Mahmoud Elhorany
email: mahmoudelhorany86@gmail.com.

Charlotte Rosso

Nader-Antoine Sourour

El-Sayed Ali

Wael Ahmed Fadel

Sonia Alamowitch

Frédéric Clarençon

Eimad Shotar

Stéphanie Lenck

Kévin Premat

Flore Baronnet-Chauvet

Sophie Crozier

Laure Bottin

Céline Corcy

Ossama Yassin Mansour

Atika Talbi

Yves Samson

Safety and Effectiveness of Mechanical Thrombectomy for Primary Isolated Distal 1 Vessel Occlusions: a Monocentric Retrospective Comparative Study 2

Keywords: Distal occlusion, middle cerebral artery, M2, M3, anterior cerebral artery occlusion, posterior 24 cerebral artery occlusion, acute ischemic stroke, mechanical thrombectomy, reperfusion, 25 Distal occlusion, middle cerebral artery, M2, M3, anterior cerebral artery occlusion, posterior 8 cerebral artery occlusion, acute ischemic stroke, mechanical thrombectomy, reperfusion, 9 Distal vessel occlusions, AIS: acute ischemic stroke, MT: mechanical thrombectomy, LVO: 50 large vessel occlusion, BMT: best medical treatment, IVT: intravenous thrombolysis, mTICI: modified 51 thrombolysis in cerebral infarction scale, AOL: Arterial occlusion lesion scale, mRS: modified Rankin 52 scale, ICH: intracranial hemorrhage, NIHSS

Distal vessel occlusions ischemic stroke management.

Tables and figures:

Tables: 2, figures: 1 11 References: 24 12 Word count: 3999

Results

37

Of 650 patients screened, 44 patients with distal vessel occlusions treated by MT+BMT were 38 selected and compared with 36 patients who received BMT alone. After PSM, 28 patients in 39 each group were matched without significant difference. Good clinical outcome defined as 40 mRS≤2 was achieved by 53.6% of the MT+BMT group and 57% of the BMT group (OR, 0.87; 41 95%CI, 0.3-2.4; P=1.00). The mortality rate was comparable in both groups (7% vs. 10.7% in 42 MT+BMT and BMT patients, respectively; OR=0.64; 95%CI, 0.1-4; P=1.00). Symptomatic The risk-benefit ratio of MT in distal vessel occlusions is still uncertain. On one hand, 65 even if AIS related to a distal occlusion may be considered less severe than a proximal 66 occlusion, the clinical outcome may not be favorable in these patients with significant 67 disability hampering the quality of life. According to the literature, the rate of good clinical 68 outcome at 3 months after a distal intracranial occlusion treated medically ranges from 20% 69 to 77%. [START_REF] Campbell | Endovascular therapy for ischemic stroke with perfusion-imaging 389 selection[END_REF][START_REF] Berkhemer | Stent-Retriever Thrombectomy for Stroke[END_REF][START_REF] Ospel | Clinical Course of Acute Ischemic Stroke Due to Medium Vessel Occlusion 393 With and Without Intravenous Alteplase Treatment[END_REF] However, our results were more favorable than those showed by the metanalysis from 287 the HERMES collaborators (mRS score≤1 in 27%, and mRS score≤2 in 46%). [START_REF] Menon | Efficacy of endovascular thrombectomy in patients with M2 segment 411 middle cerebral artery occlusions: meta-analysis of data from the HERMES Collaboration[END_REF] This may be 288 explained by the differences and non-homogenous cut off point to define distal vessel 289 occlusions; as M2 occlusion was considered to start just after the origin of the anterior 290 temporal artery, bringing no evident landmark between distal M1 segment and proximal M2 291 segment. [START_REF] Menon | Efficacy of endovascular thrombectomy in patients with M2 segment 411 middle cerebral artery occlusions: meta-analysis of data from the HERMES Collaboration[END_REF] 292 293 It is noteworthy that the distal occlusions have not been defined homogenously across 294 all the previous results. Regarding our study, we have defined distal MCA occlusions as 295 occlusions located distally to the insula's mid-height of the M2 segment according to Fischer's 296 classification. [START_REF] Fischer | Die Lageabweichungen der vorderen Hirnarterie imGefäßbild[END_REF] On contrary to other studies, considering M2 occlusions starting just by M1 297 bifurcation or the reflecting upward segment [START_REF] Miura | Endovascular therapy for middle cerebral artery M2 segment occlusion: 414 subanalyses of RESCUE-Japan Registry 2[END_REF], we chose the insula's mid-height as a 298 landmark to define distal M2 occlusions, since we think that proximal M2 occlusions are very 299 close to distal M1 occlusions and cannot be considered as distal occlusions. 300 301 Recently, a consensus regarding the precise definition of distal vessel occlusion has 302 been published, which might help to resolve this conflict and better delineating the guidelines 303 to treat such patients. This consensus sheds the light on the importance of evaluating the 304 vessel diameter beside its distal location as both represent a real challenge to navigate 305 through and recanalize safely and efficiently. [START_REF] Saver | Thrombectomy for Distal, Medium Vessel Occlusions: A Consensus Statement 385 on Present Knowledge and Promising Directions[END_REF] We did not retain the use of a size threshold 306 to define a distal occlusion in our study, since we believe that measuring the size proximal to 307 an occluded vessel is highly imprecise to evaluate the target artery's size.

309

Although the efficacy of MT+BMT in patients with distal vessel occlusion is still unclear, 310 previous retrospective studies and systematic reviews reported encouraging results. [START_REF] Wang | Efficacy and safety of mechanical thrombectomy for M2 segment of middle 416 cerebral artery: a systematic review and meta-analysis[END_REF][START_REF] Salahuddin | Mechanical Thrombectomy for Middle Cerebral Artery Division 418 Occlusions: A Systematic Review and Meta-Analysis[END_REF] 311 Of these, only a few studies compared the efficacy and safety of MT+BMT with those of 312 medical treatment. [START_REF] Miura | Endovascular therapy for middle cerebral artery M2 segment occlusion: 414 subanalyses of RESCUE-Japan Registry 2[END_REF][START_REF] Sarraj | Endovascular Therapy for Acute Ischemic Stroke With Occlusion of the Middle 420 Cerebral Artery M2 Segment[END_REF][START_REF] Rai | Endovascular therapy yields significantly superior outcomes for large vessel 422 occlusions compared with intravenous thrombolysis: is it time to randomize?[END_REF][START_REF] Qureshi | Comparison of Endovascular Treatment with 425 Intravenous Thrombolysis for Isolated M2 Segment of Middle Cerebral Artery Occlusion in 426 Acute Ischemic Stroke[END_REF][START_REF] Nagel | Distal arterial occlusions in patients with mild strokes -is endovascular therapy 428 superior to thrombolysis alone?[END_REF] Interestingly, these comparative studies are still having 313 conflicting results. Meyer et al. 2021 [START_REF] Meyer | Thrombectomy for secondary distal, medium vessel occlusions of the posterior 430 circulation: seeking complete reperfusion[END_REF], Miura et al. 2019 [START_REF] Miura | Endovascular therapy for middle cerebral artery M2 segment occlusion: 414 subanalyses of RESCUE-Japan Registry 2[END_REF] and Sarraj et al. 2016 [START_REF] Sarraj | Endovascular Therapy for Acute Ischemic Stroke With Occlusion of the Middle 420 Cerebral Artery M2 Segment[END_REF] have 314 shown significant higher efficacy of MT+BMT than the medical management. By contrary, Rai 315 et al. 2013 [START_REF] Rai | Endovascular therapy yields significantly superior outcomes for large vessel 422 occlusions compared with intravenous thrombolysis: is it time to randomize?[END_REF], Qureshi et al. 2017 [START_REF] Qureshi | Comparison of Endovascular Treatment with 425 Intravenous Thrombolysis for Isolated M2 Segment of Middle Cerebral Artery Occlusion in 426 Acute Ischemic Stroke[END_REF] and Nagel et al. 2020 [START_REF] Nagel | Distal arterial occlusions in patients with mild strokes -is endovascular therapy 428 superior to thrombolysis alone?[END_REF] did not find significant 316 difference between both strategies of management. 317 318 Denoting that only our study and Nagel et al.'s study [START_REF] Nagel | Distal arterial occlusions in patients with mild strokes -is endovascular therapy 428 superior to thrombolysis alone?[END_REF] included all distal occlusions 319 sites (M3, anterior and posterior cerebral arteries) in addition to M2 occlusions. However, 320 Nagel et al. 2020 [20] only included patients with minor strokes (NIHSS ≤5) despite those distal 321 vessel occlusions frequently cause disabling symptoms if an eloquent area is affected. [START_REF] Saver | Thrombectomy for Distal, Medium Vessel Occlusions: A Consensus Statement 385 on Present Knowledge and Promising Directions[END_REF] 322 323 Yet, there is neither consensus on clinical selection criteria nor uniform imaging criteria 324 for MT for DVOs patients. Several studies used an NIHSS threshold of ≥6, as it is currently 325 recommended for LVOs. However, a recent data analysis has shown that more than 30% of 326 DVOs patients have a baseline NIHSS <6. Moreover, the variety of clinical symptoms of DVOs, 327 which are dependent on the eloquence of the affected area, also seems to play an important 328 role in the complexity of decision-making. [START_REF] Ospel | A review of endovascular treatment for medium vessel occlusion 432 stroke[END_REF] 329 330 Furthermore, NIHSS is heavily focused on motor function and thus, unable to capture 331 the more subtle domain-specific impairment in DVOs stroke-related disability. Therefore, 332 more fine-grained scales or language and cognitive evaluations should be used in DVOs. [START_REF] Marchal | Endovascular Thrombectomy for Distal Medium Vessel Occlusions of the 434 Middle Cerebral Artery: A Safe and Effective Procedure[END_REF] 333 334 More than eighty percent of our patients underwent MT under conscious sedation, as 335 our institute protocol is to initiate MT with conscious sedation with potential conversion to 336 general anaesthesia (GA) if patient presents agitation or excessive motion. However, the 337 higher anatomical variability in distal vessels in relation to proximal vessels and the need for

  represent about 25-40% of acute ischemic stroke (AIS), either as 24 primary occlusion or secondary occlusion complicating mechanical thrombectomy (MT) for 25 large vessel occlusion. 26 Objective 27 Our aim was to evaluate safety and effectiveness of MT associated with the best medical 28 treatment (BMT) in the management of AIS patients with distal vessel occlusion in comparison 29 with the BMT alone. 30 Methods 31 Retrospective analysis was conducted on AIS patients treated by MT+BMT for primary distal 32 vessel occlusion between 2015 and 2020, and were compared with a historic cohort managed 33 by BMT alone between 2006 and 2015 selected based on the same inclusion criteria. A 34 secondary analysis was conducted using propensity score matching (PSM) including the 35 following: NIHSS, age and treatment with intravenous thrombolysis (IVT) as covariates.

