domain: sdv chim

text:
Dinesh Dhumal

Wenjun Lan

Ling Ding

Yifan Jiang

Zhenbin Lyu

Erik Laurini

Domenico Marson

Aura Tintaru

Nelson Dusetti

Suzanne Giorgio

An ionizable supramolecular dendrimer nanosystem for effective siRNA delivery with a favorable safety profile

Keywords: siRNA delivery, non-viral vector, dendrimer, supramolecular chemistry, ionizable vector, gene silencing

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Table of Contents Graphic:

An ionizable amphiphilic dendrimer self-assembles into a supramolecular dendrimer nanosystem and delivers small interfering RNA (siRNA) with potent gene silencing and effective anticancer activity, outperforming the current gold standard delivery agent while providing a favorable safety profile.

Introduction

Gene therapy based on small interfering RNA (siRNA) therapeutics is emerging as a novel and promising approach to treat diseases [START_REF] Setten | The Current State and Future Directions of RNAibased Therapeutics[END_REF][START_REF]Delivering the Promise of RNA Therapeutics[END_REF], particularly since the breakthrough success of the first siRNA drug Patirisan that was approved in 2018 [START_REF] Ledford | Gene-silencing Technology Gets First Drug Approval After 20-year Wait[END_REF] and the recently approved second siRNA drug Givosiran [START_REF] Mullard | RNAi agents score an approval and drive an acquisition[END_REF]. The advantages of siRNA therapeutics is their ability to potently and specifically silence any target gene via Watson-Crick base pairing with the corresponding complementary mRNA sequence and consequential cleavage of the mRNA. However, siRNA therapeutics are unstable in acidic/basic conditions and in biofluids because these RNA molecules are vulnerable to chemical and enzymatic hydrolysis. Additionally, because of their polyanionic nature and hydrophilicity, naked siRNAs cannot easily penetrate the cell membrane to reach the cell interior. Therefore, delivery systems that shield siRNA molecules from hydrolytic decomposition, cover their dense negative charge and promote their transport across cell barriers for effective gene silencing are highly sought after, which constitutes a major challenge for clinical use [START_REF] Kanasty | Delivery Materials for siRNA Therapeutics[END_REF][START_REF] Dong | Strategies, Design, and Chemistry in siRNA Delivery Systems[END_REF].

Identifying safe and effective systems for siRNA delivery is an actively studied topic.

Although viral vectors are generally more efficient, growing concerns over their safety and tedious manufacturing have fostered the development of non-viral vectors which, in general, are endowed with better safety profiles, easier manufacturing procedures and more flexible safe-by-design options [START_REF] Kanasty | Delivery Materials for siRNA Therapeutics[END_REF][START_REF] Dong | Strategies, Design, and Chemistry in siRNA Delivery Systems[END_REF][START_REF] Kim | Rekindling RNAi Therapy: Materials Design Requirements for In Vivo siRNA Delivery[END_REF][START_REF] Yin | Non-viral Vectors for Gene-Based Therapy[END_REF][START_REF] Ho | Biomaterials in siRNA Delivery: A Comprehensive Review[END_REF]. Lipids and polymeric systems are the two most extensively investigated non-viral vectors [START_REF] Rietwyk | Next-Generation Lipids in RNA Interference Therapeutics[END_REF][START_REF] Cooper | Polymers for siRNA Delivery: A Critical Assessment of Current Technology Prospects for Clinical Application[END_REF]. Within the latter class, dendrimers, regularly branched synthetic polymers with a ramified structure, are particularly promising by virtue of their precisely controlled composition, unique cooperative multivalence and favorable high drug loading within small nanometric volumes [START_REF] Khandare | Multifunctional Dendritic Polymers in Nanomedicine: Opportunities and Challenges[END_REF][START_REF] Liu | Dendrimers as Non-viral Vectors for siRNA Delivery[END_REF][START_REF] Cao | Molecular Engineering of Dendrimer Nanovectors for siRNA Delivery and Gene Silencing[END_REF]. However, their translation to clinical use has been hampered because of their challenging stepwise synthesis and purification [START_REF] Svenson | The dendrimer paradox -high medical expectations but poor clinical translation[END_REF].

To overcome the synthetic challenge of dendrimers and to capitalize on the advantages of lipid and dendrimer vectors, we have recently proposed innovative non-covalent supramolecular dendrimers for siRNA delivery [START_REF] Dong | A Dual Targeting Dendrimer-Mediated siRNA Delivery System for Effective Gene Silencing in Cancer Therapy[END_REF][START_REF] Liu | A Fluorinated Bolaamphiphilic Dendrimer for On-demand Delivery of siRNA, via Specific Response to Reactive Oxygen Species[END_REF][START_REF] Liu | Adaptive Amphiphilic Dendrimerbased Nanoassemblies as Robust and Versatile siRNA delivery Systems[END_REF][START_REF] Yu | An Amphiphilic Dendrimer for Effective Delivery of Small Interfering RNA and Gene Silencing in Vitro and in Vivo[END_REF]. The supramolecular dendrimers can be constructed via self-assembly of small amphiphilic lipid-dendrimer conjugates through van der Waals and hydrogen bond interactions, and combine the self-assembling feature of lipid vectors with the multivalent cooperativity of dendrimer vectors for siRNA delivery (Fig. 1A).

Figure 1B shows one of our first amphiphilic lipid-dendrimer conjugates, 1a, which was developed for siRNA delivery [START_REF] Yu | An Amphiphilic Dendrimer for Effective Delivery of Small Interfering RNA and Gene Silencing in Vitro and in Vivo[END_REF]. The structure of 1a consists of a long alkyl chain as the hydrophobic component and a small poly(amidoamine) (PAMAM) dendron carrying primary amine terminals as the hydrophilic entity. The dendrimer 1a self-assembles into supramolecular dendrimer nanomicelles that favorably bind to siRNA for effective delivery [START_REF] Chen | Mastering Dendrimer Self-Assembly for Efficient siRNA Delivery: From Conceptual Design to In Vivo Efficient Gene Silencing[END_REF]. By conjugating arginine residues at the dendrimer terminals in 1a, we further obtained 1b (Fig. 1B), which mimics arginine-rich cell penetration peptides for more effective cellular uptake and better siRNA delivery [START_REF] Liu | Promoting siRNA Delivery via Enhanced Cellular Uptake Using an Arginine-decorated Amphiphilic Dendrimer[END_REF]. In this report, we present a new ionizable supramolecular dendrimer formed via self-assemble of the amphiphilic dendrimer 1c (Fig. 1B) which, at variance with 1a and 1b, carries tertiary amine dendrimer terminals for effective siRNA delivery with a favorable safety profile. Chemical structures of the amphiphilic dendrimers 1a, 1b and 1c bearing primary amines, arginine residues and tertiary amines, respectively, at dendrimer terminals.

The rationale for developing the ionizable supramolecular dendrimer formed with 1c for siRNA delivery was based on the pKa of the tertiary amine terminals of the dendrimer surface, which will allow their protonation to generate sufficiently high positive surface charge to bind anionic siRNA through electrostatic interactions at physiological pH, while the surface potential of the siRNA/dendrimer complexes should be low enough to prevent their clearance by the immune system yet sufficient to promote cellular uptake [START_REF] Ho | Biomaterials in siRNA Delivery: A Comprehensive Review[END_REF][START_REF] Rietwyk | Next-Generation Lipids in RNA Interference Therapeutics[END_REF][START_REF] Kulkarni | Lipid Nanoparticles Enabling Gene Therapies: From Concepts to Clinical Utility[END_REF][START_REF] Cullis | Lipid Nanoparticle Systems for Enabling Gene Therapies[END_REF]. In addition, the ionizable tertiary amine terminals in 1c should be highly protonated in acidic endosomes, thereby inducing endosomal disruption for siRNA release. The first siRNA drug Patisiran is also based on ionizable cationic lipid nanoparticles (LNPs) for siRNA delivery. However, LNPs have the disadvantage of requiring specific and laborious ethanol-loading formulation techniques for preparation [START_REF] Kulkarni | Lipid Nanoparticles Enabling Gene Therapies: From Concepts to Clinical Utility[END_REF][START_REF] Evers | State-of-the-Art Design and Rapid-Mixing Production Techniques of Lipid Nanoparticles for Nucleic Acid Delivery[END_REF]. We demonstrate that the amphiphilic lipid-dendrimer conjugate 1c, which harbors eight tertiary amines at the surface and a further seven tertiary amines and one triazole ring in the interior, self-assembles into a supramolecular dendrimer nanosystem and acts as an ionizable cationic vector for siRNA delivery via strong siRNA binding at physiological pH and effective release of the siRNA within the endosomal acidic environment. Importantly, formulation of the siRNA/dendrimer complex is simple and convenient, and requires only that siRNA is mixed with 1c in aqueous solution. In addition, the siRNA/dendrimer complex is small and effectively protects the siRNA from enzymatic degradation. By virtue of its beneficial ionizable nature, the supramolecular dendrimer formed by 1c exhibited low cytotoxicity while effectively delivering siRNAs into different cancer cell lines and patient-derived cancer organoids for functional silencing of target oncogenes, leading to potent anticancer activities that outperformed the current gold standard Lipofectamine 2000 (Lipo2000). Consequently, this study offers conceptual advances in constructing supramolecular dendrimers for biomedical applications. In this study, we present the ionizable supramolecular dendrimer for siRNA delivery.

Results and discussion

We readily synthesized 1c via the amidation of the ester-terminated dendrimer precursor 1 using N,N-dimethylethylenediamine (Fig. 2A, Scheme S1) [START_REF] Wu | Polycationic Dendrimers Interact With RNA Molecules: Polyamine Dendrimers Inhibit the Catalytic Activity of Candida Ribozymes[END_REF]. The amidation reaction was maintained for 5 days to ensure complete transformation of all dendrimer terminals, and the final product 1c was obtained with an excellent yield of 86% after purification via precipitation in methanol/ether followed by dialysis in water. The structural integrity and purity of 1c was confirmed using spectroscopic analysis with 1 H-, 13 C-and two-dimensional (2D) NMR, as well as high-resolution mass spectrometry (Figs. S1-S3). For example, the peaks at 2.22 ppm in the 1 H NMR spectrum and at 45.3 ppm in the 13 C NMR spectrum correspond to the characteristic NMR signals of the methyl groups on the tertiary amine terminals present in 1c (Fig. S1).

