David Boucher

Jeppe Madsen

Liyun Yu

Qian Huang

Nicolas Caussé

Nadine Pébère

Vincent Ladmiral

Claire Negrell

Polystyrene hybrid-vitrimer based on hemiacetal ester exchange reaction

Vitrimer materials, which are permanently cross-linked but reshapable polymers, are a class of materials of growing interest due to their potential in recycling and sustainable development. More precisely, vitrimers derived from commodity polymers could have a drastic impact on plastic consumption in the upcoming years. Here, the development of a polystyrene-based vitrimer harnessing the recently discovered hemiacetal ester exchange reaction of carboxylic acid is presented. 1,4-cyclohexanedimethanol divinyl ether was reacted with 4-vinylbenzoic acid to form a divinyl species containing two hemiacetal ester functions. This dimer was then copolymerized with styrene, butyl acrylate and 4-vinylbenzoic acid to form a cross-linked material presenting pendent acid groups. This material exhibited three distinct behaviors at different temperature regimes: below 80 °C, despite the ability of exchange reactions to occur, the cross-linked material behaved as a frozen material due to hydrogen bonding formed by carboxylic acid pendent groups. Between 80 °C and 130 °C vitrimer behavior was demonstrated by rheological creep experiments. At higher temperature, the dissociation of the hemiacetal ester groups led to the loss of the covalent cross-linking.

Introduction

Vitrimers constitute a very interesting class of materials, since their inception in 2011 by Leibler's group [START_REF] Montarnal | Silica-Like Malleable Materials from Permanent Organic Networks[END_REF] as reprocessable, permanently cross-linked materials. This feat was made possible by the integration of associative exchange reactions in the matrix, namely catalyzed transesterification reactions. These associative exchanges allowed the reorganization of covalent bonds under thermal treatment. Such materials differ from most dissociative covalent adaptable networks, where exchange reactions occur through the dissociation-reformation of covalent bonds, because vitrimers possess a rheological behavior similar to that of glass (evolution of the relaxation time, or of zero-shear viscosity, with temperature following an Arrhenius equation), indicating a conservation of the amount of cross-link nodes during reshaping. [START_REF] Montarnal | Silica-Like Malleable Materials from Permanent Organic Networks[END_REF] Due to this ability to be reshaped without breaking the crosslinks, and thus keeping their mechanical properties, a high potential for sustainable economy (by recycling of cross-linked material) was attributed to these materials. Since this first proof of concept, vitrimers based on a wide range of exchange reactions have been reported. [START_REF] Denissen | Vitrimers: Permanent organic networks with glass-like fluidity[END_REF][START_REF] Guerre | Vitrimers: directing chemical reactivity to control material properties[END_REF][START_REF] Jourdain | Rheological Properties of Covalent Adaptable Networks with 1,2,3-Triazolium Cross-Links: The Missing Link between Vitrimers and Dissociative Networks[END_REF] The search for new vitrimer materials led to the classification of numerous systems as vitrimers only based on the fact that the evolution of their relaxation time, or zero-shear viscosity, with temperature followed an Arrhenius equation. [START_REF] Denissen | Vitrimers: Permanent organic networks with glass-like fluidity[END_REF][START_REF] Jourdain | Rheological Properties of Covalent Adaptable Networks with 1,2,3-Triazolium Cross-Links: The Missing Link between Vitrimers and Dissociative Networks[END_REF] However, it was recently demonstrated that some dissociative covalent adaptable networks may also be described by an Arrhenius equation. [START_REF] Jourdain | Rheological Properties of Covalent Adaptable Networks with 1,2,3-Triazolium Cross-Links: The Missing Link between Vitrimers and Dissociative Networks[END_REF][START_REF] Scheutz | Adaptable Crosslinks in Polymeric Materials: Resolving the Intersection of Thermoplastics and Thermosets[END_REF] This behavior is attributed to a faster formation than dissociation of the covalent bonds in a temperature range, leading to a quasi-constant cross-link density. [START_REF] Scheutz | Adaptable Crosslinks in Polymeric Materials: Resolving the Intersection of Thermoplastics and Thermosets[END_REF][START_REF] Hayashi | Implantation of Recyclability and Healability into Cross-Linked Commercial Polymers by Applying the Vitrimer Concept[END_REF] In addition Breuillac et al. [START_REF] Breuillac | Polybutadiene Vitrimers Based on Dioxaborolane Chemistry and Dual Networks with Static and Dynamic Cross-links[END_REF] demonstrated that a vitrimer with few cross-links can show solubility in good solvents at temperatures where the exchange reactions take place sufficiently fast. This additional understanding of dissociative covalent adaptable networks and vitrimer behavior indicates that determination of solubility [START_REF] Breuillac | Polybutadiene Vitrimers Based on Dioxaborolane Chemistry and Dual Networks with Static and Dynamic Cross-links[END_REF][START_REF] Ricarte | Linear Viscoelasticity and Flow of Self-Assembled Vitrimers: The Case of a Polyethylene/Dioxaborolane System[END_REF] or rheological behavior [START_REF] Jourdain | Rheological Properties of Covalent Adaptable Networks with 1,2,3-Triazolium Cross-Links: The Missing Link between Vitrimers and Dissociative Networks[END_REF] is not sufficient to classify a novel material as a vitrimer. [START_REF] Van Zee | Vitrimers: Permanently crosslinked polymers with dynamic network topology[END_REF] In the present article, a vitrimer is thus considered as a cross-linked system where reprocessability and glass-like rheological behavior are brought by associative exchange reactions. Vitrimer properties can also be sometimes combined with other properties that affect the material flow, such as reversible physical interactions. In this case, the material is defined as hybrid-vitrimer. [START_REF] Guerre | Vitrimers: directing chemical reactivity to control material properties[END_REF]

Scheme 1. Hemiacetal ester exchange

Due to the sometimes low compatibility of catalysts with the matrix 3,10 and potential negative environmental impact, [START_REF] Han | A Catalyst-Free Epoxy Vitrimer System Based on Multifunctional Hyperbranched Polymer[END_REF] significant effort has been devoted to developing catalyst-free vitrimer materials. [START_REF] Guerre | Vitrimers: directing chemical reactivity to control material properties[END_REF][START_REF] Scheutz | Adaptable Crosslinks in Polymeric Materials: Resolving the Intersection of Thermoplastics and Thermosets[END_REF] Another, and more recent, research trend is the development of vitrimers from commodity plastics. [START_REF] Scheutz | Adaptable Crosslinks in Polymeric Materials: Resolving the Intersection of Thermoplastics and Thermosets[END_REF] This path was first explored by Leibler's group [START_REF] Röttger | High-performance vitrimers from commodity thermoplastics through dioxaborolane metathesis[END_REF][START_REF] Ricarte | Phase Separation and Self-Assembly in Vitrimers: Hierarchical Morphology of Molten and Semicrystalline Polyethylene/Dioxaborolane Maleimide Systems[END_REF] with the development of polyethylene, poly(methyl methacrylate) and polystyrene vitrimers based on dioxaborolane metathesis. These systems were prepared in two steps: first the synthesis of linear chains with pendent dioxaborolane functions by Reversible Addition-Fragmentation chain Transfer (RAFT) polymerization and then the resulting linear polymers were cross-linked using bis-dioxaborolane cross-linker. The resulting materials showed, in addition to vitrimer behavior, similar mechanical properties as the commodity plastics used as references and reshaping did not alter their properties. The authors presented the development of such materials as an important step to shift vitrimers from academic research to industrial applications due to the position of commodity plastics in the global annual plastic production (more than 75%). [START_REF] Röttger | High-performance vitrimers from commodity thermoplastics through dioxaborolane metathesis[END_REF] Since this seminal work, other commodity vitrimer materials were developed, such as the silyl ether-based polystyrene vitrimer reported by Nishimura et al. [START_REF] Nishimura | Silyl Ether as a Robust and Thermally Stable Dynamic Covalent Motif for Malleable Polymer Design[END_REF] and the vinylogous amine polyurethane-based vitrimer materials reported by Sumerlin et al. [START_REF] Sims | Functional Diversification of Polymethacrylates by Dynamic β-Ketoester Modification[END_REF][START_REF] Lessard | Catalyst-Free Vitrimers from Vinyl Polymers[END_REF] The development of different chemistries able to induce vitrimer properties in a material affords more control over specific parameters such as the flow activation energy, the viscosity and the temperature at which the material can be reshaped. [START_REF] Guerre | Vitrimers: directing chemical reactivity to control material properties[END_REF] Here, the development of a polystyrene-based vitrimer using for the first time the hemiacetal ester exchange (Scheme 1) is presented. Hemiacetal esters were recently shown to undergo associative acid exchange at room temperature without the need for any catalyst. [START_REF] Boucher | Hemiacetal Ester Exchanges, Study of Reaction Conditions and Mechanistic Pathway[END_REF] They also possess a dissociation temperature, Tdis (depending on their structure), above which the hemiacetal ester function dissociates. [START_REF] Otsuka | Poly(hemiacetal ester)s: New class of polymers with thermally dissociative units in the main chain[END_REF][START_REF] Otsuka | Fine-tuning of thermal dissociation temperature using copolymers with hemiacetal ester moieties in the side chain: Effect of comonomer on dissociation temperature[END_REF][START_REF] Kovash | Thermoset coatings from epoxidized sucrose soyate and blocked, bio-based dicarboxylic acids[END_REF] A material based on this exchange chemistry is therefore expected to present two behaviors depending on the temperature range: belowTdis, associative exchanges would occur and lead to vitrimer behavior, whereas above Tdis the dissociation of the cross-links should lead to the loss of the material's vitrimeric properties.

Here, a poly(styrene-co-vinylbenzoic-acid-co-butylacrylate) copolymer cross-linked using a difunctional hemiacetal esters cross-linker was synthesized. Its rheological profile with temperature was examined and compared with three reference materials of similar composition but either without pendent acid groups, without crosslinks or without hemiacetal ester groups. Particular attention was given to Tdis in order to characterize the rheological behavior of the material in its crosslinked state.

Experimental section

Materials

Dichloromethane (DCM), tetrahydrofuran (THF), diethyl ether, 3Å molecular sieves, n-butyl acrylate (99 %), styrene (99 %), 1-4 cyclohexanedimethanol divinyl ether (mixture of isomers 98 %)(CDDVE), butylvinyl ether (98 %), cyano-2propyldodecyl trithiocarbonate (97 %), trifluoroacetic acid (99 %), magnesium sulfate, basic alumina and deuterated chloroform were purchased from Sigma-Aldrich. 4-Vinylbenzoic acid (VBA) (97 %) and n-dodecyl dihydrogen phosphate (90 %) were purchased from TCI Europe and Abcr GmbH, respectively.

