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Raphaël Paut

Rodolphe Sabatier

Marc Tchamitchian

Modelling crop diversification and association effects in agricultural systems

Keywords: Intercropping, Agroforestry, Diversification, Land Equivalent Ratio, LER, Modern portfolio theory, Agroecology

The need to redesign more sustainable agricultural systems able of producing more, especially through intercropping or agroforestry, cannot be achieved without taking into account the essential aspect of production variability. Yet, although many studies have focused on the effect of intercropping on overall production, the particular issue of production variability in such systems remains relatively unstudied. The approach we propose, for a shift towards sustainable intensification of agricultural systems, considers the dual dimensions of yield and risk in a combined framework for the assessment and the comparison of two diversification strategies: (i) a simple diversification strategy (SDS) considered as an increasing number of crops grown on separate plots within a farm and (ii) an intercropping strategy (IC) considered as a within-plot increased diversity, where more than one species is grown at the same time and place. The two perspectives examined here were Modern Portfolio Theory and Land Equivalent Ratio. The former quantifies the effect of diversification on risk, the latter measures the effect of association on production. This research merges both approaches in a combined framework in order to assess intercropping system performances. By applying our framework to cases selected from the literature, we explored and compared the potential benefits of these two strategies in terms of yield and risk. Results showed that intercropping, in addition to being interesting with regard to yield, can have an additional risk reduction effect compared to a simple diversification strategy. Conversely, some crop mixtures maintained or even increased yield variability. Our work contributes to a better understanding of the possible impacts of diversification strategies on trade-offs between yield and risk, but also underlines the importance of taking yield variability into account in further studies.

Introduction

Agricultural systems are facing a productive challenge, the necessity to ensure a sufficient quantity of goods for a growing population [START_REF] Fischer | Land sparing versus land sharing: moving forward[END_REF][START_REF] Tilman | Global food demand and the sustainable intensification of agriculture[END_REF][START_REF] Tscharntke | Global food security, biodiversity conservation and the future of agricultural intensification[END_REF]. However, this productivity issue cannot be decoupled, even more so today, from the necessity of stability of this provision service [START_REF] Hardaker | Coping With Risk in Agriculture: Applied Decision Analysis[END_REF]OECD, 2011). Indeed, weather and climate being significant drivers of agricultural production systems, food security seems doomed to deteriorate on a global scale, despite advances in technology and other fronts [START_REF] Matiu | Interactions between temperature and drought in global and regional crop yield variability during 1961-2014[END_REF][START_REF] Osborne | Evidence for a climate signal in trends of global crop yield variability over the past 50 years[END_REF][START_REF] Ray | Climate variation explains a third of global crop yield variability[END_REF]. In addition, the ecologization of agriculture limits the pool of management options available for farmers or their efficacy to control system variability, indicating the importance of finding farm-level responses [START_REF] Hlpe | Agroecological and Other Innovative Approaches for Sustainable Agriculture and Food Systems That Enhance Food Security and Nutrition. A Report by the High Level Panel of Experts on Food Security and Nutrition of the Committee on World Food Security[END_REF][START_REF] Reidsma | Adaptation to climate change and climate variability in European agriculture: the importance of farm level responses[END_REF].

The dual productivity-stability issue therefore seems essential for designing and evaluating sustainable agricultural systems in a situation of increasing uncertainties [START_REF] Muller | Strategies for feeding the world more sustainably with organic agriculture[END_REF][START_REF] Urruty | Stability, robustness, vulnerability and resilience of agricultural systems. A review[END_REF]. Production stability depends on scale in nested systems. Although greater stability through crop diversity can be achieved at a national level [START_REF] Renard | National food production stabilized by crop diversity[END_REF] it seems that farm-scale is a crucial level to achieve this property [START_REF] Reidsma | Analysis of farm performance in Europe under different climatic and management conditions to improve understanding of adaptive capacity[END_REF]. In this context, new farming organizations have emerged in the last decades, creating diversified systems based on the agroecological paradigm [START_REF] Krebs | Permaculture-scientific evidence of principles for the agroecological design of farming systems[END_REF][START_REF] Wezel | Agroecological practices for sustainable agriculture[END_REF][START_REF] Wezel | Agroecology as a science, a movement and a practice[END_REF]. Among these diversified systems, we can identify two diversification pathways. The first one, called Simple Diversification Strategy (SDS) corresponds to an increased cultivated diversity, that is to say, an increased number of species grown within a farm [START_REF] Malézieux | Review article mixing plant species in cropping systems : concepts, tools and models[END_REF][START_REF] Ratnadass | Plant species diversity for sustainable management of crop pests and diseases in agroecosystems: a review[END_REF]. Because different crops are grown in different fields, (i.e., spatial diversification) biological interactions between crops do not occur. The other diversification strategy is called intercropping (IC) and refers to the spatial association of crops on the same plot and involves a diversity of ecological interactions between crops [START_REF] Lovell | Temperate agroforestry research: considering multifunctional woody polycultures and the design of longterm field trials[END_REF][START_REF] Torquebiau | A renewed perspective on agroforestry concepts and classification[END_REF][START_REF] Vandermeer | The Ecology of Intercropping, Cambridge Core[END_REF]. In the case of tree-crop interactions, this system is also referred to as agroforestry.

In contrast to monocultures, both strategies lead to a more diversified production system, but while SDS relies on diversification to reduce risks, IC relies also on spatial association to benefits from the biological interactions between crops. Therefore, these two diversification strategies rely on two stacked paradigms in managing the balance between risk and yield, and a comparison of both approaches regarding risk-yield trade-offs is still lacking.

A relevant tool to address this issue is the Modern Portfolio Theory (MPT). The MPT has been initially developed in finance [START_REF] Markowitz | Portfolio theory: As I still see it[END_REF][START_REF] Markowitz | Portfolio selection[END_REF] and recently used in agricultural sciences to select relevant crop combinations in a portfolio under uncertainty [START_REF] Barkley | Wheat variety selection to maximize returns and minimize risk: an application of portfolio theory[END_REF][START_REF] Fraser | A framework for assessing the vulnerability of food systems to future shocks[END_REF][START_REF] Nalley | Using portfolio theory to enhance wheat yield stability in low-income nations: an application in the Yaqui Valley of Northwestern Mexico[END_REF][START_REF] Paut | Reducing risk through crop diversification: an application of portfolio theory to diversified horticultural systems[END_REF][START_REF] Van Noordwijk | Risk management in crop production and fertilizer use with uncertain rainfall; how many eggs in which baskets?[END_REF]. The central point of this theory is based on the fact that if productions of the two crops are not perfectly correlated with each other, combining these crops in a portfolio, which corresponds to a diversified land use, makes it possible to reduce the risk associated with crop production. Specifically, there may be large potential gains from combining varieties or species that are characterized by inverse yield responses to environmental fluctuations such as droughts, pests or diseases.

Applied to agricultural sciences, MPT makes it possible to define the best crop combinations when the expected yields, the individual risks of each crop, as well as the correlations between crops are known [START_REF] Knoke | Optimizing agricultural land-use portfolios with scarce data-A non-stochastic model[END_REF][START_REF] Paut | Reducing risk through crop diversification: an application of portfolio theory to diversified horticultural systems[END_REF]. Although other definitions of risk exist in agricultural studies the MPT considers risk as the unpredictability of yields (i.e. standard deviation of yield) rather than the probability of not reaching a stated goal [START_REF] Van Noordwijk | Reduction of risk by diversity[END_REF][START_REF] Van Winsen | Cognitive mapping: a method to elucidate and present farmers' risk perception[END_REF]. Thus the overall yield and risk of a crop portfolio, considered as a group of physically separated plots, can be assessed [START_REF] Castro | Ecuadorian banana farms should consider organic banana with low price risks in their land-use portfolios[END_REF][START_REF] Fraser | A framework for assessing the vulnerability of food systems to future shocks[END_REF]. This approach therefore seems to be more appropriate for a simple diversification strategy (SDS) rather than for the selection of intercropping systems (IC), where interactions between crops are sought and can lead to a nonlinear effect on yields [START_REF] Paul | Agroforestry versus farm mosaic systems -Comparing land-use efficiency, economic returns and risks under climate change effects[END_REF].

Besides, intercropping (IC) is another form of diversification in which interactions between crops are prevalent. Crop interactions that may lead to an increasing productivity are often assessed by the socalled Land Equivalent Ratio (LER) (Willey, 1979a). The LER compares the land-use efficiency of crops grown in intercropping systems with sole crops. It is a widespread and useful indicator to assess the benefits of intercropping in terms of production, but it is generally used in a way that does not consider a possible decrease of the yield variability that could result from intercropping [START_REF] Fukai | Intercropping -bases of productivity[END_REF][START_REF] Lithourgidis | Annual intercrops: an alternative pathway for sustainable agriculture[END_REF][START_REF] Rao | Evaluation of yield stability in intercropping: studies on sorghum/pigeonpea[END_REF]).

Yet, from a biological perspective, it seems that mixed systems are better able to buffer environmental variations. Beyond the diversification effect above-mentioned, ecological interactions can generate a combination of stabilizing mechanisms such as complementarity in resource use, i.e. the ability of one component to use resources not needed by another one, which leads to a compensation in the ecosystem functions they provide and promotes stability [START_REF] Bedoussac | Ecological principles underlying the increase of productivity achieved by cereal-grain legume intercrops in organic farming[END_REF][START_REF] Grman | Mechanisms contributing to stability in ecosystem function depend on the environmental context[END_REF][START_REF] Tilman | The ecological consequences of changes in biodiversity: a search for general principles[END_REF]. Facilitation mechanism can also occur when certain species mitigate harsh environmental conditions or provide resources for others [START_REF] Hooper | Effects of biodiversity on ecosystem functioning: a consensus of current knowledge[END_REF][START_REF] Mulder | Physical stress and diversity-productivity relationships: the role of positive interactions[END_REF][START_REF] Vandermeer | The Ecology of Intercropping, Cambridge Core[END_REF]. Such mechanisms obviously cannot occur if the crops are grown separately. However, even though biological diversity has been reported to be important in yield stability, there is still limited experimental evidence of this relationship in the specific case of crops interacting with each other [START_REF] Blandon | Analyzing risk in agroforestry systems using a portfolio approach. A case study from the United Kingdom[END_REF][START_REF] Francis | Biological efficiencies in multiple-cropping systems[END_REF]Willey, 1979b) and some conflicting reports. The early research on intercropping suggested that these systems are likely to be more stable [START_REF] Rao | Productivity and risk evaluation in constrasting intercropping systems[END_REF][START_REF] Vandermeer | Variability, Stability, and Risk in Intercropping: Some Theoretical Explorations[END_REF] but there is still little quantitative information on the extent or the operational significance of this improved performance. In some cases, intercropping systems can have more stable yields over time and can serve as a risk mitigation strategy due to the relative sensitivity of their component crops to temperatures fluctuations, drought or pests and diseases [START_REF] Francis | Biological efficiencies in multiple-cropping systems[END_REF]. Inversely, the production variability can increase due to the existence of shared pests and diseases, e.g. polyphagous insects, fungal diseases [START_REF] Schroth | Pests and diseases in agroforestry systems of the humid tropics[END_REF] or the microclimate created by one plant on the other [START_REF] Jose | Interspecific interations in temperate agroforestry[END_REF][START_REF] Monteith | Microclimatic interactions in agroforestry systems[END_REF]. In the end, in many situations it remains unclear whether increased stability can be achieved effectively and whether this stability is due to the diversification effect or to biological interactions between crops [START_REF] Blandon | Analyzing risk in agroforestry systems using a portfolio approach. A case study from the United Kingdom[END_REF][START_REF] Blandon | Agroforestry and portfolio theory[END_REF][START_REF] Paul | Agroforestry versus farm mosaic systems -Comparing land-use efficiency, economic returns and risks under climate change effects[END_REF][START_REF] Rao | Productivity and risk evaluation in constrasting intercropping systems[END_REF]Rao and Willey,1980).

