P Labastie

Distance Geometry: A Viewing Help for the Solid-Liquid Phase Transition in Small Systems

Keywords: numbers: 36.40.Ei, 64.70.dm, 05.20.Gg, 82.60.Qr

Distance geometry is the study of the arrangements of points in space using only the mutual distances between them. The basic idea in this letter is to use distance geometry for thermodynamics studies of small clusters in the microcanonical ensemble. There are constraints on these distances, which are shown to explain some characteristic features of the caloric curve in very small clusters containing 3 or 4 atoms. We anticipate that this approach could give a novel insight into the phase transitions in larger clusters as well. During these studies, we have established a very general and rather simple result for the Jacobian determinant of the change of variables from Cartesian coordinates to mutual distances, which is of wide applicability in the N -body problem.

It is well known that most of the tractable problems of physics are such because there exist a "good" choice of variables, which drastically simplifies equations to be solved. Such a good change of variable has not been found yet for the N-body problem. Actually, for computational purposes, it has been agreed that the Cartesian coordinates are the less expensive coordinates because of the simplicity of the equations of motions when written in terms of them, at least in classical mechanics. On the other hand, those coordinates introduce unnecessary invariance (the euclidean group), while physics describes mainly interactions between systems characterized by their relative position. In this respect, mutual distances should be enough to describe all physical processes. Another reason for considering mutual distances as variables is the simplicity of the potential energy when written in terms of them. It might thus be worth the try to go as far as possible with these variables in the studies of model systems described by pair potential. The work in this letter is a start in this direction and primarily of exploratory nature. It aims at bringing attention to some remarkable results (the simplicity of the Jacobian of the transformation from Cartesian coordinates to mutual distances) and to the ease of representation given by these variables.

Mutual distances have been considered by Lagrange in his study of the 3-body problem.

Several formulas for expressing various quantities of the system as a function of distances alone are known. The basic reference for these is Blumenthal's book [START_REF] Blumenthal | Theory and Applications of Distance Geometry[END_REF]. A more modern account is given in chapter 4 of [START_REF] Dattorro | Convex Optimization & Euclidean Distance Geometry[END_REF]. Note that distance geometry is an invaluable tool in the study of protein conformation, where the problem is to determine all the distances between a set of atoms, knowing just a few of them, as given by NMR or X-ray studies [START_REF] Crippen | [END_REF][START_REF] Havel | Encylopedia of Computational Chemistry[END_REF].

Several new directions can be explored and some results will be presented in forthcoming publications. In this letter, this change of variable is applied to the computation of the configurational density of states (CDOS) of a small cluster of N atoms interacting by a pair potential, in a D dimensional space for the special cases N = D or N = D + 1. In both cases, the number of distances is equal to the number s of independent variables specifying a configuration: s = DN -D(D +1)/2. Note that for N > D +1, one has to choose a subset of the distances as internal variables, which introduces technical complexities not usefull for the purpose of this letter. The general case will be presented in a forthcoming publication [START_REF] Labastie | [END_REF]. The CDOS is the fundamental quantity in classical thermodynamics. Its computation involves an s-dimensional integral. We show below that, when mutual distances are used as variables for an N = D-atom clusters, this s-dimensional integral can be approximately reduced to s convolution products. This reduction is even exact at low energy. In any case, it is much easier to compute than an s-dimensional integral.

The N = D = 3 case is especially interesting because it is possible to draw a representation of the constant energy surface in the distance space. The accidents of the density of states as a function of energy (or rather of its logarithm, the entropy S(E)) can be visualized and given a simple interpretation on this drawing. It is well known that accidents of S(E) are related to so-called phase transitions in clusters [6,7,8,9]. The solid-liquid transition (actually a phenomenon akin to this transition) has been observed both in simulations [6,10,[START_REF] Wales | Large clusters of atoms and molecules[END_REF] and in experiments [START_REF] Schmidt | [END_REF]13,14,15,16,17]. We anticipate that the simple view given in this letter could be used in understanding the solid-liquid transition in larger clusters as well.

This letter is organized as follows. First, we recall the formulas for the microcanonical thermodynamics of a set of atoms and the definition of the CDOS. Then, we make some considerations on the change of variables, and give the (very simple in our case) formula for the Jacobian determinant. We next give some properties of the space of distances. We then present drawings of the constant energy surface in this space in the case of N = 3 and explain the qualitative features of the entropy curve. Finally, we show how the CDOS can be expressed as a convolution product and present a comparison with more conventional calculations.

In the microcanonical ensemble, the system size (volume V , number of constituents N)

and energy E are fixed. The fundamental function is the density of states Ω(E, V, N).