Polymerization inhibitors contained in styrene and butyl acrylate were removed using a column of basic alumina.

Polymer synthesis

Drying of the materials 300 mg of 4-vinyl benzoic acid was dissolved in 50 ml of DCM and the solution was dried on MgSO4 for 30 minutes before filtration. The filtrate was then kept in the presence of 3Å molecular sieves and under nitrogen for 1 h before evaporation of DCM under vacuum. The resulting solid 4-vinyl benzoic acid was then used immediately. Water traces contained in styrene and n-butyl acrylate were removed using 3Å molecular sieves under inert atmosphere. n-Dodecyl dihydrogen phosphate (200 mg; 7 10 -4 mol) was dissolved into styrene (5.0 g; 4.8 10 -2 mol) and kept in the presence of 3Å molecular sieves and under nitrogen for 1 h before use. THF was dried using first MgSO4 followed by storage in the presence of 3Å molecular sieves under nitrogen.

Synthesis of the cross-linked material with pendent acid groups (CMA) 180 mg (1.2 10 -3 mol) of 4-vinylbenzoic acid, 53 mg (2.7 10 -4 mol) of 1,4-cyclohexanedimethanol divinyl ether, along with 1040 mg of the n-dodecyl dihydrogen phosphate/styrene solution [i. e. approximately 40 mg (1.4 10 -4 mol) of n-dodecyl dihydrogen phosphate and 1.0 g (9.6 10 -3 mol) of styrene] and 2.1 mg (6.1 10 -6 mol) of cyano-2-propyldodecyl trithiocarbonate, were placed in a 10 ml glass flask, dried at 130 °C in an oven for at least 5 minutes and closed with a septum. The resulting solution was purged with nitrogen for 10 minutes before being stirred for 15 h at 25 °C to convert the vinyl ether functions into hemiacetal esters. After 15 h, the conversion of VBA into hemiacetal ester was monitored using 1 H NMR in deuterated chloroform. 600 mg (4.7 10 -3 mol) of n-butyl acrylate were then added to the main solution and the RAFT polymerization was initiated by heating the solution to 100 °C. After 21 h, the glass flask was cooled down to room temperature for 1 h and then broken to recover the solid material. Residual monomers and catalyst were then extracted by successive immersion of the material in anhydrous DCM (around four times the material volume), one time overnight and twice for 2 h. After a drying step of 4 h at 100 °C in a vacuum oven the resulting material was a transparent colorless solid.

Synthesis of the cross-linked reference (CR)

This material was prepared using the protocol used for the synthesis of the CMA with some modifications. Briefly, 180 mg (1.2 10 -3 mol) of 4-vinylbenzoic acid, 106 mg (5.4 10 -4 mol) of 1,4-cyclohexanedimethanol divinyl ether, along with 1040 mg of the n-dodecyl dihydrogen phosphate/styrene solution [i. e. approximately 40 mg (1.4 10 -4 mol) of n-dodecyl dihydrogen phosphate and 1.0 g (9.6 10 -3 mol) of styrene] and 3.8 mg (1.1 10 -5 mol) of cyano-2-propyldodecyl trithiocarbonate were mixed in a 10 ml glass flask, dried at 130 °C in an oven for at least 5 minutes and closed with a septum. After 15 h of mixing, at 25 °C 70 mg (3.36 10 -4 mol) of 1,4-cyclohexanedimethanol divinyl ether and 600 mg (4.7 10 -3 mol) of n-butyl acrylate were added to the reaction medium and the resulting solution was mixed for 6 additional h. The RAFT polymerization was then carried out at 100 °C for 21 h and the materials was recovered following the protocol described above for CMA. The CR material was again a transparent colorless solid.

Synthesis of the non-cross-linked reference (NR)

This material was prepared using the protocol used for the synthesis of the CMA with some changes to reagents and their amounts, as well as some modifications of the extraction protocol. Briefly, 180 mg (1.2 10 -3 mol) of 4-vinylbenzoic acid were mixed with 200 mg (2 10 -3 mol) of butylvinyl ether, 1040 mg of the n-dodecyl dihydrogen phosphate/styrene solution [i. e. approximately 40 mg (1.4 10 -4 mol) of n-dodecyl dihydrogen phosphate and 1.0 g (9.6 10 -3 mol) of styrene], 600 mg (4.7 10 -3 mol) of n-butyl acrylate, and 3.8 mg (1.1 10 -5 mol) of cyano-2-propyldodecyl trithiocarbonate. Residual monomers and catalyst were then extracted by immersion of the material in diethyl ether, in which the material was not soluble. After a drying step of 4 h at 100 °C in a vacuum oven the recovered material was again a transparent and uncolored solid.

Synthesis of the acid dissociated reference (AR) 300 mg of the CMA were immersed in 100 ml of THF containing 4 ml of trifluoracetic acid. This reaction mixture was then stirred for 24h at ambient temperature (ca. 25 °C). After 24h, the THF was evaporated under vacuum and the material was purified by extraction with diethyl ether as NR. After a drying step of 4 hours at 100 °C in a vacuum oven the recovered material was again a transparent colorless solid.

Formation of disk samples for rheology experiments

Disk samples (8 mm in diameter and about 1.5 mm in thickness) were obtained by molding 40 mg of material under a pressure of 16.2 N/cm 2 , and at a temperature of 90 °C, for 1.5 h.

Characterization

NMR analyses

1 H NMR analyses were performed using a 300 MHz Bruker spectrometer. The acquisition parameters were as follows :

Pulse angle : 90°, acquisition time : 4 seconds, relaxation delay : 1 second and scan number : 16, temperature : 20 °C. Chemical displacement are reported in part per millions (ppm) relative to tetramethylsilane. All samples were prepared in deuterated chloroform.

In the case of CR and NR samples, 40 µl of trifluoroacetic acid (TFA) was added to the NMR tube and the mixture was agitated for 24h to ensure complete substitution of the hemiacetal ester functions by TFA.

Gel permeation chromatography analyses GPC analyses were performed using a Viscotek VE 2001 GPC instrument equipped with a refraction index detection Viscotek Model 302 and two PL gel mixed-D columns. All analyses were achieved using THF as eluent with a 1 ml/min rate flow. The instrument was calibrated with narrow polystyrene standards.

GPC sample preparation: Prior to analysis, 2 mg of copolymer was dissolved in 1 ml of THF containing 40 µl of TFA, except for AR where the absence of crosslinks and pendent acid groups made the addition of TFA unnecessary.

Differential scanning calorimetry analyses DSC analyses were performed on a Discovery DSC apparatus from TA instruments. The temperature program scanned the 20 -100 °C range with a 10 °C/min ramp in a nitrogen atmosphere.

Thermogravimetry coupled with mass spectrometry analyses TGA-MS experiments were performed using a Discovery TGA apparatus coupled with a Discovery MS instrument also from TA instruments. After an isotherm of 60 min at 100 °C aimed to evaporate all residual solvent and water, a temperature ramp of 10 °C/min was applied from 100 to 350 °C in an air atmosphere.

Isothermal thermogravimetry analyses

Isothermal TGA experiment were performed using a Discovery TGA apparatus from TA instruments. A heating ramp of 10 °C/min was applied to reach the temperature of the first isotherm. Then each isotherm was maintained for 5 h and a heating ramp of 2 °C/min was applied between each of the isotherms. These experiments were performed under air atmosphere.

Creep experiments

Creep experiments were performed using an Ares G2 rheometer from TA instruments, equipped with an 8 mm parallel plate measuring system. The experiments proceeded by successive measurements every 10 °C from 80 to 130 °C under nitrogen. Before every measurement, the temperature was allowed to stabilize for 5 min. The strain was monitored by applying a stress of 2 kPa during 600 s, and then the stress was removed.

Results and discussion

Polymer syntheses

The poly(styrene-co-vinylbenzoic acid-co-butylacrylate) copolymer cross-linked via hemiacetal linkage was synthesized using a two-step one-pot procedure. The first step consisted in the synthesis of the cross-linker containing two hemiacetal ester functions and two vinyl functions (HCr). This dimer was prepared by reacting 1,4-cyclohexanedimethanol divinyl ether (CDDVE) with 4-vinylbenzoic acid (VBA) in the presence of n-dodecyl dihydrogen phosphate as catalyst (Scheme 2) directly in the polymerization mixture. In the case of the vitrimer candidate (CMA), the aim was to convert 53% of VBA into crosslinker to keep carboxylic acid groups available for the exchange reaction. This conversion was monitored by [START_REF] Montarnal | Silica-Like Malleable Materials from Permanent Organic Networks[END_REF] H NMR (Figure S1) by comparing the signals of the hemiacetal ester and of the divinyl ether (CDDVE) (Equation S1). A conversion of 87 % of the vinyl ether functions was reached, the remaining 13% corresponding to a side reaction with water leading to the formation of alcohol and acetal (Figure S1 and S2), [START_REF] Cho | Initiation of vinyl ether polymerization by trimethylsilyl triflate, dimethyl sulfide, and adventitious water[END_REF][START_REF] Fife | Vinyl Ether Hydrolysis. The Facile General Acid Catalyzed Conversion of 2-Ethoxy-1-Cyclopentene-1-carboxylic Acid to Cyclopentanone[END_REF][START_REF] Gallucci | Reaction of hemiacetal esters, acetals, and acylals with alcohols or acetic acid[END_REF] indicating a functionalization of around 46 % of the initial VBA amount. The second step was the RAFT copolymerization (see Table S1) of styrene, n-butyl acrylate and VBA in the presence of HCr. The polymerization was promoted by self-initiation of styrene [START_REF] Khuong | The Mechanism of the Self-Initiated Thermal Polymerization of Styrene. Theoretical Solution of a Classic Problem[END_REF] at 100 °C and controlled using cyano-2propyldodecyl trithiocarbonate as the RAFT chain transfer agent (CTA). This copolymer, cross-linked by hemiacetal ester linkage and comprising available carboxylic acid functions able to engage in hemiacetal ester exchange reaction, is called "cross-linked material with acid pendent groups" (CMA, Figure 1, CMA). This material was expected to behave like a vitrimer.

