Mathieu Mougenot

Pascal Andreazza
email: pascal.andreazza@univ-orleans.fr

Caroline Andreazza-Vignolle

Raphael Escalier

Thierry Sauvage

Olivier Lyon

Pascal Brault
email: pascal.brault@univ-orleans.fr

Cluster organization in co-sputtered carbon-platinum films as revealed

Keywords: Platinum, GISAXS, Nanostructured films, Clusters, Sputtering, Fuel cells PACS : 62.23.St, 61.05.cf, 68.55.A-

Introduction

Catalytic thin film deposition is a very challenging task in many applications such as fuel cell [START_REF] Chen | Platinum-based nanostructured materials: synthesis, properties, and applications[END_REF] (Guizard, 2009) and gas sensor [START_REF] Tao | H2S sensing properties of noble metal doped WO3 thin film sensor fabricated by micromachining[END_REF] (Mitsumiya, 2003) development.

The requirements are to have catalytic clusters dispersed on a high specific area support [START_REF] Chen | Platinum-based nanostructured materials: synthesis, properties, and applications[END_REF]. Plasma sputtering deposition is demonstrated to be a particularly well suited technique for catalyst cluster growth in porous carbon in the form of catalytic thin films, especially in the case of proton exchange membrane fuel cells [START_REF] Brault | Plasma sputtering deposition of platinum into porous fuel cell electrodes[END_REF]. Indeed, magnetron sputtering deposition is known to provide various thin film morphologies ranging from dense films to columnar porous ones [START_REF] Maaza | Biaxial alignment in sputter deposited thin films[END_REF]. The main interest in these films is the expected nanostructured morphology leading to a high specific area required for efficient catalytic processes as those encountered in fuel cells, gas sensors. Such sputtered thin films can be used as substrates for depositing catalyst atoms that can grow as highly dispersed clusters on these supports [START_REF] Rabat | Carbon/platinum nanotextured films produced by plasma sputtering[END_REF]. Moreover, the interest of plasma sputtering is to localize and control the catalyst, like platinum, depth profile in the electrode region in contact with the polymer electrolyte (membrane) [START_REF] Rabat | Plasma sputtering deposition of PEMFC porous carbon platinum electrodes[END_REF]. Indeed, the platinum location and dispersion are key factors determining the fuel cell performance [START_REF] Cavarroc | Performance of plasma sputtered Fuel Cell electrodes with ultra-low Pt loadings[END_REF]. So understanding catalytic film growth is of primary importance for designing efficient fuel cell electrodes. One way for extracting the platinum atom density depth profile is to use Rutherford Backscattering Spectroscopy (Brault, 2009). However, with this technique, the catalyst morphology with respect to porous carbon matrix is not available. First attempts to determine Pt cluster size distribution have been carried out using Transmission Electron Microscopy (TEM) [START_REF] Rabat | Carbon/platinum nanotextured films produced by plasma sputtering[END_REF] [START_REF] Caillard | Plasma based platinum nanoaggregates deposited on carbon nanofibers improve fuel cell efficiency[END_REF], but extracting a cluster size depth profile is not easy and numerous statistics should be recorded. An alternative tool, Grazing Incidence Small Angle X-Ray Scattering (GISAXS), allows the determination of mean size, shape and spatial distribution of supported clusters in a near-surface region [START_REF] Penuelas | Controlling Structure and Morphology of CoPt Nanoparticles through Dynamical or Static Coalescence Effects[END_REF]) [START_REF] Andreazza | Nucleation and initial growth of platinum islands by plasma sputter deposition[END_REF]. In the Distorted Wave Born Approximation (DWBA) theoretical framework of GISAXS technique, an accurate quantitative analysis of cluster morphology is possible as described by G. [START_REF] Renaud | Probing surface and interface morphology with grazing incidence small angle X-ray scattering[END_REF]. However, characterizing cluster distributions not only at the surface but also in the thin film depth is a key challenge for the understanding and control of 3D assemblies of nanoclusters. By varying the incidence angle above the total reflection critical angle, deeper clusters will contribute significantly to the global intensity scattering from the surface to the whole film. By simulating the experimental intensity profiles, it is expected to recover the in-plane and out-of-plane clusters distribution with respect to the substrate plane, and consequently, the in-depth morphology could be fully determined. A previous attempt for analysing chemically grown Pt clusters supported on carbon powder by GISAXS [START_REF] Tsao | Small-angle X-ray scattering of carbon-supported Pt nanoparticles for fuel cell[END_REF]) yields data on cluster size distribution improving direct methanol fuel cell performances.

In this paper, we study the evolution of the in-plane and in-depth microstructure of such nanocomposite films during the growth. In particular, we use the complementary techniques of X-ray reflectivity (XRR) and small-angle scattering supported by TEM to study the formation and distribution of Pt nanoparticles within C matrix. XRR can provide nondestructive information about density, thickness, and surface and interface roughness of single-and multi-layer samples [START_REF] Gibaud | X-ray and Neutron Reflectivity: Principles and Applications, Springer, Paris Guizard C, Princivalle A[END_REF] while GISAXS explores the heterogeneities in the film structure [START_REF] Naudon | Grazing-incidence small-angle Xray scattering applied to the characterization of aggregates in surface regions[END_REF] (Maaza, 1999). This study elucidates the growth mode of co-sputtered carbon-platinum thin film, which leads to an anisotropic complex distribution of platinum clusters correlated to the textured morphology of carbon matrix.

Experimental methods

The co-sputtering deposition of platinum and carbon is achieved in a magnetron sputtering reactor which includes a 15x15cm² rotating substrate holder. The two 4 inches magnetron targets are tilted at 45° and are 12.5cm away from the substrate. Each magnetron holds a pure material target (99.995%) composed of platinum and graphite. 1 cm² pieces of (100) silicon wafer are covered by PtC thin films with different thicknesses corresponding to different deposition times. The platinum and carbon targets are sputtered with 80W DC and 1300W DC input powers respectively. The argon pressure is kept at 15.10 -3 mbar while the base pressure was 5.10 -8 mbar. These deposition parameters result in total PtC deposition rate of 15 nm min -1 . A 20% magnetron current decrease has been recorded from the carbon magnetron during deposition, while the platinum sputtering conditions remained stable.

Table .1 displays the co-sputtering conditions. I C and I Pt are the target currents. These are measuring the ion currents impinging the carbon and platinum targets respectively, resulting in the atom sputtering. The sputtered atom fluxes (F c and F Pt ) are dependent on the sputtering yields of carbon and platinum γ C and γ Pt , respectively [START_REF] Yamamura | Energy dependence of ion-induced sputtering yields from monatomic solids at normal incidence[END_REF]. They are calculated using: F C(Pt) = γ C(Pt) .I C(Pt) /e, where e is the electron charge. The argon ion kinetic energies are 750 and 380 eV when impinging respectively on C and Pt target due to bias voltage of 750 V and 380 V respectively.

In preliminary investigations, the thin film morphologies are observed using a Zeiss Field Emission Scanning Electronic Microscope (SEM) and the amount of deposited Pt atoms is calibrated by Rutherford Backscattering Spectroscopy (RBS) carried out at the CEMHTI Van de Graaff facility (Orléans, France). X-ray reflectivity (XRR) experiments were performed using the Cu-K radiation (E = 8040 eV) with a four-circle x-ray diffractometer, equipped with a parallel beam optic (mirror, slits and monochromator), to determine the thickness, roughness and density of layers constituting the stratified samples. After a global thin film morphology measurement and a determination of critical angles of total reflection  c by x-ray-reflectivity, GISAXS experiments are carried out at the SWING beamline of the SOLEIL synchrotron facility (Gif-sur-Yvette, France). The x-ray photon energy (11460 eV) was selected below the Pt L3 absorption edge to minimize Pt fluorescence and the sampledetector distance 1.2 m to adapt the 2D detector area to the size distribution range of clusters.

