Rachel Calvet
email: calvet@mines-albi.fr

Sylvie Del Confetto

Henri Balard

Eric Brendle

Jean-Baptiste Donnet

Sylvie Del Confetto

Eric Brendlé

Study of the interaction polybutadiene/fillers using inverse gas chromatography

Keywords:

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

Introduction

The reinforcing properties of carbon blacks are closely related to the interaction polymer/carbon black surface but also to the fractal nature of the particulate filler aggregates [START_REF] Kraus | Reinforcement of Elastomers by Carbon Black[END_REF][START_REF] Donnet | Carbon Black Science and Technology[END_REF] and filler networking [START_REF] Gerspacher | [END_REF]4]. So, a special attention was also given to gain insight into the relationship between disordered filler structures with different length scales and specific microscopic effects contributing to carbon black reinforcement [5,[START_REF] Klüppel | Modeling of stress softening and filler induced hysteresis of elastomer materials[END_REF]. An improved understanding of the particulate mechanisms of reinforcement is relevant in material science [START_REF] Klüppel | [END_REF].

Beside these well-investigated morphological effects, the surface activity of carbon blacks plays a key role in reinforcement of elastomers by controlling the polymer-filler phase bonding and the filler-filler interaction [START_REF] Donnet | Carbon Black Science and Technology[END_REF]. Despite this importance of the surface activity, reinforcing fillers are characterised mainly by their specific surface area related to the mean particle size and by their DBP number (adsorption of dibutyl phthalate) which is bound to the aggregate structure. Hence the surface activity of CBs cannot yet be assessed by any standard molecular parameter [START_REF] Donnet | Carbon Black Science and Technology[END_REF].

Many efforts were made to judge the level of the surface activity of carbon blacks. The first one was made, in particular, on measurements done by IGC in infinite dilution conditions [8][9][10][11][12]. Lansiger and Donnet [8,12] have revealed a relationship between the dispersive component of surface energy assessed by IGC-ID and the reinforcing properties of some carbon blacks, the higher the dispersive component, the higher the reinforcing effect. But IGC-ID analysis mainly reflects the sites having the highest energies [START_REF] Balard | Determination of acid-base properties of solids surfaces using inverse gas chromatography: advantages and limitations[END_REF], and so, delivers no quantitative information concerning the number of these latter adsorption sites.

To gain a further understanding, many investigators have employed adsorption isotherm measurement using nitrogen [START_REF] Lapra | [END_REF] and ethene [15] or IGC at finite concentration conditions [10,16,17]. They have then assessed the carbon black surface heterogeneity in terms of distribution functions of the adsorption energies. However, due to the variability of the type and origin of the carbon black samples, the nature of the adsorbate, but also of the computing method, it is difficult to extract a clear conclusion from this last data. Nevertheless, a trend is observed, the higher the specific surface area, the higher the number of sites with a high level of energy, as noted by Schröder et al. who observed that the N115 sample presents more energetic sites than samples exhibiting a lower specific surface area like the N550. Hence, adsorption analytical methods suggest that carbon blacks presenting a high energetic surface are the most efficient as regards the reinforcing power.

Surface heterogeneity can be evaluated not only in terms of surface activity, but is also dependent on their surface geometry as shown by Vilgis and Heinrich [18] and this plays an important role in the interaction of a macromolecule with a solid surface.

Upon adsorption on a surface, a polymer chain loses conformational entropy and this loss will be more important on a smooth surface than on a rough one because on the former, the polymer chain has to stretch out and accordingly its entropy loss is high, whereas on the latter the polymer chain can curve into the rough surface leading to lower entropy loss. Numerous investigations of the surface roughness of carbon black have been carried out with different techniques like gas adsorption [19][20][21][22][23][24][25], small angle X-ray and neutron scattering (SAXS and SANS) [26][27][28] or atomic force or scanning tunnel microscopy (AFM and STM) [29,30]. Based on these last techniques, Donnet and Custodero have proposed that the carbon black surface is covered by "fish scale" like structures corresponding to surface graphitic domains. The results are contradictory and range from nearly smooth carbon black particle surfaces to very rough surfaces, even with the same techniques.

It is worth pointing out that all the adsorption measurements were performed using small molecular probes [8][9][10][11][12], which are far from an actual macromolecular chain as regards both the functionality and the conformation.

Recently, it was demonstrated for talc and silica that coupling of IGC-ID analysis and progressive polymer impregnation permits a titration of the sites to which the polymer chains have access and so quantifies the affinity of a polymer to the solid surface [31,[START_REF] Hamdi | [END_REF][36][37]. Using this new approach, the surface is probed with a macromolecule and small molecules (IGC probes: alkanes and branched or cyclic alkanes), which delivers information on the interaction level between the polymer and the surface through the dispersive component of the surface energy and the surface morphology (index of nanomorphology).

In the present work, the same approach was applied to examine the interaction between a polybutadiene (PBu) and a set of carbon black samples. The variations of the dispersive component of the surface energy and of the nanomorphology index were monitored versus the degree of impregnation.

IGC-ID theory

Inverse gas chromatography (IGC) is a useful method for studying the surface properties of finely divided solids [START_REF] Balard | Determination of acid-base properties of solids surfaces using inverse gas chromatography: advantages and limitations[END_REF][START_REF] Papirer | Surface energy of silicas[END_REF]. It is based on the injection of molecules of a known nature, called probes, into a column packed with a solid to be tested. The present study has been carried out at infinite dilution (IGC-ID).

The net retention volume of the probe (V n ) is recorded which reflects the level of interaction between the solid surface and the isolated probe [START_REF] Conder | Physicochemical Measurements by Chromatography[END_REF]. The free energy of adsorption of the probe is directly related to the retention volume and to the net retention time by the following equation.

G a 0 = -RT Ln(V n ) + C = -RT Ln J × D m × T c T a × Á(T c ) Á(T a ) × t r + C ( 1
)
where J is the James Martin coefficient, which takes into account the compressibility of the carrier gas, D m the measured flow rate with an electronic flow metre, T c and T a the column and the room temperatures, Á(T) the gas carrier viscosity, t r the net retention time of the probe and C a constant depending on a reference state. Fig. 1 describes the principle of the determination of main parameters, G CH 2 a and G M a that can be acquired through IGC-ID technique and relates respectively to the non-specific interaction and the surface morphology of the solid surface.

The n-alkane line is obtained by plotting the free adsorption energy versus the number of carbon atoms of the injected alkane probes, its slope corresponds to the increment of free energy per methylene groups ( G CH 2 a ) as depicted in Fig. 1. From this parameter, the dispersive component of the surface energy was determined using the Dorris and Gray approach [40]. The dispersive component of the surface energy ( d s ) can be readily calculated according to the following equation.

d s = ( G CH 2 a ) 2 4N 2 a 2 CH 2 CH 2 (2)
where N is Avogadro's number, a CH 2 is the area covered by one methylene group (0.06 nm 2 ), and CH 2 is the surface energy of pure methylene group surface, i.e., polyethylene, CH 2 = 35.6 + 0.058(293 -T ), in mJ/m 2 (T the temperature in K). The parameter d s is directly related to the surface polarisability of the solid surface [START_REF] Balard | Determination of acid-base properties of solids surfaces using inverse gas chromatography: advantages and limitations[END_REF]. Of course, probes other than the linear apolar n-alkanes may be used to assess the interaction potential of a solid surface. Among these probes, branched or cyclic isomers of the n-alkanes that can only exchange non-specific interaction like linear ones, are of a main interest for testing the surface morphology at a molecular scale. In particular, Brendle and Papirer [START_REF] Brendlé | [END_REF] have shown that such isomers exhibit much lower retention times than the corresponding linear alkanes for solids having a lamellar structure such as crystalline silicas [START_REF] Papirer | Surface energy of silicas[END_REF] or talc [START_REF] Balard | [END_REF]. This difference of behaviour between linear and branched cyclic alkane probes was attributed to the fact that linear probes have access to the slot like adsorption sites present on the lateral surfaces of these lamellar solids whereas branched or cyclic alkanes are excluded for steric effect and so exhibit a much lower energy of interaction. These slot like sites are the result of a lack of one crystalline plane leading to a slot having a dimension of about twice the interplanar distance that means a nanometric or subnanometric dimension.

