Imane Oussakel
email: imane.oussakel@laas.fr

Philippe Owezarski
email: philippe.owezarski@laas.fr

Pascal Berthou
email: pascal.berthou@laas.fr

Experimental Estimation of LTE-A Performance

Keywords: QoS monitoring, cellular networks, bandwidth prediction, machine learning, software defined radio

In cellular networks, the emergence of machine communications such as connected vehicles increases the high demand of uplink transmissions, thus, degrading the quality of service per user equipment. Enforcing quality-of-service in such cellular network is challenging, as radio phenomena, as well as user (and their devices) mobility and dynamics, are uncontrolled. To solve this issue, estimating what the quality of transmissions will be in a short future for a connected user is essential. For that purpose, we argue that lower layer metrics are a key feature whose evolution can help predict the bandwidth that the considered connections can take advantage of in the following hundreds of milliseconds. The paper then describes how a 4G testbed has been deployed in order to investigate throughput prediction in uplink transmissions at a small time granularity of 100 ms. Based on lower layer metrics (physical and mac layers), the main supervised machine learning algorithms are used, such as Linear Regressor and Random Forest to predict the uplink received bandwidth in different radio phenomena environment. Hence, a deep investigation of the impact of radio issues on bandwidth prediction is conducted. Further, our evaluation shows that the prediction is highly accurate: at the time granularity of 100 ms, the average prediction error is in the range of 6% to 12% for all the scenarios we explored.

I. INTRODUCTION

In the scope of smart cities, where everything is connected at anytime and anywhere, a tremendous growth of services and/or applications demand is forecasted. Most applications rely on wireless networks to send their data to the corresponding remote internet servers. For instance, the Intelligent Transport System -ITS-emphasizes many services relying on car communication system. In such system, vehicular video conferencing, traffic monitoring and data uploading are handled by cellular network, thus, leading to a high amount of data to be managed by such networks. For instance, automotive data analytics [START_REF]The Connected Vehicle: Big Data[END_REF] forecasts a production of up to 30 terabytes of data per day by an average connected car to be sent to ITS servers. Therefore, contrary to the Human-to-Human applications where downlink transmission dominates, smart cities services use mostly uplink communication. Hence, the conventional roles are inverted: when all the connected nodes are sending their information, a large amount of data shall be received by their corresponding server with a high quality of service.

In wireless networks, the high performance metrics, such as throughput, change rapidly depending on the environmental situations. However, radio phenomena such as noise, pathloss and multipath fading affect the different performance metrics. It is increasingly hard to understand how such phenomena degrade the Quality of service, as they are uncontrolled. In a more general context, the QoS degradation in wireless network is mainly caused either by radio phenomena variation or when the total requested bandwidth exceeds the cells capacity [START_REF] Singh | Quality of Service Guarantees in Mobile Computing[END_REF]. In this article we focus mainly on radio issues: environment issues and radio congestion. Hence, the middle nodes are well provisioned to avoid any transport congestion in the network. 4G and 5G networks are expected to satisfy the high demand of high QoS, i.e. high throughput and low latency, not only for legacy, but also smart cities applications. Nevertheless, with the introduction of machine communication, the growth of machine traffic outpaces that of cellular networks, resulting in a quality of service degradation. In fact, several researches show that LTE suffers from congestion [START_REF] Phan | A capacity analysis for the transmission of event and cooperative awareness messages in LTE networks[END_REF], [START_REF] Vinel | 3GPP LTE Versus IEEE 802.11p/WAVE: Which Technology Is Able to Support Cooperative Vehicular Safety Applications[END_REF], especially in uplink. This congestion implies a QoS degradation and sometimes the inaccessibility to the network, which limits the envisaged applications and services.

Over years researchers were focused on enhancing downlink transmission as only downlink was challenging. They improved congestion control mechanisms while taking predicted throughput as an input parameter [START_REF] Lu | CQIC: revisiting cross-layer congestion control for cellular networks[END_REF], [START_REF] Kwan | On pre-emption and congestion control for LTE systems[END_REF]. For that, downlink cellular throughput prediction has been the ultimate goal of research, as it improves many use cases, such as video streaming quality.

Many dissimilarities between uplink and downlink lead to the insufficiency of the proposed techniques for downlink bandwidth prediction to be applied in uplink. In fact, contrary to downlink (DL) transmissions where each user predicts the incoming throughput from one base station (eNB) at a time, during uplink (UL) transmissions, the eNB will need to predict the incoming throughput of multiples users simultaneously. Moreover, the scheduling and the access techniques of DL and UL are different. For instance, the UL scheduler is synchronous while the DL is asynchronous. The UL uses SC-FDMA (Single-Carrier Frequency Division Multiple Access) as a multiple access scheme while the Orthogonal Frequency Division Multiple Access (OFDMA) is used for DL. Hence, the UL user data is affected differently by the channel variation compared to DL and the signal treatment is also different in the receiver (eNB). Such differences with downlink motivate our work on uplink performance study to investigate the possibility of estimating/predicting the bandwidth for uplink transmissions, that could be different from related work focusing on downlink.

Even though uplink and downlink are different, we forecast an improvement of uplink LTE-A schedulers in a similar way as downlink [START_REF] Margolies | Exploiting mobility in proportional fair cellular scheduling: Measurements and algorithms[END_REF], by incorporating the predicted throughput. Furthermore, a cell radio congestion avoidance can be achieved by comparing the estimated coming data rate and the base station capacity. Machine application servers may avoid saturation based on the forecasted bandwidth, hence, reducing uplink quality of service degradation. Thereby, uplink bandwidth prediction is valuable in the machine-to-machine context.

In this article, several scenarios are tested in order to understand how each radio phenomenon influences the bandwidth in cellular networks, aiming to estimate/predict the bandwidth at small time granularity. To that end, an LTE-A testbed is deployed, where the radio propagation is controlled in an anechoic room. The middle nodes are well provisioned to avoid throughput degradation caused by network bottleneck. Therefore, throughput deterioration is mainly caused by environment phenomena. We investigate our work on bandwidth prediction based on lower layers measurements/metrics over a small time granularity, from 100 ms to 1 s scales. Different machine-learning algorithms are tested to validate the impact of radio issues on bandwidth prediction.

The remaining of this paper is organized as follows. Section II points out the LTE-A background of the experimentation. The related work is reviewed in section III. Section IV describes the testbed deployment inside an anechoic room. Section V covers the prediction methodology used in this article. It includes data collection procedure, data analysis and the developed model for the prediction evaluation. Section VI presents and discusses the obtained results for uplink bandwidth prediction. Finally, section VII concludes this paper.

II. LTE-A BACKGROUND

Our work is based on 4G communications, as uplink cellular traffic has become dense with the introduction of M2M communications, especially connected cars. This section presents a quick overview of the main 4G metrics and techniques that our study is based on, motivating our focus on uplink transmissions.

Fig. 1 shows a basic architecture of the 4G network when a User Equipment (UE) is connected via the LTE access network to the Evolved Packet Core network (EPC). The EPC maintains the sessions, registration procedures and routing of UE IP-packets. The base station for LTE-A radio is named evolved NodeB (eNB); it ensures mainly Radio resource management and scheduling in both uplink and downlink. Further, to ensure the multiple users access, 4G uses OFDMA in downlink and SC-FDMA in uplink. Orthogonal Frequency Division Multiple Access (OFDMA) is a multiple carrier system, where each OFDM data symbol is transmitted over one sub-carrier. In contrast, with single-Carrier Frequency Division Multiple Access (SC-FDMA) in uplink, data symbols are transmitted in series and each symbol is carried by a wider bandwidth. Hence, contrarily to OFDMA that reduces/vanishes the intersymbol interference (ISI), SC-FDMA is prone to ISI. ISI reduces data-rate when no compensation is present. For that the eNB deploys as a first step a complex frequency equalizer to mitigate such distortion. Therefore, UL data-symbols are not only affected differently by the channel variation compared to DL, but signal treatment is also different in the eNB. Such dissimilarities with DL motivates our work on uplink performance study, that could be different from related work focusing on downlink only.

Fig. 2 exhibits the frame type used for FDD (Frequency Division Duplex) radio transmission mode, i.e. different frequency bands are used for downlink and uplink. The frame length is fixed to 10 ms, containing 10 sub-frames of 1 ms (two adjacent slots). Each slot is defined as a resource block of 0.5 ms over 12 sub-carriers spaced by 15 kHz. The LTE-A radio resource is then referred as Physical Resource Block (PRB) that forms 180 kHz x 0.5 ms. In time domain, the PRB contains seven SC-FDMA data symbols expanded over 12 sub-carriers in frequency domain.

The eNB allocates for each UE a specific number of resource blocks based on its resource-scheduling algorithm, UE's capability and the channel quality. Although schedulers are vendor specific, it is reported to consider one TTI (Transmission Time Interval) of 1 ms as the smallest scheduling time unit [START_REF] Trivedi | Comparison of Different Scheduling Algorithm for LTE[END_REF]. Thus, based on the used bandwidth, a defined number of useful resource blocks is available per slot. For example, for a 5 MHz bandwidth, 25 physical resource blocks of 180 kHz could be assigned to the users attached in the cell. In order to handle the communication between the UE and eNB, the use of these PRBs is standardized by using channels formats in the lower layers. For the uplink transmissions mainly three channels are used [27]. However, the UE activates the PRACH, Physical Random Access Channel to demand access to the network; it communicates the control signaling information using the Physical Uplink Control Channel (PUCCH). PUCCH is mainly located in the bounds of the used bandwidth in frequency domain as illustrated on fig. 2. The UE transmits its data on the Physical Uplink Shared Channel (PUSCH), which uses the remaining PRBs.

