Roxana Burghelea

Jean Philippe Mangeot

Frédéric Launay
email: frederic.launay@univ-poitiers.fr

Patrick Coirault

Safwan El

Coherent Synchronization of a Chaotic Communication System Based on a QAM-4 Transmitter

Remarkable research efforts have been invested in recent years on the chaotic communications schemes. In this paper we propose a modulation technique based on QAM-4 (Quadrature Amplitude Modulation). The data is split into I and Q channels. Each channel is spread by chaotic sequences generated over a fixed time interval by a specific digital chaotic system. There are two principal advantages of introducing the chaos in a communication scheme. The first is that it allows signals to be transmitted in a secure way, so that the signal could be perceived as a noise for a potential intruder. Secondly, the spread-spectrum characteristics of the signal improve the noise rejection propreties. A new coherent receiver structure is presented to synchronize each channel. Simulation results are presented to show the effectiveness of the proposed scheme.

Introduction

Nowadays, a great deal of research is focused on chaotic communications schemes. The potential benefits that can be gained from using chaotic signals, including robustness in multipath environments, ease of spectrum spreading, added security etc has made them promising for communications systems within various sectors on industry [START_REF] Lau | Chaos Based Digital Communication Systems[END_REF]. In this paper we propose an enhanced QAM-4 (or QPSK -Quadrature Phase Shift Keying) scheme where a chaotic encoder spreads the input sequence. The data is split into two channels and transmitted from one location to another by mapping bit sequences to symbols, and symbols to sample functions of chaotic waveforms (generated over a fixed time interval by digital chaotic systems [START_REF] Assad | Digital chaotic codec for DS-CDMA Communication Systems[END_REF]). The signals are then filtered and modulated around two quadrature carriers according to the QAM modulation technique. The resulted analog waveform passes through a band limited analog channel, where the signal is distorted and noise is added.

At the receiver, the encoded information can be extracted efficiently by means of coherent detection, where all possible sample functions are known, or by noncoherent detection, where one or more characteristics of the sample functions are estimated. First of all, the received quadrature carriers are downconverted and filtered. Furthermore the data is limited at its original spectrum by multiplying it with the original chaotic sequences and finally it is combined into the original modulating information. The proposed receiver structure is based on a coherent method. The article provides in brief the advantages and disadvantages of such a system compared with non coherent method.

The contribution of this paper is to define the characteristics of the chaotic generator in order to improve error detection, and then to propose an algorithm to correct and consequently to synchronize the receiver in a QAM scheme.

This paper is organized as follows: in Section II we focus on the conception of the QAM-4 chaotic modem. In Section III a coherent demodulation technique is proposed with a specific chaotic generator. In Section IV numerical results are provided to illustrate the effectiveness and the efficiency of the proposed scheme. Conclusions and suggestions for further works are presented in Section V.

Conception of a QAM Chaotic Modem [2]

The QAM-4 transmitter based on chaotic spread spectrum signals is composed of a QAM coder which splits the binary sequence b(t) into I and Q channels. The transmitted sequence b(t) is a periodic binary sequence of

T b period (let's rect T b a rectangle function) b(t) = ∞ k=1 b k .rect T b (t -kT b ) with b k ∈ {0, 1} (1)
Each channel is spread out by a continuous chaotic code. The chaotic generator delivers a data sequence whose value is included in the integer interval [0,254] which is converted to 8 serial bits. The coding operation is carried out by inserting the code or the opposite of the code depending if a +1 or a -1 is transmitted.

In order to product spread spectrum signal, chaotic sequences (characterized by a higher rate than that one of the symbol's component) are introduced for each symbol period as specified in the introduction part. The chaotic I and Q coded sequences can be mathematically expressed as:

c i,q (t) = ∞ k=1 c N ch i,k .rect T ch (t -kT ch ) ( 2 )
where T ch represents the sequence period and c N ch i,k , c N ch i,k are two vectors of integer values included in the [0,254] interval. Finally, the decimal chaotic sequence is converted into an 8-bit binary code by a Parallel/Series block. Another one chaotic generator based technique was studied by Galias [START_REF] Galias | Quadrature Chaos Shift Keying: Theory and Performance Analysis[END_REF]. Each channel is coded by two quadrature chaotic sequences obtained from a Hilbert filter at the output of the chaotic generator.

Then, the signals are filtered through a raised cosine Nyquist filter. After filtering operation, filtered signals are multiplied around two quadrature carriers characterized by the F c frequency.

The figure 1 represents the general diagram of the chaotic transmitter. If the chaotic coded sequence is known by the receiver, this base band signal is then filtered and de-spread by correlation operation in order to rebuild the original I and Q sequences. Finally, a level detection followed with a re-arrangement of the I and Q data entries makes it possible to recover the initial sequence b(t). However, due to the sensitivity of chaotic systems to initials conditions, permanent synchronicity between receiver and transmitter is necessary or else the trajectories of the two generators will be quickly uncorrelated. Moreover, the receiver must be able to synchronize itself on the associated generator and that in spite of the noise in the channel. A robust method to re-build the chaotics sequences is the crucial point.

Several receiver structures for chaotic spreaded system is based on a DCSK non-coherent method [START_REF] Kolumban | Differential Chaos Shift Keying: A robust coding for chaos communication[END_REF]. In this case, every transmitted symbol is represented by a pair of chaotic signal samples sent in two equal time slots:

the chaotic code sent in the first half of the symbol period serves as the reference (reference sample). -the reference code or its inverted version sent in the second half of the symbol period carries the information and represents the data sample.

Such structure can be easily implemented for QAM chaotic systems. Since the reference code is sent with the modulated data, permanent synchronization of the receiver is assured. It's the main advantage because two chaotics systems diverge quickly even if initials conditions are very close : As coded states are transmitted, the receiver does not try to synchronize its chaotic sequence from the corrupted sequence but only uses a delay to correlate modulated code with the reference one. Nevertheless since no modulated chaotic sequence is clearly sent, we can not prevent intruder to recover transmitted signal unless this one is sent in different carriers : coherent method present a better secure way.

Coder Demodulation Technique for a QAM Chaotic System

In this section we present a method to build a chaotic generator. The generator is based on Frey model [START_REF] Frey | Chaotic Digital Encoding: An Approch to Secure Communication[END_REF] but in our model the coefficients are based on maximal Hamming distance between the transmitted code and the estimated one when no more one error occured in each previous codes. Nevertheless optimal generator in the sense of Hamming may yield to a poor cyclic sequence. As a result, one may prefer degrading the Hamming distance to improve the chaotic trajectory of the generator. With Hamming distance, the demodulation technique proposed is based on statistical properties of chaotic sequence in a bounded noise. This method consists in estimating the next chaotic code according to the previous state in presence of limited noise.

The chaotic sequence c k is defined by the previous code (at least c k-1 and c k-2 ) and by a non linear function. By knowing the coder function, and thanks to the first chaotic codes, the chaotic sequence ĉk can be rebuild. Nevertheless, each code of the received chaotic sequence is corrupted by an additive Gaussian white noise. As a result, estimation of next code lead to several set of possible state named Ŝ and can quickly diverge to all possible state.

