J-B Vaney

A Piarristeguy

V Ohorodniichuck

O Ferry

A Pradel

E Alleno

J Monnier

E B Lopes

A P Gonçalves

G Delaizir

C Candolfi

A Dauscher

B Lenoir

Effective medium theory based modeling of the thermoelectric properties of composites: Comparison between predictions and experiments in the glass-crystal composite system Si10As15Te75-Bi0.4Sb1.6Te3

Keywords: A, Glass-crystal composites B, Thermoelectricity C, Effective Medium Theory

We report on the theoretical predictions of the Effective Medium Theory (EMT) and its generalized version taking into account percolation theory (GEMT) on the thermoelectric properties of composites based on Landauer and Sonntag's equations.

The results were tested experimentally on composites composed of the glassy phase Si10As15Te75 and the crystalline phase Bi0.4Sb1.7Te3. The evolution of the electrical resistivity and thermal conductivity with the fraction of crystalline phase matches very well the experimental data, although the GEMT model fails to predict the thermopower.

A better agreement between theory and experiment could be obtained by combining the principles of the GEMT and the Webman-Jortner-Cohen models. Despite the fact that the GEMT model originally predicts the possibility to optimize the dimensionless figure of merit ZT of composites by adjusting the fraction and the values of the transport properties of each phase, the new model developed rules out any beneficial influence on the ZT values. These results confirm within a different framework the early conclusions of Bergmann regarding the impossibility of improving the ZT values using multi-phased materials.

I. Introduction

In the current context of environmental concerns, energy harvesting has become a central focus in materials science research. Among all the technologies under scrutiny and development, thermoelectricity might play a role due to its numerous advantages such as the absence of gaseous emission, vibration-free character and versatility.

However, for decades, the performance and efficiency of thermoelectric (TE) materials has been limited, confining them to niche applications and markets. Since the early 1990's, new strategies aimed to increase the efficiency of TE materials mushroomed, leading to a major rejuvenation of interest in this field of research 1,2 . The thermoelectric performance of TE materials is quantitatively described by the dimensionless figure of merit defined as ZT = α 2 .T/ρ.λ, where α is the Seebeck coefficient, T is the absolute temperature, ρ is the electrical resistivity and λ is the total thermal conductivity (the sum of the electronic and lattice contributions, respectively λe and λL). Owing to the interdependence of these three transport properties, it appears extremely challenging to optimize one of them without impacting the others.

Among the various strategies explored, multi-phased materials proved to be a possible route towards high ZT values. Several studies were devoted to the investigation of materials that inherently show a microstructure at various length scales referred to as all-scale hierarchical structure in literature 3,4 . Even though the thermal stability of these phases still remains to be thoroughly investigated, very high ZT values were reported in several lead chalcogenide-based materials. For instance, Biswas et al.

reported a significant increase in ZT from 1.1 to 1.7 with the nano-precipitation of SrTe secondary phase inside an Na-doped PbTe matrix 3 . A similar strategy was applied to Na-doped PbSe compounds, with Ca-, Sr-or BaSe nano-precipitates, yielding a 30% increase 4 . Nanoparticles dispersion in a thermoelectric medium is considered as an interesting alternative to nano-precipitation and has been applied in conventional thermoelectric materials such as Bi2Te3-based alloys. In this regard, Li et al. reported on the effect of SiC nanoparticles dispersed in a Bi-Sb-Te matrix and observed an increase of 15% in ZT for as few as 0.4% of SiC particles 5 . All these optimizations arise from a significant decrease in the lattice thermal conductivity owing to numerous interfaces that act as efficient phonon diffusion centers. So far, multi-phased materials composed of two different phases (none of which is at the nanometer length-scale) have been only scarcely studied. Chalcogenides or oxide glass-ceramics are examples of such materials, which, however, exhibit weak thermoelectric performance (ZT < 0.2) 6,7 . Even though recent investigations have demonstrated the possibility to enhance the thermoelectric properties of the glassy phase through the admixture of a small fraction of crystalline phase, (REF) the corresponding pure crystalline phase still displays higher performance than the glassceramics. These results seem to conform to the long-standing idea that possible enhancement of the power factor in composites is always at the expense of the overall performance i.e. an increase in ZT values is physically impossible as argued by Bergman and Fel 8 . Yet, their model neither took precisely into account percolation mechanisms nor interface effects, both of which likely playing a role in determining the transport properties in composites.

Herein, we predict the thermoelectric properties of composite materials composed of two different phases using both the effective medium theory (EMT) and its generalized derivation (GEMT) that takes into account elements of the percolation theory. Under specific conditions on the transport properties of the two phases, these models predict that the ZT values of composites might be optimized for a given volume ratio. In order to critically assess the validity of these models, we fabricated several twophase composites with various volume ratios using the glassy Si10As15Te75 and the crystalline Bi0.4Sb1.6Te3 phases that fulfill the requirements of the model. The failure of these models to predict the evolution of the thermopower as a function of the crystalline fraction led us to introduce the Webman-Jortner-Cohen model, which yields a better agreement between theory and experiment.

The paper is organized in the following way. Experimental details concerning the fabrication and the measurements of the thermoelectric properties of the composites are described in the first paragraph. The Effective Medium Theory (EMT) is then introduced together with its generalized counterpart (GEMT), providing a theoretical basis to choose the nature of the composite phases. The last part deals with the comparison of the model predictions and the experimental data.

II. Experimental details

The raw materials (Si10As15Te75 glass and Bi0.4Sb1.6Te3 crystalline phase) were prepared by two different synthesis routes. For glassy Si10As15Te75, a total weight of 4 g of stoichiometric quantities of pure starting elements (Si 99.999% form Advent RM, As 99.99% from Goodfellow, Te 99.999% from 5NPlus) was placed and sealed in evacuated quartz ampules (6mm inner diameter) under secondary vacuum (10 -5 mbar). The ampules were then heated to 1023K at a heating rate of 9 K.h -1 and kept at this temperature for one hour. The tubes were regularly rocked in the furnace to ensure a good chemical homogenization. The tubes were finally quenched in a salt-ice-water mixture. The same procedure was rigorously followed for each sample to guarantee good reproducibility. For Bi0.4Sb1.6Te3, pure elements (Bi 99.999% from CERAC, Sb 99.999% from 5NPlus, Te 99.999% from 5NPlus) were placed and sealed in quartz ampules (14 mm inner diameter) under secondary vacuum (10 -5 mbar), which were then heated to 953 K and kept at this temperature for 6 hours in an oscillating furnace. The ampules were quenched in room-temperature water to obtain the crystalline phase. XRD patterns of both phases are shown in Figure 1. The ingots were subsequently ground into powders and finally consolidated into cylindrical pellets by SPS to ensure a better homogeneity in terms of crystal orientation.

Composites were fabricated for nine different volume ratios (crystalline/vitreous) from 10-90% up to 50-50% with an increasing step of 5%. Both materials were ground separately in agate mortars and sieved at 25 μm. For each sample, about 0.6g of mixture of the two powders was then prepared abiding by the calculated volume ratios. The powder densification was performed by Spark Plasma Sintering (SPS), allowing for fast sintering in order to prevent the glass from crystallizing. The powders were densified at 408K for 5 minutes under a pressure of 80 MPa. The evolution of the density of the composites according to the volume fraction of Bi0.4Sb1.6Te3 is discussed in the electronic supplementary information (ESI).