In the wireless system, channel is changing rapidly, resulting in bandwidth variation. In order to adapt to channel's change and guarantee a high quality of service over time in downlink and uplink communications, the UE and eNB accomplish many measurements and reports. The 3GPP has normalized the main measures and mechanisms for each side of the network. As we are interested in uplink communication, we cover in Fig. 1: Basic 4G architecture. more details the main uplink measurements/metrics used for prediction in section V -part V.A.

III. RELATED WORK

Over years, bandwidth estimation and prediction has been widely studied in wired networks and WLAN. D. Koutsonikolas et al. [START_REF] Koutsonikolas | On the feasibility of bandwidth estimation in 1xEVDO networks[END_REF] reveal the ineffectiveness of using those techniques in cellular networks as they are characterized by the large short-scale fluctuation of bandwidth. Although, [START_REF] Winstein | Stochastic Forecasts Achieve High Throughput and Low Delay over Cellular Networks[END_REF]- [START_REF] Liu | Experiences in a 3G Network: Interplay Between the Wireless Channel and Applications[END_REF] show the possibility of throughput estimation and prediction in wireless networks essentially in Downlink. Their proposed methods are based on higher layer measurements. In cellular networks, based on radio measurements, multiple performance tasks are introduced at different layers of the protocol stack. For instance, when a bad signal is received HARQ (Hybrid automatic repeat request) is triggered in the second layer. Given this, we are interested in using the lower layers measurements/metrics to predict bandwidth instead of higher layer measurements.

Even though, both client and network throughput are important in cellular networks, link bandwidth prediction related work considered only Downlink channel. F. Lu et al. [START_REF] Lu | CQIC: revisiting cross-layer congestion control for cellular networks[END_REF] use CQI (Channel quality indicator) and DRX (Discontinuous Transmission) to predict instantaneous downlink throughput in 3G networks. The study in [START_REF] Chakraborty | Coordinating cellular background transfers using LoadSense[END_REF] classifies bandwidth into high and low categories in order to increase user equipment coordination efficiency for transmission in cellular background. Margolies et al. [START_REF] Margolies | Exploiting mobility in proportional fair cellular scheduling: Measurements and algorithms[END_REF] investigate the reproducibility of signal quality over the same path to predict throughput based on user trajectory tracking. Authors of [START_REF] Yue | Cellular Link Bandwidth Prediction in LTE Networks[END_REF], [START_REF] Samba | Instantaneous throughput prediction in cellular networks: Which information is needed[END_REF] proposed the use of machine learning techniques, especially Random Forest, to predict instantaneous throughput in LTE-A networks. However, A. Samba et al. [START_REF] Samba | Instantaneous throughput prediction in cellular networks: Which information is needed[END_REF] predict throughput based on both eNB and UE information before the connection is established for content providers. They conclude that applying RF on radio measurements leads to a promising prediction: 52% of the relative prediction errors are within ±20%. C Yue et al. [START_REF] Yue | Cellular Link Bandwidth Prediction in LTE Networks[END_REF] consider a set of lower layer measurements and historical throughput to predict real-time LTE-A throughput. Accurate predictions were obtained: 69% of the relative prediction errors are within ±10% for walking scenario. For estimating/predicting UL traffic, we apply mainly Linear Regressor (LR) [START_REF] Weisberg | Applied Linear Regression[END_REF] and Random Forest (RF) [START_REF] Breiman | Random forests[END_REF] as machine learning techniques to predict the UL bandwidth based on lower layers metrics/measurements. The main reason of using LR is to investigate the linearity between the lower layers metrics related radio channel and the high QoS metric, i.e. bandwidth. In contrast, RF is well known to provide good predictions in different application domains with accurate time series and high dimensional feature spaces [START_REF] Zeki Suac | A Comparison of Machine Learning Methods in a High-Dimensional Classification Problem[END_REF], which corresponds to the case study in this article. Hence, a comparative study of the two ML techniques performance in cellular communication domain is conducted.

IV. TESTBED DEPLOYMENT

The testbed objective is to limit the uncontrolled effects during a wireless communication, and study the impact of disturbing low level features on bandwidth metric. The wireless system is LTE-A as already mentioned in the paper. R2lab [START_REF] Parmentelat | Neping: An Efficient Experiment Control Tool in R2Lab[END_REF] is used as an underlying testbed, which offers a wireless network with multiple Software defined radio (SDR) nodes inside an anechoic room. The RF propagation inside the room is controlled thanks to the microwave absorbers materiel on the walls, scattering any wireless signal that comes across. The SDR based nodes allow a full access to realistic physical metrics/measurements. With remote access to the indoor nodes, the deployment of LTE-A network is accomplished. Fig. 3 schematizes the testbed. Openairinterface (OAI) software based platform [START_REF] Nikaein | Openairinterface: A flexible platform for 5g research[END_REF] is implemented, which spans the full LTE 3GPP protocol stack (including features from LTE-Advanced and LTE-Advanced-Pro) for both radio access network (E-UTRAN) and core network (EPC). The openairinterface softmodem is connected with a hardware platform for SDR: USRP B210 (Universal Software Radio Peripheral) attached to two antennas, receiver and transmitter antennas. The USRP B210 is connected to a host computer to perform processing, and then connected to a PC running EPC network, and accessing the internet toward a distant server. The testbed works on frequency band 7 with 5 MHz bandwidth and uses FDD mode, which corresponds to the traditional and stable version of openairinterface platform with USRPB210. Further, as our objective is to observe the impact of radio phenomena on the bandwidth variation, the server and the intermediate PCs are well provisioned for not behaving as a bottleneck during communication.

To emulate 4G connected nodes, two commercial UEs are used and placed inside the anechoic room. The hardware metallic enclosure boxes scattered inside the room are considered as fixed multipath sources, which we take advantage of in investigating the impact of multipath phenomenon on bandwidth variation.

In real environments, multiple radio phenomena are scrambling the communications. Example phenomena include multipath fading leading to InterSymbol Interference (ISI) noise, pathloss and random processes such as AWGN (Additive White Gaussian Noise). These phenomena tend to attenuate aggressively the transmitted signal, which causes a significant amount of signal strength reduction and hence QoS degradation. The upcoming scenarios tend to investigate the impact of each phenomenon on the bandwidth prediction. For each test scenario, a radio phenomenon is added in the system to complexify the tests.

a) Scenario 1: Multipath fading: For the first scenario, we investigate the impact of multipath fading and pathloss on bandwidth prediction/estimation. In other words, the two UEs used in the experimentation are on different distance from the eNB, i.e. each UE has a different pathloss. Over the room, multiple metallic boxes surround the eNB (fig. 3). They introduce multipath fading leading to ISI. The two pylons, covered with absorbers are considered as shadowers for the transmissions, especially for UE2. Therefore, only pathloss, shadowing and multipath fading are present in the system as scramblers for the transmission, i.e the anechoic room is isolated from any other radio phenomena. In order to avoid any degradation/losses due to the insufficiency of radio resources the total transmitted bandwidth by the two users is inferior to the maximum network capacity. In fact, after several tests the maximum uplink data-rate achieved in the testbed, by 1 connected UE, is around 8Mbps. Hence UE 1 transmits during the whole test duration a fixed amount of data of 2Mbps and UE 2 transmits 3Mbps. A benchmark of lower layers metrics is performed to construct datasets for ML algorithms as explained further.

b) Scenario 2: Noise & Multipath fading: In this scenario, another radio phenomenon is added in the anechoic room to investigate its presence on the bandwidth prediction, i.e, noise. For that we generate in a controlled manner a specific noise profile to scramble the communication.

Noise profile: In time varying scenarios, the received signal amplitude undergoes rapid fluctuation that is often modeled as a random variable with a particular distribution. For that we consider a Gaussian distribution, AWGN, which is characterized by its amplitude that affects the signal strength. Moreover, noise (AWGN) is introduced as it causes transmission errors and may disrupt the communication with ISI production for high power noise [START_REF] Bolat | Study of OFDM Performance Over AWGNn Channels[END_REF]. Contrarily to work in literature where AWGN is often taken with constant attenuation, we introduce randomness in the attenuation in order to have attenuation fluctuations of the signal over time. For that we define noise level as noise with a given gain and amplitude. We used mainly GNUradio on USRP B210. Given a list of gain levels and an interval of maximum and minimum amplitude levels, each 10 seconds a random value of amplitude and gain are chosen. In fact, the amplitude value affects the statistical characteristics of the noise source, i.e, the standard deviation of the Gaussian noise. The gain affects the transmitted signal power. The programmed step for noise level change (10 s) is chosen as to have sufficient samples for each noise level. Therefore, low noise level values keep the channel flat, while high noise level disrupt totally the communication, with the probability of introducing ISI, not only with noise but also with metallic boxes scattered in the room. Furthermore, the abrupt changes in noise levels during the transmission tend to reflect the real complex radio environments, where the user's mobility across different shadowers leads to aggressive/alleviated signal attenuation. The bandwidth is fixed to 5Mhz to scramble the full UL bandwidth. Both users transmit at same data-rate as in the scenario 1, and same benchmark of lower layer metrics is performed.

c) Scenario 3: Radio congestion & Noise & Multipath fading: During the previous scenarios, the total transmitted bandwidth by the two UEs is much lower than the maximum available bandwidth. In this part, another radio phenomenon leading to bandwidth degradation is introduced in the testbed, i.e. radio congestion. The later occurs when the total bandwidth required by the connected UEs outpaces the maximum eNB bandwidth. Mobile application based speedtest is tested in scenario 1 for UE 1. It reaches the maximum throughput of 8 Mbps. Therefore, in order to realize radio congestion, UE1 transmits its data at a fixed amount of 4 Mbps and UE2 at 5 Mbps. Noise is introduced as described in scenario 2 to complexify the test.