During an x-ray experiment, the measured intensity is recorded in function of the vector q, which is the difference between the scattered or reflected wave vector k to the incident wave vector k i , called also momentum transfer vector q (q= 4 sin(θ)/λ where λ is the wavelength and 2θ is the angle between k and k i ). In fact, the GISAXS intensity is recorded in an adapted framework [START_REF] Revenant | Quantitative analysis of grazing incidence small-angle x-ray scattering: Pd/MgO (001) growth[END_REF], as a function of the in-plane q y and out-of-plane q z components of the q vector, respectively perpendicular and parallel to the substrate, while XRR intensity is measured only in specular geometry, in function of q z (q y =0).

As in the case of reflectivity, GISAXS technique requires convenient approaches that can explain the distribution of scattered intensity in the reciprocal space. In a first approximation, the expression of the intensity scattered by a polydispersed assembly of N particles of refraction index n p inside a medium (matrix) of refraction index n m is given below [START_REF] Renaud | Probing surface and interface morphology with grazing incidence small angle X-ray scattering[END_REF]:

2 2 2 2 4 ) ( ² 16 ) ( q n n k q I m p i    
Where ) (q  can be expressed in the kinematic approximation (Born theory) as [START_REF] Babonneau | Waveguiding and correlated roughness effects in layered nanocomposite thin films studied by grazing-incidence small-angle x-ray scattering[END_REF] [START_REF] Pedersen | Determination of size distribution from small-angle scattering data for systems with effective hard-sphere interactions[END_REF]:

dD q S q F D N q ) ( ) ( ) ( ) ( 2 0 2     S(q)
is the structure factor of the assembly, characterising the interference function between particles (Fourier transform-FT of the cluster position autocorrelation function) and F(q) is the form factor of the particles, characterising the scattering intensity from a single particle (FT of the particle shape). Considering a size distribution N(D), the Local Monodisperse Approximation (LMA) corresponds to a summation of the intensities scattered by domains in which all particles have the same scattering length, i.e. locally sizemonodisperse [START_REF] Renaud | Probing surface and interface morphology with grazing incidence small angle X-ray scattering[END_REF].

However, in grazing incidence ( i ), the scattering intensity is altered, mainly from the form factor, due to multiple scattering effects at interfaces and consequently is strongly dependent on the incidence angle. To take into account these effects, the DWBA theory adapted to particles embedded in stratified medium (carbon thin film in our case) must be used in the form factor expression for each stratum of the composite film [START_REF] Babonneau | Waveguiding and correlated roughness effects in layered nanocomposite thin films studied by grazing-incidence small-angle x-ray scattering[END_REF].

Finally, Transmission Electron Microscopy (Philips CM20 set-up) was used to check the film's morphology and the platinum particle size in the carbon layer. Cross-sections were prepared in order to observe the in-depth morphology distribution of the deposits and thus, to correlate these results with the small-angle x-ray scattering measurements.

Results and discussion

The analysis of SEM cross section images (see Fig. 1 for thick co-sputtered PtC thin films) allows the definition of an approximated model of nanostructured thin film morphology, in order to provide a fitting model for the scattering patterns analysis. It reveals a columnar carbon structure including platinum clusters over the whole thickness, as already observed in previous studies [START_REF] Rabat | Plasma sputtering deposition of PEMFC porous carbon platinum electrodes[END_REF] [START_REF] Cavarroc | Performance of plasma sputtered Fuel Cell electrodes with ultra-low Pt loadings[END_REF] (Brault, 2009). The carbon columns enlarge from the bottom to the surface with an estimated angle of 5° (for example, in the case of 110nm-thick film, the column base width=10nm and top width=30nm). This morphology often occurs in magnetron sputtering due to a shadowing effect. The highest columns grow at the expense of smaller ones and thus broaden in the course of the deposition process.

Figure 2a shows the normalized reflectivity profile as function of the out-of-plane scattering vector q z for the PtC20 sample, which was fitted by assuming a PtC homogeneous nanocomposite layer on a SiO 2 amorphous layer (see schematic drawing in inset). At small q z values, the incoming beam is totally reflected, only the sample size induced an increase of the intensity until the plateau of total reflection. Then, the Kiessig fringes expected are wellpronounced due to strong electron-density contrast between the PtC layer and the Si substrate.

Accordingly, a least square fitting procedure was applied to refine the layer parameters (density, thickness and interfacial roughness), and the roughness of the Si substrate (the density being fixed to 2.33 g/cm 3 ). This deduced roughness value of 0.3 nm is close to the classical value obtained of natively oxidized Si wafers. The best fit reveals that superposed on the interface reflectivity (Fig. 2a), an additional contribution comes from the PtC layer (indicated by a arrow in the Fig. 2a). This type of reflectivity profiles are in accordance with reflectivity measurements of heterogeneous Pt/Al 2 O 3 thin films (Maaza, 1999) with complex structure. It should be mentioned that the formalism used treats the Pt-C nanocomposite layer like a layer with a homogeneous density. Such a modelling does not take into account the heterogeneity of a metal-carbon medium, especially with a clustering behaviour and probably an out-of-plane correlation in the metal organization [START_REF] Hazra | Dependence of matrix and substrate on the morphology of nanocermet thin films[END_REF]. However the purpose of the XRR experiments is to get input values for further analysis of the GISAXS intensity, and this simple layered model is adequate to retrieve average parameters [START_REF] Babonneau | Waveguiding and correlated roughness effects in layered nanocomposite thin films studied by grazing-incidence small-angle x-ray scattering[END_REF].

The best-fit parameters obtained on the three PtC thin films selected for GISAXS analyses are reported in Table 2. Besides the thickness and average density, the most interesting parameter is the roughness that increases with the thickness, i.e. the deposition time, which could indicate that the microstructure of the metal-carbon layer changes during the growth (in agreement with SEM observation). The extracted average density is about 11 g/cm 3 , i.e. a C-Pt atom composition of about 70-30 % which is in accordance to the values obtained by RBS measurements.

Fig. 3 displays the 2D GISAXS intensity maps of the same sample (PtC100 thin film) for different x-ray incident angles. As the mutual solubility of carbon and platinum is very small [START_REF] Rutkov | Determination of the limiting solubility of carbon in platinum[END_REF], a phase separation is expected between Pt and C domains during the co-deposition, leading to metal cluster formation buried in a C matrix. However, with respect to previous works [START_REF] Gibaud | X-ray and Neutron Reflectivity: Principles and Applications, Springer, Paris Guizard C, Princivalle A[END_REF], the GISAXS patterns are not isotropic, which suggest an anisotropic shape or organization between the substrate-parallel and -perpendicular planes.

Moreover the investigation achieved at increasing incidence reveals changes in the spatial organization with the depth. Fig. 3 shows a typical reinforcement of the intensity parallel to the sample shadow (y axis) at low q z , which is characteristic to Yoneda dynamic effect amplifying the off-specular intensity whenever the emergence angle  f is close to the critical angle of the nanocomposite layer deduced from XRR measurement ( c = 0.28). Fig. 2b allows an evaluation of the analysis depth of x-ray assuming that the PtC layer is a homogeneous mix Pt 3 C 7 of density between 10 and 11 g/cm 3 in comparison to pure carbon C, pure platinum Pt and a. The crosses indicate the incidence angles used for GISAXS measurements.

The scattering intensity profile versus the scattering vector magnitude q is then obtained by a horizontal or vertical integration of the intensity (cross-sections) in q y and q z directions respectively (Fig. 4). First of all, it is interesting to note that excepted of the Keissig fringes, the small angle scattering profiles achieve in off-specular conditions (q y ≠ 0) as shown in the fig. 4b are similar to the specular curve (fig. 2a), which confirms the origin of the shoulder at q z =2.5nm -1 .