Once Hamming distance is optimal (by choosing the rigth characteristics of chaotic generator), we propose an algorithm based on correlation between estimated state from interval of error of previous state and noisy last received states: With c k-1 and c k-2 and bounded relative error, we can remove some state of Ŝ in order to reduce the set of possible states.

Definition of Chaotic Generator Characteristics

First to improve the receiver performance, we aim to design a chaotic modulator according to some assumptions.

Each code is defined over 8 bits. Let's assume that no more one error is occured by code. As a result, we aim to find the best coder function to detect and correct the wrong bit.

To improve detection of error and assumin that no more one error can occur in each code, we propose a coder function which Hamming distance is maximal between each error code. As one error can occur in the last code and in the previous one, the altered estimation code in the receiver from the last two codes should to be the more distant compared with the received code.

As a result, the coder function is defined by the following chaotic function :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x[k] = modulo (G 1 .x[k -1] + G 2 .x[k -2] + s k-1 , 255) G 1 = 9 G 2 = 6 s k-1 = 3 if x k-1 > 127 (3)
When two codes are received, an algorithm assessed the third code. This estimated code, named ĉk is compared to the transmitted one.

For example, if the first code is 4 and second code is 1 then the third code, according to (eq 7) is 33. The binary transmitted sequence is (4) -00000100, (1) -00000001, (33) -00100001. If first code received is (36) -00100100, the second one is (3) 00000011, then the estimated code is ĉk = 11110011 . Thus, compared to (33) -00100001, the hamming distance is equal to 4 as we can find in table at line 7 (bit 6 of first code is false), and at column 3 (bit 2 of last code is false)

In a general case, let's x[k-2] and x[k-1] the transmitted code. We suppose that an error occurred on x[k-1] and x[k-2]. To simplify, suppose that an error transforms a '0' to '1'. The received codes are

x r [k -2] = x[k -2] + 2 i and x r [k - 1] = x[k -1] + 2 j
where i and j represent the position of the error. The estimated code is :

x[k] = modulo G 1 .(x[k -1] + 2 j ) + G 2 .(x[k -2] + 2 i ) + ŝk-1 , 255
We estimate the error between x[k] and x[k] thus

E[k] = x [k] -x[k] E[k] = modulo G 1 .(x[k -1] + 2 j ) + G 2 .(x[k -2] + 2 i ) + ŝk-1 , 255 -modulo (G 1 .x[k -1] + G 2 .x[k -2] + s k-1 , 255) (4)
with x[k] and x[k] comprised between (0,254).

As a result, 4 can be written as

E[k] = (G 1 .(x[k -1] + 2 j ) + G 2 .(x[k -2] + 2 i ) + ŝk-1 -255.α) -(G 1 .x[k -1] + G 2 .x[k -2] + s k-1 -255.β) (5)
Finally, with modulo 255, we can write :

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ E[k] = modulo G 1 .2 j + G 2 .2 i + Δs k-1 , 255 if x[k] > x[k] E[k] = 255 -modulo G 1 .2 j + G 2 .2 i + Δs k-1 , 255 if x[k] < x[k] ( 6
)
The following table represents the Hamming distance of transmitted and estimated codes when an error occured both in the first code and in the second one and when x[k] > x [k]. The lines represent the error position of the first code and the columns represent the error position of the second code (from the least significant bit -1-to the most significant bit -8-). Of course, when no error occured (first line, first column), Hamming distance is null.

(a) when '0' become '1' of both code, (b) when '1' of x[k-1] become '0' and '0' of x[k-2] become '1' (c)when '0' become '1' of x[k-1] and '1' become '0' of x[k-2] and (d) when '1' become '0' of x[k-1] and '1' become '0' of x[k-2]

According to such a function, the Hamming error is maximal between the received code and the estimated code when one error occured in the first and/or in the second code. As a result assuming one error in the first and/or in the second code, the Hamming distance between the transmitted and estimated sequence is at least equal to 2. Thus, even if an error occurred in the third code, the Hamming distance will be greater than 1.

Modulo 255 is preferred to 256, since with modulo 256 no error of the estimated code compared with the received code is generated when the 7th bit of first and the 6th bit of the second code are '1' instead of '0' (and according with the generator characteristics : modulo( 2

7 * G 1 , 2 6 * G 2 , 256) = 0). More generally, modulo(2 7 * G 1 , 2 7 * G 1 , 256) is null if G 1 and G 2 are both odd.
Finally, all combinaisons of G 1 and G 2 with one error of the two codes can lead to an estimated code which is equal to the received code with modulo 256.

Moreover, the Hamming distance on the previous tables is estimated when [k] the Hamming distance are also comprised between 2 and 6. It's an important result since if no more one error occurred in the first code and in the second code, the Hamming distance between the estimated code and the third received code (with no more one error) will never be null.

x[k] > x[k]. Thanks to modulo 255, if x[k] < x[k],
[k] > x[k], thus if x[k] < x

Error Correction to Synchronize the Receiver

Let's c k the integer chaotic code defined between [0,254], c t k the binary transmitted code and c r k the binary received code. If information is '0' the transmitted code is the complement of the binary chaotic code. As a result, the integer chaotic transmitted code

c t k =c k if '1' is sent or c t k =modulo -c k if '0' is sent.
In the first step, we assume a transmission without error.

According [START_REF] Moshavi | Bellcore: Multi-user detection for DS-CDMA communications[END_REF], the estimated code ĉk is

G 2 .c r k-2 + G 1 .c r k-1 + s k-1 if (1,1) is transmitted, G 2 .c r k-2 + G 1 .(modulo -c r k-1 ) + s k-1 if (1,0) is transmitted, ... First, note that ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ G 2 .(modulo -c r k-2 ) + G 1 .(modulo -c r k-1 )=modulo -(G 2 .c r k-2 + G 1 .c r k-1 ) G 2 .(modulo -c r k-2 ) + G 1 .c r k-1 )=modulo -(G 2 .c r k-2 + G 1 .(modulo -c r k-1 )) if c k < 128 then modulo -c k is greater than 128. (7)
With this last condition, there exists one solution wich is equal to c k or (modulo-c k ) and as a result an unique 3-plet information bits

M k-2 , M k-1 , M k .
To determine if an error occurred, we have to compare c k with ĉk using the xor function. If there are no difference, the information bit M k is '1', if all bits are false then there are no error and the information bit M k is '0'.

When no more than one error occured in the first and second code, and according to the 4 possible states of information (00, 01, 10, 11) we extract from the table 1 the possible combinaisons of code c r k-2 and c r k-1 which can lead to this difference.

As a result, we define the following algorithm to correct the bit error.

Algorithm

First Step : First codes estimation 1. We assess ĉk with c r k-2 and c r k-1 . 2. We compare ĉk with c r k (XOR function). The number of errors E k gives us information about where the error may have occurred.

