Derry Fitzgerald
email: 1derry.fitzgerald@cit.ie

Antoine Liutkus

Zafar Rafii

Bryan Pardo

Laurent Daudet

Harmonic/Percussive Separation Using Kernel Additive Modelling

Keywords: Sound Source Separation, Kernel Additive Modelling, Harmonic/Percussive Separation

Recently, Kernel Additive Modelling was proposed as a new framework for performing sound source separation. Kernel Additive Modelling assumes that a source at some location can be estimated using its values at nearby locations where nearness is defined through a source-specific proximity kernel. Different proximity kernels can be used for different sources, which are then separated using an iterative kernel backfitting algorithm. These kernels can efficiently account for features such as continuity, stability in time or frequency and self-similarity. Here, we show that Kernel Additive Modelling can be used to generalise, extend and improve on a widely-used harmonic/percussive separation algorithm which attempts to separate pitched from percussive instruments.

I Introduction

Harmonic/Percussive (H/P) separation of mixed audio signals deals with attempting to separate pitched (harmonic) instruments from percussion instruments, which can be loosely modelled as different types of broadband noise. H/P separation has numerous applications including remixing and DJing, as well as a preprocessing tool for other tasks including automatic music transcription, chord estimation, and key signature detection. In these applications, the elimination of the percussive sources allows improved estimation of the pitched content, while elimination of the pitched sources allows improved results in applications such as rhythm analysis, beat tracking, and the automatic transcription of drum instruments.

In recent years, different techniques have been proposed for the purpose of H/P separation. Among these, a number of algorithms are based on the intuition that harmonic instruments form stable horizontal ridges across time in spectrograms, while percussive instruments form stable vertical ridges across frequency due to their broadband noise-based nature. This is illustrated in Figure 1, where the harmonics are clearly visible as horizontal lines, while the drums are visible as vertical lines in the spectrogram. Algorithms exploiting this include those based on anisotropic diffusion [START_REF] Tachibana | Comparative Evaluations of Various Harmonic/percussive Sound Separation algorithms based on anisotropic continuity of spectrogram[END_REF], as well as others based on Bayesian models of this assumption [START_REF] Duong | Multichannel Harmonic and Percussive Component Separation by Joint Modeling of Spatial and Spectral Continuity[END_REF]. Of particular interest in our context is the algorithm proposed in [START_REF] Fitzgerald | Harmonic/Percussive Separation using Median Filtering[END_REF], based on median filtering of spectrograms. In the present paper, we extend the algorithm presented in [START_REF] Fitzgerald | Harmonic/Percussive Separation using Median Filtering[END_REF] in order to refine separation through several iterations and to account for multichannel mixtures. The proposed method fits in the recently proposed Kernel Additive Modelling (KAM, [START_REF] Liutkus | Kernel Additive Models for Source Separation[END_REF][START_REF] Liutkus | Kernel Spectrogram Models for Source Separation 4th Joint Workshop on Hands-free Speech Communication and Microphone Arrays[END_REF]) framework for source separation and is consistently shown to improve over the original method [START_REF] Fitzgerald | Harmonic/Percussive Separation using Median Filtering[END_REF]. The paper is structured as follows. In a background section II, we recall the original H/P separation procedure [START_REF] Fitzgerald | Harmonic/Percussive Separation using Median Filtering[END_REF] and briefly present the recently KAM framework. In section III, we detail the proposed harmonic/percussive separation algorithm and finally evaluate it in section IV.

II Background a) H/P Separation with median filtering

The algorithm proposed in [START_REF] Fitzgerald | Harmonic/Percussive Separation using Median Filtering[END_REF] focuses on single channel mixtures. It assumes that vertical lines in a spectrogram correspond to percussion events, while horizontal lines are typically associated with the harmonics of pitched instruments. In this case, peaks due to pitched harmonics can be regarded as outliers on the vertical lines associated with percussion events. Similarly, peaks due to the percussion events can be regarded as outliers on the horizontal lines associated with pitched harmonic instruments. The algorithm goes as follows.

Let boldfaced x x x denote the power spectrogram of a monochannel mixture. It is a N f × N t matrix, where N f and N t respectively stand for the number of frequency bands and the number of frames. We define x x x (f, •) as the f th frequency slice containing the values of the f th frequency bin across time. Similarly, we define x x x (•, t) as the t th time frame. As median filters are good at eliminating outliers, then median filtering each time frame will suppress harmonics in this frame resulting in a percussion enhanced frame s s s P (•, t), while median filtering each frequency slice will suppress percussion events in this slice, yielding a harmonic-enhanced slice s s s H (f, •):

s s s P (•, t) = M{x x x (•, t) , l perc } (1)
s s s H (f, •) = M{x x x (f, •) , l harm } (2)
where M denotes median filtering, and where l perc and l harm are the median filter lengths used to generate the percussion enhanced frames and harmonic enhanced slices respectively. s s s P and s s s H are then used to generate Wiener-filter type masks to apply to the original complex valued spectrograms before inversion to the time domain.

b) Kernel Additive Modelling

Kernel Additive Modelling (KAM) is a recently proposed framework for performing source separation [START_REF] Liutkus | Kernel Additive Models for Source Separation[END_REF][START_REF] Liutkus | Kernel Spectrogram Models for Source Separation 4th Joint Workshop on Hands-free Speech Communication and Microphone Arrays[END_REF]. In contrast to well-established paradigms for separation, such as Non-negative Tensor Factorisations (NTF) [START_REF] Cichocki | Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation[END_REF], which perform a global decomposition based on superposition of fixed patterns or basis functions of the underlying sources, KAM focuses on the underlying regularities of the sources to separate them from mixtures.

In the context of audio source separation, the human ability to discriminate between sources in a mixture has been shown to depend on local features such as repetitivity, common fate and continuity [START_REF] Bregman | Auditory Scene Analysis[END_REF]. These dynamic features can be seen as depending on local regularities concerning the evolution of sources over time, frequency and space, rather than on fixed global patterns or basis functions corresponding to NTF-based approaches.

