Anderson S Nunes

Laurent Daniel

Maya Hage-Hassan

Mathieu Domenjoud

Modeling of the magnetic behavior of permanent magnets including ageing effects

Keywords: Functional fatigue, magnetic ageing, hysteresis, electrical machines, magnetization modeling, multi-scale model

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Introduction

Permanent magnets play an essential role in a wide range of aerospace, medical, energy, electronics, and industrial applications [START_REF] Gvr | electronics, industrial) by region, and segment forecasts[END_REF]. In some of these applications, such as the aerospace, permanent magnets are designed to work at extreme temperatures, from -60 up to 500 o C [START_REF] Feng | Investigation on microstructure and magnetic properties of sm2co17 magnets aged at high temperature[END_REF][START_REF] Liu | Long-term aging of Sm2(Co,Fe,Cu,Zr)17 permanent magnets at 300 and 400C[END_REF][START_REF] Hendershot | Design of Brushless Permanent-Magnet Motors[END_REF].

Typically, the equipment is required to have a lifetime in the range of tens of years and it is well known that permanent magnets lose their magnetization with time, depending on the loading level of magnetic field [START_REF] Street | Magnetic viscosity in NdFeB and SmCo5 alloys[END_REF], temperature [START_REF] Liu | Long-term aging of Sm2(Co,Fe,Cu,Zr)17 permanent magnets at 300 and 400C[END_REF][START_REF] Lekdim | Effect of the remaining magnetization on the thermal ageing of high permeability nanocrystalline FeCuNbSiB alloys[END_REF], and mechanical stress [START_REF] Mito | High pressure effects on isotropic nd2fe14b magnet accompanying change in coercive field[END_REF]. This phenomenon is known as magnetic ageing.

The most common way to estimate the ageing behavior of permanent magnets is by submitting the samples to high-temperatures during hundreds of hours and measuring the decay of the properties. In [START_REF] Lopes | Caracterizac ¸ão da estabilidade térmica das propriedades magnéticas de ímãs compósitos à base de Nd2Fe14B[END_REF] the ageing effect in NdFeB magnets processed through two methods, namely HDDR (Hydrogenation -Disproportionation -Desorption -Recombination) and melt-spinning was investigated. Distinct manufacturing processes were also studied in [START_REF] Ciuta | Une étude expérimentale de la coercivité des aimants NdFeB[END_REF] which compared the magnetic behavior of thick hard magnetic films with bulk materials submitted to an infiltration process. The reversible and irreversible magnetic susceptibilities were obtained considering five magnetic operating points, allowing to obtain the ageing factor at those specific points. Furthermore, different temperatures, compacting pressures and atmospheres were considered under the open and closecircuit conditions. A closed-circuit configuration was also adopted in [START_REF] Haavisto | Studies on the time-dependent demagnetization of sintered NdFeB permanent magnets[END_REF][START_REF] Haavisto | Time-dependent demagnetization in sintered NdFeB magnets[END_REF][START_REF] Haavisto | Magnetic behavior of sintered NdFeB magnets on a long-term timescale[END_REF] to evaluate the magnetic properties behavior of magnets under a range of applied magnetic fields. Besides that, different permeance coefficients obtained by varying the magnets dimensions were considered.

However, there are some complexities inherent to real applications that make it difficult and expensive to perform tests considering all or even representative sets of variables. For instance, the working or duty cycles of electrical machines that affect the thermal and magnetic field cycles, as shown in [START_REF] Huger | The effects of thermal cycling on aging of neodymium-iron-boron magnets[END_REF]. Also, it is well known that the temperature and magnetic field, as depicted in Fig. 1, are not uniform along the volume of permanent magnets [START_REF] Ruoho | Partial demagnetization of permanent magnets in electrical machines caused by an inclined field[END_REF][START_REF] Gutfrind | Electromechanical actuators optimization used in thermal engine air flow regulation[END_REF], suggesting that the ageing factor is a local, non-uniform, quantity.

There are other factors, such as the environment where the magnets are used, since possible gases can have an impact on the demagnetization process [START_REF] Lopes | Caracterizac ¸ão da estabilidade térmica das propriedades magnéticas de ímãs compósitos à base de Nd2Fe14B[END_REF] or crystallographic textures of the materials [START_REF] Hauegen | EBSD texture analysis of NdFeB magnets[END_REF][START_REF] Song | Crystalline and magnetic microstructures of iron-rich Sm[END_REF].

Considering the difficulties mentioned above, we propose to analyze this problem by modeling the magnetic properties, reducing the need for measurements.

The first part of the problem is to model the magnetic properties of permanent magnets. One can divide the available models into two groups: descriptive and predictive. The first one encompasses the well- known phenomenological mathematical models applied to describe the hysteresis [START_REF] Mayergoyz | Mathematical Models of Hysteresis and Their Applications[END_REF]. However, they are not based on the physical process [START_REF] Serpico | Magnetic hysteresis modeling via feedforward neural networks[END_REF] and have no interpretation in terms of energy [START_REF] Franc ¸ois-Lavet | An energy-based variational model of ferromagnetic hysteresis for finite element computations[END_REF]. Such models have been extensively applied to hard magnetic materials [START_REF] Rosu | Simulation of the magnetic hysteresis in nfeb permanent magnet using preisach's model[END_REF][START_REF] Torre | Preisach-arrhenius model for thermal aftereffect[END_REF][START_REF] Cornejo | Preisach analysis of reversible magnetization in SmCo/fe films[END_REF]. In the second group, the predictive models, one can include the multi-scale approaches such as the Stoner-Wohlfarth proposed in [START_REF] Stoner | A mechanism of magnetic hysteresis in heterogeneous alloys[END_REF] and largely applied [START_REF] Tannous | A stoner-wohlfarth model redux: Dynamic and statistical properties[END_REF][START_REF] Street | A study of magnetic viscosity, Proceedings of the Physical Society[END_REF][START_REF] Tannous | The stoner-wohlfarth model of ferromagnetism: Static properties[END_REF][START_REF] De Campos | Estimate of the anisotropy field in isotropic SmCo 2:17 magnets with the stoner-wohlfarth CLC model[END_REF][START_REF] Xu | Stoner wohlfarth model for magneto anisotropy[END_REF][START_REF] Friedman | Stoner-wohlfarth hysteresis model with stochastic input as a model of viscosity in magnetic materials[END_REF][START_REF] De Campos | Stonerwohlfarth model for nanocrystalline anisotropic SM 2 CO 17[END_REF][START_REF] Tannous | The stoner-wohlfarth model of ferromagnetism[END_REF]. However, this method assumes a single or a set of uni-axial magnetically decoupled particles. Besides that, this model assumes a coherent rotation process, which is not very common in real applications, mainly in some hard magnetic materials, Nd 2 Fe 14 B for instance, [START_REF] Skomski | Permanent Magnetism (Studies in Condensed Matter Physics)[END_REF].

PMs

Thus, we propose to use a multi-scale approach based on an energetic procedure at the single crystal scale followed by a homogenization procedure to predict the macroscopic behavior of magnetic materials based on their local properties [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF][START_REF] Daniel | A multiscale model for magneto-elastic behaviour including hysteresis effects[END_REF]. Such approaches have been extensively applied to Iron-Silicon steel [START_REF] Buiron | A multiscale model of magnetostriction strain and stress effect[END_REF][START_REF] Daniel | Homogenisation of magneto-elastic behaviour: from the grain to the macro scale[END_REF][START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF][START_REF] Van Den Berg | A mesoscopic hysteresis model based on the unconstrained minimization of the gibbs free energy[END_REF][START_REF] Daniel | A multiscale model for magneto-elastic behaviour including hysteresis effects[END_REF][START_REF] Vanoost | Magnetic hysteresis at the domain scale of a multi-scale material model for magneto-elastic behaviour[END_REF][START_REF] Ito | Energy-based magnetization and magnetostriction modeling of grain-oriented silicon steel under vectorial excitations[END_REF][START_REF] Martin | Analysis of the magnetomechanical anisotropy of steel sheets in electrical applications[END_REF] or giant magnetostrictive materials [START_REF] Armstrong | A directional magnetization potential based model of magnetoelastic hysteresis[END_REF][START_REF] Daniel | A constitutive law for magnetostrictive materials and its application to terfenol-d single and polycrystals[END_REF], but seldom to hard magnetic materials so far.

The second part of the problem is to predict the ageing, namely, the way these local properties evolve with time, depending on the experienced thermal, magnetic or mechanical loadings. To solve this part of the problem, we use the classical magnetic viscosity theory. These ageing studies based on the activation energy concept seem to have started with [START_REF] Street | A study of magnetic viscosity, Proceedings of the Physical Society[END_REF][START_REF] Street | Magnetic viscosity under discontinuously and continuously variable field conditions[END_REF]. Then the domain wall pinning by inhomogeneities and how temperature and time, i.e. the thermal activation process, affect this phenomenon in SmCo 5 alloys was presented in [START_REF] Gaunt | Domain wall pinning as a source of magnetic hardening in alloys[END_REF][START_REF] Gaunt | Magnetic viscosity in ferromagnets[END_REF][START_REF] Gaunt | Magnetic viscosity and thermal activation energy[END_REF]. A more recent application of this theory is presented in [START_REF] Lanci | Introduction of thermal activation in forward modeling of hysteresis loops for single-domain magnetic particles and implications for the interpretation of the day diagram[END_REF].

The ageing has also been discussed for permanent magnets for some years, as shown in [START_REF] Liu | Long-term aging of Sm2(Co,Fe,Cu,Zr)17 permanent magnets at 300 and 400C[END_REF][START_REF] Street | Magnetic viscosity in NdFeB and SmCo5 alloys[END_REF][START_REF] Grössinger | Determination of the magnetic viscosity in SmCo5-xCux alloys by pulsed-field method[END_REF] for SmCo and in [START_REF] Street | Magnetic viscosity in NdFeB and SmCo5 alloys[END_REF][START_REF] Martinez | Magnetic viscosity in NdFeB magnets[END_REF][START_REF] Liu | Magnetic viscosity studies of Nd16Fe77B7 permanent magnets[END_REF] for NdFeB magnets. Furthermore, some recent experimental works [START_REF] Lopes | Caracterizac ¸ão da estabilidade térmica das propriedades magnéticas de ímãs compósitos à base de Nd2Fe14B[END_REF][START_REF] Haavisto | Studies on the time-dependent demagnetization of sintered NdFeB permanent magnets[END_REF][START_REF] Haavisto | Time-dependent demagnetization in sintered NdFeB magnets[END_REF][START_REF] Haavisto | Magnetic behavior of sintered NdFeB magnets on a long-term timescale[END_REF][START_REF] Ciuta | Une étude expérimentale de la coercivité des aimants NdFeB[END_REF][START_REF] Woodcock | Understanding the microstructure and coercivity of high performance NdFeB-based magnets[END_REF] have been developed. Also the magnetic viscosity effect on the coercive field has been investigated in [START_REF] Bance | Thermal activation in permanent magnets[END_REF][START_REF] Bance | Thermally activated coercivity in core-shell permanent magnets[END_REF] in order to increase the performance of the permanent magnets.

