E Meillot

R Vert

C Caruyer

D Damiani

M Vardelle

Manufacturing nanostructured YSZ coatings by suspension plasma spraying (SPS): Effect of injection parameters

Keywords: suspension plasma spraying, shadow imaging, coating, trajectory deviation, mechanical properties, hardness, nano indentation, micro Vickers

Manufacturing nanostructured YSZ coatings by suspension plasma spraying (SPS): effect of injection parameters

E Meillot, R Vert, C Caruyer, D Damiani, Michel

Introduction

Surface treatment processes allow the deposition of various materials on manufacturing parts to improve their properties. Among them, plasma spraying is well adapted to ceramic coating deposition. The conventional plasma spraying uses micrometer size powder, injected with a carrier gas into the plasma plume, to produce thick layers, generally from 100 µm to close or more than the millimeter. To decrease the thickness of coatings, smaller size powders could be used, for instance nano size materials. In this case, the small particle mass requires the use of liquid medium, like water or ethanol, to increase the particle momentum.and ensure their penetration into the plasma jet. The nano particles are put in suspension and pushed to the plasma jet by pressure via an injector close to the torch exit. After evaporation of the medium, the particles are treated in the plasma plume and impact the substrate and so generate a coating by piling up of lamellae.

Such deposition process appears as good candidate for thermal barrier coatings. In the context of the next generation of nuclear energy systems selected by the Generation IV International Forum, studies [START_REF] Bouchard | Generation IV International Forum[END_REF] have been engaged to determine and develop materials as first insulating wall capable of operating under high temperatures (over 850°) and resisting at high pressure variations and wear (7 MPa of helium flow).

In this paper, investigations upon introduction of YSZ (Yttria-Stabilized Zirconia) suspension into DC plasma flow are presented. After a short description of the operating parameters, interactions of water jet with the plasma plume is described by two ways (numerical and experimental). Then, coatings are realized and characterized through their microstructures and properties. The first characterizations concern the mechanical coating properties to determine their wear behavior to the high pressure helium flow.

Operating conditions.

The plasma spraying conditions are summarized in Table 1, while the injection positions are represented in Figure 1. The choice of a high flow rate of Argon/Hydrogen/helium mixture and the high current intensity comes from the requirement of high velocity and temperature to get dense coatings [START_REF] Vert | Adhesion of YSZ suspension plasma-sprayed coating on smooth and thin substrates[END_REF]. However, the use of hydrogen involves a modification of the electrical arc behaviour in the torch, from a takeover mode to a restrike one due to the erratic arc root movement at the anode wall [3], [START_REF] Duan | Arc Instabilities in a Plasma Spray Torch[END_REF]. This fact implies that the downstream plasma flow is not stable but presents stretching and shrinking with hot gas puffs and modification of the gas velocity. These instabilities lead to different liquid treatments depending on the instantaneous state of the plasma [START_REF] Fazilleau | Phenomena in Suspension Plasma Spraying. Part 1: Suspension Injection and Behavior[END_REF] with more or less penetration of the liquid in the jet.

Plasma torch

Suler-Metco F4-VB

Nozzle diameter (mm) 6

Plasma gas mixture Ar/He/H2

Plasma gas flow rate (slpm) 45/45/3

Arc current intensity (A) 700

Plasma effective power (kW) 31 ± 0.2

Specific enthalpy (MJ/kg)

21.4

Injector diameter (µm) 250

Table 1: Plasma operating conditions agglomerates. The solid concentration in the suspension was 6% by weight without any dispersant since no sedimentation was observed during a time exceeding the deposition time.

Interaction between liquid jet and DC plasma flow

This study concerns the introduction of the liquid into the plasma flow. This point is of high importance because it conditions the future treatment of the liquid and particles in the plasma flow. So, two approaches have been used, the first one by experimental imaging of the material in the plasma flow, the second by modelling. A preliminary experiment has been carried out: the continuous water jet interacted with an air cross flow (Figure 2) and was analysed by shadow imaging technique

. The same experiment has been done by replacing the air cross flow by the DC plasma flow. For the plasma flow, simple calculations (without integrating compressible effect velocity and density gradients) lead to a Weber number around 500 (Figure 3) while the air cross flow leads to a Weber number around 250. According to [START_REF] Pilch | Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced brakup of a liquid drop[END_REF], this Weber number defines the low limit of a catastrophic break-up with the stripping of the matter. The fact that the fragmentation shown in the two images (highlighted by the arrows) are quite similar implies that the thermal effects are negligible upon the fragmentation and that the hydrodynamic effects are the first to take into account. Two different configurations of water injection (Figure 1) have been investigated at different injection pressures (from 0.25 to 0.65 MPa). First, the injector was located below the torch axis (this configuration is called "down to up") and the liquid flowed up to the plasma jet; then, the injector was located above the torch axis ("up to down") and the liquid fell down to the plasma jet. Instantaneous images (exposure time of 1µs) are represented in Figure 4; the three black horizontal lines represent the jet centreline and the up and down limits of the plasma torch. As expected, an increase in liquid pressure makes the liquid penetrate more easily in the jet. At this point, no significant effect of gravitation seems visible. The introduction of the continuous jet in stair-shape could be due to the unsteadiness of the plasma motion depending on the movement of the electrical arc inside the torch Each instantaneous image has been treated with a low-pass filter to keep only the minimum signal of each image. Then, a new image has been composed by superposing one hundred images to get the shape of the fragmentation zone, Figure 5 shows the resulting image and therefore the droplets treatment zone in the plasma jet. This figure illustrates the effect of the gravitation: for instance, at 0.65MPa, the "down-to-up" case shows a treatment zone completely integrated in the plasma flow while in the opposite way, the liquid shape trends to get out of the plasma by the bottom. For the other pressure cases, gravitation seems not significant because the liquid has more difficulty to reach the opposite limit of the plasma. From these images, it seems realistic to conclude that most of the particles are in the plasma jet but with a high dispersion. So the next point is devoted to determine if some differences exist between the different configurations of liquid injection.

