Mélanie Jouitteau

Free-choice and reduplication A study in Breton dependant indefinites 1

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Introduction

Indefinites are felicitous with a reading where, internally to a contextually relevant set, the particular choice of referent is irrelevant. When a magician says Pick a card, context favors an interpretation where any card from the set would be a felicitous choice, as long as it is a card from the proposed set, as illustrated for modern Breton (Continental Celtic) in (1)a. Some indefinite constructions have this free-choice reading as the only felicitous one. This paper closely investigates such a free choice indefinite (FCI) that presents a typologically unusual morphology as illustrated in (1)b. This free-choice indefinite is realized by reduplication of the head noun around what seems like a spatial proximate deictic morpheme (-mañ-). The relevant contrast with the regular indefinite ur gartenn in (1)a is loss of optionality for the free-choice reading. The sentence in (1)a is felicitous if the magician proposes only one card, whereas (1)b is not.

(1) DURING A SHOW, THE MAGICIAN SAYS: a. Trapit ur gartenn b.

Trapit kartenn-mañ-kartenn. Pick a card pick card-here-card 'Pick a card.' 'Pick a card, any card. ' Breton In this paper, I will first investigate the DP-internal syntax and morphology of the reduplication construction in (1)b. I will propose that it results from the creation of a complex head noun by reduplication in a morphological step operated between syntax and phonological form. Next, I investigate the distribution of the Breton reduplicated FCI. I show that when preceded by a specificity marker, this construction behaves like a regular indefinite.

When not preceded by this specificity marker, the noun exhibits the typical distributional restrictions of dependent indefinites. I will show that the bare use has existential quantificational force, but can acquire universal force when bound by a universal quantifier.

There is a considerable amount of variation in the reduplicated FCI across Breton dialects, both in morphology and in syntactic licensing, and its interpretations can be very subtle. I have favoured multiple in-depth elicitations with a single speaker with consistent judgments (about ten hours in five sessions). The informant, H.D., in his fifties, is a native from the Douarnenez dialect. One of his parents is a non-native but a fluent speaker, the other one a native. He also had a native caretaker as a young child. H.D. is at least trilingual, in French and Breton, as well as English. He has easy access to the Breton literature and to the written standard, for which he overtly assumes a normative attitude. Apart from the Douarnenez and standard varieties, the speaker has had little exposure to other dialects. He showed persistent judgments from one session to the other. The protocol material was built with both (i) contextually enriched translation tasks from French and (ii) corrections and judgments on

Breton examples previously collected in the literature or via Goggle searches. Unless mentioned otherwise, data in this article is from H.D. For comparison and a consistent preview of dialectal variations, the obtained data was further checked with B.R., a young native from the Vannetais variety.

I. The DP-internal distribution of reduplication

In this section I will investigate the DP internal distribution of reduplicated heads and the type of heads that can reduplicate. I will propose a morphological derivation for this morphologically complex head. In (2), the left morphemes he-and hou-, marked for gender in synthetic demonstratives, do not exist in isolation. They can however reduplicate around the bound morpheme -mañ, making (3) an occurrence of reduplication of a morpheme that is never found in isolation.

(2) hemañ, / houmañ 3SGM-here 3SGF.here 'this one.♂' 'this one.♀' (3) hen-mañ -hen Cornouillais dialect, Trépos (2001:177) 3SGM-here-3SGM 'so-and-so'

Once the complex head noun is formed, it can enter regular noun derivations, like suffixation of the default plural marker -où (yezh-mañ-yezhoù, /language-here-language-PL/, 'any languages', [B.R 11/2014]). The meaning of the reduplicated noun is indefinite, but the indefinite article is illicit before the complex noun head. The definite article is however possible, as in (4). We will see in the following sections that absence of an article is grammatical in a precise set of syntacticosemantic contexts.

(4)

{ Ar /*ur / *Ø } floder-mañ-floder a vo kastizet drouk. the / *a / _ tax.evader-here-tax.evader R be. FUT punished badly 'A tax evader (whose identification is irrelevant) will be severely punished.' Kement, 'as much/any', is a determiner that can precede the reduplicated head. It gives it a universal reading: the sentence in (4) is verified if a unique individual is punished, whereas (5) is verified if each and every tax-evader is indeed punished. Kement can appear together with an article (Stephens 1993:138), but not so before a reduplication (6).

(5)

Kement floder(-mañ-floder) a vo kastizet drouk. as.much tax.evader-here-tax.evader R be. FUT punished badly 'Any tax evader will be severely punished. ' (6) * ar c'hement floder-mañ-floder the as.much tax.evader-here-tax.evader

Most prenominal quantifiers are out before the reduplicated head structure including, as we saw, the indefinite determiner ur (at least for HD).

( The determinerless structure seems to be more accurately described in terms of bare uses: absence of determiner seems to imply absence of other prenominal elements. In the absence of a determiner, prenominal cardinals are ungrammatical (13). This is also the case for prenominal analytic singulatives, as exemplified here with pezh before the collective noun chatal (a structure equivalent to piece of furniture), and with tamm before a mass abstract noun, 'pleasure'.

(13) *(An) teir flac'h-mañ-flac'h a raio war-dro ar c'hef. the 3 girl-here girl R do.FUT P the cashier. 'Three girls (whatever girls) will work the cash register.' (14) a. Trap { ar pezh chatal-mañ-chatal / (* pezh) chatal-mañ-chatal }.

catch the piece cattle-here-cattle / piece cattle-here-cattle b. (*tamm) plijadur-mañ-plijadur piece pleasure-here-pleasure

The table in (15) summarizes the results so far and reveals systematic DP internal differences in the distribution of reduplications preceded by the definite article an, al, ar and reduplications without it. We will see in the next section that these differences correlate with meaning, but also with DP external distributional differences.

( (20)a Kement-mañ-kement e kousto deoc'h ar pred. as.much-here-as.much R cost.FUT P.2PL the meal 'The meal will cost you such and such amount.' b. {Kement-mañ / ur mousc'hoazh } e kousto deoc'h ar pred. as.much-here / a smile R cost.FUT P.2PL the meal 'The meal will cost you this much/all this/as much as this / a smile.' Only nouns thus can reduplicate, which bring in an obvious hypothesis as to the derivation of reduplication, because only nouns can be independently be followed by -mañ. Is independent compatibility with the spatial proximate deictic adverb -mañ an obligatory intermediate step in reduplication? I briefly discuss this question. As is crosslinguistically common, Breton makes use of a deictic clitic (21) in order to create demonstratives ( 22).

(21) proximity of spatial adverb and clitic deictic markers with respect to the speaker:

' amañ aze ahont -mañ -ze -hont proximal medial distal (22)a.
an dra-mañ, an dra-se, an dra-hont the thing-here the thing-there the thing-over.there 'this thing' b. kement-mañ, kement-se, kement-hont as.much-here as.much-there as.much-over.there 'this much', 'as much as this there', 'as much as all that over there' (Hemon 1995:36) Nouns thus can appear with -mañ independently of reduplication. However, there exists a morpheme that is found in reduplication but cannot independently combine with -mañ. For some speakers, the head noun hini reduplicates in hini-mañ-hini in ( 23). ( 23) hini-mañ-hini one-here-one 'so-and-so' Trépos (2001: §217) -H.D. never reduplicates pronouns However, this head can not be found in analytic demonstratives on the pattern of ( 22). The form *(an) hini-mañ is an ungrammatical outcome for 'this one'. The synthetic form of the demonstrative pronoun hemañ / houmañ (also illustrated in ( 2)) appears instead. A morphological filter with a rewriting rule can correctly obtain *hini-mañ > hemañ, but we do not want it to rule out ( 23). I propose the following scenario in ( 24), where the rewriting rule operates from an abstract input and gives a morphologically well-formed output. The filter prevents [N + DEM] to turn into *hini-mañ, and rewrites it as hen-mañ, a compound that will later be interpreted at PF as what I write hemañ, an orthographic abstraction standing for the different phonological forms in different dialect.

(24) how reduplication operates at the morphological level

input filter output final output N hini Hini √ def N
an hini an hini √ def N + DEM hen-mañ Hemañ √ reduction of *-mañ-mañ + reduplication def Thing an dra an dra-mañ-dra an dra-mañ-dra

√ N hini hini-mañ-hini hini-mañ-hini √ def N
an hini an hini-mañ-hini an hini-mañ-hini √ def N + DEM hen-mañ hen-mañ-mañ-hen hemañ-hen √ def N + DEM an hen-mañ an hen-mañ-mañ-hen an hemañ-hen √ Reduplication applies at the output of the filter, and does not have access to raw material, preventing forms like *hemañ-hini. I follow the hypothesis of Grohman and Nevins (2004) who propose, for the English reduplication process obtaining money-shmoney, that the lower copy of too local a movement has to be differentiated from its antecedent. Reduplication thus operates around the bound morpheme -mañ-that is merely needed to differentiate the two copies obtained by reduplication. This -mañ-morpheme is homophonous with, but different from the deictic marker present in demonstratives. Reduplication operates in two steps. First, the -mañ-morpheme is added, triggering morphological reduction if two occurrences of -mañ are adjacent. Second, the morpheme to the left of -mañ-is copied and reproduced to the right of it. The reduction of double occurrences of -mañ correctly obtains the grammatical form hemañ-hen (instead of *hemañ-mañ-hemañ). Postulating reduction before copying obtains that the right hand pronoun is -hen and never hemañ (*hen-mañ-hemañ). Letting the hen-mañ rewriting rule apply at every step, and especially after reduplication, is not an option because we don't want the reduplications hini-mañ-hini and an hini-mañ-(an) hini to be rewritten as *hemañ-hini or *hemañ-(an) hini. 2This scenario implies that reduplication of both an dra, 'the thing' and an dra-mañ, 'this thing' will produce the same output, both yielding an dra-mañ-tra or an dra-mañ-(an) dra (see below for tra vs. dra). I don't think this is a problem for the semantics of the construction: the common result is a specific indefinite. More importantly, this scenario implies that the -mañ-morpheme in reduplication is completely different from the deicticmañ morpheme of analytic demonstratives. I believe this is correct. The very possibility of the reduplicated form hemañ-hen might seem to suggest that reduplication is a productive process somehow building on the demonstrative paradigm. However, among the three deictic markers, only the proximal -mañ ever appears in reduplications (25). Pronominalisation of ( 22) results in hemañ, but also hennezh and henhont. If reduplication built on demonstratives, we would have to explain why and how the medial -se and distal -hont morphologically ban reduplication.

