Dr Nahla Dhib

Dr Arvind Ashta

How far can we go? Determining the optimal loan size in progressive lending

The microcredit literature indicates that progressive lending should reduce default rates but that it may lead to over-indebtedness. In this study, we show that progressive lending may be safe over a range of loan sizes, beyond which a rational borrower would indulge in a strategic default.

This range of loan sizes may be dependent on borrower characteristics (risk-taking, self-confidence, productivity, interest rates, subsistence needs) as well as the Microfinance Institution's strategy.

Many crowdfunding sites are using artificial intelligence to assess borrower risk through social ratings. We are arguing that production functions of the borrowers also need to be added.

Introduction

The World Bank reports that only 69% of adults have a bank account [START_REF] Demirgüç-Kunt | The Global Findex Database 2017: Measuring Financial Inclusion and the Fintech Revolution[END_REF]. Their report indicates that only 47% have taken a loan, and this is lower at 44% in developing countries. Indeed, one of the most important growth objectives of banks is lending to the poor and the World Bank is devoting considerable energy to gather data and report on the level of financial inclusion. It is important in this objective that appropriate loans are given to people depending on their specific characteristics and needs. For this, supply side constraints on providing customized loans may be overcome by using artificial intelligence wisely [START_REF] Makridakis | The forthcoming Artificial Intelligence (AI) revolution: Its impact on society and firms[END_REF][START_REF] Pansera | Unlocking wise digital technofutures: Contributions from the Degrowth community[END_REF].

In the past, bankers hesitated to lend directly to the poor owing to 1) information asymmetries that increase the risk; 2) lack of complementary social and human capital; and 3) small loan sizes that increase the cost of providing small loans, and microfinance institutions (MFIs) have come in to fill this market gap [START_REF] Armendàriz | The Economics of Microfinance[END_REF]. On the first issue, there is a large literature on the reasons for market failure, credit rationing or over-lending, owing to asymmetric information, focusing on adverse selection and moral hazard [START_REF] Akerlof | The Market for "Lemons": Quality Uncertainty and the Market Mechanism[END_REF]Stiglitz and Weiss 1981;[START_REF] De Meza | Does credit rationing imply insufficient lending?[END_REF]. In lending, the adverse selection problem of the lender is not being able to distinguish safe from risky borrowers b efore the loan is given; while the moral hazard increases if interest rates are high, thus leading to a backward bending supply curve. Over the last several decades, bankers have developed credit scoring tools that help bankers make decision on borrower risk, but these scores are influenced by credit histories (Bumacov, Ashta, and Singh 2017).

Microcredit is often considered a leading social innovation because it involves giving loans to poor people, thus giving them a chance to finance their venture and escape poverty, despite a lack of cr edit history [START_REF] Ashta | Microcredit as a Social Innovation[END_REF][START_REF] Armendàriz | The Economics of Microfinance[END_REF]. Microfinance Institutions were able to give loans with low default rates because they used a number of schemes to overcome the low transaction size and asymmetric problems of lending to the poor: for example, group lending [START_REF] Besley | Group lending, repayment incentives and social collateral[END_REF][START_REF] Ghatak | Group Lending, Local Information, and Peer Selection[END_REF][START_REF] Guttman | Assortative matching, adverse selection, and group lending[END_REF], lending only to women (especially in South Asia), frequency of loan repayments, and progressive lending [START_REF] Egli | Progressive Lending as an Enforcement Mechanism in Microfinance Programs[END_REF][START_REF] Singh | Dynamic Incentives and Microfinance Borrowers:A Comparative Study of Self-help Groups and Joint Liability Groups: A Case Study of Mirzapur District in India[END_REF]. As a result, many MFIs had loan repayment rates of above 99%. As borrowers moved from group lending to individual lending [START_REF] Dowla | The poor always pay back: The Grameen II story[END_REF], the use of credit scoring started being adopted by MFIs and this has indeed made the credit officers more productive, thus increasing their outreach and the financial inclusion of the poor [START_REF] Bumacov | The Use of Credit Scoring in Microfinance Institutions and Their Outreach[END_REF].

The second problem of microfinance was the small transaction size as can be seen in Table 1. The average loan size is $1839 in 2018, after filtering out the database of the Microfinance Information Exchange (MIX) for loans greater than $30000. However, the median loan size is much lower: the global median is $684 for 2018. The low transaction size, meant that operating and transaction costs were high and consequentially interest rates were high [START_REF] Rosenberg | The New Moneylenders: Are the Poor Being Exploited by High Microcredit Interest Rates? CGAP[END_REF][START_REF] Shankar | Transaction costs in group microcredit in India[END_REF]. One way to lower costs is to standardize loan products and operational procedures [START_REF] Khan | Cost control in Microfinance: Lessons from the ASA case[END_REF][START_REF] Akula | A Fistful of Rice: My Unexpected Quest to End Poverty Through Profitability[END_REF]. Considerable effort to reduce these costs and improve performance is also being made through advanced technologies [START_REF] Li | Convergence of the performance of microfinance institutions: A decomposition analysis[END_REF][START_REF] Bassem | Total factor productivity change of MENA microfinance institutions: A Malmquist productivity index approach[END_REF], notably crowdfunding to reduce financing costs (Attuel-Mendes 2014; [START_REF] Allison | Crowdfunding in a Prosocial Microlending Environment: Examining the Role of Intrinsic Versus Extrinsic Cues[END_REF]; Assadi 2016), information systems to reduce information processing costs [START_REF] Iyengar | Information Technology and Microfinance Institutions: Challenges and Lessons Learned[END_REF], and mobile banking to reduce delivery costs [START_REF] Morawczynski | Saving through the mobile: A study of M-PESA in Kenya[END_REF][START_REF] Shrivastava | Build it -Will They Come? A Study of the Adoption of Mobile Financial Services by Low Income Clients in South Africa[END_REF]. The most promising of these, mobile payments, has received global attention since many the unbanked in Africa and Asia have mobile phones. However, as mobile loans have gone up, repayment rates have fallen. The difficulty of the lenders is that they cannot anticipate the behaviour of the borrower if there are no credit histories. They are therefore turning to social credit scoring using the big data available on telephone and internet networks to analyse the personality of the potential borrower.

Insert Table 1 about here

The use of social data for credit scoring in the developing world was probably pioneered by Lenddo, a company that operated in Philippines and Columbia and has since expanded [START_REF] Hardeman | Lenddo's Social Credit Score: How Who You Know Might Affect Your Next Loan[END_REF]. These credit scoring algorithms are improving thanks the increasing use of artificial intelligence and machine learning. Many startups all over the world are bein g financed by venture capitalists including the International Finance Committee (IFC), the private investing arm of the World Bank group. For example, In Africa, Credit Suisse is trying to capture similar data from mobile telephones, using biometrics and facial recognition for identity checks as well as behavioural characteristics captured on their telephone including calling histories and financial transactions [START_REF] Crosman | Behind Credit Suisse's foray into microlending to the global poor[END_REF]. With the development of artificial intelligence and machine learning, and with the increasing use of mobile and internet banking, such customization of financial services to borrower's estimated needs is not only possible but will be essential to competitivity of financial service providers (WEF 2019).

Our economic intuition that motivates this study is that these algorithms will improve if, in addition to the borrower's social profile, their capacity to produce revenue greater than the loan repayment is also considered. This means that the borrower characteristics should include production functions appropriate to the size and sector of the borrower's business. These production functions themselves can be estimated dynamically based on machine learning.

To resume our study in a nutshell, lenders want to increase financial inclusion and make profits.

For this, they want to give more and bigger loans and yet ensure that t he borrower reimburses so that they make a profit. Our study shows that there is a range of loan sizes below which there is credit rationing and above which the borrower would make a strategic default, and this depends on the borrower characteristics, their production function as well as the strategy of the financial institution.

The next section provides a brief literature review of the borrower characteristics that may influence this progressive lending. We then introduce a simple model to understand the equilibrium position using Markovian Chains. Once the basics of our model is clear, we introduce how their perceived production function may limit the possibilities of their graduation to future stages, depending on their perceptions of optimal loan size and how the behaviour of borrowers depends on their perception of the lender's objective of financial inclusion.

Literature review

The economics of microfinance considers that the poor always pay back because they fear that nobody will give them a second chance if they do not repay [START_REF] Yunus | Banker to the Poor: Micro-Lending and the Battle Against World Poverty[END_REF]. Indeed, many microfinance authors indicate that part of the dynamic incentives provided by microfinance institutions (MFIs) is new loans, and even bigger loans, to successful entrepreneurs who pay back and threats to stop lending to those who do not [START_REF] Armendàriz | The Economics of Microfinance[END_REF][START_REF] Egli | Progressive Lending as an Enforcement Mechanism in Microfinance Programs[END_REF][START_REF] Shapiro | Microfinance and dynamic incentives[END_REF]). Yet, entrepreneurship evidence indicates that half the small business enterprises close within three to five years (INSEE 2012), indicating that microentrepreneurs who grow may be the exception rather than the rule and that dynamic incentives have a bias towards those who are confident of success.

At the same time, if one poor person does not pay back, the MFI knows that there is a risk of contagion and no one would pay back [START_REF] Bond | Borrower runs[END_REF]. Therefore, MFIs are vigilant on their solvency issues [START_REF] Schulte | Drivers of solvency risk -Are microfinance institutions different?[END_REF]. Since it is too expensive to use legal enforcement for such small loan sizes, the MFI must either coerce borrowers to pay back or it must find a valid reason for rescheduling the loan such as attributing the failure of the entrepreneur to an external shock (earthquakes, floods, typhoon). The MFI is therefore interested, financially, in increasing the loan size for successful entrepreneurs, as well as refinancing those failed micro-entrepreneurs who have a chance to be successful later [START_REF] Dowla | The poor always pay back: The Grameen II story[END_REF], which we call a second chance. It may also be interested in a social objective: to graduate people to financial inclusion where they may get even larger loans at market conditions. Indeed, some researchers consider that with progressive lending, it is normal that average loan sizes should increase.

