Ketan Lohani

Himanshu Nautiyal

Narges Ataollahi

Krishnendu Maji

Emmanuel Guilmeau
email: emmanuel.guilmeau@ensicaen.fr

Paolo Scardi
email: paolo.scardi@unitn.it

Effects of grain size on the thermoelectric properties of Cu 2 SnS 3 (CTS): An experimental and first-principles study

Keywords: Cu2SnS3, grain size effect, DFT, thermoelectric Cu2SnS3, spark plasma sintering, p-type semiconductor

Cu-Sn-based sulfides are earth-abundant and non-toxic compounds, of special interest for low-cost energy harvesting applications. In the present work, we have investigated the effect of grain-size on the thermoelectric properties of Cu2SnS3 (CTS). Three dense CTS samples with nanometric grains were produced by mechanical-alloying combined with Spark Plasma Sintering, preserving the small size of crystalline domains to 12 nm, 25 nm, and 37 nm, respectively. The experimental results show that the Seebeck coefficient (S) and electrical resistivity (ρ) decrease with decreasing domain sizes, while the thermal conductivity (k) increases. A smaller domain size correlates with a lower resistivity and a degenerate semiconductor-like behaviour due to higher carrier concentration. At the same time, our synthesis method leads to materials with very low lattice thermal conductivity thanks to the nanometric size of grains and structural disorder. As a result, the sample with the smallest grain size exhibits the highest zT of ~0.4 at 650 K. First-principles Density Functional Theory (DFT) simulations on various CTS crystallite surfaces revealed localized states near the Fermi level and the absence of bandgap, indicating the metallic nature of the surfaces. Various CTS systems were tested by DFT, showing the following order of increasing formation energy: stoichiometric CTS, Cu-vacancy, Cu-rich, Sn-vacancy, and Sn-rich.

Introduction:

Most commercially available thermoelectric (TE) devices use toxic and scarce materials, making them expensive and potentially hazardous, e.g., Sb2Te3, Bi2Te3, etc. In recent years, the search for highperformance, non-toxic, eco-friendly, and earth-abundant TE materials has led to the exploration of multinary sulfides. 1,2 Chalcogenides 3 , colusites [START_REF] Suekuni | High-Performance Thermoelectric Minerals: Colusites Cu 26 V 2 M 6 S 32 (M = Ge, Sn)[END_REF][START_REF] Guélou | Recent Developments in High-Performance Thermoelectric Sulphides: An Overview of the Promising Synthetic Colusites[END_REF] , and other metal-based sulfides [START_REF] Biswas | Tellurium-Free Thermoelectric: The Anisotropic n-Type Semiconductor Bi2S3[END_REF][START_REF] Tan | High Thermoelectric Performance in Electron-Doped AgBi 3 S 5 with Ultralow Thermal Conductivity[END_REF][START_REF] Zhai | Large Effective Mass and Low Lattice Thermal Conductivity Contributing to High Thermoelectric Performance of Zn-Doped Cu5Sn2Se7[END_REF] could be viable alternatives to existing materials. [START_REF] Powell | Recent Developments in Earth-Abundant Copper-Sulfide Thermoelectric Materials[END_REF] Cu-based sulfides have low formation energy, so that it is possible to produce them by short-period reactive milling using a planetary or vibrating mill. As also shown in the present work, high-energy reactive ball-milling, also called mechanical-alloying, can be employed with success to synthesize new and disordered phases. In addition, milling offers the advantage of facile and scalable production for industrial use. [START_REF] Baláž | Thermoelectric Cu-S-Based Materials Synthesized via a Scalable Mechanochemical Process[END_REF] The energy conversion efficiency of a TE material is determined by a dimensionless figure of merit (zT) expressed as zT=S [START_REF] Jaldurgam | Low-Toxic, Earth-Abundant Nanostructured Materials for Thermoelectric Applications[END_REF] /ρk*T, where S, ρ, k, and T are Seebeck coefficient, electrical resistivity, thermal conductivity, and absolute temperature, respectively. S 2 /ρ is referred to as power factor (PF). The Seebeck coefficient and electrical resistivity are strongly dependent of the carrier concentration (n), mobility (µ), and effective mass of the charge carriers (m*), (see Equations 1 and 2 below). A low carrier concentration and high effective mass of charge carriers can increase the Seebeck coefficient, but it also penalizes the electrical conductivity and vice versa. [START_REF] Mao | Advances in Thermoelectrics[END_REF][START_REF] Snyder | Complex Thermoelectric Materials[END_REF]

𝑆 = 𝐴𝑚 * ( " #$ ) ! " 𝑇 -1 𝜎 = 𝑛𝑒𝜇 -2 k % = '( ) - 3
where A, e, and L represent a constant, charge of an electron, and Lorenz number, respectively.

Besides high PF, TE materials also require low thermal conductivity (k), composed of a lattice (kl) and an electronic contribution (ke), combined as: k = kl + ke. The lattice contribution can be decreased by increasing the grain boundary density, vacancies, defects, impurities, strain, etc. Differently, ke shows a strong dependence on the electrical resistivity, as in the Wiedemann-Franz law (see equation 3). [START_REF] Guélou | Recent Developments in High-Performance Thermoelectric Sulphides: An Overview of the Promising Synthetic Colusites[END_REF] More recently, the determination of the thermoelectric quality factor (ꞵ) given by ꞵ=µ0mdos* 3/2 /kl (where µ0 and mdos* are the non-degenerate mobility and density of state effective mass, respectively) is in practice for the simultaneous optimization of all TE parameters. [START_REF] Beretta | Thermoelectrics: From History, a Window to the Future[END_REF][START_REF] Zhang | Electronic Quality Factor for Thermoelectrics[END_REF] Cu2+xSn1-xS3 (CTS) polymorphs are used in many applications such as photovoltaics, transistors, LEDs, and thermoelectric materials. A recent study has also suggested a potential use of CTS quantum dots for photodetector applications. [START_REF] Dias | Solvothermal Synthesis of Cu 2 SnS 3 Quantum Dots and Their Application in Near-Infrared Photodetectors[END_REF] CTS polymorphs have been extensively studied as p-type thermoelectric materials showing moderately high PF with high or low thermal conductivity, depending whether the crystal structure is ordered or disordered. [START_REF] Shen | Eco-Friendly p-Type Cu2SnS3 Thermoelectric Material: Crystal Structure and Transport Properties[END_REF][START_REF] Lohani | Ultra-Low Thermal Conductivity and Improved Thermoelectric Performance in Disordered Nanostructured Copper Tin Sulphide (Cu2SnS3, CTS)[END_REF][START_REF] Kumar | Ordered Sphalerite Derivative Cu 5 Sn 2 S 7 : A Degenerate Semiconductor with High Carrier Mobility in the Cu-Sn-S Diagram[END_REF] To increase the thermoelectric performance, different cationic substitutions have been investigated with the aim to tune the carrier concentration and to introduce structural disorder. Among the various studies, Zhao et al. have reported the highest zT ~0.9 (at T > 700 K) so far, by simultaneous cobalt and antimony doping. [START_REF] Xu | Synergistic Role of Ni-Doping in Electrical and Phonon Transport Properties of Cu2Sn1-XNixS3[END_REF] Zhai et al. [START_REF] Zhai | Structural Diversity and Electronic Properties of Cu2SnX3 (X=S, Se): A First-Principles Investigation[END_REF] and Shigemi et al. [START_REF] Shigemi | First-Principles Calculation of Cu2SnS3 and Related Compounds[END_REF] have shown that the hybridization of Cu-3d and S-3p orbitals in the valence band (VB) is responsible for the p-type nature of CTS. Heavy acceptor-doping due to unfilled d-orbitals of Co 21 , Cu 22 , Ni 18 , Fe 23 , and Mn 24 enhance the DOS effective mass of carriers (holes) and electrical conductivity, resulting in a high PF ~10 µW/cm K 2 , around 700 K. Cationic substitution also transforms the CTS crystal structure from ordered (monoclinic Cc) [START_REF] Aihara | Exciton Luminescence from Cu 2 SnS 3 Bulk Crystals[END_REF] to tetragonal I-42m [START_REF] Tan | Enhanced Thermoelectric Properties of Earth-Abundant Cu2SnS3 via In Doping Effect[END_REF] and disordered (cubic: F-43m). [START_REF] Shen | Eco-Friendly p-Type Cu2SnS3 Thermoelectric Material: Crystal Structure and Transport Properties[END_REF] In most of the cases, a mixture of the different phases is obtained. Deng et al. [START_REF] Deng | Thermoelectric Properties of Non-Stoichiometric Cu2+ XSn1-XS3 Compounds[END_REF] have shown the influence of Cu/Sn ratio in Cu2+xSn1-xS3 on the electrical and thermal properties. More recently, V. Pavan Kumar et al. [START_REF] Kumar | Ordered Sphalerite Derivative Cu 5 Sn 2 S 7 : A Degenerate Semiconductor with High Carrier Mobility in the Cu-Sn-S Diagram[END_REF] have discovered a new ordered monoclinic structure Cu5Sn2S7 by increasing the Cu/Sn ratio up to x = 0.15 leading to a specific ordering of Cu and Sn in the structure. In summary, the hole doping in CTS compounds, induced by the substitution of Cu, Sn or S by aliovalent cations, or by Cu for Sn substitution, increases the carrier concentration. Disordering phenomena, usually induced by the cationic/anionic substitution, scatters the phonon waves more effectively, resulting in a higher TE figure of merit.

