Jérôme Delplanque

David Devos

Vincent Huin

Alexandre Genet

Olivier Sand

Caroline Moreau

Cyril Goizet

Perrine Charles

Mathieu Anheim

Marie Lorraine Monin

Luc Buée

Alain Destée

Guillaume Grolez

Christine Delmaire

Kathy Dujardin

Delphine Dellacherie

Alexis Brice

Giovanni Stevanin

Isabelle Strubi- Vuillaume

Alexandra Durr

Bernard Sablonnière
email: bernard.sablonniere@inserm.fr

TMEM240 mutations cause spinocerebellar ataxia 21 with mental retardation and severe cognitive impairment

Keywords: Spinocerebellar ataxia, TMEM240, exome Abbreviations: SCA = spinocerebellar ataxia, SPATAX = network of hereditary forms of spastic paraplegias and cerebellar ataxias

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Autosomal-dominant spinocerebellar ataxia (SCA) corresponds to a clinically and genetically heterogeneous group of inherited neurodegenerative diseases that are characterized by progressive cerebellar ataxia. In some cases SCA is associated with pyramidal and extrapyramidal symptoms, cognitive impairment and peripheral neuropathy [START_REF] Klockgether | Update on degenerative ataxias[END_REF]. To date, 37 different genetic loci have been reported as being associated with SCA subtypes, and 22 causative genes have been identified [START_REF] Durr | Autosomal dominant cerebellar ataxias: polyglutamine expansions and beyond[END_REF][START_REF] Hersheson | The inherited ataxias: genetic heterogeneity, mutation databases, and future directions in research and clinical diagnostics[END_REF][START_REF] Matilla-Duenas | The spinocerebellar ataxias: clinical aspects and molecular genetics[END_REF]. In previous work, we characterized a 24-member French family pedigree, 11 members of which were affected by a new form of SCA [START_REF] Devos | Clinical features and genetic analysis of a new form of spinocerebellar ataxia[END_REF].

The disease locus was initially mapped to chromosome 7 and designated as SCA21 [START_REF] Vuillaume | A new locus for spinocerebellar ataxia (SCA21) maps to chromosome 7p21.3-p15.1[END_REF]. After examination of 12 additional family members, we reconsidered our linkage analysis. Here, we report on our use of whole exome capture to both localize and identify the causative gene. We identified a coding mutation in the TMEM240 gene in the original SCA21 pedigree. Screening of 368 French families from the SPATAX network (http://spatax.wordpress.com) with autosomal-dominant cerebellar ataxia also detected this mutation in two other unrelated families. A variety of other mutations were also identified in members of five other families in the SPATAX cohort.

MATERIALS AND METHODS

Subjects

We enrolled 37 members of Family A (the original SCA21 family), including 16 affected individuals. The age at onset ranged from 1 year to 21 years. Clinical severity was evaluated on the validated 40-point Scale for the Assessment and Rating of Ataxia (with 0 being normal) [START_REF] Schmitz-Hubsch | Scale for the assessment and rating of ataxia: development of a new clinical scale[END_REF]. A cohort of 368 additional unrelated SCA pedigrees from the SPATAX cohort (originating mainly from France) was further analysed. In all families studied, a diagnosis of SCA was determined following a neurological examination by two or more experienced neurologists. Repeat expansions in the SCA1,2,3,6,7,8,12 and 17 genes were in the normal range and the presence of conventional mutations in SCA13, SCA14 and SCA28 genes was ruled out. Written informed consent was obtained from all subjects and the present study protocol was approved by the local investigational review board (registration number: DGS 2000/0468; approval for the SPATAX cohort: RBM 01-29).

Genomic DNA was isolated from peripheral blood mononuclear cells according to standard procedures.

Genetic analysis

Linkage analysis

In Family A, linkage analysis was performed on 27 family members using Illumina 16K infinity SNPlex microarrays (Illumina). Genotypes were determined using the genotyping module of Bead Studio software (Illumina) and analysed with MERLIN 1.0. software.

Linkage analysis was performed under a 0.85 penetrance frequency for family members aged 11 years or older and 0.70 for those aged 10 years or under. We used an autosomal-dominant model and a disease frequency of 0.0001. To further narrow the candidate region, linkage analysis was refined via six microsatellite markers in the telomeric region of chromosome 1: a polymorphic dinucleotide repeat (CA22) located at 1.58 Mb, a polymorphic dinucleotide repeat (TG21) located at 2.05 Mb, D1S243 at 2.1 Mb, D1S468 at 3.58 Mb, D1S2845 at 4.3Mb and D1S2870 at 6.2 Mb. Lastly, a multipoint analysis was performed using the FastLink v4.1 software and GeneHunter v2.1 platform.

Exome sequencing

Whole exome sequencing was performed using an Agilent SureSelectXT Human Exon capture kit (Agilent Technologies). The enriched samples were sequenced on the HiSeq 2000 platform (Illumina). Three affected members and one unaffected member of Family A (Patients II-12, II-13, II-15 and III-12) were sequenced. Sequence reads were mapped to the reference human genome (UCSC NCBI37/hg19) using ELAND v2.0 software. Variant detection was performed with CASAVA v1.8 software and filtered to fit a minimum depth of 8. Variants were then annotated with Ensembl database (version 66) using the latter's Perl Application Program Interface: only non-synonymous coding, stop gain or loss, frame-shift, splice site and miRNA variants were kept for further analysis. The selected variants found in all three affected members but not in the unaffected member were annotated with dbSNP v135 and dbNSFP 2.0 beta1 (in silico functional predictions and allele frequencies from the 1000 Genomes Project).

Molecular analyses of TMEM240

Mutation analysis of the four coding exons and flanking introns in TMEM240 was performed with PCR followed by direct Sanger sequencing on an ABI3730XL sequencer (Applied Biosystems). Missense variations were analysed with Alamut 2.0 software, and protein multiple sequence alignments with Crustal Omega [START_REF] Sievers | Fast, scalable generation of high-quality protein multiple sequence alignments using Clustal Omega[END_REF]. The modified protein was also analysed according to the Uniprot database (http://www.uniprot.org/), the predicted protein structure was analysed using JPred, Tmpred, GOR4, Phyre2, and NetTurnP softwares.

Phosphorylation sites were predicted using Phospho.ELM BLAST.

RESULTS

SCA21 is linked to the telomeric region of chromosome 1 in 1p36.33-p36.32

Inclusion of additional family members and extensive clinical examination allowed us to reconstruct the family pedigree. Following genome-wide low density single nucleotide polymorphism analysis, only one chromosomal region reached the expected value of 3.0. This region spans ≈8Mb (starting from chromosome 1p) and gives a two-point logarithm (base 10) of odds (LOD) score of 3.35. Using six additional markers for fine mapping, significant linkage of the disease was found with four microsatellite markers (the CA22 repeat, TG21 repeat, D1S243 and D1S468, with LOD scores of 6.64, 2.77, 6.64 and 4.94, respectively. A maximum two-point LOD score (Zmax) of 6.64 was obtained for CA22 repeat and for D1S243 at a recombination fraction of 0.00. Three markers (CA22, TG21 and D1S243) enabled us to construct a unique haplotype shared by all affected individuals but none of the unaffected individuals. The critical interval was situated between the telomere end and the centromeric marker D1S468. In Subject III-21, the interval was defined by a recombination event distal to marker D1S243 (delimiting a 3.58Mb region). According to the GRCh37/hg19 genome reference, this locus includes 94 reference genes.

Exome sequencing identifies the causative gene for SCA21

We sequenced the exomes of four affected individuals from Family A. Using the de novo assembly of exon sequences, we detected 226 coding variants and 205 indels. After analysis and filtering, 11 heterozygous variants (six coding variants and five other variants) that mapped to 1p36.33-p36.32 were found to be shared by all affected subjects and were not present in control samples or in public databases. Sanger sequencing of these variants in the remaining 32 family members identified only one variant (c.509C4T/p.P170L in TMEM240) that completely segregated with the disease phenotype (Fig. 1). This mutation was not found in 934 French controls subjects or in public databases.

Six different mutations in the TMEM240 gene

To further confirm that the identified mutation was responsible for the SCA21 subtype, we sequenced the four coding exons of TMEM240 gene in probands of 368 additional French unrelated autosomal-dominant SCA pedigrees from the SPATAX cohort for whom known SCA gene mutations had been ruled out. Six different mutations were found in seven additional unrelated families (Fig. 1). The same c.509C4T/ p.P170L in exon 4 was found in Families B and C. A c.489C4T/p.Y163* stop mutation in exon 4 was identified in Family D.

Other missense mutations were as follows: c.346C4T/p.R116C in exon 3 in Family E, c.239C4T/p.T80M in exon 3 in Family F, c.511C4T/p.R171W in exon 4 in Family G and c.445G4A/p.E149K in exon 4 in Family H. Surprisingly, the mutations found in Families B and E seemed to be de novo events. A single founder mutation responsible for P170L in Families A, B and C seemed to be unlikely, since affected patients carrying the mutation did not bear the same haplotype (data not shown). In Families A, B and E, all subjects were available for DNA genotyping enabling us to confirm the complete segregation of the mutation with the disease. In Families D, F and G, the genotyping results (when available), showed that mutations were present in all affected subjects but in none of the unaffected relatives. In Families C and H, blood samples from siblings or children were not available for segregation analysis. Furthermore, none of the six mutations were found to be present in 396 unaffected controls of French ancestry and in thousands of control samples in public databases: [dbSNP, the 1000 Genomes Project, and the Exome Variant Server (http://evs.gs.washington. edu/EVS/)].

