Héloïse Blache

Françoise Mechin

Alain Rousseau

Etienne Fleury

Jean-Pierre Pascault

Pierre Alcouffe

Nicolas Jacquel

René Saint-Loup

Françoise Méchin

Étienne Fleury

New bio-based thermoplastic polyurethane elastomers from isosorbide and rapeseed oil derivatives

Keywords: thermoplastic polyurethanes, bio-based building blocks, isosorbide, fatty acid dimer based polyester polyol, phase segregation, water uptake

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Sustainability has become a key factor in the development of technologies in the late 20th century. As such, finding alternatives to the use of petroleum-based chemicals in the elaboration of plastics is one of the main challenges of modern-day chemistry. Easily accessible replacements usually include molecules derived from vegetable oils and sugars. In this context, isosorbide (ISO) is a green monomer from the group of 1,4:3,6-dianhydrohexitol isomers. It is obtained from starch through a three-step synthesis route [START_REF] Flèche | Isosorbide. Preparation, properties and chemistry[END_REF]. In addition to its sustainable origin, isosorbide is also a non-toxic, aliphatic, chiral and rigid molecule. These structural properties have been known to induce a high glass transition temperature, and higher hardness in several types of polycondensates such as polycarbonates, polyamides or polyesters [START_REF] Fenouillot | Polymers from renewable 1,4:3,6-dianhydrohexitols (isosorbide, isomannide and isoidide): a review[END_REF]. As such, its use as a chain extender for thermoplastic polyurethanes has a huge impact on the thermal and mechanical properties of such materials compared to more classical petroleum-sourced molecules, and increases the bio-sourced content of these polymers. The secondary nature of the hydroxyls and the H-bond formed by the endo-hydroxyl with the adjacent oxygen atom (Figure 1) decreases a lot the reactivity of isosorbide compared to classical primary di-alcohols such as 1,4-butanediol. It has however been shown that when paired with a much more reactive aromatic isocyanate function such as those displayed by phenyl isocyanate, p-tolyl-isocyanate or 4,4'-diphenylmethylene diisocyanate (MDI), both hydroxyl functions of isosorbide can be considered quasi-equireactive [START_REF] Cognet-Georjon | New polyurethanes based on diphenylmethane diisocyanate and 1,4 : 3,6-dianhydrosorbitoI, 1. Model kinetic studies and characterization of the hard segment[END_REF][START_REF] Cognet-Georjon | New polyurethanes based on 4,4'diphenylmethane diisocyanate and 1,4:3,6 dianhydrosorbitol, 2. Synthesis and properties of segmented polyurethane elastomers[END_REF][START_REF] Ionescu | New isosorbide derivatives for bio-based polyurethanes[END_REF][START_REF] Li | Reactivity and regioselectivity of renewable building blocks for the synthesis of water-dispersible polyurethane prepolymers[END_REF].

Figure 1: Spatial structure of isosorbide

The first use of isosorbide for the synthesis of polyurethanes was reported in the mid-80's [START_REF] Dirlikov | Polyurethanes based on 1[END_REF][START_REF] Thiem | Synthesis and properties of polyurethanes derived from diaminodianhydroalditols[END_REF], however the first report of a thermoplastic polyurethane containing isosorbide originates from Cognet-Georjon [START_REF] Cognet-Georjon | New polyurethanes based on 4,4'diphenylmethane diisocyanate and 1,4:3,6 dianhydrosorbitol, 2. Synthesis and properties of segmented polyurethane elastomers[END_REF] and the interest in those materials has only been renewed around the year 2010 [START_REF] Marìn | Carbohydrate-based poly(ester-urethane)s: a comparative study regarding cyclic alditols extenders and polymerization procedures[END_REF][START_REF] Tsui | Microporous biodegradable polyurethane membranes for tissue engineering[END_REF][START_REF] Park | Catalyst-free synthesis of high elongation degradable polyurethanes containing varying ratios of isosorbide and polycaprolactone: physical properties and biocompatibility[END_REF][START_REF] Kim | Synthesis of highly elastic biocompatible polyurethanes based on bio-based isosorbide and poly(tetramethylene glycol) and their properties[END_REF][START_REF] Oh | Synthesis of bio-based thermoplastic polyurethane elastomers containing isosorbide and polycarbonate diol and their biocompatible properties[END_REF]Javni et al., 2015a;[START_REF] Lee | Synthesis, characterization, and properties of polyurethanes containing 1,4:3,6-dianhydro-D-sorbitol[END_REF]. In order to obtain thermoplastic polyurethanes, it is required to synthesize linear multiblock copolymers composed of alternating hard and soft segments, the soft segments (SS) being composed of a flexible oligomer diol with a molar mass generally between 500 and 3,000 g.mol -1 , while the hard segments (HS) are composed of an alternating pattern of diisocyanates and short diols. In this prospect, poly(tetrahydrofuran) (PTMEG) [START_REF] Cognet-Georjon | New polyurethanes based on 4,4'diphenylmethane diisocyanate and 1,4:3,6 dianhydrosorbitol, 2. Synthesis and properties of segmented polyurethane elastomers[END_REF][START_REF] Kim | Synthesis of highly elastic biocompatible polyurethanes based on bio-based isosorbide and poly(tetramethylene glycol) and their properties[END_REF]Javni et al., 2015a) or polycaprolactone (PCL) diols [START_REF] Marìn | Carbohydrate-based poly(ester-urethane)s: a comparative study regarding cyclic alditols extenders and polymerization procedures[END_REF][START_REF] Tsui | Microporous biodegradable polyurethane membranes for tissue engineering[END_REF][START_REF] Park | Catalyst-free synthesis of high elongation degradable polyurethanes containing varying ratios of isosorbide and polycaprolactone: physical properties and biocompatibility[END_REF] were very often used as soft segments in combination with hard segments composed of isosorbide and either MDI [START_REF] Cognet-Georjon | New polyurethanes based on 4,4'diphenylmethane diisocyanate and 1,4:3,6 dianhydrosorbitol, 2. Synthesis and properties of segmented polyurethane elastomers[END_REF][START_REF] Marìn | Carbohydrate-based poly(ester-urethane)s: a comparative study regarding cyclic alditols extenders and polymerization procedures[END_REF]Javni et al., 2015b) or 1,6-hexamethylene diisocyanate [START_REF] Marìn | Carbohydrate-based poly(ester-urethane)s: a comparative study regarding cyclic alditols extenders and polymerization procedures[END_REF][START_REF] Tsui | Microporous biodegradable polyurethane membranes for tissue engineering[END_REF][START_REF] Park | Catalyst-free synthesis of high elongation degradable polyurethanes containing varying ratios of isosorbide and polycaprolactone: physical properties and biocompatibility[END_REF][START_REF] Kim | Synthesis of highly elastic biocompatible polyurethanes based on bio-based isosorbide and poly(tetramethylene glycol) and their properties[END_REF][START_REF] Oh | Synthesis of bio-based thermoplastic polyurethane elastomers containing isosorbide and polycarbonate diol and their biocompatible properties[END_REF], mainly for applications requiring biodegradation, biocompatibility or good mechanical properties. Isosorbide is intended to be used in such PU formulations as a new green building block and as an alternative for the very common and widely spread 1,4-butanediol (BDO), in order to bring new properties [START_REF] Marìn | Carbohydrate-based poly(ester-urethane)s: a comparative study regarding cyclic alditols extenders and polymerization procedures[END_REF]Javni et al., 2015b;[START_REF] Marìn | Carbohydrate-based polyurethanes: a comparative study of polymers made from isosorbide and 1,4-butanediol[END_REF][START_REF] Charlon | Synthesis, structure and properties of fully biobased thermoplastic polyurethanes, obtained from a diisocyanate based on modified dimer fatty acids, and different renewable diols[END_REF]. It appears that this replacement by ISO often induces smaller molar masses, but greatly increases the glass transition temperature of the hard segments [START_REF] Lee | Synthesis, characterization, and properties of polyurethanes containing 1,4:3,6-dianhydro-D-sorbitol[END_REF][START_REF] Oulame | Renewable alternating aliphatic-aromatic poly(ester-urethane)s prepared from ferulic acid and bio-based diols[END_REF]. It was moreover shown [START_REF] Marìn | Carbohydrate-based polyurethanes: a comparative study of polymers made from isosorbide and 1,4-butanediol[END_REF] that the use of ISO as a chain extender affects both crystallization degree and crystal form compared to BDO in PCL-based polyurethanes. It is expected as well that the phase segregation will be impacted as the combination of macrodiol, diisocyanate and chain extender strongly influences the microphase morphology [START_REF] Yilgör | Critical parameters in designing segmented polyurethanes and their effect on morphology and properties: a comprehensive review[END_REF]. Finally recently, isosorbide was also successfully incorporated in novel polyhydroxyurethanes, PHUs, obtained by an isocyanate-free route [START_REF] Besse | Synthesis of isosorbide based polyurethanes: An isocyanate free method[END_REF]. For this purpose the authors prepared isosorbide dicyclocarbonates and these new precursors were then reacted with various diamines. Once again the presence of isosorbide induced interestingly high Tgs by comparison with what is usually observed for PHUs, and this method could represent a promising alternative to more traditional processes once the issues related to low molar masses can be worked out.

Increasing the bio sourced content can also be achieved by using bio-sourced soft segments and diisocyanate, particularly vegetable oil-derived building blocks. Thus, fatty acid dimer-derived polyester soft segments used to synthesize TPUs were reported (Bueno-Ferrer et al., 2012a, 2012b;[START_REF] Carré | Original biobased nonisocyanate polyurethanes: solvent-and catalyst-free synthesis, thermal properties and rheological behavior[END_REF][START_REF] Carré | Solvent-and catalyst-free synthesis of fully biobased nonisocyanate polyurethanes with different macromolecular architectures[END_REF][START_REF] Carré | Synthesis and characterization of advanced biobased thermoplastic nonisocyanate polyurethanes, with controlled aromaticaliphatic architectures[END_REF]. However, most materials obtained were amorphous except when associated with MDI and BDO, resulting in poor mechanical properties. Some authors [START_REF] Charlon | Synthesis, structure and properties of fully biobased thermoplastic polyurethanes, obtained from a diisocyanate based on modified dimer fatty acids, and different renewable diols[END_REF] were able to synthesize an entirely bio-based PU containing isosorbide and a partially bio-sourced di-isocyanate obtained from fatty acid dimers, called 2-heptyl-3,4-bis(9-isocyanatononyl)-1-pentylcyclohexane (DDI). However, the mechanical properties of the obtained material were rather poor due to the lack of rigid and crystallization prone di-isocyanate. Other authors (Calvos-Correas et al., 2016) also used DDI in formulations with castor oil and isosorbide and obtained materials displaying phase segregation and a melting temperature around 60°C.

In this study, we report the synthesis of several new TPUs based on ISO/MDI hard segments associated with a rapeseed oil-issued polyester polyol, with a number average molar mass 2,200 g.mol -1 . In order to obtain insight into the effects of the hard segment content and the phase segregation quality, stoichiometry between SS and HS was varied. Model TPUs containing BDO as a chain extender were as well synthesized following the same method in order to compare the properties brought by ISO.

Materials and methods

Materials

The bio-based polyester macrodiol, Radia 7282 (FADM) used in this study was a gift from Mérylithe (France). It is produced by Oleon and made of dimeric fatty acids from hydrogenated rapeseed oil, adipic acid and hexanediol with renewable content 43% and number average molar mass (M n ) around 2,200 g.mol -1 according to the supplier. Hydroxyl and acid values as measured by the supplier were 55 and 0.5 mg.(g KOH ) -1 , respectively. The 1 H NMR characterization of the hydrogenated rapeseed oil-derived polyester polyol (FADM) can be found in the Supplementary Content, and one of the main possible chemical structures is shown in Scheme 1. 4,4'-diphenylmethylene diisocyanate (MDI) and dibutyltin dilaurate (DBTDL) were purchased from Sigma-Aldrich and used as received. Isosorbide (ISO), sold under the trade name Polysorb P, was kindly supplied by Roquette (France) and dried under nitrogen flux at 100°C. 1,4-butanediol (BDO) was purchased from Sigma-Aldrich and dried under nitrogen flux at 100°C. The hydroxyl number of both chain extenders were determined by chemical titration and found in perfect agreement with the theoretical values.

Synthesis of polyurethanes

Eight different TPUs were prepared in bulk with a slight excess of NCO (NCO/OH ratio = 1.02) and HS content ranging from 25 wt% to 50 wt%, by a two-stage synthesis process as shown in Scheme 1.

Scheme 1: Preparation of NCO-terminated prepolymers and thermoplastic polyurethanes

The synthesis was optimized in order to obtain polymers with sufficiently high molar masses to allow a subsequent correct assessment of their physical properties. In a first step, a prepolymer was obtained by reaction of the polyol with an excess of MDI for 45 min, under vacuum, in a reactor equipped with mechanical stirring and an inlet for nitrogen flushing, with an oil bath at 80°C. The chain extender was subsequently added, in liquid state with a syringe, either at room temperature in the case of BDO or at 80°C in the case of ISO. For ISO-based compositions, DBTDL catalyst was used. A solution of 0.07 wt% of DBTDL was made in the melted ISO. The reaction was then performed at 80°C under vacuum with mechanical stirring until a clouding in the reaction mixture was observed, indicating the beginning of phase segregation. At this point the viscous polymer liquid was poured from the reactor into molds to form either 2 mm-thick square sheets, or 10 mm-thick cylinders. Sheets were placed at 110°C under a 10 bar pressure for 2 hours before being moved to an oven (110°C) for 16h. Cylinders were simply placed in the oven at 110°C for 18h (higher curing temperatures cannot be considered given the thermal reversibility of the urethane bond). Samples were then removed from the oven and left to cool on ceramic benches with unopened molds. When cooled, molds were finally opened to obtain 150x150x2 mm polymer sheets

Characterization methods

Proton 1H NMR spectroscopy was used to study the structure and composition of the fatty acid dimer based macrodiol, as well as a mean to monitor synthesis. The 1H spectra were recorded at 400 MHz, in deuterated chloroform (CDCl 3 ) with a Bruker Avance III 400 (5 mm), at 298 K, acquisition time 2 s, pulse delay 4 s, and 128 scans.

Size Exclusion Chromatography was performed in a 0.01M solution of bis(trifluoromethane) sulfonimide lithium salt (LiNTf 2 ) in dimethylformamide (DMF) at 50°C as the eluent, using a Viscotek pump (VE 1122 solvent delivery and VE 5111 Injector Value Bracket), and differential refractive index (RI) detector (Viscotek VE 3580 RI detector). Three porous styrene-divinylbenzene copolymer columns from Malvern, with porosity 10 3 , 10 5 and 10 3 Å, including a precolumn, were used at a flow rate of 1.0 mL/min. Conventional calibration used polystyrene standards from Shodex. Polymer solution concentration was 3 mg.mL -1 . Injection volumes were 100 μL. The sample solutions in DMF were prepared 14 days prior to the measurement in order to obtain a good solubility. Sample solutions were filtered through 0.2 μm porous PTFE membranes prior to measurement. Concentration of free MDI in the prepolymer was obtained by using the refractometric detector, by comparison with a calibration curve of the MDI RI peak area as a function of the MDI concentration in THF.

Thermal behavior was assessed by means of a TA Instruments DSC Q20 equipment, calibrated with Indium standards. Samples of 3 to 5 mg were sealed in hermetic aluminum pans. They were subjected to a first cooling stage from room temperature to -150°C, followed by a first heating ramp from -150°C to 250°C. A second cycle of cooling and heating in the -150°C to 250°C temperature range was performed additionally. All steps were carried out at 20°C.min-1 under helium (flow rate 25 mL.min -1 ).

Dynamic mechanical analysis (DMA) was performed on an ARES G2 rheometer from TA Instruments, with temperature scans of 3°C.min -1 from -100°C to 180°C. Rectangular samples of average 2.5x11x7 mm were cut from bulk sheet samples and used to perform rectangular torsion tests. Parameters were set such that frequency was 1 Hz, and strain was 0.01 %.

Hardness tests were performed with a manual Shore D durometer from Zwick-Roell on 10 mm-thick cylindrical bulk samples, following standard DIN 53505. Shore D hardness was determined at room temperature, as the average of fifteen measurements, spaced by at least 5 mm and at a margin of 5 mm from the edges of the sample.

Electron transmission microscope samples were prepared by cryo-ultramicrotomy at -100°C. The obtained ultrathin sections of 80 nm were observed using a PHILIPS CM120 transmission electron microscope (TEM) at an acceleration voltage of 120 KV.

Atomic Force Microscopy (AFM) allowed observing phase contrast images of the smooth flat surfaces obtained by cryo-ultramicrotomy. Scans were performed in tapping mode on a Nano-observer CSInstruments equipped with a cantilever with a stiffness of 40 N/m and a resonance frequency of 300 KHz purchased from APPNANO. A setpoint of 6 V was used in order to increase the oscillation amplitude of the cantilever. This resulted in more contrast on the phase image at the expense of the topography image.

Compression set tests were made after standard NF ISO 815-1 and were performed with a simple apparatus made of two square steel plates separated by two steel spacers of dimensions 9.40x20x200 mm. Samples were obtained by piling 6 discs of diameter 29 mm, cut in the 2 mm TPU sheets. The apparatus was placed in a stove at 70°C for 24 h, at which point the samples were taken out of the stove and released from the apparatus. After 30 min, the final thickness was measured. More information about this particular measurement can be found in the Supplementary Content.

Swelling in water tests were conducted on samples of around 20x20x2 mm, which were weighed and submerged in separated bottles containing around 40 mL de-ionized water at room temperature. They were then periodically taken out, dried on paper towel and weighed in order to plot water uptake over time.

Results and discussion

Polyurethanes synthesis and physico-chemical characterization

A prepolymer with an excess of NCO functions was synthesized by reacting FAMD with an excess of MDI in bulk, producing sample named FADM-pp. The prepolymer composition is reported in Table 1. The prepolymers FADM-pp and the corresponding pure macrodiol (named FADM) were characterized by Size Exclusion Chromatography (SEC) in THF in order to obtain average molar masses and percentage of remaining free MDI. Results are reported in Table 1 as well.

Table 1: Macrodiol and prepolymer composition and molar mass

As the calibration used is the conventional method with polystyrene standards, the macrodiol's average molar mass is higher than the value of M n = 2,200 g.mol -1 given by the suppliers. Given the complexity of this polyol structure, it is difficult to precisely calculate this molar mass from the 1 H NMR spectrum; however assuming a difunctional molecule, this grossly estimated value (slightly below 2,000 g.mol -1 ) would nevertheless be closer to that deduced from the hydroxyl number (2,020 g.mol -1 ) than from these SEC results. However this technique interestingly reveals that the molar mass dispersity of the samples is impacted by the MDI functionalization: the value of Ð is increasing between FADM and FADM-pp.

Following the characterization of the prepolymers, batches of TPUs were synthesized by reaction with chain extenders isosorbide (ISO) and 1,4-butanediol (BDO). Table 2 reports the compositions and names of the studied samples along with their hard segments (HS) weight percent content and molar mass. A simple calculation shows that the biobased content of all the samples based on FADM and ISO lies around 37 to 38 wt%, i.e. a rather high value compared to traditional elastomers that are often 100% petro-based.

The prepolymer was reacted with either ISO or BDO to act as a model. The visual aspect of the samples is presented in Figure 2. Translucence increases as HS ratio decreases.

Figure 2: Visual appearance of the 2 mm thick sample sheets. First row: FMB sample series -Second row: FMI sample series. HS ratio increases from left to right

All the samples displayed a yellow hue originating from the initial FADM color. Overall, the average molar mass is at least 11 kg.mol -1 , with an average around 20 kg.mol -1 . It can be noted that the dispersity is globally high with values almost always over 3.0. Moreover, it is obvious that the curves shown in Figure 3 are shifted toward the higher retention volumes as the fraction of hard segment increases on the FMB sample series. A clear trend can be observed: an increase in fraction of hard segments is linked to a decrease in the molar mass of both series based on ISO and BDO (Table 2), which is in accordance with the results of several other studies on isosorbide [START_REF] Park | Catalyst-free synthesis of high elongation degradable polyurethanes containing varying ratios of isosorbide and polycaprolactone: physical properties and biocompatibility[END_REF][START_REF] Kim | Synthesis of highly elastic biocompatible polyurethanes based on bio-based isosorbide and poly(tetramethylene glycol) and their properties[END_REF][START_REF] Oh | Synthesis of bio-based thermoplastic polyurethane elastomers containing isosorbide and polycarbonate diol and their biocompatible properties[END_REF] and BDO (Bueno-Ferrer et al., 2012b) Moreover, in the case of the samples based on ISO, the increase in hard segment content is linked to the stagnation of the higher mass fraction values and the appearance of a small shoulder at the higher retention volumes that probably originates from pure hard segments that are not bound to any FADM soft segment and have a pronounced effect on the value of M n and dispersity for those materials. This difference is attributed to the diminished reactivity of the hydroxyl functions of isosorbide.

Influence of isosorbide on the thermomechanical behavior of the urethane elastomers

Further measurements were carried out on the TPU samples. First the Shore D hardness was plotted as a function of the HS weight ratio in Figure 4a. Overall, the samples based on ISO display a slight increase in hardness with respect to their BDO-based counterpart, in the order of magnitude of 5 to 10 Shore D units. This could be explained by a difference in phase segregation dependent on both chemical natures of the polyurethane blocks and on the thermal history [START_REF] Yilgör | Critical parameters in designing segmented polyurethanes and their effect on morphology and properties: a comprehensive review[END_REF]. From the compression set results presented in Figure 4b, the use of isosorbide chain extender seems to be favorable to a better shape retention compared to BDO. The difference between the two is however slight. Those results could be better understood by studying the thermal behavior.

Figure 4: (a) Shore D hardness and (b) Compression set results of TPU as a function of the HS weight content and sample composition

Using DSC, the glass transition temperature of the isosorbide-based series of samples FMI25, FMI35, FMI40 and FMI50 was measured around -45°C and was not influenced by the hard segment content, as can be seen in Figure 5a. All samples are thus phase-segregated. The length and concentration of the HS influences the minimum melting peak of the HS crystals: the temperature of the extremum is increasing from 197°C to 208°C for the isosorbide-based samples. It can be noted as well that no crystallization was obtained on the FMI25 sample. The presence of the HS phase can be only seen by a glass transition at 140°C.

Figure 5: Differential Scanning Calorimetry, first heating ramp measured at 20°C/min from -150°C to 250°C for the sample series based on (a) ISO; (b) BDO

This low value considering the Tg of the pure MDI/ISO HS which is around 187°C [START_REF] Cognet-Georjon | New polyurethanes based on diphenylmethane diisocyanate and 1,4 : 3,6-dianhydrosorbitoI, 1. Model kinetic studies and characterization of the hard segment[END_REF] could originate from a HS-rich phase mixed to some soft phase, or shorter HS, as the HS average length is getting lower for low HS content [START_REF] Peebles | Sequence length distribution in segmented block copolymers[END_REF][START_REF] Peebles | Hard block length distribution in segmented block copolymers[END_REF], or a mix of both situations. Lastly, an intermediate secondary transition appears quite stable as well with the hard segment content: it can be found around 55°C for all eight isosorbide-based and BDO-based samples. It is not

a b a b
present in the second heating cycle performed on the samples. In the case of the BDO-based series of samples shown in Figure 5b, the glass transition temperature is not influenced by the hard segment concentration. The melting peak extremum temperature in the case of the model BDObased samples is increasing as well with the hard segment concentration. Overall, from DSC curves, the behavior of isosorbide-based samples is remarkably similar to that of the model BDO-based samples. From Table 3, it can also be noted that all eight samples of the FADM-based series display a rather stable Tg throughout the heating cycles, which is an indicator of a kinetically favorable and stable phase segregation. ISO-based series of samples displays a slight increase in the Tg on the second heating cycle compared to BDO. Samples FMB35, FMB40 and FMB50 presented in Figure 5b, display a complex series of endothermal phenomena at high temperature (above ~130-150°C). Studies on such thermal behaviors showed a large influence of thermodynamic and structural parameters [START_REF] Koberstein | Multiple melting in segmented polyurethane block copolymers[END_REF][START_REF] Saiani | Origin of multiple melting endotherms in a high hard block content polyurethane. I. Thermodynamic investigation[END_REF][START_REF] Saiani | Origin of multiple melting endotherms in a high hard block content polyurethane. II. Structural investigation[END_REF][START_REF] Saiani | Origin of multiple melting endotherms in a high hard block content polyurethane: effect of annealing temperature[END_REF][START_REF] Fernández-D'arlas | Tailoring the morphology and melting points of segmented thermoplastic polyurethanes by self-nucleation[END_REF]. Multiple melting endotherms (MME) can be due to distinct crystal populations with different melting temperatures linked to thermal history [START_REF] Koberstein | Multiple melting in segmented polyurethane block copolymers[END_REF]Galambos, 1992, Fernández-d'Arlas et al., 2016) and DSC cooling and heating rates [START_REF] Balko | Clarifying the origin of multiple melting of segmented thermoplastic polyurethanes by fast scanning calorimetry[END_REF]. However, for high HS content, it was shown that MME were due to soft segment rich-micro-domains appearing in a hard ordered phase [START_REF] Saiani | Origin of multiple melting endotherms in a high hard block content polyurethane. I. Thermodynamic investigation[END_REF][START_REF] Saiani | Origin of multiple melting endotherms in a high hard block content polyurethane. II. Structural investigation[END_REF][START_REF] Saiani | Origin of multiple melting endotherms in a high hard block content polyurethane: effect of annealing temperature[END_REF] in model materials based on polypropylene oxide/MDI/BDO upon slow heating, and they tended to be absent upon fast heating [START_REF] Balko | Clarifying the origin of multiple melting of segmented thermoplastic polyurethanes by fast scanning calorimetry[END_REF]. The appearance of MME at temperatures over 200°C in PTMEG/MDI/BDO systems was attributed [START_REF] Li | Studies of 4,4'-diphenylmethane diisocyanate (MDI)/1,4-butanediol (BDO) based TPUs by in situ and moving-window two-dimensional correlation infrared spectroscopy: understanding of multiple DSC endotherms from intermolecular interactions and motion levels[END_REF] to the melting of ordered HS followed by the melting of microcrystalline HS domains. All samples presented here were subjected to the same thermal treatment during synthesis, and as such were all annealed for the same time and temperature. However, re-crystallization during cooling (20°C/min) and melting during the second heating scan are clearly observed only for a HS concentration over 40 wt% for ISO-based samples, while this limit is as low as 25 wt% for their BDO-based counterparts. The occurrence of this phenomenon in samples containing BDO supports the fact that MDI/BDO HS does separate and crystallize more easily from the FADM soft segment than MDI/ISO. In order to further characterize the thermal behavior, both series of FADM-based materials were tested with DMA. The obtained curves are presented in Figure 6 and Figure 7, and some numerical values are reported in Table 4 for ease of reading purposes. Overall, the DMA results are consistent with the results obtained by DSC. Again all the materials appear clearly phase-separated, with a first α relaxation well below room temperature. The values of the plateaus and Tα temperatures are very similar for ISO and BDO sample series, in spite of the lower molar masses induced by ISO. Figure 6 presents the DMA curves of the isosorbide and FADMbased samples. The temperature at the maximum of tan δ of the soft phase is very stable. However, the peak maximum decreases (from tan δ = 0.47 to 0.15) and it is as well becoming broader as the HS content increases from 25 to 50 wt%; and it is almost transforming into a plateau for samples FMI40 and FMI50. The flow limit onset of the isosorbide-based series could not be obtained because the samples broke before it was reached. Figure 7 presents the DMA of the BDO-based models. It can be noted that the α transition temperature is also very stable as the HS fraction increases. As the HS content increases, the rubber plateau is becoming less defined and the Tα transition broadens for both ISO and BDO. The flow limit onset was only obtained for the FMB50 sample as the other samples of the series broke earlier. Finally it can be noticed that the rubber moduli (or G' measured at T α + 50°C, Table 4) are generally higher for ISO-based materials than for their BDO-counterparts.

The second transition found in the DSC curves between 50°C and 70°C in both series of samples can be seen on the DMA curves of the BDO-based series. It appears as a small inflection of G' and G'' in the same range of temperature on FMB40 and FMB35 samples. This inflection is not as visible for the isosorbide-based series but may also be present. A wide array of physical phenomena could be behind this transition, including the Tg of a hard phase, H-bond interactions or annealing-induced order. This transition cannot be attributed solely to the hard domains of pure HS, as the glass transition of such HS was found for BDO/MDI to be around 100-110°C [START_REF] Cuvé | Synthesis and properties of polyurethanes based on polyolefin. II. Semi-crystalline segmented polyurethanes prepared under heterogeneous or homogeneous synthesis conditions[END_REF] and for MDI/ISO around 187°C [START_REF] Cognet-Georjon | New polyurethanes based on diphenylmethane diisocyanate and 1,4 : 3,6-dianhydrosorbitoI, 1. Model kinetic studies and characterization of the hard segment[END_REF] Moreover, partial crystallization of the soft phase occurs for segment lengths starting at 2000 g.mol-1 and above [START_REF] Klinedinst | The effect of varying soft and hard segment length on the structure-property relationships of segmented polyurethanes based on linear symmetric diisocyanate, 1,4-butanediol and PTMO soft segments[END_REF]. However, the appearance of an amorphous interphase composed of a mix of some hard segments with soft segments cannot be ruled out. It was also mentioned [START_REF] Yilgör | Critical parameters in designing segmented polyurethanes and their effect on morphology and properties: a comprehensive review[END_REF] that the breakage of hydrogen bonds between urethanes -NH and the oxygens of the ether or ester carbonyl groups from the soft segments could occur in the same temperature ranges; carbonyls are indeed found in FADM. Such H bonds were shown [START_REF] Christenson | Model MDI/butanediol polyurethanes: molecular structure, morphology, physical and mechanical properties[END_REF] to disappear and to be replaced by H bonds between urethane -NH and-CO groups when the phase segregation increased. For the moment, the exact nature of this transition is unresolved. It is however interesting to note that its temperature of appearance seems to be related to the HS nature as it is slightly higher for ISO than for BDO based samples.

More information was gathered by performing TEM on some samples (see Figure 8). The FMB35 sample is displaying a continuous soft phase (light contrast) and globular hard phase domains (dark contrast, around 2-3 μm diameter). By comparison, FMI35 is also displaying those features, but the 2 μm hard phase domains have more homogeneous size dispersion and are stacked in a hexagonal-like fashion. This stacking is even more obvious and regular for FMI40, which features smaller domains (around 1.2 μm diameter). The FMI50 is still displaying a continuous soft phase with hard phase domains. However, the hard phase domains seem to be in a coalescing dynamic, with very huge domains (up to 10 μm diameter) and broad size dispersion, suggesting that the system is getting close to phase inversion. Lastly, water uptake as a function of time was measured for samples FMI35, FMI50 and FMB35 and FMB50 (see Figure 10). First of all, one can note that the water uptake over the course of 50 days was always under 1 wt%, and that the uptake of isosorbide-based sample is on average only superior to the model BDO-based samples by 25%. By comparison of FMI50 and FMI35, one can also see that the HS content has an influence on the water uptake, as MDI/ISO HS tend to be more hygroscopic than the FADM SS. However, even if pure isosorbide is highly hygroscopic; it does not induce a large water uptake compared to model chain extenders, when in combination with a hydrophobic soft segment such as FADM. Water uptake is decreasing on FMI35 and FMB35 curves after 35 days, which might be due to a mass loss from the start of hydrolysis of the ester functions of the FADM soft segment.

Conclusion

In this work, novel thermoplastic polyurethanes based on bio-based isosorbide, ISO or 1,4butanediol, BDO, and a fatty acid dimer-based polyester derived from rapeseed oil, FADM, were investigated. Several HS contents were compared and in every instance the TPUs were successfully obtained by a two-step synthesis. In a first part, the prepolymers prepared were carefully characterized in order to validate the synthesis protocol. They were subsequently used for TPU synthesis. The lower reactivity of ISO sometimes induces lower molar masses. TPUs obtained from ISO were then systematically compared to models based on BDO, and in spite of their slightly lower molar masses were found to display a similar thermoplastic elastomer behavior in DMA, with moduli values at least equivalent, but rather higher than those of BDO-based TPUs. In agreement with the literature, ISO was moreover found to increase Tf of MDI-based HS, but also to slightly hamper phase segregation between hard and soft domains. These results were linked to a slight increase in material hardness, shape retention and in glass transition temperature of the SS domains as well as to the appearance of very characteristic phase morphologies. Finally although isosorbide is a rather hygroscopic molecule, it induces only a very slight increase in the water uptake of these fatty acidbased TPUs when compared to BDO, thanks to the rather hydrophobic nature of the macrodiol.

Although more input should be obtained from annealing and spectral analysis (such as WAXS or SAXS) studies on those samples in order to understand the nature of the thermal transitions and their relation to the phase morphology of ISO-based TPU, those elastomers might be considered as interesting and viable alternatives to entirely petro-based products.

Figure 3 :

 3 Figure 3: SEC curves of FADM based polyurethanes, in DMF 0.01 LiNTf2 -a. with ISO as the chain extender -b. with BDO as the chain extender.

Figure 6 :Figure 7 :

 67 Figure 6: Dynamic Mechanical Analysis of FADM and ISO-based samples FMI25, FMI35, FMI40 and FMI50, at a frequency of 1 Hz and 0.1% strain, with a heating ramp of 3°C/min -a. Storage (G') and loss (G'') moduli -b. Loss factor (tan δ)

Figure 8 :

 8 Figure 8: TEM images on cryo-microtomy films a. FMB35 -b. FMI35 -c. FMI40 -d. FMI50Similar morphologies were observed on AFM phase images, shown in Figure9. By comparison, the morphology of sample FMI25 is closer to FMB35 than the latter is to FMI35 in term of hard domains volume concentration. Rays on the images a. to d. are artifacts originating from the preparation method.

Figure 9 :

 9 Figure 9: AFM images on cryo-microtomy surfaces at two different magnifications (first row: 20x20μm, second row: 50x50 μm) -a. and b. FMI25 -c. and d. FMI35 -e. and f. FMI40

Figure 10 :

 10 Figure 10: Swelling in water as a function of time and TPU composition

Table 2 : Composition and molar mass of fatty acid dimers polyester (FADM)-based polyurethanes - obtained by SEC in LiNTf 2 0.01 M solution at 1 mL/min and calculated from polystyrene standards

 2

	Sample	Chain extender	%wt HS	Stoichiometry SS/MDI/Diol	M n (kg.mol -1 )	M w (kg.mol -1 )	Ð (M w /M n )
	FMI25	ISO	24	1:2:1	39.0 ±4	120 ±12	3.1
	FMI35	ISO	34	1:3:2	18.1 ±2	56.5 ±6	3.1
	FMI40	ISO	42	1:4:3	15.9 ±2	50.2 ±5	3.2
	FMI50	ISO	53	1:6:5	11.6 ±1	40.7 ±4	3.5
	FMB25	BDO	23	1:2:1	34.1 ±3	90.1 ±9	2.6
	FMB35	BDO	32	1:3:2	30.3 ±3	97.5 ±10	3.2
	FMB40	BDO	39	1:4:3	23.5 ±2	76.1 ±8	3.2
	FMB50	BDO	49	1:6:5	18.1 ±2	60.4 ±6	3.3

Table 3 : Summary of the DSC results for the sample series based on FADM

 3

					1 st Heating Cycle		2 nd Heating Cycle
	Sample	T SP	g	a) (°C) T HP	g	b) (°C)	T f (°C)	T g (°C)	T f (°C)
	FMI25		-45	140	-	-42	-
	FMI35		-43			-	197	-43	-
	FMI40		-48			-	194	-44	-
	FMI50		-45			-	208	-41	219
	FMB25		-47	128	-	-47	-
	FMB35		-47			-	169	-48	167
	FMB40		-48			-	173	-48	187
	FMB50		-49			-	209	-50	209

a) Glass transition of the soft phase -b) Glass transition of the hard phase

Table 4 : Summary of DMA results for the sample series based on FADM

 4

	Sample	T α		25°C			T α + 50°C
		(°C)	G' (MPa)	G''(MPa)	tan δ	G'(MPa)	G''(MPa)	tan δ
	FMI25	-31	5.5	0.6	0.11	5.4	0.7	0.13
	FMI35	-32	19	2.4	0.13	21	2.7	0.13
	FMI40	-34	46	4.7	0.10	51	5.8	0.11
	FMI50	-35	100	9.0	0.09	114	11	0.10
	FMB25	-33	4.9	0.5	0.10	5.3	0.6	0.11
	FMB35	-35	12	0.9	0.08	13	1.1	0.09
	FMB40	-36	25	2.0	0.08	28	2.5	0.09
	FMB50	-37	82	5.7	0.07	93	8.3	0.09

Acknowledgements

The authors would like to thank the NMR Polymer Center of the "Institut de Chimie de Lyon" (FR3023), for assistance and access to the NMR facilities. The "Centre Technologique des Microstructures, plateforme de l'Université Claude Bernard Lyon 1", CTμ is acknowledged for granting access to their equipment.


Stéphane Lallich
email: stephane.lallich@univ-lyon2.fr

Benoît Vaillant

Philippe Lenca

Parametrised measures for the evaluation of association rules interestingness

Keywords: interestingness measure, independency, indetermination

In this paper, we present a original and synthetical overview of most commonly used association rule interestingness measures. These measure usually relate the confidence of a rule to an independency reference situation. Others relate it to indetermination, or impose a minimum confidence threshold. We propose a systematic generalisation of these measures, taking into account the reference point choosen by an expert in order to apprehend the confidence of a rule. This generalisation introduces new connections between measures, leads to the enhancement of some of them, and we propose new parametrised possibilities.

Motivations

In this paper, we focus on the generalisating objective interestingness measures. We will consider association rule intersetingness measures, which aim at quantifying the quality of rules extracted from binary transactional datasets. In such datasets, each row is representing an object of the data mined, and consists of binary attributes, relating each object with properties that it may have or not. In this context, an association rule is an implication A → B, where A and B (also called itemsets) are conjunctions of attributes. We denote by n the total number of transactions in the database, n a (resp. n b , n ab , n a b) the number of transactions matching A (resp. B, A and B, A but not B), and by p a (resp. p b , p ab , p a b) the corresponding relative frequencies. Most objective measures are expressed as real valued functions of n, of the marginal frequencies p a , p b , and either p ab or p a b, i.e. as functions of n, and of the confidence (Conf) p ab /p a and marginal frequency counts of the considered rule since p a b =p a -p ab . The higher the value of the measure, the better the rule is expected to be. Considering that the more counterexamples to a rule there are, the worst it is, we restrict our set of measures to those decreasing with p a b (see table 1, references may be found in [START_REF] Lenca | [END_REF]). For a larger list of measures the reader should refer to [Guillet, 2004]. Support (Sup) and confidence (Conf) are the most famous of such measures, being the fundamentals principles of Apriori-like algorithms [Agrawal and Srikant, 1994]. These algorithms extract rules such that their Sup and Conf is above given thresholds, σ s and σ c . They are deterministic [Freitas, 2000], and produce a large number of rules which may not be interessting:

• one would expect from a rule that its Conf should be above a reference value, but the later seldom if ever equals σ c . Two main references are clearly identified as worthy from a user point of view. The first one is p b , which corresponds to the independence of the itemsets A and B. In this case the user wishes to focus on rules such that the prior knowledge of A increases the knowledge of B, i.e. rules having a confidence p b/a above the a priori frequency p b . An alternative reference sometimes used is 0.5, as in [Blanchard et al., 2005]. In our opinion, the first reference is to be taken within a targeting strategy, and the second one when considering a predictive strategy. More generally, a user may be interested in taking into account a reference value θ, 0 < σ c ≤ θ ≤ 1, and will consider only rules having a Conf greater than θ. Fukuda Gain (Fuku) is an example of such a measure, where θ = σ c . • what is more, the data mined is often subject to some sampling scheme.

In order to take that into account, a special kind of measures have been proposed. They are called "statistical" in the sense that, unlike the others (also called "descriptive" measures), their value rises with n, the relative frequencies being fixed. This consideration accounts for developping an inferential approach, and retaining only rules that are significantly well evaluated by measures, comparison to the reference choosen. Amongst the issues that arise from this approach, validating a large number of rules through the control of false rules discovery is assessed in [Lallich et al., 2004].

Various properties of interestingness measures have been investigated, in particular in [Piatetsky-Shapiro, 1991], [START_REF] Hilderman | [END_REF], [Freitas, 1999], [Lallich, 2002], [START_REF] Lallich | [END_REF], [Gras et al., 2004] and [START_REF] Lenca | [END_REF]. One of these properties deals with the reference value to which the measure compares confidence, that is to say p b (independency), 0.5 (indetermination), or some other value.

In this paper, we present a general survey of association rule interestingness measures and parametrise the reference value to which the measures will compare the confidence of a rule in order to estimate its quality. Such a consideration leads to an organised review of classical measures, the introduction of new ones, and enables us to enhance the coherence of some of them. We will first focus on descriptive measures, and then look at the statistical ones.

Descriptives measures

Reference to independency

Amongst frequently used measures added to Sup and Conf in order to capture the interestingness of a rule, are those taking the independence of the = log (Lift) IntImp (Gras et al., 1996)

P [P oi (npapb) ≥ np a b ] EII (Gras et al., 2001) ˘[(1 -h1(p ab ) 2 )(1 -h2(p ab ) 2 )] 1/4 ϕ ¯1/2 PDI (Lerman and Azé, 2003) P h N (0, 1) > ImpInd RC/B i Fuku (Fukuda et al., 1996) npa `pb/a -σc Ǵan (Ganascia, 1988) 2p b/a -1 • h1(t) = -(1 -t pa ) log 2 (1 -t pa ) -t pa log 2 ( t pa ) if t ∈ [0, pa/2[; else h1(t) = 1 • h2(t) = -(1 -t pb ) log 2 (1 -t pb ) -t pb log 2 ( t pb ) if t ∈ [0, pb/2[; else h2(t) = 1 • P oi
(OM = 1-p b p b × Conf 1-Conf ), the conviction (Conv = 1-p b 1-Conf ), whereas the information gain (IG = log Lif t) is a transformation of Lift.

Reference to indetermination

Some measures may (explicitly or not) refer to the indetermination situation, when the number of examples and counter-examples is balanced for a given n a [Blanchard et al., 2005]. This is the case of Conf and the two linear transformation: least confidence (LC = 2 × (p b/a -0.5) × pa p b ) and the Ganascia measure (Gan = 2 × (Conf -0.5)) that both additively centre Conf at 0.5. Other transformations can be listed, in particular the Sebag and Shoenauer measure (Seb = Conf 1-Conf ) and the examples and counter-examples rate (ECR = 2×(Conf-0.5)

Conf

).