Yet, there may be a limit to the loan sizes that microfinance borrowers take because it may be possible to earn less with bigger loans if the production function flattens with increasing capital, assuming a sole entrepreneur (no employee). Although there is evidence that this is the case [START_REF] Mahmood | Optimal microfinance loan size and poverty reduction amongst female entrepreneurs in Pakistan[END_REF], there is still a need to explain the existence of this optimal loan size theoretically.

The optimal loan size may differ for heterogeneous people and segments of people. For example, [START_REF] Agier | Microfinance and Gender: Is There a Glass Ceiling on Loan Size?[END_REF]) find that there may be glass ceilings from the lender's side and women are provided smaller loans, and they therefore make smaller profits. However, it is also possible that women may consider that the optimal loan size is smaller and may request smaller loans and make smaller profits.

They may, therefore, be less inclined to graduate to large loans and towards complete financial inclusion till their self-confidence and skills are improved.

We know that MFIs lend more to a village that it close to market or if there is high agricultural productivity [START_REF] Mallick | Cost effectiveness or serving the poor? Factors determining program placement of NGOs in Bangladesh[END_REF]. This may be because MFIs seek to give loans where the repayment rates are expected to be higher. In other words, MFIs intuitively look at the production function of the borrower. This is similar to banks which also give more loans in expansionary business cycles when expected production is more important [START_REF] Hyun | Heterogeneous lending behaviors and gross loan flows in developing economies[END_REF]. Therefore, in this paper we incorporate the importance of the production function into our model. This paper is prescient of the importance of this question of financial inclusion in the years to come. Indeed, as microfinance markets grow and saturate, loan sizes are increasing because MFIs try to offer bigger incentives to attract clients. MFIs may offer different kinds of loans because clients have different preferences: risk averse borrowers are less willing to take up individual loans and less selfish borrowers may prefer joint liability loans [START_REF] Baulia | Take-up of joint and individual liability loans: An analysis with laboratory experiment[END_REF]. Therefore, customization is essential for growth.

Along with growth of MFIs, default rates are increasing since screening levels are falling and borrowers may borrow from multiple lenders [START_REF] Mallick | Cost effectiveness or serving the poor? Factors determining program placement of NGOs in Bangladesh[END_REF]. As borrowers resort to multiple-borrowing, a recognized problem in growing and saturated markets, they are effectively increasing their probability of a second chance. Although credit bureaus may limit multiple borrowings, even multiplying borrowing from one MFI to two, increases the probability of a second chance. Both these factors increase the effect of default (size and probability).

In this study, we contend that MFIs could look at the production function of the MFI and use it to decide the point at which progressive loans should stop till the production process becomes more productive. [START_REF] Tedeschi | Here today, gone tomorrow: Can dynamic incentives make microfinance more flexible?[END_REF] shows that for progressive lending to work, the borrower's profitability must be high, lending costs must be low, the risky group of borrowers cannot be too risky, and the average probability of default must be sufficiently low. However, she did not look at the range of loan sizes for which progressive lending might work. Developing her perspective, we offer further insights to borrower needs and the decision process behind borrower behaviour.

Our study helps in understanding the dynamics of the interaction between success and failure probabilities perceived by microfinance borrowers as well as their perceptions of the possibilities of a second chance (rescheduling), on the one hand, and the objectives of the MFIs to offer them loans, and then graduate them from microloans to bigger loans and, eventually, to financial inclusion, on the other hand.

Research Methodology

The research question we have asked is not why people default but how to determine at what point of a progressive loan cycle a person would default so that the lender can stop lending to an otherwise good customer.

We confirmed that these are important questions based on preliminary interviews with executives from four MFIs. Since one of these respondents indicated that they were never concerned by defaults, we checked the MIX data for the most used indicator of risk in microfinance: the PAR-30, that gives the portfolio that is at risk if payments are delayed by more than 30 days, as a percentage of the total loan portfolio. We can see from the following table 2 that indeed a quarter of MFIs may not be very concerned by the default question because they may have good control systems; but the median PAR-30 is 4.07% in 2018. The average is pulled up by extreme values and is close to the third quartile. The data shows that half of the MFIs are living with high credit risk and that a quarter may be seriously facing this problem.

Insert Table 2 about here

To answer our specific research question, the methodology we use is that of a thought experiment. Thought experiments have been used in diverse fields: philosophy [START_REF] Rawls | A Theory of Justice[END_REF], ethics [START_REF] Caste | Drug Testing and Productivity[END_REF][START_REF] Wempe | Four Design Criteria for any Future Contractarian Theory of Business Ethics[END_REF]) economics [START_REF] Margolis | A thought experiment on demand-revealing mechanisms[END_REF][START_REF] Kessler | Bequest and Wealth Accumulation: Are Some Pieces of the Puzzle Missing?[END_REF][START_REF] Lucas | Macroeconomic Priorities[END_REF], political economy [START_REF] Mankiw | Defending the One Percent[END_REF]) and business [START_REF] Elkington | The Value Palette: A Tool For Full Spectrum Strategy[END_REF]. Our thought experiment starts, as a first step, with understanding why different people chose to default which helps in understanding why people may default in different situations.

We then add, as a second step, the simple economic production function to show that the same person could default as their firm becomes bigger, even if their other characteristics do not change.

For the first step, there are many studies and textbook that explain why people default.

We provide an alternative perspective to these studies using Markovian chains. A large number of high level journals in management [START_REF] Gershkov | Revenue-Maximizing Mechanisms with Strategic Customers and Unknown, Markovian Demand[END_REF][START_REF] Ibrahim | Does the Past Predict the Future? The Case of Delay Announcements in Service Systems[END_REF][START_REF] Zhang | Performance Analysis of a Queue with Congestion-Based Staffing Policy[END_REF]) have used Markovian Decision processes highlighting their importance in furthering understanding. From our study, we show that Markovian models provide insights which could be specific to the individual decision making for taking bigger loans or for defaulting.

Markovian chains are simplified systems that are used to explain different stages in a process and the possible transition from one stage to another, using probabilities that are fixed and known. They are extensively used in economics and social sciences, but most researchers use finite number of possible states. We use the stochastic process to explain the dynamic of borrower's rational behaviour and the lender's decision which affects the actions of beneficiary. If we imagine a set of poor beneficiaries of successive loans, the use of Markovian chain lets us estimate the probability and proportion of borrowers reaching inclusion.

In an information asymmetric environment, a MFI may use a pooling model to separate the quality of borrowers (bad or good) through group lending ( [START_REF] Banerjee | Syndicated lending under asymmetric creditor information[END_REF][START_REF] Tassel | Group lending under asymmetric information[END_REF]. Of course, there may be different approaches provided by MFI' to separate the type of lending under specific assumptions. For example [START_REF] Ghosh | Funding microfinance under asymmetric information[END_REF], used a separating model to show that encouraged donors to invest in microfinance as this may lead to poverty reduction and ensure a perfect equilibrium in a market where the lender compensates the higher interest rate by decreasing his funding cost. In our case, we used. Markov chain in an individual loan setting. We do not use pooling, except if individual borrowers can be grouped by similar characteristics through cluster analysis. We analyse the sequence of successive loans during transition from a state to another, potentially leading to inclusion.

For the second step, we add a simple Cobb Douglas production function. This simple function is even simpler in microfinance because most micro-entrepreneurs work alone of have a part-time helper. As a result, the labour component of the Cobb-Douglas function can be assumed to be 1 and the production function simplifies to be dependent largely on the amount of capital invested, the productivity of the capital and that of labour and capital working together. Assuming further that poor people have some small fixed savings, the production would then depend upon the microcredit that the person can avail.

While the economics of each step is simple, the novelty of combining them leads to determining, for each individual borrower, at what point a rational borrower may prefer to strategically default. Most of the simplifying assumptions that we have made can be relaxed by using computers and machine learning, in the near future, can improve the parameters that we may have taken as fixed to describe this thought experiment.

Modeling the strategies of borrowers

In this section, we build our basic model using Markov chains. We consider a beneficiary who takes a microcredit to create a small business and produces an income. If she succeeds in the small business, she receives successive larger loans. We assume that at the first state M, she invests and if she succeeds, she gets to the second state B with a bigger loan. At each stage, the behaviour of borrower is influenced by the financial capital and human capital as well as future expectations. These future expectations are based on a production function which tapers off. Costs increase linearly. Therefore, profit expectations are parabolic, initially increasing and later decreasing. To take care of this, we need three stages rather than two: the first two stages could be on the increasing phase and the third could be on the decreasing phase.

Therefore, we suppose that if she succeeds at B, she gets to inclusion state I. At any time, if she loses, she returns to exclusion state E. We define a trajectory ω = (M, B, I, E) where M is initial micro loan size, B is big loan size, I is inclusion, E is exclusion.

Additionally, we assume that the lender has a financial and a social objective and seeks to maximize the proportion of inclusion (π). This means that the lender would like the borrower to improve her financial state and succeed in her activity. This would also allow the borrower to repay the loan, ensuring the profit of the lender. The borrower's success depends on her ability to manage her small business and she identifies her strategy during the trajectory taking in account the success probability of the small activity.

