L Ponson

H Auradou
email: auradou@fast.u-psud.fr

M Pessel

V Lazarus

J P Hulin

Failure mechanisms and surface roughness statistics of fractured Fontainebleau sandstone

Keywords: numbers: 62.20.Mk, 46.50.+a, 68.35.Ct

In an effort to investigate the link between failure mechanisms and the geometry of fractures of compacted grains materials, a detailed statistical analysis of the surfaces of fractured Fontainebleau sandstones has been achieved. The roughness of samples of different widths W is shown to be self affine with an exponent ζ = 0.46 ± 0.05 over a range of length scales ranging from the grain size d up to an upper cut-off length ξ ≃ 0.15W . This low ζ value is in agreement with measurements on other sandstones and on sintered materials. The probability distributions π δz (δh) of the variations of height over different distances δz > d can be collapsed onto a single Gaussian distribution with a suitable normalisation and do not display multifractal features. The roughness amplitude, as characterized by the height-height correlation over fixed distances δz, does not depend on the sample width, implying that no anomalous scaling of the type reported for other materials is present. It is suggested, in agreement with recent theoretical work, to explain these results by the occurence of brittle fracture (instead of damage failure in materials displaying a higher value of ζ ≃ 0.8).

I. INTRODUCTION

The characterization of the geometry of fracture surfaces is of considerable interest for a broad range of practical applications ranging from the hydrology of fractured reservoirs [START_REF] Adler | Fractures and Fracture Networks[END_REF][START_REF] Zimmerman | Mechanics of Fluids-Saturated Rocks[END_REF] to the derivation of friction laws [START_REF] Chakrabarti | Modelling Critical and Catastrophic Phenomena in Geoscience: A Statistical Physics Approach[END_REF]. Their statistical analysis is also relevant to the understanding of the physics of fracture: The roughness may indeed be expected to reveal failure mechanisms occurring at the microstructure scale [START_REF] Bouchaud | [END_REF]. It is now well established that fracture surfaces satisfy a scaling invariance known as self-affinity [5,6,7,8,9,10]. A self-affine profile is one that is statistically invariant under the scaling relation: z → λz and h(z) → λ ζ h(z) where the z axis is in the mean fracture plane and normal to the direction of crack propagation, h(z) is the surface height profile and ζ is the self affine exponent. The scaling invariance implies that the probability π δz (δh) to observe a variation of height δh over an horizontal distance δz follow the scaling law:

λ ζ π λδz (λ ζ δh) = π δz (δh) (1)
The prefactor λ ζ results from the normalization of π δz (δh). As a direct consequence, the height-height correlation function ∆h(δz) satisfies:

∆h(δz) = (h(z + δz) -h(z)) 2 1/2 z ∝ δz ζ (2)
In many materials including rocks like granite and basalt [7,8,9], the roughness exponent ζ was found to be close to 0.8, suggesting at first that it was a universal value [5,6]. However, different exponents were reported for Berea sandstone ζ ≃ 0.4 -0.5 [11] and then on synthetic sandstone made of sintered glass beads [12]. In this latter case, the exponent ζ is equal to 0.40 ± 0.04 independent of the porosity in the range investigated (3 % < φ < 26 %). This difference raises important fundamental questions regarding the physics of fracture, namely whether ζ is determined by the structure of the material or by the failure mechanism.

In support of the second possibility, a recent work [13] suggests that the value of ζ depends on the existence of a process zone where damage processes take place. On the one hand, at length scales smaller than the size of the process zone, fracture surfaces would develop from the growth and coalescence of microdefaults (cracks or voids): This damage failure (also called quasi-brittle failure) would lead to an exponent ζ ≃ 0.8. On the other hand, at length scales larger than the process zone, the fracture surfaces would result from the continuous propagation of a crack. For this brittle failure, the exponent would be ζ ≃ 0.4, reflecting, at the continuum scale, the toughness distribution of the microstructure scale. Twodimensional numerical simulations realized under these two types of assumptions confirm these results by predicting lower ζ values for brittle fractures [14] than when damage is introduced [15].

While most studies dedicated to the roughness of cracks dealt with surfaces with ζ ≃ 0.8 [5,6,7,8,9,10,16,17,18,19], the present work investigates these problems through an extensive analysis of the statistical properties of the roughness of fractured Fontainebleau sandstone for which ζ ≃ 0.4 -0.5.

In addition to the self-affine exponent ζ, another important statistical characteristic is the π distribution of the surface height variations: On granite surfaces (ζ = 0.8), it was found to be Gaussian [19] and a single self-affine exponent was needed to describe the correlation of the surface heights along the crack front direction.

Another important point is the dependence of the roughness statistics on the size of the samples and on the distance to the initiation zone. In the transient propagation region of fractures initiated from a straight notch, the amplitude of the roughness is observed to depend at all length scales on the distance to the initiation [16,17,18]. This variation may be described by a so called anomalous scaling law involving a new exponent ζ global > ζ referred as the global roughness exponent. A consequence of anomalous scaling is expected in regions far from the initiation and where the roughness is fully developed [17,20]: The roughness amplitude in a window of size δz normal to the crack propagation would then scale with the width W of the fracture surface as:

∆h(δz) ∝ δz ζ W ζ global -ζ (3)
In the present study, a statistical analysis of the same type is performed on fractured Fontainebleau sandstone samples of various widths W (perpendicular to the direction of crack propagation). For each surface, the roughness exponent ζ and the range of length scales over which a self-affine description is valid are inferred from the variations of the correlation function ∆h with δz. Their dependence on the width W of the samples and on the location on the fracture surface is investigated in order to look for the anomalous scaling features observed on other materials. The probability distribution of the variations of height over given distances δz are also investigated, in order to evaluate the magnitude of multiscaling effects and their dependence on δz.

The differences between these results and the characteristics reported previously for other materials will be discussed and we shall seek to relate them to the failure mechanisms taking place at scales larger than the grain size in Fontainebleau sandstone.

II. SAMPLE CHARACTERIZATION AND EXPERIMENTAL SETUP

The X-Ray analysis of the Fontainebleau sandstone samples used in the present work indicated that they contain 99 % quartz. The size of the grains range from 100 µm to 500 µm with a mean value d ≃ 270 µm. 16 SEM micrographs covering a total area of 20 mm×20 mm were used to measure the porosity. The pictures are thresholded at a gray level adjusted so as to discriminate between grains and voids. The porosity, defined as the ratio between the void and total areas, is found to be 10 ± 1 %.

Slices of widths ranging from W = 7.8 to 51.5 mm were sawed off from Fontainebleau paving stones. Toroidal samples were obtained by using hole saws to core out the slices (see Fig. 1). This was achieved by first piercing a circular hole with a 25 mm diameter saw and then a second hole of diameter 60 mm using the same spinning axis. Four slices of widths W = 7.8, 14.1, 26 and 51.5 mm were machined and placed in a hydraulic press to perform modified Brazilian fracture tests [21]. The latter is achieved by applying a uniaxial stress at two opposite points on the outside of the torus. The load is increased until two symmetrical cracks propagate from the central hole toward the outside where the compressive forces are applied.