Clinical features of patients with TMEM240 mutations

Since the first report [START_REF] Devos | Clinical features and genetic analysis of a new form of spinocerebellar ataxia[END_REF], 13 years of medical follow-up and the examination of additional children in Family A have enabled us to specify the most striking features of the disease (Table 1). Firstly, disease onset occurs early with the delayed acquisition of cognitive and motor skills. Secondly, the most striking sign is mild to severe mental retardation early evidenced at school with a variable degree of frontal behaviour disorders (impulsivity, aggressive, apathy, etc.). Thirdly, the predominant motor sign is clumsiness, which appears at a variable age (from age 2 to 20). Fourthly, the disease progresses very slowly. There was no motor impairment, cranial nerve palsy, pyramidal sign, fasciculation or sensory loss. There was no evidence of episodic cerebellar dysfunction or epilepsy in any affected family members. Neuropsychological examinations of patients revealed a severe cognitive disorder, including moderate impairments in attention, executive function, short-term, working and episodic memory abilities and, marked impairments in action planning, abstract reasoning, language and visuospatial functions. Given that most patients had marked reading and writing difficulties, the assessment procedure was adapted to include appropriate tests [START_REF] Benton | Visuospatial judgement: a clinical test[END_REF][START_REF] Renzi | Normative data and screening power of a shortened version of the Token Test[END_REF][START_REF] Wechsler | The Wechsler intelligence scale for children[END_REF][START_REF] Korkman | NEPSY II. Administrative manual[END_REF] (Table 2).

Motor, visual and brainstem auditory evoked potentials and the results of an EMG and EEG examination were normal. In Family B, Patient III-1 presented with delayed acquisition of gait (after the age of 2 years), mild mental retardation and a severe cognitive impairment. In Family C, proband II-2, displayed moderately severe gait ataxia, writing difficulties and a mild intellectual disability. In Family D, two brothers (Patients III-2 and III-3) developed mild to moderate gait ataxia, in childhood. In Family E, proband II-2 presented delayed acquisition of cognitive and motor skills, moderate mental retardation and severe cognitive impairment.

In Families F and G, the age at onset was late (59 and 61 years, respectively) and gait ataxia was mild to moderate. The proband in Family G, Patient II-2 had mild cognitive impairment.

In Family H, proband III-1 displayed poor balance, dysarthria and gait difficulties at the age of 40. Cerebellar ataxia was usually quite severe and was associated with early or late onset cognitive impairment. MRI (3 T) of the brain revealed a cerebellar atrophy of the vermis and (to a lesser extent) the hemispheres in all families; although the brainstem was unaffected.

Mild iron overload (as measured by R2* proton relaxation rates in MRI), was noted in several cases, and appeared to be most severe in the red nucleus and the pallidum in Families B and C. In Family G, brain MRI also revealed T2 and FLAIR hyperintensities in the pons, the pallidum, the putamen and the periventricular white matter.

SCA21-associated mutations alter highly conserved amino acid residues in TMEM240

Alignment of orthologous proteins (from the zebrafish to human) revealed that the amino acids residues affected by the SCA-associated mutations were strongly conserved (Fig. 2A).

Five of the six mutations concerned the carboxyl terminal portion of TMEM240. Three of these (Y163*, P170L and R171W) are clustered near a predicted beta turn centred on Proline et al., 2012). Only T80M is located between the two transmembrane alpha helices (close to Tyrosine 77, which is predicted to be phosphorylated in human brain) (Fig. 2B).

DISCUSSION

We have identified a novel causative gene for SCA, TMEM240, with various missense mutations and a stop mutation in eight different autosomal-dominant SCA families. All the amino acids involved have been strongly conserved during evolution (from the zebrafish to the human). The original mutation (P170L, observed in three different families) was not observed in the Exome Variant Server or 1868 chromosomes from French controls. The other mutations were also unknown and not observed in 792 chromosomes from French controls. The T80M mutation (observed in Family F) is associated with a milder phenotype (i.e. the absence of cognitive impairment). These findings strongly suggest that TMEM240 is the causative gene in SCA21. TMEM240 is highly expressed in the brain in general and, in the cerebellum, dentate gyrus, putamen and caudate nucleus in particular [START_REF] Hawrylycz | An anatomically comprehensive atlas of the comprehensive human brain transcriptome[END_REF]. TMEM240 encodes a predicted transmembrane protein of unknown function but which has been detected in the mouse brain synapse membrane [START_REF] Trinidad | Global identification and characterization of both OGlcNAcylation and phosphorylation at the murine synapse[END_REF]. Another transmembrane protein (TMEM237) is involved in Joubert syndrome-related disorders characterized by a specific midhind brain malformation with hypoplasia of the cerebellar vermis [START_REF] Huang | TMEM237 is mutated in individuals with a Joubert syndrome related disorder and expands the role of the TMEM family at the ciliary transition zone[END_REF]. TMEM240 does not have any homology with TMEM237 or any known protein or motif. We did not find any mutation that would undoubtedly have resulted in the loss of gene function. On the contrary, the only identified nonsense mutation that may have resulted in a gain-of-function with production of a truncated protein is Y163*. Indeed, this mutation, located in the last exon, might not be able to activate the nonsense-mediated decay pathway. Furthermore, the fact that all other are missense mutations supports the gainof-function hypothesis in SCA21. Although mutations in TMEM240 were found to be fully penetrant in six families, de novo mutations occurred in Families B and E; this could indicate the presence of spontaneous events in this telomeric region of chromosome 1. The TMEM240 gene spans 5.6 kb on 1p36.33 and contains only four coding exons. It is lost in all pure terminal deletions and most other chromosomal rearrangements observed in 1p36 deletion syndrome [START_REF] Heilstedt | Physical map of 1p36, placement of breakpoints in monosomy 1p36, and clinical characterization of the syndrome[END_REF][START_REF] Gajecka | Monosomy 1p36 deletion syndrome[END_REF]. In autosomal dominant cerebellar ataxia, mild mental retardation was recently described in SCA13 and mild to moderate lateonset cognitive impairment has been observed in SCA14 [START_REF] Herman-Bert | Mapping of spinocerebellar ataxia 13 to chromosome 19q13.3-q13.4 in a family with autosomal dominant cerebellar ataxia and mental retardation[END_REF][START_REF] Stevanin | Mutation in the catalytic domain of protein kinase C gamma and extension of the phenotype associated with spinocerebellar ataxia type 14[END_REF]. In contrast to SCA13, no petit mal epilepsy has been observed in SCA21.

Subcortical dementia is, however, present in a small subgroup (10%) of patients with SCA1, SCA2, and SCA7 [START_REF] Rüb | Clinical features, neurogenetics and neuropathology of the polyglutamine spinocerebellar ataxias type 1,2,3,6 and 7[END_REF]. Mild cognitive impairment and behavioural problems were observed in a SCA27 pedigree [START_REF] Brusse | Spinocerebellar ataxia with a mutation in the fibroblast growth factor 14 gene (SCA27): a new phenotype[END_REF]. In SCA29 (overlapping with SCA15), a similar phenotype was observed: congenital onset, no progression of ataxia and moderate cognitive impairment in both children and adults [START_REF] Dudding | Autosomal dominant congenital non-progressive ataxia overlaps with the SCA15 locus[END_REF]. In conclusion, mutations in a novel gene (TMEM240) cause SCA21, a rare form of mild cerebellar ataxia. These mutations had a frequency of 2% (8 of 368) in the French SPATAX cohort of cerebellar ataxia families analysed in this report. The most consistent feature of the disease was mild to-severe cognitive impairment (observed in seven of the eight families).

Moreover, the associated severe cognitive impairment (with mild to moderate mental retardation in early life) is unusual; it's very slow progression sets it apart from other ataxias.

We therefore recommend testing for SCA21 in families with mild or slowly progressive ataxia, particularly when moderate to severe cognitive impairment is present. e The Tower of London Test.

TABLES

f Benton Judgement of Line Orientation in adults and the NEPSY-II Arrows subtest in children.

g A computer-controlled task Orange dots indicate missense mutations and the red dot indicates a nonsense mutation.

Figure 1 .

 1 Figure 1. Pedigree charts of the eight SCA21 families. The eight families correspond to the following codes: (A) LIL-001; (B) LIL-002; (C) AAD-617; (D) AAD-881; (E) LIL-003; (F) AAD-1005; (G) AAD-656; and (H) AAD-725. Circles denote females; squares denote males. The probands are denoted by an arrow. Filled symbols denote clinically affected members, open symbols indicate unaffected individuals. / = deceased. *Genotyped individuals.

Figure 2 .

 2 Figure 2. Nature and position of six mutations within the predicted protein TMEM240. (A) The mutated residues have been conserved throughout evolution, as shown by alignment of the protein sequences of TMEM240/C1orf70 orthologues in various organisms with Crustal Omega software. (B) TMEM240's predicted secondary showing the location of the six mutations. Predicted alpha-helices are depicted as cylinders, beta-sheets as green arrows and beta turns as red bold lines. Probable transmembrane helices are filled in purple. Yellow dots indicate the predicted phosphorylated residues Y60, Y77 and S169.

Table 1 .

 1 Clinical features in 15 affected members from the eight families

	Family code

Table 2 .

 2 Cognitive impairment test results for six patients from Family A The 10/36 test in adults and the Children's Memory Scale Dot Location test in children.