We suppose that the economic activity of the borrower follows a trajectory ω with known transition probabilities from each state (αM, αB , αI , γE ). We further consider that the dynamics of borrower behaviour depends on these success probabilities that can be captured by a Markov chain. These alphas capture not only the capacity of the borrowers but also their preferences. For example, some borrowers may prefer to remain unemployed unless they find an extremely high return project letting them become millionaires. Others may be willing to work for a small profit to avoid a situation of idleness. This may partly depend on their respective initial wealth. Similarly, the alphas also capture their desire to make a strategic default or not to do so. We assume that 1 -γ captures the expectation of the borrower to get a second chance if she loses (and defaults). This expectation may be determined with respect to stationary strategy of the lender in terms of providing refinance, the number of alternative lenders, and if the lenders have information on the previous defaults of a borrower through credit bureaus. This implies that decisions of the borrower are influenced by perceptions of lender's actions. Any borrower's actions are determined by success probability pi of a process of Markov Chain (Xt), defined at a state (i) of a set (ω) with πi > 0,

), πi = 1. This stochastic process is illustrated in Figure 1.

Insert figure 1 Referring to the property of Markov chain to solve a stationary distribution, we define a vector of lender's objectives π whereby he aims to bring his borrower to financial inclusion through a trajectory ω = (M , B, I, E). This vector summarizes the proportion of borrowers who move from a stage to the next one.

For example, the lender may have an objective of providing micro loans to a fraction of the excluded people. Later, he may graduate a fraction of this first fraction of people to a bigger loan. Then, he may graduate another fraction of these second fraction of people to inclusion. Finally, he may try to keep on servicing a fraction of the included people by matching the offer of banks.

Therefore, the borrower is facing an MFI which has its own objective for borrowers in different states (M , B, I and E). For example, it may decide that it would like 20% of the excluded to be given loans (πM ), 20% of those who get microloans to be graduated to big loans (πB ), 20% of the big loan recipients to get even larger financial inclusion loans at commercial rates (πI ). Finally, it may decide that 20% of those who default would be given a second chance (πE ). These objectives of the lender are captured in the vector,

V = (πM , πB , πI , πE ) (2)
Equilibrium distribution will result when we multiply the borrower's transition matrix P by the vector of the lender's objectives V.

To optimize the borrower's inclusion position, we maximize the previous expression (3) to provide an equilibrium state, which gives, This equilibrium stage will be achieved only for 1 -αM > 1 -αB or π* E =0

We can see from equation ( 4) that an optimal strategy for inclusion is dependent on the risk of success or failure in each State as well as the probability 1 -γ capturing the expectation by the borrower of the proportion of people to whom the lender will provide a second chance and πE is the proportion of excluded people to whom the MFI would like to offer loans. This objective may depend on the funding availability of the MFIs as well as its perception of the human and physical capital and chance of success of borrowers in that area. These do not necessarily exclude those who have lost once since the MFI will look at the whole market of the excluded.

In each case, we keep four States where a borrower can either advance one stage in a period or return to exclusion.

I. Dynamics of Interaction between Borrower Strategies and lender

Objectives Assuming all probabilities of transition are known, we could have an optimum solution which results from the interaction of the strategy of the borrower and the objective of the lender to lead the borrower to financial inclusion. What would this be? What do we learn from the stochastic process?

Each borrower is different, and we could have thousands of cases which could be regrouped. Here, we illustrate with five cases, with different probabilities of transition from one stage to the next, reflecting that borrowers are heterogeneous in capabilities and preferences. For example, in case 1 we assume equal probabilities of 0.25 of success but with larger probability of failure, but most failures lead to an expectation of a second chance (with probability of 0.75); in case 2 we increase the probability of success with large loans and maintain the high probability of second chance; in case 3, we maintain the low probability of success of case one but reduce the probability of second chance to 0.2; in case 4, we increase the probability of success further and take an intermediate perception of the probability of second chance. The Table 3 clarifies the differences between these 4 simple cases. Finally, in Case 5, we introduce more complexity where the borrower can not only succeed to the next stage but could also remain in the same stage or even go back one stage, with an intermediate probability of a second chance of 0.4.

Insert Table 3 about here

Case 1: In this situation, we can suppose that the lender is financing risky activities since the success probability is low (αM = 0.25). This means that it may lead to a difficult situation where bad debts are anticipated (αI = 0.25). Its detrimental for the MFI to lend to borrowers in such situations since it is unlikely to lead to financial inclusion. Therefore, the lender needs to provide a second chance, and this is captured by the high probability of 0.75 (1 -γE ) to try to bring the borrower back towards inclusion. This provides borrowers a situation where they can make a strategic default, knowing that they may get a second chance. In fact, they carry on their risky investments with the same probability of success (αB = 0.25).

However, the second chance means that loans are being given to refinance by deferring the loan repayment. This is usually the case where the default amount is a small part of the microloan. Otherwise, if the default is a large part of the loan, it means that if the business is to continue, the total size of the loan increases and the total risk increases for refinanced borrowers because they have not yet repaid the first series of loans and the fact of taking a second microcredit makes the borrower more vulnerable and increases her debts. To make it simple, we assume that the borrower will start a new business and start the markovian cycle with a loan (M) either from this MFI or, more probably, from another MFI who does not have access to the credit history of default.

We note that different iterations with different values for an initial starting vector V for the lender will change the final V. However, the proportions as a function of πE remain stable at those indicated in V * below. These proportions are dependent purely on the parameters of the transition matrix (1). We deduce that the best cycle for this combination of borrower strategy and lender objective is cycle 3 for Microloans, Inclusion and Exclusion, while bigger loans perform better in cycle 4. This means that success in large loans after success in microloans is difficult to achieve and, therefore, the chance of exclusion goes up and the chance of inclusion reduces. This could explain the initial optimism of microfinance. For example, Servet (2015) and Tedeschi (2008) show that successive loans can help some and increase the vulnerability for others, but they do not mention why.

We also deduce that the result of the lender at the end of the period shows the fraction of borrowers who achieve the inclusion state by trajectory of (M,B,I,E) is less than the part of exclusion. This consequence creates a disappointing result for the lender who seeks to maximize inclusion but finds that a very small part of the excluded population attains this state.

We reach an interesting conclusion which can help the lender to optimal decisions. If the lender selects risky borrowers, the lender should assure the accompanying and the controlling of borrowers to improve her business and increase the profit probability (αB > αM ), but this increases the costs for the lender. This strategy of accompaniment and coaching reduces the vulnerability of the borrower by improving her skills, thus leading to graduating to a larger loan size with less chance of failure. We examine this in case 2. Alternatively, to improve inclusion, the lender could also reduce the perception of the possibility of second chance of a borrower by increasing his γE. This would then entail a decisive step against strategic default and facilitate the selection of less risky activities by the borrower. We examine this later in case 3.

Case 2: We now study a second borrower with better capabilities for running a small business (B) and otherwise similar preferences to case 1 (see Table 3 for details). During this trajectory, compared to the borrower in case 1, the situation of borrower is the same in the first period (M to B), but is less risky in the second stage because of economies of scale or owing to better management due to better skills (B to I) and therefore we assume that the borrower increases her success probability αB > αM . However, when the business becomes too big, she is unable to manage it any better than the first case. The borrower assumes that there is the same high chance to be given a second loan of size M by the lender. Again, we find that she prefers to return to E than to carry on to I.

We find that the part of inclusion after 4 states is better for the second borrower than the first case because of the higher (αB > αM). However, the lender still finds that the stable situation is below the initial objective of inclusion. The borrower carries on her micro-business, but it never leads to inclusion since she anticipates a low probability to succeed if this business becomes too large. We can deduce that lender loses the chance for creating inclusion.

The lender may, therefore, require a strategy of providing skill training to the borrower to improve the success probability in State I. This would then promote the lender's objective of financial inclusion (we will see this in Case 4). However, we first see, in Case 3, the effects of a strategy of the lender to decrease the borrower's perception of the probability of second chance to return to the state M after losing in States M, B or I.

Case 3 : We examine what happens if the MFI reduced the borrower's perception of a second chance (see Table 3). We find, therefore, that if the perception of a second chance is low, the borrower would have an even lower preference to continue to the inclusive state, compared to the first case. Overall, it is still not very encouraging for the lender since a lot of people are excluded and they will have high bad-debts of the people who fail along the way. However, it allows them to rotate the borrowers and have new borrowers (in the following stage) who may succeed better.

Comparing case 2 to case 3, we can say that it is better for the MFI to select more competent borrowers or less risky activities than to reduce the perception of a second chance. This would then make a compelling case for using credit scoring tools.

Case 4: It is clear that if we took a combination of case 2 and case 3, the inclusion rate may go up or down. However, if we want inclusion to go up substantially, we need a bigger change in assumptions and select borrowers who are more competent or activities that are less risky and keep some intermediate chance of refinancing (see table 3).

Since the probability of success is higher than the first three cases, the borrower represents a good client for lenders who select less risky activities and more competent micro-entrepreneurs. Assuming that borrowers finance their small business by successive loans, we calculate the part of inclusion. We find from this case that it is possible to bring more than half of all potential borrowers into inclusion.

However, this may require compromises: for example, the MFI may need to discriminate against the most vulnerable persons who may be less competent and riskier. As a result, the MFI may have to adopt a topdown approach in the sense of first including the not-so-poor and then gradually moving to the very poor.

Case 5: Having understood the simple Markovian models with either/or possibilities in each case, we now introduce a slightly more realistic model where the borrower can progress, remain in the same situation, or regress. This means, for example, that one can progress (from M to B) or that one can remain still in the same state (M to M), or one can return to previous states (B to M). We suppose that α is the probability of the borrower to remain in the same state during her trajectory ω = (M , B, I, E), β is the probability when the borrower succeeds to get to the next state and λ is the probability of the borrower returning to a previous state. However, in each state, the borrower has a risk of failure and falling back to exclusion represented by (1 -αβ). For this case, we assume that the lender proposes an intermediate perception of probability of second chance (1 -γ). Based on these assumptions, the trajectory of the micro entrepreneur is illustrated by the figure 2:

Insert Figure 2 about here As in the preceding cases, the lender aims to maximize inclusion. Referring to the property of Markov chains (see equation (2 and 3), the following expression describes the stationary distribution,

We find that with such dynamic assumptions, and a good interaction between borrower strategies and lender objectives, we can have a reasonable inclusion rate.