In addition to CMA, three other copolymers were also synthesized as reference materials: 1) A cross-linked copolymer (CR, Figure 1, CR) prepared by copolymerization of styrene and n-butyl acrylate in the presence of HCr. For this material, the functionalization of the VBA into hemiacetal esters proceeded with a small excess of divinyl ether to ensure that all acid functions were converted into hemiacetal esters. A small proportion of mono-functional hemiacetal ester derivative of VBA was also likely produced under the conditions used. However, the presence of this monofunctional VBA did not impair its use as a fully cross-linked material devoid of any available carboxylic acid groups. The amount of CDDVE involved in HCr was estimated to represent between 6 and 12 wt% of the total mass of CR, with 6 wt% corresponding to the case where no mono-functional derivative of VBA was produced, and 12 wt% corresponding to the case where only mono-functional derivative was produced. 2) A non-cross-linked copolymer prepared by copolymerization of styrene, n-butyl acrylate, and a monofunctional hemiacetal ester derivative of VBA. This copolymer designated as non-cross-linked reference (NR, Figure 1, NR) possessed neither cross-links nor carboxylic acid functions. 3) A copolymer that only presented carboxylic acid functions prepared by treating CMA with excess trifluoroacetic acid (able to fully substitute the hemiacetal ester linkages) thus generating the acid-dissociated reference (AR, Figure 1, AR). Scheme 2. Preparation of dihemiacetal ester cross-linker (HCr). The copolymers were characterized by [START_REF] Montarnal | Silica-Like Malleable Materials from Permanent Organic Networks[END_REF] H NMR and gel permeation chromatography (GPC). Prior to these analyses, all the hemiacetal ester linkages were cleaved by substitution with TFA (except for AR which does not contain any hemiacetal ester functions). The 1 H NMR spectra of AR, and of the TFA-treated CR and NR are shown in Figure S3, Figure S4 and Figure S5, respectively (note that AR is the purified product resulting from the treatment of CMA with TFA). The compositions of the copolymers were calculated using Equation S2, Equation S3 and Equation S4, and are reported in Table S1 along with the molar masses measured by GPC. The GPC chromatograms of the copolymers are presented in Figure S6. [START_REF] Montarnal | Silica-Like Malleable Materials from Permanent Organic Networks[END_REF] H NMR analyses indicated molar fractions ranging from 6 to 7.5 %, from 64 to 68 % and from 26 to 28 % for vinyl benzoic acid, styrene and n-butyl acrylate respectively for the various copolymers. Table S1 indicates relatively low dispersities of the CMA and NR polymers (1.3 and 1.2 respectively). The higher value (Ɖ = 1.55) measured for CR is probably due to incomplete cleavage of the hemiacetal ester bonds. These GPC analyses thus suggest a fairly good control of the RAFT terpolymerization process. This is remarkable given the high monomer to CTA ratios used for these syntheses (ranging between 1400/1 and 2500/1).

Importantly, assuming similar reactivity for VBA and for HCr, these results show that each of the CMA chains should contain around 1.8 mol % of cross-linker. CMA thus features a fairly low cross-linking density.

Thermal characterization

All copolymers were analyzed by differential scanning calorimetry (DSC) and thermogravimetry coupled with mass spectrometry (TGA-MS) to determine their glass transition temperature (Tg), as well as Tdis. The DSC thermograms are presented in Figure 2. The DSC derivative of NR (Figure 2), that does not contain cross-links or acid functions, present one major peak at 49 °C which was attributed to its Tg. This is consistent with the Tg calculated (45 °C) from the composition determined by NMR using Fox equation (Equation S5). However, the DSC derivatives of CR, CMA, and AR (Figure 2) present several peaks which made the determination of a unique Tg impossible. The samples were therefore compared using the end of the transition phenomenon, determined by the tangents method. Although the transition upon heating seems to start at a lower temperature of around 35 °C for all samples, the upper temperature varies from one sample to the other. Thus, the upper temperature is 57 °C for NR and 65 °C for CR. This slight broadening of the transition phenomenon is ascribed to the low crosslink density of CR (amount of HCr in the structure estimated between 2 and 4 mol%), which decreases the mobility of the system at a given temperature. AR is the sample for which the broadening of the transition phenomenon is the most pronounced (end of phenomenon = 81 °C). This was attributed to the presence of numerous acid functions capable of forming a hydrogen-bonded network. [START_REF] Lewis | The Influence of Hydrogen Bonding Side-Groups on Viscoelastic Behavior of Linear and Network Polymers[END_REF][START_REF] Bershtein | The use of differential scanning calorimetry for evaluating the intermolecular reaction changes in glass-like polymers[END_REF][START_REF] Bershtein | Molecular interaction and conformation of molecules in polystyrene and styrenemethacrylic acid copolymers[END_REF] Finally, in the case of CMA (end of phenomenon = 75 °C) the broadening lies between those of CR and AR. This was ascribed to the lower amount of acid functions in CMA compared to AR.

The dissociation temperature of the hemiacetal ester function (temperature at which the hemiacetal ester function dissociate) was determined from TGA-MS experiments performed on CMA, CR and AR (Figure 3). These experiments indicated that while AR did not show significant mass loss below 300 °C, CMA and CR did show a distinct mass loss between 125 and 275 °C. Importantly, this mass loss was different for the two materials; CMA lost 3 % of its initial mass, which corresponds to the mass fraction of linked CDDVE in CMA. In comparison, CR, which contains between 6 and 12 wt% of CDDVE, lost around 9 % of its initial mass. These mass losses were thus ascribed to the evaporation of vinyl ether, which was confirmed by the analysis of the exhausted gas by mass spectrometry. As shown in Figure 3, between 125 and 275 °C, a compound with m/z = 43 was observed for CMA and CR but not for AR. This m/z value corresponds to acetaldehyde, which is a typical product of the thermal degradation of vinyl ethers. [START_REF] Gallucci | Reaction of hemiacetal esters, acetals, and acylals with alcohols or acetic acid[END_REF][START_REF] Blades | The kinetics of the thermal decomposition of vinyl isopropyl ether[END_REF][START_REF] Shimofuji | Unimolecular thermal decomposition of ethyl vinyl ether and the consecutive thermal reaction of the intermediary product acetaldehyde: shock wave experiment and ab initio calculation[END_REF] In addition, this mass loss appeared at temperatures higher than the evaporation temperature of CDDVE (Figure S8 and Figure S9) further confirming that the mass losses observed for CMA and CR were resulting from the dissociation of hemiacetal ester functions.

The feasibility of the characterization of the materials by rheology experiments at elevated temperature without significant degradation was checked by isothermal thermogravimetric analyses of CMA (Figure S10). These experiments (Table S2) showed that the relative loss at 140°C represented less than 17 % of the initial CDDVE in 5 h, or approximately 3.5 % per hour. The rheology experiments carried out under nitrogen atmosphere and at different temperatures did not last more than 15 min. The thermal degradation of CMA during these rheological characterizations was thus negligible.

Vitrimer behavior demonstration

8 mm diameter disks of the copolymer materials were prepared in a mold at 90 °C for 1.5 h prior to rheology experiment.

The formation of such a disk from CMA is illustrated in Figure 4. In contrast, this shaping step was impossible for CR at 90 °C. For CR, the shaping process required extended exposure to high temperatures (5 h at 150 °C), at which the hemiacetal ester functions dissociate. This was expected since CR is fully cross-linked and does not possess any carboxylic acid groups able to engage into exchange reactions with hemiacetal ester. Since CR could not be shaped without thermal degradation, no further characterization was performed on this material.

To demonstrate unambiguously the vitrimer behavior of CMA, creep experiments of CMA, NR and AR were performed every 10 °C between 80 °C and 130 °C. The results from the creep experiments, presented in Figure 5, show that in contrast to NR, neither AR nor CMA showed any strain at 80 °C and a 2KPa stress. These results can be linked to previous DSC results that showed broadening of the transition phenomenon at around 80 °C for AR and CMA. As AR does not present any covalent cross-links, this absence of strain and the broadening of the transition penomenon oberved in DSC were ascribed to the hydrogen bonding generated by the carboxylic acid groups. [START_REF] Lewis | The Influence of Hydrogen Bonding Side-Groups on Viscoelastic Behavior of Linear and Network Polymers[END_REF] Hydrogen bonds have already been exploited to prevent the movement of polymer chains at temperatures where exchange reactions can occur in so-called hybrid-vitrimer materials. [START_REF] Guerre | Vitrimers: directing chemical reactivity to control material properties[END_REF][START_REF] Azcune | Aromatic disulfide crosslinks in polymer systems: Self-healing, reprocessability, recyclability and more[END_REF] Above 80 °C, viscoelastic deformation was observed in the first few seconds of the experiment. This deformation was recovered when stress was released. Then, plastic deformation was observed. The plastic deformation of CMA indicates that the material can flow even though the material is cross-linked. This phenomenon was attributed to the ability of the hemiacetal ester cross-links to undergo exchange reactions with the carboxylic acid groups. Between 90 and 110 °C, the influence of temperature onto the plastic deformation was lower for CMA than for AR. However, at 130 °C, the plastic deformation of CMA was similar to that of AR. This was attributed to the acceleration of the exchanges with increasing temperature, which were not anymore the limiting factor of the plastic deformation at 130 °C. From CMA and AR data, compliance curves were also plotted (Figure S11 and Figure S12, respectively) and the zero shear viscosities were determined from linear fittings of the parts corresponding to plastic deformation (J(t) = t/η0). The determined zero shear viscosities are presented in the Arrhenius plots in Figure 6.