Powder X-ray diffraction (PXRD) was performed at 300 K using a Bruker D8 Advance instrument with CuKα1 radiation. Selected composite samples were scanned by SEM (using a Philips XL-30 SEM system) in back-scattering electron (BSE) mode (under 25kV acceleration voltage) to determine their microstructure and the chemical homogeneity of both phases. Electron probe microanalysis (EPMA) was also performed to verify the chemical composition of the different phases of the composites. These measurements were performed using a CAMECA SX-100 instrument with an acceleration voltage of 20 kV and probe current of 10 nA. Selected samples were used for X-Ray tomography using a Phoenix X-Ray Nanotom S system that enables obtaining a three-dimensional map of the distribution of the two phases within the volume of the sample. Data acquisition was realized at 90kV acceleration voltage with a current of 90µA for 153min (6 x 750 ms acquisitions + 2 x 750 ms extinctions, for 1440 angular positions with steps of 0.25° and with an approximate resolution of 2µm). The spatial reconstruction was performed with the open software ImageJ.

The three relevant transport properties were measured separately for each composition. Disk and bar-shaped samples were cut from the consolidated ingots with a diamond-wire saw. The thermal conductivity λ was obtained by measuring the thermal diffusivity a and the specific heat Cp of the composites. λ was then calculated following the formula λ=a.Cp.ρv where ρv stands for the density, calculated from the weight and geometrical dimensions of the samples and considered temperature-independent.

Thermal diffusivity was measured between 300 and 375 K on disk-shaped samples (10 mm diameter and 1 mm thickness) using the laser flash technique, with a LFA 427 system from Netzsch. Cp was measured under Ar flow using a DSC 404 F3 Pegasus system from Netzsch according to the norm ASTM e967-11.

The electrical resistivity ρ and the Seebeck coefficient α were measured simultaneously from 300 up to 375 K with a ZEM-3 (ULVAC-RIKO) system on barshaped samples of typical dimensions of 1×1.5×8 mm 3 .

III. Effective Medium Theory (EMT) and Generalized EMT for the thermoelectric properties 1). EMT and GEMT for the electrical and thermal conductivities

The effective medium theory (EMT) consists in a set of approximations that allow calculating the macroscopic properties of an inhomogeneous medium, based solely on the properties of the phases constituting the medium. For the interested reader, the theoretical and historical basis of the EMT (as well as those of the GEMT) are described in the ESI (part B).

For the electrical conductivity σ and the thermal conductivity λ, the EMT theory leads to the following formula in the presence of two phases 9,10 where φ1 and φ2 are the volume fractions of the phases 1 and 2. The subscripts 1, 2 and e correspond to the phase 1, 2 and the effective medium, respectively.

The general effective medium theory formalism (based on a phenomenological model with the addition of elements of the percolation theory to the EMT 11,12 ) can be directly applied to the electrical and thermal conductivities, yielding the equations where all the constants are the same as described in Eqs. (2), and (3). In Eqs. ( 4) and ( 5), A is a constant that depends on the actual percolation threshold φc (of the phase 2 in the phase 1) through the equation A = (1-φc)/φc, and t is a constant representing the asymmetry of the microstructure (in terms of connection between the grains).

𝜑 ! 𝜎 ! -𝜎 " 𝜎 ! + 2. 𝜎 " + 𝜑 # 𝜎 # -𝜎 " 𝜎 # + 2. 𝜎 " = 0 (2)
𝜑 ! 𝜆 ! -𝜆 " 𝜆 ! + 2. 𝜆 " + 𝜑 # 𝜆 # -𝜆 " 𝜆 # + 2. 𝜆 " = 0 (3)
𝜑 ! 𝜎 ! !/% -𝜎 " !/% 𝜎 ! !/% + 𝐴. 𝜎 " !/% + 𝜑 # 𝜎 # !/% -𝜎 " !/% 𝜎 # !/% + 𝐴. 𝜎 " !/% = 0 (4)
𝜑 ! 𝜆 ! !/% -𝜆 " !/% 𝜆 ! !/% + 𝐴. 𝜆 " !/% + 𝜑 # 𝜆 # !/% -𝜆 " !/% 𝜒 # !/% + 𝐴. 𝜆 " !/% = 0 (5)

2). EMT and GEMT for the thermopower

The thermopower appears as the last property to be described in multi-phased media to enable a direct estimation of the ZT values. In 2005 and 2006, Sonntag derived a formula valid for metals or degenerate semiconductors by applying the EMT formalism to the heat flux and chemical potential 13,14 . Few years later, replacing the heat flux by the entropy flux, a more general equation was derived, thereby extending the previous model to non-degenerate semiconductors 15 . In this model, the quantity λ/α is used as χ and is introduced in Eq.( 1) resulting in the following expression where α1, α2 and αe are the thermopowers of the phases 1 and 2 and of the effective medium, respectively. This equation requires the knowledge of λ for a set of different volume fractions prior to calculating α. We then applied the GEMT formalism in the same fashion as the two other properties by applying the equation

𝜑 ! 𝜆 ! 𝛼 ! - 𝜆 " 𝛼 " 𝜆 ! 𝛼 ! + 2. 𝜆 " 𝛼 " + 𝜑 # 𝜆 # 𝛼 # - 𝜆 " 𝛼 " 𝜆 # 𝛼 # + 2. 𝜆 " 𝛼 " = 0 (6) 𝜑 ! . 𝜆 ! 𝛼 ! / !/% -0 𝜆 " 𝛼 " 1 !/% . 𝜆 ! 𝛼 ! / !/% + 𝐴. 0 𝜆 " 𝛼 " 1 !/% + 𝜑 # . 𝜆 # 𝛼 # / !/% -0 𝜆 " 𝛼 " 1 !/% . 𝜆 # 𝛼 # / !/% + 𝐴. 0 𝜆 " 𝛼 " 1 !/% = 0 (7)
where A and t are, respectively, the same percolation-depending and asymmetry constant as in Eq . ( 4).

3). Consequences on the figure of merit ZT

Using Eqs. ( 2) to ( 7), a theoretical estimation of ZTe of a two-phased composite can be obtained, for a given volume fraction provided the transport properties of each phases are known. Based on this set of equations, several combinations of state-of-theart thermoelectric materials can be tested to determine whether the ZT values can be further optimized within this approach. By dividing the equation for a general property χ (Eq.1) by either χ1 or χ2, the function χe/χ2 (which is a function of the volume fraction of the second phase φ2) only depends on the ratio χ1/χ2. In other words, the EMT equations can be reduced to the properties of one of the two phases This procedure can be extended to the three transport properties involved in the ZT calculation and to the GEMT equations. The shape of the ZTe/ZT2 curve then only depends on the three ratios σ1/σ2, λ1/λ2, and α1/α2 as well as on the two GEMT parameters t and φc. Using these reduced equations, we calculated the evolution of ZT as a function of each ratio to determine their influence. is worthwhile to note that in order to obtain an increase in ZT of only 50%, a thermopower ratio over 8 is needed. As a consequence, assuming that the phase 2

𝜑 ! 𝜒 ! 𝜒 # - 𝜒 " 𝜒 # 𝜒 ! 𝜒 # + 2. 𝜒 " 𝜒 # + 𝜑 # 1 - 𝜒 " 𝜒 # 1 + 2. 𝜒 " 𝜒 # = 0 (8)
shows thermopower values around 100 µV/K, such ratio would then require a second phase exhibiting thermopower values of the order of 1000 µV/K, a value rarely achieved among known materials. Ratios lower than 2 have no effect on the maximum ZT.

