V Scottez
email: scottez@iap.fr

A Benoit-Lévy

J Coupon

O Ilbert

Y Mellier

Testing the accuracy of clustering redshifts with simulations

Keywords:

We explore the accuracy of the clustering-based redshift inference within the MICE2 simulation. This method uses the spatial clustering of galaxies between a spectroscopic reference sample and an unknown sample. The goal of this study is to give a preview of the redshift accuracy one can reach with this method. To do so, we first highlight the requirements of this technique in term of number of objects in both the reference and unknown samples. We also confirm that this method does not require a representative spectroscopic sample for calibration.

We estimate that a density of spectroscopic objects of 10 -5 arcmin -2 per redshift bin of width δz = 0.01 over 9000 deg 2 allows to reach 0.1 % accuracy in the mean redshift for a galaxy density compatible with next generation of cosmological surveys. This number is compatible with the density of the Quasi Stellar Objects in BOSS. Second we demonstrate our ability to measure individual redshifts for galaxies independently from the photometric redshifts procedure. The resulting individual clustering redshifts have a bias=-0.001, an outlier fraction of η = 3.57% and a scatter of σ = 0.027 to i < 25. The advantage of this procedure is threefold: i) it allows the use of clustering redshifts for any field in astronomy, ii) it allows the possibility to combine photometric and clustering based redshifts to get an improved redshift estimation, iii) it allows the use of cluster-z to define tomographic bins for weak lensing. Finally we explore this last option and build 5 clustering redshift selected tomographic bins from redshift 0.2 to 1. We found a bias on the mean redshift estimate of 0.002 per bin.

INTRODUCTION

The next generation of cosmological surveys such as Euclid (Laureijs et al. 2011;Amendola et al. 2013) , LSST (LSST Science Collaboration et al. 2009) and WFIRST (Spergel et al. 2015), whose aim is to understand the largescale accelerated expansion of the universe will perform surveys of the sky. One of the major issues in astronomy is to be able to deproject the 2D information into a 3D one.

At extragalactic scales, the study of the universe relies on our knowledge of the redshift as a tracer of distances. The ideal way of measuring resdhifts is to do spectroscopie and measure their spectroscopic redshift (spectro-z). While this method is the most accurate, it is also very timeconsumming. Another largely used approach is to estimate the redshift of an object based on its flux information through some filters. This is the so called photometric redshift (photo-z) procedure which require large and representative set of spectroscopic redshift measurements for the calibration of empirical methods (Connolly et al. 1995) and the building of representative template libraries for template-fitting techniques (Coleman et al. 1980). This procedure uses the correlation between the colours of the population of unknown redshift and the flux from a reference sample. The photo-z procedure is fast with respect to the spectroscopic one and allows to measure redshifts for millions of objects. One difficulty is that current and future spectroscopic surveys are incomplete in magnitude, redshift and galaxy properties (Cooper et al. 2006;[START_REF] Newman | Colless[END_REF], which is challenging to calibrate photometric redshifts (Masters et al. 2017). Also, projections for cosmic shear measurements estimate that the true mean redshift of objects in each photo-z bin, considering Euclid statistic, must be known to better than ∆ z = 0.002(1 + z) (Knox et al. 2006;Zhan 2006;Zhan & Knox 2006).

The method we used in this study relies on the clustering properties of galaxies. This approach is looking at the spatial correlation between galaxies at unknwon redshift and reference sources with known spectroscopic redshift. Fundamentaly this approach only require the knowledge of unknown galaxies on-sky position.

The idea of measuring redshift distributions using the apparent clustering of objects was first developed by Seldner & Peebles (1979); Phillipps & Shanks (1987) and Landy et al. (1996). This was practically forgotten mainly due to the rise of photometric redshifts. To face the challenges of future and ongoing dark energy imaging experiment [START_REF] Newman | [END_REF], Matthews & Newman (2010) and Matthews & Newman (2012) used the large scale clustering of galaxies and iteratively approximated the galaxy to dark matter bias and validated this method on simulations while McQuinn & White (2013) proposed an optimal estimator for such a measurement.

We use the method presented in Ménard et al. (2013)& Schmidt et al. (2013) compared to spectroscopic redshift by Rahman et al. (2015). This method focused on lower scales, where the clustering signal is higher, and assume that there is no or little galaxy to dark matter bias evolution when the redshift distribution is narrow. This approach was also applied to continuous fields by inferring the redshift distribution of the cosmic infrared background Schmidt et al. (2015) while Rahman et al. (016a) and Rahman et al. (016b) explored this method for near infrared data using 2MASS Extended and Point Source Catalogs as well as the SDSS Photometric Galaxies. Recently a similar approach was used to validate the redshift distribution used for weak lensing by Hildebrandt et al. (2016).

The goal of this study is to give a preview of the redshift accuracy one can reach with this method. For this, we explore the accuracy and the requirements of this technique. We also present individual clustering-based redshift measurement for a single galaxy using the MICE2 simulation. Finally we investigate the use of this method to define tomographic bins for weak lensing by building 5 clustering redshift selected bins from redshift 0.2 to 1. This paper is organised as follows. In Section 2 we review the clustering-based redshift formalism, while the MICE2 simulation and the data used to build the reference and the unknown samples are introduced in Section 3. In Section 4 we explore the evolution of the clustering redshift distribution accuracy when varying (1) the magnitude of the unknown sample, (2) the number of unknown and reference sources as well as (3) the width of the unknown distribution. Based on these results we estimate the requirements in terms of spectroscopic data needed per redshift bin for the next generation of cosmological surveys. Then we demonstrate our ability to measure clustering redshifts for individual galaxies and quantify their accuracy, in Sections 5. Based on these measurements, we investigate the possibility to directly use the clustering redshift to select tomographic bins for weak lensing. Finally, we present 5 clustering redshift selected tomographic bins for which we give an estimate of the bias. Conclusions are presented in Section 6.

CLUSTERING REDSHIFT FORMALISM

In this section we briefly review the clustering-based redshift formalism initially developed in Ménard et al. (2013), Rahman et al. (2015) and fully detailed in [START_REF] Scottez | Theses[END_REF].

The integrated cross-correlation can be written as:

ωur(z) = dz dNu dz dNr dz βu(z )βr(z ) , (1)
where dNi/dz are the redshift distributions, βi(z) the clustering amplitude with i ∈ {u,r} where lower script u,r refers to the unknown and reference samples respectively. Considering a narrow sampling of the reference population one can write:

dNu dz = ωur(z) × 1 βu(z) × 1 βr(z) × 1 Nr . ( 2
)
This expression requires the knowledge of βu(z) which is a priori not known. There are two ways to bypass this issue. First, one can estimate this term by measuring the autocorrelation functions of both the unknown and reference populations as described in [START_REF] Newman | [END_REF] and Hildebrandt et al. (2016). Or, one can build unknown subsamples localised in redshift and then consider no -or linear -evolution for βu(z) (see Schmidt et al. (2013), Rahman et al. (2015) and [START_REF] Scottez | Theses[END_REF]) which leads to:

dNu dz ∝ ωur(z) × 1 βr(z) . (3)
Where dNu/dz needs to be normalised to the number of objects in the unknown subsample.

THE MICE SIMULATION

The MICE simulation catalog (Fosalba et al. 2015;Crocce et al. 2015;Fosalba et al. 2015;Carretero et al. 2015) was generated using a hybrid Halo Occupation Distribution (HOD) and halo Abundance Matching (HAM) prescriptions to populate Friends of Friends (FOF) dark matter halos from the MICE-Grand Challenge (MICE-GC) simulation.

The catalog used as input the light-cone of the MICE-GC N-body run simulation. The input cosmological parameters are Ωm = 0.25, σ8 = 0.8, ns = 0.95, Ω b = 0.044, ΩΛ = 0.75, h = 0.7. Further details on the simulation can be found at the MICE webpage (www.ice.cat/mice).

The catalog was built to follow local observational constraints:

• The local luminosity function (Blanton et al. (2003) and Blanton et al. (2005) for the faintest galaxies).

• The galaxy clustering as a function of luminosity and colour (Zehavi et al. 2011).

• The color-magnitude diagram (NYU dr7 catalog).

It also includes:

• galaxy evolution in order to better match the luminosity function and the colour distributions at high redshift.