Figure 6 clearly shows a linear evolution of viscosity (R 2 = 0.99) for CMA. On the contrary, the evolution of the viscosity of AR was far from linear (Figure 6). This difference indicates that, contrary to AR, CMA behave as a vitrimer and that the evolution of its viscosity is governed by associative hemiacetal ester exchange reaction. The slope of the linear fit, which corresponds to the flow activation energy of the material, was 16.2 ± 0.8 kJ mol -1 . This value is low compared to values between 60 and 150 kJ mol -1 reported in other studies of vitrimer materials. [START_REF] Guerre | Vitrimers: directing chemical reactivity to control material properties[END_REF][START_REF] Han | A Catalyst-Free Epoxy Vitrimer System Based on Multifunctional Hyperbranched Polymer[END_REF][START_REF] Röttger | High-performance vitrimers from commodity thermoplastics through dioxaborolane metathesis[END_REF][START_REF] Van Herck | Covalent Adaptable Networks with Tunable Exchange Rates Based on Reversible Thiol-yne Cross-Linking[END_REF] This relatively low flow activation energy can in part be explained by the low cross-link density [START_REF] Guerre | Vitrimers: directing chemical reactivity to control material properties[END_REF] and the excess of carboxylic acid functions compared to the hemiacetal ester linkages which promote the exchange reaction. [START_REF] Han | A Catalyst-Free Epoxy Vitrimer System Based on Multifunctional Hyperbranched Polymer[END_REF][START_REF] Wu | Dynamics and reaction kinetics of coarse-grained bulk vitrimers: a molecular dynamics study[END_REF] The low value is also likely related to the fact that these acid exchange reactions can take place at ambient temperature at a significant rate as showed in a previous publication. [START_REF] Boucher | Hemiacetal Ester Exchanges, Study of Reaction Conditions and Mechanistic Pathway[END_REF] This would imply a low activation energy of the exchange reaction itself (not measured), which would in turn favor viscous flow. [START_REF] Scheutz | Adaptable Crosslinks in Polymeric Materials: Resolving the Intersection of Thermoplastics and Thermosets[END_REF] In addition, the presence of two hemiacetal ester groups per cross-link is likely to further increase the probability of encounter between carboxylic acid group and hemiacetal ester function.

Conclusion

The aim of this paper was to study the feasibility of synthesizing a vitrimer based on the recently reported hemiacetal ester exchange reaction. The material was prepared by copolymerizing a cross-linker containing two hemiacetal ester functions, with styrene, vinyl benzoic acid and n-butyl acrylate. The synthesis of the cross-linker and the copolymerization were performed using an easy and efficient, solvent-free two-step one-pot procedure. The properties of this material were then compared with materials that did not possess carboxylic acid groups able to into exchange reactions with hemiacetal ester, and with non-cross-linked materials. Although model hemiacetal ester exchange reactions were previously

shown to occur at ambient temperature in solution, [START_REF] Boucher | Hemiacetal Ester Exchanges, Study of Reaction Conditions and Mechanistic Pathway[END_REF] rheological experiments demonstrated that the vitrimer candidate material did not present any strain at and below 80 °C, likely due to hydrogen bonding formed by the pendent carboxylic acid groups. In contrast, at 90 °C and above the material presented typical vitrimer flow behavior. In addition to the room temperature vitrification by hydrogen bonding, TGA experiments demonstrated that this vitrimer material was decrosslinkable by heat treatment at elevated temperatures (significant dissociation of the cross-links was observed above 140 °C).

This work highlighted that the vitrimer was frozen due to hydrogen bonding at temperatures below 80 °C, making it a "hybrid-vitrimer". The novel hybrid-vitrimer thus presents three temperature domains: Below 80 °C, the material is frozen due to hydrogen bonding despite the ability of the exchange reaction to occur. At moderate temperatures (between 80 °C and 140 °C), hydrogen bonds dissociate, hemiacetal ester exchange reactions take place and the material show vitrimer flow behavior. Finally, at elevated temperatures (above 140°C), the hemiacetal ester cross-links dissociate and the material loses its vitrimer properties. Ongoing studies in our laboratory aim at designing hemiacetal ester linkage with higher thermal dissociation to expand the reshaping temperature range and suppress undesired degradation of the linkage entirely. In this study, a copolymer of styrene and butyl acrylate, presenting a moderate transition temperature, was used to demonstrate the vitrimer behavior of the material. However, a similar approach should be applicable for other vinylic monomers, including styrene, which is used for the production of commodity plastics and therefore relatively inexpensive.

Figure 1 .

 1 Figure 1. Schematic and ideal representation of CMA, CR, NR and AR.

Figure 2 .

 2 Figure 2. DSC thermograms and their derivatives for CMA, CR, NR and AR.

Figure 3 .

 3 Figure 3. Comparison of the TGA-MS results of CMA compared to AR, and of CR compared to AR. A temperature ramp of 10 °C/min was used in both cases.

Figure 4 .

 4 Figure 4. Shaping of CMA into a disk sample by heating at 90 °C, under a pressure of 16.2 N/cm 2 , for 1.5 h.

Figure 5 .

 5 Figure 5. Strain curves obtained from creep experiments between 80 and 130 °C for CMA, AR and NR. Plain curves representstrain when a stress of 2 kPa was applied while dashed curves represent strain when the stress was released.

Figure 6 .

 6 Figure 6. Variation of zero shear viscosity (η0) as function of temperature for CMA and AR.

Funding Sources

This project was supported by the Chimie Balard Cirimat Carnot institute through the ANR programme N°16 CARN 0008-01 and by the I-SITE MUSE (ANR managed "Investissements d'avenir" programme N° ANR-16-IDEX-0006) Explore#2 mobility grant.

ABBREVIATIONS

AR, Acid-dissociated reference; CDDVE, 1-4 cyclohexanedimethanol divinyl ether; CMA, Cross-linked material with acid pendent groups; CR, Cross-linked reference; DCM, Dichloromethane; NR, Noncross-linked reference; RAFT, Reversible Addition-Fragmentation chain Transfer; Tdis, Dissociation temperature at which hemiacetal ester functions dissociate; THF, Tetrahydrofuran; VBA, 4-Vinylbezoic acid.

ASSOCIATED CONTENT

Supporting Information. Detailed information about 1 H NMR, gel permeation chromatography, thermal and rheological characterizations. Vinyl ether side reaction with water and the equations used are also presented. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author Author Contributions

All authors have given approval to the final version of the manuscript.

ACKNOWLEDGMENT

The authors warmly thank Marc Guerre (CNRS Researcher -Paul Sabatier University Toulouse) for sharing is experience in vitrimer chemistry and for his valuable advices about rheology experiments.


Sylvie Bégot
email: sylvie.begot@univ-fcomte.fr

Muluken Z Getie

Alpha Diallo

François Lanzetta

Magali Barthès

Michel De Labachelerie

A novel model and design of a MEMS Stirling engine

Keywords: Adiabatic modelling, design guidelines, downsizing effects, engine, MEMS, Stirling

In this paper, we present a new model and design of a MEMS Stirling machine. The concept could be used to provide electricity to low level power systems. A modified adiabatic model including losses of a Stirling engine such as regenerator thermal efficiency, wall conduction, pressure drop, hysteresis, membrane finite speed and squeeze film damping losses is presented. A base design machine is proposed. A parametric study is conducted and used to derive guidelines for miniature design. Compared to macro-scale design, the same trend was observed for the influence of the thermal performance regenerator. However, different trends from macroscopic engines were observed: conduction losses are of major importance due to the low power of the miniature engines, hysteresis losses in the bounce space are also of high amplitude due to the small size if this volume, the choice of the working gas leads to hydrogen or air and not helium. We computed that membrane finite speed losses and squeeze film damping losses can be neglected in the investigated machine size. The first part of the paper presents the micromachine specific features, the second part describes the model, the last part presents the base design results and design guidelines.

INTRODUCTION

Billions of items consuming low level power may significantly contribute to global energy consumption and CO2 emissions. Therefore, the use of a renewable source to produce locally the electric power needed by these devices [START_REF] Chen | Circuit design advances for wireless sensing applications[END_REF] appears to have a significant potential for energy savings. Among renewable sources, waste heat has a large potential. Unfortunately, most of the waste heat losses are at low temperature: one-third of the waste heat is below 200°C, and another 25 % is between 200°C and 500°C [START_REF] Papapetrou | Industrial waste heat: Estimation of the technically available resource in the eu per industrial sector, temperature level and country[END_REF]. Few technologies present good technical-and-economical performances for low temperature waste heat recovery. On the other hand, MEMS (Microelectromechanical systems) technology is well suited to these temperatures and presents low costs due to batch manufacturing. Therefore, it would be interesting to investigate the use of MEMS power conversion machines operating on waste heat for the low power devices.

Among these technologies, Stirling engines seem a good candidate because of their ability to use external heat sources. The Stirling machine was invented by Robert Stirling in 1816 [START_REF] Walker | Stirling engines[END_REF]. It can operate as an engine or as a cooler by reversing the thermodynamic cycle. Since 1816, several developments led to applications in various fields such as solar converters of combined heat and power [START_REF] Bataineh | Performance analysis of stand-alone solar dish stirling system for electricity generation[END_REF] [START_REF] Piancastelli | Design and heuristic optimization of low temperature differential stirling engine for water pumping[END_REF] [START_REF] Tlili | Numerical investigation of an alpha stirling engine using the ross yoke linkage[END_REF] for example. Whereas numerous works deal with Stirling engines at a macroscopic scale [START_REF] Thombare | Technological development in the Stirling cycle engines[END_REF], only a few works concern miniaturized Stirling engines. The first prototype on a centimeter scale was developed by Nakajima et al. in 1989 [8]. It included a 0.05 𝑐𝑚 3 swept volume and delivered a power of 10 mW at 10 Hz operating between temperatures of 273 K and 373 K. In 2002, Moran patented a micro-cooler made up from a Stirling micromachine [START_REF] Moran | Micro-scalable thermal control device[END_REF]. Silicon membranes, electrostatically actuated, played the role of the pistons. Several types of regenerators were tested [START_REF] Moran | Microsystem cooler development[END_REF]. After experimental tests between 100 and 1000 Hz, no notable difference in temperature between the chambers was observed. In 2015, Formosa et al. proposed and modeled a multiphase membrane machine for thermal heat harvesting [START_REF] Formosa | Multi-physics modelling approach for oscillatory microengines: application for a microstirling generator design[END_REF]. It consisted of three coupled elementary phases, each composed of a 5 mmdiameter membrane which acted as a piston and transducer integrating a piezoelectric planar spiral, an expansion chamber, and a compression chamber linked by a regenerator.

To conclude, Stirling machines at macroscopic scale are a well known technology, but except for the centimetric engine designed by Nakajima et al., micro-scale Stirling cycles machines are not operating yet. Therefore, it seems appropriate to investigate Stirling engines at micro-scale by developing and adapting a complete thermodynamic model to the specificity of micro-scale design. This is the main topic of the paper. The novelty of the work is a comprehensive adiabatic model including not only pressure drop losses, conduction losses and regenerator thermal effectiveness but also hysteresis, finite speed and squeeze film damping losses. The use of hydrogen and helium as working gas at this scale is also analyzed. The guidelines that result from this model application are also a main point of this work.

