-Lysine ammonium ε-carbamate can be easily, rapidly, and selectively produced in quantitative yields by the solid-gas mechanochemical reaction of L-lysine with CO 2 . While the same regioselectivity could be obtained by performing the synthesis in a water/methanol mixture, synthesis by neatmilling discarded the use of the highly toxic methanol. As all atoms of the chemicals used end up in the product, neatmilling is a process that produces no waste (except an excess of CO 2 ), leading to an E factor of 0, well below the other classical processes. The process is highly robust, as the analytically pure carbamate was formed in a wide range of operational conditions (reaction time, rotational speed, and milling material). In addition, this milling approach was much faster than synthesis by bubbling CO 2 into a solution of L-lysine in water, most probably due to the absence of the competitive formation of carbonate/bicarbonate anions. Overall, the present work introduces a solid-gas reaction enabled by mechanochemistry as a rapid, regioselective, and clean method for the chemical conversion of gaseous CO 2 into L-lysine ammonium ε-carbamate via its direct and solvent-free reaction with L-lysine. This study opens doors to future uses of other innocuous amines as efficient sorbents for CO 2 sequestration under solvent-free mechanochemical conditions, which are currently under investigation in our laboratory.

■ ASSOCIATED CONTENT

* sı Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acssuschemeng.0c00217.

Experimental protocols, liquid NMR spectra, PXRD data, elemental analyses, and thermal analysis measurements (PDF)

■ AUTHOR INFORMATION

Figure 1 .

 1 Figure 1. (a) 1 H →[START_REF] Dutcher | Amine-Based CO 2 Capture Technology Development from the Beginning of 2013A Review[END_REF] C CP MAS NMR spectra of L-lysine (black spectrum) and spectra of the products obtained by the reaction of L- lysine with CO 2 gas under neat-milling conditions (red spectrum), synthesis in water/methanol (magenta spectrum), water-assisted milling (blue spectrum), bubbling CO 2 gas into an aqueous solution of L-lysine followed by water evaporation (purple spectrum), and storing L-lysine (powder) under a CO 2 gas atmosphere for 15 days (green spectrum). (b) The proposed molecular structures of L-lysine ammonium ε-carbamate (obtained by neat-milling and synthesis in water/methanol methods) and L-lysine ammonium αand εcarbamates (obtained by water-assisted milling and synthesis in water methods). Color codes: the ε-carbamate bond is shaded in red, and the α-carbamate bond is shaded in blue.

  2 O. Contrary to 1 H → 13 C CP MAS NMR experiment, the 1 H/ 13 C HMBC measurement of L-lysine ammonium ε-carbamate after solubilization in D 2 O (Figure 4) depicted two carbon signals at δ = 164 and 165 ppm, indicating the formation of two different carbamate bonds. This difference could be explained by migrations of the carbamate bonds between ε-NH 2 and α-NH 2 through the intermediate formation of solubilized CO 2 . The two cross-

Figure 2 .

 2 Figure 2. 1 H → 15 N CP MAS NMR spectra of the products obtained by the reaction of L-lysine with CO 2 gas under neat-milling conditions (red spectrum) and under water-assisted milling conditions (50 μL of H 2 O/500 mg of L-lysine; η = 0.1 μL/mg) (blue spectrum). The η parameter is defined by the ratio of the volume of the liquid (μL) to the total mass of solids (mg).

Figure 3 .

 3 Figure 3. 13 C NMR spectra of L-lysine solubilized in D 2 O (black spectrum), bubbling CO 2 (g) into the aqueous L-lysine for 15 min to 2 h (red spectrum), bubbling CO 2 (g) into the aqueous solution for 2-6 days (blue spectrum), and after the solvent evaporation (purple spectrum).

Figure 4 .

 4 Figure 4. 1 H/ 13 C HMBC spectrum of L-lysine ammonium αand εcarbamates obtained by milling. The formation of the α-carbamate ion is indicated by the two cross-peaks at δ = 3.8 (labeled Hα*) with the carbamyl carbon signal at δ = 164 ppm (labeled CN-α*) and the carboxylate carbon at δ = 181 ppm (CO*). The cross-peak between Hα (δ = 3.4 ppm) and the carboxylate carbon CO (δ = 177 ppm) is related to the formation of an α-ammonium ion. The presence of an ε-carbamate ion is confirmed by the cross-peak between Hε (δ = 2.85 ppm) and the carbamyl signal at δ = 165 ppm (CN-ε).