After the failure, one of the fractured blocks is secured to a computer controlled horizontal displacement table. The samples are moved stepwise using steps of lengths ∆z (See Tab. I for a detailed list of the actual values.) At each new position, a sensing needle is displaced vertically by a computer controlled microstep motor with a resolution of ≃ 1 µm. The contact is detected by a vertical deviation of the needle. The repeatability of the measurement, verified by scanning several times the same profile, is 1 µm. After the detection of the contact, the sensor is raised by 200 µm before the sample is moved laterally again. The sequence is repeated n z times. Then, the sample is displaced in the perpendicular direction along the x-axis by a step of length ∆x and the scanning is repeated again. Finally, one obtains surface maps including n z × n x points. Figure 2 shows one of these maps that displays troughs and bumps with, for the largest, a typical amplitude of the order of one millimeter. In the following, we discuss a detailed statistical analysis of the spatial correlations of the roughness.

III. EXPERIMENTAL RESULTS

A. Characterization of the surface roughness

We first analyze profiles parallel to the z-axis, i.e. normal to the direction of crack propagation, and located far enough from the initiation so that the roughness properties are statistically stationary. The scaling properties of these profiles are characterized by their 1D height-height correlation function ∆h(δz) defined by Eq. (2).

Figure 3 represents in a log-log scale the variations of ∆h as a function of the distance δz for one of the samples. Two distinct behaviors are observed: For δz < ξ, ∆h varies as a power law of δz (straight line in a log-log representation) while, for δz > ξ, ∆h remains roughly constant ("plateau domain"). The crossover length scale ξ is, here, defined as the abscissa of the intersection between the power law fit (solid line in Fig. 3) and the plateau variation (dotted horizontal line). The local slope of the correlation function (see inset of Fig. 3) is constant except for δz values smaller than ≃ 100 µm: This corresponds roughly to the grain size d which represents therefore a lower boundary of the self affine domain. These results indicate that, at intermediate length scales, the profiles are self-affine and characterized here by a roughness exponent ζ = 0.43. The robustness of this self-affine description was verified by using other statistical methods such as the Fourier analysis of the profiles [START_REF] Feder | Fractals[END_REF]. The exponents obtained in this way are given in Tab. I and are globally independent of the method: Regarding the dependence on W , a global increasing trend is observed but the total variation is very weak, less than 0.1, which is of the same order as the interval of confidence on the value of one exponent estimated for a given width. The self-affine exponent obtained after averaging over all data is ζ = 0.46 ± 0.05. Applying the same analysis to profiles parallel to the x direction, i.e. the direction of crack propagation, leads to a self-affine exponent equal to 0.49 ± 0.05. These observations are consistent with measurements performed on other natural and artificial sandstones [11,12,13].

For δz larger than ξ, the variations of ∆h level off indicating that the surface is flat at large scales. In order to test the robustness of the value of ξ with respect to possible bias introduced by the statistical analysis, the computation was repeated on data sets including only one out of two data points. The value of the cut-off length remained the same. Finally, the same procedure was applied in different regions of the surface. Each of the surfaces was divided into three regions of equal width W/3. Two of these subsurfaces are thus on the sides of the sample while the third one covers the central region. In Fig. 3, correlation functions ∆h computed on the central (×) and on one of the lateral (•) subregions are overlaid. All curves fall on top of each other, showing that the crossover length ξ is independent of the region of the fracture surface. Therefore, ξ is a relevant length scale of the problem which reflects neither an influence of the sides of the sample nor of the sampling period of the surface maps. Before studying the scaling behavior of ξ with the sample width, we analyze now the distribution π δz of the variations of height.

B. Statistical distribution of the fluctuations of height

For each sample, a scan including at least 30, 000 measurement points has been performed. Such a large data set allows one to compute with a good accuracy the probability distributions π δz (δh) for values of δz ranging from the typical grain size up to several times the grain diameter. Fig. 4 displays such probability distributions corresponding to several δz values ranging from 300 µm to 1.3 mm, i.e. larger than the mean grain diameter d.

For self-affine profiles, one expects π δz to satisfy Eq. ( 1), valid for any scaling parameter λ. Using the particular value λ = 1/δz, this equation becomes :

π δz (δh) = 1 δz ζ π δh δz ζ (4)
Therefore, using the normalized variable u = δh/δz ζ , all the distributions δz ζ π δz should collapse onto a same master curve π(u), at least for δz values pertaining to the self-affine regime. Such a collapse is indeed observed in Figure 4: The same data as in the insert have been plotted in these normalized coordinates using the roughness exponent ζ = 0.43 measured from the scaling of the correlation function (see Fig. 3). One of the key consequence of this precise collapse is that all moments (h(z + δz) -h(z)) p 1/p z of order p > 1 of the π δz distributions must scale as δz ζ . This is a clear evidence of the lack of multiscaling of the surface investigated. In other words, the single exponent ζ is enough to describe the full statistics of the variations of height on the fracture surface along the z-axis, at least in the range of length scales d < δz < ξ.

Let us now focus on the master curve π(u) which has the characteristic bell-like shape of Gaussian distributions. This is more precisely demonstrated in the semilogarithmic plot of Fig. 5 in which a Gaussian curve (continous line) with a second moment equal to one is observed to coincide with the experimental data sets corresponding to δz ≥ 300µm, i.e. δz ≥ d. The statistical properties of the surface are therefore fully described by this Gaussian distribution and by the single roughness exponent ζ = 0.43.

We have also represented the π δz -distributions corresponding to length scales smaller than the mean grain diameter d = 270 µm, namely δz = 75 and 150 µm. They clearly do not collapse with the other curves although their second moment was shown to scale roughly as δz ζ in this range of scales (see Fig. 3). It means that, in the domain of small distances (δz < d), the π δz distributions are not Gaussian and their moments cannot be rescaled with a single exponent. In this range of δz values, it would be possible to rescale all the moments (h(z + δz) -h(z)) (not shown here): The relevance of such a multiscaling description at length scales for which the geometry of a single grain has a predominant effect is however questionable.

These results bring new insight to a recently debated question concerning crack lines resulting from the rupture of paper sheets that were found to exhibit multiscaling [START_REF] Bouchbinder | [END_REF], at least at small scales, i.e. at length scales comparable with the length of the fibers of the paper. Other observations, performed on the same material, but over a wider range of length scales, reported that, at larger scales, there was a crossover from this multiscaling behavior towards a self-affine one [19]. These results may have similarities with our experiments, even though they correspond to the 3D rupture of a very different material: In our case, the crossover length correspond to the mean grain diameter d of the sandstone specimen.

C. Dependence of the statistical properties on the sample width

In addition to the lower limit of the order of the grain diameter d discussed above, the self-affine domain has also an upper limit ξ (see Figure 3) which generally depends on the sample size. For granite, the experimental upper limit ξ is of the order of one fourth of the specimen size [24,25]. For mortar, the geometry can be considered as self-affine only up to length scales of the order of 15 % of the specimen width W [18]. A systematic study of wood samples with various values of W leads to ξ = 0.1 W [17]. Figure 6 displays the variations of the correlation function ∆h(δz) as a function of δz for samples of different widths W . All data sets display a self-affine domain followed by a plateau region corresponding to a value of ∆h(δz > ξ) increasing with W ; the upper limit ξ of the self-affine domain also clearly increases with the system size. The variation of ξ with W is plotted in Figure 7 and is well fitted by a straight line of slope 0.15 going through the origin. A value of the upper limit of the self-affine domain for profiles parallel to the direction x of crack propagation is also plotted in Figure 7 (♦). This length is, in this case, the average of values of ξ determined for four samples of different widths and the error bar reflects the variability of the values. In order to plot this result in Figure 7, these profiles have been assumed to correspond to a value of W equal to 35 mm (i.e. the length of the samples along the x direction). The corresponding data point is consistent with the linear fit corresponding in Figure 7 to profiles parallel to the z-axis. The scaling relation between ξ and the system width W established above seems apparently also applicable in the x direction. Let us now compare the surface roughness in the selfaffine domain at length scales between d and ξ with the anomalous scaling characteristics recently observed [16,17,18]. Anomalous growth implies that, after a transient regime, the roughness amplitude ∆h(δz = cst < ξ) scales as W ζ global -ζ with the system width W (see Eq.( 3)). All the profiles analyzed correspond to the stationary propagation domain in order to make the comparisons meaningful. The extent of the transient domain near the crack initiation has been found in various materials to be of the order of few millimeters [12,18]. Here, a value close to ∼ 2 mm was measured and all profiles considered in the present work are outside this region. The height-height correlation function is plotted in Fig. 8 as a function of the specimen width W for two different window sizes δz. In contrast with the predictions of Eq. ( 3) for anomalous scaling, the roughness amplitude ∆h(δz < ξ) in the self-affine domain is independent of W over a range of variations of almost 1 to 10 (the same conclusions may already be inferred from Fig. 6 but the sensitivity is better here due to the linear scale). These characteristics differ therefore completely from those expected for anomalous scaling.