The disordered cubic CTS and Cu2SnSe3 (CTSe) phases can also be produced and stabilized without chemical alteration. [START_REF] Lohani | Ultra-Low Thermal Conductivity and Improved Thermoelectric Performance in Disordered Nanostructured Copper Tin Sulphide (Cu2SnS3, CTS)[END_REF][START_REF] Lohani | Experimental and Ab Initio Study of Cu 2 SnS 3 (CTS) Polymorphs for Thermoelectric Applications[END_REF][START_REF] Siyar | Effect of Annealing Temperature on the Phase Transition, Band Gap and Thermoelectric Properties of Cu2SnSe3[END_REF] The optical absorption measurements revealed a lower bandgap for the disordered CTS/Se polymorphs. The lower bandgap enhances the carrier concentration resulting in a higher PF compared to its ordered polymorph. Furthermore, in a theoretical study, Baranowski et al. [START_REF] Baranowski | Effects of Disorder on Carrier Transport in Cu2SnS3[END_REF] have also reported band tailing and mid-gap states for the disordered CTS phase. The structural disorder introduces potential energy fluctuations in the lattice that allow energy levels within the forbidden gap, responsible for the band tailing and reduced band gap. [START_REF] Lohani | Experimental and Ab Initio Study of Cu 2 SnS 3 (CTS) Polymorphs for Thermoelectric Applications[END_REF][START_REF] Baranowski | Effects of Disorder on Carrier Transport in Cu2SnS3[END_REF] Moreover, Dahule et al. [START_REF] Dahule | Surface Study of Cu 2 SnS 3 Using First-Principles Density Functional Theory[END_REF] have studied the electronic structure properties of (200) and (-131) surfaces of monoclinic CTS and shown their metallic character. It is worth mentioning that the metallic character was confirmed using different potentials, including the Heyd-Scuseria-Ernzerhof (HSE) potential, which results in a better estimation of the bandgap.

In terms of synthesis, the literature suggests a three-step top-down approach to produce CTS samples for TE applications. First, CTS is produced from the elemental powders (Cu, Sn, S, and dopant elements) using an extended high-temperature solid-state reaction. [START_REF] Snyder | Complex Thermoelectric Materials[END_REF] In the second step, the grain size is sometimes reduced by ball-milling. Finally, high-density pellets are produced using spark-plasma-sintering or hot-pressing. This approach results in highly crystalline materials with average domain sizes ranging from half to several microns. This synthesis method requires a long annealing time at high temperatures, which is time and resource-consuming.

Alternatively, CTS compounds, as many other Cu-based sulphides, can be produced by combined highenergy reactive ball-milling (mechanical alloying) and Spark Plasma Sintering (SPS) or hot pressing. [START_REF] Kumar | Ordered Sphalerite Derivative Cu 5 Sn 2 S 7 : A Degenerate Semiconductor with High Carrier Mobility in the Cu-Sn-S Diagram[END_REF][START_REF] Baláž | Enhanced Thermoelectric Performance of Chalcopyrite Nanocomposite via Co-Milling of Synthetic and Natural Minerals[END_REF] Mechanical alloying allows to produce fine, homogeneous, pre-crystallised and highly reactive powders, which usually lead to high purity, crystallinity and dense samples after sintering with small grain and crystallite sizes. Furthermore, a highly dense disordered CTS polymorph can be stabilized without any chemical substitution. [START_REF] Baláž | Thermoelectric Cu-S-Based Materials Synthesized via a Scalable Mechanochemical Process[END_REF] The present work aims to shed light on the degenerate semiconductor-like behaviour of nanocrystalline CTS samples. Three dense CTS samples were produced using powder synthesized by high-energy reactive ball milling and SPS. We have investigated the effects of grain size on the thermoelectric properties. Structural and microstructural analyses suggested a stoichiometry deviation in CTS phases sintered at low temperature, together with a very small crystallite and grain size. Ab initio calculations using Density Functional Theory (DFT) indicated the metallic nature of CTS surfaces, leading to a degenerate semiconductor-like trend with temperature. Further evidence is presented on the formation energies for various CTS systems with vacancies and chemical deviations.

Materials and methods:

Experimental: Elemental powders (Cu, Sn, and S) were weighted in stochiometric ratio and fed in WC vials with WC balls. A Fritsch Pulverisette-4 mill was operated for 1h, producing ~6 g of as-milled CTS powder. The entire production process was performed in a glove box with argon filled environment. During the milling, no lubricant was introduced to avoid any contamination. The production of CTS powder by high-energy ball milling of elemental powders and binary sulfides is discussed in detail elsewhere. [START_REF] Lohani | Ultra-Low Thermal Conductivity and Improved Thermoelectric Performance in Disordered Nanostructured Copper Tin Sulphide (Cu2SnS3, CTS)[END_REF][START_REF] Lohani | Experimental and Ab Initio Study of Cu 2 SnS 3 (CTS) Polymorphs for Thermoelectric Applications[END_REF] The as-milled powder was sintered using a Spark Plasma Sintering (SPS) machine (FCT HPD 25) at various temperatures and pressures to optimize the sintering conditions (Table 1). Relative density of the samples is 86, 96, and 94% for sample A, B and C, respectively.

Structural analysis was performed using X-Ray Diffraction (XRD). XRD data were collected in θ/2θ Bragg-Brentano geometry using a Panalytical Xpert Pro diffractometer equipped with Cu-Kα (λ=1.5406 Å) source.

Micrographs and Electron Dispersive X-ray (EDX) spectroscopy analyses were collected using a JEOL JSM 7200F scanning electron microscope equipped with EDX X-Flash Bruker detector.

The electrical resistivity (ρ) and Seebeck coefficient (S) were measured simultaneously from 300 K up to 700 K on bar-shaped samples of typical dimensions 2×3×10 mm 3 using an ULVAC-ZEM3 instrument under partial helium pressure. A NETZSCH LFA-457 apparatus was used for measuring the thermal diffusivity under argon flow. The thermal conductivity (κ) was determined as the product of the geometrical density, the thermal diffusivity, and the theoretical heat capacity using the Dulong-Petit approximation. The lattice contribution to the thermal conductivity (kl) was determined by subtracting the estimated electronic component (κe) from the measured total thermal conductivity, κ. The measurement uncertainties are estimated to be 6% for the Seebeck coefficient, 8% for the electrical resistivity, 11% for the thermal conductivity, and 16% for the final dimensionless figure of merit.

Hall effect measurements at RT were carried out using a Physical Properties Measurement System (PPMS; Quantum Design) under an applied magnetic field of 9 T.

Computational:

The electronic structure calculations were performed using DFT as implemented in Vienna ab initio Simulation Package. [START_REF] Kresse | Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set[END_REF][START_REF] Kresse | Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set[END_REF] The interaction between electrons and ions was described using the Projector-Augmented-Wave (PAW) method. The Generalized Gradient Approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) 34 electron exchange-correlation was used for the calculations.