Figure 5 .

 5 Figure 5. 13 C NMR spectra of L-lysine ammonium ε-carbamate assynthesized and after post-treatment (in D 2 O). L-Lysine (black spectrum) depicts the carbon signal of the carboxylate group (δ = 182 ppm). The carbamate bond of L-lysine ammonium ε-carbamate obtained by milling (blue spectrum) is cleaved after treatments by adding aqueous DCl (red spectrum) or by thermal treatment at 175 °C for 1 h (green spectrum).

Table 1 .

 1 Complete E Factors for the Synthesis of L-Lysine Ammonium Carbamates by Neat Milling, Synthesis in Water, and Synthesis in Water/Methanol a See the Supporting Information for details of the calculations. b Calculated assuming 100% recycling of CO 2 . c Calculated assuming 100% recycling of CO 2 and recycling of the solvent with 10% loss.

	type of synthesis	complete E factor (cEF)
	neat-milling	0.4 (0) b
	synthesis in water	22.5 (0.9) c
	synthesis in water/methanol	15.0 (0.9) c
	a

  Thomas-Xavier Metro -Centre National de la Recherche Scientifique (CNRS), E ́cole Nationale Supeŕieure de Chimie de Montpellier (ENSCM), Institut des Biomolećules Max Mousseron (IBMM), Univ Montpellier, Montpellier, France; orcid.org/0000-0003-2280-3595; Email: thomasxavier.metro@umontpellier.fr

	Corresponding Author

Downloaded via UNIV MONTPELLIER on August 28, 2020 at 11:56:47 (UTC).See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

https://dx.doi.org/10.1021/acssuschemeng.0c00217 ACS Sustainable Chem. Eng. XXXX, XXX, XXX-XXX B

https://dx.doi.org/10.1021/acssuschemeng.0c00217 ACS Sustainable Chem. Eng. XXXX, XXX, XXX-XXX C

https://dx.doi.org/10.1021/acssuschemeng.0c00217 ACS Sustainable Chem. Eng. XXXX, XXX, XXX-XXX D

https://dx.doi.org/10.1021/acssuschemeng.0c00217 ACS Sustainable Chem. Eng. XXXX, XXX, XXX-XXX E

https://dx.doi.org/10.1021/acssuschemeng.0c00217 ACS Sustainable Chem. Eng. XXXX, XXX, XXX-XXX F

https://dx.doi.org/10.1021/acssuschemeng.0c00217 ACS Sustainable Chem. Eng. XXXX, XXX, XXX-XXX G

https://dx.doi.org/10.1021/acssuschemeng.0c00217 ACS Sustainable Chem. Eng. XXXX, XXX, XXX-XXX H

We thank the following colleagues for technical support: Aureĺien Lebrun and Karine Parra (liquid NMR), Philippe Gaveau and Emmanuel Fernandez (MAS NMR), Amine Geneste and Romain Tavernier (thermal analysis), Dominique Granier (powder XRD), and Magalie Lefeuvre (elemental analysis).

Authors

Abdal-Azim Al-Terkawi -Centre National de

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS


Laurent Chasserat

Nicola Accettura

Keywords:

come

Short: Achieving energy efficiency in dense

LoRaWANs through TDMA Laurent Chasserat, Nicola Accettura, Pascal Berthou LAAS-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France {firstname.lastname}@laas.fr

Abstract-Long Range Wide Area Networks (LoRaWANs) have recently emerged as a hot research topic for their capability to collect sporadic data from a great number of widely spread low power devices. By enabling low-cost low-traffic wireless communications at large scale, such networks can be adopted in many application domains, including smart agriculture, logistics, and emergency detection among others. LoRaWAN employs a pure ALOHA default medium access scheme, that limits the maximum achievable throughput to 18%. Increasing the number of terminals would lead to more frame collisions, and eventually to network collapse. The easiest way to allow bigger network sizes is to synchronize devices, so that they can discretize time into slots, and use them to schedule frame transmissions. Herein, a Time Division Multiple Access (TDMA) scheme on the top of LoRaWANs is highly desired. However, since keeping synchronization can lead to heavy energy consumption, the design of such scheme should enforce energy-efficiency. This contribution describes the technical issues related to the implementation of such mechanism, that as a matter of fact represents the cornerstone on which more sophisticated random access protocols or even scheduling techniques can be designed. In dense networks, the proposed scheme combined with a very simple slotted ALOHA mechanism has also been shown to outperform the default LoRaWAN access protocol in terms of expected energy consumption.