To summarize, Modern Portfolio Theory and Land Equivalent Ratio are relevant frameworks to analyze different mechanisms associated with diversified cropping systems. MPT makes it possible to formalize the risk reduction effect of crop diversification, but does not, in its standard application, consider ecological interactions between crops neither spatially nor temporally. LER provides a standardized basis to assess mean production of intercropping systems, but does not account for the risk-reduction effect of such diversification (although it could also be applied to any parameter of a statistical distribution rather than the mean, e.g. to an X% percentile). The aim of the present study is therefore to combine these two theoretical approaches in a unified framework to formalize the effect of intercropping on both production and risk (Fig. 1). Fig. 1. Graphical summary of the approach, with μ standing for mean crop production and σ its standard deviation.

We will then apply our framework to horticultural systems, for which the issue of diversification and association is of great relevance. Horticulture, indeed, makes it possible to combine a large number of crops with a wide range of functional traits in both diversification and association configurations. Moreover, there has been a strong development of diversified horticultural systems in recent years, combining an increasing crop diversity [START_REF] Morel | Small can be beautiful for organic market gardens: an exploration of the economic viability of French microfarms using MERLIN[END_REF][START_REF] Morel | A conceptual framework for alternative farmers' strategic choices: the case of French organic market gardening microfarms[END_REF][START_REF] Navarrete | Crop management, labour organization, and marketing: three key issues for improving sustainability in organic vegetable farming[END_REF] with intercropping and agroforestry configurations [START_REF] Léger | Agroforesterie maraîchère : un choix stratégique pour garantir une durabilité en transition agroécologique? Expériences issues du projet SMART[END_REF][START_REF] Léger | Agroforestry Market Gardening: a Strategic Choice to Improve Sustainability in Agroecological Transition?[END_REF][START_REF] Sieffert | Co-design of innovative periurban horticultural agroforestry systems: case study of a pilot farm in the south of France[END_REF]. This creates a fertile ground for the application of our conceptual framework to an applied issue relevant to a growing number of farmers. For the application of our conceptual framework, we conducted a literature review from which we extracted the data necessary for the application of this model.

Material and Method

2.1 Formalizing the diversification effect through the modern portfolio theory [START_REF] Markowitz | Portfolio theory: As I still see it[END_REF][START_REF] Markowitz | Foundations of portfolio theory[END_REF][START_REF] Markowitz | Portfolio selection[END_REF] first introduced the so-called Modern Portfolio Theory (MPT) in finance in order to analyze how diversified portfolios of assets could provide a risk reducing effect, as long as assets return are not perfectly correlated. He defined risk as the portfolio standard deviation. In our case, a portfolio diversification consists in moving from a monoculture to a more diversified agrosystem. According to the MPT, the expected yield of a portfolio P is the weighted sum of each individual asset in the portfolio:

𝐸𝐸(𝑌𝑌 𝑃𝑃 ) = ∑ 𝑤𝑤 𝑖𝑖 𝐸𝐸(𝑌𝑌 𝑖𝑖 ) 𝑖𝑖 = 𝑤𝑤 𝐴𝐴 𝐸𝐸(𝑌𝑌 𝐴𝐴 ) + 𝑤𝑤 𝐵𝐵 𝐸𝐸(𝑌𝑌 𝐵𝐵 ) (1)
where wi is the weight of crop i (that is, the proportion of the cultivated area of crop i), E(Yi) is the expected yield of crop i and the sum of wA and wB is equal to 1. Notice that in this view, the yield associated with each asset is independent from the composition of the portfolio.

Furthermore, the risk of a portfolio is measured by its standard deviation. The risk of the system composed of two crops (Crop A and Crop B), σP, is defined as follows:

𝜎𝜎 𝑃𝑃 = � ∑ ∑ 𝑤𝑤 𝑖𝑖 𝑤𝑤 𝑗𝑗 𝜎𝜎 𝑖𝑖𝑗𝑗 𝑗𝑗 𝑖𝑖 = �𝑤𝑤 𝐴𝐴 2 𝜎𝜎 𝐴𝐴 2 + 𝑤𝑤 𝐵𝐵 2 𝜎𝜎 𝐵𝐵 2 + 2𝑤𝑤 𝐴𝐴 𝑤𝑤 𝐵𝐵 𝜎𝜎 𝐴𝐴 𝜎𝜎 𝐵𝐵 𝜌𝜌 𝐴𝐴𝐵𝐵 (2)
where σi is the standard deviation of yield for crop i, ρAB is the correlation coefficient of crop A and crop B. Eq. ( 2) shows that the standard deviation of the portfolio (σP) is the sum of two components: a variance term and a covariance term. It follows that the variability of a portfolio can be reduced in two ways, either by favoring crops with low yield variances, or by choosing crops with a low coefficient of correlation ρ. The effect of different values of ρ is illustrated in Fig. 2.

Formalizing the association effect through the Land Equivalent Ratio

When two crops are cultivated simultaneously on the same plot, interactions between these two crops may lead to an overall production different from the weighted sum of the production of each crop cultivated in monoculture. This association effect is classically assessed in the literature by the so-called Land Equivalent Ratio (LER) [START_REF] Bedoussac | A comparison of commonly used indices for evaluating species interactions and intercrop efficiency: application to durum wheat-winter pea intercrops[END_REF][START_REF] Mead | The concept of a 'Land equivalent ratio' and advantages in yields from intercropping[END_REF][START_REF] Vandermeer | The Ecology of Intercropping, Cambridge Core[END_REF]Willey, 1979a). LER represents the relative area needed for sole crops to produce the same yields as those obtained by an intercropping design. For a given proportion (k) of crop A in the association with crop B, the value of the LERk it is calculated as the yields of each individual crop in relation to their yield as sole crop (Eq. ( 3)).

𝐿𝐿𝐸𝐸𝐿𝐿 𝑘𝑘 = 𝐿𝐿𝐸𝐸𝐿𝐿 𝐴𝐴 𝑘𝑘 + 𝐿𝐿𝐸𝐸𝐿𝐿 𝐵𝐵 1-𝑘𝑘 = 𝑌𝑌 𝐴𝐴 𝑘𝑘 𝑆𝑆 𝐴𝐴 + 𝑌𝑌 𝐵𝐵 1-𝑘𝑘 𝑆𝑆 𝐵𝐵 (3)
Where Yk i is the yield of crop i when grown as an intercrop at the proportion of k, Si is the yield of crop i when grown as a sole crop. An LER greater than 1 means that the intercropping system mixing crop A and crop B produces more than its respective monoculture for the same cultivated area. The value 1 is the critical threshold, above which the intercrop is to be favored, and below which the monocultures may be preferred. Following [START_REF] Vandermeer | The Ecology of Intercropping, Cambridge Core[END_REF], we consider only three situations where the LER continuum might be convex (competition), concave (facilitation) or linear (no interaction), presented in Fig. 3.

Proposition of a new conceptual framework

As discussed above, despite an emerging consensus on the fact that intercropping can decrease the overall yield variability, this effect has rarely been quantified and no study clearly disentangled diversification effect from association effect on individual crop variability in horticulture (but see [START_REF] Van Noordwijk | Risk management in crop production and fertilizer use with uncertain rainfall; how many eggs in which baskets?[END_REF] for other crops). For the application of our new conceptual framework, this leads us to recompose Eqs. ( 1) and (2) into Eqs. ( 4) and ( 5)

respectively. Thus, the intercropping portfolio yield E (YAB) is considered as the weighted average of each crop yield multiplied by its relative yield in intercropping design:

𝐸𝐸(𝑌𝑌 𝐴𝐴𝐵𝐵 ) 𝑘𝑘 = 𝐿𝐿𝐸𝐸𝐿𝐿 𝐴𝐴 𝑘𝑘 𝑤𝑤 𝐴𝐴 𝐸𝐸(𝑌𝑌 𝐴𝐴 ) + 𝐿𝐿𝐸𝐸𝐿𝐿 𝐵𝐵 1-𝑘𝑘 𝑤𝑤 𝐵𝐵 𝐸𝐸(𝑌𝑌 𝐵𝐵 ) (4)
where E(Yi) k is the expected yield of crop i and LERA k is the partial LER of crop A at the proportion of k in the portfolio.

Besides, the risk of the intercropping portfolio is defined by Eq. (5) as the risk of a two crop portfolio multiplied by two factors. First, as for the yield (Eq. ( 4)), yield variability is subject to an affine transformation and is multiplied by the LER corresponding coefficient at the

Considering different effects of intercropping on risks: three hypotheses

Yield variability of each crop may be affected by the intercropping design, positively or negatively depending on the choice of intercrops and their interactions. In this study, we will successively consider three contrasted hypotheses representing different effects (β values) of intercropping on yield variability. First, the intercropping design decreases the variability of crops yields (i.e. when β < 1) compared to a simple diversification strategy (SDS), the portfolio standard deviation σP is reduced. This hypothesis integrates an additional risk reduction effect of crop association, compared to SDS. The second hypothesis states that the variability of each crop remains unchanged in intercropping compared to that of SDS. In other words, we assume that there is no biological effect of IC on yield variability of each crop. In this case, the β coefficient is therefore set at 1. Third, the intercropping design has a negative impact on crop yield stability (i.e. when β > 1), the portfolio standard deviation σP would be increased.