In the fitting procedure, we have considered that the scattered intensity is weighted by the size distribution N(D). We assume that all the particles have the same aspect ratio, which is in agreement with our previous TEM studies [START_REF] Rabat | Carbon/platinum nanotextured films produced by plasma sputtering[END_REF]. Different shapes have been considered for the simulations. The best results have been obtained with a spheroidal shape (as defined in ref 25), with a diameter D xy and height D z . The anisotropy in the scattering intensity observed between the xy plane (in-plane) and the z direction (out-of-plane) is in accordance with this morphology. Consequently, the scattering intensity in both directions have been simulated independently with different size distributions and structure factors with the dedicated software, FitGisaxs [START_REF] Babonneau | FitGISAXS: software package for modelling and analysis of GISAXS data using IGOR Pro[END_REF]. In order to describe the cluster position function (disorder) in our Pt-C composite samples, the anisotropic Percus-Yevick model [START_REF] Vrij | Mixtures of hard spheres in the Percus-Yevick approximation. Light scattering at finite angles[END_REF] has been selected as the model leading to the best fits. This model considers an interaction potential of hard spheres in order to determine the average distance between neighbour particles (inter-particle distance) at the equilibrium (Λ xy or Λ z ) which is considered as proportional to the average particles size (D xy or D z ): Λ i / D i = C i (i=xy or z) [START_REF] Evans | sland sizes and capture zone areas in submonolayer deposition: Scaling and factorization of the joint probability distribution[END_REF]. This model is particularly well adapted to particles embedded in matrix [START_REF] Naudon | Grazing-incidence small-angle Xray scattering applied to the characterization of aggregates in surface regions[END_REF]. In this formalism, we extracted the morphological parameters of the samples by fitting the q y and q z cross sections of the 2D patterns (figure 2a) [START_REF] Revenant | Quantitative analysis of grazing incidence small-angle x-ray scattering: Pd/MgO (001) growth[END_REF]. Qualitatively, the 2D GISAXS intensity maps reveal a strong intensity maximum in the in-plane direction but a weak shoulder in the out-of-plane q z direction (growth direction of the film). It means there is a broader distribution of the inter-particle distance in the direction of growth than in the substrate plane. Quantitatively, Table 3 displays the nanoscale morphological parameters of Pt-C thin films obtained from GISAXS data analysis. The mean result is that the interparticle distance in the plane increases during the growth while those out-of-plane decreases.

The used model (Local Monodisperse Approximation) which assumes that the lateral growth and inter-particle distance are correlated, is in agreement with an excluded volume effect due to the atoms surface mobility during the deposition. Indeed, the cluster growth parallel to the surface is mainly driven by surface diffusion to nucleation sites and coalescence with short distance size-position correlations (the depletion zone around clusters is related to its capture capability, which is size dependent) [START_REF] Revenant | Quantitative analysis of grazing incidence small-angle x-ray scattering: Pd/MgO (001) growth[END_REF] [START_REF] Venables | Introduction to Surface and Thin Film Processes[END_REF]. The average particles separation is higher near the surface than close to the substrate (earlier stage of the deposition). Because of the carbon columns divergence, this means that particles are located around the carbon columns instead of a homogenous 3D distribution in a carbon matrix, as found in other deposition conditions [START_REF] Babonneau | Silver nanoparticles encapsulated in carbon cages obtained by co-sputtering of the metal and graphite[END_REF]. This result can be coherent with the simplified columnar model proposed by T. Vad [START_REF] Vad | Anomalous small-angle X-ray scattering study on the nanostructure of Co-sputtered platinum/carbon layers[END_REF] which describes a disordered network of carbon columns (low electron density) and Pt/C columns (high electron density).

In addition, the experimental results show that the ratio of the average size to the average inter-particles distance is equal to the carbon flux to platinum flux ratio. It demonstrates that the perpendicular growth is due to direct deposition from the gas phase atoms and that Pt clusters have no vertical mobility. As platinum and carbon are not miscible, a phase separation occurs and the quantity of carbon separating two platinum clusters is directly proportional to the carbon flux/platinum flux ratio.

The TEM image (Fig. 5) of a typical Pt/C columns, as collected on a carbon membrane, reveals a local view of the platinum cluster distribution. Due to the high contrast between platinum and carbon, these images reveal mainly the Pt clusters, which appear as black domains. Fig. 5a shows the enlargement of carbon columns containing Pt clusters, while the average cluster size of 2nm can be deduced from the higher magnified images Fig 5b,c.

The average centre-to-centre distances between clusters in the direction parallel to the surface and in the perpendicular direction are around 2nm and 5nm, respectively (Fig. 5b,c). The average characteristic lengths of the platinum cluster distribution obtained by GISAXS are in good agreement with these TEM observations. These combined results allow us to build a consistent model for platinum-carbon columnar growth, in which platinum growing nanoclusters are decorating growing carbon column.. The model of plasma co-sputtered Pt/C thin films morphology can be schematized as shown in Fig. 6. When the sputtering time increases, carbon columns are formed with Pt clusters around them. The formation of these clusters is due to Pt atoms diffusion in the porous carbon leading to the formation of 2nm diameter clusters with a narrow size distribution.

Conclusion

In the conditions of plasma Pt-C co-sputtering, anisotropic growth mode is obtained: carbon column structure is growing and the phase separation tends to allow platinum clusters to grow around carbon columns. The platinum cluster growth and distribution in the plane of the surface are driven by nucleation, surface diffusion and coalescence phenomena. We have demonstrated that constant platinum sputtering conditions guarantee a constant average cluster dimension in the plane of the surface even after long deposition time. The average vertical distance between platinum clusters is determined by the carbon flux/platinum flux ratio of the co-sputtering conditions. A carbon structure evolution during growth (e.g. column divergence) will affect the Pt clusters organization. In order to tune the platinum cluster distribution during co-sputtering, it is crucial to control the carbon growth.

Finally, we demonstrated that GISAXS can be used as a depth sensitive method for characterising multilayer of buried metallic clusters.

Moreover, this study confirms that plasma sputtered PtC films have desirable morphology for being efficient catalytic films : small Pt clusters (2 nm) are growing on a high specific area carbon sputtered thin films. intensities in white. In the (a) pattern, the positions of the q y and q z intensity cross-sections are indicated.

   and density ρ (from XRR), platinum areal density Q Pt and atomic concentration (from RBS), critical angle of total reflection  c (at E=8040 KeV of x-ray energy) of co-sputtered PtC thin films. Table 3. Average platinum cluster sizes D and interparticle distance  in the in-plane xy (parallel) and out-off plane z (perpendicular) directions. The fitted size distribution appears very large and asymmetric (0.5-3.5nm in xy plane and 0.5-5.5 in z direction).

Fig. 1 .

 1 Figure captions:

Fig. 3 .

 3 Fig. 3. 2D GISAXS intensity patterns from PtC100 sample recorded at increasing incidence

Fig. 4 .

 4 Fig. 4. a) Extracted measured (symbols) and fitted (solid line) horizontal cross-sections (in-

Fig. 5 .

 5 Fig. 5. Cross section TEM image of a co-sputtered PtC100 sample. Platinum clusters are

Fig. 6 .

 6 Fig. 6. Shematic drawing of the PtC composite microtexture which shows the evolution in

  Fig. 1

Fig. 4

 4 Fig. 4

Fig. 5

 5 Fig. 5

Fig. 6

 6 Fig. 6

Table captions Table 1 .

 captions1 Co-sputtering conditions of platinum and carbon

Table 2 .

 2 Deposition time t s , thin film thickness d, roughness

Table . 1

 . . Co-sputtering conditions of platinum and carbon.

	I c (A) I Pt (A)	γ Pt (E=380eV)	γ C (E=750eV)	F C (at.s -1 )	F Pt (at.s -1 )	F C /F Pt
	1.5	0.2	1.01*	0.33*	3.1 10 18	1.25 10 18	2.5
	* Y. Yamamura, H. Tawara, Atomic data and nuclear data tables 62 (1996) 149-253

Table 2 .

 2 Deposition time t s , thin film thickness d, roughness σ and density ρ (from XRR), platinum areal density Q Pt and atomic concentration (from RBS), critical angle of total reflection (at E=8040 KeV of x-ray energy) of co-sputtered PtC thin films.