I. INTRODUCTION

Low Power Wide Area Networks (LPWAN) have shown their capability to interconnect a huge number of batterypowered devices [START_REF] Centenaro | Long-range communications in unlicensed bands: the rising stars in the IoT and smart city scenarios[END_REF]. Such networks are tailored for largescale sensing and actuating applications, with any device transmitting very small amounts of data per day. Among all the available LPWAN solutions, the Long Range (LoRa) [START_REF] Alliance | LoRaWAN® Specification v1[END_REF] communication technology has triggered a major interest of the research and industrial communities for its inherent ability to exploit unlicensed ISM bands for featuring robust bidirectional low-power communications over very long ranges. The LoRa modulation is proprietary and owned by Semtech, while the LoRaWAN Medium Access Control (MAC) and its open development are supported by the LoRa Alliance. An important research challenge to be addressed is the enhancement of the scaling capabilities of LoRaWANs. Indeed, access to the radio medium follows a pure ALOHA scheme [START_REF] Abramson | THE ALOHA SYSTEM: another alternative for computer communications[END_REF], with collisions naturally bounding the maximum achievable throughput to 18%. Therefore, when networks get denser, the increase of retransmission attempts leads to a significant waste of energy and eventually network collapse. Synchronization can be leveraged to alleviate this problem by allowing the use of Time Division Multiple Access (TDMA) and any contentionbased (e.g., slotted ALOHA) or contention-free (e.g., resource scheduling) technique on the top of it to increase the network capacity. However, sharing a common time reference among all devices implies additional energy consumption. In such a pictured context, the LoRaWAN Class B operational mode is enhanced by the Class S communication pattern introduced by this contribution. The synchronization induced by Class B beacons is used to define inter-beacon transmission timeslots fitting the biggest LoRaWAN frame size. This contribution additionally provides a preliminary simulationbased evaluation of Class S in the simplest case of singlegateway LoRa networks handling uplink monitoring traffic. Class S is leveraged to setup a simple slotted ALOHA access scheme, that outperforms Class A for networks connecting more than 2500 devices transmitting 1 frame per hour. Indeed, the throughput is increased by switching from pure to slotted ALOHA access [START_REF] Roberts | ALOHA Packet System With and Without Slots and Capture[END_REF], and, with a proper beacon handling, the synchronization overhead can be kept low, thus bounding the overall energy consumption. A threshold-based mechanism is then highly desirable to wisely switch between access techniques depending on the network load.

In order to give sufficient background on the technological landscape of the presented research, Sec. II overviews the LoRa modulation and LoRaWAN specification, as well as related research works on LoRa synchronization and scheduling. Then, Sec. III pictures the proposed synchronization and slotting scheme. Simulation results are presented in Sec. IV, and concluding remarks drawn in Sec. V.

II. OVERVIEW AND RELATED WORKS

LoRa is an LPWAN technology offering long range reliable communications with a large number of low power devices. While the LoRa modulation and physical layer are proprietary and owned by Semtech, the LoRaWAN specification [START_REF] Alliance | LoRaWAN® Specification v1[END_REF], [START_REF] Alliance | LoRaWAN Regional parameters v1[END_REF] is open and supported by the LoRa Alliance. In details, the LoRa modulation relies on Chirp Spread Spectrum to reach long ranges at relatively low power, while being resilient to channel fading. Information is coded over linear frequency sweeps, chirps, over the whole bandwidth (BW) of the channel, and different symbols values are represented by different offsets on the start frequency [START_REF] Knight | Decoding LoRa: Realizing a Modern LPWAN with SDR[END_REF]. One chirp carries one symbol, and the Spreading Factor (SF) is the number of bits encoded in one of these symbols. Increasing the SF lengthens the chirp Time-on-Air (ToA), thus improving the transmission range but lowering the data rate and increasing energy consumption. Based on this, developers derived 7 Data Rates (DR), corresponding to 7 compromises between range, data rate and energy consumption. As LoRa operates over unlicensed ISM bands, it has to follow region-specific regulations (e.g., in Europe this is done by the European Telecommunications Standards Institute, ETSI), which mainly consist in duty-cycle and transmission (TX) power limitations.