Comparison with table 1 if no error occur in c r

k 3-1) Assuming that M k-2 = 1 and M k-1 = 1, and with Hamming distance E k or 255 -E k according to the comparison of ĉk and c r k , we reduce the possible states ĉk-2 and ĉk-1 from the table 1 to Ŝ1,1 k-2 . According to the Table 1a, we only save the states which lead to a number of errors E k . The table 1a represents the position of the error which transforms a '0' to '1', the line is the first code and the column the second code. Let's P1(k1,k2) the position of the table 1a where Hamming distance is E k . P1(k1,k2) represents the position of a possible error of c r k-2 and c r k-1 where a '0' was transformed to a '1'. Nevertheless, suppose that at this bit position, the received bit (either on c r k-2 or on c r k-1 ) is egual to '0', thus this case is not a feasible error state. As a result this second test reduces all the possible cases : if the k1 bit of c r k-2 is '1' AND the k2 bit of c r k-2 is '1', then we extract and save a possible right code.

Same procedure with table 1b, 1c and 1d 3-2) Assuming that M k-2 = 1 and M k-1 = -1, and E k is the Hamming distance between c k and ĉk . The possible state of error from ĉk-2 and moduloĉk-1 are reduced from the table 1 to Ŝ( k -2) 1,0 (taking into account all possible errors ie, '0' become '1' and '1' became '0' and noting that if an error transform a bit '0' of moduloĉk-1 to '1' then the table of reference correspond to a '1' that becomes '0'). 4. Coming back from point 2 to point 4 assuming an error on c r k In this state, we are compelled to take into account the error on c r k to estimate c c k , the estimated corrected code. Then, according to ĉk is greater than c c k or not, the Hamming distance is E k or 8 -E k . All possible state of M k-2 = 1 and M k-1 = -1 have to be tested.

As a result, we find some combinaison of u-plet ĉk-2 , ĉk-1 , ĉk with associated bit information M k-2 , M k-1 , M k . We save these u-plet in a table T k .

Second Step : Convergence

Come back from Step 1, with c r k-1 instead of c r k-2 and c r k instead of c r k-1 . We save the u-plet in a Table T k+1 . The convergence is given by comparing the possible code ĉk-1 , ĉk with information M k-1 , M k of T k+1 and the same possible code saved at T k .

According to each state Ŝk and ĉk , we estimate ĉk+1 . Nevertheless, ĉk+1 is generated from the second bit of Ŝk and only the 8 possible states of ĉk+1 . As a result, among Ŝk couples of code,each code of the second estimated code ĉk is tested with c k+1 .

Numerical Results

Simulations have been made when no information was sent (M k ). To prove the effectiveness of our algorithm, we generated three positive consecutive errors ('0' became '1'). As a result, the number of possible case of errors is given by the number of Table 1a equal to Hamming distance, Hamming distance -1 and Hamming distance +1.

Indeed, let's the following chaotic sequence 4,1,33, 48. An error is added to the first code and to the second code at the 4th bit and in the third code to the 6th bit. The first three received code are 12, 9 and 97. According to the two first received code (12,9), the estimated third code is 153 ( 1 0 0 1 1 0 0 1). Compared to 97 (01100001), there are 5 errors. Nevertheless, as an error may have occured in the third code, we are compelled to test three Hamming distance : 4, 5 and 6.

The total number of possible error is 34. Moreover, considering that probability of Hamming distance equal to 4 is greater than Hamming distance is equal to 5, then we can conclude that there are a greater probability that an error occured in the third received code.

Nevertheless, thanks to the third point of the algorithm (First Step) we can drastically reduce the number of combinaisons. Indeed, according to table 1a, a distance of Hamming equal to 3 can be generated by an error ('0' become '1' in the third bit of the first code and in the second bit of the second code. In our exemple, the binary value of 12 is 00001100 and the binary value of 9 is 00001001. The third bit of the first code is 1 but the second bit of the second code is 0. So, since the second bit is not equal to 1 (condition of Hamming error of Three) this couple of point can not be a possibility.

As a result, we only find 5 possible errors at step 1. Let's T1 the table where the 3-plet of 5 estimated chaotic code after correction are saved ĉ1 , ĉ2 , ĉ3 .

At step 2, the chaotic sequence is shifted of one bit and launch the first step of the algorithm. We only find 6 possibles chaotic sequence after correction. Let's T2 the table of possible corrected codes ĉ2 , ĉ3 , ĉ4 .

By comparison of ĉ2 , ĉ3 in T1 and T2, we can estimate/correct and recover the chaotic sequence of the transmitter.

As a result, with no more that one error per byte and no modulated data, convergence is achieved with a few steps. nevertheless, with modulated data, more memory is necessary to test and save several ways.

Conclusion

This paper deals with a new demodulation technique based on QPSK. This schema is interesting for two reasons: on one hand the spread-spectrum characteristics of the signals authorize the usage in a multi-user system such as CDMA ( [START_REF] Yang | Chaotic Digital Code-Division Multiple acces (CDMA) Communication Systems[END_REF], [START_REF] Moshavi | Bellcore: Multi-user detection for DS-CDMA communications[END_REF]), on the other hand it makes it possible to decrease the error rate. In addition, the proposed schema does not require synchronization between the receiver and the transmitter, which constitutes a great advantage compared to the coherent detectors, especially knowing that the synchronization becomes more difficult in noise presence. Since the system is implemented in a full digital way, it makes it possible to control efficiently control all of the coder parameters. A digital signal has exact characteristics and it is possible to ensure a better interfacing with a computer data. Moreover synchronization is achieved by correcting the potential error. To prevent burst of errors we also can interleave the chaotic bits.

Nevertheless, this approach is still limited in a low noise environment, since no more than one error is accepted per code.

Fig. 1 .

 1 Fig. 1. QAM chaotic transmitter

  the Hamming distance is the complementary. As an example if Hamming distance is 3 when x[k] > x[k] (three bits over 8 are different) so, the Hamming distance becomes 5 if x[k] < x[k]. As Hamming distance in tables 1-a to 1-d are comprised between 2 and 6 when x

Table 1 .