Exploration of contrasted scenarios considering our hypotheses

In order to screen different configurations, we developed scenarios that illustrate possible patterns in a contrasting way. Within each scenario, we considered the combination of three effects: (i) the effect of crop diversification on risk (ρAB); (ii) the effect of crop association on productivity (LER) and (iii) the effect of crop association on risk (β). Depending on how crops interact with each other, the LER can take any positive value, but in most cases, it lies between 0.5 and 1.5. To show contrasting situations, we selected three LER values that represent three distinct biological interactions: competition (LER = 0.5), independence (LER = 1) and complementarities (LER = 1.5). Similarly, the correlation coefficient between crop A and crop B can take any value between -1 and 1. As an extreme situation, if yields are perfectly correlated, ρAB will be equal to 1. At the other extreme will be a situation where ρAB is equal to -1, between these two bounds will be a continuum of cases where the yields of the two crops are more or less closely related. In agroforestry systems it may be a common situation where the yields of crops and trees are not correlated [START_REF] Blandon | Agroforestry and portfolio theory[END_REF], in this case ρAB will be zero. Finally, considering our working hypotheses (Cf. supra), the association effect on risk (β) was set at 0.5, 1 and 1.5, In order to illustrate our approach, we applied our framework to real situations of intercropping. A literature review was carried out on horticultural intercropping systems.

Studies were identified from a search in the Institute for Scientific Information Web of Knowledge and Google Scholar databases. We searched for the following terms: (intercrop* OR agroforestry OR mixed crop*) AND (fruit* OR orchard OR vegetable* OR market gardening).

Among the 1,387 publications that came out of this query, we selected 74 studies that provided the data necessary to compute LER values. The reviewed papers involved vegetables and/or fruits in intercropping or agroforestry designs. It resulted in 534 experiments (an experiment is defined by a unique crop combination in site and year) from which 449 were complete, meaning that both crop yield as sole crops and intercrops were available. From each experiment, we also extracted and coded crop species, relative crop density, LER, location and other useful information from each treatment. The list of reviewed papers is provided in Appendix B.

Simulations

Among the 74 studies reviewed for LER values, our model combining MPT and LER approaches was applied to all studies for which there were inter-annual repetitions of yield measurements.

This resulted in 41 studies for which sufficient data were available: the yield of each crop E (Yi) is the mean yield of inter-annual repetitions; the standard deviation of each crop σi is calculated as the inter-annual variability of yields; the Land Equivalent Ratio LERi k is the LER of crop i at the proportion k; the weight of each crop in the portfolio wk is the proportion of crops in the intercropping design and the correlation coefficient ρAB is the correlation between crop yields through time.

Once E (Yi) and σi were computed for each intercropping design, we simulated the yield and risk combinations for 100 proportions of crop A and crop B (from wA =0 to wA = 1) considering our two hypotheses.

First, assuming intercropping had no effect on risk, Eqs. (4) and ( 5) were applied to calculate yield and risk respectively. Second, assuming intercropping had an effect on risk, we used raw data from the paper to obtain the coefficient β by solving Eq. (5bis). Finally, intercropping systems were clustered using a k-means clustering on the three model output parameters (LER, β, ρ). Clusters were projected on the three dimensions and a Tukey's HSD was used to identify differences in cluster parameters. The model was implemented and graphical outputs were realized with R software 3.5.0 (https://cran.r-project.org).

Results

Yield and risk trade-offs: comparing SDS and IC

For the 27 possible combinations of Land Equivalent Ratio (LER), beta coefficient (β) and correlation coefficient (ρ), the yield-risk relationship curves were plotted, based on Eqs. ( 4) and

(6). Points named Crop A and Crop B represent the yield and risk of sole crops (wA = 1 and wB = 1 respectively). The transition from Crop A to Crop B corresponds to the consideration of systems with an increasing area of crop B. Fig. 5 shows the possible scenarios for 27 alternative cropping systems. Moving from scenario 1 to 3 corresponds to an increasing association effect on yield, whereas moving from A to C corresponds to taking the different hypotheses of intercropping effect on risk into account. Notice that the line joining crop A to crop B in the scenario B2 also represents the weighted average of crop yields and risks. It can serve as a basis for the assessment of land-use alternatives on yield and risk dimensions. In confronting the different diversification alternatives, we can observe that some options perform better than sole crops on both dimensions (A3, B3). Some other options are only of interest in one of the two dimensions (A1, A2, B2, C2 and C3), which illustrate that intercropping can lead to a trade-off between the different expected benefits. Finally, some combinations perform poorly compared to the weighted average on both dimensions (B1 and C1). These portfolios, in addition to being in a competitive situation, exhibit greater variability than their pure crop equivalents. In addition, the combination of crops are presented considering a low risk -low yield crop combined with a high yield -high risk crop. This type of combination is very common in agriculture. However, other functional forms may be obtained when a high yield -low risk crop is combined with a low yield -high risk crop as shown by [START_REF] Paut | Reducing risk through crop diversification: an application of portfolio theory to diversified horticultural systems[END_REF]. Interestingly, this functional form adding a high-risk crop to a low-risk monoculture is still beneficial in terms of risk reduction. We do not detail these situations in this result section but we observed the same diversity of patterns. The reader may find them in Appendix C.

3.2 Application of our framework to selected cases from the literature

Characteristics of fruits and/or vegetables intercropping systems in the literature

The analysis of LER values among these observations found an average 38% increase in yield in intercropping designs compared to their monoculture components (t-test, p < 0.001). Overall, intercropping was more efficient than monocropping, since 405 out of the 449 calculated LER values were larger than 1 (Fig. 6a) but still 44 situations showed a LER < 1 corresponding to a lower yield in association than in sole crops. Moreover, the standard deviation on LER was 0.36, suggesting a high variability in LERs depending on crop species, treatments and experimental designs.

Modelling diversification and association effect on selected cases from the literature

Our model combining MPT and LER approaches was run on 41 studies for which sufficient data was available. From the reviewed papers, we could also extract the β value, which gives an indication of the relevance of the intercropping (in a given pedoclimatic context) with regard to yield variability. For illustrative purposes, two contrasting case studies have been selected among a large variety of cases, but the full set of intercropping systems is described in Appendix D.

In the first example given here (lettuce-radish intercropping, Fig. 7a, [START_REF] Cecílio Filho | Lettuce and radish productivity in intercropping systems as influenced by starting time and row spacings[END_REF], diversification led to a strong reduction in risk and association made it possible to reach the lowest level of risk for a yield similar to the one obtained with lettuce only. In this case, there is a supplementary risk reduction effect due to the intercropping design (β < 1). Thus, in addition to having a positive effect on yield (LER = 1.48), the intercropping has an additional risk-related advantage over a simple diversification.

Another interesting example of agroforestry system for which we found sufficient data in the literature is Apricot-Peanut intercropping (Fig. 7b, [START_REF] Bai | Mixing trees and crops increases land and water use efficiencies in a semi-arid area[END_REF] and it corresponds to the C3 functional form (Cf. Fig. 7). In this particular example, crop yields are strongly positively correlated, meaning that there is little effect on risk due to diversification. The yield improvement is important (LER = 1.35) but we can observe a strong increasing effect of intercropping on yields variability (β = 1.67). In this example, there is no effect of diversification on risk, but additionally, the intercropping increases yield variability.

From the reviewed papers, we observed an average positive correlation between crop yields; the average correlation coefficient was 0.25. The findings also indicate a strong positive effect of intercropping on yields, the average Land Equivalent Ratio was 1.40. We also extracted the β value, which gives an indication of the relevance of the intercropping with regard to yield variability. This coefficient provides information on whether the intercropping is profitable in terms of risk reduction, compared to a simple diversification strategy (in a given pedoclimatic context). The average value of β was 0.79. The statistical analysis (t-test) of β showed that it was significantly below 1 (p < 0.001) meaning that on average intercropping reduced individual crop variability. where intercropping is assumed to have no effect on risk (beta = 1). The solid curve represents the alternative hypothesis (H2) in the case where intercropping effect on risk is considered (β = 0.87 Fig. 7a; β = 1.67 Fig. 7b). Moving from crop A to crop B implies an increasing proportion of crop B in the intercropping design.

Distinction in intercropping types

On the basis of a cluster analysis (k-means) the 41 selected experimentations were categorized into four clusters with different patterns in the dimensions LER, β and ρ (Fig. 8). The different intercropping clusters showed important variations in their structural parameters, except from the Land Equivalent Ratio, which remained relatively similar between clusters. Cluster 1 is characterized by a low correlation coefficient (-1) and a low β (0.63). Cluster 2 shows a similar β but can be distinguished from the first cluster by a correlation coefficient equal to 1. Clusters 3 and 4 differ from the two first ones on parameter β, which is close to one (0.94 and 1.06 respectively). They differ from each other on the parameter ρ (0.18 and 1.00 respectively). Overall, the clusters are characterized by different combinations of β and ρ values. This reflects a variety of diversification and association effects on yield variability. The details of the cluster analysis and further statistical analysis are presented in Appendix E.

Discussion

A framework to comparre simple diversification strategy and intercropping

The two perspectives examined here were Modern Portfolio Theory and Land Equivalent Ratio. The former quantifies the effect of diversification on risk, the latter measures the effect of association on production. This research has merged both approaches in a combined framework in order to assess intercropping and agroforestry system performances. By applying this framework to real cases from the literature, our work contributes to a better understanding of the possible impacts of diversification strategies on the trade-offs between productivity and risk. The present model is not straightforward for deciders, but it requires some minor adaptations to serve as a decision-making tool. By formalizing the model to make it accessible and to facilitate the interpretation of its results for application purposes, it may therefore facilitate the choice of farm portfolios that results in the best agronomic performances associated with the lowest variability. The advantage of a bidimensional assessment (yield-risk) of portfolios is that it does not offer a unique better option, but rather provides a wide range of choices among which each option results in a combination of association and diversification effects.

Segregate or associate crops? Necessary trade-offs

Some authors reported that mixed cropping [START_REF] Kiaer | Grain yield increase in cereal variety mixtures: a meta-analysis of field trials[END_REF][START_REF] Reiss | Cultivar mixtures: a meta-analysis of the effect of intraspecific diversity on crop yield[END_REF] or intercropping [START_REF] Raseduzzaman | Does intercropping enhance yield stability in arable crop production? A meta-analysis[END_REF]) might be sustainable strategies for increasing agroecosystems performances, by enhancing mean yield and yield stability, associated with a lower environmental impact. Our findings seem to indicate that this is indeed the case in horticulture. By comparing different diversification alternatives (Cf. Fig. 5) we can observe that some intercropping options give better results than sole crops in both dimensions (A3 and B3, Fig. 5). Those are the intercropping designs that might be encouraged without requiring compromises. Some other options are only of interest in one of the two dimensions. They will have to be the subject of trade-offs according to the production objectives of each farmer and his/her risk aversion. From the application of our framework to existing literature, we could quantify these trade-offs for a range of crop couples. First, the high average LER (1.40) indicates that intercropping is advisable in terms of yield improvements as compared to monocultures. Second, we observed a positive correlation between crop yields (average ρ = 0.25; median = 0.83). This might be due to the low number of inter-annual repetitions that tend to result in extreme correlation values. Nevertheless, this indicates that in intercropping, diversification has a minor effect on reducing yield variability in the reviewed papers. Finally, the mean β was 0.79, suggesting that in our sample, intercropping has overall an additional risk reducing effect as compared to Simple Diversification Systems.