N = normal; + = mild; + + = moderate; + + + = severe impairment. a Assessed by performance in the Coloured Progressive Matrices and the Wechsler Intelligence Scale for Children IV's Picture Concepts for children. b The Corsi block-tapping test. c d TheToken Test in adults and the NEPSY-II Comprehension of Instructions subtest in children.

ACKNOWLEDGEMENTS

We are indebted to all the patients and family members for their participation in this study.

We are grateful to Emmanuelle Durand, Patrick Devos, Christiane Marzys, Kathy Dupont, Thomas Bardyn and Edwige Vanbrussel for excellent technical assistance; Elisabeh Vangelder and Alix de Becdelievre for their invaluable help in fine mapping; Elodie Petit and the biological resource centre at the Institut du Cerveau et de la Moelle Epinie` re for DNA from the SPATAX network. We thank Dr. David Fraser (Biotech Communication SARL, Damery, France) for helpful comments on the manuscript's English.

FUNDING

This work was funded by the University of Lille 2, the Institut National de la Santé et de la Recherche Médicale (INSERM), the European Union (6 th Framework Programme for Research and Technological Development: the EUROSCA consortium), and the Centre Hospitalier Régional et Universitaire de Lille (CHRU). The linkage and exome work was funded by the Genome Analysis Facility at the Lille Biology Institute, and University of Lille 2. Fine genotyping analysis and Sanger sequencing was performed by the Lille Biology N N N CA N CA CA CA CA CA CA CA CA Behavior troubles +


Suraju Olawale Kasali

Jose Ordonez-Miranda

Karl Joulain

Optimization of the rectification factor of radiative thermal diodes based on two phase-change materials

Keywords: Thermal rectification, heat flows, rectification factor, emissivities, radiative thermal diodes, phase-change materials

We have theoretically studied and optimized the thermal rectification of plane, cylindrical and spherical radiative thermal diodes operating with terminals of two phase-change materials, whose emissivities significantly change within a narrow interval of temperatures. Analytical expressions for the optimal rectification factors of these three diodes are derived and analyzed comparatively. Optimal rectification factors of 82%, 86% and 90.5% are obtained for the plane, cylindrical and spherical diodes made up with terminals of Ge 2 Sb 2 Te 5 and VO 2 operating with a temperature difference of 450 -300 = 150 K, respectively. The spherical geometry thus represents a suitable shape to optimize the rectification of radiative heat currents. Furthermore, it is shown that higher rectification factors can potentially be achieved by using phase-change materials with emissivities contrasts higher than those of Ge 2 Sb 2 Te 5 and VO 2 . We demonstrate that higher rectification factors could be achieved based on the combination of two phase-change materials compared to single phase-change material. The obtained results thus shed light on the phase-change materials required for optimizing the rectification factor of radiative thermal diodes with different geometries.

INTRODUCTION

Thermal diodes transporting heat in a preferred direction and blocking it in the opposite one have recently attracted significant attention due to their promising applications on heat control and nanoscale thermal modulation [START_REF] Shih | Maximal rectification ratios for idealized bi-segment thermal rectifiers[END_REF][START_REF] Li | Colloquium: Phononics: Manipulating heat flow with electronic analogs and beyond[END_REF][START_REF] Roberts | A review of thermal rectification observations and models in solid materials[END_REF][START_REF] Li | Thermal diode: Rectification of heat flux[END_REF]. The unusual behavior of thermal diodes have paved the way for controlling the rate of heat flows in a similar way as is done with their electric current counterparts in modern electronic devices. Over the last few years, the rectification effect has been theoretically studied through thermal diodes driven by electrons [START_REF] Segal | Single mode heat rectifier: Controlling energy flow between electronic conductors[END_REF], photons [START_REF] Otey | Thermal rectification through vacuum[END_REF][START_REF] Iizuka | Rectification of evanescent heat transfer between dielectric-coated and uncoated silicon carbide plates[END_REF][START_REF] Basu | Near-field radiative transfer based thermal rectification using doped silicon[END_REF][START_REF] Wang | Thermal rectification enabled by near-field radiative heat transfer between intrinsic silicon and a dissimilar material[END_REF][START_REF] Joulain | Radiative thermal rectification between sic and sio 2[END_REF], phonons [START_REF] Roberts | A review of thermal rectification observations and models in solid materials[END_REF][START_REF] Li | Thermal diode: Rectification of heat flux[END_REF][START_REF] Garcia-Garcia | Thermal rectification assisted by lattice transitions[END_REF][START_REF] Zhang | Thermal conduction and rectification in few-layer graphene y junctions[END_REF][START_REF] Pereira | Sufficient conditions for thermal rectification in general graded materials[END_REF], [START_REF] Machrafi | Thermal rectifier efficiency of various bulk-nanoporous silicon devices[END_REF][START_REF] Criado-Sancho | Theoretical analysis of thermal rectification in a bulk Si/nanoporous Si device[END_REF], quantum dots, hybrid quantum structures [START_REF] Ordonez-Miranda | Quantum thermal diode based on two interacting spinlike systems under different excitations[END_REF][START_REF] Wu | Sufficient conditions for thermal rectification in hybrid quantum structures[END_REF][START_REF] Ruokola | Thermal rectification in nonlinear quantum circuits[END_REF], and nanoporous silicon devices [START_REF] Machrafi | Thermal rectifier efficiency of various bulk-nanoporous silicon devices[END_REF]. The experimental study of the diode effect, on the other hand, was done by using quantum dots [START_REF] Scheibner | Quantum dot as thermal rectifier[END_REF], carbon nanotubes structures [START_REF] Wu | Thermal rectification in carbon nanotube intramolecular junctions: Molecular dynamics calculations[END_REF],

graphene nanoribbons [START_REF] Yuan | Tunable thermal rectification in silicon-functionalized graphene nanoribbons by molecular dynamics simulation[END_REF], nanoporous graphene [START_REF] Yousefi | Thermal conductivity and thermal rectification of nanoporous graphene: A molecular dynamics simulation[END_REF], solid-liquid phase change thermal diodes [START_REF] Meng | Numerical and experimental study of the thermal rectification of a solid-liquid phase change thermal diode[END_REF], as well as electrostatic [START_REF] Traipattanakul | A phase-change thermal diode using electrostatic-induced coalescing-jumping droplets[END_REF] and bulk oxide materials [START_REF] Kobayashi | An oxide thermal rectifier[END_REF].

One of the main methods to rectify heat flows consists of using phase-change materials (PCMs), whose optical, electrical and thermal properties significantly change with temperature, in a relatively narrow interval of temperatures. These changes are generated by the correlated interactions of phonons and electrons inside a PCM [START_REF] Joulain | Modulation and amplification of radiative far field heat transfer: Towards a simple radiative thermal transistor[END_REF] and can be applied for developing thermal memories [START_REF] Ito | Multilevel radiative thermal memory realized by the hysteretic metal-insulator transition of vanadium dioxide[END_REF], thermal memristors [START_REF] Ben-Abdallah | Thermal memristor and neuromorphic networks for manipulating heat flow[END_REF][START_REF] Ordonez-Miranda | Radiative thermal memristor[END_REF], optical switching [START_REF] Rini | Optical switching in vo 2 films by below-gap excitation[END_REF], thermal rectifier [START_REF] Mg | On the optimization of heat rectification in graded materials[END_REF][START_REF] Huang | Thermal rectification based on thermochromic materials[END_REF], thermal switch [START_REF] Gu | Thermal switch and thermal rectification enabled by near-field radiative heat transfer between three slabs[END_REF] radiative thermal diodes [START_REF] Ben-Abdallah | Phase-change radiative thermal diode[END_REF][START_REF] Nefzaoui | Radiative thermal rectification using superconducting materials[END_REF][START_REF] Nefzaoui | Simple far-field radiative thermal rectifier using fabry-perot cavities based infrared selective emitters[END_REF][START_REF] Ito | Experimental investigation of radiative thermal rectifier using vanadium dioxide[END_REF], and radiative thermal transistors [START_REF] Ben-Abdallah | Near-field thermal transistor[END_REF][START_REF] Prod'homme | Optimized thermal amplification in a radiative transistor[END_REF] operating in both the near-and far-field [START_REF] Van Zwol | Emissivity measurements with an atomic force microscope[END_REF][START_REF] Ben-Abdallah | Phase-change radiative thermal diode[END_REF][START_REF] Ben-Abdallah | Near-field thermal transistor[END_REF][START_REF] Joulain | Modulation and amplification of radiative far field heat transfer: Towards a simple radiative thermal transistor[END_REF]. This wide variety of promising applications has motivated the interest on PCMs, such as vanadium dioxide (VO 2 ), nitinol, Ge 2 Sb 2 Te 5 (GST), whose emissivities can be thermally driven [START_REF] Ben-Abdallah | Phase-change radiative thermal diode[END_REF][START_REF] Joulain | Modulation and amplification of radiative far field heat transfer: Towards a simple radiative thermal transistor[END_REF][START_REF] Van Zwol | Emissivity measurements with an atomic force microscope[END_REF]. These three thermochromic materials have recently attracted great attention in both experimental and theoretical studies due to their metal-insulator transition (MIT) at temperatures near room temperature. For example, the emissivities of GST and VO 2 undergo the MIT within the temperature ranges of 420 K < T < 430 K and 340 K < T < 345 K, respectively [START_REF] Barker | Infrared optical properties of vanadium dioxide above and below the transition temperature[END_REF]. The fact that the MIT of VO 2 occurs in a temperature interval of only 5 K has motivated its application as one of the terminals of a radiative thermal diode [START_REF] Ben-Abdallah | Phase-change radiative thermal diode[END_REF], whose rectification factor can be as high as 90%. This significant rectification is more than four times the typical ones obtained without PCMs [START_REF] Nefzaoui | Simple far-field radiative thermal rectifier using fabry-perot cavities based infrared selective emitters[END_REF], which indicates the great potential of these materials to rectify heat flows.