By combining the multi-scale approach with the ageing theory, it is possible to predict the magnetic behavior of permanent magnets as well as estimating the ageing factor. The proposed method can be embedded in numerical methods such as the Finite Element Method in order to estimate the ageing factor of permanent magnets inserted in real applications. A similar approach has been presented in [START_REF] Kurose | 3-d FEM analysis of thermal degradation in writing and reading characteristics of a perpendicular magnetic head[END_REF], but it is based on the Stoner-Wohlfarth model, which is well adapted to independent magnetic particles and offers less flexibility for describing the magnetic behavior of inhomogeneous structures of real magnets [START_REF] Skomski | Permanent Magnetism (Studies in Condensed Matter Physics)[END_REF].

To validate the proposed approach, the modeling results for ferrite and NdFeB are compared with measurements. Then, the ageing factor of a real permanent magnet at a given operating point is estimated.

Multi-scale approach

The multi-scale approach proposed in [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF][START_REF] Daniel | A multiscale model for magneto-elastic behaviour including hysteresis effects[END_REF] considers three different scales, namely :

• The magnetic domain scale, denoted by an index α, is the scale where all magnetic properties are uniform.

• The single-crystal scale, denoted by an index g is the scale where elastic properties are uniform.

• The Representative Volume Element (RVE) scale, denoted by an index m, is the macroscopic scale at which the material response is measured.

The multi-scale model is based on expressing the energy balance at the single-crystal level considering each single-crystal as a set of magnetic domains. Then, the RVE is approximated as an aggregate of single-crystals (polycrystal) oriented following an orientation distribution function corresponding to the crystallographic texture of the material.

This approach was developed to predict the magnetoelastic behavior of magnetic materials submitted to multi-axial mechanical loadings [START_REF] Bernard | Effect of stress on magnetic hysteresis losses in a switched reluctance motor: Application to stator and rotor shrink fitting[END_REF], but in this work, no stress is applied and mechanical effects are neglected.

Single-crystal model

At the single-crystal scale, the material is divided into a set of domain families. Each domain family gather all domains with the same magnetization orientation α. The magnetization M α of each family can be expressed as:

M α = M s α = M s [α 1 α 2 α 3 ] (1)
where M s is the saturation magnetization and α 1,2,3 are the cosines of the magnetization vector, Fig. 2. The magnetic equilibrium is given by the competition of the following energy terms forming the local free energy:

y x z M [001] [010] [100] [101] [110] [111] α 1 α 2 α 3
W α = W mag α + W an α + W mec α (2)
The magnetostatic energy (or Zeeman energy), W mag α , tends to align the local magnetization M α with the applied field H. µ 0 is the vacuum magnetic permeability.

W mag α = -µ 0 H • M α (3)
The anisotropy energy W an α describes the existence of preferred orientations for the magnetization (easy axes).

W an α = K 1 α 2 1 α 2 2 + α 2 2 α 2 3 + α 2 3 α 2 1 + K 2 α 2 1 α 2 2 α 2 3 ( 4
)
where K 1,2 are the magnetocrystalline anisotropy constants.

The magneto-elastic energy W mec α describes the effect of an applied mechanical stress on the magnetic equilibrium [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF]. Since no stress will be considered in this study, this term is uniformly taken as zero.

Knowing the free energy W α for all domain families, the volume fraction f α of each domain family -corresponding to the existence probability of the orientation α for the local magnetization -can be defined using a Boltzmann-type equation [START_REF] Buiron | A multiscale model for magneto-elastic couplings[END_REF][START_REF] Buiron | Modélisation multiéchelle du comportement magnéto-élastique couplé des matériaux ferromagnétiques doux[END_REF]:

f α = exp (-A s W α ) α exp (-A s W α ) dα (5)
where A s is defined from the initial anhysteretic susceptibility χ 0 of the material [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF]:

A s = 3χ 0 µ 0 M 2 s ( 6
)
χ 0 can be obtained from the anhysteretic magnetization curve or estimated at the coercive field H c of a major loop (χ 0 = ∆M/∆H| H=H c ) [START_REF] Fiorillo | Measurement and Characterization of Magnetic Materials[END_REF].

Once the volume fractions are obtained for all domain families in a crystal, the average magnetization M g at the single crystal scale can be obtained:

M g = α f α M α dα (7)
From a practical perspective, the solution of the single-crystal model can be achieved in three steps:

Step 1 -Calculate the total energy (Eq. 2) for a set of families α covering the orientations in space Step 2 -Obtain the volume fraction for each domain family (Eq. 5)

Step 3 -Calculate the magnetization M g (Eq.7) of the single-crystal In order to cover the possible orientations for the domain magnetization, it necessary to build a set of orientations distributed as uniformly as possible along the shell of a unit sphere. This distribution should be as homogeneous as possible to avoid possible artefact anisotropy due to inhomogeneity in the directions disposition. For this purpose the nodes of an icosphere can be used [START_REF] Daniel | A constitutive law for magnetostrictive materials and its application to terfenol-d single and polycrystals[END_REF][START_REF] Bernard | Multiscale approaches for magneto-elasticity in device simulation[END_REF]. An example containing 2562 directions is shown in Fig. 3.

Regarding the computation effort, indeed the lower the number of points, the lower the computational time and effort to solve the problem. Additionally, simplified methods with a lower number of orientations, as presented in [START_REF] Bernard | Effect of stress on magnetic hysteresis losses in a switched reluctance motor: Application to stator and rotor shrink fitting[END_REF][START_REF] Bernard | Multiscale approaches for magneto-elasticity in device simulation[END_REF] for instance, have been successfully tested. Another noticeable numerical aspect is the straightforward parallelization of the single-crystal calculations. A method capable of quantifying the quality of a given distribution was proposed in [START_REF] Daniel | A constitutive law for magnetostrictive materials and its application to terfenol-d single and polycrystals[END_REF], where the authors evaluated and applied a mesh composed by 10242 nodes (each node of the mesh refers to one α family direction). Once that distribution (10242 nodes) has been validated, it is used in this paper to obtain all the presented results.

Considering that the energy has to be calculated for all those directions, one can easily visualize the energy distribution. Fig. 4 shows the total energy landscapes for three different values of the applied magnetic field along the < 110 > direction as a function of the anisotropy field H a given in Eq. 13. The considered material is a ferrite, see material parameters in Table 1. One can identify the six easy axes (low energy regions) and it is possible to verify how the energy changes when a magnetic field is applied. In particular it can be seen that the field, if high enough, rotates the magnetization out of the easy axes.

Parameter M s K 1 K 2 Unit [A/m] [J/m 3 ] [J/m 3 ] Value 0.382x10 6 330x10 3 0

Polycrystal

In order to obtain the macroscopic properties of a RVE of a material, it is necessary to describe a volume large enough, i.e. containing enough single crystal orientations, to represent the average properties. The crystallographic texture can be responsible for macroscopic anisotropy effects [START_REF] Hauegen | EBSD texture analysis of NdFeB magnets[END_REF][START_REF] Martinez | Magnetic viscosity and texture in NdFeB magnets[END_REF]. It can be represented in a discrete fashion. In this study, an isotropic texture is used. It is built thanks to a regular zoning of the crystallographic orientations space, as already applied in [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF][START_REF] Buiron | Modélisation multiéchelle du comportement magnéto-élastique couplé des matériaux ferromagnétiques doux[END_REF], in terms of the Euler angles [START_REF] Engler | Introduction to Texture Analysis: Macrotexture, Microtexture, and Orientation Mapping[END_REF] presented in Table 2.

Variable Domain Number of values

Φ 1 [0, 2π] 13 cosΦ [0, 1] 7 Φ 2 [0, 2π] 6
In order to calculate the macroscopic properties of the sample, it is required to express the macroscopic loading (applied magnetic field H) at the local scale [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF][START_REF] Daniel | A constitutive law for magnetostrictive materials and its application to terfenol-d single and polycrystals[END_REF]. In this paper, we assume uniform magnetic field within the material so that the single crystal is subjected directly to the applied magnetic field H. From this local loading, the single crystal magnetization M g of each single crystal (N=13 × 7 × 6 = 546 orientations) is calculated according to the single crystal model.

Finally, the average magnetization of the material is obtained by averaging the local magnetization over all orientations:

M m = 1 N N i=1 M g (8)

Irreversible contribution

The modelling approach is anhysteretic so far, i.e., it takes into account only the reversible part of the magnetic behavior. the complete hysteresis cycle of permanent magnets implies both reversible and irreversible parts to be modelled. In this case the Hauser's method is used [START_REF] Daniel | A multiscale model for magneto-elastic behaviour including hysteresis effects[END_REF][START_REF] Hauser | Energetic model of ferromagnetic hysteresis[END_REF][START_REF] Cornejo | Hysteresis modeling of anisotropic and isotropic nanocrystalline hard magnetic films[END_REF] in order to estimate the irreversible contribution on the magnetic field H irr ,

|H irr | = δ k r µ 0 M s + c r |H| 1 -k g exp - k a k g M -M reb (9)
where δ is equal to ±1 depending on whether the applied magnetic field is increasing or decreasing, c r and k g are material parameters, and

k r = µ 0 M s H c . (10)
Based on [START_REF] Cornejo | Hysteresis modeling of anisotropic and isotropic nanocrystalline hard magnetic films[END_REF], k a is expressed by

k a = β H a H c , (11)
where

β = µ 0 M 2 s 2 χ 0 K 1 (12)
and the anisotropy field H a is given by

H a = 2 K 1 µ 0 M s (13)
Following the magnetization process in time, k g is updated at each return point depending on its previous value, k g ,

k g = 2 -k g exp - k a k g M -M reb ( 14
)
M reb is the magnetization at the previous inversion of the magnetic field, which can be detected by checking ∆H(t) × ∆H(t -∆t) < 0. This model depends on the coercive field H c . However, H c is a temperature-dependent quantity [START_REF] Feng | Investigation on microstructure and magnetic properties of sm2co17 magnets aged at high temperature[END_REF][START_REF] Bance | Thermally activated coercivity in core-shell permanent magnets[END_REF][START_REF] Bance | Thermal activation in permanent magnets[END_REF][START_REF] Bergqvist | Temperature-dependent vector hysteresis model for permanent magnets[END_REF]. At high temperatures the effects of thermal activation on the reversal process are increased, reducing the coercivity [START_REF] Bance | Thermally activated coercivity in core-shell permanent magnets[END_REF]. It is explained by the influence of temperature on the energy-barriers, which depend on K B T , where K B is the Boltzmann constant and T is the temperature. Consequently, the proposed method becomes temperature-dependent as well.