	Name	t d (s)	d ± 0.1 (nm)	σ (nm)	Q Pt ± 5% (at.cm -2 )	% Pt (at)	α c	ρ ± 0.3 (g/cm 3 )
	PtC20	90	21.7	1.6	62.10 15	29	0.43°	11
	PtC60	270	68.0	2.0	200.10 15	31	0.41°	10.4
	PtC100	450	112.0	>3	340.10 15	33	0.41°	10.4

Table 3 .

 3 Average platinum cluster sizes D and interparticle distance  in the in-plane xy

	(parallel) and out-off plane z (perpendicular) directions. The fitted size distribution appears
	very large and asymmetric (0.5-3.5nm in xy plane and 0.5-5.5 in z direction).
		 i	D xy	Λ xy	D z	Λ z
		0.28°	1.6	2.9	2.2	4.2
	PtC20
		0.35°	1.6	2.1	2.2	4.7
		0.37°	1.6	2.6	2	4.8
	PtC60
		0.51°	1.6	1.8	2	5.4
		0.27°	1.6	3.2	2.2	4.2
	PtC100	0.33°	1.6	2.9	2	4.8
		0.4°	1.6	2.2	1.9	5.1

Acknowledgements

The authors thank the SOLEIL French synchrotron committee for beam time allocation and the beamline teams for technical assistance during x-ray scattering experiments. The magnetron sputtering system PulP is made available graciously by MID Dreux Innovation.

CNRS is gratefully acknowledged for granting the project "PIE AMELi-0Pt".


Pascal Brault
email: pascal.brault@univ-orleans.fr

Amaël Caillard

Anne-Lise Thomann

Polymer electrolyte fuel cell electrodes grown by vapor deposition techniques

Keywords: fuel cell, electrode, PVD, CVD, thin film, catalyst, PEMFC

Polymer fuel cell electrode growth using vapor deposition techniques is reviewed. The supports, the nanocatalyst sizes and morphologies, the resulting electrodes are examined as a function of vapor deposition process: sputtering, CVD, PECVD, MOCVD. In each case, up-to-date fuel cell performances are highlighted. Vapor depositions are valuable techniques for designing fuel cell electrodes of various kinds with good fuel cell performances, e.g. power density and catalyst activity.

Introduction

Low temperature Solid Polymer Fuel Cells, as Proton Exchange Membrane Fuel Cells (PEMFC), Direct alcohol Fuel Cell (DAFC), Solid Alkaline Membrane Fuel Cell (SAMFC) are promising energy power supplies in the context of greenhouse gas reduction.

When powered with hydrogen and oxygen, the only by product is warm water. The fuel cell is a direct current source issued from electrochemical conversion of fuel and oxidant (e.g. pure or from air oxygen) at the two electrodes of the fuel cell. These two electrodes are separated by a either proton or anion (OH -) conducting solid polymer, forming a Membrane-Electrode Assembly (MEA). These electrochemical reactions are summarized in the equation ( The electrodes are conventionally made of a microporous catalytic layer, a microporous gas diffusion layer (GDL) supported on a carbon backing (carbon cloth or carbon paper). The proton or anion exchange membrane is a polymer allowing ion conduction in wet conditions. These components are fabricated individually and then pressed together at high temperature and high pressure. The interface between the membrane and the electrode is critical [1] . The MEA is finally sandwiched by two flow field plates (bipolar plates) for efficient current collection and gas distribution.

Part of the optimization of an electrode design is the attempt to correctly distribute the amount of volume in the catalyst layer between the transport media for each of the three phases (electrons, protons and gases) to reduce transports losses. This catalyst layer is either applied to the membrane or onto the gas diffusion layer. In either case, the objective is to place the catalyst particles (platinum, platinum alloys on non based platinum) in close proximity with the membrane, the carbon particles and the gas fed pores. Designing efficient electrodes remains a challenging tasks for increasing efficiency and stability, ensuring durability (for car application, at least 5000 h operation is required) and reducing costs (especially due to the noble catalysts).

Many methods are commonly employed to form the microporous catalytic layer [2,[START_REF]Fuel Cell Catalysis[END_REF] : chemical synthesis such as colloidal methods (Bönneman [START_REF] Sellin | [END_REF]5] , Water in Oil -Micro emulsion [6] , polyol [7,8] , carbonyl route [9] , impregnation-reduction method [10] and electrochemical impulsion method [11] from salts allows producing supported catalysts ready to be brushed on the GDL. Progresses in GDL manufacturing are also currently emerging. Especially, one can mention the use of carbon aerogel, that readily provides new carbon supports for fuel cell electrodes [12][13][14][15][16] . Beside these methods, vapor deposition methods are of growing interest for growing the catalyst support (usually carbon based), placing the catalyst nanoclusters on the upper part of the electrode and/or for growing the full electrode as a thin film. The present article will briefly describe current state of the art of these vapor techniques.

New catalyst support materials by vapor depositions

Chemical Vapor Deposition, in the field of low temperature fuel cell, is mainly used for obtaining nanostructured carbon (possibly porous) thin films as a potential alternative electrocatalyst support to replace traditional carbon black powder materials (i.e. Vulcan® XC72 and Shawinigan® carbon powders). Catalyst coating is further obtained either by salt impregnation (mainly electrodeposition or electroless deposition) or by vapor deposition (mainly plasma sputtering). Two main families of nanostructured carbon supports are obtained by vapor deposition: arrays of carbon nanotubes and nanofibers (CNT/CNF) [17][18][19][20][21][22][23][24][25][26][27] and randomly nanostructured carbon thin films [28][29][30][31][32][33] . Among the new recent materials, as carbon nanohorns, nanosheets, boron doped diamond or transition metal oxides, this short review will only discuss on transition metal oxides [34,35] .

Carbon nanotube and carbon nanofibers

Carbon nanostructures as nanotubes (single wall or multiwall) -CNT, nanofibers -CNF-are expected to provide the essential high surface area to be suitable as catalyst support in a polymer electrolyte fuel cell electrode. In most of the case, nanotubes and nanofibers are grown by catalytic CVD or PECVD. In the latter case, the CNT and CNF are grown at a lower temperature and aligned nanostructures can be easily obtained. Precise growth conditions (precursors, temperature, pressure, flows, …) depend on reactor geometry, properties of expected CNT, CNF (size, density, vertically aligned or not, …). Arrays of vertically aligned CNT-CNF are usually grown on macroporous carbon paper or microporous GDL [22] as shown on Figure 1 and Figure 2, followed by the catalyst impregnation. For simplicity and without loss of generality, we will only discuss on Pt here. The CNF/CNT array is subsequently catalysed by electroless deposition, electrodeposition (by using Pt based salts) and plasma sputtering [23] . In the two former cases, all the length of CNT-CNF are decorated by highly dispersed Pt nanoparticles with low degree of agglomeration.

Deposition of Pt alloys (Pt x Ru y , ..) was also investigated, mainly for methanol DAFC operation. Figure 3 shows the result of two studies where the Pt nanoclusters are mainly located on the top of the CNF/CNT by using plasma sputtering deposition. A Pt concentration and a nanocluster size gradients occur along the CNF/CNT. The agglomeration degree of the Pt particles increases with increasing the Pt loading. The broad range of fuel cell performances and electrocatalytic activities with CNT/ CNF as catalyst supports are strongly dependent on their synthesis methods. The CNT supported catalysts show higher resistance to carbon corrosion than the widely used carbon black without sacrificing catalytic activity (an activity enhancement is even observed in some studies). This leads to a lower loss of Pt surface area, to a lower oxygen reduction reaction activity and then improves long term durability. This suggests that (PE)CVD obtained CNT (and CNF) are promising catalyst supports for PEMFC application.

Nitrogen doped carbon nanotubes (NCNT) are also studied as catalyst support [36][37][38] . In most of the case, NCNT are grown by a modified CVD process in a nitrogen rich gas mixture. The Pt nanoparticles deposited by aerosol MOCVD, are uniformly dispersed on the NCNT, have narrower particle size distributions, whereas average Pt particle size is larger with a broader particle size distribution on CNT support as shown on Among recent intensive researches in reducing or replacing Pt based electrodes in fuel cells, arrays of boron and/or nitrogen doped CNT without metal have been announced as a good candidate and for the oxygen reduction in alkaline fuel cells [36] .

Such metal-free materials seem to have interesting electrocatalytic activity, longterm operation stability, and tolerance to crossover effect and to CO presence.

However, the exact roles of the nitrogen/boron surface species are not clear at the present stage.

Having many similarities to carbon nanotubes (CNTs) in structure and properties, including its high aspect ratio and rich electronic states, graphene is an attractive candidate for potential uses in many areas where the CNTs have been exploited as catalyst support in fuel cell. The synthesis of N doped graphene sheets by chemical vapor deposition (CVD) is a recent development [39][40][41] and has been recently exploited for oxygen reduction in fuel cell [39] . A free-standing, flexible and transparent film of ca. 4 cm 2 in size and consisting of only one or a few layers of the graphite sheets have been synthesized by a CVD process followed by a transfer process. The N-doped graphene film possesses electrocatalytic properties for the oxygen reduction reaction (ORR) similar to that of nitrogen-containing vertically aligned carbon nanotubes.

Mesoporous carbon thin film

Chemical and physical vapor depositions have been recently applied to the synthesis of three dimensional porous carbon thin film matrix to support or to embed Pt catalyst nanoclusters [28,29] . By combining these techniques with another vapor deposition technique, a large range of catalytic layers can be synthesized 42 . Among the various forms of mesoporous carbon thin films, ordered mesoporous carbon (OMC) are of great research interest, especially for electrochemical applications such as fuel cells (lithium ion batteries and capacitor) [30][31][32][33] . By using a suitable template, such films offer prominent characteristics, such as high surface area, tunable and uniform pore size, ordered pore structure, interconnected pore network, tailorable surface properties and good mechanical stabilities. CVD is one of the most used techniques to synthesize such films because of the high degree of pore filling, the formation of graphitic pores walls, the presence of crystallite-sized graphite and the possible absence of dense carbon layer on the external surfaces of the template. The use of OMC leads to higher performance in electrochemical activity than commercial carbon black particle, especially in DMFC [33] . This enhancement is attributed to the uniform mesopore network in OMC which seems to improve the diffusion of the reactants and the high electrical conductivity of the graphitic OMC.

Transition metal oxides

Transition metal oxides, such as ZrO 2 , SnO 2 , RuO 2 , TiO 2 , have been applied to improve the durability of the carbon-supported platinum catalysts and the Pt activity for the oxidation reaction of organic molecules (by lowering the adsorption energy of CO). It was recently reported that the active surface area of the platinum in the Pt/TiO 2 /C catalyst was increased with the addition of TiO 2 due to the synergy of the interface between Pt and TiO 2 [34] . TiO 2 thin film has been found to be proton-conducting for thickness below 18 nm) and the alcohol permeation of the MEA has been reduced.

Very few studies are devoted to vapor deposition of Pt/TiO 2 /C catalysts. Using RF magnetron sputtering, Yoo et al [35] have recently synthesized PtRu nanoparticles embedded TiO 2 matrix for methanol oxidation with different sizes of PtRu nanoparticles as shown on Figure 5 and indicate that the activities of the methanol oxidation reaction are dependent on the PtRu particle size. The high catalytic activity of PtRu/TiO 2 is partially attributed to the effect of the TiO 2 , on which the hydroxyl (OH) species are absorbed. This causes a shift in the surface oxidation of PtRu which facilitates the removal of CO from the PtRu nanocatalyst surface and reduces the poisoning of the Pt in the PtRu/TiO 2 thin film.

Efficient catalyst nanoclusters by vapor depositions

Two techniques are commonly used to synthesize efficient catalyst nanoclusters for low temperature fuel cell. Plasma sputtering deposition was investigated for its ability to reduce the noble catalyst content in PEMFC electrodes [43][44][45][46][47][48][49][50][51] , in order to reduce fabrication costs while preserving fuel cell performances. Metalorganic CVD (MOCVD) provides an alternative way for depositing efficient catalyst nanoclusters onto gas diffusion layer electrodes [52][53][54][55][56] .

The basic principle of plasma sputtering is to sputter a catalyst target with the plasma ions, e.g. argon [43,44] . The sputtered atoms are condensing on a substrate located a few centimeters from the target. This substrate can be the gas diffusion layer, or the polymer electrolyte for direct deposition. Plasma sputtering can be readily used for depositing only the catalyst of both the catalyst and the electron conducting matrix (mainly carbon) as described in the next section. Some references give many ways to reduce the catalyst content and eventually to increase the PEM fuel cell efficiency.

Among them, Cha et al [45] and then Haug et al [46] have deposited alternate sputtered platinum layers and painted layers of Nafion® and carbon ink (NCI) directly on the membrane. They have achieved PEMFC performance between 150 and 250 mW cm -² with a cathode Pt loading of about 30-40 μg Pt cm -2 (60-80 µg Pt cm -2 in the entire MEA) using dry (non-humidified) H 2 /O 2 at 1 atm and 60-70°C. In the mid 2000's, Brault et al have decreased the cathode Pt loading to 20-100 µg Pt cm -² on the GDL and obtained fuel cell performance of 270 mWcm -2 [51] and 400 mWcm -2 [43] (with about 100 µg Pt cm - 2 ) by using a Nafion® 115 and a Nafion® 112 membrane, respectively. Considering the efficiency per weight of catalyst, the catalyst utilization efficiency in the plasma prepared cathode reaches 2.7 kW g Pt -1 and 5 kW g Pt -1 .

Metalorganic CVD is intended to allow deposition of noble metal catalysts onto gas diffusion layer electrodes rather than using conventional chemical methods [52][53][54][55][56] . It is a suitable method for impregnating tortuous electrode supports as shown on Figure 6a. Recent experiments [52] have shown interesting oxidation reduction reaction behavior. For a Pt cathode loading of 100 µg Pt cm -2 , the maximum power density, displayed on Figure 6b, is about 800 mW cm -2 in H 2 /O 2 conditions (4 bars and 80°C) with a Nafion® 112 membrane. This leads to a Pt-specific power density of 1.7 kWg Pt -1

(cf Figure 9 in Ref. 52).

Fuel cell electrode thin film growth by vapor co-depositions

Because of the agglomeration degree of the sputtered Pt particles when increasing the Pt loading, plasma sputtering has been combined with another technique to deposit the catalyst support (mostly carbon) alternately or simultaneously [28,29,50,[57][58][59][60][61] .

Plasma co-sputtering has been used to insert additional metal into the sputtered Pt layer and to modify the seeding and the morphology of Pt deposit [50] . The platinum catalyst is more dispersed across the surface and more catalytically active sites are available. By the insertion of Cr in a 54 µg.cm -2 Pt loaded electrode, Gruber et al.

improved the fuel cell performance to 200 mW.cm -2 (in a H 2 /O 2 cell at room temperature and ambient pressure with a Nafion® 115 membrane), corresponding to the maximum power density gained at twice the Pt loading alone.

Using Pt-C co-sputtering on commercial carbon GDL [58] , highest efficient use of catalyst has been reached very recently due to efficient ultra low Pt loading (10 µg Pt .cm -2 at both anode and cathode) : 20 kW.g Pt -1 (Figure 7). This result has been Electrocatalytic hydrocarbon thin films containing Pt nano-clusters are deposited by simultaneous plasma polymerization of ethylene in a PECVD reactor and simultaneous sputtering of a Pt target [59,60] in the same reactor drawn in figure 8a.

The proposed approach is largely intended to propose an alternative PEMFC electrode (anode) fabrication method. The deposition of the carbon matrix provides the mechanical support and electronic continuity due to sufficient conduction properties. It ensures a uniform metal dispersion. The Pt catalyst is dispersed as small vertically stacked clusters with size less than 10 nm around hydrocarbon columns (250-500 nm height) of the films. PEMFC tests with plasma deposited 500 nm thick film and 0.5 mg cm -2 Pt loading as anodic electrocatalyst leads to a power density of 300 mW cm -2 . PECVD and plasma sputtering can also operate successively. Caillard et al [61] have successfully sputtered platinum nanoclusters (10-100 µg/cm²) on a array of vertically aligned 2 µm long carbon nanofibers previously grown by Helicon plasma enhanced CVD. The presence of such plasma obtained aligned carbon structures displayed on Figure 8b increases the fuel cell performance from 15% to 30% for a Pt loading in the range 10-100µg Pt cm -2 .

Conclusion

This short review is aiming to present both CVD and plasma sputtering deposition techniques in the field of improvement of fuel cell electrodes as an alternative to conventional or original chemical impregnation method. All the presented methods are successfully used to build efficient fuel cells electrodes. Plasma sputtering technique is combining the advantages for depositing pure catalyst and/or carbon atoms without any additional bonds lowering the catalytic activity. It is also advantageous for controlling catalyst distribution near the electrolyte polymer membrane and the mean size distribution. So high fuel efficiency, 20 kWg Pt -1 has been recorded.

Figure captions

Figure 1:

(a) SEM images of 2 µm long aligned CNF grown a 5 µm in diameter carbon fiber by PECVD with a close up showing a side view of the CNF array (reproduced from [22]);

(b) SEM images of 1 µm long CNF grown on a microporous GDL with a close up showing the top view of the CNF array (reproduced from [22]);

(c) Cell voltage E versus current density j curves and power density P versus current density j in a single 5 cm 2 surface area PEMFC with different Pt/C cathode structures :

(  ) cathode without GDL and with CNF , (  ) cathode with CNF and diffusion layer and (  ) cathode without CNF and with diffusion layer (reproduced from [22]).

Figure

  Figure 4a and 4b. As a result, the Pt/NCNT electrodes have a greater electrochemical

obtained with a

  Nafion® 112 membrane and in a H 2 /O 2 cell at 80°C and 3 bars. The main advantage in the successful use of plasma co-sputtering deposition techniques is to locate a very low catalyst quantity, with the correct density profile at the interface membrane-electrode: so the triple boundary point is very effective. High level performance with low Pt contents means, especially at the cathode side, that all the Pt atoms are involved in the electrochemical fuel cell reactions. Moreover, the Pt nanoclusters size distribution is very narrow and mean average size can be very low at low loading: 0.7 nm to 2 nm.

  MEA assembly: Cathode sputtered Pt loading: 20 µg Pt cm -2 , Commercial anode Pt loading 500 µg Pt cm -2 , Untreated Nafion® 115 membrane, hot-pressed MEA (130 °C for 120 s under a pressure of 40 kg cm -2 ). Operating conditions: cell temperature 80°C, H 2 /O 2 back pressure 3 atm.

Figure 2 :

 2 Figure 2:

Figure 3 :

 3 Figure 3:

Figure 4 :

 4 Figure 4:

Figure 5 :

 5 Figure 5:

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 6:

2

 2 Fig.2

  Fig.3

Fig. 4

 4 Fig. 4

Fig. 5

 5 Fig.5

Fig. 6

 6 Fig. 6

.

Fig. 7 .Fig. 8

 78 Fig. 7

  1 and 2):

	Acid fuel cell
	Anode half reaction: H 2		 2H + + 2e -(x2)	(1)
	Cathode half reaction: O 2 + 4e -+ 4H +  2H 2 O
	Alkaline fuel cell
	Anode half reaction:	2H 2 + 2OH -	 2H 2 O + 2e -(x2)	(2)
	Cathode half reaction: O 2 + 4e -+2H 2 O  4OH -
	With the net reaction: 2H 2 + O 2	 2H 2 O

Acknowledgements

We are indebted to C. Coutanceau, C. Lamy, R. Boswell and C. Charles for highlighting discussions and strong encouragements. GDR PACEM, PACS, PACTE and PIE CNRS are warmly acknowledged for continuous support.


Claudio Weidmann
email: claudio.weidmann@ensea.fr

Characterizing compressibility with Lorenz curves

published or not. The documents may come

Characterizing compressibility with Lorenz curves

I. INTRODUCTION

Sparse signal representations are at the core of state-ofthe-art lossy source coding and compressed sensing methods. Sparse vectors, in which most components are zero, can be compressed and/or sampled more efficiently. While such strict sparsity is rather straightforward to characterize, several measures have been proposed for the more general case of approximate, non-strict sparsity; see the work by Hurley and Rickard [START_REF] Hurley | Comparing measures of sparsity[END_REF], which compares a number of sparsity measures based on intuitive quantifiable criteria.

In this work, we use the term compressibility to denote non-strict sparsity, hinting at the ultimate goal, which is the lossy compression of a signal into a digital representation. We will take a probabilistic approach, modeling time-discrete signals as the outputs of memoryless random sources. Thus rate distortion theory is the appropriate tool to study the limits on achievable compression performance. The question that this work addresses is "how does compressibility of a source affect the minimal rate per sample needed to reconstruct it within a given distortion" (i.e. a lossy compression problem), and not "how many samples are needed to guarantee reconstruction" (which would be a compressed sensing type of question).

Section II introduces Lorenz curves and the associated Lorenz order as tools to display and compare the compressibility (sparsity) of sources. Section III recalls two upper bounds on the distortion rate function and shows that they can actually be obtained from a parametric curve that is equivalent to a Lorenz curve. Section IV then uses these findings to show that the Lorenz order induces an order on distortion rate bounds and hence allows to define a compressibility hierarchy in parametric families of source distributions.

II. LORENZ CURVES

In 1905, Max Lorenz proposed a simple graphical tool to display the inequality in the distribution of wealth among a finite population [START_REF] Lorenz | Methods of measuring the concentration of wealth[END_REF]; this is now known as the Lorenz curve and is still popular among economists. For a population of size n, with wealths x 1 , x 2 , . . . , x n arranged from smallest to largest (denoted by x 1:n , x 2:n , . . . , x n:n ), the Lorenz curve L x (u) is obtained by plotting the points

(u, L x (u)) = k n , k i=1 x i:n n i=1 x i:n , j = 1, 2, . . . , n,
as well as the point (0, 0), and interpolating in between. Thus L(u) corresponds to the proportion of total wealth owned by the u•100% poorest individuals. Clearly, distribution inequality is larger if the curve is more bent away from the egalitarian line running from (0, 0) to [START_REF] Weidmann | Oligoquantization in low-rate lossy source coding[END_REF][START_REF] Weidmann | Oligoquantization in low-rate lossy source coding[END_REF]. By definition, the Lorenz curve allows comparisons among populations of different size.

If two populations x and y have Lorenz curves L x and L y such that L x (u) ≤ L y (u), ∀u ∈ (0, 1), then x shows at least as much inequality as y. Many single-parameter measures of inequality have been proposed based on the Lorenz curve. Perhaps the most prominent is the Gini coefficient G, which equals twice the area between L(u) and the egalitarian line. Thus G = 0 for a population in which everyone has the same wealth, and G = 1 for a population in which a single individual owns all wealth.

Coming back to signal representations, we may consider a vector of sample or coefficient magnitudes and say that it is sparser (in non-strict sense) if its Lorenz curve is below that of an alternative representation, since fewer samples will be needed to achieve the same reconstruction fidelity. 1 Hence the Gini coefficient can be used as a measure of sparsity; in fact, [START_REF] Hurley | Comparing measures of sparsity[END_REF] found that it is one of the few sparsity measures that satisfies a range of intuitive properties that one should expect from such measures. However, the focus of this work is not on single-parameter sparsity measures, but on using the Lorenz curve as a tool to characterize the compressibility (non-strict sparsity) over the entire range of source coding rates. This motivates using the entire curve, rather than a single-parameter measure like the source entropy, which would yield only highrate characterizations.

In the following, we will model the signal representation by a real-valued memoryless source X with density f X (x) and distribution function F X (x). We may compute the expected Lorenz curve for a block X of samples generated by the source X. By letting the block size grow to infinity, one arrives at the standard definition of the Lorenz curve of a random variable X [4, Sec. 17.C]. Let L be the class of distributions of nonnegative random variables with positive finite expectation (we will slightly abuse notation and use both X ∈ L and F X ∈ L). The Lorenz curve corresponding to the distribution F in L is described by the set of points

F (x), F (1) (x) , x ≥ 0, (1)
where F (1) is the first moment distribution

F (1) (x) = x 0 z dF (z) ∞ 0 z dF (z) . (2)
Defining the right continuous inverse distribution function or quantile function by

F -1 (y) = sup{x : F (x) ≤ y}, (3)
one obtains the explicit expression for the Lorenz curve of a random variable X with distribution F ∈ L,

L(u) = u 0 F -1 (y) dy 1 0 F -1 (y) dy = u 0 F -1 (y) dy E X , 0 ≤ u ≤ 1. (4)
Using definition (4) it is possible to show that L(u) is a continuous, non-decreasing convex function on [0, 1], which is differentiable almost everywhere, and with L(0) = 0 and L(1) = 1. The Lorenz curve is scale-free by definition, i.e. L aX (u) = L X (u) for a > 0. The parametric representation ( 1) is often more convenient if no closed-form expression is available for the quantile function. Notice that a Lorenz curve can obviously be defined also for higher order moments in (2), but for technical reasons that will become clear later, we prefer to consider distributions of the powers Y = X r of the original random variable (allowing e.g. to write X r ∈ L without having to explicitly define the class of distributions with finite r-th moment). By considering the Lorenz curves associated to random variables it is possible to define a partial order in the class L (see e.g. [4, Sec. 17.C]).

Definition 1 (Lorenz order):

Let F X , F Y ∈ L, with corre- sponding Lorenz curves L X and L Y . Then X is less than Y in the Lorenz order, denoted X ≤ L Y or F X ≤ L F Y , if L X (u) ≥ L Y (u) for all u ∈ [0, 1].
Thus X ≤ L Y means that X is more equally distributed, or less compressible (sparse) than Y , that is, L X is closer to the egalitarian line than L Y .

III. DISTORTION RATE BOUNDS

This section briefly recalls two upper bounds on the meansquared error (MSE) distortion rate function of continuous random variables, which first appeared in [START_REF] Weidmann | Rate distortion behavior of thresholdbased nonlinear approximations[END_REF] and are discussed in more detail in [START_REF]Rate distortion behavior of sparse sources[END_REF]. Although from here on we focus on MSE, all results in this work can be extended to absolute r-th power distortion measures d(x, x) = |x -x| r .

The (information) rate distortion function [START_REF] Cover | Elements of Information Theory[END_REF] R(D) = min

f (x|x): E d(X, X)≤D I(X; X) (5)
is the minimal rate at which a source X can be encoded with distortion E d(X, X) ≤ D. Alternatively, it may be expressed as a distortion rate function D(R). Two important properties for MSE distortion, d(x, x) = (x -x) 2 , are the Gaussian upper bound,

D(R) ≤ σ 2 e -2R , (6)
and the Shannon lower bound (SLB),

R(D) ≥ R SLB (D) = h(X) - 1 2 log(2πeD). (7)
Note that in this work all rates are expressed in nats and all logarithms are natural, unless otherwise stated.

The following bounds are obtained by classifying the magnitudes of the source samples using a threshold t and applying the Gaussian upper bound [START_REF]Rate distortion behavior of sparse sources[END_REF] to each of the two classes. They are upper bounds on the operational rate distortion function of magnitude classifying quantization (MCQ), which sends the classification as side information and uses it to switch between two codebooks. The samples with magnitude above threshold t are called significant and are characterized by two incomplete moments, namely the probability

µ(t) = -t -∞ f (x) dx + ∞ t f (x) dx (8)
and the second moment

A(t) = -t -∞ x 2 f (x) dx + ∞ t x 2 f (x) dx, (9)
where A(0) = σ 2 is the source variance (we assume E X = 0 without loss of generality). From these we compute the conditional second moment of the significant samples,

σ 2 1 (t) = E[X 2 | |X| ≥ t] = A(t) µ(t) ,
as well as that of the insignificant samples,

σ 2 0 (t) = E[X 2 | |X| < t] = σ 2 -A(t) 1 -µ(t) .
The classification decision is sent as side information to the decoder, using

h b (µ) nats per sample (h b (p) = -p log p - (1 -p) log(1 -p)
is the binary entropy function). The encoder can now use two separate Gaussian codebooks, one for the insignificant samples with rate R 0 and one for the significant samples with rate R 1 . The average rate per sample becomes

R = h b (µ(t)) + µ(t)R 1 + (1 -µ(t))R 0 . (10)
By standard rate allocation (reverse water-filling) over the two codebooks we obtain an upper bound. Theorem 1: (High-Rate Upper Bound) For all

R ≥ R min (t) = h b (µ(t)) + 1 2 µ(t) log σ 2 1 (t) σ 2 0 (t) , (11)
the MSE distortion rate function of a memoryless source is upper-bounded by

D(R) ≤ B hr (t, R) = c(t)σ 2 e -2R , (12)
where

c(t) = exp 2h b (µ(t)) + (1 -µ(t)) log σ 2 0 (t) σ 2 + µ(t) log σ 2 1 (t) σ 2 . (13)
The best asymptotic upper bound is obtained by finding the t * ≥ 0 that minimizes c(t). Since lim t→0+ c(t) = 1, the Gaussian upper bound is always a member of this family.

Exploiting the trivial fact that (12) also upper bounds the Shannon lower bound, we obtain Corollary 2: Let µ * = µ(t * ) and A * = A(t * ) yield the tightest bound in Theorem 1. Define the pmf's

µ * = [µ * , 1 -µ * ] , a * = A * σ 2 , 1 -A * σ 2 .
Then the differential entropy h(X) is upper-bounded by

h(X) ≤ 1 2 ln(2πeσ 2 ) + h b (µ * ) -1 2 D(µ * a * ), (14)
where D(• •) is the divergence or Kullback-Leibler distance between the pmf's. A low-rate bound is obtained by upper-bounding only the significant samples, while the other samples are quantized to zero, thus yielding a distortion floor.

Theorem 3: (Low-Rate Upper Bound) The MSE distortion rate function of a memoryless source is upper-bounded by

D(R) ≤ B lr (t, R), for t ≥ 0 and R ≥ 0 (15)
where

B lr (t, R) = A(t) exp -2 R-h b (µ(t)) µ(t) + σ 2 -A(t).
For a given threshold t ≥ 0, this bound can be optimized to yield

D(R * (t)) ≤ B lr (t, R * (t)), (16)
with the locally optimal rate R * (t) given by

R * (t) = h b (µ(t)) -1 2 µ(t) 2h ′ b (µ(t)) + γ(t) + W -1 -γ(t)e -2h ′ b (µ(t))-γ(t) , (17)
where γ is the reciprocal normalized second tail moment

γ(t) = µ(t) A(t) t 2 = t 2 E[X 2 | |X|≥t] and W -1 is the second real branch of Lambert's W function, taking values on (-∞, -1]. (W(x) solves W(x)e W(x) = x.)
Proofs and a discussion of the situation when (17) has no solution appear in [START_REF] Weidmann | Rate distortion behavior of thresholdbased nonlinear approximations[END_REF] and [START_REF]Rate distortion behavior of sparse sources[END_REF]. Furthermore, a corollary shows that the low-rate and high-rate bounds coincide in the minimum of the latter, that is, as expected there is a continuous transition between the two bounds. A parametric upper bound for all rates is thus

D(R) ≤ B(R) =      B lr (t, R), t > t * , R = R * (t) < R min (t * ) B hr (t * , R), R ≥ R min (t * ), (18
) where t * is found by optimizing the high-rate bound (12).

Results by Sakrison [START_REF] Sakrison | Worst sources and robust codes for difference distortion measures[END_REF] and Gish-Pierce [START_REF] Gish | Asymptotically efficient quantizing[END_REF] imply that the operational distortion rate function δ(R) of a magnitude classifier followed by a Gaussian scalar quantizer (adapted to the class variance) will be at most a factor of πe/6 (1.53 dB) above the bound (18). Actually, this gap is even smaller at low rates, since the distortion D 0 (0) = σ 2 0 is trivially achieved for the insignificant samples.

The incomplete moments ( 8) and ( 9) can be displayed with a moment profile curve,

{(µ(t), A(t))} , t ≥ 0. (19)
In [START_REF] Weidmann | Rate distortion behavior of thresholdbased nonlinear approximations[END_REF], we pointed out that this curve is a good tool to characterize source sparsity and showed that it contains all information needed to compute the bounds (18), since dA dµ = t 2 . A simple inspection of ( 1) and ( 19) reveals that by normalizing A(t) with A(0) = σ 2 , one obtains a reverse Lorenz curve for the random variable X 2 , defined as

LX 2 (u) = 1 -L X 2 (1 -u).
That is, all properties for Lorenz curves hold for the normalized moment profile, with the appropriate direction reversals in the inequalities.

IV. LORENZ ORDER INDUCES A HIERARCHY OF BOUNDS

The above observation immediately leads to the main result of this work.

Theorem 4: Let X, Y be two continuous random variables with

E X 2 = E Y 2 < ∞. If X 2 is less than Y 2 in Lorenz order, then the associated MSE distortion rate bound B X (R) will lie above B Y (R) for all rates, that is, X 2 ≤ L Y 2 implies B X (R) ≥ B Y (R), R ≥ 0.
By the above remarks on the operational significance of the bound (18), it follows that X 2 ≤ L Y 2 implies that when using a threshold-based MCQ scheme, Y is more compressible than X. (To show that Y is more compressible with an optimal scheme would require also ordered lower bounds on D(R).)

Proof: Since the normalized moment profile {(µ(t), a(t))} t≥0 is actually identical to the reverse Lorenz curve of X 2 , defined as

LX 2 (u) = 1 -L X 2 (1 -u), it follows from X 2 ≤ L Y 2 that L X 2 (u) ≥ L Y 2 (u) and therefore a X (µ -1 X (u)) ≤ a Y (µ -1 Y (u)), ∀u ∈ [0, 1]. (20)
Here the threshold

µ -1 (u) is defined via (3) as µ -1 (u) = F -1 (1 -u). Inequality (20) may be written as a Y (µ -1 Y (u)) = a X (µ -1
X (u)) + ∆a, with ∆a ≥ 0. Hence to prove the theorem it is sufficient to show that, for a fixed u = µ(t), the bounds B lr and B hr do not increase with ∆a, that is, ∂B ∂a ≤ 0. Fixing u avoids having to deal with the optimization of the bounds.

For the low-rate bound, we have

B lr σ 2 = a exp -2 R-h b (µ) µ + 1 -a, from which ∂ ∂a B lr σ 2 = exp -2 R-h b (µ) µ -1,
which is negative for all R > h b (µ), that is the rates for which the bound actually improves over the trivial bound D(R) ≤ σ 2

(otherwise we would choose the Gaussian upper bound and have

B X (R) = B Y (R)).
For the high-rate bound, we only need to look at the divergence term in (14), since it determines (12) through c(t) (recall that µ(t) is fixed). The derivative of the divergence needs to be positive for the bound to decrease with ∆a. We have

∂ ∂a D(µ a) = ∂ ∂a µ log µ a + (1 -µ) log 1 -µ 1 -a = - µ a + 1 -µ 1 -a = a -µ a(1 -a) ≥ 0,
where the inequality a -µ ≥ 0 follows from the concavity of the reverse Lorenz curve.

An interesting application of this result draws on the existence of parametric families of distributions for which an order on the parameters induces Lorenz order, see e.g. [4, Sec. 17.D] and [START_REF] Kleiber | On the Lorenz order within parametric families of income distributions[END_REF]. Through Theorem 4, the order on the parameters therefore induces an order on the source compressibility in the operational sense outlined above. This could be used for example to optimize a transform which outputs coefficients that can be modeled as coming from a certain family of distributions (with constant variance).

As an example, we consider a distribution with density of the form

f α,ξ,θ (x) = αx αθ-1 B(ξ, θ) [1 + x α ] ξ+θ , x ≥ 0, α, ξ, θ ≥ 0, (21)
which is known as generalized F distribution, generalized beta distribution of the second kind or Feller-Pareto distribution (see [START_REF] Marshall | Inequalities: Theory of majorization and its applications[END_REF]Sec. 17.D] or [START_REF] Kleiber | On the Lorenz order within parametric families of income distributions[END_REF]; B(•, •) is the Beta function). The appeal of this family of distributions is its ability to model heavy tails, which are typical for compressible sources such as e.g. wavelet coefficients (and the fact that incomplete moments of the distribution can be expressed with hypergeometric functions, thus avoiding the need for explicit numeric integration). Theorem 5 (Kleiber 1999, [10]): Let X 1 , X 2 be in L,

X i ∼ f αi,ξi,θi , i = 1, 2. Then α 1 ≤ α 2 , α 1 θ 1 ≤ α 2 θ 2 , and α 1 ξ 1 ≤ α 2 ξ 2 imply X 1 ≥ L X 2 .
At this point, the reason for working with the standard (first moment) Lorenz curve for Y = X 2 , instead of the second moment Lorenz curve for X, becomes apparent: we will apply Theorem 5 on the distribution of X 2 and then map the resulting parametric order "down" to the corresponding source distributions (since X 1 ≤ L X 2 does not necessarily imply X 2 1 ≤ L X 2 2 ). In the example at hand, we let

Y ∼ f Y = f α Y =1,ξ Y =1.1,θ Y , establishing that θ Y1 ≤ θ Y2 implies Y 1 ≥ L Y 2 (
the source becomes less compressible with increasing θ). The density f Y determines only the density of |X|; a common choice for modeling the source would then be the symmetric density f X (x) = |x|f Y (x 2 ). Here it turns out to be a generalized F distribution with parameters α X = 2, ξ X = ξ Y = 1.1 and θ X = θ Y . Figure 1 shows three (onesided) densities from this family for different values of θ. Their reverse Lorenz curves in Figure 2 show the expected order. Finally, Figure 3 confirming the order implied by Theorem 4. The interest of this approach based on Lorenz order becomes apparent when one considers that a visual inspection of Figure 1 might lead to a correct guess about the order of high-rate D(R) curves (which is determined by source entropy alone), but that predicting behavior at low rates seems much harder.

V. CONCLUSION

Lorenz curves and the Lorenz order were shown to be useful tools to characterize the compressibility and the distortion rate behavior of parametric families of source densities. An interesting open problem is the extension to the multivariate case; there are a number of proposals for multidimensional Lorenz curves, but their properties do not always match intuition about measures of distribution inequality (sparsity), see e.g. [START_REF] Marshall | Inequalities: Theory of majorization and its applications[END_REF]Sec. 17 measuring compressibility, with lossy source coding in mind, would certainly need to have properties that are different from those of interest to economists.

5 Fig. 1 .Fig. 2 .

 512 Fig. 1. Densities f α,ξ,θ (x) of the generalized F distribution (21) with α = 2, ξ = 1.1 and θ = 0.15, 0.25, 0.5. The actual source density is symmetric with f X (x) = 0.5f α,ξ,θ (|x|). All densities scaled to unit variance.

Fig. 3 .

 3 Fig.3. Upper bounds on distortion rate for the source densities in Figure1.

Here we assume e.g. squared error distortion measure and nonlinear approximation by the k largest samples (from equal-sized vectors). Increased sparsity thus corresponds to better energy compaction.