Of course, the size exclusion effect is closely related to both the molecular shape of the probe and the surface morphology. In order to take into account the shape and structure influence, brendle and Papirer [START_REF] Brendlé | [END_REF] have put forward some topology indexes X t . The size exclusion effect leads to a decrease in the free energy of adsorption when comparing it with that of a virtual linear alkane having the same topological index (X t ) as depicted in Fig. 1 and leads to a decrement of free energy -G M a . Balard et al. [START_REF] Balard | Determination of acid-base properties of solids surfaces using inverse gas chromatography: advantages and limitations[END_REF] suggested an index of surface nanomorphology, according to the following equation:

I M (X t ) = exp -G M a RT (3)
when I M (X t ) is equal to 1, that means -G M a is equal or near zero, the linear and branched or cyclic probes have access to the same surface, the solid surface can be considered flat at the molecular level, for example for some fumed silicas [START_REF] Papirer | Surface energy of silicas[END_REF]. An I M (X t ) lower than 1 is the result of a size exclusion phenomena between linear and branched or cyclic probes due to a rough surface, it occurs with solids like talc. When the I M overcomes 1, a thick layer of polymer is deposited on the solid surface and an absorption of the probe into this layer occurs [31].

Experimental part

Carbon blacks

For the present work, five carbon blacks from the Continental Carbon Company (Texas, USA), were selected. Their main characteristics are reported in Table 1. The specific surface area was measured with nitrogen gas using an ASAP 2000 instrument from Micromeritics according to the ASTM standard procedure (ASTM D 3037) [START_REF] Astm D | Annual Book of ASTM Standards[END_REF]. The CTAB (cetyltrimethylammonium bromide) surface area, the iodide and the DBP (dibutyl phthalate) adsorption values were determined according to the ASTM standard procedures (ASTM D 3765 and ASTM D 1510) [START_REF] Astm D | Annual Book of ASTM Standards[END_REF][START_REF] Astm D | Annual Book of ASTM Standards[END_REF].

Impregnation protocol

The impregnation of carbon black were mainly done using a polybutadiene from Sigma Aldrich (Mw = 40,000 g/mol).

A polybutadiene solution having a known concentration in dichloromethane was prepared. A given amount of carbon black was dispersed in 25 ml of the same solvent. By adding increasing volumes of the PBu solution to these carbon black dispersions, a large scale of impregnation ratios ( expressed in mg/g) could be examined. After the complete evaporation of the solvent, IGC columns were filled with the PBu impregnated carbon black and conditioned at 120 • C. This impregnation protocol is well described in previous studies [31,[START_REF] Hamdi | [END_REF][36][37].

From the values, the coverage ratios (n MU ) of the surface in terms of monomer units per square nanometer (MU/nm 2 ) were computed according to the following equation:

n MU = N 1000 × M MU S × 10 18 = 602.3 M MU S ( 4
)
where N is the Avogadro number, M MU the Bu monomer unit molecular weight (54 g/mol) and S the specific surface area of the studied solid measured with nitrogen, is given in mg/g or ‰.

The monomer unit molecular area (a MU in nm 2 ) was assessed according to Eq. ( 5) from the molecular volume of the monomer unit, computed from the polymer bulk density (d in g/cm 3 ) and the monomer unit molecular weight (M MU in g/mol), considering that one monomer unit is a cube:

a MU = k M MU d × 602.3 2/3 (5)
where d is equal to 0.95 g/cm 3 for PBu.

Using this method, the molecular area (a MU ) is generally underestimated of about 20%, so the value of k was fixed to 1.2.

Knowing the MU surface area (a MU ) in nm 2 , the theoretical number of monomer unit per square nanometer of carbon black necessary to form a "so called" monomer unit monolayer could be computed, according to the following equation:

N MU = 1 a MU (6)
According to Eqs. ( 5) and ( 6), the values of both MU molecular area (a MU ) and the MU number per nm 2 for the monolayer formation (N MU ) are equal respectively to 0.25 nm 2 and 4.0 for PBu. This approach delivers a rough approximation of the area occupied by a MU on the solid surface, because the MUs are linked in the macromolecule, they are not independent entities like the nitrogen molecule in the case of BET specific area measurement.

Experimental IGC conditions

The agglomerated impregnated carbon black powder was gently crushed and sieved. Particles having diameter comprised between 200 and 400 m were retained. The powder was poured into stainless steel columns, 10 cm long and 2 mm internal diameter, which were gently taped in order to obtain a regular filling. Hence, chromatographic beds were prepared, having a convenient permeability corresponding to a pressure drop lower than 50 kPa with a flow rate of the carrier gas of about 30 ml/min.

Two Agilent 6890 gas chromatographs, each fitted with two flame ionisation detectors (FIDs), were used with helium carrier gas. The injector and detector were heated to 180 • C. All the columns were conditioned overnight at 120 • C and the analyses made at 150 • C. Net retention times were calculated by subtracting the retention time of a non-retained molecule, methane, from the retention times of the probes. Selected IGC probes -n-alkanes and cycloalkane -are purum products from Aldrich-Sigma. Firstly, three carbon black grades, N326, N330 and N347, were examined. These samples have the same mean particle size or specific surface area but differ in their structure as measured by DBP adsorption.

The variation of the dispersive component of the surface energy ( d s ) of these PBu impregnated carbon blacks with the MU coverage ratio expressed in number of MU per surface unit (n UM ), is depicted in Fig. 2.

Three main conclusions have to be given on the variation of the dispersive component with the MU coverage ratio:

1) The d s values corresponding to the initial unmodified samples differ only slightly between the 3 different carbon blacks, around 160 mJ/m 2 .

2) For the impregnated samples, the variation of d s follows the same trend for the 3 studied samples and decreases monotonously until the MU deposited covers the whole of the carbon black surface, when n UM (equal to 5 MU/nm 2 ) exceeds 20% of the theoretical value (4 MU/nm 2 ). Hence it has been showed that the PBu adsorbs at first on the more energetic sites.

3) Beyond this threshold, d s becomes constant and equal to about 30 mJ/m 2 , a value characteristic of an apolar polymer surface formed by the PBu layer.

In other words, the deposited PBu shields progressively the sites of adsorption, in which alkane probes could be inserted. This effect could be also quantified using the index of nanomorphology.

Variation of the index of nanomorphology with the impregnation ratio

The variation of the index of nanomorphology (I M ) measured with cyclooctane probe, of PBu impregnated carbon blacks of N326, N330 and N347 grades, with the MU coverage ratio expressed in number of MU per surface unit (n UM ), is depicted in Fig. 3.

Here, four main conclusions have to be done on the variation of the index of nanomorphology with the MU coverage ratio:

1) The I M values, corresponding to the initial unmodified samples are equal to 0.4, proving the presence of numerous insertion sites at the edge of the graphitic structures. 2) For the impregnated samples, a "S" shaped variation is observed for the 3 samples studied with the MU coverage ratio, proving again that these samples exhibit nearly identical surface structures. 3) Two thresholds are registered, around 1.5 MU/nm 2 and 5.5 MU/nm 2 . Obviously the second threshold is related to the formation of the continuous PBu layer. The first threshold shows that about 27% of the total surface corresponds to similar strong interacting sites presenting the same geometry, responsible of the size exclusion effect. Below 1.5 MU/nm 2 , the I M remains very low indicating that the linear alkanes have access to sites where the cyclooctane is excluded due to steric encumbrance. 4) The index of morphology I M overcomes a value of 1 MU/nm 2 slightly before the number necessary to form a continuous PBu layer and reaches a value close to 1.2 for 7 MU/nm 2 , showing preferential absorption of the cyclic probe, in comparison with linear ones. This preferential absorption starts to take place in the PBu film present at the surface.

Though the study of the variation of the index of morphology with the MU coverage ratio corroborates and completes the assumption deduced from the variation of the d s values, it also shows that the PBu at first starts to adsorb and shield the sites having the highest energy, and in which linear alkane can be inserted, and thereafter covers progressively the less energetic sites until all of the carbon black surface will be covered by the MU.

We will now examine how the grade of the carbon black studied will influence the variations of these two parameters, d s and I M , with the increase of the MU coverage ratio.

IGC-ID analysis of PBu impregnated N115, N330 and N539 carbon blacks

Variation of the dispersive component of the surface energy with the impregnation ratio

In the second part of the study, the carbon blacks examined, N115, N330 and N539, differ in mean particles size and specific surface area ranging for example with nitrogen from to 41 to 140 m 2 /g. The variation of the dispersive component of the surface energy ( d s ) of these PBu impregnated carbon blacks, with the MU coverage ratio, expressed in number of MU per surface unit (n UM ), is depicted in Fig. 4.

It immediately appears that N539 sample behaves like the N330 one as regards to the variation of d s with the MU coverage ratio. On the contrary, the initial N115 carbon black exhibits a much higher d s value than the others studied samples. Moreover the d s of the impregnated sample then decreases rapidly. Thereafter, it behaves like the other grades. Beyond 5 MU/nm 2 , the dispersive surface energy becomes equal to the PBu one. We will now consider the variation of the index of nanomorphology with the MU coverage ratio.

Variation of the index of nanomorphology with the impregnation ratio of N115, N330 and N539 samples

The variation of the index of nanomorphology (I M ) measured with a cyclooctane probe of PBu impregnated carbon blacks of N115, N330 and N539 grades, with the MU coverage ratio, expressed in number of MU per surface unit (n UM ), is depicted in Fig. 5.