(25) *(an) dra-se ('n) dra / * (an) dra-hont ('n) dra the ting-there (the) thing the thing-over.there (the) thnig Moreover, a scenario building reduplication on a demonstrative construction (26) would have to account for the reduplicated structures with a missing article (d.). Analytic demonstratives (except with kement) start with a definite determiner and would logically produce only reduplicated structures also starting with a determiner, contrary to facts. [ N-mañ-N ] non-specific dependent indefinite

One could argue that -se and especially the more rare -hont are marked, making -mañ in contrast somehow unmarked and default, appearing as a last resort during the process of reduplication. I point that if so, -mañ has been deprived of all its deictic properties. The semantic reason as to why the proximal maker would be the one preferred in order to refer to unknown entities is immediately counter-intuitive. I thus conclude that as proposed in (24), the -mañ-morpheme at the heart of the reduplicated structure is distinct from the deictic marker appearing in analytic demonstratives. It is their homophony that triggers reduction under adjacency. I now turn to the question of the module of grammar where the operation in (24) takes place. There are some indications that reduplication does not copy phonologically formed blocks. In ( 27), the definite article triggers a lenition /k >g/ on the initial of the left hand noun. The right-hand duplicate is untouched. Cases of mutation on reduplicates are also found (Jouitteau 2014a), but the important point here is that the reduplication operation can have access to unmutated forms.

(27) Livet e veze gantañ ar goantenn-mañ-koantenn hag e veze kontant. painted R was by.him the beautiful-here-beautiful and he was happy 'He used to paint some beautiful girl (whose identity is irrelevant) and was happy.'

There is also evidence that the reduplication process can be partial, and thus have access to morphological decomposition. For morphologically complex nouns, the left one can be morphologically more complex than the right one. In (28), the suffix -enn, a singulative marker on a mass noun, is absent on the right. The same partial reduplication is observed with the diminutive form (paperig-mañ-paper, vannetais Breton [B.R 11/2014]). For H.D., partial reduplication is not possible with collective nouns (29). It is possible that his rejection is triggered by a SG/PL number asymmetry.

(28) Ret-groñs eo dezho sinañ paperenn-mañ-paper. obliged-intensifier is to.3SGM sign paper.SG-here-paper 'It is mandatory for him to sign such and such paper.' (29) * ar steredenn-mañ-stered * sivienn-mañ-sivi the stars.SG-here-stars strawberries.SG-here-strawberries

There is considerable variation as to the possibility of articles in right-hand duplicates as in (30). H.D. obligatorily drops it and consistently duplicates only the head noun. The forms where the right-hand article is realized provide interesting information about the reduplication process and the syntactic domain of what can be reduplicated. Speakers who can duplicate the article of the duplicate have to have access to material outside of the head-noun.

(30) an dez-mañ ('n) dez the day-here (the) day '(on) such and such a day ' Favereau (1993: § -mañ) Stress provides an argument that the definite article is indeed not integrated to the head noun.

In dialects where word stress is on the penultimate syllable, integration of the definite article into the word structure would wrongly predict it would receive stress when preceding monosyllabics. Breton definite articles are however generally not stressed, even if the following noun is monosyllabic (Hemon 1995: §282). The definite article could however be an unstressable clitic adjoined to the head noun.

In conclusion, reduplication is a productive operation on most nouns and on the adverb kement. It operates around a dedicated -mañ-morpheme. The process can reduce morphological information on the duplicate. Reduplication does not have access to the abstract bundle of features nouns have, but to the output of morphological filters like the *hini-mañ filter and its > hemañ rewriting rule. Reduplication has access to morphological information (it recognizes deverbal nouns and reduces morphological complexity in the copying process), but it does not have access to phonologically formed material.

II. Specific indefinite vs. bare noun

In this section, I turn to the distributional and semantic differences between reduplicated head nouns with and without determiner. The definite article an, al, ar is a specificity marker. 3 In (32), the irrealis conditional is shown ungrammatical in the modifying relative of the reduplication. This is predicted if the article indeed enforces specificity. The reduplication structure with the determiner is felicitous in episodic contexts, with a specific indefinite reading (33). In (34), I show that the specific indefinite takes wide scope with respect to negation (*I didn't see anyone). Negation does not have the effect to maximize it, suggesting that the reduplication does not create a minimizer (compare with the favourite reading of I didn't see a person).

(32) Me meus c'hoant kaout ar stajiad-mañ-stajiad a zo / *vefe farsus ha jentil. 1SG I.have wish have the intern-here-intern R would.be funny and nice 'I want to have an intern that would be nice and funny.' (33)

Al labous-mañ labous a oa o nijal tost dezhi, met ne rae ket van. the bird-here bird R was at flying close P.3SGF but NEG did NEG care 'A (specific) bird was flying close to her but she didn't care.' (34) N' em eus (ket) gwelet an den-mañ-den. NEG R.1SG have not seen the guy-here-guy 'I didn't see someone (*I didn't see anyone/*I didn't see a person)'.

The precise reading associated with the reduplication form preceded by an article is somewhat hard to define. It can be specific unknown (35), or specific known by the speaker of reported discourse (36), or even specific known by both the speaker and the speaker of reported discourse (37). One could think that identification of the entity at least has to be irrelevant, but it is not always the case (38). In (39), one could even be bragging about actually knowing the name in question by refusing to reveal it.

(35) Deuet tre eo Yann en ostaleri o klask kaojeal gant ar paotr-mañ-paotr. went in is Yann in.the bar P search speak with the guy-here-guy. 'He went into the bar and started speaking to so-and-so.' (He knew the guy was here, but he didn't know him) (36) « D'an eur-mañ-eur, me a yelo, ha giz-se, bezañ pare tout an traoù. » P the hour-here-hour 1SG R go.fut & way-there be fixed all the things '(And the priest then used to decide for a time and say): "at such and such hour, I will go", and this way, things were done.' (37)

« Graet em eus mat va soñj, dimeziñ a rin gant an den-mañ-'n-den. » done R.1SF has good my mind to.marry R will.do with the guy-here-the-guy "I have made my mind, I will marry so-and-so.

"' (Naig Rozmor citing her own mother) (38) CONTEXT : THE POLICE WENT INTO A CAFE BECAUSE OF LOUD MUSIC

An amezeg-mañ-amezeg en neus douget klemm, a-dra-sur, met aet da c'hoût piv eo ! a neighbour-here-neighbour R has complained certainly, but go to know who is 'Some neighbor must have complained, that's for sure, but who knows who!' (39) Klask ez eus war ar medisin-mañ-medisin.

search R is for the doctor-here-doctor 'One searches for a certain doctor (whose name I can't find right now or won't reveal)

The semantic contribution of a free-choice item is that any element in the denotation of the restrictor is a suitable candidate for satisfying the nucleus (Jayez and Tovena 2010). To the extend that Breton reduplication brings in the free-choice meaning and the definite article brings in specificity, we should be witnessing from ( 32) to (39) a specific free-choice indefinite. Any element in the denotation of the restrictor has to be a suitable candidate for satisfying the nucleus, in a way that is fully compatible with the speaker suggesting s/he fully identifies the referent in question. The syntactic distribution of this structure is that of a specific DP.

In contrast, the determinerless reduplication structure obeys only the definition of free-choice items. Known entities are consistentlty out (40). The bare noun FCI obtains the same reading as an indefinite (ur stajiad a vefe farsus…) in non-specific irrealis contexts as in ( 41).

(

40) Mañ o vont da zimihiñ gant plac'h-mañ-plac'h (*ha teus bet tro da gejañ ganti dija).

C.is at go to marry with girl-here-girl C has had way to meet with.her already 'He is going to marry a girl (*that you have had an occasion to meet already), any girl.' (41) Me meus c'hoant kaout stajiad-mañ-stajiad a vefe farsus ha jentil. 1SG I.have wish have intern-here-intern R would.be funny and nice 'I want an intern that would be nice and funny, any intern that….' 'I want any intern that would be nice and funny.'

In the next section, I will show that the bare noun non-specific FCI created by reduplication has the restricted distribution of a dependent indefinite. It precisely shows the prototypical distribution of an existential free choice.

III. Identification as an Existential Free Choice (Ǝ-FCI)

Polarity items like dependent indefinites are elements whose distribution and interpretation are sensitive to the properties of the context of occurrence and their hallmark is exclusion from positive assertions with simple past [START_REF] Giannakidou | Negative and positive polarity items: licensing, compositionality and variation[END_REF] "-FCI can typically be rescued by subtrigging, that is by addition of a modifying relaive (Any boy ??? (who wants to know about the exam) will show up.), whereas $-FCI can not (*John chose an apple (that was red), any apple (that was red), [START_REF] Dayal | The universal force of free choice any[END_REF][START_REF] Chierchia | Logic in grammar: polarity, free choice, and intervention[END_REF]. In Breton, the bare noun reduplicated is not rescued by subtrigging.

(63) * Paotr-mañ-paotr (en dije choant da c'houzout diwar-benn an arnodenn) a zeuio. guy-here-guy 3SGM would.have wish P know about the exam R come.FUT 'A boy ??? (who wants to know about the exam) will show up, any boy.' A further contrast is between the readings of $-FCIs like Italian qualsiasi and French quelconque (64) that refer to a singular entity, and that of universal FCIs like English any book (65). Breton patterns with $-FCI.

(64) You must read any one book from the reading list. English Devi leggere un libro qualsiasi dalla lista di letture. Italian Tu dois lire un livre quelconque de la liste de lecture. French > obligation to read one book from the expressed set.

Chierchia (2013) (65) You must read any book from the reading list.

> obligation to read all books from the expressed set (66) D'an nebeutañ, lenn levr-mañ-levr diwar al listennad a zo bet roet dit.

at the least read book-here-book from the list R is been given to.2SG 'Read at least one book (any book) from the list that has been given to you.' Breton "-FCI are fine in comparative constructions, but the reduplication structure is not (67).

(67) * Hennezh 'zo pinvidikoc'h eget hini-mañ-hini/henmañ-hen/paotr-mañ-paotr. this.one is richer than N-here-N / N-here-N / guy-here-guy (intended) 'this one is richer than anybody.' "-FCIs are fine with generic readings (68)a, whereas $-FCIs are incompatible with it ((68)b, Corblin 2010). The reduplication is ungrammatical in (69).

$-FCI

Any bird flies.

*A bird flies, any bird. (69) * Labous-mañ-labous a nij.

bird-here bird R flies A prototypical universal reading is possible but requires the overt universal quantifier kement preceding the reduplication structure. The reading is then clearly universal. The structure then becomes fine for generic claims (71) and can be rescued by subtrigging.

(70)a. * Floder-mañ-floder a vo kastizet drouk. b. Kement floder-mañ-floder a vo kastizet drouk. as.much tax.evader-here-tax.evader R is. FUT punished badly 'Any tax evader will be severely punished.' '" x, x a tax evader, x will be punished.' (example translated from Muller 2007) (71) Kement studier-mañ-studier e kentañ bloavezh a zo barrek da ziluziañ ar gudenn-se.

as.much student-here-student in first year R is able to solve the problem-here 'Any first year student is able to solve this problem.' (72)Kement paotr-mañ-paotr*(en dije c'hoant da c'houzout diwar-benn an arnodenn) a zeuio.

as.much guy-here-guy would.have wish P know about the exam R come.FUT 'Any boy ??? (who wants to know about the exam) will show up.' [START_REF] Farkas | Free choice in Romanian[END_REF] postulates that the quantificational force of free-choice items "depends on whether they are caught by an existential quantifier (in which case they are existential) or not (in which case they are universal)". In Breton, the default interpretation seems the opposite. The reduplicated bare noun is an existential, except under the scope of a universal quantifier. However, if this is true for most of the collected data, there remain a few examples where the favored interpretation is universal, without any obvious universal quantification.