To model these reqularities within the spectrograms of the sources, KAM uses kernel local parametric models which have their roots in local regression [START_REF] Cleveland | Locally weighted regression: An approach to regression analysis by local fitting[END_REF]. In the audio case, it is assumed that the value s s s j (f, t) of the spectrogram of a source j at a given TF point (f, t) is close to its values as other time-frequency bins given by a sourcespecific proximity kernel I j (f, t) [START_REF] Liutkus | Kernel Spectrogram Models for Source Separation 4th Joint Workshop on Hands-free Speech Communication and Microphone Arrays[END_REF]:

∀ (f , t ) ∈ I j (f, t) , s s s j (f, t) ≈ s s s j (f , t ) , (3)
where I j (f, t) is a set containing the nearest neighbours of (f, t) from the perspective of source j. These kernels can be built using a variety of manners including the use of suitable feature spaces.

Different sources can then be modelled with different proximity kernels I j , and the KAM framework offers a large degree of flexibility in the incorporation of prior knowledge about the local dynamics of the sources to be separated. Separation of additive sources is then achieved through a variant on the backfitting algorithm [START_REF] Hastie | Generalized Additive Models[END_REF]. Many popular audio separation algorithms can be viewed as special cases of the KAM framework including Adress [START_REF] Barry | Sound Source Separation: Azimuth Discrimination and Resynthesis[END_REF], DUET [START_REF] Yilmaz | Blind Separation of Speech Mixtures via Time-Frequency Masking[END_REF], REPET [START_REF] Liutkus | Adaptive filtering for music/voice separation exploiting the repeating musical structure[END_REF] and the harmonic/percussive separation algorithm described in Section a). Furthermore, KAM provides an effective way to generate new source separation algorithms for sources which can be characterised by local features. A more detailed explanation of the framework can be found in [START_REF] Liutkus | Kernel Additive Models for Source Separation[END_REF][START_REF] Liutkus | Kernel Spectrogram Models for Source Separation 4th Joint Workshop on Hands-free Speech Communication and Microphone Arrays[END_REF].

III Model and Method a) Notation and Model

The mixture audio signal x is a set of I time series, with x(n, i) denoting the value of the i th channel of the mixture at sample n. In most popular music I = 2 for stereo signals, or 1 for mono recordings. The mixture is then assumed to be the sum of J sources sj :

x(n, i) = J j=1 sj (n, i) (4)
We then define x and {s j } j=1•••J as the Short-Time Fourier Transforms (STFT) of the mixture and the J sources respectively. These are all tensors of size N f × N t × I. s j (f, t) is then an I × 1 vector containing the values of the STFTs of source s j at time-frequency bin (f, t).

Assuming a Local Gaussian Model [START_REF] Duong | Under-determined reverberant audio source separation using a full-rank spatial covariance model[END_REF] each s j (f, t) are taken as independent vectors, each distributed according to a multivariate centered complex Gaussian distribution:

∀(f, t), s j (f, t) N c (0, s s s j (f, t)R j (f )) (5)
where boldfaced s s s j ≥ 0 is a nonnegative scalar that indicates the energy of source j at time-frequency bin (f, t). It is called the spectrogram of source j in the remaining of this paper 1 . R j is a complex I ×I positive semidefinite matrix, which is called the spatial covariance matrix of source j at frequency band f . It encodes the covariance between the different channels of s j at frequency band f . Since the mixture x(f, t) is the sum of J independent random Gaussian vectors, it is distributed as:

∀(f, t), x(f, t) N c   0, J j=1 s s s j (f, t)R j (f )   (6)
If estimates of s s s j and R j are available (termed ŝ s s j and Rj respectively), then the Minimum Mean-Squared Error estimates ŝj of the STFTs of the sources are obtained from:

ŝj (f, t) = ŝ s s j (f, t) Rj   J j=1 ŝ s s j (f, t) Rj (f )   -1 x (f, t) (7)
The source time-domain signals can then be obtained via inverse STFT.

Prior knowledge about the sources to be separated is then encoded in terms of a proximity kernel I j (f, t) for each source, which indicates which 1 s j (f, t) and x (f, t) are hence complex I × 1 vectors, while boldfaced s s s j (f, t) is a nonnegative scalar. R j (f ) is a complex I × I matrix. All estimates are denoted • Algorithm 1 Kernel backfitting for multichannel audio source separation with locally constant spectrogram models and binary proximity kernels. 1. Input:

• Mixture STFT x (f, t) • Neighbourhoods I j (f, t). • Number N of iterations 2. Initialisation • n ← 1 • ∀j, ŝ s s j (f, t) ← x (f, t) x (f, t) /IJ • R j (f ) ← I × I identity matrix 3.
Compute estimates ŝj of all sources using (7)

4. For each source j:

(a) C j (f, t) ← ŝj (f, t) ŝj (f, t) (b) Rj (f ) ← I Nt t Cj (f,t) tr(Cj(f,t)) (c) z z z j (f, t) ← 1 I tr Rj (f ) -1 C j (f, t) (d) ŝ s s j (f, t) ← median {z z z j (f , t ) | (f , t ) ∈ I j (f, t)} 5.
If n < N then set n ← n + 1 and go to step 3 6. Output: sources spectrograms ŝ s s j and spatial covariance matrices Rj (f ) to use for filtering [START_REF] Bregman | Auditory Scene Analysis[END_REF]. time-frequency points have values close to that of s s s j , as in (3). If we assume that s s s j is not observed directly but only through possibly very noisy estimates z z z j as is the case in practice, then s s s j is estimated as:

ŝ s s j (f, t) = argmin s s sj(f,t) (f ,t )∈Ij (f,t) L j (z z z j (f , t ) | s s s j (f, t)) (8)
where L j (z z z j |u) is the model cost function for source j as defined in [START_REF] Liutkus | Kernel Additive Models for Source Separation[END_REF]. In our context, it is the cost of choosing s s s j (f, t) = u when its noisy observation is z z z j . In this case we choose L j to be the absolute deviation as the observations are likely to be contaminated by outliers during the iterative backfitting process described below, and the absolute deviation is known to yield estimates that are robust to the presence of outliers. We have:

