A Nzihou
email: nzihou@enstimac.fr.

B Adhikari

R Pfeffer

Effect of Metal Chlorides on the Sintering and Densification of Hydroxyapatite Adsorbent

come

Effect of metal chlorides on the sintering and densification of hydroxyapatite adsorbent

Ange Nzihou, Benu Adhikari, R

Introduction

This work is part of a series of studies dealing with the evaluation of the effects of major elements of solid waste, especially metallic oxides, nitrates, sulfates, and chlorides, on the sintering and the densification of calcium hydroxyapatite, Ca-HAP (or HAP) adsorbent. Ca-HAP, Ca 10 (PO 4 ) 6 (OH) 2 , is a promising compound for adsorption and immobilization of heavy metals from soils, incinerator fly ashes, and hazardous industrial wastes. [START_REF] Mavropoulos | Studies on the mechanisms of lead immobilization by hydroxyapaties[END_REF][START_REF] Ma | In situ lead immobilization by apatite[END_REF][START_REF] Boisson | Evaluation of hydroxyapatite as a metal immobilizing soil additive for the remediation of polluted soils. I. Influence of hydroxyapatite on metal exchangeability in soil, plant growth and plant metal accumulation[END_REF][START_REF] Laperche | Chemical and minerological characterizations of Pb in a contaminated soil: Reactions with synthetic apatite[END_REF][START_REF] Nzihou | Calcium phosphate stabilization of fly ash with chloride extraction[END_REF][START_REF] Iretskaya | Metal leaching from MSW fly ash before and after chemical and thermal treatments[END_REF][START_REF] Da Rocha | Cadmium uptake by hydroxyapatite synthesized in different conditions and submitted to thermal treatment[END_REF][START_REF] Dong | Model apatite system for the stabilization of toxic metals: I, Calcium lead vanadate[END_REF] The immobilization of heavy metals by Ca-HAP increases significantly by calcination at temperatures ranging from 650 to 900 °C. Calcination leads to sintering and densification of Ca-HAP, which decreases the leachability of the incorporated metal ions. [START_REF] Dong | Model apatite system for the stabilization of toxic metals: I, Calcium lead vanadate[END_REF][9][START_REF] Nzihou | Mechanisms of sintering of macroporous hydroxyapatite adsorbents[END_REF] The sintering phenomenon is influenced by the composition of the solid waste considered. [START_REF] German | Surface area reduction during isothermal sintering[END_REF][12][START_REF] Raynaud | Calcium phosphate apatites with variable Ca/P atomic ratio II. Calcination and Sintering[END_REF] A number of papers describing sintering processes and mechanisms are available in the literature. [START_REF] German | Surface area reduction during isothermal sintering[END_REF][12][START_REF] Raynaud | Calcium phosphate apatites with variable Ca/P atomic ratio II. Calcination and Sintering[END_REF][START_REF] Nzihou | Effect of oxides and nitrates of lead on the sintering and densification of hydroxyapatites adsorbents[END_REF][START_REF] Bailliez | The kinetics of surface area reduction during isothermal sintering of hydroxyapatite adsorbent[END_REF][START_REF] German | Powder Metallurgy Science[END_REF][START_REF] German | Surface area reduction during isothermal sintering[END_REF][START_REF] German | Sintering and catalysis[END_REF][19][START_REF] German | Identification of the initial stage sintering mechanism using aligned wires[END_REF][START_REF] Kuczynski | Self-diffusion in sintering of metallic particles[END_REF][START_REF] Kuczynski | The mechanism of densification during sintering of metallic particles[END_REF][START_REF] Kuczynski | Self-diffusion in sintering of metallic particles[END_REF][START_REF] Coble | Initial Sintering of Alumina and Hematite[END_REF][START_REF] Coble | Sintering Crystalline Solids I: Intermediate and Final. State Diffusion Models[END_REF][START_REF] German | Surface area reduction kinetics during intermediate stage sintering[END_REF][START_REF] Ranade | The grain model applied to porous solids with varying structural properties[END_REF][START_REF] Nicholson | Variation of surface area during the thermal decomposition of solids[END_REF] Sintering is described as the bonding together of particles at high temperatures. It can occur at temperatures below the melting point by solid-state atomic transport but in many instances involves the formation of a liquid phase. On a microstructural scale, the bonding occurs due to the growth of cohesive necks at particles contacts, even though the particles are at temperatures lower than their melting temperature.

Particles sinter by atomic motions that eliminate the high surface energy associated with the powder. The surface energy per unit volume depends on the inverse of the particle diameter. Typically, the surface energy depends on the surface area. Thus, smaller particles with high specific surface areas have more energy and sinter faster. However, not all surface energy is available for sintering. The sintering mechanism describes the path of atomic motion, which produces the mass flow. For metal powders, the mechanisms are usually diffusion processes over the surfaces, along the grain boundaries, or through the crystalline lattice. The stages of sintering follow a geometric progression. They relate to the driving force and kinetics and are used to mathematically describe the process.

The structural changes associated with neck growth during sintering depend on transport mechanisms, which are mainly diffusion processes. Diffusion is thermally activated, meaning that a specific energy is necessary for atomic movement. The mechanisms of sintering are due to volume diffusion and surface diffusion, viscous flow, and evaporation-condensation. These mechanisms are distinguished by the relationship of the radius of ''neck" growth of two particles of the same size, as a function of time. These relationships have been discussed by numerous investigators and have been used to describe the sintering mechanism of different materials. [START_REF] Kuczynski | Self-diffusion in sintering of metallic particles[END_REF] The rate of growth of the neck with time depends on which sintering mechanism is prevalent.

Motion depends on an atom attaining an energy equal to or above the activation energy to break free from its current site and move into a vacant site. The geometric progression associated with sintering can be divided into three stages. During the initial stage, bonds form at the particle contacts. As densification proceeds, new contacts form; hence, there are variations in the degree of sintering from point-to-point in the microstructure due to delayed contact formation. With prolonged sintering, the pore structure becomes smoothed, leading to the intermediate stage, which corresponds to the open, continuous pore structure that exists between densities of approximately 70 and 92% of the theoretical. [START_REF] German | Powder Metallurgy Science[END_REF] In the intermediate stage, the sintering rate is continuously decreasing and is very sensitive to the pore-grain boundary morphology. Grain growth occurs in the later portion of sintering during which the pores become spherical and isolated. Elimination of isolated pores becomes difficult at that point. The final stage of sintering corresponds to closed, spherical pores that shrink slowly by vacancy diffusion to grain boundaries. Such densification is sensitive to the relative grain size and the attachment of pores to grain boundaries. Any atmosphere (gases) trapped in the pores will inhibit final densification.

These stages of sintering represent a changing driving force for mass flow at high temperature. In response to the driving force, the mechanisms of sintering represent the actual mass flow paths. The initial strain after compaction can lead to a plastic flow contribution to sintering, but generally diffusive processes are dominant. Possible diffusion paths include lattice, grain boundary, and surface regions. There is a characteristic distinction between surface and bulk processes. Bulk transport gives densification, while surface transport gives interparticle bonding without densification. It is typical to use computer simulations to quantify the sintering models.

Chemical additives that modify diffusion rates during sintering are often used. The additives can stabilize the desirable structure or, more typically, can form a liquid phase to increase the rate of sintering. In a recent paper, [START_REF] Nzihou | Effect of oxides and nitrates of lead on the sintering and densification of hydroxyapatites adsorbents[END_REF] we discussed the effect of metal oxides and metal nitrates on the sintering and densification of Ca-HAP. These results shows activated sintering for both the metal-oxides and metal-nitrates investigated. Here, we discuss the effects of chloride salts on the sintering and the densification of Ca-HAP.

Materials and Methods

Analytical-grade zinc chloride (ZnCl 2 ) and potassium chloride (KCl) were obtained from PROLABO and ChemPur (Germany), respectively. They were finely ground before use. Calcium hydroxyapatite (Ca-HAP) was prepared by mixing calcium nitrite [Ca(NO 3 ) 2 ] and ammonium hydrogen phosphate [(NH 4 ) 2 HPO 4 ] both obtained from NORSKHYDRO. The temperature was maintained at 25 °C, and the pH was adjusted to 7-8 with 10% ammonia solution according to the following reaction:

The details of the preparation and purification procedure of Ca-HAP through the route of reaction 1 is given elsewhere. [START_REF] Bailliez | The kinetics of surface area reduction during isothermal sintering of hydroxyapatite adsorbent[END_REF] The powder was finely ground using a laboratory scale grinder (IKA Works, Inc.) to obtain fine particles with a narrow size distribution. Thermal stabilization of the powder was carried out by subjecting it to 400 °C for 2 h. The stabilized sample exhibited agglomeration, and the particle size was widely altered. Hence, the powder was ground again to obtain finer particles of mean size 5 microns. This stabilized Ca-HAP powder was used throughout the study.

The structure and the composition of the dried particles was identified and quantified by X-ray powder diffractometry (Siemens D5000). XRD measurements were carried out with Cu KR radiation generated at 40 kV and 40 mA, in the 3°< 2θ < 60°range at a scan speed of 2 °C/min. The phases present were determined by comparing the patterns with JCPDS standards. The calcium and phosphorus content were determined using atomic absorption spectrometry (Varian spectrAA-400 plus).

The XRD reference of the hydroxyapatite was obtained according to the AFNOR standard29 (calcination during 15 h at 1000 °C). The calcination allows for the crystallization of the particles (as compared to the uncalcined particles) as shown in Figure 1 for Ca-HAP .Approximately 2 wt % of Ca(OH) 2 was detected in this Ca-HAP. The Ca/P ratio is of 1.7275, which becomes a mixture of lime (Ca(OH) 2 ) and hydroxyapatite at high temperature. Since the Ca/P ratio of stoichiometric hydroxyapatite is 1.6667, the Ca-HAP used for our study is clearly an hydroxyapatite with a calcium excess.