(73) Eun dizesper eo lakaad ac'hanoh da glask tra-mañ-tra a despair is put P.2PL to search thing-here-thing 'You are perfectly hopeless at finding anything.' *$ thing x, such that it is a despair to make you search for x. " x, x a thing, that it is a despair to make you search for x. (74) I REGRET I'VE HIRED THIS COORDINATOR. HE IS TOO YOUNG.

Spi em boa vefe deuet a-benn da dallañ ouzh kudenn-mañ-kudenn en e labour. hope R.1SG had would.be come at-end to front at problem-here-problem in his job 'I hoped he could solve any problem in his job.' *$ problem x, such that I had hoped he could have solved x in his job. " x, x a problem, I had hoped that he could have solved x in his job.

The literature has found that certain properties act as parameters frequently observed among free choice indefinites across languages. I briefly review here where the Breton reduplication structure stands with respect to those parameters.

(75) Ǝ-FCI Parameters:

(i) licensing by interrogatives (ii) appearance in seemingly episodic contexts (to the extent that they can be interpreted modally, Chierchia 2013).

(ii) 'no winner / no loser' constraint (Jayez andTovena 2006, 2010).

FCIs do vary across languages with respect to licensing by interrogatives [START_REF] Jayez | Description et évolution de quelque[END_REF]. Breton reduplicated structures add to the variation, because they are not felicitous in polar interrogatives, be they realized by intonation (76) or by a segmental Q head like hag eñ (77). However, wh-questions do license Breton FCIs.

(76) * Fenozh emañ ma oto ganin. C'hoant peus da vezañ kaset lec'h-mañ-lec'h ? tonight C.is my car with.me. Q wish have to be sent place-here-place 'I have my car here tonight. Do you want me to drop you of somewhere?' (77) * Hag eñ neus graet tra-mañ-tra evit lakaat anezho war e du ? Q has.3SG done thing or thing for put them on his side 'Did he do anything to convince them?' (78) Piv en deus bet tro da welet steredenn-mañ-steredenn? who R.3SGM has had chance to see stars.SG-here-stars.SG 'Who has had the chance to see a star (, any star)? (79) Piv zo bet gwelet gant archer-man-archer?

who is been seen by cop-here-cop 'Who has been spotted by a cop (, any cop)?' B: every member of R can not be P under I (no winner) [START_REF] Jayez | Epistemic determiners[END_REF] I summarize the findings in the table below. Intuitively, free-choice existentials appear when several worlds and alternatives are evoked, with at least one alternative leading to the assertion being either wrong or non-verifiable. This is not the case for universal free-choice items, which can appear in episodic contexts when modified by a relative, or in comparatives (see Jayez & Tovena 2005:23-4 for discussion of French equivalents).

(89) Distribution of the bare $-FCI: the doctor-here-doctor R can.FUT come P stay P during the vacations 'So-and-so can come to stay during the vacations.' This is of great importance for the analysis of Breton verb second orders, because in (90), the subject is the only potential satisfier for the Breton ban on verb-first. Jouitteau (2005[START_REF] Jouitteau | The Brythonic reconciliation: from V1 to generalized V2[END_REF] predicts that movement into the preverbal area, when last resort for V2, is completely transparent for semantic interpretation. Here the FCI designs a sharp contrast between a preverbal and post verbal subject in last resort V2 environments. If there is an element like conditional mood that is usually thought to be with the verb in T or C that licenses FCIs, how come the subject is not licensed if it is interpreted below it? Sensitivity to surface scope is not a property shared by all Breton dependent indefinites. Some Breton bare nouns are negative polarity items, and as such, have to be licensed by negation (Jouitteau 2014b). The immediately preverbal position is a felicitous site for licensing (92) despite overtly preceding negation.

(92) Den ne gar den. human NEG loves human '(Love has disappeared,) nobody loves nobody. ' Breton Callac, Jouitteau (2014b) The reduplication structure can precede its licensor only under reconstruction, as when the FCI is fronted for focus in (93 The adolescent-here-adolescent R will.be P.3SGM do P the children 'Such and such adolescent will have to take care of the children.' (95) * Plac'h-mañ-plac'h emañ o vont da zimihiñ ganti.

girl-here-girl C.is at go P marry with.her

IV. Conclusion

I will conclude this portrait of Breton reduplication structures by a quick discussion of its typologically outstanding morphology. In a great majority of languages, free choice indefinites are constructed around an interrogative marker and some "indefiniteness marker", which comes in four main types: wh+ever, wh+and, wh+want, and wh+be [START_REF] Haspelmath | Diachronic sources of "all" and "every[END_REF](Haspelmath :371, 1997)) Nothing in Breton prevents formation of such FCIs: this pattern even represents a productive competing strategy to reduplication. Breton bennak (illustrated in context in ( 43)) is a compound of old Breton py, the unstressed form of interrogative pronoun pe-with negative na(g) (Willis 2013:279). Breton n'eus forzh pe-X (illustrated in context in (87)) means literally 'there is no importance which-X'. Free choiceness is crosslinguistically linked to determiners. As pointed out by Jayez and Tovena (2005:66), free choiceness is "a form of irreference, and determiners use their normal resources to go about it, namely constraints on their restriction and on the intersection between restriction and scope". From this semantic point of view, it seems rather peculiar that a free choice item would lack such material. Breton however shows this outstanding reduplicative strategy that does not make use of any determiner in order to realize free-choiceness. Noun doubling around disjunction (97) or coordination ( 98) is a crosslinguistically widely spread strategy to create FCIs that is also independently represented in Breton, but as far as I know, reduplication proper is not reported a means of creating free-choiceness in human languages outside of Breton.

(97) den pe zen tra pe dra hini pe hini person or person thing or thing N or N 'one person or the other', 'one thing or the other', 'one or the other ' Hemon (1995:40) (98) Hen-ha-hen en deus prenet ur marc'h.

pron-&-pron 3SG has bought a horse 'So-and-so has bought a horse. ' Hingant (1868 :210, §202) Noun reduplication is reported as a means of creating an independent plural indefinite in Da'a. Barr (1988:5) reports a diffuseness effect ("multiple, non-singular, non-definite", "a non-specific number greater than one"). Reduplication is crosslinguistically a productive way to create plurals. However, in Breton, a singular element keeps its singular reading (across a wider set) once reduplicated.

(99) ju'a-ju'a to na-ria ni kampu bagia nu Da'a. sickness-sickness C realis-be in village area of Da'a 'The sicknesses that are in the villages in the Da'a area' Da'a, Barr (1988:5) One might explore the hypothesis that the reduplication process can be exocentric and trigger a change of category in the duplicate, morphologically creating an epistemic indefinite in a language lacking one. Reduplication can indeed be descriptively exocentric, suggesting a silent head as a category changer. In French, reduplication of the noun head 'woman' produces an adjective. In Surinamese, a reduplicated verb yields a predicative element similar to a past participle (Parkvall 2000:79), like illustrated here for Saramaccan and Ndyuka. I leave careful exploration of this proposal and its implications for further research.

( Ndyuka, Huttar & Huttar (1997:405) The reduplicated FCI also provides a typologically unique minimal pair between, on the one hand, a dependent indefinite existential FCI, and on the other hand a specific FCI whose distribution seems free. Morphologically, the difference between the two lies in that the former is a bare noun whereas the latter is preceded by a specific marker. The article an, al, ar that the Douarnenez informant can precede reduplication with enforces a specific reading on it and frees it from licensing constraints in one go. The same connection between appearance of the determiner and free distribution is provided by B.R. from the Vannetais dialect, who allows the reduplicated head to be preceded by an indefinite article (ur plac'h-mañ-plac'h, /a girl-here-girl/, 'a girl, any girl'). This structure has the same (non-specific) meaning as HD's bare structure, but it is available for her in all environments including past episodic contexts. The Breton determiner thus seems to consistently free the FCI from distributional restrictions. I leave further exploration of the cross-dialectal validity of this generalization to further research.

(

  26) a. the N-mañ specific indefinite (demonstrative) b. the [ N-mañ-N ] specific unknown/uncited indefinite c. * N-mañ d.

(

  68)a. "-FCI b.

  15) Summary of the possible pronominal orders in DPs containing reduplications:

	Reduplication seems limited to nouns, but is not available for all nouns. For H.D., deverbal
	nouns consistently fail to reduplicate (17). These are however fine for B.R. from the
	Vannetais dialect. There is also variation as to the availability of reduplication for compound
	nouns (18) and proper nouns (19), attested elsewhere but both rejected by H.D.
	(17) { galvadenn / baleadenn	/ kaozeadenn	}(*-mañ-V.adenn)
	call.action.of walk.action.of speak.action.of
	'(a) call, (a) walk, (a) discussion'
	(18) ...pe ne vez ket gwraet ken	get amzer-mañ-amzer, stumm-verb-mañ-stumm-verb.
	or NEG is NEG done anymore P tense-here-tense, form-verb-here-form-verb
	'…or that such and such tense, such an such verbal form is not used anymore.'
					Vannetais (Kistinid), Nicolas (2005:7)
	(19) Pok Nikolaz-mañ-Nikolaz ha laosk ac'hanon e peoc'h!
	kiss Nikolaz-here-Nikolaz and let P.1SG in peace
	'Kiss whatever Nikolaz and leave me alone!'	[B.R. 11/2014]
	Kement can reduplicate as in (20)a. It is homophonous to the adverb we saw in (5), but its
	nominal status is not a stipulation meant to preserve the generalization that only nouns
	reduplicate. Kement is independently found with -mañ, with which it forms an anaphora. In
	(20)b, it can be replaced by another noun.
		articles	quantifiers	analytic singulative
	√	Ø
	√	Ø	kement
	* * * √	Ø Ø Ø definite	other quantifiers numerals	pezh, tam	} X 1 -mañ-X 2
	*	definite	kement
	√	definite				pezh
	√	definite	numerals
	*	indefinite
	Let us now turn to the candidates for X 1 and X 2 in (15). Morphologically plural nouns cannot
	reduplicate even when monomorphemic (*an dud-mañ-tud, /the people.PL-here-people.PL/).
	Collective nouns are syntactically plural (they trigger verbal plural agreement and plural
	anaphora) but bear no morphological mark of plurality. They can reduplicate, and logically
	co-refer with a plural pronoun (16). Breton has complex singulars obtained by suffixation of a
	singulative -enn ending to a collective noun. Complex singulars can reduplicate.
	(16) a. Trap al logod-mañ-logod		ha skarzh anezho er-maez
	catch the mice-here-mice		and throw P.them outside
	b. Trap al logodenn-mañ-logodenn ha skarzh anezhi er-maez!
	catch the mice.SG-here-mice.SG	and throw P.her	outside

  The domain of what is visible to reduplication can include the (possibly cliticized) definite determiner outside of the word boundary.