By disentangling the effect of diversification and association on yield variability, our findings indicate that beyond the well-known effect of intercropping on yield, intercropping is also likely to be beneficial in terms of yield variability (or stability) and this beyond the simple diversification effect. However, the scarcity of long-term data leads us to be cautious about the numerical conclusions provided here. Moreover, our cluster analysis shows that risk reduction from intercropping is highly reliant on crop combinations. [START_REF] Rao | Productivity and risk evaluation in constrasting intercropping systems[END_REF] highlighted this specific issue and pointed out that intercropping stability also depends a lot on crop cultivars and agronomic management. In a wider sense, it is interesting to note that a shift toward more agroecological practices can in some cases lead to increased variability in yields. [START_REF] Knapp | A global meta-analysis of yield stability in organic and conservation agriculture[END_REF] showed that organic agriculture has, per yield unit, a higher variability compared to conventional agriculture. Similar results were reported in horticulture by [START_REF] Lesur-Dumoulin | Lower average yields but similar yield variability in organic versus conventional horticulture. A meta-analysis[END_REF]. More generally, although agroecological systems (such as intercropping or organic agriculture) may be more environmentally friendly, further attention should be paid to reducing the variability of their yields.

Finally, trade-offs also occur in practice. Whether to integrate or segregate crops is still debated [START_REF] Sida | Crop vs. tree: can agronomic management reduce trade-offs in tree-crop interactions?[END_REF][START_REF] Van Noordwijk | Segregate or integrate for multifunctionality and sustained change through rubber-based agroforestry in Indonesia and China[END_REF] and it is often a matter of balancing the benefits expected from the association with the complexity generated. A farmer intercropping multiple crops has to integrate other types of knowledge than that required by monocropping practices (e.g. coordinate plant growth, water requirements, pests and diseases management, harvest, etc.). This increased management complexity is even more critical when intercropping is integrated into highly mechanized agricultural systems (Bybee-Finley and Ryan, 2018). Nevertheless, in temperate areas, this is part of an ongoing 'path dependency' rather than an insurmountable challenge, as shown by research on mechanized agroforestry in Europe [START_REF] Dupraz | Synthesis of the Silvoarable Agroforestry for Europe project[END_REF][START_REF] Eichhorn | Silvoarable systems in Europe -Past, present and future prospects[END_REF].

Perspectives for the model improvement

In the present study, risk was assessed by the standard deviation of yields. It is the definition of risk which is brought by the portfolio theory. Standard deviation (or variance) is also a widespread metric for risk assessment in agricultural sciences [START_REF] Ben-Ari | Decomposing global crop yield variability[END_REF][START_REF] Zhong | Projection of corn production and stoverharvesting impacts on soil organic carbon dynamics in the U[END_REF]. However, other useful approaches of risk were suggested and assumed to be suitable for intercropping systems. [START_REF] Rao | Evaluation of yield stability in intercropping: studies on sorghum/pigeonpea[END_REF] suggested evaluating risk by calculating the probability of crop 'failure' to produce a specific income in comparison with single crops. The estimate of the probability of crop 'failure' can be meaningful because it provides a more consistent reflection of farmers' attitudes towards stability. It would also allow any tradeoff between mean increase and variance reduction to be assessed quantitatively [START_REF] Van Noordwijk | Risk management in crop production and fertilizer use with uncertain rainfall; how many eggs in which baskets?[END_REF]. Integrating this alternative notion of risk in our model would make it clearer whether SDS or IC can be chosen over pure crops.

Besides the issue of measuring risk, a further limitation of our application is that we limited our analysis to crop production, which does not directly reflect profitability stricto sensu. The issue of economic efficiency of agroforestry systems in a context of uncertainty has already been discussed [START_REF] Blandon | Analyzing risk in agroforestry systems using a portfolio approach. A case study from the United Kingdom[END_REF][START_REF] Blandon | Agroforestry and portfolio theory[END_REF][START_REF] Ramírez | Financial returns, stability and risk of cacao-plantain-timber agroforestry systems in Central America[END_REF][START_REF] Reeves | Reducing financial risk in agroforestry planning: a case study in Costa Rica[END_REF] but the lack of knowledge on biological interactions between crops and economics of such systems has made it difficult to develop a systemic model for the comparison of diversification schemes (SDS vs. IC) in relation to risk [START_REF] Paul | Agroforestry versus farm mosaic systems -Comparing land-use efficiency, economic returns and risks under climate change effects[END_REF]. Thus to refine our model, more information on production costs and returns (especially in the context of intercropping systems) for a wide variety of crops would be necessary [START_REF] Pande | Economic assessment of agrihorticulture production systems on reclaimed ravine lands in Western India[END_REF]. The issue of incorporating economic valuation in the model also lies in knowing the relationship between crop yields and prices. However, while the introduction of economic data can affect crop profitability, it would not generate new relationship typologies (as presented in Fig. 5). It also would have an equal impact on SDS and IC, then would not modify the relative performance of both strategies [START_REF] Paul | Agroforestry versus farm mosaic systems -Comparing land-use efficiency, economic returns and risks under climate change effects[END_REF]. In line with the issue of profitability being a suitable indicator for the assessment of such systems, we assume that alternative indicators could be integrated in the comparison between diversification strategies [START_REF] Convertino | Portfolio decision analysis framework for value-focused ecosystem management[END_REF]. Following [START_REF] Groot | Multi-objective optimization and design of farming systems[END_REF], multi-criteria approaches may be useful for the further development of our model taking into account the multiples agriculture ecosystems services. Each service can be treated separately or as proposed by [START_REF] Van Noordwijk | SDG synergy between agriculture and forestry in the food, energy, water and income nexus: reinventing agroforestry?[END_REF], as a whole multifunctionnalty indicator.

The illustrative applications of our framework to real situations were limited due to the scarcity of literature data. Further evaluations of yield and associated standard deviations in a wide range of systems are now needed to determine the conditions in which SDS or IC would be more efficient. In depth understanding of the biological interactions between crops, between trees and crops, pests and environmental conditions is still lacking, while it is often presented as a basis for farmers to design new mixtures and new ways to organize compatible crops in the field space [START_REF] Francis | Biological efficiencies in multiple-cropping systems[END_REF]. Understanding ecological functioning is indeed important and often highlighted in studies dealing with intercropping, but our data driven approach offers a complementary point of view that makes it possible to overcome the lack of knowledge on ecological processes.

Finally, the diversification approaches in this study were limited to portfolios composed of two species. Yet, considering the growing interest of farmers in combining a large number of species within complex rotations [START_REF] Galluzzi | Home gardens: neglected hotspots of agro-biodiversity and cultural diversity[END_REF][START_REF] Hoang | Farmer portfolios, strategic diversity management and climate-change adaptation -implications for policy in Vietnam and Kenya[END_REF][START_REF] Morel | Small can be beautiful for organic market gardens: an exploration of the economic viability of French microfarms using MERLIN[END_REF], it would be especially interesting to extend our approach to more diversified systems. From the theoretical point of view, extension of the framework to situations with more than two crops is straightforward and does not present any mathematical difficulty [START_REF] Paut | Benefits of diversified horticultural systems: assessment with the modern portfolio theory[END_REF][START_REF] Van Noordwijk | Risk management in crop production and fertilizer use with uncertain rainfall; how many eggs in which baskets?[END_REF]. The issue once again lies in the availability of data, since data requirement increases rapidly with the number of crops. To our knowledge, very little research has evaluated yield and standard deviations of more than two crops in associations compared to monocultures.

Conclusion

Taking risk into account in decision-making for the definition of sustainable crop rotation strategies has become essential. Our approach provides an analytical framework that can help farmers or landuse planners in the comparison of simple diversification strategies with more integrated ones such as intercropping or agroforestry. By disentangling the effects of association and diversification on risk and production, we propose an approach that makes it possible to decompose the interests and limits of each of these strategies. We show that there are trade-offs between the increase in yield generated by intercropping, the reduction of risk resulting from diversification, but also the specific effect of intercropping on risk. Taking into account the resultant of these three elements is therefore crucial for the definition of diversification strategies. Long-term field experiments will be needed to better understand biological processes that underlie these interrelationships. The example applications presented in this study illustrate the particular case of horticultural systems. Nevertheless, the approach developed here remains applicable to most agricultural systems. Indeed, our model only requires farm components (crops or animal products) yield and variance, which are commonly available data, relevant for most farming systems. In this perspective, our method and findings may have wider implications for other farming systems such as field crops or mixed croplivestock farming systems.

Fig. 2 .

 2 Fig. 2. Possible yield and risk combinations for a two-crops portfolio when: (a) crop yields are perfectly negatively correlated (ρAB = -1); (b) crop yields are not correlated (ρAB =0) and (c) crop yields are perfectly correlated (ρAB = +1).

Fig. 3 .

 3 Fig. 3. Examples of three qualitatively distinct forms of the Land Equivalent Ratio distribution. The solid curve represents a situation of facilitation (LER = 1.5); the dotted line represents a situation of independence (LER = 1) and the dashed line represents a situation of competition (LER = 0.5).

  Fig. 4. Parameter values for the development of the 27 contrasted scenarios. 2.4 Mini review on intercropping systems based on fruits and/or vegetables to identify promising intercropping systems for the design of mixed horticultural systems 2.4.1 Data collection

Fig. 5 .

 5 Fig. 5. Yield and risk trade-offs among 27 contrasting scenarios of land-use options. Each curve represents a unique combination of LER, β and ρ coefficients. Moving from crop A to crop B corresponds to the consideration of systems with an increasing area allocated to crop B.

Fig. 7 .

 7 Fig. 7. Risk-yields combinations considering our two hypotheses for two crop couples: (a) Lettuce-Radish and (b) Apricot-Peanut. The dashed curve represents the baseline hypothesis (H1) in the casewhere intercropping is assumed to have no effect on risk (beta = 1). The solid curve represents the alternative hypothesis (H2) in the case where intercropping effect on risk is considered (β = 0.87 Fig.7a; β = 1.67 Fig.7b). Moving from crop A to crop B implies an increasing proportion of crop B in the intercropping design.

Fig. 8 .