ŝ s s j (f, t) = argmin s s sj (f,t) (f ,t )∈Ij (f,t) |z z z j (f , t ) -s s s j (f, t)| (9)
This cost function is minimised by: ŝ s s j (f, t) = median(z z z j (f, t)|(f , t ) ∈ I j (f, t)) [START_REF] Barry | Sound Source Separation: Azimuth Discrimination and Resynthesis[END_REF] The kernel backfitting algorithm proposed in [START_REF] Liutkus | Kernel Additive Models for Source Separation[END_REF] for estimation of the source spectrograms ŝ s s j proceeds in an iterative manner, with separation and estimation of the parameters performed alternatively. Here, the spectrograms z z z j (f, t) of the current estimates of the source STFTs ŝj are used as noisy observations of the true value with reestimation of ŝ s s j achieved through median filtering. The kernel backfitting procedure is summarised in algorithm 1, where • denotes conjugate transpose and tr(•) denotes the trace of a square matrix. Details on the re-estimation procedure for the spatial covariance matrices can be found in [START_REF] Duong | Under-determined reverberant audio source separation using a full-rank spatial covariance model[END_REF].

b) Harmonic/Percussive Separation using KAM

The harmonic/percussive separation algorithm described in section II can be viewed as an instance of KAM. To describe this algorithm within the KAM framework we define the proximity kernel I j (f, t) for the percussive source as:

I P (f, t) = {(f + p, t) | p = -l, . . . , l} (11)
where 2l + 1 = l perc , as defined in equation ( 1), is the total number of frequency bins in the neighbourhood. We similarly define the proximity kernel for the harmonic source as:

I H (f, t) = {(f, t + p) | p = -k, . . . , k} (12)
where 2k + 1 = l harm , as defined in equation ( 2), is the number of time frames in the neighbourhood. It can clearly be seen that defining the proximity kernels in such a manner results in median filtering being applied to the same sets of time-frequency points as those in equations 1 and 2, and so the algorithm described in section II can be seen as a single iteration of KAM where the spatial covariance matrices Rj (f ) are fixed to the identity matrix which in effect forces each of the mixture channels to be modelled independently. It can be seen that KAM generalises on this algorithm by allowing iterative updating of the sources, and by the inclusion of a spatial model for the sources. It is proposed to investigate if the iterative updating of sources and the inclusion of the spatial model results in improved harmonic/percussive separation.

IV Evaluation

In order to test the effectiveness of the proposed KAM algorithm for H/P separation, a database of 10 test signals was created using excerpts from multitrack recordings where the percussion and all other instruments were available as individual recordings. Percussion-only mixes and Pitched instrument mixes (including vocals) were prepared, and used to create an overall mix for each track. These were then separated using the proposed H/P KAM algorithm. These recordings had a samplerate of 44.1kHz.

To allow direct comparison to the algorithm in [START_REF] Fitzgerald | Harmonic/Percussive Separation using Median Filtering[END_REF] an FFT size of 4096 samples, a hopsize of 1024 samples, l perc = l harm = 17 were used, ensuring that the parameters used were the same as the original algorithm. To investigate the effects of the iterative backfitting procedure, the separation performance was evaluated after each iteration, with the total number of iterations set to 10. This number was used as previous work on KAM indicated that the backfitting procedure requires only a small number of iterations to converge.

Further, to identify how much improvement in performance was due to the backfitting procedure and how much was due to the spatial model, evaluation was also carried out on a version of the algorithm where Rj (f ) is fixed to the identity matrix for all iterations. In this case, the results for iteration 1 are equivalent to those obtained from the algorithm proposed in [START_REF] Fitzgerald | Harmonic/Percussive Separation using Median Filtering[END_REF], and so this represents a baseline from which improvements in performance using KAM can be measured against. Audio examples and MATLAB code for the algorithm can be found at the paper's webpage 2Separation performance was measured using metrics from the PEASS toolkit [START_REF] Emiya | Subjective and objective quality assessment of audio source separation[END_REF]. The metrics considered are the Overall Perceptual Score (OPS), which attempts to provide an overall indication of the separation quality, the Targetrelated Perceptual Score, which attempts to measure how the spatial position of the separated source corresponds to that of the original source, the Interference-related Perceptual Score (IPS), which attempts to measure the amount of interference from other sources present in the separated source, and the Artifact-related Perceptual Score, which measures the presence of artifacts in the separated source. All these metrics have ranges from 0-100 with higher scores being better.

These metrics are plotted in Figures 2345, with red indicating the use of the spatial model and black indicating the spatial model was not used, with Rj (f ) fixed to unity throughout all iterations. Figure 2 shows the average results obtained from all 10 excerpts for the OPS for percussive separation (left) and harmonic separation (right) plotted against iteration number.

It can be seen that initially the use of the iterated approach in KAM results in significant improvements in OPS over the baseline algorithm, which consisted of a single iteration. However, at higher iteration numbers performance falls off. This is particularly evident for the harmonic sources, where 2 iterations gives a large increase in performance, followed by a large drop in performance with subsequent iterations. In contrast, the performance for percussion separation keeps increasing up until iteration 5, before starting to fall off. This suggests that to achieve optimal separation of the percussive and harmonic parts, different numbers of iterations should be used, but that if a balance between separation quality of both sources is desirable, then 2 iterations is optimal. Also of interest is that the spatial model seems to have only minor effects on the OPS, resulting in slightly improved performance for the percussive sources for most iteration numbers, but giving slightly poorer performance for the harmonic sources. It should also be noted that the phenomenon of falling OPS scores at higher number of iterations is not specific to KAM and also occurs with NTF-based methods [START_REF] Fitzgerald | On the use of Masking Filters in Sound Source Separation[END_REF]. This highlights a distinction between numerical convergence and perceptual quality of the separated sources obtained from the algorithms, and is an area that needs further research in sound source separation in general. With respect to the TPS, shown in Figure 3, it can be seen that there is a constant drop with increasing iteration numbers, with the spatial model and non-spatial model results being very similar for the percussive sources. In contrast, the use of the spatial model results in a increase in TPS over the non-spatial model for harmonic sources. The IPS values, shown in Figure 4 show the exact opposite trends to those in Figure 3, with IPS increasing constantly with iteration number. Again, the results for percussive separation are very similar regardless of whether the spatial model is used or not, while for the harmonic separation, the spatial model results in decreased separation performance. This suggests there is a trade-off in the algorithm between accurate spatial estimation of the harmonic source and the amount of interference from the percussive source which remains in the harmonic source. Finally, the APS, shown in Figure 5 again shows a constant drop with iteration number, with similar results achieved regardless of the presence or otherwise of the spatial model. This shows that the spatial model has had little effect in reducing artifacts in the separated signals.