	* * *		collective-singulative non-collective plural deverbal noun }	collective non-collective plural *-hont { deverbal noun *-se
	(31) summary of the heads that can reduplicate:
		article	original head	proximal deictic	reduplicated head
					marker
	√	def.	N		(def) N
	√		kement		kement
	√		mass noun-singulative		mass noun-singulative
	√		mass noun-singulative		-mañ-	mass noun
	√		collective-singulative		collective-singulative

  ). The two next examples show 'wrong subject constructions' with a preverbal nominal constituent corefering with a resumptive pronoun internal to IP (see[START_REF] Rezac | Building and interpreting a nonthematic A-position: A-resumption in English and Breton[END_REF][START_REF] Rezac | The Breton double subject construction[END_REF] for detailed analysis of these structures). The free-choice bare noun is ungrammatical in the preverbal area of those structures.

	(93) [ Gant plac'h-mañ-plac'h ] emañ o vont da zimihiñ _ !
	with girl-here-girl	C.is at go P marry _
	'He is going to marry a girl, ANY GIRL.
	(94) (Ar)* c'hrennard-mañ-krennard a vo dav dezhañ ober war-dro ar vugale.

  , illustrated by French, English and Greek below.4

	(96) French	Greek	English
	a. quiconque	opjosdhipote	Whoever
	b. quoi que ce soit	otidhipote	whatever
	c. à quelque moment que ce soit opotedhipote	whenever
	d. où que ce soit	opudhipote	wherever	Greek, English from Giannakidou (2001)

  100)a. C'est une femme.

		b.	C'est une femme-femme.
	'It is a woman.'	'It's a woman-feminine.'
	'It is a woman.'	'It's a feminine woman.'	French
	(101) /dí náki-náki wómi/
	DEF beat beat man
	'the beaten man'	Saramaccan, Veenstra (1996:81)
	(102) /a	fensee fika opo-opo/
	DEF window remain open open
	'The window was left open'

Both hini-mañ and an hini-mañ are uniformly rewritten as pronominal hemañ, whereas in reduplication structures, hemañ-hen and (an) hemañ-hen are both found. I could not check that the two were indeed available for the same speaker, but the anomaly remains that the article can be preserved before pronominal reduplications and never before the synthetic demonstrative pronoun hemañ. This is another difference between demonstratives and reduplications, but my proposal does not account for it.

The prenominal head an, al, ar is treated in Breton descriptive grammars and in my glosses as the definite article. It appears in analytic demonstratives (/the guy-here/, 'this guy'). This specificity marker does not have to appear in all specific nominal constituents (for example the genitive structure /guitar Yann/, 'the guitar of Yann' that requires absence of the determiner of the possessed). In modern Breton, it shows no overt morphological variation for case, number, or gender. It triggers mutation on the following noun initial consonant based on a calculation of gender and number, suggesting morphological encoding of abstract number and gender features.

In Greek, opjos is a wh-determiner used in free relatives, dhi is an invariant emphatic particle (comparable to indeed), and pote means '(n)ever'(Giannakidou 2001:3). In French quiconque, one recognizes the modal part of Old French oncques, '(n)ever', but the compound traces back to the Latin quicumque, 'whoever'.

Klask 'zo war { ur medisin / * medisin-mañ-medisin } er c'harr-nij. search is on a doctor / doctor-here-doctor in plane 'We are looking for a doctor in the plane.'

In past episodic contexts, negation is a licenses the so-called negative polarity items or affective polarity items but crucially not free-choice indefinites (Giannakidou 2001:662). Breton negation in episodic contexts licenses the bare noun den, but fails to license reduplicated structures.

(45) N' em eus gwelet { den / * den-mañ-den }. NEG R.1SG have seen person / person-here-person 'I didn't see anybody. ' (46) En noz-mañ ne oa { den / * den-mañ-den }. In.the night-here NEG was person / person-here-person 'There was nobody tonight.'

The licensing contexts for the bare noun formed by reduplication coincide with those proposed in the literature for existential free-choices [START_REF] Giannakidou | The meaning of free choice[END_REF][START_REF] Chierchia | Logic in grammar: polarity, free choice, and intervention[END_REF] Free-choice indefinites bear either existential readings ($-FCI) or universal readings ("-FCI). A set of properties of the Breton reduplication points toward an existential free-choice. I will show that the reduplication structure can bear a universal reading only when the latter is brought by another element.

(60) properties opposing existential and universal FCI (i) only an $-FCI is felicitous in non-specificity (irrealis) contexts (ii) an $-FCI is not rescued by subtrigging (iii) an $-FCI has a singular reading (iv)

an "-FCI is licensed by comparative constructions (v)

an "-FCI is licensed by generic readings First, the reduplication is felicitous in non-specificity (irrealis) contexts, which is a property of existential FCIs.

(61) Du musst irgeneinen Artz heiraten. German, Chierchia (2013) Tu dois épouser quelque docteur.

High French 'You must marry a doctor (, any one will do).' ( 62) Ret e oa dezhi dimihiñ gant medisin-mañ-medisin evit ma vefe fier he mamm. obliged R was P.3SGF marry with doctor-here-doctor for C would.be proud her mother 'She had to marry a doctor for her mother to be proud, any doctor.' (80) Pegement eo bet paieet archer-man-archer evit ober-se?

How.much is been paid cop-here-cop for do-so 'How much has a cop (,any cop) been paid in order to do so?' $-FCIs can appear in seemingly episodic contexts in German, Italian, French.

(81) Irgendjemand hat eingerufen. German 'Somebody or other called.' ( 82) Gianni e' entrato in classe e si e' rivolto ad un ragazzo qualunque. Italian Yann est entré dans la classe et s'est adressé à un garçon quelconque. French 'Gianny walked into the classroom and addressed a boy FCI.' [START_REF] Kratzer | Indeterminate pronouns: The view from Japanese[END_REF] and [START_REF] Chierchia | Logic in grammar: polarity, free choice, and intervention[END_REF] propose that crosslinguistic variation on the possible occurrence of $-FCI in seemingly episodic occurrences follows from the availability of a null 'assertoric' modal licenser. In Breton, there is evidence for a licensing operator linked to iterativity (for ( 84)and ( 85), recall that generic sentences do not license the bare reduplication structure).

(83) Bep deiz em eus desket tra pe dra, gant den-ma-den… each day R.1SG have learned thing or thing, with guy-here-guy 'Each day I've learned such and such a thing, with so-and-do…' ( 84) Eno e chom bev hiziv c'hoazh gizioù kozh a zo bet a-viskoazh gizioù ar vro here R stay alive today again customs old R is been always customs the country da festañ tra-mañ-tra, darvoud-mañ-darvoud. to celebrate thing-here-thing event-here-event 'The old customs that have always been the customs of the country to celebrate such and such thing, such and such event are still alive today here.' (85) Klevout eo bet plijet bras den-mañ-den a vez atav un allazig d'an unan. hear is been pleased big guy-here-guy R is.GEN always a comfort P the one 'to hear that so-and-so has been really pleased is always a comfort.' According to [START_REF] Corblin | Une analyse compositionnelle de quoi que ce soit comme universel[END_REF], FCIs are not compatible with a restriction of the set of values available for interpretation. He argues for instance that French indefinites un ami/quelqu'un, 'a friend, someone' are not FCIs, because the choice of a value let to the choice of the hearer can be restricted (86). Following this criterion, the Breton reduplication would not be an FCI (87).

(86) Tu peux inviter un ami / quelqu'un, mais pas n' importe qui.

you can invite a friend / someone, but not NEG matter who '?? You can invite any friend, but not just anyone.' French, Corblin (2010) (87) Gell a ri pediñ mignon-mañ-mignon, met arabat pediñ n' eus forzh piv. Can R will.do invite friend-here-friend but -invite NEG is matter who However, this seems to merely test the 'no loser constraint' from Jayez andTovena (2006, 2010) and which is subject to crosslinguistic variation.


Sindhu Preetham Burugupally
email: sindhupreetham.burugupally@wichita.edu

W Roshantha Perera

Dynamics of a parallel plate electrostatic actuator in viscous dielectric media

Keywords: Electrostatic actuator, viscous media, reduced order modeling, mechanics

Understanding the dynamics of a parallel plate electrostatic actuator in viscous dielectric media will help optimize the actuator performance for manipulating microparticles suspended in aqueous media. In this paper, we analyze the response of the actuator in a clamped-clamped configuration immersed in viscous dielectric media. We modeled the actuator as a continuous system by deriving a reduced-order model, and solving it by employing the Galerkin method and linear undamped mode shapes for a clamped-clamped beam. Our model incorporates the inertial loading effect and squeeze film damping by the media, nonlinear mid-plane stretching forces in the beam electrode of the actuator, and nonlinear contact force during the physical contact of the beam electrode with the stationary electrode. The model is utilized to study the actuator dynamics over a broad range, three orders of magnitude of viscosity and two orders of magnitude of relative permittivity of the media. We report that the actuator at lower actuation voltages and/or higher actuation frequencies can be approximated to be a linear system. We also report that at very high actuation frequencies, the beam electrode does not pull-in into the stationary electrode, and the actuator experiences a significant phase lag and displacement drift over time.

