A Crépieux

S Sahoo

T Q Duong

R Zamoum

M Lavagna

Emission Noise in an Interacting Quantum Dot: Role of Inelastic Scattering and Asymmetric Coupling to the Reservoirs

A theory is developed for the emission noise at frequency ν in a quantum dot in the presence of Coulomb interactions and asymmetric couplings to the reservoirs. We give an analytical expression for the noise in terms of the various transmission amplitudes. Including inelastic scattering contribution, it can be seen as the analog of the Meir-Wingreen formula for the current. A physical interpretation is given on the basis of the transmission of one electron-hole pair to the concerned reservoir where it emits an energy after recombination. We then treat the interactions by solving the self-consistent equations of motion for the Green functions. The results for the noise derivative versus eV show a zero value until eV = hν, followed by a Kondo peak in the Kondo regime, in good agreement with recent measurements in carbon nanotube quantum dots.

In quantum devices, the fluctuations of electrical current provide information on the dynamics of electrons [1][2][3][4], as well as on the energy-photon exchange with the measurement circuit or with the electromagnetic environment [5][6][7][8][9][10][11][12][13][14][15][16]. Understanding the nature of these fluctuations in a quantum dot (QD) is thus a crucial step insofar as this system is the elementary brick of quantum circuits. The measurement of current fluctuations in a QD is becoming more and more precise, and reliable results are now available both at zero-frequency [17][18][19] and finitefrequency [8,20,21]. Interpreting these experimental findings turned out to be a challenging task, especially in the case of a biased interacting QD with asymmetric couplings to the reservoirs. In view of the challenges, there is an increasing need to develop a theory for calculating the current noise in non-equilibrium, incorporating the inelastic scattering contributions that play a crucial role when Coulomb interactions are present. So far, most of the noise calculations in a QD connected to left (L) and right (R) reservoirs, either do not distinguish between the noise in the L-reservoir and that in the R-reservoir [23][24][25][26][27][28][29][30][31] or assume that the left coupling strength Γ L and the right coupling strength Γ R are equal [1,24,32,34], in apparent contradiction with experiments [8,20,21]. Indeed, the measured asymmetry of the couplings can be very large, e.g. a = 11 [8], where a = Γ L /Γ R is the asymmetry factor. Certainly, there are theoretical works where the distinction between left and right couplings is made, but these works are limited to the calculations of the zero-frequency noise [28] and symmetrized noise (generally not the quantity measured in experiments) both for non-interacting [2,3,[35][36][37][38] and interacting QDs [39,40]. In some other works, a linear combination of the auto-correlators and of the crosscorrelators is calculated [41,42]. In summary, develop-ing an efficient theory to calculate the finite-frequency noise in non-equilibrium, and investigating the effects of Coulomb interactions and of coupling asymmetry on the noise profile in each reservoir are important unsolved issues which we address in this Letter.

The noise considered here is the emission noise [43][44][45] at frequency ν, S αβ (ν) = ∞ -∞ ∆ Îα (t)∆ Îβ (0) e -2iπνt dt, where ∆ Îα (t) = Îα (t)-Îα is the deviation of the current from its average value (the index α (β) represents one of the two reservoirs). We calculate S αβ (ν) in an interacting QD by using the non-equilibrium Keldysh Green function technique. When the system is in a steady state, we establish the following formula:

S αβ (ν) = e 2 h γδ ∞ -∞ dεM γδ αβ (ε, ν)f e γ (ε)f h δ (ε -hν) , (1)
where f e γ (ε) and f h δ (ε) = 1 -f e δ (ε) are the Fermi-Dirac functions for electrons in the γ-reservoir and holes in the δ-reservoir respectively, and where the matrix elements M γδ αβ (ε, ν) are listed in Table I. These elements are written in terms of the transmission amplitude t αβ (ε), the transmission coefficient T αβ (ε) = |t αβ (ε)| 2 , the reflection amplitude r αα (ε) = 1 -t αα (ε), and an effective transmission coefficient defined as

T eff,α LR (ε) = 2Re{t αα (ε)} - T αα (ε) [46]
. The transmission amplitude is related to the retarded Green function in the QD for spin σ, G r σ (ε), through:

t αβ (ε) = i Γ α Γ β G r σ (ε), where Γ α = 2πρ α |V α | 2
is the coupling between the QD and the α-reservoir, V α being the electron hopping amplitude between the QD and the α-reservoir, the density of states of which is ρ α . To lighten the notations, we do not put a spin index to t αβ (ε) since we consider a spin-unpolarized QD.

M γδ αβ (ε, ν) γ = δ = L γ = δ = R γ = L, δ = R γ = R, δ = L α = L T eff,L LR (ε)T eff,L LR (ε -hν) TLR(ε)TLR(ε -hν) [1 -T eff,L LR (ε)]TLR(ε -hν) TLR(ε)[1 -T eff,L LR (ε -hν)] β = L +|tLL(ε) -tLL(ε -hν)| 2 α = R TLR(ε)TLR(ε -hν) T eff,R LR (ε)T eff,R LR (ε -hν) TLR(ε)[1 -T eff,R LR (ε -hν)] [1 -T eff,R LR (ε)]TLR(ε -hν) β = R +|tRR(ε) -tRR(ε -hν)| 2 α = L tLR(ε)t * LR (ε -hν) t * LR (ε)tLR(ε -hν) tLR(ε)tLR(ε -hν) t * LR (ε)t * LR (ε -hν) β = R ×[r * LL (ε)rLL(ε -hν) -1] ×[rRR(ε)r * RR (ε -hν) -1] ×r * LL (ε)r * RR (ε -hν) ×rRR(ε)rLL(ε -hν) α = R t * LR (ε)tLR(ε -hν) tLR(ε)t * LR (ε -hν) t * LR (ε)t * LR (ε -hν) tLR(ε)tLR(ε -hν) β = L ×[rLL(ε)r * LL (ε -hν) -1] ×[r * RR (ε)rRR(ε -hν) -1] ×rLL(ε)rRR(ε -hν) ×r * RR (ε)r * LL (ε -hν)
TABLE I: Expressions of the matrix elements M γδ αβ (ε, ν) involved in Eq. ( 1) for the noise S αβ (ν) of an interacting QD with arbitrary coupling symmetry. T eff,α LR (ε) is an effective transmission coefficient defined as T eff,α LR (ε) = 2Re{tαα(ε)} -Tαα(ε). It is important to underline that even if the noise formula for the interacting QD shows an apparent similarity to the one obtained for the non-interacting QD [1], the two formulas are distinct since the effective transmission coefficient T eff,α LR (ε) differs from T LR (ε). Especially T eff,α LR (ε) incorporates inelastic scattering contributions [47,48] which are crucial in interacting systems at finite temperature and voltage. In the absence of interactions, or when only elastic scattering processes are present, T eff,α LR (ε) simply equals T LR (ε): this result follows from the optical theorem which is verified in this case [49], and allows one to recover the formula established in Ref. [1] in the non-interacting case. The difference between the noise formula in Eq. ( 1) and its noninteracting counterpart can be seen as the exact analog of the difference between the Meir-Wingreen formula for the current valid in the presence of interactions [50] and the Landauer formula obtained using scattering theory.