We have performed the calculations for (001), (010), and (100) slabs of monoclinic CTS, and two disordered CTS cells with sulfur termination. We considered two cases for the disordered cell, one with the Cu-rich and the other with Sn-rich atomic layer below the terminating sulfur layer. For all the calculation CTS stoichiometry was preserved. A vacuum of 15 Å was created to minimize the interaction between the periodic copies. In the disordered cells, the atoms of the lower four layers were fixed in position, whereas the atoms of the upper four layers were free to move. Relaxation was performed with a plane-wave kinetic energy cut-off and an electronic convergence of 350 eV and 10 -5 eV, respectively. The involvement of core electrons was seized by freezing. Only 3d 10 4s 1 , 4d 10 5s 2 5p 2 , and 3s 2 3p 4 electrons were treated as valence electrons for Cu, Sn and S, respectively. The cell parameters were fixed, whereas the atomic position was allowed to change for relaxation. The electronic convergence was set to 10 -6 eV for DOS calculations. Brillion zone sampling was performed on a k-point grid of 9×9×1 and 5×5×1 for monoclinic slabs and disordered cells, respectively. The k-mesh was reduced for the disordered cell due to its larger size.

The formation energy calculations were performed on Cu-rich, Cu-vacant, Sn-rich, and Sn-vacant cells. For these calculations, the plane wave kinetic energy cut-off and electronic convergence were changed to 400 eV and 10 -6 eV, respectively. A Monkhorst-Pack [START_REF] Pack | special Points for Brillouin-Zone Integrations"-a Reply[END_REF] k-point mesh of 7×7×7 (centered at the gamma point) was used to integrate the Brillion zone. One atom was removed from the supercell, and the remaining atoms were allowed to relax for the vacancy calculations. The off-stoichiometry calculations were performed on a supercell of 72 atoms with stable monoclinic phase, where one Cu atom was replaced by one Sn atom and vice versa, respectively for the Sn-rich and Cu-rich simulations.

Results and discussion:

XRD patterns collected on as-milled powder with increasing milling times are shown in Figure 1. Due to its low formation energy, short milling of 15 min led to the formation of binary sulfides, especially CuS. As the milling time was increased to 30 min, the appearance of diffraction peaks of sphalerite-type structure can be noticed. However, unreacted SnS and Sn powder were still present in the sample. With the further increase of milling time to 1 h, the as-milled powder showed four broad diffraction peaks, implying the formation of a sphalerite-type crystal structure. It is worth mentioning here that, among all the elements, Sn incorporated last in the CTS system. Broad Bragg-peaks with diffuse backgrounds imply that the onehour milled powder consists of low crystalline CTS grains with a small domain size.

SEM-EDX data collected on 60 min milled powder at full-frame image showed the presence of stoichiometric CTS. On the same sample, various micrographs and EDX data were collected in different magnifications from different parts of the SEM grid. It indicated that the as-milled powder has a nonhomogeneous chemical distribution with a small domain size, which tend to agglomerate in larger particles (Figure S1).

High-density bulk CTS samples were produced by SPS under different sintering conditions (listed in Table.

1). Sintered samples maintained the same Bragg-peak positions as the as-milled powder. However, sharper Bragg peaks for sintered samples indicate an increased crystallinity and crystallite size. XRD line profile analysis was performed by the Rietveld method 36 using Topas 7 software [START_REF] Coelho | TOPAS and TOPAS-Academic: An Optimization Program Integrating Computer Algebra and Crystallographic Objects Written in C++[END_REF] . We employed the Whole Powder Pattern Modelling (WPPM) macro [START_REF] Scardi | Whole Powder Pattern Modelling Macros for TOPAS[END_REF] to simulate the peak profile. The instrumental profile was obtained from a LaB6 standard sample. [START_REF] Black | Certification of Standard Reference Material 660c for Powder Diffraction[END_REF] A nine-point Chebyshev polynomial and a 1/x-function were used to simulate the background and low-angle intensity, respectively. Subsequently, zero shift and other parameters were systematically refined [START_REF] Mccusker | Rietveld Refinement Guidelines[END_REF] . During the refinement, the variation of the Debye-Waller coefficients (thermal factors) was restricted between 0 to 2 for all atoms. For all the samples discussed below, the goodness of fit (gof) was ~1.1-1. and 160.50 Å 3 , respectively. No microstrain was observed for any sample. Average domain size and the weight fraction of CTS and secondary phases are listed in table 1 for the different sintering conditions. All the samples showed disordered cubic CTS structures. However, a small amount (weight fraction <0.5% ) of SnO2 was observed for samples A and C. For sample B (see figure 2a), two Bragg peaks around 2θ ~ 16 o and ~31 o were observed. Sample B may have a small amount of monoclinic phase. But the two peaks are quite broad which makes it hard to quantify the monoclinic phase reliably, and it is certainly below 1% in weight.

The size of crystalline domains was estimated as 12±10 nm, 25±10 nm, and 37±10 nm for samples A, B, and C, respectively. The reported average grain size for high-density CTS samples prepared by the threestep method (discussed in the introduction) is in the range of 500-600 nm, [START_REF] Tan | Enhanced Thermoelectric Properties of Earth-Abundant Cu2SnS3 via In Doping Effect[END_REF] and in some cases, micron-size grains are also reported. [START_REF] Zhao | Cobalt-Doping in Cu2SnS3: Enhanced Thermoelectric Performance by Synergy of Phase Transition and Band Structure Modification[END_REF] In contrast, traditionally sintered samples (without applying pressure) showed smaller grains, domain size ranging from ~50 nm to a few 100 nm, and low density. [START_REF] Lohani | Ultra-Low Thermal Conductivity and Improved Thermoelectric Performance in Disordered Nanostructured Copper Tin Sulphide (Cu2SnS3, CTS)[END_REF] The mechanicalalloying combined with SPS method proposed here constrained the grain size below 50 nm, [START_REF] Baláž | Enhanced Thermoelectric Performance of Chalcopyrite Nanocomposite via Co-Milling of Synthetic and Natural Minerals[END_REF] while promoting densification at relatively low sintering temperature. (i) (ii) SEM micrographs on bulk samples reveal a dense microstructure, highly intact grains with almost no porosity (shown in Figure 3, Figure S3, and Figure S4). Unlike the as-milled powder, EDX analysis on the sintered samples revealed a homogeneous chemical distribution, with the exception of sample A, where a small amount of SnS was found, likely due to incomplete reaction at 400 °C (Figure 3i, Figure S3 and S5). Some bright spots in the micrographs revealed traces of WC particles. However, WC and SnS signals were not observed in the XRD patterns because of their small weight fraction, below the detection limit (<<1%).

Comparative SEM-EDX analysis was performed on samples A and B (before and after polishing), SEM-EDX data is shown in the supporting information (Figure S3-S6). Sample A has a higher amount of SnS phase compared to sample B, probably due to a lower sintering temperature. However, the matrix composition for both samples is similar, suggesting that stoichiometric deviation in CTS grains is quite small. These micrographs also confirm that the average grain size is much smaller than the magnification of the microscope allows to see.

Figures 4(i) and 4(ii) show the absolute Seebeck coefficient (S) and resistivity (ρ), respectively, for the CTS samples of Table 1. Although there is a variation in the values of S, all the samples show a positive Seebeck coefficient, increasing with temperature. Sample A, B, and C present S ~100 µV/K, 180 µV/K, 160 µV/K, respectively, which is lower than the reported room temperature values for CTS compound (S ~500-700 µV/K). [START_REF] Kumar | Ordered Sphalerite Derivative Cu 5 Sn 2 S 7 : A Degenerate Semiconductor with High Carrier Mobility in the Cu-Sn-S Diagram[END_REF][START_REF] Deng | Thermoelectric Properties of Non-Stoichiometric Cu2+ XSn1-XS3 Compounds[END_REF][START_REF] Lohani | Experimental and Ab Initio Study of Cu 2 SnS 3 (CTS) Polymorphs for Thermoelectric Applications[END_REF] A general trend of increasing values of Seebeck coefficient with increasing the sintering temperature and grain size can be observed. The Cu-Sn-S systems exist in numerous phases, such as Cu2SnS3 16 , Cu3SnS4 41 , Cu5Sn2S7 17 , and Cu7Sn3S10 42 , etc. [START_REF] Jaldurgam | Low-Toxic, Earth-Abundant Nanostructured Materials for Thermoelectric Applications[END_REF] Mostly, these systems show p-type S1. (iii) (iv)