Introduction

Electrostatic microactuators are commonly employed for generating forces on the order of few mi-croNewtons and displacements on the order of few microns for a wide range of manipulating and probing applications. Applications include aligning the optical fibers in optoelectronic packaging [START_REF] Syms | Scalable electrothermal mems actuator for optical fibre alignment[END_REF], handling of micron-sized objects [START_REF] Lee | Fabrication of an electrothermally actuated electrostatic microgripper[END_REF], probing of biological cells [START_REF] Mukundan | Microactuator device for integrated measurement of epithelium mechanics[END_REF], actuating the miniature fins on millimeterscale rockets [START_REF] Kilberg | Mems aerodynamic control surfaces for millimeter-scale rockets[END_REF], and switching the radio frequency signals [START_REF] Rebeiz | Tuning in to rf mems[END_REF]. Parallel plate actuators are a class of electrostatic actuators known for high displacement and force resolutions [START_REF] Bell | Mems actuators and sensors: observations on their performance and selection for purpose[END_REF]. It is difficult to predict the dynamics of these actuators as they have a nonlinear dynamic response due to the nonlinear squeeze film damping, nonlinear internal stretching, and nonlinear electrostatic actuation forces [START_REF] Chu | Dynamics of polysilicon parallel-plate electrostatic actuators[END_REF] (Fig. 1). Understanding the dynamics of a parallel plate actuator will help optimize actuator performance for a specific application such as manipulating biological cells. To understand the static and dynamic response of these actuators, researchers modeled them either as lumped systems [START_REF] Rollier | The stability and pull-in voltage of electrostatic parallel-plate actuators in liquid solutions[END_REF] or continuous systems [START_REF] Younis | A reduced-order model for electrically actuated microbeam-based mems[END_REF][START_REF] Rhoads | The nonlinear response of resonant microbeam systems with purely-parametric electrostatic actuation[END_REF]. Most efforts were directed towards the design of actuators operating in air or vacuum where the fluid viscous damping is negligible or absent [START_REF] Zhang | Electrostatic pull-in instability in mems/nems: A review[END_REF][START_REF] Saghir | Approaches for reducedorder modeling of electrically actuated vonkarman microplates[END_REF][START_REF] Farokhi | Pullin characteristics of electrically actuated mems arches[END_REF]. With the recent interest in manipulating and probing of biological cells on lab-on-a-chip platform, different class of actuators -comb drives [START_REF] Mukundan | Microactuator device for integrated measurement of epithelium mechanics[END_REF], curved electrodes [START_REF] Burugupally | Performance evaluation of a curved electrode actuator fabricated without gold/chromium conductive layers[END_REF], and hybrid drives [START_REF] Hoelzle | A large displacement, high frequency, underwater microelectromechanical systems actuator[END_REF] -are being designed to operate in viscous dielectric (aqueous) media. The performance of these actuators characterized in terms of actuator displacement and frequency response depends on the relative permittivity and dynamic viscosity of the viscous dielectric media [START_REF] Burugupally | Experimental investigation of curved electrode actuator dynamics in viscous dielectric media[END_REF][START_REF] Li | Substrate effect in squeeze film damping of lateral oscillating microstructures[END_REF]. The high relative permittivity of aqueous media ( f,w =80) amplifies the generated force and displacement, while the high dynamic viscosity (µ w =8.9×10 -4 Pa-s) restricts the displacement and narrows the bandwidth frequency of the actuator [START_REF] Burugupally | Experimental investigation of curved electrode actuator dynamics in viscous dielectric media[END_REF] -hence it is important to understand the effect of media properties on the actuator performance. Some of the recent research efforts include: investigating the effect of fluid viscosity, initial electrode gap, and beam geometry [START_REF] Ahmed | Shock response of electrostatically coupled microbeams under the squeeze-film damping effect[END_REF]; determining an analytical expression for pull-in time of actuator [START_REF] Gomez | Pull-in dynamics of overdamped microbeams[END_REF]; deriving physics-based models to predict the actuator static performance [START_REF] Preetham | A curved electrode electrostatic actuator designed for large displacement and force in an underwater environment[END_REF]; optimizing actuator design parmeters to achieve large displacements in aqueous media [START_REF] Hoelzle | A large displacement, high frequency, underwater microelectromechanical systems actuator[END_REF]; and identifying means to extend the displacement range of the actuator in liquid solutions [START_REF] Rollier | The stability and pull-in voltage of electrostatic parallel-plate actuators in liquid solutions[END_REF]. In this paper, we analyze the dynamics of a parallel plate actuator in a clamped-clamped configuration immersed in viscous dielectric media (Fig. 1). Here, we model the actuator as a continuous system by deriving a reduced-order model similar to the work presented in Ref. [START_REF] Nayfeh | Reduced-order models for mems applications[END_REF]. The model is solved by employ-ing the Galerkin method [START_REF] Meirovitch | Fundamentals of Vibrations[END_REF] and linear undamped mode shapes for a clamped-clamped beam [START_REF] Gonçalves | Numerical evaluation of high-order modes of vibration in uniform euler-bernoulli beams[END_REF]. The work presented here extends the reduced-order model framework used by Younis et al. [START_REF] Younis | A reduced-order model for electrically actuated microbeam-based mems[END_REF] by modeling actuator contact physics to describe the actuator dynamics beyond the pull-in instability. Our model incorporates the inertial loading effect and squeeze film damping by the media, nonlinear mid-plane stretching forces in the beam electrode of the actuator, and nonlinear contact force [START_REF] Vyasarayani | Modeling mems resonators past pull-in[END_REF] during the physical contact of the beam electrode with the stationary electrode (Fig. 1). The model is first validated using the data available in the existing literature [START_REF] Younis | A reduced-order model for electrically actuated microbeam-based mems[END_REF] and then the model is utilized to study the actuator dynamics over a broad range, three orders of magnitude of viscosity (relates to damping or Q-factor) and two orders of magnitude of relative permittivity of the media. We describe the actuator characteristics in terms of pull-in time, and actuator displacement for both DC and AC actuation voltages. The following sections describe the design, model, and results that describe the response of the actuator in different viscous dielectric media.

Preprint

Design

The actuator is designed to achieve 1-5 µm displacement range and operate in viscous dielectric media for low actuation voltages (<20 V). The actuator comprises of two main components: a beam electrode to deliver displacement strokes and a stationary electrode to drive the beam electrode in the forward direction (Fig. 1(a)). The beam electrode is a clamped-clamped structure that is compliant, while the stationary electrode is an immovable rigid member. The actuator motion can result in axial stretching of the beam electrode. The electrodes are coated with an insulator film for electrical insulation. The nominal actuator parameters and physical properties of media are listed in Table 1. Upon energizing the actuator by supplying voltage V , the beam electrode moves in-plane closer to the stationary electrode (Fig. 1(b,c)). The actuation voltage V is given by the function V (t) = V sin2πf t, where V is the voltage amplitude, f is the frequency of the actuation voltage, and t is the time. Note that for operating the actuator in ionic viscous media (water), a pure sinusoidal voltage function can result in electrolysis, anodization, and electrode polarization [START_REF] Sounart | Frequency-dependent electrostatic actuation in microfluidic mems[END_REF]. Hence, the actuator will be supplied with an amplitude modulated actuation signal given by the function

V (t) = 1 2 V (1 + cos2πf t) Π(t) where V is the voltage am- plitude, cos2πf t is the modulation waveform, t is the time, Π(t) = 4 π ∞ j=1 1 2j-1 (-1)
j-1 cos2πf c jt is a high-frequency, zero-mean pulse signal to prevent charge shielding effects [START_REF] Sounart | Frequency-dependent electrostatic actuation in microfluidic mems[END_REF], and f and f c =500 kHz are the modulation and carrier frequencies, respectively. However, for uniformity, in this work we are restricting the actuation voltage to be a pure sinusoidal function.

Model

The actuator shown in Fig. 1 is modeled as a continuous system by deriving a reduced-order model and solved by employing the Galerkin method and linear undamped mode shapes for a clamped-clamped beam. The model incorporates the inertial loading effect and squeeze film damping by the media, nonlinear mid-plane stretching forces in the beam electrode of the actuator, and nonlinear contact force during the physical contact of the beam electrode with the stationary electrode (Fig. 1). The actuator motion is described by Eq. 1a with boundary conditions given by Eqs. 1b and 1c,

EI ∂ 4 w ∂x 4 +(ρbh + 0.785 ρ f b 2 ) ∂ 2 w ∂t 2 + ĉ ∂w ∂t = EA 2l l 0 ∂w ∂x 2 dx + N ∂ 2 w ∂x 2 + 0 f bV 2 2(d -w) 2 -k c g 3 1 + μc ∂g ∂t H, (1a)
w(0, t) = w(l, t) = 0, (1b) ∂w ∂x (0, t) = ∂w ∂x (l, t) = 0, ( 1c
)
where w is the deflection of the beam electrode, x is the coordinate along the beam electrode, b is the width of the beam electrode, h is the thickness of the beam electrode, l is the length of the beam electrode, d is the gap between the beam electrode and the stationary electrode, A is the cross-sectional area of the beam electrode, E is the Young's modulus of the beam electrode material, I = bh 3 /12 is the area moment of inertia of the beam electrode, t is the time, ρ is the density of the beam electrode material, ρ f is the density of the media, 0 is the permittivity of vacuum, f is relative permittivity of the media, ĉ is the damping coefficient by the media per unit length of the beam electrode, N is the axial tensile force in the beam electrode per unit length of the beam electrode, V is the actuation voltage at time t, g = (d -w) is the gap between the beam electrode and stationary electrode at time t, H is the Heaviside step function in terms of g, and μc =0.5 s/m models the damping and k c =10 15 N/m 4 models nonlinear spring stiffness during the contact of the beam electrode with the stationary electrode, respectively. Note that damping term ĉ for each media is calculated using the squeeze film damping model presented in Ref. [START_REF] Shih | Characterization of electrostatic gap-closing actuator arrays in aqueous conditions[END_REF], and the numerical values of μc and k c are based on the work by Vyasarayani et al. [START_REF] Vyasarayani | Modeling mems resonators past pull-in[END_REF]. Also, note that the effect of insulator film on the actuator dynamics is not considered in this work.

For analysis, the model (Eqs. 1a-1c) is nondimensionalized by applying the following scales to the variables:

W = w d , X = x l , T = t t p , G = g d (2)
where t p = ρbhl 4 EI is the reference time period. After applying these scales, the model (Eqs. 1a-1c) becomes

∂ 4 W ∂X 4 + m ∂ 2 W ∂T 2 + c ∂W ∂T = [α 1 Γ(W, W ) + N ] ∂ 2 W ∂X 2 + α 2 V 2 (1 -W ) 2 -α 3 (1 -W ) 3 1 -µ c ∂W ∂T H, (3a)
W (0, T ) = W (1, T ) = 0, ( 3b
) ∂W ∂X (0, T ) = ∂W ∂X (1, T ) = 0, (3c)
where

m = 1 + 0.785 ρ f b ρh , c = ĉl 4 EIt p , α 1 = 6d 2 h 2 , Γ 1,2 = Γ(ψ 1 , ψ 2 ) = 1 0 ∂ψ 1 ∂X ∂ψ 2 ∂X dX, N = N l 2 EI , α 2 = 6 0 f l 4 Eh 3 d 3 , α 3 = k c d 2 L 4 EI , µ c = μc d t p . (4)
To characterize the actuator dynamics, the model (Eqs. 3a-3c) is solved numerically using MATLAB. For this, Eq. 3a is rearranged by multiplying it with (1 -W ) 2 . Then, the variable W (or W (X, T )) is expressed as a summation of product of temporal U i (T ) and spatial φ i (X) variables, that is,

W (X, T ) = 2M -1 i=1 U i (T )φ i (X).
Here, i is the mode number, φ i (X) is the i th mode shape of an undamped clamped-clamped beam with negligible axial loads ( N =0) [START_REF] Gonçalves | Numerical evaluation of high-order modes of vibration in uniform euler-bernoulli beams[END_REF], and M is the total number of mode shapes and 2M -1 is the highest mode shape used to represent the beam electrode displacement, respectively. Note that only symmetric mode shapes (i ∈ {1, 3, ...2M -1}) are used in this analysis. The model is simplified by invoking the normal mode shape property 1 0 φ i φ j dX = δ ij [START_REF] Meirovitch | Fundamentals of Vibrations[END_REF], where δ ij is the Kronecker delta function. The simplified model equation becomes

m Ün -2 2M -1 i,j=1 Üi U j 1 0 φ i φ j φ n dX + 2M -1 i,j,k=1 Üi U j U k 1 0 φ i φ j φ k φ n dX + c Un + ω 2 n U n = α 1 2M -1 i,j,k=1 U i U j U k Γ i,j 1 0 φ k φ n dX + α 1 2M -1 i,j,k,l,m=1 U i U j U k U l U m Γ i,j 1 0 φ k φ l φ m φ n dX -2α 1 2M -1 i,j,k,l=1 U i U j U k U l Γ i,j 1 0 φ k φ l φ n dX + α 2 V 2 1 0 φ n dX -α 3 H 1 0 φ n 1 - 2M -1 i=1 U i φ i 3 1 -µ c 2M -1 i=1 Ui φ i dX + 2 2M -1 i,j=1 ω 2 i U i U j 1 0 φ i φ j φ n dX - 2M -1 i,j,k=1 ω 2 i U i U j U k 1 0 φ i φ j φ k φ n dX + N 2M -1 i=1 U i 1 0 φ i φ n dX + N 2M -1 i,j,k=1 U i U j U k 1 0 φ i φ j φ k φ n dX -2N 2M -1 i,j=1 U i U j 1 0 φ i φ j φ n dX + 2c 2M -1 i,j=1 Ui U j 1 0 φ i φ j φ n dX -c 2M -1 i,j,k=1 Ui U j U k 1 0 φ i φ j φ k φ n dX (5)
where n ∈ {1, 3, 5, ...2M -1}, and overdot ' ˙' and prime ' ' correspond to temporal and spatial derivatives, respectively. Eq. 5 represents a system of M nonlinear ordinary differential equations; the system of equations is solved numerically to obtain the numerical values of the temporal variable: U 1 , U 3 , U 5 , ..., and U 2M -1 .