Each image has been divided in thin slices (200 µm wide) perpendicular to droplet trajectories. In each slice, the maximum of signal intensity has been searched to determine the average maximum trajectory in the jet. The results are represented in Figure 6. For all the pressures, the liquid jet is well introduced in the plasma flow (assuming the plasma jet diameter is around the anode diameter in this area). This figure also highlights the difficulty of the liquid to penetrate into the plasma jet depending on the liquid pressure and therefore the difference of particle treatment due to the high gradient of velocity. In the "down-to-up" case, the trajectory does not reach the torch axis whatever the pressure while it does in the opposite way for the upper pressure (more than 0.50 MPa). The difference in trajectory involves different particle treatments in regard to the temperature and velocity zones because of the high gradients of the plasma flow (see Figure 10 on the left side). First, two alone droplets are injected close to the plasma jet (respectively Figure 7 and Figure 8). These classic cases correspond to a fragmented liquid jet from an injector diameter of about 40 and 110 µm respectively according to the Rayleigh-Plateau theory [6]. In Figure 7, due to the important surface tension forces, a bag is formed around the droplet and is fragmented into several secondary droplets with a mean diameter of about 8 µm. In Figure 8, the initial droplet is stretched and the formation of a thin film with surface waves is observed. As no thermal effect is taken into account in the liquid, this film stays at the surface of the gas flow without evaporation. Its thickness is about 3 µm. Then, the film is fragmented into small secondary droplets, but these droplets stay again at the boundary of the plasma flow and seem not more penetrating into the jet. For the train of droplets, several identical droplets are injected. The injection frequency is about 22 10 4 droplets/second. In Figure 9, the droplets seem to have again difficulty to penetrate the plasma but the high introduction frequency involves an increase in interaction between the droplets issued from the fragmentation and the plasma flow: this leads to trajectories closer to the plasma axis. This can be seen at the bottom-left with the smallest droplets which disperse in the flow. At this point, the thermal effects in the liquid have to be included in the model to evaporate the water involving ceramic particles flight. This image highlights the necessary mass effect due to numerous-primary-droplets injection to facilitate the penetration of the secondary droplets in the flow. For the continuous liquid sheet injection, the calculation domain was changed and it was a box of dimension 10 mm × 5 mm × 5 mm. As seen in Figure 10 right, the penetration of the liquid into the plasma is quite easier: the liquid column reaches nearly the plasma torch axis with some distortions and begins to break up in large droplets. It should be noted that the calculation domain involved more than 1.5 million of meshes to correctly model the fragmentation and get small droplets. To conclude this part, it seems easier to use a continuous liquid jet to carry on the suspension into the plasma flow core in order to get an efficient treatment of the liquid and therefore, of the ceramic particles. Big droplets, or continuous jet, bring more liquid matter which takes more time to evaporate in contrary of small droplets that evaporate rapidly and so cool down the plasma in a shorter time. So, the plasma plume properties will be completely different depending on the type of liquid injection and fragmentation. This will be confirmed in the next part with the experimental approach.

Coating properties.

Coatings have been realized under Table 1 conditions at a spraying distance of about 40 mm. The substrate surface temperature was about 400°C at the beginning of the spray process, it was monitored by an infrared pyrometer (Modeline 4, Ircon, 8-14 µm wavelength range) and controlled during the spray process by CO2 cryogenic cooling to maintain the substrate temperature close to 400 °C. 4.1. Coating architecture and thickness. The microstructures of coatings were observed by Scanning electron microscopy (SEM, Philips XL30), either with secondary electron (SE) or back-scattered electron (BSE) modes. The thickness of each coating has been estimated from SEM micrographs (ten measurements).

The architecture of coatings looks like that of Figure 11 whatever the position of the injector for the constant injection pressure.

The only effect of the liquid injection parameters concerns the coating thickness (Figure 12) with a higher deposition efficiency rate for the down-to-up position which decreases when the pressure rises up to 0.6 MPa. These different results can be compared with the mean trajectories of the particles which are presented in section 3.1. In fact, for the down-to-up injection, more particles seem to stay in the plasma flow than for the up-to-down injection. So more particles are correctly heated and accelerated and so the deposit rate is higher. Concerning the highest liquid injection pressure, a part of the particles goes out of the plasma jet, so fewer particles participate to the coating generation. 2. Surface topography. Surface roughness has been measured using a surface roughness tester (Perthometer PGK 120). The average roughness Ra, defined as the arithmetic mean of the profile irregularities from the mean line, is used to quantify the coating surface roughness. The relationship between coating surface roughness values and liquid injection pressure is illustrated in Figure 13: the average roughness decreases with the liquid injection pressure. The coating surface state is a consequence of the pilling-up of the lamellae. The decrease in roughness with pressure indicates differences in particle velocity at impact, and thus differences in particles thermal treatment in the plasma jet. As seen in Figure 6, when the injection pressure increases, more particles reach higher gas temperature and velocity zones close to the plasma jet axis leading to a more complete thermal treatment. So the coating surface is more homogeneous. But this seems contrary with the architectures which are identical for all coatings (Figure 11). The particles, which take part in the coating, result from droplets with trajectories close to the torch axis. In fact, the droplet dispersion in the plasma jet is important and only a percentage of the droplets are sufficiently treated and produce nanometre size particles that flatten on the substrate with high velocities. For the pressure range, there are always enough particles in the control zone to build a coating and the microstructures look alike. On the other hand, coating thickness and roughness which depend on the particle size are more sensitive to liquid injection pressure. Then Micro-Vickers indentation test has been performed on the top of the coating: it reveals a slight difference between the up-to-down and the down-to-up injection. This difference might be due to the difference of the coating thickness. In fact, the indentation depth is high in comparison to the thickness of coatings and so, the hardness of the substrate can play a role during the test. A mathematical model, presented in [START_REF] Chicot | Absolute hardness of the films and coatings[END_REF], gives the possibility to adjust the hardness value by the Jönsson and Hogmark model [17]:

) ( ) ) ( 2 ( 2 s f s m H H d t c d t c H H
Where H m is the measured coating hardness, H s the substrate hardness, H f the real hardness, t the coating thickness, d the Vickers print diagonal and c a constant, (0.5 for fragile materials, 1 for ductile ones). By using the value of 0.5 for c, the results are in the same range than the values obtained by nano indentation, with no effect of the suspension injection parameters (Table 3).