Considering the initial samples, it is observed that N539 and N330 samples show very close I M values, whereas N115 sample exhibits a lower value. PBu impregnation induces a similar "S" shaped variation for the three samples, but they do not follow the same trend. We have observed that the N330 and N539 samples behaves almost identically so long the MU coverage is lower 2 MU/nm 2 , thereafter they differ completely, the N539 showing a slower variation whereas the N330 now follows the same trend as the N115 sample. For the latter, the I M value remains quite constant up to a coverage ratio of 1 MU/nm 2 and then increases quickly joining the curve of the N330 sample.

Hence both trends of the dispersive component of the surface energy and of the index of nanomorphology testify that the surface structure of these three samples differs by their capacity of interaction and their surface geometry. An interpretation has to be found considering the surface model structures proposed for the carbon black surface and will be discussed now.

Discussion

We will first consider two models which were previously proposed in the literature and depicted in Fig. 6A andB.

The Donnet and Custodero model (Fig. 6A), based on STM observation, proposed a fish-like scale structure former by the graphitic domains, but gives no assumption about the structures existing at the edge of these "fish scales".

On the other hand, the Schuster et al. model (Fig. 6B), mainly from the distribution functions of the adsorption energies of ethene, suggested the existence of four types of adsorption sites bound to, (1) graphitic planes, (2) amorphous carbon, (3) crystallite edges and (4) slit shaped cavities.

The three first types of adsorption sites cannot be at the origin of any important exclusion size phenomena of the cyclooctane probe. Only the slit shaped sites could be at the origin of any size exclusion effect, but such default could not be observed by the STM.

Moreover, neither the Donnet and Custodero's nor the Schuster's model allows us to predict the correlation existing between the variation of both d s and I M parameters with the PBu impregnation ratio as it is observed, for example, for example in the N115 sample, as shown in Fig. 7.

In particular, the previous models do not allow us to explain the fact that there is a decrease in the value of d s whereas the I M value remains weak and quite constant, increasing only slowly from 0.3 to 0.4 values.

We will come back to previous studies led on crystalline silicas [START_REF] Papirer | Surface energy of silicas[END_REF], talcs [31,36,37] and graphites [START_REF] Balard | [END_REF]. These solids exhibit high d s and low I M values. In order to explain such size exclusion effect of cyclic alkanes, it was suggested that the linear alkanes might enter in the slot shaped sites present on the crystalline lateral surfaces due to gaps left by missing crystal plane, as drawn in Fig. 8A, on the left.

Depositing PEG on the talc surface, Comard and al. [31,37] had observed a very similar variation of both d s and I M parameters with the PEG impregnation ratio than that shown for the N115 sample for the PBU progressive impregnation.

Such variation could only be explained when one admits that the flexible and polar PEG chain will be absorbed at first on the lateral surfaces of the talc crystallite, shielding progressively the slot shaped site, hindering the insertion of linear alkane, inducing a rapid decrease of the d s value, which starts immediately at the first stage of impregnation process. Correspondingly, as long as the slot shaped insertion sites remain accessible, the I M parameter remains almost equal to the initial value measured on the unmodified talc. Once, these sites are completely hidden, the I M parameter start to increase until the entire surface is covered by a continuous polymer layer.

The same process will allow us to explain the variation of the carbon black surface under the controlled PBu impregnation due to the progressive shielding of the slot like sites present on the lateral surfaces of the graphitic structures. Now a question is raised, how to explain that the N115 sample behaves very differently from the N330 and N539. The answer could be found when considering the variations in the surface roughness assessed by comparing surface areas obtained with CTAB (S CTAB ), nitrogen (S N 2 ) or the iodide adsorption number N I 2 . It has been observed that under a specific surface area [START_REF] Kühner | Carbon Black Science and Technology[END_REF], the S CTAB and N I 2 are very close, but over this threshold S CTAB becomes clearly lower than N I 2 , indicating an increase of the surface roughness.

Moreover, when considering the variation in the S CTAB /S N 2 ratio, versus the mean diameter of the primary particle assessed by TEM, one can see this ratio increases quickly for carbon blacks that have a particle diameter lower than 30 nm. Over this critical value, it remains quite constant and overcomes slightly 1 as depicted in Fig. 9.

Hence, the differences in behaviour observed between the N115 CB and the other samples have to be attributed to their differences in surface roughness. This surface roughness originates from the temperature of the formation of the CB. The N115 CB is formed at a higher temperature than the N330 and, of course the N539 [START_REF] Kühner | Carbon Black Science and Technology[END_REF][START_REF] Hess | Physical properties[END_REF]. At this temperature, the graphitic domains present on the surface would be more organised leading to an increase of the number of slot shaped sites on the edges of the graphitic scales and so, leading to high energetic adsorption sites of the n-alkanes and correspondingly a more intensive exclusion size phenomenon of cyclooctane as depicted in Fig. 8B above. At a lower temperature of formation, the organisation degree of the graphitic crystallites becomes much lower, and both insertion and exclusion phenomena become less important.

This interpretation is also supported by the fact that the variations of the IGC-ID parameters depend on the nature of both the polymer and the solid. Influence of the nature of the adsorbed polymer was clearly demonstrated by Comard et al. [36] in the case of talc as adsorbant.

Conclusion

In conclusion, monitoring the variation of the dispersive component of the surface energy ( d s ) and the index of nanomorphology (I M ), is an excellent method to follow the variations in the surface properties of CB impregnated with increasing ratios of PBu.

It has been observed that there is a clear correlation between the decrease in the d s and the increase in the I M . This variation is due to the fact that the flexible, and not very polar, PBu chain will adsorb at first on the lateral surfaces of the graphitic domains present on the surface of the CB according to the models proposed in the literature. The PBu macromolecules will then shield progressively the slot shaped site, hindering the insertion of linear alkane, resulting in a rapid decrease of the d s value.

As long as the slot shaped insertion sites remain accessible, the I M parameter remains quite equal to the initial value measured on the unmodified CB. Once these sites are completely covered, the I M parameter starts to increase to one until the entire surface is covered by a continuous polymer layer corresponding to the formation of a so called MU monolayer.

However, if the N539, the N326, the N330 and the N347 CB behave similarly, the variations of the surface properties of the N115 sample are different. The literature data suggests that the N115 presents a rougher surface having a higher graphitic character due to a higher temperature of synthesis than the other studied samples.

Hence, the coupling of the IGC-ID with the controlled modification by PBu impregnation permits us:

-To get information on the surface morphology of a CB and especially on its surface heterogeneity. -To estimate quantitatively the number of sites having the highest interaction energy; sites that could not be quantified by either gas adsorption because surface roughness generates a negligible microporous volume, or by AFM that allows only to observe the sole structure of the graphitic domains. -To highlight the specific interaction that the PBu exchanges with this type of site, which may be responsible of the reinforcing capacity of the CBs towards elastomers. -To prove clearly the differences in surface properties between CBs of different grades.

Fig. 1 .

 1 Fig. 1. Variation of the free energies of adsorption of the probes with their topological indexes, Xt [39]: relation with some experimental chromatograms and principle of the determination of main IGC-ID parameters, G CH 2 a and G M a .

Fig. 2 .

 2 Fig. 2. Variation of the dispersive component ( d s ) of PBu impregnated carbon blacks of N326, N330 and N347 grades, with the MU coverage ratio (nUM). The black arrow indicates the theoretical value for the MU monolayer.

1 .

 1 Coupling of PBu controlled impregnation and IGC-ID analysis for N326, N330 and N347 carbon black grades 4.1.1. Variation of the dispersive component of the surface energy with the impregnation ratio

Fig. 3 .

 3 Fig. 3. Variation of the index of surface morphology (IM) of PBu impregnated carbon blacks of N326, N330 and N347 grades, with the MU coverage ratio (nUM).

Fig. 4 .

 4 Fig. 4. Variation of the dispersive component of the surface energy ( d s ) of PBu impregnated carbon blacks of N115, N330 and N539 grades, with the MU coverage ratio (nUM).

Fig. 5 .

 5 Fig. 5. Variation of the index of surface morphology (IM) of PBu impregnated carbon blacks of N115, N330 and N539 grades, with the MU coverage ratio (nUM).

Fig. 7 .

 7 Fig. 7. Variations of both d s and IM parameters with the impregnation ratio, observed for the N115 sample.

Fig. 6 .

 6 Fig. 6. Schematic representations of the carbon black surface topology according to Donnet and Custodero [29,30], on left, and Schuster and al. [15], on right. (1) Graphitic planes, (2) amorphous carbon, (3) crystallite edges and (4) slit shaped cavities.

Fig. 8 .

 8 Fig. 8. Schematic representation of the insertion size of linear alkanes for layered solids (A) like talc or graphite and for a "fish scale" structure (B) present on the carbon black surface.