IV. DISCUSSION AND CONCLUSIONS

Let us now discuss the origin of the statistical properties observed for the present Fontainebleau sandstone surfaces. In a recent study, Bonamy et al. [13] analyzed the fracture surfaces morphology in glassy materials, i.e. homogeneous silica glass and sintered glass. They suggest that, on the one hand, higher exponents ζ ∼ 0.8 reflect damage failure processes occuring in a zone surrounding the crack tips and are observable at length scales smaller than the size of the process zone; on the other hand, low roughness exponents ζ ∼ 0.4 would result from brittle fracture and be observable either in experiments where no process zone develops or at length scales larger than the size of the process zone. In this perspective, and as is discussed in more detail below, the low value ζ ≃ 0.45 reported here for Fontainebleau sandstone samples is the signature of brittle fracture; moreover, since it is observed at length scales down to the grain size, no process zone is present in these experiments.

In spite of these important differences, surfaces obtained from brittle and quasi-brittle failures share com-mon features. In both cases, a single exponent is enough to describe the scaling invariance of the roughness [19]: this confirms the relevance of the self-affine description of fracture surfaces for both types of materials.

The lack of anomalous scaling in the present fracture surfaces compared to materials such as concrete, wood or granite displaying damage failure is an other indication that no damage process occurs in our experiments. In these latter studies, confinment effects limit the development of the process zone: Its extension and internal structure are then set by the system size W when a stationary propagation regime is reached. As a result, from Eq. (3) [16], the roughness amplitude ∆h(δz = cst) varies with W as ∆h(δz = cst) ∝ W ζ global -ζ as is indeed observed experimentally [20]. In contrast, in the absence of a process zone, ∆h(δz = cst) is only set by the microstructural and mechanical properties of the material and does not depend on the system size (see for instance the model of Refs. [13,26]) as observed experimentally here.

The same type of approach accounts for the domains of observation of the different self-affine geometries. For damage fractures, the upper bound ξ of the self-affine domain where ζ ∼ 0.8 is set by the size of the process zone [10,13] or by the sample size W if it is smaller than the process zone. In the present case of a brittle fracture with no process zone, the lower boundary of the self-affine domain is set by the grain size. The upper cutoff length ξ increases linearly with the sample size W : in this case, no other characteristic length than W seems to determine ξ so that it might increase indefinitely with W . These results suggest that the two different self affine geometries might coexist on a same fracture surface: the characteristic exponent would have a value ζ ≃ 0.8 for lengths scales up to the size of the process zone and a value ζ ≃ 0.4 at larger length scales up to the plateau domain where ∆h(δz) becomes constant.

A key issue is whether the development of a process zone is purely material dependent. Recent experiments [27,28] suggest that fractured samples made of cemented grains, and therefore similar to Fontainebleau sandstone, do not always display brittle failure. The emission of acoustic waves was measured during the growth of a crack initiated from a chevron shaped notch under fatigue in a Permian sandstone sample. The localization of the acoustic events during failure allowed to demonstrate that a process zone was present and to determine its size ℓ pz ≃ 25 mm and its spatio-temporal evolution [27]. Moreover, the morphology of the resulting fracture surfaces [28] was found to be self-affine for length scales less than 10 mm(< ℓ pz ) and characterized by an exponent ζ = 0.75±0.03, similar to the usual value for materials displaying damage fractures. This observation supports the assertion of Ref. [13] and shows that the roughness exponent depends not only on the material, but also on the experimental conditions (geometry of initiation notch, loading mode...)

To conclude, the present experimental results and their comparison to previous works indicate that the value of the self-affine exponent ζ is a clear signature of the failure mode : damage fracture involving a process zone if ζ ≃ 0.8 or brittle fracture if ζ ≃ 0.4. Another characteristic signature of damage fracture is anomalous scaling marked by a variation of the roughness amplitude with the sample size (if the latter is smaller than the process zone). The range of length scales over which the fracture geometry is self-affine with a given exponent ζ may represent an important additional information. For ζ ≃ 0.8, this range should extend from the size of the grains to that of the process zone (which may be limited by the sample size); for ζ ≃ 0.4, it may extend from the size of the grains (if there is no process zone) or that of the process zone up to a length scaling as the size of the specimen.

An important test of these hypothesis will then be to force the development of a process zone in materials of the type used here by modifying the loading or the geometry of the experiments. For instance, one may use an initial chevron notch geometry as in refs. [27,28], where damage processes occur (so that ζ = 0.8) although the material is similar to the Fontainbleau sandstone used here. If a transition from brittle to damage fracture is induced in this way, this would result in a variation of the self affine exponent ζ or in the coexistence of two self-affine domains with ζ ≃ 0.8 below the size of the process zone and ζ ≃ 0.4 above it.

FIG. 1 :

 1 FIG. 1: Schematic view of the toroidal sample (outside radius 60 mm, inside radius 25 mm) and of forces applied during the modified Brazilian test.

20 FIG. 2 :

 202 FIG. 2: Topographic image (248 × 252 pixels) of a fractured sandstone surface with a crack propagation along the x-axis. Sample width: W = 26 mm (along the z-axis).

FIG. 3 :

 3 FIG. 3: Log-log representation of the height-height correlation function ∆h as a function of δz computed along the crack front direction (z-axis) for a sample of width W = 7.8 mm. Straight line: Linear regression of the data for δz < ξ with slope ζ = 0.43. Inset: Local slope of the correlation function.

FIG. 4 :

 4 FIG.4: Probability distribution of height variations over a distance δz normalized using Eq.(4) with ζ = 0.43 as obtained by the analysis of the second moment ∆h of the height distribution (See Sec. III A). δh and δz have been normalized by a same length scale length scale ℓ = 35 µm chosen so that the second moment of the normalized distributions is equal to 1. Inset : Plot in linear coordinates of π δz (δh) for 300 µm ≤ δz ≤ 1.3 mm.

4 FIG. 5 :

 45 FIG. 5: Probability density π δz (u) in a semi-logarithmic representation for 75 µm ≤ δz ≤ 1.3 mm. Here u = δh ℓ

FIG. 6 :FIG. 7 :

 67 FIG. 6: Log-log representation of the height-height correlation function ∆h(δz) computed along the z-axis for 4 specimens of different widths W . The straight line is a linear regression of slope ζ = 0.47.

FIG. 8 :

 8 FIG. 8: Height-height correlation function ∆h(δz) for fixed values of δz as a function of the specimen width W . (⋄): δz = 1 mm; (•): δz = 100 µm. Error bars: Standard deviation of ∆h(δz).