Additionally, mass spectrometry analysis yielded the expected signals related to the molecular weight and the corresponding isotopic pattern of the triply protonated species [1c+3H] 3+ (Fig. S3). Remarkably, 1c is soluble in water, with solubility similar to that of the dendrimers 1a and 1b. By virtue of its amphiphilic nature, 1c is able to self-assemble spontaneously in water into supramolecular dendrimer micelles of 12-17 nm in size, as shown by transmission electron microscopic (TEM) imaging (Fig. 2B). We then evaluated the critical micelle concentration (CMC) of 1c using a fluorescent spectroscopic assay with pyrene (Fig. 2C). The CMC value of 1c was around 17 µM, which is similar to that of 1a [START_REF] Yu | An Amphiphilic Dendrimer for Effective Delivery of Small Interfering RNA and Gene Silencing in Vitro and in Vivo[END_REF][START_REF] Chen | Mastering Dendrimer Self-Assembly for Efficient siRNA Delivery: From Conceptual Design to In Vivo Efficient Gene Silencing[END_REF]. To further monitor the process of supramolecular micelle formation by 1c, we used isothermal titration calorimetry (ITC) (Fig. 2D and Fig. S4). The ITC results confirmed the ability of 1c to self-assemble into supramolecular dendrimer nanomicelles at a CMC value of 19 µM at pH 7.4, in accordance with the results obtained from the fluorescent spectroscopic assay. Remarkably, 1c exhibited much lower cytotoxicity than 1a and 1b, particularly at higher dendrimer concentrations, as revealed by the toxicity assay on human kidney cells (HEK 293), mouse fibroblast cells (L929)

and Madin-Darby canine kidney cells (MDCK) (Fig. 2E and Fig. S5). The lower toxicity of 1c might be ascribed to the distinct tertiary amine terminal groups, which generated a relatively lower surface charge for micelles formed by 1c (+23 mV) when compared with those formed by 1a (+35 mV) and 1b (+40 mV) at pH 7.4 (Table S1). Collectively, these data confirm the spontaneous self-assembly of 1c to form stable supramolecular dendrimer nanomicelles with a favorable safety profile.

We next performed a potentiometric pH back titration of 1c, which showed continuous buffering between pH 4.0 and 9.0, highlighting the ionizable nature of 1c (Fig. 2F). According to this titration profile, 50% of the amine functionalities (8/16) of 1c were protonated at pH 7.4, likely corresponding to the eight tertiary amine terminals in 1c. At pH 5.0, corresponding to the endosomal environment, 72% of the amine groups (12/16) were protonated, implying that, in addition to the eight tertiary amine terminals, a further four tertiary amines located within the dendrimer interior were protonated under these conditions. Interestingly, ITC data repeatedly gave only small and constant endothermic peaks at pH 5.0 (Fig. S6 left), and the corresponding heat flow curve could not be used to determine any CMC value of 1c under these conditions. We reasoned that under acidic pH conditions 1c no longer forms stable supramolecular nanomicelles because of the growing number of positive charges populating the hydrophilic portions of the amphiphilic dendrimer. This causes stronger electrostatic repulsions within a confined volume that eventually overcome the hydrophobic attractions existing among the dendrimer hydrocarbon chains. Altogether, these results suggest that the supramolecular dendrimer formed by 1c will bind to siRNA at physiological pH while allowing endosomal release at an acidic pH.

Effectively, the supramolecular dendrimer micelles formed by 1c at pH 7.4 are sufficiently positively charged to allow the formation of stable complexes with the highly negatively charged siRNA molecules, thus protecting the siRNAs from degradation. We studied this siRNA/dendrimer complex formation using a gel mobility shift assay. As Figure 3A illustrates, complete retardation of siRNA migration in the gel was observed in the presence of 1c for N/P ratios above 2.5, suggesting formation of a stable siRNA/dendrimer complex.

TEM imaging further showed that the siRNA/dendrimer complexes at pH 7.4 were small and spherical, i.e., ~40 nm in size (Fig. 3B), in line with the results obtained from dynamic light scattering (DLS) analysis (Fig. 3C). In addition, we performed mesoscale molecular simulations to investigate the interactions between siRNA and 1c, and representative siRNA/dendrimer complexes formed are presented in Figure 3D. As clearly seen in Figure 3D, the siRNA molecules are uniformly distributed and reside on the surface of the supramolecular dendrimers, highlighting the effective interaction between siRNA molecules and the terminals of the dendrimer nanomicelles. We also assessed the thermodynamics of binding between siRNA and the supramolecular dendrimer formed by 1c using ITC (Fig. 3E). This revealed a negative enthalpic variation, reflecting favorable interactions between dendrimer and siRNA (DH = -4.99 kcal/mol), which indicates that the driving force behind stable siRNA/dendrimer complex formation is fundamentally electrostatic in nature. The related entropy variation upon formation of the siRNA/dendrimer complex was also favorable (i.e., TDS = +2.92 kcal/mol), which originated mainly from the release of water molecules and counterions into bulk solvent upon siRNA/dendrimer binding. Overall, the resulting negative Gibbs binding free energy (DG = -7.91 kcal/mol) confirms that the formation of stable siRNA/dendrimer complexes is a spontaneous and favorable thermodynamic process. Importantly, the siRNA/dendrimer complex effectively protected siRNA from enzymatic degradation (Fig. 3F). Naked siRNA was rapidly degraded within 5 min by RNase, whereas siRNA in complex with the dendrimer was not degraded readily by RNase and remained intact even after 120 min. Taken together, our results show that the supramolecular dendrimer formed by 1c is sufficiently protonated and positively charged at physiological pH to enable interaction with siRNA to form small and stable siRNA/dendrimer complexes within which the siRNA is shielded from enzymatic degradation. These are important prerequisites for successful siRNA delivery.

Benefiting the favorable ionizable properties of the tertiary amine terminals, the siRNA complexes formed with 1c exhibited a surface potential of +18 mV at pH 7.4, which is substantially lower than that of siRNA complexes formed with 1a (+31 mV) and 1b (+35 mV) (Table S1). At +18 mV, the surface potential is neither neutral nor too high to be disadvantageous for siRNA delivery; complexes with too high surface charges may disrupt cell membranes and induce eventual toxic effects, whereas neutral complexes have limited colloidal stability and aggregate into large particles that precipitate. Importantly, this positive surface potential is also sufficient and beneficial to promote cellular uptake via favorable electrostatic interactions with the cell membrane, which is negatively charged. Notably, when the pH decreased from 7.4 to 5.0, the zeta potential of the siRNA/1c complexes increased from +18 mV to +46 mV, a value similar to that of dendrimer 1c alone without siRNA at pH 5.0 (+48 mV). This indicates that siRNA dissociates from the siRNA/dendrimer complex at pH 5.0, which is further supported by the results obtained from the ITC experiment examining the interaction between 1c and siRNA at pH 5.0. As shown in the right panel of Figure S6, no stable binding was detected between siRNA and 1c under acidic conditions, highlighting that siRNA is released readily from the dendrimer complex at pH 5.0. This is in sharp contrast with the data at pH 7.4, where siRNA and dendrimer 1c form a stable complex (Fig. 3). We further evaluated the siRNA release from its dendrimer complex using the heparin-coupled ethidium bromide assay (Fig. 4). As siRNA and dendrimer 1c form a stable complex at pH 7.4, no siRNA release was detectable using heparin at the low dose of 2 unit/mL, and only 9 % siRNA release was observed when using heparin even at the high dose of 20 unit/mL. In contrast, more than 25% siRNA was released at pH 5.0 even at the low heparin dose of 2.0 unit/mL, whereas almost all free siRNA escaped from the siRNA/1c complex at the heparin dose of 20 unit/mL. There results highlight that the siRNA/1c complex was not stable at pH 5.0 and disassembled upon addition of heparin, hence releasing siRNA readily and easily. Collectively, these data support our rationale that, by virtue of the ionizable nature of the tertiary amine terminals within 1c, the siRNA/dendrimer complex forms easily at neutral pH but disassembles readily under acidic conditions. For effective gene silencing, the siRNA/dendrimer delivery system must be able to cross cell membranes to enter cells, then escape from the acidic endosomes before releasing the siRNA cargo to reach the RNAi machinery for gene silencing. We therefore studied the cellular uptake and intracellular trafficking of the siRNA/dendrimer complexes in human pancreatic cancer Panc-1 cells using confocal fluorescence microscopy. Specifically, siRNAs were labeled with the red fluorescent tag Cy3, and the green fluorescent probe LysoTracker was used to trace the acidic endosomes (Fig. 5). Fluorescence imaging revealed considerable red fluorescent signals of Cy3 within Panc-1 cells following treatment with Cy3-siRNA/dendrimer complexes after 1 h (Fig. 5A), and the uptake increased as the incubation time increased (Fig. 5E). In contrast, no obvious cellular uptake was observed when cells were treated with Cy3-siRNA without the dendrimer even after 5 h incubation (data not shown). These results demonstrate clearly the effective internalization of siRNA mediated by the dendrimer nanovector. We next studied the uptake mechanism for the observed internalization of the siRNA/1c complexes using specific inhibitors of endocytic pathways. Cytochalasin D is a macropinocytosis inhibitor, whereas chlorpromazine and genistein are specific inhibitors of clathrin-mediated endocytosis and caveolae-mediated endocytosis, respectively. Remarkably, a significant reduction in the cell uptake of siRNA/1c complexes was observed when all these inhibitors were applied accordingly (Fig. 5I). These results highlight that micropinocytosis, clathrin-mediated endocytosis and caveolae-mediated endocytosis are all actively involved in uptake of the siRNA/1c complex.

Subsequent intracellular trafficking of the siRNA/1c complex was studied using confocal Encouraged by these promising features of the siRNA/dendrimer complex, we then assessed 1c-mediated siRNA delivery and gene silencing in different human cancer cells, including pancreatic cancer Panc-1 cells, liver cancer HepG2 and Hep3B cells, and colorectal cancer HT-29 cells (Fig. 6 and Fig. S7). Two different oncogenes, Myc and Akt2, were selected as the target genes for the silencing study, because they both are promising targets for cancer treatment. Akt2 is a threonine/serine kinase frequently activated in human cancers, with essential roles in enhancing cancer cell proliferation and tumor progression [START_REF] Manning | AKT/PKB Signaling: Navigating the Network[END_REF][START_REF] Franke | PI3K/Akt: Getting it Right Matters[END_REF], whereas

Myc is an undruggable proto-oncogene that is associated with the outputs of major cellular events such as cell cycle progression, cellular transformation and apoptosis [START_REF] Franke | PI3K/Akt: Getting it Right Matters[END_REF][START_REF] Dang | MYC on the path to cancer[END_REF]. Treatment with the corresponding siRNAs delivered by the supramolecular dendrimer formed with 1c

inhibited Myc and Akt expression effectively at both the mRNA (Fig. 6A and 6D) and protein levels (Fig. 6B and6E), leading to potent anticancer activity (Fig. 6C and6F). Remarkably, such RNAi efficacy was significantly higher than that obtained with the current gold standard vector Lipo2000, and no gene silencing occurred when cells were treated with 1c alone, the target siRNA alone, or the control scrambled siRNA complexed with 1c (Fig. 6 and Fig. S7).

