Guillaume Jacques

Giorgio Arcodia

Nathan W Hill

Ken Mason

Alexis Michaud

Marc Miyake

Dmitry Nikolaev

Pavel Ozerov

Haeree Park

Vladimir Plungian

Roland Pooth

Marijn Van Putten

Graham Thurgood

Fernando Zúñiga

The spontaneous-autobenefactive prefix in Japhug Rgyalrong *

Keywords: Middle, Spontaneous, Anticausative, Autobenefactive, Japhug, Rgyalrongic, Kiranti

This paper documents the morphosyntactic and semantic properties of the autobenefactive-spontaneous prefix nɯ-. It describes the regular and irregular morphological and morphophonological alternations displayed by this prefix, as well as its three main semantic functions, namely spontaneous, autobenefactive and permansive. Finally, it discusses the historical relationship of the autobenefactive prefix with other derivations, in particular the vertitive nɯand the anticausative prenasalization.

Introduction

There is a well-attested cross-linguistic tendency for the same marker to be used for reflexive or passive and for the expression of spontaneous events without a volitional agent or anticausative (Kemmer 1993: 142-144). This closeness is best exemplified by middle constructions using a reflexive marker in Romance and Slavic languages and also in various language families.1 In Sino-Tibetan, the suffix -ɕɯ in Dulong/Rawang [START_REF] Lapolla | Reflexive and middle marking in Dulong/Rawang[END_REF] and the suffix -si in Kiranti languages are also example of quasi-prototypical middle markers (see [START_REF] Michailovsky | La langue hayu[END_REF]: 144-146, van Driem 1993: 125-129, Opgenort 2004: 260-263 and Bickel et al. 2010: 398).

Japhug presents a marker nɯthat can be used in situations referring to spontaneous events or autobenefactive actions.2 This marker differs from the reflexive as observed in Romance, Slavic and Kiranti in several important ways.

First, Japhug also has a reflexive prefix (see for instance Jacques 2010b concerning Japhug Rgyalrong) as well as distinct passive, anticausative and antipassive prefixes (Jacques 2012c) and a reciprocal construction all formally and functionally distinct from the autobenefactive. Second, although Japhug nɯsemantically belongs to the middle domain as defined by Kemmer (1993: 15), this marker can be applied to both intransitive and transitive verbs, and does not modify the valency of the verb. Third, in addition to marking autobenefactive and spontaneous actions, it also has a permansive3 aspectual value.

This paper is a description of the spontaneous-autobenefactive nɯ-in Japhug, and a contribution to the typology of middle marking systems. It comprises six sections. First, I present the morphological marking of transitivity in Japhug, and show that the spontaneous-autobenefactive marker has no influence on it. Second, I discuss the position of this prefix in the Japhug verbal template. Then, I describe the three main functions of this prefix: the expression of autobenefactive and grooming activities, spontaneous events and permansive aspect. Finally, I propose a series of hypotheses concerning the diachronic relationship of this prefix to other derivations in Japhug and other Rgyalrongic languages.

Morphological transitivity

Japhug verbs have two conjugations, transitive and intransitive. 4 The intransitive conjugation indexes the person and number (singular, dual, plural) of the S, while the transitive conjugation indexes the person and number of both A and P. The indexation of arguments on transitive verbs follows a quasi-canonical direct-inverse system (see [START_REF] Delancey | The category of direction in Tibeto-Burman[END_REF][START_REF] Sun | Caodeng Jiarongyu yu rentong dengdi xiangguan de yufa xianxiang 草登嘉戎語與「認同等第」相關的語法現象 (Empathy Hierarchy in Caodeng rGyalrong grammar)[END_REF], Jacques 2010a[START_REF] Jacques | Direct / inverse systems[END_REF][START_REF] Gong | Personal agreement system of Zbu rGyalrong (Ngyaltsu variety)[END_REF]. The person marking prefixes and suffixes of the intransitive conjugation can be combined with either direct marking (via stem alternation), inverse marking (the wɣprefix) or portmanteau prefixes (the local scenario markers kɯ-2 → 1 and ta-1 → 2).

Transitive verbs can be unambiguously distinguished from intransitive ones by three morphological criteria. First, stem alternation in the non-past 1/2/3sg→3 direct forms (including factual, imperfective, sensory, present, imperative and irrealis, see also [START_REF] Sun | Stem Alternations in Puxi Verb Inflection: Toward Validating the rGyalrongic Subgroup in Qiangic[END_REF][START_REF] Sun | The irrealis category in rGyalrong[END_REF]. Second, presence of a past transitive prefix -tin the past 1/2sg→3 forms. Third, in perfective 3→3' direct forms (without the inverse wɣprefix), a prefix aappears between the verb stem and the orientation prefixes (and phonologically merges with them).

The morphological transitivity as indexed in verb morphology is strictly correlated with ergative marking on the A argument of a transitive verb (example 2), while S and P arguments are unmarked (examples 1 and 2). The ergative clitic kɯ is obligatory with noun phrases and third person pronouns, but in the case of first and second person pronouns it is optional.

ifr-look.for

The king married a new wife. (140504 baixuegongzhu, 15) Verbal morphology, combined with ergative marking, can be used to test whether a particular derivation modifies verb transitivity or not. For instance, examples 3 and 4 show that while transitive verbs with third person arguments take the perfective 3 →3' direct aprefix, corresponding verbs with the antipassive rɤ-do not. Moreover, in example 3 the argument corresponding to the person learning would take the ergative with the base verb βzjoz if an overt A were inserted, but does not with the antipassive verb rɤ-βzjoz. Likewise, other voice derivations such as the passive a-/ɤ-, the anticausative, the reciprocal (a-/ɤ-combined with reduplication) and the reflexive ʑɣɤ-are incompatible with transitivity markers such as the perfective 3 →3' direct a-prefix, the past 1/2sg→3 suffix -t, stem III alternation, and with ergative case marking.

(3) <qiche> car kɤ-ɣɤ-βdi inf-caus-be.good On the other hand, the presence of the autobenefactive/spontaneous prefix, unlike other derivations belonging to the middle domain, has no influence on morphological or syntactic transitivity.

Examples 5 and 6 show that the prefix nɯdoes not affect stem alternation. In these two examples, stem III -ndɤm is found instead of stem I -ndo, as is expected for a verb that is in the imperative with a 2sg A and a third person P, even in 5 with the autobenefactive/spontaneous. Example 7 shows that ergative case marking on the A is not lost when the verb has the autobenefactive / spontaneous prefix, and comparing 7 with 8 reveals that the perfective direct 3→3' prefix a-5 is found even in cases when the autobenefactive nɯ-is present. The verb has stem I -ndo in these two examples as stem III is restricted to non-past tenses, and is not compatible with the perfective. ɲɯ-ŋu.

sens-be

The rider took the staff (for himself The examples above prove that transitive verb forms with the autobenefactive / spontaneous present all the morphological properties of transitive verbs: the prefix nɯhas no influence on verb transitivity. In this regard, Japhug nɯis closer to the classical Indo-European morphologically-marked middle (see for instance Pooth 2014: 19) than most constructions labelled 'middle' in the typological literature, especially Romance and Slavic.