Fig. 9 .

 9 Fig.9. Variation CTAB/nitrogen specific surface areas ratio with the mean diameter (d) of the carbon blacks primary particles. Open circles correspond to our carbon blacks samples, closed circles are extracted from literature data[START_REF] Kühner | Carbon Black Science and Technology[END_REF].

Table 1

 1 Main characteristics of the studied carbon blacks.

	Carbon blacks	BET N2 area	Mean particle	CTAB area	Iodide adsorption	DBP adsorption
		(m 2 /g)	size (nm)	(m 2 /g)	(mg/g)	(ml/100 g)
	N539 N330 N326 N347 N115	41 82 85 90 140	∼50 ∼30 ∼27 ∼30 ∼18	41 83 81 88 128	42 84 84 91 145	112 100 69 125 113


M.-P Comard

R Calvet
email: calvet@enstimac.fr

H Balard

J A Dodds

The influence of polymer impregnation ratios on the surface properties of fumed silicas as determined by inverse gas chromatography

Keywords: Silica, Inverse gas chromatography, Polymer, Surface energy, Glassy transition temperature

The evolution of the surface properties of two types of fumed silica impregnated with different amounts of different polymers (polyethyleneglycol (PEG), polyvinylchloride (PVC), polyvinylmethylketon (PVMK) or polystyren (PS)) has been studied by inverse gas chromatography at infinite dilution. IGC-ID is a convenient method for the examination of the surface properties of a solid in which molecular probes are injected in very small amounts so that they mainly interact with sites of highest energy. The impregnation of silica with increasing polymer ratios leads to the progressive killing of these sites, allowing the probes to access the less energetic free sites. The polymers were chosen on the basis of their acidic or basic properties. A basic polymer, like PEG or PS, should first interact with the silanol groups, whereas an acidic polymer should interact preferentially with siloxane bridges. What behaviour will be adopted by an amphoteric polymer, such as PVMK?

IGC-ID also allows a fine estimate of the number of monomer units per nanometer square necessary to form a polymer monolayer at the surface of the solid. Finally, we have followed the evolution of the glassy transition temperature of the adsorbed polymer, with the PVC impregnation ratios on a silica. Correlatively, the enthalpies of adsorption below and above this critical temperature were determined. Hence, we will demonstrate that it will be possible to examine the evolution of the surface properties from the point of view of the polymer layer adsorbed on the silica surface.

Introduction

The principle of IGC is very simple [1]. A column is filled with the powder material under test and molecules with clearly identified features called probes are injected into the column. The retention time and peak elution profiles of the probes give information on their interactions with the surface of the solid and on the characteristics of both the surface of the solid and the probe. The complexity of these interactions depends on the amounts of probe injected. There are two different IGC techniques: infinite dilution (IGC-ID) or at finite solute concentration. In IGC-ID, very small amounts of probe are injected into the column in such a way that the interactions between probe molecules can be considered to be negligible. Only the interactions between the surface of the solid and an isolated probe molecule exist. Depending on the nature of the probe, IGC-ID gives access to different properties of the surface of the solid. The injection of a series of linear alkanes allows the determination of the dispersive component of the surface free energy (γ d s ) of the solid contained in the column. This parameter represents the London interactions between the probe and the solid surface.

A major limitation of IGC-ID appears in the case of a heterogeneous surface because the technique only gives access to apparent thermodynamic data and is mainly sensitive to the presence of high energy adsorption sites. Here, we will try to take advantage of this limitation by coupling IGC-ID with a progressive impregnation of the sites having the highest energies by adsorption of a polymer. This polymer will progressively saturate the more active sites by preferential adsorption, causing the probes to access the less energetic free sites. So, by increasing the amount of polymer added to the solid, the probes will access different energy sites on the surface of the solid. This method has first been applied to talc [2]. The results obtained gave interesting information, in particular allowed to distinguish the two kinds of surface on talc, the basal surfaces and the lateral surfaces. In this paper, the same method is used with silica to try to distinguish two different functional groups on the same surface.

Materials and methods

Two silica samples, supplied by Wacker GmbH have been used in the experiments, and were described in previous papers [START_REF] Legrand | The Surface Properties of Silicas[END_REF][START_REF] Papirer | Adsorption on Silica Surfaces[END_REF][START_REF] Legrand | Proceedings of Conference Silica 2001[END_REF][START_REF] Barthel | Surface textural studies of an active silica filler[END_REF]. Their characteristics are presented in Table 1. As proved by the comparison of the nitrogen and CTAB adsorptions, the HDK-N20 silica can be considered to be flat at the molecular scale as evidenced by its surface fractality equal to 2. Whereas the slightly microporous HDK-T30 silica presents a surface roughness corresponding to a surface fractality equal to 2.5. This observation was confirmed by size exclusion inverse gas chromatography based on the comparison of elution behaviour of linear alkanes and their cyclic or branched isomers [START_REF] Legrand | Proceedings of Conference Silica 2001[END_REF]. Solutions of a polymer (PEG or PS or PVC or PVMK described in Table 2) at different concentrations are mixed with a given mass of silica. An ex- ample of the polymer concentration, the mass of the silica and the ratio polymer/silica is presented in Table 3 for the PEG. After evaporation of the solvent (CH 2 Cl 2 for PEG, PS or PVMK and THF for PVC) a stainless steel tube is filled with the impregnated silica. The columns are conditioned at 120 • C overnight and analysed at 100 • C. The monomer unit coverage ratios expressed in monomer unit per nanometer square were calculated using the following formula:

n MU = 6023τ M MU S
where τ (%) is the coverage ratio expressed as a percentage, n MU is monomer unit number per nanometer square of silica, M MU is molecular weight of the monomer unit of interest, S is specific area of powder in meter per gram. Two Hewlett-Packard 6890 gas chromatographs, each fitted with two flame ionisation detectors were used. The carrier gas was helium. The injector and detector were heated to 150 • C. obtained by impregnation of the two fume silicas. The curve can be divided into two parts with a break point at 8 MU for the T30 silica and slightly more for the N20 silica. The first part of the curve corresponds to the progressive formation of a monolayer of polymer at the surface of the silica. The values at the break points can be interpreted, as the amount of polymer required to form a monolayer of MU on the surface of the solid. It would appear that the structure of the silica surface slightly influences this value. A smooth surface requires a greater amount of polymer to reach the monolayer than does a microporous surface. It may be supposed that the polymer has more difficulty to penetrate roughnesses in the surface. In the second part of the curves, when the impregnation ratio is higher than the critical ratio, that is to say when the γ d s reaches a plateau or decreases more slowly, the polymer forms a thick multilayer film at the surface of the solid.

Results and discussion

Study of the influence of the impregnation ratio on the dispersive component γ

Results obtained with the T30 silica have a much wider spread than those obtained with N20 silica. The from 70 to 25 mJ/m 2 for the T30 and from 40 to 25 mJ/m 2 for the N20. We therefore concentrate on the results obtained for T30.

The same approach was used with other polymers: PS, PVC and PVMK. The first was chosen for its basic character compared to PEG, the second because it is acidic, the third as it is both acidic and basic. The evolutions of γ d s obtained by impregnation of T30 silica with these four polymers are presented in Fig. 2.

Whatever the acidic or basic character of the polymer, the γ d s curves present the same general form with one break point on each curve. The polymer is unable to interact selectively with any given kind of site without having an effect on the other [START_REF] Balard | [END_REF].

However, it can be seen that each curve is different. Specifically, the amount of polymer necessary to reach the break point depends on the polymer. This difference can be attributed to the chemical structure of the polymer and, in particular, to its conformation on the surface of the solid. Another difference is the value of γ d s for high impregnation ratios. This energy is the energy of the polymer alone because the polymer covers the whole surface of the solid.

The number of monomer unit per namometer square necessary to form a monolayer at the surface of the solid for each polymer is given in Table 4. This number will be called F, which stands for monolayer fraction. The value of F for each polymer found by IGC using the previous graph, are listed in the second column. However, if we consider that one monomer unit is a cube and that each monomer unit is independent, F can also be estimated by a simple calculation which takes the density and the molecular weight of one monomer unit into account (column 3).

F theo = 6.02d M MU 10 -2
where F theo is the monomer unit monolayer fraction, theoretical number of monomer unit per namometer square of silica necessary form a monomer unit monolayer, d is density of the polymer and M is molecular weight of the polymer monomer unit.

It is well known that this calculation underestimates this value by about 20%. So, if we compare the results obtained by IGC and those obtained by the calculation and add 20% (column 4), we find the last column of Table 4.