For static analysis, the time derivative terms in Eq. 5 are dropped, yielding a system of M polynomial equations. The polynomial equation set is solved for U 's comprising of U 1 , U 3 , ...U 2M -1 for a given DC actuation voltage V using the built-in MATLAB function fsolve. For each DC actuation voltage, a U i has two possible solutions -one results in a stable configuration and other in a unstable configuration of the beam electrode. The corresponding actuator displacement amplitudes W (1/2) can be computed using the aforementioned equation

W (X = 1/2, T ) = 2M -1 i=1
U i (T )φ i (X = 1/2). For dynamic analysis, Eq. 5 is solved for U 's comprising of U 1 , U 3 , ...U 2M -1 for a given actuation voltage V using the built-in MATLAB function ode15s for a given set of initial conditions: nondimensional actuator displacement W (X, 0) = 0 and nondimensional actuator velocity Ẇ (X, 0) = 0. The actuator displacement W (X, T ) at nondimensional time T , can be computed using the aforementioned equation

W (X, T ) = 2M -1 i=1 U i (T )φ i (X).
Water (V pi =1.52 V) _ Methanol (V pi =2.4 V) _ Air (V pi =13.6 V) _

Results and Discussion

The actuator shown in Fig. 1 with water as viscous media and parameters given in Table 1 is treated as the reference case. The actuator model (Eq. 5) is numerically solved to analyze the static and dynamic behavior of the actuator characterized in terms of pull-in time and displacement. Based on our preliminary work, we found that the first five symmetric modes (2M -1 ∈ {1, 3, 5, 7, 9}) of the beam electrode is acceptable to study the mechanics of the actuator (see Appendix). Hence for the subsequent analysis, we chose the total number of mode shapes M = 5.

Static behavior

The response of the actuator to a DC actuation voltage V in three viscous media is characterized through static pull-in analysis (Fig. 2). For all the three media, the static equilibrium curves plotted with normalized DC actuation voltages ( V / Vpi ) overlap with one another, highlighting the absence of viscous effects (Fig. 2a). Here, the DC actuation voltages ( V ) for a given media are normalized with corresponding pull-in voltages ( Vpi ). For actuation voltages below the static pull-in, that is along the lower branch, the actuator operates in a stable manner with a maximum displacement sensitivity of 0.46, 1.51, and 6.44 µm/V in air, methanol, and water media, respectively. However, beyond the pull-in voltage, the actuator quickly snaps into the stationary electrode due to the dominance of the nonlinear electrostatic force. For all the three media, the nondimensional actuator position W (1/2) at the onset of the pull-in instability is 0.46.

Dynamic behavior

The dynamic response of the actuator in three viscous media is characterized in terms of nondimensional pull-in times T pi and nondimensional actuator amplitude W (1/2). Here, the inertial, viscous, and contact force effects are considered as they are known to affect the dynamic response of the actuator [START_REF] Elata | On the static and dynamic response of electrostatic actuators[END_REF].

The effect of media properties on the actuator pullin time is determined by supplying the actuator with a step voltage input V , where the step voltage amplitude V is chosen to hold the relation V > Vpi (Figs. 3 and4). Note that the amplitudes of the step voltage input V for the media are chosen such that the generated electrostatic forces are equal for W = 0. The high relative permittivity of water ( f,w =80) amplifies the generated force, resulting in pull-in to occur at lower actuation voltage ( V =2 V) compared to air ( V =18 V); while, the high dynamic viscosity of water (µ f,w =8.9×10 -4 Pa-s) increases the pull-in time (Fig. 3a). During the initial phase of the pull-in, an actuator will have higher speed in water compared to air; however, during the later phase, the actuator will have a lower speed. This can be explained as follows: the initial higher speed is due to the higher electrostatic force -due to higher relative permittivity f -and the later lower speed is due to the higher viscous friction -due to higher dynamic viscosity µ f (Fig. 3 b-d). Moreover, as the actuator accelerates from W = 0 position, it gains speed, and this increase in speed exacerbates the viscous damping effect, resulting in lower speeds during the later phase of the pull-in. A plot of beam electrode deflection profiles at different instances in time show that after the beam electrode pulls into the stationary electrode (T = T pi ), the beam electrode zips into the stationary electrode (T = 3T pi /2), increasing the length of contact.

The actuator response near the vicinity of the pullin instability is sensitive to the actuation voltage V . For instance, the actuator in water media with a pullin voltage amplitude V =1.59 V exhibits a wide range of pull-in times -a 25% increase in actuation voltage amplitude V (from 1.60 to 2.00 V) can lower the pullin time to a third (Fig. 4a). Fig. 4b highlights the effect of viscous damping by the media on the actuator. For a fixed f and fixed actuation voltage amplitude V , an increase in viscous damping modeled by nondimensional damping term c delays the pullin time T by up to 16 times. Similarly, for a fixed actuation voltage amplitude V and damping term c, an increase in relative permittivity of the media f can cause the pull-in instability to occur and reduce the pull-in time T by up to 3 times (Fig. 4c).

The actuator displacement traces W (1/2) for different sinusoidal voltage waveforms V in water media captures the nonlinearities in the actuator (Figs. 5 and6). The actuator dynamics can be approximated to linear for lower actuation voltage amplitudes V < 1.5 V and/or high actuation frequencies f > 200 Hz. Applying a sinusoidal voltage waveform with frequency f = 100 Hz and higher voltage amplitudes ( V > 1.5 V) excites the higher harmonics f n (∀ n ∈ Z >0 ) in the actuator, as computed by applying the discrete Fourier transform (DFT) to the simulated displacement trace W (1/2) (Fig. 5). Note that the n th order harmonic frequency f n = nf and the nondimensional DFT amplitude

a n = 2 P | P -1 p=0 W (1/2) |p e -j 2πnp P | max W (1/2) |f . (6)
As can be seen in Eq. 6, a n is scaled with the actuator amplitude, max W (1/2) for voltage frequency f Hz.

Here, j = √ -1 and P is the number of sample points used in displacement measurement W (1/2). To prevent aliasing, a minimum of 1125 sample points were chosen for the computation. The presence of higher harmonics resulted in a skewed sinusoidal response instead of a pure sinusoidal response -an indication of a nonlinear system. On the contrary, for the lower voltage amplitudes ( V < 1.5 V), the higher harmonics f n (∀ n ∈ N) vanish with the appearance of a very low amplitude (< 0.2a 1 ) for second harmonic response f 2 -the actuator can be approximated to be a linear system. For high V and high f , the actuator may pull into the stationary electrode which may result in not releasing the beam electrode; an example scenario is for V =1.6 V and f =100 Hz (Fig. 6). For very high f operation (f > 500 Hz), the beam electrode does not pull-in to the stationary electrode, and a significant phase lag (θ = 39 • ) and displacement drift (∆=0.092 W ) over time is observed.

Conclusion

In this paper, we analyzed the actuator response in a clamped-clamped configuration immersed in viscous dielectric media. We modeled the actuator as a continuous system by deriving a reduced-order model, and solving it by employing the Galerkin method and linear undamped mode shapes for a clamped-clamped beam. The model is utilized to study the actuator dynamics over a broad range, three orders of magnitude of viscosity and two orders of magnitude of relative permittivity of the media. We make three important observations: (1) the actuator at lower actuation voltage amplitudes V < 1.5 V and/or higher actuation f > 200 Hz can be approximated to be a linear system; (2) the high relative permittivity of a media amplifies the generated force, resulting in pull-in to occur at lower actuation voltage amplitudes, and the high dynamic viscosity of the media increases the pull-in time; and (3) at very high actuation frequencies (f > 500 Hz), the beam electrode does not pull-in into the stationary electrode, and the actuator experiences a significant phase lag (θ = 39 • ) and displacement drift (∆=0.092 W ) over time.

Appendix

The model (Eq. 5) is validated for the static and dynamic performance of the actuator by benchmarking against the data presented in Ref. [START_REF] Younis | A reduced-order model for electrically actuated microbeam-based mems[END_REF]. The actuator shown in Fig. 1 with parameters given in Ref. [START_REF] Younis | A reduced-order model for electrically actuated microbeam-based mems[END_REF], that is, l = 610 µm, b = 40 µm, h = 2.1 µm, d = 2.3 µm, c=260, f =1, and N = -3.7 µN is treated as the reference case. Three different cases where the total number of mode shapes M ∈ {3, 4, 5} are studied and compared with data obtained from Ref. [START_REF] Younis | A reduced-order model for electrically actuated microbeam-based mems[END_REF] (Fig. 7). Our study shows that employing the first five symmetric modes (φ 1 , φ 3 , φ 5 , φ 7 , φ 9 ) in the model (Eq. 5) will be sufficient to predict actuator displacement and pull-in time (Fig. 7b). A similar study has been conducted by Ref. [START_REF] Vyasarayani | Modeling mems resonators past pull-in[END_REF] and showed that using the first five symmetric modes will result in only 10% error in contact length predictions of the beam electrode with the stationary electrode.

Figure 1 :

 1 Figure 1: Schematic sketch of the parallel plate electrostatic actuator. (a) Unenergized state. (b) Energized state. (c) A side view of the actuator in the energized stage with parameters shown.