Position in the template

The Japhug verbal template follows the general structure in Table 1 (see Jacques 2012d and Jacques 2013c).

The autobenefactive / spontaneous prefix is peculiar in that it can occupy two distinct slots depending on the verb form: it can occur in slot 12, after the passive or denominal prefix a-, but in the case of verbs without prefix a-, it occurs to the left of the reflexive prefix, as in example 11. When people we cherish come, or when we (wish) to treat ourselves, (30 macha, 74)

The prefix nɯoccurs in slot 12 for all verbs whose stem begins in a-, even when ais neither the passive or the denominal at least synchronically. For instance, in verbs such as arŋi 'be blue', whose stem is disyllabic (the aelement is not a prefix), the spontaneous prefix is infixed after the a-, as in example 12 (see also 25 below). The pine (remains) green the whole year. (07 tAtho, 51)

There is one exception to this rule. The verb atɯɣ 'meet, run into' (conjugated intransitively) has two different autobenefactive forms (also intransitive), a regular one a<nɯ>tɯɣ 'meet by oneself' (as in 13), and nɯ-ɤtɯɣ 'run into (by mistake), happen to be in' ( 14). evd.ipfv-be On the other side, he always happened to be among those who did not get anything (of the food being distributed). (28 qAjdoskAt,138)

The verb nɯ-ɤtɯɣ 'run into (by mistake)' has the autobenefactive prefix nɯin the same slot as the homophonous applicative nɯ-(see Jacques 2013a). Its meaning is not completely predictable from the base verb.

Another example of irregularity related to verbs in a-, is the verb antsɤndu 'to be exchanged by mistake' (intransitive), which derives from sɤndu 'exchange' (transitive) by a combination of the passive aand an allomorph nt-of the autobenefactive/spontaneous nɯ-. This verb is anomalous in two regards. First, there is no corresponding simple passive verb *asɤndu 'be exchanged' : sɤndu 'exchange' is morphologically the causative of andu 'be exchanged' (mainly used about money). Second, the allomorph ntis not found in any other verb; the -t-is epenthetic here, since the cluster -nsis not attested in Japhug.

Examples like nɯ-ɤtɯɣ 'run into (by mistake)' and antsɤndu 'to be exchanged by mistake' suggest the autobenefactive/spontaneous nɯ-, despite its high productivity, is better treated as a derivational rather than an inflectional morpheme, since (i) the meaning of the derived verb is not always predictable and (ii) there is not always, at least synchronically, a corresponding base verb whose only difference with the derived verb is the absence of an autobenefactive/spontaneous prefix.

Verbs in a-are not the only ones where the prefix nɯis infixed. The irregular existential verbs ɣɤʑu 'be there, exist (sensory)' and maŋe 'not exist (sensory)' also take the spontaneous marker as an infix rather than as a prefix as in 15; note that all prefixes, including the second person tɯand the generic kɯ-are infixed in the conjugation of these verbs (see Jacques 2012a, under review). Finally, an interesting property of the autobenefactive-spontaneous is that, like the inverse prefix (Jacques 2013c: 199), it is obligatorily infixed within the progressive asɯ-, as in example 16. In addition to the issue of position in the template, the autobenefactive nɯpresents two noteworthy regular morphophonological alternations. First, it can be realized as nbefore all prefixes with the shape nɯor nɤ-(denominal, applicative or tropative), as in example 17. Second, it can be optionally realized with emphatic gemination, especially in its use in the protasis of concessive conditionals (see 25 and 26 below).

Autobenefactive

The prefix nɯis commonly used with transitive verbs when the P bears a possessive prefix coreferent with the A, especially in the case of body parts and other inalienably possessed nouns, as in examples 187 and 19. It expresses that the A is affected by its own action. The autobenefactive nɯ-can also refer to an action done for the benefit of a particular referent, which can be any of the core arguments. 8 In these cases a pronoun coreferent with the referent can be placed just before the verb. This pronoun cannot bear the ergative marker even when it refers to the A, as ʑɤni 3du in example 20. It can also be the distributive pronoun ʑaka 'each one' as in 21. We are only two, we are not enough people to act separately. (The three sisters, 74)

The autobenefactive value of the prefix nɯin the imperative is used to convey a softened tone, expressing mild suggestion rather than order (examples 22 and 23). Note that the prefix nɯcan have a different semantics in the imperative, namely 'do X as much as you want (the result will be the same)' (see 28 above).

cisloc-imp-auto-buy-pl

There are all kinds of things, there are silk threads, come and buy them! (140504 baixue gongzhu, 121-122)

Spontaneous

The prefix nɯmarks spontaneous actions occurring without any external cause or against of the will of a particular referrent.

In the case of animals, plants and inanimate beings, nɯ-can be used to express their apparent spontaneous growth, as in 24. This spontaneous value of the prefix nɯ-explains its frequent presence in the protasis of alternative (25) and scalar (26) concessive conditionals (see Jacques 2014a): in these constructions, the result described in the apodosis takes place regardless of whether the condition in the protasis is fulfilled or not. The prefix nɯexpresses the fact that the resulting action is independent of the condition.

not.exist:sens

This one also grows by its root, as even if it has flowers, (I) have never seen its seeds. (17 ndZWnW, 155)

An extension of the spontaneous value of the prefix nɯis the meaning 'casually', 'at one's will', 'whatever' (Chinese 随便 suíbiàn), as in 27.

sens-be

They say 'huaguniang', they call them like that. (implied meaning: they invented their name, it is not a real name; 28 qapar, 235)

In the imperative, the spontaneous can be used to mock the addressee, telling him that all his actions will be in vain, as in 28. Note however that the prefix nɯcan also express a mild imperative, as in examples 22 and 23 below, depending on the context. It could seem to be contradictory that a single marker has such opposite semantic values. However, in both cases the action takes place against or independently of the will of a particular referent or regardless of the completion of another action. This referent can be an argument of the sentence, as in examples 31 and 32, or can be an external referent, without syntactic function in the sentence, as in 29.