Taking into account that a thermal diode is composed of two terminals and that two PCMs provide more degrees of freedom than a single PCM to tailor the heat transport between them, the rectification factor of radiative thermal diodes is expected to be optimized by the combined effect of two PCMs, as was proposed for conductive thermal diodes [START_REF] Cottrill | Analysis of thermal diodes enabled by junctions of phase change materials[END_REF][START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF]. For instance, Yang et al. [START_REF] Yang | Optimal thermal rectification of heterojunctions under fourier law[END_REF], determined rectification factors up to 50% and 80% for a conductive thermal diode made up of two PCMs, whose thermal conductivities vary linearly and quadratically with temperature, respectively. Furthermore, a high rectification ratio of 147% was also obtained by Kang et al. [START_REF] Kang | Thermal rectification via heterojunctions of solid-state phase-change materials[END_REF] for a conductive thermal diode operating with two PCMs. These rectification factors are higher than the ones obtained for conductive thermal diodes base on a single PCM [START_REF] Cottrill | Analysis of thermal diodes enabled by junctions of phase change materials[END_REF][START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF]. This fact indicates that two PCMs could also be applied to enhance the rectification factor of radiative thermal diodes, which is not reported in the literature yet, to the best of our knowledge The goal of this work is to theoretically optimize the thermal rectification effect of plane, cylindrical and spherical radiative thermal diodes based on two PCMs. This is comparatively done by considering the temperature dependence of the emissivities of each PCM within their MITs. Simple analytical expressions are derived and analyzed for the optimal rectification factors of these three diodes supporting thermal radiation generated by a relatively narrow temperature difference between their terminals. Our calculations show that the rectification factor can be raised significantly by using two PCMs instead of the single PCM.

T c T h T c T h T c 𝒒 𝑭 𝒒 𝑩 (a) (b) GST GST VO 2 VO 2

THEORETICAL MODELING

Let us consider two semi-infinite plates consisting of GST and VO 2 exchanging heat through radiation as a result of their temperature difference T h -T c > 0, as illustrated in Figs. 1(a) and 1(b), respectively. The two plates are separated by a vacuum gap of thickness d much longer than the involved thermal wavelengths to ensure the far-field regime of heat transport. In the forward configuration [Fig. 1(a)], the heat flow q F flows from GST to VO 2 , while in the backward one [Fig. 1(b)], the heat flow q B flows in the opposite direction due to the interchange of the temperatures T h and T c . This behavior of the heat flows (q F and q B ) also holds for the spherical or cylindrical radiative thermal diodes shown in Fig. 2. These heat flows are expected to be different (q F =q B ) due to the temperature dependence of the emissivities of GST and VO 2 , within their MITs. The ability of these temperature-controlled thermal diodes to rectify the heat flows can, therefore, be defined by the following rectification factor R [47]

R = |q F -q B | max(q F , q B ) . (1)
. [START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF][START_REF] Du | Control over emissivity of zero-static-power thermal emitters based on phase-changing material gst[END_REF], while lines stand for fits of Eq. (4).

VO 2 T h T c q F q B r 2 r 1 1 GST . .
Equation ( 1) establishes that a non-zero rectification is reached when the reversal of the thermal gradient induces not only the inversion of the heat flow direction but also a sizeable change of their magnitude.

Optimal rectification factor [R opt = 1 -q B,min /q F,max → 1] can thus be achieved when one of the heat flows is much smaller than the other one. According to Stefan-Boltzmann's law of far-field radiation between two bodies separated by a vacuum gap, the heat flows q F and q B in the forward and backward configurations of the plane, cylindrical and spherical radiative thermal diodes are given by [START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF]

q F = A 1 σε F (T 4 h -T 4 c ), (2a)
q B = A 2 σε B (T 4 h -T 4 c ), (2b)
where σ = 5.67×10 -8 Wm -2 K -4 is the Stefan-Boltzmann constant, A 1 and A 2 are the areas of the radiating surfaces, while ε F and ε B are their effective emissivities in the forward and backward configurations. The combination of Eqs. ( 1) and ( 2) yields the following rectification factor:

R = |A 1 ε F -A 2 ε B | max(A 1 ε F , A 2 ε B ) . (3)
Equation ( 3) explicitly establishes that the values of R are driven by the combined effect of the radiating surfaces and their effective emissivities. The rectification factors of the plane, spherical and cylindrical radiative diodes made up with terminal of GST and VO 2 are thus expected to be different due to differences in A 1 and A 2 . The emissivities of both these PCMs can be described by [START_REF] Ordonez-Miranda | Periodic amplification of radiative heat transfer[END_REF] ε

n (T ) = ε in + ε mn -ε in 1 + e -βn(T -T0n) , (4)
where ε mn and ε in are the emissivities in their respective high-and low-temperature phases, and β n is the phase-transition slope of ε n (T ) at the transition temperature T = T 0n , for n = 1 and 2. The values of these four parameters were determined by fitting Eq. ( 4) to the experimental data shown in Fig. 3 for GST and VO 2 , and they are summarized in Table I. In order to better understand the predictions of Eq. (3) for R, its particular and explicit expressions for the plane, cylindrical and spherical radiative diodes are now going to be determined and optimized.

Plane diode

For the sake of simplicity, we are going to consider that the areas of the radiating surfaces of the plane diode are equal (A 1 = A 2 = A) and therefore the effective emissivities for the forward and backward configurations can be written as follows [START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF] ε

F = 1 ε 1 (T h ) + 1 ε 2 (T c ) -1 -1 , (5a)
ε B = 1 ε 1 (T c ) + 1 ε 2 (T h ) -1 -1 , (5b)
where are ε n (T h ) and ε n (T c ) are the emissivities of the PCMs (n = 1; 2) in their high-and low-temperature phases, respectively. Given that the areas of the radiating surfaces are equal (A 1 = A 2 ), the optimal rectification factor in Eq. ( 3) can be determined by the ratio ε B,min /ε F,max between the minimum emissivity in the backward configuration to the maximum emissivity in the forward one, as follows:

R opt = 1 - ε B,min ε F,max . (6)
In order to capitalize on the MITs of both PCMs to find R opt , their temperatures should fulfill the condition

T c < T 0n < T h . According to Fig. ( 3
) and Eqs. ( 4) and ( 5), in the forward configuration

ε F = ε F,max
when the GST terminal is in its high-temperature phase (ε 1 (T h ) = ε m1 ), while the VO 2 one is in its lowtemperature phase (ε 2 (T c ) = ε i2 ). In the backward configuration, on the other hand, ε B = ε B,min when the GST terminal is in its low-temperature phase (ε 1 (T c ) = ε i1 ), while the VO 2 one is in its high-temperature phase (ε 2 (T h ) = ε m2 ). Under these conditions, Eqs. (5a) and (5b) reduce to

ε F,max = 1 ε m1 + 1 ε i2 -1 -1 , (7a)
ε B,min = 1 ε i1 + 1 ε m2 -1 -1 . (7b)
After inserting Eqs. (7a) and (7b) into Eq. ( 6), the following optimal rectification factor R opt of a radiative plane diode is obtained

R opt = 1 - ε m2 ε i2 (1 + ε i2 /ε m1 -ε i2 ) (1 + ε m2 /ε i1 -ε m2 ) . ( 8
)
Note that R opt is symmetric on the emissivities of both PCMs and increases with the ratios ε i1 /ε m1 and ε i2 /ε m2 of GST and VO 2 , respectively. Materials with a higher emissivity contrasts are thus expected to yield radiative diodes with higher rectification factors, as expected.

Cylindrical diode

In this case, the areas A 1 and A 2 of the radiation surfaces depends on their respective radii r 1 and r 2 , such that A 1 /A 2 = r 1 /r 2 . The effective emissivities for the forward and backward configurations are hence

given by [START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF] ε

F = 1 ε 1 (T h ) + 1 -ε 2 (T c ) ε 2 (T c ) r 1 r 2 -1 , (9a)
ε B = 1 ε 1 (T c ) + 1 -ε 2 (T h ) ε 2 (T h ) r 2 r 1 -1 . ( 9b
)
where r 1 and r 2 are radii of the cylindrical radiating surfaces (Fig. 2). In contrast to the corresponding effective emissivities for the plane diode, Eqs. (9a) and (9b) do depends on ratio r 1 /r 2 , which provides another degree of freedom to optimize the rectification factor of the radiative cylindrical diode. In this case, Eq. ( 3) takes the form

R opt = 1 - r 2 ε B,min r 1 ε F,max , (10)
where the effective emissivities ε F = ε F,max and ε B = ε B,min for the forward and backward configurations can be found by following a similar procedure than the one developed for the plane diode. The final results are

ε F,max = 1 ε m1 + 1 -ε i2 ε i2 r 1 r 2 -1 , (11a)
ε B,min = 1 ε i1 + 1 -ε m2 ε m2 r 2 r 1 -1 . ( 11b
)
The optimal rectification factor R opt of the cylindrical diode is then determined by the combination of Eqs. ( 10) and ( 11), as follows

R opt = 1 - ε m2 ε i2 r 1 r 2 (1 + r 2 ε i2 /r 1 ε m1 -ε i2 ) (1 + r 1 ε m2 /r 2 ε i1 -ε m2 ) . ( 12
)
The comparison of Eqs. ( 8) and [START_REF] Zhang | Thermal conduction and rectification in few-layer graphene y junctions[END_REF] indicates that the values of R opt for a cylindrical diode could be higher than the corresponding ones for a plane diode, by properly modulating the ratio r 1 /r 2 as well as the emissivity contrast of the PCMs.