With respect to computation time, the KAM H/P Separation algorithm is still very computationally efficient, if 2 iterations are used, then runtime was approximately a third of the excerpt duration on a modern laptop computer, but was 1.7 times real-time for 10 iterations. It should be noted that this was using an unoptimised version of median filtering, and that increased speed can be achieved by implementing the median filtering in a manner similar to that described in [START_REF] Robertson | Percussive Beat tracking using real-time median filtering[END_REF].

V Conclusions

In this paper we have shown how a newly-proposed framework for source separation can be used to generalise and improve on an existing H/P Separation algorithm. This new framework, termed Kernel Additive Modelling models sources through local regularities in their spectrograms. Individual time-frequency bins in these spectrograms are assumed to be close in value to other bins nearby in the spectrogram, where nearness is defined through a source-specific proximity kernel. Separation is then performed using the kernel backfitting algorithm. The performance of the KAM H/P separator improves on the original medianfiltering based H/P separation algorithm, which can be viewed as a single iteration of the KAM algorithm when the spatial model is omitted. This shows the utility of the KAM framework for the development of sound source separation algorithms.

Fig. 1 :

 1 Fig. 1: Spectrogram of Pitched and Percussive Mixture.

Fig. 2 :

 2 Fig. 2: Overall Perceptual Score for Percussive (left) and Harmonic (right) separations. Red indicates the use of the spatial model, while black indicates the spatial model was not used.

Fig. 3 :

 3 Fig. 3: Target related Perceptual Score for Percussive (left) and Harmonic (right) separations. Red indicates the use of the spatial model, while black indicates the spatial model was not used.

Fig. 4 :

 4 Fig. 4: Interference related Perceptual Score for Percussive (left) and Harmonic (right) separations. Red indicates the use of the spatial model, while black indicates the spatial model was not used.

Fig. 5 :

 5 Fig. 5: Artifact related Perceptual Score for Percussive (left) and Harmonic (right) separations. Red indicates the use of the spatial model, while black indicates the spatial model was not used.

www.loria.fr/ ~aliutkus/kamhp/


Konstantin Usevich

Ivan Markovsky

Adjusted least squares estimator for algebraic hypersurface fitting

Adjusted least squares estimator for algebraic hypersurface fitting konstantin.usevich@gipsa-lab.grenoble-inp.fr,imarkovs@vub.ac.be

We consider the problem of fitting a set of points in Euclidean space by an algebraic hypersurface. We assume that the points on a "true" hypersurface are corrupted by Gaussian noise, and we estimate the coefficients of the "true" polynomial equation. The adjusted least squares estimator accounts for the bias present in the ordinary least squares estimator. The adjusted least squares estimator is based on constructing a quasi-Hankel matrix, which is a bias-corrected matrix of moments. For the case of unknown noise variance, the estimator is defined as a solution of a polynomial eigenvalue problem. In this talk, we present new results on invariance properties of the adjusted least squares estimator and an improved algorithm for computing the estimator for arbitrary set of monomials in the polynomial equation.

K

  . Usevich a and I. Markovsky b (a) GIPSA-Lab, CNRS, UMR5216 BP.46, 38402 St.Martin d'Hères cedex, France. (b) Department ELEC, Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussel, Belgium.

This work was supported by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreements no. 258581 and 320594.


Pierre Letellier

Mireille Turmine
email: mireille.turmine@upmc.fr

Displacement of voltammetric peaks with nanoparticles size: a nonextensive thermodynamic approach

Keywords: nonextensive thermodynamics, size effect, electro-oxidation, nanoparticles a ISE member

We show here that the concepts of nonextensive thermodynamics (NET), described previously (J. Phys. Chem. B, 2004, 108, 18980) and applied to redox behaviour of nanoparticles (J. Phys. Chem. C, 2008, 112, 12116) can be used to express by a power law the variations to the electrochemical kinetics of nanoparticles and in particular to voltammetry. We proposed here a generalization of Plieth's relationship for non-spherical aggregates by assuming that the interface between the particle and its environment is fuzzy. Thus, the relations of non-extensive thermodynamics can quantitatively account for the displacements of electro-oxidation potentials of metal nanoparticles deposited on electrodes, according to their measured size. Our approach also permits to formally justify the stability of the particles may increase as their size decreases, (τ < 0). This is usually found when the aggregates are in close contact with a matrix (in the case, for example, of embedded particles).

Introduction

Several recent experimental and theoretical studies on the electro-oxidation of metal nanoparticles deposited on electrodes show the complexity of the physico-chemical behavior of the material in a divided form. Indeed, a destabilization of the reduced species with a decrease in size [START_REF] Masitas | Oxidation of Highly Unstable < 4 nm Diameter Gold Nanoparticles 850 mV Negative of the Bulk Oxidation Potential[END_REF], or rather a stabilization [START_REF] Kolb | On the unusual electrochemical stability of nanofabricated copper clusters[END_REF][START_REF] Lakbub | Electrochemical Behaviors of Single Gold Nanoparticles[END_REF], or sometimes, conditions of metastability [START_REF] Ng | Subnanometer silver clusters exhibiting unexpected electrochemical metastability on graphite[END_REF] can be observed depending on the nature of the selected systems, the size of nanoparticles and the experiment conditions. This diversity of behaviors is not specific to redox couples since it is found in other areas. Thus for indium nanoparticles, embedded in an aluminum matrix [START_REF] Lu | Melting and superheating of low-dimensional materials[END_REF], there is, according to the method of preparation (melt-spinning and ball-milling) a decrease or an increase in the value of the melting point of particles with their size. For gaseous aggregates, it is shown that nanobubbles with an average radius of 50 nm are stable for more than two weeks, whereas conventional reasoning, based on the application of Laplace's law, provide for their almost immediate disappearance [START_REF] Ohgaki | Physicochemical approach to nanobubble solutions[END_REF].

Obviously, all these changes cannot be described, let alone formalized by the classical (extensive) thermodynamics, which attaches great importance to the shape of the aggregate of nanometric size. The latter is usually assumed spherical, which is rarely the case in reality [START_REF] Zamborini | Nanoparticles in Measurement Science[END_REF].