Conclusion.

Investigations upon manufacturing of YSZ coatings by suspension plasma spraying have been carried out. The interactions between the liquid jet and the DC plasma flow were analyzed both by numerical simulation and measurements. Instantaneous and average images of the liquid by shadow imaging techniques showed the break-up modes and the difficulty of penetration depending on the injection pressure and configuration. From these images, average droplet trajectographies were extracted highlighting difference behaviors inside the plasma flow: reaching the plasma axis required high pressures and up-to-down injection configuration. By modeling, comparison of interaction between, one singular droplet, droplets train or continuous liquid jet and the plasma flow showed a better penetration into the plasma jet with continuous sheet due to a necessary mass effect. Finally, coating deposition was discussed: all the coatings showed the same microstructure while the particles treatment in the plasma depended on the penetration zone. The only parameter sensible to the penetration was the deposition efficiency rate which depended on the injection pressure: a too high pressure (higher than 0.6 MPa) led to a decrease in of the deposition efficiency rate. These investigations showed that suspension plasma spraying is a "robust" process able to generate coatings with correct mechanical properties even if the penetration of the suspension was not well controlled. This implies nevertheless that the liquid pressure is sufficient to make part of the suspension penetrate into the plasma jet.

Figure 1 :

 1 Figure 1: Schema of the suspension injection position in front of the plasma torch

3. 1 .

 1 Experimental approach Due to the injector position close to the plasma centreline, the liquid stays as a continuous jet (jet diameter = injector jet). When the Rayleigh regime is reached, the jet breaks down in uniform droplets following the Rayleigh equation (droplet diameter= 1.8 x jet diameter) [6].

Figure 2 :

 2 Figure 2: Shadow imaging of water sheet interacting with an air cross flow.

Figure 3 :

 3 Figure 3: Shadow imaging of water sheet interacting with an argon/hydrogen/helium DC plasma flow.

  0.25 Mpa -Down to up. 0.45 Mpa -Down to up. 0.65 Mpa -Down to up. 0.25 Mpa -Up to down. 0.45 Mpa -Up to down. 0.65 Mpa -Up to down.

Figure 4 :

 4 Figure 4: Instantaneous shadow photographs of a continuous water jet interacting with DC plasma flow for different pressures and injection configurations

Figure 5 :

 5 Figure 5: 100 superimposed shadow photographs of a continuous water jet interacting with DC plasma flow for different pressures and injection configurations.

Figure 6 :

 6 Figure 6: Variation of the maximum of signal intensity downstream of the torch exit3.2. Model approachAnother way to study the first interactions between the two fluids is to simulate them. This requires getting the convenient tools. Several works have been carried out to develop a multi scale model[12],[START_REF] Caruyer | Modeling the first instant of the interaction between a liquid and a plasma jet with a compressible approach[END_REF] without introducing any analytical models of break-up while other authors analyze the different existing models[START_REF] Marchand | Modelling of the plasma spray process with liquid feedstock injection[END_REF]. Here are exposed different kinds of interactions between one single droplet, a train of droplets or a continuous liquid jet and the same DC argon/hydrogen plasma flow. Due to numerous previous investigations in literature[START_REF] Marchand | Liquid precursor plasma spraying: observation of liquid feedstock break-up[END_REF],[START_REF] Marchand | Modelling of the plasma spray process with liquid feedstock injection[END_REF],[START_REF] Etchart-Salas | Influence of Plasma Instabilities in Ceramic Suspension Plasma Spraying[END_REF], these works have been carried out with Ar/H2 plasma flow for comparison and validation. Only hydrodynamic effects are taken into account, the thermal one inside the liquid with phase changing is not yet incorporated in the model. The goal is to analyze which way seems the best to introduce the suspension into the jet[START_REF] Caruyer | Modeling the first instant of the interaction between a liquid and a plasma jet with a compressible approach[END_REF]. Figure10left shows the velocity of Ar/H2 plasma flow and the analysis area where the droplets interact with the plasma. As seen in Figure10left, the interaction zone of the droplet with the plasma plume is very peripheral with rather low gas velocity, around 250 m/s. So, the individual injected droplets get low Weber number (respectively 5 for a diameter of 75 µm and a velocity of 40 m.s -1 , then 15 for a diameter of 200 µm and 30 m/s for the injection velocity).First, two alone droplets are injected close to the plasma jet (respectively Figure7and Figure8). These classic cases correspond to a fragmented liquid jet from an injector diameter of about 40 and 110 µm respectively according to the Rayleigh-Plateau theory[START_REF] Rayleigh | Proc. R. Soc. Lond[END_REF]. In Figure7, due to the important surface tension forces, a bag is formed around the droplet and is fragmented into several secondary droplets with a mean diameter of about 8 µm. In Figure8, the initial droplet is stretched and the formation of a thin film with surface waves is observed. As no thermal effect is taken into account in

Figure 7 :Figure 8 :

 78 Figure 7: Velocity field (m/s) around the water droplet at t=5.10 -6 s (We=5)

Figure 9 :

 9 Figure 9: Velocity field (m/s) at t=1.5.10 -5 s (We=5) for multi-droplet injection.

Figure 10 :

 10 Figure 10: Plasma velocity (m/s) and investigation zone for singular droplets analysis (left) and temperature field (K) for a continuous liquid jet interacting with Ar/H 2 plasma flow (right).

Figure 11 :Figure 12 :

 1112 Figure 11: SEM micrograph of coating obtained on BSE mode

Figure 13 :

 13 Figure 13: Relationship between coating surface roughness and liquid injection pressure

Table 2 .