The specific surface area of the samples was determined using nitrogen adsorption with the BET method (MICROMETRICS Gemini Vacprep 061). The solid density of the Ca-HAP was determined by helium pycnometry (MICROMETRICS, Accupyc 1330). The specific surface area and the solid density of the Ca-HAP powder were 43.48 m 2 /g and 2987 kg/m 3 , respectively. The mass loss history of the samples was determined in dynamic mode using a Themogravimetric Analyzer (TGA-DTA, NETZSCH-STA 409) at a scanning rate of 10 °C from room temperature to 1200 °C.

The shrinkage tests of the samples were carried out by a Thermomechanical Analyzer (TMA, SETARAM Setsys 16/20) with a 5 g constant load. The shrinkage, due to rise in temperature, is defined as ∆H/H 0 , where H 0 is the initial height of the sample and ∆H ) H 0 -H(t). H(t) is the height of the sample as a function of time. The relative density of the sintered specimen is expressed as F b /F s , where F b and F s are the bulk and solid density, respectively. The thermomechanical analyses (dilatometry) were carried out in dynamic mode with a scanning rate of 10 °C/min from room temperature up to 1200 °C.

The evolution of the bulk density of the specimen as a function of temperature can be determined from the measured shrinkage history. The bulk density is an important parameter and is related to the voidage of the specimen. This quantity is also important for following the relative density (F b /F s ) of the sample. The bulk density (F b ) can be determined from the equation given below

πr 2 H o [100 -shrinkage (%)] (2)
where m o and H o are the initial mass (kg) and height (m) of the sample, and r is the inner radius (m) of the crucible in the dilatometer (TMA). Equation 2was derived assuming that the variation in the sample dimension is isotropic and unidirectional and that the weight of the mass remains constant at the temperature range of interest. Calcination experiments were carried out in a furnace (AUBRY Co., France). A heating rate of 10 °C/min was imposed from room temperature to the designated isothermal temperatures; isothermal conditions were maintained for 2-300 min. The samples were placed in an alumina crucible and subjected to the above calcination regime under flowing air and cooled to room temperature at a rate of 10 °C/min. The heating (scanning) rate of 10 °C/min was imposed for TGA and TMA experiments, as well as for the calcination process in order to facilitate a comparison. The surface characteristics of the sintered agglomerate as well as the unsintered particles/aggregates were studied using an environmental scanning electron microscopy (XL 30 ESEM-FEG, Phillips). The ESEM micrographs, when combined with the mass loss (TGA) and the shrinkage (TMA), provide excellent insight into the densification process.

Results and Discussion

Effect of Chloride Salts of K and Zn on

Sintering and Densification of Ca-HAP. Figure 2 presents the variation of the specific surface area of Ca-HAP and its mixtures with 2% (w/w) of KCl and ZnCl 2 as a function of temperature. The evolution of the shrinkage rate and the bulk density of these mixtures, as a function of temperature, are presented in Figure 3 together with the evolution of the shrinkage rate and bulk density of pure Ca-HAP.

3.1.1. Effect of Potassium Chloride KCl. As shown in Figure 2 the specific surface area of the 2% mixture of KCl with Ca-HAP is much lower than that of Ca-HAP at all temperatures. As the mean particle size (5microns) of the KCl additive was of the same order as that of Ca-HAP, the much lower specific surface area (30% difference) below 800 °C means that this additive accelerates the process of surface area reduction of C-HAP.

Figure 3 shows that the densification of the mixture of Ca-HAP and KCl commences at about 800 °C, which is at about 50 °C lower than that of pure Ca-HAP. The rate of sintering of this mixture is faster when compared to Ca-HAP up to 1080 °C and then falls short and attains a lower final shrinkage. These observations imply that it is difficult to establish a direct relationship between the surface area reduction and the densification process. The results of the mixture of KCl with Ca-HAP show that surface area can reduce significantly even if the densification is negligible. The bulk density data (Figure 3) of these samples are simply another interpretation of the densification process.

The delay in densification can be explained by the slow evaporation of H 2 O from the decomposition of Ca(OH) 2 to CaO that is present in the Ca-HAP. [START_REF] Bailliez | The kinetics of surface area reduction during isothermal sintering of hydroxyapatite adsorbent[END_REF] Since melting of KCl at 771 °C can cause the particles to bind together, the evaporation of H 2 O generates a volume expansion (puffing action), which slows and delays the densification. [START_REF] Melin | Volume expansion during reaction sintering of γ-Bi12SiO20[END_REF] The result of that process could explain the low shrinkage observed even though the ESEM micrographs of the Ca-HAP mixture with 2% KCl (see Figure 4a) show surface densification at 1200 °C. A similar delay of the densification process has already been observed during the natural sintering (without load applied on the sample) of Ca-HAP containing excess calcium in the form of Ca(OH) 2 . [START_REF] Nzihou | Mechanisms of sintering of macroporous hydroxyapatite adsorbents[END_REF][START_REF] Nzihou | Effect of oxides and nitrates of lead on the sintering and densification of hydroxyapatites adsorbents[END_REF][START_REF] Bailliez | The kinetics of surface area reduction during isothermal sintering of hydroxyapatite adsorbent[END_REF] Figure 5 compares the rate of shrinkage of Ca-HAP and 2% mixtures of KCl and ZnCl 2 with Ca-HAP. This figure agrees well with Figure 2 and shows that the rate of densification of Ca-HAP with KCl is faster than that of Ca-HAP alone up to 950-980 °C, above which temperature the rate declines very sharply. The maximum densification of the mixture is achieved below 1200 °C, indicating that there is no possibility of further densification or sintering above this temperature unless the chemical structure changes due to a partial dehydroxylation of Ca-HAP and the formation of tetracalcium phosphate monoxide (Ca 4 O(PO 4 ) 2 ) and R-tricalcium phosphate [Ca 3 (PO 4 ) 2 ]. This corroborates well with the results of Figures 2 and3 that the mixture of Ca-HAP with 2% KCl undergoes a higher degree of shrinkage and surface area reduction compared to that of pure Ca-HAP.

The phenomenon that we observed with the 2% KCl mixture is referred to in the literature as "activated sintering". Activated sintering [START_REF] German | Powder Metallurgy Science[END_REF] is a process which lowers the activation energy, allowing for a lower sintering temperature and shorter sintering time by the addition of a layer of a second component (guests) to the particles undergoing sintering (hosts). In general, the activator must be either a metal or a compound, which forms a lower melting temperature phase than the metallic host particles during the sintering. The activator must have a large solubility for the base metal, while the base metal should have a low solubility for the activator. For example, KCl melts at 771 °C, which corresponds to a temperature close to the sintering temperature zone of Ca-HAP. The activator should also remain segregated at the interparticle interfaces during sintering. This segregated layer provides a high diffusivity path for rapid sintering. The lower melting point ensures a lower activation energy for diffusion, while the low solubility ensures that the activator is not dissolved into the base metal.

In two-phase systems involving mixed powders, it is possible to form a low melting phase. In such a system, the liquid provides for rapid transport and therefore rapid sintering. That is possible if the wetting liquid forms a film around the solid phase. Finally the diffusive transport for the dissolved solid atoms should be high enough to ensure rapid sintering. The liquid film provides a surface tension force to aid in densification resulting in a densification rate that is much higher than in solid-state sintering.

Activated sintering has also been used to describe the formation of composites whereby a low concentration of one material is added to another material to promote grain boundary segregation so as to increase diffusion rates. 31 The increase in diffusion gives rise to faster rates of sintering, hence a lowering of the sintering temperature of the material. 32,[START_REF] Luo | Origin of Solid-State Activated Sintering in Bi O -Doped ZnO[END_REF] Effect of Increasing the Concentration of KCl. The effect of changing the concentration of KCl on the surface area reduction of Ca-HAP is presented in Figure 6. This figure shows that the specific surface areas of a 10% mixture of KCl with Ca-HAP, at a given temperature, are always lower than those of the corresponding 2% mixture. This implies that an increase in the concentration of KCl up to 10% helps to accelerate the reduction in the surface area reduction of Ca-HAP, indicating that increasing the amount of activator could improve activated sintering

The evolution of the shrinkage of 2% and 10% mixtures of KCl, as a function of temperature, are compared in Figure 7. This figure shows that the densification of Ca-HAP + 10% KCl commences at about 700 °C compared to 800 °C for Ca-HAP + 2% KCl. This means that an increase in the amount of KCl can significantly enhance activated sintering by lowering the sintering temperature at which densification begins. The final shrinkage of Ca-HAP + 10% KCl is 6.5% compared to 11.1% for Ca-HAP + 2% KCl. Figure 8 shows that the rate of shrinkage of Ca-HAP + 10% KCl is much lower as compared to that of Ca-HAP + 2% KCl above 1043 °C. There are two densification (shrinkage) maximums at 680 and 1000 °C; the former may be due to the proximity of the melting point of KCl (771 °C).

The evolution of bulk density for 2% and 10% mixtures of KCl with Ca-HAP, as a function of temperature, is presented in Figure 9. The salient feature of this figure is that the bulk densities of the specimens with 10% additive decrease, rather than increase, as the sintering progressed. This is due to the melting of KCl that forms more bonds between particles at 10% KCl than at 2% KCl. Shrinkage is retarded due to the increased volume expansion by swelling favorable for the evaporation of water as explained previously. This also indicates that as the percent of KCl in Ca-HAP is increased, a more porous powder is produced.

3.1.2. Effect of Zinc Chloride ZnCl 2 . Figure 2 shows that the reduction in specific surface area of Ca-HAP + 2% ZnCl 2 is very close to that of Ca-HAP throughout the temperature range investigated. The average difference is 4.15%, which is within the measurement error of the equipment. This means that the addition of ZnCl 2 does not affect the surface area reduction of Ca-HAP at this concentration even though the melting temperature of ZnCl 2 is only 293 °C. Figure 3 shows that the evolution of the shrinkage of Ca-HAP + 2% ZnCl 2 is much slower compared to that of pure Ca-HAP and that the final shrinkage was much less (7.7%). Hence, it can be concluded that the addition of ZnCl 2 retards the densification of Ca-HAP. This is referred to in the literature as ''deactivated sintering", whereby the minimum sintering temperature of the host material (Ca-HAP) is increased by the addition of a layer of a second (guest) component (ZnCl 2 ).