For all the above scenarios, IPERF3 generates traffic at the UE side and IPERF3-server monitors throughput reception in the server. In order to have a fixed transmission amount of data during the whole test duration, UDP (User Datagram Protocol) is used as a transport protocol. In fact, TCP (Transport Control Protocol) changes the transmission window based on the perceived packet losses in the window. Given this, any observed bandwidth degradation is essentially due to radio environment variation.

V. PREDICTION METHODOLOGY

Given the LTE-A testbed in the anechoic room, we are able to generate 4G traffic in different scenarios as described in the previous part. In this section, we present the methodology aiming at building our datasets.

A. Data collection

The eNB performs different measurements in order to decode the received data and adapt to channel variation. With SDR at the eNB side, we are able to collect all the performed eNB measurements and metrics, especially from the lower layers (physical and mac layer). As the connected UEs communicates their control signaling information using PUCCH, and transmit their data on PUSCH channels, we perform a deep benchmark of the main metrics/measurements linked with the two channels. The OAI eNB metrics/measurements are performed as depicted by the 3GPP standard. The collected metrics are mainly extracted during the lower layers data processing.

On one hand, radio measurements are collected as they are crucial for higher layer mechanisms and reflect the channel quality. The main measures are: On the other hand, during decoding/demodulating the received channels, multiple metrics are extracted. In fact, when the UE is transmitting data via PUSCH, its data is transmitted using a Modulation Coding Scheme (MCS). The MCS is related to the modulation order Qm, e.g, QPSK (Quadrature Phase Shift Keying), 16 QAM (Quadrature Amplitude Modulation) and 64 QAM. The modulation scheme has a direct impact on the data-rate. However, when using a higher-order QAM scheme such as 64 QAM, each symbol represents a group of bits to be transmitted, i.e. each symbol has a 6-bit signature. The MCS is chosen based on the measurements exchange, reflecting the channel quality and the network capacity, between the eNB and the UE. Further, the transport block size (TBS) could be determined based on MCS value and the number of used resource blocks. TBS refers to data in the physical layer [27].

During the UE data transmission toward the network, eNB checks for data correctness using CRC (Cyclic Redundancy Check) and sends ACK/NACK to the UE. If the channel is good, errors are detected and corrected. On the other hand, when the channel quality is bad, the CRC might be insufficient. Further, LTE implements synchronous HARQ (contrarily to the asynchronous HARQ in DL) which combines ARQ error-control and a high-rate forward error-correcting coding (FEC). This way, the eNB requests a retransmission in case of error correctness inability. Such mechanism is characterized by multiple parameters, such as the round and redundancy version (rv). Is it possible to send ACK/NACK by the UE during the transmission, hence the length of ACK information in bits is considered. Moreover, the length of each received SDU/PDU (Service Data Unit/Protocol Data Unit) on the MAC layer is extracted with the buffer size when data is handled.

All of the aforementioned parameters are collected while decoding/demodulating the received PUSCH/PUCCH channels. In addition to other metrics usefull for PUSCH and UL-SCH (UpLink Shared Channel) related functions, such as decoding time, decoding iteration, beta offset, number of CRC bits and the cyclic shift. A total of 43 metric is collected during the whole test duration, i.e. 400 seconds.

One of our objectives is to compare the bandwidth predictions over different small time granularity. The bandwidth measurements are performed in a discrete time manner, each δt. As the minimum time report interval in IPERF3 is 100 ms, we fix δt = 100ms. The predictions are then made every δt. The eNB metrics are collected per subframe scale (1 ms). In order to have a representative measure per δt, we compute the maximum, minimum, mean, median and the standard deviation of each metric per δt to construct the datasets. Therefore, each metric γ is represented in the dataset as follows: {γ min , γ max , γ mean , γ median , γ std }. For each UE u, u ∈ {1, 2} and each scenario s, s ∈ {1, 2, 3}, a dataset is constructed, noted dataset s u. Each dataset s u contains all the lower layers metrics collected for the given UE u during the scenario s, including historical received bandwidth.

B. Methodology

In this article, Python scikit-learn library is used for all the tests. GridsearchCV [START_REF] Geisser | The Predictive Sample Reuse Method with Applications[END_REF] is applied to choose the optimal hyper-parameters for each estimator. It combines both gridsearch and cross-validation methods. Gridsearch consists of an exhaustive search over subset values of parameters for a given estimator and cross-validation (CV) technique [START_REF] Kohavi | A study of cross-validation and bootstrap for accuracy estimation and model selection[END_REF] estimates the prediction error of a model. CV is categorized into exhaustive and non-exhaustive categories, the former is more computational for high dimensional datasets. For that, the non-exhaustive cross-validation is chosen, mainly the recommended K-fold method, with K=10 to have a good compromise between variance and bias of the model [START_REF] Friedman | The Elements of Statistical Learning[END_REF]. Varma and Simon [START_REF] Varma | Bias in error estimation when using cross-validation for model selection[END_REF] report that the estimated prediction error from the cross-validation used to tune hyper-parameters is biased, and recommend the use of nested cross-validation instead, where an inner CV is used to select the optimized model (executed with GridsearchCV) and an outer CV to estimate the prediction error. Let denote K 1 -fold and K 2 -fold the inner and the outer CV respectively. Given an input dataset, a random split is performed to construct training and test sets. In fact, the dataset is split into K 2 -folds, one fold is used for testing and the others K 2 -1 folds constitute the training set. For each hyper-parameter combination from the gridsearch,

C. Correlation analysis

All of the aforementioned metrics/measurements depicted in section V part V-A are combined to constitute the datasets per UE and per scenario. A total of 58 metrics is then used. Each sample of the dataset contains the maximum, minimum, mean, median and the standard deviation of each metric.

When the past measurements are added (w = 1), a high dimensional features space is then considered. In this part the correlation between the collected features and the bandwidth is investigated. Because of place constraint, only some of the most correlated features with bandwidth over all scenarios are presented. Fig. 4 depicts the correlation coefficient between the selected features and bandwidth for both UEs during scenarios 2 and 3. For a given feature, when the correlation level of different statistics is pretty similar, only one bar is presented, i.e. SN R median, mean, max indicates that the SNR median, SNR mean and SNR max have pretty similar correlation level with bandwidth.

Based on the figure, the main representative statistics for a given feature are the mean, maximum, minimum and the median. Most of the features mark higher correlation level for UE 2 compared to UE 1. For example, during scenario 2, TA mean reaches 0.58 for UE 2 and only 0.28 for UE 1. Such results point out the impact of fading multipath on the features variations. It is also remarked that the correlation increases for almost all the features when scenario 3. For instance, the correlation coefficient for Qm mean (modulation order) for UE 1 increases from 0.25 in scenario 2 to 0.46 in scenario 3. This increase in correlation level sheds light on the radio congestion impact on the lower layer metrics. Further, extreme radio phenomena lead to higher correlation between bandwidth and lower layer metrics/measurements.

D. Feature selection

In order to exploit the linear relationship between the collected metrics and the bandwidth, LR is applied as an underlying ML technique. On the other hand, RF has the ability to create the linear and non linear boundaries during the trees building, which leads to accurate predictions. RF is then applied. As many features are available, the selection of the most important ones is crucial, especially for LR. To this end, the statistical technique Principal Component Analysis (PCA) is used [28]. PCA reduces the high features space dimension with variance maximization of each component. For instance, it uses orthogonal transformation to generate linear combination of orthogonal features vectors. After testing multiple component numbers, using a single component leads to the minimum prediction error. Therefore, the number of component is fixed to 1 for the upcoming tests based LR. In contrast, an approach for features importance determination is implemented in RF models. RF combines multiple decision trees to get a more accurate prediction. For each node split, it looks for the most important feature among a random subset of features. In fact, a given feature is considered as important if its perturbation leads to larger error. Hence, RF relies on features importance for building the forest. Thus, turning PCA unnecessary in the case of RF.

E. Bandwidth analysis

As the focus of this work is mainly the correlation between radio environment and bandwidth, in this part, an analysis of the impact of radio issues on the received bandwidth is conducted. For that, fig. 5 illustrates the standard deviation of the received bandwidth for each UE during each scenario. Each column is scenario specific. During scenario 1, the two UEs transmit at a fixed low level data-rate. From the figure, UE 1 bandwidth distribution is straight compared to the UE 2 received bandwidth. In fact, the amount of received bandwidth for UE2 is spread out over much wider range. Such issue could be mainly explained by the radio phenomena experienced by UE 2. The UE 2 transmissions face a high level of multipath fading and shadowing compared to UE1 transmissions. UE 1 has a direct line of sight with the eNB. This observation points out the sensitivity of the bandwidth to multipath fading phenomenon.

For scenario 2, noise profile is injected in the testbed. The noise source is placed between the UE1 and the eNB. On the second column of fig. 5, it is clear that noise has added variance to the UE 1 distribution, with the curve becoming right skewed. Such impact is expected as the noise variation introduces errors and ISI, especially for UE1 transmissions. In contrast, surprisingly the UE2 bandwidth variance is decreased in this scenario. The minimum received bandwidth is increased too. Such result exhibits the positive impact of the noise profile on the unwanted multipath signals. This interesting result need to be investigated deeply in further work.