LoRaWAN defines the MAC layer in details, as well as the server interfaces, security considerations and regional parameters. In LoRaWAN deployments, large numbers of end devices communicate via gateways using LoRa modulation. Gateways are in charge of forwarding the messages to the server through a TCP/IP ethernet or cellular backhaul. The LoRaWAN specification defines three main device classes.

Class A is the default operational mode that all LoRaWAN devices should implement. The radio medium is accessed by the device in a pure ALOHA fashion. Then, two receiving (RX) slots, RX1 and RX2, are opened at fixed delays after the uplink transmission (1 and 2 seconds by default for Europe). RX2 is only opened if nothing was received during RX1. In the case of a confirmed message, if no downlink acknowledgement was received during RX1 or RX2, a collision must have occurred: the message will be retransmitted after a random delay. Receiving windows are sufficiently long to let the device detect a LoRa preamble (30ms). If a preamble is detected, then the window will be kept open to demodulate the whole message. A downlink generated by the server may only be transmitted synchronously with the RX1 (or RX2) opened by the recipient end device after this latter has sent an uplink.

Class B devices still implement Class A for uplink communications but must open additional RX slots, namely ping slots, to enable server initiated exchanges, thus enhancing bidirectionality. Ping slots are 30ms long, as RX1 and RX2, to correctly capture a LoRa preamble. Devices stay synchronized to the network by listening to periodic beacons broadcast by gateways. The opening of reception windows both for beacons and ping slots clearly implies an additional energy consumption for end devices. As shown in Fig. 1a, each beacon is sent at the beginning of a 128s long BEACON_PERIOD composed of: (i) a 2.120s-long BEACON_RESERVED interval, with the beacon being broadcast at the beginning of such an interval; (ii) a 122.880s long BEACON_WINDOW containing exactly 4096 ping slots; and (iii) a 3s BEACON_GUARD interval, permitting the transmission of the biggest frame started during the last ping slot of the BEACON_WINDOW. More specifically, a given device opens its first ping slot after pingOffset slots (randomly chosen), and then periodically after pingPeriod slots until the end of the beacon window. Fig. 1a pictures a device opening 2 ping slots, with a downlink received in the second one. All devices join a LoRaWAN using Class A and can switch to Class B after negotiation with the network server.

Finally, Class C devices are always listening when not transmitting. This allows to reduce processing delays to the minimum, but at the cost of a considerable energy consumption. In fact, Class C targets actuators or repeaters.

The LoRaWAN specification is quite recent and the networking research community is giving momentum to its development at large scale. In more details, several studies on LoRaWAN synchronization and scheduling have been published in the last few years. In [START_REF] Rizzi | Using LoRa for industrial wireless networks[END_REF], Rizzi et al. propose a first attempt at using Time Slotted Channel Hopping (TSCH) within LoRaWAN, relying on very large slots encapsulating Class A operation to target soft real time industrial applications, but no synchronization strategy is stated. The authors of [START_REF] Reynders | Improving Reliability and Scalability of LoRaWANs Through Lightweight Scheduling[END_REF] addressed the capture effect problem by scheduling large timeslots in which transmissions have similar Received Signal Strength and SF requirements. Yet, achieving scalability was not a design guideline for these works, hence intra timeslot collisions were not prevented. Offline scheduling algorithms are proposed in [START_REF] Islam | Duty-cycle-aware real-time scheduling of wireless links in low power wans[END_REF] to reach real-time constraints with dutycycle requirements. In [START_REF] Leonardi | Industrial LoRa: A Novel Medium Access Strategy for LoRa in Industry 4.0 Applications[END_REF] the beacon window is split onto sub-periods with and without contention to satisfy different quality of service expectations. However, these approaches are not directly applicable to online and dynamic networks. In [START_REF] Abdelfadeel | FREE -Fine-grained Scheduling for Reliable and Energy Efficient Data Collection in LoRaWAN[END_REF] and [START_REF] Zorbas | Offline scheduling algorithms for time-slotted lora-based bulk data transmission[END_REF], Zorbas et al. explore the solutions in which data is stored within the nodes and then periodically collected in bulk, preparing the ground for embedding gateways in drones or satellites. Other synchronization techniques have also been investigated. For instance, clock-based solutions are featured in [START_REF] Polonelli | Slotted ALOHA on LoRaWAN-Design, Analysis, and Deployment[END_REF] and [START_REF] Haxhibeqiri | Low Overhead Scheduling of LoRa Transmissions for Improved Scalability[END_REF], which additionally presents a full scheduling mechanism built on top of Class A. Besides, the possibility to use low-power wake-up receivers has been investigated in [START_REF] Piyare | On-Demand TDMA for Energy Efficient Data Collection with LoRa and Wake-up Receiver[END_REF]. Yet, exploiting the existing beacons would help lightening enddevice complexity, which is crucial in LPWANs.