In addition, neither the dendrimer 1c alone nor the scramble siRNA/dendrimer complex showed any notable cytotoxicity (Fig. S8). These data highlight that dendrimer 1c is capable of delivering siRNA efficiently and inducing effective gene silencing without adverse cytotoxicity. siRNA alone, or the scrambled siRNA/1c complex using 50 nM siRNA at an N/P ratio of 10.

The commercially available vector Lipo2000 was used as a reference control. Akt2 and Myc expression at the mRNA (A, D) and protein (B, E) levels were determined using qRT-PCR and western blotting, respectively. (C, F) Cell viability of HepG2 and Panc-1 cells upon treatment with the siRNA/1c complex was assessed using the MTT test.

We further validated the efficacy of 1c-mediated siRNA delivery and consequent anticancer activity in pancreatic cancer patient-derived organoids using siRNA targeting the oncogene Myc (Fig. 7). Notably, patient-derived organoids are emerging as robust and effective preclinical models that can be compared with traditional genetically engineered animal models and patient-derived xenograft models (PDXs) [START_REF] Tuveson | Cancer modelling meets human organoid technology[END_REF]. More importantly, human cancer organoids constitute reliable models for personalized medicine, as they can be used to accurately predict responses to therapeutic treatments [START_REF] Vlachogiannis | Patient-derived Organoids Model Treatment Response of Metastatic Gastrointestinal Cancers[END_REF]. Myc is an undruggable oncogene that promotes cancer cell proliferation and tumor progression [START_REF] Dang | MYC on the path to cancer[END_REF][START_REF] Chen | Targeting Oncogenic Myc as a Strategy for Cancer Treatment, Signal Transduct[END_REF]. Therefore, using Myctargeting siRNA constitutes a promising strategy to combat pancreatic cancer [START_REF] Wirth | Concepts to Target MYC in Pancreatic Cancer[END_REF], for which there is no effective treatment. In this study, from the PaCaOmics cohort (clinical trial registration number NCT01692873) [START_REF] Dusetti | Organoids from Pancreatic Ductal Adenocarcinoma[END_REF][START_REF] Bian | Pancreatic Cancer Organoids for Determining Sensitivity to Bromodomain and Extra-Terminal Inhibitors (BETi)[END_REF], we specifically selected two pancreatic cancer patient-derived-organoid models, PDAC087T and PDAC115T, as both exhibit elevated expression levels of Myc. Treatment of organoids PDAC087T and PDAC115T with 1cdelivered siRNA targeting Myc produced significant inhibition of Myc expression at both mRNA and protein levels, with subsequent inhibition of cancer cell proliferation (Fig. 7). The gene silencing efficiency of the siRNA/dendrimer 1c complex in patient-derived cancer organoids was lower than that observed for the cancer cell lines. This observation can be ascribed to the intrinsic three-dimensional (3D) organization of the cancer cells within the organoids. Cell proliferation is often slower and lower in 3D organoids because of space restrictions and limitations when compared with that of cancer cells seeded on plates as two-dimensional (2D) organizations. Additionally, penetration and delivery of the siRNA/vector complexes will be less favorable within the interior of the 3D organoids because of the dense cellular organization in 3D when compared with that of cancer cells cultured on 2D plates. All these critical factors can reduce the efficiency of gene silencing in cancer organoids when compared with that of cancer cells cultured on 2D plates. Nevertheless, our supramolecular dendrimer formed by 1c showed significantly higher gene silencing and anticancer activity when compared with the gold reference Lipo2000. These results demonstrate that the supramolecular dendrimer formed by 1c is a promising system for functional siRNA delivery. In addition, siRNA-based treatment in combination with human cancer organoid models can also offer a reliable strategy for developing personalized treatment.

Conclusions

In summary, we report the ionizable supramolecular dendrimer nanovector formed by selfassembly of the amphiphilic dendrimer 1c and its effective delivery of siRNA for RNAi-based gene silencing in treating cancer. By virtue of its amphiphilic nature, 1c readily self-assembled into a supramolecular dendrimer nanovector, which acted as an efficient and safe ionizable vector for effective siRNA binding and delivery at physiological pH thanks to the slightly positive surface charge of the resulting siRNA/dendrimer complex. Contextually, it is also able to induce effective endosomal escape and subsequent cytoplasmic siRNA release at an acidic pH, as supported by its pH titration profile and the experimental data for siRNA binding and endosomal release. Remarkably, the siRNA/dendrimer complexes protected the siRNA from degradation and delivered the siRNA effectively into cancer cells to target the corresponding oncogenes for gene silencing. The siRNA/dendrimer 1c complex showed considerable anticancer activity in different human cancer cell lines and in patient-derived-organoid cancer models, outperforming the current gold standard Lipofectamine 2000. In contrast to our previous dendrimers [START_REF] Yu | An Amphiphilic Dendrimer for Effective Delivery of Small Interfering RNA and Gene Silencing in Vitro and in Vivo[END_REF][START_REF] Liu | Promoting siRNA Delivery via Enhanced Cellular Uptake Using an Arginine-decorated Amphiphilic Dendrimer[END_REF], the 1c dendrimer nanosystem displayed almost no discernible cytotoxicity when tested on human kidney cells (HEK 293), mouse fibroblast cells (L929) and Madin-Darby canine kidney cells (MDCK). With effective and non-toxic delivery features, combined with simple formulation, the supramolecular dendrimer nanosystem formed by 1c holds great potential in siRNA delivery for both genomic and therapeutic applications, in particular for primary immune cells, which are very sensitive to delivery materials and therefore require especially effective and non-noxious delivery [START_REF] Ellert-Miklaszewska | Efficient and Innocuous Delivery of siRNA to Microglia Using an Amphiphilic Dendrimer Nanovector[END_REF][START_REF] Garofalo | Natural Killer Cells Modulate Motor Neurons-Immune Cells Cross Talk in Amyotrophic Lateral Sclerosis[END_REF][START_REF] Chen | Synthesis and use of an amphiphilic dendrimer for siRNA delivery into primary immune cells[END_REF]. The present study highlights that imparting ionizable properties via tertiary amine functionalities is an effective and practical approach for siRNA delivery with a favorable safety profile. This study also offers a new perspective for a supramolecular dendrimer in biomedical applications.

Figure 1

 1 Figure 1 Schematic illustration of the supramolecular dendrimer formed via self-assembly of

Figure 2

 2 Figure 2 Synthesis, self-assembly, cytotoxicity and ionizable features of the amphiphilic

Figure 3

 3 Figure 3The supramolecular dendrimer issued from 1c formed small and stable complexes

Figure 4 :

 4 Figure 4: Release profile of siRNA from the siRNA/1c complexes at N/P 10 as a function of

Figure 5

 5 Figure 5 Cellular uptake of the siRNA/dendrimer 1c complexes in human pancreatic cancer

  microscopic study. Co-localization of the majority of the red fluorescent signals of Cy3-labeled siRNA and the green fluorescent signals of LysoTracker after 1 h of treatment (Fig. 5B-5D) revealed trafficking of siRNA/1c complexes into endosomes. We further observed, after 5 h treatment, considerable dispersion of the red Cy3 fluorescent signals of siRNA and separation from the green fluorescent signals of LysoTracker (Fig. 5F-5H). These dispersed red fluorescent signals represent the effective release of Cy3-labeled siRNAs from endosomes into the cytoplasm. In combination, our results showed that the siRNA/dendrimer delivery complex entered cells via endocytosis and then escaped from the endosomes before effectively releasing siRNA.

Figure 6

 6 Figure 6 Supramolecular dendrimer-mediated siRNA delivery and gene silencing of the

Figure 7

 7 Figure 7 Delivery of Myc siRNA in patient-derived cancer organoids by the supramolecular

Acknowledgments

This work was supported by the Ligue Nationale Contre le Cancer (LP, ZL), China Scholarship Council (WL, LD), Italian Association for Cancer Research (IG17413) (SP), the French National Research Agency under the frame of the H2020 Era-Net EURONANOMED European Research projects 'Target4Cancer', 'NANOGLIO', 'TARBRAINFECT', 'NAN-4-TUM' (LP), and H2020 NMBP 'SAFE-N-MEDTECH' (LP). This article is based upon work

Associated content

Supporting information: Table S1, Scheme S1 and Figures S1-S8, as well as all detailed experimental methods and protocols. This information is available free of charge via the Internet.


M Stiti

Guillaume Castanet
email: guillaume.castanet@univ-lorraine.fr

Alexandre Labergue

F Lemoine

Icing of a droplet deposited onto a subcooled surface

Keywords:

come

Icing of a droplet deposited onto a subcooled surface

Introduction

Icing phenomena occurring on flying aircrafts is a major hazard in aeronautics. Ice accretion is caused by the impingement and the freezing of supercooled water droplets on the aircraft surface. The resulting security problems are very hazardous: reduced visibility, pitot tube blockage, and reduced flight performance by modification of the aerodynamic performances [START_REF] Cebeci | Aircraft icing[END_REF] .

Given the importance of this issue, studies of the droplet icing are undertaken to deepen the understanding of the freezing process. Besides, these studies will contribute to identify the factors that play a role in the freezing process such as the temperature of the subcooled surface [START_REF] Jin | An experimental investigation into the icing and melting process of a water droplet impinging onto a superhydrophobic surface[END_REF] , the size of the droplet [START_REF] Inada | Ice nucleation in water droplets on glass surfaces: from micro-to macro-scale[END_REF] , the roughness and wettability of the surface [ 4 , 5 ], the angle between the surface and velocity vector of the impinging droplet [START_REF] Jin | The impact, freezing, and melting processes of a water droplet on an inclined cold surface[END_REF] , the heat transfer [START_REF] Jin | Quantification of unsteady heat transfer and phase changing pro-cess inside small icing water droplets[END_REF] and the flow [START_REF] Kawanami | Solidification characteristics of a droplet on a horizontal cooled wall[END_REF] within the impinging droplet. Ultimately, researches aim at developing innovative surfaces that can alleviate this icing problem, for instance surfaces that can sufficiently reduce water adhesion and slow down the icing process [START_REF] Schwarzer | Supercooled Water Drops Do Not Freeze during Impact on Hybrid Janus Particle-Based Surfaces[END_REF] .