The results obtained by IGC are in very good agreement with those obtained by calculation if the chains of the polymer are linear and flexible as for PEG or if the volume of the branch is not too great as in PVC. In spite of their opposing chemical characters, these two polymers adopt the same conformation at the surface of the solid. These macromolecules can form a uniform film of thickness quite equal to the monomer unit dimension. This is obviously not the case for PVMK and PS. The greater the volume of the branch, the bigger is the difference between the IGC results and the calculation. Because of the bulkiness of the monomer unit, these polymers form a film with a thickness greater than the thickness of 1 MU. For example, all the cycles of PS cannot be adsorbed flat on the surface, so they probably form loops (Fig. 3).

The comparison between F found by IGC and determined by calculation demonstrates that PS and PEG do not adopt the same conformation at the surface of the silica. The evolution of the specific surface of the T30 silica impregnated with different ratios of PEG and PS leads to the same conclusion (Fig. 4). These evolutions depend on the polymer, and even if at 8 MU/nm 2 the specific surfaces are the same for both polymers, the evolutions are different when the impregnation ratio becomes higher than this critical ratio. If the evolution observed for PEG is relatively easy to understand: the deposited PEG smoothes at first the surface irregularities of the surface of the primary particles or bound to the junction points between partially coalesced primary particles, leading to a rapid decrease of the specific surface area. Then, when all the surface roughnesses are filled, the contribution of the increase of a smooth thick film of polymer contributes much slightly to the decrease of the specific surface area.

On the contrary, the particular behaviour of the PS can only be explained if we admit that this polymer is unable to fit closely the silica surface leading to a more rapid plugging of the internal porosity of the aggregate. Hence a rapid decrease of the specific area over the critical coverage ratio will be observed.

Study of the evolution of the polymer glassy temperature with the impregnation ratio

The dispersive component is mainly sensitive to the progressive surface coverage of the silica surface. On the contrary, when having a look to the glassy temperature of the adsorbed polymer, one will have access to the way whom the solid surface influences the mobility and/or the conformation of the polymer alone.

IGC is a convenient method for the determination of the glassy transition temperature (t g ) of a thin film of polymer deposited on a surface of the solid [8]. At a given impregnation ratio, it is possible to measure the retention time of a given probe at different temperatures. The chosen probe was n-heptane. So the variation of the impregnation ratio allows plotting the variation of the free energy as a function of the temperature. In the case of PVC, the variations of -G • a /T versus the inverse of the temperature show two straight lines with a break point in all the curves obtained with the different impregnation ratios of polymer (Fig. 5).

This break point appears at a temperature of 81 • C, the normal glassy transition temperature for PVC. The t g is observed for all the impregnation ratios studied and is independent of the amount of polymer. So, even if the polymer does not cover the whole surface of the solid it is possible to measure the glassy transition temperature.

After a least square linearisation of the experimental points, the slope of each straight line was measured that are related to the enthalpies of adsorption of the n-heptane before and after the glassy transition temperature. The evolution of the adsorption enthalpies of n-heptane with all the different impregnation ratios studied are shown in Fig. 6.

The plot (Fig. 6) of the H values with the ratio of polymer shows three types of behaviour which are the same for the temperatures below and above the glassy transition temperature.

In the first zone, called A, the amount of PVC deposited on the surface is not enough to form a continuous film on the surface, so we observe the adsorption of an isolated macromolecule on the surface of the solid (Fig. 7). In zone B, the macromolecules form a 'monolayer of macromolecules', that says that the macromolecules are now in close contact at the surface forming a continuous film.

Finally, in zone C, the impregnation ratios are so high that the macromolecules form a thick film at the surface. The thickness of this film is considered to be constant at any point on the surface. The H is nearly constant. The same observation were done recently by Hamdi and co workers. that have studied the behaviour of PPMA adsorbed on the same T30 fumed silica [9].

Conclusion

IGC-ID allows the study of the surface properties of two fumed silicas impregnated with different polymers. The interactions between the polymers and the silica are affected not only by the surface of the solid but also by the structure and conformation of the polymer on the surface of the solid. This conformation depends on the bulkiness of the branch of the polymer. IGC-ID allows an estimate of the number of monomer unit per nanometer square necessary to form a monolayer at the surface of the solid.

IGC-ID is also a useful method for examining the behaviour of a polymer adsorbed on a solid surface and in particular to determine glassy transition temperature. This study highlights three kinds of adsorption corresponding to an isolated macromolecule, close-packed macromolecules and the formation of a thick film of polymer on the surface of the solid.

To conclude, coupling IGC-ID and controlled surface modification through polymer adsorption is a powerful method for obtaining information on the surface of a solid.

Fig. 1 Fig. 1 .

 11 Fig. 1 displays the variation of γ d s with respect to the number of monomer units of PEG 20 M per nm 2 (MU/nm 2 )

Fig. 2 .

 2 Fig. 2. Variation of the dispersive component of the surface energy (γ d s ) of the T30 silica impregnated with different ratios of PEG, PS, PVC and PVMK.

Fig. 4 .

 4 Fig.[START_REF] Papirer | Adsorption on Silica Surfaces[END_REF]. Evolution of the specific surface of the T30 silica impregnated with different ratios of PEG and PS as measured with a micromeritics ASAP 2010 using nitrogen.

Fig. 3 .

 3 Fig. 3. Schematic representation of the loops of PS at the surface of the silica T30.

Fig. 5 .

 5 Fig. 5. Variation of the -G • a /T with 1/T for T30 silica at 3 different molecular impregnation ratios: 4.01, 10.14 and 20.27 MU/nm 2 .

Fig. 6 .

 6 Fig.[START_REF] Barthel | Surface textural studies of an active silica filler[END_REF]. Variation of the enthalpies of adsorption with molecular impregnation ratios (n MU ) of PVC impregnated on T30 silica samples.

Fig. 7 .

 7 Fig. 7. Schematic representation of the three modes of adsorption.

Table 1

 1 Surface area and fractal dimension of fumed silica samples[START_REF] Legrand | The Surface Properties of Silicas[END_REF]

	Silica	S N2	S CTAB	R CTAB/N2	D CTAB/N2	OH/nm 2 *
	samples	(m 2 /g)	(m 2 /g)
	N20	199	200	1.01	2.0	1.8
	T30	300	245	0.82	2.5	1.8

*

The number of OH/nm 2 was determined by acid-base titration.

Table 2

 2

	Mean characteristics of the polymers
	Polymer	Abbreviation	Supplier	Molecular weight (g/mol)	Density	Purity
	Polyethyleneglycol	PEG	Fluka	20,000	1.24	Purum
	Polystyren	PS	Aldrich	23,0000	1.05
	Polyvinylchloride	PVC	Aldrich	43,000	1.42	99%
	Polyvinylmethylketon	PVMK	Aldrich	50,0000	1.12	99%
	Table 3
	Preparation of the samples with T30 silica and PEG
	Concentration of the	Volum of PEG in	Mass of	τ, ratio	τ, ratio	n UM of PEG
	PEG in CH 2 Cl 2	CH 2 Cl 2 added to	silica (g)	polymer/silica	polymer/silica	on the surface
	solution (mg/l)	the silica (l)		(mg/g)	(%)	(UM/nm 2 )
	0.01	0	1.0143	0.000	0.000	0.000
	0.01	200	1.0156	1.973	0.197	0.135
	0.01	600	1.0403	5.778	0.578	0.395
	0.04	250	1.0457	9.575	0.958	0.655
	0.01	2,000	1.0315	19.425	1.943	1.330
	0.01	3,000	1.0613	28.320	2.832	1.938
	0.01	5,000	1.0453	47.922	4.792	3.280
	0.01	8,000	1.0799	74.218	7.422	5.080
	0.01	10,000	1.0060	99.587	9.959	6.816
	0.01	15,000	1.0578	142.066	14.207	9.723
	0.01	20,000	1.0485	191.102	19.110	13.080
	0.01	30,000	1.0119	297.020	29.702	20.329
	0.04	12,500	1.0439	479.590	47.959	32.825
	0.01	40,000	0.6101	656.843	65.684	44.956
	0.04	25,000	1.0455	957.712	95.771	65.549

Table 4

 4 Number of monomer unit per nanometer square necessary to form a monolayer at the surface of the solid for each polymer (PEG, PVC, PVMK and PS) found by IGC and calculation

	Polymer F IGC (MU/nm 2 )	F theo (MU/nm 2 )	F theo + 20% (MU/nm 2 )	F IGC / (F theo + 20%) (MU/nm 2 )
	PEG	8.0	6.6	7.9	1.01
	PVC	6.6	5.7	6.8	0.97
	PVMK 7.7	4.5	5.4	1.42
	PS	8.0	3.3	4.0	2.00


Bertrand Neveu
email: bertrand.neveu@enpc.fr

Martin De La Gorce
email: martin.delagorce@gmail.comd

Pascal Monasse
email: pascal.monasse@enpc.fre

Gilles Trombettoni
email: gilles.trombettoni@lirmm.fr

An Interval Branch and Bound Algorithm for Parameter Estimation and Application to Stereovision

Abstract. The parameter estimation problem is a challenging problem in engineering sciences consisting in computing the parameters of a parametric model that fit observed data. The system is defined by unknown parameters and sometimes internal constraints. The observed data provide constraints on the parameters. The parameter estimation problem is particularly difficult when some observation constraints correspond to outliers and/or the constraints are non convex. For dealing with outliers, the RANSAC randomized algorithm is efficient, but non deterministic, and must be specialized for every problem. In this work, we propose a generic interval branch and bound algorithm that produces a model maximizing the number of observation constraints satisfied within a given tolerance. This tool is inspired by the IbexOpt Branch and Bound algorithm for constrained global optimization (NLP) and is endowed with an improved version of a relaxed intersection operator applied to the observations. The latest version of our B&B follows the Feasible diving strategy to visit the nodes in the search tree. Experiments on a stereovision problem have validated the approach.