The particle shape is generally difficult to determine from standard geometrical dimensions (length, area, volume) and the concept of interfacial tension for material as a whole loses its meaning. It is for these reasons that we introduced in 2004 the bases of a "non-extensive" thermodynamics [START_REF] Turmine | Nonextensive approach to thermodynamics: Analysis and suggestions, and application to chemical reactivity[END_REF] adapted to the study of the reactivity of nanoparticles (NP) which does not involve the knowledge of the shape of the particle but only its mass. The interface between the particle and its environment is supposed to be fuzzy. The use of this approach to describe the reactivity of various interfacial systems [START_REF] Letellier | Drop size effect on contact angle explained by nonextensive thermodynamics. Young's equation revisited[END_REF][START_REF] Letellier | Melting point depression of nanosolids: Nonextensive thermodynamics approach[END_REF][START_REF] Letellier | Solubility of nanoparticles: nonextensive thermodynamics approach[END_REF][START_REF] Letellier | Micellar aggregation for ionic surfactant in pure solvent and electrolyte solution: nonextensive thermodynamics approach[END_REF], has been particularly fruitful.

The electrochemistry of surfaces is a field in constant evolution [START_REF] Kolb | Electrochemical surface science: past, present and future[END_REF] and with regard to redox systems, we introduced in 2008, a study of the electrochemical properties of metal nanoparticles [START_REF] Letellier | Redox behavior of nanoparticules: Nonextensive thermodynamics approach[END_REF] at equilibrium. This work presented a generalization of the relationship proposed by Plieth [START_REF] Plieth | Electrochemical properties of small clusters of metal atoms and their role in surface enhanced Raman-scattering[END_REF] to non-spherical aggregates.

Theory

Without repeating all the bases of non-extensive thermodynamics, which can be found in detail in our previous publications, we present below some theoretical elements to easily follow our approach.

The conceptual bases of non-extensive thermodynamics, notion of fuzzy interface

In most cases, the nanoparticles are aggregates with geometric borders difficult to specify the. Under these conditions, it is unrealistic to set that the interfacial energy relative to the aggregate is written as the product of interfacial tension, , by an area, A. Therefore, the use of the Laplace's law giving the pressure difference between the inside of the particle and its environment is artificial. In the approach we propose, we consider an aggregate of mass M, whose interface between the particle and its environment is not defined. By convention, we will name this type of interface a "fuzzy interface" [START_REF] Letellier | Drop size effect on contact angle explained by nonextensive thermodynamics. Young's equation revisited[END_REF][START_REF] Setz | Fuzzy set approach for fitting a continuous response surface in adhesion formulation[END_REF][START_REF] Gerson | The role of fuzzy interfaces in the nucleation, growth and agglomeration of aluminum hydroxide in concentrated caustic solutions[END_REF]. The fact that we cannot describe the geometry of the object does not prevent it from having an interfacial energy, E. We agreed to write its variations as the product of a tension (intensive magnitude), τ, and an "extensity", , which varies with the mass of the system according to a homogeneous function (Euler's function) of degree equal to m, m can be different from one.

  d dE  (1)
This is why our thermodynamic approach can be described as "nonextensive" as opposed to Gibbs thermodynamics, in which the extensities are Euler functions of degree one of the system mass. Gibbs thermodynamics is then called "extensive". The formal choice we have made is purely conventional. It has the advantage of using the same state functions as those involved in the classical thermodynamics, to give a simple operational form to the chemical potential, especially to admit the extensive classical thermodynamics as a particular case of the nonextensive thermodynamics.

In this expression, can be positive or negative. One could be amazed at this possibility whereas the surface tension  is always positive.

On the theoretical level, when a thermodynamics that is no longer based on the extensivity is developed, it is necessary to redefine all variables of the system and clarify its meaning. They may have the same mathematical definitions as in extensive thermodynamics but with different properties. Thus, in some developments of nonextensive thermodynamics, stemming from the work of Tsallis [START_REF] Tsallis | Possible generalization of Boltzmann-Gibbs statistics[END_REF], the temperature is shown as a magnitude whose value depends on the system mass. In this case, this variable is not intensive (homogeneous function of degree 0 of the system mass) [START_REF] Abe | Nonextensive thermodynamic relations[END_REF]. This shows that it is always difficult to transpose to nonextensive approaches, patterns of thought inherited from the Gibbs thermodynamics.

Concerning τ, there was no theoretical reason, a priori, to fix the sign and we did not do it.

Experimentally, from the analysis of data from different fields of study, we showed that both situations ( > 0 and  < 0) are met. Thus, in terms of the change in melting points with the size of metal nanoparticles, we showed [START_REF] Letellier | Melting point depression of nanosolids: Nonextensive thermodynamics approach[END_REF], from literature values, the relationships that we propose allow fully describing the evolution of considered systems. Most often, there is a decrease of the value of the melting point with decreasing particle size. This case corresponds to the evolution expected by the Gibbs-Thompson relation (ie, with  > 0), and regards us a positive value of  . However, when the considered particles are embedded or placed within a matrix, the opposite effect is observed. This corresponds in our approach to a negative value of  . We can find another example of this feature of  considering the behavior of charged micelles [START_REF] Letellier | Micellar aggregation for ionic surfactant in pure solvent and electrolyte solution: nonextensive thermodynamics approach[END_REF]. In micellar solutions, the aggregates formed by the amphiphiles and their counter -ions can be treated as non-extensive interpenetrating phases. The activities of charged amphiphiles are fully described is by adopting a negative value of . In contrast, the activities of the counterions against is represented by a positive value of .

By convention, we assume that χ is an Euler's function of degree m of the particle mass, or its volume, V. We recall here that in classical thermodynamics, the extensities are strictly extensive magnitudes, that is to say, by convention, an Euler's function of degree 1 of the mass. In our approach, the degree m can be different from 1 while being lower ( 1  m ).

Among the possible Euler's functions, we chose to link χ to the volume of the particle by a power law. The parameter m is called the "thermodynamic dimension" of the system.

m aV   dV V m dV maV d m     1 (2)
The coefficient a characterizes the considered system. The variation of the interfacial energy is then written

dV V m d dE       (3)
It can be introduced in the expression of the internal energy U. For a system defined by its content (n1, n2... ni moles), in contact with an external phase at the pressure P and at the temperature T, we can write

   d dn PdV TdS dU i i      (4)
Since χ is not an extensive quantity, U is not either.