Modelling the freezing of a droplet impinged onto a subcooled surface remains very challenging due to the inherent complexity of obtaining reliable and quantitative data on the ice fraction, the shape and position of the ice front in a droplet from representative experiments. Basically, the freezing of a supercooled droplet can be divided into two phases. In a first phase of recalescence, dendrites appear and propagate rapidly causing a significant heat up of the liquid. As the temperature cannot exceed the melting temperature, the growth process of the dendrites is interrupted. Recent studies reveal that the ice fraction in the droplet after this first phase is only dependant on the initial droplet temperature before the freezing [START_REF] Hindmarsh | Using magnetic resonance to validate predictions of the solid fraction formed during recalescence of freezing drops[END_REF] . Also, the initial droplet temperature controls the propagation velocity of the dendrites [ 11 , 12 ]. Recently, a new optical technique based on total internal reflection [START_REF] Pallav Kanta | Fast-freezing kinetics inside a droplet impacting on a cold surface[END_REF] was used for the direct visualisation of the nucleation event in a droplet impact. This technique was used to highlighted some of the interaction mechanisms between the droplet impact and phase transition. For instance, it revealed that the freezing morphology can be dramatically changed due to the advection of the ice crystals, which occurs from the centre to the edge of the droplet in conjunction with the spreading.

The second phase of the freezing process is the propagation of an isothermal freezing front. The ice front remains permanently at the melting temperature and its propagation is mainly controlled by heat conduction in the freezing medium as in the classical Stefan's model [START_REF] Thiévenaz | Solidification dynamics of an impacted drop[END_REF] . The dynamic of the ice front can be characterized Weber number with the help of side view shadow images of the droplet [ 15 , 16 ]. However, this approach, although relatively straightforward, is not able to reconstruct the shape of the ice front in all situations. In fact, only the upper border of the ice front is visible from side view images, since the ice front takes a curved shape oriented downwards at the end of the freezing process, as illustrated in Fig. 1 (D).

The final shape of a droplet freezing on a subcooled surface has been extensively investigated. Under specific conditions, a droplet of water deposited onto a subcooled plate freezes into an ice droplet with a "pointy tip" as illustrated in Fig. 1 (A,B,C). The pointy tip appears at the top of the droplet, which is the last region to freeze. This phenomenon is quite fascinating, given that surface tension of the liquid is expected to smooth out any sharp points on the air-droplet interface. Nonetheless, there is still no satisfactory explanation. Some have attributed the formation of a conical tip to the expansion of water upon freezing, due to the lower density of the solid phase than the liquid one. A model relying on geometrical consideration, in particular a horizontal ice front during the solidification process, suggested the density ratio ψ between the liquid and solid phases must be under a threshold limit of 0.75 for observing a pointy tip [START_REF] Snoeijer | Pointy ice-drops: how water freezes into a singular shape[END_REF] . However, this threshold is not satisfactory, since water droplets exhibit a pointy shape with a density ratio ψ = 0 . 92 [START_REF] Marín | Universality of tip singularity formation in freezing water drops[END_REF] . Recent observations from Marín et al. [START_REF] Marín | Universality of tip singularity formation in freezing water drops[END_REF] suggest that the tip angle is independent of the substrate temperature and wetting angle. Based on several measurements of the contact angle in a Hele-Shaw cell, they pointed out that the ice front develops a spherical shape that ends perpendicularly to the solid-air interface. However, these measurements need to be confirmed for other freezing conditions. In contrast, some authors like [START_REF] Anderson | The case for a dynamic con-tact angle in containerless solidification[END_REF] have argued that the contact angle at the solid/liquid/air tri-junction point could depend on the freezing rate. Recently, Zhang et al. [START_REF] Zhang | Shape variation and unique tip formation of a sessile water droplet during freezing[END_REF] claimed that a time-varying value of the contact angle at the tri-junction would be consistent with the experimental observations, but in their case, observations were only based on side view images of the freezing droplet, thus without any possibility to monitor the actual shape of the ice front. Experiments performed in Hele-shaw cells ( Fig. 1. D) are currently the only ones that can to be used reliably to follow the evolution of the ice front shape inside a droplet [START_REF] Marín | Universality of tip singularity formation in freezing water drops[END_REF] , but they suffer from some limitations, in particular they are not compatible with the observation of the freezing processes occurring during the drop impact onto a solid wall. The aim of the present study is to develop a new approach based on Laser Induced Fluorescence (LIF) to reconstruct the ice front inside a freezing droplet. This novel method is applied to droplets deposited with a significant impact velocity impinging onto two different solid substrates already covered with a thin frost layer. The technique can provide a direct measurement of the angle between the liquid and the solid ice front at the solid/liquid/air tri-junction point.

Experimental study of the icing

This section describes the development of a novel experimental approach to reconstruct the evolution of the ice front within a single droplet deposited on a subcooled solid surface. The actual position of the ice front is determined by measuring the thickness of the liquid layer that is located above the ice front during the freezing process, using Laser Induced Fluorescence (LIF). Basically, a small amount of an organic fluorescent dye (typically a few μg/L) is dissolved into water prior to the droplet deposition. Recent experiments [ 19 , 20 , 21 ] revealed a quite significant effect of icing on the fluorescence signal. A much larger quenching of the fluorescence is observed when the fluorescent dye is present in the solid phase than in the liquid phase. According to [START_REF] Stiti | Temperature measurement and state determination of supercooled droplets using laser-induced fluorescence[END_REF] , the difference in signal intensity is typically in a ratio 1:10 for fluorescent dyes such as rhodamine 560 or kiton red. Similar behaviours were also observed for some coumarin dyes [START_REF] Strausky | Sliding plastics on ice: fluorescence spectroscopic studies on interfacial water layers in the μm thickness regime[END_REF] . Thus, the fluorescence signal that originates from the solid phase can be safely neglected, which makes it possible to quantify the thickness of the liquid phase using the intensity of the LIF signal from a deposited freezing droplet.

Experimental setup

The experimental setup is depicted in Fig. 2 . To generate the water droplet, a 400 μm diameter needle is connected to a micro- pump. The droplet is detached from the needle when surface tension force exceeds the droplet weight. This occurs in our experiments when the droplet diameter is equal to 2.6 mm. The impact velocity ( u ) of the free-falling droplet increases with the distance h between the needle and the solid substrate ( u = 2 gh ). Hence, changing the distance h allows controlling the Weber number (defined by We = ρu 2 d/σ where d is the drop diameter, ρ the liquid density and σ the surface tension) and thus the spreading of the drop onto the solid surface. In this study, an impact velocity u = 1 . 35 m/s, corresponding to We = 65, is considered. The temperature of the solid surface is controlled by means of a heat exchanger and a liquid coolant flowing in a closed loop. The solid substrate onto which the droplet impact takes place is inserted directly into a hole drilled on the surface of the heat exchanger. To study the influence of the surface material and the quality of the surface state, the substrate can be replaced. In this study, the focus will be placed on two kinds of materials: duralumin and optical glass (N-BK7) which have very different thermal properties ( Table 1 ). Both substrates have a thickness of 5 mm. The temperature of the liquid coolant (mineral oil Thermal HL 80) can be decreased down to -60 °C by means of a cooling unit (Julabo Presto®) delivering 1.8 kW of cooling power. A thermocouple allows controlling the temperature of the substrate and the experiment begins when the temperature is stable for several minutes. The experimental system (droplet and substrate) operates in air at room temperature without humidity control. Because of this, a thin layer of ice quickly develops on the substrate before each impact experiment. Far from being a drawback, the presence of this frost layer allows obtaining a great reproducibility of the freezing phenomenon. The freezing starts immediately after the droplet impinges onto the solid surface. Thus, there is no supercooling of the liquid water, as there is no time delay for the liquid to be cooled below the melting point before the icing. No recalescence events can be observed in the experiments. The frost covering the substrate is removed regularly, so that its thickness remains negligible when the droplet impinges on the substrate. Typically, the initial frost layer does not exceed a few tens of micron in thickness.

ρ [ kg. m -3 ] k [ W. m -1 . K -1 ] Cp [ J.k g -1 . K -1 ] D [ m 2 . s -1 ] e [ J . K -1 m -2 s -

Observation of the droplet impact and freezing

Shadow imaging was used to visualize the evolution of the droplet during the impact and the freezing process. As an example, Fig. 3 shows the droplet evolution for two similar impinging conditions, and two substrates made of different material: duralumin and N-BK7 glass. For these two impacts, the initial wall temperature was T 0 = -10 °C and the impact velocity was u = 1 . 35 m/s. During the initial stage of the impingement, liquid inertia is mainly controlled by the droplet spreading ( We 1) and the same spreading evolution is observed for both materials due to the initial presence of frost. At the end of spreading process (roughly about 5 ms after the beginning of the impact), the spreading diameter is approximately 2.5 times the initial droplet diameter. After having reached the maximum spreading, no noticeable recoiling of the droplet occurs. Due to the initial frost layer, freezing can start very early near the triple contact line as the spreading of the liquid decelerates. This subsequently blocks any motion of the triple line [START_REF] Schremb | Normal impact of supercooled water drops onto a smooth ice surface: experiments and modelling[END_REF] .

The freezing process takes a much longer time than the droplet spreading. There are two orders of magnitude between the duration of these two processes, which makes it possible to investigate separately the freezing process and the droplet spreading. The duration of the drop freezing is mainly a function of the thermal properties of the solid wall. Experimentally, the measured freezing time is 22 s for N-BK7 and 2.2 s for duralumin, which has a much larger heat conductivity. A noticeable difference between the experiments carried out on the two substrates, is the formation of a pointy tip at the end of the solidification phase in the case of N-BK7 ( Fig. 3 ). This tip is absent in the case of the duralumin substrate. The tip formation is clearly related to the duration of the freezing process, and thus the propagation velocity of the ice front. Previous studies [ 16 , 23 , 24 ] reveal that even in the case of highly conductive substrates a pointy tip can be observed. One difference with these studies relates to the freezing conditions. Presently, a frost layer already covers the solid surface before the drop impact. This frost layer triggers the freezing immediately when the droplet touches the surface and stops the droplet spreading after a few ms. The contact line with the solid is pinned with no possibility for the droplet edge to recede. The droplet has thus a very flat shape compared to many studies reported so far. The contact angle with the solid is about 17 °at the end of the spreading process. More explanations will be given in the last section. Although the experimental set-up makes it possible to study the effect of the impact velocity, it was decided to maintain the same impact velocity for the entire study ( u = 1 . 35 m/s and W e = 65 ). For higher Weber numbers (typically on the order of 130), remarkably different freezing behaviours were reported, such as the formation of an 'egg shape' ice cap [START_REF] Thiévenaz | Solidification dynamics of an impacted drop[END_REF] . These situations will deserve further investigations in the future.