To the best of the authors' knowledge, this paper is the first work taking advantage of Class B beacons to define timeslots that tightly fit the biggest LoRaWAN frame size, and then using this configuration to setup a slotted ALOHA access while thoroughly evaluating the impact of such a change on energy consumption. This solution can substantially improve the network throughput and is easily applicable to online dynamic networks with very few changes.

III. PROTOCOL DESIGN

This proposal relies on the LoRaWAN Class B synchronization scheme to slice time into timeslots large enough to match the Time-on-Air (ToA) of the longest LoRa frame allowed. As the Class B access scheme enhances Class A to permit asynchronous downlink transmissions, the introduced Class S ("slotted") scheme is meant to enhance Class B to increase the uplink traffic throughput through TDMA. In that, the length of a Class S timeslot is set to be multiple of the 30ms duration of a Class B ping slot, thus facilitating backward compatibility with the current LoRaWAN standard. Any Class S timeslot can be used for either uplink or downlink transmissions. As an example, Fig. 1b shows a downlink transmission happening within the second slot, and an uplink during the fourth.

The ToA of a frame [16] is the number of preamble and payload symbols (respectively s pb and s pl ) multiplied by the ToA of a symbol. s pb is 12.25 in Europe, and:

s pl = 8 + max 8P -4SF -20H + 44 4(SF -2D) (C + 4), 0 , (1)
with P the number of payload bytes, H being 0 or 1 whether a header is present, D being 0 or 1 whether the Adaptive Data Rate mechanism is enabled, and C the coding rate ranging from 1 to 4. Finally, the duration of a symbol is:

ToA symbol = 2 SF BW (2)
Herein, the slot size computation has been done for SF7 with BW125 in Europe (DR5), as it offers the smallest power consumption (Data Rate adaptation is thus disabled). Yet, similar considerations can be drawn to handle several orthogonal SFs in parallel. All parameters are set to configure the longest frame ToA. The maximum MAC payload length allowed for DR5 has been considered, i.e., 250 bytes [START_REF] Alliance | LoRaWAN Regional parameters v1[END_REF], to which the 5 bytes of MAC header and Message Integrity Code are added [START_REF] Alliance | LoRaWAN® Specification v1[END_REF]. CR is set to 4, and explicit header is enabled.

All in all, the maximum frame ToA is 626.94ms. Hence, 21 ping slots would be needed to fit the corresponding transmission. For the sake of generality, using 22 ping slots to achieve 660ms long Class S timeslots would allow an end device to sense the radio medium for possible incoming downlinks before starting an uplink. Indeed, this feature can be safely exploited to design advanced scheduling techniques for dense networks. Following such design guidelines, 187 timeslots can be started during a BEACON_WINDOW, with the last one ending during BEACON_GUARD interval.

IV. PRELIMINARY SIMULATION RESULTS

The TDMA access scheme described above is general enough to be used by any scheduling mechanism, and for uplink or downlink traffic indifferently. This section describes a preliminary performance evaluation of such a strategy in the particular case of unconfirmed uplink communications randomly initiated at the beginning of any Class S timeslot. In this way, the resulting access scheme is slotted ALOHA.