  43 intracranial hemorrhage (ICH) was seen in only one patient treated by MT+BMT (3.6%). 44 Introduction 55 Despite distal vessel occlusions (DVOs) represent about 25% to 40% of acute ischemic 56 stroke (AIS), whether as primary occlusion or complicating mechanical thrombectomy (MT) 57 for large vessel occlusion (LVO), they were underrepresented by the main randomized trials 58 of MT.[1] MR CLEAN (Multicenter Randomized Clinical Trial of Endovascular Treatment for 59 Acute Ischemic Stroke in the Netherlands) and EXTEND IA (Extending the Time for 60 Thrombolysis in Emergency Neurological Deficits -Intra-Arterial) were the only among the five 61 known randomized trials that included patients with isolated M2 occlusions per protocol, and 62 this subgroup of patients represented about 8% in relation to other occlusion sites.[2, 3] 63 64

Declaration

  hand, the risks of MT in distal occlusions are well-known. The special 72 characteristics of distal vessels of smaller diameters, increased tortuosity, and longer 73 distances of navigation, along with the iterative advances in endovascular devices necessitate 74 to pay more attention for the management of these types of occlusions. 75 76 The aim of this study was to evaluate effectiveness and the safety of mechanical 77 thrombectomy in the management of AIS patients with distal vessel occlusion in comparison 78 with the best medical treatment alone. was primarily conducted on AIS patients who were treated by 91 MT + BMT for primary and isolated distal vessel occlusion at our institution between January 92 2015 and June 2020. These patients were compared with a historic cohort of patients from 93 the same Institution managed by best medical treatment (BMT) alone including intravenous 94 thrombolytic therapy (IVT) between April 2006 and January 2015 and selected based on the 95 same inclusion criteria. The study was approved by the institutional review board that waived 96 the patients' consents. This retrospective study was conducted in accordance with the 97 isolated primary distal vessel occlusion proven by vascular 102 imaging were included except patients with secondary distal vessel occlusion as a 103 complication of MT for LVO who were excluded. Distal vessel occlusion was identified as 104 occlusion of the distal aspect of the M2 segment and M3 segment of the middle cerebral 105 artery, any segment of the anterior cerebral artery, or any segment of the posterior cerebral 106 artery. Occlusion of the distal aspect of the M2 segment was defined as an occlusion above 107 the mid-height of the insula, while M3 occlusion was defined as occlusion at any point from 108 the parietal operculum till emerging from the Sylvian fissure at the lateral surface of the brain 109 according to the Fischer's classification. Proximal M2 occlusions were excluded.[6, 7] 110 111 Clinical severity was assessed using the NIHSS at admission, 24 hours and one week by 112 a vascular neurologist. The modified Rankin Scale (mRS) was evaluated at three-months. 113 114 Pre-procedure and Post-procedure Imaging Assessments 115 All the patients underwent a pre-procedure brain magnetic resonance imaging (MRI) 116 including diffusion-weighted imaging (DWI) and 3D Time of Flight (TOF) MR angiography 117 (MRA). Brain computed tomography (CT) scan was performed if there was contraindication 118 for MRI, like cardiac pacemakers. An unenhanced brain CT scan and/or MRI was performed 119 within 24 hours after the endovascular procedure to exclude any haemorrhagic complications. patients received IVT within 4.5 h of stroke symptoms' onset. 124 Otherwise, an anti-platelets agent was the reference. Afterward, the individual decision to 125 perform a MT was a multidisciplinary consensus by the neurologist and 126 neurointerventionalist. Arguments to perform MT were a relevant clinical deficit. Moreover, 127 it was upon the operator's discretion to weigh the specific procedural risk against the potential 128 benefit of recanalization; and given the current absence of evidence for MT in such population, 129 the decision was not based on predefined fixed criteria. 130 131 For MT, all the procedures were systematically performed in a biplane 132 neuroangiography suite using a tri-axial system with a 6F long sheath, an intermediate distal 133 access catheter (5F or 6F) and a microcatheter. When combined to IVT, endovascular 134 preparation was initiated simultaneously with or soon after IVT administration was started 135 without waiting for evaluation of its initial response (bridging paradigm).136 137 Mechanical thrombectomy was performed with approved devices, using stent-138 retrievers with maximal diameter of 3 mm or 3MAX aspiration catheter (Penumbra, Alameda, 139 CA, USA). The choice was left to the attending neuroradiologist's discretion.

140 141

 140 Baseline parameters, technical features, and angiographic were noted.

  point was the rate of good clinical outcome defined as mRS ≤ 2 at 3-148 months. Secondary efficacy criteria included the rate of excellent clinical outcome, defined as 149 a 3-month mRS ≤ 1 and early neurological improvement, defined as a complete resolution of 150 the neurological deficit or an improvement from baseline in the National Institutes of Health 151 Stroke Scale (NIHSS) by 4 or more points 24 hours after the onset of stroke.[8] 152 153 Moreover, the rate of successful reperfusion, defined as mTICI ≥2b (Modified 154 Thrombolysis in Cerebral Infarction) at the end of the procedure for the MT + BMT group; and 155 defined as AOL ≥ 2 (Arterial Occlusion Lesion scale) on 24 hours MRI and MRA for the BMT 156 group.[9] 157 158 The safety criteria were the rate of symptomatic intracranial hemorrhage (ICH), 159 according to the trial of thrombolytic therapy with intravenous alteplase in acute ischaemic 160 stroke (ECASS II) study criteria [10], asymptomatic ICH, the frequency of distal emboli, 161 procedural arterial dissection and perforation in the MT + BMT group. 162 163 As an additional analysis, we developed a propensity score matched cohort in order to 164 explore the robustness of our analyses. The variables for the propensity score included the 165 following: baseline NIHSS score, age, and treatment with IVT. The patients in the BMT group 166 were matched to those in the MT group using a 1:1 matching technique. We compared 167 primary outcomes and secondary outcomes of the propensity score matched cohort between 168 the MT+BMT and BMT groups. The effects of MT+BMT relative to BMT are displayed as ORs 169 and 95% CIs.

  performed using (IBM Corp. Released 2017. IBM SPSS Statistics for 178 Windows, Version 25.0. Armonk, NY: IBM Corp.) χ2 or Fisher's exact test was used to compare 179 frequencies, when appropriate. Comparison of means was performed using Student-t test or 180 Mann-Whitney test, depending on the data distribution. Results were considered statistically 181 significant when P value <0.05. Propensity score matching was used with a calliper width of 182 0.1 based on age, initial NIHSS and treatment with IVT as covariates. The patients in the BMT 183 group were matched to those in the MT+BMT group using a 1:1 matching technique. For 184 outcome variables, unadjusted odds ratios (ORs) were calculated by using univariate logistic 185 regression models, including treatment by MT+BMT as independent variable. patients treated by mechanical thrombectomy in our centre during the 205 predetermined period screened (2015-2020), 44 patients with distal vessel occlusions who 206 met the inclusion criteria were treated by MT + BMT and compared with 36 patients who 207 received BMT alone (2006-2015). (Figure 1 shows the study flow chart) 208 209

  results were comparable with a recent large multicenter study by 281 Anadani et al. 2021, studying primary distal vessel occlusions (M3, and anterior and posterior 282 cerebral arteries) treated by MT.[12] This study showed results of 78 % successful 283 recanalization of mTICI ≥ 2B, 45% good outcome of 3-months mRS ≤ 2, 19% 3-months mortality 284 rate, and 5 % significant and symptomatic ICH.285 286

  338an excellent roadmap to navigate through distally located clots seem to bring GA more 339 beneficial by complete elimination of the patient's movement. Thus, further trials 340 investigating impact of GA on functional outcome in DVOs are warranted. 341 342 From a technical perspective in M2 and M3 occlusions, despite lack of statistical 343 evidence in our cohort, it was observed that the navigation through the superior division was 344 more challenging with excessive microwire manipulation than the inferior one. A possible 345 explanation for this is that superior divisions are usually more angulated and tortuous than 346 the inferior ones. A recent retrospective study compared MT in superior and 347 inferior division occlusions had found that perforations were non-significantly more frequent 348 in superior division occlusions compared to inferior division occlusions.[24] 349 350 Distal vessel occlusions are still managed for now subjectively on case-by-case basis, 351 according to the dispersion of the operator, the degree of vessel tortuosity and the different 352 institutional protocols. Indeed, distal vessel occlusions management warrant randomised 353 clinical trials to better precise the selecting criteria of patients who could benefit from 354 MT+BMT in a safe manner, especially those having disabling symptoms despite of the small 355 calibre or the distality of the occluded vessel.

Figure. 1 Figure. 2

 12 Figure captions

Table 1

 1 shows baseline characteristics. Compared with patients in the BMT group, 210 those in the MT + BMT group had a higher baseline NIHSS score (median [IQR], 11 [7-21] vs. 6 211 [3-10], p<0.001) and were older (median [IQR], 78 [64 -85] vs. 70 [57-78], p<0.04); and tended 212 to have higher rates of IVT treatment (63.6 % vs. 41.7 %; P = 0.07). There was no difference

	213
	214	among the groups for stroke comorbidities.
	215
	216	Regarding MT, stent retrievers (with or without aspiration) was used in 33 cases (75%)
	217	and 7 MT procedures (16%) were performed under general anaesthesia.
	218
	219	After the propensity score matching, 28 patients in each group were matched. Baseline
		characteristics of the two groups after propensity score matching are shown in

Table 1 .

 1 No 220 significant difference was noted between the two groups.

	221
	222
	223

Clinical outcomes in the entire cohort

	234	Comparing both groups regarding the final recanalization state, the MT + BMT group
	235	had significantly higher successful recanalization defined as AOL ≥ 2 (90.9% vs. 47.2% in
	236	MT+BMT and BMT groups; p= <0.001) (Figure 2 shows an illustrative case)
	237
	238	For the safety outcome, mortality rate at 90 days occurred in 18.2% and 13.9% (p=0.76)
	239	of MT+BMT and BMT patients.
	240
	241	Symptomatic ICH occurred in two patients (4.5%) in the MT+BMT group with no
	242	observed symptomatic ICH in the BMT group (P=0.5). However, asymptomatic ICH occurred
	243	in further 4 patients in the MT+BMT group (9%), and one patient in the BMT group (2.8%,
		p=0.37) (
	224
	225
	226	Good clinical outcome (defined as mRS ≤ 2 at 3-months) was observed in 47.7% and
	227	58.3% (p=0.38) of MT + BMT and BMT patients, respectively; while excellent clinical outcome
	228	(defined as mRS ≤ 1 at 3-months) was achieved by 29.5 % and 38.9% (p=0.48) of MT + BMT
	229	and BMT patients, respectively. Early neurological improvement was observed in 45.5 % and
	230	38.9 % (p=0.65) of MT + BMT and BMT patients, respectively.
	231
	232	The overall successful reperfusion rate in the MT group defined as mTICI ≥ 2B was
	233	77.3%.

Table 2 ).