Laser-induced fluorescence signal within a freezing droplet

To obtain quantitative data on the icing, the ice front is reconstructed based on Laser Induced Fluorescence (LIF) signal. Basically, the method consists in dissolving an organic fluorescent dye in water prior to the droplet generation. A laser excitation tuned on the In practice, a high-speed camera (phantom V710) is required given the short duration of the icing phenomenon. A tilted mirror is placed at the top of the droplet in order to obtain top view images of the droplet ( Fig. 2 ). A CW frequency-doubled Nd-YAG laser (Ventus Laser Quantum®, @532 nm; P = 2.5 W) is used to induce fluorescence. The laser beam is transmitted by a light guide consisting of a bundle of optical fibres to illuminate the droplet after it was deposited on the solid surface. Rhodamine 6 G (R6G) has been selected as fluorescent marker for many reasons. It has a high solubility in water and its dependence in temperature is quite low when excited at 532 nm [START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF] . Furthermore, the high quantum efficiency of R6G (about 0.95 [START_REF] Kubin | Fluorescence quantum yields of some rhodamine dyes[END_REF] ) and its strong absorption at 532 nm are advantageous for maximizing the fluorescence signal. Additionally, the high solubility of R6G allows assuming that thermophoresis is negligible for this chemical component [START_REF] Marcellini | Time-lapse, in situ imaging of ice crystal growth using confocal microscopy[END_REF] . Hereafter, the concentration of R6G will be considered constant and uniform in both liquid and solid phases. A long-pass edge filter, transmitting longer wavelengths than 542 nm, is mounted in front of the camera lens to eliminate the contribution of laser scattering and to detect only the contribution of the LIF signal.

Fig. 4 presents fluorescence images of the droplet freezing onto the duralumin substrate. During the initial stage of the impact (between t = 4 . 5 ms and t = 9 . 5 ms ), spatial variations in the fluorescence intensity can be observed. They result from variations of the liquid thickness during the spreading process. At t = 4 . 5 ms , it is possible to identify the rim on the droplet edge and some capillary waves that are travelling at the surface of the liquid lamella in the central region. After about 10 ms, surface oscillations stop and the drop takes an equilibrium shape. Afterwards, the fluorescence field evolves at a much slower pace. The most noticeable evolution is a reduction in the area of the fluorescence emission. This latter indicates that the liquid inside the droplet is gradually freezing with a freezing front that advances toward the centre of the droplet. An important feature for the measurement is that there is a negligible fluorescence signal from the frozen regions of the droplet. This makes it possible to measure the thickness of the remaining liquid water above the ice front based on the intensity of the fluorescence signal. It is, however, an indirect reconstruction process of the ice front which requires a fine analysis of the fluorescence images.

Modelling of the fluorescence signal

The fluorescence signal emitted into a solid angle d from an elementary volume dV around a point M located in the liquid phase of the droplet, can be written as:

d F ( M ) ∼ I 0 ( M ) φ ε C d V d 4 π , ( 1
)
Where I 0 ( M ) is the local intensity of the laser excitation, φ the fluorescence quantum yield, ɛ the absorption coefficient at the laser wavelength, C the dye concentration. Paraxial approximation is made to describe the transmission of fluorescence emission to the camera. As depicted in Fig. 5 , only light rays emerging parallel to the optical axis of the camera are assumed to reach the sensor of the camera.

I f = t C D ∫ C dF ( M ) , ( 2
)
where t C is the transmission coefficient at point C and dF ( M ) is the elementary fluorescence signal emitted in a given point M expressed by Eq. ( 1) .

The relatively flat shape of the deposited droplets allows making further simplifications. Due to the initial drop spreading at the impact, the contact angle is about 13 °. Under these conditions, application of Fresnel formulas shows that t C is close to 1 at any position of the droplet surface. The slightly flat shape of the droplet also makes it possible to consider that the lens effect due to the curvature of the free surface is very limited. Thus, the distance between C and D is roughly equal to the liquid thickness Z L under point C.

Eq. ( 2) assumes a negligible reabsorption of fluorescence over the ray path in the droplet. To secure this assumption, the concentration of R6G was set at a value of 5 × 10 -6 mol.L -1 . The reabsorption can be calculated using the Beer-Lambert law. The maximum of the absorption coefficient in the detection band is ε=7 × 10 -4 L/mol.cm at 542 nm [START_REF] Nag | Solvent effect on two-photon absorption and fluorescence of rhodamine dyes[END_REF] . Typically, for a 1 mm path length, the re-absorption is less than 1 %. As the droplet thickness is less than 0.5 mm, in practise the error due to reabsorption never goes above 0.5 %. It is worth to note that the liquid thickness decreases with the time due to freezing. This further mitigates the effect of re-absorption. Absorption also affects the laser light (fluorescence excitation field). A similar calculation at the excitation wavelength ( ε=12 × 10 -4 L/mol.cm at 532 nm [START_REF] Nag | Solvent effect on two-photon absorption and fluorescence of rhodamine dyes[END_REF] ) leads to com- parable values for the attenuation of the laser light. Following the above discussion, the fluorescence signal collected by the camera can be expressed:

I f ( x, y ) = K I 0 ( x, y ) φ ε C Z L ( x, y ) (3)
In this expression, K is an optical constant related to the transmission of the optics. This parameter does not change during the experiments. I 0 ( x, y ) denotes local value of the laser excitation intensity. Taking a reference I f, ref where the liquid thickness is Z L, ref , the liquid thickness can be evaluated at any time during the freezing as followed:

Z L ( x, y ) = Z L,re f ( x, y ) . I f I f,re f . ( 4
)
Hence, it is possible to determine the liquid thickness Z L by using a reference image taken under the same spatial distribution of laser excitation I 0 ( x, y ).

Processing of the fluorescence images

The main steps of the image post processing are illustrated in Fig. 6 . They involve:

Step 1: The raw fluorescence image is subtracted from the background noise.

Step 2: According to Eq. ( 3) , the fluorescence intensity depends on the local laser intensity. An image of the laser intensity distribution can be obtained by positioning a thin fluorescent PMMA slab of uniform thickness at the same location as the droplet, prior to the measurement. The non-uniformity of the laser illumination is corrected by dividing the droplet image by the reference image (obtained on the fluorescent PMMA slab) from the plate. In principle, after this stage, grey level of the resulting normalized image I norm is only dependant on the liquid thickness.

Step 3: The radial distribution of I norm is extracted from the previous image. An axysimetric averaging is performed, which helps in improving the SNR except near the droplet centre.

Step 4: A geometrical scaling is required as I norm only provides a relative measurement of the liquid thickness. This scaling factor, denoted by S F , can be determined from the initial droplet volume ( V 0 ). The value of SF can be determined as follows:

S F = + ∞ 0 2 π r I norm ( r ) dr V 0 ( 5
)
As a validation, the previous method was used to reconstruct the shape of a droplet deposited onto an isothermal sur- face ( Fig. 7 ). The thickness profile obtained applying the previous method is compared to the one extracted directly from side-view shadow images. The maximum deviation is observed on the edge of the droplet. It may be attributed to the reflection of the fluorescence light on the substrate surface, since some signal is unexpectedly detected outside the droplet contour.

Reconstruction of the ice front

The previously described method provides only the thickness of the liquid top layer within the droplet. As an approximation to determine the position Z I of the ice front, the volume expansion due to the freezing can be neglected. Knowing the droplet profile Z d at t = t 0 ( t 0 is taken just after the dumping of the surface oscillations, i.e. just after 20 ms), the height Z I of the ice front can be evaluated as follows:

Z I ( R, t ) = Z d ( R, t 0 ) -Z L ( R, t ) (6)
Since the icing did not have time to propagate significantly during the impact process (which lasts only a few tens of ms), Z d ( R,t 0 ) can be deduced from the fluorescence signal just like in Fig. 7 knowing the initial drop volume V 0 .

In order to account for volume expansion in the reconstruction of the ice front, an additional camera is required. This allows obtaining sideview images of the droplet from which the actual shape and position of the droplet surface can be inferred. Denoting Z d the height of the droplet surface on the sideview images, the position of the ice front Z I can be determined as follows:

Z I ( R, t ) = Z d ( R, t ) -Z L ( R, t ) (7)
Eq. ( 7) is similar to Eq. ( 6) except that Z d ( R,t ) is the height of the free surface at the time corresponding to the LIF image.

Fig. 8 presents the propagation of the ice front in the case of a droplet deposited on duralumin. The icing conditions are the same as in Fig. 3. a. The reconstruction of the ice front based on Eqs. ( 6) and [START_REF] Jin | Quantification of unsteady heat transfer and phase changing pro-cess inside small icing water droplets[END_REF] give comparable results at the beginning of the freezing process, but progressively, the ice front is moved to a higher position when volume expansion is accounted for. The error made by neglecting the volume expansion in the reconstruction becomes significant during the second half of the icing process.

Modelling of the droplet freezing

The problem of freezing belongs to the class of Stefan's problems, in which an interface between two phases moves with time. The location of the front between the solid and liquid is not known beforehand but must be determined as part of the solution. To begin, Stefan's one-dimensional two-phase problem is recalled because an analytical solution can be found. Then, this solution is used to validate a two-dimensional numerical approach, which is implemented to further take into account the geometry of the droplet and the heat transfer conjugated with the wall.

The one-dimensional two-phase Stefan problem

The one-dimensional Stefan problem will be considered in first. The one-dimensional Stefan problem can be represented as a semiinfinite liquid occupying the domain x > ∞ at the initial temperature T ∞ Fig. 9 ). In the present case, the two-phase Stefan problem, which takes place when the initial liquid phase is at a temperature T ∞ greater than the melting point T m is considered The liquid is cooled from the side at x = 0 where the temperature is imposed at T 0 < T m . As a consequence, the temperature decreases to reach the melting point and the liquid starts freezing. To address this problem, the following usual assumptions are adopted: ( (1) The heat distribution and heat transfer is described by the heat conduction equation, (2) a clear interface (or phase change boundary) between the two phases can be considered, (3) the temperature of the interface is a priori known, (4) two-phase medium is considered as semi-infinite. Furthermore, any volume change due to the solidification is ignored [START_REF] Oberli | Condensation and freezing of droplets on superhydrophobic surfaces[END_REF] .