 8 Fig. 8. Representation of the studies reviewed as a function of the three dimensions (LER, β, ρ) considered in our model. K-means clustering on the three dimensions distinguished four types of intercropping systems.

  proportion of k in the intercropping design. Then, we introduce the coefficient β for the specific 𝐴𝐴 𝑤𝑤 𝐵𝐵 𝛽𝛽 𝐴𝐴 𝐿𝐿𝐸𝐸𝐿𝐿 𝐴𝐴 𝑘𝑘 𝜎𝜎 𝐴𝐴 𝛽𝛽 𝐵𝐵 𝐿𝐿𝐸𝐸𝐿𝐿 𝐵𝐵 𝑘𝑘 𝜎𝜎 𝐵𝐵 𝜌𝜌 𝐴𝐴𝐵𝐵 (5) βA and βB are the partial coefficient of crop A and B respectively, that represent the effect of intercropping on yield variability. Due to the lack of sufficiently accurate data in literature to evaluate βA and βB, we will consider βA and βB being equal in the present application, but different values of βi are explored in Appendix A. Beta being strictly positive, we can factorize

	effect of intercropping on risk:
	𝜎𝜎 𝑃𝑃(𝐴𝐴,𝐵𝐵) 𝑘𝑘 2 + 2𝑤𝑤 into Eq. 5bis: = � 𝑤𝑤 𝐴𝐴 2 �𝛽𝛽 𝐴𝐴 𝐿𝐿𝐸𝐸𝐿𝐿 𝐴𝐴 𝑘𝑘 𝜎𝜎 𝐴𝐴 � 2 + 𝑤𝑤 𝐵𝐵 2 �𝛽𝛽 𝐵𝐵 𝐿𝐿𝐸𝐸𝐿𝐿 𝐵𝐵 𝑘𝑘 𝜎𝜎 𝐵𝐵 �
	𝜎𝜎 𝑃𝑃(𝐴𝐴,𝐵𝐵) 𝑘𝑘	= 𝛽𝛽 � 𝑤𝑤 𝐴𝐴 2 �𝐿𝐿𝐸𝐸𝐿𝐿 𝐴𝐴 𝑘𝑘 𝜎𝜎 𝐴𝐴 � 2 + 𝑤𝑤 𝐵𝐵 2 �𝐿𝐿𝐸𝐸𝐿𝐿 𝐵𝐵 𝑘𝑘 𝜎𝜎 𝐵𝐵 � 2 + 2𝑤𝑤 𝐴𝐴 𝑤𝑤 𝐵𝐵 𝐿𝐿𝐸𝐸𝐿𝐿 𝐴𝐴 𝑘𝑘 𝜎𝜎 𝐴𝐴 𝐿𝐿𝐸𝐸𝐿𝐿 𝐵𝐵 𝑘𝑘 𝜎𝜎 𝐵𝐵 𝜌𝜌 𝐴𝐴𝐵𝐵	(5bis)

Acknowledgments

The present work was carried out with the support of the Fondation de France (grant n°00064844) which is hereby thanked.

Declaration of Competing Interest

None.

Appendix A, B, C, D and E

Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.agee.2019.106711.


Li Zhang

Franck Dumeignil

Sébastien Paul

Benjamin Katryniok
email: benjaminkatryniok@centralelille.fr

Supported Rb-or Cs-containing HPA catalysts for the selective oxidation of isobutane

Keywords: Keggin-type HPA, selective oxidation, isobutane, heterogeneous catalysis

Silica-supported catalysts based on Keggin-type heteropolyacids (HPAs) containing rubidium or cesium as counter cations have been prepared by the impregnation method and evaluated in the selective oxidation of isobutane to methacrolein and methacrylic acid. The catalysts were characterized by various techniques such as XRD, N2 physisorption, TGA, Raman spectroscopy, H2-TPR, and NH3-TPD in order to study their thermal stability, structural, and textural properties, acidity and reducibility.

It was evidenced that the reducibility of the Keggin type HPAs was improved by supporting the active phase on SiO2. A loading of 40 wt.% was the optimum for the selective oxidation of isobutane (IBAN) to methacrylic acid (MAA). The selectivities to MAA and methacrolein (MAC) at given conversion were increased when Cs + was used as counter cation compared to Rb + . The same trend was observed for mono-and di-vanado-substituted phosphomolybdic acid, whereby the performance followed the order: CsV1/SiO2 > RbV1/SiO2 > CsV2/SiO2 > RbV2/SiO2. The density of acid sites was correlated to the catalytic activity, which underlines the importance of the acid sites for alkane activation.

Introduction

Selective oxidation of light alkanes gives access to valuable oxygenated products such as acrylic acid [1,2], methacrylic acid [3], acetic acid, acrolein, etc. [3][4][5], which are widely used as monomers in the petrochemical industry [6]. Today, the activation and selective conversion of light alkanes (C1-C4) is not only an important research topic for the utilization of natural gas components, but also of practical significance to produce high value-added products. Therefore, the development of new efficient catalysts for the selective oxidation of light alkanes is still of scientific and industrial interest [7,8], notably due to the large availability and cheap costs of the raw materials.

Methacrylic acid (MAA) is an important chemical intermediate, which has two functional groups: a carbon-carbon double bond and a carboxylic acid group. It is used to prepare methyl methacrylate (MMA) for coating, rubber, adhesive, resin, polymer material additives and functional polymer materials. MMA, as a specialty monomer for poly-methyl-methacrylate (PMMA), has been widely used in the fields of architecture, medicine and fine chemicals [9][10][11][12]. According to the source of raw materials, the main production for MMA includes the acetone-cyanohydrin (ACH) route and the light hydrocarbons route, from ethylene, propane, propylene, propyne, isobutene, and isobutane. Reacting acetone and hydrogen cyanide (HCN) yields MMA (ACH route), which is the most widely used route in Europe and North America, where it is still the dominant commercial process. However, the intermediates (hydrogen cyanide, and acetone cyanohydrin) produced by the ACH process are toxic and source of environmental hazard [13]. Synthesis of MMA through esterification of MAA is an alternative route. Light hydrocarbons, such as C2, C3 and C4 hydrocarbons, were widely developed as feedstocks for MAA. Nevertheless, industrial application was restricted by the harsh reaction conditions, the expensive reactants, the low yields and the relatively high number of reaction steps [14]. Due to the increasing demand for PMMA, the price of MMA monomer increased by 10% annually [12], therefore, to overcome the environmental and production issues linked to the acetone cyanohydrin process, the isobutane oxidation method, which would provide a one-step route to form methacrylic acid, has attracted extensive attention because of the easiness of the process and decreased by-product generation [14,15]. Moreover, it is inexpensive and has a lower environmental impact than other fossil feedstocks-based routes and optimally converts isobutane to MAA in a single step [16][17][18][19]. With respect to the reaction mechanism, the first step is the catalytic dehydrogenation of isobutane to form a C=C double bond. In the following steps, the adjacent methyl group is partially oxidized to a carboxyl group (allylic oxidation). Bifunctional catalysts with acidic and redox properties are necessary to activate the C-H bond [20][21][22][23]. Redox properties also have an effect on the oxygen insertion reactions, which generate the oxidized products. Therefore, bifunctional catalysts simultaneously possessing acid and redox properties are necessary.

Heteropolyacids, also known as polyoxometalates (POMs), are nanoscale inorganic metal-oxygen clusters formed by oxygen bonding of pretransition metals (Mo, W, V, Nb, Ta, etc.) [24]. Based on the above composition and structural characteristics, Keggin heteropolyacids usually exhibit two properties: Brønsted acidity and oxidative reducibility [25]. The strong Brønsted acidity is related to its highly symmetrical spatial structure. Due to symmetry, cage structure and nanometer size, the negative charge is highly delocalized in the Keggin anion, so that the charge density on the surface is low, the binding ability to proton is weak, and the proton activity is quite high [26][27][28]. The coordination atoms in Keggin heteropolyacids are generally in their highest oxidation state. Thus, they can be used as multi-electron acceptors with certain oxidation properties. Notably Mo-based heteropolyacids (HPA) accept easily electrons and are reduced to a blue color, named as "heteropoly blue". The initial anion structure is kept after the heteropoly acid is reduced to "heteropoly blue".

Heteropolyacid can be reoxidized to recover their initial oxidation state, hence showing a reversible redox property [29].

As aforementioned, selective oxidation of isobutane for the production of methyl methacrylate and methacrylic acid has attracted extensive research interest. Selective oxidation of isobutane to produce MAC and MAA requires C-H activation, oxidative dehydrogenation, and oxygen insertion.

Therefore, the design and development of the catalytic system must focus on the optimization of the parameters influencing the steps of this mechanism. Given that vanadium is known to play a crucial role for both the dehydrogenation and selective oxidation steps of the reaction network, our strategy is focused on the development of a supported Keggin-type HPA catalyst for the selective oxidation of IBAN to MAC and MAA. The heteropoly acid chosen was phosphomolybdic acid H3PMo12O40, which was substituted with V 5+ . The corresponding vanado-phosphomolybdic acid was furthermore partially neutralized by Cs + counter cations which forms an heteropoly salt presenting a high surface area.

Cesium substitution, also increases the hydrophobicity, thus increasing the tendency to expel MAA and MAC, which are polar, resulting thus in increased catalytic performance [30]. Silica was chosen as a support with respect to its larges specific surface area and the fact that it does not have negative impact on the acidity and structure of HPAs [31][32][33].

To confirm the best loading and optimize the properties of the Cs2H2PMo11VO40/SiO2 catalyst, a series of catalysts with 10-50 wt.% loading of Cs2H2PMo11VO40 (CsV1) on SiO2 (CARiACT) was prepared. Similarly, two other series of catalysts Cs2.5H2.5PMo10V2O40 (CsV2) on SiO2 and Rb2.5H2.5PMo10V2O40 (RbV2) on SiO2 were prepared with the same loadings. Their physicochemical properties and catalytic performances in isobutane oxidation were studied. To the best of our knowledge, substitution of H + with rubidium (Rb + ) in heteropolyacid has never been investigated before in the selective oxidation of IBAN to MAC and MAA.

Experimental section

Materials syntheses

The formation of H4PMo11VO40 and H5PMo10V2O40 starts from stoichiometric mixtures [34]. For example, for H4PMo11VO40, 1.42 g of Na2HPO4 (Sigma-Aldrich) were dissolved in a solution of 1.22 g of NaVO3 (Fluka) in 50 mL of boiling water (temperature at 100 °C) under stirring. Then, after cooling to room temperature, 1 mL HCl (34-37%, was added and the color turns reddish brown.