We have developed in detail the consequences of this property by posing as convention that T and τ are intensive magnitudes (Euler's function of degree 0). The volume and numbers of moles are by definition extensive variables (which corresponds to Euler's function of degree one of mass). The result (relation 4) is that the entropy is no more extensive.

Pressure and chemical potentials are no more intensive magnitudes, that is to say that their values vary with the mass (or volume) of the considered system, at constant composition.

One of the most original results of this "non-extensive approach of thermodynamics" is to show that the value of the pressure, NE P , in an aggregate made of n moles of a compound, considered as a non-extensive system, of volume V, of extensity χ, and of dimension m, is different from that of its environment, P.

There is a relationship between the pressure of the non-extensive system and the external pressure, such as

V m dV d P P NE        ) ( (5)
According to the sign of , this difference can be positive or negative.

Laplace's equation is a particular case of application of this equation for spherical systems.

Thus, in the case of a liquid drop in contact with a gas, of Liquid-gas (Lg) interfacial tension Lg  and radius r, the pressure difference between the inner of the drop, Pg, (P NE = Pg) and the external pressure, P of the gas atmosphere (unlimited phase), is obtained by identifying  = Lg  and χ = A Lg . The area is an Euler's function of degree m = 2/3 with respect to the mass or volume of the drop. This becomes

r 2 V A 3 2 V m ) P P ( Lg Lg Lg g        - (6)
In the case of using the Laplace's law, the surface tension is always a positive quantity, which means that the value of the pressure in the spherical particle is always larger than the one of the environment. This restriction is lifted in the case of the non-extensive thermodynamics,  can be negative. The fact that the pressure within an aggregate may be less than that of its surroundings may seem troubling in the case of nanoparticles, but it is an accepted concept in studies concerning the behavior of fluids separated by concave meniscus, for example in capillary systems [START_REF] Tas | Capillarity induced negative pressure of water plugs in nanochannels[END_REF].

This approach allows locating the origin of the change in behavior of the matter when finely divided. The pressure inside the nanoparticles is different from the ambient pressure, which involves modifying the chemical potential of the species. The variation of the chemical potential of component i with pressure, at constant temperature and composition, satisfies the following relation

' i n , T i V P i          (7)
' i V is the partial molar volume of the species i.

This equation can be integrated between the pressure of the non-extensive phase, NE P , and the ambient pressure, P, of the "unlimited phase" in contact, by supposing that the value of the partial molar volume of i does not vary with P (condensed phases)

dV d V ) P P ( V ' i un i NE ' i un i NE i           ( 8
)
NE i
 is the chemical potential of i in the non-extensive phase, un i  is the chemical potential of i in the unlimited phase at the pressure P, un stands for unlimited.

The partial molar volumes of the species involved in these phenomena are most often positive.

The extensity χ varies in the same way that the volume. According to the sign of , the chemical potential of i can be higher or lower than the chemical potential of the unlimited phase. When it is higher, the species is destabilized by the reduction in size and becomes more reactive. On the contrary, it is stabilized, then less reactive, when the value of  is negative. The value of the tension  can be small and then no effect of the size on the behavior of the particle will be observed.

Eq.8 allows justifying the variety of observed behaviors on the nanometric systems.

Conditions of the redox equilibrium

The behavior of redox couples is, by convention, referred to the couple

 H H / 2
. For a redox couple throwing in n electrons, the behavior is characterized by the conventional

Nernst equation R R nH red H n ox , 2 2     (9)
In order to simplify the notations, we do not show the number of charges (+) carried by the oxidized species. At the considered temperature, hydrogen, R H 2 , is taken in this standard state (perfect gas at the pressure of 1 bar,

* 2 2 H H R   
). The protons R H ,  are taken in their hypothetic standard state at infinite dilution (values of activity and concentration are equal to one,

    H H R   ,
). Let consider the case in which the reduced species is the non-extensive phase at the pressure P NE and where the oxidized species is in the solution (the unlimited phase) at the external pressure P.

The free energy of the reaction, equilibrated in charge, is written as a function of the chemical potentials of the participating species

* H P ox P red H redox 2 NE 2 n n G            (10)
In these conditions, the standard free energy of the conventional reaction of electronic exchange is written according to

NE H ox P red H P redox nFE n n G NE NE , 0 * , 0 , 0 2 2               (11)
By taking into account the previous relations, the standard chemical potential of the reduced species in unlimited phase can be introduced in this last expression.

NE , 0 * H ox 0 red un , 0 red H P , 0 redox nFE 2 n dV d V n G 2 NE                  (12)
For the redox reaction implying oxidized and reduced species in unlimited phase, the standard redox potential is

ill , 0 * H ox un , 0 red H ill , 0 redox nFE 2 n n G 2               (13)
The variation of the redox potential with the volume can be deduced

dV d nF V E E 0 red un , 0 NE , 0     (14)
If the value of the tension, , is positive, the standard redox potential decreases with the volume of the object and then the reduced species is more oxidizable.

In the case of a spherical particle of metal, of radius r and solid-liquid (SL) interfacial tension,  SL , the relationship proposed by Plieth (which comes from Gibbs-Thompson equation) is found

r 2 nF V E E SL 0 red un , 0 NE , 0    (15)
In this case, as  SL is always positive, a decrease of the standard redox potential with the particle radius is expected. These results can be used for analyzing the electrochemical behavior of redox couples in cyclic voltammetry.

Cyclic voltammetry 2.1. Unlimited phases

We may first recall the main relationships that characterize the behavior of a reversible redox couple studied in cyclic voltammetry for reactions in unlimited phases.

For a redox reaction involving n electrons, the half-peak potential Ep/2 is simply expressed as a function of redox potential and the diffusion coefficients of the reduced and oxidized species.

2 / 1 ox red un , 0 2 / 1 D D ln nF RT E E           (16)
If for a given redox couple, the value of E° varies with the state of confining of the compounds involved in the reaction, without changing the diffusion coefficients, then the current peak position changes.

Reversible reaction in non-extensive thermodynamics

Usually, the electrochemical behavior of metal nanoparticles with their size is characterized by studying their electro-oxidation after their deposition on electrodes.