It exists for any finite system with a classical Hamiltonian H(P, X). Here, the notation X represents the set of Cartesian coordinates of the constituents and P the corresponding momenta. The Hamiltonian H is the sum of the kinetic energy T (P) and the potential energy E p (X), so that the density of states is actually the convolution of the kinetic density of states and the configurational density of states Ω c (E, V, N)

Ω c (E, V, N) = ∂ ∂E Ep(X)≤E d 3N X. (1)
Since the kinetic density of state can be computed analytically, the configurational density of state is the quantity we focus on. The configurational entropy S c (E) is defined by Boltzmann formula

S c (E, V, N) = k B ln Ω c (E, V, N), (2)
with k B the Boltzmann constant. In the remaining of the paper, the configurational entropy will be called just entropy for short.

It can be shown [START_REF] Ruelle | Satistical Mechanics: Rigorous Results[END_REF] that, under mild conditions on the potential energy function E p (X), the entropy is extensive in the thermodynamic limit, and that in this limit, the second derivative of the entropy is always negative. This is not true in a finite system: the entropy is not extensive because surface effects are important, and furthermore it can show portions of positive curvature, which are the accidents of the CDOS we mentioned above.

We now describe shortly the change of variables from Cartesian coordinates to mutual distances. The Cartesian coordinates of an atom a, (a = 1 . . . N), are noted x ia , (i = 1 . . . D).

Rather than the distances, it is more convenient to define the variables y ab by

y ab = D i=1 (x ia -x ib ) 2 2 . ( 3
)
When going from Cartesian coordinates to internal coordinates, one should define a "moving frame" (see for example Ref. [START_REF] Littlejohn | [END_REF]). Amazingly enough, the Jacobian determinant for the transformation does not depend on this choice. We suppose that the cluster is contained in a (hyper)spherical volume of radius R, with the center of mass fixed at the center of the sphere. This eliminates the need to integrate on the translation degrees of freedom. The integration on the rotation coordinates gives a constant irrelevant multiplicative factor, the volume of the group SO(D), noted V rot . Eventually, it can be shown that the configurational volume element can be written

rotation i,a dx ia = V rot J 1≤a<b≤N dy ab , (4)
with

J = 1 if N = D and J = 1/V if N = D +1
, where V is the volume of the parallelotope built on the D + 1 points. This volume can be computed as the square root of a determinant involving the y ab 's only [START_REF] Blumenthal | Theory and Applications of Distance Geometry[END_REF]: Let us now think of the set of distances y ab as defining the Cartesian coordinates of a point in an abstract space D, which we call the 'Distance Space'. There are several conditions to be verified by this point. They amount to ensure that all the determinants similar to that of Eq. ( 5), for any subset of the N atoms, are positive. It can be shown (see e.g. Ref.

V 2 = -det 0 1 1 1 • • • 1 1 0 -y 12 -y 13 • • • -y 1N 1 -y 12 0 -y 23 • • • -y 2N 1 -y 13 -y 23 0 • • • -y 3N . . . . . . . . . - . . . . . . . . . 1 -y 1N -y 2N -y 3N • • • 0 . ( 5
)
2) that these conditions restricts the representative point in the distance space to be inside a convex cone. In other words, D is a space with a frontier.

We have not yet included the condition that the atoms are confined inside a sphere of radius R. Actually, the distance r 2 a from atom a to the center of mass can be expressed in terms of the y variables alone:

r 2 a = 2 N 2 (N -1) b =a y ab - b<c b,c =a y bc . (6)
Imposing that r 2 a ≤ R 2 amounts to adding the further constraints that the distance space is limited by a set of N (hyper)planes. Those planes, together with the cone defined above define a finite volume in the distance space. Figure 1 shows a view of the cone and the planes in the case N = 3. Now, What is the shape of the constant energy surface (CES) in distance space? We suppose that the potential is E p = a =b v(y ab ), where v(y) has the usual shape of a pair potential: one minimum at y = 1 with value -1, the limit when y → ∞ is 0 (that is the dissociation energy is 1) and the limit when y → 0 is +∞. The CES in the case N = 3 is shown in the insets of Fig. [START_REF] Dattorro | Convex Optimization & Euclidean Distance Geometry[END_REF]. At low energy (inset (a), E p = 0.2), the surface is close to a spheroid around the absolute minimum of the potential energy, which is y 12 = y 13 = y 23 = 1/2. Notice that this surface is far from the limiting cone. At higher energy (inset(b), E p = 0.5), the surface becomes elongated in the direction parallel to the (distance) axes. More precisely, it is easy to see that the intersection of the surface with the axis y 12 = y 13 = 1/2 for example, occurs for y 23 such that v(y 23 ) = E p . This value of y 23 increases rapidly with increasing E p . There is of course a symmetry between the three axes.