 2 4.3. Mechanical properties.The mechanical properties of the coatings have been evaluated by two techniques: nano-indentation and classical Vickers Hardness test. The nano-indentation has been performed on the polished cross section of samples, in the conditions described in[START_REF] Vert | Adhesion of YSZ suspension plasma-sprayed coating on smooth and thin substrates[END_REF]. The results are presented in Whatever the injection configuration and parameters are, the values are in the same range showing no difference of hardness and Young Modulus between the coatings. This agrees with the coating morphology observations.

			Up to down injection			Down to up injection
	Liquid injection pressure (MPa)	0.35	0.45	0.55	0.65	0.35	0.45	0.55	0.65
	Coating hardness (GPa)	9.1 ± 1.4	9 ± 0.8	8 ± 1.4	8.5 ± 0.6	8.8 ± 1	10.1 ± 1.1	9 ± 1.2	8.5 ± 1.2
	Coating Young's modulus (GPa)	140 ± 14	136 ± 8	139 ± 9	135 ± 8	141 ± 9	158 ± 11	139 ± 10	137 ± 12

Table 2 :

 2 Coating mechanical properties from nano-indentation technique

Up to down injection Down to up injection

	Injection
	pressure	0.35	0.45	0.55	0.65	0.35	0.45	0.55	0.65
	(MPa)
	Hardness (GPa)	8.89	8.98	8.84	9.33	10.27 10.42 9.86	10.79
	Figure 14: Coating hardness from
	Vickers indentation test on the top of
	the coating

Table 3 :

 3 Vickers Hardness values adjusted by using the Jönsson & Hogmark model


Roger ; Collet

Elise ; Boucq

Jean-Loup Madre

Laurent Hivert

Long term dynamics of inequalities between French households concerning automobile

Keywords: panel, automobile, car ownership, car use, income inequalities, saturation, France

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION

In many countries like in France, most taxes on automobile were progressive, when they had been established. They have become neutral or even regressive because of the social diffusion of car ownership and use ( [START_REF] Madre | Les interventions publiques dans les transports de voyageurs et leurs effets redistributifs[END_REF], [START_REF] Purwanto | The Dynamics of Inequality in Transportation Expenditure between Private Car Users in the Paris Region[END_REF]). Has multi-car ownership made this evolution slower? The answer to this question is probably different in low density areas, where there is no alternative to automobile, and in large conurbations, where several destinations can be reached by foot or by bicycle and where public transport are available. Expenditure for car purchase are still concentrated in high income groups, as well as for toll motorways or parking, because cheaper (e.g. second hand cars) or free alternatives exist ( [START_REF] Madre | Quelques éléments sur les effets redistributifs du péage urbain, Entretiens Jacques Cartier[END_REF], [START_REF] Berri | Dynamiques de la motorisation et des dépenses de transport des ménages Analyses sur données individuelles et semi-agrégées[END_REF], [START_REF] Berri | Transport consumption inequalities and redistributive effects of taxes: a comparison of France, Denmark and Cyprus, 26èmes[END_REF]).

First, the data from two panel surveys, on which rely the calculation of annual time-series since the waves of year 1974 for each quartile of income distribution, will be presented in section 2. Then, the methodologies used for the calculation of these time-series (interpolated quartiles, optimised estimate of time-series from panel surveys) and the inequality indicators will be described in section 3. A descriptive overview will give the first results (section 4). Finally, modelling in terms of logistic curves [START_REF] Røed-Larsen | Distributional Effects of Environmental Taxes on Transportation : Evidence from Engel Curves in the United States[END_REF]) will be implemented in section 5: do poor/medium/rich households follow the same trajectories for car ownership and use according to the evolution of their income? What results can be derived in terms of relationship between cross-sectional and longitudinal income-elasticities [START_REF] Madre | Why Long term dynamic elasticities differ from Crosssection elasticities ?[END_REF]?

MORE THAN 30 YEARS OF ROTATING PANEL SURVEY DATA

This research is based on two annual nationwide household surveys describing both car ownership and use: -the "Enquête de Conjoncture Auprès des Ménages" (ECAM) conducted by the National Institutes of Statistics (INSEE) among a sample of dwellings drawn from the French census: 10,000 to 13,000 households responding by interview each year, of which about one third had been also interviewed one year before ; the period from 1974 to 1994 (year of end of this survey) is covered by the data files available at INRETS; th WCTR, July 11-15, 2010 -Lisbon, Portugal 3 -the "Parc-Auto" (Car Fleet) survey is a postal survey conducted by the private marketing research company TNS-Sofres; each annual wave sample includes 6,000 to 7,000 volunteer respondent households, of which about 3/4 have already responded the year before (even if having moved, contrary to ECAM survey); data files are available at INRETS for all waves since 1984 and this survey is still on-going ( [START_REF] Hivert | Inégalités d'accès à l'automobile, disparités d'équipement et d'usage entre les ménages les plus pauvres et les plus riches[END_REF], [START_REF] Hivert | L'automobile en Europe, Changements de comportements d'équipement et d'usage[END_REF]). For a short description of data, see also [START_REF] Kalinowska | Comparability and comparison of French and German passenger car purchase, ownership and utilization : emphasizing the diverging developments between petrol and diesel driven vehicles, mission report for COST[END_REF] and [Papon, Hivert, 2007, 2008]. Despite these differences in survey methodology, we have checked that these data source show consistent results for the period 1984-94, when both datasets are available at INRETS.

Both questionnaires contain:

-the annual income of the household in about 10 brackets, -a description of the household (socio-economics, demographic structure, place of residence, etc.), -a description of cars (age, type of fuel, main driver, etc.) at permanent disposal of the household (up to 2 cars in ECAM, to 3 cars in Parc-Auto), -an estimate of the annual mileage for each car described (rounded and heaped ( [START_REF] Hivert | Le parc automobile des ménages, étude en fin d'année 1999 à partir de la source "Parc Auto" SOFRES[END_REF], [START_REF] Yamamoto | A random heaping model of annual Vehicle Kilometres Travelled considering heterogeneous approximation in reporting[END_REF], but unbiased according to odometer reading), as well as an information on main purposes of vehicle use (used/non used).