  Physical parameter (units) Value Length of the beam electrode, l (µm) 2000 Thickness of the beam electrode, h (µm) 5 Width of the beam electrode, b (µm) 45 Gap between the electrodes, d (µm) 5 Density of the electrode material, ρ (kg/m 3 ) 2329 Young's Modulus, E (GPa) 130 Axial tensile force/unit length, N (N/m) 0 Relative permittivity of air, f,a 1 Relative permittivity of methanol, f,m 32 Relative permittivity of water, f,w 80 Dynamic viscosity of air, µ a (µPa-s) 18.1 Dynamic viscosity of methanol, µ m (µPa-s) 700 Dynamic viscosity of water, µ w (µPa-s) 890 Nondimensional damping term c for air, c a 10 Nondimensional damping term c for methanol, c m 120 Nondimensional damping term c for water, c w 200 Density of air, ρ f,a (kg/m 3 ) 1.2 Density of methanol, ρ f,m (kg/m 3 ) 790 Density of water, ρ f,w (kg/m 3 ) 1000

Figure 2 :Figure 3 :

 23 Figure 2: Static equilibrium curves for the actuator in air, methanol, and water media. Curves plotted for all the media against (a) normalized DC actuation voltage V / Vpi with nondimensional actuator displacement amplitude W (1/2) (b) DC actuation voltage V with dimensional actuator displacement amplitude w(l/2).

Figure 4 :

 4 Figure 4: The effect of media properties on the actuator pull-in dynamics. (a) In water media for different actuation voltage amplitudes V and nondimensional damping term c= 200. (b) For actuation voltage amplitude V =2 V, relative permittivity f,w =80 and different nondimensional damping terms c. (c) For actuation voltage amplitude V =2 V, nondimensional damping term c=200, and different relative permittivities f .

Figure 5 :

 5 Figure 5: Actuator dynamic response to a sinusoidal voltage input V in water media. (a) Displacement trace of the actuator W (1/2) for different actuation voltage amplitudes V and voltage frequency f =100 Hz. Note that the time period Tp refers to the voltage waveform, where Tp = 1/(tpf ). (b) A DFT of the actuator displacement trace W (1/2) simulated for different actuation voltage amplitudes V and voltage frequency f =100 Hz. The nondimensional DFT amplitude an is scaled with the actuator amplitude W (1/2) at the first harmonic frequency f 1 =100 Hz.

Figure 6 :

 6 Figure 6: Actuator dynamic response to a sinusoidal voltage input V in water media. (a) Displacement trace of the actuator W (1/2) for different actuation voltage frequencies f and voltage amplitude V =1.6 V. Note that the time period Tp refers to the voltage waveform, where Tp = 1/(tpf ). (b) A DFT of the actuator displacement trace W (1/2) simulated for different actuation voltage frequencies f and voltage amplitude V =1.6 V. The nondimensional DFT amplitude an is scaled with the actuator amplitude W (1/2) at the first harmonic frequency f 1 .

Figure 7 :

 7 Figure 7: Our model validation. (a) Static equilibrium curves for the actuator computed for M =3 (solid), M =4 (dashed), M =5 (dotted) modes and compared with results from Ref. [9] (Younis 2003, circles). Note the Young's modulus E of the beam used in this static analysis is 149 GPa. (b) Dynamic response of the actuator for a step input voltage amplitudeV computed for M =3 (solid), M =4 (dashed), M =5 (dotted) modes and compared with results from Ref.[START_REF] Younis | A reduced-order model for electrically actuated microbeam-based mems[END_REF] (Younis 2003, circles). Note the Young's modulus E of the beam used in this dynamic analysis is 166 GPa.

Table 1 :

 1 Actuator parameters and physical properties.

Acknowledgment

This work was supported in part by Wichita State University capitalization funds.


Sindhu Preetham Burugupally

Mechanics of a curved electrode actuator operating in viscous dielectric media: simulation and experiment

Keywords: Curved electrode electrostatic actuator, Finite element method, Frequency response, Viscous dielectric media, Microelectromechanical systems (MEMS)

Curved electrode electrostatic actuators have been shown to have the potential to manipulate microparticles at high speeds in viscous dielectric media. However, there are no multiphysics models that fully describe the mechanics of these actuators in viscous dielectric media. This work is the first step in the direction of describing the actuator mechanics through Finite Element Method (FEM) simulations and validate it with the experiments. Simulations are performed using the commercial software Coventor MEMS+ R and the results are found to be in good agreement with experiments. Through a simple lumped-parameter model of the actuator, we report that the viscous damping effect by the substrate on the actuator submerged in methanol or water media becomes significant for actuation frequencies >100 Hz.

Introduction

Electrostatic actuators are the physical systems comprising of mechanical and electrical components that generate mechanical motion at micron scale O (10 -6 ) m. These actuators have the potential to physically manipulate microparticles suspended in a viscous media [1,2,[START_REF] Shen | Eppell[END_REF]4]. To manipulate microparticles at a fast rate of O (10 2 ) particles/s, there is a need for a high speed actuator that can oscillate at a frequency O (10 2 ) Hz and achieve a displacement on the order of few microns in the viscous microenvironment. The most common electrostatic actuator configurations such as parallel plate [2] and comb-type [5] have shortcomings -the requirement of large actuation voltages in comb drives [6] and the displacement limitation in parallel plates [START_REF] Senturia | Microsystem Design[END_REF]. To overcome these shortcomings, a curved electrode electrostatic actuator in a clamped-clamped beam configuration was designed that operates in a stable manner above the pull-in voltage and achieves large displacement at low actuation voltages (Fig. 1) [START_REF] Burugupally | [END_REF]. The curved electrode actuators have been experimentally shown to have the potential to manipulate microparticles at high speeds in viscous dielectric media [9]. However, there are no multiphysics models that fully describe the mechanics of these actuators in the viscous dielectric media. An in-depth understanding 1 for the more details.

of the actuator mechanics in the viscous dielectric media is very beneficial, especially for designing actuators meant for biological applications such as cell probing and sorting [2,[START_REF] Burugupally | [END_REF]10].

In this work, we analyze the mechanics of the curved electrode actuator in viscous dielectric media for the nonzipped operation mode, where the beam electrode does not come in physical contact with the curved electrode. Here, the actuator performance is analyzed in terms of displacement and frequency range through Finite Element Method (FEM) simulations and later validated with experiments. FEM simulations are performed using Coventor MEMS+ R and found to be in good agreement with the experiments. In addition, a simple lumped-parameter model of the actuator is derived to analyze the effect of viscous damping by the substrate on the actuator. This work is the first step in understanding the effect of viscous media on actuator performance characterized in terms of the cutoff frequency. The following sections describe the actuator operation, model and experiment setup, and results that describe the actuator behavior in viscous environments where the quality factor (Q) is at least two orders of magnitude lower than in air.

Materials and Methods

The curved electrode actuator shown in Figs. 1 and 2 is fabricated on a 100 mm diameter silicon-on-glass (SOG) wafer (purchased from PlanOptik, Elsoff, Germany) made by anodic bonding of a silicon wafer (<100> p-type boron doped to 100 ppm) to a 500 µm thick Borofloat33 glass wafer and then thinning and polishing the silicon wafer to a thickness of 45 µm. The actuator components are micromachined from the silicon layer with parameters given in Table 1, while the glass layer acts as a handling substrate and provides optical access for viewing with an inverted microscope. The actuator fabrication details are presented in the Appendix section (Fig. A1).

The actuator is characterized in terms of actuator displacement amplitude w(L) at different voltage amplitudes V ∈ [1,4] V in the increments of 1 V and actuation frequencies f ∈ [0, 4096] Hz using the test setup shown in Fig. 3. For the static tests, the actuator is supplied with a high frequency (500 kHz)

(t) = 1 2 V (1 + cos 2πf t)Π(t) and V B (t) = 1 2 V (1 -cos 2πf t)Π(t)
, where cos 2πf t and -cos 2πf t are the modulation waveforms, t is the time, Π(t) = 4 π ∞ j=1 1 2j-1 (-1) j-1 cos 2πf c jt, and f and f c (500 kHz) are the modulation (also called actuation) and carrier frequencies, respectively (Fig. 1). Note that the voltages V A and V B are 180 • out-of-phase to drive the actuator in forward and backward directions. For given actuation voltages V A and V B , the cinephotomicrography was recorded at frame rates greater than 6 times the corresponding actuation frequency f so as to prevent aliasing. A custom image processing MATLAB R script is used to create time traces of the actuator displacement for follow-up analyses of frequency content and peak-to-peak displacement amplitude 2w(L).

The 3-dimensional Finite Element Method (FEM) model of the actuator is built and analyzed using the commercial software Coventor MEMS+ R comprising of mechanical and electrical submodels (Fig. 2b). Here, the beam electrode, curved electrode, and reinforcing beam are individually constructed from 120 Bernoulli beam elements. Each beam element has 6 nodes, where each node has one degree of freedom that corresponds to the node displacement in the x -y plane with direction perpendicular to the beam element length. To determine the appropriate number of beam elements, a convergence test is performed on the actuator model; here, first an FEM model of the actuator with a different number of beam elements ranging from 20 to 200 is constructed, then a fixed DC voltage (4 V) is applied to one set of the curved electrodes located on one side of the beam electrode, and corresponding static displacements w(L) of the actuator are obtained (Fig. A2). Based on the convergence test (Fig. A2), in this work, we chose 120 beam elements to construct the individual actuator components. The model does not include the viscous fluid damping by substrate. The inertial loading effect by the viscous media trapped between the reinforcing beams is incorporated into the model -as a lumped mass m l made of silicon -located at the midpoint of the reinforcing beam (Fig. 2b). To prevent electrical shorting between the beam electrode and curved electrode, a 20 nm thick Al 2 O 3 insulator film is added to the curved electrodes. To reduce the computation burden and assist in the solution convergence, the electrostatic and fluidic (squeeze film damping) forces are linearized; here, the linearization value is set to 0.25 µm. The source code for simulating the actuator mechanics is made available on GitHub [START_REF] Burugupally | Curved electrode electrostatic actuator simulation in DI water and methanol media[END_REF].

A lumped-parameter model of the actuator operating in the viscous dielectric media is derived using the Newton's second of motion (Fig. 2c). The actuator is idealized as a spring-mass-damper system with a single degree-of-freedom. The frequency response of the actuator is given by the transfer function T [15]

T = F e (k e -4m e π 2 f 2 ) 2 + (2πf b e ) 2 (1)
where k e is the overall stiffness, m e is the effective mass [START_REF] Rollier | [END_REF], b e is the overall damping coefficient [START_REF] Shih | [END_REF], and F e is the amplitude of the electrostatic force acting on the actuator. The mathematical expressions for these terms are given by

m e = 2ρ f h 0.398dL 1 + π 2 h (L 1 + L) +0.796ρh (tL + 2t 1 L 1 ) (2a) b e = 4µ f h 3 L (g + s(x) -w(x)) 3 + 4µ f h 3 L 1 (g 1 -w(x)) 3 Γ (2b) k e = 2Eht 3 L 3 + 4Eht 3 1 L 3 1 ( 2c
)

F e = 0 f Lh V 2 (g + s(x) -w(x)) 2 (2d)
where µ f is the dynamic viscosity of the media, w(x) is the displacement of the beam electrode or reinforcing beam at any location x along the beam length, f is the relative permittivity of the media, and ρ f and ρ are the densities of the media and silicon, respectively. Note the first parenthesis term of the effective mass m e (Eq. 2a) corresponds to the inertial effect of the media, while the second parenthesis term corresponds to the effective mass of the beam electrode and reinforcing beam. The square parenthesis term of the damping coefficient b e models the squeeze film damping by the media on the actuator (Eq. 2b).