Following the addition of an aqueous solution of 26.6 g Na2MoO4•2H2O (Alfa Aesar) in 50 mL of water under stirring, the mixture was acidified with 40 mL HCl (34-37%) and the color changed to orange-red. The acidified mixture was kept at room temperature for 5 h and poured into a dividing funnel and then extracted with 70 mL of diethyl ether. The extraction was accomplished by repeated shaking, after which the layers separated. The HPA etherate in the lowest layer was isolated, 7 mL of water was added and kept for crystallization at room temperature. The yield was 20 g, calculated for H4PMo11VO40•30.5H2O.

The CARiACT ® SiO2 (Q-10), commercially available from Fuji Silysia Chemical Company, was used as a support. The preparation of 10-50 wt.% Cs2H2PMo11VO40, Rb2H2PMo11VO40, Cs2.5H2.5PMo10V2O40 and Rb2.5H2.5PMo10V2O40 supported on SiO2 was carried out in two steps [35]:

1) SiO2 was impregnated with Cs2CO3 (Fluka) or Rb2CO3 (Aldrich). The support and 30 mL of 100% ethanol were stirred at 60 °C. Then, the desired amount of Cs2CO3 or Rb2CO3 was added. After 30 min, the solvent was evaporated under vacuum at 60 °C for 1.5 h. 2) The as-prepared support was then impregnated with HPA: after suspending in 30 mL of 100% ethanol was subsequently added the required amount of HPA (H4PMo11VO40 or H.5PMo10V2O40) dissolved in a minimum quantity of ethanol. After 30 min, the solvent was evaporated under vacuum at 60 °C until the obtained solid was dry.

In order to make comparisons easier between the different catalysts, the formulation of the series of catalysts Cs2H2PMo11VO40/SiO2, Cs2.5H2.5PMo10V2O40/SiO2 and Rb2.5H2.5PMo10V2O40/SiO2 are labelled in the followings as CsV1/SiO2, CsV2/SiO2 and RbV2/SiO2, respectively, where CsV1 or CsV2 or RbV2, is the active phase of the catalyst and SiO2 is the support of the catalyst. Catalysts with different mass of active phase on SiO2 support x wt.% Cs2H2PMo11VO40/SiO2, x wt.% Cs2.5H2.5PMo10V2O40/SiO2 and x wt.% Rb2.5H2.5PMo10V2O40/SiO2 are labelled in the following as xCsV1, xCsV2 and xRbV2, respectively, where x ranges from 10 to 50.

Characterization

Optical microscope images were obtained using a ZEISS Axio Lab. A1 equipped with a 5x objective (ZEISS EC EPIPLAN) and 10x eyepieces, a differential interference contrast (C-DIC) and a professional microscope camera AxioCam ERc 5s equipped with a 5MP CMOS Sensor giving a resolution of 2560 (H) × 1920 (V). Thermogravimetric analysis (TGA) was performed using a SETARAM instruments to study the thermal decomposition behavior of the catalysts. The catalysts were heated from 20 °C to 700 °C at a rate of 3 °C/min under air flow. X-ray diffraction patterns (XRD) were recorded in ambient conditions using the CuKα radiation (λ = 1.5418 Å; 40 kV, 30 mA) on a Siemens D5000 diffractometer, with a 0.02° scan step and 1 s time step in the 10-80° range. The interpretation of the diffractograms was performed using the Diffrac Eva software and by comparison by JCPDS reference files. IR-Raman spectra were collected on a Horiba Xplora using a laser beam of 532 nm wavelength for excitation in confocal mode. A 50x long working distance objective (Olympus) was used for both focusing the excitation beam on the sample and collecting the scattered light. The latter was dispersed using an 1800 grooves spectrometer grating after having passed through a confocal hole of 300 µm. A Peltier-cooled CCD detector was used for recording the Raman spectra.

The Brunauer-Emmett-Teller (BET) model serves as the basis for the measurement of the specific surface area of materials. N2 physisorption isotherms at liquid nitrogen temperature were obtained on a TriStar II Plus gas adsorption analyzer (Micromeritics) after outgassing at 130 °C in vacuum overnight. Specific surface areas were evaluated with the BET model over the following range: P/P0 = 0.05-0.30. The total pore volume was measured from the volume of N2 adsorbed at P/P0 = 0.95.

Temperature-programmed reduction (TPR) analyses were conducted with an automated

AutoChem II 2920 Chemisorption Analyzer from Micromeritics. 100 mg of catalyst were inserted into a glass reactor. The sample was flushed with a gas mixture comprising 5 Vol.% H2 in Ar at room temperature with a flow rate of 50 mL/min STP. The temperature of the sample was increased linearly at a ramp rate of 5 °C/min up to 1000 °C. The consumption of hydrogen was monitored via a TCD signal converted by use of a calibration factor, to get quantitative information on the species present in the catalyst. Ammonium temperature-programmed desorption (NH3-TPD) with mass spectrometry (MS) was performed to quantify the amount and strength of the acidic sites of the catalysts. A typical test was as follows: 50 mg catalyst were pre-treated in a He flow (30 mL/min) at 130 °C for 2 h in order to remove the physisorbed water. Then, NH3 was absorbed at the surface by pulsed injections at 130 °C until saturation was stated from the MS signal. The TPD profiles were monitored by MS and thermal conductivity detector and recorded from 130 to 650 °C at a heating rate of 10 °C/min.

Catalytic oxidation of isobutane

The catalytic reactions for the selective oxidation of isobutane to methacrolein and methacrylic acid were performed in a fixed-bed reactor under atmospheric pressure. The reactor was loaded with 0.8 g of catalyst sandwiched by carborundum (SiC, 0.21 mm diameter) to keep the catalytic bed in the isothermal zone of the reactor. The reactor was heated to reaction temperature at a 5 °C/min heating rate. The reactants mixture was fed at a constant total flow-rate of 10 mL/min STP (molar ratio 28% IBAN; 12% O2; 10% H2O; 50% He), corresponding to a WHSV (weight hourly space velocity) of 0.75

L•g -1 •h -1 .
Water was introduced via a saturation system keeping 10% partial pressure of water in the gas stream. All products were analyzed on-line by gas chromatography (GC). The reaction data were obtained under stream at steady state conditions. were formed, which maintain similar uniformity. Upon increasing the amount of active phase, the colors of the surfaces of the CsV1/SiO2 spheres change progressively from white to yellow. For the 10 wt.%-20 wt.% loadings (Fig. 1B andC), a bright yellow shell is observed, while for the 30 wt.% -40 wt.% loadings (Fig. 1D andE) the yellow color is more homogeneous. For the 50 wt.% loading (Fig. 1F), the shell became too thick and detaches from the silica surface, whereby small pieces of debris (CsV1) were clearly visible on the micrograph. Similar results, were obtained for CsV2/SiO2 and RbV2/SiO2 catalysts (not shown).

Thermal decomposition of the fresh catalysts TGA

According to the thermogravimetric curve in Fig. 2, the weight loss of the catalysts can be divided into two stages. The first stage is at temperatures below 100 °C. Here, the weight loss is caused by the evacuation of physisorbed water. The second weight loss -observed in the 150 °C to 340 °C range -is caused by the loss of crystal water which is in agreement with literature [36], reporting that phosphomolybdic acid loses its constitution water at 375 °C. Furthermore, these results evidence that below 400 °C, the catalyst only loses crystal water, but no constitution water, whereby the acidic protons are retained, which is important with respect to the chosen reaction temperature (< 400 °C). Fig. 3 shows the typical XRD patterns of SiO2, CsV1/SiO2 catalysts (Fig. 3A), CsV2/SiO2 catalysts (Fig. 3B) and RbV2/SiO2 catalysts (Fig. 3C). The patterns exhibit diffraction peaks at 2θ =

(A) (B) (C) (D) (400) (d)
compounds were largely enhanced, suggesting the presence of large amounts of active phase on SiO2.

As a matter of fact, the interplanar spacing was then expanded as Cs + has a larger ionic radius than Rb + , ascribed to d-spacing decrease when the protons were substituted by Rb cations in comparison with by Cs cations. The d-spacing of the (222) plane increased from 0.3362 nm for Rb2H2PVMo11O40 to 0.3393 nm for Cs2H2PMo11VO40. Therefore, the diffraction peaks of Rb-containing catalysts were slightly shifted to the higher angles in comparison with the diffraction peaks of Cs-containing catalysts, could be clearly observed in Fig. 3D. Fig. 4 shows the XRD patterns of the catalysts with 30 wt.% before and after reaction. It is worth mentioning that there was no new diffraction peak in the spent catalysts, which indicated that the catalysts did not decompose into metal oxides during the reaction process, which also indicated that the catalysts were quite stable under the reaction conditions. However, the diffraction peak of the spent catalyst is slightly shifted towards a small angle, by 0.2°. As shown for the lattice plane of (222), the diffraction peak of the Cs-containing catalyst is shifted from 26.2° to 26.0°, same for the Rb-containing catalyst (also from 26.2° to 26.0°). This was accounted for the loss of crystal water which led to a small modification of the lattice parameter, which is also consistent with the results of TG analysis [37]. It should also be noted that the diffraction peaks of the spent catalyst become narrower compared to those of the fresh one, implying an increase in crystallinity of the catalyst after reaction, probably due to sintering. This proposal is consistent with that observed in the BET results, with a slight decrease in the specific surface area of the spent catalysts compared to the fresh ones (Table S1).

BET specific surface area and pore volume of the catalysts

The specific surface areas of the SiO2 (CARIACT) support and of the catalysts are gathered in respectively. Theoretical values obtained from the calculation (assuming that the active phase was well and homogeneously dispersed as referred to [14]) is also given in brackets in Table 1. When comparing the experimental and theoretical BET values, it can be seen that all the catalysts display a higher specific surface area than the calculated theoretical one. This indicates that the active phase (CsV1, CsV2, RbV2) partially contributes to the catalysts surface area and that new surface is created due to the porous structure of the active phase.