semiconducting nature due to the unfilled Cu 3d-orbitals. Additionally, Cu-S bonds in Cu-Sn-S-based systems form a 3-dimensional conduction network which enhances their electrical conductivity. [START_REF] Shen | Eco-Friendly p-Type Cu2SnS3 Thermoelectric Material: Crystal Structure and Transport Properties[END_REF] It is well known that undoped CTS is highly resistive with a non-degenerate semiconductor-like trend in temperature owing to its low carrier concentration. [START_REF] Lohani | Ultra-Low Thermal Conductivity and Improved Thermoelectric Performance in Disordered Nanostructured Copper Tin Sulphide (Cu2SnS3, CTS)[END_REF] However, samples A and B show a degenerate semiconductor-like nature, whereas sample C, with the largest S (~320 µV/K, above 650 K), shows a nondegenerate trend up to ~650 K. As expected, samples with a higher Seebeck coefficient have higher electrical resistivity and vice versa, but a trend of decreasing electrical resistivity can also be observed with the decrease in crystalline domain size. It is worth mentioning here that off-stoichiometry in CTS compounds can lead to similar results, although such deviations should have a Cu/Sn ratio >2. In fact, Deng et al. [START_REF] Deng | Thermoelectric Properties of Non-Stoichiometric Cu2+ XSn1-XS3 Compounds[END_REF] show that an excess of Cu in Cu2+xSn1-xS3 (x ≥ 0.016) changes the electrical resistivity from a non-degenerate to degenerate behaviour. This suggests that CTS phase, especially when sintered at low T (400°C, sample A), is non-stoichiometric.

In general, a small grain size leads to a higher Seebeck coefficient due to the scattering of charge carriers by the grain boundary, otherwise known as Energy-filtering. [START_REF] Narducci | Impact of Energy Filtering and Carrier Localization on the Thermoelectric Properties of Granular Semiconductors[END_REF] This, however, decreases the mobility, in turn, increases the electrical resistivity. The lower grain size is also crucial in the suppression of thermal conductivity by scattering phonon waves effectively. The results shown here contrast with the energy filtering mechanism. The samples with smaller domains are more conductive, and we observe a trend of increasing resistivity with an increase in grain size. Carrier concentration measurements on these samples revealed that the higher conductivity of samples with small domains correlates with a higher carrier concentration, 8.6×10 20 cm -3 and 7.4×10 19 cm -3 for samples A and B, respectively. These values are one to two orders of magnitude higher than the reported values for the CTS [START_REF] Lohani | Experimental and Ab Initio Study of Cu 2 SnS 3 (CTS) Polymorphs for Thermoelectric Applications[END_REF] . However, other Cu-Sn-S based compounds with higher Cu/Sn ratio presented increasing carrier concentration with increasing Cu/Sn ratio, i.e., 2.6×10 21 cm -3 and 5.57×10 21 cm -3 for Cu7Sn3S10 44 and Cu5Sn2S7 17 , respectively. Sample A sintered at low sintering temperature has a lower carrier mobility (1.2 cm 2 V -1 s -1 ) than sample B (2.1 cm 2 V -1 s -1 ) which is explained by the larger carrier concentration (increased probability of charge carrier collision) and lower crystallite size in sample A.

Due to the decreased electrical resistivity, sample A shows the highest PF values of ~ 4.5 µW/K 2 cm, above 650 K, which is 3-fold and 9-fold higher than cubic (disordered) CTS (PF ~1.5 µW/K 2 cm) [START_REF] Lohani | Experimental and Ab Initio Study of Cu 2 SnS 3 (CTS) Polymorphs for Thermoelectric Applications[END_REF] and monoclinic (ordered) CTS ( PF ~0.47 µW/K 2 cm) [START_REF] Shen | Eco-Friendly p-Type Cu2SnS3 Thermoelectric Material: Crystal Structure and Transport Properties[END_REF] , respectively. Samples B and C have comparatively lower PF of ~1.5 µW/K 2 cm and ~1.2 µW/K 2 cm, respectively. The thermal diffusivity measurements are shown in Figure 4(iv). It is evident from the data that samples with smaller domains (more conductive) have a higher diffusivity. From the above considerations, it appears that the electrical properties are strongly dependent on the sintering temperature and grain size. Additional results following the same trend of Seebeck coefficient and electrical resistivity with grain size support this statement (Figure S7). As mentioned above, a slight stoichiometric deviation can occur in the small grains due to the uncomplete reaction and crystallisation at low temperature, leading to high carrier concentration and electrical conductivity. The fast kinetic of reaction and crystallisation during SPS may also lead to variation of compositions at the grain surfaces and boundaries, which could be highly conductive and possibly injecting additional charge carriers in the system.

To investigate this further, three ordered and two disordered CTS surfaces (shown in figures 5 and 6, respectively) were studied. The first three images in figure 5 show ordered CTS slabs with (001), (010), and (100) planes. The other two images show disordered structures consisting of a Cu-rich layer and a Snrich layer, just below the terminating sulfur layer. Please note that the overall chemistry of CTS system was respected while making all the structures shown in figure 5 and6. For each structure in figure 5 and figure 6, corresponding DOS are shown with the structures. After minimization, large distortions were observed for the surface atoms of all the structures. The reason is that surface atoms form incomplete bonds due to the change in the coordination number caused by the surface termination. The DOS on the VB side is higher than the CB side showing p-type nature. However, no bandgap was observed suggesting a metallic character of all the studied structures. The localized states are visible near the Fermi level due to the dangling bonds on the surface. These dangling bonds provide additional charge carriers (holes), which increase the carrier concentration leading to the degenerate semiconductor-like behaviour observed in Figure 4. We are aware that the calculated structures are not an exact model of the grain boundary effects. Since the grains could terminate with any plane or orientation (favoured by the local environment), these calculations only make plausible the hypothesis that CTS grains should have dangling bonds, which would lead to localized states at the Fermi level and increased electrical conductivity. These results are in agreement with a recent report by Dahule et al. [START_REF] Dahule | Surface Study of Cu 2 SnS 3 Using First-Principles Density Functional Theory[END_REF] , although limited to the monoclinic phase. The thermal conductivity and figure of merit are shown in Figures 7 (i) and 7 (ii), respectively. Since we are considering the heat capacity well above the Debye temperature (θD ~213 K), the total thermal conductivity was calculated considering a constant value of Cp ~0.44 J/g K for all samples. The total thermal conductivity for all samples decreases with temperature, due to an increased phonon-phonon interaction (Umklapp process or U-process). The electronic part of thermal conductivity (κe) was estimated according to the Wiedemann-Franz law (see equation 3), where the Lorentz number (L) was calculated using equation 4 [START_REF] Kim | Characterization of Lorenz Number with Seebeck Coefficient Measurement[END_REF] .

𝐿 = 1.5 + 𝑒 *( |$| %%& ) - 4
where L is in 10 It is evident from figure 7(i), at higher temperatures, the lattice part of the thermal conductivity is similar for all the samples, kl ~0.6 W/m K, above 673 K. This value is comparatively lower than the diamond-like CuFeS2+x. system, where kl ~1.0 W/m K is reported at 650 K. [START_REF] Xie | Thermoelectric Performance of CuFeS2+2x Composites Prepared by Rapid Thermal Explosion[END_REF] However, other Cu-Sn-S based compounds such as Cu3SnS4 and Cu7SnS10 show a similar kl around 650 K, which further decreases to kl ~0.4 W/m K at 750 K. [START_REF] Yang | Enhancing the Thermoelectric Performance of Cu 3 SnS 4 -Based Solid Solutions through Coordination of the Seebeck Coefficient and Carrier Concentration[END_REF][START_REF] Deng | A Low-Cost and Eco-Friendly Br-Doped Cu 7 Sn 3 S 10 Thermoelectric Compound with ZT around Unity[END_REF] The electronic contribution of thermal conductivity is the highest for sample A, and decreases for samples B and C, respectively. This agrees with the corresponding resistivity and carrier concentration measurements, as samples A, B, and C have increasing resistivity (decreasing carrier concentration) in the same order. The figure of merit of samples A, B, and C is equal to 0.40, 0.17, and 0.14 at 673 K, respectively. Overall, the two-step production method can be used to produce high-density nanostructured pellets. The experimental results indicate that a similar zT is reported for Cu2+xSn1-xS3 systems with Sn substitution (x = 0.2) by Zn 15 , Ni 18 , In 26 , etc.