The proof of Eq. ( 1) is the following (see Ref. [49] for details): we start from Eqs. (A11)-(A15) of Ref. [1], obtained in the flat wide band limit for the conduction band after having factorized the two-particle Green function in the QD into a product of single-particle Green functions. Provided that the system is in a steady state, we have [2,3]:

G ≷ σ (ε) = G r σ (ε)Σ ≷ tot,σ (ε)G a σ (ε), where Σ ≷ tot,σ (ε) is the total self-energy [2, 4]: Σ ≷ tot,σ (ε) = Σ ≷ L,σ (ε) + Σ ≷ R,σ (ε) + Σ ≷ int,σ (ε), with Σ ≷ α,σ (
ε), the selfenergy brought by the coupling with the α-reservoir, and Σ ≷ int,σ (ε), the additional self-energy brought by the interactions in the QD. Making use of these relations and noticing that the linear and quadratic terms in Σ ≷ int,σ (ε) cancel in the steady state, one derives Eq. ( 1). In the same way that in the Landauer approach the current is interpreted in terms of transmission of electrons from L-reservoir to R-reservoir, the auto-correlator S αα (ν) can be interpreted in terms of transmission of e-h pairs or their constituents through the QD, from all possible initial locations, before the pairs recombine leading to the emission of an energy hν in the α-reservoir. To get S αα (ν), we thus have to identify the whole set of such physical processes for each given initial state, determine their transmission amplitudes t i , and take the quantum superposition | i t i | 2 to calculate the transmission probability. The processes contributing to S LL (ν) are six in number as depicted in the top row of Fig. 1. We restrict the discussion to S LL (ν) because one can straightforwardly deduce S RR (ν) by interchanging L and R indices.

When the e-h pair is initially located in the L-reservoir, there are three possibilities to emit energy in the Lreservoir by recombination of e-h pairs: (i) through process P 1 in which one electron of energy ε (green sphere) and one hole of energy ε -hν (blue sphere) both experience an excursion into the QD and come back to the L-reservoir, corresponding to the transmission amplitude t 1 = t LL (ε)t * LL (ε -hν); (ii) through process P 2 in which the electron experiences an excursion into the QD and comes back to the L-reservoir, whereas the hole is reflected by the left barrier, corresponding to the transmission amplitude t 2 = t LL (ε)r * LL (ε -hν); and (iii) through process P 3 in which the hole experiences an excursion into the QD and comes back to the L-reservoir whereas the electron is reflected, corresponding to the transmission amplitude t 3 = r LL (ε)t * LL (ε -hν). By taking the quantum superposition of these three processes,

|t 1 + t 2 + t 3 | 2 ,
we get a contribution to the noise which is equal to the matrix element M LL LL (ε, ν) of Table I [49]. Note that even if the amplitudes t 1,2,3 involve the L-index only, we use the subscript LR in the notation for the effective transmission coefficient, T eff,L LR (ε), for the reason that it gives back T LR (ε) when the optical theorem holds [49].

When the e-h pair is initially located in the R-reservoir, both particles cross the entire structure to emit energy in the L-reservoir by recombination, as depicted in Fig. 1(P 4 ), giving rise to the transmission amplitude t 4 = t LR (ε)t * LR (ε -hν), which leads to the matrix element M RR LL (ε, ν) of Table I after taking |t 4 | 2 . When the electron is initially located in the L-reservoir and the hole in the R-reservoir, as depicted in Fig. 1(P 5 ), the electron is reflected and the hole transmitted, giving rise to the transmission amplitude t 5 = r LL (ε)t * LR (ε -hν) which leads to the matrix element M LR LL (ε, ν). By symmetry, the transmission amplitude in process P 6 is t 6 = t LR (ε)r * LL (ε -hν), leading to the matrix element M RL LL (ε, ν). We do not need to take any quantum superposition for the three processes P 4 -P 6 as each of them corresponds to a different initial state. To get the cross-correlators, one needs to consider the interference terms between the processes accompanied by an emission of energy in both reservoirs [2,3,32].

Our study shows that the sum S LR (ν) + S RL (ν) corresponds to the interference between the processes P 5 and P 11 as regards the term proportional to

f e L (ε)f h R (ε -hν), since M LR LR (ε, ν) + M LR RL (ε, ν) = t 5 t * 11 + t * 5 t 11
, and to the interference between the processes P 6 and P 12 as regards the term proportional to

f e R (ε)f h L (ε -hν), since M RL LR (ε, ν) + M RL RL (ε, ν) = t 6 t * 12 + t * 6 t 12 .
These interference terms can be either positive or negative according to the relative values of ε and ν, but become strictly negative at zero-frequency due to charge conservation. As far as the contributions proportional to f e α (ε)f h α (ε -hν) are concerned, they are given by the interference between the process P 7 and the set of processes P 1 -P 3 when α = L, and between the process P 4 and the set of processes P 8 -P 10 when α = R. The noise, given by Eq. ( 1) with M δγ αβ (ε, ν) of Table I, is completely determined once the retarded Green function G r σ (ε) in the QD is known. For the non-interacting single energy level QD, we take the Breit-Wigner form:

G r σ (ε) = [ε -ε 0 + i(Γ L + Γ R )/2] -1
where ε 0 is the QD energy level. For the interacting single energy level QD, we use the self-consistent renormalized equation-of-motion approach, as developed in Refs. [5][6][7], which applies to both equilibrium and non-equilibrium and allows one to determine G r σ (ε) [49]. It has been successfully used [57] to quantitatively explain the experimental results [58] about the interplay of spin accumulation and magnetic field in a Kondo QD, and is well adapted to describe the Kondo regime in which the noise measurements are performed [8]. In Fig. 2, we report the noise derivative, dS αα (ν)/dV , as a function of the voltage V for two values of a = Γ L /Γ R and U (with ε 0 = -U/2), as well as the derivative of the sum of the cross-correlators d[S LR (ν) + S RL (ν)]/dV . For completeness, we also plot the derivative of the total noise, dS tot (ν)/dV , where S tot (ν) = [S LL (ν) + a 2 S RR (ν) -aS LR (ν) -aS RL (ν)]/(1 + a) 2 , following recent theoretical works which show, by using a current conservation argument along with the Ramo-Shockley theorem, that this is the quantity which is measured in experiments [37,38,40]. A common point to all the curves is the presence of a plateau of value zero at voltage smaller than frequency, here |V | < hν/e = 0.32 mV, since hν = 78 GHz. The origin of this plateau is related to the fact that the system cannot emit at a frequency higher than the energy provided to it, i.e. the voltage, in full agreement with experiments [8,21]. In the absence of interaction (Figs. 2(a) and(b)), the noise derivatives present a broad peak at |eV | > hν. Its intensity is larger for dS RR (ν)/dV than for dS LL (ν)/dV for both symmetric and asymmetric couplings, due to the fact that the L-reservoir is grounded (µ L = 0). The effect of the coupling asymmetry is to shift the position of the broad peak towards lower values of V . Note that in both cases, the derivative of S LR (ν) + S RL (ν) is sign negative (green curves in Figs. 2(a) and(b)). In the presence of interactions, the electronic transport through a QD is strongly affected. In the Kondo regime, when the number of elec- trons in the QD is equal to 1 and T T K (T K being the Kondo temperature), the differential conductance shows a Kondo peak around V = 0, in addition to the broad peaks resulting from the Coulomb blockade [49]. These effects have their counterparts in the noise. Indeed, the noise derivative shows two clear features (Figs. 2(c) and (d)): a Kondo peak above |eV | = hν, and a secondary broad peak in the proximity of |eV | = U/2, corresponding to the boundaries of the Coulomb blockade structure, in full agreement with experiments [8]. We observe that the noise intensity is reduced in the presence of interactions, as expected in the Kondo regime [24]. This is related to the fact that T eff,α LR (ε) > T LR (ε) (curves not shown), leading to a decrease of M LR αα (ε, ν) which provides the dominant contribution at low temperature. Moreover, the derivative of S LR (ν) + S RL (ν) changes sign at |eV | = U/2, going from positive to negative values with increasing V . It explains why for symmetric couplings (a = 1) the total noise derivative becomes smaller below |eV | = U/2. We also notice that the height of the Kondo peak in dS LL (ν)/dV is larger than in dS RR (ν)/dV when a = 1. This relative order in magnitude is reversed when Γ L > Γ R : in this case the Kondo peak becomes more prominent for the more weakly-coupled reservoir. The explanation is the following: when a = 1, (i) the more pronounced Kondo resonance in the density of states is pinned at the chemical potential (µ L ) of the more strongly-coupled reservoir (orange curve in Fig. 3(a)), and (ii) at low temperature, the main process contributing to dS LL (ν)/dV is P 5 with a probability equal to T LR (ε -hν) at low transmission, whereas the main process contributing to dS RR (ν)/dV is P 11 with a probability equal to T LR (ε) at low transmission. In the process P 5 of Fig. 3(c), there is a transfer of holes from the R-reservoir to the L-reservoir at energy close to µ R = -eV , in the vicinity of which a relatively smaller Kondo resonance is observed, whereas in the process P 11 of Fig. 3(c), there is a transfer of electrons from the L-reservoir to the R-reservoir at energy close to µ L = 0, in the vicinity of which a stronger Kondo resonance is observed (orange curve in Fig. 3(a)). Since P 11 contributes to dS RR (ν)/dV , the Kondo peak is more visible in dS RR (ν)/dV . In the same way, Fig. 3(b) illustrates how the height of the Kondo peak in dS LL (ν)/dV is larger than that in dS RR (ν)/dV when a = 1. We have established a general formula for the emission noise in an interacting QD asymmetrically coupled to reservoirs taking the inelastic scattering contributions into account, and we have given a physical interpretation of the results in terms of the transmission of an e-h pair through the QD with an emission of energy. Combining the theory with the equation-of-motion approach to determine the transmission amplitudes entering the noise formula, we have discussed the profile of the noise derivative. The obtained results explain most of the distinctive features recently observed for the noise in a carbon nanotube QD, specially, the presence or the absence of a narrow peak in dS x (ν)/dV versus V in the vicinity of ±hν/e, and why the Kondo peak in the noise derivative is more prominent in the more weaky-coupled reservoir. The theory developed in this Letter can be applied to treat other realistic systems. Acknowledgments -We would like to thank H. Baranger, In this Supplemental Material (SM), we first present the detailed calculation of the finite-frequency noise for an interacting quantum dot (QD) with asymmetric couplings to the reservoirs (Section A), secondly we prove the relation

dS x (ν)/dV (a) (b) (d) (c) a = 1 a = 4 U = 0 U = 0 a = 1 x = LL x = RR x = LR+RL x = tot
M LL LL (ε, ν) = |t 1 +t 2 +t 3 | 2 (Section B)
, where the second formulation appears when we take the coherent superposition of the processes P 1 , P 2 and P 3 . Thirdly, we give the relation between the various transmission amplitudes and coefficients in the case of a non-interacting QD, or when only elastic scattering is present, for which the optical theorem holds, and we derive the noise matrix elements in that case (Section C). Fourthly we give the expression of the self-consistent equations of motion used to determine numerically the retarded Green function in the case of an interacting QD (Section D), and end up by discussing the results obtained for the differential conductance in the case of symmetric and asymmetric couplings (Section E).

A -CALCULATION OF THE CURRENT NOISE FOR AN INTERACTING QD WITH ARBITRARY COUPLING SYMMETRY

We start from the expression for the current noise in a QD given by Eqs. (A11-A15) in Ref. [1] obtained in the flat wide band limit (FWBL) for the conduction band. We get S αβ (ν) =

5 i=1 C (i) αβ (ν) with C (1)
αβ (ν) = e 2 h δ αβ ∞ -∞ dε G < σ (ε)Σ > α,σ (ε -hν) + Σ < α,σ (ε)G > σ (ε -hν) , (S1) C (2)
αβ (ν) = - e 2 h ∞ -∞ dε G r σ (ε)Σ < α,σ (ε) + G < σ (ε)Σ a α,σ (ε) G r σ (ε -hν)Σ > β,σ (ε -hν) + G > σ (ε -hν)Σ a β,σ (ε -hν) , (S2) C (3)
αβ (ν) = e 2 h ∞ -∞ dε Σ < α,σ (ε)G r σ (ε)Σ r β,σ (ε) + Σ a α,σ (ε)G < σ (ε)Σ r β,σ (ε) + Σ a α,σ (ε)G a σ (ε)Σ < β,σ (ε) G > σ (ε -hν) , (S3) C (4)
αβ (ν) = e 2 h ∞ -∞ dεG < σ (ε) Σ r α,σ (ε -hν)G r σ (ε -hν)Σ > β,σ (ε -hν) +Σ r α,σ (ε -hν)G > σ (ε -hν)Σ a β,σ (ε -hν) + Σ > α,σ (ε -hν)G a σ (ε -hν)Σ a β,σ (ε -hν) , (S4) C (5)
αβ (ν) = - e 2 h ∞ -∞ dε G < σ (ε)Σ r β,σ (ε) + G a σ (ε)Σ < β,σ (ε) G > σ (ε -hν)Σ r α,σ (ε -hν) + G a σ (ε -hν)Σ > α,σ (ε -hν) , (S5) where
Σ r,a,≷ α,σ (ε) = k∈α |V α | 2 g
r,a,≷ k,α,σ (ε) is the contribution to the self-energy brought by the tunneling between the α-reservoir and the QD. G r,a,≷ σ (ε) and g r,a,≷ k,α,σ (ε) are the retarded, advanced, Keldysh greater and lesser Green functions in the QD and in the disconnected α-reservoir respectively. The approximation made to get Eqs. (S1-S5) amounts to having factorized the two-particle Green functions in the QD into a product of two single-particle Green functions. Provided that this approximation is made together with the FWBL assumption, the latter expression for the noise is general and valid for any value of the Coulomb interactions in the QD and any symmetry of the tunneling couplings of the dot to the two reservoirs. We remark that even if the spin index σ appears in the Green function and self-energy notations, it does not appear as an index in the noise notation since the system we consider in this Letter is spin unpolarized. When the system is in a steady state, G ≷ σ (ε) is simply given by [2, 3]