Acknowledgments

We are indebted to G. Chauvin, R. Pidoux (FAST) and J.P. Villote, V. Godard (IDES) for their assistance in the realization of the experimental set-up. We are grateful to E. Bouchaud, D. Bonamy, J.P. Bouchaud, A. Hansen and K.J. Måløy for their enlightening comments. This work is supported by the CNRS and ANDRA through the GdR FORPRO -the EHDRA (European Hot Dry Rock Association) -the PNRH programs and the Paris XI university through a PPF. L. P. is partially supported by the French Ministry of Foreign Affairs (Lavoisier Program).


Dr Julie Broggi
email: julie.broggi@univ-amu.fr

Dr Marion Rollet

Jean-Louis Clément

Dr Gabriel Canard

Prof Thierry Terme

Dr Didier Gigmes

Prof Patrice Vanelle
email: patrice.vanelle@univ-amu.fr

Dr J-L Clément

Polymerization Initiated by Organic Electron Donors

Keywords: Chain-growth polymerization, Electron transfer, Organic electron donor, Reduction

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Polymerization Initiated by Organic Electron Donors

Julie Broggi,* [a, †] Marion Rollet, [a, ‡] Jean-Louis Clément, [a, ‡] Gabriel Canard, [b] Thierry Terme, [a, †] Didier Gigmes, [a, ‡] and Patrice Vanelle* [a, †] Abstract: Polymerization reactions using organic electron donors (OED) as initiators are presented herein. The metal-free polymerization of various activated alkene and cyclic ester monomers was performed in short times, under mild conditions, using small amounts of organic reducing agents and without the need for co-initiators or activation by photochemical, electrochemical or other methods. Hence, OED-initiators allow the development of an efficient, rapid and room temperature process, responding to energy-friendly, cost-efficient and secure technical specifications. Mechanistic investigations support an electron-transfer initiation pathway leading to the reduction of the monomer.

Reducing agents with totally neutral organic structures and exceptionally negative redox potentials have gained astonishing advances and a renewed interest (Figure 1). [1] Strong organic electron donors (OEDs) are capable of single-or doubleelectron transfers to organic substrates under mild conditions, promoting bond formations through the generation of radical or anionic intermediates. Among others, Murphy and coworkers made major contributions by demonstrating the efficiency of OEDs in the reduction of challenging substrates, such as aryl halides or arenes, usually achieved by means of metallic reducers (Scheme 1). [ 1 ,2] In order to further study the potential of OEDs, we decided to tackle the reduction of hitherto uninvestigated functional groups and explore unprecedented applications. Notably, we focused on the initiation of chain-growth polymerizations by electrontransfer. Typical redox initiating systems combine a reducing agent (metal) and a radical priming source. [START_REF]Principles of Polymerization[END_REF][START_REF] Sarac | [END_REF] Examples where the monomer itself acts as one component of the redox pair are rare and generally use constraining alkali-metal systems. [5,6] In 1977, Tsuda and co-workers reported a case of free-radical polymerization via formation of an electron-transfer complex between an amine and a vinyl monomer. [ 7 ] Considering that aminoethylene derivatives are more powerful reducers than amines, [ 1 a] we assumed that they could be remarkable initiators. We report herein an original concept of chain-growth polymerization induced by spontaneous and intrinsic electron transfer between organic reducers and various monomers. As our OED-system could lead to radical or anionic priming species, mechanistic investigations were also undertaken to better understand the initiation and chain propagation pathways. The polymerization process was first studied using ethyl methacrylate (EMA) as monomer and various organic electron donors as initiators (Table 1). In the presence of the bispyridinylidene OED-2 (10 mol%), [8] EMA was totally converted into the expected poly(ethyl methacrylate) (PEtMA) in less than 3 min (Entry 5). The weight-average molar mass (Mw) and the number-average molar mass (Mn) were of 13 100 and 9900 g.mol -1 , respectively, indicating a dispersity (Đ) of 1.33. The spontaneous and immediate initiation of the polymerization reaction was slightly exothermic (35°C) and accompanied by a change of color (from dark purple to blood red), attributed to the formation of a stabilizing complex between the anionic propagating species and the oxidized form OED-2 2+ . The experimental difficulties (side reactions, early chain termination) often encountered with ionic polymerizations of alkyl methacrylates were not experienced here. [START_REF]Principles of Polymerization[END_REF] Our simple methodology did not require particular precautions such as special vessels and conditions, secure handling or meticulous monomer and solvent purifications. Neither solvent or additive, nor light-or thermal-activation was necessary. = -1.24 V], the lower reduction potentials of these electron donors could explain their lack of reactivity [E1/2 = -0.62 V and E1/2 = +0.32, +0.71 V, respectively (Figure 1)]. This hypothesis was consistent with the results observed using benzimidazolebased donor OED-1 which exhibits an intermediate redox potential [E1/2 = -0.76, -0.82 V]. [START_REF]OED-1 was in-situ generated (See SI)[END_REF] At 25°C, no reaction was observed, while at 120°C, the electron-transfer occurred but oligomers were recovered (Entries 3-4). Formation of radical species and chain transfer reactions could explain the low molecular weights observed with OED-1. Unlike OED-2 able to initiate double-electron transfers at r.t. to generate carbanions, OED-1 is known as a single-electron transfer reagent allowing the formation of radicals under thermal activation. [ 9 ,10] Table 1. Polymerization of EMA initiated by OEDs. [a] Entry Initiator (mol%) Solvent Time Conv(%) [b] Mn Mw Ð

1 (2) neat 4 h 95 28000 161000 5.76 [c] 8 13 THF 24 h 0 [d] ---14 C6D6 24 h 0 [d] ---

TDAE (10) neat 24 h 0 - - - 2 TTF (10) neat 24 h 0 - - - 3 OED-1 (10) DMF 24 h 0 - - - 4
[a] EMA (1 equiv.), initiator (1-10 mol%), neat or solvent (1M), 25°C, Ar. Mn and Mw determined by SEC analysis (Eluent THF). [b] 1 H NMR conversion. [c] multipopulated. [d] Full conversion reached upon solvent evaporation.

The performance of OED-2 was then compared at different loadings and in different solvents (Table 1). Significantly, we found that the well-defined and solid form of OED-2 allowed an accurate control of the quantity of initiator introduced, enabling to tune the length of the polymer chains. The number-average molar mass (Mn) increased (up to 28 000 g.mol -1 ) as the number of equivalents of initiators decreased (Entries 5-7), following a conventional controlled polymerization behavior. The initiator loading could be decreased to 2 mol% without change in the rate of conversion (95%) though the reaction time increased to 4h and led to a multipopulated distribution (Entry 7). While the reaction was sluggish at -30°C (Entry 10), polymerization proceeded at 0°C with a good conversion of 95% in 3h and increased Mn (Entry 9). Kinetic study of the polymerization revealed a process without inhibition period and a linear increase in number-average molar mass (Mn) rate versus conversion (Figure 2). The observation of the rapid priming and the linear growth of the polymer chain through the course of the reaction indicated a pseudo-controlled process, even if the dispersities are quite large. The rate of the chain-growth polymerization was also strongly influenced by the solvent. The reaction only took place in highly polar aprotic solvents and the conversion rate increased with the polarity of the solvent. In DMF (1M), the molecular weights were higher than the results obtained without solvent, for similar conversion rate (96%) and reaction time (30 min) (Entry 11). On the other hand, the polymerization was very slow in acetonitrile, forming only oligomers, and null in tetrahydrofuran or benzened [6][7][8][START_REF]OED-1 was in-situ generated (See SI)[END_REF][10][START_REF]See SI for details and cyclic voltammetric data[END_REF][START_REF] Zmolek | [END_REF][13][14]. Surprisingly, complete removal of the solvent, by evaporation after 24h, led to full conversion into the expected polymer in few minutes. Solvents can dramatically affect polymerization rates due to preferential solvation interactions with either the initiator or the propagating species. [START_REF]Principles of Polymerization[END_REF] Equivalent reduction potential values (≈-1.25 V) were obtained for OED-2 in DMF, THF or CH3CN and suggest that EMA should be reduced equally in these solvents. [START_REF]See SI for details and cyclic voltammetric data[END_REF] Hence, the lack of reactivity in THF is unlikely to be a problem of electron-transfer initiation but rather of propagation. In ionic polymerization, increased solvent polarity favors the formation of solvent-separated ion pairs, more reactive than the tight ion pairs that predominate in solvents of low polarity. [START_REF]Principles of Polymerization[END_REF] We thus attributed it to a solvent cage effect on the complex formed between the anionic intermediate and OED-2 2+ , preventing the chain propagation. This particularity offers an interesting concept of latent chain-propagation system that could be activated upon evaporation of the solvent.