 1 Table of Hamming distance when x[k] > x[k]

	0 1 2 3 4 5 6 7 8	0 1 2 3 4 5 6 7 8	0 1 2 3 4 5 6 7 8	0 1 2 3 4 5 6 7 8
	0 0 2 2 2 2 2 2 2 4	0 0 6 6 6 6 6 6 2 6	0 0 2 2 2 2 2 2 2 4	0 6 6 6 6 6 6 6 6 4
	1 2 4 2 3 4 4 4 2 4	1 2 6 6 4 6 5 4 4 6	1 6 2 2 4 2 3 4 4 2	1 6 4 6 5 4 4 4 6 4
	2 2 3 4 2 3 4 4 4 4	2 2 2 6 6 4 6 5 4 2	2 6 6 2 2 4 2 3 4 6	2 6 5 4 6 5 4 4 4 4
	3 2 2 3 4 2 3 4 4 6	3 2 4 2 6 6 4 6 5 2	3 6 4 6 2 2 4 2 3 6	3 6 6 5 4 6 5 4 4 2
	4 2 4 2 3 4 2 3 4 6	4 2 4 4 2 6 6 4 6 3	4 6 4 4 6 2 2 4 2 5	4 6 4 6 5 4 6 5 4 2
	5 2 4 4 2 3 4 2 3 6	5 2 5 4 4 2 6 6 4 4	5 6 3 4 6 2 2 2 4 4	5 6 4 4 6 5 4 6 5 2
	6 2 4 4 4 2 3 4 2 2	6 2 6 5 4 4 2 6 6 4	6 6 2 3 4 4 6 2 2 4	6 6 4 4 4 6 5 4 6 6
	7 2 3 4 4 4 2 3 4 2	7 2 4 6 5 4 4 2 6 6	7 6 4 2 3 4 4 6 2 2	7 6 5 4 4 4 6 5 4 6
	8 2 2 3 4 4 4 2 3 3	8 2 6 4 6 5 4 4 2 6	8 6 2 4 2 3 4 4 6 2	8 6 5 4 4 4 4 6 5 5
	(a)	(b)	(c)	(d)


Alexis Front

Djahid Oucheriah

Christine Mottet

Hakim Amara

Melting properties of Ag x Pt 1-x nanoparticles

At the nanoscale, materials exhibit unique properties that differ greatly from those of the bulk state. In the case of Ag x Pt 1-x nanoalloys, we aimed to study the solid-liquid transition of nanoparticles of different sizes and compositions. This system is particularly interesting since Pt has a high melting point (2041 K compare to 1035 K for Ag) which could keep the nanoparticle solid during different catalytic reactions at relatively high temperatures, such as we need in the growth of nanotubes. We performed atomic scale simulations using semi-empirical potential implemented in a Monte Carlo code at constant temperature and chemical composition in canonical ensemble. We observed that the melting temperature decreases with the size (pure systems and alloys) and the composition. We show that the melting systematically passes through an intermediate stage with a crystalline core (pure platinum or mixed PtAg depending on the composition) and a pure silver liquid skin, which strongly questions the idea of having a faceted solid particle in catalytic reactions for carbon nanotubes synthesis.

I. INTRODUCTION

The reduction of materials to the nanoscale has a profound impact on their structure and characteristics. In this context, metallic nanoparticles (NPs) are widely studied for their particular reactivity and/or rather unique physical properties in many scientific fields. Indeed, their nanometric dimension confers them different specificities compared to their bulk counterparts [START_REF] Guo | Mechanical properties of nanoparticles: basics and applications[END_REF][START_REF] He | Novel behaviors/properties of nanometals induced by surface effects[END_REF] . These specificities are enhanced in bimetallic alloys (or nano-alloys) where the association of two metals within a NP can exalt or combine certain features making their scope of application even broader [START_REF] Ferrando | Nanoalloys: From theory to applications of alloy clusters and nanoparticles[END_REF][START_REF] Alloyeau | Nanoalloys: Synthesis, Structure and Properties[END_REF][START_REF] Alloyeau | Size and shape effects on the order-disorder phase transition in CoPt nanoparticles[END_REF][START_REF] Amara | Effect of size on the surface energy of noble metal nanoparticles from analytical and numerical approaches[END_REF] .

A typical example is the use of bimetallic NPs as a catalyst for the synthesis of carbon nanotubes (CNTs) with a defined structure. Over the last few decades, CNTs have been the subject of intense research efforts, highlighting their exceptional mechanical, electronic, optical and thermal properties [START_REF] Yang | Carbon nanotube-and graphene-based nanomaterials and applications in high-voltage supercapacitor: A review[END_REF][START_REF] Wu | Application-driven carbon nan-otube functional materials[END_REF] . The main bottleneck in the development of a CNTbased technology is the difficulty of obtaining large quantities and high purity of a single CNT structure. Indeed, depending on its structure, a nanotube can be metallic or semiconducting. In terms of synthesis, they are commonly produced by catalytic Chemical Vapor Deposition (CVD) at medium temperature (800 to 1500 K) using metal NPs that catalyze the decomposition of the carbon-bearing gas precursors and make the growth of nanotubes possible [START_REF] Jourdain | Current understanding of the growth of carbon nanotubes in catalytic chemical vapour deposition[END_REF][START_REF] Amara | Modeling the Growth of Single-Wall Carbon Nanotubes[END_REF][START_REF] Rao | Carbon nanotubes and related nanomaterials: Critical advances and challenges for synthesis toward mainstream commercial applications[END_REF] . However, the complex interaction of catalyst NPs exposed to reactive carbon makes it a challenge to achieve tube growth with a specific structure [START_REF] Magnin | Entropy-driven stability of chiral singlewalled carbon nanotubes[END_REF][START_REF] He | Growth kinetics of single-walled carbon nanotubes with a (2¡i¿n¡/i¿, ¡i¿n¡/i¿) chirality selection[END_REF][START_REF] Yang | Chirality pure carbon nanotubes: Growth, sorting, and characterization[END_REF] . For the past few years, use of bimetallic catalysts is presented by different groups as a key to achieve a structural or property control [START_REF] Forel | Tuning bimetallic catalysts for a selective growth[END_REF] . Typically, true metallic W-Co alloy NPs, seem to favour the formation of semiconducting tubes with defined chiralities [START_REF] Yang | Chirality-specific growth of single-walled carbon nanotubes on solid alloy catalysts[END_REF][START_REF] Yang | Water-assisted preparation of high-purity semiconducting (14,4) carbon nanotubes[END_REF] . To explain these striking results, it has been proposed that the presence of an alloying element within the catalyst with a high melting point tends to keep the particle solid during synthesis [START_REF] Yang | Chirality-specific growth of single-walled carbon nanotubes on solid alloy catalysts[END_REF][START_REF] Zhang | Arrays of horizontal carbon nanotubes of controlled chirality grown using designed catalysts[END_REF][START_REF] Li | Metallic catalysts for structure-controlled growth of single-walled carbon nanotubes[END_REF] . Among proposed growth mechanisms, it has been argued that the structure of the solid catalyst, such as a facets and steps, can affect the produced tubes [START_REF] Zhang | Arrays of horizontal carbon nanotubes of controlled chirality grown using designed catalysts[END_REF][START_REF] Zhu | Atomic-resolution imaging of the nucleation points of single-walled carbon nanotubes[END_REF] . In this approach, catalyst NPs serve as a template that could control the structure of grown carbon layers via planar or perpendicular interactions between carbon sp 2 networks and the catalyst facet, specifically relying on the epitaxial relationship. Although elegant, this explanation is open to debate as there are only few experimental studies to determine the state of the catalyst during synthesis. Experimental investigations at this level are usually done by Transmission Electron Microscopy (TEM) performed post mortem outside the CVD reactor [START_REF] Zhang | Arrays of horizontal carbon nanotubes of controlled chirality grown using designed catalysts[END_REF][START_REF] Hua An | Atomic-scale structural identification and evolution of co-w-c ternary swcnt catalytic nanoparticles: High-resolution stem imaging on sio¡sub¿2¡/sub¿[END_REF][START_REF] Zhang | Solid supported ruthenium catalyst for growing single-walled carbon nanotubes with narrow chirality distribution[END_REF][START_REF] Li | Kinetics-controlled growth of metallic single-wall carbon nanotubes from corex nanoparticles[END_REF] . Besides, in situ TEM reveals more details about the role of the catalysts during the CNT nucleation-growth [START_REF] Hofmann | In situ observations of catalyst dynamics during surface-bound carbon nanotube nucleation[END_REF][START_REF] Picher | Nucleation of graphene and its conversion to single-walled carbon nanotubes[END_REF][START_REF] Zhang | Growth termination and multiple nucleation of single-wall carbon nanotubes evidenced by in situ transmission electron microscopy[END_REF] . However, direct evidence of the state of the NP using this approach is lacking and this even if a perfectly crystalline particle seems to be highlighted very recently during the CNT nucleation and thus appearing to confirm the model proposed so far. Although these techniques are very advanced, the precise state of the surface (solid, liquid or a mixture) is not so easy to identify and are very scarce. This is particularly true in the typical temperature range of CNT synthesis. In order to address this issue, we have chosen to study the Ag-Pt system, which is particularly interesting in the field of catalysis in general [START_REF] Esfandiari | Synthesis, characterization and performance determination of an ag @ pt/c electrocatalyst for the orr in a pem fuel cell[END_REF] . Although this alloy is not used for carbon nanotube growth, it is an ideal model system since it involves two metals with very different melting temperatures, i.e. 1235 K and 2041 K for Ag and Pt, respectively.