Based on the above, we can see that the MFI and the borrowers can play on the different success rates and perceptions, as well as on the final inclusion objectives of the MFI. The use of Markovian model lets us understand the impact of each of these in the different states and the MFI can play on these to select its borrowers as well as help them develop appropriate skills for greater success. This explains why many MFIs and crowd lending sites are building complicated credit scoring models to incorporate borrower characteristics.

However, we can go one step further, and see how Markov chains help us learn more about the fragility of customers falling from inclusion to exclusion or rising from exclusion to inclusion by introducing an additional element: the expected cash flow produced from small business financed by small loans.

Insert Figure 3 about here

II. Introducing Expected cash flows into the Markov Chain analysis

We introduce a rationality constraint incorporating that success means earning an income from the business. We assume that cash flows and income are the same since it is only working capital that is usually financed in thAis setting. It is only if borrowers cover all charges in each state that they can grow in the small business from M to B and from B to I by increasing the size of loan. In terms of our diagram (figure 3), we suppose that a borrower can provide her irreducible consumption at a non-zero intercept but that as income increases owing to loans, she pays more interest to finance her small business. Therefore, the consumption is upward sloping. The borrower produces her net income (gross of interest expense) according to the concave curve. We are effectively assuming that borrowers have no other income.

The where the lender offers his borrower a lower interest rate (r -< r) to stay with the MFI, rather than going to a bank, in the inclusion state.

Using our Markov chains of figure 1, we calculate the expected net cash flow (Z) on transition from actual state to next one by the following equation,

Z(xt, xt+1) = E(f (xt+1)|xt) ( 14
)
What can we then say about the optimal loan size as a percentage of Z and how does the set of probabilities of success influence it? To answer this question, we discuss the optimal cash flow expected (Z) on transition from one state to another which leads to financial inclusion depending to the size of microcredit.

Z(xt, xt+1) = αx t * g ( xt, xt+1)(K, r) (15)
To link the two parts of our discussion, we illustrate in the following cases how this ties together. For each case, we take three scenarios with loan sizes of 1000, 2000 and 3000 corresponding to states M, B and I.

Although meant to be purely illustrative, these are reasonable in view of the world average loan size as can be seen from Table 1: the microloan is smaller than the world average, the big loan is slightly bigger than the world average and the borrower included has a much larger loan size. We assume the expected cash flows for the business model are the same for any business, borrowers are heterogeneous and their success probabilities (α) are different, as stated in equation

We develop the five cases of part IV and for this we take their respective αs and the γ. Figures 4 to 8 show the expected profits for the five cases in each of the four states (M,B, I E) as determined by the assumed production function.

Insert Figures 4 to 7 about here * Case 1: At first glance, expected profits are increasing in each state and we would expect the rational borrower to be incentivized by bigger loans. However, if the borrower has a high expectation of a second chance, we note that in exclusion she has an expectation of Z(E, M ) = 49.47. Therefore, rather than go from M to B, she would prefer to default and go to E. Our first insight is that there are situations where only the borrowers who are straight away given loans of size B that feel incentivized to want bigger loans of size I. Therefore, we can conclude that for this kind of borrower, we may need to give bigger loans from the beginning. Alternatively, in opposition to part I, the lender needs to reduce the perception of the second chance of the borrower. This additional insight would not have materialized if we had not introduced the production function.

*Case 2: This kind of borrower has higher expectations in the intermediate Big loan than in the biggest I size loan or in the exclusion state. So, for this borrower the best would be to stay in the B state rather than opt out to go to exclusion or try to progress to inclusion. However, in the Markov chain we defined that she cannot stay in the same state. The MFI will try to push her to I, and she prefers this to exclusion where she expects only Z(E, M ) = 49.47. Our second insight is that there are situations where full inclusion may not be the best outcome for poor borrowers faced with such probabilities of success.

Alternatively, as discussed in the previous part, the lender must provide training so that the borrower increases her perception of success in the last state and wants to go towards financial inclusion.

*case 3: αM = 0.25 < αB = 0.5 > αI = 0.25: As shown in Figure 6, if the borrower carries on her small business on trajectory (M to B to I), she expects earning (Z(M , B) = 16.49 < Z(B, I ) = 47.84 < Z(I, I ) = 63.69. Therefore, normally she is incentivized to continue towards financial inclusion. Moreover, since we have drastically reduced her perception of second change to 0.2, we find that her expectation in exclusion is very low Z(E, M ) = 13.19 . This confirms our insight that the threat of stopping lending is a great incentive to continue towards financial inclusion, if the borrower perceives that the threat is realistic.

*case 4: Although the numbers differ, we see that the results are similar to those of case 3. This kind of borrower is more confident and has higher expectations of success and would also not be tempted to opt out.

*case 5: As mentioned earlier, the first four cases provide simple insights that can be gained by using the Markov chain, while the fifth case is more realistic and the new Markov Chain illustration (figure8) provides a very rich understanding. The total expected probabilities seem to give a simple result, comparable to cases 3 and 4. However, the devil is in the detail because no one has the possibility to do all the different things: they must choose between progressing, staying in the same state, regressing or defaulting. As shown in figure 8, the person who decides to progress seems to be most rational since all these probabilities are increasing Z(M , B) < Z(B, I ) < Z(I, I )) and more than the expectations with a second chance Z(E, M ). However, what happens to someone who fails in her quest for progress? We find Z(M , M ) < Z(B, B)) if staying at same state. This means that those who fail in the M stage, would be tempted to default (Z(M , M ) < (E, M )), while those who are in the B Stage would be happy to stay in that state (Z(B, B) > Z(E, M )). However, if the loss is too high, then the borrower needs to come back and the expectations reduce (Z(B, M ) < Z(I, B)) if returning to previous state. Here, we find that the borrower would rather default than be given a smaller loan since these expectations are lower than Z(E, M). Thus, the insight we get from all this is that once such a person is financially included, it is difficult for her to borrow less. She would be happier to default than get smaller loans.

Insert Figure 8 about here Depending on the risk preferences of the borrower, her expected incomes, the perception of possibility of a second chance, and her ability to manage a small business at different scales, she may either continue her business or exercise a default option.

III. The impact of optimal loan size on financial inclusion objectives

This decision to continue or to default may depend on the optimal loan size. If the borrower is below the optimal loan size, she would try to stay along the trajectory of financial inclusion. However, if she perceives that taking a bigger loan is beyond the optimal loan size, she may not proceed further, and may even default. Therefore, it is important for the lender to understand what each borrower considers her optimal loan size, depending on her productivity and in each stage (M,B,I,E).

The lender can then decide his loan offerings according to the optimal loan size for each borrower type, which in turn depends on their human capital and their level of consumption. As illustrated in Figure 9, continuing from equation ( 23), in each state, g = Y -C needs to be positive. Therefore, there is a minimum capital size, Kmin and a maximum capital size Kmax, which determine the range of viable alternatives. However, we can see that there is an optimal capital size which maximizes the net gain (g), i.e., it maximizes the difference between Operating Income (Y) and Consumption (C). We can see that the more the size of the loan increases, the borrower anticipates more charges (KB > KM => r * KB > r * KM ) during the following transition state. In fact, she should produce a net income g(KB , r) > g(KM , r). This will be true if the production function Y is still greater than the consumption function C.

Insert figure 9 about here

We can deduce the optimal loan size k * for any productivity fraction of θ provided by the borrower from the equation 23.

K * = (η * θ/r) 1/(1-θ) (16)
The additional insight that we can get from this is that since this gain (g) reduces after k * , the borrower will not want a loan bigger than K * . Therefore, if the financial inclusion loan size (I) is more than K*, the borrower may prefer to stay at a lower level (M or B) or even default (E). Thus, the lender needs to either ensure that the loan size for financial inclusion (I) is smaller than the borrower's k * or that he finds way to increase the borrower's k * by either lowering interest rates or by suggesting more productive business lines to the micro entrepreneur, thus increasing the productivity of capital (θ) or by increasing the productivity of human capital through training (η).

Discussion, Practical implications, and Concluding Remarks

Part 3 of this study has shown that Markovian models are useful to understand the interaction between borrower strategies and lender objectives by clarifying that there are diverse kinds of borrowers and lender strategies need to adapt to this heterogeneity. Part 4 shows that these different borrowers may choose to default if they are not provided the correct loan sizes and appropriate training or if they consider that they are highly likely to get a second chance. One recommendation that follows is that MFIs need to provide training to borrowers to improve their human capital. Of course, this would need to be modelled since the costs of the MFI would increase.

We note that our model has links to what is happening today in the real world of using artificial intelligence and data analytics for credit scoring. Table 4 provides examples of some lenders who are using credit scoring, often aimed at small businesses, microcredit or financially excluded people. Many of them indicate on their website that they are using alternative data to the traditional financial or operating data. This has been made possible with new technologies. We also take an excerpt from each website to indicate the use or impact of their work, as claimed by them.

Insert table 4 about here Moreover, most researchers indicate that the lender proposes successive loans with increasing size as an incentive mechanism to improve the profit. However, our analysis in part 5 shows that this incentive will not motivate the borrower if the production function does not allow greater income, i.e., if the business is not scalpel beyond K * . Therefore, borrowers may prefer not to graduate to successive lending

states. Formally, we propose

Proposition 1: There is a range of loan sizes where progressive lending works. The upper limit of this range depends upon the borrower characteristics as well as the production function.