On the other hand, at direction 45 • from the axes, the maximum value of the y's is such that v(y) = E p /2, which is much closer to the point (1/2, 1/2, 1/2). At an energy close to 1, the shape is that of 3 "tentacles" along the directions parallel to the axis, and the surface reaches the enclosing cone for E p ≃ 0.969 (inset (c)). This last value is specific of the Lennard-Jones potential; it is always very close to one for any potential. For E > 1 (inset (d)), the CES is open. However, the volume inside the cone is still finite. The configurational entropy is shown on the same figure. At low energy, S c (E) ∝ ln(E), which is the usual behavior. At higher energy, the entropy departs from the logarithm, and would even diverge when E → 1 (the dashed curve). However, the limitation imposed by the cone results in a discontinuity of the slope when the CES meets the cone. It is important to understand that there is a balance between the tendency of the pair potential to allow a distance to become large, which results in a positive curvature of S c (E), and the limitation of pure geometric nature due to the limiting cone, which bends in the negative direction. Such "bumps" of the entropy curve are characteristic of a phase transition in a finite system. We give here a new geometric view of what happens, at least in this simple system.

Let us now compute the CDOS. The simplest case is N = D. When ǫ < 1, the equation v(y) = ǫ has two solutions, which we denote y max (ǫ) and y min (ǫ). When ǫ → 1, y max → ∞ and disappears for ǫ > 1, while y min exists for any value of epsilon. To begin with, we forget about any constraint on the distances, and we suppose that N = D. It is easy to see that the CDOS can be written in this case the integral can be computed in O(n 2 s) time, to be compared to the O(n s ) necessary for a general s-dimensional integral. Equation ( 7) is valid as long as the energy is low enough for the CES to not intersect the limiting cone. Above that, an approximate density of states can be computed if one supersedes Eq. ( 8) by

Ω c (E) = P s i=1 ǫ i =E s i=1 ω(ǫ i ) dǫ i , (7)
ω(ǫ) = d dǫ [y max (ǫ) -y min (ǫ)] if y max (ǫ) ≤ Y , (9)
= d dǫ [ -y min (ǫ)] if y max (ǫ) > Y . (10)
Y = (N -1)/N -2) is the value that zeroes the determinant ( 5) with all the y's equal to 1/2 except one, which is equal to Y . The result of this approximation is shown on Fig. 2 for N = 3 and Fig. 3 for N = 8. Notice that the good agreement with the exact CDOS extends far beyond the expected range, which is E < 1.

When N = D + 1, the integral in Eq. ( 7) contains one more factor, the Jacobian determinant, so that it is no more a convolution. On the other hand, the CES develops essentially along directions parallel to the axes, hence all the y's are close to 1/2. It can thus be included Lennard-Jones potential in dimension 2.The dotted curve is the result of a Monte-Carlo calculation, the plain curve is the approximation from Eq. [START_REF] Wales | Large clusters of atoms and molecules[END_REF].

in the integral by changing one of the ω's with

ω ′ (ǫ) = 1 V [y max (ǫ)] dy max dǫ - 1 V [y min (ǫ)] dy min dǫ , (11)
where V (y) is the volume of Eq. ( 5) with all the y ab 's equal to 1/2, save for one, which is y. Of course, when y max > Y , only the term containing y min is retained as above. This approximation together with the exact Monte-Carlo result is shown on Fig. 4 for 3 atoms in dimension 2. The agreement is still very good, although the approximation is somewhat crude.

As a conclusion, mutual distances have been used as variables for the study of some N-body problems. It has been shown that it is possible to make efficient approximations for the calculation of classical densities of states and that considering the space of distances is a good help for viewing what happens at phase transitions in small systems.

I thank Ms. Ch. Lauzeral for help on the initial stages of this project. Many calculations have been done or checked with the help of the Maple software from Maplesoft, a division of Waterloo Maple Inc. * pierre.labastie@irsamc.ups-tlse.fr

FIG. 1 :

 1 FIG. 1: (color on line) The distance space for N = 3 atoms. The axes for the coordinates (y 12 , y 13 , y 23 ) are shown by the intersection of two planes. An acceptable point can only be inside the cone. Furthermore, if the cluster is contained in a sphere, the accessible space is further limited by 3 planes, which interset on the (1, 1, 1) direction. On the figure, one of the planes has been made transparent in order to show the cone interior.