 2

	244
	245
	246	As procedural complications, distal embolization to the same territory occurred in two
	247	patients (4.5%). Moreover, vessel perforation occurred during the procedure in two other
	248	patients (4.5%); one of them suffered from symptomatic ICH.
	249
		Clinical

outcomes in the propensity score-matched cohort

	250
	251
	252	After balancing of both groups by using NIHSS score, age, and treatment with IVT as
	253	covariates to develop a propensity score matched model; 3-months good clinical outcome was
	254	achieved by 53.6% of the MT+BMT treated patients and by 57% of the BMT group (odds ratio
	255	[OR], 0.87; 95% CI, 0.3-2.4; P=1.00). Thirty two percent and 39.3% of patients treated by
	256	MT+BMT and BMT, respectively, had achieved excellent clinical outcome at 3-months (odds
	257	ratio [OR], 0.73; 95% CI, 0.24-2.2; P=0.78). Early neurological improvement had been achieved
	258	by 35.7% and 39.3% in MT+BMT and BMT groups, respectively (odds ratio [OR], 0.86; 95% CI,
	259	0.29-2.5; P=1.00).
	260
	261	For the safety outcome, the mortality rate was comparable in both groups (7% in
	262	MT+BMT treated patients vs. 10.7% in patients with BMT; odds ratio [OR], 0.64; 95% CI, 0.1-
	263	4; p=1.00). Vessel perforation during the procedure occurred in one patient and leaded to
	264	symptomatic ICH (3.6%). Asymptomatic ICH occurred in 3 patients in the MT+BMT group
		(10.7%), and one patient in the BMT group (3.6%, p=0.61) (

Table 2 ).

 2

	265

The research leading to these results has received funding from "Investissements d'avenir

Disclaimer

Table.1 Patients' baseline demographics and procedural parameters


Mariya Borovikova
email: mariya.borovikova@sorbonne-nouvelle.fr

Loïc Grobol
email: lgrobol@parisnanterre.fr

Anaïs Lefeuvre-Halftermeyer

Sylvie Billot

A Methodology for the Comparison of Human Judgments With Metrics for Coreference Resolution

We propose a method for investigating the interpretability of metrics used for the coreference resolution task through comparisons with human judgments. We provide a corpus with annotations of different error types and human evaluations of their gravity. Our preliminary analysis shows that metrics considerably overlook several error types and overlook errors in general in comparison to humans. This study is conducted on French texts, but the methodology should be language-independent.

Introduction

Coreference resolution is still one of the most challenging tasks in Natural Language Processing. Several metrics have been proposed to evaluate the task, each of them meant to rectify the weaknesses of the previous ones. However, neither their correctness nor their ability to reflect the real quality of algorithms is easily be provable from their mathematical definition. Consequently, some additional tests should be conducted in order to confirm their pertinence. This work aims to compare the evaluation measures used for coreference resolution task with human judgments, i.e. to study them in terms of interpretability. More precisely, B-CUBED [START_REF] Bagga | Algorithms for scoring coreference chains[END_REF], LEA (Moosavi and Strube, 2016), CEAFe and CEAFm [START_REF] Luo | On coreference resolution performance metrics[END_REF], CoNLL-2012 (MELA) [START_REF] Denis | Global joint models for coreference resolution and named entity classification[END_REF], BLANC [START_REF] Recasens | BLANC: Implementing the Rand index for coreference evaluation[END_REF] and MUC [START_REF] Vilain | A modeltheoretic coreference scoring scheme[END_REF] metrics will be analysed.

Related work

Although some properties of coreference resolution quality measures have already been studied in [START_REF] Lion-Bouton | Comment arpenter sans mètre : les scores de résolution de chaînes de coréférences sont-ils des métriques ? (do the standard scores of evaluation of coreference resolution constitute metrics ?)[END_REF], Moosavi (2020), [START_REF] Kummerfeld | Errordriven analysis of challenges in coreference resolution[END_REF] and others, to the best of our knowledge, there are no works dedicated to the comparison between automatic measurements and human evaluation of performance for this task. However, very few similar studies were conducted in other domains. [START_REF] Doshi | Towards a rigorous science of interpretable machine learning[END_REF] study the interpretability of machine learning models, in general, using application-grounded, human-grounded, and functionally-grounded approaches.

Foster ( 2008) describes an experience of evaluating a non-verbal behaviour of an embodied conversational agent. People were asked to choose the most appropriate talking head among the two generated using different strategies. Then β inter-annotator agreement measure [START_REF] Artstein | Inter-coder Agreement for Computational Linguistics[END_REF] was calculated.

In [START_REF] Plank | Do dependency parsing metrics correlate with human judgments[END_REF], the correlation between metrics for the dependency parsing task and human judgments was examined. Several models were tested for different languages. The annotators had to choose the best of the two annotations predicted by two different models without knowing the correct option. The obtained results were normalised using Spearman's ρ and compared with standard metrics. [START_REF] Novikova | Why we need new evaluation metrics for NLG[END_REF] explore Natural Language Generation (NLG) evaluation measures. The annotation process is organised as follows: an annotator should score an example using three Likert scales from 0 to 6 based on informativeness, naturalness and quality criteria. The obtained results were normalised using Spearman and intra-class correlation coefficients and compared with NLG metrics.

Considering these studies, for the present research, we will use an approach similar to [START_REF] Novikova | Why we need new evaluation metrics for NLG[END_REF], where the annotators evaluate a system on a Likert scale. Despite possible difficulties with the Likert scale treatment (too many mid-point answers, a broad spectrum of responses for one question, etc.), this method seems more appropriate for our purposes. Two main reasons make us choose this approach: (1) we do not test particular systems and, therefore, have no alternative annotations and (2) a scaled approach is more accurate and exact while evaluating a system.

Methodology

This section is dedicated to the theoretical description of the methods used in the experiments within the scope of this study.

Errors typology

In order to correctly evaluate the quality of the algorithm, it is necessary to consider all the types of errors it can produce and, therefore, to define those types.

For our purposes, we have chosen the typology of Landragin and Oberle (2018):

1. Border errors occur when limits of referential expressions are marked inaccurately;

2. Type errors occur when a referential expression is assigned to a false chain;

3. Noise errors occur when irrelevant linguistic expressions are marked as a part of a coreference chain;

4. Silence errors occur when a system ignores referential expressions which are included in a relevant coreference chain;

5. Tendency of irrelevant coreference chains construction occurs when a system composes a new chain from several unrelated mentions.

We use this typology because it is more comprehensive than others and reflects the semantic aspect of the problem. However, we need to introduce an additional error type which we call "chain absence". This error may be regarded as a form of the "silence" error, and it occurs when the whole coreference chain (entity) is missing. The necessity of introducing a new error type arose after the experimentation phase of this study as it allowed to explain some patterns in the behaviour of the metrics. You can find the examples for each error type in the appendix section 1.

Corpus creation

Our corpus consists of a series of texts, each with two coreference annotations: one is a manual gold annotation, and the other is a purposefully erroneous annotation, one or more manually introduced errors of one of the types defined in section 3.1. There are also a few examples with errors of different types. Two existing coreference resolution corpora for French were used as a basis for the corpus. 52 texts were taken from the DEMOCRAT corpus [START_REF] Landragin | LIVRABLE L2 "Manuel d'annotation du corpus et organisation de formations sur l'annotation" du projet DEMOCRAT[END_REF] and 4 examples -from the ANCOR corpus [START_REF] Muzerelle | ANCOR_Centre, a large free spoken French coreference corpus: description of the resource and reliability measures[END_REF]. More precisely, we have selected the self-standing passages that are understandable out of context. The corpora are collected in the CoNLL-2012 format [START_REF] Sameer Pradhan | CoNLL-2012 shared task: Modeling multilingual unrestricted coreference in OntoNotes[END_REF] 1 . The final dataset consists of 127 passages of 90-130 words each. 108 examples contain only one error, allowing us to analyse to what extent each error reduces the overall system quality. The rest of the samples are needed to adjust the annotations . Coreference chains lengths vary from 2 to 20 mentions. The mentions to contain an error were chosen at random. The total number of each error in the 108 samples varies between 16 and 28. The total number of each error varies between 44 and 97.

Evaluation scale

As the primary goal of this study is to evaluate the interpretability of the metrics, it is necessary to compare them to humans opinions about the correctness of the system's responses. Even though the metrics' output values are between 0 and 1, we will not use this range as it is more natural for people to evaluate the quality on an integer scale.

For our study, we use a Likert scale [START_REF] Likert | A technique for the measurement of attitudes[END_REF] with an even number of choices in order to avoid too many mid-point answers. Usually, coreference resolution is only a part of a pipeline of a more complex system, and the way of evaluation depends on the resolved task. In this study, an information retrieval task has been chosen as a global framework. These conditions require some changes in the classic scale; namely, we introduce a notion of the "importance" of an element. We distinguish two types of elements: peripheral elements and key elements. Peripheral elements can be removed from a text without severe consequences in its general sense. Key elements constitute the core of a text, so their removal will lead to the total loss of meaning. Thus, the gravity of an error and the importance of an element with an error is taken into account.

This scale also contains two points to allow differentiation between similar examples with little nuances: (0) The presumed system's annotation contains significant errors on key elements; (1-2) The presumed system's annotation contains significant errors on peripheral elements; (3-4) The presumed system's annotation contains insignificant errors on key elements; (5-6) The presumed system's annotation contains insignificant errors on peripheral elements; (7) The presumed system's annotation does not contain any errors.

Annotation

Every annotation sample contains a correct annotation and an annotation with mistakes. In order to detect inconsistent annotators, three samples appear twice. The objective given to the annotators is to evaluate coreference resolution samples as a part of an information retrieval system using the Likert scale described in section 3.3. General instructions given before the annotations explain all the necessary concepts 2 .

As an inter-annotator agreement measure, Krippendorff's alpha [START_REF] Krippendorff | Bivariate agreement coefficients for reliability of data[END_REF] has been chosen and used to identify annotators whose answers differ much from the others using a new algorithm (see algorithm 1 in the appendix). The coefficients obtained by the algorithm (hereinafter the trust coefficients) allow us to detect outliers (an annotator is considered an outlier if their score is less than or equal to 0.5).

In order to interpret the reasoning of each respondent, regressors have been trained to imitate the annotators' and metrics' behaviours. Each model should predict a score having the number of occurrences of each error type as input features. We have trained one model for each annotator and metric. Once the models are trained, the weights assigned to each feature (error type) are extracted and used for further interpretation.

Experiments and results

Human evaluation analysis. Since participation in this study was not rewarded and contained many questions, it involved only 12 participants, 9 of whom were linguists and 8 of whom have already worked with coreference. The analysis of the three duplicated questions showed that no one answered at random among the annotators. Krippendorff's alpha is rather low, so we supposed that some questions in our questionnaire raised more confusion among the respondents than others. Therefore we eliminated the questions that contained more than three different answers from the annotators and computed the results only for the remaining simple questions. The total number of questions used in the main analysis is 97. We also decided to compute the inter-annotator agreement on a reduced scale from 0 to 4 points (0 → 0, 1 and 2 → 1, 3 and 4 → 2, 5 and 6 → 3, 7 → 4) and on the gravity (no errors -insignificant error(s) -significant error(s)) and elements importance (no errors -error(s) on peripheral element -error(s) on key element) scales. These agreements are presented in table 1.

2 You can find the google form with the instructions at https://forms.gle/cgpsfZvKg5zasnqd6. Human-machine correlation analysis. In order to compare the obtained scores with human judgments, we calculated an average and a mode of human evaluations having previously transformed to a scale from 0 to 1. Every metric was compared with the annotators' assessment on the standard scale, on the reduced scale and on the scale with errors gravity evaluation only. According to the data distributions, in general, the difference between a metric and humans is about 0.33. The averages of differences for all the examples are given in table 2.

Scale

Analysis by error type. In order to analyse the influence of a particular error type on a score, we train a linear regression model with the number of errors of each type as the input features and the reversed scores3 as the outputs. All the input features were centered and reduced in order to obtain more stable results. The coefficients that were assigned to each input feature (and which correspond to one of the error types) during the training have been used as a measure of the importance of an error in the process of deciding the example's score (see tables 3 and 4).

Discussion

Human evaluation analysis. Table 1 reports the interannotator agreement on different scales, with several interesting properties about the task. Firstly, we may observe that the reduced scale results are better than those on the standard scale. It can be explained by the fact that even if people agree on the characteristics of the suggested categories, all of them have their own bias about the task, so they pay attention to different annotation nuances. Secondly, the inter-annotator agreement increased when we eliminated the annotators indicated as outliers by the trust coefficient.

Human-machine correlation analysis. One may notice that the average scores of all annotators are relatively high (see table 2). The average difference between all metrics and the annotators is usually above 0 and varies from 0.2 to 0.4 after normalisation, which shows that, generally, metrics tend to overestimate the actual quality of a model significantly.

Analysis by error type. In order to perform the analysis regarding the error types, we modified the table 4 by removing all positive and null coefficients as they mean either the absence of answers considering a particular error type or insufficient training quality of some models. These modifications can be justified by the fact that every coefficient of the model should be negative. Otherwise, it would mean that the presence of an error improves a score.

As our analysis shows, the border, silence and irrelevant chains construction errors are treated correctly. It could be proven by the fact that metrics coefficients are similar to the human ones. The type, noise and chain absence errors are underestimated by the metrics, as their scores are usually higher for the metrics than for the humans coefficients (see correspondent columns of the table 3).

We can analyse each metric separately as well. Firstly, we have noticed that the MUC metric considerably underestimates all types of errors except for the "silence" and the "irrelevant chains" ones. Secondly, the B-CUBED measure put relevant scores only to the examples which contain "border" and "silence" errors. The CEAFe score estimates correctly only the examples with "border" and "irrelevant chains" errors. Similarly, the CEAFm metric also underestimates all examples where any errors except for "border" and "irrelevant chains" ones were made. The BLANC measure treats properly only texts with "silence" errors. We ob-serve that the CoNLL-2012 metric tends to overstate the results of a model when the examples contain any errors except for "border" errors. Likewise, the LEA metric considerably underestimates all error types except for "border", "silence" and "irrelevant chains" errors (see correspondent lines of the table 3).

Conclusion

This study aims to investigate the extent to which we may understand the results produced by the coreference resolution metrics. The preliminary results on the limited corpus show that metrics underestimate errors gravity compared to humans and add approximately 0.33 points to the final score on the scale from 0 to 1. However, these results need to be proven on a more significant number of annotators. This work's contribution consists in creating the corpus with various errors types and its annotation with the human judgments about the gravity of these errors, the proposal of the new automatic outlying annotator identification algorithm and the suggestion of a methodology of comparison of human evaluations with automatic metrics. All the code and corpus are available at https://github.com/ project178/coref-metrics-vs-humans.

Possible future work directions may consist in involving more people in the annotation process of the proposed corpus in order to verify the obtained results and in the development of a new metric that will take into consideration the identified shortcomings of the existing measures.

A Appendix

Table 1 :

 1 Krippendorff's alphas. An arrow shows that there are outlier annotators on the particular scale and set of examples. A value on the right of an arrow is an alpha after removing outlier annotators.

		All examples Simple examples
	Standard	0.11	0.25 → 0.27
	Reduced	0.16	0.34
	Gravity	0.24	0.48
	Importance	0.16	0.34 → 0.42

Table 2 :

 2 Differences between humans evaluations and metrics on the scale from 0 to 1.

	Scale	Method	MUC	B-CUBED CEAFm CEAFe BLANC	LEA	CoNLL
	Standard	Average Mode	0.289 0.294		0.321 0.326	0.308 0.313	0.269 0.283	0.281 0.285	0.231 0.24	0.291 0.299
	Reduced	Average Mode	0.314 0.312		0.346 0.347	0.333 0.333	0.29 0.292	0.305 0.304	0.253 0.255	0.315 0.316
	Gravity	Average Mode	0.43 0.43			0.463 0.464	0.45 0.451	0.405 0.408	0.422 0.422	0.368 0.369	0.432 0.434
	Name		Border	Type	Noise	Silence Irrelevant chains Chain absence
	MUC		-0.242	-0.249	-0.121	-0.58	-0.345	-0.076
	B-CUBED	-0.662	-0.15	-	-0.889	-	-0.264
	CEAFm	-0.325	-0.34	-0.139	-0.408	-0.353	-0.101
	CEAFe		-0.458	-0.283	-0.322	-0.447	-0.222	-
	CoNLL	-0.382	-0.217	-0.083	-0.556	-0.179	-0.187
	BLANC	-0.174	-0.385	-0.233	-0.973	-0.074	-0.56
	LEA		-0.425	-0.22	-0.207	0.73	-0.432	-
	Humans	-0.343	-0.629	-0.598	-0.513	-0.467	-0.727

Table 3 :

 3 Coefficients of errors importances. "Humans" is the average of all the coefficients of models trained on humans' evaluations. See a more detailed version in the appendix (table4).

  Borders errors. Whales are marine mammals . instead of Whales are marine mammals . 2. Type errors. John likes his brother because he is funny instead of John likes his brother because he is funny. 3. Noise errors. The dog barked. It 's time to go. instead of The dog barked. It's time to go. 4. Silence errors. A phone is on the table. It rings. I pick it up instead of A phone is on the table. It rings. I pick it up. 5. Tendency of irrelevant coreference chains construction. A cat and a dog are playing together instead of A cat and a dog are playing together. 6. Chain absence. A phone is on the table. It rings. I pick it up instead of A phone is on the table. It rings. I pick it up.

					Figure 1: Error types examples.
	Name	Border Name	Type	Name	Noise Name	Silence Name	Irrelevant chains Name

1.

https://github.com/boberle/coreference_ databases.git

We replaced 7 by 0, 6 by 1, 5 by 2, etc.

Acknowledgements

This work was funded by Région Centre-Val-de-Loire through the RTR DIAMS.

Chain absence REDUCED_A6 -0,033 B-CUBED -0,15 CoNLL-2012 -0,083 GRAVITY_A6 -0,011 GRAVITY_A6 -0,018 MUC -0,076 GRAVITY_A10 -0,057 CoNLL-2012 -0,217 GRAVITY_A7 -0,105 STANDARD_A12 -0,262 BLANC -0,074 STANDARD_A10 -0,099 STANDARD_A7 -0,097 LEA -0,22 MUC -0,121 GRAVITY_A8 -0,265 STANDARD_A11 -0,08 CEAFm -0,101 GRAVITY_A7 -0,118 GRAVITY_A7 -0,239 REDUCED_A1 -0,122 STANDARD_A11 -0,274 REDUCED_MEAN -0,097 GRAVITY_A11 -0,124 GRAVITY_A11 -0,152 GRAVITY_A8 -0,24 CEAFm -0,139 STANDARD_A9 -0,329 GRAVITY_A11 -0,164 STANDARD_A7 -0,17 STANDARD_A10 -0,166 MUC -0,249 STANDARD_A9 -0,146 GRAVITY_A11 -0,391 STANDARD_A4 -0,167 CoNLL-2012 -0,187 BLANC -0,174 CEAFe -0,283 GRAVITY_A11 -0,167 STANDARD_A10 -0,406 CoNLL-2012 -0,179 GRAVITY_A7 -0,247 STANDARD_A6 -0,182 STANDARD_A10 -0,31 LEA -0,207 CEAFm -0,408 CEAFe -0,222 B-CUBED -0,264 STANDARD_A8 -0,212 CEAFm -0,34 BLANC -0,233 CEAFe -0,447 GRAVITY_MEAN -0,319 GRAVITY_A8 -0,265 MUC -0,242 STANDARD_A12 -0,362 STANDARD_A7 -0,316 GRAVITY_A1 -0,49 STANDARD_A8 -0,335 STANDARD_A6 -0,283 CEAFm -0,325 GRAVITY_A3 -0,372 CEAFe -0,322 GRAVITY_A7 -0,547 MUC -0,345 STANDARD_A11 -0,289 CoNLL-2012 -0,382 BLANC -0,385 STANDARD_MEAN -0,356 STANDARD_A6 -0,548 CEAFm -0,353 STANDARD_A12 -0,411 GRAVITY_A3 -0,388 STANDARD_A8 -0,458 STANDARD_A11 -0,364 CoNLL-2012 -0,556 GRAVITY_A7 -0,375 BLANC -0,56 LEA -0,425 GRAVITY_A11 -0,507 STANDARD_A10 -0,532 MUC -0,58 STANDARD_A6 -0,407 STANDARD_A8 -0,634 CEAFe -0,458 STANDARD_A9 -0,563 GRAVITY_A3 -0,566 GRAVITY_A10 -0,586 GRAVITY_A8 -0,409 GRAVITY_A1 -0,737 B-CUBED -0,662 STANDARD_MEAN -0,607 GRAVITY_A1 -0,691 STANDARD_A7 -0,629 LEA -0,432 STANDARD_A9 -0,756 GRAVITY_MEAN -0,848 STANDARD_A11 -0,625 REDUCED_MEAN -0,712 LEA -0,73 GRAVITY_A3 -0,453 STANDARD_A4 -0,919 REDUCED_A1 -1,529 GRAVITY_A1 -0,663 GRAVITY_A4 -0,911 REDUCED_MEAN -0,883 STANDARD_A9 -0,491 GRAVITY_A10 -0,932 STANDARD_A7 -0,686 GRAVITY_A10 -0,954 B-CUBED -0,889 STANDARD_A7 -0,64 REDUCED_MEAN -0,943 STANDARD_A6 -0,741 GRAVITY_MEAN -1,184 BLANC -0,973 REDUCED_A6 -0,819 GRAVITY_A6 -1,149 GRAVITY_A6 -0,741 GRAVITY_A6 -1,197 GRAVITY_MEAN -1,565 STANDARD_A12 -0,859 REDUCED_A1 -1,419 STANDARD_A4 -0,818 STANDARD_A6 -1,238 STANDARD_MEAN -0,904 REDUCED_A6 -1,473 GRAVITY_A4 -0,869 GRAVITY_A4 -0,911


Marie Corpart

Julien Dervaux

Christophe Poulard

Frederic Restagno

François Boulogne

Evaporation of liquid coating a fiber

Introduction. -The coating of liquid on a solid is a common operation in everyday life as well as in industrial processes (e.g. lubrication, painting). The first theoretical description of the deposited layer has been made almost simultaneously by Landau and Levich [START_REF] Landau | Dragging of a liquid by a moving plate[END_REF] and Derjaguin [START_REF] Derjaguin | On the thickness of the liquid film adhering to the walls of a vessel after emptying[END_REF]. The coated thickness h is found to be proportional to the radius a of the fiber and depends in particular of the entrainment velocity V , the liquid surface tension γ and the fluid viscosity η through the so-called capillary number Ca = ηV /γ, which compares the velocity to a fluid intrinsic velocity. Thus, the thickness h can be written as h = a f (Ca), where f is a function which can be simplified under certain conditions on the capillary ranges. Coating of complex fluids also received some attention, such as non-Newtonian fluids or water-surfactant solutions [START_REF] Quéré | Fluid coating on a fiber[END_REF][START_REF] Rio | Withdrawing a solid of a bath: how much liquid is coated?[END_REF].

Due to the fiber curvature, the liquid film is unstable; a mechanism that was first characterized by Plateau [START_REF] Plateau | Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires[END_REF] and rationalized by Lord Rayleigh in the 19th century [START_REF] Rayleigh | On the instability of jets[END_REF]. The liquid surface tension leads to the minimization of its surface area by breaking the film into a series of regularly spaced droplets. The characteristic growing time of the Rayleigh-Plateau instability τ = 12ηa 4 /(γh 3 ) depends significantly on the coating thickness. For non-Newtonian fluids, or for colloidal suspensions, the Rayleigh-Plateau instability can be delayed or suppressed [START_REF] Boulogne | Annular cracks of thin films of colloidal silica particles coating a fiber[END_REF][START_REF] Bauer | Suppressing the rayleigh-plateau instability in fielddirected colloidal assembly[END_REF][START_REF] Boulogne | Suppression of the Rayleigh-Plateau instability on a vertical fibre coated with wormlike micelle solutions[END_REF].

Carroll [START_REF] Carroll | The accurate measurement of contact angle, phase contact areas, drop volume, and laplace excess pressure in drop-on-fiber systems[END_REF] described the shape of axisymmetric drops on a fiber, which depends on the wetting properties of the material and the droplet volume with respect to the fiber size. Based on Carroll's results, the drop height and the length of the wetted area can be used to characterize the wettability of the liquid on the fiber [START_REF] Mchale | Wetting of a high-energy fiber surface[END_REF][START_REF] Mchale | Estimation of contact angles on fibers[END_REF][START_REF] Mchale | The shape and stability of small liquid drops on fibers[END_REF][START_REF] Mchale | Global geometry and the equilibrium shapes of liquid drops on fibers[END_REF]. Beyond the axisymmetric shape, the equilibrium conformation of a drop on a fiber can also be a clam-shell: a drop sitting on a curved surface. Experimental observations have evidenced that both barrel and clam-shell conformations can coexist and Chou et al. studied in detail the phase diagrams of droplet-on-fiber with or without gravity [START_REF] Chou | Equilibrium phase diagram of drop-onfiber: Coexistent states and gravity effect[END_REF].

The aim of the paper is to describe the different evaporation regimes of the fiber coatings when the evaporation is dominated by the diffusion of water in the atmosphere. Despite the fact that this is a model system with many industrial applications, there is a lack of theoretical rationalization on this subject.

In 1918, Langmuir [START_REF] Langmuir | The evaporation of small spheres[END_REF] explained for the first time that the total diffusive evaporation flux over a spherical droplet of size R is not proportional to R 2 , which means that, due to the curvature of the surface of the droplet, the evaporation flux is not proportional to the surface of the drop. The case of a sessile drop evaporation (see [START_REF] Cazabat | Evaporation of macroscopic sessile droplets[END_REF][START_REF] Brutin | Droplet Wetting and Evaporation: From Pure to Complex Fluids[END_REF] for a review) is more complex because of the diverging evaporative flux at the triple line [START_REF] Lebedev | Special functions and their applications[END_REF][START_REF] Picknett | The evaporation of sessile or pendant drops in still air[END_REF][START_REF] Deegan | Capillary flow as the cause of ring stains from dried liquid drops[END_REF] and several studies have been devoted to the nature of the substrate. For instance, the case of a drop on a tilted surface [START_REF] Timm | Evaporation of a sessile droplet on a slope[END_REF], on crossed fibers [START_REF] Boulogne | Mechanical tuning of the evaporation rate of liquid on crossed fibers[END_REF], on superhydrophobic surfaces [START_REF] Stauber | Evaporation of droplets on strongly hydrophobic substrates[END_REF] or with complex wetting patterns [START_REF] Sáenz | Dynamics and universal scaling law in geometrically-controlled sessile drop evaporation[END_REF] can be cited. The situation of a droplet on a curved surface such as a convex or concave surface [START_REF] Paul | Substrate concavity influenced evaporation mechanisms of sessile droplets[END_REF][START_REF] Shen | Theoretical analysis of a sessile evaporating droplet on a curved substrate with an interfacial cooling effect[END_REF] has been studied as well, the latter corresponding to a clam-shell on a fiber.

In this paper, we discuss the two limit cases of the diffusive evaporation of a liquid deposited on a fiber. In a first part, we study the evaporation of a liquid cylinder deposited on a fiber that we call the sleeve configuration. This configuration corresponds to a deposited layer observed at a time much smaller than the Rayleigh-Plateau characteristic time or when the instability is inhibited. To do so, we will develop an analytical model to calculate the evaporation, valid for small and large aspect ratios. These predictions are supplemented by a full 3D numerical simulation using finite element method. In the second part, we study the evaporation of a droplet on a fiber which corresponds to destabilized coating on a fiber. More precisely, we restricted the study to the axisymmetric barrel situation which is the common case, in particular for small contact angles of the liquid on the fiber. We first numerically calculate the shape of a droplet around a fiber using Surface Evolver [START_REF] Brakke | The surface evolver[END_REF], a surface minimization algorithm, whose results can be compared to the prediction by Carroll [START_REF] Carroll | The accurate measurement of contact angle, phase contact areas, drop volume, and laplace excess pressure in drop-on-fiber systems[END_REF] and Chou et al. [START_REF] Chou | Equilibrium phase diagram of drop-onfiber: Coexistent states and gravity effect[END_REF] in absence of gravity. By using finite element computations, we calculate the evaporation flux around the droplet, and show that except for very small liquid volume, the evaporation rate of the droplets is not significantly affected by the presence of the fiber, whatever the liquid contact angle is.

Sleeve. -We consider a sleeve of liquid of length 2L and of radius a as depicted in figure 1(a), such that the liquid cylinder has the same radius as the fiber. A natural geometrical parameter is λ = L/a. We assume that the liquid evaporation occurs in a diffusion-limited process in the stationary regime. Thus, the concentration field c in the gas phase is the solution of the Laplace equation △c = 0, which reads in cylindrical coordinates We provide in Supplementary Materials the derivation of analytical solutions to equation 1 in the limits of small and large aspect ratios λ. The difficulty associated with this linear boundary-value problem (BVP) resides in the discontinuity of the boundary conditions (BC) at the inner boundary r = a. More precisely, two different types of BCs, namely a Dirichlet-type BC and a Neumann-type BC are applied on disjoint complementary subdomains of the cylindrical surface located at r = a. This difficulty prevents the use of the classical Fourier-Hankel analysis of the problem and, instead, the BVP is reduced to a set of dual integral equations. Following the theory of Sneddon [START_REF] Sneddon | Fourier transforms[END_REF], these coupled integral equations are then reduced to a single integral equation with a weak, logarithmic singularity that is solved analytically in the asymptotic cases λ ≪ 1 and λ ≫ 1. From the resulting concentration field c(r, z), we can compute the local evaporative flux j(z) defined as j(z) = -D ∂c ∂r r=a , where D is the diffusion coefficient of the vapor in the gas phase.

We also propose to solve equation 1 by using finite element method implemented in the proprietary software COMSOL multiphysics using Transport of Dilute Species physics in axisymmetric 2D geometry. To compute accurately j(z) while keeping the computational time reasonable, we divide the atmosphere in three concentric domains centered on the liquid. The maximum mesh size in these domains is chosen according to the variations of concentration: the area near the three-phase contact is meshed with more refinement to capture properly the divergence of the concentration gradient at the contact line, and as the distance to the fiber increases, the maximum mesh size allowance is increased. The size of the box describing the atmosphere is at least one hundred times larger than the largest of the lengths of the system in order to be considered as infinite with respect to the above boundary conditions. With this approach, the computational time is reduced to few hours for the largest aspect ratios. The numerical computations are performed for a vapor saturating concentration c sat = 1.8 • 10 -2 kg/m 3 , a vapor concentration far from the liquid c ∞ = 0 and a diffusion coefficient of vapor in air D = 2.36 • 10 -5 m 2 /s [START_REF] Rankin | Handbook of chemistry and physics[END_REF]. These values are chosen to be those of water evaporating at 20 • C and zero relative humidity R H = c ∞ /c sat . The evaporative flux being proportional to Dc sat (1 -R H ), these results can be easily generalized for any value of R H , D and c sat . Sleeves of different aspect ratios are generated by changing independently the length and the radius.

Results. In the limit λ ≪ 1, we obtained analytically the local evaporative flux

j small (z) = j small 0 1 - z 2 L 2 -1/2 , (2)
where j small 0 is the local flux at z = 0 defined as

j small 0 = D(c sat -c ∞ ) L(1 -2γ e -ln λ 4 ) , (3)
with γ e ≈ 0.577, the Euler gamma constant. Because the geometry is not reduced to a single lengthscale, equation 3 indicates that the flux depends both on the sleeve length L and the aspect ratio λ, which is found numerically as shown in the Figure 2a. We find a good agreement between the numerical results and equation 3.

Equation 2 indicates that the flux diverges at the edge between the liquid and the solid. This behavior is confirmed numerically as observed in the Figure 2b where numerical results obtained for various λ are compared to the analytical prediction of equation 2 represented in solid black line. This divergence is also obtained for sessile droplets, which indeed exhibit the same diverging expression in the limit θ → 0 [START_REF] Deegan | Capillary flow as the cause of ring stains from dried liquid drops[END_REF].

We also solve analytically equation 1 for infinitely long sleeves i.e. λ → ∞. Under this hypothesis, the system is invariant by translation along the fiber axis such that the local flux is uniform. We can write

j large 0 = D(c sat -c ∞ )π 2a 2 -2γ e + ln 2 + π 2 ln λ . ( 4
)
We extend this result to large but finite aspect ratios by arguing that the local evaporative flux remains mostly uniform, such that j large (z) ≃ j large 0 . Again, we observe a good agreement between numerical and analytical results for the local flux at the center of the sleeve j 0 as shown in Fig. 2a. As for small λ, because of the cylindrical geometry, numerical results (Fig. 2a) and equation 4 indicate that the flux depends on both the sleeve radius a and the aspect ratio λ.

For λ > 1, Figure 2b shows that the increase of λ leads to a localization of the divergence at the contact line. For the largest tested aspect ratios, the flux per unit surface is nearly constant along the sleeve except close to the contact line. However, as shown in the inset of Figure 2b, we still find close to the contact line the classical minus one-half power-law divergence even for large aspect ratio. Indeed, mathematically, this divergence with a power -1/2 is the only one that is solution of equation 1 to describe the divergence of the local flux at the contact line. For λ = 5 • 10 3 , numerical uncertainties prevent us from concluding on the expression of the divergence, but we expect that the classical square root law holds.

The total evaporative flux is defined as Q = j(z) dS where the integral is taken over the liquid-vapor surface area. In the limit of small aspect ratios, the total flux writes

Q small = 2D(c sat -c ∞ )aπ 2 1 -2γ e -ln λ 4 . (5)
In the limit of large aspect ratio, we neglect the contribution of the divergence. Indeed, this divergence is localized and contributes weakly once integrated to the total flux [START_REF] Boulogne | Coffee-stain growth dynamics on dry and wet surfaces[END_REF]. Thus, the total evaporative flux writes Q large = j large 0 4πaL, which gives

Q large = 2D(c sat -c ∞ )λaπ 2 2 -2γ e + ln 2 + π 2 ln λ . ( 6
)
As shown in figure 3, where we plot the dimensionless total flux Q/(2D(c sat -c ∞ )aπ 2 ) as a function of the aspect ratio λ, we observe an excellent agreement between the numerical results and the analytical model. The good agreement between numerical results and theory at large aspect ratio demonstrates the small effect of the edge contribution to the local flux, which is due to the localization of the divergence when λ increases. Indeed, the more localized the divergence, the more constant the local flux j can be considered and the less effect the variation of j with z has on the total integrated flux Q.

As a final note, the model described here is valid asymptotically for λ ≪ 1 and λ → ∞ but still provides an excellent description of both the local flux in the center of the liquid j 0 (Fig. 2a) and the total evaporative flux Q (Fig. 3) for λ ∼ 1.

In the next section, we present the method to perform the numerical computation of equation 1 with the associated boundary conditions for axisymmetric droplets on fibers. We then discuss these results and compare them to those obtained for sleeves and spherical drops.

Axisymmetric drop on a fiber. -Due to the curvature of the substrate, a droplet of perfectly wetting liquid put on a fiber does not necessarily adopt a sleeve morphology and a macroscopic axisymmetric drop with a vanishing contact angle can exist [START_REF] Carroll | The accurate measurement of contact angle, phase contact areas, drop volume, and laplace excess pressure in drop-on-fiber systems[END_REF][START_REF] Mchale | Wetting of a high-energy fiber surface[END_REF][START_REF] Mchale | Estimation of contact angles on fibers[END_REF][START_REF] Mchale | The shape and stability of small liquid drops on fibers[END_REF][START_REF] Mchale | Global geometry and the equilibrium shapes of liquid drops on fibers[END_REF][START_REF] Chou | Equilibrium phase diagram of drop-onfiber: Coexistent states and gravity effect[END_REF]. This morphology is called a barrel-shaped droplet and is stable for low contact angle θ and quite large volume with respect to a 3 . For small volume and/or large contact angle a droplet on a fiber adopt a clam-shell morphology. Here, we choose to focus on barrel shaped droplets. To study the evaporation of such droplets, we vary independently the drop volume Ω at constant fiber radius, a = 125 µm, and the contact angle θ. In contrast to the sessile droplet for which the system is completely defined by the choice of only two parameters, Ω, and θ, the equilibrium profile of a drop on a fiber depends on Ω, θ and a. Once two dimensionless parameters, say θ and Ω/a 3 are chosen, the liquid adopts its equilibrium shape as represented in figure 4a, where the profile of the drop is defined by h(z). We denote the height of the liquid at the apex h 0 = h(z = 0) and the wetted length 2L. We define two dimensionless parameters, the dimensionless length λ = L/a and the drop aspect ratio H = h 0 /L. The barrel shape morphology exists for a limited dimensionless parameter space (λ, H), which restricts the studied range of these parameters. Due to the complexity of the problem arising from the drop shape, this part of the study is performed numerically.

λ = 5E-03 λ = 1E-02 λ = 2E-02 λ = 5E-02 λ = 1E-01 λ = 2E-01 λ = 5E-01 λ = 1E+00 λ = 2E+00 λ = 5E+00 λ = 1E+01 λ = 2E+01 λ = 5E+01 λ = 1E+02 λ = 2E+02 λ = 5E+02 λ = 1E+03 λ = 2E+03 λ = 5E+03 10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4 L -|z| (m) 10 1 10 2 j (kg.m -2 .s -1 ) -1/2
In the next section, we present the method to get numerically the drop equilibrium profile and then to solve equation 1 for this system. Numerical procedure. In contrast to the numerical resolution of the sleeve configuration, our numerical procedure for the drop on fibers is decomposed in two steps. First, we used Surface Evolver [START_REF] Brakke | The surface evolver[END_REF], a computer program that minimizes the energy of a surface subject to constraints, to obtain the meshed surface of the drop on a fiber. The parameters are the dimensionless drop volume Ω/a 3 and the liquid contact angle θ. The convergence of the minimization process is testified by comparing the drop height h 0 and length L (Fig. 4) with the analytical predictions made by Carroll [START_REF] Carroll | The accurate measurement of contact angle, phase contact areas, drop volume, and laplace excess pressure in drop-on-fiber systems[END_REF]. Illustrations of the resulting shapes are presented in figure 4(a-c). The second step consists in computing the vapor concentration field with COMSOL multiphysics in a similar manner as for the sleeves except that the simulations are performed in 3D in order to easily import the profiles obtained with Surface Evolver. With COMSOL, the Surface Evolver meshes are converted in 2D to keep only the boundary surface. A 3D drop is then reconstructed to ensure the compatibility with other elements of the geometry and to mesh properly the contact between the drop and the fiber. The boundary conditions are the same as for the sleeve, the vapor concentration is equal to the saturating vapor concentration near the interface and is constant far from the droplet i.e. c(r = h, z) = c sat , lim r→∞ c = c ∞ . Initial conditions are c = c ∞ for the concentration in the atmosphere and c = c sat inside the drop. There is no vapor flux normal to the surface of the fiber ∂c ∂r r=a = 0 for |z| > L (Fig. 4a). As for the sleeve case, the dimension of the box describing the atmosphere is taken to be at least one hundred times larger than the longest lengths of the system to ensure c = c ∞ far from the droplet.

Results. First, we want to understand how the presence of the fiber influences the evaporation speed. To do so, we obtained numerically the total evaporative flux Q of barrel-droplets on fibers of radius a = 125 µm. Numer- ical simulations are performed for various drop volumes Ω and contact angles θ. We recall that we limit ourselves to cases of barrel-shaped droplets, which only exists for drops of sufficiently large volumes that are placed on hydrophilic substrates [START_REF] Chou | Equilibrium phase diagram of drop-onfiber: Coexistent states and gravity effect[END_REF].

The results are shown in figure 5a where we plot Q as a function of Ω 1/3 and in figure 5b representing the dimensionless evaporative flux as a function of the contact angle. The total evaporative flux of a spherical drop [START_REF] Langmuir | The evaporation of small spheres[END_REF],

Q sphere = (48π 2 ) 1/3 D (c sat -c ∞ ) Ω 1/3 , (7)
is also represented in dashed black line for comparison. We observe that the evaporation rate of a barrel-shaped drop on a fiber is similar to that of a sphere for all the contact angles studied except for the smallest volumes tested. This means that varying the contact angle has a negligible effect on the evaporative flux except for small volume drops. We also note that, for small volumes, the deviation from the sphere is increasing as the contact angle is getting smaller, the largest variation being observed for the 0.01 µL drop having a contact angle of 10 • whose profile is given in figure 4b. From Fig. 5b, we can quantify the deviation from the sphere, which varies from a few percent for Ω ≥ 0.1L to around 35% for θ = 10 • , Ω = 0.01 µL We can also note that decreasing contact angle has a small effect on the drop aspect ratio H except for small volume drops. This is due to the curvature of the substrate that allows the existence of a macroscopic drop (H ∼ 1) even for small contact angles.

To understand these observations in more details, we consider the local evaporative flux of barrel-shaped droplets. First, we focus on the effect of the contact angle. In figure 6a, we plot the local flux j as a function of the position along the interface z for a droplets of volume 1 µL with various contact angle corresponding to aspect ratios H ≃ 1. This figure shows that the flux j diverges in close vicinity of the drop contact line. The decrease of the contact angle has a very small effect on the local evaporative flux and this effect is significant only near the contact line.

We also compare the results obtained for barrel drops for which λ ∈ [5; 10] with those of the sleeves having the same dimensionless length. These sleeves have an aspect ratio λ > 1, which means that the divergence of the local flux has already the localization effect described in the previous Section. The comparison between the drops and the sleeves highlight the effect of the drop profile curvatures. The longitudinal curvature of the surface is estimated by H, with H = 1 corresponding to a spherical drop whereas H = 0 is a sleeve. Fig. 6a demonstrates that, due to the curvature of the drop surface, the divergence of the evaporation flux of a drop on a fiber is even more localized at the triple line than for the sleeve. We propose a phenomenological equation for the local flux j(z) of drops on fibers defined as

j(z) = j 0 β 1 - z 2 L 2 -α + (1 -β) , (8)
where α and β are positive adjustable parameters. Finally, to understand the difference observed between the total evaporative flux of a spherical drop and the one of a small volume wetting barrel-drop on a fiber, we plot in figure 6 the local evaporative flux as a function of the position along the interface for drops of small contact angle θ = 10 • and different volumes. This difference is significant for small volumes and small contact angles, i.e. when H vanishes. In this case, the geometry is similar to a to liquid cylinder and we obtain results comparable to those expected for a sleeve. Nevertheless, the drop, which has a dimensionless length λ ≈ 2.3 can be well compared to the sleeve for a greater aspect ratio, λ = 5. We interpret that although h 0 tends to 0, the liquid thickness still has a significant effect on the evaporative flux.

Conclusion. -In this paper, we studied the curvature effect on the evaporation rate of a liquid deposited on a fiber when the evaporation is isothermal and purely diffusive. Two particular situations where analyzed: a sleeve of liquid of size L deposited on fiber of radius a and a droplet of volume Ω and wetting contact angle θ on the same fiber. The sleeve is obtained when a liquid fiber is coated by a liquid layer before the onset of destabilization of the Rayleigh-Plateau instability. The droplet, and more particularly the axisymmetric barrel shape that we studied here, is encountered in the late stage of destabilization when the liquid coating has been destabilized in a series of liquid pearls.

For the evaporation of the sleeve we obtained an analytical calculation of the local evaporation rate along the sleeves and the full evaporation rate in the case of small and large aspect ratios. For large aspect ratios, the evaporation rate is almost uniform along the sleeve except near the edges, where we recover the same power divergence than the one observed for sessile droplets at low contact angles. For small aspect ratio, the role of the edges progressively becomes more and more important, the evaporation rate is varying significantly along the sleeve since the power law divergence invades all the sleeve. We compared our analytical calculation to finite element computations and shown that our asymptotic calculations captures quantitatively the simulations even in the regime where the aspect ratio is close to one.

For the evaporation of the liquid axisymmetric barrels, we performed numerical simulations in order to calculate precisely the effect of the fiber on the evaporation rate. We have evidenced that, for drops of volume Ω ≥ 0.1 µL corresponding to dimensionless volume Ω/a 3 ≥ 50, the evaporation rate is almost independent of the wetting contact angle and that the droplet evaporates as a sphere of the same volume. More precisely, the evaporation flux diverges near the triple line, but due to the barrel shape, the divergence is strongly localized close to the edges and the evaporation rate is nearly constant. Indeed the divergence close to the edge compensates fortunately the fact that the longitudinal curvature of the surface at the apex, estimated by H, is not equal to the one of a spherical drop.

This study provides precise calculations that capture the drying dynamics of the two important morphologies on a fiber under the assumption of an isothermal evaporation process. They both emphasize the localization of the evaporation divergence close to the triple line, in contrast to sessile drops. To complete this analysis of the evaporation of a liquid coated on a fiber, it is necessary in a future work to focus more precisely on what happens after the destabilization of the sleeve into a series of regularly spaced drops. Indeed, additional studies would be necessary to understand and quantify the mutual influence of the drops on each other.

1 r ∂ ∂r r ∂c ∂r -k 2 c = 0. (6)
This equation can be solved by a standard separation of variable and has two independent Bessel-like solutions but only one of them vanishes in the limit r → ∞ and thus satisfies [START_REF] Derjaguin | On the thickness of the liquid film adhering to the walls of a vessel after emptying[END_REF]. This solution can be written as

c(k, r) = C(k)K 0 (kr), (7)
where K 0 is the modified Bessel function of the second kind of integer order 0. The remaining task is to find the function C(k) such that (3) and ( 4) are satisfied. Plugging the transform (5) together with the solution (7) into the two boundary conditions ( 3) and ( 4) leads to the following system know as dual integral equations

∞ 0 C(k)K 0 (k) cos(kz)dk = 1, for z < λ, (8)
∞ 0 kC(k)K 1 (k) cos(kz)dk = 0, for z > λ, (9)
where K 1 is the modified Bessel function of the second kind of integer order 1. Introducing the function

κ(k) = K 0 (k) K 1 (k) , (10)
as well as the function

D(k) = k C(k)K 1 (k)
, the dual integral equations (8, 9) can be recast in the canonical form:

∞ 0 κ(k) k D(k) cos(kz)dk = 1 for z < λ, (11)
∞ 0 D(k) cos(kz)dk = 0, for z > λ. (12)
In order to solve [START_REF] Mchale | Wetting of a high-energy fiber surface[END_REF][START_REF] Mchale | Estimation of contact angles on fibers[END_REF], we choose the singular integral equation approach and start by noticing that the second equation ( 12) is automatically satisfied if we define the auxiliary function g(u) as:

D(k) = λ 0 g(u) cos(ku)du (13)
= 1 0 λg(u) cos(λku)du, (14)
where g(u) = g(λu). The result above follows directly from Fourier's inversion theorem. Substituting ( 14) into (11) leads to a double integral equation, which reads, after inverting the order of integration and introducing the reduced variable z ′ = λz:

1 0 g(u) ∞ 0 κ(k) k cos(λkz ′ ) cos(λku)dk du = 1 λ , for z ′ < 1. (15)
Using usual trigonometric relations and the fact that g is an even function, we may recast (15) as:

1 -1 g(u)L(λ(z ′ -u)) = 2 λ for |z ′ | < 1, (16)
where the kernel L(x) is given by the following integral:

L(x) = ∞ 0 κ(k) k cos(kx)dk. (17)
Unfortunately, the kernel ( 17) cannot be expressed in closed form. In order to make further progress, let us focus on the asymptotic behavior of equations ( 16)- [START_REF] Cazabat | Evaporation of macroscopic sessile droplets[END_REF].

1.3 Small sleeve λ ≪ 1

In the limit where the length of the sleeve is much smaller than the radius of the sleeve, i.e. when λ ≪ 1, we also have λ(z ′u) ≪ 1 since {z ′ , u} ∈ [-1, 1]. We may thus use an approximation of L(x) valid at small x. In order to obtain such an approximation, we first split the kernel [START_REF] Cazabat | Evaporation of macroscopic sessile droplets[END_REF] into two integrals:

L(x) = 1 0 κ(k) k cos(kx)dk + ∞ 1 κ(k) k cos(kx)dk (18)
Next we replace κ(k) by its small-k approximation (resp. large-k approximation) in the first (resp. second) integral above. Using the following approximations at small and large k:

κ(k) = -γ e -ln k 2 + O k 2 ln k 2 for k → 0 κ(k) = 1 + O 1 k for k → ∞
we obtain the following approximate expansion for the kernel L(x)

L(x) ≈ -ln |x| + n=∞ n=0 b n |x| n for x ≪ 1 (19)
where the first few terms of the development are given in table 1.3 and γ e = 0.577... is the Euler gamma constant. Note that the value of the coefficient b n 's depends on the underlying approximation for κ. We now look for a series solution of ( 16) in the following form:

g(x) = 1 λ ∞ n=0 λ n g n (x, ln λ) (20)
Inserting the previous relations [START_REF] Lebedev | Special functions and their applications[END_REF] and ( 20) into the singular integral equation, we obtain the following recurrent chain of integral equation defined in the domain |z ′ | < 1:

1 -1 g n (u, ln λ) ln |z ′ -u| -b 0 + ln λ du = f n (z ′ , ln λ), (21)
where the right hand side of ( 21) is given by:

f 0 (z ′ , ln λ) = -2, f n (z ′ , ln λ) = n m=1 1 -1 b m |z ′ -u| m g n-m (u, ln λ)du.
Introducing the parameter β = -b 0 + ln λ as well as the following transforms:

z ′ = e -β w, u = e -β τ, g n (u, ln λ) = G n (τ, ln λ), f n (z ′ , ln λ) = e -β F n (w, ln λ),
we obtain the following recurrent chain of singular integral equations defined for all w in the interval [-e -β , e -β ]:

e β -e β G n (τ, ln λ) ln |w -τ |du = F n (w, ln λ) (22)
that we recognize as the Carleman integral equation used in various problems of hydrodynamics, elasticity, etc. Its solution is given by:

G n (τ, ln λ) = 1 π 2 (τ + e β ) (
We can find that the first terms in the solution of the original equations ( 16)-( 17) are

g(z ′ ) = 2 λπ(1 -2γ e -ln λ 4 ) √ 1 -z ′2 + λ (7 + 6γ e -6 ln 2)((2z ′2 -1)(2γ e + ln λ 4 ) -2z ′2 ) 18π √ 1 -z ′2 1 -2γ e -ln λ 4 2 + O(λ 3 ). ( 24
)
We only provide here the first two terms as higher-order terms rapidly become rather lengthy expressions but we note that the exponent of the divergence of the flux near the edge of the sleeve is preserved in the development. We can now find the function D(k) using ( 14):

D(k) = J 0 (λk) 1 -2γ e -ln λ 4 -λ 2 (7 + 6γ e -6 ln 2)(J 0 (λk) 2-2γe-ln( λ 4 ) 1-2γe-ln( λ 4 ) -2J 1 (λk) λk ) 36 1 -2γ e -ln λ 4 + O(λ 3 ). ( 25
)
The quantity of interest being the total flux integrated over the surface of the sleeve, let us first recast the flux j small (z) per unit of surface of the sleeve in physical units,

j small (z) = D(c sat -c ∞ ) a ∞ 0 D (k) cos k z a dk, (26)
which becomes, after integration (we only provide here the integration of the first term in the expansion of D(k)):

j small (z) = D(c sat -c ∞ ) (1 -2γ e -ln λ 4 ) √ λ 2 a 2 -z 2 . ( 27
)
Now the total flux Q small is found by integrating the flux per unit surface over the surface of the sleeve

Q small = λa -λa dz 2π 0 adθj small (z), (28)
and we find immediately that 1.4 Large sleeve λ ≫ 1

Q small = 2D(c sat -c ∞ )aπ 2 1 -2γ e -ln λ 4 . (29)
The asymptotic result for the total diffusive flux in the limit of large sleeves λ ≫ 1 can be found in an approximate fashion. Let us first introduce the scaled variables u ′ = λu and z ′′ = λz ′ and rewrite ( 16) as:

1 0 g(u ′ /λ)L(z ′′ -u ′ )du ′ + λ 1 g(u ′ /λ)L(z ′′ -u ′ )du ′ = 1 for |z ′′ | < λ (30)
Noting that the kernel L(x) has the following expansions at small and large x:

L(x) ≈ 1 -2γ e -ln |x| 2 for x → 0 L(x) ≈ π 2|x| for x → ∞,
we now replace the kernel L(x)) appearing in the first (resp. second) integral above by its small x (resp. large x) approximation to obtain:

1 0 g(u ′ /λ) 1 -2γ e -ln |z ′′ -u ′ | 2 du ′ + λ 1 πg(u ′ /λ) 2|z ′′ -u ′ | du ′ = 1 for |z ′′ | < λ. (31)
In the limit of an infinitely long sleeve, we expect the evaporative flux to become invariant by translation along the axis of the fiber. In order to find this constant value, we set z ′′ = 0 in (31) and we look for a constant solution for g, which simply reads after integration

g(z ′ ) = 1 2 -2γ e + ln 2 + π 2 ln λ . ( 32
)
Inserting this solution into [START_REF] Mchale | Global geometry and the equilibrium shapes of liquid drops on fibers[END_REF], we find the function D(k) in the limit of large sleeves:

D(k) = λ sinc(λk) 2 -2γ e + ln 2 + π 2 ln λ , (33)
where sinc(x) = sin(x)/x is the cardinal sine function. This leads to the following expression for the constant evaporation flux:

j large (z) = D(c sat -c ∞ )π 2a 2 -2γ e + ln 2 + π 2 ln λ . ( 34
)
Integrating the flux per unit surface over the surface of the sleeve, we finally find the total flux in the large sleeve limit:

Q large = 2λD(c sat -c ∞ )aπ 2 2 -2γ e + ln 2 + π 2 ln λ . ( 35
)

Summary

To summarize, we have the two following limits for the total diffusive flux:

Q =    2D(csat-c∞)aπ 2 1-2γe-ln( λ 4 ) , λ ≪ 1 2λD(csat-c∞)aπ 2 2-2γe+ln 2+ π 2 ln λ , λ ≫ 1. (36)
As a final note, let just add that the results above are valid in the limit where the thickness of the diffusive layer is much larger than any other length scale of the problem (namely a and λa). If this thickness becomes comparable to any of those lengths, we expect a saturation of the diffusive flux and the emergence of a different scaling involving this new length.