In details, an ad-hoc simulation environment, LoRaWANsim, has been used to mimick single gateway scenarios. Networks operating in Class A are compared with ones using Class S. For these simulations, only one radio channel has been used and each deployment is evaluated for a 1 day duration. In compliance with European regulations, end devices and gateways are subject to duty-cycling limitations of 1 and 10% respectively. Each device pseudo-randomly generates Poisson traffic with an average of 1 frame per hour, matching typical LoRaWAN-based application requirements. Defining "unit area" the squared surface having the coverage range as side, the density of end devices per unit area has been varied until 3000 (corresponding to ∼ 9420 end devices in the coverage area around a gateway). For each of the considered density values, 10 seeds have been used to feed the random number generator, thus obtaining 10 different realizations of the same network configuration. The network throughput and the energy efficiency have been evaluated for each scenario, by averaging the results over those 10 realizations.

Fig. 2 shows the throughput curves for Class A and Class S. The maximum achievable throughput is 0.18 at a density of 800 per unit area for the pure ALOHA Class A. Instead, the slotted ALOHA access implemented on the top of Class S permits a maximum achievable throughput of ∼ 0.34 at about 1600 devices per unit area. Such a value is lower than the theoretical e -1 ≈ 0.36 for slotted ALOHA networks, because no uplinks are allowed during the BEACON_RESERVED and most of the BEACON_GUARD intervals. As a consequence, synchronization allows to increase network capacity by better exploiting the available bandwidth.

Instead, the energy efficiency index plot in Fig. 3 is:

Energy efficiency = E useful E all + E synchro , (3)
where E useful is the per-device energy spent for successful transmissions, E all is the per-device overall energy spent for Fig. 3: Energy efficiency of devices using Class A and Class S.

any transmission attempt, and E synchro is the energy spent for keeping synchronization. It is important to specify that the energy consumed for a transmission attempt is evaluated as the product of the frame ToA, the current drawn by a typical SX1276 LoRa transceiver in TX mode [17], 20mA, and a typical battery voltage, 3.6V. Instead, the energy spent for synchronization is evaluated as the product of the total time a device keeps its radio on for listening to and receiving beacons, the typical current drawn by a SX1276 transceiver in RX mode, 10.8mA, and the same typical battery voltage. Devices having a clock drift of +/-30ppm RTC XTAL will drift to a maximum of 4ms per beacon window. As specified in [START_REF] Alliance | LoRaWAN® Specification v1[END_REF], end devices shall be able to maintain a beacon-less Class-B operation for 2 hours, i.e., without listening to 56 consecutive beacons. This specification detail can be wisely exploited on end devices to intentionally skip beacons, thus increasing their energy efficiency. For instance, listening to 1 out of 5 beacons bounds the drift to 20ms, which is reasonable for 660ms-wide Class S timeslots. Fig. 3 highlights that a beacon-skipping mechanism is crucial for energy efficiency. For low densities, Class A is more efficient as collisions are rare and spending energy for synchronization is energywasting. However, when 80% of the beacons are skipped (i.e., when 1 beacon out of 5 is still listened to by end devices) and the density of devices increases beyond 1200 per unit area, the TDMA access results more energy-efficient than the default Class A communication pattern. More interestingly, when end devices keep listening to beacons once every 10, and the density of end devices is the one that maximizes the throughput when they are operating in Class A (i.e., around 700 end devices per unit area, see Fig. 2), the TDMA-based Class S communication pattern outperforms Class A both in terms of throughput and energy-efficiency.

As matter of fact, the point where the Class A and Class S curves cross clearly shows the reason for which a mechanism to switch between communication patterns is extremely needed to build scalable and energy-efficient LoRaWANs.

V. CONCLUSIONS

In this paper, Class S has been introduced, featuring a novel slotting scheme enabling TDMA access in LoRaWANs. Such a scheme is the milestone on which more complex medium access techniques will be designed. It has been shown through simulations that even a very simple slotted ALOHA access scheme outperforms Class A in dense deployments, improving the network capacity while being more energy efficient. Further than the impact of clock drift on synchronization, future works will tackle the design of smart threshold-based mechanism allowing a seamless switch between LoRaWAN classes depending on the network load. More interestingly, a thorough design of scheduling techniques will be done even for very complex scenarios, including multi-gateway LoRaWAN deployments and bidirectional communications. As a major achievement, the goodness of this design will be tested on real-world experimental setups as well.

  (a) Class B. (b) Class S.

Fig. 1 :

 1 Fig. 1: Usage of the BEACON_WINDOW.

Fig. 2 :

 2 Fig. 2: Throughput of Class A and Class S LoRa deployments.