Ageing phenomena

If the magnetic field acting on a magnetic material changes, the magnetization will keep evolving over a certain time longer than the short times associated with the eddy current effects [START_REF] Gaunt | Magnetic viscosity and thermal activation energy[END_REF]. This time dependence of the magnetic properties, such as magnetization, is known as magnetic viscosity or ageing [START_REF] Skomski | Nucleation field and energy product of aligned two-phase magnets-progress towards the 1mj/m3 magnet[END_REF].

This behavior is caused by thermal activation of magnetization reversals over the energy barriers and the waiting time before an energy barrier is overcome is referred to as the relaxation time τ, which is given by [START_REF] Gaunt | Magnetic viscosity and thermal activation energy[END_REF]:

τ -1 = C exp E a K B T ( 15
)
where E a is the activation energy of a given energy barrier and C is a frequency constant of order 10 10 -10 12 Hz.

As can be noticed from Eq. 15, the energy landscape is key to the ageing behavior. This energy landscape highly depends on the microstructure of the material [START_REF] Skomski | Magnetic aging[END_REF]. The energy landscape models are often based on the assumption of single-domain particles, as presented in [START_REF] Lanci | Introduction of thermal activation in forward modeling of hysteresis loops for single-domain magnetic particles and implications for the interpretation of the day diagram[END_REF], which may not describe the pinning effects in permanent magnet materials.

It was observed experimentally [START_REF] Street | A study of magnetic viscosity, Proceedings of the Physical Society[END_REF] that the magnetization changes linearly with the natural logarithm of time. This decay can be described by [START_REF] Grössinger | Determination of the magnetic viscosity in SmCo5-xCux alloys by pulsed-field method[END_REF][START_REF] Skomski | Simple Models of Magnetism[END_REF]

M(t) = M(t 0 ) -S ln t t 0 ( 16
)
where

S = 2 M s K B T f (E a ) (17)
f (E a ) is an energy-barrier distribution which depends on the activation energy E a .

In order to separate the field-dependent irreversible susceptibility χ Irr from the magnetic viscosity problem, it is convenient to express S as [START_REF] Skomski | Permanent Magnetism (Studies in Condensed Matter Physics)[END_REF]:

S = K B T χ Irr v a µ 0 M s ( 18
)
where v a is the activation volume. Thus, the magnetic viscosity coefficient S v is defined as

S v = K B T v a µ 0 M s ( 19
)
S v is shape independent, unlike χ Irr and S [START_REF] Martinez | Magnetic viscosity in NdFeB magnets[END_REF]. S can finally be expressed as:

S = S v χ Irr ( 20
)
The magnetic viscosity is attributed to irreversible changes of the magnetization across the energy barriers [START_REF] Bance | Thermal activation in permanent magnets[END_REF][START_REF] Wohlfarth | The coefficient of magnetic viscosity[END_REF], thus it affects only the irreversible variation of the magnetization that can be described by the irreversible susceptibility χ Irr (χ Irr = χ T ot -χ Rev ). This parameter can be obtained experimentally following the procedures presented in [START_REF] Street | Magnetic viscosity in NdFeB and SmCo5 alloys[END_REF][START_REF] Ciuta | Une étude expérimentale de la coercivité des aimants NdFeB[END_REF].

Considering the above-presented formulae and that the activation energy, Eq. 17, depends on the material properties discussed in Section 2, one can find out that the magnetic ageing depends on the material properties, applied magnetic field vector, time, and temperature. This ageing methodology is applied based on the following steps:

Step 1 -Calculate the irreversible susceptibility (χ Irr = χ T ot -χ Rev ) for a given number of operating points. This can be done from small loops included in a major loop. The major loop provides χ T ot at a given excitation level while the slope of the minor loop provides χ Rev Step 2 -Obtain the S parameter for the same operating points. This is done by measuring the change in magnetization over time at a given fixed excitation. S is then obtained using Eq. 16

Step 3 -Express S as a function of the applied magnetic field. It can be done in two different ways:

(a) Interpolate S and apply it directly into Eq. 16 for any time and magnetization.

(b) Calculate S v (Eq. 20) and then v a (Eq. [START_REF] Serpico | Magnetic hysteresis modeling via feedforward neural networks[END_REF]. Interpolate v a . The S parameter is then obtained by Eq. 18.

The procedures for steps one and two are detailed in [START_REF] Ciuta | Une étude expérimentale de la coercivité des aimants NdFeB[END_REF]. It was assumed that the applied magnetic field and temperature are constant over time.

Experimental setup

A specific set up was designed and built to investigate the magnetic properties of permanent magnets.

The standard [START_REF] Standard | Magnetic materials -Permanent magnet (magnetically hard) materials -Methods of measurement of magnetic properties[END_REF] suggests a magnetic circuit with one air gap, but in this work, the magnetic circuit is composed of two U-shaped Fe-Si yokes to create two air gaps, and four excitation coils, as depicted in Figures 5 and 6. Therefore two magnets are characterized at the same time, one at each air gap of the device, which helps to ensure the robustness of the measurements. The depth of the magnetic yokes is 52 mm. The dimensions of the transverse section of the magnetic circuit are chosen to impose a magnetic field as uniform as possible in the air gaps.

The magnetic induction is measured by a coil wounded around each magnet and the magnetic field in the air gaps is measured by two Brockhaus BGM 201 gaussmeters with two transverse probes BTP 201-F75.

The control of the applied current and the acquisition of the signals are performed by a dSPACE processor board DS 1006 connected to a computer. The A/D and D/A boards are, respectively, the DS2004 and the DS2103.

More information about the control system is available in [START_REF] Domenjoud | Characterization of giant magnetostrictive materials under static stress: influence of loading boundary conditions[END_REF], also providing information about demagnetization process, drift in B, current control and acquisition system. The experimental setup is used to measure the hysteresis loops of the permanent magnets and, based on these loops, the saturation magnetization M s , the coercive field H c , and the initial susceptibility χ 0 .

M s is the maximum measured magnetization, H c is obtained at the intersection of the curve with the x axis, i.e., M = 0, and χ 0 is estimated by ∆M ∆H | H=H c [START_REF] Fiorillo | Measurement and Characterization of Magnetic Materials[END_REF]. The applied signal is sinusoidal at 2 Hz frequency (controlled in current).

Results

First, the results obtained with the multi-scale model are presented for two different materials, namely ferrite and NdFeB.

Then, the measurement data given in [START_REF] Ciuta | Une étude expérimentale de la coercivité des aimants NdFeB[END_REF] are used to obtain the ageing parameters mentioned in Section 3 and the results obtained with this procedure are presented.

Multi-scale approach

Ferrite

In this case, the material parameters were obtained using the in-house measurement setup presented in Section 4, except for K 1 which was found in [START_REF] Skomski | Permanent Magnetism (Studies in Condensed Matter Physics)[END_REF], and are presented in Table 3. The anhysteretic curve modeled with the multi-scale approach (Sections 2.1 and 2.2), the hysteresis cycle obtained by adding the reversible field portion (Section 2.3) and the measurement data are shown in Fig. 7.

A good match between the results is obtained, except along the "knees" in the upper-right and lower-left curves. Nonetheless, typically in the case of permanent magnets, the second-quadrant "knee" is used and along this region, the model presents a good correlation with measurements.

NdFeB

The same procedure is followed for the NdFeB material, whose properties are given in [START_REF] Ciuta | Une étude expérimentale de la coercivité des aimants NdFeB[END_REF] and presented in Table 4. The results are shown in Fig. 8.

Parameter

M s K 1 K 2 H c χ 0 Unit [A/m] [J/m 3 ] [J/m 3 ] [A/m] - Value
1.3x10 6 3.75x10 6 0.4x10 6 5.28x10 5 5.5

Ageing

Considering the difficulties of achieving a magnetic field intensity capable of saturating the NdFeB permanent magnets, the results of Nd 2 Fe 14 B TMC bulk at 400K given in [START_REF] Ciuta | Une étude expérimentale de la coercivité des aimants NdFeB[END_REF] are used. The procedure adopted is given in the three steps, with sub-item (b) presented in Section 3.

The S factor and the χ 0 measured are presented in Figures 9 and10 From Eq. 18 it is possible to obtain the activation volume v a , which is an important parameter since smaller the volume, more susceptible to thermal degradation the material is [START_REF] Kurose | 3-d FEM analysis of thermal degradation in writing and reading characteristics of a perpendicular magnetic head[END_REF]. For this specific case and temperature (400 K), from Fig. 11 one can see that v a does not vary drastically in terms of the applied field, suggesting that it would be reasonable to assume it as a constant.

With the second quadrant of the hysteresis loop depicted in Fig. 8 and by interpolating these ageing parameters, it is possible to estimate the decay of magnetization in time for any value of the magnetic field for the complete working range of the permanent magnet.

An example is depicted in Fig. 12, where the second quadrant of the hysteresis loops are compared and one random operating point is chosen. In this case, it is possible to estimate that for this given Nd 2 Fe 14 B at 400K and after 2000 hours, the magnetization will be reduced to about 85% of its initial value.

CONCLUSION

A multi-scale approach was used to describe the macroscopic behavior of permanent magnets. It was shown that this approach, using only four to five material parameters, was able to describe with satisfactory accuracy the major hysteresis loops of ferrite and Nd-FeB magnets.

Then, an elementary but well-established ageing model was used. The model is based on the measurement of two parameters, namely the ageing factor and the irreversible susceptibility, for a set of magnetic excitations.

Combining the two approaches proved to be an efficient way to estimate the ageing of permanent magnets working at any magnetic operating point. It was shown for instance that a NdFeB magnet placed at 400 K during 2000 h experiences a reduction of magnetization of about 15 %.

It was also found that the ageing factor appears to be proportional to the irreversible susceptibility, which means that the S factor reaches its maximum at the coercive field.

The proposed approach is a first attempt to describe in a predictive way ageing effects in permanent magnets. It is restricted so far to constant magnetic field and temperature loadings. The effect of stress was not considered in this study, but it is naturally incorporated in the multi-scale approach. Experimental validation, however, would be required. The study also considered isotropic magnets, but the multi-scale approach provides a straightforward way to take into account crystallographic textures through the description of grain orientations. The effect of the operating environment and notably aggressive gases can also be significant and was not studied here.