• magnified positions due to gravitational lensing effects computed using projected mass density maps (in HEALPIX format) of the MICE-GC light-cone simulation.

We apply the following evolutionary correction for all magnitudes to get the right number counts at higher redshifts: mag ev = mag cat -0.8 × (arctan(1.5 × z cgal ) -0.1489) , (4) where mag ev is the magnitude corrected for the evolutionary correction, mag cat is the original one (that does NOT include any correction) and z cgal is the true redshift of the galaxy. In this paper, we explore and quantify the accuracy of clustering redshift using the state of the art simulation currently available. Nevertheless, because of the incompleteness at irz ∼ 24.5 and the limited redshift range to z < 1.4, it is not possible to create, for example, a Euclid like scenario.

Data selection

In this exploratory work, since our goal is not to explore the maximum limit accuracy of this method nor to reach any given requirement but to give a preview of the achievable accuracy for individual redshift and mean redshift of a distribution, we choose for simplicity to work on a subsample of 100 deg 2 in the region 10 o < RA < 20 o , 10 o < Dec < 20 o . The application to a larger area for a given survey and the scientific application correspond to a full study that we will investigate in a next paper. The parent sample results in 8.5M galaxies for which we corrected the magnitudes according to Equation 4 and for which we selected the following fields: RA,DEC, ZSPEC and ugrizYJH-bands. This parent sample will be used to build both the reference and the unknown samples.

Reference sample

We choose to build an optimistic reference sample made of ∼ 200 000 sources corresponding to ∼ 0.55 gal.arcmin -2 . These sources are randomly selected from the parent sample to have ir < 22.5 and a flat distribution in redshift as show in Figure 1. This choice allows us to reduce statistical fluctuation in the detection due to the variation in the number of sources over the range of references slices. One can note that the number of reference sources decreases at high redshift. This is directly due to the decreasing number of sources available with i < 22.5 at these redshifts from the MICE2 simulation. Therefore, we do expect a higher noise level between redshift 1.2 and 1.4.

The evolution with redshift of the clustering amplitude of the reference sample βr is shown in Figure 2. We choose to normalize βr at z0 = 0.07.

Unknown sample

The unknown sample is made from the previous parent sample from excluding the sources selected to be in the reference population and contains 8.3M galaxies corresponding to a density of 23 gal.arcmin -2 .

The top panel of Figure 3 shows the i-band magnitude distribution of the unknown sample. This sample is complete to iu = 23.5. The following analysis is made using subsamples of this one. The break at iu = 22.5 is due to the objects removed from the parent sample to build the reference population. This figure also shows the redshift distributions for different bins in magnitude. The first bin in purple has iu < 22.5 and has the same magnitude limit than the reference sample. The next two bins (in blue and cyan) are fainter than the reference sample, but are still complete while the fourth bin (green) starts to be incomplete. Finally the last two magnitude bins (yellow and red) are more and Figure 3. TOP PANEL:i-band magnitude distribution of the unknown sample. This sample starts to be incomplete at iu = 23.5. The break at iu = 22.5 is due to the objects removed from the parent sample to build the reference population. BOTTOM PANEL: redshift distribution of each selected bin in i-band magnitude. The two fainter bins are incomplete in redshift due to the incompletness of the simulation at these magnitudes. From these 6 bins, we randomly select galaxies matching ∆zu = 0.02 and Nu = 2 000 to build the tomographic subsamples.

more incomplete. This incompletness of the simulation for these samples induce unrealistic variations in redshift. This choice of binning will allow to study the evolution of clustering redshift accuracy with fainter magnitude and when the unknown sample is incomplete in Section 4.1.

Cleaning the reference sample

As highlighted by Rahman et al. (2015), spurious correlation at low redshift (z < 0.2) can appear due to the inhomogeneous sampling arising from the clustered distribution of the objects in the reference sample. To address this issue, they removed all galaxies in regions where the measured densities are 2σ from the mean. This procedure homogenises the spatial correlation of the reference sample, but also decreases the number of sources. To prevent spurious correlation at low redshift and optimise the statistical power of clustering-based redshift inference we weight each reference galaxy based on its local density.

To illustrate this, we select unknown tomographic subsamples according to the following criteria:

• true spectroscopic width of the unknown bin, ∆zu = 0.005.

• number of reference sources, Nr = 200 000.

• number of unknown objects in one bin, Nu = all available objects. Here, this number is not a constant. It can vary from bin to bin as shown in the middle left panel of Figure 3.

• magnitude of the unknown sample, iu < 22.5.

Then, for each unknown subsample we measure the corresponding clustering redshift distribution with and without cleaning the reference sample by individually weighting each reference galaxy by its local density. Both plots are shown in Figure 4. The left panel shows the clustering redshift distributions obtained with the original reference sample. One can note the existence of horizontal stripes caused by the inhomogeneity of the reference sample. These stripes disappear in the right panel where we homogenize the reference population by individually weighting each reference galaxy by its local density. These weights are directly measured on the reference population and do not depend of the unknown sample. One can still note that the stochastic fluctuations beyond z clust = 1.2 are higher than bellow. This is due to the decreasing number of reference sources at these redshifts (see Figure 1).

There is also a vertical stripe in both panels from zspec 1.2 to 1.4. There are different effects at play in this region. First, this increased noise is mainly due to the low number of unknown objects per bin at these redshifts. Indeed, in the central part of the figure there is ∼ 10k unknown galaxies while in the vertical stripes between redshift 1.2 to 1.4 this number drops to ∼ 100. Moreover, this range also corresponds to the redshift range where there are few reference galaxies, leading to a lower signal-to-noise ratio (SNR). To understand how each of these effects can affect the clustering based redshift estimation we present some tests in the next section.

TESTS ON INPUT PARAMETERS

This section aims to highlight the behaviour of the clustering-based redshift inference with respect to the magnitude of the unknown sample, the number of objects in the unknown/reference samples and the width of the unknown distribution.

To do so, we consider an idealistic case in which we select galaxies of the unknown sample using theur true redshift to reduce the influence of the galaxies to dark matter bias. In each of the following subsections we fix a set of parameters and we vary one. The parameters are: the true width of the unknown tomographic bin ∆zu, the number of reference sources Nr, the number of unknown objects in the bin Nu, the magnitude of the unknown sample iu. Each time, even when we vary Nr, we keep a flat redshift distribution for the reference sample.

Evolution with the magnitude of the unknown sample: iu

In this part we fix the following set of parameters:

• ∆zu = 0.02.

• Nr = 200 000 (∼ 0.55 gal.arcmin -2 ).

• Nu = 2 000 (∼ 0.55 10 -2 gal.arcmin -2 ).

We then vary the magnitude of the unknown sample to be:

• iu < 22.5.

Figure 5. Cluster/Spectro-z map for iu < 22.5. Regions where there is no galaxies available from the unknown sample are dashed.

• 22.5 iu < 23.

• 23 iu < 23.5.

• 23.5 iu < 24.

• 24 iu < 24.5.

• 24.5 iu < 25.

The unknown samples selected here, correspond to the magnitude bins visible on Figure 3, their redshift distribution are visible on the bottom panels of this figure. This choice of binning allows to study the evolution of clustering redshift accuracy with fainter magnitude and when the unknown sample is incomplete.

For each of these 6 samples we build tomographic subsamples according to ∆zu = 0.02 and Nu = 2 000. Therefore, due to this choice of criteria and depending on the magnitude bin there are some regions in redshift where there is no data.

For each magnitude bin we compute the clustering redshift distribution of all subsamples. The cluster-z/spectro-z plot for the bin iu < 22.5 is shown in Figure 5. The right panel shows the clustering versus spectroscopic redshifts for iu < 22.5. The dashed area beyond zspec = 1.2 corresponds to redshifts where there is no galaxy available from the unknown sample. For each clustering redshift distribution (vertical lines) we compute the mean cluster-z zclust . Then we compare this value to the mean spectroscopic redshift zspec from the selected tomographic sample ∆z = zclust -zspec, visible on the top left panel. From this distribution, we compute an estimate of the standard deviation as: σ = σ∆z/(1 + zspec), where σ∆z = 1.48 × median(|∆z|). The bottom left panel shows the histogram of redshift densities where no signal is expected. µnoise and σnoise then reflect the background noise. We also compute the mean signal to 1. This table summarises the accuracy of the clusteringbased redshift reconstruction on the mean redshift when varying the i-band magnitude from 22.5 to 25 by step of 0.5. From these tests it appears that the method is stable and does not depend on the i-band magnitude at a level higher than a few 0.1%.

noise ratio per magnitude bin, SNR defined as: SNR = mean(SNRi), where SNRi is the signal to noise ratio of each clustering redshift distribution estimated by fitting a Gaussian and taking the ratio between the amplitude and its uncertainty. We give all plots as complementary online supplement for the reader and we summarize all results in Table 1. From this table it appears that the clustering based redshift inference does not intrinsically depend on the i-band magnitude of the unknown population at a level higher than few 0.1% in the measurement of the mean redshift.