Furthermore, the higher the thermopower ratio, the lower the required crystalline fraction. This piece of information is particularly relevant regarding synthesis issues.

As for the thermal conductivity, no clear trend emerges from these calculations.

This ratio affects mainly the position of the maximum, converging to 33% as the ratio decreases. Once more, in a very counterintuitive way, the presence of one phase with a very low thermal conductivity does not affect the final ZT but rather the volume fraction φ2 at which the maximum occurs.

The fact that a maximum ZT higher than ZT1 and ZT2 is achieved may be rather surprising at first glance since this theory should normally yield results of effective properties that are bounded by the properties of the original phases. However, the difference in the evolution of ρ, λ and α as a function of the crystalline fraction gives rise to enhanced ZT values.

Taken as a whole, these results provide interesting insights into the transport properties the two phases should possess in order to possibly achieve higher ZT values.

In the frame of this model, one phase should display large thermopower values together with low thermal conductivity while the second phase should show the opposite properties, that is, low thermopower and high thermal conductivity. In spite of the fact that the electrical resistivity of the first phase is expected to have virtually no influence, the electrical resistivity of the second phase should be low enough to reach high ZT2

values. The former combination can be obtained by considering chalcogenide glasses, which exhibit extremely low thermal conductivity (< 0.3 W m -1 K -1 ) due to the inherent structural disorder, and high thermopower values (>> 500 µV K -1 ) [START_REF] Mott | Electronic processes in non-crystalline materials[END_REF] .

IV. Experimental test of the models with glass-crystal composites

Because amorphous systems are usually thermally stable in a limited temperature range above room temperature, we prepared composite samples using a glassy Si-As-Te phase with crystalline Bi-Sb-Te compounds that show excellent thermoelectric properties near room temperature. This type of glassy system exhibits a very low vitreous transition temperature Tg. The chosen composition Si10As15Te75 shows a Tg at

383K (see ESI, Figure D).

A low-Tg glass presents the main advantage of facilitating the fabrication process.

Above this characteristic temperature, the viscosity of the glass drastically decreases while maintaining its solid state. Hence, under pressure, it can then be severely strained.

Mixing powders of the two phases, with heating and pressure, allows producing a composite of crystalline particles (phase 2) embedded in a glassy matrix (phase 1).

The composition of the crystalline phase was chosen in order to obtain a high contrast between both thermopower and thermal conductivity ratios. Moreover, the two phases are p-type, as this type is more easily obtained in chalcogenide glasses [START_REF] Kolobov | [END_REF] . The properties of the two phases at 300K as well as the corresponding ratios are summarized in Table 1. 1), the composites may be expected to show some degree of anisotropy in their transport properties which might be a function of the crystalline fraction. For this reason, we carried out the calculations in both directions taking into account the ρ, α and λ values associated to each direction.

1). Characterization of the BiSbTe-SiAsTe composites

SEM analysis describes a very poly-disperse and homogeneous distribution of the crystalline phase (light-grey phase) in all composites. Crystalline grains of few microns "connecting" larger grains could be observed. Although the crystalline powder was sieved at 25 µm, larger features are also present, emphasizing agglomeration of that phase. An example of a set of SEM pictures for one of the composites (60-40%) is shown in the ESI (Figure F1). In addition, EPMA measurements (ESI, Figure F2) revealed a slight migration of Sb (around 1 at%) from the crystalline phase to the glassy phase.

In the classical EMT (ESI, part B), the percolation threshold is implicitly predicted to be around 33%. However, this is not to say that this value is indeed achieved in the present series of samples. To shed light on the possible actual percolation threshold, we performed X-ray tomography to determine the crystalline phase distribution in the sample's volume. Of note, it should be kept in mind that the spatial resolution of this technique is around 2 microns, while features at the micrometer and sub-micrometer scales are clearly observed (see Figure F1 in ESI). Hence, this method only allows setting an upper limit to the percolation threshold in the present case.

Figure 3 shows the reconstructed 3D maps of the crystalline phase in the composites with crystalline fractions of 10, 20 and 30%. The left-hand image (10%)

clearly shows the absence of connection between the grains, while on the right-hand image (30%), clear connections between the grains are evidenced. In the last image (20%), the situation seems intermediate with connections between some grains and unconnected areas. Even though an exact determination of the percolation threshold is somewhat difficult, the actual percolation threshold in this series of composites is estimated to be below 20%. This estimation is consistent with the percolation theory which predicts a value between 15 and 17 % [START_REF] Aharony | Introduction To Percolation Theory[END_REF] . With such a constraint on φc, Eqs.

(4),( 5) and ( 7) of the GEMT can be used to fit the experimental data with t and φc as free parameters, at every measured temperature in the range 300 -375K.

2). Comparison between the predictions of the GEMT model and the measured properties

Figure 4 presents the results of the fitting procedure for both the electrical resistivity and thermal conductivity at 300 and 375 K. The best fits to the data were obtained for t = 2 and φc = 16 % (a percolation threshold close to the 16.6 % predicted by the percolation theory) and are displayed in Figure 4A. From early fitting tests to the electrical resistivity, it was noticed that the experimental trend in ρ at the asymptotic limit (i.e. 100% crystalline fraction) was not in agreement with the value of 7μΩ.m measured for Bi0.4Sb1.6Te3. A possible origin of this discrepancy might be related to variations in the chemical composition of the crystalline phase. Chemical analyses indicated an enrichment of the glassy phase in Sb (normally not affecting its properties considering its very low concentration [START_REF] Mott | Electronic processes in non-crystalline materials[END_REF] ), being indirectly a sign of Sb depletion in the original crystalline phase. As pointed out by Scherrer et al. [START_REF] Rowe | Thermoelectrics handbook: macro to nano[END_REF] , the evolution of the Bi/Sb ratio could be responsible for an increase in the electrical resistivity of that magnitude. A modified electrical resistivity for the crystalline phase (25 µΩ.m instead of 7 μΩ.m in the perpendicular direction, obtained from the asymptotic limit at pure crystalline phase in the experimental data) was subsequently used in the calculations. As can be observed in Figure 4, the fitted curve matches very well the experimental data. For the thermal conductivity, the measured λ values of Bi0.4Sb1.6Te3 were decreased accordingly using the Wiedemann-Franz law. However, the resulting fit still overestimates the measured data. As phonon diffusion at interfaces is not taken into account in EMT-type models and considering the microstructure of the composites, the reduction in the thermal conductivity by these mechanisms could result in such a decrease [START_REF] Bhandari | [END_REF] .