Proposition 2: The use of Markovian chains is essential to determining this upper limit.

We have noted that the rational borrower of our type may prefer to default in many situations.

However, owing to moral and social considerations, they may not default, leading to high stress and even suicides emanating from shame [START_REF] Ashta | Does Microfinance Cause or Reduce Suicides? Policy Recommendations for Reducing Borrower Stress[END_REF][START_REF] Karim | Demystifying Micro-Credit: The Grameen Bank, NGOs, and Neoliberalism in Bangladesh[END_REF]. Future research can factor in these considerations for understanding why borrowers default. Part 6 shows that this action of the borrower may affect the microfinance institution's objective of financial inclusion.

Certainly, the qualitative evaluation of the probability to default or reimburse depends on many variables. However, if we have data, a future researcher could run a Cox regression which would validate our model by estimating the probability of default or reimbursement connected to different variables (gender, family conditions and obligations, the objective to improve business, education and many other).

With such a model, the MFI can draw a prototype of ideal borrowers who would get to inclusion and detect the weak points in the chain which deserve attention to avoid default.

We have already indicated that with the development of artificial intelligence and machine learning, these issues will increasingly be incorporated in credit scoring models for decision -making. Our study suggests that in addition to these borrower characteristics, the lender may also be able to incorpor ate borrower's production function in his lending strategy, which should also be possible with the development of artificial intelligence and machine learning. In fact, AI and machine learning may also refine the estimated production functions over time.

We note that our model is limited to entrepreneurial lending where successive loans can be repaid by higher incomes: it is not applicable to consumer microlending.

  about here Drawing all probabilities in Figure 1, we define the transition matrix (P) where the rows indicate the current state and the columns indicate the transition state, as shown below (equation 1).

  borrower aims to maximize her earnings (g = Y -C) after taking care of her reimbursement and her consumption needs (C). The income (Y) is assumed to follow a Cobb-Douglas function, and we accordingly suppose that the borrower produces an income Y (K, L) = K θ * L 1-θ , where K is capital and L is Labour. In the case of our micro-entrepreneurs (L = 1) since they do not employ anyone else, they each produce a total income Y (K) = η * K θ ) where η > 0 includes the human capital added by the coaching and training of lender. The borrower expects financial charges (K * r) which represents the reimbursement of loan and the consumption needs(C = K * (1 + r) + c). Note that, g(K, r) = η * K θ -(K * (1 + r) + c) (12)Combining this constraint with our earlier discussion on Markov chains in part 1, we can say that the continuation or progress of a borrower in states M, B or I assume that she expects a positive result g in each state. Otherwise, she would default and go to E. Thus, the alphas of success in M, B and I are a function of g being positive, which itself depends on Y and C, where Y depends on the productivity of capital and the coaching provided by the lender.To calculate the optimum contract on this scenario, we suppose a function f for any borrower (x) who has a trajectory of states (M , B, I, E),

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Markov chain describes the dynamic of the borrower's strategy depending on

Figure 8 :

 8 Figure 8: Expected profit (z) with possibility of advancing, staying in the same state retreating to a

Table 1 :

 1 Average loan size in microfinance(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018) in US dollars

	Average loan balance per	2010	2012	2014	2016	2018
	borrower (USD)
	Africa	743	830	1055	937	1351
	East Asia and the Pacific	1055	1380	1251	1810	1777
	Eastern Europe and Central Asia	3299	3422	3726	2415	2802
	Latin America and The Caribbean	1720	1946	2236	2269	2555
	Middle East and North Africa	933	839	905	877	1077
	South Asia	237	226	223	276	347
	Grand Total	1410	1553	1714	1541	1839
	Source: Based on MIX data downloaded on March 18, 2020, after filtering out balances over
	$30,000. The number of MFIs reporting this data declined from 1381 in 2010 to 684 in 2018.

Table 2 Portfolio at risk (30 days) of microfinance institutions (2010-2018)

 2

		2010	2012	2014	2016	2018
	Sample Size
	(number of MFIs)	1058	906	932	716	644
	Minimum	0.00%	0.00%	0.00%	0.00%	0.00%
	First Quartile	1.51%	1.26%	1.19%	1.73%	1.60%
	Median	4.18%	3.81%	3.70%	4.56%	4.07%
	Average	8.00%	7.05%	5.83%	8.39%	7.17%
	Third Quartile	9.33%	7.68%	6.59%	8.64%	7.66%
	Maximum	99.40%	97.74%	98.57%	98.70%	99.64%

Note: Based on MIX data downloaded on March 18, 2020, after filtering out MFIs that report 100% or higher PAR 30

Table 3 : Alternative probabilities of success and second chance Success probability Second Chance Probability

 3

		Low	Intemediate	High
	Low	Case 3
		(0.25, 0.25, 0.25, 0.20)
	Intermediate		Case 4
			(0.5, 0.7, 0.8, 0.4)
	High	Case 1	Case 2
		(0.25, 0.25, 0.25, 0.75)	(0.25, 0.50, 0.25, 0.75)
	Source: Authors

Table 4 :

 4 Credit scoring applicationsSource: Authors, different websites last accessed on June 16, 2020.Kreditech has proven the predictive power of its proprietary algorithm by recently performing a backtest on two million scored customers to show that big data combined with traditional data produces default rates lower than the industry standard.FarmDrive collects and aggregates alternative datasets from multiple sources, in Kenya and around the world, to build credit scores for smallholder farmers in Africa.

							Website/blog
	Firm	Countries/Base Kind of data	Aimed at	Technologies	Excerpt from website/blog	accessed on 25 to 28 May
							2020
		Singapore-	Psychometric data, financial
		based,	transactions, form filling
	LenddoE FL	Philippines, Columbia, Mexico, India, S. Korea (now	analytics, e-commerce transaction, social networks, mobile data, browser data, telecom data, internal data,	Financial inclustion, reaching new clients	1I and advanced analytics	The LenddoScore complements traditional underwriting tools, like credit scores, because it relies exclusively on non-traditional data derived from a customer's social data and online behavior.	https://www.le nddo.com/abou t.html
		over 20	credit bureau data, applicant
		countries)	data
		US-based,	Daily life data from smart	Anyone with a		Tala offers the world's most accessible consumer credit
	Tala	Kenya, Tanzania, Phillipines,	phone, ncluding financial transactions, savings, network diversity, and geographic	mobile phone, regardless of their credit	Analytics	product, instantly underwriting and then disbursing loans to people who have never had a formal credit history, all through a smartphone app. Loans range	https://tala.co /about/
		Mexico, India	patterns	history.		from $10 to $500.
							https://www.k
							reditech.com/
	Kredittec h	Germany-based, Poland, Spain, Romania, Russia, India	Alternative data	Access to finance for those with no credit score	Big Data, Machine learning algorithms		news/post-by-the-ceo-redefined-scoring-logics-from-5-to-15-
							000-data-
							points
						ID Finance provides financing for those who cannot get
	ID Finance	Spain-based, Mexico, Brazil	Large data arrays	Microcredit	Machine learning, AI	funding from traditional financial institutions, by identifying non-trivial patterns in unstructured data	https://idfinance.com/sustain
						that banks typically ignore.
	FarmDriv e	Kenya	individual and social data from mobile phones, key satellite, agronomic, economic, environmental data	Smallholder farmers	Machine learing algorithm		https://farmdriv e.co.ke/credit-scoring
						Juvo builds highly predictive Financial Identities
	Juvo	US-based, Latin America, now worldwide	mobile transactions and interactions	Unbanked with a telephone	Machine learing technologies	through mobile network operator billing, usage, and airtime lending data. Lenders can apply those insights to target qualified leads, screen applicants, and	https://juvomo bile.webflow.io /
						optimize loan portfolio management.
	Branch	US-based, Kenya, Tanzania, India, Mexico,	Alternative data	Those with little or no credit histories	Machine learning algorithms	Applying machine learning, Branch has created an algorithmic approach to determine credit worthiness via customers' smartphones.	https://branch. co/about
			Smartphone data, Social media			Colendi takes into account a variety of real-time, non-
	Colendi Turkey	data, Transaction data, Blockchain credit history data,	Unbanked/unde rbanked	AI and blockchain	financial information which is transformed into a financial credit score using machine-learning	https://www.co lendi.com/faq
			Personal data.			algorithms.
						Since 2007, OnDeck has been transforming the way
						small businesses access capital, using our proprietary
						technology and small business credit scoring system,
			Data range from cash flow and			the OnDeck Score®, to efficiently evaluate a business'
			transactional data to public			creditworthiness and make targeted lending decisions	https://www.o
			records to our own extensive			in real time. The OnDeck Score is a better alternative	ndeck.com/res
			internal historical performance		big data &	to personal credit scores for assessing Main Street's	ources/ondeck
	0ndeck US-based	data.	Small business	analytics	credit health.	score


Alberto Alvarez-Fernandez

Frédéric Nallet

Philippe Fontaine

Cian Cummins

Georges Hadziioannou

Philippe Barois

Guillaume Fleury

Virginie Ponsinet

Large area Al 2 O 3 -Au raspberry-like nanoclusters from iterative block-copolymer self-assembly

In the field of functional nanomaterials, core-satellite nanocluster have recently elicited great interest due to their unique optoelectronic properties. However, core-satellite synthetic routes to date are hampered by delicate and multistep reaction conditions and no practical method has been reported for the ordering of these structures onto a surface monolayer. Herein we show a reproducible and simplified thin film process to fabricate bimetallic raspberry nanoclusters using block copolymer (BCP) lithography. Inorganic raspberry nanoclusters fabricated consisted of a ~ 36 nm alumina core decorated with ~15 nm Au satellites after infusing multilayer BCP nanopatterns. A series of cylindrical BCPs with different molecular weights allowed us to dial in specific nanodot periodicities (from 30 to 80 nm). Highly ordered BCP nanopatterns were then selectively infiltrated with alumina and Au species to develop multi-level bimetallic raspberry features. Microscopy and x-ray reflectivity analysis were used at each fabrication step to gain further mechanistic insights and understand the infiltration process. Furthermore, grazing-incidence small-angle X-ray scattering studies of infiltrated films confirmed the excellent order and vertical orientation over wafer scale areas of Al2O3/Au raspberry nanoclusters. We believe our work demonstrates a robust strategy towards designing hybrid nanoclusters since BCP blocks can be infiltrated with various low cost salt-based precursors. The highly controlled nanocluster strategy disclosed here could have wide ranging uses, in particular for metasurface and optical based sensor applications.