Thus, this information has to be homogenised mainly for the income, which has been coded in brackets using different grids of nominal income over time (see below).

CALCULATION OF TIME-SERIES FOR EACH QUANTILE OF INCOME, METHODOLOGICAL ASPECTS

For the ECAM surveys conducted by the National Institute of Statistics, about 70% of households give for each person detailed information on their resources, including different sources of income (wages, retirement pensions, social benefits, etc.). About 25% give a global information on a pre-coded grid, and 5 to 10% refuse. Thus, we chose to use it, particularly because the rate of non-response is much lower when using this pre-coded information.

Most of models rely on quantitative variables, but responses given by households are often rounded for income as well as for annual mileage. Moreover inflation, which was important before mid-80's, make comparisons difficult over time. That is why income grids have been revised in 1977, 1983, 1987 and 1997. The grids have been also typically revised in 2002 in order to be converted from Franc to Euro.

Rather than conventional methods ( [START_REF] Bhat | Imputing a continuous income variable from grouped and missing income observations[END_REF]) like simulated residuals ( [START_REF] Lollivier | D'une variable discrète à une variable continue : Une application de la méthode des résidus simulés dans Mélanges Economiques, Essais en l'honneur d'Edmond Malinvaud[END_REF]), we preferred a more robust method: interpolated quantiles from the middle of each bracket ( [START_REF] Madre | Des quantiles interpolés pour suivre l'évolution des inégalités, XXXIXème colloque de l'ASRDLF[END_REF]). It could be any quantiles (e.g. terciles or 4 quintiles) but we have first to check that the number of brackets is much larger than the number of quantiles.

Let us consider the distribution of a variable of interest (e.g. the number of cars in the household) by income bracket. In order to locate the limits of each income quantile, the distribution is interpolated, and the average number of cars per household is calculated in each quantile with the rather strong hypothesis, that car ownership is constant inside each bracket containing a limit between two quantiles (e.g. first quartile Q1, median or third quartile Q3).

This method has been tested on the respondents of the 1993-94 French National Travel Survey, which gave a precise answer in terms of income. The result is obviously better, when the upper and lower limit of a quantile are nearer to the thresholds of brackets on the grid. But surprisingly enough, splitting the sample into a larger number of brackets when dividing the middle of each class by the number of consumption units does not improve the quality of the interpolation, because even inside a bracket the income level is correlated with the composition of the household. Thus the interpolation is more precise for total household income than for the income per consumption unit, which is a more adequate measurement of the standard of living.

After obtaining homogenized income groups through quantiles, how to estimate time-series accurately enough despite of the small sample size of our panel survey data ? Referring to [START_REF] Cochran | Sampling Techniques[END_REF], we have implemented the method optimising the accuracy of the timeseries for the most recent periods. This optimisation is crucial for the estimation of models using variables affected with small annual changes (often 1 or 2%). However we had to smooth several time-series by moving averages over three consecutive years at national level and over five years for Paris region, which represents about 15% of the total household sample.

For the measurement of inequalities, we have adopted three indicators: -the Gini index [START_REF] Gini | Measurement of inequality of income[END_REF], -the ratio "Q4/Q1", i.e. between the means of the variable of interest for the extreme quartiles (Q1 representing the poorest and Q4 the richest), -the cross-sectional income-elasticity, with usual references (1 when the variable of interest increases proportionally with income and 0 when it is not influenced by income).

More sophisticated indicators could be considered, for example those proposed by [START_REF] Atkinson | On the Measurement of Inequality, Faculty of Economics and Politics, University of Cambridge, England[END_REF], [START_REF] Atkinson | Lectures and Public Economics[END_REF][START_REF] Atkinson | The Comparison of Multi-dimensioned Distributions of Economic Status[END_REF] (also cited in [START_REF] Madre | Les interventions publiques dans les transports de voyageurs et leurs effets redistributifs[END_REF], [START_REF] Berri | Dynamiques de la motorisation et des dépenses de transport des ménages Analyses sur données individuelles et semi-agrégées[END_REF]). In order to avoid heterogeneity over time due to different income grids, these indicators have been calculated from only four points, which are the mean values for each quartile.

DESCRIPTIVE ANALYSIS

For each quartile of the distribution of income per consumption unit, time-series have been estimated since 1974 for : -the average number of cars per adult (i.e. aged > 18, which is the minimum for driving license in France), -the proportion of equipped households, -the proportion of multi-car households among equipped households, -the average annual mileage per household or per car, separately nationwide, and for the Metropolitan Area of Paris (Paris region) in order to show the specificity of a high population density area, where the diffusion of automobile has started earlier than elsewhere in France. Generally, the figures for intermediate quartiles Q2 and Q3 are quite close, thus they have been represented by a single curve.

Nationwide (figure 1), the inequalities of income per household have decreased between mid-70's and mid-80's, then have remained almost constant till 2000 except a temporary increase during the 1993-94 recession. Since 2001, high incomes are growing faster than those of medium or low income groups. For the whole country, the ratio Q4/Q1 was of 2.1 for the average number of cars per adult in 1974 (figure 3). It has remained almost constant around 1.7 between 1982 and 1993, then it has decreased to 1.4 in 2005. The cross-sectional income-elasticity of the number of cars per adult has followed the same pattern, from 0.6 in mid-70's to 0.2 around 2005, questioning the hypothesis that cross-sectional elasticities represent long term elasticities. In the special case of Paris region, the ratio Q4/Q1 was only 1.8 in the mid-70's and has remained almost constant at 1.5 since 1982. Indeed, car ownership has been higher in the conurbation of Paris than in the rest of France since the mid-60's, but high population density with good public transports limits the expansion of car ownership. Thus, multi-car ownership has not reversed the social diffusion of automobile, but it has stopped it in Paris region since the early 80's and nationwide between mid-80's and mid-90's.