The first part of the paper presents the principle of the Stirling cycle, and the micro-machines specific features. The second part describes the model. The last part of the paper presents the base design characteristics and results. Parametric studies are presented. From these results, design guidelines for miniature engines are provided.

STIRLING ENGINE CHARACTERISTICS

Ideal Stirling cycle

The ideal Stirling cycle [START_REF] Walker | Stirling engines[END_REF] includes 4 thermodynamic transformations:

• an isothermal compression where heat Qc is rejected to the cold sink at a temperature Tc, • an isochoric heating where the heat Qr stored in a special heat exchanger called the regenerator is transmitted to the working gas, • an isothermal expansion where heat Qe is absorbed from the external hot source at a temperature Te, • an isochoric cooling where the gas transmits heat Qr to the regenerator. The efficiency of this ideal cycle is the Carnot efficiency:

𝜂 = 1 -𝑇 𝑒 /𝑇 𝑐 (1)

Mechanical configuration

An Alpha configuration Stirling engine includes two cylinders and two pistons (Figure 1) [START_REF] Walker | Stirling engines[END_REF]. A regenerator connects the expansion and compression spaces. The movements of the pistons allow the gas to follow the Stirling cycle. The regenerator stores the heat of the fluid during the transfer of the gas from the hot part to the cold part and restores it in the opposite direction. Two heat exchangers, one at the hot source and one at the cold sink, are also necessary. A part of the machine, not represented on Figure 1 contains the mechanical drive that links the pistons. The volume of this part can also be pressurized and is called the bounce space.

Micro-machine specific features

At micro-scale level, the Stirling machine design has to be adapted for technological reasons as MEMS fabrication relies on techniques that are used in integrated circuit technology rather than mechanical engines technology. Generally, a silicon substrate is used, then the process adds or removes layers before the final assembly and packaging [START_REF] Madou | Fundamentals of microfabrication: the science of miniaturization[END_REF]. These technological choices have an impact on the model equations, for this reason, the model is detailed after defining the microscale design main features. The general layout of the machine is presented in Figure 2a as an exploded view. We choose an Alpha configuration because the symmetry of the compression and expansion chambers are adapted to batch fabrication. The micro-machine includes two cylinders mechanically coupled. The cylinder volumes are swept by membranes. Both cylinders are closed in their upper part by plates used as heat exchangers (Figure 2c). Channels are etched on the heat exchangers plates to promote heat transfer. A regenerator connects the hot and cold heat exchangers (Figure 2b). In macro-scale design, pistons are used to but are not practical at microscale [START_REF] Moran | Micro-scalable thermal control device[END_REF], therefore the membrane is a hybrid membrane made from a center silicon disk embedded in RTV silicone (Figure 3). More details on the design and fabrication are given in Diallo et al. works [13][14] where the technological feasibility of this membrane was established. In the same way, the regenerator in macro-scale design is usually made from woven screens, metal felts or involute foils. In our case, these technologies are not applicable, therefore we choose to use an array of pillars to fulfill the function of regeneration (Figure 2b). The machine materials that can be used are those adapted to clean rooms facilities and batch fabrication. Therefore, the chosen base materials are glass and silicon that are usually available as wafers [START_REF] Madou | Fundamentals of microfabrication: the science of miniaturization[END_REF]. Due its higher thermal conductivity, silicon is used where heat exchange is needed: heat exchangers plates and regenerator pillars. Glass is used when heat exchange is detrimental: regenerator walls to limit the heat transfer from the hot part of the machine to the cold part. The fabrication of the regenerator includes a first step where a glass wafer and silicon wafer are assembled by anodic bonding, then the pillars are etched by Deep Reactive Ion Etching (DRIE). More details on the design, fabrication and CFD modeling of the regenerator can be found in reference [START_REF] Dellali | Design, fabrication and CFD modeling of a Stirling engine microregenerator[END_REF] (Figure 4).

ADIABATIC MODEL INCLUDING LOSSES

Ideal adiabatic analysis adapted to micromachines

In real Stirling cycle machines, the compression and expansion processes tend to the adiabatic rather than isothermal. Therefore, we use the following assumptions to describe the cycle [START_REF] Urieli | Stirling cycle engine analysis[END_REF]:

• the working gas is ideal,

• the kinetic and potential energies of gas streams are negligible,

• all processes are performed at steady-state condition,

• the compression and expansion spaces are adiabatic,

• the temperature in the hot and cold heat exchangers are constant,

• heat is transferred to the working fluid only in the hot and cold heat exchangers,

• the temperature change of the working fluid in the regenerator is linear.

The analysis is based on the development presented by Urieli and Berchowitz [START_REF] Urieli | Stirling cycle engine analysis[END_REF] where all the derivation details can be found. The machine is divided into 5 volumes (Figure 5).

Figure 5. Adiabatic model

The analysis leads to the differential equations for pressure P, masses in the different volumes m, heat exchange in the heat exchangers Q, temperature T in the various volumes, and expression for the adiabatic work 𝑊 𝑎𝑑 . They are briefly recalled here:

𝑚 𝑐 + 𝑚 𝑘 + 𝑚 𝑟 + 𝑚 ℎ + 𝑚 𝑒 = 𝑚 𝑡 (2)
𝑑𝑚 𝑐 = 𝑃𝑑𝑉 𝑐 + 𝑉 𝑐 𝑑𝑃 𝛾 𝑅𝑇 𝑐𝑘 (3
𝑑𝑄 𝑟 = 𝑉 𝑟 𝑑𝑃𝐶 𝑣 𝑅 -𝐶 𝑝 (𝑇 𝑘𝑟 𝑚̇𝑘 𝑟 -𝑇 𝑟ℎ 𝑚̇𝑟 ℎ ) (7)
𝑑𝑄 ℎ = 𝑉 ℎ 𝑑𝑃𝐶 𝑣 𝑅 -𝐶 𝑝 (𝑇 𝑟ℎ 𝑚̇𝑟 ℎ -𝑇 ℎ𝑒 𝑚̇ℎ 𝑒 ) (8)
𝑑𝑇 𝑒 = 𝑇 𝑒 ( 𝑑𝑃 𝑃 + 𝑑𝑉 𝑒 𝑉 𝑒 - 𝑑𝑚 𝑒 𝑚 𝑒 ) ( (9)
) 10
𝑑𝑇 𝑐 = 𝑇 𝑐 ( 𝑑𝑃 𝑃 + 𝑑𝑉 𝑐 𝑉 𝑐 - 𝑑𝑚 𝑐 𝑚 𝑐 ) ( 11
)
𝑊 𝑎𝑑 = ∮ (𝑃𝑑𝑉 𝑐 + 𝑃𝑑𝑉 𝑒 ) 𝑐𝑦𝑐𝑙𝑒 ( 12
)
This set of differential equations is solved iteratively by a Runge-Kutta method. This model was validated against the macroscopic GPU3 engine results [START_REF] Thieme | Baseline performance of the GPU 3 Stirling engine[END_REF].

Modified ideal adiabatic analysis adapted to micromachines

This ideal model is not accurate enough to predict the engine performances. It is necessary to compute various losses to enhance the model accuracy. Among the losses that are usually accounted for in macroscopic Stirling engines [START_REF] Urieli | Stirling cycle engine analysis[END_REF][18] [START_REF] Timoumi | Design and performance optimization of GPU-3 Stirling engines[END_REF], shuttle heat losses, mass leakage losses and piston friction losses are not relevant due to the use of membranes instead of pistons. On the other hand, losses due the non-ideal heat transfer in the regenerator, direct conduction, pressure drop, hysteresis, piston finite speed are relevant at both scales. The micro-scale design also leads to analyze squeeze film damping losses [START_REF] Bao | Analysis and design principles of MEMS devices[END_REF]. Non-ideal heat transfer of the regenerator is included in the set of the differential equations. It leads to correct the temperatures in the hot and cold heat exchangers following the "simple" algorithm proposed by Urieli and Berchowitz [START_REF] Urieli | Stirling cycle engine analysis[END_REF]. The other losses are subtracted to the overall engine power. For a Stirling cycle engine having a non-ideal regenerator, when the working gas flows from the cold heat exchanger to the hot heat exchanger, the gas do not recover all the heat ideally stored in the regenerator. The thermal energy regenerator losses 𝑄 𝑟𝑙 are given by:

𝑄 𝑟𝑙 = 𝑚𝐶 𝑝 (1 -𝜖)(𝑇 𝑐 -𝑇 𝑒 ) (13)
where ϵ denotes the regenerator efficiency. This efficiency is defined by using the Number of the Transfer Units (NTU) as:

𝜖 = 𝑁𝑇𝑈 𝑁𝑇𝑈 + 1 (14)
The NTU is related to the Stanton number St through:

𝑁𝑇𝑈 = 𝑆𝑡 𝐴 𝑤𝑔,𝑟 2𝐴 𝑟 (15)
where 𝐴 𝑤𝑔𝑟 and 𝐴 𝑟 denote the regenerator wetted area, and free flow area respectively. The Stanton number is computed from the friction factor following the Reynolds analogy:

𝑆𝑡 = 𝑓 𝑟 2𝑅𝑒𝑃𝑟 (16)
where 𝑅𝑒, 𝑃𝑟, 𝑓 𝑟 are the Reynolds number, Prandtl number and Fanning friction factor. The correlation used to determine the friction factor is detailed in §3.2.2. Thus, the net heat absorbed from the hot heat exchanger and net heat released to the cold heat exchanger at their respective wall temperatures by the working gas per cycle are given by:

𝑄 𝑘 = 𝑄 𝑘,𝑖𝑑𝑒𝑎𝑙 -𝑄 𝑟𝑙 = ℎ 𝑘 𝐴 𝑤𝑔,𝑘 (𝑇 𝑤,𝑘 -𝑇 𝑘 ) 𝑓 (17)
𝑄 ℎ = 𝑄 ℎ,𝑖𝑑𝑒𝑎𝑙 + 𝑄 𝑟𝑙 = ℎ ℎ 𝐴 𝑤𝑔,ℎ (𝑇 𝑤,ℎ -𝑇 ℎ ) 𝑓 (18)
where Qh,ideal, Qk,ideal denote the heat absorbed or released of the ideal adiabatic analysis, i.e. without taking into account the regenerator thermal effectiveness, f denotes the frequency of operation. From Eq. ( 17) and Eq. ( 18), the gas temperatures in the heat exchangers can be computed.

Heat losses due to wall conduction in the regenerator

In a Stirling cycle machines, regenerators are physically located between the hot heat exchanger and cold heat exchanger. In a micromachine, the distance between the hot and cold part of the machine are small, therefore, direct conduction heat losses are of primary importance. The amount of energy losses Qwrl caused by heat conduction in the regenerator walls is found as follows [START_REF] Timoumi | Design and performance optimization of GPU-3 Stirling engines[END_REF]:

𝑄 𝑤𝑟𝑙 = 𝑘 𝑟 𝐴 𝑤,𝑟 𝐿 𝑟 𝑓 (𝑇 𝑤,ℎ -𝑇 𝑤,𝑘 ) (19)
where kr is the regenerator wall conductivity (Table 1), Awr is the surface of the wall, Lr is the regenerator length, Twh and Twk are the hot and cold heat exchanger wall temperature and f is the frequency of operation. Other conduction losses in the machine are supposed to be negligible compared to the regenerator wall conduction losses.

Friction factors and heat exchange coefficients in heat exchangers

The regenerator is made of an array of micro pillars. Each pillar is approximated as a cylinder of elliptic cross-section. The Fanning friction factor is defined from the experimental work of Vanapalli et al. [START_REF] Vanapalli | Pressure drop of laminar gas flows in a microchannel containing various pillar matrices[END_REF]:

𝑓 𝑟 = 6.745𝑅𝑒 -0.82 ( 20
)
where Re is the Reynolds number based on the pillars array hydraulic diameter dH,r and average gas velocity over half a cycle.

The hydraulic diameter of the regenerator is defined as:

𝑑 𝐻,𝑟 = 4𝜓𝑉 𝑟 𝐴 𝑤𝑔,𝑟 (21)
where ψ is the regenerator porosity and Vr its volume.

The Stanton number and heat exchange coefficients are computed using the Reynolds analogy as explained in §3.2.1.

The cold and hot heat exchangers friction factors and heat exchange coefficients are chosen referring to non-circular ducts correlations in laminar flow:

𝑓 𝑟 = 24 𝑅𝑒 (22)
𝑁𝑢 = 7.54 (23)

Losses due to pressure drop in heat exchangers

The pressure drop losses in heat exchangers are calculated to evaluate the amount of power losses due to fluid friction losses. The amount of energy dissipated Wfr due to pressure drops at the hot heat exchanger, regenerator and cold heat exchanger is evaluated as:

𝑊 𝑓𝑟 = ∫ (𝛥𝑃 𝑑𝑉 𝑒 𝑑𝜃 ) 𝑑𝜃 𝑐𝑦𝑐𝑙𝑒 (24)
where:

𝛥𝑃 = 𝛥𝑃 ℎ + 𝛥𝑃 𝑟 + 𝛥𝑃 𝑘 (25)
and

𝛥𝑃 𝑖 = (4𝑓 𝑟,𝑖 𝐿 𝑑 𝐻,𝑖 + 𝐾 𝐷,𝑖 ) 𝜌 𝑖 𝑣 𝑔,𝑖 2 2 (26)
where vg,i is the mean gas velocity observed on half a cycle in the exchangers, and i=h,r,k.

Minor losses are also considered. At the interface of the regenerator and heat exchangers, a loss coefficient for a sudden contraction or expansion is defined as:

𝐾 𝐷,𝑖 = 0.3 (27)

Losses due to finite speed of membrane

In the regenerative Stirling cycle machine, the working spaces are periodically compressed and expanded by the pistons or in our case by membranes. Based on the finite speed thermodynamic principles, the instantaneous pressure of compression and expansion spaces differs from the pressure in the respective membranes' surfaces. The work losses due to finite speed of the membranes over a cycle is evaluated from this pressure drop as [START_REF] Costea | The effect of irreversibilities on solar stirling engine cycle performance[END_REF][22]:

𝑊 𝑓𝑠 = ∮ 𝛥𝑃 𝑓𝑠 𝑑𝑉 𝑐𝑦𝑐𝑙𝑒 ( 28
)
and

𝛥𝑃 𝑓𝑠 = 1 2 (𝑃 𝑎𝑣 𝑚,𝑐 𝑐 𝑐 + 𝑃 𝑎𝑣 𝑚,𝑒 𝑐 𝑒 ) ( 29
)
where vm is the membrane mean speed, c is the average molecular speed 𝑐 = √3𝑅𝑇 , and 𝑎 = √3𝛾 . The membrane mean speed is defined as:

𝑣 𝑚 = 2𝑠𝑓 (30)
where s is the stroke of the membrane at its maximal deformation and f the frequency of operation.

Gas spring hysteresis losses

If the bounce space is closed, the gas in this volume is compressed and expanded periodically, thus acts as a gas spring. This thermodynamic process is not recoverable and may introduce additional losses in the engine that could be in the form of the dissipation of the gas spring fluid. This hysteresis power losses are modelled using the following expression :

𝑊 ̇ℎ𝑦𝑠 = √ 1 32 𝜔𝛾 3 (𝛾 -1)𝑇 𝑤𝑏 𝑃𝑘 𝑔 ( 𝛥𝑉 𝑏 𝑉 𝑏 ) 2 𝐴 𝑤𝑔,𝑏 (31)
where 𝑊 ̇ℎ𝑦𝑠 , 𝜔 , 𝑇 𝑤𝑏 , 𝑃 , 𝛥𝑉 𝑏 , 𝑉 𝑏 and 𝐴 𝑤𝑔,𝑏 denote the hysteresis power losses, the angular speed of the machine, the bounce space wall temperature, the bounce space mean pressure, the variation of the bounce space volume, the total bounce space volume, and the wetted area of the bounce space respectively. We make the hypotheses that the bounce wall temperature is the same as the cold heat exchanger wall temperature 𝑇 𝑤𝑘 and that the mean pressure is the same as the machine mean pressure.

Squeeze film damping losses

As the volume forces that work on a machine vary in direct proportion to the cube of the length, while surface forces such as viscous force vary in direct proportion to the square of the length, surface forces that may be negligible at a macroscopic scale may play an important role in a micromachine [START_REF] Bao | Squeeze film air damping in MEMS[END_REF]. Therefore, when miniaturizing Stirling engines, the issue of squeeze film damping has to be analyzed. The aim is to check that during the displacement between the expansion and compression spaces, the gas can escape the chambers. The squeeze number 𝜎 is defined as [START_REF] Bao | Squeeze film air damping in MEMS[END_REF]:

𝜎 = 12𝜇𝑟 𝑚 2 𝜔 𝑃𝑧 2 (32)
where 𝑟 𝑚 is the characteristic length of the device, in our case the radius of the membrane, and 𝑧 is the height of the film. In our case, 𝑧 is chamber height. When 𝜎 << 1, as it is the case in the base design, the Reynolds equation holds and the squeeze film damping force coefficient for a circular plate can be written as:

𝐷 = 3𝜋 2𝑧 3 𝜇𝑟 𝑚 4 (33)
The damping force is computed as:

𝐹 𝑠𝑓 = -𝐷𝑥̇𝑗 (34)
where 𝑥̇ is is velocity of the membrane, 𝑥 its position and 𝑗 = 𝑐, 𝑒 . The work of the damping force over a cycle is computed as:

𝑊 𝑠𝑓 = ∮ F 𝑠𝑓 𝑑𝑥 𝑗 𝑐𝑦𝑐𝑙𝑒 (35)
We derive the expressions of velocity from the sinusoidal variation of the swept volume:

𝑑𝑥 𝑐 𝑑𝜃 = 1 2 𝑉 𝑠𝑤𝑐 𝜋𝑟 𝑚 2 𝑠𝑖𝑛𝜃 (36)
Integrating leads to the expression of the loss due to squeeze film damping for the compression space:

𝑊 𝑠𝑓,𝑐 = 𝜔 4 𝐷𝑉 𝑠𝑤𝑐 2 𝜋𝑟 𝑚 4 ( 37
)
The loss is identical for the expansion space. Thus:

𝑊 𝑠𝑓 = 𝜔 2 𝐷𝑉 𝑠𝑤𝑐 2 𝜋𝑟 𝑚 4 (38)

Power and efficiency

The power 𝑊 ̇ delivered by the engine and its efficiency are computed as:

𝑊 ̇= 𝑊 ̇𝑎𝑑 -𝑊 ̇𝑓𝑟 -𝑊 ̇𝑓𝑠 -𝑊 ̇𝑠𝑓 -𝑊 ̇ℎ𝑦𝑠 (39) 𝜂 = 𝑊 Q̇𝑖 𝑛 (40)
where 𝑄 ̇𝑖𝑛 = 𝑄 ̇ℎ + 𝑄 ̇𝑟𝑙 + 𝑄 ̇𝑤𝑟𝑙 (41)

MODEL RESULTS AND ANALYSIS

In this section, a base design characteristics and model results are described. Then a parametric study is performed with a detailed analysis of the miniaturization effects. Design guidelines appropriate to micro-scale engine are derived.

Base design model characteristics

In this section, a proof of concept design is described, the various parameters are presented in Table 1. In this base design, the target power is 2 mW. The hot source temperature is supposed to be 200°C (473 K), therefore it is within the temperature range of the considered low temperature waste heat. First, to fulfil the target power, the chamber diameter is chosen at 5 mm and the mean pressure at 0.15 MPa. The membrane thickness is 0.2 mm and its maximal deformation is 0.1 mm as it was demonstrated that this value is repeatable in the pressure range [START_REF] Diallo | Contribution to the design and construction of a thermal micromachine with Stirling cycle[END_REF]. The total maximal deformation of the membrane is 0.2 mm. However, the membrane does not sweep the entire volume cylinder defined by the membrane surface and maximal deformation. In our base design case, the swept volume is considered to be 70 % of this cylinder volume. So, the swept volume is 2.75 𝑚𝑚 3 . The chamber height is chosen at 0.23 mm. On the opposite side of the membrane, the bounce space volume is closed to protect the membrane. As in macroscopic engine, the bounce space volume is larger than the chamber volume and 20% of this volume is supposed to be swept. Its pressure is also the machine mean pressure. As in all Stirling engines, the volume of the heat exchangers is a trade-off between the heat exchange requirements, the pressure drop and dead volumes. In this design, the length of the heat exchangers ducts is limited by the chamber diameter, the number and depth of ducts by technology. A design of 6 ducts was chosen. The regenerator design is also a trade-off between the same elements. Its porosity is chosen at 0.8, a common value in macroscopic engines, and its length is 2 mm. The regenerator walls link the hot part of the machine to the cold part. Therefore, wall thickness is reduced to 0.05 mm to avoid conduction losses.