The proposed material model is a first brick to the development of ageing models. Indeed, practical devices are subjected to external loadings significantly varying in time. Scaling rules are required to predict the evolution of magnetization under such variable loadings from the knowledge of the response under constant loadings. This latter point is still a very open field of research.

Figure 1 :

 1 Figure 1: Magnetic field distribution inside the permanent magnets of a synchronous machine in the no-load condition.

Figure 2 :

 2 Figure 2: Representation of the magnetization vector in the single crystal frame considering a cubic anisotropy.

Figure 3 :

 3 Figure 3: Unit sphere showing the distribution of the families α (2562 points).

Figure 4 :

 4 Figure 4: Total energy distribution for three different values of the applied magnetic field. H a is presented in Eq. 13.

Figure 5 :

 5 Figure 5: Experimental setup schematic.

Figure 6 :

 6 Figure 6: Picture of the experimental setup.

Figure 7 :

 7 Figure 7: Comparison between the multi-scale model and the measurement for the ferrite.

Figure 8 :

 8 Figure 8: Comparison between the multi-scale model and the measurement for the NdFeB.

Figure 9 :Figure 10 :

 910 Figure 9: S factor interpolated from [9].

Figure 11 :Figure 12 :

 1112 Figure 11: Activation volume.

Table 1 :

 1 Properties of ferrite[START_REF] Skomski | Permanent Magnetism (Studies in Condensed Matter Physics)[END_REF].

Table 2 :

 2 Euler angles distribution for an isotropic crystallographic texture.

Table 3 :

 3 Properties of ferrite.

	Parameter	M s	K 1	H c	χ 0
	Unit	[A/m]	[J/m 3 ]	[A/m]	-
	Value	0.416x10 6 330x10 3 257x10 3 50

Table 4 :

 4 Properties of NdFeB.

Acknowledgment

This work was supported by the Automotive Mechatronics Chair, a cooperation between Faurecia, Cen-traleSupélec and Esigelec.


PROF. DR Judith Rainhorn

THE COLOUR OF CONTROVERSY. SCIENTIFIC KNOWLEDGE, THE MARKET AND INDUSTRIAL RISK REGULATION IN THE NINETEENTH CENTURY

Should white paint use lead-based or zinc-based pigments? This seemingly arcane question actually rattled Europe's scientific, industrial, public health and policymaking circles for a century and a half -from the French Revolution until the 1920s. The economic competition between white lead and zinc white (the two most common white pigments used in building paint) cannot be separated from the scientific controversy that they sparked due to their harmful effects: white lead (also known as ceruse) was accused of being an insidious poison that harmed the workers who manufactured and used it, whereas zinc white, a latecomer to the market, was promoted as an innocuous substitute. The story of this conflict, which intermingled scientific and public health controversy with economic rivalry, is particularly suited for examining the very timely issue of industrial risks in our contemporary societies. This is a history at the crossroads of the economy, society, science and politics.

The long-run history of this controversy between two white pigments is an excellent vantage point for observing, over the long 19 th century, the mechanisms whereby white leadan indisputably toxic substance -dominated its non-toxic rival on the market. This story stretches across the millennia, alternating highs and lows, with brief periods of fame and longer periods in the shadows -before disappearing from the scene mid-20 th century. White lead was known to the Persians and in Ancient Greece. After a lengthy "childhood" during which actors and prostitutes used it to paint their faces, later becoming a celebrated element of pharmacopeia for whitening birthmarks and soothing the painful nipples of nursing women, white lead finally came of age in the early 19 th century. At that time, lead oxide was mainly produced as a pigment for building paint -as the building sector grew along with the widescale urbanization of Europe -and for many other industrial sectors (porcelain, wallpapers, glossy paper, artificial flowers, etc.). The method of producing it did not change considerably over time; from the recipe described by Pliny the Elder (1 st century BCE) to the process used by Northern European industrialists in the early 19 th century, what changed was mostly the scale of production. The industrial manufacturing process -known as the "Dutch process" developed in workshops of

Amsterdam and Rotterdam in the late 18 th century -mainly consisted of a slight improvement to the ancient technique, streamlining the space needed for production and increasing the size of workshops: rolled sheets of lead were oxidized by contact with acetic acid (vinegar), heated steadily by horse manure. The process created a very fine white powder, which, mixed with oil, became the white paint that would soon be commonly used for the inner and outer walls of buildings and homes across Europe, as well as carriages and ships. 1 White lead production became significant in Holland in the 1770-80s, then in England and Belgium in the 1800s, in France in the early 1820s and in the U.S. just later. [START_REF] Warren | Brush with Death. A Social History of Lead Poisoning[END_REF] Considering this long-term economic cycle, two contradictory biographies of the substance can be written, a light narrative and a dark one: manufactured on an industrial scale, white lead was attributed all the virtues of a paint pigment that enjoyed booming commercial success thanks to its intrinsic properties (whiteness, opacity, resistance, ease of drying and low cost); but it was also, at the same time, criticized as a poison responsible for the sickness and death of thousands of people. In work spaces, white lead is responsible for lead poisoning, a slow intoxication that affects the nervous system, brain and kidneys, causing serious and recurring attacks that can lead to death. In the archives of chemical industrialists, world's fair brochures, and paint manufacturers' accounts, full of orders, we can trace its bright face. Meanwhile, the dark legend loomed in hospital admissions registers, the records of mutual insurance societies that compensated sick workers, physicians' accounts denouncing the ravages of lead poisoning in the working classes, or the workers themselves describing the "slaughterhouses" where they manufactured the deadly substance. This paper is thus a portrait of white lead as a "serial killer" that remained on the run despite its acknowledged deadly effects. I have addressed elsewhere the comparison between France and the U.S. for the denunciation of the toxicity of white lead and the politicisation of its use. [START_REF] Rainhorn | Banning white lead: French and American Experiences in a comparative perspective (early 20 th century)[END_REF] There is no doubt that France had a special role in this global picture as this material attracted such scientific, political and public attention early on that France quickly became the largest consumer of white lead in Europe. Therefore, the French case is both exceptional and emblematic, as most other countries that used the substance did not make it a political product until its disappearance from the market in the 1950-60s.

Not until the early 20 th century did white lead begin to cede its place to zinc white, with the French parliament banning it in 1909 -a ban that was never fully enforced -and the newlyfounded International Labour Organization (ILO) passing an international convention on it in 1921, encouraging ILO member states to abandon it in favour of innocuous pigments. Nevertheless, in this defeat of white lead, it is quite difficult to separate out the influence of legal bans, economic competition, changing uses or technical progress: each of these arguments played a part in this narrative, and I will endeavour to pick them apart. I argue that the government's actions, over the 19 th century, to respond to and regulate the health hazards of white lead were driven less by medical knowledge (lead toxicity was already known in the late 18 th century) than by technical progress, market forces, and the balance of economic and social powers. In addition, this scientific controversy was inherently linked to mechanisms for manufacturing ignorance, or at the very least, for instilling doubt and scientific confusion. During this period, economic and social stakeholders used medical and chemical expertise and evidence as persuasive arguments. While these mechanisms were unable to silence other stakeholders completely, they nevertheless kept the debate over white lead out of the public eye, "compartmentalizing" it to restricted arenas of experts and policymakers.

For more than a century, the manufacture of doubt was working to create uncertainty about the hazards of white lead or the innocuousness of zinc white, even though both these facts had been scientifically proven a number of times since the late 18 th century. In a striking example of how history repeats itself, "merchants of doubt" [START_REF] Oreskes | Merchants of Doubt: How a Handful of Scientists Obscured the Truth on Issues from Tobacco Smoke to Global Warming[END_REF] were at work in the public health debate, in a process that looks similar to the mechanisms that would later obliterate knowledge or even manufacture ignorance about the toxicity of asbestos or tobacco in the 20 th century. Just like white lead in an earlier era, asbestos and tobacco were substances whose toxicity was scientifically proven, yet they were sold on the market for decades, generating significant profits and social acceptance of their risks, both in an occupational setting and in daily life.

The shadows and unspoken parts of this story are quite eloquent if we shine the spotlight of historical investigation on them. Over the very long 19 th century covered here, the history of the economic and scientific conflict between white lead and zinc white adheres to a chronology in which scientific and technical innovations, along with political and social events, reshape the long-term economic cycle of the two substances.

Two fundamental facts must be noted. First, white lead's harmful effects upon the workers who manufactured it or the painters who used it were known in the late 18 th century and widely documented from the 1830s. Second, an alternative substance, harmless for the workers who manufactured or used it -zinc oxide -was invented in the late 18 th century and came into fierce competition with white lead from the mid-19 th century after a new production process made it cheaper and competitive with the latter. Why did these two facts not hamper the massive growth of European and global white lead markets throughout the long 19 th century (1780-1920)? Why did the zinc substitute fail to take over the chemical pigment market and to undermine the commercial domination of white lead on the paint market? Why were health concerns pushed aside by industrialists, by the vast majority of paint manufacturers and users, and by consumers in the general public? More specifically, how did technical innovation interact with risk acceptance and the rules governing the market and competition? The history of white lead forces us to set aside a teleological explanation whereby the social body and the government slowly but inexorably evolved to take into consideration health and safety concerns. Instead, it shows the complex mechanisms at work in this conflict that intermingled scientific and technical knowledge on the one hand, and price and economic interests on the other hand. The history of this conflict, mixing public-health debates and economic rivalry, is a perfect prism for analysing industrial risk and its weight in the economic competition between products with diverging qualities. Nevertheless, it is not easy to disentangle which mechanism -legal bans, economic competition or technical advances -prompted the defeat of this long-lived material. I argue that technical factors and the invention of new and improved pigments played as important a role as the development of restrictive legislation against the toxic substance.

ENTANGLED WHITES: THE MESSY CHRONOLOGY OF A SCIENTIFIC CONTROVERSY

When it entered into competition with zinc white, white lead raised a long controversy that threatened its lifetime as a commercial product. The first phase occurred in late 18 th century. Paduan physician Bernardino Ramazzini had briefly described the use of white lead as a pathogen for studio painters, and French chemist Antoine-François Fourcroy, translating Ramazzini's opus in 1777, blamed the material in a lengthy note on the etiology of lead poisoning. [START_REF] Ramazzini | De Morbis Artificum Diatriba[END_REF] During this period, the famous chemist and encyclopaedist Guyton de Morveau, assisted by pharmacist Courtois, worked on creating a zinc-based white pigment that would be harmless and could replace white lead. Guyton de Morveau advocated substituting zinc for lead for both health and technical reasons, as lead paint tended to darken when exposed to sulphur vapours, whereas zinc paint appeared unaffected. Their research led to the first large-scale production of zinc white in France, while English chemist John Atkinson claimed this promising innovation as his own and patented zinc oxide in paint as a substitute to lead oxide (1796). Thus, at the end of the 18 th century, in France and Britain, the two nations that used the most white lead -due to their early urbanization, as well as their sizeable naval and merchantmarine fleets which used white lead on a massive scale to paint ships -scientists were aware of the scientifically-proven toxicity of the material, and the existence of a safe alternative. Such eminent and renowned scientists as Fourcroy, Berthollet and Vauquelin all enthusiastically supported zinc white, the defects of which "are so slight compared to the disadvantages of using white lead, that its adoption cannot be reasonably refused." [START_REF] Messrs | Berthollet and Vauquelin to the National Insitute[END_REF] However, the relatively high price of zinc white -three times more expensive than white lead at the time -overshadowed the health arguments and prevented the chemical innovation from becoming an industrial success. Conversely, white lead was widely promoted by industrial concerns (Société d'encouragement pour l'industrie nationale, Comité consultatif des Arts et Manufactures, etc.) and by the highest public health bodies in France (Conseil d'hygiène et de salubrité, as well as the prefectures and municipalities, which were responsible for enforcing health rules), in order to foster domestic production over imports from Holland, Belgium and England that weighed on the national budget. As a result, white lead, albeit acknowledged as a deadly poison to experts, kept its dominant position in the paint pigment market during the first half of the 19 th century.

In the mid-19 th century, another French innovation rekindled the economic rivalry and the scientific controversy. After several years of experiments, Jean-Edme Leclaire (1801-1872), a former painter who owned a paint company and was influenced by Saint-Simonianism, perfected a manufacturing process for zinc white that would allow for industrial-scale production and filed for a patent in 1845. Leclaire had witnessed the workers in Paris write the word "slaughterhouse" on the white lead workshop door, and had at first stood by powerless to prevent the health effects of the "white poison" on his own workers. But from 1849, Leclaire's factory on the River Seine, a few miles downstream from Paris, was able to produce 6,000 kilograms of zinc white a day. Leclaire's zinc white was higher quality than previous versions, more weather resistant, and its lower production cost made it a contender for industrial-scale use -which turned the economic equation upside down because it was sold at the same price as white lead. Leclaire soon formed a partnership with the Société des Mines et Fonderies de la Vieille-Montagne, [START_REF]Subsequently called[END_REF] the world leader in zinc mining based in Liege, Belgium, that had a powerful distribution network. He brought an industrial innovation to European markets that was a true competitor to white lead producers:

"It was a revolution in the habits of the trade; people were divided into two sides: those who remained attached to their old routine and used the poison, with gaunt, pale and sickly workers; and on the other side, the advocates of zinc white, whose personnel were plump, hearty men, with rosy cheeks and a winning appearance." 8 Of course, this description, appearing in an etching given to Jean Leclaire by his grateful workers in 1851, was biased (fig. 1). The competition between the two products threatened the undisputed domination of white lead, depicted in this caricature as the sickly painter heading to the hospital, whereas his rival, who uses zinc paint, is hearty, haughty and healthy, proudly showing off the tools of his trade. The mid-19 th century saw an increase in publications by reputed hygienists and professionals, such as Alphonse Chevallier (a member of the Paris Hygiene Committee) and the famous architect Viollet-le-Duc, in favour of zinc white. [START_REF] Chevallier | Rapport fait à la Société d'encouragement pour l'industrie nationale au nom du Comité des arts chimiques, sur la substitution du blanc de zinc au blanc de plomb et que couleurs à base de plomb et de cuivre[END_REF] In 1849, Jean Leclaire was awarded a gold medal by the Société d'Encouragement pour l'Industrie nationale, and the following year, he received the Prix Montyon from the Academy of Sciences, followed by the Legion of Honour, the highest civil distinction in France. Ambitious public action coincided with this series of awards. In August 1849, while hygienist periodicals agreed that the manufacture and use of zinc white were harmless, a Decree, soon followed by a government circular in February 1852, called for zinc white to be used for all public building works at the département and municipal levels.

At mid-century, all events suggested that white lead would thus soon disappear. Yet, a cloak of silence fell again on the dangers of white lead, which, like the many-headed Hydra, regained its strength even though it had been beheaded. I argue that the economic sector of the white lead chemical industry invented a new narrative to resist competition during this period and succeeded to make the issue dormant for nearly 50 years, before the issue of industrial poisons returned to the political debate at the turn of the 20 th century.

Just after the 1898 Act establishing employers' liability for work-related accidents, the workers' movement to obtain compensation for occupational illnesses focused its efforts on industrial poisons. This chapter of the biography of white lead did not revolve around technical innovations, but was driven much more by changes in the labour movement, which enjoyed unprecedented ties with the political authorities under the French Republic. From 1900 to 1909, when the first law banning white lead paint was passed after years of parliamentary meanderings, white lead's public notoriety was at a peak, becoming a symbol. Labour leaders and journalists, high-ranking politicians such as Viviani and Clemenceau, public health experts and members of parliament worked together to make the perennial issue of industrial poisoning a major political "cause." In the French law passed just after World War I (Act of 25 October 1919), lead poisoning in painters was the occupational disease par excellence and its causes and effects were perfectly known by the general public, as newspapers were full of the health risk of using it and of economic rivalry of the two whites.

Back in the spotlight in 1921 in a transnational context, the conflict between lead-based and zinc-based paint briefly flared up again, as the 3 rd International Labour Conference in Geneva debated an international convention banning white lead in building paint. France was clearly a pioneer on the topic, as the first country to have (partially) banned lead paint in buildings in 1909, and it was a leader in the ILO -whose first director was socialist Albert Thomas. The vigorous debates on white lead in Geneva read like a retrospective of the previous century: all the medical, technical, economic, health and humanist arguments were repeated in a single time and place. This repetition betrayed a collective amnesia making it necessary to review all the data already debated over the prior century: white lead was not dead yet. The 13 th International Convention banning lead paint (1921), ratified by France, Belgium and a dozen other countries by 1926, gave a final twist to this debate, showing the powerful "manufacture of doubt" when assessing whether materials in the workplace are hazardous or innocuous.

BETWEEN INVISIBILITY AND COMPARTMENTALIZATION

For a century, the risk of industrial poisoning attracted sporadic attention from scientists, political authorities, experts and public opinion. Sometimes carefully documented, proven, debated, and partially dealt with, the risk was in other times overshadowed or denied. The agnotological approach has been enhanced in recent social science research in the field of public health, and has shown that the invisibility of a public health issue is due not necessarily to deliberate concealment efforts, but most often to compartmentalization that prevents information from circulating outside limited, or even marginal, social spaces. The latter is a valuable perspective for casting light on the economic and scientific battle between white lead and zinc white, as periods of bright light and deep knowledge on the one hand and periods of limited circulation of the information on the other hand alternate during a century. [START_REF] Proctor | Origins of the Cigarette Catastrophe and the Case for Abolition[END_REF] Thus, rather than speaking of periods of silence, one can identify periods of compartmentalization. Each time the white lead controversy returned to centre stage during the 19 th century, stakeholders' apparent amnesia regarding the material was quite surprising, as if the scientific and technical knowledge from previous episodes had been forgotten, as if new experts, chemists and physicians had to be brought onboard to prove the dangers of the material, and new paint experts to demonstrate that zinc white could replace white lead.

The first "silent period" on the issue of lead poisoning (1800-1845) in France was definitely part of the "erasure of the worker's body," a historical process identified by Le Roux in the dismantling of worker protections during the revolutionary period, giving rise to an industrialist mindset. [START_REF] Roux | L'effacement du corps de l'ouvrier. La santé au travail lors de la première industrialisation de Paris[END_REF] Prior to 1820, however, the economic stake was weak: production in France was still on a small scale and most white lead was imported from Holland, Belgium and (to a lesser extent) England. As a result, a small number of physicians and chemists regretted a health issue affecting a small number of people. The public authorities gave their full support to the first attempt to produce white lead on an industrial scale and looked favourably on the factory built in 1809 in Clichy, on the outskirts of Paris. Given the French Empire's enthusiasm for industrial and chemical innovation, the white lead factory in Clichy fuelled France's hopes of asserting its economic independence from rival countries. The support given to this enterprise from the outset was indicative of the overt collusion between politicians, industrialists and eminent public health experts, such as Chaptal, Thenard and Roard (the director of the Clichy white lead factory). [START_REF] Roux | Le laboratoire des pollutions industrielles[END_REF] Despite the development of white lead manufacturing in the 1820s and the subsequent increase in lead poisoning cases, which became apparent in the corridors of hospitals -especially in Paris and Lille, the two cities were the centres for France's white lead industry -the message of the material's dangers was restricted to the narrow spheres of public health. These health hazards were documented in the Annales d'hygiène publique et de médecine légale but found few echoes outside this specialist publication. Furthermore, the fact that no alternative material could compete with white lead in terms of quality or price meant that the "white poison" dominated the paint industry: despite a few isolated cases of experts deploring the "white poison", it was not until Leclaire developed his industrial process for zinc white in the 1840s that this public health issue gained strong visibility.

Once again, after the mid-century flurry of public action to establish a ban (1849-1852), a second period of heavy silence began, lasting a half-century. Three major factors were behind this period of waning controversy. First, the white lead industry formed a powerful lobby group under the Second Empire. Mainly operating behind the scenes, this pro-lead coalition became very active in the mid-19 th century to fend off the government's attempts to ban white lead. French white lead manufacturers were few in number (around a dozen nationwide), but were very concentrated geographically (Lille, Paris and Tours). These industrial powerhouses were strengthened with cross-shareholdings and often backed by family ties -especially with the powerful textile industry in Northern France -and were well-organized to exert their influence on decisive regional and national stakeholders such as the Chambers of Commerce and Industry. In a market that was opening up and becoming more international, they constantly called on the government to change its customs duties on raw and finished materials in order to fend off imports and protect the national market, notably during the negotiation of free trade agreements in 1860-61. These industrialists viewed these free trade agreements as the death knell for their material, which would lose out to British and German competitors. The lobby's influence on the Second Empire's economic institutions was undoubtedly strong enough to explain why white lead production continued in the latter half of the 19 th century despite opposition to the material.

The Third Republic opportunistically supported the white lead industry, thanks to the efforts of Charles Expert-Bezançon, who was the leading white lead industrialist in Paris, an influential member of French industrialist circles and senator for Paris département (1900-1909). He was a spokesman, or even a tribune, for the powerful white lead lobbying group over several decades, which invested considerable efforts in publications and public events to foster doubts about the toxicity of white lead and the capacity of zinc white to be an effective substitute. Refusing to consider the rosters of sick workers treated in hospitals, challenging any attempts to quantify the lead poisoning epidemic, denying that white lead could be dangerous when it was handled with all the "desirable" precautions, the white lead industrialists echoed hygienist arguments that workers were negligent in protecting themselves. They also skilfully fuelled confusion about other toxic materials or other sectors responsible for lead poisoning that did not use white lead (printing, plumbing, battery manufacturing, etc.). Boasting their industry's technical advances since the mid-19 th century, they endeavoured to cast doubt on the validity and the hidden motives of the campaign against white lead, whose assertions were considered baseless and in no way related to epidemiological reality. [START_REF] Fleury | Blanc de zinc et blanc de céruse, historique complet de la question, réfutation du rapport de la commission législative[END_REF] From a technical standpoint, the white lead industry sought to prove that lead-based paint was superior to its zinc-based rival by shifting the debate to industry experts and away from physicians or the general public, considered uninformed about opacity, ease of application, drying, and resistance of the paint over time. From a political standpoint, the approach repeatedly sparked suspicion about the true motivations of the opponents to white lead, accusing physicians, unionists and politicians of being corrupt and "anti-patriotic," allegedly being paid by the Belgian or German zinc industry to destroy the French white lead industry. This argument occasionally drifted into an anti-parliamentarian, xenophobic and anti-Semitic diatribe. [START_REF] Dubreuil | Blanc de zinc contre blanc de céruse[END_REF] Lastly, from a medical standpoint, the white lead industry attempted to manipulate the nascent clinical knowledge about lead poisoning. Substantial research focused on lead poisoning after 1830, but the findings were largely limited to a scientific audience. Following Tanquerel des Planches (1834, 1839) and Grisolle (1835), clinical knowledge in the late 19 th century specified the many symptoms and pathologies all connected to lead -designated without a doubt as a violent poison. Similar statements were made in Britain, where the growing white lead industry in Newcastle-upon-Tyne caused waves of illness and even death among women workers: Dr Charles T. Thackrah had early on denounced the serious intoxications linked to the use of lead in industry and described the main symptoms allowing the disease to be diagnosed and patients to be distanced from the source of poisoning (1831), as had Henry Burton, who stated that a greyish coloration of the gums was an unmistakable symptom of lead poisoning, then known as the "Burtonian line" (1840). More attempts to give a conceptual and practical framework to the disease were made in the same period in Italy and Germany. These facts illustrate how knowledge was being built up on a European scale about what was the main industrial work-related disease at that time. However, this clinical knowledge and the medical controversies were inherently technical, remaining somewhat invisible to the general public. Only at the turn of the 20 th century did the medical and political fields partly become synchronized, notably through networks of republicans and Free Masons, such as Paul Brouardel, a friend and follower of Louis Pasteur and professor of forensic medicine and an eminent member of the Academy of Medicine. During the 1900-1910 decade, medical expertise on occupational lead poisoning found an audience outside the hospital sphere: in the political arena of the Third Republic, the interconnected but contradictory demands of personal freedom and social justice brought scientific experts back into the political spotlight that they had partially deserted during the 19 th century. [START_REF] Topalov | Laboratoires du nouveau siècle. La nébuleuse réformatrice et ses réseaux en France (1880-1914[END_REF] The labour movement in the latter half of the 19 th century was not yet mature enough to publicize the health issues of industrial poisons and to demand about occupational health conditions in the chemicals industry. There were no official labour unions in white lead factories. In Paris, where white lead factories employed around a hundred workers in the 1880s, there are no traces of a union in the sector. In Lille, white lead workers were three or four times greater in number, there are no signs of a trade organisation or of workers' demands regarding health conditions in this small but powerful part of the local chemical industry. Over the previous decades, hygienist journals had published a fairly large number of articles about arsenic, lead and mercury poisoning, but the topic was nearly ignored by the nascent labour movement, which was mostly focused on achieving political and doctrinal unity, and emphasized fundamental demands for higher wages and reduced working hours. It was not until 1900, when house painters began to organize as a trade (they were the main users of white lead), that labour unions started to refer to lead poisoning as a "cause." It remained nevertheless a secondary topic, quickly set aside when not considered useful for building a unified labour movement. [START_REF] Rainhorn | Workers Against Lead Paint: How Local Practices Go Against the Prevailing Union Strategy (France, at the Beginning of the 20th c.)[END_REF] In the end, labour movements making demands for the "victims" joined forces with medical professionals to bring the issue of lead paint poisoning into the public sphere, just as French republican officials were focusing on social matters. The distance separating the various arenas was reduced: no longer confined to the side lines after a half-century, white lead entered the spotlight thanks to the convergence of the labour movement, scientists speaking out about public health, social policy being promoted by government leaders, and the press's newfound interest in publicizing this cause, designing a new "regime of perceptibility". [START_REF] Murphy | Murphy M. Sick Building Syndrome and the Problem of Uncertainty: Environmental Politics, Technoscience, and Women Workers[END_REF] This convergence of influences gave rise to new national rules, with the 1909 act that banned the use of white lead in building paint and attempted to eliminate the risk for house painters. This was only partially successful, however, because the ban, scheduled to take effect on January 1, 1915, was overtaken by the outbreak of World War I.

After the Armistice, the efforts to fuel doubts about zinc white as a possible substitute for white lead reappeared during the heated debates at the 3 rd International Labour Conference, starting the international career of the product. The recently-founded ILO (1919) quickly addressed the issue as a sign of its determination to set standards for sanitary working conditions in order to draft an International Convention banning white lead. However, even a century and a half after Guyton de Morveau's demonstrations, the qualities of the two materials were again addressed for several weeks in Geneva, suggesting that a scientific debate was still legitimate and unsettled. As Albert Thomas, director of the ILO, noted at the time, the advocates of white lead insisted on "undermining the authority of the scientific demonstrations on which [the convention] was based! It is therefore important […] that the experts […] defend the truth against those who wish to adulterate it." [START_REF]Speech by the Director of the ILO[END_REF]

WHEN POISON ENTERS THE MARKETPLACE

As we have seen, scientific and technical arguments held great sway in this century-long battle. This conflict was chiefly an economic confrontation between two materials on a very lucrative market, governed by rules that were upended by new health and moral considerations. Endorsing the liberal anti-interventionist credo that the marketplace should act as the "site of veridiction," [START_REF] Foucault | The Birth of Biopolitics[END_REF] the advocates of white lead opposed government intervention for the sake of open economic competition, which they claimed revealed its true value and thus should be the sole determinant: "When the railways were built, the stage coaches disappeared; they died a timely death. If zinc white is truly superior to white lead, it will kill us in the marketplace, but the government should not intervene." [START_REF] Breton | Rapport supplémentaire fait au nom de la commission chargée d'examiner le projet de loi sur l'emploi des composés du plomb dans les travaux de la peinture en bâtiments[END_REF] These were the words of Expert-Bezançon, in his February 1903 deposition to the parliamentary committee examining the bill for banning leadbased pigments in paint. Among the industrialists, the debate was essentially about political economy and free enterprise, and the denied right of the State to remove a product due to drawbacks extrinsic to the market. However, several factors disrupted the competitive landscape shaped by the invention of the manufactured zinc white as an alternative product to the poisonous white lead. White lead's longer presence in the paint market was a major advantage for the material. It had been a luxury good, before being manufactured on an industrial scale inundating the Western European markets for paint for artists and decorations. When Leclaire's zinc white entered the market as of 1845, white lead held a dominant position, having reigned supreme in the building paint markets under its many avatars: Venetian ceruse, Meudon, Spanish, Champagne or Troyes whites, all derivatives of the original pigment, named according to the proportion of white lead content. Zinc white would have to overtake white lead. Beginning in the 1840s, Belgian firm Vieille-Montagne -leader in the European zinc market -implemented a clever strategy to win over the paint market. It acquired and filed for patents, built subsidiaries, carried out various experiments, made verbal and secret agreements to buy out competitors, all in an attempt to defeat white lead. However, the white lead industry enjoyed substantial private and political support under the Second Empire, even though the French government put a legal -but never implemented -ban in public works projects in 1849. Despite the financial firepower of the Vieille-Montagne, which manufactured and sold zinc white in the French, Belgian and German markets, zinc white struggled to challenge white lead in the paint market until the 1920s. Similar in price to its competitor, zinc-based paint nevertheless required painters to work differently. It was slightly more difficult to handle than lead paint, requiring the worker to give "greater attention to even out his painting. A certain amount of training [was] needed, as for any change in occupation", a "turn of the wrist" that took some time to learn, although specialists said it came rather quickly. [START_REF] Finance | Peintures au blanc de zinc. Rapport présenté par M[END_REF] While many painters were opposed to this change, this is also because it favoured skilled workers, with more technical experience, whereas in the final decades of the 19 th century, growing numbers of unskilled labourers took up painting on building sites, in a proletarianization process of the trade. These unskilled workers commanded lower wages and were preferred by many unscrupulous business owners. However, such workers struggled to use zinc-based paint, which "lathered up" less with the paint brush. Lastly, the switch to zincbased paint made it more difficult to "cheat" on the quality of paint because zinc white is harder to mix with adulterated substances (such as chalk, flour and other white powders, often used in the mixing instead of pigments) than white lead. Since such fraudulent practices were common among building companies, this also explains professional painters' opposition to zinc-based paint. Thus, despite its harmlessness and because of this unwanted technical turn in the workers' know-how, zinc white was rejected by most paint professionals.