Comparison between analytical description and numerical results

Figure 1 shows numerical results of the local evaporative flux of sleeves of radius a = 125 µm and various lengths. In black lines are plotted the predictions obtained by the model developed here, equation 27 for small aspect ratio sleeves (solid line) and equation 34 for large aspect ratio sleeves (dashed line). obtained from numerical computations for a = 125 µm. The black solid line corresponds to the local flux described by equation ( 27) made dimensionless by its value at the center of the fiber (z = 0). The black dashed line corresponds to j = j 0 as predicted for infinitely long sleeve (Eq. ( 34)).

10 0 10 1 1 -z 2 L 2 -1/2 10 0 10 1 j/j 0 λ = 5E-03 λ = 1E-02 λ = 2E-02 λ = 5E-02 λ = 1E-01 λ = 2E-01 λ = 5E-01 λ = 1E+00 λ = 2E+00 λ = 5E+00 λ = 1E+01 λ = 2E+01 λ = 5E+01 λ = 1E+02 λ = 2E+02 λ = 5E+02 λ = 1E+03 λ = 2E+03 λ = 5E+03 Theoretical prediction λ 1 Theoretical prediction λ 1
2 Evaporation of axisymmetric drops on fiber 2.1 Comparison between j 0 and j sphere In the figure 2a, we show comparison between the local evaporative flux of barrel-shaped droplets on fiber of radius a = 125 µm and local evaporative flux of a spherical drop (j sphere in black dashed line). The volume of the drops is Ω = 1 µL. In the figure 2b, we plot j/j sphere as a function of Ω -1/3 for three different contact angles. The spherical case is in black dashed line.

Fitting expression for j = f (z)

Figure 3 shows the dimensionless local flux as a function of the dimensionless position along the interface for barrel-shaped droplets on fiber. The volume of the drops is Ω = 1 µL and the fiber radius is a = 125 µm. We propose to fit curves of the figure 3 (black dashed lines) with :

j(z) = j 0 β 1 - z 2 L 2 -α + (1 -β) , (37)
where α and β are positive adjustable parameters.

Fig. 1 :

 1 Fig. 1: (a) Notations used to describe a sleeve of length 2L on a fiber of radius a. The dimensionless length is defined as λ = L/a. Boundary conditions used to solve equation 1 as well as the coordinate system are also represented schematically. Vapor concentration field for (b) λ = 5 • 10 -3 and (c) λ = 500, obtained from COMSOL for sleeves of radius a = 125 µm.

( 1 )

 1 In this geometry, the boundary conditions are (a) a satu-rated vapor concentration c sat in the vicinity of the interface, i.e. c(r = a, z) = c sat for |z| < L, (b) the absence of evaporative flux at the surface of the fiber, i.e. ∂c ∂r r=a = 0 for |z| > L, and (c) a constant concentration far from the liquid, lim r→∞ c = c ∞ .

a

  = 12.5 (µm) a = 39.5 (µm) a = 125 (µm) -1.00 -0.75 -0.50 -0.25 0

Fig. 2 :

 2 Fig. 2: (a) Local evaporative flux the center of the sleeve aj0 as a function of the dimensionless sleeve length λ = L/a obtained from numerical computations for various sleeves lengths L and three different radii a (see caption). The black solid line corresponds to equation 3 and the black dashed line corresponds to equation 4 both multiplied by a. In the inset, the local flux at the center of the sleeve j0 is plotted as a function L -1 for the three sleeves radii mentioned above. (b) Dimensionless flux density j/j0 as a function of dimensionless position z/L along the sleeve. The points are obtained from numerical computation for a = 125 µm and various sleeves lengths. The black solid line corresponds to equation (2) nondimensionalized by equation 3. The black dashed line corresponds to j = j0 as predicted for infinitely long sleeve (Eq. 4). In the inset, the flux j is plotted as a function of the distance to the contact line for a = 125 µm and various sleeves lengths. The curves are arbitrarily shifted for clarity. The color gradient represents the length L of the sleeve on both graphics.

Fig. 3 :

 3 Fig. 3: Total dimensionless flux as a function of the dimensionless sleeve length λ = L/a obtained from numerical computations for a = 125 µm. The solid black line corresponds to equation 5 and the dashed black line to equation 6.

Fig. 4 :

 4 Fig. 4: Surface Evolver profiles of axisymmetric drops on fiber of radius a = 125 µm. (a) Drop of volume Ω = 1 µL and contact angle θ = 10 • . The wetted length is L and the height of the drop h0 is defined between the surface of the fiber and the apex of the drop. (b) Drop of volume Ω = 0.01 µL and contact angle θ = 10 • . (c) Drop of volume Ω = 1 µL and contact angle θ = 45 • . (d) Vapor concentration field from COMSOL for profile given in (a) a = 125 µm, Ω = 1 µL, θ = 10 • .

  01 µL Ω = 0.1 µL Ω = 0.25 µL Ω = 0.5 µL Ω = 1.0 µL Ω = 10.0 µL Ω = 100.0 µL Ω = 1000.0 µL

Fig. 5 :

 5 Fig. 5: (a) Total evaporative flux of droplets on fiber as a function of Ω 1/3 . (b) Total dimensionless evaporative flux as a function of contact angle. The points are numerical results for evaporating drops on fiber for different contact angles. Colors indicate the variation of the droplet aspect ratio H = h0/L. The dashed black line is the evaporative flux of a sphere (Eq. 7) in the same condition as the numerical simulations.

Hθ = 10 • 5 θ 5 Fig. 6 :

 10556 Fig. 6: Dimensionless evaporative flux per unit surface j/j0 as a function of 1 -(z/L) 2 . The points are obtained by numerical simulations for droplets of volume Ω and contact angle θ placed on a fiber of radius a = 125 µm. The dimensionless wetted length λ = L/a of the droplets are also given and the results for sleeves with the same λ are plotted for comparison. The colors represent the droplet aspect ratio H = h0/L. (a) Results obtained for Ω = 1 µL and contact angle varying from 10 to 80 • corresponding to λ ranging from 5 to 8 and aspect ratio H between 0.5 and 0.8. (b) Results obtained for θ = 10 • and droplets volumes varying from 0.01 µL to 1 µL corresponding to λ ranging from 2 to 8 and aspect ratio H varying from 0.1 for the smallest volume tested to 0.5.

Fig- ure 4

 4 provided in the Supplementary Materials shows fitted data for 1 µL drops for different contact angle. The good agreement between numerical results and fitted curves shows that, unlike sessile drops, the local evaporative flux of a drop on a fiber cannot be described by a simple power law. If we compare the local flux of barrelshaped droplets to the local flux of a spherical droplet (Fig 2a-b in Supplementary Materials) we see that the local flux at the center of the barrel-shaped droplets is about 1.5 times smaller than the evaporative flux of a sphere. Thus, barrel-shaped drops are evaporating at the same speed as spherical droplets because the difference in local fluxes at the center of the drop compensates the localized divergence of the local flux of barrel-shaped droplets.

F

  n (s, ln λ)(s + e β ) (e βs)ds .

Figure 1 : 2 L 2 - 1 / 2

 12212 Figure 1: Dimensionless flux density j/j 0 as a function of 1 -z 2 L 2

- 1 .Hθ = 10 •HθFigure 2 : 3 1/ 3 Dθ = 10 • 5 θ = 20 •θθθ 3 θ

 110233105203 Figure 2: (a) Local flux j as a of dimensionless position along the air/liquid interface z/L for barrel-shaped droplets on fibers, a = 125 µm, Ω = 1 µL and θ is varied between 10 and 80 • . (b) Local flux at the center of the drop j 0 divided by local flux of a sphere evaporating under the same experimental conditions as a function of Ω -1/3 . Black dashed lines represent local flux of a sphere j sphere = 4π 3 1/3 D (c satc ∞ )Ω -1/3 . Colors represent the aspect ratio H = h 0 /L of barrel-shaped drops.

Figure 3 :

 3 Figure 3: Dimensionless flux density j/j 0 as a function of 1 -(z/L) 2 . The points are obtained from numerical computation for drops of volume 1 µL placed on a fiber of radius a = 125 µm and various contact angles (see captions). The results are fitted by equation (37) (black dashed line) where α and β are given in the caption for each contact angle.

  e βτ )

	×	e β -e β	(s + e β ) (e β -s) ∂Fn ∂w | w=s s -τ	ds +	1 ln (2e β /4)	e β -e β

Table 1 :

 1 Values of the first b n 's coefficients in equation[START_REF] Lebedev | Special functions and their applications[END_REF].

	b 0 1 -2γ e + ln 2 b 1 0
	b 2	7+6γe-6 ln 2 36
	b 3 0
	b 4	-21-20γe+20 ln 2 2400

We thank Saint-Gobain and ANRT for funding this study and J. Delavoipière, M. Lamblet, and B. Sobac for useful discussions. We are also grateful to C. Dubois from COMSOL for providing support.

Supplementary materials: Evaporation of a liquid coated on a fiber In these materials, we report the details of our calculations of the Laplace equation in Section 1. Then, we show additional data obtained numerically on the evaporative flux in Section 2.

Evaporation of liquid sleeves

In this Section, we solve the diffusion equation of an evaporating liquid sleeve with the boundary conditions specified in the main article that we recall briefly here.

Problem

All lengths are normalized by the radius a of the fiber. The dimensionless total length of the sleeve is denoted 2λ. In physical units, its total length is thus 2λa. We also introduce the rescaled concentration c = (cc ∞ ) / (c satc ∞ ) which now has values in the interval [0, 1]. We thus solve the following boundary value problem:

= 0, for z > λ.

The difficulty associated with this linear boundary value problem resides in the discontinuity of the boundary condition at r = 1. More precisely, two different types of boundary conditions, namely a Dirichlet-type boundary condition (3) and a Neumann-type boundary condition (4) are applied on disjoint complementary subdomains of the surface defined as r = 1. This difficulty prevents the use of the classical Fourier-Hankel analysis of the problem and a dual integral approach must be employed. (

Solution -theoretical details

Substituting the transformation (5) in the Laplace equation [START_REF] Landau | Dragging of a liquid by a moving plate[END_REF] results in an ordinary differential equation for c(k, r),