To further support the versatility of our system, polymerization of a range of monomers was examined using 5 mol% of initiator (Table 2). Acrylates and acrylonitrile polymerized smoothly and full 1 H NMR conversions were reached in few minutes. As well, the cyclic α,β-unsaturated cyclohexenone was converted into the corresponding polymer in less than 1 minute with high Mw and Mn. The high dispersity (Đ = 3.59) of the fast polymerizing poly(butyl acrylate) could be decreased to 1.5 under more diluted conditions. As the poly(acrylamide) precipitated upon its formation in DMF, the reaction was run for 24h and the conversion (97%) determined after drying the crude. Ringopening polymerization of the D,L-lactide was also achieved although it required a longer reaction time to reach total conversion and led to a relatively low Mn.

Table 2. Scope of the polymerization initiated by OED-2. [a] Monomer Solvent Time Polymer Conv(%) [b] Mn Mw Ð >99 5000 9200 1.84 [d] O Cyclohexenone neat <1 min O n >99 15800 22900 1.45 [d] Acrylamide O NH 2 DMF (1M) 24 h [c] O NH2 97 1400 2000 1.42 [e] O C S S DMF (1M) 2 h [c] -S-C(S)-S->99 21200 35600 1.68 [d] Styrene neat 24 h 0 --- Usually prepared upon photolysis or under high pressure, [START_REF] Zmolek | [END_REF] the carbon disulfide polymer, an important class of organic conductor, was smoothly obtained with high molar masses from liquid carbon disulfide (CS2). Thus, the organic reducer OED-2 was able to initiate the polymerization of a lactone and diverse alkenes bearing activating substituents. Unfortunately, less activated alkenes, such as styrene or vinyl acetate, could not be polymerized. The reduction potentials of methyl and ethyl methacrylate were measured at -2.45 and -2.53 V vs SCE in DMF, [START_REF]See SI for details and cyclic voltammetric data[END_REF] i.e. in the array of redox potentials that the effective reducing power of OED-2 can reportedly attain. [13] In contrast, at potentials up to -3.1 V vs SCE, no reduction wave was observed for styrene and vinyl acetate by cyclic voltammetry. The inability of OED-2 to reduce these monomers certainly explains the absence of polymerization.

To evaluate the nonterminating character of the propagating anion stabilized by OED-2 2+ , we explored the synthesis of welldefined block copolymers by sequential addition of monomers. After complete MMA consumption to form PMMA, a second monomer (EtMA) was added to the stirred reaction and also polymerized quantitatively (Scheme 2). The blood red color, characteristic of propagating anions, persisted until the solution was exposed to air. 1 H NMR and SEC analysis confirmed the formation of a PMMA-b-PEtMA copolymer as represented by increased molecular weights compared to the PMMA homopolymer. Several experiments were performed to gain insight into the mechanism of polymerization initiated by OED-2. First, we focused on the initiation mode since organic electron donors can act as nucleophile, as base or as reducing agent. [14] Initiation by attack of the nucleophilic double bond of OED-2 on the monomer would involve the incorporation of the reducer as an end group of the polymer chain. Yet, no characteristic proton of OED-2 was observed on the 1 H NMR of the purified polymer PEtMA. [START_REF]See SI for details and cyclic voltammetric data[END_REF] Moreover, no polymerization in the presence of other electron-rich tetraminoethylenes, such as TDAE, was observed (Table 1, Entry 1). Although not reported for OED-2, another possible source of nucleophilic initiator comes from the existence in solution of equilibrium between aminoethylenes and the corresponding carbenes. [15] N-heterocyclic carbenes (NHC), such as the 1, 3-bis(2,4,6-trimethylphenyl)imidazol-2-ylidene (IMes), are well known organocatalysts in polymer synthesis. [16] However, co-initiators are usually required to avoid the formation of non-reactive single-addition product. [ 17 ] Under our reaction conditions, IMes-mediated polymerization of EMA gave small oligomers (200<M<3400) with a low conversion rate of 27% (Table 3,entry 9). NHC-initiated polymerization reactions are quenched by adding carbon disulfide in order to irreversibly trap the NHC within a zwitterionic NHC-CS2 adduct. [18] In our case, the polymerization of CS2 by OED-2 (Table 2) irrefutably excluded the intervention of a carbene intermediate.

Since OED-2 contains non-bonded electron pairs on its nitrogen atoms, it is in principle sensitive to protons. We postulated that if OED-2 behaved like a basic amine in catalyzing the polymerization, similar results should be obtained with the structurally related 4-(dimethylamino)pyridine (DMAP). In the presence of a nucleophilic initiator such as an alcohol, DMAP is able to catalyze polymerization processes like the ringopening polymerization of L-lactide. [ 19 ] Under our alcohol-free conditions, DMAP did not catalyze the polymerization of the EMA (Table 3, Entry 8) nor the polymerization of the D,L-lactide (not in the table). It was also observed that the polymerization process initiated by OED-2 was inhibited by the addition of ethanol (Entry 7). We could thus reasonably discount the basic properties of OED-2 and affirm that the activation of the monomer did not imply the formation of a hydrogen bond.

The documented performance of OED-2 as a powerful electron donor in organic synthesis, [ 1 , 8 ] suggests that the initiation of the polymerization was induced by electron-transfers. As previously stated, the reactivity of the monomers, varying according to their reduction potentials, and the restriction to highly polar aprotic solvents both support this hypothesis. A change of color, as observed during the reaction, is also typical in electron-transfer reactions and corresponds to the formation of the donor-acceptor charge-transfer complex. Moreover, OED-2 2+ derivatives were observed by 1 H NMR at the end of the reaction. Oxygen (O2), Copper(II) chloride (CuCl2), or paradinitrobenzene (p-DNB) are commonly used as inhibitors in the study of electron-transfer mechanism. [ 20 ] Under our reaction conditions, the polymerization was totally inhibited by CuCl2, p-DNB or O2 (Entries 1-3). The inhibition took place regardless of when addition occurred, i.e. at the outset or after 5 minutes of stirring (Reaction stopped at 65% conversion). These results provide good evidence of a chain initiation pathway via electron transfers.