Evolution with the number of objects in both

the unknown and reference sample:

In the first part we fix the following set of parameters:

• ∆zu = 0.02.

• Nr = 200 000 (∼ 0.004 gal.arcmin -2 .(δzr/0.01) -1 ).

• iu < 22.5. We then vary the number of unknown objects in each tomographic bin to be:

• Nu = 2 000 (∼ 0.55 10 -2 gal.arcmin -2 ).

• Nu = 1 500 (∼ 0.41 10 -2 gal.arcmin -2 ).

• Nu = 1 000 (∼ 0.27 10 -2 gal.arcmin -2 ).

• Nu = 500 (∼ 0.14 10 -2 gal.arcmin -2 ).

• Nu = 250 (∼ 0.07 10 -2 gal.arcmin -2 ).

Then, to study the evolution in the number of reference sources we fixed Nu = 2 000 and start from the optimistic reference sample. We go to lower and lower pessimistic spectroscopic densities :

• Nr = 200 000 (∼ 0.004 gal.arcmin -2 .(δzr/0.01) -1 ).

• Nr = 150 000 (∼ 0.003 gal.arcmin -2 .(δzr/0.01) -1 ).

• Nr = 100 000 (∼ 0.002 gal.arcmin -2 .(δzr/0.01) -1 ).

• Nr = 50 000 (∼ 0.001 gal.arcmin -2 .(δzr/0.01) -1 ).

• Nr = 25 000 (∼ 0.0005 gal.arcmin -2 .(δzr/0.01) -1 ).

In both cases we follow the same procedure than in Section 4.1 and summarise the results in Table 2. Figure 6 shows that the SNR evolves as √ NuNr. This is in agreement with the expected behaviour of the integrated crosscorrelation function. Indeed a crosscorrelation signal evolves as the number of pairs of objects NuNr while in regions where there is no signal, the noise is stochastic and is expected to evolve as √ NuNr.

Evolution with the width of the unknown distribution: ∆zu

In this part we fix the following set of parameters:

• Nr = 200 000 (∼ 0.004 gal.arcmin -2 .(δzr/0.01) -1 ).

• Nu = 2 000 (∼ 0.55 10 -2 gal.arcmin -2 ).

• iu < 22.5.

Finally, we vary the redshift width of the unknown sample to be: ∆zu = 0.02, 0.04, 0.08, 0.16, 0.32, 0.64. Wider distributions progressivly break our hypothesis on the evolution of the unknown clustering amplitude: dβu/dz = 0. This can also be seen as a photometric redshift selection with ∆zu ∼ 2σ phot where σ phot is the photo-z scatter. This is correct considering no or few catastrophic failures. The results are shown in Figure 7 where the red dashed line shows the expected offset considerring a linear evolution with redshift of the galaxy to dark matter bias. This is done assuming a Gaussian redshift distribution following Ménard et al. (2013). This confirm that working with localised unknown samples reduced the non linear evolution of the galaxy to dark matter bias to a linear evolution. The agreement between our measurements and the theoretical expectations also shows that if a survey does not allow to preselect localised distributions narrower than ∆zu = 0.2 then considering no evolution of the galaxy to dark matter bias does not allow to reach 0.2% accuracy in the mean redshifts. Instead, it is highly recommended to consider a linear evolution.

Requirements for clustering redshift

Based on these studies we can now higlight the statistical requirement on the clustering redshift method to reach ∆z ∼ 0.001 × (1. + z) in few photo-z bins. Then depending on the possibility to preselect narrow distributions one can consider sufficiant to assume no or linear evolution for the galaxy to dark matter bias in the measurement (see Section 4.3). Let's consider a realistic reference sample like the BOSS DR12Q (Pâris et al. 2017). We note that there is at least 2000 QSOs over 9376 deg 2 per δz = 0.05 from redshift 0.5 to 4. This gives an estimate of the available reference density of:

10 -5 qso.arcmin -2 . δz 0.01 -1

.

(5)

In the previous study, we reached the desired accuracy of 0.1% on the bias with Nr=100k, Nu=2000 over 100 deg 2 . The reference density was then:

2.10 -3 gal.arcmin -2 . δz 0.01

-1 , (6)
while the unknown one was: 5.10 -3 gal.arcmin -2 .

Using BOSS as reference sample implies a requirement on the minimal unknown density of galaxies of 5 gal.arcmin -2 . Consequently a BOSS-like spectroscopic survey over an area of 9000 deg 2 with ∼ 200 reference sources per reference slices δzr = 0.005 allows to reach ∆z ∼ 0.001 × (1. + z) for tomographic bins.

INDIVIDUAL REDSHIFT MEASUREMENT

This section aims at demonstrating that it is possible to get accurate PDF(z) measurement for each individual galaxy of the unknown population. Following [START_REF] Scottez | Theses[END_REF] we present a straightforward approach based on colours.

In the general case the sample distribution is the sum of all distinct PDFs:

dN dz = N i PDFi(z) . (8)
In the limit case where dN dz = δ D (z -z0), then:

PDFi(z) = 1 N × δ D (z -z0) . (9)
When the redshift distribution is not a δ D (z) function, this approximation will lead to an error in the measurement. The narrower the distribution, the smaller the error.

In this section we quantify the accuracy of individuals clustering redshifts by computing: the bias= (z clustzspec)/(1 + zspec), the outlier fraction η = |z clust -zspec|/(1 + zspec) > 0.15 and the scatter σ = 1.48 × median(|z clust -zspec|)/(1 + zspec).

Sampling the true colour space

To preselect narrow redshift distribution, we subsample the parent unknown sample defined in Section 3.1.2 based on galaxy colours using: u, g, r, i, z, Y, J, H photometric bands. One can note that these values are true values since there is no error on magnitude within the MICE2 simulation. Each of these samples correspond to a 7-dimensional colour space volume element of width ∆ colour = 0.1. We can then measure the clustering redshift distribution of these samples with Nu 1000. Due to this selection the fraction of lost objects is important and is around 67%. Nevertheless, this effect is expected to decrease with the increasing size of the unknown sample.

Considering these distributions are sufficiently narrow, we take the median of each distribution and claim that this is our estimate of the individual redshift for all galaxies in this volume element. Applying this on the full color space we can estimate the redshift of each galaxy (see Figure 8). This gaussian kernel density map shows the clustering redshift as a function of the spectroscopic redshift for each galaxy down to i < 25.

We remind the reader that here we are using perfect photometry. We will explore a realistic case in the next section. However this study demonstrate that our clustering based redshift estimator is not intrinsically biased at a level higher than 0.01%.

Sampling the real colour space

In this section we added realistic error on the magnitude. To do so, we simulate the photometry by assuming megacam instrumental characteristics for ugriz. For NIR depth we assum the WIRCam instrument characteristics. The exposure times are tuned to match the following depths at 5σ for 2 diameter apertures u = 24.2, g = 24.5, r = 23.9, i = 23.6, z = 23.4, Y = 23, J = 23, H = 23 and modeling the error as a gaussian random error on the true flux and a gaussian random background noise from the sky from 2 diameter aperture. Figure 9 shows the variation with magnitude of the resulting error for the i-band.

Adding noise to magnitudes induces a scatter of galaxies in colour space. This leads on average to a decrease in the number of galaxies per cell. For this reason we choose to use a resolution of ∆ col = 0.15. Then, we apply the same procedure than in Section 5. We show the resulting individual clustering redshifts compare to their true spectroscopic redshifts in Figure 10. These objects have iu < 25 with a median at iu = 23. While the quality of these measurements obviously degrades with respect to Figure 8 they are still quite competitive with photometric redshifts goals of next generation of cosmological surveys in terms of scatter, outlier rate and bias. At this point it seems important to remind the reader that this approach is independent from the photometric redshifts procedure since the colour information is only used in the preselection step and not to extract the redshift information. The redshift information come only from the clustering of objects. The only observables used to extract this information are then the right ascension (RA) and the declination (DEC) of the unknown population.

The advantage of this procedure is threefold: i) it allows the use of clustering redshifts for any field in extragalactic astronomy, ii) it allows the possibility to combine photometric and clustering based redshifts to get an improved redshift estimation, iii) it allows the use of cluster-zs to define tomographic bins for weak lensing. We explore this last option in the next section.

Clustering redshifts based tomographic sampling

Finally, we investigate the accuracy in the mean of the redshift distribution one can reach in the context of tomographic weak lensing. To reduce the effect of the evolution of the galaxy to dark matter bias of the unknown sample, we selected 30 clustering redshifts bins from 0.2 to 1 based on individual estimates from Figure 10. For each of these 30 bins we re-measured the redshift distribution to benefit from the increase in Nu. These bins are then combined to build 5 tomographic bins as shown in Figure 11 where the redshift resolution is δzr = 0.03 for clarity. This figure shows the true and measured redshift distributions of the five clustering redshift selected tomographic bins. The lower panel shows the bias of each bin (black) while the dashed area corresponds to ∆ z 0.002(1 + z). Please note that the bias is estimated on the mean redshift of the distribution: bias = ( z clust -zspec )/(1+ zspec ). We remind the reader that this is done with a sample complete to iu = 23.5 due to the simulation. Even if our results in Section 4.1 indicate no evolution with the magnitude of the unknown sample, this could be affected by faintest objects at redshifts higher than the simulation limite.

SUMMARY

In this study, we aimed at giving a preview of the redshift accuracy one can reach on the redshift estimation using the clustering of galaxies.

We explored the accuracy of clustering redshifts and we estimate that a density of spectroscopic objects -galaxies or QSOs -of 10 -5 sources.arcmin -2 per redshift bins of width δz = 0.01 allows to reach the 0.1% accuracy in the estimate of the mean redshift for a galaxy density compatible with next generation cosmological surveys. This number is compatible with the density of Quasi Stellar Objects in BOSS.

We also demonstrated that it is possible to get an estimate of the redshift for each galaxy in a fully independent way from photometric redshifts. The resulting clustering redshifts have a bias= 0.001, an outlier fraction of η = 3.57% and a scatter of σ = 0.027 with a sample complete to i = 23.5 with maximum magnitude of i = 25. Clustering redshifts are then competitive compared to photometric redshifts.

These individual measurements: i) allow the use of clustering redshifts for any field in extragalactic astronomy, ii) allow the possibility to combine photometric and clustering based redshifts to get an improved redshift estimation, iii) it allows the use of cluster-zs to define tomographic bins for weak lensing.

Based on these results we investigated the reachable accuracy in the measurement of the mean of the redshift distribution which is currently still an issue for tomographic weak lensing. In this context, we demonstrated our ability to build 5 clustering redshift selected tomographic bins from redshift 0.2 to 1 with a bias of 0.002 per bin with a sample complete to i = 23.5 and imax = 25.

Finally, even if Section 4.1 indicates no evolution with magnitude and allow us to be confident in the generalisation of this result to fainter complete sample. Nevertheless, this need to be confirmed. Also, the lack of redshift > 1.4 which leads to an additional degeneracy in color space resulting in additional noise in the detection. This effect and the use of clustering redshifts for weak lensing will be explored in a future work within the full Flagship simulation (Potter et al. 2016).

Figure 1 .

 1 Figure 1. Redshift distribution of the reference sample. The decreasing number of sources from z = 1.2 directly reflect the lack of sources in the MICE2 simulation at these redshifts for i < 22.5.

Figure 2 .

 2 Figure 2. Clustering amplitude of the reference population normalised to 1 at z 0 = 0.07. The solid green line shows a smoothed version of the measurement.

Figure 4 .

 4 Figure 4. LEFT PANEL: clustering redshifts obtained with the full reference sample. One can note some horizontal stripes caused by the inhomogeneity of the reference sample. RIGHT PANEL: clustering redshifts obtained with the cleaned reference sample, where the stripes disappear.

Figure 6 .

 6 Figure 6. Evolution of the signal-to-noise ratio of the clusteringbased redshift when varying the i-band magnitude of the unknown sample (black) the number of unknown objects (green) and the number of objects in the reference sample (blue). As expected the SNR evolves as the square root (red dashed line) of the number of pairs in the reference and unknown populations.

Figure 7 .

 7 Figure 7. Offset in the mean clustering based redshift, when varying the width of the unknown redshift distribution ∆zu from 0.02 to 0.64. This plot also shows the 2σ region around the best fit of the measurement (blue), the Euclid requirement of 0.2% accuracy (yellow) and the expected offset considerring a linear evolution with redshift of the galaxy to dark matter bias (red dashed).

Figure 8 .

 8 Figure8. Individual clustering redshifts as a function of the spectroscopic redshifts for each galaxy when using perfect photometry. The red dashed lines correspond to the limit beyond which the redshift of a galaxy is considered as catastrophic.

Figure 9 .

 9 Figure 9. Simulated error for the i-band versus the true magnitude. This takes into account a gaussian random error on the true flux and a gaussian random error from the sky background through 2 diameter aperture.

Figure 10 .

 10 Figure 10. Density map of individual clustering redshift as a function of the spectroscopic redshift considering realistic error on magnitude measurements. The red dashed lines correspond to the limit beyond which the redshift of a galaxy is considered as catastrophic.

Figure 11 .

 11 Figure 11. True and measured redshift distributions of the five clustering redshift selected tomographic bins. The lower panel shows the bias of each bin (black) while the dashed area corresponds to the Euclid requirements. Please note that here the bias is estimated on the mean redshift of the distribution.

Table

	Magnitude	∆z	σ	µ noise	σ noise	SNR
	iu < 22.5	0.001	0.002	0.001	0.007	20
	22.5 iu < 23	0.001	0.002	0.001	0.007	19
	23 iu < 23.5	0.001	0.002	0.001	0.007	19
	23.5 iu < 24	0.001	0.002	0.001	0.007	20
	24 iu < 24.5	0.001	0.002	0.001	0.008	19
	24.5 iu < 25	0.001	0.002	0.001	0.007	22

Table 2 .

 2 This table summarises the accuracy of the clustering based redshift reconstruction on the mean redshift when varying the number of unknown objects in each selected bins for Nu = 2000 down to 250 and when varying the total number of reference objects: Nr = 200k down to 25k.

	Number of objects	∆z	σ	µ noise	σ noise	SNR
	Nu = 2000	0.001	0.002	0.001	0.007	20
	Nu = 1500	0.001	0.002	0.001	0.008	18
	Nu = 1000	0.001	0.002	0.001	0.010	16
	Nu = 500	-0.017	0.002	0.001	0.013	12
	Nu = 250	-0.077	0.003	0.001	0.019	9
	Nr = 200k	0.001	0.002	0.001	0.007	20
	Nr = 150k	0.001	0.002	0.001	0.007	18
	Nr = 100k	0.001	0.002	0.001	0.009	17
	Nr = 50k	0.002	0.002	0.002	0.010	14
	Nr = 25k	0.002	0.002	0.002	0.018	9

ACKNOWLEDGMENTS VS acknowledges funding from the Centre National d'tudes Spatiales (CNES) through the Convention CNES/CNRS N o 140988/00 on the scientific development of VIS and NISP instruments and the management of the scientific consortium of the Euclid mission. VS acknowledges the Euclid Consortium and the Euclid Science Working Groups. The MICE simulations have been developed at the MareNostrum supercomputer (BSC-CNS) thanks to grants AECT-2006-2-0011 through AECT-2015-1-0013. Data products have been stored at the Port d'Informació Científica (PIC), and distributed through the CosmoHub webportal (cosmohub.pic.es). Funding for this project was partially provided by the Spanish Ministerio de Ciencia e Innovacion (MICINN), projects 200850I176, AYA2009-13936, AYA2012-39620, AYA2013-44327, ESP2013-48274, ESP2014-58384, Consolider-Ingenio CSD2007-00060, research project 2009-SGR-1398 from Generalitat de Catalunya, and the Ramon y Cajal MICINN program.