In this two-phase problem, the temperature distribution in the two phases is controlled by the thermal conduction, but the position X ( t ) of the liquid/solid interface is unknown. The problem can be described by the following set of equations:

∂T ∂t = D L ∂ 2 T ∂ x 2 , t > 0 and x > X ( t ) (8)
∂T ∂t

= D I ∂ 2 T ∂ x 2 , t > 0 and 0 < x < X ( t ) (9)
where D L and D I are respectively the thermal diffusivities of the liquid and solid phases. Initially, the interface X ( t ) is positioned at x = 0 . The initial condition can be written as:

T ( x, t = 0 ) = T ∞ > T m (10) X ( 0 ) = 0 (11)
The associated boundary conditions are:

T ( 0 , t ) = T 0 < T m (12) lim x →∞ T ( x, t ) = T ∞ (13) T X ( t ) + , t = T X ( t ) -, t (14)
Additionally, writing the Stefan condition, is needed to close the problem. The physical constraint is that the local velocity of the interface depends on the heat flux discontinuity at the free boundary. This yield:

-ρ L dX dt = -k I dT dx X ( t ) - + k L dT dx X ( t ) + , t > 0 ( 15
)
where L is the latent heat of phase change, ρ the density of the solid or liquid (the volume change occurring during the solidification being ignored), k I and k L are the thermal conductivity of the liquid and the solid phases respectively. An analytical solution can be obtained by introducing the similarity variable ξ = x / √ t into Eqs. ( 8) - [START_REF] Marín | Universality of tip singularity formation in freezing water drops[END_REF] .The propagation of the ice front is described by:

X ( t ) = 2 λ D I t , ( 16
)
where λ is a parameter depending on the difference between T 0 and T ∞ and of the thermal properties of liquid water and ice. The parameter λ is obtained solving the transcendental equation derived from Eq. ( 15) : S t L ν. e ( νλ) 2 erfc ( νλ)

+ S t I e λ 2 er f ( λ) = λ √ π, ( 17
)
where

ν = D L / D I , S t L = C p L ( T m -T ∞ ) /L and S t I =
C p I ( T m -T 0 ) /L are the dimensionless Stefan numbers for the liquid and solid phases respectively. Finally, the temperature profile in the liquid phase is given by:

T ( x, t ) -T 0 ( T m -T ∞ ) = er f ( η) er f ( λ) with η = x 2 D L t (18)

Numerical simulation of the two-dimensional freezing problem

When the geometry of the domain is two-or threedimensional, the resolution of the Stefan problem has to be performed numerically. The impact velocity changes the spreading diameter of the droplet, but its effect on the internal flow becomes rapidly negligible. The icing process lasts several seconds, while the initial kinetic energy is dissipated over a much shorter period of time. A viscous boundary layer develops progressively during the spreading process from the solid surface because of the no-slip condition [START_REF] Lagubeau | Spreading dynamics of drop impacts[END_REF][START_REF] Roisman | Inertia dominated drop collisions. II. An analytical solution of the Navier-Stokes equations for a spreading viscous film[END_REF][START_REF] Eggers | Drop dynamics after impact on a solid wall: Theory and simulations[END_REF] . Typically, it would take 40 ms for the viscous boundary layer to reach the top of the droplet, which is short in comparison to the duration of the icing process of the order of a few seconds. During the spreading phase, a large part of the liquid is also going into the rim formed due to capillary forces on the droplet edge ( Fig. 3 A at t = 5 . 5 ms, and Fig. 4 at t = 4 . 5 ms). In this edging rim, the flow is highly rotational and kinetic energy is dissipated very efficiently. In our experiments, no receding of the droplet edge is observed. The pinning of the triple contact line occurs when the spreading diameter has reached its maximum. In such situation, only a small part of the surface energy stored during the spreading phase, can be restored afterwards through surface oscillations. No evidence of surface oscillations can be effectively observed after 20 ms ( Fig. 3 ). The droplet adopts an equilibrium shape, which evolves slowly with time only due to the volume expansion induced by the icing. Based on the above discussion, it appears that the hydrodynamic problem associated with the drop impact and the ice front problem can be treated separately.

Neglecting liquid convection inside the liquid phase, the heat equation can be written in the cylindrical coordinates ( r, θ , z ) as followed:

dT dt = D 1 r .

∂ ∂r r

∂T ∂r

+ ∂ 2 T ∂ z 2 , ( 19
)
where D = k ρCp is the thermal diffusivity. The thermophysical properties of the liquid water or ice are considered independent on temperature. Their values are given in Table 1 . A parameter, playing the role of a phase indicator, is introduced to discriminate between the liquid and solid phases, based on the local temperature:

μ( T ) = 0 i f T < T m 1 i f T ≥ T m ( 20
)
The thermophysical properties of the liquid/ice mixture are expressed using the phase indicator μ. The thermal diffusivity D is calculated as follows:

D ( μ) = D I . ( 1 -μ) + μ • D L ( 21
)
To account for the heat release when freezing takes place, the following expression is used for the specific heat capacity:

C p ( μ) = C p I . ( 1 -μ) + μ • C p L + L dμ dT ( 22
)
The volume change induced by the icing is not considered in the numerical resolution. In other words, the drop geometry is not updated between the time steps of the numerical resolution. Consequently, it is not possible to account for the velocity field induced by the volume change in the heat equation. Zhang et al. [START_REF] Zhang | Simulation and experiment on supercooled sessile water droplet freezing with special attention to supercooling and volume expansion effects[END_REF] developed a theoretical model for the droplet shape, but their simulations were still restricted to the initial and final droplet shapes with no update of the droplet geometry during the simulation time steps. To the best of our knowledge, only Vu et al. [START_REF] Vu | Numerical investigations of drop solidification on a cold plate in the presence of volume change[END_REF] have implemented a variable droplet geometry using a front track-tracking method. Nonetheless, they only partially succeed in reproducing the pointy tip.

In practice, to simulate the droplet freezing, the finite element method was used using COMSOL Multiphysics® software. An unstructured triangular mesh and a constant time step were employed. The main difficulty associated with the numerical resolution is to capture the motion of the liquid-solid interface. In the above description, the ice front is an infinitesimally thin surface that separate adjacent phases; therefore, the solutions of the underlying partial differential equations and its derivatives may suffer from discontinuities across the freezing interface. To mitigate this 16) and [START_REF] Anderson | The case for a dynamic con-tact angle in containerless solidification[END_REF] and the 2D numerical solution for the temperature profile (A) and the interface position (B).

problem, a continuous piecewise linear function is considered for μ:

μ = 0 , if T < ( T m -T / 2 ) μ = T -T m T + 1 / 2 if | T -T m | ≤ T / 2 μ = 1 if T > ( T m + T / 2 ) ( 23
)
The value of T is chosen equal to 1.2 °C for the resolution. In practice, this value has been found to minimize the difference between the numerical results and the analytical solution. The nonlinear time-dependent solver is based on a backward differentiation formula (BDF) and a damped Newton method ensuring a proper treatment of the specific heat discontinuity. Fig. 10 (A) presents a comparison between the analytical solution of the twophase 1D Stefan's problem described by Eqs. ( 16) and ( 18) and the 2D axisymmetric numerical approach described above. For this test, the computational domain has a length of 100 cm and a width of 50 cm. The temperature was fixed at T 0 = -10 °C on the left side of the domain ( x = 0 ) , while the liquid water was initially at T ∞ = 18 °C . Convergence is obtained for elements size of 10 -5 m or smaller, as there is no longer noticeable change in the numerical solution. The average error is less than 1% in the position of the ice front as shown in Fig. 10 (B). As a conclusion, the 2D axisymmetric numerical simulation is congruent with the 1D analytical solution of the Stefan's problem.

Thermal interaction between the solid wall and the droplet

As shown in Fig. 3 , the thermal properties of the wall material have a very significant effect on the final shape of the frozen droplet and the freezing time. To capture this effect in the modelling, it is necessary to consider the combination of heat transfers in the solid substrate and in the deposited droplet. Fig. 11 summarizes the conditions applied at the different boundaries of the media that participate to heat transfer. A zero heat flux is imposed at the air interface with the droplet and the solid substrate (purple lines in Fig. 11 ):

∂T ∂n = 0 , ( 24
)
where n is the normal to the surface directed to the outside of the computational domain. This condition is justified by the smallness of the heat transfer associated with natural convection in comparison to the heat flux involved in the thermal contact between the drop and the solid surface. The continuity of the heat flux and temperature is applied at the interface between the droplet and the solid substrate, as well as between the substrate and its holder, which is temperature regulated (red lines in Fig. 11 ):

k + ∂T ∂ n + = k -∂T ∂ n - (25)
T + = T - (26)
The outer edge of the holder is maintained at the initial substrate temperature T 0 (green lines in Fig. 11 ):

T = T 0 ( 27
)
It should be pointed out that the underlying physics of the problem is not modified with the addition of one or several solid layers under the freezing droplet. For the equivalent semi-infinite 1D problem (also presented in Fig. 11 ), a self-similar solutions remains valid, and the position X ( t ) of the ice front can be determined by:

X ( t ) = D e f f t (28)
The parameter D eff is function of the thermal properties and thickness of the different layers that compose the solid wall and the phase-change medium [START_REF] Thiévenaz | Solidification dynamics of an impacted drop[END_REF] , but an analytical expression can be hardly provided. The value of D eff is obtained numerically by performing a 1D simulation. A liquid domain that is sufficiently extended in space is considered in order to obtain a solution comparable to the semi-infinite case over a considered period of time. The time evolution of the ice front position obtained from the simulation is fitted by a square root of time and the value of D eff is evaluated. Using this approach, D eff was found to be respectively 7 . 5 × 10 -8 m 2 . s -1 for the duralumin substrate and 2 . 5 × 10 -8 m 2 . s -1 for the N-BK7 substrate in the case of a substrate thickness of 5 mm, T 0 = -10 • C and T ∞ = 18 °C . Hence, the freezing velocity is very different for the two substrates. The value of D eff obtained in the case of duralumin, is rather close to the solution of the classical 1D two-phase Stefan's problem ( Eq. ( 16) ), which is about 8 × 10 -8 m 2 . s -1 . This can be attributed to the fact that the thermal effusivity of duralumin is much higher than that of the ice (see Table 1 ) which contributes to keep the temperature close to T 0 at the contact between the duralumin and the ice during the freezing process. This is not the case for the NBK-7 substrate ( D eff is about 3 times lower) because this material has a thermal effusivity slightly lower than ice.