The aim of the present work is to investigate at the atomic scale the state of Ag x Pt 1-x nanoparticles (solid, liquid, mixed?) of varying sizes and compositions during the different stages of their melting. For this purpose, we performed atomic scale simulations using a semi-empirical potential implemented in a Monte Carlo code to lead the structure towards its thermodynamic equilibrium [START_REF] Frenkel | Understanding Molecular Simulations[END_REF] and to analyse in details the solid-liquid transitions. Obviously, there are many works in the literature seeking to characterize the state of the particle during the solid-liquid transition for pure metals [START_REF] Delogu | Structural and energetic properties of unsupported cu nanoparticles from room temperature to the melting point: Molecular dynamics simulations[END_REF][START_REF] Omid | Melting enthalpy and entropy of freestanding metallic nanoparticles based on cohesive energy and average coordination number[END_REF][START_REF] Fu | Size-and shape-dependent melting enthalpy and entropy of nanoparticles[END_REF][START_REF] Samantaray | Molecular dynamics simulation study on thermodynamical properties of cu and ni nanocluster[END_REF] or alloys [START_REF] Kim | The solidto-liquid transition region of an ag-pd bimetallic nanocluster[END_REF][START_REF] Omar | Structural and thermal properties of elementary and binary tetrahedral semiconductor nanoparticles[END_REF] . Indeed, Buffat and Borel revealed in the 1970s a significant decrease in the melting temperature of very small gold NPs, explaining that the surface-to-volume ratio was at the origin of this phenomenon [START_REF] Ph | Size effect on the melting temperature of gold particles[END_REF] . Regarding the case of bimetallic NPs, the study of systems with two differ-ent melting points is badly lacking, not allowing to show if the crystalline structure of the nanoparticle is preserved at relatively high temperature by the combination of these two metals within the NP. Moreover, the analysis of the surface during the solid-liquid transition has never been addressed. Based on our atomic scale study, we confirmed that the melting temperature decreases with the size of the NPs (pure systems and alloys). However, our detailed analysis shows that the melting systematically passes through an intermediate stage with a crystalline core (pure Pt or AgPt depending on the composition) and a layer of liquid Ag, which strongly questions the idea of having a faceted solid particle in catalytic reactions at high temperature.

II. METHODOLOGY A. Ag-Pt system

The bulk Ag-Pt system presents an atypical phase diagram in the Ag-based family, with a large miscibility gap between Ag 2 Pt 98 and Ag 88 Pt 12 at low temperature [START_REF] Okamoto | Ag-pt (silver-platinum)[END_REF][START_REF] Ph | A revision of the binary system ag-pt[END_REF] and a L1 1 phase for an extremely narrow composition range around Ag 50 Pt 50 [START_REF] Gus | Revisiting the revised ag-pt phase diagram[END_REF] . This phase is quite scarce in the face-centered cubic alloys with an alternation of pure planes in the (111) direction instead of the more usual L1 0 phase. FIG. 1. Core@shell L1 1 /Ag/Pt@Ag structure of a Truncated Octahedron (TOh) nanoalloy of 1289 atoms. The Ag-shell (left) is composed of vertex, edge, (100) and (111) facets. The core (right) is characterized by a L1 1 phase surrounded by a subsubsurface enriched in Ag and a subsurface enriched in Pt, named L1 1 /Ag/Pt.

At the nanoscale, Ag-Pt system has been recently investigated experimentally by High-Resolution TEM. In case of Ag 3 Pt, Truncated Octahedron (TOh) NPs up to 3 nm exhibit a L1 1 ordered phase in the core and a silver surface shell whereas for larger ones the ordered phase breaks in multiple domains or is interrupted by faults [START_REF] Pirart | Reversed size-dependent stabilization of ordered nanophases[END_REF] . With increasing particle size, the Ag-shell causes internal stress in the L1 1 domains leading to subsurface enrichment in Pt coupled to a "subsubsurface" enrichment in Ag, that allows to release the inhomogeneous stress [START_REF] Front | Stress effect on segregation and ordering in pt-ag nanoalloys[END_REF] . As a result, the core@shell L1 1 /Ag/Pt@Ag structure (Fig. 1) is therefore stabilized for size bigger than 3 nm or 807 atoms. In the present work, pure Ag and Pt NPs as well as Ag 3 Pt, AgPt and AgPt 3 NPs will be considered to characterize the effect of Pt concentration on the melting temperature of NPs of different sizes (from 201 atoms to 2951 atoms).