When the transmitted data outpaces the network capacity in scenario3, two modes are highly distinguishable with low variance for the both users: the maximum received bandwidth is increased. Such result is expected, as during the transmission, a high level of packet loss is observed and retransmission was highly active. It is worth mentioning that multiple metrics values were missed during that scenario. It reflects the real complex wireless systems, where the presence of multiple radio phenomena leads to severe bandwidth degradation.

Overall, this illustration sheds light on the high correlation between radio phenomena and high level QoS, i.e. bandwidth.

VI. UPLINK BANDWIDTH PREDICTION EVALUATION

This part presents and discusses the results obtained on bandwidth prediction using machine learning techniques, i.e. LR and RF. Six datasets are tested with two ML techniques RF and LR, i.e. one dataset for each UE per given scenario. The evaluation is based on RMSE as a performance metric.

A. Prediction accuracy

In order to compare the prediction performance of the two ML techniques, fig 6 exhibits error prediction based RMSE for each UE over all the scenarios. With the two MLs, the prediction error doesn't exceed 13 Kbytes. In general RF outperforms LR, except for dataset 1 1. For instance, for dataset 2 1 (scenario 2, UE1), the observed RMSE using RF is 6.44 Kbytes and 12.04 Kbytes using LR. It is to be noted that the maximum received bandwidth in scenario 2 is 82.3 Kbytes and 64 Kbytes for UE 1 and UE 2 respectively. Hence, RF leads to accurate predictions, around 7.8% of errors in an environment where only multipath and noise are present. For more complex scenarios such as scenario 3, the maximum received bandwidth is 92 Kbytes and 105 Kbytes for UE 1 and UE 2 respectively. The prediction error in such scenario reaches 13 %, but remain acceptable. Therefore, this results prove the possibility of predicting accurately uplink bandwidth using lower layer metrics/measurements in different radio scenarios.

B. Window forecast and lag size sensitivity

In order to exhibit the influence of window forecast size on predictions, we evaluate each dataset over iδt, with i ∈ {1, 3, 5, 7, 10} and w = 1. That is, we predict bandwidth based on the instantaneous and the past lower layers measures/metrics, including the past received bandwidth. In addition, the importance of having numerous past radio measurements for good predictions is analyzed. The main lag sizes w tested are 3, 7 or 10, i.e. using the past three, seven and ten δt measurements. Fig. 7 plots in 3D the variation of the perceived RMSE over iδt with i ∈ {1, 3, 5, 7, 10} and per lag size w, w ∈ {1, 3, 7, 10} for UE 2 prediction on dataset 2 2 and dataset 3 2 using RF as an underlying ML for predictions. Based on dataset dataset 2 2, the RMSE doesn't exceed 6.47 Kbytes for all the forecasted windows. The prediction at a granularity of δt = 3 leads to a smooth amelioration, i.e. the RMSE decreases by 1.49 Kbytes. Generally, when increasing the forecast window, a negligible increase of RMSE is observed based on both datasets. Thus exhibiting the insensitivity of the models to the forecast windows.

The introduction of lag size for predictions at granularity of 1δt doesn't improve necessarily the performance, i.e. using dataset 3 2, the RMSE increases from 6.46 Kbytes to 6.78 Kbytes when using w = 10. Such result points out the insensitivity to lag size when predicting at a granularity of 1δt. On the other hand, a remarkable improvement is observed when predicting larger forecast windows and using lag sizes of w = 3 or w = 7. This indicates the sensitivity to lag size for predictions of higher forecast windows. Considering that larger w may not necessary improve the prediction accuracy while leading to a higher computation overhead, we fix w = 1 in the rest of the paper.

C. Sensitivity to training data length

Machine learning techniques leverage training sets to give accurate predictions. The size and variance of a given training set should then impact the prediction performance. In this part, we investigate the sensitivity of our model to the training dataset size. For that, we train RF model with different datasets lengths. In order to have a significant variance in each training

VII. CONCLUSION

In this article, we investigate the uplink bandwidth prediction in cellular network and the radio phenomena impact on high-level QoS metrics, i.e. bandwidth. For that, a testbed is deployed in an anechoic room where the radio phenomena are controlled. This allows a clear analysis of encountered behaviors in bandwidth variation. Exhaustive benchmark of lower layers measurements/metrics is performed to constitute dataset reflecting the network response to radio environment variation. In order to predict the received bandwidth at a granularity of 100 ms, machine-learning techniques are used, mainly random forest and linear regression. Nested cross-validation is used for each case study to generalize the obtained error predictions. Different radio scenarios are tested, where the testbed complexity is increased gradually. The analysis of bandwidth variation per radio scenario exhibits the strong correlation between bandwidth and radio environment variation. For each scenario, the sensitivity of the models to forecast window, lag size and training data length are investigated. The model shows insensitivity when predicting. The two machine learning techniques lead to accurate predictions, especially random forest model. Therefore, the obtained results confirm the ability of the developed method to provide accurate uplink predictions in a real environment with very versatile devices and users.

Fig. 2 :

 2 Fig. 2: Uplink resource grid for one slot.

Fig. 3 :

 3 Fig. 3: LTE-A Testbed deployment.

Fig. 4 :

 4 Fig. 4: Features correlation with bandwidth.

Fig. 5 :

 5 Fig. 5: Bandwidth standard deviation over all scenarios.

Fig. 6 :

 6 Fig. 6: Error prediction for forecast window i=1δt.

Fig. 7 :

 7 Fig. 7: Sensitivity window forecast and lag size for UE 2.

Fig. 8 :

 8 Fig. 8: Sensitivity to training dataset length.

•

  Timing Advance (TA): As the propagation delay of different UEs is different based on their positions, TA is introduced to ensure synchronization between uplink and downlink at the eNB side. TA is a negative offset at the UE. Based on the UE transmitted PRACH, the eNB estimates the initial TA. Once the UE is connected, the eNB keep estimating TA and adjusting it by transmitting to the UE the required value, TA update.

• SNR: SNR: Signal to Noise Ratio compares the level of the desired received signal to the level of noise. Taking P signal and P noise as the average power of the received signal and noise respectively, SNR is defined (in decibels) as follows: SN R dB = 10.log 10 (P signal /P noise ). It is measured for each received PUSCH holding UE's data.

• received UL CQI: Uplink Channel Quality Indicator. It is computed at the eNB based on the observed SNR. • PUCCH received power and noise power: the two measures are estimated principally for each PUCCH handling a scheduling request. • PUCCH threshold: It is the threshold to detect the pucch format1.

Algorithm 1 Bandwidth prediction from dataset s u Input:

Historical metrics/measures with lag size w, historical received bandwidth, M: optimized prediction model. Output: BW =[(BW 1 (1δt,w) , .., BW 1 (i,w) ), .., (BW m (1δt,w) , .., BW m (i,w) )] : the predicted bandwidth.

for each(w,i) do for t in range (1, m) do

test is the value of a feature in time t. end for end for return BW K 1 -fold is applied. It divides the training set into K 1 equal folds; K 1 -1 folds are used for training and the remaining fold for evaluation. It computes the prediction error and iterates until all the folds are used for both training and validation, then the prediction error is averaged over all the K 1 cases of CV. The hyper-parameter combination achieving a minimized prediction error is selected as the best optimized model. In order to generalize the selected model, the outer loop CV is used where the model is tested K 2 times on unseen data, i.e. the test set. Then, the generalized prediction error is the average of the estimated prediction error over the tested sets.

Algorithm 1 represents the prediction model for bandwidth. Given the dataset, we apply the algorithm summarized in the following. Let N be the size of the considered dataset. K 2fold CV is chosen for having 25% of the dataset for testing which we denote X test ={X 1 test , .., X m test } with size m, and 75% for the training set denoted X train ={X 1 train , .., X n train } with size n, where X p train and X p test are the p th samples in their corresponding datasets. GridsearshCV is applied on the X train to select the optimized model, referred by M . Let w be the rolling window, it represents the past w time units, and i denotes the forecast window, where i ∈ {1δt...10δt}, with δt=100 ms. The maximum forecast window is then 1 second. For each lag w and forecast window i, M uses the historical metrics (X t test , X t-1 test , .., X t-w test ) to predict for each sample X t test from X test the corresponding bandwidth BW t (i,w) , i ∈ {1..10}. For example, with i=1 and w=2, M uses the current and two past time units of X t test : [X t test , X t-1 test , X t-2 test ] to predict the upcoming bandwidth in 100ms. Therefore, for predictions evaluation, we compare the predicted bandwidth BW t (i,w) with the received BW t (i,w) bandwidth, based on RMSE (Root Mean Squared Error) metric. RMSE is attractive from a statistical and scientific perspective. It represents the average error prediction in the model, expressed in the units of the variable of interest. It is computed as follows: RM SE = (1/n * n i (y i -ŷi ) 2 ) where y 1 ...y n are the actual values and ŷ1 ... ŷn the predicted ones. By squaring the error, a high weight is given to the large errors. RMSE score is negatively oriented, hence lower values are better.