Introduction

The appeal of nanostructured bimetallic surfaces is expanding in many fields of research and technology, due to their interesting specific properties (catalytic, magnetic, ferroelectric, mechanical, optical and electronic). Many applications are now possible based on bimetallic (or trimetallic) features such as analytic spectroscopy, [1][2][3] data storage, 4 optics and electromagnetism control, 5,6 or control of wetting and antifouling. 7 The search for diversity and expanded functionalities leads to an ever-revived search for new fabrication methodologies, giving access to nanostructures including metal, inorganic or hybrid materials, with controlled morphologies and characteristic sizes. Self-assembly inducing spontaneous organization in soft condensed matter is undoubtedly a promising tool. 8,9 Some self-assembled structures, which spontaneously organize due to intermolecular interactions of energy close to thermal energy kT, present characteristic sizes at the nanoscale and variable degrees of order, including liquid crystals and block copolymers (BCPs). BCP nanopatterning is only one of the many opportunities offered by macromolecules at interfaces or other selfassembled nanostructures. But they indisputably present many advantages 10 compared to alternative methods for the production of nanopatterned surfaces, including the tunability of domain size and morphology, the ease of processing on many types of surfaces and the possibility to smarten the systems up with chemical functionalities. BCPs in their simplest form i.e. a diblock copolymer A-B (di-BCP), are comprised of two distinct polymer chains that are covalently joined. BCP self-assembly is a well-known strategy to design and create ordered structures in thin films with tunable size and morphology through the control over the BCP macromolecular parameters. di-BCP material characteristics such as the relative volume fraction of the blocks (φ), the degree of polymerization (N), the Flory-Huggins interaction parameter (χAB) as well as the thin film process (i.e., deposition and annealing conditions) enable the fine control of morphology and feature sizes. 11,12 Subsequently, the production of large scale arrays of nanoparticles (NP) has been demonstrated by the selective hybridization with metallic or dielectric species of the nanostructured BCP scaffolds. [13][14][15][16][17] However, the energy minimization during self-assembly process for common di-BCPs limits the number of accessible NPs patterns such as lines or dots. 11,12 Multi-BCPs represent a possible alternative, giving access to many more morphologies, however accessing multi-BCPs relies on complex synthesis. 18,19 Recently, pioneering works from Rahman et al 20 have proposed that this bottleneck can be overcome using a methodology based on the iterative assembly of BCP layers. Indeed, multilayered BCP self-assembly was used to produce a library of three-dimensional structures that are absent from the native BCP phase diagram, i.e. sphereson-lines or lines-on-lines, paving the way to the production of on-demand 3D structures from BCP self-assembly. [21][22][23] This process, applied with creativity and ingenuity, will give access to a wide array of original final morphologies. It contributes to the other appealing characteristics of BCPs outlined above. The transfer of the morphology to materials able to provide solidstate properties (e.g. magnetism, plasmonics or catalysis), requires another clever processing step. Patterning of metals has been extensively reported [24][25][26] and infiltration of oxides is increasingly studied 27 and has reached a level of maturity to understand growth kinetics and thermodynamics. 28,29 However, hybrid nanoclusters, combining more than one material, remain a challenge. 30 Core-shell colloids have been proposed to tackle the manufacturing of hybrid nanoclusters, [31][32][33] but reports on combining multiple materials and complex structure are scarce due to a lack of reproducible fabrication methodologies. Core-satellite raspberry-like nanocluster architectures, in which satellite nanoparticles surround a central core nanoparticle are recently of great interest due to their unique optoelectronic properties. 34 Designing core-satellite raspberrylike nanoclusters has impacted a broad range of application areas, such as surface-enhanced fluorescence, 35,36 Raman scattering, [37][38][39][40] optical magnetism, 41 drug delivery 42 and sensing. [43][44][45] Bottom-up approaches based on colloidal chemistry have been revealed as possible strategies in the synthesis of such objects. 46 However, tedious chemical pathways associated to complex nanofabrication processes significantly hinder both scalability and development. Furthermore, even if the self-assembly of these complex structures into different 3D structures has been reported through different processes, i.e. microfluidics 47 , DNA interactions 48 or non-specific electrostatic force 49 , the ordering of such structures into a surface monolayer still constitutes a scientific challenge. [START_REF] Nanz | Optics InfoBase Conference Papers[END_REF] Thus, simplified routes, i.e. methodologies that are based on rapid and controllable processing, are needed to overcome the significant roadblocks of colloidal chemistry strategies. In this regard, the inherent characteristics of BCPs provide a possible solution to forming raspberry-like nanosurfaces.

In this work, we demonstrate the robust fabrication of coresatellite Al2O3/Au raspberry nanoclusters using multi-layered self-assembled BCP films. This methodology leads to welldefined arrays of bimetallic raspberry nanoclusters obtained from simple processing steps using conventional wet chemistry practices. 46,47 Different molecular weight poly(styrene)-blockpoly(2-vinylpyridine) (PS-b-P2VP) and poly(styrene)-blockpoly(4-vinylpyridine) (PS-b-P4VP) were synthetized and selfassembled into hexagonal out-of-plane cylindrical arrays with different structural dimensions. Next, metallic-dielectric hybrid NPs were generated using the sequential self-assembly of the BCP layers selectively impregnated with different metallicdielectric combinations. Large scale ordered monolayers have been obtained for the first time using a one-step process. Moreover, the approach described here allows control over the shape, size and density of particles owing to the flexibility of BCP nanopatterning and metal infiltration methods that can be expanded to create a library of bi-or tri-metallic species.

Experimental Materials

Sec-Butyllithium (sec-BuLi, 1.2 M in cyclohexane), dibutyl magnesium (1 M in heptane) and calcium hydride (CaH2) were purchased from Sigma Aldrich and used as received. Anhydrous lithium chloride was purchased from Fisher Scientific and used as received. Styrene (S) from Sigma-Aldrich was first distilled over CaH2 and then stirred over dibutyl magnesium for 2 hours. 2 and 4-Vinylpyridine (2VP and 4VP, Sigma Aldrich) as well as 1,1-diphenylethylene were distilled twice over CaH2. Tetrahydrofuran (THF, Sigma-Aldrich), dried over a Braun MB-SPS-800 solvent purification system, was additionally distilled over sodium benzophenone ketyl prior to use. Acid tetrachloroauric (HAuCl4) (99.999% trace metals basis) and propylene glycol monomethyl ether acetate (PGMEA) (Reagent Plus, ≥ 99.5%) were purchased from Sigma-Aldrich and Merck respectively, and used without further purification. Silicon wafers (100) were purchased from Si-Mat silicon materials and cut to appropriate dimensions (approximately 1×1 cm 2 ).

BCPs synthesis

The PS-b-P2VP (Mn (PS) = 150.0 kg/mol, Mn (P2VP) = 32.0 kg/mol) and PS-b-P4VP (Mn (PS) = 14.7 kg/mol, Mn (P4VP) = 6.3 kg/mol) BCPs were synthesized by living anionic polymerization according to the standard procedures reported in the literature. [START_REF] Varshney | [END_REF]52 In a 500 mL flamed dried round flask equipped with magnetic stirrer, THF (400 mL) was introduced. The solution was cooled to -78°C before the sequential addition of Sec-BuLi and styrene. The reaction mixture was stirred for 30 min and the living polystyryl lithium anions were end-capped with 1,1-diphenylethylene. VP monomers were added after 30 min and the reaction mixture was kept stirring for 30 min. Finally, the reaction was terminated by the addition of degassed methanol, concentrated, precipitated in cyclohexane and dried under vacuum at 35°C. The different BCPs were characterized by 1 H NMR ( (ppm), 400 MHz, THF-d8), and size exclusion chromatography (SEC) in THF using the universal calibration technique.

PS-b-P2VP BCP self-assembly

Hexagonal out-of-plane cylindrical arrays formed from the BCP self-assembly in thin films were produced by spin coating a 2.5 wt.% solution of PS150K-b-P2VP32k (BCP1) in toluene and a 0.5 wt.% solution of PS14.7K-b-P4VP6.3k (BCP2) in PGMEA respectively, onto bare silicon wafers (4000 rpm, 30s). While BCP2 samples were used as-spun, a subsequent solvent vapour annealing (SVA) treatment in THF for 16 hours was used to improve the BCP ordering for BCP1.

Out-of-plane PS-b-P2VP lamellae were obtained directly by spin-coating of a 1 wt.% solution of PS102K-b-P2VP97K (BCP3) in PGMEA onto the modified Si wafers. 15

Atomic layer deposition

Sequential infiltration synthesis (SIS) was performed using an ALD (Ultratech SAVANNAH G2) tool in exposure mode. 53 This mode allows the SIS of Al2O3 dots in the VP domains using an alternating exposure of the BCP thin films to trimethylaluminum (TMA) and deionized water at 85°C with a N2 purge after each exposition step. The exposure and purge times used in this study were 60 s and 300 s, respectively, and ten SIS cycles (TMA/purge/H2O/purge) were performed to obtain the Al2O3 dots.