As seen in figure 4, the decrease in inequality for car ownership during the first period is mainly due to the poorest quartile (Q1) catching up with the upper ones: the proportion of households without car has dropped from 55% in mid-70's to 35% in mid-90's. During the same decades, the proportion of multi-car households among equipped households has From mid-90's, different zones can be identified. In the most densely populated areas, the income-elasticity of multi-car ownership dropped from 0.7 during the late 90's to 0.1 in 2005:

Q 1 Q 2 Q 3 Q 4 0,2 0,3 0,4 0,

8

Q1 curve is increasing, while those for Q2 to Q4 are decreasing. In Paris region at least since 1995, there is almost no differentiation of multi-car ownership according to income. In low density areas the elasticity decreases more continuously, as well as the ratio Q4/Q1.

Mid-90's, the increase of ownership in terms of cars per adult (especially visible for Q4) can be explained by incentives introduced by two successive Governments in France (see below).

A new grant for scrappage ("prime à la casse") has been recently introduced in France (early 2009). It has succeeded to previous incentives for scrapping old cars implemented in the middle of the nineties :

-decided by the Balladur Government, a scrap bonus of 5,000 francs (762 euros) was implemented from February 1994 to June 1995 to purchase a new car, when scrapping a vehicle more than ten years old. This "balladurette" measure aimed to help a depressed market and boost consumption, but also cleaned up somehow the French car fleet.

-under the next Juppé Government, from October 1995 to October 1996, another measure allowed any owner to benefit a bonus when scrapping a vehicle more than 8 years old and under 3.5 tonnes This premium amounted to 5,000 francs to buy a new car of economic or low range, and to 7,000 francs (1,067 euros) for higher-end vehicles. This premium, known as "Juppette" was also intended to support the Government decision of toughening automobile technical inspections (originally introduced in 1992).

The annual mileage per household (figure 5) has increased till 2000, then decreased, due to a lower mileage per vehicle induced by a more expensive fuel (peak of fuel price in 2000, then a more continuous increase from 2004 to mid-2008. This inflection is described in detail for example in [START_REF] Hivert | Multimodalité et tendances démographiques -CADMIUM : Comparative Analysis of Demographic Modifications and Impact on the longitudinal Use of Modes, Deufrako-Predit3 report[END_REF] and in [START_REF] Kemel | How do French motorists react to a multi-annual fuel price increase ? An econometric analysis based on 1999-2007 panel data[END_REF]. This decline is more important in urban areas than in low density areas, especially for medium and high income groups (Q2 to Q4). th WCTR, July 11-15, 2010 -Lisbon, Portugal 9 In terms of annual mileage per vehicle, the ratio Q4/Q1 has declined from 1.8 in 1984 to 1.1 in 1995, because of a slower increase of car use in high than in low income groups. This period corresponds to decreasing fuel price: sharp decline of oil price between mid-1985 and mid-1986, followed by an increasing proportion of diesel vehicles using a 30% cheaper fuel.

Between mid-90's and 2005, the ratio Q4/Q1 has decreased in low density areas, while it has increased in urban areas, because car use can be reduced by low income households only where an alternative exists (public transport or slow modes).

Thus, after a period of status quo between mid-80's and mid-90's, the social diffusion of automobile continues like during the 70's, especially in low density areas, whose inhabitants are more car dependant even if their income is low.

The two next figures present, for each quartile by consumption unit, the evolution of car ownership (per adult) and car use by household income, in constant 2006 Euros (considering all the annual waves together).

Globally, the scatter of points in figure 6 shows that, as expected, household car ownership has increased with their income. Indeed, the number of cars per adult has been about 0.45 for a real income of €10,000 while it is steady about 0.7 for incomes over €25,000 (constant 2006 Euros). The shape of the scatter of points suggests a concave augmentation of car equipment with income. However, the slope seems different when comparing each of the four quartile. Indeed, it is decreasing when the household has a higher social position.

Particularly, car ownership is increasing faster with real income for the household Q1 than for the household Q4. Over the years, this has induced a reduction of social inequalities concerning car ownership, as it is shown in figure 3. Considering car use in figure 7, analog conclusions roughly emerge. The average mileage of households has been about 12,500 km/year for an annual income of €10,000. Over €25,000, it has globally ranged between 15,000 and 20,000 km/year. Even if it seems less obvious than in figure 6, the scatter of points in figure 7 also suggests a concave growth of household car use with their real income. Regarding the quartile specific households, the slope of the annual mileage over real income also seems to decrease with their position in the income scale. The concave shapes in figures 6 and 7 suggest that the diffusion of automobile can reach saturation thresholds for both car ownership and use when the households are getting wealthy. Theoretically, the social diffusion of a good (either in time or in the income scale) can be represented using a sigmoid curve, ended by a saturation level. Refering to figures 6 and 7, our data seem to cover only the upper part of an assumed sigmoid. It is modelled in 11 the following section. In comparison, time (figures 4bis and 5) is clearly less relevant than income to fit a sigmoid curve. However, it is also tested as explanatory variable.

ESTIMATION OF LOGISTIC CURVES TO MODEL CAR OWNERSHIP AND USE

The average households for each of the four quartiles of annual income per consumption unit are observed annually, during 33 years from 1974 to 2006. Let Q1, Q2, Q3 and Q4 refer to these households by increasing order of resource. For both car ownership and use, the figures in section 4 suggest the upper part of a sigmoid curve. In this section, we assume that they are generated by logistic functions. The logistic model, which is exposed below, is applied on the data of each quartile specific household and on the pooled dataset.

The model

Let refer either to the number of cars per adult or to the annual mileage in 10 4 kilometres for the household Qi at period t. Both these variables are modelled separately assuming a logistic specification. The explanatory variable is denoted and stands successively for the time (section 5.2) and the real income (section 5.3). Thus, the model is given by: ,

where is assumed to be i.i.d. along a , and where are the parameters to be estimated. For positive, the formulation (1) implies that is increasing with along a symmetrical sigmoid, bounded by two horizontal asymptotes: the lower plateau is fixed at while the upper plateau, corresponding to a saturation level, is located at . The inflection point, for which the second derivative of with respect to is zero, is located at ( ; ).