Permansive

In addition to the two previous meanings, which are relatively straightforward for a middle marker, the spontaneous-autobenefactive prefix is also used with an aspectual function. It expresses the continuity of an action or a state, despite the occurrence of another action which could have been expected to stop it (as in examples 33, 34), or despite the fact that a long time has passed (see 12 above) like the adverb 'still' in English.

It can be used to insist on the fact that a particular state is maintained without change, as in 35.

The permansive reading of the prefix nɯis only possible in nonperfective verb forms, in particular factual, imperfective, past imperfective evidential and sensory. It is just that it is small when it comes out, (otherwise) it is already like that. (24 zwArqhAjmAG,19) The permansive use of nɯis not without typological parallels. One of the clearest cases is the Russian pronominal element себе, originally the dative form of себя 'oneself', and which alongside its autobenefactive value, is used in certain contexts with a permansive value ('continue to ...'), generally with repetition of the verb as in example 36 taken from a song.9 The time goes on and on... This construction is not fully grammaticalized in Russian, but it is nevertheless a good parallel to the permansive value of nɯ-in Japhug. Since it is clear in Russian that the original meaning of this marker can only have been autobenefactive, not permansive or spontaneous, this fact suggests that the directionality of grammaticalization is more likely to be from autobenefactive to permansive in Japhug too. The following pathway in four stages can be proposed to account for this evolution; note that all four stages represent attested uses of Japhug nɯ-.

1. Do X for oneself (autobenefactive).

2. Do X on one's own.

Autobenefactive and other derivations

The spontaneous-autobenefactive nɯin Japhug has cognates in other Rgyalrongic languages, in particular in Khroskyabs (its cognate Nis discussed in Lai 2013: 157-160). Since this prefix presents irregularities in Japhug and complex morphophonological alternations in Khroskyabs, and since no obvious lexical source can be proposed as its lexifier in either language, it is reasonable to hypothesize that it can be reconstructed to proto-Rgyalrongic with at least the spontaneous and autobenefactive functions (the permansive function has not been documented elsewhere).

No nasal prefix with semantics comparable to Japhug nɯhas been described in any other Sino-Tibetan language. Hence, it is likely that it is one of the many common innovations of the Rgyalrongic languages. 10This raises the question of the source of the prefix nɯand its historical relationships to phonetically similar prefixes in Japhug. While the ultimate origin of the autobenefactive nɯis not yet clearly established, we show that it is likely to be related to two other derivations: the vertitive11 and the anticausative.

Vertitive

There are three derivational verbal prefixes homophonous with the spontaneous-autobenefactive (Jacques 2013a): the applicative nɯ-/ nɯɣ-, the denominal nɯ-and the vertitive nɯ-. The first two are unlikely to be historically related to the autobenefactive. In particular, while other voice derivations, including the antipassive rɤand the applicative nɯhave been shown to originate from denominal derivations (see Jacques 2014b), the autobenefactive nɯ-and the denominal derivations in nɯare semantically too different for such a hypothesis to be possible: there are no denominal verbs in nɯin Japhug with an intrinsic spontaneous or autobenefactive meaning. Therefore, this section only focuses on the vertitive.

The vertitive nɯ-is exclusively attested with a restricted set of motion verbs, indicated in Table 2. It implies a motion back to the origin point. Vertitive verbs can be combined with the spontaneous-autobenefactive, as in example 37. q Do you drive them (the cows) over there, and then they come back by themselves? (Conversation 2003, 19) All vertitive verbs are homophonous with the corresponding spontaneous-autobenefactive forms. Example 38 shows the use of the vertitive form of tsɯm 'take away', while 39 illustrates its spontaneous form. It is clear in the case of 39 that -nɯ-tsɯm cannot be interpreted as 'take back' (since a river flows in one direction and does not take back floating objects to its source); here nɯindicates that the water took away the object against the will of the speaker -it is also present with the previous verb βde 'throw' to express the meaning 'throw by mistake'. q-be:fact Is it the one that I lost? Is it the one that the water took away (spontaneous)? (140427 bianfu jingji he shuiniao, 29)

There is no evidence of vertitive prefix in Khroskyabs, and it is thus probably a Rgyalrong-proper extension of the autobenefactive prefix.

It is conceivable that the vertitive meaning developped out of the autobenefactive 'take for oneself' → 'take to one's home' → 'take back home'. More precise data on the use of the autobenefactive in all Rgyalrongic languages is however necessary to confirm this hypothesis.

Anticausative

Japhug is one of the few languages with a specifically anticausative derivation, distinct from the passive, the reflexive and other middle markers (Jacques 2012c). This derivation turns a transitive verb into an intransitive one. The S of the intransitive verb corresponds to the P of the base verb. The anticausative is in direct concurrence with the agentless passive prefix a-. The semantic different between the two is that in the former, the agent is completely deleted semantically, while in the latter it is still recoverable.

The anticausative in Japhug is not marked by a prefix, but by a morphophonological alternation: the prenasalization of the onset of the verb stem (it only applies to monosyllabic verbs).

The anticausative in Japhug is only marginally productive: it only applied to one single Tibetan loanword χtɤr 'scatter' (Tibetan gtor), whose anticausative is ʁndɤr 'be scattered'. No Tibetan verb comparable to ʁndɤr 'be scattered' exists, and in any case the phonotactics of the onset of this verb itself (fricative+prenasalized voiced stop) is incompatible with that of any Tibetan language. Hence, this verb can only have been created within Japhug from its transitive counterpart χtɤr 'scatter' which is a borrowing.

This example is of considerable importance, as it proves that the directionality of derivation is from the transitive verb to the intransitive one, and not the other way round. 12All known examples of anticausative alternations in Japhug are presented in Table 3. The anticausative radically differs from autobenefactive in several ways. First, it is an intransitivizing derivation, while the autobenefactivespontaneous does not modify the verb valency. Second, the two derivations are formally different (prefix vs prenasalization). Yet, there is some degree of semantic overlap between the two derivations: both can be used to express spontaneous events occurring without any external agent. Moreover, while they are phonetically different, it is conceivable that onset prenasalisation is the regular phonetic development of nasal prefixes before stops and affricates, while the tautosyllabic nɯ-represents a regularized allomorph occurring before onsets that cannot undergo prenasalization and later generalized everywhere. Hence, it is possible that these two derivations came from a common origin: a nasal prefix expressing spontaneous / non-volitive actions.

The anticausative is widespread throughout the Sino-Tibetan family, even in Chinese [START_REF] Sagart | Reconstructing the s-prefix in Old Chinese[END_REF] and Tibetan (Jacques 2012b), and can safely be reconstructed to the proto-language. 13 The question is whether the primary meaning of this derivation was spontaneous action or specifically anticausative (with loss of transitivity); this can only be effectively tested in Sino-Tibetan languages that have morphological transitivity.