The proximity of the actuator to the glass substrate results in the additional damping on the actuator, and is modeled by Γ, where Γ = [START_REF] Shih | [END_REF]. Here, g o is the clearance between the actuator and glass substrate. In the nonzipped mode of operation, the actuator displacement is small (<5 µm) and hence the overall damping coefficient b e and electrostatic force F e are linearized. The numerical values of the media properties are given in Table 2.

4g 3 t+2g 3 o h g 3 t+2g 3 o h

Results and discussion

Through FEM model and experiment, we analyze the mechanics of the actuator in the nonzipped operation. The nonzipped operating mode occurs for voltages below the pull-in voltage, and found to be around 4 V for deionized water media [START_REF] Preetham | Napa Microsystems Workshop[END_REF]. The actuator displacement w(L) increases with actuation voltage amplitude V (Fig. 4). The actuator achieves displacement w(L) on the order of few microns at low actuation voltage with high displacement sensitivity (∼0.28 µm/V). The static displacement of the actuator in deionized water media for actuation voltages V ∈ [1,4] V predicted using the FEM model is in good agreement with the experiment data collected from three actuators (Fig. 4). Similarly, the dynamic response of the actuator obtained using the simulation for deionized water media at low actuation voltage ( V =4 V) and low actuation frequency (f =1 Hz) agrees well with the experiment (Fig. 5).

The FEM simulation results are compared with the experimentally obtained frequency response plots in the nonzipped mode for deionized water and methanol media (Fig. 6). Since deionized water has higher relative permittivity than methanol (Table 1), the actuator peak-to-peak displacements in deionized Here, the actuator operates in the nonzipped mode where the peak-to-peak displacement amplitudes are less than 5 µm. Inset shows the frequency response of the actuator with (w/ substrate) and without (w/o substrate) considering the viscous damping effect by the substrate. The displacement measurement has a quantization error of ± 1/2 pixel = ± 0.15 µm.

water are higher for a given voltage amplitude V due to the stronger electrostatic force. The measured and simulated frequency responses of the actuator for both media show similar trends-the actuator displacements begin to roll-off beyond 300 Hz actuation frequency. For actuation frequencies higher than 100 Hz, the actuator displacement values obtained from the FEM model begin to deviate from the experiment. This is due to the increase in difference between the actual and simulated damping force on the actuator. As earlier mentioned, the viscous fluid damping on the actuator by the substrate is neglected in the FEM model, while in reality (experiment), the actuator experiences this viscous damping force. Hence, the FEM model gravitates towards over predicting the actuator displacement and frequency bandwidth (Fig. 6). This inference has been verified by modeling the actuator as a lumped model whose transfer function is given by Eq. 1. As shown in Fig. 6, the actuator modeled without the viscous damping by the substrate (w/o substrate) has slightly higher displacement compared to the actuator modeled with the viscous damping by the substrate (w/ substrate). In contrast, for actuation frequency f < 100 Hz, the effect of viscous damping by the substrate on the actuator is negligible.

A proper choice of media characterized in terms of dielectric constant (relative permittivity) and dynamic viscosity is essential for the optimal performance Figure 8: Simulated frequency response of the actuator in different media using the FEM model. Here, the actuation voltages are chosen such that the peakto-peak actuator displacements at 1 Hz are equal (3.7 µm). The hypothetical fluid has a relative permittivity of the air and dynamic viscosity of the silicone oil. This presents a scenario where the media is a high viscous fluid with low permittivity. See Table 2 for fluid properties.

of the actuator. Different media namely, silicone oil, water, methanol, and hy-pothetical fluid are investigated to understand the effect of the media properties on actuator performance (Figs. 7 and8, Table 2). For a given media and low actuator displacements, the cutoff frequency of the actuator is found to be independent of the actuation voltage and/or actuator displacement amplitude (Fig. 7). For instance, the cutoff frequency in water is 1000 Hz for voltages V =2, 3, and 4 V. This implies the low actuator displacements (< 5 µm) will not result in invoking the nonlinear effect of the squeeze film damping that can affect the actuator's cutoff frequency. An actuator immersed in a high viscous media will have higher squeeze film damping and a lower cutoff frequency. For instance, the cutoff frequency f c =1000 Hz reduces to f c =300 Hz when the media is changed from water to silicone oil (Fig. 8).

Conclusions

Currently, there are no multiphysics models that fully describe the mechanics of the curved electrode actuators submerged in the viscous dielectric media. In this paper, we developed an FEM model of the actuator using a commercial simulation software Coventor MEMS+ R , and validated the model using experiments for the nonzipped operating mode. The simulation results are found to be in good agreement with experiments. Two observations are reported in this study: (1) for a given media and low actuator displacements, the cutoff frequency of the actuator is independent of the actuation voltage and/or actuator displacement; and (2) an actuator immersed in a high viscous media will have higher squeeze film damping and a lower cutoff frequency. Through a simple lumped-parameter model of the actuator, we report that the viscous damping effect by the substrate on the actuator submerged in methanol or water media becomes significant for actuation frequencies >100 Hz. then in methanol, and then dried at room temperature (f). Next, the actuator is coated with 10 nm thick Al 2 O 3 insulator film using atomic layer deposition (ALD) process (processed at 120 • C using TMA/O 2 precursors) (g). Finally, the 10 nm thick Al 2 O 3 insulator film is anisotropically etched using inductively coupled plasma assisted reactive ion etching process (RIE-ICP; using BCl 3 /Cl 2 gases) to unmask the Au/Cr electrical contact pads that were coated with Al 2 O 3 during the ALD process (h).

1 Figure 1 :

 11 Figure 1: A stereomicroscope image of the actuator with actuation voltage waveforms (V A , V B ) and displacement signal (w(L)) shown. The inset shows a portion of the actuator with geometric parameters. The actuator comprises of a beam electrode, set of curved electrodes, and a reinforcing beam. The beam electrode and reinforcing beam together form the compliant clamped-clamped structure. The curved electrodes are immovable rigid members whose contour s(x) = δ m (x/L) 2 . See Table1for the more details.

Table 2 :Relative permittivity of hypothetical fluid 1

 21 Nominal viscous dielectric media properties. Density of deionized water (kg/m 3 ) 1000 Density of methanol (kg/m 3 ) 800 Density of silicone oil (kg/m 3 ) 900 Density of hypothetical fluid (kg/m 3 ) 900 Dynamic viscosity of deionized water (Pa-s) 0.00089 Dynamic viscosity of methanol (Pa-s) 0.00054 Dynamic viscosity of silicone oil (Pa-s) 0.00459 Dynamic viscosity of hypothetical fluid (Pa-s) 0.00459 square wave voltage signal at different voltage amplitudes (V A or V B = V ) to one set of the curved electrodes located on one side of the beam electrode. For the dynamic tests, the actuator is supplied with two amplitude-modulated voltage signals V A

Figure 2 :

 2 Figure 2: Curved electrode actuator. (a) A scanning electron microscope (SEM) image of the actuator. (b) A 3-dimensional FEM model of the actuator built in Coventor MEMS+ R software. (c) A schematic sketch of the actuator and its equivalent lumped model.

Figure 3 :

 3 Figure 3: (a) Actuator evaluation experiment setup. (b) A photomicrograph of the actuator submerged in viscous media (deionized water) captured using high-speed camera and inverted microscope. The test setup comprises of an inverted microscope (Zeiss Observer A1 microscope, Germany) and high-speed camera (Miro M110, Vision Research, USA) for photomicrography of the moving actuator in the region of interest (ROI), and LabVIEW/NI PCIe-6343 data acquisition system and function generator (Tektronix AFG3022C, Beaverton, OR, USA) for triggering the high-speed camera to initiate cinephotomicrography and generating two amplitude-modulated voltage waveforms V A and V B .

Figure 4 :

 4 Figure 4: Comparison of actuator static displacement amplitude w(L) in deionized water media for different actuation voltage amplitudes V obtained using the FEM model and experiment. The displacement measurement has a quantization error of ± 1/2 pixel = ± 0.15 µm.

Figure 5 :Figure 6 :

 56 Figure 5: Measured and simulated displacement-time trace of the actuator in deionized water media over five successive cycles for V =4 V and f =1 Hz. The displacement measurement has a quantization error of ± 1/2 pixel = ± 0.15 µm.

Figure 7 :

 7 Figure 7: Simulated frequency response of the actuator in deionized water and methanol for different voltages V ∈ {2, 3, 4} using the FEM model.

Figure

  Figure A1: Fabrication scheme of the actuator. [Figure adapted from Ref. [8] with permission from Springer Nature Microsyst. Technol. 24, 3479 (2018). Copyright 2018 Springer-Verlag GmbH Germany, part of Springer Nature.]

  2

Table 1 :

 1 Nominal and measured actuator parameters and physical parameters. Measured parameters given as mean ± standard deviation for four measurements.

	Parameter	Numerical value
	Beam electrode half-length, L	1000 µm
	Reinforcing beam half-length, L 1	670 µm
	Beam electrode thickness, t	4.7 ± 0.2 µm
	Reinforcing beam thickness, t 1	3.8 ± 0.2 µm
	Electrode gap, g	3.5 ± 0.1 µm
	Substrate clearance, g o	3 µm
	Minimum beam gap, g 1	30 µm
	Reinforcing beam spacing, d	17 µm
	Beam electrode height, h	45 ± 0.4 µm
	Throw of the curved electrode, δ m	10 µm
	Overall actuator stiffness, k e	10 N/m
	Young's modulus of Si [11], E	130 GPa
	Relative permittivity of Al 2 O 3	9

Acknowledgements

The author would like to thank David Hoelzle at The Ohio State University for granting access to research facilities to conduct experiments shown in this work. Also, the author would like to that Coventor MEMS+ R for providing the software application. This work was supported in part by University of Notre Dame Walther Cancer Foundation IITP Fellowship, University Research/Creative Projects award at Wichita State, and Wichita State and Notre Dame capitalization funds. Technical consultation was given by the Notre Dame Nanofabrication Facility.

Appendix

Actuator Fabrication Recipe: On the SOG wafer, a composite metal film comprising of 30 nm Cr and 200 nm Au is deposited on the silicon side by ebeam evaporation and patterned by photolithography to make electrical contact pads (a, b). Next, an etch mask to be used for bulk micromachining of actuator components is defined by depositing a 1000 nm thick SiO 2 layer using a plasma enhanced chemical vapor deposition (PECVD) process (processed at 250 • C using SiH 4 /N 2 O gases), patterned using a photoresist mask, and then etched using inductively coupled plasma assisted reactive ion etching process (RIE-ICP; using CHF 3 /Ar gases) (c, d). The high-aspect ratio actuator features (curved electrodes, beam electrode, reinforcing beam) are machined by DRIE using a standard Bosch process in 5s:2s step cycles; the glass substrate acts as an etch stop (e). After DRIE, the movable structures of the actuator are released by timed HF (49% HF:H 2 O::1:1 volume ratio) wet etch process for approximately 2 minutes at standard room temperature, triple rinsed in water,