G ≷ σ (ε) = G r σ (ε)Σ ≷ tot,σ (ε)G a σ (ε) . (S6)
In the presence of interactions, the total self-energy can be put in the form [2,4] Σ

r,a,≷ tot,σ (ε) = Σ r,a,≷ L,σ (ε) + Σ r,a,≷ R,σ (ε) + Σ r,a,≷ int,σ (ε) , (S7)
where Σ r,a,≷ int,σ (ε) is the additional contribution brought by the interactions residing in the central region. In the FWBL, Σ r α,σ (ε) = -iΓ α /2, Σ a α,σ (ε) = iΓ α /2, Σ < α,σ (ε) = iΓ α f e α (ε), and Σ > α,σ (ε) = -iΓ α f h α (ε), where Γ α = 2πρ α |V α | 2 , ρ α being the density of states of the α-reservoir and V α the electron hopping amplitude between the QD and the α-reservoir.

In the absence of interactions, Σ int,σ (ε) = 0 and the self-energy contains only the tunneling contributions. As a result, the following relation holds:

G r σ (ε) -G a σ (ε) = -iG r σ (ε)(Γ L + Γ R )G a σ (ε)
, ensuring the optical theorem to be satisfied; the expression for the noise in this case is given in the Section C of this SM.

In the general case, when interactions are present, the latter relation no longer holds due to the contribution of the interaction self-energy. However the calculation of the noise can still be done according to the procedure presented below provided that the system is in a steady state. By incorporating Eqs. (S6) and (S7) into the expression of the noise given in Eqs. (S1-S5), one gets

S αβ (ν) = e 2 h Γ α δ αβ ∞ -∞ dε f h α (ε -hν)G r σ (ε) γ Γ γ f e γ (ε)G a σ (ε) + f e α (ε)G r σ (ε -hν) γ Γ γ f h γ (ε -hν)G a σ (ε -hν) + e 2 h Γ α Γ β ∞ -∞ dε G r σ (ε) γ Γ γ f e γ (ε)G a σ (ε)G r σ (ε -hν) δ Γ δ f h δ (ε -hν)G a σ (ε -hν) -f e α (ε)f h β (ε -hν)G r σ (ε)G r σ (ε -hν) -f e β (ε)f h α (ε -hν)G a σ (ε)G a σ (ε -hν) -i f e α (ε)G r σ (ε) -f e β (ε)G a σ (ε) G r σ (ε -hν) γ Γ γ f h γ (ε -hν)G a σ (ε -hν) +i f h α (ε -hν)G a σ (ε -hν) -f h β (ε -hν)G r σ (ε -hν) G r σ (ε) γ Γ γ f e γ (ε)G a σ (ε) . (S8)
The r.h.s. of Eq. (S8) results from the contribution proportional either to Σ

< γ,σ (ε)Σ > δ,σ (ε-hν) or to Σ > γ,σ (ε)Σ < δ,σ (ε-hν) in Eqs. (S1-S5
), once we have inserted Eqs. (S6) and (S7). We have checked that the remaining contributions coming from the linear and quadratic terms in Σ ≷ int,σ (ε), Σ ≷ int,σ (ε -hν) cancel in the steady state.

Expression for the auto-correlators SLL(ν) and SRR(ν)

From Eq. (S8), taking α = β = L and rearranging the various terms, we get the expression of the auto-correlator noise associated with the L-reservoir, that is

S LL (ν) = e 2 h ∞ -∞ dε γ Γ L Γ γ G r σ (ε)G a σ (ε)f e γ (ε)f h L (ε -hν) + γ Γ L Γ γ G r σ (ε -hν)G a σ (ε -hν)f e L (ε)f h γ (ε -hν) + γ,δ Γ 2 L Γ γ Γ δ G r σ (ε)G a σ (ε)G r σ (ε -hν)G a σ (ε -hν)f e γ (ε)f h δ (ε -hν) -Γ 2 L G r σ (ε)G r σ (ε -hν) + G a σ (ε)G a σ (ε -hν) f e L (ε)f h L (ε -hν) -i γ Γ 2 L Γ γ G r σ (ε) -G a σ (ε) G r σ (ε -hν)G a σ (ε -hν)f e L (ε)f h γ (ε -hν) +i γ Γ 2 L Γ γ G a σ (ε -hν) -G r σ (ε -hν) G r σ (ε)G a σ (ε)f e γ (ε)f h L (ε -hν) . (S9)
Introducing the transmission amplitude

t αβ (ε) = i Γ α Γ β G r σ (ε) and the transmission coefficient T αβ (ε) = |t αβ (ε)| 2 = Γ α Γ β G r σ (ε)G a σ (
ε) in this expression, and performing the sum over γ and δ, we get

S LL (ν) = e 2 h ∞ -∞ dε T LR (ε)T LR (ε -hν)f e R (ε)f h R (ε -hν) + T LL (ε) + T LL (ε -hν) + T LL (ε)T LL (ε -hν) + t LL (ε)t LL (ε -hν) + t * LL (ε)t * LL (ε -hν) -2Re{t LL (ε)}T LL (ε -hν) -2Re{t LL (ε -hν)}T LL (ε) f e L (ε)f h L (ε -hν) + 1 -2Re{t LL (ε)} -T LL (ε) T LR (ε -hν)f e L (ε)f h R (ε -hν) + 1 -2Re{t LL (ε -hν)} -T LL (ε -hν) T LR (ε)f e R (ε)f h L (ε -hν) . ( S10
)
We have used this result to write the expression of the matrix elements M γδ LL (ε, ν) along the first row of Table I introducing an effective transmission coefficient defined as: T eff,L LR (ε) = 2Re{t LL (ε)} -T LL (ε). The expression for the auto-correlator noise in the R-reservoir, S RR (ν) given along the second row of Table I, is obtained from S LL (ν) by interchanging the indices L and R.