Base design model results

Applying the base design parameters to the modified adiabatic model leads to a 2.2 mW machine with an efficiency of 6% (Table 2) at 100 Hz. The various losses values are presented in Table 2 for an operating frequency of 100 Hz.

Squeeze film damping

𝑊 ̇𝑠𝑓,𝑑𝑙𝑒𝑠𝑠 = 𝑊 ̇𝑠𝑓 /𝑊 ̇𝑎𝑑 1.05x10 -6 In order to derive some general analysis, dimensionless losses are computed: mechanical losses values are divided by the power issued from the adiabatic analysis 𝑊 ̇𝑎𝑑 , and the thermal losses are divided by the hot heat power 𝑄 ̇𝑖𝑛 . The resulting dimensionless variables and results are displayed in Table 3.

Compared to macroscopic machine losses distribution, such as those observed in the GPU3 engine [START_REF] Urieli | Stirling cycle engine analysis[END_REF], we observe that the regenerator thermal efficiency is also a main contributor to the losses, and so are the thermal conduction losses. Pressure drop is also a major concern at both scales. Hysteresis losses are very high in miniaturized Stirling engines, compared to macroscopic engines, therefore careful design should be used to minimize their impact. Finite speed and squeeze film damping lead to very small losses, finite speed losses being larger than squeeze film damping.

Parametric analysis

Influence of operational parameters: hot source temperature, frequency and working gas

On the considered hot source temperature range 373 K-573 K, the machine delivered power raises from 0.4 to 5.4 mW ( Figure 6). Therefore, the power density per cm 2 of heated surface varies from 1.8 mW.cm -2 to 27.2 mW.cm -2 . The increase in the efficiency is limited by the increase in regenerator conduction and inefficiency losses (Figure 7) as the difference between the cold source temperature and hot sink temperature increases (Eq. ( 13);( 19)). Hysteresis and friction dimensionless losses decrease as the power of the machine raises with temperature. The evolution of efficiency and delivered power with frequency (Figure 8) presents a raise, then a maximum at 100 and 200 Hz respectively, and a decrease for both curves. Therefore, maximal efficiency and maximal power are not observed at the same operating point. The observation of the main losses (Figure 9) reveals that as frequency raises, the losses due to pressure drops, that are mainly due to the regenerator, sharply raise. Moreover, the losses due to inefficiency of the regenerator also raise. Therefore, the engine miniaturization does not lead to a large increase in the optimal operational frequency compared to macroscopic engines that operate at about 50 Hz, and this is mainly due to thermofluidic losses. Finally, we studied the influence of helium and hydrogen as working gases. Helium is a frequent working gas for macroscopic engines due to its high thermal conductivity. Hydrogen properties are also interesting and miniature engines need a small mass of gas, therefore hydrogen security aspects can be easier to handle. As explained in the preceding paragraph, hydrogen and helium lead to high hysteresis losses if the bounce space volume is too low. At a ratio of 20%, the hysteresis losses amount to 8.8 mW and 5.7 mW for helium and hydrogen respectively and would be dissipating more than the adiabatic power. Therefore, to compare these gases with air, we use a design where the bounce space volume variation is 5 %. The efficiency and power for the three gases are plotted in Figure 10. Hydrogen appears to be the better choice, with a higher efficiency and an equivalent power compared to a dry air engine. Air engine efficiency is reduced by high regenerator efficiency losses (Figure 11). Helium is the worst choice due to high hysteresis and friction losses that have a strong impact on the power. Geometric parameters that may have an influence on the main losses mainly concern the regenerator as the regenerator inefficiency and conduction losses are the first two main losses of the base design model (Table 2). The efficiency variation with the regenerator length (Figure 12) presents a raise, a maximum at 3 mm, then decreases. The observation of the losses (Figure 13) shows that a short regenerator leads to high thermal inefficiency and high conduction losses. Indeed, a short regenerator does not enable all heat exchange between the gas and the regenerator pillars to take place during the cycle. The conduction losses increase as the regenerator wall thermal conductance is inversely proportional to the regenerator length. On the other hand, the higher length values lead to an increase in the friction losses. These phenomenon account for the bell shape or the efficiency curve (Eq. (39)).

The same analysis was made with the variation of the regenerator porosity. For a 2 mm-long regenerator, the efficiency is nearly constant over a porosity range of 0.7 to 0.9 (Figure 14). On this range, the power slightly raises. The observation of the main losses (Figure 15) shows that the regenerator thermal efficiency losses raise when the porosity is higher but on the other hand friction losses decrease. This accounts for the shape of the efficiency curve. Hysteresis losses in the bounce space are also one of the major losses (Table 2). They depend on the bounce space volume relative variation (Eq. (31)). So, they greatly contribute to the overall losses if this volume variation is high. Therefore, in macroscopic engines, the bounce space is usually designed large. However, this can be difficult at micro-scale level, as the aim is to have a miniature machine. Therefore, we studied the efficiency and power of the engine with the bounce space volume variation (Figure 16). Clearly, a large bounce space volume leading to small relative variations during the cycle is beneficial. We observe that, for the base design using air as working gas, the hysteresis losses overcome the pressure drop losses for a volume variation of 6% (Figure 17). This ratio is also a key parameter when using hydrogen or helium as working gases in the engine due the high values of specific heat ratio of and conductivity for these gases. regenerator wall thickness should be minimized due to the impact of conduction losses at low swept volumes, • bounce space volume need to be carefully optimized due to the impact of hysteresis losses, • hydrogen is a good choice as a working gas, air is second choice and helium is not a good choice if the bounce space is not purposefully designed, • the raise in frequency expected for micro-scale devices is limited by thermofluidic losses, • squeeze film damping and finite speed losses can be neglected.

CONCLUSION

In this paper, we have proposed a new model and design of a MEMS miniaturized Stirling engine. This engine could be used to locally provide electric power to systems of low energy demand. A modified adiabatic model including losses such as regenerator thermal efficiency, wall conduction, pressure drop, hysteresis, membrane finite speed and squeeze film damping losses was developed. A base design machine was proposed, a parametric study was conducted and used to derive guidelines for miniature design. Compared to macro-scale design, the same trend was observed for the influence of the thermal performance regenerator. However, different trends from macroscopic engines were observed: conduction losses are of major importance due to the low power of the miniature engines, hysteresis losses in the bounce space are also of high amplitude due to the small size if this volume, the choice of the working gas leads to hydrogen or air and not helium. The raise in frequency expected for micro-scale devices is limited by thermofluidic losses. Squeeze film damping and finite speed losses can be neglected at this scale.

NOMENCLATURE

Figure 1 .

 1 Figure 1. Alpha configuration

Figure 2 .

 2 Figure 2. Design of the Alpha micromachine

Figure 3 .

 3 Figure 3. Membranes of the micromachine

Figure 4 .

 4 Figure 4. Scanning Electron Microscopy images of the regenerator

Figure 6 .Figure 7 .

 67 Figure 6. Variation of efficiency and power with temperature

Figure 8 .

 8 Figure 8. Variation of efficiency and power with frequency

Figure 9 .Figure 10 .Figure 11 .

 91011 Figure 9. Variation of main losses with frequency

Figure 12 .Figure 13 .Figure 14 .Figure 15 .Figure 16 .Figure 17 .

 121314151617 Figure 12. Variation of efficiency and power with regenerator length

  ) 𝑇 𝑐𝑘 𝑚̇𝑐 𝑘 -𝑇 𝑘𝑟 𝑚̇𝑘 𝑟 )

										𝑑𝑄 𝑘 =	𝑉 𝑘 𝑑𝑃𝐶 𝑣 𝑅	-𝐶 𝑝 (
	𝑑𝑚 𝑒 =	𝑅𝑇 ℎ𝑒 𝑃𝑑𝑉 𝑒 + 𝑉 𝑒 𝑑𝑃 𝛾	(4)
	𝑃 =	𝑉 𝑐 𝑇 𝑐	+	𝑉 ℎ 𝑇 ℎ	+ 𝑚 𝑡 𝑅 𝑉 𝑟 𝑇 𝑟 +	𝑉 𝑐𝑟 𝑇 𝑐𝑟	+	𝑇 𝑒 𝑉 𝑒	(5)
	𝑑𝑃 = -	𝑉 𝑐 𝑇 𝑐𝑘	+ 𝛾( 𝑉 𝑘 𝑇 𝑘 𝑃𝛾(	+ 𝑑𝑉 𝑐 𝑉 𝑟 𝑇 𝑟 𝑇 𝑐𝑘 + + 𝑑𝑉 𝑒 𝑉 ℎ 𝑇 ℎ 𝑇 ℎ𝑒 ) ) +	𝑇 ℎ𝑒 𝑉 𝑒	(6)

Table 1 .

 1 Base design main parameters

	Parameter	Symbol	Unit	Values
	General
	Gas	-	-	Air
	Mean pressure	𝑃	MPa	0.15
	Hot source	𝑇 ℎ	K	473
	temperature
	Cold sink	𝑇 𝑘	K	288
	temperature
	Frequency	𝑓	Hz	100
	Phase angle	𝛼	degrees	90
	Chambers
	Membrane diameter	𝑑 𝑚	mm	5
	Membrane stroke	𝑠	mm	0.4
	Chamber height	𝑧	mm	0.23
	Swept volume	𝑉 𝑠𝑤𝑐	mm 3	2.75
	Bounce space
	Volume	𝑉 𝑏	mm 3	13.7
	Volume variation	𝛥𝑉 𝑏 /𝑉 𝑏	%	20
	Height	-	mm	0.7
	Cold and hot heat
	exchangers
	Number of ducts	-	-	6
	Length	𝐿 ℎ,𝑘	mm	2.5
	Width	-	mm	0.4
	Height	-	mm	0.075
	Regenerator
	Length	𝐿 𝑟	mm	2
	Porosity	𝜓	-	0.8
	Internal width	-	mm	0.5
	Internal height	-	mm	0.1
	Wall thickness	-	mm	0.05
	Wall thermal	𝑘 𝑟	𝑊𝑚 -1 𝐾 -1	1
	conductivity

Table 2 .