Conclusions:

We have studied the effects of crystalline domain size on the thermoelectric properties using experimental analyses and first-principles simulations. A CTS powder was produced from elemental powders (Cu, Sn, and S) by high-energy reactive ball-milling, subsequently, high-density pellets were produced using sparkplasma-sintering. This method adds a distinct advantage by maintaining the nanostructure while reaching the required high density. Rietveld refinement of the XRD patterns revealed average domain sizes below 50 nm for samples sintered in different conditions. Low temperature SPS sintering of mechanically alloyed powders allows to maintain structural disorder of the CTS phase, while resulting in nanostructured ceramics. This leads to extremely low lattice thermal conductivity. On the other hand, the sample with the smallest average domain size also presents the lowest resistivity and Seebeck coefficient, resulting in the highest power factor. Experimental evidence suggests that S and ρ correlate with the sintering temperature which induces stoichiometric deviations and different grain sizes. First-principles (DFT) simulations of three ordered and two disordered surfaces revealed the presence of dangling bonds and localized states near the Fermi level. DOS at the Fermi level was continuous i.e., bandgap was absent, suggesting that CTS grain boundary is conductive. Formation energies for stoichiometric CTS, Cu-vacant, Cu-rich, Sn-vacant, and Sn-rich CTS are in increasing order, implying stoichiometric CTS is energetically the most favourable, followed by Cu-vacant and Cu-rich systems. The Cu-vacant, Cu-rich, and Sn-vacant systems increase the p-type behaviour and carrier concentration, thus explaining the observed low resistivity of the sintered components. Samples with average domain sizes of 12±10 nm, 25±10 nm, and 37±10 nm displayed a figure of merit zT of 0.40, 0.17, and 0.14, respectively, around 673 K. The results presented here for various CTS surfaces, formation energies, and doping effects, can be generalized to a broad family of Cu-Sn-S-based systems. Our synthesis method also provides an interesting route for many materials to synergistically tune their electrical and thermal properties. TOC:

3 .Figure 1 .

 31 Figure 1. XRD patterns for the powder milled for 15, 30, and 60 min.

Figure 2 .

 2 Figure 2. (i) XRD pattern of samples sintered on various conditions.; (ii) Riet1veld refinement data for Sample A (collected (Yobs), calculated (Ycal), difference (Difference) pattern. See Figure S2 for the Rietveld refinement data for samples B and C.

Figure 3 .

 3 Figure 3. (i) SEM micrograph of the surface of SPS sintered sample A showing compact sample with almost no pores and the presence of SnS.; (ii) WC particles on one of the samples (bright spots in the micrograph, identified by EDX); (iii) micrographs of polished samples A and (iv) B. EDX data are provided in TableS1.

Figure 4 .

 4 Figure 4. Temperature-dependent (i) absolute Seebeck Coefficient, (ii) Resistivity, (iii) power factor (PF), and (iv) diffusivity for sintered samples A, B, and C.

Figure 5 .

 5 Figure 5. Monoclinic (ordered) CTS slabs with orientation (001), (100), and (010), and corresponding Density of states (DOS) plots. Here, the Fermi level is set to zero. Cu, Sn, and S atoms are represented by blue, grey, yellow colours, respectively.

Figure 6 .

 6 Figure 6. Structures and corresponding Density of states (DOS) for two disordered structures, where one has Cu-rich and other has Sn-rich layer, just below the terminating sulfur layer. Here, the Fermi level is set to zero.

  data on 60-min milled CTS powder. Rietveld refinement data. Corresponding EDX data for figure 2. Comparative SEM-EDX data for unpolished and polished sample A and B. Temperature dependent absolute Seeebeck coefficient and resistivity for various CTS samples with increasing average domain size. Density of States (DOS) for Stoichiometric, Cu-vacant, Cu-rich, Sn-vacant, and Sn-rich CTS systems.

Table 1 .

 1 Sample names, sintering conditions, densities, phase purity, and average grain size for various CTS samples.

	Sample	Sintering	Sintering	Sintering	Dwell	Density	Weight Fraction	Average
	name	Die	Pressure	Temperature	Time	(g/cm 3 )	(±1%)		Domain
			(MPa)	( o C)	(min)		Disordered-CTS	Secondary phase	Size (±10 nm)
								name
	A	WC	500	400	60	4.06	99	SnO2-1	12
	B	Graphite 64	512	30	4.52	100		25
	C	Graphite 64	500	30	4.45	99	SnO2-1	37

Table 2 .

 2 Energy of system, formation energy, and formation energy per unit atom for various CTS systems

	System	Configuration	Energy of the	Formation	Formation energy
			System	Energy	per unit atom
			eV	eV	eV/atom
	CTS	24-Cu, 12-Sn, and 36-S	-308.0450	-23.8016	-0.3306
	Cu-vacant 23-Cu, 12-Sn, and 36-S	-303.7652	-23.2520	-0.3275
	Cu-rich	25-Cu, 11-Sn, and 36-S	-307.5523	-23.42630	-0.3254
	Sn-rich	23-Cu, 13-Sn, and 36-S	-306.7465	-22.38570	-0.3109
	Sn-vacant	24-Cu, 11-Sn, and 36-S	-302.2878	-21.8920	-0.3083

-8

WΩ/K 2 and S in µV/K.

Acknowledgement:

The authors are thankful to C. Bilot, J. Lecourt, and Dr. Mirco D'Incau for technical support.

REFERENCES:


Elaine Chew
email: echew@usc.edu

Jie Liu Epstein

Alexandre R J François
email: afrancoi@usc.edu

ESP: Roadmaps as Constructed Interpretations and Guides to Expressive Performance

Keywords: H.5.5 [Information Interfaces and Presentation]: Sound and Music Computing -methodologies and techniques; H.5.2 [Information Interfaces and Presentation]: User Interfaces -input devices and strategies Design, Experimentation, Human Factors, Performance Expression synthesis, driving interface, expressive performance, musical interpretation, motion metaphor

The Expression Synthesis Project (ESP) literalizes the driving metaphor to allow users to create expressive performances through a driving interface. In musical performance, controlled variations in tempo, dynamics, and articulation, lead to musical gestures that clarify structural ambiguities and create musical affect. Guided by the road in ESP, the user changes the tempo by depressing the gas or brake pedals; dynamics are coupled with acceleration, and articulation is controlled independently. Straight road segments suggest opportunities to accelerate, while varying degrees of road bends prompt slowdowns of different levels. The road serves as a representation of, and guide to, a particular interpretation of a piece. This paper focuses on the concepts underlying the design of the road in ESP, providing case studies on mapping diverse interpretations to roadmaps. We describe the design of two roadmaps each for Brahms' Hungarian Dances Nos. 2 and 3. We show that the road has a distinct influence on the way a driver traverses a path, presenting empirical results of driving/performance outcomes using the roads designed, and analyzing the tempo time series. General trends of slowdowns near curvature extremities and accelerations are observed. Statistics show that a particular roadmap has the effect of constraining the performances to a certain style, with specifically placed gestures, while allowing the driver latitude in expressive control. Performances from the same paths are shown to be more similar to each other than those from different paths, having average correlation coefficients of 0.39 vs. 0.16 for Dance No.2, and 0.61 vs. 0.38 for Dance No.3. Overall, the interpretation maps resulted in expressively pleasing performances made available for further examination.

INTRODUCTION

This paper focuses on musical interpretation and the encoding of an interpretation in the mapping of a road within the Expression Synthesis Project's (ESP's) driving interface for generating expressive performances. The ESP driving interface is shown in Figure 1.

Musical interpretation has been the subject of much research (see [START_REF] Gabrielsson | Music Performance[END_REF]). When a composer creates and documents a composition for subsequent re-creation, the written score is an approximation of the idea of the piece. The approximation is incomplete, leaving ample room for interpretation when a performer realizes the score in a performance. Through the shaping of tempo, dynamics, and articulation in the performance of a musical piece, an interpretation serves to convey musical meaning by clarifying structural ambiguities, communicating expressive gestures, and evoking emotional effect (see p.118-121 in [START_REF] Palmer | Music Performance[END_REF]). An interpretation is the result of analysis, as well as decisions on gesture and affect. A well-crafted interpretation can focuses attention, create interest, and clarify important structural patterns in the music. It is important to point out that there is not just one solution to an interpretation of a score. There exist multiple interpretations of viable and effective grouping boundaries, and patterns of tension, relaxation, and motion, in the reading or hearing of any composition.

