Andr Es Ortigosa

Selena Gimenez-Ibanez

Nathalie Leonhardt

Roberto Solano
email: rsolano@cnb.csic.es

Design of a bacterial speck resistant tomato by CRISPR/Cas9-mediated editing of SlJAZ2

Keywords: coronatine, CRISPR/Cas9, JAZ2, pseudomonas, stomata, tomato

Due to their different lifestyles, effective defence against biotrophic pathogens normally leads to increased susceptibility to necrotrophs, and vice versa. Solving this trade-off is a major challenge for obtaining broad-spectrum resistance in crops and requires uncoupling the antagonism between the jasmonate (JA) and salicylate (SA) defence pathways. Pseudomonas syringae pv. tomato (Pto) DC3000, the causal agent of tomato bacterial speck disease, produces coronatine (COR) that stimulates stomata opening and facilitates bacterial leaf colonization. In Arabidopsis, stomata response to COR requires the COR co-receptor AtJAZ2, and dominant AtJAZ2Djas repressors resistant to proteasomal degradation prevent stomatal opening by COR. Here, we report the generation of a tomato variety resistant to the bacterial speck disease caused by Pto DC3000 without compromising resistance to necrotrophs. We identified the functional ortholog of AtJAZ2 in tomato, found that preferentially accumulates in stomata and proved that SlJAZ2 is a major co-receptor of COR in stomatal guard cells. SlJAZ2 was edited using CRISPR/Cas9 to generate dominant JAZ2 repressors lacking the C-terminal Jas domain (SlJAZ2Djas). SlJAZ2Djas prevented stomatal reopening by COR and provided resistance to Pto DC3000. Water transpiration rate and resistance to the necrotrophic fungal pathogen Botrytis cinerea, causal agent of the tomato gray mold, remained unaltered in Sljaz2Djas plants. Our results solve the defence trade-off in a crop, by spatially uncoupling the SA-JA hormonal antagonism at the stomata, entry gates of specific microbes such as Pto DC3000. Moreover, our results also constitute a novel CRISPR/Cas-based strategy for crop protection that could be readily implemented in the field.

Introduction

Pseudomonas syringae is a widespread bacterial pathogen that causes disease on a broad range of economically important plant species. Among them, P. syringae pv. tomato DC3000 (Pto DC3000) is the causative agent of the bacterial speck disease of tomato (Solanum lycopersicum) [START_REF] Blancard | Tomato Diseases[END_REF]. Susceptible tomato cultivars include the variety Moneymaker, which is agronomically and economically important. Outbreaks of bacterial speck on tomatoes occur in moderate temperatures (15-25°) and wet conditions [START_REF] Jones | Compendium of tomato diseases[END_REF]. Under favourable environmental conditions, disease symptoms appear as small brown necrotic spots (specks) in leaf and fruits [START_REF] Bender | Reduced pathogen fitness of Pseudomonas syringae pv. tomato Tn5 mutants defective in coronatine production[END_REF]. This disease affects negatively the productivity and marketability of the tomatoes [START_REF] Jones | Compendium of tomato diseases[END_REF] and causes economic losses all over the world [START_REF] Gitaitis | Incidence and development of Pseudomonas syringae pv. syringae on tomato transplants in Georgia[END_REF][START_REF] Schneider | Bacterial speck of tomato: sources of inoculum and establishment of a resident population[END_REF].

The initial infective process of P. syringae relies on natural openings and accidental wounds on the plant surface to colonize internal tissues [START_REF] Melotto | Role of stomata in plant innate immunity and foliar bacterial diseases[END_REF]). Stomata are an example of such openings, providing one of the main routes through which the foliar pathogen P. syringae can penetrate the leaf epidermis and start multiplying aggressively in the apoplast [START_REF] Melotto | Stomatal defense a decade later[END_REF]. Stomata are pores present on the surface of the leaves that are involved in gas exchange, regulating photosynthesis and water loss through transpiration stream [START_REF] Kim | Guard cell signal transduction network: advances in understanding abscisic acid, CO2, and Ca2+ Signaling[END_REF]. Besides this, stomata also play an active and dynamic role in defence against pathogens being an integral part of the plant innate immunity system [START_REF] Liu | RIN4 functions with plasma membrane H + -ATPases to regulate stomatal apertures during pathogen attack[END_REF][START_REF] Melotto | Plant stomata function in innate immunity against bacterial invasion[END_REF][START_REF] Melotto | Role of stomata in plant innate immunity and foliar bacterial diseases[END_REF][START_REF] Zhang | The plant innate immunity response in stomatal guard cells invokes G-protein-dependent ion channel regulation[END_REF]. Upon microbial perception, achieved by the specific recognition of conserved microbial associated molecular patterns (MAMPS) such as bacterial flagellin, via surface-localized receptors, plants rapidly close stomata [START_REF] Melotto | Plant stomata function in innate immunity against bacterial invasion[END_REF]. This closure requires the salicylic acid (SA) and abscisic acid (ABA) plant hormones, and inhibits the entry of P. syringae restricting host tissue colonization [START_REF] Du | Closely related NAC transcription factors of tomato differentially regulate stomatal closure and reopening during pathogen attack[END_REF][START_REF] Gimenez-Ibanez | JAZ2 controls stomata dynamics during bacterial invasion[END_REF][START_REF] Melotto | Plant stomata function in innate immunity against bacterial invasion[END_REF][START_REF] Zeng | A prominent role of the flagellin receptor FLAGELLIN-SENSING2 in mediating stomatal response to pseudomonas syringae pv tomato dc3000 in arabidopsis[END_REF][START_REF] Zhang | The plant innate immunity response in stomatal guard cells invokes G-protein-dependent ion channel regulation[END_REF]. Once in the apoplast, P. syringae encounters apoplastic plant immunity, which also relies on a plethora of plant hormones including SA and jasmonic acid (JA). In general terms, SA defences positively regulate resistance to biotrophic and hemi-biotrophic microbes such as P. syringae, whereas a combination of JA and ethylene (ET) pathways activates resistance against necrotrophic pathogens such as the fungus Botrytis cinerea [START_REF] Robert-Seilaniantz | Hormone crosstalk in plant disease and defense: more than just JASMONATE-SALICYLATE antagonism[END_REF]. SA and JA/ET defence pathways generally antagonize each other, and consequently, elevated resistance to biotrophs is often correlated with increased susceptibility to necrotrophs, and vice versa [START_REF] Glazebrook | Contrasting mechanisms of defense against biotrophic and necrotrophic pathogens[END_REF].

Pseudomonas syringae strains such as Pto DC3000 have evolved a refined strategy for manipulating hormonal crosstalk by producing coronatine (COR), a mimic of the bioactive JA hormone, JA-isoleucine (JA-Ile) (Fonseca et al., 2009a). By activating the JA pathway, COR inhibits SA-dependent defences and thus stimulates the reopening of stomata to facilitate bacterial invasion and growth in the apoplast [START_REF] Brooks | The Pseudomonas syringae phytotoxin coronatine promotes virulence by overcoming salicylic acid-dependent defences in Arabidopsis thaliana[END_REF][START_REF] Gimenez-Ibanez | JAZ2 controls stomata dynamics during bacterial invasion[END_REF][START_REF] Joardar | The Arabidopsis thaliana JASMONATE INSENSITIVE 1 gene Is required for suppression of salicylic acid-dependent defenses during infection by Pseudomonas syringae[END_REF][START_REF] Melotto | Plant stomata function in innate immunity against bacterial invasion[END_REF][START_REF] Melotto | Role of stomata in plant innate immunity and foliar bacterial diseases[END_REF][START_REF] Zheng | Coronatine promotes pseudomonas syringae virulence in plants by activating a signaling cascade that inhibits salicylic acid accumulation[END_REF]. The signalling cascade triggered by COR and JA-Ile is well established. COR is perceived by a co-receptor formed by the F-box protein COI1 (CORONA-TINE-INSENSITIVE 1) and JAZ (JASMONATE ZIM DOMAIN) repressor proteins [START_REF] Chini | The JAZ family of repressors is the missing link in jasmonate signalling[END_REF][START_REF] Katsir | COI1 is a critical component of a receptor for jasmonate and the bacterial virulence factor coronatine[END_REF][START_REF] Sheard | Jasmonate perception by inositol-phosphatepotentiated COI1-JAZ co-receptor[END_REF][START_REF] Thines | JAZ repressor proteins are targets of the SCFCOI1 complex during jasmonate signalling[END_REF][START_REF] Xie | COI1: an Arabidopsis gene required for jasmonate-regulated defense and fertility[END_REF]. JAZ co-receptors are COI1 substrates that negatively regulate the JA-signalling pathway by directly interacting with and repressing transcription factors (TFs) such as MYC2/3/4 that control JA-regulated genes [START_REF] Chini | The JAZ family of repressors is the missing link in jasmonate signalling[END_REF][START_REF] Andez-Calvo | The arabidopsis bHLH transcription factors MYC3 and MYC4 are targets of JAZ repressors and act additively with MYC2 in the activation of jasmonate responses[END_REF][START_REF] Lorenzo | JASMONATE-INSENSITIVE1 encodes a MYC transcription factor essential to discriminate between different jasmonate-regulated defense responses in Arabidopsis[END_REF][START_REF] Sheard | Jasmonate perception by inositol-phosphatepotentiated COI1-JAZ co-receptor[END_REF][START_REF] Thines | JAZ repressor proteins are targets of the SCFCOI1 complex during jasmonate signalling[END_REF]. Repression of TFs by JAZ is mediated by recruitment of the TOPLESS (TPL) corepressor complex, through the adaptor protein NINJA. JAZ also prevent the interaction of the TFs with the MEDIATOR complex through MED25 [START_REF] Cevik | MEDIATOR25 acts as an integrative hub for the regulation of jasmonate-responsive gene expression in arabidopsis[END_REF][START_REF] Chen | The arabidopsis mediator subunit MED25 differentially regulates jasmonate and abscisic acid signaling through interacting with the MYC2 and ABI5 transcription factors[END_REF][START_REF] Pauwels | NINJA connects the co-repressor TOPLESS to jasmonate signalling[END_REF][START_REF] Zhang | Structural basis of JAZ repression of MYC transcription in jasmonate signalling[END_REF]. Under stress conditions, COR or JA-Ile promote the formation of JAZ-COI1 complexes, triggering JAZ degradation via the 26S proteasome [START_REF] Chini | The JAZ family of repressors is the missing link in jasmonate signalling[END_REF][START_REF] Sheard | Jasmonate perception by inositol-phosphatepotentiated COI1-JAZ co-receptor[END_REF][START_REF] Thines | JAZ repressor proteins are targets of the SCFCOI1 complex during jasmonate signalling[END_REF]. This leads to derepression of the TFs that initiate the transcription of JAdependent genes, and repression of SA-dependent defences against the bacteria (Fonseca et al., 2009b;[START_REF] Gimenez-Ibanez | Nuclear jasmonate and salicylate signaling and crosstalk in defense against pathogens[END_REF]. Thus, COR acts as a potent virulence factor in plants by triggering the degradation of JAZs. Acquisition of COR by bacterial pathogens has been of tremendous adaptive importance during host-pathogen evolution because it has allowed bacteria to manipulate the host hormonal network to promote susceptibility.

Among JAZ repressors, we recently showed that AtJAZ2 is a major COR/JA-Ile co-receptor in Arabidopsis controlling stomata dynamics during bacterial invasion. JAZ2 is constitutively expressed at the stomata and hijacked by bacterially produced COR to suppress SA-dependent stomatal closure and promote bacterial penetration [START_REF] Gimenez-Ibanez | JAZ2 controls stomata dynamics during bacterial invasion[END_REF]. Arabidopsis jaz2 loss-of-function mutants are partially impaired in pathogeninduced stomatal closing and more susceptible to Pseudomonas. In contrast, truncated JAZ2 forms lacking the C-terminal Jas domain (JAZ2Djas) act as gain-of-function mutations that prevent stomatal reopening by COR and are highly resistant to bacterial penetration. The C-terminal Jas domain is responsible for the interaction with COI1 in the presence of hormone and thus, JAZ forms lacking this conserved domain are resistant to COR-induced degradation, overcoming the action of the phytotoxin during the entry process of P. syringae [START_REF] Gimenez-Ibanez | JAZ2 controls stomata dynamics during bacterial invasion[END_REF]. The fact that JAZ2 function is mostly restricted to the stomata makes that alterations in this gene do not affect the SA-JA crosstalk in other tissues and, therefore, jaz2Djas mutants still retain unaltered resistance against necrotrophs. Our work suggested that gain-of-function mutations in JAZ2 could be used as a general strategy to spatially uncouple SA-JA hormonal antagonism and to block the entry of P. syringae strains that produce COR without compromising resistance to necrotrophs, which is mostly apoplastic.

Genome editing technologies such as CRISPR/Cas9 (Clustered Regularly Interspaced Short Palindromic Repeats/CRISPR-associated endonuclease 9) enable precise and directed modifications of DNA sequences in vivo [START_REF] Schiml | Revolutionizing plant biology: multiple ways of genome engineering by CRISPR/Cas[END_REF]. CRISPR/Cas is emerging as an accurate method to improve crops because of its specificity, simplicity and versatility. The Cas9 protein functions as a nuclease and is directed to a target site by a specific guide RNA (gRNA), inducing site-specific double-strand breaks. This damage can be then repaired by non-homologous end-joining (NHEJ) or homologous recombination (HR), but in the case of NHEJ, the process of repair is error prone, which results in disruptive insertions or deletions at targeted loci that can result in translational frame shifts, amino acid replacements or deletions [START_REF] Mahfouz | Genome engineering via TALENs and CRISPR/Cas9 systems: challenges and perspectives[END_REF].

Genetic manipulation of defence pathways has succeeded in enhancing resistance to specific kinds of pathogens. However, due to the antagonistic interactions between the SA and JA defence pathways, efforts to develop plants with broad-spectrum resistance by manipulation of hormonal signalling genes has had limited success so far because enhancement of the SA-dependent defences leads to a reduction in JA-based resistance and vice versa [START_REF] Robert-Seilaniantz | Hormone crosstalk in plant disease and defense: more than just JASMONATE-SALICYLATE antagonism[END_REF][START_REF] Pieterse | Hormonal Modulation of Plant Immunity[END_REF]. Solving this trade-off is a major challenge in agriculture and requires uncoupling the antagonism between hormonal pathways in crops.

In this study, we report on a solution to this trade-off by spatially uncoupling the SA-JA antagonism at the stomata and generating a tomato (Moneymaker) resistant to the bacterial speck disease caused by the pathogen Pto DC3000, without compromising resistance to necrotrophic pathogens. We identified the functional ortholog of the COR stomatal co-receptor AtJAZ2 in tomato (SlJAZ2) and edited this gene with the CRISPR/ Cas9 system to generate truncated JAZ2 forms lacking the Cterminal Jas domain (SlJAZ2Djas). This edited gain-of-function SlJAZ2Djas mutant fully prevented stomatal reopening by COR and reduced bacterial entry through the stomata, increasing resistance to Pto DC3000 infection. Stomatal aperture in Sljaz2edited plants remained unaltered during transpiration. Moreover, since our strategy does not affect JA-signalling outside the stomata, Sljaz2Djas plants showed unaltered levels of resistance to the necrotrophic fungal pathogen B. cinerea, causal agent of the tomato gray mold. In addition to uncouple the SA-JA defence antagonism in a crop, our results constitute a novel strategy for crop protection using CRISPR/Cas that could be readily implemented in the field.

Results

SlJAZ2 is the ortholog of AtJAZ2

The AtJAZ2 protein belongs to the TIFY super-family that includes 12 canonical members in Arabidopsis. Similarly, 12 JAZ proteins have also been identified in tomato [START_REF] Chini | Identification of TIFY/JAZ family genes in Solanum lycopersicum and their regulation in response to abiotic stresses[END_REF][START_REF] Ishiga | Jasmonate ZIMdomain (JAZ) protein regulates host and nonhost pathogen-induced cell death in tomato and nicotiana benthamiana[END_REF][START_REF] Sun | Systemin/jasmonate-mediated systemic defense signaling in tomato[END_REF]. However, correlation between specific versions of JAZ genes among plant species is unclear. In order to identify the ortholog of AtJAZ2 in tomato we performed phylogenetic analysis using protein sequences of all JAZs from both plant species. AtJAZ2 protein grouped into a clade that included AtJAZ1, and the SlJAZ1 and SlJAZ2 proteins as the closest tomato orthologs (Figure 1a and Figure S1). SlJAZ3 and SlJAZ4 were also closer to AtJAZ2 than to any other tomato or Arabidopsis protein. Among proteins in this clade, AtJAZ1, AtJAZ2 and SlJAZ2 showed a remarkable similar length (Figure 1a). This phylogenetic analysis indicated that the ortholog of AtJAZ2 in tomato was likely SlJAZ2 or SlJAZ1, although SlJAZ3 and SlJAZ4 cannot be discarded.

JAZ2 is strongly expressed at stomatal guard cells compared to other JAZs in Arabidopsis [START_REF] Gimenez-Ibanez | JAZ2 controls stomata dynamics during bacterial invasion[END_REF]. Thus, we next analysed if any of the closest tomato candidate orthologs of AtJAZ2 were expressed at the stomata. To do this, we performed quantitative RT-PCR analyses that compared the expression levels of these genes between tomato whole leaf tissue and epidermal peels, which are enriched in guard cells. SlJAZ2 was the only gene enriched in the stomata-abundant epidermal peels fraction whereas SlJAZ1, SlJAZ3 and SlJAZ4 relative levels were similar among them (Figure 1b). This pinpointed SlJAZ2 as the functional tomato ortholog of AtJAZ2.

SLJAZ2 editing through CRISPR/Cas9

In Arabidopsis, a truncated form of AtJAZ2 lacking the Jas domain (AtJAZ2DJas) is resistant to AtCOI1-dependent degradation after COR treatment. Arabidopsis plants overexpressing this degradation resistant JAZ2 dominant form are insensitive to the phytotoxin COR and more resistant to DC3000 [START_REF] Gimenez-Ibanez | JAZ2 controls stomata dynamics during bacterial invasion[END_REF]. We used CRISPR/Cas9 genome-editing technology to generate sequence-specific mutations that would disrupt the Jas domain of SlJAZ2, leading to a JAZ2DJas tomato variant in the commercial variety Moneymaker. To do this, we targeted the start site of the Jas domain in the SlJAZ2 locus using a dual gRNA strategy that facilitates the generation of homozygous deletions [START_REF] Brooks | Efficient gene editing in tomato in the first generation using the clustered regularly interspaced short palindromic repeats/CRISPR-associated9 system[END_REF]. The two selected gRNA targets within SlJAZ2 were positioned on opposite DNA strands, and overlapped 7 nucleotides (Figure 2a). DNA sequence analysis of primary transformants identified two homozygous lines for SlJAZ2 containing deletions of seven and four nucleotides in the designated area (Figure 2b). Both deletions were predicted to disrupt the Jas domain of SlJAZ2 (Figure 2b). We further explored if these genome edited Sljaz2Djas mutations were specifically being expressed in our tomato lines. To do this, we performed quantitative RT-PCR experiments amplifying wild-type (WT) SlJAZ2 or each one of the two specific Sljaz2Djas mutations generated in both tomato lines. WT, Line 1 and Line 2 amplified exclusively their expected transcripts (Figure 2b), indicating both lines carry homozygous mutations at the SlJAZ2 locus generating forms of SlJAZ2Djas that are effectively being transcribed. Further analysis of T1 and T2 Sljaz2Djas lines by DNA sequencing of SlJAZ2 showed that the homozygous mutations were stably transmitted to the offspring.

Gain-of-function Sljaz2Djas prevent stomatal reopening by COR

Pathogen perception induces stomatal closure to limit pathogen penetration. Some P. syringae strains produce COR as a critical virulence strategy to re-open stomata and cause disease. Thus, we next evaluated the ability of both Sljaz2Djas tomato lines to close stomata upon perception of the highly conserved flg22 peptide of bacterial flagellum [START_REF] Chinchilla | A flagellin-induced complex of the receptor FLS2 and BAK1 initiates plant defence[END_REF], and to reopen them in the presence of COR. We incubated leaves of WT plants and Sljaz2Djas mutants with flg22, flg22 plus COR or a mock solution as a control. In mock conditions, the aperture range of the stomata was wide and similar between WT and both Sljaz2Djas tomato lines. As expected, flg22 induced stomatal closure in all tomatoes, whereas COR induced stomatal reopening in WT plants (Figure 3a). In contrast to WT, both Sljaz2Djas tomato lines were fully impaired in COR-mediated stomatal reopening (Figure 3a). This indicates that, similar to Arabidopsis, SlJAZ2 has a key role regulating stomata dynamics in response to bacteria and that COR-induced stomata reopening requires inhibition of SlJAZ2, which cannot be achieved in the dominant Sljaz2Djas mutants.

Stomata are the major regulators of water transpiration in plants [START_REF] Brodribb | Evolution of the stomatal regulation of plant water content[END_REF]. We next evaluated whether the rate of transpiration was affected in Sljaz2Djas lines compared to WT tomato plants. Time-course measurements showed that the rate of water loss in all plants was similar (Figure 3b). We also analysed leaf temperature, which is an indirect measure of stomata opening and water transpiration, using infrared thermography and found no differences related to stomatal apertures (Figure S2). Overall, this indicates that Sljaz2Djas tomato lines are not affected in the aperture of the stomata during the process of transpiration, but are rather specifically affected in the response to bacteria.

Dominant Sljaz2Djas mutants are resistant to P. syringae infection

In order to evaluate the resistance of adult Sljaz2Djas plants towards bacterial pathogens such as P. syringae, we next compared bacterial replication of Pto DC3000 on WT and both Sljaz2Djas lines infected by surface inoculation (dipping) or syringe infiltration. Surface infection mimics natural infection conditions and it is a very common technique to assess plant resistance to pathogens [START_REF] Zipfel | Bacterial disease resistance in Arabidopsis through flagellin perception[END_REF]. In contrast, the infiltration technique overcomes the early stomatal level of regulation measuring mainly apoplastic cell-based defences. WT plants infected by surface inoculation with Pto DC3000 showed extensive chlorosis and specks, typical symptoms of the bacterial speck disease in tomatoes (Figure 4a). In contrast, Sljaz2Djas plants did not display these typical symptoms of disease suggesting their enhanced resistance (Figure 4a). Disease symptoms correlated well with bacterial titres. Sljaz2Djas plants surface inoculated showed significantly lower bacterial titres than those in its respective WT control plants (Figure 4b). In the case of plants infected with Pto DC3000 by infiltration, bacteria grew to similar levels on both WT and Sljaz2Djas lines (Figure 4c). Consistently, inoculation with the coronatine-deficient Pto DC3000 strain AK87 (Pto DC3000 CORÀ) showed similar bacterial titles in WT and Sljaz2Djas plants (Figure S3). These results supports the idea that SlJAZ2 is a key regulator of stomata dynamics during the penetration process of Pto DC3000 and indicates that Sljaz2Djas plants promote bacterial resistance by blocking the action of the phytotoxin COR at the stomata.

Dominant Sljaz2Djas tomato plants show unaltered levels of resistance against the necrotrophic pathogen Botrytis cinerea

The antagonism between JA and SA pathways is a barrier to improve plant immunity and generate broad-spectrum resistance.

To evaluate if the insensitivity to COR generated in the edited Sljaz2Djas tomato lines affects resistance against necrotrophic fungal pathogens, we selected B. cinerea, causal agent of the gray mold of tomatoes. Thus, we measured the susceptibility of WT and Sljaz2Djas lines to B. cinerea by quantifying leaf infected area. Lesions caused by the fungus indicated similar progression of disease symptoms (Figure 5a) and area affected by the pathogen in WT and Sljaz2Djas lines (Figure 5b). These results indicate that SlJAZ2 function seems restricted to stomata guard cells and it plays a very limited or inexistent role in apoplastic defence responses against necrotrophic fungi.

Discussion

One of the great challenges for food security in the 21st century is to improve yield through the development of disease-resistant crops. However, novel strategies to improve plant immunity are particularly challenging. In vascular plants, hormonal crosstalk between JA and SA, the key two hormonal pathways controlling resistant to vastly all type of pathogens, often antagonize each other, and thus, enhanced resistance against biotrophic pathogens normally leads to increased susceptibility to necrotrophs, and vice versa [START_REF] Glazebrook | Contrasting mechanisms of defense against biotrophic and necrotrophic pathogens[END_REF]. Solving this trade-off is a major challenge for obtaining broad-spectrum resistance and requires uncoupling the antagonism between these hormonal pathways in crops.

Pto DC3000 is the natural causal agent of the bacterial speck disease of tomato. Bacterial spots on tomato fruits have been reported to cause up to 52% loss of fruit weight [START_REF] Jones | Survival of Xanthomonas campestris pv. vesicatoria in Florida on tomato crop residue, weeds, seeds, and volunteer tomato plants[END_REF], whereas in the field, yield losses due to bacterial speck vary from 75% in plants infected at an early stage of growth to 5% in plants infected later in the season [START_REF] Yunis | Weather dependence, yield losses and control of bacterial speck of tomato caused by Pseudomonas tomato[END_REF]. Therefore, it becomes necessary to uncover novel sources of resistance against this pathogen. Pto DC3000 uses COR as a virulent factor to induce stomata opening and facilitate entrance into the leaf apoplast. We have recently identified the specific JAZ co-receptor of COR at the stomata in Arabidopsis [START_REF] Gimenez-Ibanez | JAZ2 controls stomata dynamics during bacterial invasion[END_REF]. A dominant mutation of this stomataspecific AtJAZ2 lacking a C-terminal Jas regulatory domain (Atjaz2Djas) is insensitive to COR and cannot re-open stomata after bacterial infection. This mutation confers resistance to biotrophic Pto DC3000 without affecting susceptibility to necrotrophs, due to its specific action at guard cells [START_REF] Gimenez-Ibanez | JAZ2 controls stomata dynamics during bacterial invasion[END_REF].

In this study, we demonstrate that similar Djas mutations in stomata-accumulating JAZ proteins can be used as a general strategy to increase resistance to COR-producing bacteria in crops. Thus, as a proof-of-concept, we identified the tomato ortholog of AtJAZ2 and successfully generated a bacterial speck resistant tomato in the commercial variety Moneymaker by CRISPR/Cas9-mediated SlJAZ2 editing. Sljaz2Djas plants were significantly more resistant to Pto DC3000 when plants where surface inoculated, but equally susceptible by infiltration. This indicates that this SlJAZ2 variant is mainly affecting the stomatal defence layer by counteracting the effect of COR. Consistently, Sljaz2Djas tomato lines were fully impaired in COR-mediated stomatal reopening. Remarkably, since the Sljaz2Djas mutation only manipulates the JA-dependent defences at the stomata, the JA-SA antagonism is not affected in the mesophyll and, therefore, apoplastic resistance to the necrotrophic pathogen B. cinerea remains unaffected. These results suggest that similar to Arabidopsis, SlJAZ2 plays a major role in regulating stomatal aperture during biotic stresses, and that COR produced by Pto DC3000 hijacks SlCOI1-SlJAZ2 co-receptor to promote the entry of the bacteria to the internal tissues of tomato leaf. Moreover, these results prove that is possible to uncouple SA-JA antagonism spatially to design novel strategies for protection against CORproducing P. syringae strains in crops, without affecting resistance to necrotrophs.

Alteration of the regulation of plant stomatal aperture can lead to deleterious effects. A number of environment factors, including CO 2 level, light, water and other abiotic stresses also regulate stomatal dynamics. In all these cases, ABA functions as a chemical messenger that induces stomata closure [START_REF] Lim | Function of ABA in stomatal defense against biotic and drought stresses[END_REF][START_REF] Vishwakarma | Abscisic acid signaling and abiotic stress tolerance in plants: a review on current knowledge and future prospects[END_REF]. However, several lines of evidence suggest that Sljaz2Djas tomato lines are not affected in these ABA-mediated physiological processes. First, in Arabidopsis, gainof-function mutations in JAZ2 prevent stomatal reopening by COR by inhibiting SA-dependent stomatal closure triggered upon specific microbial perception without affecting ABA signalling at the stomata [START_REF] Gimenez-Ibanez | JAZ2 controls stomata dynamics during bacterial invasion[END_REF]. Second, the rate of stomatal aperture in normal light conditions was not affected in Sljaz2Djas tomato mutants. Third, neither water transpiration rate or leaf temperature was affected in Sljaz2Djas compared to WT. Finally, Sljaz2Djas plants displayed a complete phenotypically normal development compared to WT tomatoes (Figure S4). Taken together, these results suggest that the Sljaz2Djas mutation impact on the regulation of stomatal aperture exclusively during biotic conditions, when SA is produced to execute resistance against biotrophic pathogens trying to invade the plant.

Current methods to control bacterial speck disease of tomato are based on removal of plant debris and weeds, crop rotation, and the use of non-infected seeds and transplants. Chemical sprays based on copper are also commonly used, but these may not be effective in environmental conditions favourable for infection [START_REF] Somodi | Relationship between the hypersensitive reaction and field resistance to tomato race 1 of Xanthomonas campestris pv. vesicatoria[END_REF]. Moreover, copper can also kill plant cells if absorbed in sufficient quantities and can produce problematic toxicity in soil. In contrast, tomato resistant varieties offer the most effective means of management. P. syringae strains vary in their ability to establish and maintain epiphytic populations on the surface of the leave before infection of internal plant tissues. In this sense, it is becoming clear that transition from epiphytic to endophytic lifestyles is a critical process to establish a successful infection cycle. This transitional process is enhanced by the production of COR. In the case of Sljaz2Djas, these plants would impede the entry on the bacteria, elongating its epiphytic phase where P. syringae encounters a harsh environment with limited nutrients. This elongated epiphytic phase on Sljaz2Djas tomatoes should reduce survival of the bacteria further enhancing resistance.

Current resistant varieties are based on the introduction of specific disease-resistant genes that recognized pathogenic molecules intracellularly. However, pathogens frequently adapt to and overcome genetic resistance especially when it is determined by major resistant genes [START_REF] Brown | Durable resistance of crops to disease: a darwinian perspective[END_REF]. The ability of Sljaz2Djas tomato plants to interfere with the penetration of the pathogen represents a new strategy for providing resistance that may be achieved by a low energetic cost for the plant and in a clean way for the environment. Ultimately, a hierarchical design containing multiple layers of resistance in a crop may be the most feasible way to achieve durable resistant in the field [START_REF] Fuchs | Pyramiding resistance-conferring gene sequences in crops[END_REF][START_REF] Pilet-Nayel | Quantitative resistance to plant pathogens in pyramiding strategies for durable crop protection[END_REF].

Recent results demonstrate that novel techniques for genome editing could be successfully applied in crops to introduce directed resistance. For example, CRISPR/Cas has been recently used to generate powdery mildew resistant tomatoes [START_REF] Nekrasov | Rapid generation of a transgene-free powdery mildew resistant tomato by genome deletion[END_REF] and engineer resistance to geminiviruses in tobacco [START_REF] Ali | CRISPR/Cas9-mediated viral interference in plants[END_REF]. Public policies are forcing researchers to develop transgene-free improved crops. In this context DNA editing technologies (also called New Breeding Techniques) have emerged as a novel strategy to overcome this challenge. The status of genetically edited crop varieties is still debated by regulatory authorities, for example countries such as US consider transgene-free genetically edited crops as non-GMO. We hope that plant varieties such as the one presented here could be adopted worldwide to enhance plant productivity and resistance against pests. These superior varieties have the potential to fight against agricultural losses in the field due to pests, reducing the chemical inputs towards a more sustainable agriculture for the environment.

Altogether, we demonstrate the feasibility to create bacterial speck resistant tomatoes through CRISPR/Cas9-mediated editing of SlJAZ2. Moreover, we also define a novel strategy to overcome the penetration of COR-producing P. syringae strains through the stomata by spatially uncoupling SA-JA antagonism.

Experimental procedures

Phylogenetic analysis

JAZ proteins sequences from Arabidopsis thaliana and Solanum lycopersicum (tomato) were aligned with Dialing software (http:// www.genomatix.de/cgi-bin/dialign/dialign.pl). Phylogenetic tree was represented using the online tool 'Phylodendron', (http://iub io.bio.indiana.edu/treeapp/treeprint-form.html).

RT-qPCR analysis

Leaf tissue or epidermal peels were harvested from 4-to 6-weekold plants and frozen in liquid nitrogen. RNA was extracted using a Plant total RNA purification mini kit (Favorgen â ). One microgram of RNA was used for cDNA synthesis using a MultiScribe Reverse Transcriptase (Applied Biosystems). PCR was performed on a 7500 thermocycler using SYBR-green (both from Applied Biosystems). SlActin was used as endogenous control. Oligonucleotides used in this study are described in Table S1. Data analysis shown was done using three technical replicates from one biological sample; similar results were obtained with at least two additional independent biological replicates.

Plasmid construction

Designed gRNAs (Table S2) targeting the Jas domain of SlJAZ2 (Solyc12 g009220) were cloned into the vector Bsa_Bbs_tandem gRNA_pBS. This vector confers resistance to ampicillin and gRNAs are expressed under different versions of the ubiquitin promoter. Then, a fragment containing both gRNAs and their promoters were cloned into the vector Pk7_CAS9-TPC_polylinker pBKS. This vector confers resistance to spectinomycin in bacteria and kanamycin in plants. Cas9 is expressed under an ubiquitin promoter. Annealing of gRNAs and cloning was performed by ordinary methods using BsaI and BbsI restriction enzymes. Fragment with the two gRNAs together with their promoters were clone using the KpnI restriction enzyme. gRNAs were designed with the online tool 'Breaking Cas' [START_REF] Oliveros | Breaking-Cas-interactive design of guide RNAs for CRISPR-Cas experiments for ENSEMBL genomes[END_REF] (http://bioinfogp.cnb.csic.es/tools/breakingcas/). gRNAs were unique and specific for the target region (Jas domain sequence of SlJAZ2). 1°gRNA had a score of 96.3% with the best possible off-targets having a 0.2%. 2°gRNA had a score of 98.4%, with the best possible off-targets having a 0.5%.

Plant growth and transformation

A tomato cultivar Moneymaker was transformed with the pK7_CAS9-TPC_polylinker pBKS, containing both gRNAs for SlJAZ2, as previously described [START_REF] Wittmann | Development of a tomato plant resistant to Clavibacter michiganensis using the endolysin gene of bacteriophage CMP1 as a transgene[END_REF]. Plants were grown under a 16-h-light/8-h-dark cycle in a growth chamber or greenhouse. Around 120 cotyledons (60 tomato plants) were initially used for transformation with Agrobacterium. Many cotyledons generated a callus and produced shots, which were transferred for root generation. In total, 23 regenerated plants were transferred to the greenhouse. PCR amplification of SlJAZ2 and DNA sequencing identified mutations in these plants and indicated that 8/23 were wild-type, 12/23 were heterozygous ª 2018 The Authors. Plant Biotechnology Journal published by Society for Experimental Biology and The Association of Applied Biologists and John Wiley & Sons Ltd., 1-9 or chimeras and 3/23 contained an homozygous mutation. T1 and T2 progeny of selected Sljaz2Djas Lines (1 and 2) was further obtained and analysed by DNA sequencing of SlJAZ2 to confirm that homozygous mutations were stably transmitted to the offspring.

Measurements of stomatal aperture

After 2 days of stratification of seeds at 4°C, tomato WT and Sljaz2Djas Lines 1 and 2 were grown on soil in a chamber with an 8 h light period at 23°C and a 16 h dark period at 19°C, relative humidity of 75%. The abaxial side of leave from 4-to 6-week-old tomato plants were collected and stuck on cover slips (peeled). Samples were incubated in a solution of 10 mM MES/Tris pH 6.0 and 30 mM KCl (working buffer). For promotion of closure assays, compounds (10lm flg22, 10lm flg22 + 2lm COR or a mock solution) were directly diluted in the working buffer in contact with epidermal peels during 2 h. Aperture of stomata and hydathode pores were determined under an optical microscope fitted with a camera lucida and a digitizing pad linked to a computer as described [START_REF] Leonhardt | Evidence for the existence of a sulfonylurea-receptor-like protein in plants: modulation of stomatal movements and guard cell potassium channels by sulfonylureas and potassium channel openers[END_REF]. Values reported are the means of at least three independent experiments, for which 60 aperture widths were measured each time. Error bars represent standard deviations of the means. One-way Anova with Tukey-HSD was used for statistical analysis. Calculations were performed using a statistical calculator (http://astatsa.com/OneWay_Anova_with_Tuke yHSD/).

Pseudomonas bacterial strains

Pseudomonas strains used in this study were Pseudomonas syringae pv. tomato (Pto) DC3000 and the coronatine-deficient Pto DC3000 strain (Pto DC3000 CORÀ) which is a Pto DC3000 AK87 mutant that carries mutations in cmaA (coronamic acid A) and cfa6 (coronafacic acid 6) [START_REF] Brooks | Identification and Characterization of a Well-Defined Series of Coronatine Biosynthetic Mutants of Pseudomonas syringae pv. tomato DC3000[END_REF].

Pseudomonas syringae DC3000 infection assays

Tomato plants for infection assays were grown in a cycle of 14 hours light and 10 hours darkness with 45-60% humidity for 6-7 weeks. Bacterial growth assays in tomato were performed as described previously with some modifications [START_REF] Balmuth | Genetic and molecular requirements for function of the Pto/Prf effector recognition complex in tomato and Nicotiana benthamiana[END_REF]. For surface infection assays, plants were dipped in a bacterial suspension containing 10 8 colony-forming units m/L (cfu/mL) bacteria (OD 600 = 0.2) with 0.02% Silwet L-77. For bacterial growth assays by syringe infiltration, leaves were syringe infiltrated with a bacterial suspension containing 5x10 5 cfu/mL bacteria (OD 600 = 0.001). Leaf disks were collected 2 days (infiltration) and 6/7 days (surface infection) postinfection and bacterial growth was quantified as described previously [START_REF] Gimenez-Ibanez | AvrPtoB targets the LysM receptor kinase CERK1 to promote bacterial virulence on plants[END_REF]. Error bars indicate SEM (n = 7). These experiments were repeated at least three times with similar results, and representative results are shown.

Botrytis cinerea infection assays

Botrytis cinerea infection assays were performed as described previously [START_REF] Monte | Rational design of a ligand-based antagonist of jasmonate perception[END_REF] with some modifications. Five-to six-week-old tomato leaves were inoculated with 20 lL of a suspension of 5 9 10 6 spores m/L PDB (Difco, Le Pont de Claix, France). At least eight leaves were inoculated per treatment. Disease symptoms were scored 4 days postinoculations. Area of the lesion was measured using jimage software (https://imagej. nih.gov/ij/). This experiment was repeated twice with similar results.

Leaf temperature and water transpiration ratio

For leaf temperature assays, 3-4 weeks tomato plants were examined under Thermacam infrared camera (FLIR A655sc 25°, 50°Hz) during different days. Images were saved and analysed on a personal computer using the ResearchIR4 Max Software provided by FLIR. For the calculation of water transpiration rate, plants were stratified 2 days at 4°C in dark and grown in soils from 1 to 2 weeks. Then plants were transferred to pots with soil at 20% humidity (all pots weight the same and humidity content was calculated to be 20%) and keep some days for acclimatization. During at least 10 days plants weight was measured every day, after measuring water was added to restore 20% humidity in pots, a control (without plant) was used to measure the amount of water lost by exchange of the soil. The difference of the water lost compare to the negative control (without plant) is a measure of the water transpiration rate of the plants through the stomata. This experiment was repeated twice with similar results.

Statistical methods

Statistical significance based on Students's t test analysis was calculated using Excel software (*P < 0.05; **P < 0.01). Statistical significance based on one-way Anova with Tukey-HSD (*P < 0.05; **P < 0.01) was calculated using a statistical calculator (http://astatsa.com/OneWay_Anova_with_TukeyHSD/).

ª

  2018 The Authors. Plant Biotechnology Journal published by Society for Experimental Biology and The Association of Applied Biologists and John Wiley & Sons Ltd., 1-9

Figure 1 Figure 2

 12 Figure 1 SlJAZ2 is the ortholog of AtJAZ2. (a) Phylogenetic tree of JAZ1, JAZ2, JAZ3 and JAZ4 from Arabidopsis and tomato. SlJAZ1 and SlJAZ2 are the closest proteins to AtJAZ2. The schematic representation indicates length and domains of each JAZ protein. (b) Enrichment ratio of RT-PCR analyses comparing expression levels of SlJAZ1, SlJAZ2, SlJAZ3 and SlJAZ4 in tomato whole leaf tissue and epidermal peels, which are enriched in guard cells. This experiment was repeated twice with similar results.

Figure 3 Figure 4

 34 Figure 3 Sljaz2Djas mutants prevent stomatal reopening by COR. (a) Stomatal aperture in tomato WT and Sljaz2Djas mutants (Line 1 and Line 2) after 2 h of incubation with flg22, flg22 plus COR or a mock control. Error bars indicate standard error of the mean (SEM; n = 60). Letters above bars represent statistically distinct groups (P < 0.01, one-way Anova with Tukey-HSD). Similar results were obtained in three independent biological replicates (b) Water transpiration rate (water loss) in WT and Sljaz2Djas (Line 1 and Line 2) tomato plants during a time course experiment of 10 days. Error bars indicate standard deviation (SD; n = 3). Not statistical difference was found between WT and mutant lines (one-way Anova with Tukey-HSD). Results are representative of two independent experiments.

Figure 5

 5 Figure 5 Sljaz2Djas mutants retain WT resistance to necrotrophic fungi. (a) Botrytis cinerea disease symptoms on WT and Sljaz2Djas mutants 4 days postinoculation (dpi) with 5 9 10 6 spores m/L. (b) Lesion Area produced by Botrytis cinerea on WT and Sljaz2Djas mutants 4 days postinoculation (dpi) with 5 9 10 6 spores m/L. Error bars indicate standard error SEM (n = 8). (Student's t-test, P < 0.05; ns, not significant). Results are representative of two independent experiments.

ª 2018 The Authors. Plant Biotechnology Journal published by Society for Experimental Biology and The Association of Applied Biologists and John Wiley & Sons Ltd., 1-9

Acknowledgements

We thank Dr. Michael Nicolas and Dr. Pilar Cubas (CNB-CSIC), for developing and sharing CRISPR/Cas vectors. We also thank Caterina Brancato and Dr. Kenneth Berendzen (Center for Plant Molecular Biology (ZMBP), University of T€ ubingen) for advice on tomato transformation. A.O. was supported by a 'Severo Ochoa' CNB PhD fellowship. S.G-I. was supported by the Spanish Ministry of Science and Innovation Grant BIO2014-55884-JIN. Work in N.L.' s laboratory is funded by ANR program. Work in R.S.' s laboratory is funded by the Spanish Ministry for Science and Innovation grant BIO2016-77216-R (AEI/FEDER, UE) and Fundaci on UAM Grant 2015007.

Conflict of interest

Supporting information

Additional supporting information may be found online in the Supporting Information section at the end of the article.


F Doumenc

V Janeček

V S Nikolayev

Thin wedge evaporation/condensation controlled by the vapor dynamics in the atmosphere

Evaporation or condensation in the vicinity of the immobile (pinned) contact line in an atmosphere of some inert (noncondensable) gas is considered here in a partial wetting configuration. Such a problem is relevant to many situations, in particular to a drop or a liquid film drying in open air. The thermal effects are not important and the mass exchange rate is controlled by the vapor dynamics in the gas. By following previous works, we account for the weak coupling between the diffusion in the gas and flow in the liquid through the Kelvin effect. Such a problem is nonlocal because of the diffusion in the gas. For generality, we consider a geometry of a liquid wedge posed on a flat and homogeneous substrate surrounded by a gas phase with a diffusion boundary layer of uniform thickness Λ. Similarly to the moving contact line problem, the phase change leads to the hydrodynamic contact line singularity. The asymptotic analysis of this problem is carried out for the liquid wedge of the length L ≫ Λ. Three asymptotic regions are identified: the microscopic one (in which the singularity is relaxed, in the present case with the Kelvin effect) and two intermediate regions. The Kelvin effect alone turns to be sufficient to relax the singularity. The scaling laws for the interface slope and mass evaporation/condensation flux in each region are discussed. It is found that the difference of the apparent contact angle (i.e. interface slope in the second intermediate region) and the equilibrium contact angle is inversely proportional to the square root of Λ and square root of the microscopic length, whatever is the singularity relaxation mechanism.

Introduction

Evaporation and condensation phenomena are met widely both in everyday life and in various industrial processes. From physicist's point of view, there are two practically important limiting cases of the mass exchange,depending on which mechanism limits the phase change rate. The first is the phase change controlled by the heat transfer in the liquid phase, where the liquid is in contact with its pure vapor [1][2][3]. During evaporation, the liquid-covered solid substrate is overheated with respect to the saturation temperature and the heat coming from the substrate is consumed as the latent heat at the liquid-gas interface. During condensation, the heat flow is opposite. This case of strong mass exchange is met in various heat exchangers (boiling heat exchangers, heat pipes, etc.). The second limiting case is that of a generally weaker mass exchange with an atmosphere of some inert (noncondensable) gas [4,5]. A common example is the drop drying in the open air. The thermal effects are much less important a Present address: Department TSIE: Microgravity Payloads, Airbus Defence & Space, 88039 Friedrichshafen, Germany because the slowest (thus limiting) process is the vapor dynamics in the diffusion layer formed in the gas phase. Many researchers worked in this important field. A significant progress has been achieved by Deegan et al [4] who have obtained the stationary diffusion problem solution in the half-space above a spherical-cap-shaped sessile liquid droplet posed on a solid substrate with the known macroscopic (i.e. apparent) contact angle. Many theories based on this solution have been developed afterwards, one can cite [6][7][8][9][10][11] among many others. The success of the Deegan et al solution is due to the integrability of the resulting local evaporation flux so that the total evaporation rate can be calculated and compared to the experiment. However, an incoherency appears in the liquid phase hydrodynamic problem. As discussed in sec. 3 below, such an evaporation flux would lead to the hydrodynamic stress that diverges at the contact line in a non-integrable way, similarly to what happens in the isothermal contact line motion problem [12]. Such a divergence is nonphysical. On the one hand, the total viscous dissipation becomes infinite, which implies the infinite energy. On the other hand, in a problem with a free interface it would lead to the infinite interface slope. One knows that the singular-ity is possible to relax by introducing some microscopicscale phenomena like precursor film [10], hydrodynamic slip [13] or Kelvin effect [14,15]. The macroscopic scale solution is obtained by matching the solutions in microscopic and macroscopic regions [16]. The introduction of an intermediate length scale turns out to be necessary. A remarkable feature of such a problem is that the form of the macroscopic solution (called Cox-Voinov formula) is independent of the microscopic mechanism that conditions only two constants, the Voinov length and angle.

In the case of pure vapor and latent-heat-controlled evaporation, the microscopic mechanisms have been extensively studied by many authors starting from Potash and Wayner [1]. These studies showed that the apparent contact angle grows with the evaporation intensity (see, e.g. [2,3,[17][18][19][20] among many others). The theoretical results have been confirmed experimentally [21,22]. The independence of the results of the microscopic model has been shown numerically by comparing the apparent contact angle for different microscopic mechanisms but the same fluid parameters [23]. We do not discuss this evaporation regime below. Instead, we address the diffusioncontrolled evaporation regime that was less studied.

It has been shown experimentally [24] that the increase of apparent contact angle with evaporation rate exists in this case too. In earlier approaches [6][7][8][9], the evaporation flux distribution along the interface was imposed independently of the interface shape. The full coupling of the problems in the vapor and liquid phases through the Kelvin effect is studied in more recent works [10,11,25]. The problem is more complicated in this case because of the non-locality of evaporation flux: the evaporation rate at one point of the interface depends on the vapor concentration distribution over the whole liquid-vapor interface. However the Kelvin effect is important as it provides zero evaporation flux at the immobile contact line [20] in the case of partial wetting (the case considered here), which conforms to the complete wetting case.

In the continuity of the above cited works, our purpose is to study the effect of the apparent contact angle increase under evaporation. This objective was targeted by several groups. Berteloot et al [8] proposed an approximate solution for an infinite liquid wedge on a solid substrate using the expression for the evaporation flux given by Deegan et al [4] thus decoupling the diffusive and hydrodynamic parts of the problem. The singularity is avoided by assuming a finite liquid height at a microscopic cut-off distance, imposed a priori. Eggers and Pismen [10] have considered the vapor phase and liquid phase problems coupled via the concentrational Kelvin effect for an axisymmetric drop geometry. They considered the complete wetting conditions and the precursor film.

By following our preceding work [26] devoted to the moving contact line, we consider a partial wetting configuration with a 2D geometry suitable, e.g., for drying of the liquid films deposited on the Wilhelmy plate [27]. We consider a diffusive layer of thickness Λ above the thin liquid wedge-like film. In a real process the value of Λ would depend on the characteristic of the flow (diffusion, natural convection or forced convection). We consider also the kinetic interfacial resistance effect that appears to be important in many cases. Using the Kelvin effect as microscopicscale phenomenon to relax the contact line singularity, we develop a complete description of the micro and intermediate regions (cf. fig. 1 and sec. 2). The influence of Λ is analyzed, and a law for the apparent contact angle is deduced. Although the regularisation by the Kelvin effect may not apply to common fluids, (the micro-region size being too small for a continuum hydrodynamic approach [26]), the scalings characterising the intermediate region are independent of the mechanism chosen to relax the singularity. We thus draw results of general significance for the scaling of the apparent contact angle, since they are valid for any microscopic-scale phenomenon.

Problem statement

The problem to be considered is a thin liquid wedgeshaped film posed on a flat and homogeneous substrate in a situation of partial wetting. The atmosphere of a non-condensable inert gas surrounds the substrate and the condensation or evaporation mass exchange with it is controlled by the vapor transfer in the gas. The gas supersaturation with vapor causes condensation onto the film, and under-saturation causes its evaporation. The fluid is assumed to be isothermal, which may be justified when the substrate has a good thermal conductivity and is maintained at room temperature. Such a situation is common, e.g. in the case of slow drying of liquid sessile drops. The contact line is assumed pinned, with a source/sink of liquid at x = ∞ that compensates the evaporation/condensation mass flux (the problem of contact line motion under saturation conditions has been solved in our preceding work [26]). The model considers the vapor diffusion in the gas phase, as well as the kinetic interfacial resistance.

First, some assumptions about the characteristic scale hierarchy need to be made. One can identify a macroscopic length scale L that concerns the liquid phase. It is related to the macroscopic-level interface curvature, e.g. the interface curvature radius. Another macroscopic length scale concerns the gas phase description. It is the diffusion boundary layer thickness Λ. It is millimetric (for natural convection) or smaller, for forced convection. For simplicity, in this work we consider the diffusion boundary layer of homogeneous thickness (band-like). The idea of our approach is to find a stationary solution for the vapor diffusion that can be representative of real microscopic situation. For this we assume that

L ≫ Λ ≫ h ( 1
)
where h is the wedge thickness, see fig. 1a,b. It is evident that the inequality (1) is violated far enough from the wedge apex (contact line) for any wedge slope. However, if the slope is small, the region of validity of such a geometry is large. At a distance Λ from the substrate, a deviation from the concentration corresponding to saturation is imposed. The main rationale of such a geometry is to decouple the scaling related to the macroscopic scale L from dependencies on other length scales that exist in this problem so that our approach is applicable for different macroscopic geometries (drop, film, bubble, etc.), provided that inequality (1) is verified. Consequently, the liquid domain length L must be much larger than the boundary layer thickness Λ. This excludes, for instance, the case of a sessile drop evaporating in quiescent air (no convection) [7], because in that case Λ ∼ L.

A hydrodynamic singularity that exists in our problem is solved at the microscopic scale that we call ℓ which is typically nanometric (fig. 1d). Because of the existence of several characteristic lengths, intermediate regions (fig. 1c) are needed to match micro and macro regions. We will see later that, generally, two intermediate regions need to be introduced.

Liquid phase equations

Stokes equations (negligible inertia) are solved in lubrication approximation (small interface slopes) and neglect-ing the gravity. With no-slip boundary condition at the substrate and stress-free boundary condition at the interface, the horizontal component of the liquid velocity for 0 < y < h, h being the liquid height, reads

v x (y) = σ µ (hy - 1 2 y 2 ) dκ dx , (2)
where σ is the surface tension and κ = d 2 h/dx 2 is the local interface curvature, µ the liquid viscosity and the coordinate system is shown in fig. 1d.

The fluid volume flux in the liquid layer vertical crosssection, q(x), is

q(x) = h 0 v x (y)dy = σh 3 3µ dκ dx . (3)
The governing equation in the liquid phase is obtained by writing the mass conservation

ρ dq dx = -j, ( 4
)
where ρ is the liquid density and j is the interfacial mass flux (j > 0 for evaporation). From eqs. (3-4) one deduces

d dx σh 3 3µ dκ dx = - j ρ . (5)
The interfacial mass flux is assumed to be controlled by the vapor diffusion in the inert gas:

j = -D g ∂c ∂y y=0 , (6)
where c(x, y) is the vapor concentration (expressed in kg/m 3 ) and D g is the vapor diffusion coefficient. Boundary conditions for the liquid phase are the following:

-For x = 0 (contact line):

h = 0, dh dx = θ, ( 7
)
where θ is the equilibrium contact angle imposed by intermolecular interactions at microscale.

-For x → ∞, the interface curvature vanishes,

κ = 0. ( 8
)

Gas phase model

In the gas phase, the equation for vapor diffusion reads

∂ 2 c ∂x 2 + ∂ 2 c ∂y 2 = 0. (9)
In the framework of the small wedge slope approximation, the liquid gas interface seen from the large scale of the gas atmosphere is assumed to coincide with the line y = 0, x ≥ 0, as shown in fig. 1a. This assumption is valid for a liquid height much lower than the boundary layer thickness, i.e. for x ≪ Λ/θ. As θ is a small angle, the model is valid over a distance to the contact line much larger than Λ.

The boundary conditions in partial wetting configuration are:

-For y = Λ: c(x, Λ) = c eq + ∆c, (10)
where c eq is the vapor concentration at thermodynamic equilibrium for a flat liquid-gas interface, ∆c is the deviation from the equilibrium vapor concentration (∆c < 0 corresponds to evaporation, while ∆c > 0 results in condensation). -For y = 0 and x < 0:

∂c ∂y = 0. ( 11
)
-The vapor concentration c i = c(x, y = 0) at the liquidgas interface can deviate from c eq because of two reasons. First, the local equilibrium value c i,eq can be different from c eq because of the interface curvature (Kelvin effect). It will be considered later on. Second, c i can differ from c i,eq because of the interfacial resistance R i , so that the following condition is valid for x ≥ 0:

c i = c i,eq -jR i . (12)
The interfacial resistance appears because of the molecularkinetic effects and is given by the Hertz-Knudsen relation [28,[START_REF] Carey | Liquid-Vapor Phase Change[END_REF],

R i = 2 -f 2f 2πM R g T , ( 13
)
where f is the accommodation factor close to unity [28], R g the ideal gas constant, T the temperature, and M the molar mass.

Hydrodynamic contact line singularity related to the mass exchange

To analyze the contact line singularity due to the internal flow, we drop temporarily the microscopic (Kelvin) effect so that c i,eq = c eq in eq. ( 12). First, consider the simplest case dominated by the kinetic resistance, where the concentration difference caused by the diffusion in the gas phase is much smaller than the kinetic resistance effect. The concentration gradient in the gas is then negligibly small and the concentration c i at the liquid interface is equal to the concentration at the upper edge of the diffusion boundary layer y = Λ: c i = c eq + ∆c; c i is thus constant. As a consequence, the evaporation flux j = ∆c/R i is constant [cf. eq. ( 12)].

From eqs. (2-4), the tangential stress at the substrate reads

µ ∂v x ∂y y=0 = - 3µjx h 2 ρ . ( 14
)
Using h ∼ θx, it follows that the tangential stress at the substrate scales as x -1 for the configuration dominated by the kinetic resistance. It diverges at the contact line. The integral, i.e. the force acting on the substrate, diverges as ln(x), which is nonphysical. In reality, this quantity should be finite. In addition, the curvature diverges as x -1 , which means that the resulting slope (=apparent contact angle, the integral of curvature) is infinite, which is also nonphysical.

A similar analysis can be performed for the integral viscous dissipation

µ h 0 ∂v x ∂y 2 dy = 3µ(jx) 2 h 3 ρ 2 , (15)
which diverges as x -1 like the tangential stress. This is also nonphysical because it leads to the infinite energy losses. These non-integrable divergences both for the tangential stress and viscous dissipation are similar to those of Huh and Scriven [12] in their seminal article about the moving liquid wedge with no evaporation.

Consider now the opposite case, i.e. negligible kinetic resistance and evaporation limited by diffusion in the gas phase. The solution for such a 2D problem [equations (9)(10)(11)(12) with c i,eq = c eq ] can be obtained by analogy with an electrostatic problem, cf. Appendix A. From eq. (A.12) [cf. its asymptotics (42) below], it follows that the evaporation flux j(x) scales as x -1/2 in the vicinity of the contact line, in agreement with the Deegan's 3D axi-symmetric solution [4] applied for the vanishing contact angle. Accordingly, the tangential stress at the substrate scales as x -3/2 and the total force diverges as x -1/2 . For the viscous dissipation and its integral, the scalings are x -2 and x -1 , respectively. The curvature is non-integrable either (scales like x -3/2 ).

We thus observe non-physical singularities in these two limiting cases, similarly to classical moving contact line problem. Note that the interfacial kinetic resistance alone does not relax the singularity in the hydrodynamic problem. Saxton et al [START_REF] Saxton | [END_REF] showed that regularization can be achieved by considering both interfacial resistance and hydrodynamic slip. In the present study, similarly to the moving contact line problem considered in our preceding work [26], the Kelvin effect is used to relax the contact line singularity.

Kelvin effect and dimensionless formulation

The Kelvin effect consists in the dependence of the equilibrium interfacial vapor concentration c i,eq on the interface curvature κ. In the first approximation this dependence reads [10] c i,eq -c eq = -M c eq ρR g T σκ.

(

) 16
The microregion size ℓ (cf. fig. 1) for the case of the singularity relaxation with the Kelvin effect can be obtained with a scaling analysis [26]. For the purely diffusion regime (i.e. for R i = 0) the characteristic size of a region dominated by the Kelvin effect is

ℓ = 1 ρ 3µM c eq D g θ 3 R g T . (17)
In fact the diffusion regime corresponds to R i ≪ ℓ/D g . Note that in the opposite limit (that we call kinetic regime hereafter) where

R = R i D g /ℓ ≫ 1, the characteristic size is ℓ R = ℓ/R = 3µM c eq θ 3 ρ 2 R g T R i . ( 18
)
Estimations given in Table 1 of our preceding article [26] show that R varies strongly from one fluid to another and with the contact angle. For example, R = 0.72 for ethanol and θ = 1 • , while R = 166 for glycerol and θ = 5 • . Throughout the rest of the paper, we choose ℓ to make the equations dimensionless. The dimensionless abscissa is then x = x/ℓ and, accordingly to the wedge geometry, the liquid height scales as H = h/(θℓ). The dimensionless curvature is K = ℓκ/θ.

The characteristic scale J for the mass flux ( j = j/J) is J = θ 4 σρ/(3µ) and the dimensionless lubrication equation reads

d dx H 3 dK dx = -j. ( 19
)
Concentration deviation is reduced as c = (c-c eq )/C, with

C = ρσθ 4 ℓ 3µD g , (20)
as obtained from eqs. (5)(6). The dimensionless counterpart of eq.( 16) reads ci,eq = -K.

The two remaining dimensionless equations are the mass flux and interfacial concentration expressions:

j = - ∂c ∂ ỹ ỹ=0 , (22) c
(ỹ = 0) ≡ ci = -K -jR. ( 23
)
They serve as boundary conditions for the diffusion problem.

Weak evaporation approximation

The weak evaporation approximation ∆c ≪ C is considered here. The variables are expanded in a regular perturbation series in ǫ = ∆c/C. At the zero order corresponding to thermodynamic equilibrium (ǫ = 0), K 0 = 0, H 0 = x, j0 = 0 and c0 = 0, thus

K = ǫK 1 + • • • , H = x + ǫH 1 + • • • , j = ǫ j1 + • • • , c = ǫc 1 + • • • .
Note that ǫ is negative during evaporation and positive during condensation while j1 , H 1 and K 1 are always negative.

With the scaling defined in sec. 4, the first order lubrication equation reads

d dx x3 dK 1 dx = -j1 , (24)
with the following boundary conditions:

-For x = 0 :

H 1 = 0, H ′ 1 = 0, -For x → ∞ : K 1 = H ′′ 1 = 0.
The first order problem for the diffusion in the gas phase is

∂ 2 c1 ∂ x2 + ∂ 2 c1 ∂ ỹ2 = 0, (25)
with the boundary conditions

-For ỹ = Λ ≡ Λ/ℓ: c1 = 1. ( 26
)
-For ỹ = 0 and x < 0:

∂c 1 ∂ ỹ = 0. (27)
-For ỹ = 0 and x ≥ 0:

j1 = - K 1 + ci,1 R = - ∂c 1 ∂ ỹ ỹ=0 . ( 28
)
In the following sections we focus on two limit cases corresponding to situations dominated either by kinetic interfacial resistance (sec. 6) or by diffusion of vapor in the gas (sec. 8). We are looking for a regular solution at ỹ → 0, with no divergencies.

Kinetic regime (R → ∞): analytical solution

We first consider the mass exchange controlled by kinetic resistance. In practice, this assumption applies for R ≫ Λ ≫ 1. These two inequalities ensure that ci,1 ≃ 1.

The problem defined in sec. 5 can be solved analytically in the kinetic regime. Equation (28) reduces to

K 1 = -1 -R j1 , (29)
and eq. ( 24) reads

d dx x3 d j1 dx = j1 R . ( 30
)
With the boundary condition lim x→∞ j1 (x) = -1/R [stemming from eq. ( 29)] and the condition of a regular solution at x → 0, the solution of eq. ( 30) where K n are the modified Bessel functions of the second kind.

is j1 = - 2 R 2 x K 2 2 √ Rx , (31)
The curvature K 1 is then obtained from relation [START_REF] Carey | Liquid-Vapor Phase Change[END_REF],

K 1 = 2 Rx K 2 2 √ Rx -1, (32)
and the slope dH 1 /dx by integration of K 1 ,

dH 1 dx = 2 x R K 1 2 √ Rx -x. ( 33
)
The asymptotic behavior of the slope at infinite x (i.e. in the intermediate region) is obtained from a series expan-sion of eq. ( 33)

dH 1 dx ≃ - 1 R ln x ξ , (34)
where ξ = e 2γ-1 /R and γ ≃ 0.577216 is the Euler -Mascheroni constant. An illustration of the kinetic regime is given for R = 100 in figs. 2 both for the flux and slope. Kelvin effect regularizes the solution. The flux does not diverge any more but drops over the dimensionless distance ∼ ξ from the contact line (i.e. dimensional distance ∼ ℓ R as expected for the kinetic regime). We obtain an expression similar to that of Cox-Voinov for the moving contact line [26]; the spatial variation of curvature is also similar. The characteristic length of the microscopic region is ℓ R e 2γ-1 . One notes that:

this length is exactly the same as the Voinov length obtained for the moving contact line problem in the R ≫ 1 limit [26] ; -The dependence [START_REF] Grinberg | Selected Problems of the Mathematical Theory of Electric and Magnetic Effects[END_REF] is invariant of the microscale relaxation mechanism; only the expression for ξ depends on it.

7 Numerical solution for arbitrary R For a finite R, there is no analytical solution and the problem is solved numerically. Assuming that x3 dK 1 /dx → 0 at x → 0, integration of eq. ( 24) gives

x3 dK 1 dx = - x 0 j1 dx. ( 35
)
The interfacial concentration ci,1 can be expressed as a function of the mass flux (see Appendix B):

ci,1 (x) = 1 - ∞ 0 G 0 (x, x′ ) j1 (x ′ )dx ′ , (36)
with G 0 (x, x′ ) given by eq. (B.14). Equation ( 28) thus reads

K 1 = -1 + ∞ 0 G 0 (x, x′ ) j1 (x ′ )dx ′ -R j1 . ( 37
)
By its substitution into eq. ( 35), one gets the governing equation

x3 ∞ 0 ∂G 0 (x, x′ ) ∂ x j1 (x ′ )dx ′ -R d j1 dx = - x 0 j1 (x ′ )dx ′
(38) that can be solved only numerically. Details of the numerical method are given in Appendix C. Once j1 is known, concentration ci,1 and curvature K 1 are obtained through equations ( 36) and (37), see Appendix D; dH 1 /dx and H 1 are computed by successive integrations of K 1 .

This numerical approach is used in the following section to study the situation dominated by diffusion of vapor in the gas.

analyze in Λ on the contact line behavior, restricting the simulations to the purely diffusive case that corresponds to a situation such that R ≪ 1 ≪ Λ.

Because of the presence of the lengthscale Λ ≫ ℓ, one identifies three asymptotic regions instead of two: the microregion dominated by the Kelvin effect and two intermediate regions where this effect is negligible. The first intermediate region ranges from 1 to Λ in the dimensionless units, while the second goes from Λ to infinity ( L in reality).

To describe both intermediate regions, the problem defined in sec. 5 is solved neglecting the Kelvin effect, which means ci,eq = 0. As mentioned in sec. 3, the solution (A.12) can be obtained by analogy with an electrostatic problem, cf. Appendix A. The asymptotic expressions of this solution are obtained for both intermediate regions, assuming that the boundary layer thickness Λ is much larger than the microregion size. The development obtained for x → 0 within the ceq,1 = 0 assumption describes the first intermediate region x ≪ Λ, while the limit x → ∞ describes the second intermediate region such that Λ ≪ x. It is however impossible to solve the microregion analytically so that numerical solution for all three regions will be found. Some scaling proper to the microregion is discussed.

Asymptotic solutions for curvature in two intermediate regions

We begin with the second intermediate region x ≫ Λ, where the mass flux can be approximated by the expression stemming from eq. (A.12),

j1 (x ) ≃ - 1 Λ . ( 39
)
By using this expression in eq. ( 24), one obtains the curvature

K 1 = - 1 Λx + α 1 x2 + α 2 , (40)
with α 1 and α 2 two integration constants. Vanishing curvature at infinity imposes α 2 = 0. Considering that eq. ( 40) is an asymptotic solution for x → ∞, we get

K 1 (x) ≃ - 1 Λx ( 41
)
at a leading order.

We turn now to the first intermediate region x ≪ Λ, where the mass flux can be also obtained from eq. (A.12), j1 (x) ≃ -

1 π Λx . ( 42
)
This scaling was obtained previously for the thin 3D axisymmetric drop and infinite boundary layer [4]. However the present solution is different because the geometry is 2D and the liquid film is semi-infinite. For these reasons, the asymptotic solution for Λ → ∞ is always Λ-dependent (see eq. (A.12) and the asymptotics (39,42), where j 1 → 0 when Λ → ∞). Instead of Λ, the solution in ref. [4] contains the drop diameter.

The flux diverges at the contact line, as expected: it cannot describe the microregion dominated by the Kelvin effect. One gets the curvature after two successive integrations of eq. ( 24) where the expression (42) is used for j1 :

K 1 (x) ≃ - 4 3 1 π Λx 3 + α 1 x2 + α 2 , (43)
containing the integration constants α 1 and α 2 . Before going further, we need to determine which term in the rhs of eq. ( 43) is the leading order of the asymptotic development in the subregion x ≪ Λ. The leading term order of magnitude estimated at the cross-over scale,

M ≡ max 1 Λ2 , |α 1 | Λ2 , |α 2 | , (44)
should match the order of magnitude of the solution (41), also estimated at the cross-over scale, i.e.

M ∼ 1 Λ2 for x ∼ Λ. ( 45
)
First, we show that α 2 can be neglected in eq. ( 43). The matching condition (45) cannot be satisfied if |α 2 | ≫ 1/ Λ2 , since the definition (44) implies either M ∼ |α 2 | ≫ 1 Λ2 or M ≫ |α 2 |. Indeed, the first case is contradictory with eq. ( 45), which means |α 2 | 1/ Λ2 . An immediate consequence is α 2 ≪ 1/ Λx 3 for x ≪ Λ (because, for x ≪ Λ, 1/ Λ2 ≪ 1/ Λx 3 ). In other words, the first term in the r.h.s. of eq. ( 43) prevails asymptotically over the last which thus vanishes from the asymptotic solution.

Consider now the following asymptotic relation between the remaining two terms in eq. ( 43): 1/ Λx 3 ≪ |α 1 |/x 2 . In this case, eq. ( 43) leads to |K 1 (x)| ∼ |α 1 |/x 2 , and the matching condition (45) imposes |α 1 | ∼ 1. We get |K 1 (x)| ∼ 1/x 2 at leading order. This alternative is non-physical since the curvature is independent of Λ when Λ → ∞ (see the remark above on this issue). The only alternative is thus |α 1 |/x 2 ≪ 1/ Λx 3 , which implies the condition α 1 = 0 (we consider the subregion x ≪ Λ). The asymptotic solution for the curvature in the first intermediate region is thus

K 1 (x) ≃ - 4 3 1 π Λx 3 . ( 46
)
One can see that the curvature (i.e., pressure) divergence is non-integrable at x → 0 as expected for an intermediate region.

to the expression for the slope in the first intermediate region (x ≪ Λ),

dH 1 dx ≃ 8 3 1 π Λ 1 √ x - 1 √ η , (47)
with η an integration constant. It is worth noting that a similar expression has been obtained by Berteloot et al [8]. However, both the "microscopic cut-off" length η and the prefactor imposed ad hoc in ref. [8] are obtained rigorously within the present approach (see the next section for the η determination).

The slope in the second intermediate region (x ≫ Λ) is obtained by integration of relation (41):

dH 1 dx ≃ - 1 Λ ln x Λ + C , (48)
with C = -Λ dH1 dx (x = Λ) an integration constant. When Λ ≫ η, the slopes (47) and ( 48) should match at the crossover x ∼ Λ. This results in C = k Λ/η with k a positive constant of the order unity. Both η and C are determined by matching to the microregion model in the next section.

Figures 2 shows the full numerical solution together with both asymptotic solutions (47, 48) to illustrate their validity.

Numerical determination of constants in the asymptotic expressions

The microregion asymptotic solution cannot be obtained analytically. Instead, the constants η and C that enter Eqs. (47, 48) are determined for R = 0 by solving numerically eq. ( 38) valid in all (microscopic and both intermediate) regions. The procedure described in Appendix C is used to solve such a complicated equation.

The slope perturbation dH 1 /dx is zero at the contact line since the contact angle is imposed at x = 0. In the microregion, dH 1 /dx ∼ -x/ Λ [see fig. 3b]. The Λ-1/2 scaling is important to match the asymptotics (47) coming from the first intermediate region. The departure from the linear behavior corresponds to the end of microregion. Its size is of the order 1 in the dimensionless units (ℓ in the dimensional), as expected.

The first intermediate region is described by eq. ( 47) that involves a characteristic length η. It can be obtained numerically by plotting the quantity

Γ = 3/8(π Λ) 1/2 dH 1 dx -x-1/2 (49)
as a function of x. This quantity is derived from eq. ( 47) and, according to it, is expected to be constant in the first intermediate region. According to the numerical solution (fig. 3a), this quantity shows the x-independent region (where Γ ≃ -1.37) only for Λ ≫ 1. This is exactly the criterion of existence of the first intermediate region, cf. sec. 8. From the relation Γ = -η -1/2 that comes from Eqs. (47,49), one gets η ≃ 0.53, which is of the order of 1 as expected.

To analyze the large scale behavior (in the second intermediate region) one needs to renormalize the slope data of fig. 3. It is done in fig. 4a. A logarithmic variation of dH 1 /dx is observed for x/ Λ > 1, as predicted by eq. ( 48). The integration constant C is obtained by fitting eq. ( 48) to the numerical results [see dashed lines in fig. 4a ] and taking the intersection of each fit with the dash-dotted line x = Λ. The parameter C is given in fig. 4b as a function of Λ. For Λ ≫ 1, we get C = β Λ with β ≃ 2 and the law C = k Λ/η indeed holds, as expected (cf. sec. 8.2). By using η ≃ 0.53 obtained above, we get k ≃ 1.46.

Apparent contact angle

It is interesting to analyze the behavior of the apparent contact angle θ app , i.e. that observed at the macroscopic length scale L. Strictly speaking, the slope depends on x so that θ app can only be determined after asymptotic matching to a particular macroscopic geometry (drop, bubble, etc.). Such a procedure has been already performed, e.g. for the capillary dewetting [31]. However, some useful information about the apparent contact angle can be obtained from the present analysis. It can be taken from the slope in the second intermediate region.

Indeed, in the case of the thin thermal boundary layer and long film, x can be in the second intermediate region Λ ≪ x ≪ L, where, from eq. ( 48), the slope varies logarithmically:

dh dx ≃ θ - 3µD g ∆c ρσθ 3 Λ β Λ ℓ + ln x Λ . ( 50
)
This expression means that the actual apparent contact angle value will slightly depend on the curvature in the macroscopic region, similarly to the moving contact line problem [31]. Neglecting the weak logarithmic dependence in eq. ( 50) yields an approximate expression of the apparent contact angle,

θ app ≃ θ - 3β √ ℓ µD g ρσθ 3 ∆c √ Λ . ( 51
)
On the other hand, eq. ( 51) can also be obtained by tending x → ∞ in eq. ( 47), i.e. within the first intermediate region. In this case one gets instead of 3β ≃ 6 the coefficient 8/ √ π η ≃ 6.2. Two values are very close, which is not surprising because both intermediate regions are matched (in the asymptotic sense) for x → ∞ for the first and x → Λ for the second. The agreement shows simply that the matching has been performed correctly. Note that eq. ( 51) is the result of the intermediate regions where microscopic effects are negligible (see eqs. ( 47)-( 48)). Therefore, the form of eq. ( 51) is independent of the microscopic singularity relaxation mechanism. The microscopic details impact only the microregion length scale ℓ. For instance, if the singularity was relaxed with the hydrodynamic slip length ℓ s , the characteristic length 3ℓ s /eθ (where e = 2.71 . . . ) [32] would need to be used instead of ℓ. If the singularity was relaxed with the precursor film, ℓ would scale on the Israelachvili length (A/6πσ) 1/2 , where A is the Hamaker constant. Equation ( 51) is very close to the result of Berteloot et al [8] (one needs to drop the independent Cox-Voinov term in their eq. ( 13) to conform to the present problem statement; in addition, it should be noted that the boundary layer thickness Λ is included in their parameter J 0 ). There are however differences between our approach and x that of ref. [8]. Berteloot et al introduced the microscopic scale ℓ phenomenologically, and found the numerical prefactor 3β of the second term equal to 8. In our approach, the scales are matched rigorously, and a more precise value of 6 is obtained. In the work of Berteloot et al, only the first intermediate region x ≪ Λ is considered. The slope logarithmic variation in the second intermediate region is thus implicitely neglected. The last but not least, within our approach, the local curvature and heat flux are obtained at all length scales including the microregion. Their behavior is discussed in the next section.

/ Λ Λ dH 1 /dx Λ =500 Λ =300 Λ =100 Λ =30 Λ =10 Λ =3 (

Mass flux and curvature behavior

Figures 5 displays the mass flux and curvature for Λ varying over several orders of magnitude. First, one can see that both the mass flux and the curvature are regular at the contact line due to the account of the Kelvin effect.

The microregion corresponds to the domain x 1, where the Kelvin effect is dominant and regularizes the singularity. Numerical results show that the flux j1 varies in the microregion as x2 Λ-1/2 [dash-dotted line in fig. 5a].

In the first intermediate region 1

x Λ, -Λ1/2 j1 and -Λ1/2 K 1 plotted as functions of x gather on master curves [see figs . 5]. Similarly to the slope, one can see that the Λ-1/2 scaling of other microregion variables matches that of their intermediate region counterparts [cf. equations (42) and ( 46)].

In the second intermediate region x ≫ Λ, the flux goes to the constant value -1/ Λ and the curvature behaves as -1/( Λx) (figs. 6). As predicted by equations ( 39) and ( 41), the quantities -j1 Λ and -K 1 Λ2 plotted as functions of x/ Λ gather on master curves. A cross-over to this constant flux regime occurs at a distance of about Λ to the contact line, as expected.

Finally, it is interesting to compare the behavior of the flux and slope for the purely kinetic and purely diffusive regimes, as depicted in figs. 2. The two parameters, R = 100 and Λ = 100, have been chosen in order to obtain the same flux far from the contact line. First, the behavior of the evaporative flux is different. Diffusion in the gas induces a large peak close to the contact line, while a monotonic decrease is observed for the kinetic case. At large scale, both regimes show a logarithmic variation of the slope, but the diffusive regime exhibits also an additional (the first) intermediate region that does not exist for the kinetic regime.

Conclusions

Evaporation or condensation controlled by vapor diffusion in the atmosphere causes a hydrodynamic singularity in the vicinity of the edge of the liquid divergence of contact line kinetic resistance does not lead singularity relaxation (this was first shown in ref. [START_REF] Saxton | [END_REF]). Another microscopic phenomenon thus needs to be introduced into the modeling of such a phenomenon to relax the singularity.

The singularity relaxation by Kelvin effect (a dependence of saturation vapor concentration on the interface curvature) is used to solve in a closed form a 2D problem of a thin liquid wedge in the atmosphere at given super-saturation ∆c in the presence of diffusive boundary layer of a finite thickness (the negative ∆c causes undersaturation of the atmosphere and thus liquid evaporation). We use the approximations of weak mass exchange, i.e. small ∆c, and small wedge slope.

The numerical estimates for the size ℓ of the contact line vicinity, in which the Kelvin effect is important (the Kelvin length) can be found elsewhere [26]. They lead to the same conclusion as for the moving contact line case: for common fluids, the liquid height in the microregion is never much larger than 1 nm, and much lower in some cases, which is not consistent with the continuum hydrodynamic approach. The macroscopic scaling laws are independent of the precise mechanism that ensures singularity relaxation. Due to scale separation, the intermediate region scalings remain valid if the singularity is relaxed with another microscale effect like hydrodynamic slip [3] or flow in a thin adsorbed liquid film adjacent to the wedge [10]; In the asymptotic expressions, only ℓ value depends on the microscopic details.

We consider two limiting cases: the kinetic case where the mass exchange rate is entirely controlled by the interfacial kinetic resistance (1 ≪ Λ ≪ R), and the diffusive case where it is entirely controlled by the diffusion in the gas phase (R ≪ 1 ≪ Λ). The kinetic case is solved analytically. The result is very similar to the Cox-Voinov solution for the moving contact line case, with exactly the same characteristic size ℓ of microscopic region.

u II IV a IV III A 0 B -π/2 C -π E π F π/2 D(∞) b II B π/2 A(∞) F -π/2 D III C(-∞) E(-∞)
In a general problem, there is another length scale: the thickness Λ ≫ ℓ of the diffusion boundary layer. Because of this, two intermediate regions can be identified. For the first of them, ℓ ≪ x ≪ Λ. For the second, Λ ≪ x ≪ L, where L is the macroscopic length scale (drop or bubble size, capillary ridge width, etc.). The asymptotic scaling laws describing the mass flux, curvature and interface slope are derived for both intermediate regions. They agree with the numerical solutions of general problem.

It is found that the apparent contact angle is close to the value obtained by Berteloot et al. [8]. The prefactor in their equation should be however corrected. The precise value of the apparent contact angle depends weakly (logarithmically) on the macroscopic curvature (that of the drop or curved meniscus). This work has been financially supported by the LabeX LaSIPS (ANR-10-LABX-0040-LaSIPS) managed by the French National Research Agency under the "Investissements d'avenir" program (ANR-11-IDEX-0003-02).

Author contribution statement

All the authors were involved in the problem statement, interpretation of results and manuscript review. FD and VJ performed the numerical simulations. VSN performed the analytical calculations.

A Vapor diffusion from thin liquid wedge to the boundary layer One aims at solving the 2D Laplace equation for the vapor concentration c produced by the evaporating zero thickness liquid wedge within the diffusive boundary layer, cf. fig. 1(a). The geometrical setup of the reduced problem is adapted to the already solved equivalent electrostatic

D [ς(r)∆ r G(r, r ′ ) -G(r, r ′ )∆ς(r)]dr = L [ς(r)∇ r G(r, r ′ ) -G(r, r ′ )∇ς(r)] • ndl r , (B.1)
where G is the Green function, ς = c 1 -1, and L is the boundary of the domain D with the outward unit normal n. The boundary consists of the x axis L x and the line L Λ defined by y = Λ (the integrals over the infinitely distant vertical parts of the contour are zero, which can be easily checked with the Green function derived below). The differentiation in all differential operators is assumed hereafter to be performed over the components of the vector r rather than those of r ′ .The equation for ς is The solution is easy to construct by the mirror reflection method, by placing two point sources at locations (u ′ , v ′ ) and (u ′ , -v ′ ), and two point sinks at locations (-u ′ , v ′ ) and (-u ′ , -v ′ ). One gets

G(u, v, u ′ , v ′ ) = 1 4π ln [(u -u ′ ) 2 + (v -v ′ ) 2 ][(u -u ′ ) 2 + (v + v ′ ) 2 ] [(u + u ′ ) 2 + (v -v ′ ) 2 ][(u + u ′ ) 2 + (v + v ′ ) 2 ] . (B.11)
The solution G(x, y, x ′ , y ′ ) can be readily obtained by substitution of the relations (B.8, B.9).

ki+1 ki x 1 x 2 x i x i+1 x N ki-1 x i-1 c L 1 O i U (x)
(r ′ ) = Lx [ς(r)∇ r G(r, r ′ ) -G(r, r ′ )∇ς(r)] • ndl r .
(B.12) To obtain the governing equation connecting the reduced vapor concentration at the interface and the mass flux, one needs to consider the solution for y

′ = 0, x ′ > 0, ς(x ′ , y ′ = 0) = - ∞ 0 G 0 (x, x ′ )j 1 (x)dx, (B .13)
where

G 0 (x, x ′ ) = G(x, y = 0, x ′ , y ′ = 0) = 1 π ln exp πx 2Λ -exp πx ′ 2Λ exp πx 2Λ + exp πx ′ 2Λ . (B.14)
By using the relation ς = c 1 -1, one obtains eq. (36).

C Solution method for the integro-differential equation

The tilde denoting dimensionless quantities is omitted in this appendix for clarity. Equation ( 38) is solved numerically. The unknown function j 1 (x) is interpolated in the domain [0, L c ] with L c an arbitrary cut-off such that L c ≫ Λ. The interpolation in the domain [0, L c ] is performed by splitting this interval into N subintervals of length k i (i = 1 to N ), and by using the interpolation functions U The approximate expression for j 1 (x) then reads

i (x) = H(x i -k i /2) -H(x i + k i /2),
j 1 (x) = N i=1 j (i) U i (x) + H(x -L c )f (x). (C.1)
where f (x) is an asymptotic expansion of j 1 (x) at large x, which can be obtained a priori. Indeed, at a large x, the concentration field in the gas phase is such that |∂c 1 /∂x| ≪ |∂c 1 /∂y|, which allows to consider a lubricationlike approximation for the diffusion equation (a sufficient condition for this approximation to be valid being x ≫ Λ).

The Laplace equation ( 25) thus reduces to ∂ 2 c 1 /∂y 2 = 0 and eq. ( 28) to j 1 = (c i,1 -1)/Λ. Using relation (28) It is worthwhile to point out that the asymptotics (C.3) is different from the asymptotics (39) for the second intermediate region obtained from equation (A.12). Indeed, the model developed in Appendix A ignores the microscale effects. They however are accounted for here. An important consequence is that the numerical solution of the equation (38) can describe the whole liquid wedge, i.e. from x = 0 to infinity. While the Kelvin effect declines outside the microregion, the interfacial resistance persists and thus contributes to the relation (C.3). One notes however that, in reality, Λ ≫ R by many orders of magnitude.

There are N unknowns in the expression (C.1), thus requiring N equations. The discretized version of equation (38) is

D i = 1 (Λ + R) ∞ Lc ∂G 0 ∂x (x i , x ′ )dx ′ - 1 (Λ + R) 2 ∞ Lc 1 x ′ ∂G 0 ∂x (x i , x ′ )dx ′ . (C.6)
The integral in eq. (C.5) and the first integral in eq. (C.6) are computed analytically. Due to the interchangeability of x and x ′ in the expression for G 0 (B.14), the antiderivative with respect of x ′ of ∂G 0 /∂x is simply equal to -G 0 . Note that the kernel of the integral (C.5) diverges when x ′ = x i , which happens when m = i. In this case, one can easily check that the integral has a zero Cauchy Principal Value, so C ii = 0. The last integral in eq. (C.6) is computed numerically, by adding N a additional nodes in the D Numerical integration to get K 1 from j 1

The tilde denoting dimensionless quantities is omitted in this appendix for clarity. Computing the curvature K 1 (x i ) from eq. (37) requires numerical evaluation of an integral which involves the Green function G 0 (x i , x ′ ). Using the discretization of j 1 given by eq. (C.1), the integral is evaluated from

∞ 0 G 0 (x i , x ′ )j 1 (x ′ )dx ′ ≃ N m=1 j (m) xm+km/2 xm-km/2 G 0 (x i , x ′ )dx ′ +α ∞ Lc G 0 (x i , x ′ )
x ′p dx ′ . Because of the of an absolute value in eq. (B.14), two cases must be considered when computing the antiderivative of G 0 (x i , x ′ ) (defined as usually within a constant). For x ′ < x i ,

G 0 (x i , x ′ )dx ′ = Λ π 2 -4 Li 2 exp π 2Λ (x ′ -x i ) + Li 2 exp π Λ (x ′ -x i ) , (D.3)
where Li 2 is the dilogarithm function. For

x i < x ′ , G 0 (x i , x ′ )dx ′ = - 2Λ π 2 Li 2 1 -exp π 2Λ (x i -x ′ ) + Li 2 -exp π 2Λ (x i -x ′ ) + 1 π (x ′ -x i ) ln 1 -exp π 2Λ (x i -x ′ ) . (D.4)
Notice that G 0 (x i , x ′ ) diverges when x ′ = x i , but the integral converges.

Fig. 1 :

 1 Fig. 1: Hierarchy of scales considered in the article and geometries for the vapor diffusion (a) and hydrodynamic (b-d) problems. Note that the radius of curvature in (b) is assumed to be much larger than the diffusive boundary layer width Λ shown in (a).

Fig. 2 :

 2 Fig. 2: (Color online) Comparison of kinetic [eqs. (31,33) with R = 100, in red] and diffusive regimes (R = 0 and Λ = 100); (a) flux curves and (b) slope curves. The dashed lines are the asymptotes coming from eq. (34) for the kinetic regime, eqs. (47) and (48) for the first and second intermediate subregions in the diffusive regime. In the latter case, panel (b) illustrates also the asymptotic matching of the first and second intermediate subregions.

Fig. 3 :

 3 Fig. 3: Matching of microregion and first intermediate region. (a) Solid lines represent Γ (x) function obtained numerically; dash-dotted line is the asymptotic value Γ ≃ -1.37 of the first intermediate region. (b) -Λ1/2 dH 1 /dx; solid lines: numerical results; dashed line: asymptotic microregion behavior. The arrows show the directions of the Λ increase.

Fig. 4 :

 4 Fig. 4: Matching of the first and second intermediate regions. (a) ΛdH 1 /dx for different values of Λ; solid lines: numerical results; dashed lines: asymptotic equation (48); vertical dash-dotted line: x = Λ. Note that the microregion behavior cannot be seen on these curves (the scale is too large). (b) Integration constant C as a function of Λ.

Fig. 5 :

 5 Fig. 5: Behavior of mass flux (a) and curvature (b) for different values of Λ. The arrows show the directions of the Λ increase. The scaling is appropriate for microregion and first intermediate region. Dashed line: first intermediate region asymptotics (42) (a) and (46) (b). Dash-dotted lines: microregion behavior (fits to numerical curves).

Fig. 6 :

 6 Fig. 6: Behavior of mass flux (a) and curvature (b) for different values of Λ. The arrows show directions of the Λ increase. The scaling is appropriate for the second intermediate region. Dashed lines: asymptotics (39) (a) and (41) (b) for the second intermediate region. Vertical dashdotted lines correspond to x = Λ.

Fig. A. 1 :

 1 Fig. A.1: Conformal mapping: (a) z domain, (b) w domain. The corresponding points A,B,C,D,E,F and areas I,II,II,IV of the both domains are shown.

Fig. B. 1 :

 1 Fig. B.1: Domain for the Green function problem in the z plane (a) and its conformal map, the w plane (b).

Fig. C. 1 :

 1 Fig. C.1: Mesh and interpolation function U i (x).

  with H(x) the Heaviside function and x i the center of the i th subinterval (see figure C.1).

, one gets K 1 = 1

 11 -(Λ + R)j 1 -1. Combining this relation with equation (24) leads to (x) = -1/(Λ+R) that corresponds to zero curvature at infinity. The truncation of the solution for large x yields f (x) =be used in equation C.1.

C 1 m=1j

 1 im j (m) -Rx 3 i j (i+1) -j (i) x i+1 -x i + i-(m) k m +j (i) k i 2 = x 3 i D i for i = 1,2, . . . N, (C.4) with x N +1 = L c and j (N +1) = -(Λ+R) -1 +(Λ+R) -2 L -1 c . The coefficients C im and D i are obtained with the expression (B.14) for the Green function: , x ′ )dx ′ , (C.5)

L c =10 3 L c =3×10 3 ,

 33 10 4 , 3×10 4

Fig. C. 2 :L

 2 Fig. C.2: Convergence test for the numerical treatment when increasing the cut-off L c for Λ = 100 and R = 0.

(D. 1 ) 2 G 0

 120 The second integral in the rhs of relation (D.1) is approximated by adding N a additional nodes in the range [L c , L ∞ ] with L c ≪ L ∞ , as already done in eq. (C.7):∞ Lc G 0 (x i , x ′ ) x ′p dx ′ ≃ L∞ Lc G 0 (x i , x ′ ) x ′p dx ′ (x i , x ′ )dx ′ . (D.2)

problem where the potential is defined as ϕ = (c-c eq )/∆c. The reduced coordinates are (u, v) and the problem ∆ϕ(u, v) = 0 , (A.1) ϕ| v=π/2 = 1, (A.2) ∂ϕ ∂v v=0, u>0 = 0, (A.3)

Unlike the initial problem statement, the wedge situates to the left of the point u = 0 so that the relations between the actual and reduced coordinates are x = -2uΛ/π and y = 2vΛ/π. A symmetric equipotential plate can be added at v = -π/2 and the domain can be extended to Such a domain may be conformally mapped to a halfband in the plane (a, b) shown in figure A.1a, where ϕ = 1 is imposed at the side a = 0, -π < b < π, and ϕ = 0 is at the sides b = ±π, 0 < a < ∞. The corresponding conformal mapping function [START_REF] Korn | Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review[END_REF] is

where z = a + ib and w = u + iv. It is noteworthy that the inverse function is identical. The solution

of the Laplace problem for such half-band domain is known [START_REF] Grinberg | Selected Problems of the Mathematical Theory of Electric and Magnetic Effects[END_REF] ( §7.4, p. 47). From (A.6), one obtains

where the function arctan(x ′ , y ′ ) gives the polar angle -π ≤ α ≤ π of a point (x ′ , y ′ ) (in programming languages it is often called atan2).

The substitution of equations (A.8,A.9) into (A.7) results in the complete solution of the problem (A.1-A.4).

We are interested in the evaporative flux (i.e. in the concentration gradient) at the liquid-gas interface v → +0, u < 0 (line C-D in figure A.1b). Since both arguments in eq. (A.9) are negative in this case, eq. (A.9) reduces to b = -π -arctan sin v sinh u , (A.10)

where arctan(x ′ ) is now for -π/2 < x ′ < π/2 as usually. One can see now that b → -π when v → 0.

To find the derivative ∂ϕ/∂v| v→+0 at u < 0, one writes In the notation of sec. 5, the flux j1 reads

For x ≫ Λ, eq. (A.12) results in eq. ( 39), which is simply a constant gradient solution for the boundary layer of the thickness Λ. For x ≪ Λ, one obtains eq. ( 42).

B Governing equation for the vapor concentration field

The tilde denoting dimensionless quantities is omitted in this appendix for clarity.

The starting point for the derivation is Green's second identity