Secondly, the pathway of the chain propagation was examined. The cyclic voltammogram of OED-2 exhibits a single two-electron redox wave [START_REF]See SI for details and cyclic voltammetric data[END_REF] and the formation of principally anionic intermediates promoted by OED-2 has been observed on many occasions. [ 8 ] To confirm the predominance of the anionic over radical chain propagation, the polymerization of EMA was conducted in protic solvents. Addition of ethanol or degassed water, [ 21 ] at the outset or after 3 min of stirring, inhibited the reaction by quenching the anionic chain propagation via proton transfer termination (Entries 6-7). In contrast, the use of a large excess of 2,2,6,6-tetramethyl-1-piperidinyloxy (TEMPO) as radical trap did not inhibit the polymerization process (Entries 4-5). Finally, unlike certain monomers that can polymerize either by radical or anionic pathways, D,L-lactide is known to polymerize only by anionic chain propagation. Table 3. Mechanistic considerations for the polymerization of EMA. [a] Entry Initiator (mol%) Conditions Time Conv(%) [b] Mn Mw Ð [a] EMA (1 equiv.), OED-2 (5 mol%), solvent (1M), 25°C, Ar. Mn and Mw determined by SEC analysis (Eluent THF). [b] 1 H NMR conversion.

Scheme 3. Proposed polymerization mechanism initiated by OED-2. [22] As depicted in scheme 3, these mechanism considerations support the hypothesis of a transfer of electrons from the organic donor OED-2 to the monomer leading to its reduction. The generated anionic intermediate could then initiate the polymerization via anionic chain propagation. Even though several indicators are pointing out the anionic pathway, competing processes cannot be totally excluded and could explain the modest result of our dispersities. We are currently investigating the structures of the initiating and terminating fragments incorporated into the polymer chains by mass and NMR spectroscopy.

In conclusion, we report here the first chain-growth polymerization using small amounts of organic reducing agents as initiators. The metal-free polymerization of various activated alkene and cyclic ester monomers was performed in short times and under mild conditions, providing the expected polymers with good polydispersities and without the need for co-initiators or external activation. Organic electron donors are remarkable polymerization initiators, allowing the development of an efficient, simple and room temperature process, responding to energyfriendly, cost-efficient and secure technical specifications. Their high group tolerance makes them fully compatible with the synthesis of a large range of polymers and bio-polymers of wide industrial importance. Our mechanistic considerations support both the initiation through electron transfer and the anionic chain propagation. Moreover, these results open new horizons for OEDs, never previously used in the reduction of alkene or lactone functional groups and intermolecular processes.

Figure 1 .Scheme 1 .

 11 Figure 1. Selected OEDs with their redox potentials (E1/2 (vs SCE)) and Redox equilibria of OED-2.

Figure 2 .

 2 Figure 2. Kinetic of the polymerization of EMA using OED-2 (5 mol%), sequence of 1 min intervals over 10 min.

Scheme 2 .

 2 Scheme 2. Synthesis of a PMMA-b-PEtMA copolymer.

  conditions, using organic electron donors (OED) as initiators and without the need for co-initiators or external activation methods. Hence, OED-initiators allow the development of an efficient, simple and room temperature process, responding to energy-friendly, cost-

Acknowledgements

This work was supported by the CNRS and Aix-Marseille Université. Dr Maurice Médebielle (ICBMS) is thanks for the cyclic voltammetry studies conducted on monomers. Our thanks to Dr Omar Khoumeri and Dr Hervé Clavier for valuable discussions.


Nicolas Audibert
email: nicolas.audibert@univ-paris3.fr

Cécile Fougeron
email: cecile.fougeron@univ-paris3.fr

Cédric Gendrot
email: cedric.gendrot@univ-paris3.fr

Martine Adda-Decker
email: martine.adda-decker@univ-paris3.fr

DURATION-VS. STYLE-DEPENDENT VOWEL VARIATION: A MULTIPARAMETRIC INVESTIGATION

Keywords: Style, vowel duration, French, reduction, acoustic metrics

This study examines phonetic variation in the acoustic properties of the French /i,e,a,o,u/ sub-system as a function of vowel duration and speech style in order to better understand the interplay between these two well-known factors of vowel reduction. Over 1000k vowels extracted from three large corpora of continuous French including read speech (BREF), partly scripted journalistic speech (ESTER) and casual conversations (NCCFr) were split into duration classes (short, medium, long). Six metrics based on F1 & F2 frequencies were used to capture variation in multiple dimensions. Style and duration show comparable effects in terms of reduction in the acoustic working space (overall and in F1 or F2 dimensions), centralization of vowel categories and variability within vowel categories. However, interesting differences are found in terms of overlap between vowel categories: more neutralization of vowel contrasts independent from vowel duration is found in casual conversational speech.

INTRODUCTION

Phonetic forms of segments are known to be modulated according to speech style and communication situation, and to be function of segment duration, two factors which are not independent from each other. Differences in speech style, a general term embedding very different aspects, can be conceived as differences along a continuum ranging from clear speech (where acoustic cues are made to be maximally informative for the ease of intelligibility and segmental distinctiveness) to casual interactional speech (where situational, performance, and communicative requirements may be met at the expense of clarity at the segmental level). This continuum can also be seen as a modulation between hyper-and hypo-articulated segmental outputs. Differences in phonetic forms according to speech styles have been reported along different acoustic dimensions, including variations in the acoustic properties of vowel segments with changes in both temporal and spectral properties. Casual speech, for instance, has been described to be prone to reduction in vowel duration and in vowel spectral properties [START_REF] Duez | On spontaneous French speech: Aspects of the reduction and contextual assimilation of voiced stops[END_REF][START_REF] Gendrot | Impact of duration on F1/F2 formant values of oral vowels: an automatic analysis of large broadcast news corpora in French and German[END_REF][START_REF] Johnson | Massive reduction in conversational American English Proceedings of the Workshop on Spontaneous Speech: Data and Analysis[END_REF][START_REF] Meunier | Vowel reduction in conversational speech in French: The role of lexical factors[END_REF], while clear speech has been characterized by an increased duration of vowels [START_REF] Ferguson | Talker Differences in Clear and Conversational Speech: Acoustic Characteristics of Vowels[END_REF][START_REF] Ferguson | Vowel intelligibility in clear and conversational speech for normal-hearing and hearing-impaired listeners[END_REF][START_REF] Moon | Interaction between duration, context, and speaking style in English stressed vowels[END_REF][START_REF] Picheny | Speaking clearly for the hard of hearing. II: Acoustic characteristics of clear and conversational speech[END_REF] as well as an increased size of the vowel acoustic space [START_REF] Johnson | The hyperspace effect: Phonetic targets are hyperarticulated[END_REF][START_REF] Picheny | Speaking clearly for the hard of hearing. II: Acoustic characteristics of clear and conversational speech[END_REF][START_REF] Torreira | The Nijmegen corpus of casual French[END_REF]. This link between temporal and spectral changes according to speech style is usually accounted for by the biomechanical trade-off between articulation time and articulatory precision. Undershoot is then explained by a lack of time and dynamic adjustment to reach articulatory, and thus acoustic, segmental targets [START_REF] Lindblom | Explaining phonetic variation: A sketch of the H&H theory[END_REF][START_REF] Moon | Interaction between duration, context, and speaking style in English stressed vowels[END_REF]. Different large corpora studies on vowel reduction in French [START_REF] Gendrot | Impact of duration on F1/F2 formant values of oral vowels: an automatic analysis of large broadcast news corpora in French and German[END_REF][START_REF] Meunier | Vowel reduction in conversational speech in French: The role of lexical factors[END_REF] have indeed reported a progressive acoustic vowel space reduction when moving from long to short vowel durations, within a given speech style.