Expression for the cross-correlators SLR(ν) and SRL(ν) From Eq. (S8), taking α = L and β = R and rearranging the various terms, we get the expression for the crosscorrelator noise, that is

S LR (ν) = e 2 h Γ L Γ R ∞ -∞ dε γ,δ Γ γ Γ δ G r σ (ε)G a σ (ε)G r σ (ε -hν)G a σ (ε -hν)f e γ (ε)f h δ (ε -hν) -G r σ (ε)G r σ (ε -hν)f e L (ε)f h R (ε -hν) -G a σ (ε)G a σ (ε -hν)f e R (ε)f h L (ε -hν) -i γ Γ γ f e L (ε)G r σ (ε) -f e R (ε)G a σ (ε) G r σ (ε -hν)G a σ (ε -hν)f h γ (ε -hν) +i γ Γ γ f h L (ε -hν)G a σ (ε -hν) -f h R (ε -hν)G r σ (ε -hν) G r σ (ε)G a σ (ε)f e γ (ε) . (S11)
Introducing the transmission amplitude and coefficients, and performing the sum over γ and δ, we get

S LR (ν) = e 2 h ∞ -∞
dε T LR (ε)T LL (ε -hν) -t LL (ε)T LR (ε -hν) -T LR (ε)t * LL (ε -hν) f e L (ε)f h L (ε -hν)

+ T LR (ε)T RR (ε -hν) -t * RR (ε)T LR (ε -hν) -T LR (ε)t RR (ε -hν) f e R (ε)f h R (ε -hν) + T LR (ε)T LR (ε -hν) + t LR (ε)t LR (ε -hν) -t RR (ε)T LR (ε -hν) -T LR (ε)t LL (ε -hν) f e L (ε)f h R (ε -hν) + T LR (ε)T LR (ε -hν) + t * LR (ε)t * LR (ε -hν) -t * LL (ε)T LR (ε -hν) -T LR (ε)t * RR (ε -hν) f e R (ε)f h L (ε -hν) .
(S12)

We have used this result to write the expression of the matrix elements M γδ LR (ε, ν) along the third row of Table I. The cross-correlator S RL (ν), given along the fourth row of Table I, is obtained from the expression of S LR (ν) by interchanging the indices L and R. From Table I, we have

M LL LL (ε, ν) = |t LL (ε) -t LL (ε -hν)| 2 + T eff,L LR (ε)T eff,L LR (ε -hν) , (S13)

FIG. 1 :

 1 FIG. 1: Illustration of the six physical processes contributing to SLL(ν) with the emission of an energy hν in the L-reservoir (top row with orange background devices), and the other six physical processes contributing to SRR(ν) with the emission of an energy hν in the R-reservoir (bottom row with pink background devices). The transmission amplitude ti of the e-h pair for each process with i ∈ [1, 12] is indicated at the bottom of each diagram. A green (blue) sphere represents an electron (a hole) and a yellow wavy arrow represents the emission of energy hν in one of the reservoirs.

FIG. 2 :

 2 FIG. 2: Noise derivative dSx(ν)/dV as a function of V (with µL = 0, µR = -eV ) at T = 80 mK, ν = 78 GHz (chosen such that hν < kBTK ) for ε0 = -U/2 (middle of the Kondo ridge). (a) and (b): U = 0. (c) and (d): U = 3 meV. (a) and (c): ΓL,R = 0.5 meV (a = 1). (b) and (d): ΓL = 0.8 meV, ΓR = 0.2 meV (a = 4). A Kondo peak is observed close to eV = hν when U = 0. Plots for V < 0 are not shown since dSx(ν)/dV is an odd function in V .

8 FIG. 3 :

 83 FIG. 3: (a) Spectral density A(ε) = -π -1 Im{G r σ (ε)} for ΓL,R = 0.5 meV (a = 1), and ΓL = 0.8 meV, ΓR = 0.2 meV (a = 4), at U = 3 meV, ε0 = -U/2, and T = 80 mK. The vertical lines indicate the positions of µR = -eV = -0.35 mV and µL = 0. The a = 1 curve has been vertically translated for clarity. The spectral density at U = 0 is shown in blue in inset. (b) and (c) Schematic representation of the relative importance of the transmission processes at low temperature, low transmission and for eV ≈ hν: P5 is dominant over P11 for a = 1, and P11 is dominant over P5 for a = 4.

B

  -PROOF OF THE RELATION: M LL LL (ε, ν) = |t1 + t2 + t3| 2

H. Bouchiat, R. Deblock, R. Delagrange, M. Guigou, T. Martin, F. Michelini and X. Waintal for valuable discussions. For financial support, the authors acknowledge the Indo-French Centre for the Promotion of Advanced Research (IFCPAR) under Research Project No.4704-02.

with T eff,L LR (ε) = 2Re{t LL (ε)} -T LL (ε), thus

which can be factorized under the form

S16) where we have used the definitions of the reflection amplitude: r LL (ε) = 1 -t LL (ε), and the reflection coefficient: R LL (ε) = r LL (ε)r * LL (ε). Note that we have also the relation: R LL (ε) = 1 -T eff,L LR (ε). The terms appearing in Eq. (S16) correspond precisely to the terms appearing in

In this Section, we show that when we neglect the interactions in the QD, or when we have only elastic scattering, the Table I of Ref. [1] can be derived from the Table I given in the main text of this Letter. Indeed, in that case there are specific relations between the transmission and reflection coefficients which can be obtained from the optical theorem as summarized below.

-Optical theorem

We define the S-matrix of a QD connected to a L-reservoir and a R-reservoir as S = 1 + iT, where the T -matrix is given by

in the {L, R} basis, where τ αβ (ε) = it αβ (ε) with t αβ (ε) the transmission amplitude from the α-reservoir to the βreservoir. For a QD, we have τ αβ (ε) = -Γ α Γ β G r σ (ε), and thus τ RL (ε) = τ LR (ε). At low temperature, when only elastic scattering of electrons are considered, S is a unitary matrix, SS + = 1. Consequently the T -matrix must fulfill the optical theorem:

where we have defined the transmission coefficient as

Eqs. (S19) to (S21) can equivalently written in terms of transmission amplitudes as

Note that these relations are automatically verified if the relation

(S25)

-Matrix elements appearing in the noise

When the optical theorem holds, which is the case in the non-interacting limit or in the elastic scattering processes limit, the matrix elements involved in Table I of this Letter can be rewritten as shown below in Table II. Indeed, starting from the definition of the effective transmission coefficients, we have

thanks to Eqs. (S19) and (S20). Note that for symmetric couplings, i.e. Γ L = Γ R , the transmission amplitude does not depend on the reservoir index any longer, and is simply denoted t(ε), which allows one to derive the matrix elements given in Table I of Ref. [1].

TABLE II: Expressions of the matrix elements M γδ αβ (ε, ν) for a non-interacting QD, or in the presence of elastic scattering processes only.