Reference at θ

In order to generalise the expression of interestingness measures with respect to θ, i.e. rules such that 1 ≥ Conf (A → B) ≥ θ(A → B), we will alternatively consider the quantities Conf -θ, Conf θ and Conf-θ 1-θ . Descriptive interestingness measures are generalised as follows:

CenConf |θ = Conf -θ Gan |θ = Conf-θ 1-θ = Loe |θ = 1 1-θ CenConf |θ F uku |θ = P S |θ = np a (Conf -θ) Lif t |θ = Conf θ IG |θ = log(Lif t |θ ) Conv |θ = 1-θ 1-Conf OM |θ = Seb |θ = Conf θ × 1-θ 1-Conf = Lif t |θ × Conv |θ LC |θ = Conf -θ 1-θ × pa p b = Loe |θ × pa p b
Some measures in table 1 are particular instances of several generalised expression:

OM |θ=p b = Seb |θ=0.5 , Gan |θ=0.5 = Loe |θ=p b , F uku |θ=σc = P S |θ=p b 3 Statistical measures 3.

Intrinsics of statistic and probabilistic measures

As mentioned previously, a statistic measure takes into account the size of the sampling scheme. It is qualified of "probabilistic" when expressed as the complement of the p-value of the test under p b/a ≤ p b hypothesis. Classical approaches use the independence of itemsets A and B hypothesis as reference.

The modelling of this hypothesis realised in [Lerman et al., 1981] can be done in three different ways, with respectively 1, 2 and 3 hazard levels. We introduce model 1 which is an alternative to model 1 where p a is fixed, rather than n a (table 2). We denote by N ab the random variable generating n ab , and H and B refer respectively to the hypergeometric and binomial laws. The statistic and probabilistic index based on n a b are built as follows: by establishing the law of N ab et N ab under null hypothesis (H 0 ) following the choosen modelling, we can express a centered and reduced index under H 0 , noted N CR ab . Under standard conditions, the law of this index can be approximated to the normal distribution, leading to the definition of a probabilistic measure, defined as the surprise of observing such a high value of the index under H 0 . The choosen modelling does not affect the expectation, but does modify the variance. [Gras, 1979] and [Lerman et al., 1981] prefer the third modelling, that dissociates most rules A → B and B → A whereas the first modelling makes no dinstinction between these rules. The measure hence obtained is the implication intensity (IntImp), which is most satisfying on properties one expects a measure should have [START_REF] Lenca | [END_REF], [Gras et al., 2004].

N ab -npap b √ npapap b p b = -r √ n 1.1' N ab -npap b √ npap b p b N ab -npap b q npap b (1-pap b ) IndImp N ab -npap b √ npap b Probabilistic index P (N (0, 1) > N CR ab )
1.1 P (N (0, 1) < r) IntImp P (N (0, 1) > IndImp)

Table 2. Modelling of the various statistical and probabilistic index

Retaining the discriminating power

Although having many good properties, one of the major drawbacks of IntImp (drawback shared by the other statistic and probabilistic measures) is the loss of discriminating power. By its definition, it will evaluate rules significantly different from independency between 0.95 and 1. If n becomes important, which is particularly true in a data mining context, the slightest divergence from an independency situation becomes highly significant, thus leading to high and homogeneous values of the measure, close to 1.

In order to counter-balance this loss in discriminating power, [Lerman and Azé, 2003] introduce a contextual approach where ImpInd is centered and reduced ( CR notation) on a case database B, thus leading to the definition of the probabilistic discriminant index, P DI = P N (0, 1) > ImpInd CR/B .

[ [START_REF] Gras | [END_REF] propose an alternative solution by wheighting IntImp through the use of an inclusion index. This index is based on the entropy of experiments B/A and A/B. We denote by H(X) = p x log 2 p x + p x log 2 p x the entropy associated with an event X. In [START_REF] Blanchard | [END_REF] the most general form of the inclusion index is given as:

i(A ⊂ B) = (1 -H * (B/A) α ) 1 -H * (A/B) α 1 2α
where H * (X) = H(X) if p x > 0.5, H * (X) = 1 otherwise. The α parameter is choosen by the user. The value α = 2 is advised if one wants that this index should be tolerant to initial counter-examples, and we will use this value from now on. Hence, [START_REF] Gras | [END_REF] define the entropic intensity of implication as

EII = [IntImp • i(A ⊂ B)] 1 2
The shift from H(X) to H * (X) aims at discarding uninteresting situations, such as p b/a < 0.5 or p a/b < 0.5, and complies with a predictive strategy. In a targeting strategy, the value of p b/a should have been compared to p b , and the value of p a/b to p a .

The wheighting of the implication of intensity by the inclusion index, although effective, is problematic. The inclusion index is a measure of the distance to indetermination based on entropy, thus being null when p b/a = 0.5, and so is EII. Still, IntImp values 0.5 at independency. Hence EII is not always null at independency:

EII = 8 (1-H(A) 2 )(1-H(B) 2 )
16 if p a < 0.5 and p b > 0.5, and is null otherwise.

Revised entropic intensity of implication

We will now propose two adaptations if EII in order to cope with the above mentioned issues: REII (Revised EII) et TEII (Troncated EII). Our first proposal consists in replacing IntImp by IntImp * = max{2IntImp -1; 0} in EII. This will solve the issues pointed out, but has the inconvenient of modifying the entire spectrum of values by EII:

REII = [IntImp * • i(A ⊂ B)] 1 2
Our second proposal solely nullifies the values of EII when nanb n ≤ n ab ≤ min{ na 2 , nb 2 }, whithout modifying its values otherwise. To achieve this, we introduce an adequate version of H(X). In order to take into account both predictive and targeting strategies, a rule will have a null evaluation by the inclusion index, and hence by TEII when the following conditions are jointly met:

• p b/a > 0.5 (prediction) and p b/a > p b (targeting); i.e. p b/a > max(0.5, p b ) • p a/b > 0.5 (prediction) and p a/b > p a (targeting); i.e. p a/b > max(0.5, p a ) With these new conditions, TEII is null whenever the number of counter-examples is above min nanb n ; na 2 ; nb 2 .

T EII = [IntImp(A → B) × i t (A ⊂ B)]
1 2 , with:

• i t (A ⊂ B) = (1 -H * t (B/A) α ) 1 -H * t (A/B) α 1 2α , • H * t (B/A) = H(B/A) if p b/a > max(0.5, p b ), H * t (B/A) = 1 otherwise, • H * t (A/B) = H(A/B) if p a/b > max(0.5, p a ), H * t (A/B) = 1 otherwise.
3.4 Measures making reference to indetermination [Blanchard et al., 2005] propose IPEE, a probabilistic measure of deviation from equilibrium. The authors implicitly use modelling 1 since they consider N ab ≡ B(n a , 0.5) under indetermination hypothesis, i.e. N CR ab =

N ab -0.5na 0.5 √ na . They introduce IP EE = P B(n a , 0.5) > n ab ≈ P N (0, 1) > n ab -0.5na 0.5 √ na . Under normal approximation, IPEE equals 0.5 at indetermination. This measure corresponds to the probalistic index associated to modelling 1 (see table 2), where p b is replaced by 0.5. IPEE will hence inherit of the weak discriminating power of this kind of measures, thus leading the authors to propose that it should be modulated by the inclusion index, which is all the most coherent, since both index make reference to indetermination.