Spherical diode

In this case, the ratio of the areas A 1 and A 2 of the radiating surfaces is given by the square of the one of their radii (A 1 /A 2 = (r 1 /r 2 ) 2 ) and the effective emissivities for the forward and backward configurations are given by [START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF] ε

F = 1 ε 1 (T h ) + 1 -ε 2 (T c ) ε 2 (T c ) r 1 r 2 2 -1 , (13a)
ε B = 1 ε 1 (T c ) + 1 -ε 2 (T h ) ε 2 (T h ) r 2 r 1 2 -1 , (13b)
Equations ( 13a) and (13b) are similar to Eqs. (9a) and (9b) for the cylindrical diode, but the only difference is that the ratio r 1 /r 2 is now squared. The spherical geometry is thus expected to have a different impact than the cylindrical one on the rectification factor of a radiative diode. According to Eq. ( 3), the optimal rectification factor can be written in terms of the effective emissivities for the spherical diode as follows:

R opt = 1 - r 2 2 ε B,min r 2 1 ε F,max , (14)
where, according to Eqs. (13a) and (13b), the effective emissivities ε F = ε F,max and ε B = ε B,min for the forward and backward configurations are given by Finally, the explicit expression for R opt is determined by the combination of Eqs. ( 14) and [START_REF] Wu | Sufficient conditions for thermal rectification in hybrid quantum structures[END_REF], which 2, while the dashed one corresponds to a system with interchanged terminals. The inset in (a) shows a zoom-in of the first maximum and minimum values of q B . Calculations were done for Tc = 300 K and

ε F , max = 1 ε m1 + 1 -ε i2 ε i2 r 1 r 2 2 -1 , (15a)
ε B , min = 1 ε i1 + 1 -ε m2 ε m2 r 1 r 2 2 -1 . (15b)
r 1 /r 2 = 0.8. yield R opt = 1 - ε m2 ε i2 r 1 r 2 2 (1 + r 2 2 ε i2 /r 2 1 ε m1 -ε i2 ) (1 + r 2 1 ε m2 /r 2 2 ε i1 -ε m2 ) . (16)
One can see that R opt is symmetrical on the emissivities of both PCMs and has a stronger (squared) dependence on the ratio r 1 /r 2 than that of a cylindrical diode. The spherical diode is thus expected to yield higher rectification factors than the corresponding cylindrical one, as shown below.

RESULTS AND DISCUSSIONS

Figure 4(a) shows the heat flows exchanged by the terminals of a plane diode operating in the forward and backward configurations, as functions of the temperature T h . Note that both heat flows q F and q B exhibit a nearly linear increase with T h and change of slope at temperatures around the transition temperatures (T h ≈ T 0n with n = 1, 2) of the PCMs emissivities. According to Fig. 3, the transition of q B at T h = 340.6

K is related to the MIT of VO 2 , while that of q F at T h = 343.9 K is driven by the MIT of GST. For T h < 340 K, both PCMs do not undergo any phase transition, q F and q B are equal, and therefore there is no rectification, as shown in Fig. 4(b). By contrast, the rectification factor reaches its first plateau R 30% just above the VO 2 MIT (T h > 340 K). For higher temperatures (T h > 423.9 K), both terminals undergo their MITs and the rectification factor increases to R 82%, which agrees with the predictions of Eq. ( 8).

This fact indicates that the combination of the MITs of GST and VO 2 is able to maximize the rectification factor of a plane diode to values higher than the corresponding ones obtained with a single PCM [START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF][START_REF] Nefzaoui | Radiative thermal rectification using superconducting materials[END_REF].

Figure 1 :

 1 Figure 1: Scheme of a plane radiative thermal diode made up with terminals of GST and VO 2 operating in the (a) forward and (b) backward configurations. Hot and cold thermal baths are set at the temperatures T h and Tc, respectively.

Figure 2 :

 2 Figure2: Scheme of a spherical (or cylindrical) radiative thermal diode made up of GST and VO 2 operating in the forward and backward configurations. The radii r 1 and r 2 are set at the temperatures T h and Tc for the forward configuration, while they are interchanged for the backward one. In practice, a spherical diode could be implemented by using the classical Langmuir-Blodgett law[START_REF] Langmuir | Currents limited by space charge between concentric spheres[END_REF].

5 - 1 Figure 3 :

 513 Figure 3: Effective emissivities of GST and VO 2 as functions of temperature within their MITs. Dots represent experimental data reported in the literature[START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF][START_REF] Du | Control over emissivity of zero-static-power thermal emitters based on phase-changing material gst[END_REF], while lines stand for fits of Eq. (4).

  t f l o w , q ( W ) T e m p e r a t u r e , T h ( t i f i c a t i o n f a c t o r , R ( % ) T e m p e r a t u r e , T h ( K ) 3 4 0 . 6

Figure 4 :

 4 Figure 4: Temperature dependence of the radiative heat flows for the (a) forward and backward configurations of a plane radiative diode operating with terminals of GST and VO 2 . Inset: Zoom-in of the first maximum and minimum values of q B . (b) Rectification factor as a function of T h . Calculations were done for Tc = 300 K.

Figure 5 :

 5 Figure 5: Temperature dependence of the radiative heat flows for the (a) forward and backward configurations of a cylindrical diode operating with terminals of GST and VO 2 along with (b) its corresponding rectification factor. The solid line stand for the thermal diode shown in Fig.2, while the dashed one corresponds to a system with interchanged terminals. The

  e r i c a l d i o d e C y l i n d r i c a l d i o d e P l a n e d i o d e R e c t i f i c a t i o n f a c t o r , R ( % ) T e m p e r a t u r e , T h ( K )

Figure 7 :

 7 Figure 7: Comparison of the rectification factors obtained for the plane, cylindrical, and spherical diodes operating with terminals of GST and VO 2 . Calculations were done for Tc = 300 K and r 1 /r 2 = 0.8.

Figure 8 :

 8 Figure 8: Temperature dependence of the rectification factor of plane, cylindrical, and spherical diodes with terminals of (a) VO 2 and a black body and (b) GST and black body. Calculations were done for Tc = 300 K and r 1 /r 2 = 0.8.

Table 1 :

 1 Material properties involved in Eq. (4).

	Materials ε in	ε mn β n (K -1 ) T 0n (K)
	GST	0.13 0.52	0.46	426.9
	VO 2	0.79 0.22	1.6	342.4

q F q B H e a t f l o w , q ( W )

T e m p e r a t u r e , T h ( K ) 2, while the dashed one corresponds to a system with interchanged terminals. The inset in (a) shows a zoom-in of the first maximum and minimum values of q B . Calculations were done for Tc = 300 K and

The temperature dependence of the heat flows q F and q B for the forward and backward configurations of a cylindrical diode along with its rectification factor are shown in Figs. 5(a) and 5(b), respectively.

Note that the values of the heat flow generated in the cylindrical diode exhibits a similar behavior than the corresponding ones in the plane diode, although their values are relatively higher than these latter ones. The diode geometry has thus a significant impact on the values of the heat flows, whose behavior is determined by the PCMs' emissivities. One can see that both q F and q B increase linearly with T h , such that their difference is greater than zero for temperatures T h ≤ 342.9 K. The sharp increase of q F around T h = 423.9 K is generated by the GST emissivity and allows increasing the rectification factor up to R 86%. This value is consistent with Eq. ( 12) and is higher than the one obtained for a plane diode (Fig. 4(b)).

The heat flows q F and q B of a spherical diode operating with terminals of GST and VO 2 are shown in Fig. 6(a). Note that the linear increase of q F and q B with T h is consistent with the one reported for the plane and cylindrical diodes, such that their values are relatively higher than the corresponding ones for these two latter diodes. Note also that the temperature interval (3.6 K in width) between the first maximum and first minimum of q B of the three diodes is the same, which confirms that the behavior of the heat flux is driven by the PCMs' emissivities. Taking into account that these extreme values of q B appear at temperatures close to the those determining the borders of the metal-insulator transition of VO 2 (Fig. 3), the local maximum of q B is related to the appearance of this phase transition, while the local minimum of q B is associated with its disappearance, as the temperature is raised. On the other hand, the two plateaux of the rectification factor occur at R 63% and R 90.5% (Fig. 6(b)), which are higher than the corresponding ones of the

T e m p e r a t u r e , T h ( K ) ( c ) plane and cylindrical diodes. This fact indicates that the spherical geometry is better than the other two diode geometries to maximize the rectification of radiative heat flows, as shown in Fig. 7. Note that if the terminals of the thermal diode shown in Fig. 2 are interchanged, the rectification factors of the cylindrical and spherical thermal diodes are smaller than the corresponding ones obtained for the system in Fig. 2, as shown in Figs. 5(b) and 6(b). This is expected given that both diode geometries have a strong dependence on the effective emissivities ratio and area of the radiating surfaces. However, this is not the case for the plane diode rectification factor, as the areas of the radiating surfaces are equal.