The mean value of the coverage of active phase on the support for the different loadings was calculated based on a Keggin unit occupancy of 1.44 nm 2 [31,38]. In Table 1 the average coverage of the support expressed as the number of active phase layers stacked on the surface is reported. Based on the surface area of the CARiACT®-SiO2 (264.2 m 2 /g), a theoretical CsV1 monolayer coverage (0.8-1.3 layers) was reached for a loading of 30-40 wt.%. This theoretical monolayer indicates a homogenous distribution of the active phase on the support, as evidences by optical microscopy. ST = SBET-SiO2 × (100 -x) with x in wt.%. The CsV1 series of catalysts has been studied using Raman spectroscopy (Fig. 5 and Fig. S1) to further confirm the integrity of the Keggin structure in the salt of the HPA samples. The observed bands corresponded to the characteristic Keggin feature bands at the following vibrations [14,39]: νs Mo=Od at 983 cm -1 , νas Mo-Ob-Mo at 871 cm -1 , νas Mo-Oc-Mo at 600 cm -1 and δ(Mo-O-Mo) 255 and 235 cm -1 , respectively. The P-O vibration in Keggin units is Raman-inactive [40]. This also indicates that the Keggin structure of the catalyst has been effectively formed. Nevertheless, the signal of V-O bonds was not observed, which may be explained by the low amount of vanadium in the catalysts which is in agreement with the literature [14]. Fig. 6 shows the reducibility of the three series of catalysts, which were examined by H2-TPR measurements. The reduction at lower temperatures starting from 550 °C is ascribed to the first step of the reduction of the Mo 6+ species (from tetrahedral Mo 6+ to Mo 4+ ). The reduction peak at the higher temperature of approximately from 800 °C was attributed to the reduction of the tetrahedral Mo 4+ to Mo 0 [41]. The broad peaks observed hereby indicated a rather complex reduction process composed of one or more transition states [14]. Concerning vanadium, the content was too low to observe its reduction which should occur at a lower temperature. As one can see from Fig. 6D, the H2 consumption increased linearly with the loading, indicating that all Mo was accessible for reduction. It is noteworthy that with the increase in the loading (10-40 wt.%), the reduction peak shifted to lower temperature, before shifting back when the loading reached 50 wt.%, indicating that the catalyst with 40 wt.% exhibited the highest reducibility.

Structural features of the catalysts observed by Raman spectroscopy

Reducibility measurements using TPR

(A) (B) (C) (D)
The amount and strength of acid sites of the three series of catalysts (xCsV1/SiO2, xCsV2/SiO2 and xRbV2/SiO2 catalysts) were determined by temperature-programmed desorption of ammonia (NH3-TPD). The desorption temperature profiles are plotted in Fig. 7 and the quantitative calculation for the NH3 uptake are listed in Table S2. For silica, a small desorption peak was observed, which indicates that the support has no acidity. For the three series of silica-supported catalysts, four NH3 desorption peaks could be distinguished: weak acid sites (100-300°C), medium acid sites (300-450°C), strong acid sites (450-560°C) and very strong acid sites (560-700°C) [38]. However, no clear tendency regarding the distribution of acid strength was found in these series. On the other hand, concerning the total amount of acid sites, the latter increased with the amount of active phase. For Cs2H2PMo11VO40/SiO2 catalysts, for example, from 10 wt.% to 50 wt.%, the ammonia uptakes were 0.16, 0.28, 0.34, 0.46 and 0.50 mmol/g, respectively. However, for lower loadings, the number of acid sites of the catalysts increased significantly, but only moderately for higher loadings from 40% to 50 wt.% (Fig. 8A). This was ascribed to overloading with active phase leading to non-accessible sites and the formation of isolated detached bulk CsV1, as evidenced in the microscope images. By comparing the three series of catalysts, we found that the CsV1/SiO2 series exhibited the higher amount of acid sites, followed by the CsV2/SiO2 series catalysts, and the RbV2/SiO2 series ((Fig. 8B). The catalytic performance of all the series of catalysts was evaluated in the selective oxidation of isobutane to MAC and MAA in the same operating conditions (Fig. 9). The lowest IBAN conversion (XIBAN = 3.7 %) was found for 0.24 g of bulk CsV1, this mass corresponding to the amount of active phase in the 30CsV1 catalyst (Table S3). This low conversion was related to the rather low specific surface area (70.4 m 2 /g) of the sample. Furthermore, it was found that the bulk catalyst was unstable, becoming over-reduced under the reaction conditions as evidenced by a color change to green-blue after reaction, whereas the supported catalyst remains yellow-green. After supporting the CsV1 active phase over CARiACT ® -silica, the catalytic activity was increased to XIBAN = 6.0 % for the 30CsV1 catalyst.

Effect of the active phase and reaction temperature

The catalytic performances of the catalysts with a constant amount of 30 wt.%. of active phase on SiO2 support, namely 30CsV1, 30CsV2, 30RbV1 and 30RbV2 were studied and reported in Fig. 10 and Tables S4 to S7. In addition, in all cases, the carbon balance was >99.5%, which means that at least 95% of the product was detected even at low isobutane conversion. The isobutane conversion over the four kinds of catalysts increased with the reaction temperature, as expected. Maximum conversions of 8.7% for the 30CsV1 catalyst at 340 °C, 7.7% for the 30CsV2 catalyst, 7.3% for the

MAC+MAA Selectivity (%)

IBAN Conversion (%) (a) (b)
30RbV1 catalyst and 5.5% for the 30RbV2 catalyst at 350 °C were observed. As one can see, the reaction temperature is a key factor that affects the catalytic activity. At higher temperature, the IBAN conversion increased, but this increase was accompanied by the over-oxidation of the products. In fact, large selectivity to CO2 was found whereas the selectivity to the main products of interest, namely MAC and MAA, decreased. For the by-products acetic acid and acetone, the selectivity has an opposite trend with respect to temperature according to Tables S4 to S7. Acetone also decomposes further with increasing temperature to form C2 and C1. This result, is in agreement with Busca et al., [START_REF] Centi | Selective Oxidation by Heterogeneous Catalysis[END_REF][START_REF] Busca | [END_REF] postulating a reaction network for the formation the by-products, where the oxidative degradation of MAA and MAC to AA and CO2 is responsible for the decrease in desired products. The reaction network was also proposed by Paul et al. [44,45], who claimed that MAC, MAA and degradation products were formed through parallel reactions from isobutane directly. At a high reaction temperature, more over-oxidation products are produced, resulting in the formation of less desired products since the degradation of MAC was around 50 times faster than isobutane activation. Therefore, maintaining the high selectivity of the desired product requires that the temperature of the oxidation operation should be appropriately lowered. Comparing the different counter-cations, the conversions of isobutane over the 30CsV1 catalyst at different reaction temperatures were much higher than those obtained over the 30RbV2 catalyst. For example, at 340°C, the conversion of isobutane was 8.7% for 30CsV1, while for 30RbV2, it was 5.1%.

Hence, although there have been some reports about rubidium-doped catalysts in alcohol catalytic conversion [46,47], rubidium is not the best choice for the catalytic oxidation of isobutane. Fig. 11 shows the IBAN conversion as a function of the MAC+MAA selectivity for 30CsV1, 30CsV2, 30RbV1 and 30RbV2 catalysts. The 30CsV1 catalyst exhibits the highest selectivity in the main products at a given conversion, meanwhile, it possesses the highest isobutane conversion at a given selectivity. The trend was observed for the performance followed the order: 30CsV1 > 30RbV1 > 30CsV2 > 30RbV2. This clearly evidenced that the use of cesium as counter cation is more favorable than rubidium and that replacing molybdenum by one vanadium is better than by two vanadium. In fact, it is known that di-vanado substituted phosphomolybdic acid is not a stoichiometric phase, since compounds with vanadium stoichiometry > 1 cannot be obtained in the pure state, but are a mixture of polyanions, which is disadvantageous for the catalytic activity [12,48,49]. Another postulate is that the decrease in activity of a catalysts at a stoichiometric vanadium content of 2 would be related to the formation of VO2 species [48,50,51]. The difference in the catalytic performance between CsV and RbV can be explained by taking a closer look to their respective acidities. In fact, according to the NH3-TPD analysis, a correlation was attempted between acidity and catalytic activity, defined as follows [38]:

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 𝐹𝐹 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 × 𝑋𝑋 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑆𝑆 𝐼𝐼𝐵𝐵𝐵𝐵 × 𝑚𝑚 𝑐𝑐𝑐𝑐𝑐𝑐.
where FIBAN is the molar flowrate of fed isobutane, XIBAN the isobutane conversion (320 °C), SBET the specific surface area and mcat. the mass of catalyst. By plotting activity vs. acidity, a linear relationship was obtained (Fig. 12). Therefore, the number of acid sites in the 30CsV1 catalyst is higher than that in the other three catalysts, confirming that the acidic sites are mainly required for the C-H bond activation step [START_REF] Busca | [END_REF].

Effects of active phase amount

Following the test of different kinds of active phases, catalysts with different amounts of CsV1, CsV2 and RbV2 were prepared on SiO2 and tested at 320 °C, 330 °C and 340 °C.

The catalytic performances of the three series of catalysts with different loadings are summarized in Tables S8 to S10. For 10CsV1, the conversion of isobutane was quite low. The most plausible explanation is the low loading of active phase resulting in low amounts of active sites -in agreement with the low acidity found by ammonia TPD. On the other hand, it is also possible that the catalyst with low loading has poor stability, and that the catalyst is thus decomposed under the reaction conditions [31]. For the other catalysts containing 20 wt.%-50 wt.% of CsV1 on SiO2, the isobutane conversion increased (from 5.0% for 20CsV1 to 10.1% for 50CsV1), obviously with the increase in loading of active phase, indicating that the conversion of isobutane is proportional to the amount of active phase and the quantity of acid sites. The catalytic performances of the other two series of catalysts, CsV2/SiO2 and RbV2/SiO2, showed a similar trend. The best IBAN conversion was 8.9% for 50CsV2 and 7.7% for 50RbV2. However, the selectivity to by-products also increased with the increased amount of active phase, especially concerning the selectivity to CO2 and acetic acid (AA). catalysts at a given selectivity or conversion are in the order of: CsV1/SiO2 (Cs2H2PMo11VO40/SiO2) > CsV2/SiO2 (Cs2.5H2.5PMo10V2O40/SiO2) > RbV2/SiO2 (Rb2.5H2.5PMo10V2O40/SiO2). It is well known that the first step in selective isobutane oxidation, the dehydrogenation of the tertiary carbon atoms, is the rate determining step of the reaction [14]. The acidic sites on the surface effectively promote the hydrocarbon activation and lead to the C=C bond formation by oxidative dehydrogenation. Therefore, the density of acidic sites on the catalyst surface plays a key role in the activation of isobutane. Fig. 14 actually shows the relationship between the IBAN conversion and the surface acid density of these three series catalysts (CsV1/SiO2 catalysts, CsV2/SiO2 catalysts, RbV2/SiO2 catalysts). As one can see, the isobutane conversion increases when increasing the surface acidity. For lower loadings, a significant increasing was observed for cesium catalysts (10 wt.% to 20 wt.%) and rubidium catalysts (20 wt.% to 30 wt.%). However, for higher loadings, > 30 wt.%, the conversion of isobutane only moderately increased with the surface acid density which is in agreement with previous reports [14]. In order to determine the optimal loading of active phase, the selectivity to the main products is depicted at the same level of IBAN conversion in Fig. 15 and summarized in Tables S11 andS12.

According to the discussion about the effect of reaction temperature and catalytic performances on the catalytic activity in the last section, the selectivity for these series of catalysts have been compared

with catalysts loaded with 30 wt.% of active phase. At the first glance one can see that, except for 10-20 wt.% catalysts, the best selectivity to the desired products (MAC+MAA) was observed for the CsV1/SiO2 and RbV2/SiO2 series with 39.1% (IBAN conversion at 6.0%) of 40 CsV1, and 32.9%

(IBAN conversion at 5.0%) of 40 RbV2. Thus, the optimal loading is 40 wt.%. This conclusion is consistent with the characterization results showing that the 40 wt.% samples had high acidity, highest reducibility and a good dispersion of the active phase on the surface of the silica support. Finally, the long-term catalytic performance was evaluated for the 30CsV1 at 320 °C. From the results, Fig. 16, it can be noticed that catalyst exhibits a quite stable performance during 70 h under stream. The conversion of isobutane was stable at 6 % after 4 h from the start of the reaction.

Long-term catalytic performance of catalyst

Only slight decrease in the conversion of isobutane from 6.2 % at the very beginning to 5.7% after 70 h was found, which is probably due to the sintering of the catalyst to form larger crystallites as evidenced by XRD and BET (108 Å for fresh and 164 Å for spent analyzed; specific surface area of 220 m 2 /g for fresh and 207 m 2 /g for spent). catalysts) with 10-50 wt.% loadings of active phase on SiO2 were prepared by the impregnation method. They were characterized different techniques (XRD, BET, TGA, Raman spectroscopy, H2-TPR, and NH3-TPD), and their catalytic activity in the selective oxidation of isobutane to methacrylic acid and methacrolein were evaluated. First, catalysts loaded with 30 wt.% of heteropoly salt were selected to assess the effect of temperature on the catalytic performances. We found that the counter cation and the degree of vanadium substitution had significant impact on the catalytic performance: Rubidium-containing catalysts require higher reaction temperatures to convert isobutane, which did not favor the selectivity to the desired product. Cs + as a counter-ion in the presence of one V achieved the higher MAA and MAC selectivities at a given level of isobutane conversion. In the following, the performance of the CsV1/SiO2, CsV2/SiO2 and RbV2/SiO2 catalysts with active phase contents ranging from 10 to 50 wt.% were studied. It followed the order: CsV1/SiO2 > CsV2/SiO2 > RbV2/SiO2. Moreover, it was shown that using higher contents of active phase -up to a certain extent -led to an increase in conversion. This could be explained by the fact that acidity is the key to activate the C-H bond: with the increase in the active phase loading, the surface acid sites density increased, which led to an increase in catalytic activity. Hence, the best performance was observed over the 40 wt.% samples, which not only had high acidity, but also a highly homogeneous dispersion of the active phase on the support. Within all the catalysts tested, through the above test comparison, 40CsV1 has shown the best catalytic performance.

Fig. 1 .

 1 Fig. 1. Typical optical microscope images of SiO2 and xCsV1 spheres with x = 10 to 50. (A) SiO2, (B) 10CsV1, (C) 20CsV1, (D) 30CsV1, (E) 40CsV1, (F) 50CsV1.

Fig. 1

 1 Fig.1shows the typical optical microscope images of SiO2 and xCsV1/SiO2 spheres with x = 10 to 50. It indicates that the CARIACT SiO2 spheres possess good uniformity with an average size of ~200 µm as expected (Fig.1A). After coating of CsV1 active phase, core-shell CsV1/SiO2 spheres

Fig. 2 . 3 Fig. 3 .

 233 Fig. 2. TGA curves of the series of catalysts (A) CsV1/SiO2 catalysts, (B) RbV2/SiO2 catalysts.

Fig. 4 .

 4 Fig. 4. XRD patterns of fresh and spent catalysts: (A) 30CsV1 and 30CsV2 catalysts, (B) 30RbV1 and 30RbV2 catalysts.

Fig. 5 .

 5 Fig. 5. Raman spectra for the series of Cs2H2PMo11VO40/SiO2 catalysts (a) 10 wt.%, (b) 20 wt.%, (c) 30 wt.%, (d) 40 wt.%, (e) 50 wt.%, (f) bulk active phase.

Fig. 6 .

 6 Fig. 6. TPR profiles of the catalysts: (A) CsV1 and CsV1/SiO2 catalysts, (B) CsV2 and CsV2/SiO2 catalysts, (C) RbV2 and RbV2/SiO2 catalysts with various loadings: 10-50 wt.%, (D) H2 consumption as a function of active phase loading for the three series of catalysts (a) CsV1/SiO2 catalysts, (b) CsV2/SiO2 catalysts, (c) RbV2/SiO2 catalysts.

3. 1 . 7 Fig. 7 .

 177 Fig. 7. NH3-TPD profiles for: (A) CsV1 and CsV1/SiO2 catalysts, (B) CsV2 and CsV2/SiO2 catalysts, (C) RbV2 and RbV2/SiO2 catalysts with various loadings: 10-50 wt.%, (D) NH3-TPD deconvolution curve of 40CsV1 sample taken as an example.

Fig. 8 .Fig. 9 .

 89 Fig. 8. (A) Amount of acid sites as a function of the active phase loading. (B) Relationship between the active phase loading and the surface acid density: (a) CsV1/SiO2 catalysts, (b) CsV2/SiO2 catalysts, (c) RbV2/SiO2 catalysts.

Fig. 10 .

 10 Fig. 10. Effect of the reaction temperature on the catalytic performances of (A)30CsV1 catalyst, (B) 30RbV1 catalyst, (C) 30CsV2 catalyst, and (D) 30RbV2 catalyst. (IBAN = isobutane, MAC = methacrolein, MAA = methacrylic acid).

Fig. 11 .

 11 Fig. 11. Relationship between IBAN conversion and selectivity (MAC+MAA) of (a) 30CsV1, (b) 30CsV2, (c) 30RbV1 and (d) 30RbV2 catalysts at different temperatures.

Fig. 12 .

 12 Fig. 12. Correlation between the activity in isobutane oxidation and the acidity of catalysts.

Fig. 13

 13 Fig.13 shows the selectivity to the main products of interest (MAA+MAC) for the three series of catalysts as a function of isobutane conversion. It clearly shows that the catalytic performances of

Fig. 13 .

 13 Fig. 13. Relationship between conversion and selectivity of (a) CsV1/SiO2 catalysts (320 °C), (b) CsV2/SiO2 catalysts (330 °C), (c) RbV2/SiO2 catalysts (340 °C) with various loadings.

Fig. 14 .

 14 Fig. 14. Relationship between the IBAN conversion and the surface acid density over (A) CsV1/SiO2 catalysts (320 °C), (B) CsV2/SiO2 catalysts (330 °C), (C) RbV2/SiO2 catalysts (340 °C).

Fig. 15 .

 15 Fig. 15. Relationship between active phase loadings and selectivity of (a) CsV1/SiO2 catalysts (310-320 °C), (b) RbV2/SiO2 catalysts (330-340 °C) at the same level of IBAN conversion.

Fig. 16 .

 16 Fig. 16. Catalytic stability test over the sample 30CsV1 at 320 °C.

  , three series of Keggin-type catalysts (xCsV1/SiO2, xCsV2/SiO2 and xRbV2/SiO2

SiO 2 22 23 24 25 26 27 28 29 30 31 32 33

		(222)							(222)
		(211)	(400)					(211)		(400)
	Intensity (a.u.)			(332)			40CsV 1 CsV 1 50CsV 1 30CsV 1	Intensity (a.u.)			(332)			CsV 2 50CsV 2 40CsV 2 30CsV 2
							20CsV 1							20CsV 2
							10CsV 1							10CsV 2
							SiO 2								SiO 2
	10	20	30	40	50	60	70	80	10	20	30	40	50	60	70	80
			2θ (degree)						2θ (degree)
		(222)
	Intensity (a.u.)	(211)	(400)	(332)			RbV 2 40RbV 2 30RbV 2 50RbV 2		(b) (c)
							20RbV 2		(a)
							10RbV 2
	10	20	30	40	50	60	70	80
			2θ (degree)						2θ (degree)
	18.5°, 23.9°, 26.2°, 30.4° and 35.8° assigned to the lattice planes of (211), (310), (222), (400), and
	(332) of both cubic CsV1/SiO2 and CsV2/SiO2 catalysts, respectively, which are characteristic for
	Keggin-type compounds. For RbV2/SiO2 catalysts, the observed diffraction peaks at 2θ = 18.6°, 24.1°,

26.5°, 30.7° and 36.2° can be assigned to the lattice planes of (211), (310), (

222

), (400), and (332), respectively. With the increase in active phase amounts, the intensities of the diffraction peaks corresponding to the lattice planes of (211), (310), (

222

), (400), and (332) belonging to Keggin-type Intensity (a.u.)

(222)

Table 1 .

 1 The surface area of the catalysts decreased with increasing the amount of active phase, from

	264.2 m 2 /g (SiO2 support) to 150.9 m 2 /g (50CsV1), 197.8 m 2 /g (50CsV2) and 194.2 m 2 /g (50RbV2),

Table 1 .

 1 Textural properties of the supported catalysts and coverage of active phase. theoretical values were calculated according to the specific surface area of SiO2 measured in the same conditions:

	Catalysts	SiO2 (CARIACT)	10CsV1	20CsV1	30CsV1	40CsV1	50CsV1	CsV1
	SBET, m 2 /g	264.2	233.0 (237.8) a	226.0 (211.4) a	219.8 (184.9) a	179.4 (158.5) a	150.9 (132.1) a	70.4
	Vpore, cm 3 /g	1.12	0.94	0.81	0.69	0.49	0.4	0.05
	Catalysts		10CsV2	20CsV2	30CsV2	40CsV2	50CsV2	CsV2
	SBET, m 2 /g	--	242.3 (237.8) a	234.9 (211.4) a	217.5 (184.9) a	215.0 (158.5) a	197.8 (132.1) a	122.3 -
	Vpore, cm 3 /g	-	0.96	0.81	0.66	0.6	0.46	0.09
	Catalysts		10RbV2	20RbV2	30RbV2	40RbV2	50RbV2	RbV2
	SBET, m 2 /g	--	236.1 (237.8) a	232.3 (211.4) a	215.2 (184.9) a	206.3 (158.5) a	194.2 (132.1) a	120.1 -
	Vpore, cm 3 /g	-	0.94	0.82	0.68	0.65	0.48	0.08
	Coverage, layer -	0.2	0.5	0.8	1.3	1.8

a

Acknowledgements

This work has benefited from the support of the CSC-Ecole Centrale de Lille PhD scholarship program.

Chevreul Institute (FR 2638), Ministère de l'Enseignement Supérieur, de la Recherche et de l'Innovation, Hauts-de-France Region and FEDER are also acknowledged.