Selective Au and Pt impregnation

1 wt.% solutions of HAuCl4 and H2PtCl6 respectively in mili-Q water were used to impregnate the BCP films by immersion of the sample in the solution for 30 min. The samples were exposed to an O2 RIE plasma treatment in a PE-100 chamber (plasma conditions: 60 W, 10 sccm O2, 60 s) in order to etch the polymer scaffold and reduce the gold salts to metallic Au and Pt respectively. 15

AFM Characterization

AFM (Dimension FastScan, Bruker) was used in tapping mode to characterize the surface morphology of the different films. Silicon cantilevers (Fastcsan-A) with a nominal tip radius of 5 nm were used. The resonance frequency of the cantilevers was about 1.25 kHz.

GISAXS measurements

Grazing-incidence small-angle X-ray scattering (GISAXS) experiments were performed on SIRIUS (Soft Interfaces and Resonant Investigation on Undulator Source) station at the SOLEIL synchrotron in Gif-sur-Yvette (France) (8 keV). 54 The incidence angle was set in the range of 0.12° -0.19°, which is between the critical angle of the BCP film and the silicon substrate. The beam illuminates the samples with a typical footprint of 150 mm 2 . 2D scattering patterns were collected with a PILATUS 1M Dectris detector with a vertical beam stop in front of the detector's window. The sample-to-detector distance was set to 4459 mm. The beam center position, the sample-to-detector distance and the resulting angular range were calibrated using a silver behenate standard sample. GISAXS patterns were reduced using the FitGISAXS software. 55

Reflectivity measurements

The reflectivity measurements were carried out at Centre de recherche Paul-Pascal, using a custom-made instrument. X-rays were produced by a Rigaku MM007_HF rotating copper-anode generator (λ = 0.154nm) equipped with a multilayer collimating monochromator from Osmic. The samples were mounted vertically on a three-circle Huber goniometer and the reflected signal was collected with a Peltier-cooled solid-state detector XR-100CR from Amptek Inc (residual noise = 10 -2 cnts/s). The beam at sample position was h0.2×v2mm² and the resolution along the (horizontal) normal to the substrate was Δqz = 0.018 nm -1 FWHM. The data were analysed by the GenX 2.4.10 software, 56 available online at https://genx.sourceforge.io/.

Results and discussion

Ordered monolayer distributions of raspberry-like bimetallic nanoparticles were produced by the sequential multi-layered self-assembly of PS150K-b-P2VP32k (BCP1) and PS14.7K-b-P4VP6.3k (BCP2), following the methodology sketched in Figure 1.

In a first step, a 2.5 wt. % solution of BCP1 in toluene was spincoated onto bare silicon wafers resulting in a BCP layer of 90 nm thickness. Considering the low mobility of the high molecular BCP1 chains, an ill-defined self-assembled structure is obtained as shown on the AFM topographical image (see Figure 2A). [57][58][59] Therefore, a subsequent solvent vapour annealing (SVA) treatment using THF was used to improve the BCP ordering. GISAXS experiments were performed at different SVA durations to elucidate and to define an optimal SVA BCP ordering process window. The GISAXS pattern of the as-cast BCP1 sample (Figure 2C) presents a unique broad Bragg rod, revealing the out-ofplane orientation of the BCP structure. The lack of higher order reflections suggests a poor in-plane ordering of the selfassembled structure obtained directly after casting corroborating the AFM characterization. For solvent-annealed BCP1 films, higher order Bragg rods are evident on the GISAXS patterns, indicating an improved translational order of the BCP hexagonal structure (see Supporting Information, Figure S1).

After 16 h of SVA, the first order Bragg rod is positioned at q* =0.091 nm -1 (corresponding to a center-to-center distance of dC-C = 80 nm between cylinders in accordance with the AFM characterization) and higher order Bragg rods at q/q* = √3 and √7 are clearly visible (see Figure 2D) confirming the hexagonal packing. Using the GISAXS intensity curves of the first order Bragg rod (see Figure 2E), the full width at half-maximum (FWHM) was estimated for the different annealing time in order to extract the correlation length, ξ, (typical length scale over which the self-assembled morphology has conserved orientational and positional order) using Scherrer analysis. 60 For longer SVA duration, larger ξ were observed (see Supporting Information, Figure S2). Optimal results were obtained after 16 h of SVA, with a ξ of ≈ 350 nm. These results are further confirmed by the nanostructured BCP1 film AFM image recorded after 16 h of SVA which shows a well-ordered hexagonal out-of-plane cylindrical arrays of P2VP domains in a PS matrix (Figure 2B). The sequential infiltration synthesis (SIS) of Al2O3 within the cylindrical VP domains was subsequently carried out to replicate the self-assembled BCP structure into hexagonally packed Al2O3 dots. After 10 SIS cycles, Al2O3 was selectively introduced into the P2VP domains [61][62][63][64] and a subsequent etching step using UV/O3 in order to remove around 30 nm of the BCP film thickness revealed the Al2O3 dots localized at the BCP/air interface (see Figure 3A). Accordingly, this step introduced a topographical field to guide the self-assembled BCP2 structure formed through the spin coating of a 30 nm thick layer as shown in Figure 3B. By a subsequent immersion of the multi-layered structure into an aqueous metallic salt solution (HAuCl4, 1 wt. % in H2O) for 30 minutes, the gold salts were selectively incorporated into the P4VP domains. In a final step, the samples were exposed to an O2 RIE plasma (30 s, 10 sccm and 60 W) in order to trigger the reduction of the gold salts into metallic gold while removing the BCP2 scaffold. 15 The AFM topographical view of the final structure is presented in Figure 3C and consists in hexagonally packed Al2O3 dots (diameter around 36 nm) decorated by smaller gold dots (diameter around 15 nm). SEM image confirms the ordered structure created with distinct Au dots atop the larger diameter Al2O3 dots (see Supporting Information, Figure S3). GISAXS experiments were performed at each step of the process to follow the iterative self-assembly process and verify the out-of-plane orientation of features and their degree of ordering. Figure 4A shows the GISAXS pattern obtained for the samples composed of Al2O3 dots made by SIS from the nanostructured BCP1 film. A sequence of Bragg rods (q/q* = 1, √3, √7) consistent with an out-of-plane hexagonal packing of the P2VP cylindrical domains is retrieved and the centre-tocentre distance (4π/(√3q1*) = dC-C = 83.3 nm) extracted from the GISAXS pattern is in good agreement with the AFM characterization (i.e. 79.4 nm, see Supporting Information, Figure S4). The GISAXS pattern obtained after the deposition of the BCP2 layer on top of the Al2O3 array differs drastically as shown in Figure 4D. An additional intense Bragg rod appears at q = 0.24 nm -1 while the GISAXS pattern line-cut along qy integrated around the Yoneda band confirms that the morphological characteristics of the Al2O3 array remains unchanged (Figure 4B and4E). The additional Bragg rod observed at q = 0.24 nm -1 is attributed to the Au arrays formed on top of the Al2O3 layer even if the position of the Bragg rod is modified as regards to the neat BCP2 nanostructured film due to the topography swelling induced by the Al2O3 array (see Figure 4G and4H). It is noteworthy that the apparent broadening of this particular Bragg rod hints to a poor translational order of the Au 0 dots array confirmed by the absence of higher order Bragg rods at higher q values. Nevertheless, the GISAXS data are in excellent agreement with the formation of ordered arrays of bimetallic nanoclusters decorating the substrate surface over a large scale through this iterative self-assembly strategy.

We have additionally studied the internal stratification between the components (i.e., perpendicular to the substrate) using Xray reflectivity (XRR). It constitutes one of the most efficient techniques to characterize the structure of thin films with high sensitivity, as it offers a high spatial resolution over large penetration depths and allows us to further corroborate the large scale nature of the Au@Al2O3 raspberry features formed. [START_REF] Zaumseil | Materials for Information Technology[END_REF] Figure 5 presents the reflectivity curves obtained at different steps of the fabrication process: neat BCP1 nanostructured film (Figure 5A), Al2O3 dots array (Figure 5B) and the final Au@Al2O3 bimetallic raspberry-like nanoclusters (Figure 5C). The reflectivity fringes obtained for the BCP1 film (Figure 5A) suggest a homogeneous film of low roughness while more complex structures are detected for the Al2O3 dots array and the final bimetallic raspberry-like nanoclusters since some destructive interferences can be observed.

In order to gain some insights on the internal structure, different structures (taking into account geometrical data derived from the AFM and GISAXS characterizations) were created using the GeniX software to model the samples by fitting with the experimental results. 56 Theoretical values of SLD for the different elements were calculated using the procedure presented in the supporting information, and are listed in Table 1. SLD values are expressed in Å -3 , obtained by dividing the SLD value by the Thomson radius (re).

In case of the neat BCP1 layer, the best fit with a figure of merit (FOM) of 4.26×10 -2 (black data and red fit lines in Figure 5A, respectively) correspond to one homogeneous layer with some roughness at the surface. The scattering length density (SLD) profile was then extracted as a function of the height z (Figure 6A). For negative z, a SLD value of 0.71 Å -3 is obtained in good agreement with the theoretical value of Si. From z = 0, the SLD rapidly decreases to a uniform SLD value of 0.31 Å -3 . Since the theoretical SLD value expected for the polymer is 0.33 Å -3 , the experimental value is consistent with a homogeneous BCP film. At the upper surface of the film, the SLD decreases with a certain interface thickness, due to the topography of the BCP1 layer (see Figure 6B). Such behaviour can indeed be related to the quasi-hemispherical shape of the P2VP cylinders protruding from the PS matrix. The thickness of the sample is found to be 103 nm, including the native SiO2 layer between the silicon and the polymer. The fabrication process of the nanoclusters continues with the incorporation of Al2O3 into the P2VP domains by ALD. In this case, the model that fits best the XRR data is more complicated, as it consists of four different layers.