In the end, however, the health argument won out -at least in public opinion. The health argument or, again, a matter of price? The social cost of illness became a strong argument for zinc white in the court of public opinion, unsettling market rules. While estimates of the number of occupational lead poisoning victims had been imprecise and very incomplete, by the mid-19 th century, several publications revealed the financial cost of these diseases for society, in addition to the images of people suffering. Figures showed the growing number of white lead workers and painters suffering from poisoning, giving them a new-found visibility in hospital wards: 948 patients hospitalized in Paris for lead poisoning over a period of 942 days (from September 1849 to March 1852), costing the Paris hospital administration some 19,000 days of care to lead poisoned patients. [START_REF] Richelot | De la substitution du blanc de zinc au blanc de plomb dans l'industrie et les arts[END_REF] Crippled men with arm and leg paralysis unable to work and depending on charity for life, weak and disabled children born to lead-poisoned mothers, families forced into penury by the early death of fathers: it eventually appeared that including the health and social cost of chronic lead poisoning in the calculation made white lead much more expensive than its substitute. In the latter half of the 19 th century, references to the cost of treating lead poisoning victims and taking care of the disabled for life became a leitmotiv of those arguing that zinc white should be preferred "in the interests of humanity and the state." 23 Ultimately, this socio-technical controversy raises the basic question of competition in the marketplace between two materials whose qualities and value were built under different criteria, along different timelines. On the one hand, technical knowledge determined the chemical makeup, utility and price of each material. On the other hand, medical knowledge and expertise assessed each material's harmfulness for human health. Thus, the purely industrial and commercial motives were overshadowed by the health issues, and the resulting regulatory standards impacted the markets by changing the notion of quality and thus also market conditions. Consumers' "choice" -the famous routine -had little to do with defining quality. [START_REF] Stanziani | La définition de la qualité des produits dans une économie de marché[END_REF] White lead's commercial resilience was attributable less to its supposed intrinsic technical qualities (ease of application and drying, opacity, weather resistance, etc.) than to the qualities attributed to it in a specific historical context: its centuries-old reputation, national production capacity, and "traditions" in the painter's trade, all underpinned by an unprecedented public lobbying campaign and much concerted effort by stakeholders. Conversely, during the periods when the health argument prevailed as a driver of public action (namely, late 18 th century, 1849-52 and 1900-1930s), the material's alleged merits were completely overshadowed by its harmfulness that somehow swept away the economic conflict by asserting that human life was more important than the market. As Justin Godart, French delegate to the ILO, claimed in 1921:

"Faced with all these private interests defending themselves […], we must consider a higher interest: human interest. I believe that by protecting human life […], we serve the general economy and commonwealth more usefully than by allowing white lead to be used, even if it is an established fact that it provides longer protection to the materials it covers, even if its cost is lower." 25 In these circumstances, public health and free competition appear to be two irreconcilable facets of economic growth. Nevertheless, a transnational system of standards was crafted in an attempt to reconcile these two values in favour of public health -but proved to be widely unsuccessful.

CONCLUSION

The controversy between lead and zinc whites in the 19 th and 20 th centuries exemplifies the strong interconnections between economic, social, technical and political factors that determine the life and the identity of materials in the long run. White lead came to be seen as an industrial poison rather than as a useful and common chemical product after a complex process involving scientific research (in medicine and chemistry), technical debates, economic rivalries and political sparring. For more than a century, self-proclaimed experts who would later be known as "merchants of doubt" maintained a fog of social and political doubt about the dangers of white lead and the innocuousness of zinc white. Even though the latter had been scientifically established by the end of the 18 th century, they did not hinder the continued commercial dominance of white lead for more than a century. In order to disentangle the mesh of converging interests interweaving the alliance between white lead and industrial societies, scientific evidence may be necessary but is by no means sufficient. Industrial and social stakeholders use mechanisms for manufacturing ignorance or, at the very least, perpetuating doubt about medical and chemical expertise thus creating scientific confusion and thus breaking the market rules. Powerful public and private actors thus participated in the elaboration of a regulation of the toxic substance which, while relying on the elaboration of a protective legal norm for people, permitted the resilience of the material and the social acceptance of a risk inherent in the industrial society of "progress". On this topic, the case of the white paint pigments used on a massive scale during the rapid urbanization of Europe is a valuable historical angle to identify long-lasting mechanisms which can be seen even today, when certain political and economic interests vocally challenge scientific knowledge. * * * 1 Homburgand E, De Vlieger JH. A Victory of Practice over Science: The Unsuccessful Modernization of the Dutch White Lead Industry (1780-1865). History and Technology. 1996; 13: 33-52.

Fig. 1 -

 1 Fig. 1 -Lithograph given to Jean Leclaire by his workers, 1851, all rights reserved.


Valeria Daniele

François-Xavier Legrand

Patrick Berthault

Jean-Nicolas Dumez

Gaspard Huber

Single-scan multidimensional NMR analysis of mixtures at sub-millimolar concentrations by using SABRE hyperpolarization

Keywords: analytical methods, hyperpolarization, NMR spectroscopy, para-hydrogen, ultrafast experiments

Introduction

Nuclear magnetic resonance is a very powerful but particularly insensitive analytical technique. Hyperpolarization techniques aim at improving its sensitivity by increasing the population difference between nuclear spin states far beyond the Boltzmann value. One of these techniques, Para-Hydrogen Induced-Polarization (PHIP), [START_REF] Green | [END_REF][2][3] relies on the total spin alignment of parahydrogen (p-H2) molecules, which can be generated with no need for an expensive and complex apparatus. [4,5] Signal Amplification By Reversible Exchange (SABRE) is a promising PHIP method [6,7] based on the reversible binding of parahydrogen and the substrate of interest to a metal center, usually an iridium complex: the resulting transient coupling network can transfer the spin order of p-H2 to the nuclear spins of the substrate where observable polarization is formed. [8] This method has the potential to vastly broaden the application field of parahydrogen-based hyperpolarization, since it does not require the existence of an analogue of the substrate of interest containing unsaturated bonds. It has for example been applied to two drugs, pyrazinamide and isoniazid, used in the treatment of tuberculosis. [9] It has also been applied to nicotinamide which acts as an active microbial agent against Mycobacterium tuberculosis and Human Immunodeficiency Virus. [10] In the first SABRE experiments, a rather high substrate concentration, in the millimolar range, was necessary to observe optimal hyperpolarized signals for the free substrate: at low concentration the solvent may compete with the substrate on the catalyst. This limitation has been overcome by the addition of a concentrated co-substrate, [11] i.e. a suitable ligand which can enter the coordination sphere of the metal center. In separate studies, the use of a deuterated co-substrate has been shown to yield higher signal enhancement factors. [12,13] The SABRE technique has also been applied to the quantitative analysis of a dilute molecule in a complex mixture: in this case, the other molecules in the mixture, which are in large excess, act as cosubstrates. [14] One-dimensional 1 H NMR spectra of complex mixtures, however, typically suffer from extensive overlap and may not be sufficient for analytical purposes. [15] The spectral overlap problem is even more acute when a concentrated co-substrate is used, owing to the very intense signals of the co-substrate itself. Multidimensional (nD) NMR spectra are often essential for the analysis of mixtures, but the short-lived nature of hyperpolarized signals -mainly determined by their longitudinal relaxation times -is not compatible with conventional approaches of multidimensional NMR. One possibility is the use of ultrafast NMR, which yields nD spectra in a single scan [START_REF] Frydman | Proc. Natl. Acad. Sci[END_REF][START_REF] Frydman | [END_REF] and is becoming a powerful analytical tool. [18,19] The ultrafast COSY pulse sequence has been used to detect parahydrogenbased hyperpolarization of quinoline at 10 mM. [20,[START_REF] Lloyd | Developing SABRE as an Analytical Tool in NMR[END_REF] In this article, we describe an approach to obtain clean, SABRE-hyperpolarized 1D and ultrafast 2D NMR of mixtures of small molecules at sub-millimolar concentrations. We first show that the use of a deuterated co-substrate virtually suppresses unwanted intense signals of the co-substrate itself, thus simplifying the NMR spectra, particularly those of sample mixtures. Then, we illustrate the use of ultrafast 2D in combination with a deuterated co-substrate to detect a substrate at sub-millimolar concentrations. Finally, we show how the proposed approach can help in the analysis of a complex mixture of small molecules.

When molecules of interest have a total concentration of less than 3 times the catalyst concentration, addition of a cosubstrate to the sample is necessary to achieve significant hyperpolarization with current SABRE catalysts. [11] An example of this approach, the hyperpolarization of 4,4'-bipyridyl (bpy, Fig. 1), is shown in Figure 2. These spectra were obtained on a dilute solution of the substrate bpy (CS = 0.5 mM, which means 0.25 µmol in the NMR tube), in the presence of the metal complex precursor [Ir(COD)(IMes)Cl] (1, CM = 0.13 mM) and pyridine (either standard or perdeuterated) as a co-substrate (CC = 10 mM). Perdeuterated pyridine is a commercially available molecule, widely used as a solvent for NMR. Activation of complex precursor 1 by molecular hydrogen leads to the formation of several possible metal complexes of the formula [Ir(IMes)(H)2(Si)(Sj)(Sk)]Cl (2) where i, j and k are either the substrate or the co-substrate. [START_REF] Cowley | [END_REF] After degassing the NMR tube, fresh 51%-enriched p-H2, prepared at liquid nitrogen temperature, was introduced and the tube was shaken at 7 mT, in the fringe field of the spectrometer, then quickly introduced into the spectrometer magnet. Comparison of Fig. 2a and2c shows that the signal of o-py (abbreviations in Fig. 1) is enhanced by a factor ca. 900. However, it is obvious in Fig. 2c that the intense signal of the hyperpolarized co-substrate is detrimental for the analysis of the molecule of interest, as it obscures the peak corresponding to the beta protons of bpy (see Table S1 for chemical shifts). In contrast, the spectra shown in Fig. 2b and 2d were obtained using deuterated pyridine as a co-substrate, deuteration rate being measured at 99.5, 99.3 and 99.0% at ortho, meta and para positions, respectively. In this case, the two signals of bpy can be observed. In particular, b-bpy can be distinguished from residual protonated p-py signal. Hence, deuteration of the co-substrate removes the need to find a co-substrate with adequate chemical shifts, which would not overlap with those of the molecules of interest. This is only partly true for pyridine, as a 1 H signal is still observed for the ortho positions. This may be explained by the fact that iridium(I) complexes such as 2 are known to catalyze ortho-directed hydrogen isotope exchange. [23][24][25] Using a deuterated co-substrate can also have advantages in terms of sensitivity, in addition to yielding a cleaner spectrum. First, a higher receiver gain may be used in the absence of strong signals from the co-substrate. Then, a deuterated cosubstrate can improve the enhancement factor for the substrate. [12,13] In particular, for the alpha protons of bpy (noted a-bpy in Fig. 1), the enhancement factor increases from 86-fold in the presence of protonated pyridine (py) to 140 in the presence of deuterated pyridine (py-d5). This higher enhancement is a consequence of the fact that polarization is not efficiently transferred into the 2 H-nuclei of the deuterated ligand, because of the greatly reduced hydride-2 H J-couplings compared to the hydride-1 H J-couplings and their frequency difference. One-dimensional 1 H NMR may not be sufficient to identify or quantify analytes in a complex mixture. Ultrafast 2D COSY spectra can be obtained in a single scan with a COSY pulse sequence using continuous spatial encoding. [26] Figure 3 shows an ultrafast COSY spectrum recorded after SABRE hyperpolarization of bpy (see Experimental Section and Fig. S1 for details of the sequence). The use of a co-substrate here makes it possible to observe a good-quality spectrum of a substrate at sub-millimolar concentration. In addition, deuteration of the co-substrate removes the overlap issue between the para position of pyridine and the beta positions of bpy. The asymmetry of the ultrafast COSY spectrum might be due to the asymmetric spatial encoding used. [27] As ultrafast 2D NMR suffers from a sensitivity penalty, [28] SABRE hyperpolarization broadens the range of systems that may benefit from the single-scan acquisition of 2D spectra. As an illustration of the potential of our approach for obtaining clean, hyperpolarized 2D spectra of low-abundant molecules in a single scan, we analyzed a mixture of small molecules. The mixture consisted of five different substrates, namely bpy, ni, dmni, 2-epy and 3-epy (Fig. 1), all at a concentration on the order of 0.5 mM (see Table S2 for the concentration values). The corresponding ultrafast COSY spectrum (Fig. 4) has a very good quality, especially considering the unfavorable conditions in which it was acquired, i.e. after shaking the NMR tube and without optimized shims. The only molecule not visible is 2-epy, apparently not polarized: this might come from a steric hindrance of the ethyl group in ortho position of the nitrogen atom, which is the bound part of the molecule to the catalyst. [29] For the other molecules, the enhancement factors, estimated from the corresponding 1D SABRE spectrum, are reported in Table S3. Importantly, almost all correlations can be assigned; several of them correspond to resonances that are not resolved in the 1D spectrum, such as 6-ni and a-bpy. In this case, 2D correlations allow full characterization of the mixture. Very few of these correlations are observed when the ultrafast COSY spectrum is recorded at thermal equilibrium (Fig. S2). At such low concentrations, some correlations that are seen in Fig. 4 require an acquisition of several minutes to be observable in a conventional COSY spectrum (Fig. S3). Signal-to-noise ratio has been evaluated to ca. 85 and ca. 7, for the intense correlation 5/4-dmni and the weak correlation 4/2-ni. The presence of high signal-to-noise ratios prompted us to record an ultrafast COSY spectrum at even lower concentrations, less than 0.1 mM. It is given in Supporting Information (Fig. S4). Note that all COSY spectra are displayed here in magnitude mode, as is usual for spatially encoded experiments. [19] While the SABRE process may create multi-spin longitudinal order terms, [8] their contribution to the signal is suppressed by the coherence selection gradients. Other ultrafast 2D NMR experiments are compatible with SABRE hyperpolarization. In particular, the recently reported direct polarization of 15 N spins [30] should be compatible with HSQC-type experiments [31] . It should be noted that, in the case of a sample mixture, deuteration of the co-substrate again significantly improves the quality of the spectrum, as it removes almost completely interferences by signals of the co-substrate. The isotopic exchange involving protons in ortho position of pyridine-like molecules, however, limits the time scale during which the cosubstrate has minimal contribution to the spectrum. However, this time is sufficient to record good-quality spectra.