In order to single out the effect of a PCM on the rectification factor of the three radiative diodes under consideration, we have replaced the GST terminal by a black body (emissivity = 1) and plotted the temperature dependence of the rectification factors thus obtained in Fig. 8(a). Note that R undergoes a single transition generated by the VO 2 MIT, such that it reaches a plateau for high enough temperatures. A similar behavior is observed when the VO 2 terminal is replaced by a black body, as shown in Fig. 8(b). In


domain: nlin phys

text:
Suraju Olawale Kasali

Jose Ordonez-Miranda

Karl Joulain

Conductive thermal diode based on two phase-change materials

Keywords: Thermal rectification, conductive thermal diode, phase-change materials

The thermal rectification of a conductive thermal diode made up of two phase-change materials, whose thermal conductivities significantly change within a narrow interval of temperatures, is theoretically studied and optimized. This is done by deriving analytical expressions for the temperature profiles, heat fluxes and rectification factor. An optimal rectification factor of 60% is obtained for a thermal diode operating with terminals of VO 2 and Polyethylene with a temperature difference of 250 K spanning the metal-insulator transition of both materials. It is shown that this high rectification of a conductive thermal diode could be maximized even more by increasing the thermal conductivity variations of both diode terminals. The obtained results can thus be useful to guide the development of phase-change materials capable of optimizing the rectification of conductive heat fluxes.

INTRODUCTION

Thermal diodes allowing the heat exchange between two terminals in a given direction and blocking it in the opposite one, have attracted significant attention over the past decade, due to their potential applications in heat control [START_REF] Shih | Maximal rectification ratios for idealized bi-segment thermal rectifiers[END_REF][START_REF] Li | Colloquium: Phononics: Manipulating heat flow with electronic analogs and beyond[END_REF][START_REF] Roberts | A review of thermal rectification observations and models in solid materials[END_REF]. This thermal rectification is analogous to the electrical one driven by electronic diodes and has been studied through thermal diodes operating with photons [START_REF] Otey | Thermal rectification through vacuum[END_REF][START_REF] Iizuka | Rectification of evanescent heat transfer between dielectric-coated and uncoated silicon carbide plates[END_REF][START_REF] Basu | Near-field radiative transfer based thermal rectification using doped silicon[END_REF][START_REF] Wang | Thermal rectification enabled by near-field radiative heat transfer between intrinsic silicon and a dissimilar material[END_REF], phonons [8, 9, 10, 11? , 3], electrons [START_REF] Segal | Single mode heat rectifier: Controlling energy flow between electronic conductors[END_REF] and quantum structures [START_REF] Wu | Sufficient conditions for thermal rectification in hybrid quantum structures[END_REF][START_REF] Ordonez-Miranda | Quantum thermal diode based on two interacting spinlike systems under different excitations[END_REF]. This phenomenon was first experimentally observed by Starr [15] at the interface between copper/cuprous oxide and then observed in carbon nanotubes structures [START_REF] Wu | Thermal rectification in carbon nanotube intramolecular junctions: Molecular dynamics calculations[END_REF][START_REF] Chang | Solid-state thermal rectifier[END_REF], dielectric quantum dots [START_REF] Scheibner | Quantum dot as thermal rectifier[END_REF], phase-change materials (PCMs) [START_REF] Van Zwol | Tuning near field radiative heat flux through surface excitations with a metal insulator transition[END_REF][START_REF] Van Zwol | Emissivity measurements with an atomic force microscope[END_REF][START_REF] Van Zwol | Fast nanoscale heat-flux modulation with phase-change materials[END_REF][START_REF] Tso | Solid-state thermal diode with shape memory alloys[END_REF] and bulk materials [START_REF] Kobayashi | An oxide thermal rectifier[END_REF].

Thermal rectification of conductive heat fluxes is generally generated by the temperature dependence of the thermal conductivities of the diode terminals [START_REF] Li | Non-reciprocal geometric wave diode by engineering asymmetric shapes of nonlinear materials[END_REF] and therefore it can, in principle, be achieved with arbitrary solid materials subjected to a sufficiently large temperature gradient. Based on the Fourier's law of heat conduction, Yang et al. [START_REF] Yang | Optimal thermal rectification of heterojunctions under fourier law[END_REF] modeled and optimized the rectification ratio of a conductive thermal diode made up of two terminals, whose thermal conductivities follow either a linear or a quadratic dependence on temperature. For the linear variation, they obtained a maximum rectification ratio of 0.5, which increases to 0.86, for the quadratic dependence. These relatively high rectification ratios required a temperature difference greater than 100 K between the diode terminals, which could be difficult and expensive to achieve in practice. This drawback can be overcome by means of PCMs, such as vanadium dioxide (VO 2 ), nitinol, and polyethylene (PE), whose thermal conductivities strongly change in a relatively narrow interval of temperatures [START_REF] Ordonez-Miranda | Transistorlike device for heating and cooling based on the thermal hysteresis of vo 2[END_REF][START_REF] Ito | Experimental investigation of radiative thermal rectifier using vanadium dioxide[END_REF][START_REF] Rini | Optical switching in vo 2 films by below-gap excitation[END_REF][START_REF] Ito | Multilevel radiative thermal memory realized by the hysteretic metal-insulator transition of vanadium dioxide[END_REF][START_REF] Ben-Abdallah | Thermal memristor and neuromorphic networks for manipulating heat flow[END_REF][START_REF] Qazilbash | Electrostatic modification of infrared response in gated structures based on vo 2[END_REF][START_REF] Driscoll | Dynamic tuning of an infrared hybrid-metamaterial resonance using vanadium dioxide[END_REF]. These PCMs with a strong phonon-electron correlation have recently attracted a lot of experimental and theoretical interest, due to their metal-insulator transition (MIT) at temperatures close to room temperature. For instance, the thermal conductivities of VO 2 and PE typically change within a relatively small ranges of temperature 340 K < T < 345 K and 395 K < T < 400 K, respectively [START_REF] Zhang | Giant thermal rectification from polyethylene nanofiber thermal diodes[END_REF][START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF]. The fact that the MIT of VO 2 occurs in a temperature range of only 5 K has motivated its application in combination with a non-PCM, as the terminal of a conductive thermal diode, however, the maximal rectification factor obtained was lower (∼ 20%) [START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF] than that reported by Yang et al. [START_REF] Yang | Optimal thermal rectification of heterojunctions under fourier law[END_REF] This indicates that the operation of a conductive thermal diode with both terminals made up of PCMs could enable not only to maximize its rectification factor, but also to reduce the temperature difference between its terminals, as was recently proposed [START_REF] Cottrill | Dual phase change thermal diodes for enhanced rectification ratios: Theory and experiment[END_REF]. Based on an alternative definition of the rectification factor, Hyungmook et al. [START_REF] Kang | Thermal rectification via heterojunctions of solid-state phase-change materials[END_REF] recently determined a rectification factor of 147% for a thermal diode with the terminals of VO 2 and PE, whose thermal conductivities significantly change with temperatures. This relatively high rectification was obtained by describing the thermal conductivities of both PCMs as step functions, which is similar but not equal to their real behaviors, as the phase transitions of VO 2 and PE occur in a temperature interval greater than 5 K. Therefore, a more realistic description of the thermal conductivities of the rectification of thermal diodes based on two PCMs is thus required.

In this work, we theoretically study the optimization of the rectification factor of a conductive thermal diode made up of a junction of VO 2 and PE. This is done by taking into account the temperature dependence of the thermal conductivities of these PCMs within their phase transitions. Simple analytical expressions for the temperature profiles, heat fluxes and optimal rectification factor are derived and analyzed for the steady-state heat conduction. The present study is thus based on the combination of two PCMs, which are expected to yield higher rectification factors than a thermal diode based on a single PCM, which we addressed in a previous work et al. [START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF]. Hot and cool thermal shells are set at the temperatures T h and Tc, respectively.

THEORETICAL MODELING

Let us consider a system of two phase change materials exchanging heat through steady-state conduction as a result of their temperature difference T h -T c > 0, as illustrated in Figs. 1(a , the heat flux q B occurs in the opposite direction, as a result of the interchange of the temperatures T h and T c . The values of q F and q B are driven by the thermal conductivities of κ 1 (T ) of PCM1 and κ 2 (T ) of PCM2, and are not expected to be equal (q F =q B ), due of the temperature dependence of κ 1 (T ) and κ 2 (T ).

For temperatures within the MIT of VO 2 and PE. Therefore, the asymmetry of κ n (T ) around its transition temperature T 0n , with n = 1, 2, is much larger. It allows optimizing the heat flux difference q F -q B by setting the terminals' temperatures as follows: T c < T 0n < T h . The ability of this temperature-controlled thermal diode to rectify heat fluxes can thus be defined by the following rectification factor R [34]

R = |q F -q B | max(q F , q B ) . (1)
Equation ( 1) establishes that a nonzero rectification is achieved when the reversal of the temperature gradient induces not only the inversion of the heat flux direction, but also a significant variation of the heat flux magnitude. Optimal rectification (R = 1) is reached when one of the heat fluxes is zero. triangles stand for experimental data reported in the literature [START_REF] Oh | Thermal conductivity and dynamic heat capacity across the metalinsulator transition in thin film vo 2[END_REF][START_REF] Zhang | Giant thermal rectification from polyethylene nanofiber thermal diodes[END_REF], while lines represent the theoretical predictions of Eq.

(3).