Regarding the prediction of the thermopower, the results obtained for the same set of t and φc parameters are shown in Figure 5. With these t and φc factors, the agreement between the measured and calculated data is far from being satisfactory. Even though the drop in α at low crystalline fractions is predicted, its magnitude is not well captured.

Noteworthy, varying the fitting factors did not improve the quality of the fit.

3). Discussion on the prediction failure by EMT for the thermopower

Although the GEMT yielded some reliable results for both the electrical resistivity and thermal conductivity, this model clearly fails to predict the thermopower based on Sonntag's equation. One may argue that the formalism applied to ρ and λ in the GEMT may not be fully valid for λ/α. However, this discrepancy is likely more deeply rooted in the approximations of the model itself. Interfaces and/or microstructure might play an important role and are not taken into account in Sonntag's model. As for the derivation of the equations themselves, nothing can really be argued, not to mention their experimental validation on the conducting composite systems 14,15 . However, having in mind the simple short circuit concept, it seems rather unphysical that electrical conductivity should not play any direct role in the equation of the thermopower for a composite system, especially with a combination of conducting and insulating components. Among the thermopower models we subsequently compared to these results, the Webman Jortner and Cohen model 21 seemed the most adequate with a very different equation derivation (not based on Eq.1 contrarily to Sonntag's model), while remaining in the frame of the EMT.

4). Webman-Jortner-Cohen (WJC) model for the thermopower

Webman, Jortner and Cohen derived a different equation for the thermopower by applying the EMT to Onsager equations. They assumed that corrections for both σe and λe are small due to thermoelectric effects 21,22 and wrote αe as follows:

where Δ & = (2𝜎 " + 𝜎)(2𝜅 " + 𝜅). If the system is limited to two components, the development, in a similar fashion as the above-mentioned equations of the EMT, yields where all the constants are as in part III. The limiting cases of this equation are well described by Snarskii et al. in Ref. 22. In Figure 6, we present four cases for large and small electrical conductivity ratios, with thermopower and thermal conductivity ratios similar to those of our composites. These results are compared to those obtained with the model developed by Sonntag.

Considering the comparison presented in Figure 5, the WJC model seems more consistent with our experimental results than the Sonntag's model i.e. in the case of very high ρ1/ρ2 ratio. Both models agree only when this ratio is low. Compared to the experimental thermopower values obtained for our composites presented in Figure 5, the WJC model reproduces well the steep decrease followed by a plateau. However, the curve shape predicted by Eq.( 10) suffers from the same drawbacks as the original EMT theory: the percolation threshold is irrevocably pinned at 33% while it can considerably

𝛼 " = 〈𝛼𝜎/Δ & 〉 〈𝜎/Δ & 〉 (9)
𝜑 ! 𝜎 ! (𝛼 " -𝛼 ! ) (2𝜎 " + 𝜎 ! )(2𝜆 " + 𝜆 ! ) + 𝜑 # 𝜎 # (𝛼 " -𝛼 # ) (2𝜎 " + 𝜎 # )(2𝜆 " + 𝜆 # ) = 0 (10)
differ in real materials. Hence, we combined the WJC model within the framework of the GEMT by introducing the factors t and A in Eq.10, leading to the following equation:

Figure 7A presents the results obtained with Eq.11, using the t factor and percolation threshold φc previously obtained by fitting the electrical resistivity and thermal conductivity (i.e. t = 2 and φc = 16%). As can be seen, this modified equation accounts well for the variations in the thermopower and the ZT values as a function of the crystalline fraction (Figs. 7A and7B).

The better match between experiment and theory obtained with the WJC model emphasizes that both electrical and thermal conductivity should be taken into account when modeling the Seebeck coefficient in composites. Yet, does this model leave room for enhanced ZT in composites? To answer this question, we tested numerically several combinations with transport properties and t and φc parameters spanning the range of physically meaningful values for thermoelectric composites (ρ1/ρ2 ranging from 0.1 to 10 4 , α1/α2 from 0.1 to 20, λ1/λ2 from 0.01 to 10, t from 1 to 3 and φc from 10 to 50%).

None of the combination tested led to an optimized ZT value. Thus, it can be safely concluded that within the frame of the GEMT equations, the association of two or more compounds with different thermoelectric properties cannot lead to enhanced ZT values. This is particularly clear when the electrical resistivity ratio is of several orders of

𝜑 ! 𝜎 ! !/% . (𝛼 " !/% -𝛼 ! !/% ) (𝐴. 𝜎 " !/% + 𝜎 ! !/% )(𝐴. 𝜆 " !/% + 𝜆 ! !/% ) + 𝜑 # 𝜎 # !/% . (𝛼 " !/% -𝛼 # !/% ) (𝐴. 𝜎 " !/% + 𝜎 # !/% )(𝐴. 𝜆 " !/% + 𝜆 # !/% ) = 0 (11)
magnitude. This result follows the observations of Bergman 8 , although this conclusion was reached in the present case using a very different theoretical framework. However, it should be underlined that these theoretical models use some simplifications. More specifically, interface effects such as electron filtering (able to increase the thermopower in specific cases [START_REF] Rowe | Thermoelectrics handbook: macro to nano[END_REF]23 ), or the influence of thermoelectric effects on both the electrical resistivity and thermal conductivity (that may trigger peculiar phenomena close to the percolation threshold 22 ) are not taken into account.

V. Conclusion

We reported on the modeling of the thermoelectric properties of two-phase composites using various models based on the effective medium theory. The predictions of these models were compared to experimental results obtained on crystals of Bi0.4Sb1.6Te3 embedded in a Si10As15Te75 glassy matrix with various fractions. While the generalized version of the EMT predicts a maximum in ZT at a particular crystalline fraction, our results did not show such an increase. A modified version of the GEMT that includes the WJC equation was found to provide a better agreement between theory and experiment. This model enables predicting the transport properties of composite materials with two phases exhibiting high electrical and thermal conductivity contrasts.

In addition, this model demonstrates that this approach yields no significant increase in ZT values, which agrees with the work of Bergman despite the different theoretical frameworks used. However, considering the huge interface effects on the thermal and electronic properties, and the behavior near percolation threshold, it cannot be ruled out that increased ZT values might be achieved in two-or more components compounds. Crystalline-crystalline composite systems, for which Sonntag's equation still apply and for which electron filtering may occur, would be worth investigating to further test this model. Finally, the presented set of equations could be used to predict the limit at which secondary phases, frequently appearing in the synthesis process of thermoelectric materials, deteriorate significantly their transport properties.

Figure 2

 2 Figure 2 displays the evolution of the ZT of a composite as a function of the volume

Figure 1 .

 1 Figure 1. XRD patterns of glassy Si10As15Te75 and crystalline Bi0.4Sb1.6Te3 phases.

Figure 2 .

 2 Figure 2. Evolution of ZT with the crystalline fraction of phase 2 embedded in a matrix

Figure 3 .

 3 Figure 3. X-ray tomography reconstruction of a sample volume (typically 4x0.5x0.5

Figure 4 .