Conclusions

In summary, we have obtained clean, SABRE-hyperpolarized 1D and ultrafast 2D NMR spectra of mixtures of small molecules at low concentrations. It is worth noting that by using a 100%enriched p-H2 instead of a 51% one, enhancements should be increased roughly by a factor of 3, resulting in decreasing the lower concentration limit by the same factor. This approach could help in the use of hyperpolarized NMR for the analysis of complex mixtures. In particular, ultrafast 2D NMR has been shown to be suitable for quantitative applications [15] and its use for hyperpolarized samples may be assessed for this purpose, for which promising results have been obtained with 1D NMR. [14] Another area of applications is the use of hyperpolarized ligands in binding studies, which can stimulate the screening and discovery of new drugs [32] . Obviously, further improvements in the detection threshold will be obtained with the use of a higher enrichment of hydrogen in the para isomer.

Experimental Section

Chemicals

Complex precursor [Ir(COD)(IMes)Cl] (1) (IMes = 1,3-bis(2,4,6trimethylphenyl)imidazole-2-ylidene, COD = cyclooctadiene) was purchased from the Centre for Hyperpolarization in MRI of the University of York (UK). Pyridine (py), 4,4'-bipyridyl (bpy), nicotinamide (ni), N,Ndimethylnicotinamide (dmni), 2-ethylpyridine (2-epy), 3-ethylpyridine (3epy) and 4-ethylpyridine (4-epy) were purchased from Sigma-Aldrich. Methanol-d4, pyridine-d5 (py-d5) and 3-(trimethylsilyl)propionic-2,2,3,3-d4 acid sodium salt (TSP-d4) were purchased from Cambridge Isotope Laboratories. All chemicals are used as supplied. Relevant proton chemical shifts are provided in Table S1.

Para-hydrogen (p-H2) was produced by cooling normal hydrogen down to 77 K in the presence of hydrous ferric oxide (Molecular Products, Inc.). The resulting 51% p-H2, this percentage being controlled by Raman spectroscopy with experimental error of 2%, was transported to a glass container: in the absence of paramagnetic impurities, this percentage of p-H2 remains stable for several hours.

Protocol and NMR experiments

All spectra were acquired at room temperature with a Bruker AVANCE II 600 MHz spectrometer operating at 599.82 MHz and equipped with an unshielded Oxford magnet and a 5 mm TCI cryo-cooled probe. Samples were collected into 5 mm Young pressure valve NMR tubes and degassed through two freeze-pump-thaw cycles. 1 mM of TSP-d4 was added into each sample as a concentration reference.

A few minutes after its preparation, p-H2 was introduced at a pressure of ca. 4 bar and the sample was vigorously shaken for several seconds, in order to saturate the solution, allowing the formation of metal complexes [Ir(IMes)(H)2(Si)(Sj)(Sk)]Cl (2) via hydrogenation of the COD ligand of 1.

After full activation, checked by monitoring the saturation of the cyclooctane signal intensity, the volume above the solution was evacuated and p-H2 was introduced at a pressure of ca. 4 bar. Samples were shaken for 15 seconds at a fixed position in the stray field of the spectrometer, resulting in polarization transfer field strength of approximately 7 mT. This magnetic field has been shown to be optimal for pyridine. [START_REF] Cowley | [END_REF] , although recent simulations performed in continuous re-hyperpolarization suggest an optimum at 5.5 mT. [33] Obviously the optimum may be different for each substrate. As the magnet is not actively shielded, the sample experiences limited magnetic field gradient during this step. The sample was then rapidly inserted into the bore; then a 90° rf pulse was applied to obtain a SABRE spectrum ca. 8 s after the end of the shaking step. Reference spectra for signal enhancement determination were recorded in quantitative conditions, i.e. with a prescan delay of at least 5 times the longest proton longitudinal relaxation times, and at least 10 minutes after the SABRE experiments. Signal enhancements were obtained by dividing the integrals of the SABRE signals by the corresponding ones on the reference spectra. Signal-tonoise ratios were estimated from the sino function of Topspin 3.2 software, applied to rows and columns at maximum height of the chosen correlations.

The ultrafast COSY sequence, described elsewhere, [18,34] is reported in figure S1. Spatial encoding was performed using a constant-time encoding scheme. [26] Smoothed chirp encoding pulses of 15 ms with a sweep width of 8 kHz were applied for spatial encoding. The amplitude of the spatial encoding gradients Ge was 0.8 G/cm. The amplitude of the coherence selection gradients before and after the mixing pulse was 43.2 G/cm with a duration of 1.25 ms. The anti-echo gradient Gp consisted of a 0.3 ms sine-bell gradient of maximal intensity 16.2 G/cm. Acquisition gradients Ga of duration 312.8 µs and strength 32.4 G/cm were used. 128 gradients pairs {Ga;-Ga} were used during acquisition. Ultrafast 2D spectra were processed with the ufproc program provided by the University of Nantes. [35] Entry for the Table of Contents (Please choose one layout) Layout 1:

ARTICLE

Here we describe an approach to obtain clean, SABRE-hyperpolarized 1D and 2D NMR of mixtures of small molecules in a single scan. By relying on a deuterated co-substrate together with ultrafast 2D NMR sequences, the discrimination of molecules at sub-millimolar concentrations can be achieved.

Figure 1 .

 1 Figure 1. Structures and nomenclature of the complex precursor 1, the activated complex 2, the co-substrate and substrates used in this study. The abbreviations indicate: 4,4'-bipyridyl (bpy), pyridine (py), nicotinamide (ni), N,N-dimethylnicotinamide (dmni), 2-ethylpyridine (2-epy), 3-ethylpyridine (3epy) and 4-ethylpyridine (4-epy).

Figure 2 .

 2 Figure 2. Aromatic region of 1 H NMR spectra of 0.5 mM 4,4'-bipyridyl with SABRE hyperpolarization (c, d) and after equilibration of the sample polarization (a, b). In addition to the substrate, the samples contained 0.13 mM complex precursor 1, 10 mM py (a and c) or 10 mM py-d5 (b and d) as a cosubstrate and 4 bar of p-H2 in methanol-d4. The insert in d) shows the discrimination between b-bpy (solid line) and residual p-py (dashed line) signals. The vertical scale was adjusted in c and d.

Figure 3 .

 3 Figure 3. One-scan ultrafast COSY spectrum of 0.5 mM 4,4'-bipyridyl acquired in less than 1 s, with SABRE hyperpolarization. In addition to the substrate, the sample contained 0.13 mM complex precursor 1, 10 mM py-d5 as the cosubstrate and 4 bar of p-H2 in methanol-d4.

Figure 4 .

 4 Figure 4. One-scan ultrafast COSY spectrum of a mixture composed by bpy, ni, dmni, 2-epy and 3-epy at concentrations on the order of 0.5 mM, with SABRE hyperpolarization. In addition to the substrates, the sample contained 0.13 mM complex precursor 1, 10 mM py-d5 as a co-substrate and 4 bar of p-H2 in methanol-d4. The ultrafast 2D spectrum was acquired in less than 1 s. The spectrum displayed on top of the 2D map has been recorded in the absence of SABRE hyperpolarization.

  Valeria Daniele, François-Xavier Legrand, Patrick Berthault, Jean-Nicolas Dumez*, Gaspard Huber* Page No. -Page No.Single-scan multidimensional NMR analysis of mixtures at submillimolar concentrations using SABRE hyperpolarization

Acknowledgements

We thank Hana Kourilova, Bruno Coltrinari and Estelle Léonce for help with the setup of parahydrogen experiments. Support from the CEA Nanosciences program is acknowledged (HYPARE Project). We thank the EBSI group from the University of Nantes for providing tools for the acquisition and processing of NMR spectra.