 2 Base design main results

		Symbol Unit	Values
	Power and
	efficiency
	Heat power in	𝑄 ̇𝑖𝑛	mW	36.52
	Heat power out	𝑄 ̇𝑘	mW	14.32
	Power out	𝑊 ̇	mW	2.21
	Efficiency	𝜂	%	6.04
	Losses
	Regenerator	𝑄 ̇𝑟𝑙	mW	9.43
	inefficiency
	Regenerator	𝑄 ̇𝑤𝑟𝑙	mW	7.77
	conduction
	Pressure drop	𝑊 ̇𝑓𝑟	mW	0.64
	Hysteresis	𝑊 ̇ℎ𝑦𝑠	mW	2.17
	Piston finite speed	𝑊 ̇𝑓𝑠	mW	1.59x10 -4
	Squeeze film	𝑊 ̇𝑠𝑓	mW	5.28x10 -6
	damping

Table 3 .

 3 Base design dimensionless losses

	Dimensionless	Symbol	Values
	losses
	Regenerator	𝑄 ̇𝑟𝑙,𝑑𝑙𝑒𝑠𝑠 = 𝑄 ̇𝑟𝑙 /𝑄 ̇𝑖𝑛	0.49
	efficiency
	Regenerator	𝑄 ̇𝑤𝑟𝑙,𝑑𝑙𝑒𝑠𝑠 = 𝑄 ̇𝑤𝑟𝑙 /𝑄 ̇𝑖𝑛	0.40
	conduction
	Pressure drop	𝑊 ̇𝑓𝑟,𝑑𝑙𝑒𝑠𝑠 = 𝑊 ̇𝑓𝑟 /𝑊 ̇𝑎𝑑	0.13
	Hysteresis	𝑊 ̇ℎ𝑦𝑠,𝑑𝑙𝑒𝑠𝑠 = 𝑊 ̇ℎ𝑦𝑠 /𝑊 ̇𝑎𝑑	0.43
	Piston finite	𝑊 ̇𝑓𝑠,𝑑𝑙𝑒𝑠𝑠 = 𝑊 ̇𝑓𝑠 /𝑊 ̇𝑎𝑑	3.2x10 -5
	speed

ACKNOWLEDGMENT

This work has been supported by the EIPHI GraduateSchool (contract ANR-17-EURE-0002) and the Region Bourgogne-Franche-Comté."


Book Review

Stephen H. Whiteman, Where Dragon Veins Meet. The Kangxi Emperor and His Estate at Rehe Seattle: University of Washington Press, 2020.

Full reference for quotation: Catherine Jami, "Rezension von: Stephen H. Whiteman: Where Dragon Veins Meet. The Kangxi Emperor and His Estate at Rehe, Seattle: University of Washington Press 2020," in Sehepunkte 21 (2021), Nr. 3 [15.03.2021], URL: http://www.sehepunkte.de /2021/03/34337.html Located in present-day Chengde 承德 in Hebei province, the Mountain Estate to Escape the Heat (Bishu shanzhuang 避暑山莊 hereafter Mountain Estate), together with the temples that surround it, has been a UNESCO World Heritage Site since 1994. It was long thought of merely as a summer resort of the Manchu emperors in the eighteenth century. However, this changed following the development of the New Qing History movement, which during the last decades has highlighted the need to take into account sources in Manchu to write the history of East Asia from the seventeenth to the twentieth centuries. In so doing, it has brought about a new view of China as being embedded in a wider Qing empire whose rulers foregrounded their own ethnicity and the multiple non-Han Chinese who lived in their vast territories. This contrasts with the earlier view of the Qing dynasty as simply the final period of imperial China. As part of this historiographic shift, the Mountain Estate has attracted attention as an essential centre of imperial power, from which relations with the Inner Asian dominions and with Russia were largely orchestrated. [1] The Mountain Estate was built in two stages: in 1702, the Kangxi emperor (r. 1662-1722) ordered the construction of a summer residence; the works lasted for about a decade. His grandson, the Qianlong emperor (r. 1736-1795), had the site extended throughout his sixtyyear reign. In particular, he had the Eight Outer Temples (wai ba miao 外八廟) built around the estate, including one modelled after the Potala Palace in Lhasa. Most of the buildings that exist today thus date to the Qianlong period.

Where Dragon Veins Meet is an analysis of the Mountain Estate that focuses on the Kangxi period and aims to historicize the site and contribute to an understanding of the early Qing period in its own terms. It relies on an unprecedented range of sources, from the site itself to the poems that the Kangxi emperor wrote about it, including a variety of visual and textual materials. Although these are mostly if not exclusively in classical Chinese, the work also builds on the historiographical achievements of the New Qing History. It draws from fields as varied as art history, historical geography, landscape history, and history of science to reveal the significance of the Mountain Estate, devised by the Kangxi emperor as a visual and spatial artefact that partook in the construction of imperial ideology.

The first feature of the book that strikes the reader on opening it is that it is a beautiful object, evidently the result of careful design and production. It is richly illustrated with maps and plans as well as reproductions of paintings and prints, and includes translations of a number of literary sources. This wealth of visual and textual materials helps the reader (especially the reader who, like the present reviewer, is unfamiliar with art history) follow and consider the argument made by the author.

The book is divided into four parts that altogether contain six chapters and two translated texts.

Part I, entitled "Recovering the Kangxi landscape", guides the reader through a process of reconstruction of the estate as it was before the Qianlong period. A translation of the account written by Zhang Yushu 長玉書(1642-1711), a high official, of his visit to the Mountain Estate in 1708 precedes a chapter devoted to reconstituting the park as it was at the time of his visit. Both the experience of the space as one moved through the park, and the ways in which this experience changed over time between 1703 and 1713 are discussed. The building of the Palace of Righteousness in 1711, which added an element associated with Confucian governance to the Mountain Estate, contributed to such change. It became the venue for coming to court (laichao 來朝) at the Mountain Estate for officials like Zhang, but also for leaders of the Mongolian Bannermen. These visitors thus experienced an imperially constructed landscape that, Whiteman argues, reflected a vision of Qing emperorship.

Part II, entitled "Allegories of empire", discusses the spatial symbolism of the Mountain Estate. Kangxi's choice of the Rehe 熱河 site to build his Mountain Estate is analysed in terms of its location, and interpreted as part of a centring of Qing geography; evidence is drawn from accounts of imperial touring and ritual performances. But also, and most importantly, Kangxi's interpretation of Chinese geomancy turned Mount Changbai 長白, the mythical cradle of the imperial clan, into the origin of the "dragon vein", embodying the power of Mount Tai, China's main sacred mountain. Mount Changbai thus became the origin of the empire's geomantic force. Whiteman argues that, for the emperor, a branch of this dragon vein extended to Rehe, and then further West into the Altai Mountains (hence the book's title). Secondly, starting from the imperial essay entitled "Imperial Record of the Mountain Estate to Escape the Heat" (1711), the author also proposes a reading of the Mountain Estate as a miniature representation of the whole Qing empire encompassing both the provinces of Ming dynasty China, and the territories beyond the Great Wall under his control.

Part III, entitled "Space and Pictoriality", focuses on pictorial representations of the Mountain Estate produced during the Kangxi reign. Leng Mei's 冷枚 1709 View of Rehe, central to the first part of the discussion, is shown to be an example of an "auspicious landscape", in which concentric iterations of the shape of the highly auspicious ruyi 如意 sceptre are embedded. Whiteman also analyses the painting's spatial construction and compares it to the measured views of European cities produced in the early modern period. The View of Rehe, he concludes, is both early modern, in that it includes some features that circulated across Eurasia in its time, but also Qing, in that it is constructed from varied modes of representation. The second part of the discussion focuses on the sources and the making of the Thirty-Six Views of the Mountain Estate, an album in which views are combined with poems composed by the emperor. Here the author looks at the circulation of images of imperial and royal palaces between Europe and China. [2] Part IV, entitled "The Metonymic Landscape", again examines the Thirty-Six Views, this time from the viewpoint of its circulation and reception. Two versions were made: one from woodblocks that were cut by two Chinese engravers, the other from copperplates, which were engraved by the Italian missionary Matteo Ripa (1682-1746), who was then serving at the court. Identifying a number of Ming dynasty precedents for garden albums, and constructing merged views from two or more of the views, Whiteman argues that, through their printing and circulation, Kangxi enabled the happy few who were given a copy to partake both in the private imperial landscape of his Mountain Estate, and in the emperor's emotions upon viewing them.

In a couple of passages of the book, I was not entirely convinced by the author's arguments. For example, in part II, he presents a visualisation of the assemblage of all 41 sheets that make up the Map of a Complete Survey of Imperial Territories (the outcome of the huge cartographic project carried out from 1708 to 1718) (figure 2.1, 61), and proposes reading this map as a geographical landscape in which, he states, geographic distinctions between the various territories that constituted the Qing empire dissolved, leaving a single entity. But such a reading does not stand the test of looking at the original document: the assembled map is more than 3 meters high and 4 meters wide. Even imagining that the assembled map is laid out on the floor, just by looking at it over its edge, an observer would immediately see the distinction between the Inner Territories (Ming China) and the rest of the empire, as it is marked by the use of the Chinese script for toponyms in the former and the Manchu script for the toponyms in the latter. The book-page sized visualisation corresponds to a bird's eye view of the map that would never have been seen by a beholder of the Qing period.

Such minor reservations aside, Whiteman's book is a valuable contribution to the history of the early Qing period, as well as to art history. Combining textual, visual and material sources, it reveals a new facet of Kangxi's construction of the Qing as universal rulers, whose self-display drew on the large variety of resources at their disposal in the various parts of the empire, and beyond. These resources were then incorporated into a cultural idiom that the emperor shared with Chinese scholars. When closing the book, one feels that its beautiful design and production, striking on first opening it, contribute to bringing to the fore the original and thought-provoking arguments put forward within it.

  Philippe Forêt: Mapping Chengde. The Qing Landscape Enterprise, Honolulu 2000; James A. Millward / Ruth W. Dunnell / Mark C. Elliott / Philippe Forêt (eds): New Qing Imperial History. The Making of Inner Asian Empire at Qing Chengde, London / New York 2004. [2] Richard E. Strassberg / Stephen H. Whiteman: Thirty-Six Views. The Kangxi Emperor's Mountain Estate in Poetry and Prints, Washington, D.C. 2016. Catherine Jami