8 )FIG. 2 :

 82 FIG. 2: Configurational Entropy S as a function of energy E for a 3 atom cluster interacting via Lennard-Jones potential. The insets represent the CES in distance space (see text): (a) E = 0.2, (b) E = 0.5, (c) E = 0.969, (d) E = 1.2. The dashed curve is the entropy without constraint, the dotted curve is the result of a Monte-Carlo calculation, the plain curve is the approximation by a convolution product (see text).

FIG. 3 :FIG. 4 :

 34 FIG. 3: Configurational Entropy S as a function of energy E for an 8 atom cluster interacting viaLennard-Jones potential. The dotted curve is the result of a Monte-Carlo calculation, the plain curve is the approximation by a convolution product.


Agathe Hubau
email: a.hubau@brgm.fr

Solène Touzé
email: s.touze@brgm.fr

Stéphane Brochot
email: s.brochot@caspeo.net

Philippe Wavrer
email: p.wavrer@caspeo.net

Estimation of uncertainty for measuring metals content in waste printed circuit boards

The amount of waste printed circuit boards (WPCBs) currently represent a fast-growing issue that must be properly managed to limit their impact on the environment and human health. Due to their metal content, they can also be considered as a resource. Their characterization is a key point for evaluating different valorisation processes. The conventional methodologies to characterize wastes and/or metal resources are hardly applicable to such waste, as they are highly heterogeneous, difficult to micronize, and their individual components (plastics, glass, ceramics, and metals) are hard to liberate. Thus, in parallel to developing analytical tools that allow their accurate characterization, a sampling strategy suitable for WPCBs must also be established. In this study, an empirical approach was developed that aims at estimating the uncertainty arising from sampling WPCBs. To do so, the duplicate method of uncertainty estimation was followed, to compare the metal content in different sub-samples and to determine confidence intervals.

Introduction

Waste printed circuit boards (WPCBs) are gaining increasing importance due to their high metal content (around 30 to 40 wt%) and the increase of volumes that are produced. They are thus considered as a secondary resource, but also as potentially harmful wastes requiring to be handled with caution. While many studies deal with the development of new processes to valorise these wastes, there is very few studies dealing with the establishment of a sampling strategy enabling to obtain representative and reproducible samples. Three challenges are arising when developing a WPCBs sampling strategy. First, it is difficult to decrease the particle size of WPCBs. This issue is mainly linked to the presence of ductile metals, preventing the use of most of the conventional shredding and grinding equipment. The wet grinding processes produce wastewater that must be treated, while dry grinding processes produce dust, causing a potential loss of precious metals, as well as an increase of temperature that probably leads to the partial degradation of resins and plastics [START_REF] Kaya | Electronic Waste and Printed Circuit Board Recycling Technologies[END_REF] . Secondly, these wastes are very heterogeneous, with a composition that changes over time. Some of the constituting materials (plastics, resins, glass, ceramics and metals) are embedded or diluted (as alloys) in other materials, preventing their complete liberation. There is also a large range of metal content, with metals that are highly concentrated (as Cu, with a content reaching up to 20 %w/w) or that are only present in PCBs as trace elements (as Pt, with less than 10 mg.kg -1 ). Finally, the last point concerns the lack of reference material, preventing to assess analytical uncertainty.

To the authors' knowledge, as of date, there is no adjustment of the Theory of Sampling (TOS) for WPCBs. The sampling strategies that are available in the literature on this material are mainly based on very small samples (around ten kilograms of WPCBs from a single batch at most) and are evaluating the quality of the last sampling step (production of the test portion used for the analysis). They are also based on pyrometallurgical approaches to reduce grinding issues and sample mass. However, this last strategy is not applicable when aiming at obtaining representative samples of WPCBs containing intact non-metal fractions (plastics and resins). Moreover, the quality of sampling is not evaluated in most of these studies. Due to the high heterogeneity of this material and the difficulty to obtain a complete liberation of constituting elements, the modelling approach to quantify the uncertainty, such as the one developed by Pierre Gy, is not applicable to the best of current knowledge, as there are no data available in the literature on the liberation factor, shape factor or constitution factor.