Forward configuration

According to the Fourier's law of heat conduction and the principle of energy conservation, the steadystate heat flux q F in the forward configuration of the thermal diode shown in Fig 1(a), is given by

q F = -κ 1 (T 1 ) dT 1 dx = -κ 2 (T 2 ) dT 2 dx , (2)
where T 1 (x) and T 2 (x) are the temperatures within the PCM1 (0

≤ x ≤ L 1 ) and PCM2 (L 1 ≤ x ≤ L 1 + L 2 ),
respectively. In this work, we consider that the thermal conductivities κ 1 (T ) and κ 2 (T ) belong to VO 2 and PE, respectively, and their temperature variations are shown in Fig. 2, within their MITs. Note that the thermal conductivity contrast of PE (25/5 = 5) is higher than that of VO 2 (6/3.6 = 1.7), such that they reach their maximum thermal conductivities for temperatures T < 395 K and T > 345 K, respectively. This fact along with Eq. ( 2), indicates that q F can be maximized by properly setting the temperatures T h and T c to fulfill these temperature conditions.

The thermal conductivities of both VO 2 and PE (Fig. 2) can be described by

κ n (T ) = κ in + κ mn -κ in 1 + e -βn(T -T0n) , (3)
where κ mn and κ in are the thermal conductivities in their respective high-and low-temperature phases, and β n is the phase-transition slope of κ n (T ) at the transition temperature T = T 0n , for n = 1 and 2. The Material κ in (Wm The temperatures T 1 (x) and T 2 (x) can now be determined by integrating Eq. ( 2) for the thermal conductivities in Eq. ( 3), which yields

-1 K -1 ) κ mn (Wm -1 K -1 ) β n (K -1 ) T 0n (K) VO 2 3.
T n (x) -T 0n + 1 β n γ n ln [1 + M n (T n (x))] = C n κ mn - q F κ mn x, (4)
where C n is an integration constant, γ n = κ mn /(κ mn -κ in ), and

M n (T ) = exp [-β n (T -T 0n )].
The values of the three parameters q F , C 1 and C 2 are determined by the boundary conditions

T 1 (0) = T h , T 2 (L 1 + L 2 ) = T c , and T 1 (L 1 ) -T 2 (L 1 )
= ρq F , with ρ being the interface thermal resistance between the PCM1 and PCM2. Equation ( 4) thus yields the following expressions for the heat flux and temperature profiles

q F = 1 α 1 ln Z 1 (T h ) Z 1 (T 1 (L 1 )) , (5a)
q F = 1 α 2 ln Z 2 (T 1 (L 1 ) -ρq F ) Z 2 (T c ) , (5b)
Z 1 (T 1 (x)) = [Z 1 (T h )] 1-x/L1 [Z 1 (T 1 (L 1 ))] x/L1 , (5c)
Z 2 (T 2 (x)) = [Z 2 (T 1 (L 1 ) -ρq F )] 1-(x-L1)/L2 [Z 2 (T c )] (x-L1)/L2 , (5d)
where α n = β n γ n ρ mn , ρ mn = L n /κ mn , and

Z n (T ) = [1 + M n (T )] / [M n (T )]
γn . The substitution of Eq. (5a) into Eq. (5b) results in a strong non-linear dependence of the temperature T 1 (L 1 ) = θ on T h and T c , as follows:

Z 1 (T h ) Z 1 (θ) α 2 α 1 = Z 2 θ -ρ α1 ln Z1(T h ) Z1(θ) Z 2 (T c ) . (6)
After numerically solving Eq. ( 6) for the interface temperature θ, the forward heat flux and temperature profiles can be determined through Eqs. (5b)-(5d), for a given set of temperatures T h and T c .

Backward configuration

The backward heat flux (q B ) and the temperatures [T 1 (x) and T 2 (x)] for the backward configuration [Fig. 1(b)], can be determined by following a similar approach than the one developed for the forward configuration. The appropriate boundary conditions are

T 2 (0) = T h , T 1 (L 1 +L 2 ) = T c , and T 2 (L 2 )-T 1 (L 2 ) = ρq B .
The final results are

q B = 1 α 1 ln Z 1 (T 1 (L 2 )) Z 1 (T c ) , (7a)
q B = 1 α 2 ln Z 2 (T h ) Z 2 (T 1 (L 2 ) + ρq B ) , (7b)
Z 1 (T 1 (x)) = [Z 1 (T 1 (L 2 ))] 1-(x-L2)/L1 [Z 1 (T c )] (x-L2)/L1 , (7c)
Z 2 (T 2 (x)) = [Z 2 (T h )] 1-x/L2 [Z 2 (T 1 (L 2 ) + ρq B )] x/L2 . ( 7d
)
The combination of Eqs. (7a) and (7b) yields the following transcendental equation for the interface temperature T 1 (L 2 ) = φ, as a function of T h and T c :

Z 1 (φ) Z 1 (T c ) α 2 α 1 = Z 2 (T h ) Z 2 φ + ρ α1 ln Z1(φ) Z1(Tc) . (8)
The numerical solution of Eq. ( 8) for φ, fully describes the heat conduction in the thermal diode through Eqs. (7b)-(7d).

Rectification factor

In order to maximize q F and minimize q B for optimizing the rectification factor R in Eq. ( 1), we have conveniently rewrite Eqs. (5a), (5b), (7a), and (7b), as follows:

q F = T h -θ ρ m1 + (ρ m1 ) -1 β 1 γ 1 ln 1 + e -β1(T h -T01)
1 + e -β1(θ-T01) , (9a)

q F = θ -T c ρ + ρ m2 + (ρ + ρ m2 ) -1 β 2 γ 2 ln 1 + e β2(T02+ρq F -θ) 1 + e β2(T02-Tc) , (9b)
q B = φ -T c ρ m1 - (ρ m1 ) -1 β 1 γ 1 ln 1 + e β1(T01-Tc) 1 + e β1(T01-φ) , (9c)
q B = T h -φ ρ + ρ m2 - (ρ + ρ m2 ) -1 β 2 γ 2 ln 1 + e -β2(φ-T02+ρq B ) 1 + e -β2(T h -T02) . (9d)
Equations ( 9a) and (9b) establishes that when the temperature T h (T c ) is high (low) enough for fulfilling the conditions

β 1 (T h -T 01 ) 1, β 1 (θ -T 01 ) 1, β 2 (T 02 -T c ) 1, and β 2 (ρq F + T 02 -θ) 1, they reduce to q F = T h -θ ρ m1 = θ -T c ρ + ρ i2 , (10)
where ρ i2 = L 2 /κ i2 and ρ m1 = L 1 /κ m1 . Equation [START_REF] Pereira | Sufficient conditions for thermal rectification in general graded materials[END_REF] explicitly indicates that, under these conditions, the whole terminal 1 (PCM1) is in its high-temperature (metallic) phase, while the entire terminal 2 (PCM2) is in its low-temperature one. According to Fig. 2, both of these phases are characterized by the highest thermal conductivities of PCM1 and PCM2, and therefore they maximize the heat flux q F = q F,max determined by

q F,max = T h -T c ρ m1 + ρ + ρ i2 , (11a)
T 1,max (L 1 ) = (ρ + ρ i2 )T h + ρ m1 T c ρ m1 + ρ + ρ i2 , (11b)
T 2,max (L 1 ) = ρ i2 T h + (ρ m1 + ρ)T c ρ m1 + ρ + ρ i2 , (11c)
where T 1 (L 1 ) = T 1,max (L 1 ) and T 2 (L 1 ) = T 2,max (L 1 ) are the interface temperatures associated to q F,max .

On the other hand, Eqs. (9c) and (9d) show that when the temperature T h (T c ) is high (low) enough for satisfying the conditions

β 1 (T 01 -T c ) 1, β 1 (T 01 -φ) 1, β 2 (T h -T 02 ) 1, and β 2 (ρq B + φ -T 02 ) 1,
they turn into

q B = φ -T c ρ i1 = T h -φ ρ + ρ m2 , (12)
where ρ i1 = L 1 /κ i1 and ρ m2 = L 2 /κ m2 . Equation ( 12) explicitly points out that, under these conditions, the whole terminal 1 (PCM1) is in its low-temperature (insulating) phase, while the entire terminal 2 (PCM2) is in its high-temperature one. According to Fig. 2, both of these phases correspond to the lowest thermal conductivities of PCM1 and PCM2, and hence they minimize the heat flux q B = q B,min given by Eq. ( 12), as follows

q B,min = T h -T c ρ i1 + ρ + ρ m2 , (13a)
T 1,min (L 2 ) = ρ i1 T h + (ρ + ρ m2 )T c ρ i1 + ρ + ρ m2 , (13b)
T 2,min (L 2 ) = (ρ i1 + ρ)T h + ρ m2 T c ρ i1 + ρ + ρ m2 , (13c)
where T 1 (L 2 ) = T 1,min (L 2 ) and T 2 (L 2 ) = T 2,min (L 2 ) are the interface temperatures related to q B = q B,min .