The FOM of the fit is 5.68×10 -2 showing that the proposed model matches very well the experimental results (Figure 5B).

The SLD profile obtained is therefore more complex (Figure 6D). First, like in the previous case, a high SLD value of 0.71 is obtained at z = 0, which is assigned to the silicon substrate. Then the SLD value decrease until 0.33 Å -3 (expected value for a polymer film). The following layer has a higher SLD (0.39 Å -3 ), which we interpret as the introduction of Al2O3, since the presence of the Al2O3 is expected to lead to an increase of the electron density of the film. It is noteworthy that the incorporation of Al2O3 by SIS does not occur homogeneously in the whole cylinder height (see Figure 6E), the sample has a thickness around 100 nm and the alumina penetration is found to be 50 nm deep into the polymer film, in agreement with previous studies where homogeneous distributions of the alumina precursors have been only observed in thinner samples (≈ 30 nm thick). [START_REF] Ruiz | [END_REF] This is clearly observed in the SLD profile,

Material Formula

Density SLD (Å

Where 𝑆𝐿𝐷 𝐵𝐶𝑃 = 0.31 Å -3 , 𝑆𝐿𝐷 𝐴𝑙 2 𝑂 3 = 1.15 Å -3 and 𝛷 𝐵𝐶𝑃 + 𝛷 𝐴𝑙 2 𝑂 3 = 1. A volume fraction of 7% of Al2O3 is calculated at 70 nm thickness, increasing to 13% close to the film surface. In the next step, the sample was exposed to a UV/O3 treatment in order to remove partially the BCP layer. Then the second film (i.e. BCP2) was spin-coated on top of the Al2O3 array and impregnated with the gold salt. An O2 RIE treatment was carried out to remove this BCP2 layer and reduce the gold salt to metallic gold. Bimetallic nanostructures composed of an Al2O3 core covered by metallic Au satellites were obtained on the top of the surface. In this case, the model requires to appropriately correlate with the reflectivity data is even more complex and 12 different layers were necessary to obtain a satisfactory fit of the XRR data (Figure 4C). The FOM obtained with this model is 3.07×10 -2 and the SLD profile obtained is presented in Figure 6G.

At z < 0 and z = 0, the silicon substrate is observed as well as a thin layer of SiO2 corresponding to a SLD value of 0.71 Å -3 , in agreement with the theoretical values listed on Table 1.

Between z = 0 and z = 20 nm, three thin layers with SLD values of 0.37 Å -3 , 0.41 Å -3 and 0.37 Å -3 , respectively were used to satisfactory fit the XRR data. The SLD values are higher than expected for the pristine polymer (0.33 Å -3 ), suggesting the presence of some metallic impurity. One possible explanation could be related to the presence of the residual BCP1 layer Above, the SLD value increases again due to the presence of the Al2O3. A constant SLD value of 0.44 Å -3 is observed from 22 to 44 nm, corresponding to a 𝛷 𝐴𝑙 2 𝑂 3 of 11.3%. Therefore, a densification of the Al2O3 layer is observed (from the previous 7% to the current 11.3%). This can be attributed to the partial removal of the BCP1 layer by UV/O3. 66,67 A higher SLD value (0.53 Å -3 ) is observed near the surface of the film due to the presence of the gold NPs (also the imaginary part decreases to highly negative values as shown in Figure 6I). Finally, the decrease of the SLD is consistent with the presence of Au NPs on top of the larger alumina dots, corroborating the formation Au@Al2O3 raspberry-like nanoclusters (Figure 6H).

In order to generalize this strategy, and show the high versatility of this approach, another complex nanostructure was studied.

In this case, a first layer of metallic platinum nanowires (Figure 7B) was obtained after replication of an out-of-plane lamellar structure obtained from BCP3 self-assembly (see Figure 7A).

Once the platinum nanowires were obtained, a 0.5 wt.% solution of BCP2 in PGMEA was spin coated on top of it. After a subsequent immersion in the gold precursor solution, a hexagonal array of gold nanodots was obtained, decorating the platinum nanowires and creating, in this case, a spheres-on lines bimetallic complex structure (Figure 7C). These two examples show the high versatility of this strategy to produce different hybrid structures, combining not only different materials but also different morphologies giving rise to a wide variety of metallic-metallic or metallic-dielectric combinations.

Conclusion

We have presented here a new, versatile and straightforward strategy to obtain on-demand bimetallic Au@Al2O3 raspberrylike nanoclusters using multi-layered self-assembled BCP layers. For the first time, highly ordered monolayers of an Al2O3 core with Au satellites have been obtained. Several techniques, i.e., AFM, SEM, GISAXS and XRR, have been used to confirm the high degree of order of the designed structures, exemplifying the robustness and reproducibility of the Au@Al2O3 nanocluster fabrication process. Indeed, we believe a wide variety of metallic-metallic or metallic-dielectric combinations are accessible with this approach, giving rise to the possibility of exploring unique surface properties. The versatility and highly controlled nature of the nanocluster strategy disclosed here surpass traditional wet chemistry methods, and therefore could find use for wide ranging applications. For example, such bimetallic features could be applied as optical metasurfaces, catalytic active sites for material growth or label free biosensors where nanodimensional control are all-important.

Figure 1 .

 1 Figure 1. Schematic representation of the raspberry nanoclusters fabrication: (A) Formation of an out-of-plane cylindrical structure obtained from BCP1 self-assembly. (B) SIS of Al2O3 by ALD into the cylindrical PVP domains. (C) Controlled removal of 30 nm polymer film by UV/O3 etching. (D) Formation of a second out-of-plane cylindrical structure obtained from BCP2 selfassembly on top of the first layer. (E) Au metal impregnation into the PVP domains followed by (F) Au metal reduction by RIE plasma.Figure 2. AFM topographical images of (A) as-cast BCP1 layer and (B) self-assembled BCP1 layer after 16h SVA. GISAXS patterns of BCP1 films at different solvent vapour annealing (SVA) time: (C) as-cast and (D) 16h. (E) Corresponding GISAXS line-cut along qy integrated around the Yoneda band.

Figure 3 .Figure 4 .

 34 Figure 3. AFM topographical images of the different steps of the process to obtain bimetallic raspberry-like nanoclusters: (A) Alumina dots obtained after SIS and 30 min of treatment with UV/O3. (C) self-assembled BCP2 layer on top of the Al2O3 dots generated from the SIS of the self-assembled BCP1 film and (C) hexagonal ordered raspberry-like nanoclusters (Au@Al2O3) monolayers obtained by iterative self-assembly. Inset in (C) correspond to the FFT of the AFM image. Two different periodicities are clearly identified, corresponding to the Al2O3 dots and Au NPs respectively.

Figure 5

 5 Figure 5 XRR reflectivity data (black line) obtained for A) nanostructured BCP1 film, B) Al2O3 dots array and C) the final Au@Al2O3 bimetallic raspberry-like nanoclusters. Red lines correspond with fitting obtained by GenX.

Figure 6 .

 6 Figure 6. SLD profile (left) and imaginary SLD profiles (left: Real part; right: Imaginary part) obtained after fitting the experimental XRR data for: (A-C) neat BCP1 film, (D-F) BCP1 layer impregnated with Al2O3 and (H-J) bimetallic raspberry-like nanoclusters consisting in a Al2O3 core decorated with Au satellites. Graphical representations of the structures (B, E and H) are also presented to help the data interpretation.

Figure 7 .

 7 Figure 7. AFM topographical images of the different steps of the process to obtain decorated bimetallic nanowires (A) out-ofplane lamellar structure obtained from BCP3 self-assembly (B) Pt nanowires obtained by selectively impregnation out-of-plane BCP3 lamellar films and (C) bimetallic Au-Pt nanowires obtained by self-assembly of BCP2 on top of the Pt nanowires.

Table 1 :

 1 SLD theoretical values for the materials of interest in this study which increases progressively from around 50 nm from the surface of the substrate until reaching a maximum value of 0.46 Å -3 at 100 nm. Thereby, the volume fraction, 𝛷 𝐴𝑙 2 𝑂 3 , of the Al2O3 though the film is not constant and can be estimated with the following expression.𝑆𝐿𝐷 𝐶𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 = 𝑆𝐿𝐷 𝐵𝐶𝑃 𝛷 𝐵𝐶𝑃 + 𝑆𝐿𝐷 𝐴𝑙 2 𝑂 3 𝛷 𝐴𝑙 2 𝑂 3

				-2 )	SLD / re
					(Å -3 )
	Silicon	Si	2.3	2.0×10 -5 -i	0.71 -i
				4.6×10 -7	1.6×10 -2
	BCP	C15H15N	1.02	9.3×10 -6 -i	0.33 -i
				1.3×10 -8	4.6×10 -4
	Gold	Au	19.6	1.25×10 -4	4.44 -i
				-i 1.3×10 -5	0.46
	Alumina	Al2O3	3.95	3.25×10 -5	1.15 -i
				-i 3.8×10 -7	1.3×10 -2

Acknowledgments

This work was performed within the framework of the LabExAMADEus ANR-10-LABEX-0042-AMADEUS and the EquipExELORPrintTec ANR-10-EQPX-28-01 with the help of the Frenchstate Initiative d'Excellence IdEx ANR-10-IDEX-003-02. We acknowledge SOLEIL for provision of synchrotron radiation facilities at the SIRIUS station for the GISAXS experiments 20170241.

Notes and references

Conflicts of interest

There are no conficts to declare.