Results and discussion

In this section, the experimental results are compared with the 1D semi-infinite ( Eq. ( 28) ) and the 2D axisymmetric (2D-axi) simulations of the freezing. The size of the substrate considered in the 2D-axi simulations is the same as in the experiments (a diameter of 25.4 mm and a thickness of 5 mm). The time step of the resolution is 10 ms. The largest size of the triangular mesh elements is 1 μm in the drop and 15 μm in the solid substrate and the holder. It has been verified that increasing or decreasing the sizes of the mesh elements by a factor of 10 does not significantly modify the results of the simulations.

Icing of the droplet deposited on the duralumin substrate

As already shown in Fig. 8 , the ice front takes a concave shape during the freezing. In the following development, point A will refer to the position of the ice front at the droplet centre, and point B at the tri-junction between the ice front, the liquid phase and the air ( Fig. 5 ). Fig. 12 (A) presents the time evolution of the vertical position Z B of point B for the initial wall temperature T 0 = -10 • C and a droplet initial temperature T ∞ = 18 • C . The analysis of the experimental results shows that Z B increases following approximately as a square root of the time, which is congruent to results concerning the freezing of sessile droplets in the literature [START_REF] Zhang | Shape variation and unique tip formation of a sessile water droplet during freezing[END_REF] . As presented in Fig. 12 (A), the 1D and 2D-axi simulations are able to reproduce quite well this temporal evolution of Z B . However, the motion of Z B in the simulation is slower than observed experimentally. The difference can be attributed to the volume expansion due to the freezing, which adds an additional velocity to the ice front. The best fit of the experimental data is D e f f, exp = 1 . 08 × 10 -7 m 2 . s -1 while D eff, m is about 8 × 10 -8 m 2 . s -1 for the semi-infinite 1D without volume expansion. It can be noted that ( D eff, m / D eff, exp ) 0.5 ≈0.88, which is not far from ψ = ρ L / ρ S = 0 . 917 . If the ice front is reconstructed assuming that the droplet volume is not changing (i.e. using Eq. ( 6) instead of Eq. ( 7) for the reconstruction of the ice front), the agreement with the 2D-axi simulation is better. Fig. 12 (B) also presents the evolution of the radial position R B of point B. Considering that the shape of the deposited droplet can be assimilated to a spherical cap of radius R, R B is expected to follow a R 2 -law [START_REF] Marín | Universality of tip singularity formation in freezing water drops[END_REF] :

R B = R 2 -( Z B -Z c ) 2 , ( 29
)
where Z c is the vertical position of the centre of the spherical cap at the droplet free surface. Replacing Z B by D e f f .t yields:

R 2 B -R 2 = - D eff t -Z c 2 (30)
The temporal evolution of R B predicted by Eq. ( 30) is plotted in Fig. 12 (B) using D e f f = 1 . 08 × 10 -7 m ²/s, which is the value obtained experimentally for Z B (taking into account the volume expansion in the ice front reconstruction). Eq.( 30) fits well with the experimental data. The 2D-axi simulation is also able to mimic rather accurately the evolution of R B despite the deviations previously observed for Z B in Fig. 12 (A). Fig. 13 shows the time evolution of the ice front at point A (on the droplet axis at R = 0 ) for the same icing conditions as in Fig. 12 . The analysis of the experimental data reveals that Z A evolves as t 0.75 . The 2D-axi simulation is able to reproduce this temporal evolution of Z A quite well, but in the very last instants of the freezing, typically for t > 2.1 s, a sudden acceleration of the ice front is observed. The volume expansion inherent to the freezing is certainly responsible for it since this acceleration is not captured by the 2D-axi simulation. If the ice front is reconstructed experimentally assuming no volume expansion ( Eq. ( 6) ) like in the simulations, it is worth noting that the agreement between the experimental data and the simulation is a bit improved.

The scaling of Z A as t 0.75 is related to the tridimensional nature of heat transfer. For example, increasing the droplet size in the 2Daxi simulations, heat transfer becomes more and more one dimensional at the centre of the droplet ( R = 0). The evolution of Z A tends to evolve as t 0.5 like in the semi-infinite 1D model ( Eq. ( 28) ). Thus, the exponent of 0.75 is not universal. It depends on the thermal properties of the solid substrate, as it will be illustrated hereafter with the results obtained for the N-BK7 substrate. It is also a function of the size and thickness of the deposited droplet . To illustrate that point, several simulations have been carried out. In Fig. 14. A, the size of the deposited droplet was changed while maintaining the ratio h/R (with h the maximum droplet height and R the droplet radius) constant (i.e. the contact angle with the solid substrate is fixed). As it can be observed, Z A tends to follow an evolution as t 0.5 when h is sufficiently large. From an experimental point of view, increasing the droplet size with the experimental system is not obvious, since the droplet size results from a competition between the droplet weight and the capillary forces that keep the droplet attached to the tip of the needle. Further experiments were undertaken to determine the effect of the wall temperature T 0 ( Fig. 14. B). The results seem to indicate that T 0 has very little influence on the exponent of the time evolution of Z A . However, more experiments would allow covering a wider range of surface temperatures.

The emphasis was also placed on the shape of the ice front and the mass m s of ice. Fig. 15 shows the position of the ice front determined by the 2D-axi simulation and LIF measurements at different times. As already noticed, the position of the ice front is rather well predicted by the simulations, especially if the volume expansion is ignored in the experimental reconstruction of the ice front. However, in the simulation results, the ice front is flat in most of the droplet central region while its actual shape is approximatively a spherical surface. This can be assessed by doing a fitting of the ice front by a circle ( Fig. 16 ). The observed spherical shape is consistent with the observations of Marín et al. [START_REF] Marín | Universality of tip singularity formation in freezing water drops[END_REF] in a Hele-Shaw cell ( Fig. 1 ). Fig. 17 shows the time evolution of the mass of ice. This mass is determined by calculating the volume under the ice front doing an integration in the cylindrical coordinates. For the simulations, the density of ice is equal to that of the liquid due to the fact that the volume expansion due to the solidification cannot be taken into account in the model. A good agreement can be observed between the numerical results and the experimental data.

To perform the above comparisons with the 1D and 2D-axi models, it was necessary to shift temporally the beginning of the freezing for both models. However, this temporal shift, about 15 ms, is quite moderate and it can be explained by the presence of the thin frost layer already present on the substrate before the droplet impact. Applying Eq.( 28) , it was estimated that the distance travelled in 15 ms by the ice front is about 50 μm, which is comparable to the thickness of the frost layer before the droplet impact.

Icing of the droplet deposited on the N-BK7

Fig. 18 presents numerical and experimental results obtained in the case of the N-BK7 substrate under the same icing conditions as in the previously described experiments with the duralumin substrate ( T 0 = -10 • C and T ∞ = 18 • C) . Two main differences from the previous case of duralumin can be pointed out: the freezing dynamics is much and the ice front forms a much deeper cavity.

These noticeable differences can be explained by the heat transfer between the solid wall and the droplet. Immediately after the droplet deposition, the temperature at the interface between the droplet and the wall reaches the thermal contact temperature T c defined by: where e L and e S are the thermal effusivity of the liquid water and the solid ( e = k ρ C p ). The application of Eq. ( 31) provides a contact temperature T c = 7 °C in the case of the N-BK7 substrate. Therefore, this positive contact temperature prevents the formation of ice for a while. In Fig. 19 (A), the results from the 2D-axi numerical simulation show that a time of about 10 s is necessary for the surface temperature of the substrate to reach 0 °C at R = 0 . In contrast, the surface temperature of the substrate never becomes positive on the droplet edge ( Fig. 19 B). This time delay at the onset of freezing between the centre and the edge of the droplet makes that the ice front propagation is mainly radial during the first 10 s after the impact. Consequently, the vertical propagation velocity of the ice front is null or extremely low in the central region ( Fig. 18 ). This process results in a more pronounced concavity of the ice front (and thus of γ angle in Fig. 21 ), which ultimately contribute to the formation of the pointy tip. Such a depth is not observed in the case of duralumin for which γ is much lower. For comparison, the contact temperature T c is about -8 °C for duralumin under the same freezing conditions, which allows the freezing to start immediately at the droplet centre just after the drop impact. Fig. 20 (A) shows the temporal evolution of Z B in the case of the N-BK7 substrate. At the shortest times, the evolution of Z B is comparable to a square root of time, but after t = 10 s the velocity of the point B increases. This acceleration occurs during the final stage of the freezing when the pointy tip is forming at the top of the droplet. Obviously, since the 2D-axi simulation does not consider the volume expansion due to the freezing, this effect is not captured in the simulation results. Even though the simulation does not perfectly predict the shape of the ice front, the mass of ice is rather correctly evaluated in Fig. 20 (B).

T c = T 0 e S + T ∞ e L e L + e S , (31)
The tip formation can be described taking into account the spherical shape of the ice front [START_REF] Marín | Universality of tip singularity formation in freezing water drops[END_REF] . For both substrates, it was verified that the shape of the ice front was spherical after t = 10 s ( Fig. 16 B). Denoting V L and V I respectively the liquid and ice vol- ume ( Fig. 21 ) and ψ the density ratio between liquid water and ice, the mass conservation yields:

d dz ( V L + ψ V I ) = 0 (32)
The liquid volume V L can be decomposed into a spherical cap of angle θ and a downward volume V d : The ice volume V I is a section of a spherical cap between z = 0 and Z B , from which the volume V d must be subtracted:

V L = R 3 f ( θ ) + V d (33)
V I = z B ∫ 0 π R z 2 dz -V d , ( 34
)
where R ( z ) and θ ( z ) are respectively the local radius and angle of the frozen droplet (see Fig. 21 ). From geometrical consideration, regarding the spherical cap, the function f ( θ ) can be expressed as:

f ( θ ) = π 3 2 -3 cos ( θ ) + cos 3 ( θ ) sin 3 ( θ ) (35)
Assuming that the ice front takes a spherical shape and denoting φ = γθ :

V d = r 3 f ( γ -θ ) (36)
Using d r/d z = tan (θ ) , the following relationship between θ and γ can be deduced:

f ( γ -θ ) + f ( θ ) = ψ f ( γ -θ ) + π 3 tan ( θ ) . ( 37
)
Assuming that the pointy tip is a cone of angle α, the previous equation can be used to determine the cone angle α = π -2 θ of the pointy tip at the end of the freezing process [ 15 , 16 ]. However, an additional condition at the solid/liquid/air tri-junction (point B) is required for closing the problem. Different possibilities were discussed in the literature, but they usually neglect that the ice front is spherical. Snoeijer and Brunet [START_REF] Snoeijer | Pointy ice-drops: how water freezes into a singular shape[END_REF] considered that no inflexion point can exist at the three-junction point. This assumption combined to a flat horizontal ice front, allows them to derive an expression for the angle θ depending solely on the density ratio ψ.