In this study, an empirical approach was chosen to estimate the uncertainty arising from sampling WPCBs. To do so, the duplicate method was applied to a unique batch of around 500 kg of WPCBs. As recommended in the Eurachem guide on measurement uncertainty 2 , the measurements of metal contents in WPCBs samples were replicated after successive shredding and subsampling steps (see Figure 1) and uncertainties were estimated at different levels of the sampling plan. To do so, the content of six metals (Cu, Fe, Ni, Zn, Pb and Co) was determined for two laboratory samples (LS) of 4 kg of -10 mm WPCBs. These two laboratory samples were ground to minus 750 µm and two 40 g test samples (TS) were taken for each. Each duplicate test sample was divided into 8 test portions (TP) which were analysed (Figure 1). The sampling quality of "between laboratory samples (LS)", "between test samples (TS)" and "between test portions (TP)" was evaluated. Some statistical tools were used to determine if the mean values of the test samples were significantly different. Variances and standard deviations were calculated and the contributions of "between test portions" sampling, "between test samples" sampling and "between laboratory samples" sampling to the total uncertainty were determined for each metal. Finally, intervals for the metal content values at 95 % confidence level were determined. The goal was to provide insights in the quality of sampling and to assess the reproducibility of the composition of sub-samples for further work on the development of WPCBs recycling technologies. From this study, a new sampling plan for extra batches of WPCBs could be designed.

Materials and methods

WPCBs sample, sampling and characterization

WPCBs used in this study were provided by a French recycling company, from the small waste electrical and electronic equipment category (mix of appliances such as computer, audio and video equipment, toys, personal care products, small kitchen appliances, etc.). The whole methodology used for sampling the 526 kg batch of WPCBs was previously detailed in in Hubau et al. (2019) [START_REF] Hubau | Recycling oriented methodology to sample and characterize the metal composition of waste Printed Circuit Boards[END_REF] and is given in Figure 1. Photographs of samples are given in Figure 2. To determine metal contents, each test portion was digested with aqua regia (HNO3:HCl 1:3) with a solid to liquid ratio of 1:11 w/v. The digestions were performed using a Kjeldatherm-Gerhardt digestion system, maintaining the aqua regia at reflux for 2 hours. After solid-liquid separation, leachates were analysed by flame atomic absorption spectrometer (Varian SpectrAA-300). The efficiency of aqua regia digestion for such wastes has been previously demonstrated [START_REF] Hubau | Recycling oriented methodology to sample and characterize the metal composition of waste Printed Circuit Boards[END_REF][START_REF] Touzé | Sampling waste printed circuit boards: Achieving the right combination between particle size and sample mass to measure metal content[END_REF] (less than 2 wt.% of Cu, Zn, Sn, Pb, Co and Pd are undissolved, and less than 7 wt.% for other metals, except for Ta). The same operator performed all the grinding, quartering, digestion and analysis operations for all of the 32 test portions. In this study, the observed variability both includes the uncertainty arising from the sampling steps (TSE) as well as the total analytical error (TAE estimation is described elsewhere [START_REF] Brochot | Experimental determination of the constitutional heterogeneity for sampling of Waste Printed Circuit Boards[END_REF] ).

Statistical tools

After determining the standard deviation on the 32 measurements for each metal, it appears that the relative standard deviation is very large for Co (22.3 %) (Table 4). It is thus better to consider the natural logarithm of metal content values for the calculation of the uncertainty factors and for the application of statistic tests, as recommended by Eurachem guide 2 . Shapiro-Wilk tests were performed on these values to ensure that the hypothesis of log-normal distribution for metal contents cannot be rejected. For this study, the significance level for the p-value was fixed to 0.05. As demonstrated in Table 1, hypothesis of log-normal distribution for metal contents in all the 32 test portions cannot be rejected. The hypothesis of normal distribution cannot be rejected too, while it is more difficult to conclude for Fe. These tests were also performed within each test sample (n = 8, data not shown) and similar results were obtained. The calculation of variance was performed following the Eurachem guide methodology 2 . Variance can be calculated when a reasonable number of values are studied. However, as a misuse of language, Eurachem guide uses the term of variance for low number of samples (even 2) and this terminology was thus used in our study.

The formulas used to estimate the variance arising from "between test portions" sampling, from "between test samples" sampling and from "between laboratory samples" sampling are given in Table 2, with xi,j,k referring to the natural logarithm of the metal content in the test portion k, from test sample j, from laboratory sample i. For this study, NTP, NTS and NLS refers to the number of test portions (8), test samples (2) and laboratory samples (2) respectively, with the indices k, j and i respectively, 𝑥 𝑖𝑗𝑘 being the logarithm of content in the TP k of TS j of LS i.

From this, the standard deviation (SD), the relative standard deviation (RSD), the expanded uncertainty factor ( F U) and the limits for confidence interval for each contribution were calculated as follow (example given for TP sampling):

𝑈 𝐹 𝑇𝑃 = 𝑒 2𝑆𝐷 𝑇𝑃 𝐿𝑜𝑤𝑒𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑙𝑖𝑚𝑖𝑡 = 𝑥̅ 𝑈 𝐹
and 𝑈𝑝𝑝𝑒𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑙𝑖𝑚𝑖𝑡 = 𝑈. 𝑥̅ 𝐹 For the expanded uncertainty factor, the coverage factor of 2 enables approximately a 95 % confidence on the calculated interval. When the calculated variance estimator is negative, it is supposed to be null.