PARAMETER ESTIMATION

Parameter estimation is a difficult problem widely studied by engineering sciences. It consists in determining the n numerical parameters of a model based on m observations. Calibration or geolocation can be viewed as specific parameter estimation problems.

A parameterized model is defined by an implicit equation f (x, p) = 0, p = (p 1 , . . . , p n ) being the n-vector of parameters to be determined. An observation o i is a d-dimensional vector of observed data (values) for x. Given a finite set of observations {o 1 , . . . , o i , . . . , o m }, we search for a parameter vector that fits the observations, i.e. that satisfies the observation constraints f (o i , p) = 0. Because the model is not perfect or due to uncertainties on the observations o i , there exists generally no model fitting all the observation constraints exactly. In addition, the answer of least square methods or numerical local methods (for nonlinear observation constraints) is poor in the presence of outliers. Outliers can have numerous origins, including extreme values of the noise, erroneous measurements and data reporting errors.

That is why we consider an approach where the observation constraints are handled within a tolerance value τ:

-τ ≤ f (o i , p) ≤ +τ.
The observations are partitioned into a group of outliers and a group of inliers. The inliers correspond to the consensus set (Consensus(p)), i.e. the set of observations compatible with p:

Consensus(p) = {o i | -τ ≤ f (o i , p) ≤ +τ}. (1)
Outliers do not satisfy the corresponding relaxed observation constraints.

The problem is to compute a model fitting a maximum number of observations. More precisely, a parameterized model can be defined by the observation constraints and internal constraints between parameters C(p), so that the estimation parameter problem handled by the interval Branch and Bound proposed in this paper is defined as follows:

arg max p (card(Consensus(p))) s.t. C(p).
(2)

PARAMETER ESTIMATION FITTING A MAXIMUM NUMBER OF INLIERS Interval Arithmetic and Contractors

We denote by [x i ] = [x i , x i ] the interval/domain of the real-valued variable x i , where x i , x i are floating-point numbers.

A Cartesian product of intervals like the domain

[x] = [x 1 ] × ... × [x N ] is called a (parallel-to-axes) box. width(x i ) denotes the size or width x i -x i of an interval [x i ].
The width of a box is given by the width x maxx max of its largest dimension x max . The perimeter of a box, given by i x ix i , is another size measurement.

Interval methods also provide contracting operators (called contractors), i.e. methods that can reduce the variable domains involved in a constraint or a set of constraints without loss of solutions. Let us consider one observation o i and a subspace of the parameter space given by a box [p]. Given an observation o i , we denote V i the set of parameter vectors that are compatible with that observation, i.e.

V i = {p | -τ ≤ f (o i , p) ≤ +τ}.
The set of parameter values inside this box that are compatible with o i is given by V i ∩ [p]. A contractor is able to reduce the input box [p] while keeping all the points in V i ∩ [p], i.e. without losing any solution. An important contractor used in our optimization code is the well-known HC4-revise [START_REF] Benhamou | Revising Hull and Box Consistency[END_REF][START_REF] Messine | Méthodes d'optimisation globale basées sur l'analyse d'intervalle pour la résolution des problèmes avec contraintes[END_REF], also called forward-backward. This contractor traverses twice the expression tree corresponding to a given constraint to contract the domains of its variables.

Another contractor is applied to the m boxes obtained after having contracting the box studied using the m observation constraints. The q-intersection operator relaxes the (generally empty) intersection of the m boxes by the union of all the intersections obtained with q boxes (q < m). More formally:

Definition 1

Let S be a set of boxes. The q-intersection of S , denoted by ∩ q S , is the box of smallest perimeter that encloses the set of points of R n belonging to at least q boxes. The q-intersection of boxes is a difficult problem and has been proven DP-complete in [START_REF] Carbonnel | Q-Intersection Algorithms for Constrained-Based Robust Parameter Estimation[END_REF] where an exact algorithm based on the search of q-cliques has been proposed. In this paper, we simply resort to a non optimal q-intersection operator often used in parameter estimation, called here q-proj, that solves the problem on each dimension independently [START_REF] Chew | Masking failures of Multidimensional Sensors[END_REF]. The overall complexity of q-proj is O(nm log(m)).

Interval Branch and Bound for Parameter Estimation

We have designed an interval B&B algorithm for parameter estimation that computes a model maximizing the consensus of a parameterized model. It appeared that the standard best-first search strategy has difficulties to find a good solution and that a depth first search strategy gets stuck in a local optimum area. Therefore we have used a mixed strategy based on best-first search where, at each node, a feasible diving procedure is performed [START_REF] Neveu | [END_REF].

The main EstimB&B algorithm returns the best solution found (best is the node where the best solution has been found), with a number q min of validated inliers, and an interval [q min , q max ]. Either the precision ∆ ob j required on the number of inliers is obtained, i.e. q maxq min ≤ ∆ ob j , or q maxq min > ∆ ob j , which means that "small" boxes with width inferior to sol remain open (unexplored) with a number of inliers at most equal to q sol max . The EstimB&B algorithm achieves a best-first search by maintaining a set of open nodes nodeHeap. This set is implemented by a heap data structure having at its top the node with the largest number of possible inliers (i.e., an upper bound q max of the objective function value to maximize). EstimB&B selects iteratively the node having the greatest q max and calls the FeasibleDiving function on it.

Feasible Diving

The FeasibleDiving function is called with the model as parameter: the parameters p of the model, i.e. the variables to be determined using the internal constraints C and the observation constraints C obs = {-τ ≤ f (o i , p) ≤ +τ}. From the selected node, FeasibleDiving, described in Algorithm 1, builds a tree in depth-first order and keeps only the most promising node at each iteration. More precisely, the box in the node is bisected along one dimension and the FeasibleDiving (node, C, C obs , best, nodeHeap, sol , q sol max ) while node.box ∅ and width(node.box) > sol and not node.isLea f do (node 1 , node 2 ) ← Bisect (node) // Bisection of node.box (node 1 , best, q sol max ) ← Contract&Bound (node 1 , C, C obs , ob j , ∆ ob j ,best,q sol max ) (node 2 , best, q sol max ) ← Contract&Bound (node 2 , C, C obs , ob j , ∆ ob j ,best,q sol max ) (node better , node worse ) ← Sort (node 1 , node 2 ) if node worse .box ∅ and width(node worse .box) > sol and not isLea f then nodeHeap ← Push(node worse , nodeHeap) node ← node better return (nodeHeap, best, q sol max ) Algorithm 1: The Feasible Diving procedure run at each node of our interval B&B algorithm. two sub-boxes are handled by a Contract&Bound procedure. The sub-box with the largest q max is handled in the next step while the other sub-box is pushed into the heap nodeHeap of open nodes. Thus, FeasibleDiving is a greedy algorithm that does not perform any backtracking.

Contract and Bound

The Contract&Bound function handles every open node. It contracts the node box and tries to prove (guarantee) that more than best.q min + ∆ ob j observation constraints are satisfied inside the box. Recall that the best node found so far has at least q min valid inliers, so that the next best node (if any) is searched for with a cost at least equal to best.q min + ∆ ob j + 1.

1. The box is first contracted by the constraints C between parameters, if any: first, the HC4 constraint propagation algorithm is performed. We add calls to a linear programming solver for improving variable interval bounds, using a linear relaxation of the constraints based on affine arithmetics [6]. 2. The q-intersection projection algorithm, using q = best.q min + ∆ ob j +1 is then called on the n parameter directions. We also perform a q-projection on an additional direction where we hope to obtain small intervals, thus favoring a failure of the q-intersection. To this end, we linearize and relax every observation constraint, and project the parallelograms obtained on the direction corresponding to the mean normal vector of the "parallelogram" gradients. Details can be found in [START_REF] Neveu | Improving a Constraint Programming Approach for Parameter Estimation[END_REF]. 3. An upper bound node.q max of the number of inliers in the node is given by the minimum over all dimensions of the maximum number of intersected intervals found by the q-projection procedure in a dimension. If node.q max < q, then the node box becomes empty and the corresponding branch in the tree will be pruned. 4. To improve the lower bound, one has to find a feasible point in the current box, i.e. a point that satisfies all the constraints C, if any, and a greater number of observation constraints C obs than the current solution. When C is not empty, the search of the solution validating inliers is a difficult task in itself. We have designed a variant of an algorithm used in IbexOpt (see [START_REF] Araya | [END_REF]) that tries to build an inner polytope inside the box where all the points satisfy the constraints C. If such a polytope is built, a point (vertex) is picked inside it and we compute the number of inliers at this feasible point for updating the new current solution.