D -NUMERICAL CALCULATION OF G r σ (ε) IN THE PRESENCE OF COULOMB INTERACTIONS

When Coulomb interactions U are present in the QD, it is necessary to take the spin degree of freedom into account. Indeed the Hamiltonian describing our system is the single-site Anderson Hamiltonian including on-site Coulomb interaction and reads as

where d † σ (c † kασ ) and d σ (c kασ ) are the creation and annihilation operators of an electron in the QD (α-reservoir) respectively, and n σ = d † σ d σ . Following Refs. [5][6][7], we numerically calculate the retarded Green function using the expression

In the FWBL, Σ 0 σ (ε) is independent of ε and takes the value -iΓ σ . Π

where for i ∈ [1,4],

with

kασ = 1, and

kασ ) * . γ σ and γ D are calculated by using the Fermi golden rule up to the fourth order with V α [6,7]. The numerical calculations are performed self-consistently. S1(a)) when the bias voltage profile is symmetric, or two parallel branches (eV = ε 0 and eV = ε 0 + U ) when the bias voltage profile is asymmetric (see Fig. S1(c)). The introduction of an asymmetry in the couplings (weakening Γ R over Γ L , keeping Γ R + Γ L constant) induces the following two changes in the conductance: (i) the value of the conductance decreases due to the reduced transmission through the QD, as it can be clearly seen by comparing the intensity along the Kondo ridge; (ii) at a given value of ε 0 , the relative height of the two broad peaks is changed, e.g., in case of asymmetric bias voltage profile (Fig. S1(d)), the two sub-branches, eV = ε 0 + U < 0 and eV = ε 0 > 0, becomes more prominent than the two other sub-branches eV = ε 0 + U > 0 and eV = ε 0 < 0. Additional remark: the profile of the bias voltage through the QD and the asymmetry of the left and right couplings are closely linked. It is generally expected that the profile is symmetric at a = 1, and asymmetric at a = 1. However only the consideration of the electrodynamics of the whole system can help in determining the link. Here, we rather consider all the possible cases in Fig. S1, but choose to focus on an asymmetric bias voltage profile to be able to compare the curves of Fig. 2 given in the main text of this Letter with experiments of Ref. [8].


Jean-Benoît Le Cam

Strain-induced crystallization in rubber: A new measurement technique

Keywords: calorimetry, crystallinity, infrared thermography, natural rubber, strain-induced crystallization ORCID

The crystallinity of stretched crystallizable rubbers is classically investigated using X-ray diffraction. In this study, we propose a new method based on temperature measurement and quantitative calorimetry to determine rubber crystallinity during mechanical tests. For that purpose, heat power density are first determined from temperature variation measurements and the heat diffu-sion equation. The increase in temperature due to straininduced crystallization is then deduced from the heat power density by subtracting the part due to elastic couplings. The heat capacity, the density, and the enthalpy of fusion are finally used to calculate the crystallinity from the temperature variations due to strain-induced crystallization. The characterization of the stress-strain relationship and the non-entropic contributions to rubber elasticity is not required. This alternative crystallinity measurement method is therefore a user-friendly measurement technique, which is well adapted in most of the mechanical tests carried out with conventional testing machines. It opens numerous perspectives in terms of high speed and full crystallinity field measurements.

INTRODUCTION

The strain-induced crystallization (SIC) of rubber is classically investigated using X-ray diffraction (XRD). Since the pioneering work by Katz in 1925, [1] which provided the first XRD pattern of a stretched natural rubber (NR), numerous studies on SIC were carried out, dealing with the crystalline phase structure, [2][3][4][5] chain orientation, [6] and kinetics of crystallization, [7,8] nonexhaustively.

On a practical point of view, specific equipments and tests conditions are required to measure crystallinity from the XRD technique. On a metrological point of view, quantifying small crystallinity changes is difficult, especially when crystallinity is inferior to about 2%. [8] Moreover, in case of heterogeneous fields, for instance in crack tip vicinity, each point of the crystallinity field are obtained successively, by scanning the zone under study. [9][10][START_REF] Demassieux | Structural changes in the process zone of a cyclic fatigue crack in filled natural rubber[END_REF] Finally, the crystallinity is determined from relatively large measurement zones. As an example, the beam size in [10] was 300×200 μm,whichislarge compared to the stretch (crystallinity) gradient at the crack tip. In this context, developing an alternative measurement method providing

• the crystallinity field instantaneously, • at higher resolutions (temporal, spatial and on the value of the crystallinity), and

• with a conventional testing machine is an important issue to go further on the characterization of crystallization/melting process in crystallizing rubbers.

Change in the XRD is not the only effect of SIC. SIC is also well known to be strongly exothermal, and temperature or calorimetry measurements should be therefore a relevant alternative to determine the crystallinity. In the 1970s, Göritz and co-workers showed that crystallinity can be quantified accurately during stretching through calorimetric measurements and carried out numerous studies on the calorimetric response of stretched natural rubber (see for instance Göritz and Müller [START_REF] Göritz | [END_REF] ). 1 This remarkable work on unfilled natural rubber provided the crystallinity in relation with the stretch during a mechanical loading and during stress relaxation tests carried out at different stretch levels. The authors showed that crystallinity deduced from quantitative calorimetry is measured with a higher accuracy than using XRD, especially at low stretches where XRD does no longer enable us to detect any change in crystallinity. Despite this, calorimetry under stretching was no longer used to measure crystallinity. Recently, Spratte et al. have taken up the idea that crystallinity can be measured from the thermal activity at the surface of rubber specimens. [14] Authors used an infrared camera recording thermal field at the surface of crystallizing rubber, filled and unfilled, tested during cyclic mechanical loadings. The approach proposed by the authors is not a calorimetric one: Crystallinity was deduced from the temperature increase due to crystallization T cryst . To do so, the authors invoked the kinetic theory due to Meyer and Ferri, [15] which considers that the whole work done on the system by the external force 2 is transferred into microscopic kinetic energy and is measurable as temperature. In other words, energetic effects are negligible. Therefore, only two contributions to temperature changes were distinguished by the authors: crystallinity change and entropy change. Authors measured the temperature variation and calculated the heat corresponding to the work done on the system. T cryst was determined from the previous quantities by removing the effect of heat lost by convection at the specimen surface. The crystallinity 𝜒 was deduced from T cryst by considering that the crystallization energy of natural rubber can be approximated by the enthalpy of fusion ΔH cryst :

𝜒(t)= 𝜌C p T cryst (t) ΔH cryst (1)
The authors assumed that the sample density 𝜌 and heat capacity C p were independent of strain and temperature. For filled materials, this ratio was weighted by a factor 1

1-Φ filler

,whereΦ is the filler volume fraction. Results obtained enabled the authors to qualitatively compare the effects of the nature and surface treatment of fillers on the relationship between crystallinity and stretch. Nevertheless, a systematic deviation from the thermomechanical model was obtained in all the crystallinity measurements and led to a negative crystallinity at low stretches for which the material is not crystallizing. This was attributed by the authors themselves to the effect of nonentropic contributions (mainly rotation of carbon bonds, network bonds effects, and interactions between chains). The negative crystallinity they observed shifted therefore the onset of crystallization towards higher stretches and decreased the crystallinity.