B. Tight-Binding potential

The energetic model is based on a semi-empirical manybody potential derived from the density of states of the d-band metals approximated to its second moment (width of the dband) in the tight-binding framework [START_REF] Ducastelle | Module élastique des métaux de transition[END_REF][START_REF] Rosato | Thermodynamical and structural properties of f.c.c. transition metals using a simple tight-binding model[END_REF] (TB-SMA). It leads to an attractive term with a square root dependence on the coordination number for the band term at site i:

E band i = - ∑ j,r i j <r cut ab ξ 2 ab e -2q ab r i j r 0 ab -1 (1)
where (a, b) is the nature of the metal atom, r i j is the distance between the atom at site i and one neighbor at site j, r cut ab is the cutoff distance, and r 0 ab is the first-neighbor distance depending on the nature of the atoms. An empirically repulsive term of the Born-Mayer type is added to simulate the ionic and electronic repulsions: (2) Ag-Pt system has the particularity to form an ordered L1 1 phase at the equiconcentration which is stabilized by taking into account the contribution of the second-neighbors [START_REF] Front | Stress effect on segregation and ordering in pt-ag nanoalloys[END_REF] . Consequently, the TB-SMA potential has been adapted by adding an attractive term with a gaussian shape centered on the second-neighbor distance r 2nd

E rep i = ∑ j,r i j <r cut
ab :

E gauss i = -∑ j,r i j <r cut ab a =b G ab e - (r i j -r 2nd ab ) 2 2σ ab 2 (3)
Therefore, the total energy of an atom i is given by:

E tot i = E band i + E gauss i + E rep i (4)
The homo-atomic (a = b) parameters are fitted to the cohesive energies, lattice parameters and elastic constants calculated by density functional theory (DFT). For the heteroatomic interactions (a = b), the parameters p ab and q ab are the average of the pure metal ones. Finally, A ab , ξ ab , G ab and σ ab are fitted to the formation enthalpies of the L1 0 and the L1 1 phases as well as the solution energies of one impurity in the matrix of the other metal. More details on the fitting procedure can be found in Ref. [START_REF] Front | Stress effect on segregation and ordering in pt-ag nanoalloys[END_REF]

C. Monte Carlo simulations

This atomic interaction model is then implemented in a Monte Carlo (MC) code, based on the Metropolis algorithm [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF] using the canonical ensemble. This procedure makes it possible to relax the structure at finite temperature according to a Boltzmann type probability distribution. In the canonical ensemble, standard MC trials correspond to randomly exchange the positions of two atoms with a different nature or randomly choosing an atom and its displacement with an amplitude 0.05 √ T (ξ -0.5). T is the temperature and ξ is a random number between 0 and 1. We performed 10 3 MC macrosteps for equilibration then the average quantities are calculated over 10 3 macrosteps. One macrostep consists in proposing randomly either n dep * N displacements of atom or n ex * N atomic exchanges, N being the total number of atoms of the system. In practice, we chose n dep = 10 and n ex = 1 to ensure a quick convergence of the total energy. This quantity is taken as a sum of local terms, this avoids recalculating the total energy of the whole system at each Monte Carlo step making efficient the implementation of our TB-SMA model. As a result, the total energy is only recalculated at each MC trial for atoms impacted by the displacement or the exchange of an atom i as it is the case here. This approach is then perfectly adapted to deal with large systems. We performed heating (increasing temperature) Monte Carlo simulations which means the simulation at the next temperature starts from the last converged configuration of the previous temperature.

III. MELTING OF PURE NANOPARTICLES

First, the finite size effect of pure Ag and Pt NPs on the melting temperature (T m ) is investigated. In this context, we considered TOh shapes from 201 to 2951 atoms corresponding to magic numbers. The solid-liquid transition is characterized by performing simulated annealing where the typical trend of total energy as a function of temperature (see Fig. 2a). Starting from a crystalline structure, these caloric curves display an initial rapid increase in energy with temperature. At the melting point, a large jump is observed indicating the phase change corresponding to the melting of the NPs. Then, a steady increase in energy with temperature is observed where the NP is liquid. Most of the time, in the past, these curves have been obtained by molecular dynamic simulations [START_REF] Ercolessi | Melting of small gold particles: mechanism and size effects[END_REF][START_REF] Lewis | Melting, frezzing, and coalescence of gold nanoclusters[END_REF][START_REF] Qi | Melting and crystallization in ni nanoclusters: the mesoscale regime[END_REF][START_REF] Delogu | Structural and energetic properties of unsupported cu nanoparticles from room temperature to the melting point: Molecular dynamics simulations[END_REF][START_REF] Mottet | Single impurity effect on the melting of nanoclusters[END_REF] and further analyzed in terms of phase coexistence in finite systems [START_REF] Wales | Coexistence in finite systems[END_REF][START_REF] Hendy | Stability of phase coexistence in atomic clusters[END_REF] to be compared to first-order melting transition in bulk systems. Here the same behavior is obtained by canonical Monte Carlo simulations and clearly indicates a two-state model (solid/liquid) at the phase transition where T m can be obtained.

Using this approach, we determined T m for Ag and Pt clusters with different size (from 201 to 2951 atoms) in the truncated octahedral shape. T m is plotted against 1/R (Fig. 2b) where R is the radius of the nanoparticle defined as R = (N/4) 1 [START_REF] Ferrando | Nanoalloys: From theory to applications of alloy clusters and nanoparticles[END_REF] , with N the number of atoms. The melting temperature exhibits a linear decrease as a function of 1/R, in agreement with the first experimental observation by Buffat and Borel [START_REF] Ph | Size effect on the melting temperature of gold particles[END_REF] of gold nanoparticles in a size range of 2 to 20 nm in diameter. Later on, Frenken and van der Veen 51 observed the melting on a (110) lead surface at lower temperature than the lead melting point, called also "surface premelting", which can be correlated to the liquid phase formed at the surface of nanoparticles. It has been accepted that beyond a certain size, where surface and core can be distinguished, the formation of a liquid surface skin was a precursor effect of the melting of NPs. [START_REF] Ercolessi | Melting of small gold particles: mechanism and size effects[END_REF][START_REF] Qi | Melting and crystallization in ni nanoclusters: the mesoscale regime[END_REF][START_REF] Delogu | Structural and energetic properties of unsupported cu nanoparticles from room temperature to the melting point: Molecular dynamics simulations[END_REF] Therefore the surface/volume ratio variation with cluster size determines the decrease of melting temperature with cluster size.

From the two curves of melting temperature as a function of cluster size in Fig. 2 we can extrapolate the melting temperature of the bulk, leading to 1280 K for Ag and 2170 K for Pt, in excellent agreement with the experimental values, 1235 K and 2041 K respectively. These results show that the interatomic potential fitted on bulk data is perfectly transferable to nanoparticles and thus well adapted to our problem where we seek to characterize the solid-liquid transitions of nano-alloys involving elements with two very different melting points. As seen in Fig. 3, three different regimes are observed. The first zone is characterized by a gradual increase of the total energy with temperature where the NP is in the solid state. From ∼ 900 K, a change of slope is observed until a jump feature of the melting temperature (here around 1500 K). Lastly, the NP is liquid and a regular increase of the total energy is again recorded. To go beyond, local analyses are performed by addressing the concentration and energy of the different sites during the MC simulations. From the site concentrations curves we notice that the initial configuration exhibits a Pt pure core and a mixed Ag-Pt surface configuration with Ag pure edges and vertices, and mixed Pt-Ag (111) and (100) facets with an Ag composition of 0.60 and 0.33 respectively. [START_REF] Front | Amazing chemical ordering predicted in pt-ag nanoalloys[END_REF][START_REF] Front | A theoretical study of pt-based nanoalloys: Co-pt, typical ordering system, and pt-ag, hybrid system between orderig and demixion[END_REF] . It is worth to be noticed that usually the surface segregation (here Ag segregation) is larger for less coordinated sites. The exceptional inversion of hierarchy between the (111) and (100) facets is due to the chemical ordering on the (100) facets which forms atomic rows of alternatively Ag/Pt. Because of the size of the facet, three by three atoms square, and because the edges are populated by Ag atoms, we get two atomic rows in Pt and one in Ag (see Fig. 4a) leading to the composition of 1/3 in Ag. This composition is lower than the one of the (111) facets which are fully disordered. [START_REF] Front | Amazing chemical ordering predicted in pt-ag nanoalloys[END_REF][START_REF] Front | A theoretical study of pt-based nanoalloys: Co-pt, typical ordering system, and pt-ag, hybrid system between orderig and demixion[END_REF] In the solid regime, the concentrations remain constant and decrease up to the nominal concentration (25% of Ag) starting with the vertices, the edges and the facets. However the two types of facet do not evolve in the same way. Whereas the (111) facet concentration decreases regularly, after the vertices and edges, the (100) facet concentration, first increases to reach the same value as the other ones, and then decreases with the other ones in order to get an homogeneous Ag concentration on the whole surface around 1300 K. Also at that temperature the core begins to incorporate Ag atoms and at the end of the melting process, in the liquid phase, the core concentration is just slightly lower that the surface site one.