In Kiranti languages, which like Rgyalrongic languages, have distinct transitive and intransitive conjugations, example of voicing alternations are relatively few (in Khaling for instance, only eight examples are attested, see Jacques 2013b). The voiced counterpart of such verb pairs is nearly always intransitive, with a clear anticausative meaning identical the Japhug examples in Table 3.

Yet, there is at least one example of voicing alternation where the voiced remains morphologically transitive: Khaling plum-'rinse in water' vs blum-'sink in water' are both transitive; the voiced form blumis conjugated intransitively, and takes an agent marked with the ergative (the liquid in which one sinks), as in 40.

(40) ku-ʔɛ water-erg ʔi-blʉm-ʌtʌ inv/2-sink-1sg:S/O:pst I sank in the water.

Such an example may suggest that the primary function of the anticausative was that of spontaneous action, and not intransitivization. In this view, Japhug and other Rgyalrongic languages have reshaped the original spontaneous derivation into two distinct morphological processes with specialized semantics, which most Sino-Tibetan languages have only preserved the spontaneous derivation in its more reduced anticausative function.

If this hypothesis is correct, more examples such as Khaling plum-vs blum-, with voicing alternation without intransitivation, should be discovered in morphologically rich languages of the family.

Conclusion

The spontaneous-autobenefactive prefix nɯin Japhug has cognates in most Rgyalrongic languages. Further reasearch will be needed to determine to what extent the use of the cognate prefixes in the other languages are similar or different from that of Japhug nɯ-. In particular, it is unclear whether the permansive value of nɯis restricted to Japhug or found elsewhere, and whether the cognate prefixes have the same values in the imperative.

Additional research on the historical origin of the spontaneousautobenefactive will also require data from other languages. Within Rgyalrongic, it is especially important to collect all examples of irregular spontaneous-autobenefactive forms, such as Japhug ɣɤ<nɯ>ʑu <auto>exist:sens. Outside of Rgyalrongic, progress may be achieved by looking for traces of cognate spontaneous markers. In particular, it could be fruitful to search for cases of voiced / unvoiced verb pairs in which both verbs are morphologically transitive, like Khaling |plum-| 'rinse' and |blum-| 'sink'.

Finally, the data presented in this paper is of interest to typologists working on middle marking systems, as Japhug is one of the few languages with specific markers to express reflexive, passive, reciprocal, antipassive, anticausative and autobenefactive derivations, in other words, a language in which all subsections of the middle domain are clearly kept separate by distinct morphology.

  -go-1sg Take the staff (to hit the animal), I am going to the toilet. -bare.inf:CP:keep a-tɤ-tɯ-ɣɤ-βdi irr-pfv-2-caus-be.well ma lnk Take this, you will have to keep it well because... (140428 mu e guniang, 24)

  pst.ipfv-not.be Their marriage was arranged by their parents, they did not get together by themselves. (elicited)

  every year, and it is still there.(20 grWBgrWB, 51)

  3sg.poss-hand pjɤ-k-ɤ<nɯ>sɯ-χtɕi-ci ifr.ipfv-evd-prog<auto>-wash-evd When I arrived (there), he was washing his hands. (elicitation, Chen Zhen)

  He slept on the road. (140505 liuhaohan zoubian tianxia, 120)

  He saved his own life. (140512 yufu yu mogui, 127)

  -diluted.alcohol ɲɯ́-wɣ-jtsʰi-a-ndʑi pfv-inv-give.to.drink-1sg-du pɯ-ɕti pst.ipfv-be.assert 'Your two sisters (...) ate the tasty food and gave me the burned part of the bread, drank the alcohol and gave me diluted alcohol to drink.' (The three sisters,

  -nɯ-ndze transloc-imp-auto-eat[III]Go and eat nice food there! (140426 jiagou he lang, 64)

  some say that it comes from the navel, I don't know, it grows by itself.(21 mdzadi, 54-55)

  -give:fact You can cry as much as you like, nobody is afraid of you, I won't give (my daughter) to you (in marriage).(The frog1, 149) With a human S/A, nɯcan indicate a action performed of one's own volition, without being forced by anything or anyone, as in 29, or without help from anybody else ('by oneself'), as in 30. (29) aʑo 1sg pjɯ-kɯ-ɣɤrat-a-nɯ ipfv:down-2→1-throw-1sg-pl mɤ-ra neg-fact:need ma because aʑo 1sg pjɯ-nɯ-mtsaʁ-a neg-ipfv:down-auto-jump-1sg jɤɣ fact:be.possible You don't need to throw me in there, I will jump of my own free will.to I have to go to have a look by myself. (140507 tangguowu, 139) Somewhat paradoxically, nɯcan also indicate that an action occurs by mistake or against the volition of the S/A, as in 31. (31) ɯ-qom 3sg.poss-tear ci indef pɯ-a-nɯ-ɕlɯɣ pfv-3→3'-auto-drop ɲɯ-ŋu, sens-be She shed a tear (involuntarily). (Kunbzang 228) The verb jmɯt 'forget' almost always appears with nɯin the corpus (in 23 examples out of 28). In the first person, the autobenefactive / spontaneous can be combined with the inferential to insist on the non-volitionality of the action, as in 32. -forget-1pl I don't know, I forgot those things. (Conversation)

  to) comfort the girl, but it was for nothing, the girl was still crying. (Bean and linen, flower) falls down, it dries up, and its seed is still wrapped in it. (13 tCamu, 59) like the ʑmbɯlɯm and the Matsutake, which are spherical when they come out and progressively open towards the exterior.

  3. Do X on one's own, disregarding external conditions. 4. Continue to do X, despite (adverse) external factors. (permansive) At the present moment, we have no evidence that cognates of Japhug nɯin other Rgyalrongic languages have developed a permansive value, but further research might reveal similar examples elsewhere.

  ifr-vert-take.away qʰe lnk The prince took the girl back (vertitive) to his kingdom. (140504 baixue gongzhu

Table 1 :