According to Eqs. ( 1), (11a) and (13a), the optimal rectification factor R opt = 1 -q B,min /q F,max is then determined by

R opt = 1 - ρ m1 + ρ + ρ i2 ρ i1 + ρ + ρ m2 = 1 - κ i1 κ m1 (1 + ρ/ρ m1 )r + κ m1 /κ i2 (1 + ρ/ρ i1 )r + κ i1 /κ m2 . ( 14
)
where r = L 1 /L 2 . Note that R opt is symmetric on the thermal resistances of both PCMs and increases with the thermal conductivity contrasts κ m1 /κ i1 and κ i2 /κ m2 between the metallic and insulator phases of the PCMs. In contrast to the optimal rectification factor of a conductive thermal diode operating with a single PCM [START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF], R opt in Eq. ( 14) does depend on the interface thermal resistance ρ between the terminals, such that lower values of ρ yield higher values of R opt . The optimization of the rectification factor of heterojunction based on two PCMs thus requires high values of κ m1 /κ i1 and κ i2 /κ m2 and low ones of ρ. T e m p e r a t u r e , T h ( K ) H e a t f l u x , q ( W m m -2

) 0 1 0 2 0 3 0 4 0 5 0 q F q B R R e c t i f i c a t i o n f a c t o r , R ( % )

RESULTS AND DISCUSSION

Figure 3 shows the temperature evolution of the forward (q F ) and backward (q B ) heat fluxes of a conductive thermal diode operating with terminals of VO 2 and PE. Note that both q F and q B exhibit a nearly linear increase with T h , such that their difference is positive (q F -q B > 0), for most temperatures higher than the VO 2 transition one (T 01 = 342.5 K). The slope change of q F is due to the MIT transition of VO 2 at T h = T 01 , such that the difference q F -q B increases for T h T 01 . The heat fluxes q F and q B shown in Fig. 3 also increase with the interface temperatures T 1 (L 1 ) and T 2 (L 2 ), as shown in Figs. 4(a) and 4(b), respectively. This fact is consistent with Fig. 3, Eqs. (11a) and (13a) as well as with theoretical predictions [START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF][START_REF] Zhu | Nonlinear heat radiation induces thermal rectifier in asymmetric holey composites[END_REF] and experimental data [START_REF] Cottrill | Dual phase change thermal diodes for enhanced rectification ratios: Theory and experiment[END_REF]. The metallic phase of VO 2 is thus able to enhance the rectification factor of the proposed conductive thermal diode when the temperature difference of its terminals is high enough.

The interface temperatures for the forward [T 1 (L 1 ) and T 2 (L 1 )] and backward [T 2 (L 2 ) and T 1 (L 2 )] configurations of the conductive thermal diode are respectively shown in Figs. 5(a) and 5(b), as functions of the temperature T h . The expected increase of these four temperatures with T h is driven by the combined effect of the thermal conductivities of VO 2 and PE, while the temperature differences T 1 (L 1 )-T 2 (L 1 ) = ρq F and T 2 (L 2 ) -T 1 (L 2 ) = ρq B are determined by the forward and backward heat fluxes shown in Fig. 3, respectively. Larger interface temperature jumps are obtained for higher temperatures T h , such as they tend to constant values (> 5 K) for T h > 380 K. This fact along with Fig. 3, indicates that the optimization of the rectification factor of the conductive thermal diode requires relatively high temperature jumps at

3 3 0 3 4 0 3 5 0 1 0 0 2 0 0 3 0 0 q F q B
H e a t f l u x , q ( W m m -2

) T e m p e r a t u r e , T 1 ( L 1 ) ( K ) ( a )

3 3 0 3 4 0 3 5 0 3 6 0 1 0 0 2 0 0 3 0 0 q F q B
H e a t f l u x , q ( W m m -2

) T e m p e r a t u r e , T 2 ( L 2 ) ( K ) ( b ) Calculations were done for Tc = 300 K and ρ = 0 mm 2 KW -1 . the interface of its terminals, for a given interface thermal resistance ρ > 0. These jumps, of course, also increase with ρ and vanish for ρ = 0, as shown in Fig. 6. For a typical ρ = 1 mm 2 KW -1 , temperature were done using R = [1 -min(q F , q B )/max(q F , q B )] and Tc = 300 K.

jumps of T 1 (L 1 ) -T 2 (L 1 ) = 65 K and T 2 (L 2 ) -T 1 (L 2 ) =
interface thermal resistance ρ, for three values of the temperature T h of the hotter terminal. Note that R increases as ρ decreases and/or T h increases, such that its value for T h = 550 K is well predicted by the analytical expression in Eq. ( 14). This verify the reliability of Eq. ( 14) to describe the optimal rectification factor, for different interface thermal resistance, and it indicates that the rectification factor reaches its maximum value for T h = 550 K, as confirmed by the density plot in Fig. 8. Given that ρ is the interface thermal resistance between VO 2 and PE, its reduction to optimize the rectification factor will thermal resistance ρ. Calculations were done using R = [1 -min(q F , q B )/max(q F , q B )] and Tc = 300 K. Figure 9: Rectification factor R of the conductive thermal diode as a function of the temperature T h . Calculations were done by using the alternative definition Ra = [max(q F , q B )/min(q F , q B )] -1 for three representative interface thermal resistances and Tc = 300 K.

at T h = 550 K differs from the optimal one by only 3%, which reduces for higher values of T h . The optimal rectification factor R opt = 60% is reached for ρ = 0 mm 2 KW -1 and a temperature difference of T h -T c = 550 -300 = 250 K between the diode terminals. Similar temperature and thermal rectification factors have been reported for heterojunction made up of graphene-hexagonal boron nitride [START_REF] Chen | Thermal rectification and negative differential thermal resistance behaviors in graphene/hexagonal boron nitride heterojunction[END_REF], graphene-graphene phononic crystals [START_REF] Hu | A series circuit of thermal rectifiers: an effective way to enhance rectification ratio[END_REF], graphene nanoribbons [? ] and carbon nanotube-graphene [START_REF] Yang | Ultrahigh thermal rectification in pillared graphene structure with carbon nanotubegraphene intramolecular junctions[END_REF], but by involving temperatures relatively higher than the ones determined here for a VO 2 /PE heterojunction. Furthermore, we note that the maximum rectification factor obtained for a conductive thermal diode operating with two (VO 2 and PE) PCMs is about 3 times the corresponding one obtained with a single PCM [START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF], which shows the advantage of properly combining the operation of two PCMs to enhance the thermal performance of conductive thermal diode. It is worth mentioning that in terms of the alternative definition of the rectification factor R a = [max(q F , q B )/min(q F , q B )] -1, the optimal rectification factor generated by the junction of VO 2 and PE is around 240%, which is greater than the 147% reported by Kang et al. [36]. This improvement is due to the difference in the thermal conductivity of tungsten-doped VO 2 used by Kang et al. [START_REF] Kang | Thermal rectification via heterojunctions of solid-state phase-change materials[END_REF] with respect to the one of pure VO 2 utilized in this work. This is reasonable, given that thermal diodes with higher rectification factors require larger thermal conductivity contrasts (κ high /κ low ) for both of its terminals, as established by Eq. ( 14).

CONCLUSIONS

We have theoretically analyzed and optimized the rectification factor of a conductive thermal diode made up of a junction between two phase-change materials, whose thermal conductivities significantly change in a narrow interval of temperatures. This has been done by deriving analytical expressions for the temperature profiles, heat fluxes and rectification factor. A rectification factor up to 60% have been determined for a conductive thermal diode operating with terminals of VO 2 and polyethylene set with a temperature difference of 250 K from either side of the MIT temperatures. Higher rectification factors in a narrower temperature interval could be obtained with phase-change materials exhibiting higher thermal conductivity contrasts and faster phase transitions. The proposed model could thus be useful for guiding the development of phasechange materials capable of optimizing the rectification of conductive thermal diodes, which represent a fundamental building block for developing thermal circuits based on heat currents.

Figure 1 :

 1 Figure 1: Scheme of a dual PCMs conductive thermal diode operating in the (a) forward and (b) backward configurations.

  ) and 1(b), respectively. In the forward configuration [Fig 1(a)], the heat flux q F flows from PCM1 to PCM2, while in the backward one [Fig 1(b)]

2 (W m - 1 K - 1 )Figure 2 :

 2112 Figure 2: Thermal conductivities of VO 2 and PE, as functions of the temperature within their corresponding MIT. Dots and

Figure 3 :

 3 Figure 3: Rectification factor, forward and backward heat fluxes of the conductive thermal diode of VO 2 and PE, as functions of the temperature T h . Calculations were done for Tc = 300 K and ρ = 0 mm 2 KW -1 .

Figure 4 :

 4 Figure 4: Temperature evolution of the forward and backward heat fluxes, as functions of (a) T 1 (L 1 ) and (b) T 2 (L 2 ).

T 2 (T 1 (

 21 L 1 ) T e m p e r a t u r e , T ( K ) T e m p e r a t u r e , T h ( K ) L 2 ) T e m p e r a t u r e , T ( K ) T e m p e r a t u r e , T h ( K ) ( b )

Figure 5 :

 5 Figure 5: Interface temperatures for the (a) forward [T 1 (L 1 ) and T 2 (L 1 )] and (b) backward [T 2 (L 2 ) and T 1 (L 2 )] configurations, as functions of the temperature T h . Calculations were done for Tc = 300 K and ρ = 1 mm 2 KW -1 .

1 ρ=1 mm 2 Figure 6 :

 126 Figure 6: Temperature profiles within the terminals of the conductive thermal diode operating in the forward and backward configurations. Arrows stand for the direction of heat flux and calculations were done for Tc = 300 K, T h = 550 K, and two values of the interface thermal resistance ρ.

Figure 7 :

 7 Figure 7: Rectification factor of the conductive thermal diode, as a function of the interface thermal resistance ρ. Calculations

Figure 8 :

 8 Figure 8: Density plot of the conductive thermal diode rectification factor as a function of the temperature T h and interface

ρ=0. 6 1 ρ=0

 61 mm 2 KW -1 ρ=0.28 mm 2 KW -

Table 1 :

 1 Material properties involved in Eqs. (3).