Indeed, nonentropic contributions can be significant in rubber elasticity. Deviations to the classical kinetic theory of the rubber elasticity have already been experimentally evidenced by thermoelasticity studies by Allen et al., [16][17][18] Shen, [19] and Le Cam. [20] As discussed in Treloar, [21] the nonentropic (energetic) contribution to rubber elasticity is due to not only changes in intermolecular internal energy but also to changes in the network chain conformational energies, that is, in intramolecular internal energy. This is confirmed by the fact that the energetic contribution to rubber elasticity does not change when diluents are incorporated. [22] In this study, we propose to determine crystallinity from heat power density instead of temperature, by using infrared thermography during the mechanical tests. In practice, any temperature measurement technique can be used, typically a pyrometer. This method takes intrinsically into account both the nonentropic (energetic) and the entropic contributions to rubber elasticity. The measurement of the stress is therefore not required. Moreover, this technique provides a full thermal field, meaning that crystallinity heterogeneity can be characterized, at several thousands frames per second, each value of the field being obtained at the same time. Section 2 presents the methodology used to determine the crystallinity during rubber deformation. Section 3 presents the application of the methodology to experimental data available in the literature for unfilled NR. Concluding remarks close the paper.

METHODOLOGY FOR CRYSTALLINITY MEASUREMENT

This section aims at presenting the thermomechanical framework of quantitative calorimetry, which enables us to define the heat power density from temperature measurements. Then, the methodology giving crystallinity from the heat power density evolution is detailed.

Calorimetry under stretching: Thermodynamical framework

Most of mechanical tests are conducted under nonadiabatic conditions. The temperature measured is therefore affected by heat diffusion during the tests, possible temperature gradients at the surface of the undeformed specimen, and external heat sources (for instance radiations). Therefore, changes in temperature are not only due to the material deformation, and the heat diffusion equation is used to determine the corresponding heat power density from temperature measurement. This quantity is intrinsic to the material deformation. In such conditions, any temperature measurement technique can be used as a calorimeter, an infrared camera in this study. To do so, any thermodynamical system out of equilibrium is considered as the sum of several homogeneous subsystems at equilibrium. The deformation is considered as a quasistatic thermodynamic process. The equilibrium state of each volume material element is defined by n state variables: the absolute temperature T, the deformation gradient tensor F,andm (= n -2) internal tensorial variables 𝝃 𝜶 . The specific heat capacity, denoted C in the following, is the energy required to produce a unit temperature increase of a unit reference volume of a body while maintaining the deformation and internal variables fixed. It is defined by

C =-T ( 𝜕 2 Ψ 𝜕T𝜕T ) F,𝝃 𝜶 > 0( 2 )
where Ψ is the free energy function. In unfilled natural rubber, volume changes are small [23,24] and the specific heat does not vary significantly, even when crystallization occurs [START_REF] Vogt | The Chemistry and Technology of Rubber[END_REF][START_REF] Boissonnas | [END_REF][27] ; therefore, the product 𝜌C can be assumed to be constant. The local form of the heat diffusion equation writes as follows:

𝜌C Ṫ -Div(𝜿 0 Grad T)= int + T 𝜕P 𝜕T ∶ Ḟ + T m ∑ 𝛽=1 𝜕A 𝜷 𝜕T ∶ ξ 𝜷 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ s + R (3)
where 𝜿 0 is a positive semidefinite tensor characterizing the thermal conductivity of the material and s denotes the overall heat power density induced by the deformation process. The term  int corresponds to the intrinsic dissipation. The term T 𝜕P 𝜕T ∶ Ḟ is the heat power density due couplings between temperature and strain, where P is the nominal stress tensor. The term T 𝜕A 𝜷 𝜕T ∶ ξ 𝜷 corresponds to the other thermomechanical couplings (for instance related to phase change in the material). Let us denote 𝜃 the temperature variation with respect to the equilibrium temperature T ref in the reference state, corresponding to the undeformed state (T ref is constant and equal to the ambient temperature). In case where changes in ambient temperature occur, T ref has to be corrected accordingly. After some calculations, which are not detailed here, the heat diffusion equation can be rewritten in case of homogeneous heat power density field [28] :

𝜌C ( θ + 𝜃 𝜏 ) = s ( 4
)
where 𝜏 is a parameter characterizing the heat exchanges between the specimen and its surroundings. It can be easily identified from a natural return to room temperature after a heating (or a cooling) for each testing configuration (machine used, environment, stretch level, etc.). For instance, in case where the material is beforehand heated, the exponential formulation of the temperature variation is used to determine parameter 𝜏: 𝜃 = 𝜃 0 e -(t-t 0 ) 𝜏

. Further details are provided in Samaca Martinez et al. [29]

Determination of the crystallinity

Figure1 illustrates the methodology proposed. It is composed of four steps. Input data are the temperature variation 𝜃, parameter 𝜏, and the thermophysical parameters 𝜌, C,andΔH.

• The first step consists in calculating the heat power density by using Equation 4. The diagram illustrates the typical calorific response of unfilled NR, which exhibits a strong increase in the heat power density when crystallization starts. [START_REF] Göritz | [END_REF]29] Here, 𝜆 c and 𝜆 m stand for the stretches at which SIC starts and melting is complete, respectively. It should be noted that all the work done by the system may not systematically increase the specimen temperature. [16][17][18][19][20][21][22] Therefore,

FIGURE 1 Methodology summary

the strain power density and the heat power density are not systematically superimposed, whatever the material is crystallizing or not; 3

•

Step 2 deals with the prediction of the heat power density due to elastic couplings only. The thermal energy due to SIC is indeed the area located between the heat power density (curve A) and the part of the heat power density due to elastic couplings (curve B). The latter is not known once crystallization starts but is known before. In this study, we assume that the elastic couplings evolve in a similar way after crystallization starts. The corresponding heat power density is predicted by using a polynomial form such as C 1 (𝜆 -𝜆 -2 )+C 2 (𝜆 -𝜆 -2 ) 2 + C 3 (𝜆 -𝜆 -2 ) 3 , which parameters are identified from the experimental heat power density before SIC starts. It should be noted that the heat power density can be different from zero at the lowest stretches (typically inferior to 1.1), due to energetic effects, but this does not affect the prediction of the heat power density due to elastic couplings at higher stretches; • The heat power density due to SIC (A-B) is then computed in Step 3 for determining the temperature variation due to SIC T cryst . In practice, T cryst is calculated as the primitive of the heat power density due to SIC. The integration constant is determined considering that T cryst is equal to zero before crystallization starts (see the numerical scheme provided in Figure 1, Step #3). It should be noted that the numerical scheme presented is not centered, but another numerical scheme can be used; • The fourth step consists in calculating the crystallinity by applying Equation 1.

This method is very simple and does not require measuring the nominal stress variations nor characterizing possible nonentropic effects.