ESP [START_REF] Chew | ESP: A Driving Interface for Expression Synthesis[END_REF][START_REF]ESP -Expression Synthesis Project[END_REF], takes the metaphor of driving for expressive performance literally to create a driving interface for the control of expressive parameters such as tempo, loudness, and articulation. Anecdotal evidence suggests that performing a piece of music also shares many similarities with the act of driving a car. Motion in an automobile possesses the added feature of machine actuation of acceleration and deceleration. The advantage of a literal reading of the metaphor of driving for expressive performance is that in both driving and expressive control, smooth tempo (speed) control is a hallmark of an expert operation. The ESP interface allows novices to create expressive performances without having to master an instrument, it allows performers to experiment with interpretations prior to mastering the notes in a piece, and it provides a means for composers who may not be expert performers on numerous instruments, to record their preferred interpretation of their compositions for these instruments.

The focus of this paper is the mapping of musical interpretations systematically to the road in the virtual environment in ESP, so as to guide the user toward particular expressive choices. Todd [START_REF] Todd | The kinematics of musical expression[END_REF] has shown that crescendo (increasing loudness) and decrescendo (decreasing loudness) shapes indicate structural groups, and the level of embedding determines the dynamic range; in addition, dynamics are coupled to the tempi. Therefore, one can infer from these observations (and experience) that tempo modulation is an important tool in clarifying structural ambiguities. In ESP, we exploit patterns of driving behavior (such as slowing down at a bend to maintain control and speeding up when the road is straight) to guide the user towards particular expressive gestures that can help resolve certain structural ambiguities.

In ESP, the user manipulates tempo and dynamics parameters by depressing the gas or brake pedals. While the function of the pedals in the ESP interface is clear, the wheel and road play an equally important, although less immediately apparent, role in expression synthesis. By mapping an interpretation, a particular sequence of motion patterns and grouping boundaries, to a road that suggests certain driving patterns, we use the road to influence the performance of a piece of music. The user's actions are guided by the road in the virtual environment. Straight segments suggest opportunities to accelerate, while varying degrees of road bends prompt different levels of slowdowns. It is the steering wheel and the road that allows the user to interact with the road designed so as to create a pleasing performance. It is important to note that the road is simply a suggestion of an interpretation. Nothing stands in the way of the user who wishes to accelerate into, and speed through, a curve, nor one who decides to slow down during a straight section. As in live expressive performance on a traditional instrument, the choice is entirely up to the user/performer. In the case of the ESP interface, the road not only serves to guide the expression, the resulting map represents a particular interpretation of the piece. This paper introduces the concepts underlying the design of the road, and case studies on mapping interpretations to roadmaps and vice versa. The purpose of this paper is to make clear the connections between the road and a particular interpretation, and to show that such a strategy indeed guides the user towards particular expressive interpretations over others. The remainder of the paper is organized as follows: in Section 2, we demonstrate the mapping of two interpretations each to Brahms' Hungarian Dances Nos. 2 and 3. We show the roadmaps constructed from each of these four interpretations of the two pieces. In Section 3, we show and analyze empirical data collected from the "performances" created on these roads in ESP. Graphs of tempo/acceleration over musical time (fractions of a beat) are compared to the corresponding curvature graphs and analyzed for similarity across performances on different roadmaps. We present conclusions and discussions in Section 4.

About ESP

In this section, we quickly outline the capabilities and mappings of the ESP interface. The details of the system design and mapping strategies can be found in [START_REF] Chew | ESP: A Driving Interface for Expression Synthesis[END_REF], [START_REF] François | An Architectural Framework for Interactive Music Systems[END_REF] and [START_REF] Liu | User Interface Design in ESP: Ensuring Tempo Smoothness in the Mapping of Driving to Music Performances[END_REF]. In ESP, the pedals control the speed of the car. Like most driving interfaces, depressing the right pedal causes the car to speed up. Depressing the left pedal activates the brakes, and causes the car to slow down. Tempo control on a prescribed road is a focus of this paper. The system design and the software architecture style for enabling real-time interaction is described in [START_REF] François | An Architectural Framework for Interactive Music Systems[END_REF]. The mapping strategies to ensure tempo smoothness are described in detail in [START_REF] Liu | User Interface Design in ESP: Ensuring Tempo Smoothness in the Mapping of Driving to Music Performances[END_REF].

The road, the other central topic of this paper, represents the music, and one particular interpretation of its grouping structure. Prior research has shown that tempo slowdowns tend to occur at group boundaries, for example, at the edge of phrases. These interpretative cues are embedded in the twists and turns in the road. Bends in the road suggest places to slow down, and straight sections encourage speedups. By careful placement of these bends and straight sections, we design of the road in such a way as to guide the user towards a particular, hopefully desirable, interpretation.

The car can be thought of as a pointer to the music file. Fast travel on the road results in a fast tempo replay, and vice versa for slower speeds. Extra buttons on the wheel control the articulation (note lengths). Presently, the dynamics (loudness) can be coupled with the acceleration or independently engaged (through button control). In the examples in this paper, dynamics are linked directly to the acceleration. In previous work, we have mentioned only in passing the mapping of Brahms' Hungarian Dance No. 5 to a road in ESP. This paper will systematically describe multiple interpretations of Dances Nos. 2 and 3 from the same set of pieces, using them as detailed case studies of how an interpretation is created and mapped to a road in the ESP interface.

Related Work

Conductor programs, tools or software that control accurate playback of notes while allowing the user to focus only on features of interpretation (tempo, dynamics, etc.) have been around since the turn of the century (see online review by Malinowski, [START_REF] Malinowski | The Conductor Program www[END_REF]). Some of the most recent additions to this distinguished line of work is the Air Worm [START_REF] Dixon | The PERFORMANCE WORM: Real-Time Visualization of Expression based on Langner's Tempo-Loudness Animation[END_REF], which allows a user to control the tempo and loudness of an audio recording of a piece of music that has been stripped of its expressive nuances, and a literal gestural conducting interface developed in the Personal Orchestra project [START_REF] Borchers | A Music Stand for Interactive Conducting[END_REF]. Control of expressive parameters are also part of accompaniment programs (see for example the Music Plus One system described in [START_REF] Raphael | Orchestral musical accompaniment from synthesized audio[END_REF]), in which the soloist acts as the conductor.

The link between locomotion and expressive performance is a well-established one (see for example, [START_REF] Todd | The kinematics of musical expression[END_REF]). Although expressive performance has primarily been linked to human gait and dance steps [START_REF] Friberg | Music from motion: Sound level envelopes of tones expressing human locomotion[END_REF], anecdotal evidence suggests that performing a piece of music also shares many similarities with the act of driving a car. Entertainment companies focusing on music games have also used the metaphor of movement through space for travel through musical time. See, for example, Harmonixmusic's [8] PS2 games Amplitude, Frequency, and Guitar Hero. Their focus is on generating music as a path that is traversed repeatedly in a loop. Such music games differ from ESP, in which no notes are altered, added or deleted in the process of a performance. Instead, ESP focuses on the synthesis of musical expression. The only parameters under the user's control are those that pertain to the rendering of expression, i.e. tempo, dynamics, and articulation.

[2] contains further descriptions of projects and research related to ESP.

FROM INTERPRETATIONS TO ROADMAPS

In musical interpretation, the choice of structural groupings can exist on multiple levels, and exhibit complex and embedded patterns. Such levels of phrasing and embedded groupings can be made apparent by the manipulation of the tempo and loudness. As Todd [START_REF] Todd | The kinematics of musical expression[END_REF] discovered, crescendo and decrescendo shapes of varying dynamic ranges indicate the level of embedding in a system of phrases. Furthermore, the dynamics increase and decrease with the tempi.

In ESP, the road guides users to show the grouping structures through a combination of straight and curved sections. For each interpretation, we create a road that encourages the particular set of grouping structures. Bends are inserted at the grouping boundaries to encourage slowdowns of varying degrees. The curvature of a road bend corresponds to the level of embedding in the phrase structure, and the degree of slowdown desired. Because the dynamics are coupled with the acceleration in ESP, the dynamics do increase and decrease with the tempi.