In conclusion, with the implementation of our EstimBB code, we are close to provide a generic tool for parameter estimation taking into account outliers.

EXPERIMENTS

Our EstimBB code has been implemented in the Interval Based EXplorer (Ibex) [9], a free C++ library devoted to interval computing. We made preliminary experiments on the shape (plane or circle) detection problem studied in [START_REF] Neveu | Improving a Constraint Programming Approach for Parameter Estimation[END_REF], replacing the search of all solutions with at least q inliers by the optimization problem presented in this paper. These problems have a parameter space of dimension n = 3 and have no additional (internal) constraints. Second, we tried to solve the more challenging problem of essential matrix estimation in stereo vision having n = 5 parameters. We tested our method on the 6 real instances for essential matrix estimation from USAC library [10], named test1 to test6, with between m = 534 and m = 2245 pairs of 2D points. The test2, ..., test5 instances are solved to optimality within a 1% precision on the number of inliers w.r.t the total number of correspondences in several hours. The test6 instance seems more difficult because the percentage of inliers is small, but we succeeded in computing a model to optimality by solving a partial instance with 320 correspondences. The test1 instance is the most difficult one, since the tolerance is smaller and one has to go deeper in the search tree to validate the solution. The algorithm did not terminate in 100,000 seconds. The figure below shows the set of inliers found on test5.

CONCLUSION

This paper has presented an exact interval B&B approach implemented in IBEX for parameter estimation taking into account outliers. The corresponding code is generic. First experiments on different computer vision problems suggest that the current interval B&B algorithm provides good results in medium dimension, i.e. up to 4 or 5 in number of parameters (depending on the inlier rate), whereas most exact approaches cannot cope with dimension 3.


domain: math shs

text:
Bernard Parzysz
email: parzysz.bernard@wanadoo.fr

Using spreadsheets to teach probability in French high school

Keywords: Probability, simulation, continuous distribution, spreadsheet. Thematic Working Group 05

Including a 'frequentist' point of view has resulted in experimentation becoming an important issue in the teaching of probability in high school. Spreadsheets are now widely used, but the status of the results produced and how to use them are not always clear for the students, since two domains are at play in turn: statistics and probability. Through the French examplebut this can also be applied to the teaching of probability in other countriesthis paper reviews some questions about spreadsheets, namely simulating random experiments and shifting from discrete to continuous distributions.

In France as in many other countries, probability has become a prominent subject in the teaching of mathematics, in link with the growing importance of numerical data in everyday and professional life and the development of technologies allowing to process them. Conversely, the point of view on probability has evolved, taking into account a 'frequentist' point of view implying students making experiments. Among the technological tools spreadsheet has become much used in the teaching of probability, the main reason being that it makes getting a large number of tries of a random experiment very easy and fast. This paper is based on the French current situation, but it certainly can apply to many other countries all over the world. Its aim is to give some insights on how new didactic questions have occurred and have now to be tackled by teachers in their classes. As it has already been noticed, "it is not good enough to only consider which technology to use, but (…), in order for effective learning to take place, it is how the technology is integrated into the curriculum and learning process and how the teacher uses it that are vital" (Pratt, Davies & Connor, 2008, p. 98), the more so that "most teachers have little experience with probability and share with their students a variety of probabilistic misconceptions" (Batanero et al., 2005, p.1). I shall discuss some questions about the use of spreadsheet in high school, namely:

-the nature of simulation, implying using a model of the random experiment and making in turn intervene probabilistic and statistical paradigms; -possible purposes of a simulation: visualize the law of large numbers, make conjectures, bring out the notion of stochastic model, help solving probability problems… -the suitability of spreadsheet for introducing continuous distributions (exponential, Gaussian…).

For this, I shall use a theoretical framework including Kuzniak's Mathematical Working Space (MWS), Kuhn's paradigms and Duval's semiotic registers.

Theoretical framework

In order to get a holistic view of the work undertaken by somebody solving a mathematical problem, one has to take into account not only the domains at play but also the cognitive processes involved. The Mathematical Working Spaces, or MWS, framework [START_REF] Kuzniak | L'espace de Travail Mathématique et ses genèses[END_REF] considers two "planes" -epistemological and cognitive-, each one having three components:

in the epistemological plane: a set of representations ('real space'), a set of artefacts (instrume nts) and a theoretical reference system; -in the cognitive plane, three processes: visualization, construction and proof An important feature of the model is the interaction between these two planes according three dimensions, semiotic, instrumental and discursive, linking each component of one plane to a corresponding component of the other (Figure 1). The model also assumes that efficient mathematic work results from involving the 3 dimensions together with interactions between them. -reference MWS, defined by the syllabus, -suitable MWS, planned by the teacher to be implemented in his/her class, -personal MWS of the student.

Kuhn [START_REF] Kuhn | The structure of scientific revolutions[END_REF] defined scientific paradigms as "universally recognized scientific achieveme nts that, for a time, provide model problems and solutions for a community of researchers," (page X of the 1996 edition). This notion was adapted by Kuzniak to taught mathematics, regarding the epistemological plane. In the case of probability several paradigms can be distinguished [START_REF] Parzysz | Quelques questions didactiques de la statistique et des probabilités[END_REF]:

a realistic paradigm (R), i.e. the real ('concrete') random experiment itself; -a paradigm (P1) resulting from a first ("light") modelling of the real experiment by establishing a precise protocol, a list of issues and assigning a probability to each of them; -a paradigm (P2), in which notions of random experiment and probability are defined, together with properties of probability which can be used for solving problems. -a paradigm (P3) of the axiomatic type, taught in university.

N.B. In France, at secondary level, only P1 and P2 paradigms are considered, the latter being possibly extended with some elements of calculus at the end of high school (P2+) for the study of continuo us distributions.

Regarding the semiotic dimension I shall refer to the notion of 'semiotic register', i.e. a coherent semiotic system allowing 3 cognitive activities: produce identifiable elements (representatio ns), transform an element into another of the same register, convert an element into an element of another register [START_REF] Duval | Sémiosis et pensée humaine[END_REF]. For Duval, a better knowledge is obtained through the use of several registers interacting one with another. He also indicates that the shift from one representation to another one is more efficient when there is a 'semantic congruence' between them, i.e. when there is a one-to-one correspondence between signifying elements of the two representations.

Simulation

In the beginning of its being studied in high school, probability was taught as an application of combinatorics, through Laplace's formula (probability = number of favorable issues / number of total issues). This 'cardinalist' point of view implies that all the issues have the same chance to appear, and then other random phenomena had to be left aside. For instance this is the case for drawing pins: when tossed, they may come down in two ways, like coins, but no argument of symmetry can help and one cannot assign a plausible a priori probability to each of them. In such a case you have to observe the relative frequencies of the issues, assuming that they will 'converge' toward their probability when the number of tries grows 'to infinity'. This is the 'frequentist' point of view, theorized by the law of large numbers. This point of view was introduced in French high school 20 years ago. Anyway, whatever is the point of view on probability, one has to decide which probability will be allocated to each issue of the experiment, the difference being that the decision is made:

-either on a priori ground (e.g. 'symmetry' of the issues) in a cardinalist approach; -or on a posteriori ground (frequencies of the issues) in a frequentist approach.

In past days, teachers were reluctant to let their students perform sequences of a random experiment, mostly because it was noisy and requested too much time, but the coming of computers in classrooms, namely spreadsheet including a so-called 'random' generator (see for instance [START_REF] Kroese | Handbook of Monte Carlo methods[END_REF], provided them with an alternative path (although starting with real experimentation remains necessary to materialize the link with reality). Spreadsheet is now widely used to simulate random experiments, with various purposes.

1) Spreadsheet can be used to visualize the compatibility of the cardinalist and frequentist points of view, and finally get the students confident in the generator. For this purpose one has to introduce into the machine a probability for each issue. Then the evolution of the relative frequencies on fairly large numbers of tries can be observed (Figure 2), this process being in fact a visualization of the 'law' of large number (belonging to the P3 paradigm).