APPLICATION TO AN UNFILLED NATURAL RUBBER

The methodology proposed for measuring crystallinity is now applied to experimental data issued from the literature. Here, data given in other literature [29,30] are used. They were obtained with an unfilled natural rubber submitted to cyclic uniaxial tensile loading. The mechanical loading applied corresponded to four sets of three cycles at a maximum stretch of 2, 5, 6, and 7.5 at ±100 mm/min. The stretch is defined as the ratio of the current length over the initial length. 𝜌 and C are equal to 936 kg/dm 3 and 1768 J/(kg.K). 𝜏 depends on the stretch and is defined as 𝜏(𝜆)=40.48 -3.25𝜆.

Mechanical and calorimetric responses

Figure 2 presents the mechanical response obtained in terms of the nominal stress in relation to the stretch. As expected, no stress softening is observed, and all the curves are superimposed for cycles at the same maximum stretch.

A significant hysteresis loop is observed for cycles at 𝜆>5, that is, at stretch superior to the stretch at which crystallization starts (about 4). These results are in good agreement with those reported in the literature for unfilled natural rubber. The size of the hysteresis loop increases when the maximum stretch is increased. This is consistent with an increase in the crystallinity with the maximum applied stretch. As assumed by Flory [31] and widely highlighted experimentally, once crystallization occurs, relaxation is induced in the amorphous phase, and a plateau is observed in the stress-strain curve. [7,8] Figure 3 depicts the heat power density in relation to the stretch. During loading, the heat power density is positive and increases with the stretch. During unloading, the heat power density is negative. This highlights preponderant entropic effects in the thermal response. Before SIC starts, the load-unload curves are symmetrical, meaning that the heat produced during the loading phase is equal to the heat absorbed during the unloading phase. Once SIC starts, the heat power density evolution for loading and unloading are no longer symmetrical. During loading, the heat power density evolves in a quasilinear manner until a stretch close to 4 is reached. A dissymmetry is observed for stretches higher than 4; the stretch level at which SIC starts. During unloading, the heat power density rate first increases in absolute value until reaching a stretch equal to 4, is constant until a stretch equal to 3.5 is reached, and then decreases. Hence, crystallization continues during unloading. It should be noted that no intrinsic dissipation was detected in Samaca Martinez et al. [29] The hysteresis loop is therefore due to SIC without any viscous effects (in the range of the low stretch rates applied).

Determining crystallinity

As explained in Section 2.2, the thermal energy due to SIC is the area located between the heat power density measured (curve A in Figure 1) and the predicted part of the heat power density curve due to elastic couplings (curve B). Figure 3 gives the heat power density measured (Step #1) and the predicted heat power density due to elastic couplings (Step #2). The difference between the two curves, that is, the heat power density due to SIC, is processed to determine the corresponding temperature variation T cryst (see Figure 4(a)). In practice, a primitive calculation of the heat power due to SIC is carried out. The integration constant is determined considering that T cryst before crystallization starts is equal to zero. Only cycles at a maximum stretch applied equal to 6 and 7.5 for which a significant hysteresis loop is observed are considered. The results being the same for cycles conducted at the same maximum stretch, only cycles #9and#12 are presented.

The crystallinity is obtained by dividing T cryst by the enthalpy of crystallite fusion ΔH cryst and multiplying by 𝜌C.The crystallization energy of natural rubber is approximated by the enthalpy of fusion, which is assumed to be independent of temperature and stretch. [31] Here, as a first approach, we use the value determined by, [32] that is, 15,3 cal. per gram of poly- mer, which corresponds to 59,9 J.cm -3 . The crystallinity versus stretch is given in Figure 4(b). For cycle #9at𝜆 = 6, SIC starts approximately at 𝜆 = 4.5, slightly increases before increasing linearly with the stretch. The maximum crystallinity reached is about 9%, which is consistent with XRD measurements performed at lower stretch rates. [8] During unloading, crystallinity is higher than during loading at any stretch, meaning that crystallization continues during unloading. Crystallite melting seems to be complete at 𝜆 = 3. Before 𝜆 = 3 is reached, the slope curve decreases. For cycle #12 at 𝜆 = 7.5, SIC starts at a higher stretch (𝜆 = 5). This slight shift in the stretch at which SIC starts is explained by a slight accommodation effect due to previous mechanical cycles. Taking into account the residual strain in the calculation of the initial length will decrease the level of the characteristic stretches 𝜆 c and 𝜆 m . Crystallinity increases linearly with the stretch until reaching about 9%, then its rate decreases. This is consistent with the decrease in the heat power density rate observed in Figure 3 from 𝜆 = 6 on. The maximum crystallinity reached is about 20%.Duringtheunloading,thesameevolution as for the lower stretch is observed except for the end of the unloading. Indeed, from 𝜆 = 3 on, the crystallinity does not return to zero. The magnification in the boxed zone shows that the curve slope decreases and the crystallinity decreases linearly with the stretch until reaching a value close to zero. This questions a possible melting kinetics and echoes previous work by Müller [START_REF] Müller | Calorimetric investigations of deformation processes[END_REF] and Samaca Martinez et al. [29] This is currently studied by the author, and no further discussion is provided in this paper.

CONCLUSION

This study proposes a new measurement technique to determine the crystallinity in stretched crystallizing rubber using IR thermography. For this purpose, a methodology based on quantitative calorimetry is proposed. This methodology enables us to define the temperature variation due to SIC from the corresponding heat power density. The characterization of the stress-strain relationship and the nonentropic contributions to rubber elasticity is not required. It is therefore well adapted to characterize heterogeneous crystallinity field. Crystallinity obtained as a function of the stretch provides comparable results as those obtained by XRD measurements. The method proposed is therefore a realistic alternative to measure crystallinity of stretched crystallizing rubbers from any temperature measurement technique and quantitative calorimetry. It is a user-friendly measurement technique, accurate, and well adapted to most of the mechanical tests performed with conventional testing machine. Measuring temperature from infrared thermography should open numerous perspectives in terms of high speed and full crystallinity field measurements.

FIGURE 2 FIGURE 3

 23 FIGURE 2 Mechanical response obtained during cyclic loading of unfilled NR in terms of the stress-stretch relationship

FIGURE 4

 4 FIGURE 4 Temperature change due to (a) crystallization melting (see Step #3) and (b) crystallinity (see Step #4) in relation to the stretch

ACKNOWLEDGEMENTS

The author thanks Prof. Gert Heinrich and Dr. Eric Robin for the fruitful discussions and the "Manufacture Française des Pneumatiques Michelin"for providing the samples. The experimental data used in this paper are provided in Samaca Martinez et al. [29] and Le Cam. [30]

ENDNOTES 1 It should be noted that Boonstra was the first who suggested that crystallinity could be determined from the induced change in internal energy and the heat of fusion per repeating unit. [13] Using calorimetry is in the same line, because change in the internal energy can be taken into account with a "stretch calorimeter." 2 𝜎d𝜖 under uniaxial loading, with 𝜎the Cauchy stress and 𝜖 the engineering strain. 3 This might explain why negative crystallinity was found in Spratte et al. [14] when the material was not crystallizing;