Deactivated sintering can be defined as a process whereby the surface of the host particles is covered with a layer of another material (guest) to delay and reduce the sintering of the host material, hence causing an increase in the minimum sintering temperature. Since the melting temperature of ZnCl 2 is 293 °C, the Ca-HAP may be covered by a discrete layer of melting ZnCl 2 . The deactivated sintering observed is discussed below.

The delay in sintering might be caused by an increase in the activation energy of the system, which gives rise to slower rates of diffusion. The question arises as to what is the importance and/or application of increasing the sintering temperature of particulate materials. Recently, a number of papers have appeared in the literature [START_REF] Alonso | Powder Coating in a Rotary Mixer with Rocking Motion[END_REF][START_REF] Alonso | Mechanism of Combined Coating Mechanofusion Processing of Powders[END_REF][START_REF] Ramlakhan | Promotion of deactivated sintering by dry-particles Coating[END_REF][START_REF] Dukhin | Deactivated sintering by particle coating: the significance of static and dynamic surface phenomena[END_REF] describing the phenomenon of deactivated sintering which was observed after applying a discrete coating of metallic oxide powders onto the surface of a variety of host powders to change their surface properties. This caused a reduction in the sintering rate and an increase in the minimum sintering temperature of the host particulate materials.

Figure 5 further shows that the rate of densification of a Ca-HAP-ZnCl 2 mixture is always smaller than that of Ca-HAP and that it retards the densification process. This result is confirmed by the ESEM micrographs in Figure 4b showing the mixture of ZnCl 2 with Ca-HAP sintered at different temperatures. It can be seen that, the surface of the agglomerates of the mixture of Ca-HAP + 2% ZnCl 2 at 1200 °C is porous and not completely coalesced. This indicates that the addition of ZnCl 2 retards the densification process and fails to bring a reduction in surface area.

The explanation for the low densification and deactivated sintering observed for the 2% ZnCl 2 additive could be the following. For Ca-HAP containing 2 wt % of Ca(OH) 2 , the shrinkage begins at 600 °C; this temperature corresponds to the decomposition of lime Ca(OH) 2 to CaO. The reduction in length observed in the porous structure of Figure 4b could be explained by the effect of the generation of the porosity by evaporation of water. [START_REF] Nzihou | Mechanisms of sintering of macroporous hydroxyapatite adsorbents[END_REF][START_REF] Nzihou | Effect of oxides and nitrates of lead on the sintering and densification of hydroxyapatites adsorbents[END_REF][START_REF] Bailliez | The kinetics of surface area reduction during isothermal sintering of hydroxyapatite adsorbent[END_REF] This phenomenon may increased by the evaporation of ZnCl 2 at 732 °C through the pores. So, under the load of the thermomechanical analyzer this porous medium shrinks and leads to the reduction in length observed.

An increase in the concentration of ZnCl 2 to 10% could possibly further improve the deactivated sintering observed for the mixture Ca-HAP + 2% ZnCl 2 . But, the results obtained with a mixture of Ca-HAP with 10% ZnCl 2 were not relevant because of the presence of a significant amount of molten ZnCl 2 that prevented accurate measurement of the shrinkage using the dilatometer.

Conclusions

The effect of chloride salts of potassium and zinc on the sintering and densification of calcium hydroxyapatite (Ca-HAP) was studied by monitoring the reduction in specific surface area and shrinkage. The effect of concentration of these additives was also investigated using 2% and 10% (KCl only) of these additives solid mixtures with Ca-HAP. The addition of 2% w/w of KCl in Ca-HAP produced a substantial reduction in surface area and a further increase to 10% produced an even greater reduction in the specific surface area. The 2% w/w KCl activates the sintering by lowering the densification temperature by 50 °C; the 10% w/w lowered the densification temperature by 150 °C. However, the final shrinkage of the mixture containing 10% KCl was much lower than the 2% KCl mixture with Ca-HAP. The addition of 2% of ZnCl 2 in Ca-HAP did not show a reduction in specific surface area of the Ca-HAP and served to deactivate the sintering by retarding the densification process and lowering the rate of shrinkage as well as the final shrinkage. Hence, there does not appear to be any correlation between surface area reduction and the volume shrinkage. The rate of shrinkage of the mixture had two or more peaks indicating that the additives bring about multiple speeds in the densification process.

Literature Cited

Figure 1 .

 1 Figure 1. XRD pattern of not calcined and calcined Ca-HAP during 15 h at 1000 °C.

Figure 2 .

 2 Figure 2. Specific surface area of Ca-HAP and mixtures of 2% KCl and ZnCl2 as a function of temperature.

Figure 3 .

 3 Figure 3. Variation of shrinkage (%) and bulk density (m 2 /g) of Ca-HAP and the mixtures of Ca-HAP with 2% KCl and 2% ZnCl2 as a function of temperature.

Figure 4 .

 4 Figure 4. Environmental scanning electron micrographs of (a) Ca-HAP + 2% KCl and (b) Ca-HAP + 2% ZnCl2 sintered at different temperatures for 2 h.

Figure 5 .

 5 Figure 5. Shrinkage rate (%/min) of Ca-HAP and mixtures of Ca-HAP with 2% KCl and 2% ZnCl2 as a function of temperature.

Figure 6 .

 6 Figure 6. Variation of the specific surface area (m 2 /g) of Ca-HAP, a 2% and 10% mixture of KCl with Ca-HAP as a function of temperature (time ) 2 h).

Figure 7 .

 7 Figure 7. Variation of the shrinkage (%) for a 2% and 10% mixture of KCl with Ca-HAP as a function of temperature.

Figure 8 .

 8 Figure 8. Evolution of rate of shrinkage (%/min) for a 2% and 10% mixture of KCl with Ca-HAP as a function of temperature.

  [START_REF] Bailliez | The kinetics of surface area reduction during isothermal sintering of hydroxyapatite adsorbent[END_REF]

  10Ca(NO 3 ) 2 + 6(NH 4 ) 2 HPO 4 + 8NH 4 OH f Ca 10 (PO 4 ) 6 (OH) 2 + 20NH 4 NO 3 + 6H 2 O (1)


Abderrahim Ait Wakrime

J Paul Gibson

Jean-Luc Raffy

Formalising the Requirements of an E-Voting Software Product Line using Event-B

Keywords: Software Product Lines, Feature Model Configuration, Formal Modelling, Event-B, E-Voting

A Software Product Line (SPL) is a tool/method used to generate a family of program/system variants for a specific domain, and to support a more efficient software development of future products within the same domain. A Feature Model (FM) is a popular graphical/textual representation used in SPL requirements specification; it is used to capture commonality and variability information existing in an SPL as a set of interrelated and configurable features. A concrete model of an SPL instance is obtained by binding the variation information in the FM with a configuration that meets a specific set of feature requirements. Since configuration decisions are taken prior to instantiation, invalid configurations should be detected/avoided before design begins. This paper addresses the problem of the verification of the correctness (validity) of FM instances and FM configuration during requirements modelling. It proposes a requirements model based on Event-B contexts, allowing us to check the correctness of a given configuration, before starting the correct-by-construction design and implementation process, based on refinement.

I. INTRODUCTION

A Software Product Line (SPL) is [START_REF] Clements | Software product lines[END_REF]:". . . a set of softwareintensive systems that share a common, managed set of features satisfying the specific needs of a particular market segment or mission and that are developed from a common set of core assets in a prescribed way." A Feature Model (FM) is a language used to model the commonality and variability between product variants. Several approaches have been proposed for combining features, where each combination corresponds to a configuration or a program in a SPL [START_REF] Liang | Sat-based analysis of large real-world feature models is easy[END_REF], [START_REF] Thüm | Analysis strategies for software product lines[END_REF]. Such approaches verify only that a specific instance respects the configuration rules specified by the feature model, so that the system can be configured. They do not verify that the features, as configured, are compatible (coherent). The use of a formal specification of feature models would facilitate the verification of both types of properties -valid configuration and coherent configuration.

Once a configuration is verified as correct, then we must start the development (design and implementation). A SPL generally incorporates a general architecture which all such instances must be built upon. Specific instances will have different combinations of features, each of which must be added incrementally to the system. Adding a feature's behaviour can be thought of as a system refinement, and so we chose to use a formal specification language -Event-B [START_REF]Modeling in Event-B: system and software engineering[END_REF] -that supports refinement-based development. Our approach is based on modelling using Event-B contexts. These contexts are used to model three inter-related types of entities: domain specific concepts, product line features, and product line instantiation.

The structure of the paper is as follows. Section 2 puts our contribution into context and reviews relevant related work. Section 3 provides an overview of our generic formal SPL approach. Section 4 illustrates how our general approach can be applied to the problem of e-voting. Section 5 provides more details concerning the formal verification of the instantiation process, using a concrete e-voting system as an example. It also comments on the notion of feature interaction, and how such interactions can be detected during different stages of the formal development. The 6th and final section concludes the paper, and comments on future extensions to the research.

II. RELATED WORK

In this section, we will focus on the related works of existing approaches of analysis of feature models (FMs) using formal methods and the use of formal methods during e-Voting service construction. In [START_REF] Mazo | Constraints: The heart of domain and application engineering in the product lines engineering strategy[END_REF] product line models are represented as constraint programs in order to specify configuration requirements and provide support for the product configuration activity. In [START_REF] Karatas | From extended feature models to constraint logic programming[END_REF] constraint logic programming is also used to translate and formalize extended FMs over finite domains in order to analyze the different relationships. In another work [START_REF] Ripon | Modeling and analysis of product-line variants[END_REF], an approach to modelling and analyzing SPL variant feature diagrams using first-order logic is presented. This formalization is based on logical expressions that can be built by modelling variants and their dependencies by using propositional connectives. In [START_REF] Shen | Feature modeling and verification based on description logics[END_REF] the authors adopt a description logics-based FM, including the feature class and the constraints of the features. Some constraints rules are introduced to verify the consistency and completeness of FM instances.