Results

Pairwise comparison of mean values

The pairwise comparison of mean values was used to compare the metal contents of the test samples and determine if significant differences exist between mean values that could reveal a bias during the "between laboratory samples" sampling. To express the results of this Tukey test, mean values are associated to a group: two values are not in the same group if their mean values are significantly different (see Table 3). For example, Cu, Ni and Co mean values are all in the same group, meaning that there is no significant difference between mean values for the 4 test samples.

𝑆𝐷 𝑇𝑃 = √𝑉𝑎𝑟 𝑇𝑃 𝑅𝑆𝐷 𝑇𝑃 = 𝑆𝐷 𝑇𝑃 𝑥̅

Estimation of uncertainty for measuring metals content in waste printed circuit boards

(n=8) 1 1-2 1 1 1 1 LS1-TS2(n=8) 1 1 1-2 2 1 1 LS2-TS1 (n=8) 1 1 2 2 1 1 LS2-TS2 (n=8) 1 2 1-2 3 1 1
For Fe and Zn, two different groups are observed, which means that there are some significant variations between mean values of some samples. However, the distribution between groups is not the same for both metals and some mean values are belonging to both groups. For Pb, there is a significant difference of mean values for all samples, except for LS1-TS2 and LS2-TS1 mean values. This distribution between groups is not linked to the laboratory sample from which they are originated, thus enabling to conclude that there is a sufficient degree of reproducibility between the laboratory samples (4 kg samples with particle size less than 10 mm). The variations observed between test samples are not caused by a bias during "between laboratory samples" sampling.

Variance calculations

Following the duplicate methodology, it was possible to partition the total variance, to determine the contribution of all the studied sampling steps to the total variance. The total variance, given in Table 4, increases with decreasing metal content: Cu total variance is 0.00281 while its mean content is around 15.4 %(w/w), while it reaches 0.0485 for Co, which is much less concentrated in WPCBs (mean content around 350 mg.kg -1 ). The contribution of each sampling step variance to the total variance is given in Figure 3. Three classes of metals can be distinguished, correlated to their contents in WPCBs.

Figure 3. Components of the variance of the logarithm of content for LS, TS and TP sampling stages contributions for the six metals.

For Cu and Fe, the uncertainty arising from "between laboratory samples" sampling is calculated as zero, meaning that it is negligible compared to other contributions. This may be due to their high content in WPCBs and that for each piece of board, thus enabling a good distribution between the 4 kg samples at 10 mm. On the contrary, the "between test portions" sampling mainly contributes to the total variance. Two main hypotheses can explain this result: (i) first, this variance both includes the variance from analysis and from the "between test portions" sampling (this hypothesis is valid for all metals). This hypothesis and the corresponding uncertainty analysis is investigated and presented in another paper of WCSB10 proceeding 5 . (ii) Secondly, Fe and Cu are mainly present as coarse particles, with very high content, which are not easy to grind and are thus largely distributed in the -800 + 400 µm size fraction of the samples [START_REF] Hubau | Conception d'un procédé de biolixiviation pour la valorisation des métaux contenus dans les déchets de cartes électroniques (Design of a bioleaching process for the recovery of metals contained in spent Printed Circuit Boards)[END_REF] . With these large particle sizes highly liberated, their distribution between the 5 g samples (test portions) may not be homogeneous, thus leading to the high contribution of the "between test portions" sampling step into the total variance.

For Zn and Pb, with mean contents reaching around 1.5 % (w/w), the contribution of the three sampling steps to the total variance are of the same magnitude. As previously observed with pairwise comparison of mean values, their distribution between the 4 kg sub-samples (laboratory samples) may already differ.

For Co and Ni, it appears that the "between test samples" sampling does not largely contribute to the total variance. The "between test portions" sampling is the main contributor of the variance, while there is also a contribution from the "between laboratory samples" sampling. Ni and Co are used as small batteries in WPCBs, which are supposed to be removed during the depollution of WPCBs (however not the case in this study). Ni may also be originated from stainless steel. In the -10 mm laboratory samples, there might be intact small batteries that are not equally distributed between samples, creating some variations due to nugget effect. That could cause the high values of "between laboratory samples" variances observed for Ni and Co compared to other metals. In the 5 g test portions, previous study [START_REF] Touzé | Sampling waste printed circuit boards: Achieving the right combination between particle size and sample mass to measure metal content[END_REF] showed that erratic points might be observed for Ni and Co, maybe also due to the "nugget" effect, i.e., the occurrence of one or several tiny single flakes of concentrated metal in one test portion (however not observed at 750 µm in their study). These erratic points, which were not observed for Zn, Cu and Pb, might explain the large variance arising from "between test portions" sampling for Ni and Co.