Position of the peak in unlimited phase

Suppose that the reduced species is a pure non-extensive phase on the electrode. The main consequence (eq. [START_REF] Letellier | Redox behavior of nanoparticules: Nonextensive thermodynamics approach[END_REF] is that the values of the peak and half-peak potentials will be modified according to the nanoparticle volumes.

dV d nF V D D ln nF RT E D D ln nF RT E E * red 2 / 1 ox red un , 0 2 / 1 ox red NE , o NE 2 / p                        (17)
which can be also written

dV d nF V E E E E * red NE 2 / p un 2 / p NE p un p       (18)
By taking into account that the extensity is an Euler's function of order m of the mass (or of the volume), we show the general relationship

V m nF V E E E E * red NE 2 / p un 2 / p NE p un p       (19)
This relation can be applied when the aggregates are spherical particles of radius r and their interfacial tension, , is known.

r 2 nF V E E E E * red NE 2 / p un 2 / p NE p un p      (20)
In such a case, the peak potential varies with the inverse of the radius. The value of the peak potential will decrease with r. This relationship has been proposed by Plieth in 1982 [START_REF] Plieth | Electrochemical properties of small clusters of metal atoms and their role in surface enhanced Raman-scattering[END_REF]. It serves as a theoretical reference to current studies.

Peak position for non-extensive phases

By writing as previously that the extensity is an Euler's function of order m of the mass, we obtain

1 m * red NE 2 / p un 2 / p NE p un p V m a nF V E E E E       (21)
The peak potential varies according to a power law of the volume (or of the mass) of the particles of reduced species. The relevance of this approach can be checked from the experimental results available in the literature.

However, in most of the literature, we do not have the volume of aggregates, but an experimental size parameter, D, usually measured by spectroscopy. This can be a radius, a diameter, a length, a thickness, etc. We can transform the above equation by writing that the extensity, χ, and the volume are both homogeneous functions of the variable D, such as

p D    q D V   (22) Then dD pD d 1 p    dD qD dV 1 q   (23) which leads to q p D q p dV d      ( 24
)
The variation of the peak (and half-peak) potential is written

q p * red NE 2 / p un 2 / p NE p un p D q p nF V E E E E         ( 25
)
This relation shows that the potential difference  

NE p un p E E 
is linked to the measured size of the particles by a power law. Then, the use of experimental results can be realized graphically from the relationship

D ln ) q p ( ) q p nF V ln( ) E E ln( * red NE p un p        (26)

Discussion

There are, in the literature, many data on the variation of the peak potential with the nanoparticles size of the reduced species. Generally, these changes are interpreted from the Plieth's relationship that most often does not well describe the experimental reality, primarily for three reasons.

The first is that the value of the interfacial tension between the metal and its environment is not experimentally accessible. It must therefore be estimated from energy considerations, which makes it uncertain.

The second is that the Plieth's relationship relates to a homogeneous population in size and in shape of the nanoparticles. The micrographies reported in the literature show that it is rarely the case. The dispersions are sometimes very large.

The third reason is that nanoparticles are not spherical and thus the Plieth's relationship does not apply.

We extracted several examples from the literature to illustrate the relationships of the nonextensive thermodynamics outlined above. In particular, we will discuss two studies of Ivanova and Zamborini. The first considers silver particles [START_REF] Ivanova | Size-Dependent Electrochemical Oxidation of Silver Nanoparticles[END_REF]. These authors studied by voltammetry the oxidation of silver nanoparticles deposited on a glass electrode coated with indium-tinoxide. The nanoparticles are characterized by microscopic techniques (AFM and SEM). The oxidations are carried out in 0.1 mol L -1 H2SO4 aqueous solution. The values of the potential peaks are reported in Table 1 of their publication. We used the values in this table, adopting 799 mV/SHE as the redox potential value of silver in unlimited phase. To be consistent with the experimental values of potential referred to the Ag/AgCl electrode (172 mV/SHE according to the authors), we corrected the standard redox potential of silver and adopted the value of E 0,un = 627 mV/Ag/AgCl to realize our exploitation. Verification of the eq. 26 was made using the diameter values determined by SEM that lead to a consistent set of points.

The graphical evaluation leads to an excellent straight-line of slope -0.295 and intercept equal to 6.621.

Fig. 1 From these data, the experimental variations of the potential can be properly described. These results indicate that the decrease in standard potential with the size does not vary as assumed by Plieth's relation in 1/d but in 1/d 0.295 . It is difficult to go further in the analysis, but if we assume that the volume is a quantity that varies as the particle size cubed (geometric dimension, q = 3), this means that the extensity varies as the diameter to the 2.7 th power. If the interface had the geometric dimension of a surface, it would vary according to the square of the size. This proves that the interface between the nanoparticles and their environment is much more complex than a simple surface, as micrographies of particles also show.

The second study by the same authors, reported here, considers gold nanoparticles [START_REF] Ivanova | Electrochemical Size Discrimination of Gold Nanoparticles Attached to Glass/Indium-Tin-Oxide Electrodes by Oxidation in Bromide-Containing Electrolyte[END_REF]. In this paper, the authors describe the electrochemical oxidation of an assembly of gold nanoparticles (Au NPs) attached to glass/indium-tin-oxide (ITO) electrodes as a function of particle size.

They show that the potential of electro-oxidation peak decreases of approximately 200 mV when the particle diameter decreases from 250 to 4 nm. The problem is more complex than previously because several oxidation degrees of gold are likely to be affected.

We have exploited data reported by these authors for the electrodeposition made at 0.8V, taking as the value of the peak potential, 926 mV, and as E° those corresponding to the one-electron exchange (i.e., 963 mV) [START_REF] Ivanova | Electrochemical Size Discrimination of Gold Nanoparticles Attached to Glass/Indium-Tin-Oxide Electrodes by Oxidation in Bromide-Containing Electrolyte[END_REF]. The graphical exploitation of the eq. 26 leads to an excellent straight-line of slope -0.4428 and intercept 5.8984.

It was reported on Figure 3, the experimental data and the calculated curve. The agreement is excellent.

Fig. 3 As before, the reduction in potential does not match that described by the Plieth's relation, probably because the particles are not spherical, as shown by the SEM images. The extensity varies with the measured particle diameter to the power of 2.5.

Conclusion

The development that we described shows that the relations of non-extensive thermodynamics can quantitatively account for the displacements of electro-oxidation potentials of metals deposited as nanoparticles on electrodes, according to their measured size.