An alternative approach, proposed in [START_REF] Soltani | Automated planning for feature model configuration based on functional and non-functional requirements[END_REF], applies an artificial intelligence planning technique to automatically select suitable features that satisfy both the stakeholders preferences and constraints especially with regard to non-functional properties. In addition, the research in [START_REF] Cordy | Beyond boolean product-line model checking: dealing with feature attributes and multifeatures[END_REF] presents a model checking technique for SPsL, precisely modelling the FM with non-Boolean features (numeric attributes) and multi-features. Petri nets [START_REF] Mennicke | Automated verification of feature model configuration processes based on workflow petri nets[END_REF] can be used to model a FM configuration. This approach, based on workflow Petri nets, allows a formal operational model for staged configuration that makes explicit causal dependencies among feature selections. Petri nets are also integrated as an established formalism for modelling systems with a high degree of variability in an SPL [START_REF] Muschevici | Feature nets: behavioural modelling of software product lines[END_REF]. To achieve this, a Petri net extension, named Feature Nets, is used to provide modelling dynamic SPLs. Variability is often modelled using transition systems. In [START_REF] Beek | Modelling and analysing variability in product families: model checking of modal transition systems with variability constraints[END_REF], [START_REF] Lauenroth | Model checking of domain artifacts in product line engineering[END_REF], modal transition systems are used to model the behaviour in product families, in order to define and derive a valid product behaviour starting from product family behaviour.

Alternative approaches are used that provide a precise and rigorous formal interpretation of the feature diagrams. A binary-search based approach to FM verification is presented in [START_REF] Zhang | Binary-search based verification of feature models[END_REF] in order to detect deficiencies in FMs. Similar analysis are possible in [START_REF] Andersen | Efficient synthesis of feature models[END_REF] using an efficient technique for synthesis of models from respectively CNF and DNF formulas. In another work [START_REF] Mendonca | Sat-based analysis of feature models is easy[END_REF], automated analyses of FMs is adopted by translating models to propositional logic and using satisfiability (SAT) solvers.

With respect to our validation case-study, formal methods have been previously used for e-Voting; for example, firstorder logic is used for the analysis and development of voting schemes to provide a formal specification and verification [START_REF] Beckert | Verifying voting schemes[END_REF]. In [START_REF] Kremer | Election verifiability in electronic voting protocols[END_REF] a formal symbolic definition of election verifiability, based on π-calculus, to precisely identify which parts of a voting system need to be trusted for verifiability. In [START_REF] Davies | Technological foundations of electronic governance[END_REF], formal techniques are exploited to build technical solutions for electronic governance in order to specify desirable functionality, build an implementation model and verify that the implementation satisfies the specification. In [START_REF] Poppleton | The single transferable voting system: Functional decomposition in formal specification[END_REF], a formal specification is presented using the Z language for the Single Transferable Vote form of election. This specification exemplifies a functional decomposition style supporting the validation of requirements.

Generally, the purpose of FMs modelling and verification is to detect deficiencies in FMs, in order to avoid having these deficiencies enter into the process of product development. In our work, we provide an Event-B formal approach to check the consistency between a FM and its configurations to manage variability in e-voting services as well as the scalability of automatically generating and validating configurations.

III. GENERAL APPROACH

Our approach aims to formally specify a SPL based on FMs at a high level of abstraction using the Event-B method based on a sequence of correct-by-construction refinements. This formal specification provides a way to verify correctness of such properties of FM configuration instances with respect to product validity.

Our approach is structured as the following sequence of activities: The user specifies their requirements through configuration of the feature tree model. The validity of the configuration is checked by the Event-B tool set and the user can animate the high-level abstract behaviour. Once they validate the specified requirements using the Rodin theorem prover that supports the generation of Proof Obligations belonging to Event-B models, design and implementation are done through refinement. The whole process focuses on the client trusting that their requirements are properly specified, and that correctness of implementation is guaranteed because the requirements are formally specified. Fig. 1 depicts the overall approach. It should be noted that this paper reports on only the requirements engineering phase of devlopment, where we model the chosen features using only the Event-B contexts. The next phase of development (refinement-driven design) combines the context specifications with Event-B machines. An example of this correct-by-construction design approach can be found in [START_REF] Gibson | Applying a dependency mechanism for voting protocol models using event-b[END_REF].

IV. E-VOTING FEATURE MODEL

Before presenting our e-voting FM, we need to recall some basic concepts and definitions.

A. E-Voting Model

A FM definition was first introduced by [START_REF] Kang | Feature-oriented domain analysis (foda) feasibility study[END_REF] in the FODA report in 1990. A FM represents the information of all possible products of a SPL regarding features and relationships between them. A FM is a graphical/textual representation that is widely used in SPL engineering. The FMs are presented as trees in which each node is a feature and each edge can have five possible values:

• And: all sub-features must be selected.

• Alternative: only one subfeature can be selected.

• Or: one or more can be selected.

• Mandatory: features that required.

• Optional: features that are optional. A FM may also have constraints that cannot be easily expressed hierarchically and graphically. These constraints are named cross-tree constraints. Cross-tree constraints are propositional formulae using features as variables. A feature can be concrete or abstract. A concrete feature is a terminal feature (a leaf. of the tree). Contrastingly, an abstract feature is a non terminal and compound feature. Fig. 2 depicts a simplified FM to better understand the relationships between its different entities. An extract of our e-voting FM (tree) can be seen in Fig. 3. All the concepts which appear in the tree have been formally specified in an e-voting domain ontology. This ontology has been validated by a domain expert. A government that wishes to procure a particular instance of a voting system specifies a valid instance by removing branches from the SPL FM.

A FM configuration is a set of concrete features. A configuration is valid if the selection of all features contained in the configuration and the deselection of all other concrete features is allowed by the FM. If a concrete feature is selected, its parent must also be selected. If a parent is selected, all the Mandatory sub-features must be selected, one subfeature in each of its Alternative groups must be selected and at least one of its sub-features in each of its Or groups must be selected. It is necesssary that each valid configuration satisfies propositional constraints. For example in Fig. 2, a valid FM configuration is: {SubF eature1 1, SubF eature2 1, SubF eature2 2}.

B. Transformation E-Voting Model using Event-B

In this section, we introduce our Event-B formalism of the FM and its configuration correctness. The consistency of the model is ensured by formal proofs, see section IV-C.

The Event-B model, defined in a static Context, is used primarily to fix the definitions and formalizations of the concepts in the FM. It also defines a static part where all the relevant properties and hypotheses are formalised. This model is generated automatically from the graphical feature model specification.

The properties considered in our approach formalize the FM, that contains a sets of relationships between a parent feature and its child features. The FM nodes are represented by a FEATURE set that it is divided into three subsets of the nodes and they are classified as follows: Root, Node. A Root represents the initial and single parent which has no parent and identifies the SPL. The Node type is the node that has a child and one parent. These three nodes types are used to define a part of tree structure of FM (axm1, axm2 and axm3 of Listing 1).

The relationships between a feature and its sub-features are represented using four constants named operators: And, Or, Xor and Null. The And operator is used to mention that all sub-features are mandatory. When the Or operator is bound to a node, this node has at least one sub-feature. When the Xor operator is bound to a node, that represents the alternative between sub-features. Howeover, the Null operator indicates that the feature has no sub-features i.e. this feature is a leaf. The successors of a given feature are specified by the SuccOf relation (axm7 of Listing 1).

= SuccOf -1 ∪ (SuccOf -1 ; SuccOf T rans -1 ) axm13: theorem (F EAT U RE id) ∪ SuccOf T rans -1 = ∅ axm14: ∀s.(s ⊆ SuccOf [s] ⇒ s = ∅) Listing 1.
An Event-B Context for describing a FM structure: a Context T reeSpec.

The structure of a FM is represented by a graph without cycle i.e. a tree. For that, we explicitly express the property that verifies if a FM is exactly a tree in which any two features are connected by exactly one path. Then, we define a relation SuccOfTrans to check that a FM does not contain cycles (axm12, axm13 and axm14 of Listing 1).

As mentioned before, the different relationships of a FM are formalized by a set of operators as: And, Or, Xor and Null (Listing 2). The And operator is used when it is bound to a node. The children of a given feature are obtained using the ChildrenOf relation which is defined in Event-B The operator And allows one to select all children that are also the successors (axm16 of Listing 2). When the operator Or is bound to a feature, this feature has at least one child and all of its children are also successors of that feature (axm17 and axm18 of Listing 2). When the operator Xor is bound to a feature, this feature has at least and at most a child which is also one of its successors (axm19 and axm20 of Listing 1.2).

C. Verifying E-Voting Configuration

This section describes the verification and validation of our development. In the verification step, we have the static properties of the system that can be formally verified. We verify the static properties, which are expressed in terms of Contexts, using formal proofs (proof obligations). In order to check a given instance that represents a FM configuration, we use the Context T reeSpec that defines the types and structure FM (Listing 1 and Listing 2). The general e-voting FM is transformed into an Event-B context to include it in the verification process of each instance i.e. configuration. For space reasons,we represent only a part of the complete generic e-voting FM in Fig. 4, and how it is transformed to an Event-B Context (Listing 3) named F M V oting. In addition, each configuration to be verified can be translated to a specific Event-B Context. We use the Rodin tool that supports the application of the Event-B formal method, providing core functionality for syntactic analysis and proof-based verification of Event-B models [START_REF] Abrial | Rodin: an open toolset for modelling and reasoning in eventb[END_REF].

CONTEXT F M V oting SETS F EAT U RE CONSTANTS V ote V oteV isibility EncryptedV ote nonEncrypted V AccessRights V Adm V P ublic V N one V ADuring V AAf ter V P During V P Af ter axm1: F EAT U RE = {V ote, V oteV isibility, EncryptedV ote, nonEncrypted, V AccessRights, V Adm, V P ublic, V N one, V ADuring, V AAf ter, V P During, V P Af ter} axm2: {Root} = {V ote} axm3: N ode = {V oteV isibility, EncryptedV ote, nonEncrypted, V AccessRights, V Adm, V P ublic, V N one, V ADuring, V AAf ter, V P During, V P Af ter} axm4: And = {V ote} axm5: Or = {V AccessRights, V Adm, V P ublic} axm6: Xor = {V oteV isibility} axm7: N ull = {V ADuring, V AAf ter, V P During, V P Af ter} axm8: M andatory = {V oteV isibility, V AccessRightsy} An example of the proofs, we established, concerns the e-voting configuration F M V otingConf OK represented in Listing 4. We have to prove the correctness of this instance using the Rodin platform with its automatic prover. For this, the proof obligations were automatically discharged without interactions to help the provers to find the right rules, as shown in Fig. 5. Furthermore, concerning the e-voting configuration F M V otingConf KO represented in Listing 5, proof obligations are discharged automatically (see the proving perspective Rodin shown in Fig. 6).