 4 Figure 4. Electrical resistivity (A) and thermal conductivity (B) of the BiSbTe-SiAsTe

Figure 5 .

 5 Figure 5. Thermopower of the BiSbTe-SiAsTe composites measured at 300 and 375 K.

Figure 6 .

 6 Figure 6. Predicted evolution of the thermopower with the crystalline fraction for α1/α2

Figure 7 .

 7 Figure 7. (A) Comparison between the results of the WJC model for the thermopower

Figure 1 25Figure 2 26Figure 3 Figure 5 29Figure 6 30Figure 7

 123567 Figure 1

Table 1 :

 1 Properties of the two phases (crystalline Bi0.4Sb1.6Te3 and glassy Si10As15Te75) at room temperature, and ratio of the two properties as expressed in Section III. Properties of the Bi0.4Sb1.6Te3 phase are described in both directions i.e. perpendicular and parallel to the pressing direction in SPS.

		Bi0.4Sb1.6Te3	Bi0.4Sb1.6Te3
	Phase type	(measurement parallel to the pressing	(measurement perpendicular to the pressing	Si10As15Te75	Ratio (vitreous/crystalline)
		direction)	direction)
	Electrical
	resistivity	10.1	7.3	3.39 10 8	4.6 10 7
	(µΩ.m)
	Seebeck
	coefficient	+166	+ 164	+ 1400	8.5
	(µV/K)
	Thermal
	conductivity	1.1	1.4	0.17	0.12
	(W/m.K)
	Due to the well-known intrinsic anisotropy of Bi2Te3 compounds 18 clearly evidenced in
	Bi0.4Sb1.6Te3 (Table

Acknowledgements

The authors would like to thank financial support from the French National Agency (ANR) in the frame of its program "PROGELEC" (Verre Thermo-Générateur "VTG"). We acknowledge C. Merlet and M.Boyer for their help in EPMA experiments, M. Bigot for help in glass synthesis and J. Couve for his help in the DSC measurements.


J-B Vaney

J Carreaud

G Delaizir

C Morin

J Monnier

E Alleno

A Piarristeguy

A Pradel

A P Gonçalves

E B Lopes

C Candolfi

A Dauscher

B Lenoir

Low temperature transport properties of Bi substituted β-As2Te3 compounds

β-As2Te3 belongs to the family of Bi2Te3-based alloys, a well-known class of efficient thermoelectric materials around room temperature. As2Te3 exists in two allotropic configurations: α-and β-As2Te3, of which only the latter crystallizes in the same rhombohedral structure as Bi2Te3. Herein, we report on the substitution of Bi for As in the system As2-xBixTe3 with x = 0.0, 0.015, 0.025 and 0.035. These samples have been characterized by X-ray diffraction and scanning electron microscopy. The transport properties have been measured at low temperatures (5 -300 K) in both directions, parallel and perpendicular to the pressing direction. The results are compared to those obtained in a previous study on samples substituted by Sn. Compared to Sn, Bi allows for a clear decrease in electrical resistivity while maintaining the thermal conductivity below 1 W/m.K over the whole temperature range. As a result, a comparable peak ZT value near 0.2 was obtained at room temperature

Introduction

In the current context of environmental concerns, energy harvesting has become a central focus in materials science research. Among all the technologies under scrutiny and development, thermoelectricity might play a role due to its numerous advantages such as the absence of gaseous emission, vibration-free character and versatility. However, for decades, the performance and efficiency of thermoelectric (TE) materials has been limited, confining them to niche applications and markets. Near room temperature, Peltier coolers are used as solid-state heat pumps for electronic device cooling for example [START_REF] Rowe | Thermoelectrics handbook: macro to nano[END_REF].

The thermoelectric performance of TE materials is quantitatively described by the dimensionless figure of merit ZT defined as ZT = α 2 .T/ρ.λ, where α is the Seebeck coefficient or thermopower, T is the absolute temperature, ρ is the electrical resistivity and λ is the total thermal conductivity. Achieving high ZT on a broad temperature range still represents one of the most challenging aspect in thermoelectricity [START_REF] Goldsmid | Thermoelectric refrigeration[END_REF][START_REF] Snyder | [END_REF].

The ternary or quaternary compounds based on A2 V B3 VI (A = Bi or Sb and B = Te or Se) compounds remain the most efficient thermoelectric materials to date, for applications close to room-temperature. When properly optimized, both p-type and n-type compounds show ZT values close to unity near 300K.

β-As2Te3 is one of the two allotropic forms of As2Te3 (together with α-As2Te3, monoclinic, space group C2/m) and, as Bi2Te3, belongs to the family of compounds crystallizing in the rhombohedral space group 𝑅3 # 𝑚. β-As2Te3 has been reported to be a high-pressure phase, with a transition from α-to β-As2Te3 near 7

GPa [START_REF] Yakushev | DOKLADY AKAD NAUK SSSR[END_REF]. Further studies unveiled that quenching could be an efficient method to produce these phases at ambient pressure [START_REF] Shu | [END_REF][6] although they were not extensively studied from the thermoelectric point of view so far. Recently, we demonstrated that β-As2Te3 could be stabilized under ambient conditions and that the carrier concentration could be tuned by alloying with Sn, leading to enhanced thermoelectric performance (ZT ~ 0.6 at 423K) [7].

In this article, we report on the effect of the substitution of Bi for As on the thermoelectric properties of β-As2Te3 at low temperatures (5 -300 K). We evidence that Bi is another element that might lead to interesting thermoelectric properties in p-type β-As2-xBixTe3 (up to x = 0.035) above room-temperature.

Experimental

All β-As2-xBixTe3 (x = 0.0, 0.015, 0.025, 0.035) samples were prepared by direct reaction of stoichiometric amounts of pure elements (As (Goodfellow, 99.99%, shots), Te (5N+, 99.999%, shots) and Bi (Strem Chemicals, 99.9999%, shots)) sealed under secondary vacuum in quartz tubes. The tubes were heated slowly up to 923 K at a rate of 10 K min -1 , dwelt at this temperature for 2 hours and finally quenched in icy water. The resulting ingots were hand-ground into micron-sized powders and cold-pressed into 10 mm diameter cylindrical pellets under a pressure of 750 MPa. Remarkably, all the obtained ingots showed an experimental density higher than 95% of the theoretical density situation reminiscent to what is also observed in (Bi,Sb)2(Se,Te)3 compounds [START_REF] Rowe | Thermoelectrics handbook: macro to nano[END_REF].

Noteworthy, for this phase, sintering techniques such as hot-pressing or spark plasma sintering could not be used as the required temperatures trigger the phase transition of β-to α-As2Te3 that sets in at X K.

The crystal structure was determined by powder X-ray diffraction (PXRD) at 300 K with a Bruker D8 Advance instrument using monochromated CuKα1 radiation. The chemical homogeneity of the samples was checked by scanning electron microscopy (SEM) and energy dispersive x-ray spectroscopy (EDXS) using a Quanta FEG (FEI).