 1 The Japhug verbal template

	nɯ-ɣɤ-noun Σ -t -a -nɯ rɯ--ndʑi	etc.	10. Causative sɯ-/z-/sɯɣ-/ɕɯ-/ɕ-/ɕɯɣ-/ʑ-, tropative nɤ-, ap-nɯ-plicative	11. Passive or Intransitive thematic marker a-/ Deexperi-sɤ-, causative encer ɣɤ-	12. Autobenefactive-spontaneous nɯ-	13. Other derivation prefixes nɯ-ɣɯ-rɯ-nɤ-ɣɤ-rɤ-	14. Noun root	15. Verb root	16. Past 1sg/2sg transitive -t (aorist and evidential)	17. 1sg -a	18. Personal agreement suffixes (-tɕi, -ji, -nɯ, -ndʑi)
	a-mɯ-ɕɯ-tɤ-tɯ-wɣ-ʑɣɤ-sɯ-rɤ-nɤ-a-mɤ-ɣɯ-pɯ-sɤ-	etc.	1. Irrealis a-, Interrogative ɯ́-, conative jɯ-	2. Negation ma-/ mɤ-/ mɯ-/ mɯ́j-	3. Translocative / Cislocative ɕɯ-and ɣɯ-	4. Directional prefixes, apprehensive ɕɯ-	5. Second person (tɯ-, kɯ-2→1 and ta-1→2), direct →3' a-perfective	6. Inverse -wɣ-/ Generic S/O prefix asɯ-. nɯ-is infixed within kɯ-, Progressive The Autobenefactive-spontaneous asɯ-.	7. Reflexive ʑɣɤ-	sɯ-8. Causative sɯ-, Abilitative	9. Antipassive sɤ-/ rɤ-

Table 2 :

 2 The vertitive prefix nɯin Japhug

	Base verb Meaning	Derived verb Meaning
	ɕe ɣi tsɯm ɣɯt no zɣɯt	go come take away bring chase (cattle) nɯno nɯɕe nɯɣi nɯtsɯm nɯɣɯt arrive nɯzɣɯt	go back (home) come back (home) take back (home) bring back (home) chase back (home) arrive back (home)

Table 3 :

 3 Examples of anticausative in Japhug

	basic verb	derived verb
	ftʂi	to melt (vt)
		ndɯ	to appear (rainbow),
	χtɤr tʂaβ qraʁ qia qlɯt sɤpʰɤr pri xtʰom tɕɣaʁ kɤɣ qrɤz cʰɤβ cɯ pʰaʁ	to build (road, bridge) to spill to cause to roll to tear to tear to break to shake off, to wipe off mbɤr ʁndɤr ndʐaβ ɴɢraʁ ɴɢia ɴɢlɯt to tear mbri to put horizontally ndom to squeeze out ndʑɣaʁ to bend ŋgɤɣ to shave ɴɢrɤz to flatten, to crush ɲɟɤβ to open ɲɟɯ to split mbaʁ	to be built (road, bridge) to be spilled to roll (vi) to be torn to get loose to be broken wiped off to be torn to be horizontal to be squeezed out to be bent to break (of hair, dry leaves etc) to be crushed, flattened to be opened to split, break

ndʐi to melt (vi) kio to cause to drop ŋgio to slip kra to cause to fall ŋgra to fall plɯt to destroy mblɯt to be destroyed prɤt to cut mbrɤt to be cut pɣaʁ to turn over (vt) mbɣaʁ to turn over (vi) qɤt to separate ɴɢɤt to be separated qʰrɯt to completely scratch ɴɢrɯt to be completely scratched qrɯ to cut, to tear, to break ɴɢrɯ to break (vi), be torn tɕɤβ to burn (vt) ndʑɤβ to be burned tʰɯ to pitch (tent),

The use of the term 'middle' in[START_REF] Kemmer | The Middle Voice[END_REF] and subsequent typological work differs from the morphologically-based definition of the middle in Indo-European linguistics. In this paper, I use 'middle domain' to refer to all constructions and derivations (such as passive, anticausative, reciprocal, autobenefactive etc) that can overlap with reflexive marking in particular languages.

A similar marker exists in all core Rgyalrong languages, for instance nəin Tshobdun (glossed as 'spontaneous' in Sun 2014: 634) and also in Khroskyabs(Lai 2013: 157-160).

The term permansive or remansive refers to an aspect expressing the continuation of a process that has already started(François 2003: 130).

A small class of labile verbs can be conjugated either transitively or intransitively(Jacques 2012c). All examples exhibit agent-preserving (i.e. A=S) lability.

The prefix ais restricted to transitive non-local scenarios without inverse marking.

Note that the homophony between the autobenefactive-spontaneous prefix and the perfective 'towards west' prefix nɯis fortuitous. The latter only appears verbs that lexically select this particular orientation prefix, or verbs of motion when the orientation 'towards west' is appropriate. In 9 and 10, we know that the nɯin nɯ-βzu-t-a pfv-dopst:tr-1sg must be perfective and not autobenefactive because the transitive past suffix -tcannot occur in a verb form without a perfective orientation prefix.

This sentence would however still be grammatical without the nɯprefix.

For benefactive towards non-core arguments, two constructions are available in Japhug, either marking the beneficiary with the genitive ɣɯ or as a possessive prefix with the noun ndʐa 'reason' (for instance nɤ-ndʐa 2sg.poss-reason 'for your sake'). These two strategies are attested in many other languages of the world (seeZúñiga & Kittilä 2010: 7-10).

I would like to thank Dmitry Nikolayev and Pavel Ozerov for pointing out this fact to me, providing examples and suggesting the grammaticalization path proposed in this section.

However, one cannot completely exclude the possibility that cognate prefixes have been lost without traces in some languages.

This term is adopted from Siouan linguistics, cf[START_REF] Taylor | On Verbs of Motion in Siouan Languages[END_REF].

In addition, we observe that several phonological contrasts found in the transitive verbs, such as aspiration, are neutralized in the corresponding anticausative verbs, a fact that strengthens the idea that the intransitive verbs are derived from the transitive ones in this voice alternation.

Although some authors appear to confuse the anticausative and causative sderivations[START_REF] Mei | The causative *s-and nominalizing *-s in Old Chinese and related matters in Proto-Sino-Tibetan[END_REF], the general consensus is that these are completely distinct phenomena[START_REF] Lapolla | An overview of Sino-Tibetan morphosyntax[END_REF][START_REF] Hill | A note on voicing alternation in the Tibetan verbal system[END_REF]): languages as diverse as Japhug, Tibetan, Rawang and Jingpo have both distinct causative (marked by a coronal fricative prefix) and anticausative (marker by voicing alternation) derivations. This fact is difficult to reconcile with the idea that voicing alternations are due to the devoicing of voiced initials caused by the prefix *s-.

research was funded by the HimalCo project (ANR-12-CORP-0006) and is related to the research strand LR-4.11 ''Automatic Paradigm Generation and Language Description'' of the Labex EFL (funded by the ANR/CGI).


Denis Rochais

Gilberto Domingues

Franck Enguehard

Numerical simulation of thermal conduction and diffusion through nanoporous superinsulating materials

Keywords:

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Nanoporous superinsulating materials (NSMs) are getting more and more attractive for various applications (particularly in the aerospace and construction industries) due to their extraordinary power of thermal insulation [1][2][3][4][5]. Indeed, their thermal conductivity falls down to a few when they are placed under primary vacuum at ambient temperature.