V. FORMAL SPECIFICATION AND INTEGRATION OF FEATURES

In this section we describe the problem of feature interactions, and the utility of a formal approach (as presented in this paper) for detecting possible interactions as early as possible in the development process; and to avoid future interactions by updating the formal FM appropriately.

A. Feature Requirements

The feature requirements are represented as Event-B Contexts using sets and relations from the Event-B domain model. Each abstract feature is represented by a Context that has a behavior. The mismatch between FM configurations and program variants is primarily caused by abstract features. In this context, the interactions between abstract features can be realized, if they share constants or/and sets. When two abstract features interact in a coherent manner, we can classify these abstract features as being Friends when they can always work together in a coherent manner, or Politicians when they must co-operate in a particular way in order for their behaviours to be coherent. Otherwise, if they cannot interact cohernetly, then we can classify them as Enemies [START_REF] Hamilton | Composing fair objects[END_REF]. We give an example of these different interactions levels.

Example 1. Let F 1 and F 2 be two abstract features represented by two different Contexts and nbreCandidates a constant that belongs to both Contexts. When nbreCandidates ∈ {1, 2, 3} in the F 1 and nbreCandidates ∈ {1, 2, 3} in the F 2 , the two abstract features F 1 and F 2 are Friends. When nbreCandidates ∈ {1, 2, 3} in the F 1 and nbreCandidates ∈ {3, 4, 5} in the F 2 , the two abstract features F 1 and F 2 are Politicians as they can agree on a value of 3. On the other hand, When nbreCandidates ∈ {1, 2, 3} in the F 1 and nbreCandidates ∈ {4, 5} in the F 2 , the two abstract features F 1 and F 2 are Enemies.

The formal event-B models allow us to automatically identify friendly feature requirements. In this case, each feature can be developed independently, and its correctness guaranteed by refinement. Irrespective of the design decisions taken during the refinement process, friend machines are guaranteed to inter-operate correctly when they are composed. In contrast, when we detect two abstract feature requirements as Enemies, it is mandatory to change the FM by adding a constraint that prohibits the use of these two features, or by manual modification of the FM structure. The 3rd case is the most challenging: when feature requirements are Politicians we must be careful when we develop each of them in parallel. Although each formal refienement guarantees the correctness of the individual feature's behaviour, it risks breaking the coherency with the other feature(s). We currently have no automated technique for managing the complexity that arises when we have to develop a SPL instance in which there are a large number of Politician features.

B. Analyses of Interactions

As mentioned above, each abstract feature has a single Event-B Context that describes its behavior. This feature can have an potential incoherent interactions with other feature during refinement process. For example, the abstract feature V oteCounting, in the Fig. 4, allows to calculate the number of votes for each candidate in the elections. For that, we consider the presidential elections second round with two candidates and the result will be a sorted list generated automatically, that is represented with the configuration {V oteCounting, CountingAlgorithm, V alidityChecking,-Automatic}. Listing 6 represented the Event-B Context of this configuration.

In this Context, the Urn set is part of the domain model common to all e-voting systems. A Vote set (as can be recorded in an urn) is part of the domain model common to all e-voting systems. In the 2nd round of the election this is the first of two candidate options is represented by OPTION1 constant. OPTION2 constant represents the second of two candidate options in the 2nd round of the election. An empty urn is part of the domain model common to all e-voting systems is defined by EMPTY URN. A valid vote, ValidVote constant, is part of the domain model common to all e-voting systems. but the invalid one is represented by InvalidVote constant. The function permitting the addition of a vote to an urn is described by addVoteToUrn. The result of counting the votes in the urn is stored in result constant. count constant represents the function which calculates the result of counting the votes in the urn. In the 2nd round of the election this is the candidate option having the most votes recorded in the urn. when the count is carried out (or an equality if no unique winner exists), elected constant record the candidate in question. countVotesInUrnForOption constant is used as function to calculate the number of votes for a specific option. For the exceptional case when there is no clear winner after the count i.e. we have an equality, this case is representd by TIE constant. Now, we will describe the internal behavior of our valid configuration. Votes can be valid or invalid, but not both, this constraint is part of the domain model common to all evoting systems, axm1. In the 2nd round of the election, only 2 candidates correspond to a valid vote, axm2. At the beginning of the election we need an urn with no votes recorded, axm3. The function permitting the addition of a vote to an urn, axm4. The result of counting the votes in the urn, provides a function mapping valid votes to their final count, axm5. Axm6 describes the function which calculates the result of counting the votes in the urn. In the 2nd round of the election this is the candidate option having the most votes recorded in the urn. when the count is carried out (or an equality if no unique winner exists), axm7. Axm8 defines an utility function to calculate the number of votes for a specific option. When there are no valid votes in an empty urn, it is represented by axm9. Taking a single vote from the urn and incrementing the count for the valid candidate option associated to it (if the vote is for the specified candidate option), this behavior is defined using axm10. Axm11 is used to remove a single invalid vote from the urn, and do not count it. Axm12 is exploited to take a single vote from the urn and do not increment the count as the candidate option is not the one we are lokking for. A utility function to calculate the number of votes for a specific option is defined by axm13. Axm14, axm15 and axm16 are used, respectively, to describe if option1 wins, option2 wins and if we have equality between the two options. This Event-B Context of the Listing 6, shows explicitly the interactions between three abstract features namely: V oteCounting, CountingAlgorithm and V alidityChecking. These features are Politicians, the combined requirements can be met in a coherent manner but we have to be careful during refinement that we do not introduce inconsistency. It is beyond the scope of this paper to detail the way in which the Event-B context requirements models are integrated into an Event-B machine, ready for refinement-driven design.

VI. CONCLUSION AND FUTURE WORK

In this paper, we have presented a formal approach to verifying the correctness of a FM configuration based on variability management and modelling using the Event-B method. We have also exploited requirements engineering and domain analysis of e-voting to propose a general approach, from design to implementation of SPLs, by strengthening and underpinning the correct-by-construction process.

In future work, we plan to extend our formal approach to make it more dynamic. In other words, to permit changes to the generic SPL and specific configurations during execution of the system. The challenge is to use the dynamic (machine) part of Event-B, in order to validate the insertion/deletion/replacement of a concrete/abstract feature.

Fig. 1 .

 1 Fig. 1. Feature Models Transformation and Verification Approach.

Fig. 2 .

 2 Fig. 2. A sample Feature Model and its Legend.

  CONTEXT T reeSpec SETS F EAT U RE CONSTANTS Root N ode SuccOf And Or Xor N ull M andatory SuccOf T ransChildrenOf axm1: f inite(F EAT U RE) axm2: partition(F EAT U RE, {Root}, N ode) axm3: partition(F EAT U RE, And, Or, Xor, N ull) axm4: partition(N ode, And, Or, Xor, N ull) axm5: M andatory ⊆ F EAT U RE axm6: Root ∈ M andatory axm7: SuccOf -1 ∈ N ode F EAT U RE \ N ull axm8: f inite(SuccOf ) axm9: SuccOf [And] ⊆ M andatory axm10: theorem SuccOf [Xor] ∩ M andatory = ∅ axm11: theorem SuccOf [And ∩ M andatory] ⊆ M andatory axm12: SuccOf T rans -1

Fig. 3 .

 3 Fig. 3. Feature Model extract for e-voting Product Line.

  ChildrenOf ⊆ SuccOf axm16: theorem (And ChildrenOf ⊂ And SuccOf ) ∧ (And SuccOf ⊂ And ChildrenOf ) axm17: theorem Or ChildrenOf ⊆ Or SuccOf axm18: theorem Or ⊆ dom(ChildrenOf ) axm19: theorem Xor ChildrenOf ⊆ Xor SuccOf axm20: theorem ¬(Xor ChildrenOf ∈ Xor → F EAT U RE) axm21: N ull ChildrenOf = ∅ axm22: theorem ∀x•(ChildrenOf [{x}] = SuccOf [{x}]) ∧x ∈ Xor ⇒ card(SuccOf [{x}]) = 1 axm23: theorem SuccOf M andatory ⊆ ChildrenOf axm24: theorem ran(Or SuccOf ) ∩ M andatory ⊆ ran(Or ChildrenOf ) END Listing 2. A part of Event-B Context for describing a FM operators: a Context T reeSpec.

Fig. 4 .

 4 Fig. 4. Feature Model extract for e-voting Product Line to model.

  CONTEXT F M V otingConf KO SETS CONSTANTS axm1: ChildrenOf [{V ote}] = {V oteV isibility} axm2: ChildrenOf [{V oteV isibility}] = {nonEncrypted, EncryptedV ote} axm3: ChildrenOf [{nonEncrypted}] = {V AccessRights} axm4: ChildrenOf [{V AccessRights}] = {V Adm} axm5: ChildrenOf [{V Adm}] = {V ADuring, V AAf ter} END Listing 5. An Event-B Context for describing a e-voting FMVotingConfKO configuration.

Fig. 5 .

 5 Fig. 5. Proving FMVotingConfOK configuration in Rodin.

Fig. 6 .

 6 Fig. 6. Proving FMVotingConfKO configuration in Rodin.

  CONTEXT presidentialElectionSecondRound SETS U rn V ote CONSTANTS OP T ION 1 OP T ION 2 EM P T Y _U RN V alidV oteInvalidV ote addV oteT oU rn result count elected countV otesInU rnF orOption T IE axm1: partition(V ote, V alidV ote, InvalidV ote) axm2:V alidV ote = {OP T ION 1, OP T ION 2} axm3: EM P T Y _U RN ∈ U rn axm4: addV oteT oU rn ∈ V ote × U rn → U rn axm5: result ∈ V alidV ote → N axm6: count ∈ U rn → result axm7: elected ∈ result → {OP T ION 1, OP T ION 2, T IE} axm8: countV otesInU rnF orOption ∈ U rn × V alidV ote →N axm9: ∀option•option ∈ V alidV ote ⇒ countV otesInU rnF orOption(EM P T Y _U RN → option) = 0 axm10:∀option, urn•option ∈ V alidV ote ∧ urn ∈ U rn ⇒ countV otesInUrnF orOption(addV oteT oU rn (option → urn) → option) = 1+ countV otesInU rnF orOption(urn → option) axm11:∀option, urn•option ∈ InvalidV ote ∧ urn ∈ U rn ⇒countV otesInU rnF orOption(addV oteT oU rn (option → urn) → option) = countV otesInU rnF orOption(urn → option) axm12:∀option1, option2, urn•option1 ∈ V alidV ote ∧ option2 ∈ V alidV ote ∧ urn ∈ U rn ∧ option1 = option2 ⇒ countV otesInU rnF orOption( addV oteT oU rn(option1 → urn) → option2) = countV otesInU rnF orOption(urn → option2) axm13:∀opt, urn, c•opt ∈ V alidV ote ∧ c ∈ N ∧ urn ∈ U rn ⇒countV otesInU rnF orOption(urn → opt) = c axm14:∀option1, option2, urn• option1 ∈ V alidV ote ∧ option2 ∈ V alidV ote ∧ urn ∈ U rn ∧ (countV otesInU rnF orOption(urn → option1) > countV otesInU rnF orOption(urn → option2)) ⇒elected(count(urn)) = option1 axm15:∀option1, option2, urn• option1 ∈ V alidV ote ∧ option2 ∈ V alidV ote ∧ urn ∈ U rn ∧ (countV otesInU rnF orOption(urn → option1) < countV otesInU rnF orOption(urn → option2)) ⇒elected(count(urn)) = option2 axm16:∀option1, option2, urn• option1 ∈ V alidV ote ∧ option2 ∈ V alidV ote ∧ urn ∈ U rn ∧ (countV otesInU rnF orOption(urn → option1) = countV otesInU rnF orOption(urn → option2)) ⇒elected(count(urn)) = T IE END Listing 6. An Event-B Context for vote counting of presidential elections.


Somia Chikouche
email: chikouchesoumia@yahoo.frbouziane.abderraouf@gmail.com

Abderraouf Bouziane

Salah Eddine Bouhouita-Guermech

Messaoud Mostefai

Mourad Gouffi

Innovation Diffusion in Social Networks: a Survey

Keywords: Diffusion of innovation, Mathematical models, Social networks models

The innovations diffusion is a process of communication by which a new idea spreads among a population. A successful propagating leads researchers to seek the elements, which approve and consequently contribute to its spread, or otherwise, find the elements that prevent it. In fact, many efforts were made to models this social phenomenon; however, each one had its strengths and weaknesses. Therefore, this paper aims (1) to discuss and to analyze existing models showing their utility and limitation, (2) highlighting the detail of their applications, and (3) suggesting a taxonomy, which resumes the state of art. Then, to address the current social networks models, it seems necessary to begin by presenting the diffusion of innovation theory, then to detail models, which do not incorporate the social structure, more precisely the mathematical models.

Introduction

The advent of the Internet, and its universally emerging use, by its Democratic policy, allows any person to be connected to the network, and therefore anyone can benefit, distributes information, and proposes its own services or product as a marketing process. As a matter of fact, this free exchange of information involves unrestricted spread.

It has also encouraged academics to reflect, what are the reasons that encourage the propagation or the diffusion, in a specific case, if it carries a certain innovation. In any social system, the diffusion notion takes a chief place, as its influences the individuals' behavior [START_REF] Steyer | Influence social et diffusion de l'innovation (Social influence and Diffusion of innovation)[END_REF]. Thus, this process is the heart of the society evolution. Studying the innovations diffusion is analyzing and identifying mechanisms that allow the social system to change [START_REF] Rogers | Diffusion of innovation[END_REF]. It is also to recognize the way that encourages a population to adopt an innovation whose advantage is not immediately observable [START_REF] Rogers | Diffusion of innovations[END_REF].

In other words, the research in diffusion is mainly to provide more possibility to comprehend how and why this phenomenon do well [START_REF] Rogers | Diffusion of innovations[END_REF], [START_REF] Rogers | Diffusion of innovation[END_REF], [START_REF] Rogers | A prospective and retrospective look at the diffusion model[END_REF], [START_REF] Rogers | Diffusion of innovations[END_REF], [START_REF] Valente | Social networks and health[END_REF], [START_REF] Valente | Evaluating health promotion programs[END_REF]. The study of the innovation diffusion affords an insight to the adoption prediction. To expound and analyze this social phenomenon, the diffusion of innovation theory was introduced as a general framework. This theory is the ultimate prominent behavioral application of network investigation [START_REF] Valente | Social networks and health[END_REF], [START_REF] Valente | The origins and development of the diffusion of innovations paradigm as an example of scientific growth[END_REF].

Coming back in history, the psychologist used the mathematical models as a behavior quantitative theory [START_REF] Suppes | Models of data[END_REF]. After a duration, an answer is wanted, how much could mathematical models support social science? [START_REF] Apolloni | A study of information diffusion over a realistic social network model[END_REF]. Epstein [START_REF] Epstein | Why model?[END_REF] maintains that these models may possibly explain the social phenomenon but have not the possibility to predict its future. Therefore, Silverman and Bryden [START_REF] Silverman | From artificial societies to new social science theory[END_REF] recommend simulation because it offers an optimistic solution to social science. The current development of technologies affords the simulation with more opening and capability to study complex systems comparing with mathematical models [START_REF] Jong | Evolutionary Computation A Unified Approach[END_REF]. Axelrod [START_REF] Axelrod | Advancing the art of simulation in the social sciences[END_REF] explains that the fact that in social science, the simulation helps to take into account the interaction and its influence. In other word, the simulation takes into account the social network structure. Models and simulation approaches provide a better understanding of this social process and give convincing answers to these questions [START_REF] Thiriot | Vers une modélisation plus réaliste de la diffusion de l'innovations à l'aide de la simulation multi-agents (Towards more descriptive models of innovation diffusion using multi-agents simulation[END_REF]. Geroski submits "We use models to help illuminate phenomena that we find difficult to understand or to solve problems which are too difficult to think through. These benefits come because models simplify reality" [17, p. 621].

The richness of literature prompted the study to propose a taxonomy that summarizes the state of the art on this subject. The objective is to provide analysis around the social networks models. To correctly place this problem, some basic concepts will be first defined, then in Section 2; the mathematical models will be explained and analyzed. Moreover, Section 3 will dealt the social networks models by presenting the necessary details. Finally, section 4 will conclude the work's findings.

2

The Diffusion of Innovation Rogers [6] defines the innovation diffusion as "the process by which an [START_REF] Steyer | Influence social et diffusion de l'innovation (Social influence and Diffusion of innovation)[END_REF] innovation is communicated through certain (2) channels over (3) time among the members of (4) a social system".

(1) The innovation is anything perceived as new by the potential adopters.

(2) The communication channels are the medium whereby the message is transferred and exchanged. (3) The time is key factors, because whatever the innovation contribution, it diffuses slowly in the population. Thus, it requires time. (4) The social system is a set of interdependent units that are got together to achieve a common objective.

The following definition and assumption are important to the understanding of the subsequent models.

Adoption rate

It presents the level of innovation adoption by the population as a function of time.

Initially, the diffusion rate goes up pretty much slowly. Then, rises over time, which leads to a period of rapid adoption from the first period, until saturation. It draws an "S" curve, which should be guaranteed by the diffusion models [START_REF] Mahajan | Models for innovation diffusion (quantitative spplications in the social sciences)[END_REF].

Cycle of innovation diffusion

Rogers [START_REF] Rogers | Diffusion of innovation[END_REF] in the book of 1962 defined different types of attitudes towards the innovation. Its idea is to associate the different types corresponding adopters to the different phases of adoption of a novelty where some people are more open to newness than others do. These stages and rates draw a curve in "bell".

Innovators (2.5%).

Courageous people, aim change. They are individuals with a strong ambition towards new ideas or technologies "the taste of adventure", they prefer to be the first to possess them. They present innovation to their social system, they have very significant communication mechanisms.

Early Adopters (13.5%). They follow very closely the innovators. They try new ideas, but in a careful way. They are educated and more integrated into the social system than innovators, their opinions matter and can facilitate the diffusion process. They are the opinion leaders and with whom other members get information and seek advice on innovation.

Early majority (34%). They are thoughtful, pragmatic and do not seek changes but developments that improve life. They have many social contacts. They accept change more quickly than others do. They adopt innovation as a group with the accumulation of positive opinions. It represent alone a third of individuals. The one by which the innovation can be considered to be in diffusion on a large scale. Because it serves as a transmission link between members who relatively adopted the innovation early, and those whose decision is tardy. Therefore, it is the privileged target of companies because of its quantity and its ease to convince.

Late Majority (34%). They are conservative skeptics; they accept it only when the majority has already adopted it. They are the ones who undergo social pressure before deciding. Indeed, the late majority adopts innovation more by social pressures than by actual predisposition. They also represent 1/3 of the members of the system. If it resembles the early majority, it differs from them in the motivations for its adoption of innovation.

Laggards (16 %).

They are traditional people, very skeptical, they like to be glued to "old ways", and will admit them only if the new idea has become the mainstream or even the tradition. Sometimes they are critical about new ideas and they expect a great reliability of innovation, they adopt only by constraint or absolute necessity. They are volunteers who refuse to innovate.

Mathematical Models

Several mathematical models have been suggested to model the spread of information among the population. In this section, three well-known models are detailed:

Logistic model

It is based on the cumulative distribution patterns. It follows a growth model, which has proven to be a consistent pattern by empirical research on the diffusion of innovation [START_REF] Mahajan | Models for innovation diffusion (quantitative spplications in the social sciences)[END_REF]. It is approximated by a logistic function and described by the following equation

F (t) = P/ r * e -s t ( 1
)
Where t is the time P and s are positive real numbers, r any real. This function was proved by Pierre François Verhulst in 1845 [START_REF] Verhulst | Recherches mathématiques sur la loi d'accroissement de la population (mathematical researches into the law of population growth increase[END_REF]. He proposed as a model of evolution of the non-exponential population with s steepness and capacity P.

Gompertz model

It is a mathematical model, where the development begins and finish slowly. This sigmoid function is come close gradually to  value. This function refers to Benjamin Gompertz [START_REF] Gompertz | On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies[END_REF].It is an exceptional case of the logistic function. Gompertz function is described by

F (t) = e -et (2)
Where ,  and  are positive real numbers.

Bass model

A remarkable expansion took place in the marketing literature on the diffusion 1970s and the largest enhancement to this scientific explosion is a model for predicting of the spread of new consumer products proposed by Frank Bass [START_REF] Bass | A new product growth model for consumer durables[END_REF] in 1969. Bass prediction model has become so important in the field of marketing because it offers some plausible answers to the uncertainty linked to the introduction of a new product in the market. Some of the largest US companies have used the Bass model [START_REF] Rogers | Diffusion of innovations[END_REF].

To better describe the distribution, particularly to take into account the effect of social pressure Bass defines two main forces driving adoption: innovation and imitation. These explanatory factors reflect the fact that part of individuals adopts by social pres-sure of individuals who have already adopted, while some adopt by interpersonal persuasion. Diffusion speed is proportionate to (a) the population that has already adopted the service, denoted by f(t), and (b) the remaining market potential represented by M F(t).

From the 19th century, the sociologist Gabriel Tarde [START_REF] Tarde | The laws of imitation[END_REF] has well highlighted the importance of social influence and the way it is at the base of the formation of a value of the invention, based on its diffusion [START_REF] Rogers | Diffusion of innovation[END_REF]. He used the word "imitation" to describe the adoption; in his book, he called "the laws of imitation" which published it in 1903. The Bass model is a predictive model of the future adoption of a product. To apply, you must first define the market size M, then the coefficients p innovation and imitation q, and it can estimate the number of adoptions f (t) in any specified time. The equation is f tM p+ (qp) F (t)q/M F (t) 2 (3)

Discussion

For the scientist who attempts to predict the future, there is always a need for simple models that describing events. Logistic and Gompertz models could be used to compare the growth rates for different innovations [START_REF] Valente | Network models and methods for studying the diffusion of innovations» Models and Methods in Social Network Analysis[END_REF]. Each such model should be based upon easily understood assumptions; that support the forecaster in his efforts to make the future what he wants it to be. The models aforementioned are simply based on clearly coefficient. The best-known diffusion models used for technology diffusion purposes are the Bass model, the logistic family models [START_REF] Bewley | A Flexible Logistic Growth Model with Applications in Telecommunications[END_REF], as well as the Gompertz model [START_REF] Gompertz | On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies[END_REF] .

For several decades, great attention turned to the Bass diffusion model [START_REF] Rogers | Diffusion of innovations[END_REF]. The estimation of accurate parameters was the most important point for getting a proximate forecasting [START_REF] Wang | Implementing a diffusion model optimized by a hybrid evolutionary algorithm to forecast notebook shipments[END_REF]. To deal with this problem, the evolutionary approach is a practical solution. For example, Venkatesan and Kumar [START_REF] Venkatesan | A genetic algorithms approach to growth phase forecasting of wireless subscribers[END_REF] joined the diffusion model of Bass with the evolutionary technique precisely the genetic algorithms (GA). They aim to predict future sales in the telecommunications sector. Three main genetic operations used: reproduction, crossover, and mutation. By using cellular phone adoption data, they found out that the integration of genetic algorithm gives more robustness and produces more consistency comparable to non-linear least squares(NLS), ordinary least squares(OLS), and time series model. A further illustration is a work of Wang et al. [START_REF] Wang | Implementing a diffusion model optimized by a hybrid evolutionary algorithm to forecast notebook shipments[END_REF], where a hybrid approach that combines the genetic algorithms with the particle swarm optimization (PSO) were proposed. By comparison with NLS, GA and PSO, the consequence of this conjunction provides better predictions, and more accurate parameters for forecasting the notebook shipment.

A significant number of existing models based on the use of parameters that determine the process of adoption of innovation and simple mathematical functions concentrated on observation and description of patterns of diffusion. These models allow more explicit diffusion pattern, but require the estimation of diffusion coefficients, obtained from historical data or time series. This raises the problems of the application in contexts where there is no data or the data are insufficient [START_REF] Sampaio | Diffusion of innovation simulation using an evolutionary algorithm[END_REF].

The problem with these mathematical models that is impossible to reproduce the failure of the diffusion of innovation, except by artificially placing the potential adopters at 0, while potential adopters may exist but not be reached during the propagation. The aggregative approach itself does not take into account network effects, geographical boundaries, or the heterogeneity of people. Precisely, the difference between Rogers's categories of adopters is overlooked.

Modeling at this macro level, however, is imprecise because it assumes the highconnected social network, in which each person interacts with all world [START_REF] Valente | Network models and methods for studying the diffusion of innovations» Models and Methods in Social Network Analysis[END_REF] [START_REF] Granovetter | Threshold models of collective behavior[END_REF]. Pelc [START_REF] Pelc | Diffusion of innovation in social Networking, selected concepts, issues and cases[END_REF] aimed at encouraging researchers to integrate the social network where interactivity level is so important. Epstein [START_REF] Epstein | Why model?[END_REF] addressed this issue, stating different reasons for building social models: mathematical models could explain (not predict) the emergence of collective phenomena, or capture qualitative behavior of phenomena. Such criticism strongly limits the description of the mathematical models. To take into account the heterogeneity of the population and the network effect, the models are moving towards approaches called individual-oriented models where the social structure and the communication channels are taken into account. The following section probes into the social networks models.

4

Social networks models

Threshold model

The way that the social network influences the diffusion process, is the principal focus of threshold model [START_REF] Valente | Social network thresholds in the diffusion of innovations[END_REF]. This model has been assumed to understand the process of innovation diffusion [START_REF] Valente | Network models and methods for studying the diffusion of innovations» Models and Methods in Social Network Analysis[END_REF].The threshold is the fraction of adopters in a system needed for an individual to be an adopter. Valente [START_REF] Valente | Social network thresholds in the diffusion of innovations[END_REF] applied this concept as an extension of Granovetter's [START_REF] Granovetter | Threshold models of collective behavior[END_REF] work, where, the individual's decision depends on the behavior of others in the group or the system. For the reason that the individuals are more effected by facts received from personal network. Valente [START_REF] Rogers | Rise of the classical diffusion model[END_REF] proposes using the direct communication network rather than a system to which the individual belongs. The exposure Ei is the proportion of the adopters in the personal networks. It is measured by

Ei ∑ wij yi/ ∑wi (4)

The principle of threshold models is as follows: an individual changes its state if a sufficiently large proportion of his neighbors are in this state. This proportion constitutes the threshold of the individual. The diversity of these thresholds, as well as the position of the individual on the social network, create a heterogeneity in the population. The linear threshold model is derived from threshold model.

Linear Threshold Model. Suggested by Kempe et.al [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF]. LTM works as follow: the individual is influenced by each neighbor according to a weight. A uniform random threshold from 0 to 1 are assigned to each individual I. This threshold is imperative to I to became an adopter. For each step of simulation, an activated sequence of individuals activate similarly their neighbors' and so on. The propagation ends when there are no further inactive person.

Epidemic Approach

It is easy to observe the relation between epidemic disease and the diffusion of information. The Influenza for example, is a contagious disease that spread from person to person. Epidemics can pass through a population, or they can persist over long time periods at low levels. As epidemics can spread between people, ideas and information can also spread from person to person, across similar kinds of networks that connect people. The Diffusion of innovation is usually treated as social contagion [START_REF] Ronald | Social contagion and innovation: Cohesion versus structural equivalence[END_REF]. Similar to the spread of disease, as a reaction chain phenomenon. First there are some adopters, while members of their networks adopt, then they move on to their own networks and so on. First slowly, then faster and faster and then slows down again because potential adopters are reduced. This approach assumes that exposure to information enough to become informed, and potentially transmit the information to someone else. However, we know that all individuals do not have the same propensity to adopt and transmit the information immediately after its receipt. Mathematically, this phenomenon is best described by an S curve, it can be easily constructed from an array of single frequency of adoption time. The epidemic approach assumes that meetings between individuals can appear randomly in the population. To simulate the diffusion of innovation, where the main channel of communication is the word-of-mouth, this approach is the most appropriate for this case [START_REF] Geroski | Models of technology diffusion[END_REF]. Indeed, it is generally used to describe the transmission of innovations [START_REF] Easley | Networks, Crowds, and Markets:Reasoning about a Highly Connected World[END_REF].

The SIR Epidemic Model. The state of individuals changes through three possible stages:

Susceptible. The individual is suspected to catch the disease from its neighbors.

Infectious. If the individual catch the disease, he became infectious and has some probability to infect his neighbors.

Removed. After some period of infection, the individual became removed from consideration.

Ba et al. [START_REF] Ba | Étude de la diffusion d'innovations en milieu rural à l'aide de simulations multi-agents[END_REF] have designed a model that is based on the epidemic approach. To take into account the heterogeneity of the population, which is a real fact, they inspired by the Granovetter threshold models. They simulated the innovation diffusion in Senegalese rural environment by using Netlogo development software. The simulation was done using multi-agent systems. The network on which the diffusion took place was generated using an interaction network generator. This simulation has led to some interesting conclusions: on the first part, the geographical disparity that separates individuals does not in itself constitute a barrier to the wide dissemination of information; on the other part, an important factor is the social structure of the environment. The social network and links on through which the influence of external sources happens is the central interesting for Myers, Zhu and Leskovec [START_REF] Myers | Information diffusion and external influence in networks[END_REF]. They investigate how information spread in the network from node to node. They apply the contagion and exposure models. The results confirms that the effects of external sources on the information diffusion. Namely, by quantifying this degree of influence over time, they argue that the information is likely to "jump" across the network.