H Zaidaoui
email: hajar.zaidaoui@cnrs-orleans.fr

T Boushaki

J C Sautet

C Chauveau

I Gökalp

Effects of CO 2 -H 2 O dilution on the characteristics of CH 4 -air-O 2 flames

• Freely propagating methane-air flames.

• One-dimensional premixed flame.

• GRI-mech 3.0 mechanism.

• Atmospheric pressure and 300K.

• Equivalence ratio : from 0.8 to 1.2

• Oxygen enrichment : from 21% to 30%vol.

• Dilution : (0-20%vol) for both CO 2 and H 2 O.

The numerical computations were conducted with COSILAB software -With O 2 : notable increase in flame temperature and laminar burnig velocity.

-With dilution: significant decrease in laminar burning velocity, the effect of CO 2 is greatest.



  Study of non-premixed turbulent flames stabilized by a swirler  Control of pollutant emissions / emissions standards  Improve performances of combustion plants  Investigations on flame characteristics  Effects of oxygen enrichment  Effects of steam and CO 2 dilution  Effects of swirl intensity

→

  The authors are grateful for the financial support of this work provided by the French Government's Investissement d'Avenir program: LABEX CAPRYSSES, under GrantANR-11-LABX-0006-01. NO X and CO emissions and exhaust gas temperature are reported as a function of H 2 O and CO 2 dilution. Case: Sn = 0.8, ɸ = 0.8, and 25% oxygen enriched methane/air mixture at atmospheric pressure. Effect of CO 2 dilution is stronger than that of steam dilution. OH* intensity distributions of methane/air swirling flames diluted by CO 2 , H 2 O and EGR, in the case of 21% O 2 , for Sn=0.8 and ɸ=0.8. → The flame becomes taller and unsettled with dilution. → CO 2 has greater effect on the flame lift-off heights. Flame lengths as a function of CO 2 and H 2 O diluted for 25% oxygen enriched methane/air mixture for Sn=1.4 and ɸ=0.8. → EGR dilution increases lift-off heights about 60% compared to the case without dilution. → The flame becomes much taller with CO 2 dilution. Axial velocity and RMS profiles at z=30mm with & without CO 2 and water vapor dilution for Sn=1.4 and ɸ=0.8.→ With CO 2 and H 2 O dilution, the maximum velocity is higher and the flow is narrower. → Root Mean Squares are higher in the case of dilution.


Experimental investigation of the evaporation of droplets in diamagnetic levitation

Sergey Semenov [START_REF] Chetouani | Diamagnetic levitation with permanent magnets for contactless guiding and trapping of microdroplets and particles in air and liquids[END_REF] , Vincent Haguet 2 , Christian Jeandey 3 , Mickaël Antoni [START_REF] Chetouani | Diamagnetic levitation with permanent magnets for contactless guiding and trapping of microdroplets and particles in air and liquids[END_REF] 1 Aix-Marseille Université, CNRS, MADIREL, Marseille, France 2 Univ. Grenoble Alpes, CEA, Inserm, BIG-BGE, Grenoble, France 3 Univ. Grenoble-Alpes, CEA INAC-SBT, Grenoble, France

The time evolution of levitating pure water and water-ethanol droplets is investigated. Droplets are trapped in stable diamagnetic levitation in the magnetic well produced by a cylindrical permanent magnet. Radius of evaporating droplets and their temperature differences with air are monitored over time using a thermographic camera.

Experimental set-up

A cylindrical permanent magnet made by 8 NdFeB sectors with radial magnetization and covered by a pole piece (Figure 1(a)) is used to trap diamagnetic droplets in levitation. A screwed brass base maintains all the NdFeB sectors together. In addition, a cylindrical hole of 1.6 mm diameter is realized along the symmetry axis of this set up. The gap between the sectors and the pole piece is adjusted in order to produce a potential well where diamagnetic levitation can be achieved [1,2].

Droplets are generated by a piezoelectric jetting device. It consists of a tiny glass tube covered with a piezoelectric ceramic (Figure 1(b)). A capillary transports the liquid solution from a 15 mL test tube to the glass tube. The applied potential in the piezoelectric ceramic is computercontrolled. As the ceramic shrinks, pressure builds up in the glass tube and a jet of microdroplets (i.e. droplets with a radius of few tens of micrometers) is ejected from the tube nozzle (Figure 1(b)). The ejection energy depends upon the applied potential and therefore determines the properties of the jet. The main difficulty in this procedure consists in the requirement for microdroplets to fall into the magnetic well. As the jet is extremely sensitive to disturbances of environment, it tends to behave randomly. A confinement facility is used to limit this drawback. Once microdroplets are trapped in the magnetic well, they spontaneously coalesce after few seconds and form larger droplets. When millimeter-sized droplets are produced (i.e., with a radius larger than 200 m), the jetting is stopped. A thermographic camera placed above the magnetic levitation site captures infrared (IR) radiation emitted by the droplet in the spectral wavelength range from 7 m to 10 m. It records a field of view (FOV) of ~2.0 mm x 1.6 mm and allows measuring the droplet size as well as the intensity of emitted IR radiation over time. Images are grabbed on a 160 x 128 matrix of sensors and post treated with the ImageJ software. The spatial resolution is about 12.8 m. Thermography measurements depend on the IR emissivity and observation angles. Accessing to the temperature field in the full FOV is therefore not straightforward (although sensitivity of the camera is less than 20 mK), especially for ethanol that is transparent in the considered spectral range.

Used liquid solutions comprise pure water and water/ethanol mixtures (20%, 40%, 60% and 80%, 96% v/v). Water is deionized. In order to achieve a better reproducibility, experiments were carried out several times for each solution. Room temperature T and humidity H are 203 °C and 305%, respectively.

Results

Depending upon composition and size, droplets evaporate in periods of time ranging from seconds to hours. Evaporation time may be strongly affected by working conditions and in particular by the saturation of ambient air with the solution vapor. Controlling experimental conditions is therefore of fundamental importance and that is why experiments were repeated up to ten times.

Figure 2(a) presents a typical image of a levitating droplet right after creation. As temperature conditions are almost isothermal, color contrasts are low but the droplet is still clearly visible. Its diameter can therefore be evaluated and followed over time. For water/ethanol mixtures initial contrasts are usually worse due to transparency of ethanol. But, as time runs, evaporative cooling comes into play and temperature difference with the environment increases. Droplets become then clearly observable and their diameter easy to measure as illustrated in Figure 2 Figure 3 shows the time evolution of the volume V(t) of a water and a water/ethanol (40% v/v) droplet in comparable working conditions with initial volume V(0)≈0.04 L. Evaporation time tevap is defined from a linear fitting as illustrated in this figure. As expected, ethanol mixtures evaporate faster than pure water droplets (tevap ≈ 470 s for 40% ethanol, tevap ≈ 610 s for pure water).

Conclusions

Diamagnetic levitation makes possible contactless experiments with evaporating droplets of deionized water and water/ethanol mixtures. Thermography has been used to follow the time evolution of droplet volume. Evolution of the IR signals captured by the camera indicates clear qualitative changes in the droplet temperature. However, they have not been fully exploited so far. Main limitation here is the non-trivial relation between angle-dependent IR emissivity of the droplets and their temperature. Despite this limitation, occurrence of evaporative cooling regimes could be detected and their influence on the overall mass flow quantitatively addressed.

Figure 1 .

 1 Figure 1. (a) Top view of the permanent magnet. (b) Experimental set-up made of a piezoelectric nozzle, the magnet and a thermographic camera. The microdroplet jet from the piezoelectric nozzle to the magnetic levitation site is schematically plotted with a full blue line.

  (b).

Figure 2 .

 2 Figure 2. Thermographic images of levitating droplets. (a) A levitating water droplet (r≈500 m) right after formation. Dashed circle indicates the border of the 1.6 mm-large magnet hole. (b) A levitating water/ethanol droplet (96% v/v ethanol) 21 minutes after its formation.