These results show that the uncertainty is mainly due to the last sampling step (TP), except for Pb. It is therefore necessary to work on this step to reduce significantly the total uncertainty. Besides working on analysis uncertainties, it is possible to increase the mass of TP and/or decrease its particle size.

As observed by other authors [START_REF] Lyn | The duplicate method of uncertainty estimation: are eight targets enough?[END_REF] , the use of less than eight duplicates may increase the uncertainty on variance estimates. In this study, it could be very useful to supplement the data with other laboratory and test samples characterization. It is nevertheless interesting to see that for most metals that are studied, the highest uncertainty contribution originates from the sampling step with the larger number of replicates.

Uncertainty factor and 95 % confidence interval limits From the total variance, the expanded uncertainty factor was calculated, then enabling to calculate the limits of the interval in which 95 % of the values of metal contents should be comprised. Results are given in Table 4. These intervals are not centred on the mean value as a log-normal distribution was used. As the objective of this study is to characterize the reproducibility of samples to use them for further developments of metal recovery processes, the knowledge of metal contents intervals helps to better characterize the initial content and thus the efficiency of the processes that are developed. The width of these intervals is limited for Cu, Fe and Zn. However, for Pb, Ni and Co, these intervals are quite large. Two options emerge. For valuable metals that are in the scope of the process development, mass balances based on fine characterization of the outputs of the developed process are required to obtain accurate efficiency yields of the process. For other elements that are not targeted in the process but that may affect its efficiency, the knowledge of these intervals may be sufficient.

Conclusion

This study gives an overview of the application of variance analysis to the sampling of WPCBs which are highly heterogeneous materials. From the pairwise comparison of the mean values, the calculation of variances and the determination of 95% confidence interval limits, this study assessed the quality of the WPCBs sampling plan that was used.

It appears that the "between laboratory samples" sampling (4 kg with particle size lower than 10 mm) partially contributes to the uncertainty. However, the distribution of metals in the two different laboratory samples that are studied was not highly affected by this sampling step. It would be of interest to increase the number of replicates at this step to confirm this result.

For most metals, the sampling of the test samples (40 g with particle size lower than 750 µm) does not highly contribute to the uncertainty on the metal content compared to the sampling of the test portions (5 g with particle size lower than 750 µm) as expected. The uncertainty calculated both arose from sampling and analysis. Based on a bootstrap strategy, previous work based on this last step evaluated the interest of replicating the test portions to reduce the uncertainty on metal content determination 4 . To do so, three test portion masses were used (0.5 g, 2 g, and 5 g) as well as three top particle sizes (200 µm, 750 µm and 2 mm). It concluded that for 5 g test portions, a 200 µm fineness drastically reduces the width of intervals in which metal content values are comprised, while a 750 µm fineness is a good compromise (when considering the time and energy consumption of the pulverizing step from 750 to 200 µm). That is consistent with the fundamentals of TOS: particle size, or diameter of the largest particles, must be reduced before reducing the sample mass if one do not want to increase the Fundamental Sampling Error (FSE). Nevertheless, the observed differences in variability cannot be only explained with this FSE approach [START_REF] Brochot | Experimental determination of the constitutional heterogeneity for sampling of Waste Printed Circuit Boards[END_REF] .

To go further, it could be interesting to consider trace elements that are valuable in WPCBs (critical metals such as Ga, Ta, Nd, La, Au, Pt, Pd). This will be in the scope of the MetroCycleEU project 8 funded by EMPIR EURAMET initiative, in which a similar sampling strategy will be developed and evaluated.

Figure 1 .

 1 Figure 1. Sampling plan used in this study.

Figure 2 .

 2 Figure 2. WPCBs samples (raw material, -30 mm, -10 mm, -2 mm and -750 µm, from left to right).

Table 1 . p-values obtained with normal and log-normal distributions of metal contents when considering the 32 test portions.

 1 Wilk tests were also performed on the residuals (difference between measured values and their estimates with the log-normal distribution), for which the hypothesis of normal distributions cannot be rejected, allowing to use ANOVA tool (Excel Stat 2019.3.2). Tukey tests of pairwise comparison were used to compare the mean values of each test sample. The significance level for Tukey test was fixed to 0.05.