Evolutionary Model

Despite the promising outcomes and recent interest in modeling social systems, it was pointed out that in many cases; models lack realism and do not take the realities of the field into account [START_REF] Apolloni | A study of information diffusion over a realistic social network model[END_REF]. In recent years, numerous publications have appeared dealing with the incorporation of the evolutionary perspective to the social science, this integration known as evolutionary psychology. This combination gets a chief place within psychological science [START_REF] Buss | Adaptations, exaptations, and spandrels[END_REF]. The evolutionary psychology is a field that apply the evolutionary concepts in psychology arena. Buss et al. [START_REF] Buss | Adaptations, exaptations, and spandrels[END_REF] highlight the importance of incorporating the evolutionary perspective to social science. The evolutionary principal base on the notion that the individuals with more effective characteristics have more probability to replicate and live [START_REF] Buss | Adaptations, exaptations, and spandrels[END_REF]. The evolutionary computation need only a little specific problem knowledge. Thus, they can be applied to deal with a varied range of problems [START_REF] Streichert | Introduction to evolutionary algorithms[END_REF].

Nevertheless, a little available literature on innovation diffusion that applies techniques based on evolutionary computation [START_REF] Sampaio | Diffusion of innovation simulation using an evolutionary algorithm[END_REF]. An example of the evolutionary algorithm is the paper of Sampaio et al. [START_REF] Sampaio | Diffusion of innovation simulation using an evolutionary algorithm[END_REF]. They recommend the evolutionary algorithm to simulate the diffusion of innovation process. The evolutionary computation is suitable to model the progress of adoption decision process. Their central support is Rogers' innovation diffusion theory and the threshold approach. Another important factor that was taken into account, is the knowledge function of Verhulst [START_REF] Verhulst | Recherches mathématiques sur la loi d'accroissement de la population (mathematical researches into the law of population growth increase[END_REF]. This function is used to simulate the evolution accumulation acquaintance by time. The result was compared to Logistic, Gompertz, and Bass models. The effect of first adopters and the social structure are studied.

In a previous work [START_REF] Chikouche | Evolutionary knowledge based on human interaction model for the innovation diffusion in social networks[END_REF], we proposed a novel evolutionary model based on the human interactions as an evolutionary learning process, because the decision does not occur directly from the first exposition to the innovation, but it follows an evolutionary process affected by the other neighbors already been adopters. We used three social networks type: regular ring network, lattice network on a torus and random network. The simulation proved the capability of the model to produce the S-shaped diffusion pattern, it is built on two associated factors: the individual decision process and the social learning influence. Consequently, the paper may offer a significant outline to better understand the subject of human behavior.

Yavaş and Yücel [START_REF] Yavaş | Impact of homophily on diffusion dynamics over social networks[END_REF] addressed the way the homophily in the social network can affect the diffusion process. An agent-based simulation implemented model demonstrates that the homophily is self-reinforcing especially in its early increases. It is, in other words, more supporting at the beginning of the diffusion rather than being so later. In addition, by applying the evolutionary homophilous network, the extent of macro high homophily degree gets more influence than the one of individual's local neighborhood preference. Furthermore, Cowana and Jonard [START_REF] Cowana | Network structure and the diffusion of knowledge[END_REF] focused on the relationship between the network topology and the diffusion process. They examined the effects of incremental innovations diffusion by means of three different network architectures. Additionally, they implemented heterogeneous agents connected exchanging information over time. This interaction involves the knowledge, which is represented as an evolutionary vector.

4.4

Discussion: The elements of the innovation diffusion are differently considered in the process of modeling. Rogers [START_REF] Rogers | Diffusion of innovations[END_REF] mentioned four important elements that are: (1) the innovation, (2) the time, (3) the social system and (4) the communication channels. The threshold models showed particular interest on the social structure, making distributed probability to create the social heterogeneity. The question is what about a personal network with a minimum of communication? However, the impact of communication channels was the focus of epidemiology approach and the evolutionary models. Another problem, is our choices depend only on the social pressure? While all the mentioned models ignored the various innovations characteristics, the need of a model that take into account the innovation features' is vital.

Summary

The innovation diffusion models converge to similar findings that are the "S" diffusion curve. Fig. 1 illustrates a taxonomy that resumes the state of art in the subject of innovation diffusion.

    Gompertz     Bass     Threshold     Epidemic     Evolutionary    
The table. 2 gives a summary of the innovation diffusion elements more or less considered by the abovementioned models.

Conclusion

When new idea, activity or object are invented, diffused, adopted or rejected, social changes occur. Research in this field has produced various theoretical models. A survey of mathematical models and social networks models for innovation diffusion was presented. Rogers's theory suggests that the adopters of any innovation can be classified into five categories: innovators, early adopters, the early majority, the late majority, and laggards. This theory describes the mechanisms that drive the diffusion of innovation [START_REF] Rogers | Rise of the classical diffusion model[END_REF]. The diffusion of innovation leads to the S-shaped curves representations, this object is guaranteed by the most of the models, the logistic and Gompertz models are good examples, which is much more used to demonstrate the cumulative effect. In 1969, inspired by Rogers and other theorists of the domain, Frank Bass developed a quantitative model that was as a reference to quantify the rate of adoption of an innovation in a population [START_REF] Rogers | Diffusion of innovations[END_REF]. The mathematical models concentrate on the rate of adopters. A time ago, the innovation diffusion was only measured by the rate of adoption.

However, taking into account the interactions between individuals is critical [START_REF] Pelc | Diffusion of innovation in social Networking, selected concepts, issues and cases[END_REF], because, they are the main driver of the evolution of individuals [START_REF] Rogers | Diffusion of innovations[END_REF], which the mathematical models do not take into account. The individual-oriented approach was a solution to integrate the social network. Later, several models have been introduced, in order to better comprehend the process and to incorporate the essential elements. These social network models confirm hypotheses about the causes of success or failure. The designed models still need some improvements like the integration of the innovation characteristics in the process of conceptualization. A taxonomy of models discussed and analyzed in this paper. These existing models greatly differ in their goals, as well as in their approach. The paper aims to offer some advance on this area and helps the novel researchers to understand the field of the innovation diffusion models.

Fig. 1 .

 1 Fig. 1. Represent a taxonomy of the innovation diffusion models.

Table 1 .

 1 The elements of innovation diffusion theory considered by the mentioned models.

	Models	Innovation	Social network	Communication	time
	Logistic


Hector Uriel

Rodriguez Vera

Fabien Baillon

Philippe Accart

Fabienne Espitalier

Olivier Louisnard

Hector Uriel Rodriguez Vera

Crystallization of α-glycine by anti-solvent assisted by ultrasound

Introduction

In this work, we studied the crystallization of α-glycine by addition of an antisolvent (ethanol) assisted by ultrasound. The glycine is an amino acid presenting three polymorphs: α-glycine, γ-glycine and β-glycine. At room temperature, the γ-glycine is the most stable [START_REF] Yang | Polymorphism in the crystallization of glycin[END_REF][START_REF] Bazi | [END_REF]. It is soluble in the water and insoluble in ethanol.

Previous works on cooling crystallization of glycine with ultrasound in aqueous phase showed that a purer crystalline phase with moderate power is formed with more regular faces [3] and a decrease of the induction time when the ultrasonic power increases [START_REF] Baillon | Power ultrasonics: Applications of high intensity ultrasound[END_REF].

Materials and Methods

Pure α-glycine is used for experiments (VWR CHEMICALS, 99,9 % purity). The experiments of crystallization are conducted in a solution of 150 ml shaken by a magnetic rod. A solution of concentration 0.26 kg α-glycine / kg water is prepared at 35 °C (5 °C above the temperature of saturation). This solution is then cooled at 30 °C (temperature close to the temperature of saturation of the initial solution). The ultrasound is then applied to various powers thanks to a plunging transmitter (generator Bioblock, 20 kHz). Once the temperature is stabilized at 30 °C, ethanol is added at a constant flow rate by means of a syringe pump. The addition rate varies between 0.04 in 0.36 g of ethanol/min. The measurement of refractive index of solution permits to follow the concentration of glycine C glycine during the experiment. The supersaturation ratio (S) is calculated by the ratio of C and C α-glycine , eq . C α-glycine , eq is the equilibrium concentration of α-glycine measured for this study. At the end of the experiments, the suspension is filtered, the solid washed with ethanol and dried at 60 °C. The final crystals are characterized by their final size distributions measured by laser granulometry, their morphologies observed by scanning electronic microscope (SEM) and their crystalline structures by differential scanning calorimetry (DSC). The influence of ultrasonic power (continuous or pulsed modes) measured by calorimetry method is studied for different addition rates. The maximal supersaturation ratio S max and the induction time (time between S = 1 and S max , Fig. 1) are observed.

Results and Conclusion

Analysis by DSC with a heating rate of 5 °C/min between 20 and 200 °C show no endotherm of transformation of α-glycine towards γ-glycine [START_REF] Yang | Polymorphism in the crystallization of glycin[END_REF]. This means that pure metastable α-form is obtained whatever the operating conditions. A decrease of induction time and of S max is observed in the presence of ultrasound (Fig. 1). In presence of ultrasound, smaller crystals with a monomodal size distribution are obtained at the end of the crystallization (final supersaturation ratio equal to 1). The rate of desupersaturation is higher in presence of ultrasound inducing a higher nucleation rate or a higher growth rate or both. From experimental results, modeling of crystallization will permit to conclude on crystallization rates. The use of pulsed ultrasound (50 one/50 off) seems to be interesting because similar results are obtained: same size distributions and same concentration profiles.

Figure 1 .

 1 Figure 1. Supersaturation ratio as function of time for experiment conducted with an addition rate of 0.19 g ethanol/min in silent condition or with ultrasound (13, and 40 W)

Figure 2 .

 2 Figure 2. Size distributions of final crystals for experiment conducted with an addition rate of 0.19 g ethanol/min in silent condition or with ultrasound (13 and 40 W)