Agnès Gorczyca

Dr Pascal Raybaud

Dr Virginie Moizan

Dr Yves Joly

Dr Céline Chizallet
email: celine.chizallet@ifpen.fr

Dr A Gorczyca

Atomistic Models for Highly-Dispersed PtSn/γ-Al2O3 Catalysts: Ductility and Dilution Affect the Affinity for Hydrogen

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Introduction

The Pt/γ-Al2O3-like systems, where platinum is in a highly dispersed form (sub-nanometric particles), are of paramount important in heterogeneous catalysis, [1,2] in particular in the presence of H2. For some importantoften at the industrial scaleapplications like hydrogenation of unsaturated hydrocarbons, [3] dehydrogenation of light alcanes [4][5][6][7][8] or catalytic reforming of naphta, [START_REF] Sinfelt | Catalytic reforming[END_REF] tin is added to improve the selectivity, stability and regeneration properties of the catalysts. [START_REF] Bond | Metal-Catalyzed Reactions of Hydrocarbons[END_REF] Yet these effects remain poorly understood at the molecular scale due to some resolution limitations of characterization techniques when undertaken operando. To make progress in this area and allow a better control of the active phase, the elaboration of structural models for bimetallic platinum-tin subnanometric particles supported on γ-alumina, is of fundamental and practical importance. Density functional theory (DFT) calculations are useful in that respect, due to the challenging characterization of supported clusters, that are often subnanometric and non-regular in shape. [5,[START_REF] Siri | [END_REF][12][13][14][15][16] Significant amount of work has been devoted so far for the proposal of relevant dispersed monometallic Pt/Al2O3 models, [17] in particular by some of us in the case of the gamma polymorph of alumina. [18][19][20] The ductility of small platinum clusters (typically, Pt13) appears to be very strong, [19,21] in particular in the presence of hydrogen, [22,23] oxygen [24] and hydrocarbons [25][26][27] despite the presence of a support. Moreover, it was shown that Pt13 cluster exhibit a higher adsorption capacity than the infinite Pt(111) surface. [22,23] Experimental findings support these theoretical observations. [23,28] However, only a few theoretical studies deal with bimetallic PtSn nanoclusters, [14,[29][30][31]32] despite their high practical interest. None of these studies includes at the same time support and environment effects. To the best of our knowledge, with the exception of an earlier study of H2 interaction with isolated Pt-Sn dimers, [33] hydrogen adsorption has been studied on ideal and infinite alloyed Pt2Sn surfaces only. [34][35][36] The surface properties of bimetallic systems with respect to pure compounds were extensively discussed in terms of electronic (ligand) versus dilution (also called ensemble or geometric) effect. [37,38] The former corresponds to the change induced by electronic transfers between metals of different electronegativities, whereas the later means that the dilution of one metal by the other affects the adsorption modes and coverage, in the case where one metal interacts more strongly than the other with a given molecule. In what follows, we will preferentially use the expression dilution (or ensemble) instead of geometric effect which may stand for a broader range of structural phenomena. For extended surfaces, both kinds of effects lead to a lower affinity of platinum-tin for hydrogen as compared to platinum. [35,36,38] It remains thus of crucial interest to combining this effect of Pt-Sn alloying with the nanosize effect as previously studied for Pt13 clusters, [22,23] to predict how the structural, electronic, and hydrogen adsorption properties will be affected in the case of a "ductile" cluster, different from a "rigid" extended surface. The present work proposes a theoretical investigation of the PtxSny/γ-Al2O3 system, in the case where tin in formally reduced (Sn 0 ). These species are likely to exist for PtSn/γ-Al2O3 catalysts, in particular when synthesized by surface organometallic chemistry. [8,[START_REF] Siri | [END_REF]12,14,[39][40][41] Models for platinum-tin sub-nanometric clusters, for thirteen metal atoms, are proposed (PtxSn13-x with 0 ≤ x ≤ 13). This particle size (close to 0.9 nm as shown in our previous work devoted to Pt13 [22] ) is relevant with respect to experimental dispersions. [6,13,15,42,43] Moreover, 13 is the smallest number of atoms needed to build an icosahedron or a cuboctahedron. Isolated clusters are simulated thanks to a simulated annealing method, as well as clusters supported on γ-Al2O3, thanks to the support model established in ref. [44] A detailed investigation of the interaction of the supported Pt10Sn3 cluster with hydrogen is then reported, providing a mapping of the hydrogen coverage as a function of the operating conditions (T, P(H2)), and giving first insights into the understanding of the reactivities of platinum-tin and platinum catalysts, in terms of geometric and electronic effects.

Computational Methods

Periodic density functional theory (DFT) calculations were performed using a plane-wave method as implemented in the Vienna Ab initio Simulation Package. [45] The exchange-correlation functional was treated within the generalized gradient approximation (GGA) parameterized by Perdew, Burke and Ernzerhof [46] (PBE) and the electron-ion interaction was described by the projector augmented wave (PAW) scheme [47] with an energy cutoff of 400 eV. Note that corrections for dispersion forces were not included in the present work for two reasons: i) for the sake of consistency with previous studies that we want to make comparison with, [19,22] ii) and because such refinements to the PBE energies and forces were shown to be rather low in the case of PtSn surfaces in interaction with hydrogen. [36] The situation might be more critical for the quantification of the interaction with hydrocarbons, [27,36,48] which we do not consider here, or for the description of the metal-support adhesion in some cases. [49] Spin-polarized calculations were performed for non-supported clusters and used the interpolation formula of Vosko, Wilk and Nusair. [50] For isolated metallic clusters, order 1 Methfessel-Paxton smearing with σ = 0.1 eV was applied. [51] Gaussian smearing with σ = 0.02 eV was used for clusters supported on gammaalumina models. The criterion for the convergence of the self-consistent cycles was set to 10 -6 eV. Bader charges [52] were calculated at the same level of theory. Geometry optimizations were run until forces on relaxed atoms were lower than 2.10 -2 eV.Å -1 . Dipolar correction was added for nonsymmetric slabs in the direction perpendicular to the slab. Isolated PtxSn13-x clusters (0 ≤ x ≤ 13) were optimized in a 20x20x20 Å 3 cubic cell and the corresponding Brillouin-zone k-point mesh sampling was restricted to the gamma point. In the same spirit as our previous works on monometallic clusters, [22,23,53] a simulated annealing method was applied to find the most stable PtxSn13-x clusters by scanning a large structural configuration space. This simulated annealing method is based on first-principles NVT velocity scaled molecular dynamics (MD) followed by quenches at 0 K. The whole simulation time was at least 20 ps and the time for each step was set to 10 fs. Several simulation temperatures were tried (between 400 and 1200 K), as well as several starting geometries (Pt cuboctahedron or biplanar structure, in which 0 to 13 atoms of Pt were substituted by Sn). The binding energy of isolated PtxSn13-x clusters (per metallic atom) is defined by equation (1).

  13 / x)E(Sn) - (13 - xE(Pt) - ) Sn E(Pt ) Sn (Pt E x 13 x x 13 x b    (1)
where E(PtxSn13-x), E(Pt) and E(Sn) are the energies of the PtxSn13-x cluster and of a free Pt/Sn atom, respectively. According to this definition, the more negative the binding energy, the more stable the cluster. The formation energy per metallic atom ΔfE(PtxSn13-x) is defined by equation ( 2), by analogy with the definition of the standard enthalpy of formation of solid Pt-Sn alloys. The Pt13 and Sn13 clusters are used as references, while for solid state thermodynamics the references are the stable states of the elements in standard conditions at 25°C. [54]   (2) Supported PtxSn13-x clusters (0 ≤ x ≤ 13) were modeled, using the dehydrated γ-Al2O3 (100) surface model described previously. [44] Indeed, from previous studies about monometallic or bimetallic catalysts, [13,[22][23][24]55] models involving theγ-Al2O3(100) dehydrated surface already provided fruitful detailed information. Moreover, this surface exhibits pentacoordinated Al 3+ ions, which were experimentally suggested to stabilize Pt-Sn clusters efficient for propane dehydrogenation. [43] Periodic slabs consist of four alumina layers separated by a vacuum thickness corresponding to more than two equivalent empty layers. The two topmost alumina layers were relaxed, as well as all metallic atoms, whereas the two other ones were frozen. For the investigation of the supported PtxSn13-x in the absence of hydrogen, the dimensions of the cells are 11.14x16.79x21.80 Å 3 (see supporting information S1), and a 2x1x1 k-points mesh was used. Two strategies were adopted to obtain models for the PtxSn13-x/γ-Al2O3(100) systems: (i) the most stable PtxSn13-x non supported structures found after simulated annealing were deposited on the γ-Al2O3(100) surface model, close to the most stable cluster location found for the Pt13 cluster in reference [19] , and geometry optimizations were performed; (ii) starting from the most stable morphology of the biplanar Pt13 cluster supported on γ-Al2O3(100), 1 to 13 Pt atoms were replaced by Sn, studying several locations for substituted Sn atoms, and geometry optimizations were performed. The binding energy of supported PtxSn13-x clusters, given per metallic atom, is defined by equation (3).

        ) E(Sn 13 x) - ( 13
  13 / ) O E(Al - x)E(Sn) - (13 - xE(Pt) - ) O /Al Sn E(Pt ) O /Al Sn (Pt E 3 2 3 2 x 13 x 3 2 x 13 x b    (3)
Where E(PtxSn13-x/Al2O3) and E(Al2O3) are the energies of the supported clusters and of the alumina slab, respectively. Similarly to the isolated clusters, we also define the formation energy per metallic atom ΔfE(PtxSn13-x/Al2O3) by equation (4).

          ) O E(Al - ) O /Al E(Sn 13 x) - ( 13
The metal-support interaction energy of a PtxSn13-x cluster on the γ-Al2O3 support is calculated by equation (5).

  13 / ) O (Al E' - ) Sn (Pt E' - ) O /Al Sn E(Pt ) O /Al Sn (Pt E 3 2 x 13 x 3 2 x 13 x 3 2 x 13 x Support - Metal     (5)
E'(PtxSn13-x) and E'(Al2O3) are respectively the energies of the cluster and the alumina slab with their respective geometry when they are interacting with each other. They differ from their most stable geometry when they are separated. The structure of the supported Pt10Sn3 system was further refined before investigating hydrogen adsorption, by extending the cell to 16.71 Å in the x direction, by adding new initial states before geometry optimization, and adding velocity scaled MD steps (similar as for non-supported clusters), as shown in supporting information S1. The most stable structure found and reported in the following, results from two of these independent MD (the first one starting from the non-doped supported cluster in which 3 atoms were substituted by tin, the second one starting from the non-doped isolated cluster in which 3 atoms were substituted by tin, then supported) runs followed by a quench at 0 K. Hydrogen adsorption was modeled on the Pt10Sn3/γ-Al2O3(100) system. We did not consider in detail the adsorption of H2 on the alumina support, as it was shown that the (100) surface orientation only weakly interacts with hydrogen. [56] The H coverage on the cluster was varied between 2 to 20 hydrogen atoms per cluster, which appeared to be sufficient to saturate the cluster, contrary to the case of the monometallic Pt system which required to simulate up to 36 atoms (see section 3.4.). In a first approach, and in the same spirit as ref. [22] , the most stable Pt10Sn3/γ-Al2O3(100) system was used as a starting structure, on which hydrogen atoms were adsorbed, and the system was subjected to a velocity scaled MD study. The cluster morphology showing strong modifications at H coverage higher than 10 atoms per cluster, other investigations were made on a cuboctahedral cluster morphology (see Supporting Information S2) saturated by hydrogen atoms, and then with decreasing hydrogen loading. The cuboctahedron was put in contact with the alumina support, at a similar location as that of the optimal Pt10Sn3 cluster in the absence of hydrogen. At each coverage, velocity scaled MD runs followed by quench steps were undertaken. The average adsorption energy ΔadsE for the adsorption of n hydrogen atoms is given by H2 molecule and calculated thanks to equation (6). It includes the possible contribution of the reconstruction of the cluster upon H2 adsorption.

) E(H 2 n - ) O /Al Sn E(Pt - ) O /Al Sn Pt E(H E Δ 2 n 2 3 2 3 10 3 2 3 10 n ads  (6)
The phase diagram depicting the stability of the system in terms of hydrogen coverage as a function of the temperature and the hydrogen partial pressure, was built by the same method as the one used in references [22,23] . The evaluation of the chemical potential for H2, assuming an ideal gas behaviour, leads to the estimation of the temperature -pressure domains where each system at a given hydrogen coverage is the most stable.

Results and Discussion

Non-supported PtxSn13-x cluster models

The simulated annealing sequence described in the methods section leads to a large set of structures, the most stable one for each composition are shown in Figure 1. All clusters exhibit distorted 3D morphologies and only low coordination metal atoms (no central atom contrary to a cuboctahedron or an icosahedron) thus resulting in cage-like structures. Up to 7 tin atoms in the PtxSn13-x cluster (6 ≤ x ≤ 13), the intercalation of Sn atoms in Pt-Pt bonds is favored, without any segregation of tin. The same holds true for platinum, intercalated in Sn-Sn bonds for 1 ≤ x ≤ 6. Some analogy can be seen with the bulk platinum-tin compounds (supporting information S3). In particular, the bulk Pt2Sn3 structure exhibits cage-like sub-structures similar to the Pt6Sn7 and Pt5Sn8 clusters, whereas the PtSn4 bulk structure reveals layers of platinum atoms between Sn layers, as for the Sn-richer clusters (see supporting information S3 for the extraction of cages and layers from the bulk alloy structures). These observations are compatible with previous works for some of the compositions addressed here. [32 , 57]

(c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) Figure 2.
(a) Binding energies for non-supported and supported PtxSn13-x, (b) formation energies of the bimetallic clusters with monometallic clusters as reference, together with standard formation enthalpy from the literature (in green) for bulk alloys. Thermodynamic calculations from ref. [54] , experimental data from ref. [54,[START_REF] Ferro | Matematiche e Naturali[END_REF] The binding energy (equation ( 1)) and the formation energy (equation ( 2)) are reported in Figure 2. The sharp increase of the binding energy starting from 7 tin atoms reveals the low Sn-Sn cohesion as compared to the Pt-Pt one. A shallow minimum in binding energy is however observed close to 5 tin atoms, corresponding to maximal cohesion, indicating the existence of a driving force for the formation of an alloy from isolated atoms. This is confirmed by the evolution of the formation energy (Figure 2-(b)), taking Pt13 and Sn13 clusters as references. It is minimal close to 50% Sn, following the same trend as bulk alloys.

Bader charges for the non-supported clusters are reported in section S4. The trends being similar to the one calculated in section 2 for the supported clusters, the reader is referred to the next section for a detailed description. In line with previous reports, [29,30,59,60] we find that Pt holds a negative charge in the alloy, making the platinum atoms being in repulsive coulomb interaction one with respect to the other, which likely explains why they do not aggregate in the alloy. [59] 2. PtxSn13-x / γ-Al2O3(100) models Starting either from the most stable supported Pt13 geometry (by Sn substitution) or from the most stable PtxSn13-x structures reported in section 1, we obtained the most stable supported structures reported in Figure 3. Their binding energy follows roughly the same trend as the isolated clusters (Figure 2-(a)), except for the lowest tin loading, for which the systems are significantly more stable thanks to the support. This suggests a significantly weaker Sn 0 -support interaction with respect to the Pt 0 -support interaction. Indeed, for 13-x < 6, the cluster exhibits a biplanar-like morphology, which is thus strongly reminiscent of the monometallic Pt13 cluster morphology, whereas at higher Sn loading, the cluster shape is 3D-like, inherited from the simulation of non-supported clusters. For all these models except the tin richer ones, tin atoms are well intercalated between the Pt atoms, and prefer to be far away from the support, as compared to platinum, in line with MD reported for supported Pt10Sn10 and Pt15Sn5 clusters. [30] From a structural point of view, the shortest Pt-O and Pt-Al bonds follows three regimes (Figure 4-(a)): up to 6 tin atom per cluster, the values of these minimal distances are close to the one measured in the case of the pure supported platinum cluster (2.1 -2.2 Å and 2.5 -2.7 Å for the Pt-O and Pt-Al distance, respectively). These become slightly longer (up to 2.4 Å for Pt-O and 3.3 Å for Pt-Al) for tin richer supported clusters. Indeed, from 7 to 11 Sn in the cluster, the morphology is inherited from the gas phase one, and although a platinum atom is pointing towards the support, a disconnection of the cluster with respect to the support starts to take place. Caballero et al. reported long Pt-O bonds (2.24 Å) for supported PtSn alloys on gamma-alumina, [12] which do not appear for the related monometallic Pt system. One may propose that such a contribution is related to the increased metal-support distance due to tin. For the supported Pt1Sn12 cluster, the platinum atom is occluded inside the cluster by the surrounding Sn atoms so that the distance to the support sharply increases. The weakening of the metal-support interaction upon increasing tin loading is further quantified by the metal-support interaction energy, reported in Figure 4-(b), with a nearly zero-interaction starting from 8 tin atoms in the cluster. This can also be seen in Figure 2-(b), where the formation energy of the supported cluster is lower than that of the isolated cluster for low tin loading, the reverse trend being observed at high tin loading. Despite these differences, the formation energy roughly follows the same trend than for other types of alloys.

From an electronic point of view (Figure 5), the clusters are negatively or very slightly positively charged over the whole composition range, which indicates that the support gives electrons to the cluster through the γ-Al2O3(100) facet, as this was already the case for pure platinum. [18,19] Pt atoms are negatively charged, tin atoms are positively charged, which indicates a charge transfer from Sn atoms to Pt atoms, consistently with the respective electronegativity values of the atomic species (1.8 for Sn and 2.2 for Pt in the Pauling scale [START_REF] Pauling | The nature of the chemical bond and the structure of molecules and crystals : an introduction to modern structural chemistry[END_REF] ), with previous computational studies for clusters and surfaces, [29,30,59,60] and with XANES experiments. [12] Our results also shows that tin tends to be partially oxidized in the nano-alloy. When the tin loading increases, Pt atoms are individually more and more negatively charged, contrary to tin atoms which are less and less positively charged. In other words, the maximal oxidation state per Sn atom is found when one single Sn atom is present in the cluster. The trend is of course symmetrical in the case of Pt for which the maximal charge gain per Pt atom occurs when the cluster contains only one Pt atom. For 13-x > 8, the cluster tends to a neutral state. This can be seen as a consequence of the loss of the metal-support interaction for high loading of tin. Notably, the cumulated charge of Pt and Sn atoms together reaches a minimum, respectively maximum for 7 tin atoms in the cluster, meaning that the overall intensity of the charge transfer is maximal for the close to stoichiometric alloy.

Refinement of the Pt10Sn3/γ-Al2O3(100) model

Often, during the catalysts synthesis, the Sn and Pt loading are of the same order of magnitude. Nevertheless, previous works suggest that a lower amount of tin is conjectured to be present in the reduced nano-alloy. [2,5,[START_REF] Siri | [END_REF]41] We thus chose the supported Pt10Sn3 cluster, whose stoichiometry is the closest one to the bulk Pt3Sn one, to simulate hydrogen adsorption. The structure of this system was thus refined (see methods), generalizing the MD procedure to each step of the calculation, whereas preliminary models (section 2) were obtained by geometry optimization only for the supported states. From this approach, one particular cluster morphology showed a superior stability (Figure 6), with a binding energy lowered by 10 kJ.mol -1 with respect to the previous geometry optimization (Figure 2-(a)). The resulting refined morphology is closer to the one of the Pt13 cluster optimized in gas phase (Figure 1-(a)). Compared with the biplanar geometry of the monometallic supported particle (Figure 3-(a)) and that of the Pt10Sn3 supported cluster before MD investigation (Figure 3-(d)), the bimetallic one is more three-dimensional and exhibits 3 stacked atomic layers (5-6-2) instead of the initial bilayer (7-6). The 3 tin atoms are located on the two upmost layers (further from the support) and are surrounded by platinum atoms. The cluster exhibits 6 Pt-Sn bonds below 2.7 Å, corresponding to an average of 2.66 Å and an average Pt-Sn coordination number of 0.6 (by platinum atom, supporting information S5). Six additional Pt-Sn bonds exist between 2.7 and 3 Å, if we include these in the average calculation, the Pt-Sn bond length becomes 2.71 Å for a coordination number (CN) of 1.2. From EXAFS, Siri et al. [START_REF] Siri | [END_REF] reported Pt-Sn length of the order of 2.6-2.7 Å for CN between 2.1 and 2.7. Iglesias-Juez et al. [5] report Pt-Sn length of 2.8 Å for a CN (per platinum) of 0.7, which appears to be compatible with our calculations. Caballero et al. [12] fitted longer (2.9 Å) Pt-Sn bonds with lower (1.2) CN, which could be related to the presence of Sn II species. Regarding Pt-Pt bonds, focusing on the 17 bonds shorter than 3 Å (thus an average CN of 1.7), the model exhibits an average length of 2.67 Å, in excellent agreement with several sets of EXAFS experiments. [START_REF] Siri | [END_REF]12] Note however that EXAFS was fitted in ref. [5,[START_REF] Siri | [END_REF] without invoking Pt-O nor Pt-Al bonds, which we have in the model. This may be explained by their amount by platinum atoms (average CN lower than one, see supporting information S5). Other authors need Pt-O bonds to be introduced in the fitting of EXAFS data, [12] with Pt-O length close to 2.24 Å, in agreement with our average value of 2.22 Å. The absence of Sn-Sn bonds below 3 Å (the shortest in the model is at 3.7 Å) is compatible with EXAFS recorded at the K edge of Sn. [START_REF] Meitzner | [END_REF] These structural features result in a higher (lower in terms of absolute value) metal-support interaction (Figure 4-(b)) as compared to the cluster obtained before MD. The enhanced stability of this new model is thus not due to the metal-support interaction, but to a lower deformation energy (85 instead of 246 kJ.mol -1 for the refined and non-refined supported Pt10Sn3 models respectively, both referenced to the isolated Pt10Sn3 cluster). In both cases, the interaction between the cluster and the support remains strong to keep the the shortest Pt-O and Pt-Al bond lengths almost unchanged (Figure 4-(a)). The difference in Bader charges before and after the MD procedure is also negligible (Figure 5), suggesting that the main driving force for the charges of Pt and Sn is the alloying effect rather than the support effect. q(Pt x Sn 13-x ) q(Pt x ) q(Pt x )/x q(Sn 13-x ) q(Sn 13-x )/(13-x)

Simulation of hydrogen adsorption on the Pt10Sn3/γ-Al2O3(100) cluster

The hydrogen adsorption study was undertaken on the refined structure of the Pt10Sn3/γ-Al2O3(100) cluster obtained previously. For that, we chose various starting configurations for the MD simulations for each hydrogen coverage, as explained in the computational section. The most stable structures are reported exhaustively in Supporting information S6, a selection of these being shown in Figure 7. It systematically appears that tin atoms are not favorable sites for hydrogen adsorption, in line with results obtained on flat alloyed surfaces. [34,36] For the system investigated here (in particular, with the (100) alumina surface orientation), H spillover from the cluster to the support was not found to take place spontaneously. The cluster morphology evolved significantly with H adsorption, and is close to the one of a cuboctaedron for 20H. It is remarkable that such a cuboactahedron morphology appeared for Pt13 in the same range of H atom number adsorbed per cluster. [22,23] For a hydrogen coverage from 2H to 16H, tin atoms are always diluted between Pt atoms, and stay at a significant distance from the support, the shortest Metal-O or Metal-Al bonds always being Pt-O and Pt-Al bonds (Figure 8-a). Thus, the whole cluster keeps connected to the support thanks to Pt-Al (close to 2.4 Å) and Pt-O (close to 2.1 Å) bonds. For 20 H, the distance between the cluster and the support become larger (Figure 8-a cluster, in comparison with the monometallic Pt13 cluster (data from [22] ). Hollow sites are omitted because they are absent from the Pt10Sn3 cluster and almost absent from the Pt13 cluster.

The number of top and bridge sites occupied on the most stable systems for a given H coverage are depicted on Figure 8-(b). At low H coverage, excepted for a single hydrogen atom which is preferably adsorbed on top position, the bridge position is largely preferred. For more than 16H adsorbed, there are no bridge sites left due to the intercalation of Sn atoms, then H atoms occupy top sites. For monometallic Pt13 clusters, [22] the trend was different with competing top and bridge sites at hydrogen coverages inferior to 20 H, and with predominant bridge sites at higher hydrogen coverages. However, these high coverages are never reached on the bimetallic cluster simulated here, which is a clear effect of nano-alloying. The lack of bridge sites reveals the geometric effect of tin in the evolution of the properties from pure Pt to Pt alloyed with Sn. Notably, the cuboctaedron obtained at 20 H (Figure 7) contains 12 Pt-Pt edges accessible to hydrogen (symmetrically 12 Sn-Pt edges, i.e. 4 edges per Sn atom). So in principle 12 H atoms can be adsorbed at Pt-Pt bridge sites for the Pt10Sn3 cuboctaedron, versus 24 H for the Pt13 cuboctaedron. These Sn-Pt bridge position are actually inactive sites for H adsorption. This is also the case for 3 top sites located on Sn. Thus the saturation level is reached for about 20 H atoms, where it was about 36 H atoms. For H coverages higher than 20 H, we observed hydrogen desorption in the course of the MD run (which was not the case for Pt13), which also justifies the lower saturation level found for Pt10Sn3.

Considering the important role of bridge sites for both types of clusters, this is likely a key feature to explain the differ ence between them.