The thermoelectric properties were measured simultaneously between 5 and 300 K on parallelepipedshaped samples cut with a diamond wire-saw (typical dimensions of 2×3×7 mm 3 ) using the thermal transport option of a physical property measurement system (PPMS, Quantum Design). Fine Cu wires were brazed to the samples ensuring good electrical and thermal contacts, in a four collinear contacts set-up. Owing to the anisotropic crystal structure of the studied compounds, the transport properties were measured on samples cut both parallel and perpendicular to the pressing direction. The Hall coefficient was measured at 300 K on the perpendicular samples (same as used for TTO option) with the ac transport option of the PPMS. The magnetic field was swept between -0.5 and +0.5 T. The hole concentration and Hall mobility were estimated from and .

Results and discussion

Figure 1 presents the PXRD patterns (Fig. 1-A) obtained for the illustrative x = 0.025 sample (all diffraction patterns being very similar) together with the evolution of the lattice parameter (Fig. 1-B) and a sketch of the crystal structure of this compound (Fig. 1-C). All reflections can be indexed in the 𝑅3 # 𝑚 space group with an additional peak ascribed to non quantifiable AsTe impurities (𝐹𝑚3 # 𝑚, a = 5.778 Å) existing for all compositions. Compared to the effects of As/Sn substitution where a strong increase in both the electrical resistivity and the thermopower is observed up to an Sn content of 0.035, these two properties hardly evolve upon isoelectronic substitution of Bi for As at 300K.

Table 1 presents the Hall carrier concentration (p) and Hall mobility (μH) measured at 300 K for x = 0, 0.015 and 0.025 and 0.035. Only a very subtle evolution can be observed between the three Bi-substituted samples while the value of p for the non-substituted compound lies at slightly higher values, below 10 20 cm -3 . Mobility values at 300K firstly increase from the initial value of ~ 49 cm 2 .V -1 .s -1 for the nominal compound to the very high value of ~ 81 for x = 0.015. For upper x, the electronic mobility constantly decreases down to ~ 58 for x = 0.035.

Figure 4-A shows the temperature dependence of the total thermal conductivity λ. The anisotropic behavior is similar to that displayed by the electrical resistivity data, even though being opposite, with a thermal conductivity in the parallel direction considerably lower than in the perpendicular direction. Compared to the As2-xSnxTe3 compounds, the decrease in the λ values due to Bi substitution is less marked. Given the monotonic increase in electrical resistivity upon Sn substitution in contrast to the moderate evolution observed here, the electronic contribution to λ is considerably lowered in the former with respect to the latter, resulting in a stronger decrease in λ when substituting Sn for As. The values range between 0.7 and 1 W.m -1 .K -1 in the parallel direction while decreasing around 0.5 W.m -1 .K -1 for the Sn-substituted compounds at 300 K [7]. Figure 4-B shows the lattice thermal conductivity as a function of the temperature calculated by subtracting the electronic contribution estimated by the Wiedemann-Franz law (𝜆 ! = 𝑇𝐿/𝜌, where L = 2.44x10 -8 W W K -2 is the Lorenz number). The observed lattice thermal conductivities range from 0.8 W/(m K) to slightly above 0.4 W/(m K) at 300 K, bearing in mind that due to the experimental setup, room-temperature thermal conductivity is always slightly overestimated (due to difficulties in estimating radiation losses in a direct measurement).

A remarkable feature rarely observed in the Bi2Te3 family of compounds is the absence of a dielectric peak at low temperatures, mimicking the temperature dependence of a glass [8]. The structural transition βà β' is also evidenced in the thermal conductivity curves by a sudden decrease near 175 K owing to enhanced disorder in the β' structure compared with the β structure.

Figure 4-C depicts the temperature evolution of ZT in this series of compounds. The β à β' structural transition is perceivable around 175 K for all compositions. Owing to compensation effects between the electrical and thermal transport properties, the ZT values appear to vary negligibly with x within experimental uncertainty. A peak ZT value of 0.22 at 300 K is obtained concomitantly for x = 0.015, 0.025 and 0.035. Yet, the anisotropy of the transport properties is however not compensated in ZT giving rise to higher values by 25 % in the perpendicular direction with respect to the parallel direction. Interestingly, all the Bi-substituted compounds show ZT values very similar to those obtained in the Sn-doped compounds [7], suggesting that similar thermoelectric performances may emerge at higher temperatures (i.e. ZT > 0.6).

Conclusion

We reported on the low-temperature thermoelectric properties of polycrystalline samples of β-As2-xBixTe3 for x = 0.0, 0.015, 0.025 and 0.035. Similar ZT values of 0.22 were obtained at 300 K for all the Bi-substituted samples.

Remarkably, these values are similar to those unveiled in the Sn-doped compounds suggesting that high ZT values could be achieved above 400 K. Concomitantly, As substitution in the Bi2Te3 system remains so far scarcely studied and should be evaluated in relation to the present results on Bi substituted β-As2Te3 phases. In the same manner as in the Bi2Te3 system, the influence of several substitutions and the thermoelectric performances of other solid solutions based on β-As2Te3 are still to be investigated to fully assess its potential as a thermoelectric material for room-or mid-range temperature applications.

Figure and table captions

Figure 2 .

 2 Figure 2.

Fig 1 :

 1 Fig 1: A) Powder X-Ray diffraction pattern of β-As1.975Bi0.025Te3. The observed data are marked as open red circles. The calculated profile is represented by a solid black line while the difference between experimental and numerical patterns is shown by the solid blue line (RF = 4.024, χ 2 = 8.04). The positions of the Bragg reflections are indicated as green bars. The peak ascribed to AsTe impurity is marked by a black open square. B) Evolution of the hexagonal lattice parameters a and c (respectively blue circles and red squares) as a function of the Bi content. For information, the cell volume (orange diamonds) and the c/a ratio (green triangles) are also represented. C) Perspective view of the hexagonal crystal structure β-As2Te3 (the As atoms are represented as small red balls and the Te atoms by large blue balls).

Fig 2 :

 2 Fig 2: Scanning Electron Microscope (SEM) elemental contrast picture of the sample β-As1.965Bi0.035Te3 in Back Scattering Electrons (BSE), together with the corresponding element mapping of As, Bi and Te. The distribution of the elements is overall homogeneous.

Fig 3 :Fig 4 :

 34 Fig 3: Temperature dependence of the electrical resistivity ρ A) and the thermopower B) of the polycrystalline β-As2-xBixTe3 samples. The measurements performed parallel and perpendicular to the pressing direction are represented by filled and open symbols respectively. The composition color code is identical in both panels

Figure 1 Figure 4

 14 Figure 1

Table 1 :

 1 Hall carrier concentrations (p) and mobility (μ), in the system β-As2-xBixTe3.at 300K deduced from samples measured perpendicular to the pressing direction.

	x	0	0.015	0.025	0.035
	p (10 19 cm -3 )	8.4	7.1	6.8	8.0
	μ (cm 2 .V -1 .s -1 )	48.6	81.2	69.5	57.8


Ghazanfar Shah

Ali

Polette Arnaud

Pernot Jean-Philippe
email: jean-philippe.pernot@ensam.eu

Monti Marina

Sensitivity analysis for simulated annealingbased fitting of CAD models to point clouds of digitized parts

Keywords: Sensitivity analysis, CAD model parameters, simulated annealing, reverse engineering, digital twins

This paper addresses the way a simulated annealing-based fitting strategy can be enhanced by leveraging a sensitivity analysis able to characterize the impact of the variations in the parameters of a CAD model on the evolution of the deviation between the CAD model itself and the point cloud of the digitized part to be fitted. The principles underpinning the adopted fitting algorithm are briefly recalled. The applied sensitivity analysis is described together with the comparison of the resulting sensitivity evolution curves with the changes in the CAD model parameters imposed by the simulated annealing algorithm. This analysis suggests several possible improvements that are discussed. The overall approach is illustrated on the fitting of single mechanical parts but it can be directly extended to the fitting of parts' assemblies. It is particularly interesting in the context of the Industry 4.0 to update digital twins of physical products and systems.

Introduction

Today, being able to reconstruct or update 3D representations of existing products and systems has become mainstream to support the creation and exploitation of digital twins in the context of Industry 4.0 [START_REF] Lu | Industry 4.0: a survey on technologies, applications and open research issues[END_REF]. Clearly, the relative ease of access to more or less sophisticated 3D acquisition devices has certainly accelerated the demand while extending the range of possible applications. However, the point cloud treatment and the reverse engineering of CAD models are not yet fully au-tomatized. Most of the existing reverse engineering techniques follow a time-consuming patch-by-patch reverse engineering strategy that does not satisfy the Industry 4.0 requirements [START_REF] Falcidieno | Processing free form objects within a product development process framework[END_REF]. Indeed, in most applications, the structure of the products or systems does not change, and only the position and orientation of its constitutive parts is to be updated. Thus, the update problem can be transformed in a fitting problem for which the optimal parameter values of CAD models have to be found to minimize the deviation between the digitized point clouds and the digital twins.

To support the fitting of parts and assemblies through the reverse engineering process, a simulated annealing (SA) based fitting algorithm has been developed [START_REF] Shah | Simulated annealing-based fitting of CAD models to point clouds of mechanical parts' assemblies[END_REF]. It considers as input the digitized point cloud, the CAD models to be fitted and the set of its parameters to be modified by the SA algorithm to minimize the deviation. This metaheuristic algorithm has a stochastic behavior [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF], with few knowledge on how the CAD parameters' values are to be changed throughout the optimization. A possible way to understand the influence between the output and parameters' evolution is sensitivity analysis [START_REF] Iooss | A review on global sensitivity analysis methods[END_REF][START_REF] Hamby | A review of techniques for parameter sensitivity analysis of environmental models[END_REF]. It has been used in various contexts, as, for example, for the parameter choices of SA for continuous network design [START_REF] Yang | Sensitivity Analysis of Simulated Annealing for Continuous Network Design Problems[END_REF], and various indices have been specified [8][9][10]. This paper studies the possibility of coupling a sensitivity analysis to the SA algorithm. The sensitivity analysis allows the evolution of the influence of the CAD model parameters to be monitored throughout the fitting process. Comparing the obtained sensitivity evolution curves to the changes of the parameters directly imposed by the SA algorithm provides hints to improve the fitting process.

The paper is organized as follows. Section 2 briefly reviews the previously developed fitting framework, and the new sensitivity analysis is introduced in section 3. Section 4 discusses the results on a L-like shape and how such an analysis could be intelligently exploited throughout the fitting process. The last section concludes the paper and discusses the future works.

Simulated annealing-based fitting of CAD models

Recently, a breakthrough has been achieved in the definition of part-by-part reconstruction or update of editable CAD models fitting the point cloud of a digitized mechanical part or assembly avoiding the patch-by-patch reconstruction [START_REF] Shah | Simulated annealing-based fitting of CAD models to point clouds of mechanical parts' assemblies[END_REF] At an iteration 𝑗 of the SA algorithm, the energy function to be minimized is evaluated as follows:

𝐸(𝑝 1,𝑗 , ⋯ , 𝑝 𝑁 𝑝 ,𝑗 ) = ∑ d 2 (PC[𝑖], ℳ 𝑗 ⊳ (𝑝 1,𝑗 , ⋯ , 𝑝 𝑁 𝑝 ,𝑗 )) 𝑁 𝑖=1
, with j ≥ 0

wherein ℳ 𝑗 ⊳ represents the tessellation of the CAD model ℳ 𝑗 after its update by the CAD modeler at the jth iteration of the SA algorithm, and d(point, mesh) is the distance function that returns the closest distance between a point and a mesh.

Sensitivity analysis

The previous section has introduced the newly developed fitting algorithm that exploits a SA algorithm to minimize the overall deviation between the point cloud and the CAD model to be fitted. Even though this process has proved to generate goodquality results when compared to the traditional patch-by-patch fitting strategy, the path followed to reach the minimum remains under the control of the SA algorithm, which works as a black box and that neither takes into account the semantics associated to the various parameters 𝑝 𝑘 nor the way their variations may influence the evolution of the energy function step after step. Indeed, while acting on the variables to be optimized, the SA algorithm does not distinguish for instance the lengths, widths, thicknesses and radius that parameterize the features, and all the variables are considered in the same way. It also does not take into account the fact that the influence of a parameter variation on the energy may vary a lot depending on whether the parameter is associated to a structural feature (e.g. the length L of the L-like shape in Figure 1) or to a detail feature (e.g. the radius r in Figure 1). To better compare the influence of the parameters, a sensitivity estimator 𝑆 𝑘,𝑗 is here introduced to compute at each step j of the optimization process the sensitivity of the energy function with respect to each parameter 𝑝 𝑘 :

𝑆 𝑘,𝑗 = 𝜕𝐸 𝜕𝑝 𝑘 | (𝑝 1,𝑗 ,⋯,𝑝 𝑁 𝑝 ,𝑗 ) , with j ≥ 0 (2)
Since the energy function is computed using a nearest point algorithm, it cannot be differentiated. Thus, at the jth iteration, the sensitivities are approximated using a first-order finite difference scheme (either forward or backward) directly controlled by the step ℎ 𝑘 chosen for the corresponding parameter 𝑝 𝑘 :

𝑆 𝑘,𝑗 + = 𝐸 𝑗 (𝑝

Results and discussion

The core of the fitting algorithm has been implemented in MATLAB®, which is able to call the built-in functions of SolidWorks® to perform the successive CAD model updates and ensure the consistency of the resulting B-Rep model during the optimization loops. All the steps ℎ 𝑘 used to compute the sensitivities are equal and only depend on the type of parameter they are associated to. Thus, step ℎ 𝑘 = 0.1mm in case 𝑝 𝑘 is a length, and ℎ 𝑘 = 0.1° in case it is an angle. Those values have been chosen in coherence with the accuracy of the adopted CAD modeler.

In order to compare the sensitivity of the parameters to the way the SA algorithm modifies the parameters values step after step, the evolution of the parameters can be tracked in an absolute or relative manner, as follows:

𝛿𝑝 𝑘,𝑗 = 𝑝 𝑘,𝑗+1 -𝑝 𝑘,𝑗 (5)
∆𝑝 𝑘,𝑗 = 𝛿𝑝 𝑘,𝑗 𝑝 𝑘,𝑗 (6)
Thus, the value 𝛿𝑝 𝑘,𝑗 characterizes the absolute evolution of the parameter 𝑝 𝑘 between iterations 𝑗 and 𝑗 + 1, whereas ∆𝑝 𝑘,𝑗 refers to its relative evolution. At each iteration 𝑗 of the SA algorithm, it is therefore possible to compare the sensitivity 𝑆 𝑘,𝑗 of each parameter 𝑝 𝑘 , to the decision taken by the SA algorithm on how to modify the parameter value for the next step 𝑗 + 1. This is illustrated in Figure 2 for the fitting of the L-like shape introduced in section 3. For this example, the initial temperature of the SA algorithm has been set up to 𝑇 0 = 10 following the initialization procedure suggested in [START_REF] Shah | Simulated annealing-based fitting of CAD models to point clouds of mechanical parts' assemblies[END_REF]. For the sensitivities, the forward finite difference scheme 𝑆 𝑘,𝑗 + has been used when the absolute evolution 𝛿𝑝 𝑘,𝑗 > 0, and the backward scheme 𝑆 𝑘,𝑗 -otherwise. From this experimentation, one can clearly see that the sensitivities of the five parameters defining the L-like shape are relatively different at the beginning, and that they slightly evolve as the fitting process goes on (Figure 2 right). On the contrary, the changes imposed by the SA for each parameter at each iteration 𝑗 are very close. For sake of clarity, the sensitivity of the five parameters at the beginning and the average of first ten iterations are given in Table 1. This table also shows the desired values, 𝑝 𝑘,𝐷 , and the final values 𝑝 𝑘,𝐹 for the object's parameters. One can also see that the successive changes imposed by the SA algorithm to the parameters' values (Figure 2 left) cannot be easily correlated to the evolution of the sensitivities. Thus, several ways of improvement can be sketched:

-the parameters 𝑝 𝑘 could be grouped according to their levels of sensitivity, so as to treat them in several batches for which the sensitivity is comparable; -the way the SA algorithm changes the parameters values could be further optimized while considering the sensitivity analysis all along the process;

-the sensitivity analysis could be performed at a lower level, while decomposing the energy function and tracking which parts of the energy function are affected by the changes to save time when computing the overall energy. The first idea has been implemented, but due to space limitation results cannot be shown here. On the example of the L-like shape, two main categories of parameters can be distinguished when comparing their sensitivities at the beginning of the fitting process: {𝐿, ℓ, 𝑒} and {𝑅, 𝑟} gather together parameters having comparable sensitivities. Such a decomposition in two groups gives rise to two successive optimization loops. Overall, doing this way the accuracy of the fitting is improved compared to the use of a single optimization loop.

Conclusion

This paper has explored the possibility to integrate a sensitivity analysis within a SA algorithm used to fit a CAD model in the point cloud of a digitized part. Such an approach helps grouping the CAD model's parameters so as to treat batches having comparable sensitivities. Other improvements have also been sketched to be more accurate and to reduce optimization times.

  . The idea is to work directly at the level of the part whose parameters are modified by a SA algorithm until a good-quality fitting is obtained. The inputs are the point cloud PC to which a parameterized CAD model ℳ 0 is to be fitted. The point cloud is composed of N points PC[𝑖], with 𝑖 ∈ [1. . 𝑁], and the CAD model is parameterized by 𝑁 𝑝 control parameters 𝑝 𝑘 , with 𝑘 ∈ [1. . 𝑁 𝑝 ]. During the prearrangement step, the user locates in a coarse manner the part inside the point cloud, this initializes the parameters values 𝑝 𝑘,0 (Figure 1.a). At each iteration 𝑗, the SA algorithm then tries to modify the parameters values 𝑝 𝑘,𝑗 until the updated CAD model perfectly fits the point cloud (Figures 1.b and 1.c). The quality of the fitting is assessed thanks to an energy function which characterizes the overall deviation between the point cloud and the CAD model. The process stops when the energy function no longer decreases with respect to a given accuracy 𝜀 𝑠 , or when a max number of iterations 𝑀 𝑖𝑡𝑒𝑟 is reached.

Fig. 1 .

 1 Fig. 1. Global fitting of a L-like shape defined by 5 controlled parameters: (a) initial configuration after the prearrangement step, (b) fitted part after 50 iterations of the SA algorithm, (c) final fitted part minimizing the deviation with the point cloud (𝜀 𝑠 = 10 -1 and 𝑀 𝑖𝑡𝑒𝑟 = 1000).

Fig. 2 .

 2 Fig. 2. Comparison between the evolution of the sensitivities 𝑆 𝑘,𝑗 (right) and the changes 𝛿𝑝 𝑘,𝑗 imposed by the SA algorithm (left) at each iteration 𝑗 of the fitting process.

  1,𝑗 ,⋯,𝑝 𝑘,𝑗 +ℎ 𝑘 ,⋯,𝑝 𝑁 𝑝 ,𝑗 )-𝐸 𝑗 (𝑝 1,𝑗 ,⋯,𝑝 𝑘,𝑗 ,⋯,𝑝 𝑁 𝑝 ,𝑗 )

		ℎ 𝑘	+ Ο(ℎ 𝑘 ), with j ≥ 0	(3)
	𝑆 𝑘,𝑗 -=	𝐸 𝑗 (𝑝 1,𝑗 ,⋯,𝑝 𝑘,𝑗 ,⋯,𝑝 𝑁 𝑝 ,𝑗 )-𝐸 𝑗 (𝑝 1,𝑗 ,⋯,𝑝 𝑘,𝑗 -ℎ 𝑘 ,⋯,𝑝 𝑁 𝑝 ,𝑗 ) ℎ 𝑘	+ Ο(ℎ 𝑘 ), with j ≥ 0	(4)

Table 1 .

 1 Values of the comparison metrics for the initial iteration (𝑗 = 0) and average of first ten iterations (𝑗 ∈ [0. .9]) .

	k	1	2	3	4	5
	𝑝 𝑘,0	25.0	20.0	2.0	3.0	2.0
	𝑆 𝑘,0	-8.0E+4	3.0E+4	4.0E+4	0.0E+00	1.0E+4
	𝛿𝑝 𝑘,0	0.56	-0.45	-0.22	0.15	0.16
	∆𝑝 𝑘,0	0.02	-0.02	-0.11	0.05	0.09
	𝑆 ̅ 𝑘,𝑗∈[0..9]	4.4E+4	2.9E+4	4.1E+4	6.0E+3	6.0E+3
	𝛿𝑝 ̅̅̅ 𝑘,𝑗∈[0..9]	2.55	1.25	1.76	1.79	1.60
	∆𝑝 ̅̅̅̅ 𝑘,𝑗∈[0..9]	0.10	0.06	0.88	0.60	0.80
	𝑝 𝑘,𝐷	35	30	5	6	5
	𝑝 𝑘,𝐹	34.92	29.84	4.89	5.93	5.49