This value can be compared to the one of air (generally regarded as an excellent thermal insulator), equal to at ambient temperature and pressure.

-1 -1 mW.m .K -1 -1 25 mW.m .K Such a level of thermal insulation of NSMs finds its explanation in the microstructure of these materials. Very porous (their porosity is of the order of 90%) and made of extremely fragmented solid matter (the main solid constituents are generally brought down to nanometric scales), they force the conduction heat flux to travel through very tortuous routes made of a multitude of elementary thermal resistances located at the coalescences of neighboring nanoparticles. Furthermore, these NSMs include very small quantities of micrometric-scale fibers and particles. Whereas the fibers provide some mechanical reinforcement to the NSMs, the role of the particles, made of opaque materials in the [5µm -80µm] infrared spectrum, is clearly to cut down the infrared thermal radiation transfer in the course of its progression within the nanoporous structures.

In this contribution, we want (i) to develop very simple models to quantify the level of conduction heat transfer traveling through a NSM in relation with its nanostructure, and then (ii) to use these models to study the impact of the nanoparticle volume fraction on the effective thermal conductivity and diffusivity of a NSM.

To this purpose, we will first describe in detail the micro and nanostructure of a typical NSM.

Second, we will present the computer generation code that we use to construct our virtual NSM structures. Third, we will describe two models which allow us to perform either a steady-state thermal conduction numerical experiment or a time-resolved thermal diffusion numerical experiment. Finally, we will exploit these models to study the influence of the nanoparticle volume fraction on the effective thermal properties of a NSM.

Description of the structure of a typical NSM

As introduced previously, a NSM is typically made of (i) nanometric scale particles joined to one another to form a nanoporous matrix, the dominant constituent of the material in terms of volume fraction (near 100%), (ii) a small volume fraction (of the order of 1%) of micrometric scale particles expected to provide infrared opacity to the NSM, and (iii) a small volume fraction (of the order of 1%) of micrometric scale fibres for a mechanical reinforcement of the nanoporous structure. The nanoparticles, constituting the nanoporous matrix, form the dominant population in terms of solid volume fraction, therefore they must be made of a material that is a good thermal insulator. In a large majority of cases, this material is chosen to be amorphous silicon oxide a-SiO 2 . The matrix is extremely porous (its porosity is more than 90%), and a TEM observation focused on this matrix reveals its nanotexture, as shown in figure 2. This image being at a scale of the order of 100 nm (the surface covered by the image is approximately

), stacks of coalesced a-SiO 1.8 µm 2.2 µm × 2 nanoparticles appear (dark structures) as well as the 3D nanostructure that they constitute. A further TEM analysis allows an approximate evaluation of (i) the average diameter of these a-SiO 2 particles (10 nm) and (ii) the average size of the pores (between 50 and 100 nm). A convection heat transfer cannot occur in such small pores [START_REF]Thermal transmission measurements of insulation[END_REF]. Furthermore, the thermal conductivity of the confined fluid surrounding the nanoparticles is much lower than its thermal conductivity in an unconfined environment. Indeed, Griesinger et al [START_REF] Griesinger | [END_REF] give the following correlation for the thermal conductivity of a confined gas:

( ) ( ) ( ) o g g T T , P , d T , P 1 2 d λ λ = + β (1)
where T is the temperature, P is the pressure, d is the pore size, is the mean free path of the molecules, β is a dimensionless parameter depending on the nature of the gas and is the thermal conductivity of the gas in a unconfined state. For example, if we take T , and d , then we obtain which is already smaller than the value of the thermal conductivity of air in a unconfined state:

. Now, if the nanoporous matrix is placed under primary vacuum

(for example at 1 mbar) at ambient temperature, then the confined gas thermal conductivity falls down to : the gas becomes a nearly perfect insulating material. This participates of course to the insulating power of a NSM.

( )

o g T λ 300K = P 1bar = 100 nm = ( ) -1 -1 g T , P , d 6 mW.m .K λ ≈ ( ) o g T 25mW.m .K λ ≈ -1 -1 ) ( g T , P , d λ -1 -1 6 µW.m .K
As described previously, a typical NSM can be regarded at 2 different characteristic scales: a micrometric one, where the material can be considered as a dispersed phase of micrometric scale particles and fibers embedded within a homogeneous matrix, and a nanometric one, where stacks and chains of coalesced nanoparticles appear. For each characteristic scale of the material, a particular strategy of conduction heat transfer modeling can be envisaged. At micrometric scale, classical homogenization laws (Maxwell-Garnett, Bruggeman …) can be used to calculate the NSM effective thermal conductivity, provided the microparticle volume fraction is small enough (which is the case here) and the thermal properties of the matrix and the microparticles are known. At nanometric scale, we have to develop a numerical model in order to analyze the conduction heat transfer within the 3D matrix structure. In the following, we will focus on these models which allow us to determine the effective thermal conductivity and diffusivity of the matrix.

Computer generation of a virtual nanoporous matrix

To be representative of the conduction heat transfer within a NSM, our simulations must take into account its real structure. We resort to computer generation to construct matrix structures by the location in space of solid spheres meant to represent the nanoparticles. We have started in this field with very simple periodic structures (square-shaped, diamond-shaped, brickshaped …). Although these structures are far from realistic, many interesting results can be obtained from their study, such as the influence of the structure on the heat flux travel. We will illustrate this point in the 4 th part of this paper. Here, in order to generate more realistic nanostructures, we propose to apply the fractal method [START_REF] Mandelbrot | The fractal geometry of nature[END_REF].

In reference [START_REF] Legrand | The surface properties of silicas[END_REF], Legrand explains that nanoporous structures can be regarded as elementary schemes reproduced in the 3 space directions according to a mass evolution law having the following form:

( ) f D M R R ∝ ( 2
)
where R is the radius of the sphere limiting the nanoporous structure, is the mass within this sphere and indicates the fractal dimension associated to the nanoporous structure.

( )

M R f D
The procedure adopted to generate the nanostructure is based on a cluster-cluster aggregation process the rules of which are the following [START_REF] Jullien | [END_REF]:

-disperse a set of nanoparticles randomly in space, -move each nanoparticle or aggregate in a random direction, -if the nanoparticle or aggregate hits a neighbor, both are stuck to each other and will subsequently be moved together, -application of boundary conditions.

Although these rules are simplistic, they are sufficient to assure the physics of the model if relevant parameters (space dimension of the structure, rigid stick and simultaneous diffusion of aggregates) are respected. The fractal dimension expected for such a generating process is of the order of 1.4-1.8 in 2-D or 3-D spaces [11].