Our approach generalizes the one proposed by Plieth to particles with non-spherical shape and it allows describing by power laws the experimental variations of peak potential with the aggregate size. The non-extensive thermodynamics also permits to formally justify the stability of the particles may increase as their size decreases, (τ < 0). This is usually found when the aggregates are in close contact with a matrix (in the case, for example, of embedded particles).

We cannot be certain that the particle/matrix interaction is not largely responsible for this behavior.

Our thermodynamic approach has also its application limits. Thus, there is in the literature many data that we have not been able to exploit, especially those published in the works Brainina et al. Indeed, in order to have a thermodynamic meaning, our relations can only be applied on homogeneous populations of particles. Even if we admit that their shape is fuzzy, this shape must be "uniform" with similar characteristics in size and structure. Brainina et al. studies [START_REF] Brainina | Mathematical modeling and numerical simulation of metal nanoparticles electrooxidation[END_REF][START_REF] Brainina | Bismuth nanoparticles electrooxidation: theory and experiment[END_REF][START_REF] Brainina | Gold nanoparticles electrooxidation: comparison of theory and experiment[END_REF][START_REF] Brainina | Silver nanoparticles electrooxidation: theory and experiment[END_REF][START_REF] Brainina | Electrochemistry of metal nanoparticles: the effect of substrate[END_REF][START_REF] Brainina | The effect of the system polydispersity on voltammograms of nanoparticles electrooxidation[END_REF] show very well the shifts of the peaks that are consistent with the positive values of the tension, τ. But the considered systems are usually too widely polydisperse for properly applying eq. 26.

In the examples that we have exploited and given here, the SEM images showed that the particles were geometrically ill-defined. But this may not be always the case. Indeed, an abundant literature reports nanoparticles with well-defined interfaces in terms of crystallography, such as the (111) plane of gold or copper [START_REF] Kolb | Electrochemical nanostructuring[END_REF]. The non-extensive thermodynamic approach also allows describing such systems, assuming that the particle exhibits to the environment several different types of interfaces. The pressure difference between the non-extensive phase and unlimited phase is then written

V m dV d P P i i i i i i i NE          ( 27
)
Each interface is characterized by extensity, i  , a dimension, mi, and a tension, i, which can account for the anisotropy of the studied crystal. This is the way we used to treat the equilibrium conditions of a liquid drop put onto a substrate [START_REF] Letellier | Drop size effect on contact angle explained by nonextensive thermodynamics. Young's equation revisited[END_REF]. It would be interesting to have quantitative studies of well-defined particles according to their structural characteristics and size.

Another limitation that we have not mentioned is the assumption that we implicitly do on the fact that the value of  does not vary with the mass of the nanoparticle. This may not be true for the particles of weaker masses. Probably, the non-extensive thermodynamics has the same limitations as classical thermodynamics.

On this point, we must recall the remark we made in our study about the redox potential whose the lower limit of validity is placed around the nanometer. We can assume that below this size, particle stability essentially involves quantum phenomena [START_REF] Kubo | Electronic properties of metallic fine particles.1[END_REF]. However, we used measures of Tang et al. [START_REF] Tang | Electrochemical Stability of Nanometer-Scale Pt Particles in Acidic Environments[END_REF][START_REF] Tang | Electrochemical Stability of Elemental Metal Nanoparticles[END_REF] on the electrochemical stability of platinum nanoparticles of very small size in acidic medium. The experimental values published by these authors respond to the depression of standard redox potential proposed in eq. 26 with a power law in 1/r 0.7 and not 1/r as assumed by Plieth's relation.

We have plotted on Figure 4 the experimental data extracted from the figure of ref [START_REF] Tang | Electrochemical Stability of Nanometer-Scale Pt Particles in Acidic Environments[END_REF] and the calculated values. The agreement is excellent.

Fig. 4 The exploitation of voltammetry data and more generally of electrochemistry at equilibrium

shows that, as in other areas, the relations of non-extensive thermodynamics can adequately describe the change in the behavior of matter when it is in the shape of nanoparticles.

This reinforces our belief that the behavior of "small systems" and in particular that of nanoparticles does not respond to classical extensive thermodynamics as assumed by Du [START_REF] Du | Thermodynamic properties and equilibrium constant of chemical reaction in nanosystem: An theoretical and experimental study[END_REF],

and it is necessary to introduce the non-extensivity to account for it.

Another way of expressing this conclusion is to say that thermodynamics, in general, is not extensive and the extensivity is a particular case, which allows accounting for the behavior of unlimited systems.

Lists of Captions

Figure 1 : application of eq. 26 to the data of table 1 of ref [START_REF] Ivanova | Size-Dependent Electrochemical Oxidation of Silver Nanoparticles[END_REF]. The sizes correspond to the one measured by SEM (diameter D in nm). The redox potential value (E un ) against Ag/AgCl is taken equal to 0.627V.

Figure 2 : experimental variation of peak potentials for the system (points) described in ref. [START_REF] Ivanova | Size-Dependent Electrochemical Oxidation of Silver Nanoparticles[END_REF]. The solid curve is calculated from eq. 26.

Figure 3 : experimental of peak potentials for the system (solid points) described in ref. [START_REF] Ivanova | Electrochemical Size Discrimination of Gold Nanoparticles Attached to Glass/Indium-Tin-Oxide Electrodes by Oxidation in Bromide-Containing Electrolyte[END_REF]. The solid curve is calculated from eq.26.

Figure 4 : redox potential of Pt/ Pt 2+ couple in respect with the nanoparticle radius according to ref. [START_REF] Tang | Electrochemical Stability of Nanometer-Scale Pt Particles in Acidic Environments[END_REF]. The solid line curve corresponds to E = 1.188exp(-0.9656)* r -0.7

Graphical Abstract

The nonextensive thermodynamic approach is applied to the electro-oxidation of metal nanoparticles to describe the displacement of voltammetric peaks. We showed that, for many examples, the interface can be characterized by a function  which varies according to a power law. The interface of the nanoparticle is then defined as "fuzzy".

nF V V q p nF V E E           

Fig. 2

 2 Fig.2

  Fig.1

Acknowledgements

The authors thank K. Fajerwerg and M. Comtat from université Paul Sabatier of Toulouse for the fruitful and instructive discussions.