Generalised intensity of implication

Using the same approach as with descriptive measures, we can generalise statistical measures and evaluate the interestingness of a rule by comparing its Conf to θ. This is done by considering in table 2 that for each modelling under H 0 , the probability of an example, conditionally to n a , of an example is θ: N ab ≡ B(n a , θ).

The results of the hence adapted modelling 1 is immediate, and those of modelling 2 and 3 are easily obtained through the use of the probability generating functions. If X ≡ B(m, p), its generating function then is G(s) = E(s X ) = (1 -p + ps) m , and if

X ≡ P oi(λ), it is G(s) = E(s X ) = e -λ(1-s) .
• In modelling 2, n is fixed, N a ≡ B(n, p a ) and N ab /(N a = n a ) ≡ B(n a , θ).

Since 1-s) , we have:

G N ab (s) = E(s N ab ) = E(E(s N ab /N a )) = E (1 -θ + θs) Na ,
As G Na (s) = E(s Na ) = E(E(s Na /N )) = E((1-p a +p a s) N ) = e -npa(1-s) , then N a ≡ P oi(np a ). Similarly, since G N ab (s) = E(s N ab ) = E(E(s N ab /N a )) = E((1 -θ + θs) Na ) = e -nθpa(
N ab ≡ P oi(nθp a ) and N ab ≡ P oi(n(1 -θ)p a )
From these results, we propose a range of generalised measures (see table 1), and will focus on two of them. The first one, GIP E |θ , associated to modelling 1 and generalises IPEE. It corresponds to the χ 2 adjustment of B/A distribution and (θ; 1 -θ). The second one, GIntImp |θ , associated to modelling 3 generalises IntImp.

Law N ab 1.1 H(n, n a , 1 -θ) 1.1' B(n a , 1 -θ) B(n, (1 -θ)pa) P oi(np a (1 -θ)) Statistical index N CR ab 1.1 N ab -npa(1-θ) √ npapaθ(1-θ) 1.1' N ab -npa(1-θ) √ npaθ(1-θ) N ab -npa(1-θ) √ npa(1-θ)(1-pa(1-θ)) GIndImp |θ N ab -npa(1-θ) √ npa(1-θ) Probabilistic index P (N (0, 1) > N CR ab ) 1.1' GIP E |θ GIntImp |θ = P (N (0, 1) > GIndImp |θ )
Table 3. Modelling of the various generalised index

Discriminant power of the generalised measures

The generalised statistical or probabilistic measures have, as the original ones do, a weak discriminating power. In order to enhance these measures, we will consider two approaches, one being contextual, like [Lerman and Azé, 2003], the other one relying on a weighting through the use of an inclusion index, like [START_REF] Gras | [END_REF].

In the contextual approach, GIndImp |θ (or its equivalent following modelling 1 and 2) is centred and reduced on a case database B, and thus define a generalised probabilistic discriminant index, GIPD |θ , as follows.

GIP D |θ = P N (0, 1) > GIndImp

CR/B |θ

This way, we also define the generalised entropic intensity of implication, GEII |θ , as the product of GIndImp |θ and an inclusion index. In order to remain coherent, we think advisable to use a generalised inclusion index i |θ , using θ as reference value and not 0.5. This can be achieved by replacing in the original formula H(B/A) by H |θ (B/A) and H(A/B) by H |θ (A/B) where:

• H |θ (B/A) is expressed as H(B/A), in which we replace p b/a by p b/a defined as follows:

p b/a = p b/a 2θ if p b/a ≤ θ, p b/a = p b/a + 1 -2θ 2(1 -θ) otherwise • H |θ (A/B)
p a/b = p a/b 2θ if p a/b ≤ θ, p a/b = p a/b + 1 -2θ 2(1 -θ) otherwise
This first possibility generalises the inclusion index proposed in [START_REF] Gras | [END_REF], and can be found back using θ = 0.5. -or using 1 -

Conclusion

Following modelling and coherence principles, we proposed in this paper an innovating framework, from which a unified view of a large number of interestingness measures can be drawn, and which clarifies some of the links between these measures. Moreover, this framework is at the basis of the definition of new measures, namely the generalised intensity of implication, generalised probabilistic discriminant index, generalised entropic intensity of implication and the generalised probabilistic measure of deviation from equilibrium, that all compare the confidence of a rule to a reference parameter.

  npa `pb/ap b ´= npap b (Lift -

  stands for Poisson and N (0, 1) for the standard normal distribution • ImpInd CR/B corresponds to ImpInd, centred reduced (CR) for a rule set B Table1. List of measures itemsets A and B as reference. This is the case of many linear transformation of Conf: the centered confidence (CenConf), Piatetsky-Shapiro (PS), Loevinger (Loe), the implication index (ImpInd), and the lift (Lift). All these measures additively centre confidence on p b from p b/a -p b , save Lift for which the centring is multiplicative and based on p b/a p b . Other monotonically increasing transformations of confidence making reference to independency are the odd multiplier

  Na≡ B(n, p a ) 2.2 /N a = na, Nab≡ B(n a , p b ) 3.1 N ≡ P (n) 3.2/N = n, Na≡ B(n, p a ) 3.3 /N = n, Na= na , Nab≡ B(n a , p b ) Law Nab under H0 1.1 H(n, na, pb) 1.1' B(na, pb) B(n, p a pb) P oi(np a pb) Law N ab under H0 1.1 H(n, n a , p b ) 1.1' B(n a , p b ) B(n, p a p b ) P oi(np a p b )

  we have: N ab ≡ B(n, θp a ) and N a b ≡ B(n, (1 -θ)p a ) • In modelling 3, we have N ≡ P oi(n), N a /(N = n) ≡ B(n, p a ), and N a /(N = n and N a = n a ) ≡ B(n a , θ).

  pa p b × (1 -θ) as reference, since p a/b = 1 -pa p b × (1 -p b/a ). In this case, when considering independancy (i.e. θ = p b ), the reference value for H |θ (A/B) is p a . H * |θ (B/A) and H * |θ (A/B), are defined as:H * |θ (X) = H |θ (X) if p x > θ, H * |θ (X) = 1 otherwiseand i |θ as:i |θ = 1 -H * |θ (B/A) α 1 -H * |θ (A/B) α 1 2α , with α = 2.From this, we deduce GEII |θ asGEII |θ = IntImp |θ × i |θ 1 2, which is a more discriminating version of GIntImp. A similar approach leads to the definition of a generalised probabilistic measure of deviation, GEIPE |θ , asGEIP E |θ = GIP E |θ × i |θ 1 2. Their behaviour, compared to their original counterparts, is represented figure1. They were obtained using 3 different values for θ, θ = 0.1 (thus targeting at independency), θ = 0.2 (targeting for situations such that B happens twice more often when A is true) and θ = 0.5 (prediction).

Fig. 1 .

 1 Fig. 1. Behaviour of the measures, in function of p b/a for n = 1000, pa = 0.05 and p b = 0.10