Figures 3 and4 show nanostructures obtained via the scheme presented above. They confirm the expected 2-D and 3-D fractal dimensions as determined by Jullien [11].

X (arbitrary units) Y (arbitrary units)

Mass(R)=R Df

Conduction heat transfer model

As discussed before, we want to study the conduction heat transfer traveling within a heterogeneous material made of nanoparticles surrounded by a gas, both constituents being assumed thermally isotropic. Two types of models will be developed: (i) a steady-state thermal conduction model to determine the effective thermal conductivity, (ii) a time-resolved thermal diffusion model to determine the effective thermal diffusivity and possibly other thermal properties.

Steady-state model

The knowledge (i) of the thermal conductivities of the 2 constituents of the matrix and (ii) of its spatial structure must allow us to determine the thermal conductivity tensor of its equivalent homogeneous material. This thermal homogenization step will be achieved with the help of a steady-state thermal conduction numerical experiment. A cube-shaped nanoporous structure is submitted to a temperature difference between 2 of its 6 boundary surfaces (see figure 5

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (3)
where is the heat flux ( ϕ )

T : Fourier's law ϕ = -λ ∇ and n corresponds to the direction normal to the considered sample boundary surface. The thermal conductivity λ being not constant from a point to another within the matrix, we have to use a numerical technique to solve this problem. We have chosen a classical finite difference method to calculate the temperature field within the matrix. The considered material volume is discretized into elementary cubic elements. The mesh is made fine enough so that each element can be considered to be homogeneous and isotropic. Then, each element is attributed local thermal properties corresponding either to a nanoparticle or to the gas. The system of equations resulting from (3) is linear with a symmetric, positive definite matrix. A lot of numerical techniques exist that can achieve the matrix inversion. We have chosen a conjugate gradient one.

Once the temperature field is known, one can calculate the thermal power traveling through a plane section S parallel to the 2 imposed temperature boundary surfaces. This thermal power can be expressed by:

S n dS Φ = ϕ ⋅ ∫∫ (4)
For example, in the case illustrated by figure 1, the expression (4) of the thermal power becomes:

x S S

x dS dS

Φ = ϕ ⋅ = ϕ ∫∫ ∫∫ (5)
where is the unit vector normal to the considered section S.

x

Now, if we replace the heterogeneous material by its equivalent homogeneous material in the direction, we can rewrite the expression of the thermal power Φ in the following way: x

( h c S T T X λ Φ = -) (6)
where λ is the effective thermal conductivity of the heterogeneous material in the x direction and X is the distance between the 2 imposed temperature boundary surfaces.

Finally, eliminating Φ between ( 5) and ( 6), one reaches the following expression for the effective thermal conductivity λ in the x direction:

( ) ( ) x h c h c S S X X dS dS S T T S T T x ∂ ⎛ ⎞ λ = ϕ = -λ ⎜ ⎟ - - ⎝ ⎠ ∫∫ ∫∫ T ∂ (7)
The numerical experiment described above, including the resolution of the temperature field, the calculation of the thermal power Φ (expression ( 5)) and finally the evaluation of the effective thermal conductivity λ (expression ( 7)), will have to be repeated in each space direction for the determination of the effective thermal conductivity tensor of the heterogeneous material.

To illustrate this approach, let us present results collected with periodic geometries. We have generated 3 very simple NSM matrix structures: a 2-D brick-shaped one, a 2-D diamondshaped one, and a 3-D brick-shaped one. For these 3 material geometries, the porosity remains equal to 95%, the matrix is considered to be under vacuum (i. e. the thermal conductivity of the surrounding gas is taken to be 0), and the thermal conductivity of a nanoparticle is set equal to 1 W.m np λ -1 .K -1 , corresponding to the bulk value for silica. We have solved the steady-state problem for each case and have calculated the resulting effective thermal conductivity of the matrix .

Time-resolved model

To study the thermal diffusion through a nanoporous matrix, we have developed a model inspired from a very classical thermal characterization technique, usually referred to as the "flash technique", for the experimental evaluation of the thermal diffusivity [12]. This time (see figure 7), the cube-shaped nanoporous structure is submitted to an instantaneous heat pulse on one boundary surface ( term), this boundary surface and the one located at its opposite are also submitted to permanent linearized heat losses (coefficient h, assumed

Q (t) δ 2-D brick- shaped 2-D diamond- shaped 3-D brick- shaped λ np = 1 W/m/K λ np = 1 W/m/K λ np = 1 W/m/K ↓ ↓ ↓ λ m =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ϕ ⋅ = -λ = ⎪ ∂ ⎩ (8)
where C is the volume specific heat (in J.m -3 .K -1 ), Q is the energy per unit surface absorbed by the irradiated boundary surface (in J.m -2 ), δ(t) is the Dirac delta function, and the notations n and refer to the normal direction oriented outwards of the calculation domain. n A finite difference discretization scheme is applied to the system of equations ( 8), and the space and time resolved temperature field is calculated with the help of an implicit algorithm in time. At each time iteration, a linear system of equations is solved with the conjugate gradient algorithm. At the end of this procedure, an average temperature rise of the opposite boundary surface versus time curve is collected: this curve is called a thermogram.

Now, if we replace the heterogeneous sample by a homogeneous one of thermal conductivity λ, thermal diffusivity a and volume specific heat C / a = λ , then the analytical expression of the thermogram of the rear side of the sample is known:

( ) ( ) ( ) 2 2 2 n n n n 2 2 n 1 n sin Bi t T x X,t K exp τ Bi Bi 2 Bi +∞ = α α α + ⎡ ⎤ α = = - ⎢ ⎥ α + + ⎣ ⎦ ∑ (9)
where has units of temperature, is the thermal diffusion characteristic time, and is the dimensionless Biot number which indicates the level of the heat

K Q / C X = 2 X / a τ = Bi h X / = λ
exchange between the sample and its environment. Finally, the dimensionless coefficients are the positive roots of the transcendental equation:

n , n 1 α ≥ 2 2 Bi tan Bi α α = α -2 (10)
It appears that expression [START_REF] Legrand | The surface properties of silicas[END_REF] involves the time t plus the 3 parameters τ, Bi and K. In order to analyze experimental thermograms collected from homogeneous samples on our thermal diffusivity measurement "flash" setup, we have developed a least squares minimization routine that allows us to identify τ, Bi and K. The values of Bi and K can not be further exploited, because (i) the parameters Q and h of the experiment are not perfectly known, and

(ii) the temperature rise curve on the rear side of the sample is not measured quantitatively.