René Fulchiron

Beatriz Olalla

Rene Fulchiron

Relaxation of loose agglomerates of magnesium hydroxide in a polymer melt

The study of agglomerates of magnesium hydroxide badly dispersed in an olefinic polymer by means of oscillatory shear flow in the melt indicates that the agglomerates behave as a distinct phase. Optical and rheo-optical measurements carried out on these agglomerates also show that they are wetted by the melt and that they can be easily deformed under shear. However, the relaxation of shape in oscillatory shear is also the trace of the existence of an effective interfacial tension between the agglomerates containing high amounts of filler and the polymer matrix. The analogy with the behavior of polymer blends and the modeling by the Palierne model enables the calculation of such an interfacial tension. The surface free energy of magnesium hydroxide that can be calculated from this result by a simple mixing law is 95 mN m -1 . This value is compared with data from the literature on brucite and similar minerals.

INTRODUCTION

Magnesium hydroxide is widely used as flame retardant in olefinic polymers such as polyethylene, polypropylene and their copolymers. Upon heating, the strong endothermic dehydratation reaction which occurs at 580 K with water as a product makes it particularly efficient for the improvement of fire resistance properties of plastic materials. Nonetheless, this efficiency is only interesting provided that other technological properties such as mechanical resistance are not impaired by sideeffects connected to the bad dispersion of the mineral in the organic medium especially regarding the occurrence of agglomerates.

In a previous work [1], we studied the dispersion mechanism of a single agglomerate of magnesium hydroxide immersed in a molten polymer matrix and subjected to shear. At low shear stress, erosion was the main mechanism responsible for the dispersion while when the shear stress increases, the mechanism switches to rupture. This has been reported many times in the literature and for various fillers mostly for particle agglomerates with high cohesive strength. In the case of magnesium hydroxide the cohesion is much less therefore, the transition occurs at lower shear stress in comparison to silica or carbon black for example. This is partly due to the high porosity of the agglomerates. Because of this porosity, the molten polymer is able to infiltrate the space between the elementary particles. For this reason, it was observed that the rupture of agglomerates occurs upon shearing in a ductile manner, with a large deformation of the wetted agglomerates before breaking. This means that for infiltrated agglomerates, the stress criterion is not meaningful enough and that a "new" criterion mixing stress and strain such as energy must be used instead.

The fact that an infiltrated agglomerate behaves as a separate phase with internal cohesion that differs from that of the matrix implies that this phase must have its own surface properties. The modification of the surface tension of polymers by mineral or organic fillers has already been reported. The issue is generally addressed in view of the compatibilization of polymer blends by fillers that, in particular conditions, may locate at the interface between the phases. This is an extension of phenomena already reported in low viscosity liquids where such emulsions are often referred as Pickering emulsions [2] when colloidal particles are used. These particles act to reduce the coalescence of droplets but reduction of interfacial tension is not the only mechanism [3]. In the case of highly viscous liquids, the situation seems more complex since both reduction of interfacial tension and hindering of coalescence are reported. Ellias et al. [4] found that depending on their hydrophilic or hydrophobic nature, silica particles may act by one and the other mechanism in PS/PP blends. Hydrophilic silicas were preferentially located in the PS phase and reduce the interfacial tension from 2.15 until to 0.085 mN m -1 . On the contrary in this study, hydrophobic silicas were either located at the interface preventing coalescence or in the PP phase without effect on the interfacial tension. According to the same authors [5][6][7], in PP/EVA blends, silica particles decrease the effective interfacial tension independent from the nature of the surface of the silica, hydrophobic (from 0.92 to 0.18 mN m -1 ) or hydrophilic (from 0.75 to 0.30 mN m -1 ). The hydrophilic silicas remain within the EVA phase whereas the hydrophobic silica stays in the EVA phase or migrate to the interface depending on the viscosity of the EVA. Effects of silica in SBR/NBR elastomer blends were also reported [8]. Hong et al. [9] showed an interfacial tension reduction in PBT/PE/clay nanocomposite from 57.6 to 1.4 mN m -1 . Similarly, the apparent interfacial tension of PE/PA blends, drops from 10 mN m -1 to 4.5 mN m -1 by addition of 1% of cloisite C30B and tends to a value of 3 mN m -1 for 2% and 4% of added clay [10]. In similar cases with PS/PP [11], the reduction of the interfacial tension has been attributed mainly to the preferential location of the clay at the interface. In other cases, it was supposed to be due to penetration of the polymer chains in the space between layers in the case of lamellar organo-clays in PS/PP blends [12], in PS/PEMA [13] or PC/PMMA blends [14][15]. In theses situations however, interfacial tension is an apparent property that should rather be considered as a property of the fillerpolymer system than a property of the polymer couple.

In this paper, we report about the apparent interfacial tension that may exist between a polymer melt and the same melt filled with a mineral component. In particular, it was observed that when a mineral such as magnesium hydroxide is badly dispersed in a melt of polyethylene-octene where it forms wetted agglomerates, these agglomerates behave as a distinct phase from the rheological point of view. In particular, they deform in oscillatory shear flow and show a shape relaxation similar to that observed in polymer blends. Such a behavior can be modeled by the Palierne model and owing to the fitting, an apparent interfacial tension between the loose agglomerates and the melt can be determined. The value of free surface energy of the mineral that can be calculated from the interfacial tension is discussed in view of data of the literature regarding the surface energy of minerals commonly used as fillers in polymers.

It is worth noting that in most of the studies mentioned before, the particles were nanometer in scale, at least in one dimension. In particular, for the studies dealing with silica, the primary particles were nanometer in scale with aggregate dimensions ranging from 10 to 100 nm and mostly around 10 nm. In the present work, the primary particles are sub-micronic and agglomerates are hundred times larger. Though the dispersion problems should clearly be discussed in a multi-scale approach, the mechanisms that lead to the infiltration of the polymer in the agglomerates are not discussed in details since they were addressed in a previous paper [1]. Therefore, the present paper assumes that continuum-scale models can be applied.

EXPERIMENTAL

Materials

Magnesium hydroxide Mg(OH) 2 Hydrofy® (Nuova Sima) was dispersed in an ethylene-octene copolymer Engage® 8401 (Dow Chemicals). Magnesium hydroxide is natural and non-treated with a uniform shape factor. The size of the particles measured on the powder in a Mastersizer®2000 device using a 3 bar gas flow showed 10 vol-% of elemental particles having a diameter lower than 0.67 μm, 50 vol-% of particles having a diameter lower than 2.90 μm and 10 vol-% of particles having a diameter higher than 7.85 μm. Therefore, these are the minimum sizes that can be achieved by mechanical dispersion. The distribution of sizes is given on Figure 1. The olefinic copolymer contains 30% of octene co-monomer, its weight average molecular weight is 70000 g/mol with a polydispersity of 2. The rheological behavior is Newtonian until 1 s -1 and the zero-shear viscosity at 140°C is 1250 Pa.s.

Mixing

Two set of samples were prepared. In the first set, samples containing 0 to 60 wt-% (0 to 33.5 vol-%) of filler were prepared in the same mixing conditions. The desired amount of powder was dispersed in the matrix in a Haake Rheomix batch mixer at 140°C. The polymer granules and the powder were previously tumble-blended during 5 minutes before feeding then blending was carried out at 32 rpm (approximately 16 s -1 ) during 10 minutes. In all cases, the actual content of filler was checked by weighting the amount of minerals left after one hour at 700°C using the decomposition reaction of the magnesium hydroxide. No significant loss of mineral can be noted.

The second set of samples was prepared in the same way but only 10 wt-% of powder (3.6 vol-%) was used and the mixing time was varied from 2 to 10 minutes. In the selected shear rate and temperature conditions, the shear stress within the polymer matrix is less than 20 kPa. According to our previous work [1] on diluted systems, the main dispersion mechanism is erosion for agglomerates with diameters below 80 µm. For larger diameters, this mechanism switches to a rupture mechanism.

State of dispersion

The state of dispersion of the filler was analyzed by SEM microscopy coupled with X-ray analysis of the magnesium element in a Hitachi S3000N microscope. Observations were performed on cryofractured samples after coating with Au-Pd alloy at 15kV. Mapping of the magnesium in the sample was used to assess the state of dispersion. Acquisition time was about 20 min for a 100µm x 100 µm typical picture. The pictures were then treated using the imageJ software. The observed pictures were threshold and binarized, eventually including some despeckling. All images presented in this paper show the mapping of magnesium (in black) in the matrix (in white) without effects of the topography of the samples.

In addition, direct observations of the individual agglomerates and of the reduction of size during shearing were obtained in the counter-rotating transparent rheometer developed by Seyvet and Navard at CEMEF [16][17]. Particles were disposed in between two polymer films with less than 1 mm thickness and then between two 40 mm glass plates rotating in opposite directions in the rheometer. The observations of an immobile particle were made with an optical microscope coupled with a video camera at 140ºC.

Rheological behavior

Measurements in oscillatory shear flow were performed at 140°C in a Paar-Physica MCR301 rheometer using a plate-plate geometry with a diameter of 25 mm and a gap of 1.5 mm. Frequency sweeps were carried out in the range 0.01 rad/s to 100 rad/s. The maximum strain amplitude was chosen to operate within the linear viscoelastic conditions. The strain values range from 1% for the pure polymer to 0.01% for the highly filled materials. For each composition, the experiments were repeated three times and reproducibility was good. In particular, no slip could be detected even at high filler loadings and the phase angle signal was stable.

RESULTS

Evidence of agglomerates at all concentrations from the hydrodynamic interactions

Figure 2 shows the trace of agglomerates of magnesium hydroxide obtained from the SEM pictures after X-ray mapping of the magnesium in the first set of samples with 0 to 33.5 vol-% of filler. After threshold and binarization, the image displays some areas with large concentration of magnesium that correspond to the agglomerates. In this polymer matrix, though the residence time in the mixer was long for these samples, the dispersion efficiency is limited due to the low viscosity of the melt.

These agglomerates are thus made of elementary particles surrounded by the polymer but the shear stress is not high enough to induce the breaking of the agglomerates especially if their size becomes too small [1]. Our previous work on very diluted systems, and particularly the derivation of the relation between the critical stress and the size (Equation 2), show for the shear stress generated in the mixing conditions (20 kPa), the particles with diameter larger than 80 µm cannot be broken. Of course the situation might be clearly more complicated in concentrated systems since collisioninduced breaking can also occur. In this previous work, we have shown an evidence of the penetration of the melt inside the agglomerates through the observation of the effect of the hydrodynamic contribution of the filler to the overall rheological behavior of the polymer filled with various amounts of magnesium hydroxide. Indeed, the function (C) describing the hydrodynamic interactions can be obtained from the storage and loss moduli measured in oscillatory shear in the form proposed by Krieger and Dougherty [18]:

M C ] η [ M ) C KC 1 ( = ) C ( ψ   (1)
Where [] is the intrinsic viscosity of the particles, C M is the maximum packing fraction,

In this equation, we have added the K parameter to account for the effective volume that is actually occupied by the particles if they are infiltrated by the matrix [19]. This accounts for the amount of polymer that is trapped between the elementary particles into the agglomerates. For dense spherical particles with uniform sizes, the packing fraction is 0.62, [], C M = 2, and K =1 [19]. On the opposite, when polymer is entrapped between the elemental particles in the agglomerate, the effective volume is larger and K>1. The figures show that for the compounds, a secondary plateau on the storage modulus appears for volume fraction above 13 vol-%. A secondary plateau also exists on the loss modulus. For high concentration, such an effect is related to gel effect for well dispersed particles but also to internal reorganizations within the agglomerates [20][21]. Whatever their nature, these inter-particular interactions are generally observed at low frequencies and only a slight effect is expected at high frequency, especially regarding the loss modulus. This is evidenced in Figure 4 giving the ratio of the loss modulus at 100 rad/s for a given filler content to that of the pure matrix as a function of the volume fraction of filler.

Evidence of agglomerates in semi-concentrated suspensions during mixing

Figure 5 shows the trace of the agglomerates of magnesium hydroxide obtained from the SEM 10 pictures after X-ray mapping of the magnesium in the case of samples containing 3.6 vol-% of filler at various mixing times. After threshold, despeckling and binarization, the images show an apparent area connected to magnesium that is larger than the actual amount of filler. This evidences that the agglomerates are infiltrated and swollen by the molten polymer and that they are not compact objects. The volume fraction of the agglomerates was calculated from the pictures. It is indicated in Figure 6 and it decreases with the mixing time. Simultaneously, the mean diameter of the observed particles also decreases. These results provide confirmation that, in the present case, the main mechanism for dispersion is erosion. Erosion leads to small elementary particles that are progressively detached from the parent agglomerates. The reduction of the number of particles inside the agglomerates decreases their size and the elementary particles are transferred into the melt. The agglomerates with the smallest size are eliminated and their overall volume fraction is also reduced upon shearing. Direct visualization of the agglomerates was also obtained in the counter-rotating rheometer as shown on Figure 7 for a large agglomerate with a diameter of 100 µm subjected to a shear rate of 9 s -1 . These images show, that under this low shear rate, the main dispersion mechanism is rupture in agreement with our previous work [1]. However, the most interesting feature in the frame of the present work is the viscoelastic behaviour of the agglomerate that shows large deformation upon shearing. This is consistent with a large amount of polymer infiltrating the loose space between particles. For such particles and diluted systems, the relation between the critical shear stress for rupture and 12 the particle diameter obtained in our previous work follows Equation 2 in the form proposed in [1]:

32 . 1 R 3300R > σ  (2)
with  R is the shear stress in kPa, R is the radius of the agglomerates in µm.

The exponent of Equation 2 is also characteristic for porous agglomerates since for high porosity, the number of bonds between the particles becomes independent of the size and the critical shear stress is inversely proportional to the square of the agglomerate radius.

Relaxation of loose agglomerates in semi-concentrated suspension in oscillatory shear

Figure 8 shows that in oscillatory shear flow, the loose and deformable agglomerates behave as a distinct phase and that the relaxation of shape under oscillation is similar to that reported in blends of molten polymers. The effect of this long relaxation time is visible on the storage modulus as a shoulder at low frequency without great influence on the loss modulus. The results in Figure 8 also show that the amplitude of the phenomena is connected to the amount and size of agglomerates.

Increasing the mixing time improves the dispersion and conversely reduces the amplitude of the low frequency relaxation. The phenomenon is best viewed on the relaxation spectra of Figure 9. These results show that the relaxation time of the agglomerates is quite constant and that only the intensity of the mode attributed to the agglomerate is changing in agreement with the decrease of the volume fraction of agglomerates. Because of the similarity of the observed behaviour with that of molten incompatible polymer blends, the Palierne model was used as a theoretical frame for a tentative explanation of the low frequency relaxation. The Palierne model [22] considers the behaviour of emulsions of incompressible viscoelastic materials in which the interfacial tension gives rise to an excess of elasticity due to the variation of the interfacial area upon deformation. The average complex shear modulus of the medium accounts for the mechanical interactions between inclusions and may be written as:

) ( H 2 1 ) ( H 3 + 1 ) ( G = ) ( G i i i i i i * m *          (3) with: ] ) ( 19G + ) ( 16G )][ ( 3G + ) ( 2G [ + ) ( G + ) ( G [ R α 40 ] ) ( 19G + ) ( 16G [ ) ( G ) ( G [ + ) ( 5G + ) ( 2G [ R α 4 = ) ( H * d * m * m * d * d * m i * d * m * m * d * d * m i i              
Where G* d , G* m are the complex moduli of the dispersed phase and matrix,  i are the volume fraction of the inclusions with radius R i ,

   is the interfacial tension.
The model was used with the behaviour of the pure polymer as the modulus of the matrix. For sake of simplicity, only average values of the radius and volume fraction were used instead of the distribution. The average radius of the agglomerate was taken from Figure 6 and the volume fraction was assumed to be that of the agglomerates accordingly. The next problem was to choose the viscoelastic behaviour of the agglomerates. For that, we tried various complex moduli of filled polymers such as those given in Figure 3 which were considered to correspond to the most homogeneous systems because the mixing time was longer. This is not absolutely true since even in these conditions of long mixing times, agglomerates were observed. Tests using the behavior of the samples containing various amounts of filler until C=0.335 was carried out in the case of the sample with the actual 3.6 vol-% of filler and a short mixing time of 2 minutes. The interfacial tension that enables the reproduction of the experimental results was set at 1.5 mN m -1 . A zero interfacial tension does not generate any long relaxation mode. Results are presented in Figure 10 for two selected concentrations C within the agglomerates. The comparison shows that systems containing a large amount of filler lead to non-deformable agglomerates and in this case the shape relaxation does not exist since the agglomerate can never be deformed. Agglomerates containing C=0.077 of filler yields the best modeling of the experimental results. In the model the hydrodynamic factor in the form of Equation 1 was also used because part of the filler is already dispersed as elemental particles and not present in the agglomerates. The packing fraction was taken at 0.62, K was set to unity because this refers to elemental dense particles and the exponent was -2. Significance of the interfacial tension between the agglomerates and the pure polymer.

The results indicate that there must exist an interfacial tension between the pure polymer and the filled polymer constituting the agglomerates. Therefore, an effective surface tension of the filled polymer can theoretically be calculated as suggested by Elias et al. in their work on blends of olefinic polymers containing hydrophobic or hydrophilic silica [4][5][6]. The value of interfacial tension between pure polymer and polymer containing C=0.077 of magnesium hydroxide is 1.5 mN m -1 . Assuming that the surface tension of the matrix is that of polyethylene,  PE =35.3 mN m -1 [START_REF] Bandrup | Polymer handbook[END_REF][START_REF] Wu | Polymer Interface and Adhesion[END_REF], the surface tension of PE containing C=0.077 of homogeneously dispersed magnesium hydroxide can be calculated by the Girifalco-Good Equation [START_REF] Girifalco | [END_REF]:

2 1 2 1 12 γ γ 2 γ + γ = γ  (3)
This yields the value of 39.9 mNm -1 . If the argument based on free energy developed by Elias et al.

is valid, the effective surface tension is an averaged value combining the surface properties of both the polymer and the mineral. The average must be calculated by accounting for the fraction of area occupied by the polymer and the filler on a given area. A calculation using a simple blending law with the volume fraction is therefore possible assuming that the dispersion is homogeneous. For C=0.077, the calculated surface energy of the magnesium hydroxide is 95 mN m -1 . Value for comparison can not be found in the literature and can hardly be measured on the powder surface because of the porosity. However, the high value of the surface energy tends to indicate a hydrophilic surface as it is generally recognized for untreated hydroxides and other minerals [26].

In comparison, surface energies range from 32 to 80 mN m -1 for hydrophobic and hydrophilic silicas. Of course, in this rough analysis, no separation can be made between dispersive and polar components of the surface energy. Nevertheless, for polyethylene, the polar component is zero. On the opposite, it must also be taken into account that for solid minerals, the prevalent theory used in surface chemistry for the calculation of surface tension is the Van Oss-Good-Chaudhury equation [27]. In this theory, the surface tension is divided into three components (γ LW : Lifshitz Van der Wals component, γ + : polar electron acceptor component, γ -: polar electron donor component) related by:

- + LW S γ γ 2 + γ = γ (4)
In Table 1, we indicate some values of surface tension components of minerals from the literature especially oxides from magnesium and aluminium, hydrophobic and hydrophilic silica and calcium hydroxide or other minerals commonly used in polymer applications. Other values can be found in the review of Lewin and al. [26]. The surface energy of the {1000} plane of natural magnesium hydroxide (brucite) was calculated ab initio by many authors and reported values ranges from 3.3 to 8 meV Å -2 [30][31] or 53 mN m -1 to 128 mN m -1 in the units used in this paper. The energy is much larger for the other planes but the plane is considered to be the most open and the largest surface. Therefore, the experimental value obtained indirectly in this work by experimental means is consistent with this range.

CONCLUSIONS

Various states of dispersion of magnesium hydroxide in an olefinic polymer were obtained by mixing the mineral with the polymer in the molten state and by varying the mixing time. In particular, agglomerates can be obtained at short mixing times. These agglomerates show a low frequency shape relaxation in oscillatory shear flow in the melt indicating that they behave as a distinct phase. Optical and SEM measurements show that this phase is much more concentrated in minerals than the matrix. Magnesium hydroxide is wetted by the melt and this mix forms loose and deformable agglomerates. The relaxation is all the more pronounced than agglomerates are large and vanishes for well-dispersed materials.

The system looks like a polymer blend and by analogy, the Palierne model for the relaxation of a viscoelastic liquid droplet within a viscoelastic liquid medium was tested. The shape relaxation not only implies a two-phase system with different viscoelastic behaviour but also the existence of an interfacial tension between the phases. This was found to be 1.5 mN m -1 . From the experimental conditions and from the knowledge of the surface tension of the liquid olefinic polymer, the surface free energy of magnesium hydroxide was calculated to be 95 mN m -1 .

This value is consistent with ab-initio calculations on the {1000} plane of brucite, the natural magnesium hydroxide mineral, reported in the literature. Also it indicates a highly hydrophilic mineral and it is comparable with the value for hydrophilic silica or calcium hydroxide.

Figure 1 :

 1 Figure 1: Particle size distribution for magnesium hydroxide particles.

Figure 2 :

 2 Figure 2: Threshold X-ray mapping of magnesium on SEM images showing the dispersion of magnesium hydroxide (in black) in the polymer matrix (in white). From left to right and top to bottom, volume fraction is: 0.036, 0.077, 0.126, 0.183, 0.252, 0.335.

Figure 3 :

 3 Figure 3: Storage and loss modulus of ethylene-octene copolymer filled with 0 to 33.5 vol-% of magnesium hydroxide.

Figure 4

 4 also shows the variation of the function  from the experimental data and the fit by Equation 1 considering a dispersed phase with uniform size and the random packing fraction C  =0.62[19]. These two extreme cases show that, even after long mixing times, the apparent volume of the dispersed phase is always larger than the magnesium hydroxide content indicating agglomerates swollen by the molten polymer. The concentration of filler within the agglomerates is 1/K = 0.84 if it is calculated from the assumption of uniform size. The corresponding fitted value of the intrinsic viscosity is 5 with this assumption.

Figure 4 :

 4 Figure 4: Hydrodynamic function (C), symbols: experimental data from loss modulus, lines: fit with Equation 1 with C  =0.62.

Figure 5 :

 5 Figure 5: Apparent area of agglomerates at mixing times: 2, 6, 10 min (image size is 100x100µm).

Figure 6 :

 6 Figure 6: Variation of the apparent volume fraction and diameter of the agglomerates with mixing time.

Figure 7 :

 7 Figure 7: Shearing of an individual agglomerate at 9 s -1 and times 0 s, 4.8 s, 8.72 s (Image width is 500 µm).

Figure 8 :

 8 Figure 8: Storage and loss modulus of ethylene-octene copolymer filled with 3.6 vol-% of magnesium hydroxide

Figure 9 :

 9 Figure 9: Relaxation spectrum of ethylene-octene copolymer filled with 3.6 vol-% of magnesium hydroxide

Figure 10 :

 10 Figure 10: Moduli of ethylene-octene copolymer filled with 3.6 vol-% of magnesium hydroxide at mixing time 2 min. Experiments versus model for two selected concentrations (C) inside the agglomerates.

Figure 11

 11 Figure11shows the results in the case of the various mixing times using the same sets of values and data of Figure6for the amount and size of agglomerates. For the long mixing time, however the amount of agglomerates measured on the SEM pictures was found to be too high to reproduce the experimental results and 2 vol-% of agglomerates were used in the model. The discrepancy can be explained by the lack of accuracy of the SEM measurements because at very long mixing time the agglomerates have almost disappeared.

Figure 11 :

 11 Figure 11: Moduli of ethylene-octene copolymer filled with 3.6 vol-% of magnesium hydroxide at various mixing times. Experiments versus model with concentrations C=0.077 inside the agglomerates.

Table 1 :

 1 Surface tension components of minerals used in polymer applications for comparison withmagnesium hydroxide (values in mN m -1 ).

	Materials	γ S	γ d	γ p	γ LW	γ +	γ -	Ref
	Al 2 O 3	41.3			35.1	0.6	16.1	[28]
	Al 2 O 3	39.7			31.6	0.6	27.2	[26]
	SiO 2	32	30	2				[24] (*)
	Hydrophobic
	SiO 2	80	29.4	50.6				[24] (*)
	Hydrophilic
	SiO 2	50.7			39.2	0.8	41.4	[26]
	MgO	-	95			-		[29] (*)
	Ca(OH) 2	70						[26]

(*) Only dispersive component or polar components are given.

Acknowledgements

This work was granted and supported by the Région Rhône-Alpes, France and by the Nexans Research Center in Lyon (France).


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