The local energies (still in Fig. 3) follows globally the concentration behaviors. The lowest energies around -5.50 eV/at. correspond to the core Pt atoms which have 12 neighbors. This is very near the total energy per atom of a face-centered cubic bulk Pt in our TB-SMA model (-5.53 eV/at. [START_REF] Front | Stress effect on segregation and ordering in pt-ag nanoalloys[END_REF] ). The second population corresponds to the surface atoms with higher energy. It is noteworthy that the site energy hierarchy according to the coordination number (Z i ) is partially respected because of chemical composition. In a pure system, the larger Z i is, the higher the cohesion energy leading to the following hierarchy: e (111) < e (100) < e edge < e vertex . As already mentioned, because of the surface chemical ordering on the (100) facet with alternative Ag/Pt rows, the (100) facets are 2/3-rich in Pt, which makes it more stable by the presence of a high concentration of Pt whose cohesion energy is lower leading to a lower energy for the (100) facets as compared to the (111). When the temperature increases, going through the premelted regime, we can see the destabilization of the most sub-coordinated atoms leaving their equilibrium positions. Indeed, the site energies of the edges and vertices decrease strongly contrary to the more coordinated sites as core and facet ones which are less affected. In the liquid regime, vertices, edges, and facets no longer exist leading to an energy distribution relatively homogeneous and an average Ag concentration equal to the nominal one. Consequently, this local analysis shows us that the intermediate zone between 900 K to 1600 K corresponds to a destabilization of the surface which thus becomes less crystalline rather quickly before the complete melting of the NP.

The analysis of the density profiles along the radius of the NP displayed in Fig. 4 confirms in a more quantitative way such observation and helps to move a step further. At low temperature, the solid NP is characterized by an alternance of peaks with a full Pt core and above 14 Å a mixed Ag-Pt surface. This chemical arrangement is clearly illustrated in Fig. 4 where snapshots of the TOh 1289 AgPt 3 NPs and its slice view are presented. At 1000 K, the chemical configuration (Pt core and Ag-Pt surface) is kept with slightly wider peaks due to the higher temperature. More precisely, the peaks are broadening near the surface, whereas they remains narrow in the core. Notably, the Ag peak at the surface is quite large which is the signature of a less pronounced crystallinity as seen on the slice view where the surface is strongly distorted. When increasing T , no more sharp peaks (Ag and also Pt) are present mainly between 14 Å and 19 Å corresponding to a melted surface. At higher temperature (T = 1550 K), the melting process diffuses within the NP where only the center of the NPs (to 8 Å) is still solid whereas the remaining NP is completely liquid. Finally, a complete melting of the NP is achieved at T > T m where no peak is observed anymore but Ag surface segregation remains as in the solid state.

Thanks to our very detailed analysis, we have shown that the solid-liquid transition goes through a premelted phase of the surface which propagates along the NP. In the present case, the complete melting of the nanoparticle is estimated at a temperature of 1600 K. Interestingly, at much lower temperature (here around 1300 K), the surface is completely distorted so that no facets are present anymore. Note that similar conclusions were obtained whatever the size of the NP.

B. TOh 1289 Ag 3 Pt

We extend the previous analysis to the Ag 3 Pt NPs where all results for TOh 1289 are represented in Fig. 5 and Fig. 6. The initial configuration at 100 K is a core@shell L1 1 /Ag/Pt@Ag as described in section II A and illustrated in Fig. 6-a. When increasing the temperature, we again found the three regions discussed before, namely the solid, premelted and liquid ones. Compared to the AgPt 3 stoichiometry, the premelted threshold is lowered: ∼700 K vs ∼900 K. This is mainly due to the absence of Pt atoms on the surface with higher melting point. Consequently the melting temperature is also strongly impacted with a decrease of about 300 K. Regarding the local analysis, the hierarchy of the site is well respected in that case since the shell is fully composed of Ag atoms. However, during the simulation annealing, in the premelted zone, (100) facets are first impacted with a slight Pt enrichment (see Fig. 5). Then the low-coordinated atoms (vertices and edges) and the (111) facets leave their equilibrium positions strongly modifying the structure of the surface. As found in case of AgPt 3 NP, we get a completely liquid state above T m where the notion of site is no longer relevant with an average Ag concentration equal to the nominal one, i.e. of about 0.75. To go further, the atomic density profiles of Ag and Pt along the radius of the NP at different temperatures are presented in Fig. 6. The initial configuration reveals quite sharp peaks as expected from a crystalline structure. More precisely, the presence of a pure Ag surface is perfectly clear with alternating Ag and Pt peaks typical of the core@shell L1 1 /Ag/Pt@Ag structure detailed earlier. At 1000 K, the surface is no longer Ag pure with a slight Pt enrichment on the (100) facets seen in Fig. 5. The slice views highlight a disordered core, keeping a subsurface enriched in Pt and a subsubsurface enriched in Ag. When increasing the temperature (T = 1200 K), atomic density profiles display broad peaks in the surface area (between 16 Å and 19 Å). This particular state corresponds to a surface shell containing (111) facets partially melted coupled to other sites fully melted. Meanwhile, the premelted surface propagates within the NP through the (100) facets leading to disordered layers beneath. At 1250 K, the core of the NP is fully cristalline whereas no more peaks are observed between 11 Å and 20 Å , signature of a melted surface and subsurface. When the solid-liquid transition is achieved, at 1300 K, all peaks have disappeared from the atomic density profiles revealing a homogeneous structure along the particle. More precisely, the liquid state is composed of an Ag enriched surface with Pt atoms underneath. Thus, it seems that the chemical arrangement observed in the liquid state with Ag segregation and Pt subsurface segregation remains from the initial crystalline structure.

As highlighted in case of AgPt 3 NPs, the same melting mechanisms are observed with first of all a premelted surface which propagates within the NP. Here again, the presence of Pt obviously increases the T m of the NP as compared to Ag pure ones. However, the melting of the surface is observed far before the complete melting leading to a facet loss at relatively low temperature (a few hundreds of Kelvin below the melting temperature) and this whatever the size of the NP considered.