	Distribution	Cu	Fe	Zn	Pb	Ni	Co
	Normal	0.657	0.067	0.884	0.200	0.489	0.204
	Log-normal	0.662	0.172	0.802	0.614	0.210	0.766
	Shapiro-Variance, standard deviation and uncertainty factor calculations

Table 2 . Formula used to calculate the sum of squares, degree of freedom and variance for TP, TS and LS sampling stages.

 2

	Formula	"Between test portions"		"Between test samples"	"Between laboratory sam-
		sampling			sampling					ples" sampling
	Mean value SS Sum of squares	𝑥 𝑖𝑗 ̅̅̅̅ = 𝑆𝑆 𝑇𝑃 = ∑ ∑ ∑(𝑥 𝑖𝑗𝑘 -𝑥 𝑖𝑗 1 𝑁 𝑇𝑃 ∑ 𝑥 𝑖𝑗𝑘 𝑁 𝑇𝑃 𝑘=1 ̅̅̅̅) 𝑁 𝐿𝑆 𝑁 𝑇𝑆 𝑁 𝐿𝑆	2	𝑥 𝑖 ̅ = 𝑆𝑆 𝑇𝑆 = ∑ ∑ ∑(𝑥 𝑖𝑗 1 𝑁 𝑇𝑆 ∑ 𝑥 𝑖𝑗 ̅̅̅̅ 𝑁 𝑇𝑆 𝑗=1 ̅̅̅̅ -𝑥 𝑖 ̅ ) 𝑁 𝐿𝑆 𝑁 𝑇𝑆 𝑁 𝐿𝑆	2	𝑥̅ = 𝑆𝑆 𝐿𝑆 = ∑ ∑ ∑(𝑥 𝑖 𝑁 𝐿𝑆 1 ∑ 𝑥 𝑖 ̅ 𝑁 𝐿𝑆 𝑖=1 ̅ -𝑥̅ ) 2 𝑁 𝐿𝑆 𝑁 𝑇𝑆 𝑁 𝐿𝑆
		𝑖=1	𝑗=1	𝑘=1		𝑖=1	𝑗=1	𝑘=1					𝑖=1	𝑗=1	𝑘=1
	Degree of free-	𝑑𝑓 𝑇𝑃 = 𝑁 𝐿𝑆 𝑁 𝑇𝑆 (𝑁 𝑇𝑃 -1)		𝑑𝑓 𝑇𝑆 = 𝑁 𝐿𝑆 (𝑁 𝑇𝑆 -1)			𝑑𝑓 𝐿𝑆 = 𝑁 𝐿𝑆 -1
	dom df
	Variance Var	𝑉𝑎𝑟 𝑇𝑃 =	𝑆𝑆 𝑇𝑃 𝑑𝑓 𝑇𝑃		𝑉𝑎𝑟 𝑇𝑆 =	1 𝑁 𝑇𝑃	( 𝑆𝑆 𝑇𝑆 𝑑𝑓 𝑇𝑆	-	𝑆𝑆 𝑇𝑃 𝑑𝑓 𝑇𝑃	)	𝑉𝑎𝑟 𝐿𝑆 =	1 𝑁 𝑇𝑃 𝑁 𝑇𝑆	( 𝑆𝑆 𝐿𝑆 𝑑𝑓 𝐿𝑆	-	𝑆𝑆 𝑇𝑆 𝑑𝑓 𝑇𝑆	)

Table 3 . Mean value of metal content and results of the pairwise comparison of mean values from test portions with Tukey test.

 3

	Metal	Cu	Fe	Zn	Pb	Ni	Co
	Mean value (n=32)	15.4 %	12.5 %	1.68 %	1.23 %	3800 mg.kg -1	350 mg.kg -1
	Test sample				Groups
	LS1-TS1

Table 4 . Mean content, relative standard deviation (RSD), total variance, total uncertainty factor and 95 % confi- dence interval limits for the six metals.

 4

			Metal content in %(w/w)		Metal content in mg.kg -1
		Cu	Fe	Zn	Pb	Ni	Co
	Mean value of metal content	15.4	12.5	1.68	1.23	3752	350
	RSD (%)	5.8	6.9	10.3	12.5	13.3	22.3
	Total variance*	2.81.10 -3	3.55.10 -3	1.22.10 -2	1.94.10 -2	1.75.10 -2	4.85.10 -2
	Total uncertainty factor	1.11	1.13	1.25	1.32	1.30	1.55
	95 % confidence interval limits	13.8 -17.1	11.1 -14.1	1.34 -2.08	0.93 -1.62	2850 -4850	220 -530

* The total variance is calculated with the natural logarithm of metal content values, as explained in the Material and Methods section.