The mapping from an interpretation to a road follows a few basic principles: (1) adjacent phrases at the lower levels (smaller groupings) are delineated by a bend in the same direction as the general curvature of the road; (2) larger groupings at higher levels are delineated by curve that marks a change in direction; and, (3) larger boundaries, such as those between sections, have steeper bends than elsewhere.

The positions of the curves are adjusted to account for variability in response time to road bends. Human response time to the bends vary with the degree of curvature. In driving, the user tends to decelerate before entering a bend. The distance before the bend at which this deceleration first occurs depends on the sharpness of the curve. Gentler bends require less slowdown, and hence the deceleration occurs closer to the curve. Sharp bends demand more care in driving, and encourage slowdowns well before the curve.

In the following sections, we show examples of how different interpretations of Brahms' Hungarian Dances Nos. 2 and 3 are mapped to different roads in ESP. We choose Brahms' Hungarian Dances as example pieces because of the extreme ranges of tempi and dynamics (as well as articulation) encapsulated within each piece in the set. They form an excellent test set for the ESP interface.

Mapping Study I: Roadmaps for Brahms' Hungarian Dance No. 2

In this section, we present two expressive interpretations of the first 32 bars of Brahms' Hungarian Dance No. The second interpretation, shown in Figure 3, stresses melodic peaks (shown by single grey-lined mounds in bars 3 and 7) and areas of harmonic tension (indicated by squiggles) and release (forward arrows), which lead to musical gestures that often do not correspond to the symmetric phrase structures.

The resulting roads corresponding to the symmetric phrasing and gesture-based interpretations are shown in Figures 4 and5 respectively. The maps on the green backgrounds are the birds' eye views provided to the user; the color-segmented maps on the right show the AABB sections of the excerpt mapped, indicated by red, orange, green, and blue segments respectively. Table 1 shows an example of a curvature table from which the roadmap in Figure 4 is constructed. In this example, we chose to alternate the direction of the curves between left (L) and right (R).

Phrase boundaries at the lowest level are marked by small curves, and higher level phrase boundaries correspond to larger bends. In the first interpretation, we take the most obvious (naïve) phrase grouping arising from strict adherence to the bar-lines (see Figure 1). The lowest level grouping follows the barline, one group per bar. This results in a march-like two-step rhythm in each bar in Section A (the first 12 bars, counting repeated bars). Each set of three bars form the next level grouping, and are shown by larger phrase arcs in Figure 6. In Section A, there is the option to create a middle level grouping of 1:2 or 2:1 within each three-bar set.

We have chosen the 2:1 mid-level grouping for Section A. In the B section, again, the lowest level grouping resides at the bar, and the high level grouping consists of three bars each. Due to the continuous flow of this new material, we do not introduce a midlevel grouping here.

The resulting roadmap for this first (naïve) interpretation is shown in Figure 8. As before, the map on the green background shows the user's bird's eye view, while the color-segmented map indicates the ABA segments (red, green, and red again). The red segments are identical, and cross each other in the two-dimensional map. In the color-coded map, the yellow header contains the upbeat note and marks the beginning of the piece.

Due to the brevity of the lowest level phrase in this piece, the junction between each phrase pair at the bar level is marked by small curves in the same direction as the preceding segment. If this junction corresponds to a phrase boundary at a higher level, then the road changes direction. The B section is constructed in a similar fashion. Between the A and B sections, we increased the curvature of the bend to mark the large structural boundary. To prevent too much of a slowdown (resembling stalling) in the second set of three bars, the curve here is made more gentle to encourage forward motion.

In a second, more nuanced, interpretation (shown in Figure 7), we choose structural groupings that do not adhere to the barlines; instead, they follow the melodic grouping principles of longer notes and larger interval leaps marking boundaries, short upbeats being grouped with the longer notes, and of repeated patterns and symmetries signaling groups. For example, the opening upward gesture (C, CDE, F) is followed by another upward gesture (F, Bb, D, C) in the right hand, forming the first two low level groups. At the next level up, the second set of four eighth notes follows a pattern similar to the first set of four eighth notes, forming two higher level groups. As in the previous interpretation, the larger phrases consist of three bars each, with the option to subdivide into 1:2 or 2:1, the difference being that the groupings in this section are not aligned with the barlines. For a change, we have chosen the 1:2 mid-level grouping in Section A.

In Section B, again the phrases are offset from the barlines by an eighth note. In this more nuanced interpretation, we add a mid-level grouping that subdivides the high level three-bar phrase into two parts, each lasting six eighth notes, cutting across the barlines, and resulting in a pleasing symmetry.

This more nuanced interpretation encouraged us to examine variance in grouping significance for structural boundaries at the same level. We found that not all boundaries at the same level should be treated the same way. For example, even though the boundaries between the four sets of three bars in section B (green)

were at the same level, we chose to make the middle one less then the others so as to continue the forward motion in the music.

The resulting roadmap for this second (nuanced) interpretation is shown in Figure 9. As for the naïve interpretation map, the lowest level phrase boundaries were marked by same-direction bends; the road direction changes for higher-level boundaries. Because this interpretation does not follow the bar lines, the distances of the phrase groupings are not as symmetrical as that for the naïve interpretation. We also found that because of the range of curvature types and proximity of curved sections, we had to be more careful with planning the placement of the curves to adjust for different deceleration patterns when approaching a curve.

FROM ROADMAPS TO PERFORMANCES

In this section, we present and analyze the driving (performance) patterns resulting from the maps of the different interpretations of the two Hungarian Dances. One of the authors (Jie Liu), an experienced user of the ESP system, created ten performances each on every one of the roads described in Section 2. The driving style was careful, slowing down and bends as necessary in order to minimized collisions with the road edge. The numerical data sets are presented in this section; the MIDI files of the performances can be obtained from wwwrcf.usc.edu/~mucoaco/ESP/2006-AMCMM for validation. The effect of an expressive performance cannot be fully comprehended by simply examining tempo graphs. Thus, the reader is strongly encouraged to listen to the examples posted by visiting the url given above.

Performance Results I: Brahms' Hungarian Dance No. 2

The resulting tempo curves for the two interpretation maps, symmetric phrasing and gesture-based, for Brahms' Hungarian Dance No. 2 are shown in Figure 10. The corresponding MIDI files are named Brahms_HD2_symmphrasing_m.mid and Brahms_HD2_gesture_n.mid respectively in the MIDI repository, where m, n = 1, …, 10. In Figure 9, Each line in the two graphs shows the result of one drive through a path -there are ten in total for each map, all produced by the same driver (Jie Liu). The ten iterations through the same path show the patterns of tempo variance over different iterations over the same road. For this data set, the driver took a break after each set of five performances. On the graphs, each section of the map is indicated by A or B labels, with color-codes that parallel the road segments. The tempi are given in beats per second; for example, a tempo level of 1 corresponds to 60 beats per minute.

In general, the gesture-based roadmap resulted in more extreme speedups and slowdowns, as intended. In other words, the gesture-based map resulted in larger musical gestures. For example, each section A is marked by a steep slowdown near its beginning, a gesture at the right place, but of magnitude perhaps larger than intended; and each section B begins with an acceleration in tempo, as intended in the road design. The musical effect can best be examined through hearing playbacks of the MIDI files recorded from these performances.

The systematic variations in the symmetric phrasing map are less easy to define by inspection. Recall that even though bends encourage slowdowns, the user ultimately still has control over the expressive gestures through their dexterity at the wheel and real-time auditory feedback of the unfolding piece. The latitude afforded the user in making performance decisions makes it difficult to map every slowdown to a particular bend. To demonstrate empirically that the two sets of tempo time series are significantly different, we calculated, for each pair of performances, the correlation coefficient and the sum of squared errors (SSE) between their tempo curves. The equations for the correlation coefficient and the SSE measures for two tempo sequences, {x i } and {y i }, are as follows:

r = 1 n -1 x i -x s x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ y i -y s y ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ i ∑ ,

Performance Results II: Brahms' Hungarian Dance No. 3

The performance (driving) results (tempo graphs) for the naïve and nuanced interpretations of Brahms' Hungarian Dance No. 3 are shown in the two graphs in Figure 12. The corresponding MIDI files are named Brahms_HD3_naive_m.mid and Brahms_HD3_nuanced_n.mid respectively in the MIDI repository, where m, n = 1, …, 10. As before, the same driver (Jie Liu) drove on each path ten times to create the twenty performances shown on the two graphs.