Figure 2: Relative frequencies of heads in 1000, 2000, …50000 tries of heads and tails

Thus performing a simulation implies constructing at least a simple probabilistic model (within the P1 paradigm), in order to implement it in the machine. Hence simulation is a ternary process: what is implemented in the software is not the real experiment but a theoretical model of it (Figure 3). In such a task several registers are appealed to in turn: natural language, symbolic language (software) and Cartesian graphs. The semiotic-instrumental plane of the MWS is at play, involving the initial P1 paradigm (finding a model of the experiment), then shifting to another paradigm: descriptive statistic (DS) (results of the simulation). It is only when a conjecture about the experiment is asked that the discursive dimension appears (within P2 paradigm).

Both as teacher and teacher trainer, I could observe that some students find it difficult to distinguis h between statistical and probabilistic paradigms (the more so than some notions are similar), somewhat analogous with geometrical paradigms. Here, like with GDS, the dynamic feature of spreadsheet, allowing an easy and fast observation of many samples -and consequently many different results (Figure 4)-, can help distinguishing the P2 paradigm (theoretical value) from SD (observed value). But French textbooks do not put the stress on the distinction between the two domains, in particular using the notions in a very loose way (e.g. confusion average / expectation), this probably reflecting actual teaching in classrooms. Similarly, about the instrumental dimension a tendency of textbooks to 'overguide' the students, in order to help them deal with the software, must also be noticed.

Figure 4: Relative frequencies of heads in 50 samples of 100 tries of heads and tails

2) Simulation can also be used to estimate the value of a probability. For instance, if various models of a same experiment give different probabilities for a given event, a simulation mimicking the concrete experiment can tell which model(s) can be discarded. This is the case with the following problem, which was the basis of an action research with high school students introducing the frequentist approach [START_REF] Parzysz | Expérience aléatoire et simulation: Le jeu de croix ou pile. Relecture d'une expérimentation déjà un peu ancienne[END_REF]:

Toss a well-balanced coin; if head (H) happens you win and if tail (T) happens you toss the coin again; then if H happens you win and if T happens you lose. What is your probability of winning?

In 1754, Jean Le Rond d'Alembert thought that one has 2 chances against 1 to win, at variance with "all authors" claiming that one has 3 chances against 1; Mimicking the process with spreadsheet leads to some difficulty, since one has to distinguish between the two possible results of the first toss. For an easier implementation in the computer the teacher may decide that the coin would be tossed in every case. From a probabilistic point of view the two processes are equivalent but all students are not convinced. (Historically, a similar argument opposed Blaise Pascal with Gilles Roberval in 1654). The reason for such a reluctance is that the second process is not semantically congruent with the real experiment.

Table 1 shows the corresponding spreadsheets. When comparing the sheets, one can see that putting anything (H or T) in the empty boxes of the left sheet has no influence on the final result. After that one can forget the "Again?" column and have a second toss in all cases, i.e. replace the initial procedure by the second one without any inconvenie nce (Figure 5). 3) A most widespread type of activity in French 10 th and 11 th grades describes a random experiment and then asks the student to simulate it a number of times with the spreadsheet, observe the results and formulate a conjecture about the probability of one of the issues or the possible value of a parameter. Then he/she is asked to solve the problem using the probability theory and compare the theoretical results with the initial conjecture.

In this process several paradigms are at play. As seen above, starting from reality (R), the student shifts to probability (generally P1), then moves to descriptive statistic (DS) to extract informatio n from the spreadsheet (frequency, mean, etc.) and back to P1 to formulate a conjecture; solving the problem within P2 will imply the discursive dimension (Figure 6).

N.B. In such activities the spreadsheet is used as a multi-purpose tool: it intervenes in turn as logica l tool (instructions for software), random generator (simulation), copying machine (results of tries), calculator (statistical parameters) and plotter (diagram). The French syllabus for 12 th grade includes an introduction to continuous probability laws, and the official resource document for that level suggests starting with a statistical continuous variable, in order to approximate the histogram of a sample by a continuous curve "which fits the histogram, the area under the curve being equal to 1". The general idea is to link a random variable X, not with a set of isolated probability values as was previously the case, but with a function f (density) verifying :

P(a ≤ X ≤ b) = ∫ 𝑓(𝑥)𝑑𝑥 𝑏 𝑎
for any a and b with a ≤ b. This modelling process, which seems quite sensible, implies also a shift from statistics (DS paradigm) to probability (here P2+) and an essential, though transitional, point is histogram. This rises a difficulty, since spreadsheet cannot produce histograms, or rather what it calls histogram is in fact a bar chart. In order to overcome this problem one may widen the bars till they become contiguous (Figure 7), but this trick is restricted to cases in which all intervals have the same width. But the notion of density gets sense only with unequal intervals, since in a histogram the basic notion is area, not height. Thus in this case spreadsheet appears to be of no help if the software does not permit producing real histograms. , where Xn follows the binomial law B(n, p). Then the bar chart for Zn is approximated by the curve of a function of the x → λ.exp(-ax 2 ) type. As above a shift from discrete to continuous law occurs, but this time spreadsheet can help since, contrary to the general case, the values of Zn are equidistant (the distance being 1 √𝑛𝑝(1 -𝑝) ⁄ ) and then a pseudo histogram is suitable.

All textbooks follow this scheme, in which three types of diagrams are at play in turn: first bar chart (for Zn), then pseudo histogram, then bell curve. Spreadsheet is necessary at every stage of the process, first to get the values of P(Xn = k) for 0 ≤ k ≤ n) and various values of n and p, then the corresponding bar charts of Zn, then its pseudo-histogram and finally the (pseudo-)curve of the standard normal law (in fact a polygon). The main difficulty comes from the histogram and the curve looking proportional but not equal (Figure 8) because the distance between the values of Zn is different from 1 (see above).

Figure 8: Pseudo-histogram and pseudo-curve

This problem of scale is tackled rather awkwardly in textbooks, as is the standardization of the binomial variable. However, the question appears when one wants to compare the shapes of the bar charts for several binomial distributions (Figure 9); one may then think of a 'calibration', i.e. changing the units on the axes, in order to get diagrams with the same average and height. In the process the semiotic and instrumental dimensions of MWS are much appealed to, but the discursive dimension is not much present, due to the students' lack of knowledge.

Conclusion

The current French high school curriculum starts with descriptive statistics (from 6 th grade on) and later goes on with probability (at 9 th grade), introduced through a dual, frequentist and cardinalis t, point of view involving several mathematical paradigms (DS, P1, P2, P2+). Experimentation has become a central issue in teaching probability and in this process spreadsheet extends real tries, for the reason that it is incomparably faster once its use (language, gestures) is mastered. It is now included in the semiotic-instrumental plane of the MWS and can play an important role in many ways and for multiple purposes. Some points are of importance for teaching with simulation, namely pay attention to the model subjacent to the 'real' random experiment (even when it does not clearly appear), help students distinguish between the statistical and probabilistic paradigms, bring out the idea of stochastic model… When coming to continuous probability a sensible way to introduce it consists of approximating a histogram by a continuous curve. Unfortunately usual spreadsheet cannot produce general histograms -i.e. with unequal classes-but only bar charts, becoming possibly 'pseudo histograms', and histograms should have to be produced with another software. However class experimentation showed that a pseudo histogram may prove useful as a transitory artefact in the particular case of shifting from binomial to normal law.

On the whole, although spreadsheet was not conceived for educational but for professional purposes, it has now become a quite appreciable, if not indispensable, tool for the teaching of probability in high school.

Figure 1 :

 1 Figure 1: The MWS model (after Kuzniak, 2011) Kuzniak distinguishes 3 main MWSs:

Figure 3 :

 3 Figure 3: The ternary process of simulation

Figure 5 :

 5 Figure 5: From experiments to model Thus a visual comparison within the register of double entry tables, in the semiotic-discursive plane, can be a means for deciding if two models are equivalent. And for younger students this can be a possible path towards the bnotion of stochastic model.

Figure 6 :

 6 Figure 6: The paradigms and processes involved in the activity Continuous distributions N.B. This point is based on a recent research (Derouet & Parzysz, 2016).

Figure 7 :

 7 Figure 7: From bar chart to pseudo histogram This same syllabus includes the study of normal law and recommends introducing the standard law N(0, 1) from the observation of the distribution of Zn= 𝑋 𝑛 -𝑛𝑝 √𝑛𝑝 (1-𝑝)

Figure 9 :

 9 Figure 9: Comparison of binomial distributions (B(100, .1) and B(50, .4))

Try n° 1st toss Again? 2nd toss

					Result	Try n° A coin B coin	Result
	1	H	no		won	1	H	T	won
	2	T	yes	H	won	2	T	H	won
	3	T	yes	T	lost	3	H	T	won
	4	T	yes	T	lost	4	T	T	lost
	5	T	yes	H	won	5	H	T	won
	6	H	no		won	6	T	T	lost
	7	H	no		won	7	T	H	won

Table 1 : Spreadsheets of the two simulations

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