C. Generalization to the Ag x Pt 1-x system

In the following, we extend the melting temperature determination to nanoparticles with sizes from 201 atoms to 2951 atoms and also of equiatomic composition AgPt. In this particular case, the TOh structure is characterized by a Ag shell and a Pt-rich core containing (111) Ag-planes [START_REF] Front | Stress effect on segregation and ordering in pt-ag nanoalloys[END_REF] . The calculated T m for different compositions and sizes, including bulk phases, are presented in Fig. 7. First of all, we can note that our TB-SMA model correctly reproduces the melting temperature of the bulk phases which increases with the platinum concentration. This good agreement makes relevant our approach to characterize the T m of NPs at different compositions. At a given concentration, the melting temperature decreases with size as observed in case of pure Ag and Pt. Typically, the melting temperature is almost divided by a factor two from bulk phase to TOh 201 NPs. For the largest sizes (here 2951 atoms), T m is close to that of the bulk whatever the concentration. Moreover, the detailed analyses presented before have clearly shown that the melting starts from the surface and then propagates within the NP. As for pure metals, the surface/volume ratio drives T m which therefore decreases with the size. Meanwhile, the melting temperature clearly increases with Pt concentration at a given size. However, our calculations emphasize that the variation is not strictly linear. Indeed, two zones are perfectly defined, with a frontier at the equiconcentration. On the Ag-rich side, T m shifts significantly towards higher temperatures when the Pt concentration is raised. Indeed, it increases on average by about 30% when moving from pure Ag to Ag 3 Pt NPs. However, increasing the amount of Pt atoms on the Pt-rich side has less impact on the melting temperature. Thus we observe an improvement of less than 10% from AgPt 3 to pure Pt. It is worth noticing that c Ag varies as a function of cluster size in order to fit the concentration of the L1 1 @Ag phase.

Obviously, the presence of Pt in the NP enhances the melting temperature, but the effect is more sensible in the Ag-rich side than in the Pt-rich side, so it is not necessary to consider alloys with a very high Pt concentration to delay the solid-liquid transition.

IV. CONCLUSIONS

Through this study on the solid-liquid transition of Ag x Pt 1-x NPs, several significant insights have been highlighted that may be of interest to the research community dealing with the control of nanotube structure grown from metal particles and in the field of nanoalloys in a more general way. Our atomic scale simulations clearly show that the presence of Pt, which has a high melting point, results in a higher solidliquid transition than in the case of a pure Ag NP. This conclusion is therefore in direct line with the experimental interpretations put forward to justify the use of bimetallic particles with a high melting point element to grow tubes on NPs within the solid state. Nevertheless, our very detailed study has also provided more precise information on the structure of the nanoparticle before its complete melting. Indeed, some complex transitions have been revealed from the surface to the core of the nanoparticle whatever the size and the composition. More precisely, we systematically observed a liquid Ag layer forming at the surface at temperatures well below the complete solid-liquid transition whatever the size and the composition of the NP. In other words, although Pt tends to delay the melting point of the NP, it cannot prevent the presence of a liquid surface where the rest of the particle remains crystalline. In such a core(solid)/shell(liquid) structure, the facets or the steps are no longer present in contradiction with the idea of a possible epitaxial growth of CNTs.

Although Ag x Pt 1-x system was clearly not considered for the growth of carbon nanotubes, our study shows that the assumptions proposed by the community to explain the excellent tube selectivity from bimetallic NPs is not so simple. Obviously, these conclusions are very dependent on the system studied here. Typically, the strong tendency of the element with lower melting point (here Ag) to segregate necessarily influences the mechanisms of melting by the surface. As a result, it is not the purpose of the present work to generalize our results to the different bimetallic alloys used in the CVD growth of carbon nanotubes. Each catalyst being unique, a complete and precise study of its physico-chemical state combining both atomic scale simulation [START_REF] Qiu | Contact-induced phase separation of alloy catalyst to promote carbon nanotube growth[END_REF] and experimental analysis is therefore essential to engineer nanocatalyst with defined properties.

FIG. 2 .

 2 FIG. 2. (a) Caloric curves of pure Ag and Pt NPs. (b) Melting temperature of pure Ag (in grey) and pure Pt (in blue) nanoparticles of TOh morphologies as a function of the inverse radius.

A. TOh 1289 AgPt 3

 3 FIG. 3. Total energy (top), site concentration of Ag (middle) and site energy (bottom) as a function of the temperature of truncated octahedron of 1289 atoms at AgPt 3 composition. The structures corresponding to the temperatures pointed by (a-e) are represented in Fig. 4. Solid, premelted and liquid zones are separated by dashes lines.

FIG. 4 .

 4 FIG. 4. Atomic density of Ag (in grey) and of Pt (in blue) as a function of the distance from the center of TOh 1289 nanoparticles of AgPt 3 composition at 250 K (a), 1000 K (b), 1450 K (c), 1550 K (d) and 1600 K (e) as reported in Fig. 3. The middle column represent shell views, and the right column shows cutting views of the middle column. Ag atoms are in grey and Pt atoms in blue.

FIG. 5 .

 5 FIG. 5. Total energy (top), site concentration of Ag (middle) and site energy (bottom) as a function of the temperature of TOh 1289 at the composition Ag 3 Pt. The structures corresponding to the temperatures pointed by (a-e) are represented in Fig. 6. Solid, premelted and liquid zones are separated by dashes lines.

FIG. 6 .

 6 FIG. 6. Atomic density of Ag (in grey) and of Pt (in blue) as a function of the distance from the center of TOh 1289 nanoparticles of Ag 3 Pt composition at 100 K (a), 1000 K (b), 1200 K (c), 1250 K (d) and 1300 K (e) as reported in Fig. 5. The middle column represent shell views, and the right column shows cutting views of the middle column. Ag atoms are in grey and Pt atoms in blue.

FIG. 7 .

 7 FIG.7. Melting temperature as a function of size and composition of TOh Ag x Pt 1-x NPs and their bulk counterparts. In case of bulk structures, the experimental data[START_REF] Okamoto | Ag-pt (silver-platinum)[END_REF] are given in bracket.

TABLE I .

 I Parameters of the Tight-Binding SMA potential. Ag 11.7240 2.8040 0.0748 1.0064 --Pt-Ag 11.2600 3.0008 0.1456 1.5920 0.0440 0.1393

	a-b	p ab	q ab	A

ab (eV) ξ ab (eV) G ab (eV) σ ab Pt-Pt 10.7960 3.1976 0.1993 2.2318 --Ag-

ACKNOWLEDGEMENTS

The authors thank ANR GiANT (N • ANR-18-CE09-0014-04). The authors acknowledge networking support from the International Research Network "Nanoalloys" of CNRS. This work was granted access to the HPC resources of IDRIS under the allocation 2017-096829 made by GENCI. H.A. thank C. Bichara for fruitful discussions.