Again, by inspection, the two different roads produced different tempo trajectory patterns. One prominent difference is the marked tempo slowdown exhibited by all performances on the nuanced interpretation road in the return from the B to A section, as shown by the local minima of the entire bunch of tempo curves around the second dotted vertical lines in Figure 12.

As before, we show the distributions of the correlation coefficients and the SSE's of performances from the same and different roads in Figure 13. The correlation coefficients of same and different road performances come from distributions with different means. The mean correlation coefficient value for same road performances is 0.61, while that for different road performances is 0.38. The SSE values ranged from 0 to 41. Although the SSE peaks for same and different road performances are not visibly different, their underlying means are 9.4 and 13.0, respectively, for same and different road performances. Again, both measures show that, overall, performances resulting from the same road patterns are more similar to one another than those from different roads.

CONCLUSIONS AND DISCUSSION

We have demonstrated, in this paper, examples of multiple interpretations of the same scores, and how this could map to different roads in the ESP interface. We have presented empirical results of driving/performance outcomes using these roads in ESP, and analyzed their tempo time series. The general trends of slowdowns near curvature extremities (which correspond to structural boundaries or gestural slowdowns) and accelerations when the road is straight can be observed in many cases. Statistics show that the road has the effect of constraining the performances to a particular style, with specific gestures, although the user (driver) still maintains some latitude in expressive control. Overall, the interpretation maps for the two Hungarian Dances resulted in expressively pleasing performances.

A natural question arises on the role of auditory feedback in the performance/driving of a piece. For an expert driver, the speeds will likely be much higher, with greater variance, without the moderating effects of auditory feedback. Thus, the tempo trajectory recorded in each performance is not only the result of the road mapped according to an interpretation of the score, it is also guided by the user's hearing of the piece. Further experiments will have to be designed to determine the extent of the effect of auditory feedback on expressive control.

Although many curvature maxima and minima, which correspond to right or left turns, induced tempo slowdowns, the actual local tempo minima are sometimes a little before, sometimes a little after, the extreme points. Occasionally, little slowdown, or even speeding up, is observed at a bend. We observed that the link between deceleration and curvature extremity is more pronounced in the cases when the curves can be more reliably predicted through adequate lead-ins or periodic spacing. When there is little time between curves, their slowdown effect is reduced, and the performance will show low correlation between tempo slowdowns and curvature peaks.

Aural inspection (listening) shows that the performances generated on the roadmaps constructed from particular interpretations of the score did indeed achieve the desired effects marked in the interpreted scores. The sense that the guided performance worked as planned is not always readily captured in the tempo data (which is correlated with the loudness because of the link between loudness and acceleration). For example, a positive tempo change of 0.2 beats per second, is sometimes smoothed over by the human ear and regarded as stable, even though visually the graph shows a significant tempo increase. The gap between perception and signal leads us to postulate that perhaps the instantaneous tempo should be considered a fuzzy quantity in the extraction and understanding of musical gesture.

More work remains to be done to devise new and more precise ways of evaluating the effectiveness, and degree of influence, of the ESP interface. The challenges include the subjectivity of music performance evaluation, and the fact that unique and outlier performances can often be desirable ones. Systematic evaluation in the future will likely include investigations such as (1) the impact of a road pattern on driving (performance) behavior; (2) the impact of auditory feedback on driving (performance) choices; and, (3) the effect of repetition on expressive choices over the same material. It would be desirable to validate the experimental results across multiple users.

One way that music creates interest is by generating expectations and meeting them through repetition. For example, in Brahms' Hungarian Dance No. 2 example, the structure is AABB, where the A and B sections are each repeated; in the Hungarian Dance No. 3 example, the A section returns at the end of the ABA structure. A real performer will likely play the second time a section appears a little differently than the first. The same is hypothesized to be true in performances generated through the ESP interface. Evidence for this hypothesis can be found, for example, in the two iterations each of A and B sections in performances of Brahms' Hungarian Dance No. 2 produced from the gesture-based map (shown in the lower half of Figure 10). Future experiments will focus on the intentional expressive choices exercised by the user when a section is repeated. Such investigations will also reveal the nature of the effect of listening on performance choices.

Figure 1 .

 1 Figure 1. The ESP driving interface for creating expressive performances

Figure 2 .Figure 3 .

 23 Figure 2. Hungarian Dance No. 2, symmetric phrasing.Figure 3. Hungarian Dance No. 2, gestures.

2

 2 and their corresponding roadmaps. With the two repeats, we have a total of 64 bars of music in an AABB form. The first interpretation is symmetric and square, grouping bars by powers of two. The symmetric grouping at different hierarchical levels, indicated by phrase arcs, is shown in Figure 2. The bar groupings can be represented numerically (by bar numbers) as: [ (1,4)(5,8) ] { [ (9,10)(11,12) ][ (13,14)(15,16) ] } … .

Figure 4 .

 4 Figure 4. Hungarian Dance No. 2 roadmap, symmetric.

Figure 5 .

 5 Figure 5. Hungarian Dance No. 2 roadmap, gestures.

  . The third Hungarian Dance is marked by an atypical 3-bar phrase structure, which gives rise to asymmetries and many possibilities for alternate groupings. The first 24 bars (counting the repeated bars) are shown in Figures6 and 7. Bars 25-36 (not shown) repeat material in bars 1-12, resulting in an ABA structure.

Figure 6 .

 6 Figure 6. Hungarian Dance No. 3, naïve interpretation.

Figure 7 .

 7 Figure 7. Hungarian Dance No. 3, nuanced interpretation.

Figure 8 .

 8 Figure 8. Hungarian Dance No. 3 roadmap, naïve interpretation.

Figure 9 .Figure 10 .

 910 Figure 9. Hungarian Dance No. 3 roadmap, nuanced interpretation.

Figure 12 .Figure 11 .

 1211 Figure 12. Tempo curves for Hungarian Dance No.3 from (top) naïve and (bottom) nuanced interpretation maps.

Figure 13 .

 13 Figure 13. Hungarian Dance No.3: histograms of (top) correlation coefficient and (bottom) SSE values for performances from same and different roads.

Table 1 . Curvature data for constructing roadmap for Brahms' Hungarian Dance No. 2, symmetric phrasing

 1

	Curve	Length	Degree Turn	Direction
	Number	(beats)
	1	5.5	1	L
	2	3.5	100	R
	3	4.5	10	L
	4	3.5	120	R
	5	4	80	L
	6	4	120	R
	7	4	100	L
	8	4	150	R
	9	4.5	1	L
	10	3.5	100	R
	11	4.5	10	L
	12	3.5	120	R
	13	4	80	L
	14	4	120	R
	15	4	100	L
	16	4	150	R
	17	4	80	L
	18	4	100	R
	19	4	80	L
	20	4	120	R
	21	4	80	L
	22	4	100	R
	23	4	80	L
	24	4	120	R
	25	4	80	L
	26	4	100	R
	27	4	80	L
	28	4	120	R
	29	4	80	L
	30	4	100	R
	31	4	120	L
	32	4	150	R
	2

.2 Mapping Study II: Roadmaps for Brahms' Hungarian Dance No. 3 In

this section, we propose two grouping solutions for the first 36 bars of Brahms' Hungarian Dance No. 3, shown in Figures

6 and 7

  and y are the means, and s x and s y are the standard deviations of {x i } and {y i } respectivelyThe distributions of these values are shown in Figure11. The correlation coefficient values for same road and different road performances come from distributions with different means, 0.16 for performances from different roadmaps versus 0.39 for performances from the same roadmap. The SSE values ranged from 0 to 65. The SSE distributions for same and different road performances are each bimodal with significantly different peaks, at 20 and 55 for performances from different roadmaps versus 0 and 25 for performances from the same roadmap. The mean SSE for different road performances is 39.2, while that for same road performances is 28.4. Both the correlation coefficient and the SSE measures show that, in general, performances resulting from the same road are more similar to each other than those from different roads.

	and SSE =	∑	x i -y i ( ) 2	,
		i
	where x

ACKNOWLEDGMENTS

We thank Dennis Leung for his help in encoding the Brahms Hungarian Dances in Sibelius format for generating expressionless MIDI and score information. This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 0347988. The work made use of the Integrated Media Systems Center Shared Facilities supported by NSF under Cooperative Agreement No. EEC-9529152. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF.