Alain Nadaï
email: nadai@centre-cired.fr

Keywords: Climate-energy, local initiatives, entanglements, justice issues

Local initiatives in the energy climate field have gained importance, visibility and support in many countries. Political agendas have been flagging localism as a key to success for energy transitions. While encouraging the involvement of multiple constituencies in energy transition processes on various scales, policy frameworks have been evolved to increasingly mirror a neoliberal agenda, relying on business models and the construction of new markets. The composition of and the direction taken by energy transition processes under such a new configuration, as well as their implications as regards to social justice and sustainability on various scales, still have to be explored. This special issue proposes longitudinal explorations of local energy transition processes in time, in order to critically examine their entanglements as well as their social and environmental implications.

In their concluding statement to the recent Paris scientific conference -Our common future under Climate Change (OFCC) (July 2015)2 -which preceded the Paris Summit on Climate Change (CoP 21), scientists from around the world have acknowledged our entrance into a new phase as regards to climate change issues : Climate Change and the 2°C threshold could be considered as part of (stabilized) scientific facts and the time had come to exploring actual solutions to GHG mitigation. The recent Paris agreement3 has confirmed this entering into the time of action, which the energy transitions are part of.

The way in which these energy transitions are approached by governments and nongovernmental actors has itself been transformed within less than a decade. The inability of States, during the Copenhagen Summit (2009), to agree on a legally binding agreement to meet the 2°C threshold had triggered a renewed commitment from non-state actors and networks. Local initiatives in the energy climate field have gained importance, visibility and support in many countries. Political agendas have been flagging localism as a key to success for energy transitions in several countries. They encouraged the involvement of multiple constituencies in energy transition processes on various scales.

In the meantime, as we hereafter detail it, policy frameworks were evolved to increasingly mirror a neoliberal agenda, relying on business models and the construction of new markets in order to foster the development of new energy technologies.

The composition of and the direction taken by energy transition processes under such a new configuration, as well as their implications as regards to social justice and sustainability on various scales, still have to be explored.

This special issue follows suit with debates in this and other journals about the reach and limits of local energy transition processes, their social and environmental implications and their entanglements. It proposes longitudinal explorations of such processes in time, in order to critically examine these issues and discuss the potential and the 'just sustainability' of these processes.

In what follows, we quickly introduce the reader to these changes and issues, before presenting the papers composing this special issue and discussing their outcomes.

The rise of local initiatives

Since the beginning of the nineties, so-called 'community', 'local' or 'territorial' initiatives in the domain of climate energy have been developed. In certain countries, such as UK and Germany, the number of rural initiatives has become very significant [START_REF] Burch | In pursuit of resilient, low carbon communities: An examination of barriers to action in three Canadian cities[END_REF][START_REF] Seyfang | Community action for sustainable housing: Building a low-carbon future[END_REF][START_REF] Seyfang | Grassroots innovations for sustainable development: Towards a new research and policy agenda[END_REF][START_REF] Seyfang | A thousand flowers blooming? An examination of community energy in the UK[END_REF][START_REF] Middlemis | Building capacity for low-carbon communities: The role of grass roots initiatives[END_REF][START_REF] Walker | Community renewable energy: What should it mean?[END_REF]. In other countries, such as France, the number is lower, albeit growing rapidly [START_REF] Yalçın-Riollet | Energy autonomy in Le Mené: A French case of grassroots innovation[END_REF][START_REF] Nadaï Alain | French Policy Localism: Surfing on 'Positive Energie Territories' (Tepos)[END_REF].

National and transnational networking of local initiatives -such as the Convention of Mayors (EU) [START_REF] Com | Covenant of Mayors[END_REF], the CLER -Network for the Enegry Transition (France), the Network for 100% ReN Regions (DeEnet 100% Regionen -Germany) -have also developed since the nineties and gained institutional recognition. Aside from their input in devising policy frameworks, they provide 'local authorities' with arenas for networking, exchanging practices, returns on experience and means for capacity building. New ways of 'commoning' around energy are emerging [START_REF] Timothy | Whose energy transition is it, anyway? Organisation and ownership of the Energiewende in villages, cities and regions, Local Environment[END_REF]. They renew the way in which energy is approached and practiced, bringing it into relation with an enlarged, more collective responsibility and closer to market initiatives and practice.

The rise of policy localism

While local initiatives have gained political status and a role, policy discourse has also evolved, sometimes asserting the role of local communities / territories / authorities as central actors in the energy transition. Policy programmes have bounced on the ongoing political structuration of rural territories in this domain, sometimes calling for the replication of innovative experiences through standardization, exemplification or demonstration. The role of local authorities in contributing to innovative ways of addressing climate energy issues has been both acknowledged and debated [START_REF] Amin | Local community on trial[END_REF][START_REF] Geoghan | Community development and the contested politics of the late modern agora: of, alongside or against neoliberalism?[END_REF][START_REF] Marvin | Creating myths rather than sustainability: The transition fallacies of the new localism[END_REF][START_REF] Purcell | Against the local trap: scale and the study of environment and development[END_REF][START_REF] Walker | Community renewable energy: What should it mean?[END_REF][START_REF] Walker | Trust and community: Exploring the meanings, contexts and dynamics of community renewable energy[END_REF]. Pointed to has been a purported policy of 'localism' that is being developed in certain countries, including France and the United Kingdom, yet with without a steady resolve with respect to political changes and cycles (Nadaï et al. 2014[START_REF] Catney | Big society, little justice? Community renewable energy and the politics of localism[END_REF][START_REF] Walker | Community renewable energy: What should it mean?[END_REF], 2010). The term indicates a policy rhetoric and practice that praises and foregrounds the 'local' as the optimal scale for climate energy action.

The rise of policy localism has been accompanied by a change in policy programmes and modes of support to local energy projects. In some countries, such as the United Kingdom, this has resulted in a market-based approach to energy policy and a reduction in the overall financial support to local climate energy initiatives [START_REF] Catney | Big society, little justice? Community renewable energy and the politics of localism[END_REF][START_REF] Jung | Fostering community energy and equal opportunities between communities[END_REF]North, 2011). Policy programmes embracing a neo-liberal approach to energy policy have framed local communities as individualised, structured, competent if not competitive entities, morally responsible and accountable for their destiny, and possessing a capacity for selfregeneration [START_REF] Amin | Local community on trial[END_REF][START_REF] Geoghan | Community development and the contested politics of the late modern agora: of, alongside or against neoliberalism?[END_REF]. Feed-in tariffs, for instance, started to be replaced by other market-based instruments such as primes over price or repayable loans for supporting local experiences [START_REF] Catney | Big society, little justice? Community renewable energy and the politics of localism[END_REF][START_REF] Fr | LOI n° 2015-992 du 17 août 2015 relative à la transition énergétique pour la croissance verte[END_REF]. In certain cases, an emphasis has also been placed on citizen /third-part financing of new energy technologies, such as cooperative developments, crowdfunding or other ways of collective financing, etc. While these evolutions have taken place at a moment when EU authorities invoked the (market) maturity of some renewable energy technologies (wind power) in order to evolve ReN policy (from feed-in tariff to auction or prime over price -EU, 2013b and c), they may also be regarded as a way for states to cope with the reduction in their resources and financial capacity to support the development of climate energy projects.

A new phase coming up?

To put it in a nutshell, these evolutions may result in what could be regarded as an ongoing new phase, in which: 1) local initiatives become multiple; 2) their extension and their institutional recognition take place in countries other than UK and Germany, where they were first broached; 3) local initiatives develop new articulations to market and market-making, and 4) local initiatives develop new ways of 'commoning' around energy. Put differently, we may wonder whether a change is occurring in the institutionalisation of local initiatives by which the 'local' is supposed to endorse a new role in the domain of climate-energy action, passing through business/economic models and market, and weaving them together with innovative ways of commoning around energy. Public policies and local initiatives have to find their way in these new articulations.

Justice and equity issues

Such a change cannot come about without raising issues of justice and equity. Such issues have recently been addressed in the field of energy (Sovacool, 2013, Hall et al., 2013;[START_REF] Jenkins | Energy justice: A conceptual review[END_REF]. They have been analysed in relation to the recent rise of policy localism in UK energy policy (see supra), indicating a redefinition of what 'local', 'social cohesion' or 'community' mean, which has paved the way to a reduction in financial and policy support to community structuration and increased inequalities in the conditions for access to this support. It therefore seems worth exploring the extent to which these apply to other policy contexts. For instance, [START_REF] Nadaï Alain | French Policy Localism: Surfing on 'Positive Energie Territories' (Tepos)[END_REF] have suggested that a framing slightly different from the UK neo-liberal and market framing might be at work in emerging French policy localism, one that seizes French local climate energy initiatives as potential levers for green growth and technological demonstration policies.

Exploring entanglements and temporalities

While such differences in policy / political approaches to localism are meaningful, there is always the possibility that local initiatives will make their way in the midst of adverse policy settings, leaning on multiple scales of action, support and networking in order to develop themselves. It has been suggested that the structuring of such initiatives has a history going back to a time before the rise of climate energy issues [START_REF] Nadaï Alain | French Policy Localism: Surfing on 'Positive Energie Territories' (Tepos)[END_REF] and relies on processes that weave together multiple dimensions (spatial, material, socio-institutional, political, market, etc.) in rather singular and contingent ways [START_REF] Rydin | The Financial Entanglements of Local Energy Projects[END_REF]. In-depth longitudinal case studies are useful in understanding both the processes of emergence of local initiatives, the re-articulation between policy support and market, and how we (and governments) might learn from them in a period in which these initiatives are becoming numerous, a process of learning that might not start from or through the transfer of solutions or outcomes as if they were generic recipes.

The Special Issue takes a close look at the entanglements and temporalities of some emergence processes of local initiatives, and at the recent history of two networks of local initiatives, in order to explore these issues. It gathers together five papers.

Three of them are cases studies of local development of renewable energy projects in France and Germany. Béatrice Cointe traces the emergence of a successful local photovoltaic project in South-West France (les Fermes de Figeac, operational since 2011), which is driven by a local agricultural cooperative relying on a tradition of mutualisation. The project comprises about a hundred of local farmers, who rent their roofs for installing PV panels and so have become energy producers and shareholders of the PV-project company. The author explores how supporting policies -the feed-in tariff and its associated procedures -have been progressively turned into a tool for territorial innovation and development, and how the mutualisation that is constitutive of the local cooperative is organised at the core of the project and has become part of the construction of market-access, profit maximisation and redistribution. The case study thus follows the detailed technical and relational work of constructing politics and local development out of a market instrument through negotiation and trials at all steps of project development. As the author points out, while most of the elements that have been identified as critical success factors for community projects in the literature pre-existed this project and made it possible, it did not emerge directly from these pre-existing capacities and resources. Instead, the trials that shaped the project also transformed these capacities and resources, and created new ones.

Most interesting is the fine-grained description of the way in which the concepts of justice, solidarity and equity associated with 'mutualisation' were translated into the project by the local cooperative. As the author states, this work amounted to a reinterpretation of the feed-in tariff as a 'policy device intended to accelerate the large-scale deployment of photovoltaics without incorporating concerns over territorial redistribution or equity'. Reinterpretation is at work in all the tensions with established procedures that the author describes (selection of participants, grid connection, negotiations with banks). These tensions also testify to the extent to which the project is entangled in multiple scales and networks (grid-connection, purchase agreement, mass market for electricity, national and European policies and policy objectives, etc.). This shows that the project's 'local character […] is, in fact, protected and made possible by these wider entanglements'. Ultimately, as the French feed-in tariff rests upon a longstanding conception of electricity as a public service embedded in price-setting practices (including per-equation, the equalisation of electricity prices across the French territory), the tension between the French feed-in tariff and mutualisation could as well be interpreted as being about reconciling two different conceptions of equity and modalities of redistribution. Thus the case study brilliantly highlights and problematises the entanglements through which a specific conception of equity is enacted in a project and redistributive concerns are incorporated into an economics-centred policy device.

Both Edith Chezel and Alain Nadaï, and Antoine Fontaine and Olivier Labussière, focus on case studies of scaling up of renewable energy projects, addressing the relation between scaling up renewable energy development and justice. Chezel and Nadaï focus on the successful collective development of citizen wind parks in the German district of Northern Friesland, a well-known example of citizen-funded wind power development, and on the failed attempt at expanding this approach through planning in the federal state of Schleswig-Holstein (of which Northern Friesland is a part). The authors follow in fine detail the careful, step-by-step, collective structuring of public participation and financing in the development of local wind parks in Northern Friesland. They point out a shared sense of 'fairness' that results from these practices and is part and parcel of what they term a 'wind power assemblage'. Fairness, which weaves together and balances the three dimensions of justice (distributive, procedural and recognition), is characterised as deeply empirical. It results from a balance between practical capacities and distribution of opportunities. Turning to Schleswig-Holstein, the authors point out justice issues in the government attempt to scale up this experiment through planning. Tensions arose in certain communities over the right to take part in the development of wind power that blocked the overall planning process. These difficulties evince issues in the distribution of opportunities and in the recognition of potential differences between territories. The case study thus shows how hard it is to strike the balance between the different dimensions of justice in the scaling up of renewable energy, and hence how such a balance is embedded in local practices and socio-spatial configurations. It also suggests the challenge that these tensions present when we seek to derive good practices from local experiments.

Fontaine and Labussière describe the development of community-based solar experiments, the 'centrales villageoises' in south-eastern France (Rhône-Alpes region). Developing collective PV projects in France is challenging because French institutions -that is, the procedures associated with the granting of feed-in tariff, grid connection authorisation, bank loans, etc. -are made for individual projects (one person, one roof, one production unit). In this context, projects that pursue more-than-individual development and investment strategies have to experiment with their assembling in innovative ways, as already illustrated in Cointe's paper (cf. above). The authors propose a slightly different perspective on the development of collective renewable energy projects. They focus on what they call the assembling of the solar resource -the practices of assembling all the afferent materialities (panels, roofs, buildings, grid connection, landscapes, and so on) -engaged in turning solar radiation into usable energy. With fine-grain attention, they describe these practices, particularly their potential to associate the solar resource with the dimension of a common good by granting materialities a shared status (shared roofs, shared investments, shared risks and gains). Relying on a Deleuzian distinction between ways of assembling (growth/decrease) that are based on dimensions and relations (aggregating) versus ways that are based on metrics and directions (granulating), they show that the former, because they prioritize relations over individualisation, leave more flexibility for succeeding in both collective ventures and the assembling of multi-dimensional, more sustainable PV projects. In a way, this paper echoes Cointe's results about the challenge of collective ventures in ReN development in France and the multiple, multi-scalar, entanglements on which 'local' energy projects rely in addressing this challenge. However, in placing the emphasis on the solar resource, the authors show how assembling collective projects is closely articulated with assembling a just landscape [START_REF] Mitchell | Cultural Landscapes: Just Landscape or Landscape of Justice?[END_REF] that fairly redistributes benefits from PV development and incorporates dimensions of equity, justice and sustainable development; what Agyeman and colleagues have called 'just sustainabilities' [START_REF] Agyeman | Just Sustainabilities, Development in an Unequal World[END_REF].

Cyria Emelianoff and Carol Wernert propose a more historical and political perspective. They analyse the factors that have motivated local energy production and transition in the border city of Metz (France), which inherited a municipal energy company founded during the German annexation. The case study analyses the local geopolitical and historical process by which energy became a municipal resource and its production was expanded, in the teeth of the French centralized energy system. Geopolitics is here defined as '[the] rivalry for power over territories or populations […]' in relation to contested infrastructure projects, either because of their environmental, landscape or economic impacts, or because they serve electoral strategies. While the municipalisation (municipal management by a local company that is entirely or partially public owned) of the production and distribution of energy has chiefly been analysed at the national/international level, and staged as an attempt of public ownership and energy democracy (energy justice, distributed power, energy self-sufficiency, etc.) in the face of neoliberal globalization, the case of France, and of Metz as a border municipality, sounds peculiar because the opposition to municipalisation comes from French public operators, defending equality of access and low energy prices. In this context, the authors show that the driving force for renewable energy production in Metz, and of the resistance to energy centralization, was the safeguarding of municipal energy management because it has always underlain the local finances, the capacity for the municipality to act and the tenure of the mayors. They also show that, through political and municipal networks, the struggle plays itself out at a local and a national level, fostering institutional changes, and that the power and benefits conferred by the control of energy production, a central dimension of energy justice, have not been better shared than at the national level. This contribution brings a historical and transnational perspective to the special issue, showing how important the underlying national institutions and politics are when it comes to understanding local energy change. It also underlines the fact that local territories often engage in climate energy action for reasons and motivations other than sustainability, a point which had been emphasized in other case studies [START_REF] Nadaï Alain | French Policy Localism: Surfing on 'Positive Energie Territories' (Tepos)[END_REF] but not from such a geopolitical angle. The emphasis on the quest for political power through energy developments and control, justifying this original geo-political perspective on a 'local' case study and its consequences for the (limited) justice dimensions of local energy developments should warn us again against idealising the 'local' as a scale for political action.

Pia Buschman, Peter Moser, Alain Nadaï and Yannick Régnier propose contributions based on two practitioners' viewpoints (Moser and Régnier, with the help of two academics, Buschman and Nadaï) on the way in which they manage a national network of local initiatives (the 100%-Regions network [Germany] and the CLER-network for the energy transition [France] respectively), describe its build-up process and consider of issues of justice. The close-up of these two networks supports, if only to a limited extent, the view that local initiatives are entering a new period. It shows important differences between the two countries, and points out a remaining unevenness and a stop-and-go in public policy support for local initiatives. In this context, the paper proves how decisive the art of these intermediaries is, their tinkering with multiple types of supports and scales of networking in order to maintain continuity. While the description displays networks that somewhat crisscross academic categories and borrow from both transition management and grassroots descriptions (especially for the German network), they also show that these networks have inspired or are inspiring national policies in multiple ways: directly, by accessing policy arenas and contributing to policy making; or indirectly, by publicising beacon experiences that stretch boundaries in how far territories can go in taking charge of climate-energy issues. Interestingly, as the authors point it out, motives or ends of action are only reflectively, after probing, related by the practitioners to ideas of justice, and only then can they be identified as markers of action or ends. This is consistent with analyses of other networks (community energy projects, transition network initiatives) in the literature, underlining that justice issues appear to be seldom explicitly discussed or flagged in practice because of fear of division or confrontation. This restraint, however, contrasts with the centrality of dimensions such as 'mutualisation' and 'fairness' described in other papers of the special issue, suggesting that it may be more the ideas/ language rather than the dimensions of justice (except for distributive justice) that fail to consort easily with the level of practice. Justice issues thus seem in any case bound up with a politics of assembling, either as a marker of federative action (all fighting for justice) or as a potentially divisive issue.

Taken together, the papers in this special issue afford an unprecedented view of the joint construction of economic and political engagements around local energy projects and the related justice dimensions. They first show in great detail how these entanglements are developed as these 'local' initiatives emerge. The multi-dimensional and multi-scalar character of these entanglements has been pointed out in the literature (Rydin et al. 204), but this has mainly been adduced as a feature of local initiatives and in a static perspective. The case studies gathered here provide a new perspective as they follow the making of these entanglements in time and space, allowing for a critical perspective on the intersection between market and political engagement at work.

First, it is in the detailed description of the trials and probings that these initiatives have to face that we can see the way in which shared political values are upheld and eventually integrated at the core of socio-material assemblages. It is because the papers propose longitudinal case studies that we can see and follow the work and practices of project making, which allows for a political dimension to be inscribed in the core of these local initiatives.

Secondly, so-called 'market-based' approach to energy policy (cf. supra) is here described in somewhat greater depth than usual, allowing us to see their detailed influence and impact on the making of ReN projects. The market approach under consideration here appears multi-dimensional, so that speaking of 'market' or 'market instrument' in the singular, as has been underlined elsewhere, makes little sense: there are many types of markets and markets can convey multiple values [START_REF] Grandclément | Transitioning through markets[END_REF]Geiger at al., 2015). So too for economic instruments such as the feed-in tariff, the main 'economic instrument' for renewable energy policy. The work of the French feed-in tariff shows that it is a double-edged sword. On the one hand, it clearly prioritizes individual over collective ventures, hence conveying a neo-liberal type of framing. On the other hand, it endows local actors with power of negotiation by granting them with a long term, stable revenue from ReN projects. It thus imposes on local projects an individual and economic framing while granting them the power of negotiating its curbing in order to construct a political engagement. Such a double dimension reflects the struggle between free-market ideology and innovation-based policy that overarched its political adoption at the EU level [START_REF] Cointe | Feed-in tariffs in the European Union -Renewable energy policy, the internal electricity market and economic expertise[END_REF]Nadaï and Cointe, In Press), testifying again to the multi-scalar entanglements of local processes.

Thirdly, these longitudinal case studies also put the territorial dimension of local initiatives in a new light. We can follow the ways in which the collective or landscape dimensions of these projects are maintained throughout their development and eventually inscribed (or not) in the core of these initiatives. The care and the determination necessary to carry through the articulation between the economic, political and territorial dimensions, and to write them into the materialities of the projects, explains the difficulty incurred in attempts to scale up these initiatives. Tsing has defined scalability as the 'ability to make projects expand without changing their framing assumptions', which usually calls for reducing things or processes to one or very few of their dimensions. Scalable accounts, she argues, are decisive in performing an 'interplay between scalable and nonscalable in forms of capitalism in which scalable accounting allows nonscalable labour and natural resource management', that is, forms of capitalism 'in which wildly diverse forms of work and nature are made commensurate -for capital' (Lowenhaupt Tsing, 2015: 38). The accounts presented here are sufficiently fine-grained to display 'the work it takes to create scalability' (Lowenhaupt Tsing, 2015:38), be it in attempts at augmenting the size of a project or at scaling up the process of producing projects through institutionalisation and planning. Such a perspective presents a profound challenge to the purported notion of 'global niche' based on the idea that learning from local experiences could be scaled up by being standardised and detached, so as to travel and be applied elsewhere [START_REF] Franck | Local and global dynamics in technological development: a socio-cognitive perspective on knowledge flows and lessons from reinforced concrete[END_REF]. It militates for a type of scaling up through encounters and dissemination more respectful of the multidimensionality and the singularities of each situation, that is, one closer to (Deleuzian) 'aggregating'.

Last but not least, the case studies point out a peculiarity of justice issues in these processes and socio-technical assemblages. Justice issues are in many ways central in the making of these initiatives and networks of initiatives. They are present in the construction of their economic, territorial, spatial and landscape dimensions, and also in the articulation of these dimensions. They are essential for the success, the cohesion and the durability of these initiatives. Yet they are not asserted in the language and through the idea of justice, or are done so only reflectively, after probing. The observation has already been made that justice issues, even when they are marked by the network discourse, are not always made explicit on the ground (for example, Bailey and Darkal 2018). One assumption is that justice considerations, since they presuppose the existence of differences, might convey a divisive potential unless they are flagged as a federative marker of the network. In any case, this suggests that there is a tension between the invocation of justice issues and the politics of assembling a network, and thus that justice issues seem bound up with a politics of assembling. A point that needs to be explored further.

CFCC15 Scientific Committee, 2015, Our Common Future under Climate Change Outcome Statement, July 10? Paris, CFCC15 Scientific Committee, http://poolo.kermeet.com/Data/kmewexV7/block/F_bedaa0dbe3d01a517f0fa7eb11d4b1a4559fae1ae868b.pdf

https://unfccc.int/sites/default/files/english_paris_agreement.pdf

Acknowledgement

This work was supported by the French Agency for the Environment and the Energy (ADEME) under the Program 'Mettre l'innovation sur la trajectoire du facteur 4' [Grant 11 10 C 0079, projet "CLIMENCORED"]; the French National Research Agency (ANR) under the program 'Sociétés innovantes' [Grant number 2011-SOIN-003-01, projet "COLLENER].


Nelly Herault

Lilian Olivet

Laurence Pirault-Roy

Catherine Especel

Maria A Vicerich

Carlos L Pieck

Florence Epron
email: florence.epron@univ-poitiers.fr

Nelly Hérault

María A Vicerich

Controlled preparation and characterization of Pt-Rh/Al2O3 bimetallic catalysts for reactions in reducing conditions

Keywords: Bimetallic supported catalysts, Pt-Rh, cyclohexane dehydrogenation, FTIR of CO, FTIR of NO-CO

Controlled preparation and characterization of Pt-Rh/Al2O3 bimetallic catalysts for reactions in reducing conditions

Introduction

Pt-Rh bimetallic systems were demonstrated in the late 1980's as efficient bimetallic three-way catalysts for reducing NO and oxidizing CO. While both metals are active for these two reactions, a synergetic effect between the two metals was proposed to explain the remarkable stability and higher activity of the aged bimetallic catalysts supported on alumina compared to the monometallic ones [1]. Recently, Pt-Rh bimetallic catalysts were presented as active and selective systems for the ring-opening of cyclic hydrocarbons [2][3][4], whereas rhodium and platinum supported monometallic catalysts present a poor selectivity to ring-opening products (ROP). This specific behavior of the bimetallic Pt-Rh catalysts was demonstrated as depending on the preparation method, and was attributed to the presence of bimetallic Pt-Rh alloys, i.e.

Pt-Rh entities in strong interaction, at the metal particle surface, as identified by CO chemisorption followed by Fourier Transformed Infra-red (FTIR) spectroscopy. However, it was not possible to quantify, by this technique, the amount of Pt, Rh and Pt-Rh alloy at the catalyst surface.

The objective of the present study was to prepare bimetallic Pt-Rh catalysts supported on γalumina with various types or degrees of interaction between rhodium and platinum and to characterize them by various techniques, and more particularly by FTIR of NO and CO, which was demonstrated as an efficient way to quantify the amount of accessible Pt, Rh, and Pt-Rh alloys at the metallic particle surface [5], in order to rationalize the effect of the preparation method on both the Pt-Rh interactions and the particle surface composition. For this purpose, we chose three different techniques known for favoring or not the formation of bimetallic entities and tried to control the metal particle size (< 4 nm expected):

-Classical coimpregnation, used as a reference. By this procedure, a random deposition of both metals (Rh and Pt) may occur on the support.

-Surface modification of a parent monometallic catalyst by addition of the second metal by organometallic grafting. According to this procedure, core-shell particles should be obtained.

The result strongly depends on the monometallic catalyst particle size, which may vary in the reaction medium.

-Formation of bimetallic particles in solution, by a colloidal way, using a polymer for stabilizing metallic particles. This type of synthesis leads to homogeneous particle sizes with generally strong metallic interactions. The two key-points of this technique are the deposition on the support and the removal of the polymer or the surfactant.

The effect of the interactions between platinum and rhodium will be evaluated on two test reactions, cyclohexane dehydrogenation and methylcyclopentane ring-opening. Cyclohexane dehydrogenation is a simple reaction for which the activity of a monometallic catalyst will mainly depend on the nature of the metal and on its accessibility, whereas for bimetallic catalyst, it will be a function of the amount of each metal at the surface if their activity is very different. This reaction could be then an effective means to estimate the amount of rhodium and platinum at the surface of bimetallic particles. As far as methylcyclopentane ring-opening reaction is concerned, strong interactions between these two metals, with the formation of alloyed entities, will lead to synergetic effects and then to higher performances than those of the monometallic catalysts [2][3]. Consequently, this reaction is used in the present study to verify the formation of alloyed species depending on the preparation method.

Experimental

Catalysts preparation

An alumina (AluminumOxidC (Degussa)) with a specific surface area of 100 m² g -1 was used as support. It was crushed and sieved in order to retain grains with diameters between 0.1 and 0.25 mm. Finally, the support was calcined for 4 h at 773 K under air (flowing 60 cm 3 min -1 ) and then reduced with hydrogen for 4 h at 773 K (same flow-rate as above).

Hexachloroplatinic acid (H2PtCl6.6H2O, Alfa Aesar) and rhodium (III) chloride (RhCl3.3 H2O, Johnson Matthey) were used for the preparation of monometallic and bimetallic catalysts, except for the organometallic grafting performed in organic medium, for which rhodium (III) acetylacetonate (Rh(acac)3, Alfa Aesar) and platinum (II) acetylacetonate (Pt(acac)2, Alfa Aesar) were chosen. For all catalysts, the Rh and Pt nominal contents were of 0.5 wt% and 1.0 wt%, respectively, which corresponds to the same atomic content. Monometallic catalysts were prepared by simple impregnation. Different preparation techniques were used for bimetallic catalysts, namely coimpregnation, organometallic grafting and microsuspension.

Preparation by coimpregnation (CI)

The bimetallic catalyst named (PtRh)/CI was prepared by coimpregnation of hexachloroplatinic acid and rhodium chloride on the support.

In this procedure, 30 mL of water acidified by addition of hydrochloric acid and 1.5 mL of precursor salts were mixed. Then, the support was covered by the mixture and left one night under stirring. Water was then evaporated at 343 K and the catalyst was dried overnight at 393 K, treated under air at 573 K during 4 h and then under flowing hydrogen at 573 K for 4 h.

Preparation by Organometallic Grafting method (OG)

Unlike the others, this preparation was carried out in organic media. This technique was based onto grafting of the second metal on the preformed Pt particles surface. The monometallic catalyst was prepared by impregnation of platinum 2,4-pentanedionate on alumina. The metal particle size of the parent monometallic catalyst is <d> = 2.0 nm.

The preparation was carried out in a fixed-bed reactor equipped with an addition funnel, according to a procedure similar to the one described in [6]. After the pretreatment of 2.5 g of Pt/Al2O3 under hydrogen at 673 K during 2 h to ensure the complete reduction of metal particles, 15 mL of toluene were introduced at ambient temperature on the catalyst under hydrogen for 5 minutes. Then, the solution was purged under nitrogen to remove the excess of hydrogen and 15 mL of a solution containing toluene and rhodium 2,4-pentanedionate (0.97 mmol.dm -3 of Rh(acac)3), previously purged, were introduced onto the catalyst. The mixture (catalyst and solution) was heated at 343 K under argon flow for 6 h with a cooling system at the top of the reactor to avoid the evaporation of toluene. After that, toluene was removed by filtration; the catalyst was dried under argon at 393 K and reduced under hydrogen flow at 473 K. This final catalyst was named Pt-Rh/OG.

Preparation in Microsuspension (MS)

The catalyst was prepared in microsupension following the procedure described by Venezia et al. [7]. The reaction was carried out in a water/ethanol mixture (1:1 v/v) containing polyvinylpyrrolidone (PVP, MW: 10 000, Aldrich) as steric agent with a concentration of 3.75 mg.cm -3 and both chlorinated metallic salts with a total concentration of 0.75 mg.cm -3 . This solution was refluxed at 356 K overnight after introduction of the support at this temperature.

Immediately, the suspension turned in dark due to metal reduction. The solvent was removed by centrifugation, and then the catalyst was washed five times with 50 mL of water and two times with 50 mL of ethanol and removed by centrifugation too. The efficiency of washes was checked by measuring the total organic carbon amount in remaining solution after each wash.

The solid was finally dried at 343 K overnight. This catalyst was named (PtRh)/MS.

As the objective of the study was to compare the effect of the preparation methods, it was chosen to not calcine the samples for MS and OG methods since it is known that, for bimetallic particles, oxidizing treatments lead to an enrichment phenomenon or metal segregation, risking to mask the effect of the preparation method.

Characterization

Determination of metal content

The metal content was determined by ICP-OES (Perkin Elmer Optima 2000 DV) and X-ray fluorescence techniques (Bruker AXS S4 Explorer). Before ICP analysis, samples were mineralized in strong acids with microwave heating.

Chemisorption of hydrogen

The hydrogen chemisorption capacity was determined by a static volumetric vacuum technique at room temperature. First of all, the catalyst was cleaned under ultra high vacuum overnight to eliminate the adsorbed molecules. Then, the catalyst was reduced under 750 mbar of H2 at 573 K for 1 h, and purged under vacuum until a low pressure (P < 10 -5 mbar) was reached. Two series of hydrogen adsorption isotherms were performed with an increasing hydrogen pressure, from 10 mbar to 70 mbar with an increment of 10 mbar. Between the two series, the sample was submitted to ultrahigh vacuum in order to remove weakly bound hydrogen. The amount of adsorbed hydrogen was calculated by extrapolating the isotherm to P = 0. The first isotherm allows determining the total amount of adsorbed hydrogen and the second one the amount of physisorbed hydrogen. The amount of chemisorbed hydrogen was calculated by difference between these two values. The metal particle size was calculated considering a stoichiometry of H/M = 1, and assuming that particles correspond to cubes deposited on the support and exposing 5 faces [8], with an equal distribution between (111), (110) and (100) planes.

Adsorption of probe molecules followed by FTIR

The adsorption of probe molecules on catalyst surface was followed by Fourier Transformed Infra-Red (FTIR) spectroscopy using a Nicolet 6700 FT-IR spectrometer. The catalyst was prepared in the form of a self-supported wafer of 15-20 mg with a diameter of 16 mm and a surface of 2 cm 2 . This pellet was placed in a cell and reduced at 573 K for 1 h under H2 and purged at this temperature under vacuum (P < 10 -5 mbar). Reference spectrum of the clean surfaces was then collected at room temperature (RT). The spectra of CO and/or NO adsorbed on the metal surface were obtained by subtraction of the reference spectrum to the collected spectrum after probe molecule adsorption. This allowed eliminating the influence of the characteristics of the sample, such as particle size and specific surface area of the support and density of the wafer [9].

* Chemisorption of CO (CO/FTIR)

After purging under ultra-high vacuum, CO was introduced in excess at RT to reach 27 mbar of CO in the cell, which was checked to permit a full coverage of the metal surface by CO.

Then, the sample was submitted to ultra high vacuum and a spectrum was recorded. In order to determine the accessibility of platinum in bimetallic catalysts, a calibration curve was used, obtained by using well-known monometallic platinum catalysts with various different dispersions, determined by hydrogen chemisorption and checked by TEM, and by measuring the corresponding surface area of the linear CO band. This allowed determining the molar extinction coefficient for linear CO adsorbed on Pt according to Beer's law [9]. * Chemisorption of NO, then CO (NO-CO/FTIR) After purging under ultra-high vacuum, 240 mbar of 3%NO/He (corresponding to 15 mbar of NO) in the cell were introduced on the sample heated at 473 K. The pellet was maintained for 10 min at this temperature and was then placed under vacuum for 45 min at room temperature before taking a spectrum. Then, 27 mbar of CO was introduced for 3 min. The gas was evacuated and another spectrum was collected. As previously performed for determining the molar extinction coefficient of linear CO on Pt for direct CO adsorption, calibrations curves were plotted after NO adsorption, for Rh, and NO then CO adsorption for platinum, using rhodium and platinum monometallic catalysts of known dispersion values.

Temperature Programmed Reduction

Temperature programmed reduction (TPR) was carried out in a Micromeritics AutoChem II equipped with a CryoCooler II. First of all, the catalyst (200 mg) underwent an oxidation pretreatment during 1 h at 673 K under O2 and was flushed at the same temperature under N2 for 2 h. Then, the catalyst was cooled to 253 K and hydrogen was passed on the catalyst up to 773 K and analyzed by GC.

Transmission Electron Microscopy

The sample was dispersed in ethanol and deposited on a copper grid. Then, particle shape and size were determined by Transmission Electron Microscopy (TEM) analysis (TEM/STEM JEOL 2100 UHR) with a resolution of 0.19 nm. This apparatus was coupled with an energy dispersive X-ray spectrometer (EDX) used in the nanoprobe mode for the analysis of Pt and Rh.

The particle size distribution was obtained from TEM pictures calculating the average particle diameter from the measurement performed on at minimum 100 particles.

X-ray diffraction

X-ray diffraction (XRD) measurements were performed at room temperature in a Bruker AXS D5005 X-ray diffractometer, working with CuKα radiation (λ = 1.54184 Å), generated at 40 kV and 40 mA. Signal is recorded for 2θ between 10° and 90° with a step of 0.04° and a step time of 6 s. For sample identification, diffraction patterns were compared to the ICDD (International Center for Diffraction Data) database integrated in the EVA software.

Catalytic tests a. Cyclohexane dehydrogenation

The cyclohexane dehydrogenation reaction was performed in a fixed-bed reactor at atmospheric pressure. 20 mg of catalyst was reduced for 1 h at 573 K and the temperature was decreased to 543 K for reaction. Cyclohexane was injected using a calibrated motor-driven syringe (0.03 cm 3 min -1 ) in a hydrogen flow (60 cm 3 min -1 ). Gases were taken every 10 min for 2 h and analyzed by GC with a flame ionization detector (Varian 3400X) on a HP-PLOT Al2O3 "KCl" column. The only detected product was benzene.

b. Methylcyclopentane ring-opening

Methylcyclopentane (MCP) ring-opening reaction was performed in a fixed-bed reactor at atmospheric pressure. Before the reaction, 135 mg of catalyst were reduced with H2 at 773 K for 1 h. The conditions used were: H2 flow rate = 36 cm 3 min -1 , MCP(liq) flow = 0.362 cm 3 h -1 , reaction temperature = 533 K, reaction time = 2 h. The reaction products were analyzed on a Varian CX 3400 gas chromatograph after separation on a PONA column.

Results and discussion

TEM and H2 chemisorption

For all catalysts, the measured actual composition corresponds to the nominal values, i.e. 0.5 ± 0.1 wt.% of Rh and 1.0 ± 0.2 wt.% of Pt (Table 1).

Table 1 shows that the H/M ratio varies according to the preparation method. The (PtRh)/CI sample presents the best dispersion leading to d = 1.3 nm, whereas Pt-Rh/OG and (PtRh)/MS present diameters higher than 3.0 nm. The small size of CI catalyst may be linked to the presence of hydrochloric acid [4], which may play an essential role in metallic dispersion because chloride ions may compete with precursor adsorption during coimpregnation procedure, and increase the platinum and rhodium dispersion on alumina [10]. For the CI catalyst, the size determined from the H/M values corresponds to the one measured on the TEM pictures (Fig. 1a), whereas for the two others (Figs. 1b and1c), the size deduced from TEM pictures is lower than the one determined from chemisorption experiments. The large particle size obtained with the OG technique is directly linked to the preparation procedure, which involves an enlargement of particles size by deposition of rhodium on the preformed platinum particles with an initial diameter of 1.3 nm. In this sample, the analysis by EDX showed that small particles are composed of both Pt and Rh, and then are bimetallic, with a composition around the nominal composition (Pt/Rh around 50/50), whereas the largest ones are richer in platinum. This result is consistent with the preparation procedure, since the accessibility of small particles, which are majority on this sample, is higher than the one of large particles.

Consequently, there was more Rh deposited on small Pt particles than on the largest ones. For (PtRh)/MS, the overestimation of the particle size as calculated from H2 chemisorption results may be due to the presence of some polymer (polyvinylpyrrolidone) residues at the particle surface, which decreases the chemisorption capability. The presence of the polymer is evidenced by characteristic bands in infrared spectra (Fig. 2) performed before and after reduction. In the literature [11], spectrum of pure polyvinylpyrrolidone has shown 6 series of bands: one at 3500 cm -1 corresponding to the stretching vibration of water, two at 2910 and 2960 cm -1 attributed to stretching of CH2 or CH groups, one at 1660 cm -1 corresponding to stretching vibration of carbonyl group (C=O) and two at 1460 and 1430 cm -1 for bending vibration of CH2. As shown in Fig. 2 before and after reduction of (PtRh)/MS, a shift of the vibration band corresponding to carbonyl to lower wavenumber (≈1600 cm -1 ) is observed, which may be explained by the adsorption of the carbonyl function on particle surface, the small peak at 1660 cm -1 being attributed to the carbonyl function of free PVP [12]. It can be also seen that the reduction modifies some bands, especially those attributed to water, CH2 and C-N vibrations, but the band corresponding to carbonyl adsorbed on the metal function is still present. For the MS sample, the EDX analysis demonstrated that all the particles are bimetallic with a composition similar to the nominal one. On the contrary, on the CI sample, in some places monometallic Pt particles were identified, whereas in other places, Rh and Pt were analyzed together.

Temperature programmed reduction by H2

To characterize Pt-Rh interactions, temperature programmed reduction studies were carried out on catalysts preoxidized at 673 K. If platinum and rhodium are in strong interaction, the metal oxide that is reduced at the lowest temperature may catalyze the reduction of the other one and permits to decrease its reduction temperature [13][14][15]. In Fig. 3, TPR profiles of bimetallic catalysts are compared to those of the corresponding chlorinated (for CI and MS bimetallic catalysts) and non-chlorinated (for OG bimetallic catalyst) monometallic catalysts. Monometallic Rh catalysts are reduced at lower temperature compared to their monometallic Pt counterparts, since Rh and Pt catalysts prepared with organic precursors present a maximum of reduction peak at 343 and 383 K, respectively, and the Rh and Pt ones prepared with chlorinated salts at 451 and 493 K, respectively. The higher reduction temperature observed for catalysts prepared from chlorinated precursors may be explained by the lower reducibility of oxychloride species in stronger interaction with the support compared to the oxide species. For (PtRh)/CI catalyst (Fig. 3d), two reduction peaks are detected: one at 388 K with a much higher H2 consumption compared to the second one at 498 K. In comparison with Pt and Rh monometallic catalysts (Fig. 3f and3g), the first peak may be attributed to the reduction of rhodium oxide alone with a small part of platinum oxide, but as the reduction temperature is lower than that of the monometallic chlorinated Rh, these species are probably in weak interaction with alumina. The second peak at high temperature may be attributed to the reduction of platinum oxychloride. This result indicates that only a small part of Pt and Rh atoms are in contact, in line with the preparation method that favors the random deposition of Pt and Rh precursors on alumina. For Pt-Rh/OG (Fig. 3c), only one single peak is present at 343 K, at a temperature similar to that of the reduction of Rh (Fig. 3a), and the hydrogen consumption corresponds to the amount needed to completely reduce the rhodium and platinum oxides, proving the strong interaction between rhodium and platinum atoms. These results confirm that this preparation method favors the preferential deposition of rhodium on platinum particles and not on the alumina support, in agreement with the literature [6,16,17]. For (PtRh)/MS (Fig. 3e), the TPR profile displays two reduction peaks, a large one at 284 K and a small one at 750 K, with a H2 consumption indicating a total reduction of both oxides. It should be noted that monometallic Rh and Pt catalysts, prepared in the same conditions as (PtRh)/MS, show reduction peaks at 331 K and 268 K, respectively (results not shown). These reduction temperatures are much lower than those of the monometallic catalysts prepared by impregnation. This may be explained by the low metal-support interaction resulting from the preparation method, consisting in the reduction of the metal precursor in suspension in the presence of a polymer and the subsequent deposition on the support. The main reduction peak of the bimetallic (PtRh)/MS catalyst is between those of the two corresponding monometallic catalysts, characteristic of the co-reduction of Pt and Rh, indicating that these two metals are present in the same metal nanoparticles.

In conclusion, from TPR results it was shown that Pt and Rh are together in bimetallic particles for catalysts prepared by organometallic grafting and microsuspension, whereas they are present in both mono and bimetallic particles for catalyst prepared by coimpregnation.

Characterization by XRD

XRD analysis was used for characterizing all the samples but given the low amount of metal and the small particle size, only the alumina support was detected. For the MS sample, the metal particles were also analyzed before their deposition on the support. The XRD pattern, reported in Fig. 4, demonstrates that particles are probably made up of PtRh alloy and then in strong interaction, thus confirming the results obtained by TPR.

CO/FTIR on Pt, Rh and Pt-Rh supported catalysts

CO is widely employed as probe molecule to characterize Pt alumina catalysts. By infrared spectroscopy, two series of bands are detected [18][19][20][21]. The first one near 2095-2000 cm -1 corresponds to linear Pt 0 -CO and the other one, between 1800 and 1875 cm -1 , corresponds to bridged species Pt 0

x-CO. As more than 95 % of CO is linearly bonded to Pt 0 , the accessibility of platinum can be determined from the molar extinction coefficient determined from this linear CO band. In our case, this coefficient was 0.315 cm.µmol -1 . The frequency of this band depends on the Pt atoms position at the particle surface (on various types of faces, on edge or corner)

and increases with the coordination number of platinum [18][19][20][21]. A summary is shown in Table 2.

For Rh catalysts supported on alumina, in addition to the bands corresponding to CO linearly and bridged bonded to Rh 0 , two others bands are identified and attributed to symmetric and anti-symmetric gem-dicarbonyl species on Rh +I (Rh I -(CO)2) [22]. The presence of Rh +I is explained by the oxidation of Rh 0 in the presence of CO involving hydroxyl groups of the alumina support 23 due to the migration of Rh onto the support in the presence of CO. The positions of these various bands are also summarized in Table 2. The Rh I -(CO)2/ Rh 0 -CO ratio fluctuates with rhodium particle size and also with contact time with CO. Therefore, contrary to platinum, there is no linear dependence between the amount of CO linearly bound to Rh 0 and the rhodium dispersion, with no possibility to accurately determine the amount of Rh atoms at the metal particle surface. Thus, for bimetallic PtRh/Al2O3 catalysts, only the amount of CO adsorbed on Pt can be determined, after decomposition of the spectra, and the band frequencies may give information on the platinum location on metallic surface. Another problem for bimetallic catalysts is that there is an overlap of bands corresponding to linear CO adsorbed on Pt 0 or Rh 0 between 2070 and 2040 cm -1 . In the following, only bands at frequencies higher than 2070 cm -1 will be unambiguously attributed to Pt 0 -CO. Another band, around 2060 cm -1 may be attributed to CO adsorbed on Pt alloyed with Rh, according to Lévy et al. [5].

As shown in Fig. 5, the spectra are deeply different depending on the preparation method used for obtaining bimetallic catalysts. On the three catalysts, after decomposition of the spectra with a Gaussian function, bands corresponding to Pt 0 -CO, around 2081±2 cm -1 , and Rh +1 -(CO)2, around 2025 and 2100 cm -1 , are clearly identified. Also, one band at 2069±2 cm -1 which could be attributed either to Pt 0 -CO or to Rh 0 -CO is observed for the three catalysts. In the case of the (PtRh)/CI catalyst (Fig. 5a) one band is observed at 2054 cm -1 , which could correspond to Rh 0 -CO or Pt 0 -CO. An additional band is detected at 2060±2 cm -1 for (PtRh)/MS and Pt-Rh/OG catalysts, which may be attributed to the linear CO adsorption on one Pt atom alloyed with Rh [3,5]. This is in accordance with the preparation methods used favoring Pt-Rh interactions and with TPR results. As proposed by Lévy et al. [5], the molar extinction coefficient of Pt 0 -CO after CO adsorption was used to quantify the number of Pt atoms constituting the Pt-Rh alloy at the surface. Due to the complexity of the spectra obtained for bimetallic catalysts, the direct quantification was performed only for peaks clearly corresponding to Pt alone (2081 cm -1 ) or in alloys (2060 cm -1 ). The amount of the other species leading to the other observed bands was quantified by difference, using the total amount of accessible metals as determined by chemisorption of NO and then CO, followed by FTIR, which will be discussed in the following paragraph. One have to keep in mind that these other species may correspond either to rhodium (vibrations of Rh(CO)2 identified on the 3 catalysts), or to platinum or rhodium, since it was impossible to unambiguously attribute the bands at 2054 and 2069 cm -1 to Pt 0 -CO or Rh 0 -CO.

The results are reported in Table 3. It can be seen that the amount of Pt alloyed with Rh is more significant for the Pt-Rh/OG than for (PtRh)/MS. Also, one has to keep in mind that the Pt-Rh/OG catalyst was prepared by deposition of Rh atoms on monometallic Pt particles by organometallic grafting. As the amount of added Rh atoms is much higher than the amount of accessible Pt atoms, i.e. Pt atoms at the surface, after preparation the Pt surface should be completely covered by a Rh overlayer. As CO adsorption followed by FTIR has demonstrated that Pt is present at the surface of the bimetallic particles, it means that some Pt atoms were not covered by Rh during the preparation.

NO-CO/FTIR on Pt, Rh and Pt-Rh supported catalysts

At moderate temperature, NO is able to completely oxidize Rh 0 to Rh +1 and leads to the formation of Rh I -NO + . Thus, its FTIR spectrum presents a characteristic vibration band completely isolated from the others [23]. Hence, it is possible to determine the quantity of rhodium at the surface of metallic particles by NO adsorption. In addition, CO adsorption may be carried out after NO adsorption to characterize the platinum surface. According to the literature, many NOx species adsorbed on rhodium or on alumina may be observed by FTIR, as shown in Table 4.

In the present study, the NO adsorption at 473 K, temperature used by Lévy et al. [5] to oxidize all rhodium in the +I state at the surface, was at first studied on Rh monometallic catalyst with various contact time to follow the evolution of adsorbed NOx species as a function of time. As shown in Fig. 6, after a short exposure time to NO at 473 K, between 5 and 11 min, a large and isolated band at 1910 cm -1 , attributed to Rh I -NO + formation, is observed accompanied with multiple bands in the 1800-1400 cm -1 range, corresponding to NOx species on alumina. For longer times, these latter bands increase considerably, and are associated with a smaller band corresponding to Rh I -NO + species, which decreases as a function of time. The reaction between NO and the oxygen of the alumina support leads to the formation of nitrates/nitrites.

Consequently, the experimental conditions, especially the contact time, were chosen in order to avoid formation of nitrates as far as possible. Therefore, the molar extinction coefficient was determined after 10 min of contact time from Rh I -NO + band using Rh/Al2O3 catalysts with various dispersions. This coefficient was equal to 0.237 cm.µmol -1 . Moreover, it is important to note that, when this exposure to NO is followed by an exposure to CO at room temperature, a new band may appear at 2115-2130 cm -1 corresponding to Rh 2+ -CO species [24] showing that, in some cases, NO oxidizes Rh 0 not only to +I state but also to +II oxidation state (Fig. 7). As shown in this figure, after CO adsorption, spectrum shows also rhodium gem-dicarbonyl species at 2030 cm -1 (antisymmetric gem-dicarbonyl) and 2100 cm -1 (symmetric gemdicarbonyl), and the linear NO band adsorbed on Rh +I decreases proportionally. Linear adsorption of CO on Rh 0 is not observed, proving that all Rh 0 species were oxidized during the treatment under NO.

Contrary to rhodium, the oxidation of Pt to Pt + by NO is difficult at 473 K. NO adsorption on Pt atoms may form linear Pt 0 -NO and non-linear Pt 0 -NO species [25], at respectively 1750-1760 cm -1 and 1400-1650 cm -1 . Freysz et al. [26] observed linear Pt 0 -NO at 1797 cm -1 and another species at 1833 cm -1 attributed to Pt + -NO species. On Pt/Al2O3 after contact with NO, only bands attributed to the formation of nitrates and nitrites on alumina are observed (Fig. 8).

When CO is introduced in the cell, a new band appears in the 2000-2100 cm -1 range, corresponding to Pt 0 -CO species. The molar extinction coefficient determined for this Pt 0 -CO band after NO-CO adsorption is equal to 0.114 cm.µmol -1 . This coefficient is three times smaller than the one determined after direct CO adsorption, as also observed by Lévy et al. [5].

Moreover, for Pt/Al2O3 catalyst, a band at 2115-2150 cm -1 may also be detected corresponding to Pt 2+ -CO species [24], which is not observed in the present case (Fig. 8).

Thus, the advantages of submitting a Pt-Rh catalyst to NO at moderate temperature and then to CO at room temperature, are that (i) there is not Pt 0 -NO band superimposed with that of Rh I -NO + (1910 cm -1 ) and (ii) there is not Rh 0 -CO band interfering with that of Pt 0 -CO, thus favoring the quantification of Rh and Pt species at the particle surface.

As shown in Fig. 9, the Pt-Rh/Al2O3 spectra after NO and NO-CO adsorption are deeply different depending on the bimetallic catalysts. The quantification of rhodium at the surface was deduced from Rh I -NO + band after NO adsorption (1910 cm -1 ) and that of platinum from the Pt 0 -CO band after NO-CO adsorption (2000-2100 cm -1 ). Results (Table 3) for the amount of Pt alone obtained from NO-CO/FTIR are similar to those obtained from CO/FTIR considering the same global metal accessibility. However, the band attributed to Pt alloyed with Rh has disappeared from the spectra of (PtRh)/MS and Pt-Rh/OG. This may be due to a surface enrichment by Rh at the expense of Pt in the Pt-Rh alloyed entities, in the presence of NO at 473 K [5]. These Pt species in strong interaction with Rh, correspond to Pt surrounded by Rh, at the surface and subsurface layer. It means that an exchange of this type of Pt atom at the surface with one Rh atom at the subsurface is sufficient to perform the enrichment and lead to the disappearance of the Pt atom from the surface. This driven force of this enrichment phenomenon is due to the much stronger Rh-O bond strength compared to the Pt-O one, the oxygen originating from NO adsorption and dissociation [27]. The free oxygen atoms, can access Rh atoms in the first two layers, drawing them to the surface [28]. It was shown that this reconstruction only affects the two first atom layers, the surface layer being composed of mainly Rh, the second one being Pt enriched [29]. The segregation of Rh under oxidizing conditions for Pt-Rh bimetallic catalysts is a very well known phenomenon, which has been the subject of many studies [27][28][29][30][31][32][33][34][35][36]. It is also often explained by the higher stability of rhodium oxides compared to the one of platinum oxides [31,33]. Consequently, it can be inferred that the surface composition determined after oxidation under NO and then CO chemisorption, is characteristic of the surface under oxidizing conditions, whereas the one determined directly after reduction and then CO adsorption will be more in accordance with the surface composition under reducing conditions.

Cyclohexane dehydrogenation and methylcyclopentane ring opening

Catalysts were evaluated in cyclohexane dehydrogenation and methylcyclopentane ringopening. Results are reported in Table 5.

For cyclohexane dehydrogenation, the nature of the metal is a crucial parameter for the catalytic activity [37]. The results of cyclohexane dehydrogenation of the mono and bimetallic catalysts (Table 5) show that the Turn-Over Frequency (TOF) value measured on the platinum catalyst are higher than the one obtained on rhodium, in agreement with results of the literature [2,24,38]. As TOF values for the bimetallic catalysts are between the values of the monometallic Rh and Pt catalysts, this indicates that both metals are at the surface of the metal particles, in accordance with the results obtained by FTIR of CO or NO-CO (Table 3). Taking into account the amount of Pt, alone or alloyed with Rh, as determined directly by CO chemisorption followed by FTIR, and considering that Pt-Rh interaction has a negligible effect on the activity of Pt or Rh atoms for cyclohexane dehydrogenation, TOF values of the bimetallic catalysts can be calculated by linear combination of the TOF values of the corresponding monometallic catalysts modulated by the proportion of Pt and Rh at the particle surface [39].

The calculated TOF values are similar to the experimental ones for (PtRh)/CI and Pt-Rh/OG catalysts, thus validating for these catalysts the surface composition determined by FTIR of probe molecules. On the contrary, for the (PtRh)/MS catalyst the TOF value is higher than the experimental one, 0.51 s -1 against 0.33 s -1 . This may be explained by the much lower metal accessibility, as determined from FTIR results (9.4 μmol g -1 ) compared to the one determined by chemisorption of hydrogen (32.2 μmol g -1 ), which was also lower than the one deduced from TEM pictures. Consequently, the quantitative metal distribution at the surface determined using probe molecules should be considered with caution for the (PtRh)/MS catalyst, for which polymer remains at the metal particle surface, probably disrupting the adsorption capacity of the metallic particles.

Methylcyclopentane selective ring opening is a reaction that was initially studied to characterize the metal function [40]. At low conversion, the formation of ring-opening products (ROP), namely n-hexane (nC6), 2-methylpentane (2MP) and 3-methylpentane (3MP) is favored. Other reactions may occur such as ring enlargement by isomerization, and then the production of cyclohexane that may be dehydrogenated to benzene, and cracking or deep hydrogenolysis, yielding C1 to C5 molecules. When the conversion increases, the selectivity to ROP decreases and among monometallic catalyst, iridium is the most active and selective one, since it allows maintaining high ROP selectivity for high MCP conversions [41] contrary to the other tested metals. Also for high MCP conversions, Rh is more active than Pt but both present low ROP selectivities [2]. Results, displayed in Table 5, obtained with monometallic catalysts, confirm that Rh presents higher C-C breaking activity than Pt [2,42], with a TOF ~3 times higher. The bimetallic catalyst prepared by coimpregnation presents an activity roughly similar to that of the monometallic Rh catalyst. Surprisingly, the Pt-Rh/OG and (PtRh)/MS catalysts present TOF values much superior to those of the monometallic catalysts. High TOF values for bimetallic catalysts, higher than those of the two monometallic catalysts, were already observed [4] for Pt-Rh systems on alumina support , for MCH ring opening at high pressure, but only for very high Rh contents (Pt/Rh ≈ 0.17, corresponding to 0.6 wt.% of Pt and 0.85 wt.% of Rh). For lower Rh contents, TOF values were between those of the two reference catalysts, as for the (PtRh)CI catalyst. It was observed in this study that this latter catalyst presented the weakest interactions between Pt and Rh, with the presence of monometallic particles, as characterized by TPR. As far as formation of Ring-Opening Products (ROP) is concerned, the selectivity to ROP is higher for bimetallic catalysts than for the monometallic ones. The most selective catalysts are Pt-Rh/OG and (PtRh)/MS, the two catalysts for which the presence of Pt-Rh alloy species, favoring high ROP selectivities [4], was demonstrated by CO-FTIR. It is known [43] that the selective ring opening of MCP may occur according to (i) a selective mechanism, leading only to 2-methylpentane (2MP) and 3-methylpentane (3MP), with 2MP/3MP = 2, without n-hexane (nC6) (ii) a statistical mechanism, where the probability of breaking an endocyclic C-C bond is independent of its position, leading to a statistical product distribution, Bimetallic Pt-Rh catalysts supported on alumina were prepared by three different ways, namely coimpregnation (CI), organometallic grafting (OG) and microsuspension (MS), these latter two methods favoring the formation of bimetallic entities. The better dispersion associated with the weakest Pt-Rh interaction was obtained with the catalyst prepared by coimpregnation, which is composed by monometallic and bimetallic particles. Lower metal accessibilities were obtained for the two other bimetallic catalysts, but in this case, the presence of strong Pt-Rh interaction was identified by TPR. For the catalyst prepared by the colloidal route, even being carefully washed, some stabilizing polymer remains at the metal particle surface, which slightly hinders chemisorption of molecules such as H2, CO and NO. For this reason, the chemisorption of CO or NO-CO followed by FTIR gives only partial information on the surface composition for this catalyst. However, under CO, the presence of bimetallic entities, corresponding to Pt alloyed with rhodium, i.e. surrounded by Rh, was evidenced. These Pt-Rh entities are also present in the bimetallic catalyst prepared by organometallic grafting. Two conclusions can be drawn from the results for cyclohexane dehydrogenation. In one hand, Pt and Rh interaction does not play a role in the activity of the catalysts for this reaction, as expected. On the other hand, the calculated TOF is in agreement with the experimental one for the CI and OG catalysts, validating the FTIR results. This calculated TOF is obtained from amount of surface Pt and Rh atoms determined by FTIR and based on the individual performances from monometallic catalysts. The performances of the catalysts for MCP ring-opening showed that, in agreement with previous results, the presence of strong Pt-Rh interactions, evidenced at the surface of bimetallic particles for the OG and MS catalysts, favors high selectivity to ring-opening products, and in addition, increases the TOF value compared to the one of the monometallic catalysts. In conclusion, controlled preparation of bimetallic catalysts by organometallic grafting or microsuspension leads to strong Rh-Pt interactions, which favor high activity and selectivity for methylcyclopentane ring-opening. a not determined by CO-FTIR b determined by difference between the total amount of metals at the surface and the amount of Pt or alloyed metal calculated from CO-FTIR spectra. c)

Catalyst

with 2 /

 2 5 of nC6, 2/5 of 2MP and 1/5 of 3 MP, and (iii) a partly selective mechanism, intermediate between the two previous ones. In the present study, a selective RO reaction occurs only on Pt-Rh/OG and (PtRh)/MS, for which the formation of nC6 is negligible and 2MP/3MP ≈ 2. Monometallic Pt catalyst opens MCP via a statistical mechanism, according to the results of the literature [4,44,45], whereas Rh monometallic catalyst and the bimetallic CI catalyst are partly selective. In conclusion, the formation of bimetallic Pt-Rh particles with very strong interactions, forming Pt-Rh alloys as identified by CO-FTIR during the preparation by organometallic grafting and microsuspension, leads to very selective catalysts for MCP ringopening.

Fig. 1 :Fig. 2 :Fig. 3 :

 123 Fig. 1: TEM pictures and size distribution of the Al2O3 supported bimetallic PtRh catalysts a) (PtRh)/CI

Fig. 4 :Fig. 6 :Fig. 7 :Fig. 8 :

 4678 Fig. 4 : XRD patterns of metallic particles before deposition on the Al2O3 support during the preparation of Pt-Rh bimetallic particles in microsuspension.

Fig. 9 :

 9 Fig. 9: Infrared spectra of the Al2O3 supported bimetallic PtRh catalysts (= ) after NO

Table 1 :

 1 Properties of monometallic and bimetallic catalysts: metallic content, particles size and metallic accessibility.

	Catalyst	Metal content (wt.%)	H/M	<d> a (nm)	<d> b (nm)
		Pt	Rh
	Pt	1.1	0	0.72	1.3	-
	Rh	0	0.6	0.64	1.4	-
	(PtRh)/CI	0.9	0.4	0.71	1.3	1.3
	Pt-Rh/OG	1.0	0.6	0.29	3.1	3.0
	(PtRh)/MS	1.2	0.6	0.27	3.4	2.6

a Calculated from H2 chemisorption results (with H/M = 1) b Calculated from TEM pictures

Table 3 :

 3 Metallic surface composition of the Al2O3 supported bimetallic PtRh catalysts determined by NO-CO/FTIR, CO/FTIR.

Table 4 :

 4 Rh-NO, Pt-NO and Al2O3-NO species and their associated wavenumber in IR spectroscopy

	22,24

Positions Dimick et al 24 Lévy et al 22

	1400-1650 cm -1	No-linear NO on Pt	Pt 0 -NO
		Species	Structure 24
	1230 et 1310 cm -1	bidentate bridged nitrite
			N(AlO)2
	1440 cm -1	monodentate nitrite	(AlO)NO
	1530 cm -1	coordinate nitrite
			(AlO)NOAl
	1585 cm -1	bidentate chelated nitrate
			(AlO2)NO
	1650 -1660 cm -1	bidentate bridged nitrate		Rh 0 -NO
			(AlO)2NO
	1695 et 1770 cm -1	NO anionic on Rh	Rh 0 -NO -	Rh(NO)2
	1800 -1830 cm -1	neutral NO on Rh	Rh 0 -NO	Rh(NO)2
	1905 -1910 cm -1	nitrosyl Rh	Rh I -NO  andRh + -NO 	Rh I -NO  
	Not observed	dinitrosyl Rh	Rh(NO)2
	1750 and 1760 cm -1	Linear NO on Pt	Pt 0 -NO

Table 5 :

 5 Results of cyclohexane dehydrogenation (CH) and methylcyclopentane ring-opening (MCP) obtained with mono and bimetallic catalysts and selectivity in ring-opening (RO) products and in various RO products (2MP, 3MP, nC6). : based on CO-FTIR results on Pt and Rh atomic surface and assuming a linear combination of the TOF values of the corresponding monometallic catalysts modulated by the proportion of Pt and Rh at the particle surface.

	Catalyst	Experimental	Calculated a TOF	TOF for	Selectivity (%)
		TOF for CH	for CH	MCP	RO 2MP 3MP nC6
		(s -1 )
			(s -1 )	(h -1 )
	Pt	0.60	-	27	78 34	15 29
	Rh	0.17	-	72	61 30	18 13
	(PtRh)/CI	0.30	0.31	68	88 46	29 13
	Pt-Rh/OG	0.53	0.48	228	93 56	28	9
	(PtRh)/MS	0.33	0.51	179	93 63	27	3

a

Acknowledgments

The authors gratefully acknowledge the ECOS-MINCyT program n• A12E02 (2013)(2014)(2015) which allowed the collaboration between the two research teams. N.H. acknowledges the Region Poitou-Charentes for its financial support.


Stefan M Moser

Ligong Wang

Michèle Wigger

Asymptotic High-SNR Capacity of MISO Optical Intensity Channels

This paper derives the asymptotic capacity for the multiple-input single-output free-space optical intensity channel in the regime of high signal-to-noise ratio (SNR). The asymptotic result is proven via upper and lower bounds on capacity at finite SNR.

I. INTRODUCTION

Optical wireless communication is a form of communication in which visible, infrared, or ultraviolet light is transmitted in free space (air or vacuum) to carry a message to its destination. Recent works suggest that it is a promising solution to replacing some of the existing radio-frequency (RF) wireless communication systems in order to prevent future rate bottlenecks [START_REF] Kahn | Wireless infrared communications[END_REF]- [START_REF] Khalighi | Survey on free space optical communication: A communication theory perspective[END_REF]. Particularly attractive are simple intensity-modulation-direct-detection (IM-DD) systems. In such a system, the transmitter modulates the intensity of optical signals coming from light emitting diodes (LEDs) or laser diodes (LDs), and the receiver measures incoming optical intensities by means of photodetectors. The electrical output signals of the photodetectors are essentially proportional to the incoming optical intensities, but are corrupted by thermal noise of the photodetectors, relative-intensity noise of random intensity fluctuations inherent to low-cost LEDs and LDs, and shot noise caused by ambient light. In a first approximation, noise coming from these sources is usually modeled as being additive Gaussian and independent of the transmitted light signal; see [START_REF] Kahn | Wireless infrared communications[END_REF], [START_REF] Hranilovic | Wireless Optical Communication Systems[END_REF].

The free-space optical intensity channel has been extensively studied in the literature. In the single-input singleoutput (SISO) scenario, where the transmitter employs a single transmit LED or LD, and the receiver a single photodetector, the works [START_REF] Wigger | Bounds on the capacity of free-space optical intensity channels[END_REF], [START_REF] Lapidoth | On the capacity of free-space optical intensity channels[END_REF] established upper and lower bounds on the capacity of this channel that are asymptotically tight in both high-signal-to-noise-ratio (high-SNR) and low-SNR limits. Improved bounds at finite SNR have subsequently been presented in [START_REF] Mckellips | Simple tight bounds on capacity for the peak-limited discrete-time channel[END_REF]- [START_REF] Thangaraj | Capacity bounds for discrete-time, amplitude-constrained, additive white Gaussian noise channels[END_REF]. For the multiple-input and multipleoutput (MIMO) optical intensity channel, where the transmitter is equipped with multiple LEDs or LDs, and the receiver with multiple photodetectors, the recent work [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF] has derived several asymptotic capacity results in the high-SNR limit; see details below. Previous to [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF], various code constructions for this setup have been proposed in [START_REF] Haas | Space-time codes for wireless optical communications[END_REF]- [START_REF] Mroueh | Quadratic extension field codes for free space optical intensity communications[END_REF]. When there is no crosstalk so the MIMO channel can be modeled through a diagonal channel matrix, bounds on capacity were presented in [START_REF] Thangaraj | Capacity bounds for discrete-time, amplitude-constrained, additive white Gaussian noise channels[END_REF], [START_REF] Chaaban | Fundamental limits of parallel optical wireless channels: capacity results and outage formulation[END_REF].

The main contributions of [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF] are the asymptotic high-SNR capacity for the MIMO channel in the following cases:

• The channel matrix is of full column rank, i.e., its rank equals the number of transmit LEDs, and the inputs are subject to any peak-and average-power constraints; • The channel is multiple-input and single-output (MISO), and the inputs are subject to only a peak-power constraint, or only an average-power constraint, or both constraints but with the average-power constraint being sufficiently loose.

In the current work, we consider the MISO channel in the regime where there are both average-and peak-power constraints, and where the average-power constraint is not "sufficiently loose." Together with the MISO results in [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF], this completely characterizes the asymptotic high-SNR capacity of any MISO channel for all parameter ranges.

The basic tools for proving our new result are similar to those used in [START_REF] Lapidoth | On the capacity of free-space optical intensity channels[END_REF], [START_REF] Thangaraj | Capacity bounds for discrete-time, amplitude-constrained, additive white Gaussian noise channels[END_REF], [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF]: our capacity lower bound is derived using the Entropy Power Inequality (EPI), and our upper bound is based on duality [START_REF] Lapidoth | Capacity bounds via duality with applications to multiple-antenna systems on flat fading channels[END_REF]. However, the proofs are much more involved. On the one hand, the optimal input distribution for the lower bound turns out to be more complex than those in [START_REF] Lapidoth | On the capacity of free-space optical intensity channels[END_REF], [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF]: it involves LED cooperation (compared to independent signaling in the MIMO full-columnrank case [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF]), and, with certain probabilities, assigns to each LED a truncated exponential distribution (compared to either exponential or uniform distribution in the special MISO cases in [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF]), whose parameters must be optimized. On the other hand, the output distribution chosen in the duality-based upper bound depends on the capacity-achieving input distribution, and its analysis is based on a new lemma that bounds the capacity-achieving probability measure on a given interval.

II. CHANNEL MODEL AND MAIN RESULT

A. Channel Model

Consider a communication link where the transmitter is equipped with n T LEDs (or LDs) and the receiver with a single photodetector. The photodetector receives a superposition of the signals sent by the LEDs, and we assume that the crosstalk between LEDs is constant. Hence, the channel output is given by

Y = h T x + Z, (1)
where the n T -vector x denotes the channel input, whose entries are proportional to the optical intensities of the corresponding LEDs, and are therefore nonnegative:

x i ∈ R + 0 , i = 1, . . . , n T ; (2)
where the length-n T row vector h T is the constant channel state vector with nonnegative entries, which, without loss of generality, we assume to be ordered:

h 1 ≥ h 2 ≥ • • • ≥ h nT > 0; (3)
and where Z ∼ N 0, σ 2 is additive Gaussian noise that is independent of X. Note that, in contrast to the input x, the output Y can be negative.

Inputs are subject to a peak-power (peak-intensity) and an average-power (average-intensity) constraint:

Pr X i > A = 0, ∀ i ∈ {1, . . . , n T }, (4)
nT i=1

E X i ≤ E, (5)
for some fixed parameters A, E > 0. Note that the averagepower constraint is on the expectation of the channel input and not on its square. Also note that A describes the maximum power of each single LED, while E describes the allowed average total power of all LEDs together. We denote the ratio between the allowed average power and the allowed peak power by α:

α E A , (6)
where 0 < α ≤ n T . For α = n T the average-power constraint is inactive in the sense that it is automatically satisfied whenever the peak-power constraint is satisfied. Thus, α = n T corresponds to the case with only a peak-power constraint. On the other hand, α 1 corresponds to having a dominant average-power constraint and only a very weak peak-power constraint.

We denote the capacity of the channel (1) with allowed peak power A and allowed average power E by C h T ,σ 2 (A, E). The capacity is given by [START_REF] Shannon | A mathematical theory of communication[END_REF]

C h T ,σ 2 (A, E) = sup Q I(Q, W ) (7)
where the supremum is over all laws Q on X satisfying (2), (4), and (5). When only an average-power constraint is imposed, capacity is denoted by C h T ,σ 2 (E). It is given as in [START_REF] Farid | Channel capacity and non-uniform signalling for free-space optical intensity channels[END_REF] except that the supremum is taken over all laws Q on X satisfying (2) and ( 5).

B. Previous Results

Denote

X h T X. (8)
Because X --X --Y form a Markov chain, and because X is a function of X, we have [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF] I(X; Y ) = I( X; Y ).

Hence the MISO channel ( 1) is equivalent to the SISO channel whose input is X, while the constraints ( 4) and ( 5) on X are transformed to a set of admissible distributions for X.

Define the following quantities:

h sum nT i=1 h i ( 10
)
α th 1 2 + 1 h sum nT k=1 h k (k -1). (11)
In the case where there is no peak-power constraint, or where α ≥ α th , the asymptotic high-SNR capacity of the above MISO channel is given by [10, Cor. 7 & Thm. 8]:

Theorem 1 ([10]): When α ≥ α th , lim A→∞ C h T ,σ 2 (A, αA) -log A = 1 2 log h 2 sum 2πeσ 2 . ( 12
)
Without a peak-power constraint,

lim E→∞ C h T ,σ 2 (E) -log E = 1 2 log eh 2 1 2πσ 2 . (13)

C. Main Result

The main result of this paper characterizes the asymptotic high-SNR capacity of the MISO channel in the regime that is not covered by Theorem 1.

Theorem 2: If α < α th , then

lim A→∞ C h T ,σ 2 (A, αA) -log A = 1 2 log h 2 sum 2πeσ 2 - inf λ∈(0,min{ 1 2 ,α}) log µ(λ) 1 -e -µ(λ) -1 + µ(λ) e -µ(λ) 1 -e -µ(λ) + inf p : n T k=1 p k (k-1)=α-λ D p h h sum , (14)
where p denotes a probability vector, i.e., p 1 , . . . , p nT are nonnegative and sum to one; where D(p q) denotes relative entropy between the probability vectors p and q:

D(p q) nT k=1 p k log p k q k ; (15)
and where µ(λ) is the unique positive solution for µ to

1 µ - e -µ 1 -e -µ = λ. ( 16
)
Theorem 2 is proven via lower and upper bounds on the capacity. The proof will be outlined in Section III. Before doing so, we make some remarks about this result. Optimization in [START_REF] Mroueh | Quadratic extension field codes for free space optical intensity communications[END_REF]. For a fixed λ, the optimal choice for p = (p 1 , . . . , p k ) is (see, e.g., [START_REF] Cover | Elements of Information Theory[END_REF]Problem 12.2])

p * k = h k a k nT j=1 h j a j , k = 1, . . . , n T , (17)
where a ≥ 0 is obtained by solving the following equation:

nT k=1 h k ka k nT j=1 h j a j = α -λ + 1. (18)
The optimization over λ can easily be performed numerically.

Asymptotically Optimal Input Distributions. As our capacity lower bound will show, in the high-SNR limit, the optimal strategy in terms of capacity is to maximize the differential entropy h( X). One can show that the optimal distribution for X in this sense is given by the following: with probability p k , k ∈ {1, . . . , n T },

X j = A w.p. 1, j ∈ {1, . . . , k -1}, (19a) X k ∼ truncated exponential of parameter µ, (19b) X j = 0 w.p. 1, j ∈ {k + 1, . . . , n T }, (19c)
where p and µ are the optimal choices of p and µ(λ), respectively, in [START_REF] Mroueh | Quadratic extension field codes for free space optical intensity communications[END_REF]. Note that, when α ≥ α th , [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF] shows that the truncated exponential distribution in (19b) should be replaced by a uniform distribution, and the optimal p is given by p k = h k /h sum , k ∈ {1, . . . , n T }. These choices yield a uniform distribution on [0, h sum A] for X. When α < α th , it is no longer possible to achieve a uniform distribution for X without violating the average-power constraint [START_REF] Lapidoth | On the capacity of free-space optical intensity channels[END_REF]. Instead, the optimal distribution for X is a concatenation of n T truncated exponentials.

An Example. Consider a MISO channel with two transmit LEDs and with channel parameters h 1 = 3 and h 2 = 1. We plot the outer infimum in [START_REF] Mroueh | Quadratic extension field codes for free space optical intensity communications[END_REF] against α in Fig. 1. Note that this infimum characterizes the capacity gap to the case with no average-power constraint, in the high-SNR limit. As expected, the gap becomes zero when α reaches α th = 0.75, and approaches infinity when α tends to zero.

III. PROOF OUTLINE A. Achievability: Lower Bound

We prove the following lower bound on capacity, from which the achievability part of Theorem 2 follows by optimization over the parameters.

Proposition 3: Assume α < α th . For any probability vector p such that

0 < α - nT k=1 p k (k -1) < 1 2 (20)
we have

C h T ,σ 2 (A, αA) ≥ log A + 1 2 log h 2 sum 2πeσ 2 -log µ * 1 -e -µ * + 1 - µ * e -µ * 1 -e -µ * -D p h h sum ( 21
)
where µ * is the unique positive solution to

1 µ * - e -µ * 1 -e -µ * = α - nT k=1 p k (k -1). ( 22
)
Proof: Choosing a particular input distribution (specified below) and using the Entropy Power Inequality (EPI) [START_REF] Cover | Elements of Information Theory[END_REF]Thm. 17.7.3] we obtain

C h T ,σ 2 (A, αA) ≥ I( X; X + Z) (23) = h( X + Z) -h(Z) (24) ≥ 1 2 log e 2 h( X) + e 2 h(Z) -h(Z) (25) = h( X) - 1 2 log 2πeσ 2 + 1 2 log 1 + 2πeσ 2 e 2 h( X) (26) ≥ h( X) - 1 2 log 2πeσ 2 . ( 27
)
Now we fix some probabilities p 1 , . . . , p nT summing to one and satisfying (20), and we define a random variable (RV) U ∈ {1, . . . , n T } with Pr[U = k] = p k . Conditional on U = k, we choose

X j cond. on U = k = A if j = 1, . . . , k -1, 0 if j = k + 1, . . . , n T , (28)
and we choose X k to be distributed according to the law

X k cond. on U = k ∼ 1 A µ * 1 -e -µ * e -µ * x A I{0 < x ≤ A}, ( 29
)
where µ * is the unique positive solution to (22). Note that

E   nT j=1 X j   = nT k=1 p k nT j=1 E[X j |U = k] (30) = nT k=1 p k (k -1)A + 1 µ * - e -µ * 1 -e -µ * A (31) = αA, (32)
where the last equality holds because of (22). Thus, this choice of input satisfies both (4) and [START_REF] Lapidoth | On the capacity of free-space optical intensity channels[END_REF]. Also note that conditional on U , only one antenna has a random input, while the others are deterministic. Therefore it is possible to deduce from X the value of U :

U = k ⇐⇒ X ∈ k-1 j=1 h j , k j=1 h j . (33)
Thus,

h( X) = h( X) -h( X|U ) + h( X|U ) (34) = I(U ; X) + h( X|U ) (35) = H(U ) -H(U | X) = 0 + h( X|U ) (36) = H(p) + nT k=1 p k h( X|U = k) (37) = H(p) + nT k=1 p k h h k X k U = k (38) = H(p) + nT k=1 p k log h k + log A -log µ * 1 -e -µ * + 1 - µ * e -µ * 1 -e -µ * , (39)
where in (38) we use that conditional on U = k, only X k is random, the other RVs are deterministic and can therefore be subtracted from X.

Combining (39) with ( 27) now proves the claim.

B. Converse: Upper Bound

We first derive the following upper bound on capacity. Proposition 4: For every k ∈ {0, . . . , n T }, define

s k k j=1 h j . (40)
Let p k , k ∈ {2, . . . , n T }, denote the probability that X ∈ (s k-1 A, s k A], and p 1 that X ∈ [0, s 1 A], under the capacityachieving input distribution. For any choice of parameters µ, δ, η ∈ (0, ∞), β ∈ (0, h nT ), and n T -dimensional probability vector q, we have

C h T ,σ 2 (A, αA) ≤ max 1, log 2πeσ 2 • Q δA σ + δA √ 2πσ e -δ 2 A 2 2σ 2 -log 1 -2 Q δA σ -2(n T -1) δA η + log η nT k=1 Q (h k -δ)A σ -Q (h k + δ)A σ + Q (h k+1 -δ)A σ -Q (h k+1 + δ)A σ + 2n T log η • log 2 + 1 e + 1 2 log 1 + β 2 A 2 4σ 2 1 2 log 1 + A 2 2πeσ 2 • Υ(α, δ) + 2 log η • Q (β -δ)A σ -Q (β + δ)A σ + Q δA σ • nT k=1 max 0, log A h k -2δ 1 -2 Q δA σ -2(n T -1) δA η + log A - 1 2 log 2πeσ 2 + Q δA σ nT k=1 µs nT h k -log q k + µσ A √ 2π nT k=1 1 h k + nT k=1 p k log h k q k -log µ + log 1 -e -µ + µ α - nT k=1 p k (k -1) , (41)
where

Υ(α, δ)    e 2αδ 1-e -δ δ 2 if α ∈ 0, 1 2 , 1 if α ≥ 1 2 . ( 42
)
Proof Outline: We apply the duality-based bounding technique [START_REF] Lapidoth | Capacity bounds via duality with applications to multiple-antenna systems on flat fading channels[END_REF]

C h T ,σ 2 (A, αA) ≤ E Q * - ∞ -∞ 1 √ 2πσ e -(y-X) 2 2σ 2 log f (y) dy - 1 2 log 2πeσ 2 , (43)
where Q * is the capacity-achieving input distribution, and f is any probability density function on the real line. Here we define the parameters

s k s k + δ, k = 1, . . . , n T -1; (44) s k s k -δ, k = 1, . . . , n T -1; (45) s 0 s 0 -δ;
(46) s nT s nT + δ;

(47)

µ k µ h k , k = 1, . . . , n T , (48)
and choose

f (y) =                                              1 √ 2πσ e -y 2 2σ 2 if y ≤ s 0 A, q k µ k 1 -2 Q δA σ -2(n T -1) δA η A(e -µ k s k-1 -e -µ k s k ) e -µ k A y if s k-1 A < y ≤ s k A, k ∈ {1, . . . , n T }, 1 η if s k A < y ≤ s k A, k ∈ {1, . . . , n T -1}, 1 √ 2πσ e -(y-sn T ) 2 2σ 2 if y > s nT A. (49)
Note that, on the interval [0, h sum A], our choice of f resembles the asymptotically optimal distribution for X, the latter being a concatenation of truncated exponential distributions. But, for technical reasons, we add "buffers" between neighboring truncated exponentials, as well as Gaussian tails on both sides of the interval. Calculations are omitted due to space limitations.

To use Proposition 4 to derive the desired asymptotic upper bound, we choose

q k = Ap k + 1 A + n T , k ∈ {1, . . . , n T }. (50)
Also, denote λ(p) α-nT k=1 p k (k -1), fix some ζ ∈ (0, 1), and choose

µ =      µ(p) if A -(1-ζ) < λ(p) < 1 2 , A 1-ζ if λ(p) ≤ A -(1-ζ) , 1 A if λ(p) ≥ 1 2 , (51)
where µ(p) is the unique positive solution for µ to

1 µ - e -µ 1 -e -µ = λ(p). (52)
We then choose

δ = log(1 + A) A , β = log 2 (1 + A) A , η = log 2 (1 + A), (53)
and let A → ∞. After some further calculations, which are again omitted, we obtain that the right-hand side of ( 14) is an upper bound on its left-hand side.

IV. CONCLUDING REMARKS

We have shown that, in the high-SNR limit, the capacity of the peak-and average-intensity limited optical intensity channel is given by the maximum value of h(h T X) minus the differential entropy of the Gaussian noise. There may be alternative ways to show this asymptotic result. For example, if one can show that, at high SNR, the optimal input distribution has a bounded density, then one may be able to come to the above conclusion via convergence properties of the differential entropy; see, e.g., [START_REF] Godavarti | Convergence of differential entropies[END_REF]. On the other hand, such an approach, if successful, would not provide finite-SNR bounds on capacity as our Propositions 3 and 4.

We have also identified the input distribution that maximizes h(h T X). This input distribution has the following structure: whenever a weaker transmit LED is switched on, all the stronger transmit LEDs must be transmitting at maximum intensity with probability one. Furthermore, conditional on the transmit signal of a specific LED being "random," this signal has a truncated exponential distribution, with a parameter that is the same for all transmit LEDs.

Recall that [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF] has established the high-SNR asymptotic capacity of the MIMO optical intensity channel when the channel matrix has full column rank. A close look at the results in [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF] confirms that, in the full-column-rank case, the asymptotic capacity is also given by the maximum differential entropy of HX, H being the channel matrix, minus that of the noise vector.

With the current work and [START_REF] Moser | Asymptotic capacity results for MIMO wireless optical communication[END_REF], the only MIMO optical intensity channels whose asymptotic high-SNR capacities are not yet known are those with more than one receive antennas (photodetectors), and with channel matrices that do not have full column rank. It is natural to conjecture that, for those channels, the asymptotic high-SNR capacity is again given by the maximum of h(HX) minus the differential entropy of the noise.

Fig. 1 .

 1 Fig.1. The outer infimum in[START_REF] Mroueh | Quadratic extension field codes for free space optical intensity communications[END_REF] as a function of α, for MISO channel with two transmit LEDs, h 1 = 3 and h 2 = 1. This infimum is the asymptotic capacity gap to the case with no average-power constraint C h T ,σ 2 (A, A).