However, one may wonder whether this temporal/spatial trade-off is the sole responsible of vowel reduction in casual speech. Several studies have shown interesting interactions between style and duration on acoustic reduction in French. In their study based on MFCC-based parameters, [START_REF] Rouas | Comparison of Spectral Properties of Read, Prepared and Casual Speech in French[END_REF] have shown that differences in spectral reduction could be found between read and conversational speech only for short segments. Comparison between the patterns of acoustic space reduction observed according to vowel duration in [START_REF] Ferguson | Talker Differences in Clear and Conversational Speech: Acoustic Characteristics of Vowels[END_REF] and [START_REF] Meunier | Vowel reduction in conversational speech in French: The role of lexical factors[END_REF] also suggests styledependent interaction. Looking at variation between the same duration classes (long vs. mid vs. short vowels), the first study is based on scripted journalistic speech, while the later is based on vowels produced in a casual interaction between friends. If both studies report an overall reduction of the vowel acoustic space, vowels in casual speech ( [START_REF] Meunier | Vowel reduction in conversational speech in French: The role of lexical factors[END_REF]) present a global centralization of most vowel categories, while journalistic speech ( [START_REF] Ferguson | Talker Differences in Clear and Conversational Speech: Acoustic Characteristics of Vowels[END_REF]) features most reduction along the F1 dimension, due to a massive reduction on F1 for /a/.

The present paper further explores the relationship between duration-dependent and style-dependent vowel reduction in three large corpora of continuous French, produced in distinct situations: laboratory read speech, radio broadcast news, casual face-to-face conversations. They therefore vary in many aspects: type of speakers (professional or not), interactive/communicative goals, degree and type of planning (reading vs. scripted vs. spontaneous), and linguistic content (written press, broadcasted news, debate between friends). A further original aspect of our study consists in looking at various possible dimensions of variation of a vowel system (here a subset of the French vowel system), capitalizing on the fact that reduction or expansion of a vowel space is "neither uniform nor simple" ( [START_REF] Ferguson | Talker Differences in Clear and Conversational Speech: Acoustic Characteristics of Vowels[END_REF], see also [START_REF] Ferguson | Vowel intelligibility in clear and conversational speech for normal-hearing and hearing-impaired listeners[END_REF][START_REF] Harmegnies | A study of styleinduced vowel variability: Laboratory versus spontaneous speech in Spanish[END_REF]). For this, various metrics based on F1/F2 are used to describe and quantify variation in terms of working space reduction (overall, F1-and F2-dimensions), of centralization of vowel categories, of exemplar dispersion within vowel categories, and of neutralization of contrasts between categories.

SPEECH MATERIAL AND METRICS

In order to explore style-and duration-based vowel variation, naturally/un-controlled continuous speech produced by a rich number of speakers is studied. Productions extracted from three publicly available French corpora with various content are used. Read speech comes from the French BREF corpus [START_REF] Lamel | BREF, a Large Vocabulary Spoken Corpus for French[END_REF] composed of read parts of the newspaper Le Monde produced by non-professional speakers in a laboratory setting. This reading task was considered as difficult for most of the speakers. Prepared speech corresponds to partly scripted speech samples originated from broadcasted news and political/societal debates found in various radio and TV programs within the ESTER corpus [START_REF] Galliano | Corpus description of the ESTER evaluation campaign for the rich transcription of French broadcast news[END_REF][START_REF] Galliano | The ESTER 2 evaluation campaign for the rich transcription of French broadcasts[END_REF]. Professional speakers produce the largest part of this quite careful speech material. Finally, conversational speech produced in a casual face-to-face setting comes from the NCCFr corpus [START_REF] Torreira | The Nijmegen corpus of casual French[END_REF], which includes partly free and partly guided discussions on societal topics between friends.

A sub-system of French vowels extracted from these corpora is investigated. It included exemplars of the oral vowels /i, e, ɛ, a, o, ɔ, u/. Vowels /e, ɛ/ and /o, ɔ/ were merged into single classes, hereafter labelled /e/ and /o/, to account for their frequent neutralization and regional variation in French. A total of 1 143 941 vowels were included in the analysis. Vowels were subdivided into three duration classes: short (up to 50 ms), mid (up to 80 ms), and long (up to 300 ms), following the criteria applied in [START_REF] Gendrot | Impact of duration on F1/F2 formant values of oral vowels: an automatic analysis of large broadcast news corpora in French and German[END_REF] and [START_REF] Meunier | Vowel reduction in conversational speech in French: The role of lexical factors[END_REF]. They are produced by about 50 different speakers in each corpus (74 for BREF, 61 for ESTER, 45 for NCCFr) with a balance between male and female speakers. Speakers were selected based on the amount of vowel exemplar available (at least 15 for the least frequent vowel, /u/, in each duration class).

While the ESTER and NCCFr corpora show a similar distribution of vowels into the three duration classes (52%-30%-18% of short, mid and long vowels for ESTER, 50%-29%-21% for NCCFr), short vowels are slightly under-represented in the BREF corpus (22% short, 44% mid, 34% long).

F1/F2 formant values were extracted using Praat [START_REF] Boersma | Praat: doing phonetics by computer[END_REF] on a forced automatic alignment, taking the average of values extracted at 1/3, 1/2 and 2/3 of each vowel. A filter was applied to discard aberrant formant values according to the same criteria used in [START_REF] Gendrot | Impact of duration on F1/F2 formant values of oral vowels: an automatic analysis of large broadcast news corpora in French and German[END_REF]. In the three corpora, for each speaker and each duration class, the acoustic metrics described in Figure 1 were computed from F1 and F2 frequencies. pVSA: Vocalic Space Area (pentagon). Surface of vocalic shape defined by the /i,e,a,o,u/ pentagon in F1/F2 plane. Vowel positions are computed as weighted centroids of measured tokens.

F1RR: First formant range ratio. Designed to measure the ambitus in the F1 dimension (jaw/tongue height), see [START_REF] Sapir | Formant Centralization Ratio (FCR): A proposal for a new acoustic measure of dysarthric speech[END_REF].

F2RR: Second formant range ratio.

Designed to measure the ambitus in the F2 dimension (front-back tongue dimension and/or lip rounding), see [START_REF] Sapir | Formant Centralization Ratio (FCR): A proposal for a new acoustic measure of dysarthric speech[END_REF].

DistCentroid: Vowel Distance to Centroid

A common metric computing the average distance of the 5 vowels' centroids to the overall centroid of the speaker's vowel space.

V-Dispersion: Vowel Dispersion Area Averaged intra-category vowel dispersion: area of the 95% confidence ellipses.

ContrastLoss: Contrast between vowel categories (or system distinctiveness) Misclassification rate in a linear predictive model of vowel class membership based on F1

and F2 values (see [START_REF] Harmegnies | A study of styleinduced vowel variability: Laboratory versus spontaneous speech in Spanish[END_REF]).

RESULTS

A linear mixed effect model, with corpus (BREF, ESTER, NCCFr) and duration class (short, intermediate, long) as fixed effects and speaker as a random effect, is used to predict variation on acoustic metrics. Significant main effects are found for both duration and corpus predictors (all p<10 -5 ) on all the metrics. Statistical results are summarized in Table 1 and differences are illustrated in the boxplots of

                                                    Figure 2.
In order to compare the influence of duration and style on metrics values, effects sizes (estimated by χ 2 values in the likelihood ratio tests, given in Table 1) are compared, indicating a larger effect of duration class for all metrics except ContrastLoss.