Time as explanatory variable

In this section, the explanatory variable of model ( 1) is the time. Precisely, is given by the index t, with t=0 for the year 1974, t=1 for 1975... and t=33 for 2006. Tables 1 and2 report the estimates for car ownership and use respectively. As expected, the results show that car ownership and use have increased along time. Indeed, the parameter is found to be positive and significant in every case on tables 1 and 2. For both car ownership and use, the inflection has occurred in the mid 80's for the two lower quartiles (Q1 and Q2), in the 70's for Q3 and in the 60's for the higher income quartile Q4. The highest saturation level is for Q2, then it decreases till Q4 because higher income groups live in more densely populated areas. Saturation thresholds are quite low for Q1.

Considering the car ownership model, pooling the data does not seem illegitimate since the confidence intervals for saturation level overlap around 0.74 cars per adult (table 1). The same stands for the car use model. Indeed, the confidence intervals for the saturation level in table 2 overlap around an annual mileage of 20,000 km per household. However, the resources of households are a more robust factor than time to explain the diffusion of automobile. Let's now consider income growth as determinant in model (1).

Real income as explanatory variable

In this section, household car ownership and use are modelled using real income as explanatory variable. Particularly, in model ( 1) refers here to the real annual income of the household Qi at period t, expressed in 10 4 constant 2006 Euros. th WCTR, July 11-15, 2010 -Lisbon, Portugal 13 Pooling the 132 corresponding points to fit the model (1), the saturation levels are estimated at 0.72 cars per adult and 17,800 yearly km per household (tables 3 and 4). These thresholds are not very far from the levels already reached. However, when adjusting the model separately for each quartile, these estimated thresholds are higher. Indeed, the saturation threshold for the number of cars per adult ranges from 0.73 for the wealthiest household (Q4) to 1.23 for the poorest one (Q1). This later value seems quite high, but the pertained confidence interval is very wide. Indeed, the estimation of the saturation threshold is much more accurate for higher income groups, which are currently near to their estimated asymptote.

Here, pooling the data is more questionable since the confidence intervals for the four quartile households overlap just at 0.74, which is the upper limit of the quite narrow confidence interval obtained from pooled data. Thus, while the estimates of the saturation level are less accurate using income instead of time as explanatory variable, we obtain a very accurate estimate of the saturation threshold from the income model when assuming that these data can be pooled.

In the meantime, the estimated saturation mileage ranges from 18,200 km/year for the household Q4 to 24,800 km/year for the household Q2. This is higher than in the pooling case. Therefore, the pooling estimates might be affected by an heterogeneity bias. But it is not sure, because the four confidence intervals overlap, and the estimate for the pooled data (17,800 km/year) is the lower limit of the intersection of these four intervals.

Regarding the saturation levels, the estimates show a regular decrease of the number of cars per adults with the economic position of households in the income scale. The same stands for the saturation threshold of car use, except for the household Q1 which has a greater estimated threshold (22,300 km/year) than the household Q2. These results can be explained by a location effect. Indeed, richest households can afford to live in dense urban areas where public transports are an alternative to car use. At the opposite, poorest households are more frequently living in outer suburbs or rural areas, where car is often the only possible mode. For a better representation in the left side of the logistic curves, the model (1) has also been fitted using the log of real income, ), instead of real income in level. The logistic adjustment does not perform better using this alternative specification: pooling the data, the mean log-likelihood slightly decrease to 1.76 for the car ownership model and 0.27 for the car use model. The estimated saturation levels are higher: 22,700 km per household for annual car use, and 1.05 cars per adult. These estimates are closer to those obtained from the previous household-specific adjustments using real income in level (in tables 3 and 4). However, [START_REF] Hivert | L'automobile en Europe, Changements de comportements d'équipement et d'usage[END_REF] showed that the number of cars in a household does not exceed the number of driving license holders in Germany and in France. The contrary is observed in the US, where there are much more specialized vehicles (SUV, vans, urban cars, etc.).

The model (1) assumes the symmetry of the adjusted logistic curves with respect to the inflection point. This can be viewed as a strong hypothesis and asymmetric curves might be more relevant in our context (for an example of application, see [START_REF] Ricketts | A five-parameter logistic equation for investigating asymmetry of curvature in baroreflex studies[END_REF]). Particularly, the estimated income of inflection for the household Q4, when assuming symmetry, is found to be negative for both car ownership and use. This unrealistic result pleads for an asymmetric shape of the logistic curves. However, the assumption of symmetry is made necessary when considering our data. As we mentioned above, the scatter of points of the annual mileage as well as the one of the number of cars per adult against the annual income when pooling observations suggest a strictly concave shape, corresponding to the upper part of a sigmoid (figures 6 and 7). In other words, the data do not cover the lower part of the sigmoid, at the left side of the real inflection point. As there is no empirical information about this part, its representation has to rely on some assumptions. A solution would have been to extend the observation period farther in the past, back to the end of World War II. Unfortunately, detailed data about car use and equipment between 1945 and 1973 are not available.

CONCLUSIONS

The social diffusion of car is a major feature of economic growth in Occidental Europe after World War II. This paper describes this phenomenon in France for four income groups from 1974 to 2006. Multi-car ownership has interrupted but not reversed this long term trend : after a period of status quo between mid-80's and mid-90's, the social diffusion of automobile continues like during the 70's, especially in low density areas, whose inhabitants are more car dependant even if their income is low.

However, the growth of car ownership becomes slower and slower, which shows that saturation levels are not far from already reached levels. In order to determine these thresholds, we have adjusted logistic curves separately for each income quartile, then on pooled time-series. Geographic factors (as density, distance to city-centre, size of conurbation) probably explain why it is for the second quartile that saturation levels are the highest, which could be checked by estimating the same models only for the inhabitants of the Metropolitan Area of Paris. Moreover, pooling these data does not seem illegitimate because the saturation level for pooled series just belongs to the intervals determined by the four quartiles. It lets us suspect some heterogeneity, mainly due to the geographic factors mentioned above. It is only through more narrow confidence intervals obtained from longer time-series or more adequate functional forms that we could get a real proof of this heterogeneity.

This heterogeneity opens theoretical discussions about cross-sectional versus longitudinal estimates. Another example is an important change over time of cross-sectional income elasticities, which makes impossible to consider them as a proxy for long term longitudinal elasticities. There is still much to do with these data. For instance, introducing fuel price as explanatory factor for car use should allow to derive saturation thresholds for the annual mileage at different fuel price levels, which could give an important information for building scenarios for sustainable development. However we should take into account that even fuel price elasticity depends on income level [START_REF] Kemel | How do French motorists react to a multi-annual fuel price increase ? An econometric analysis based on 1999-2007 panel data[END_REF].

Figure 1 :

 1 Figure 1: Evolution of inequalities for income per household (ratio of extreme quartiles Q4/Q1) -France

Figure 2 :

 2 Figure 2: Evolution of inequalities for income per household (ratio of extreme quartiles Q4/Q1) -Paris region

Figure 3 :

 3 Figure 3: Evolution of inequalities for the average number of cars per adult -France

Figure 4 :

 4 Figure 4: Evolution of car ownership with time: share of equipped households by quartile of income per consumption unit (Q2 and Q3 together, smoothed by mobile averages MA3) -France

Figure 5 :

 5 Figure 5: Evolution of average annual mileage per household by quartile of income per consumption unit -France

Figure 6 :

 6 Figure 6: Evolution of car ownership (average number of cars per adult) with household income -France

Figure 7 :

 7 Figure 7: Evolution of car use (annual mileage per household) with household income -France

of inequalities for the average number of cars per adult -France ratio Q4/Q1 and income-elasticity

	3,5 developed more slowly in Q1 than in richer quartiles, especially Q2 and Q3 catching up with Evolution of inequalities of income per household -Paris region 3,3 ratio Q4/Q1 Q4.
	2,7 2,9 3,1 Then, figure 4bis presents the evolution of ownership in terms of average number of cars per ratio Q 4/Q 1 adult. Taking into account that average number of adults per household remains higher in
	2,5 lower quartiles (e.g. from 2.29 in 1994 to 1.90 in 2006 for Q1, versus from 1.77 to 1.61 in
	2,3 Q4), the average number of cars per households is finally almost the same in 2006 (1.05 in
	Q1 versus 1.06 in Q4), so the Q4/Q1 ratio decreased from 1.24 in 1994 to 1.0 in 2006.
	2,1
	1976		1978	1980	1982	1984	1986	1988	1990	1992	1994	1996	1998	2000	2002	2004
										y ear
	1,2 1,5 1,8 2,1 2,4			Evolution income-elasticity
																	ratio Q4/Q1
	0,9
	0,6
	0,3
	0
	1975	1977	1979	1981	1983	1985	1987	1989	1991	1993	1995	1997	1999	2001	2003	2005
										7

Table 1 :

 1 Estimates for the car ownership model as a function of time dependant variable is the number of cars per adult in households. The explanatory variable is time. All the parameters are significant at the 90% level, except those indicated by *.

	Parameters	Pooling	Hh.Q1	Hh.Q2	Hh.Q3	Hh.Q4
		0.07	0.06	0.06	0.08	0.10
		-0.05*	-0.66	-0.52	0.04*	0.66
		-0.29	-0.27	-0.17	-0.26	-0.31
		-4.83	-7.58	-7.03	-7.84	-6.98
	Mean log-likelihood	1.00	2.37	2.09	2.40	2.07
	Nb. of observations	132	33	33	33	33
	Saturation threshold (# of cars per adult) [90% conf. interval]	0.74 [0.63, 0.87]	0.76 [0.61, 0.96]	0.84 [0.72; 0.98]	0.77 [0.73, 0.81]	0.74 [0.71, 0.77]
	Date of inflexion	1975	1986	1982	1973	1967
	Note: the

Table 2 :

 2 Estimates for the car use model as a function of time

	Parameters	Pooling	Hh.Q1	Hh.Q2	Hh.Q3	Hh.Q4
		0.06	0.07	0.07	0.07	0.05
		-0.32*	-0.83	-0.79	-0.19*	0.66
		0.71	0.56	0.81	0.74	0.65
		-2.70	-4.75	-3.27	-3.94	-4.27
	Mean log-likelihood	-0.06	0.95	0.21	0.55	0.71
	Nb. of observations	132	33	33	33	33
	Saturation threshold
	(km/year)

Table 3 :

 3 Estimates for the car ownership model as a function of real income

	Parameters	Pooling	Hh.Q1	Hh.Q2	Hh.Q3	Hh.Q4
		1.31	0.97	0.84	0.85	0.93
		-0.62	-1.46	-0.90	-0.29	0.15*
		-0.32	0.21*	-0.06	-0.23	-0.31
		-6.52	-7.64	-6.80	-7.65	-6.87
	Mean log-likelihood	1.84	2.40	1.98	2.40	2.02
	Nb. of observations	132	33	33	33	33
	Saturation threshold (# of cars per adult) [90% conf. interval]	0.72 [0.71, 0.74]	1.23 [0.56, 2.71]	0.94 [0.71, 1.24]	0.79 [0.74, 0.85]	0.73 [0.70, 0.75]
	Income of inflexion (in 2006 Euros)	4700	15000	10700	3500	-1700
	Note: the dependant variable is the number of cars per adult in households. The explanatory variable is their income expressed in 10 4 constant 2006 Euros. All the parameters are significant at the 90% level, except those
	indicated by *.

th WCTR, July 11-15, 2010 -Lisbon, Portugal