On the other hand, the knowledge of τ results in an immediate evaluation of the thermal diffusivity a of the sample via the relation:

2 X a = τ (11)
When dealing with "numerical experiments", the situation is quite different because all quantities are known: the parameters Q and h are inputs of the calculation, and the thermogram on the opposite boundary surface is quantitatively evaluated. Then, the identification of τ, Bi and K from the "numerical" thermogram gives us the possibility to determine respectively the effective thermal diffusivity a , the effective thermal conductivity λ and the effective volume specific heat C of the heterogeneous material via the following relations:

2 X hX a , a n d C Bi K X = λ = = τ Q (12)
To illustrate this approach, we have performed a "flash" numerical experiment in the z direction on the 3-D fractal NSM representation shown on figure 4. The sample thickness X was equal to 230 nm, the energy surface density Q was set equal to 1 J.m -2 , and the coefficient h was given the value of 10 4 W.m -2 .K -1 (i. e. a very high value, in order to compensate for the very low value of the thickness X and hence to produce a Bi value of the order of unity). The matrix was considered to be under vacuum, and the thermal properties retained for a nanoparticle were the following: thermal conductivity 1 W.m -1 .K -1 , volume specific heat 2.94 10 5 J.m -3 .K -1 (corresponding to the bulk value for silica). We have calculated the thermogram on the opposite boundary surface and identified the effective thermal diffusivity, thermal conductivity and volume specific heat of the 3-D fractal NSM representation. Figure 8 shows the resulting thermogram and its related fit: a good agreement is observed between the two curves, which is a first indication that the heterogeneous sample behaves as a homogeneous material in the course of this transient numerical experiment. The minimization procedure allows us to identify τ, Bi and K and to derive the values of a , λ and C . We have found: τ = 1.12 10 -6 s, Bi = 0.93, K = 83.1, a = 4.72 10 -8 m 2 .s -1 , λ = 2.47 mW.m -1 .K -1 and C = 5.23 10 4 J.m -3 .K -1 . At this point, it is interesting to note that the 3 identified thermophysical properties a , λ and C perfectly verify the relation a

C λ = × ,
which is a further indication that the heterogeneous sample responds as a homogeneous material to the transient numerical experiment applied to it. To confirm the identified thermal conductivity value, we have performed a steady-state thermal conduction numerical experiment on the same material, with the same thermal conductivities and in the same heat flux traveling direction. The effective thermal conductivity value was found equal to 2.49 mW.m -1 .K -1 , which is very close to the one determined with the "flash" numerical experiment.

Application

The models presented above can reveal helpful for the analysis of a priori surprising experimental results. Thermal measurements made in our laboratory on NSMs of different densities indicated that the increase in the effective thermal diffusivity was small when the NSM density was multiplied by a factor of 2, and consequently that the effective thermal conductivity behaved quasi-proportionally to the density. In order to validate these experimental observations, we have used our models to study the impact of the nanoparticle volume fraction on the effective thermal properties of the matrix. For this parametric study, we have chosen to simulate the heat transfer within a simple 3-D periodic brick-shaped structure (see figure 6), the nanoparticle volume fraction being easier to vary with this geometry than with a fractal structure. Figure 9 illustrates the effective thermal diffusivity evolution with the volume fraction as obtained via the time-resolved model, and figure 10 illustrates the effective thermal conductivity evolution with the volume fraction as obtained via the 2 models. These 2 graphs confirm our experimental characterizations: (i) the effective thermal diffusivity is multiplied by a factor of only 1.4 when the nanoparticle volume fraction is multiplied by a factor of 10, and (ii) the effective thermal conductivity evolves quasilinearly with the nanoparticle volume fraction.

Conclusions

In this work, we have presented numerical models that allow us to quantify the level of conduction heat transfer traveling through NSMs. The first step of our simulations consists in the generation of a fractal representation of the matrix structure. Then, 2 types of simulations can be performed: either a steady-state thermal conduction numerical experiment, which allows us to calculate the effective thermal conductivity of the matrix, or a time-resolved thermal diffusion numerical experiment, which allows us to calculate the effective thermal diffusivity of the matrix as well as its effective thermal conductivity. These simple models have been applied to a study concerning the impact of the nanoparticle volume fraction on the effective thermal properties of a nanoporous matrix. The simulations have permitted to understand the experimental results derived from "flash" measurements performed on NSMs of different densities.

For the moment, these models produce quite good approximations of the effective thermal properties of NSMs. Nevertheless, they need to be improved to integrate the specificities of heat transfer at the scale of a nanoparticle. Indeed, Domingues et al [13] have demonstrated that the heat transfer between 2 nanoparticles at a distance smaller than a nanoparticle diameter is 2 to 3 orders of magnitude more efficient than when they are in contact, which means that we need to ask ourselves about the separation between conduction and radiation at these length scales.

Figure 1

 1 Figure 1 is a TEM image at a micrometric scale (the surface covered by this image is approximately 30 µm 40 µm × ) of a particular NSM made of a nanoporous a-SiO 2 matrix and

Figure 1 :

 1 Figure 1: TEM image at a micrometric scale of a particular NSM made of a nanoporous a-

Figure 2 :

 2 Figure 2: TEM image of the nanoporous a-SiO 2 matrix appearing in figure 1. Coalesced

Figure 3 :

 3 Figure 3: Representation of a nanoporous matrix as a 2-D fractal structure. The fractal

Figure 4 :

 4 Figure 4: Representation of a nanoporous matrix as a 3-D fractal structure. The fractal

Figure 5 :

 5 Figure 5: Schematic representation of the steady-state problem to be solved with its boundary

Figure 6

 6 shows the results: very interesting are (i) the orders of magnitude of the calculated effective thermal conductivities (they are comparable to the ones measured experimentally), (ii) the impact of the geometry on the value of the effective thermal conductivity (the 2-D diamond-shaped structure yields a value which is significantly lower than the one obtained with the 2-D brick-shaped structure), and (iii) the impact of the space dimension on the value of m

Figure 6 :

 6 Figure 6: Comparison of the effective thermal conductivities calculated for 3 different

  8 mW/m/K λ m = 13 mW/m/K λ m = 20 mW/m/K identical on the 2 surfaces), and the temperature elevation versus time curve is calculated on the boundary surface located at the opposite of the irradiated one. From this curve, it is possible to identify the effective thermal diffusivity of the heterogeneous material in the direction normal to the 2 boundary surfaces.

Figure 7 :

 7 Figure 7: Schematic representation of the transient "flash" problem to be solved for the

Figure 8 :

 8 Figure 8: A numerical thermogram obtained with our transient model and its related fit.

Figure 9 :Figure 10 :

 910 Figure 9: Effective thermal diffusivity variation versus the nanoparticle volume fraction.