Corpus effect: All duration classes pooled, the corpus is found to be a significant predictor of acoustic variation for all the metrics studied with twoor three-ways distinctions between corpora. In particular, the NCCFr corpus is always distinct from the other two. In all duration classes the corresponding vowel space is more reduced, both overall (pVSA) and in specific dimensions (F1RR, F2RR), in the casual-speech corpus (NCCFr) than in the more formal speech corpora (ESTER and BREF not being distinguished when compared by duration classes). Vowel categories are also more centralized (DistCentroid) and vowel tokens within each vowel categories are more variable (V-Dispersion) in casual speech compared to more formal speech (ESTER and BREF).

The measure of within-system distinctiveness, assessed as the rate of token misclassification (ContrastLoss), shows a similar pattern with more systematic differences along the casualness continuum. For each duration class, vowel tokens are more accurately classified (i.e. less misclassified, less overlapping in acoustic shape across vowel categories) in read speech (BREF) than in journalistic speech (ESTER), while misclassifications are the most frequent for vowels produced in the conversational speech corpus (NCCFr). Duration-class effect: All corpora pooled, vowel duration is also a significant predictor of vowel variation on all metrics. A three-way distinction between short, mid and long vowels is found showing a larger acoustic space reduction (pVSA, F1RR, F2RR) and centralization (DistCentroid) from the long to the mid to the short vowel classes in the three corpora. Variability within vowel categories (V-Dispersion), however, significantly increases only for short vowels, as compared to the other duration classes, in the three corpora. A smaller effect of duration is found on distinctiveness between vowel categories (ContrastLoss), with a larger misclassification rate found for short vowels compared to long vowels only.

DISCUSSION

The five metrics measuring reduction and centralization of the vocalic system, as well as intraclass dispersion, show that similar reduction patterns are observed according to style and vowel duration. Vowels produced in the NCCFr casual speech corpus and vowels with shorter durations are more prone to reduction in the F1/F2 acoustic space, are less peripheral, have more unstable acoustic targets, and have more overlapping distributions reducing the accuracy of their classification into their respective vowel categories. However, interactions between style and duration factors show that variations differ in magnitude when vowels of the same duration class are compared across corpora. Effect size comparisons also show that style is a predominant predictor of variation as compared to vowel duration.

The effect of style appears mainly between two categories of corpora: the casual interactive speech corpus (NCCFr) and the more 'clear speech' corpora (BREF and ESTER). Some distinctions appear between BREF and ESTER when all vowel durations are pooled, but disappear when controlling for duration class (for pVSA and DistCentroid). To conclude that there is a difference between interactive vs. non-interactive speech situation is tempting. However, if we can ascertain that in the NCCFr corpus the interlocutors are conversing which each other, the content of the ESTER corpus would need a further comparison between journalistic monologues and broadcasted debates (even though such debates generally leave little place for interactions!).

The inclusion of a metric assessing the degree of distinctiveness between vowel categories (ContrastLoss) is particularly interesting for a multidimensional investigation of reduction. Indeed, it is the only metric showing a three-way distinction between the three corpora for each duration class. ContrastLoss is also found to be marginally affected by duration, as compared to the leading role played by this particular aspect in speaking style differences. Examination of misclassification scores between vowel categories shows that for all corpora and duration classes, overlap is largest for vowel /u/, which also has the largest intra-class dispersion values. However, inter-corpus variation is mainly linked to misclassification of /o/ and /i/ vowels.

In this study, we examined the effect of speech style and segmental duration on vowels produced in a natural environment rather than relying on instructed laboratory-induced variations as done in numerous studies (see e.g. [START_REF] Ferguson | Talker Differences in Clear and Conversational Speech: Acoustic Characteristics of Vowels[END_REF][START_REF] Huet | Contribution à la quantification du degré d'organisation des systèmes vocaliques[END_REF][START_REF] Johnson | The hyperspace effect: Phonetic targets are hyperarticulated[END_REF][START_REF] Moon | Interaction between duration, context, and speaking style in English stressed vowels[END_REF]). This large corpusbased approach makes the analysis of naturally occurring reduction phenomena feasible, but it does not allow for the control of all variables known to affect vowel variation. In the context of acoustic metrics computation, a strict control of segmental context would imply selecting only speakers with the same context distribution in each corpus and duration class. Examination of left and right segmental context distributions in each corpus and duration class indicates similar distributions between BREF and ESTER, but a different distribution in the NCCFr corpus (with an underrepresentation of alveolar contexts for vowels /a,e,i,o/ compared to the two other corpora). Although such differences might account for part of the variation measured between casual and formal speech, they are unlikely to explain all of it, particularly concerning the within-system distinctiveness (ContrastLoss) for which three-way distinctions are found between the three corpora.

CONCLUSION

Altogether, the present results outline the need for a multidimensional approach to quantify the various qualitative changes that a vocalic system can undergo when studying phonetic reduction. Style-and duration-dependent vowel variation was assessed in this study on over 1000k French vowels on dimensions linked to acoustic working space reduction, dispersion and distinctiveness. In each duration class, results indicate more reduction of the acoustic space, a larger degree of vowel centralization, and more intra-category dispersion in casual speech as compared to the other two more formal speech styles. More overlap between vowel categories is also found in casual speech and in journalistic speech compared to read speech. Duration-dependent variations modulate the vowel system in the same way for most dimensions, showing a three-way or two-way distinction between short and longer vowels, but to a smaller extent.

Figure 1 :

 1 Figure 1: Description of the six acoustic metrics used to quantify vowel space variation.

Figure 2 :

 2 Figure 2: Boxplots for the 3 corpora NCCFr, ESTER, BREF (casual/prepared/read) and 3 duration classes (short, medium, long) for the 6 acoustic metrics: pVSA, DistCentroid, F1RR, F2RR, V-Dispersion, ContrastLoss.

Table 1 :

 1 Summary of main effects and interaction for linear mixed models with each of the 6 metrics as dependent variable. Model specification: corpus and duration class as fixed effect, speaker as random effect. Stars indicate significant differences (p<10 -4 ). χ 2 values with two degrees of freedom are reported as effect size estimates.

		14
	pVSA (bark 2 )	6 8 10 12
		4
		2
	V-Dispersion (bark 2 )	6 8 10 12 14
		4
		short	mid	long	short	mid	long short	mid	long	short	mid	long	short	mid	long short	mid	long
			NCCFr		ESTER			BREF			NCCFr		ESTER		BREF
						Main effects					Interactions
	Metrics		Corpus effect			χ 2	Duration class effect	χ 2	Duration class effect per corpus	Corpus effect per duration class
	pVSA	NCCFr<*ESTER<*BREF	68	short<*mid<*long 1009
	DistCentroid NCCFr<*ESTER<*BREF F1RR NCCFr<*ESTER=BREF F2RR NCCFr<*ESTER=BREF	69 39 119	short<*mid<*long 1045 short<*mid<*long 950 short<*mid<*long 689	in the 3 corpora: short<*mid<*long	in 3 duration classes: NCCFr<*ESTER=BREF
	V-Dispersion NCCFr<*ESTER=BREF		64	short<*mid=long	332	in the 3 corpora: short<*mid=long
	ContrastLoss NCCFr>*ESTER>*BREF 1290 short>*mid>*long	25	in the 3 corpora: short>*long		in 3 duration classes: NCCFr>*ESTER>*BREF

Acknowledgments: This work was supported by the French ANR project Typaloc (ANR-12-BSH2-0003)

and by the Labex EFL program (ANR-10-LABX-0083).