The electrostatic charges were calculated with a Bader analysis for the hydrogenated Pt10Sn3/γ-Al2O3(100) (Figure 9). The bimetallic cluster (plus hydrogen atoms) is slightly negatively charged, which shows that it receives electron density from the support. This charge (in absolute value) decreases with increasing the H coverage, so that the whole cluster becomes almost neutral for 20H. This general evolution is similar to that of the Pt13/γ-Al2O3(100) cluster, [18,19] which corresponds to the gradual loss of the covalent metalsupport interaction. However, the monometallic cluster (plus H atoms) was intrinsically more negatively charged (-1.1 and -0.7 on the same interval of H) due to the presence of tin which is positively charged. Pt atoms are negatively charged, much more strongly than in the monometallic cluster, which corresponds to the charge transfer from tin atoms, as a consequence of the respective electronegativity values of platinum and tin (already seen in Figure 5). The charge of platinum atoms is decreasing with the H coverage. H atoms exhibit a weak hydridic nature, with a small negative charge. This result was found also for the biplanar morphology of Pt13 clusters, in the "low" coverage regime. [22] As Sn atoms are positively charged with increasing H coverage, we can deduce that a charge transfer takes place, concomitantly from the support and Sn atoms to Pt and H atoms. A phase diagram was built according to the method described in the computational section, which depicts the evolution of the optimal H coverage as a function of the hydrogen partial pressure and of the temperature (Figure 10). It is compared with the one obtained for the monometallic system, [22] and for the Pt2Sn/Pt(111) extended surface as modeled by Gautier and Sautet. [36] The maximal number of hydrogen atom appearing in the diagram is 16 H by cluster, meaning that the cluster will likely not be disconnected from the support, contrary to what was calculated for Pt13 supported clusters. [22,23] This could be related to the redispersion role of Sn seen experimentally, although a role of oxidic forms of Sn (not simulated here) can also be thought of in that respect. [6] Comparing the PtSn cluster and the extended surface (Figure 10 (a) and (c) respectively), higher loadings of the clusters are described for given thermodynamics conditions. The zero coverage domain is much larger for the surface, while its maximal loading above 200 K is 0.67 ML (6/9 on the figure). If we define the monolayer on the cluster as the number of hydrogen atom per metallic atom (Pt+Sn), the maximal coverage appearing on the diagram is 1.23 ML. 100), adapted from ref. [22] , (c) the Pt2Sn/Pt(111) extended surface (figure reprinted from ref. [36] with permission). For (a) and (b), the number of hydrogen atoms per cluster is given in the corresponding zones. For (a) and (b), all systems within a free adsorption energy difference of 10 kJ.mol -1 are indicated in the given zone. For (c), the coverage is given in monolayer. Comparing the pure platinum and alloyed clusters, for the high temperature (above 500 K) / low H2 pressure (below 1 bar) domain, the number of hydrogen atoms on Pt10Sn3 is generally higher than that of Pt13. For example, the zero coverage domain is larger in the case of platinum. This may be linked with the experimental observation according to which small amount of tin increase the uptake in H2 titration experiments. [63] This can already be seen from the respective ranking of average adsorption energies between both cluster compositions (Supporting information S7), demonstrating a stronger adsorption of hydrogen for low coverage on Pt10Sn3 than on Pt13. This is in contradiction with the electronic effect observed for extended surfaces, which makes the interaction lower for PtSn with respect to Pt. [35,36,38] Moreover at low coverage the dilution effect does not apply. The explanation may raise from the larger structural ductility of the Pt10Sn3 with respect to Pt13. Indeed, the geometries of the supported Pt10Sn3 clusters after adsorption of 0, 2 and 8 hydrogen atoms are very different one to another (Figure 7), while for Pt13 the cluster remains biplanar in this range of hydrogen coverage. [22] Such a reconstruction cannot take place on infinite surface which may explain why this trend was not observed on Pt2Sn surface. [36] However, when lowering the temperature or increasing the hydrogen pressure, the presence of tin drastically reduces the amount of adsorbed hydrogen on the cluster, as compared to pure supported platinum clusters. For example, under T = 300 K and P(H2) = 1 bar, the H/Pt is at most 1.6 (H/(Pt+Sn) = 1.2) for the bimetallic cluster against 2.6 for the monometallic one. This limitation can be directly linked with the dilution of platinum by tin mentioned above. This behavior is compatible several chemisorption and microcalorimetry experiments made for Pt and PtSn systems on various supports. [15,39,63,64,65] Regarding hydrogen adsorption energies (Supporting information S7), a crossing of the curves between Pt13 and Pt10Sn3 indeed takes place. The two effects (ductility and dilution effects by PtSn) may have some consequences on the catalytic properties of the cluters. The ductility effect has been recently invoked to be involved in the dehydrogenation mechanisms of Pt13 clusters supported on alumina. [25,27] The formation of the PtSn nano-alloy thus further enhances this ductile character of the cluster with a potential impact on the dehydrogenation reactivity thanks to the stabilization of H atoms at low coverages. On the one hand, the stronger affinity at low coverage may enhance the dehydrogenation steps. On the other hand, since the maximal number of H atoms is diminished (with respect to monometallic cluster) by dilution effect, it is expected that the reverse hydrogenation reaction is limited. Although this qualitative interpretation must be further quantified by dedicated theoretical investigations, we may understand better why alumina supported PtSn catalysts are preferentially used in dehydrogenation and dehydrocyclisation reactions. [4][5][6][7][8] To compare the calculated amounts of hydrogen with experiments, let us focus on the total amount of chemisorbed hydrogen measured evaluated by Singh et al. [15] for dispersed Pt and PtSn catalysts supported on alumina. The authors start chemisorption after in situ reduction under H2. Prior adsorption of hydrogen, the samples were evacuated under vacuum at 473 K and then cooled down at 303 K. In these conditions, estimating the initial hydrogen coverage is very difficult, as for both catalysts composi tions, the phase diagrams suggest a very high sensitivity of the hydrogen coverage upon temperature and pressure. This means that the initial coverage will depend on the vacuum level, and probably also on the duration of the thermal treatments. Let us take the coverage given on the diagram at T = 473 K and 10 -4 bar of pressure. The expected coverage are close to 20 H for Pt (Figure 10-(b)) and 8 H for PtSn (Figure 10-(a)). The adsorption is then performed at 303 K. [15] In these conditions, the maximal number of H atom per cluster is expected to be 36 for Pt and 16 for PtSn. Differentially, this means that Pt13 and Pt10Sn3 clusters will host close to 16 and 8 H atoms, corresponding to H/Pt ratios of 1.23 and 0.8, in remarkable agreement with the experimental values of 1.25 and 0.75. [15] Similar ranking was obtained by Merlen et al., [65] although the absolute values differ a little bit probably due to variations in the treatment conditions before hydrogen chemisorption. Although the evaluation from ab initio calculations is made after several sets of hypotheses, the fact that the order of magnitude is right means that we have captured the main feature leading to the drop of hydrogen adsorption capacity after alloying with tin.

Conclusions

Thanks to ab initio calculations consisting of geometry optimizations and simulated annealing sequences, we have established a set of PtxSn13-x and PtxSn13-x/γ-Al2O3 models (with 0 ≤ x ≤ 13), in the case where tin in formally reduced (Sn 0 ). A considerable diversity of morphologies are described, dependent on the composition of the cluster (Sn/Pt), the effect of the support, and on the hydrogen coverage (investigated for the Pt10Sn3 stoichiometry). This illustrates the ductility of such sub-nanometer objects. Sn 0 is reluctant to undertake interactions with the alumina support, and does not aggregate for moderate Sn/Pt ratios, giving rise to alloyed systems. A charge transfer results from the formation of the alloy, from Sn to Pt, reinforced on alumina by electron transfer from the support, and electron withdrawing by hydrogen atoms adsorbed on the cluster. The analysis of hydrogen adsorption geometries and of the adsorption thermodynamic diagram lead us to conclude that at high hydrogen coverage, the availability of bridge adsorption sites is key to explain the lower adsorption capacity of PtSn nano-alloy with respect to Pt. By contrast, at low hydrogen coverage, the higher ductility of the Platinum-Tin cluster is playing a key-role, making PtSn more prone to adsorb hydrogen than Pt. This effect is specific to sub-nanometric particles. These combined effects open the route to the interpretation of the beneficial role of tin in PtSn catalyzed reactions, providing the right balance in terms of Pt/Sn ratio and hydrogen coverage is found. In any case, the behavior of PtSn nano-alloy with respect to H is different from the one of extend surface particularly at low H coverages. As a first perspective, it may be relevant to investigate how this modulation of the nano-alloy cluster affinity for H adsorption may impact the reactivity and mechanisms of dehydrogenation reactions. Moreover, one should also underline that oxidized forms of tin (Sn II and Sn IV ) are known to be present on such catalysts, depending on synthesis parameters such as platinum and tin content and precursors nature, activation/reaction conditions or the presence of heteroatoms. [5-7, 11-14, 40, 41, 43, 62, 65, 66] Sn species remaining isolated at the surface support are indeed suspected to play a role in PtSn particles (re)dispersion. [6] Our charge analysis reveals that tin is indeed partially oxidized in the alloy, but stronger oxidation may be reached by considering the interaction of additional oxygen with tin. As a second perspective to this work, accounting for the presence of these species will be necessary in the future to provide a more realistic description of the system. The reduced systems simulated here can be a good starting point for such investigation, in the spirit of previous work done for the description of oxidized forms of supported platinum. [24]

Figure 1 .

 1 Figure 1. Most stable geometry found for PtxSn13-x isolated clusters, after simulated annealing and quenches. x is equal to: (a) 13, (b) 12, (c) 11, (d) 10, (e) 9, (f) 8, (g) 7, (h) 6, (i) 5, (j) 4, (k) 3, (l) 2, (m) 1, (n) 0. Blue: Pt ; Grey: Sn.

Figure 3 .

 3 Figure 3. Most stable geometry found for supported PtxSn13-x clusters. x is equal to: (a) 13, (b) 12, (c) 11, (d) 10 (before molecular dynamics investigations), (e) 9, (f) 8, (g) 7, (h) 6, (i) 5, (j) 4, (k) 3, (l) 2, (m) 1, (n) 0.

Figure 4 .

 4 Figure 4. (a) Shortest Pt-O and Pt-Al distances and (b) Metal-Support interaction energy in the models of supported PtxSn13-x clusters. The plain squares correspond to the supported Pt10Sn3 cluster after molecular dynamics.

Figure 5 .

 5 Figure 5. Bader charges on the supported PtxSn13-x/γ-Al2O3(100) models. The bigger plain squares depict the Bader charge of the supported Pt10Sn3 cluster after molecular dynamics.

Figure 6 .

 6 Figure 6. Most stable morphology obtained for Pt10Sn3/γ-Al2O3(100) cluster after the simulated annealing sequence: (a) top view; (b) side view. Same color code as in figure 3.

Figure 7 .

 7 Figure 7. Selected most stable structures found for increasing hydrogen coverage on the Pt10Sn3/γ-Al2O3(100) cluster. Side (along x and y) and top (along z, bottom) views.

Figure 8 .

 8 Figure 8. (a) Shortest Pt-O and Pt-Al distances and (b) H adsorption mode, function of the number of H atoms adsorbed on the Pt10Sn3/γ-Al2O3(100) bimetallic cluster, in comparison with the monometallic Pt13 cluster (data from[22] ). Hollow sites are omitted because they are absent from the Pt10Sn3 cluster and almost absent from the Pt13 cluster.

Figure 9 .

 9 Figure 9. Bader charges on the HnPt10Sn3/γ-Al2O3(100) models. All data are drawn for the most stable cluster structures for a given n value.

Figure 10 .

 10 Figure 10. Thermodynamic diagrams depicting the most stable hydrogen coverage as a function of the temperature and the hydrogen partial pressure for (a) Pt10Sn3/γ-Al2O3(100), (b) Pt13/γ-Al2O3(100), adapted from ref.[22] , (c) the Pt2Sn/Pt(111) extended surface (figure reprinted from ref.[36] with permission). For (a) and (b), the number of hydrogen atoms per cluster is given in the corresponding zones. For (a) and (b), all systems within a free adsorption energy difference of 10 kJ.mol -1 are indicated in the given zone. For (c), the coverage is given in monolayer.

Acknowledgements

The authors are grateful to Prof. Carlo Lamberti (University of Turin), who passed away abruptly on February 1 st , 2019,for his encouraging and fruitful comments about this work. This work was performed using HPC resources from GENCI-CINES (Grant A0020806134) and from IFP Energies nouvelles. The authors thank S. Lacombe and P. Avenier (IFP Energies nouvelles) for fruitful discussions.

Keywords: Nanoparticles • platinum • tin • density functional theory • reconstruction