Other conditions at the tri-junction were considered by Anderson et al. [START_REF] Anderson | The case for a dynamic con-tact angle in containerless solidification[END_REF] still in the case of a flat ice front. They include a constant contact angle θ , a fixed contact line, a constant or a dynamic growth angle γ . The latter was considered by these authors as being the most suitable condition for realistic density ratios. Recently, observations by Marín et al. [START_REF] Marín | Universality of tip singularity formation in freezing water drops[END_REF] suggested that the conical tip angle α is independent of the substrate temperature and of the wetting angle. For water droplets, they measured a value of α equal to about 140 °, whatever the freezing conditions. Fig. 22 presents current experimental results for the contact angle γ at the tri-junction (point B) as well as the radius of curvature R s of the ice front. For both substrates, freezing dynamics induces a significant change in these two parameters. A decrease in γ and R s can be observed during the first stage of the freezing for both duralumin and N-BK7 substrates. In the case of duralumin, γ reaches an end value of about 27 is observed experimentally. In the case of N-BK7, the initial value of γ is significantly higher due to the fact that freezing begins at the droplet edge. The pointy tip begins to appear around t = 17 s after γ has reached its minimum value of about 70 •. The pointy tip formation is associated with a sharp increase of γ . At the end of the freezing process, γ is close to 100 °which is not far from γ = 90 • observed by [START_REF] Marín | Universality of tip singularity formation in freezing water drops[END_REF] . An accuracy of ±5 • is expected in the current estimation of γ . With this end value of γ , a cone angle α of 125 • can be predicted by Eq. ( 37) . This value is in good agreement with the measurement of α obtained from sideview images (about 120 °±5 • ). It should be noticed an inflection point at the tri-junction was not observed. Further experimental investigations will be necessary to determine if this could serve as a general condition at the tri-junction point for closing the ice front problem.

Conclusions

A new experimental approach based on laser-induced fluorescence has been developed to reconstruct dynamically the ice front within a water droplet impinging on a subcooled solid substrate. This method benefits from the significant loss of fluorescence emission for certain fluorescent dyes when they are incorporated into an iced phase. As the fluorescence signal from the ice part is negligible, top view images of the deposited droplet can be safely considered for measuring the thickness of the liquid phase. The grey levels in the images closely correspond to an integration of the fluorescence signal over the thickness of the liquid layer. A second camera is necessary to provide side view images of the liquid/gas interface to account for the increase of the droplet volume during the freezing. This novel technique has made it possible to observe how the freezing dynamics influences the shape of the ice front. Experimentally it has been observed that the ice front takes a spherical shape. The study of two materials, namely duralumin and N-BK7 glass, indicates a very clear influence of the thermal properties of the solid substrate on the evolution of the ice front, most notably thermal diffusivity and effusivity. The freezing rate is radically different for these two substrate and large differences in the radius of curvature of the front and in the angle made of the ice front with the free liquid surface at the top of the droplet are also observed. Due to the low thermal effusivity of N-BK7, a rise in the temperature of the solid surface is observed just after the droplet impact. It leads to a significant delay in the onset of icing at the centre of the droplet in comparison to its edge. The ice front turns into a spherical cavity with a much deeper aspect than in the case of duralumin. This geometry is appropriate for the development of a pointy tip as the volume expands during the freezing. The modelling of these different phenomena remains difficult; one major challenge lies in incorporating volume expansion into theoretical models and simulation. Without any specific treatment for the volume expansion, 1D and 2D-axi models are able to reproduce some of the experimental observations. These simplified models are relatively good at predicting the mass of ice, the freezing time, the position of the tri-junction point and the depth of the cavity in the case of highly thermally conductive substrates such as duralumin. However, they completely fail to reproduce the spherical shape of the ice front and the angle between the ice front and the free surface. Further experimental results and more advanced approaches in the modelling will be necessary in order to provide a comprehensive model to close the problem at the tri-junction point and to provide information of the effect of the surface temperature, Weber number, and contact angle on the geometry of the ice front.

  . L -1 ] Cp : thermal heat capacity [ J . K -1 ] D : thermal diffusivity [ m 2 s -1 ] e : thermal effusivity [ J . K -1 m -2 s -1 2 ] I f : intensity of fluorescence signal [ W . m -2 ] k : thermal conductivity [ W . m -1 . K -1 ] L : latent heat [ J . k g -1 ] tension [ N . m -1 ]γ :angle at the tri-junction point [ • ]

Fig. 1 .

 1 Fig. 1. Experimental observation of the "pointy tip" with a cone angle α [15] . The experiment were conducted for different aspect ratio H/R and different substrate temperature T 0 . Case A) α = 147 • , T 0 = -34 . 4 • C and H/R = 4 . 96 . Case B) α = 141 • , T 0 = -44 . 1 • C and H/R = 1 . 16 . C) α = 139 • , T 0 = -20 . 8 • C and H/R = 0 . 46 . D) represent the ice front in a droplet observed in a Hele-Shaw cell.

Fig. 2 .

 2 Fig. 2. Experimental setup.

Fig. 3 .

 3 Fig. 3. Images of the freezing of a droplet impinging a subcooled ( T = -10 °C) surface at We = 65 . (a) Duralumin and (b) N-BK7 glass.

Fig. 4 .

 4 Fig. 4. Fluorescence images from a droplet deposited onto the duralumin substrate subcooled at -10 °C. The initial droplet temperature is 18 °C. Dotted lines represent the contact line between the droplet and the substrate.

Fig. 5 .

 5 Fig. 5. Sketch of the freezing droplet and definition of the main geometrical parameters.

Fig. 5 (

 5 left side) displays a fluorescence light ray emerging at the droplet liquid-air interface surface at a point C in the line of sight of the camera. In fact, this ray corresponds to fluorescence along the optical path between C and D, where point D is located on the ice front. Below point D, the fluorescence emission from the solid phase can be totally neglected. Neglecting the reabsorption of fluorescence by the fluorescent dye on the optical path DC, the signal collected by a given pixel of the camera can be determined by:

Fig. 6 .

 6 Fig. 6. The different steps of the image post processing.

Fig. 7 .

 7 Fig. 7. Comparison of the liquid profile measured by LIF and extracted from a sideview shadow image. Case of a droplet deposited onto an isothermal surface with a mirror finish.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Comparison between of the ice front profile obtained with or without taking into account the volume expansion during the solidification process. Same icing conditions as in Fig. 4 .

Fig. 10 .

 10 Fig. 10. Comparisons between the 1D analytical model Eqs. (16) and[START_REF] Anderson | The case for a dynamic con-tact angle in containerless solidification[END_REF] and the 2D numerical solution for the temperature profile (A) and the interface position (B).

Fig. 11 .

 11 Fig. 11. Boundary conditions considered in the numerical resolution.

Fig. 12 .

 12 Fig. 12. Evolution of the ice thickness A) and radial position B) of the ice front at the air/liquid/solid tri-junction (point B) in function of the time for T 0 = -10 • C and T ∞ = 18 • C .

Fig. 13 .

 13 Fig.[START_REF] Pallav Kanta | Fast-freezing kinetics inside a droplet impacting on a cold surface[END_REF]. Evolution of the ice thickness at the droplet centre R = 0 (point A). h d0 is the droplet height before freezing. Same freezing conditions as in Fig.12.

Fig. 14 .

 14 Fig.14. Evolution of the ice front thickness Z A in the droplet centre ( R = 0) for different droplet thickness while maintaining the same contact angle with the substrate (A), for different substrate temperatures while maintaining the droplet size and impact velocity (B).

Fig. 15 .

 15 Fig. 15. Position of ice front determined with the two reconstruction methods (II.3.c). Comparison with 2D-axi simulation. Same icing conditions as in Fig. 12 .

Fig. 16 .

 16 Fig. 16. Experimental ice front profile. A) Duralumin, t = 1 s and T 0 = -10 • C , B) N-BK7 substrate at t = 13 s and T 0 = -10 • C . R S is the radius of curvature of the fitting circle.

Fig. 17 .

 17 Fig. 17. Temporal evolution of the ice of mass. Same icing conditions as in Fig. 12 .

Fig. 18 .

 18 Fig. 18. Evolution of the ice front profile in the case of the icing on the N-BK7 surface at T 0 = -10 • C and an initial drop temperature T ∞ = 18 • C .

Fig. 19 .

 19 Fig. 19. Evolution of the surface temperature of the N-BK7 substrate: (A) at the droplet centre, (B) on the droplet edge ( R = 3.2 mm). Same icing conditions as in Fig. 18 .

Fig. 20 .

 20 Fig. 20. Evolutions of A) the thickness at the air/liquid/solid tri-junction (point B), and B) the total mass of ice.

Fig. 21 .

 21 Fig. 21. Scheme and main parameters of the spherical ice front.

Fig. 22 .

 22 Fig. 22. Evolutions of the radius of curvature R S and of the angle γ for (A) Duralumin and (B) N-BK7 substrates, cooled at -10 °C.

Table 1

 1 Thermophysical properties of the substrates and water.


Alkiviadis Gourgiotis

Arnaud Mangeret

Gérard Manhès

Pascale Blanchart

Lucie Stetten

Guillaume Morin

Pierre Le Pape

Pierre Lefebvre

Mathieu Le Coz

Charlotte Cazala

New Insights into Pb Isotope Fingerprinting of U-Mine Material Dissemination in the Environment: Pb Isotopes as a Memory Dissemination Tracer

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

New insights into Pb isotope fingerprinting of U-mine

1 material dissemination in the environment: Pb isotopes as a 2 memory dissemination tracer.

  1* , A. Mangeret 1 , G. Manhès 2 , P. Blanchart 3 , L. Stetten

  Pb isotope, a precise characterization of the Pb isotope composition of 40 the U-ore was performed without the use of U-ore samples. The well-defined end-members 41 with the help of a reevaluated isotope mixing model allowed the accurate determination of 42 the radiogenic Pb percentages in the cores that were overall found to be >50%. Non-43 correlated distributions of radiogenic 206 Pb and U are observed in several of the wetland soil 44 samples studied. They reveal post-depositional U redistribution explained by major U 45 speciation changes due to redox cycling in the wetland. On the contrary, the radiogenic 206 Pb 46 showed no or small post-depositional mobility and thus can be considered to be a memory 47 tracer of the dissemination of U-rich radioactive material: even after an important U loss, 48 the radiogenic206 Pb is able to reveal past contamination by U-rich materials.

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	34	Abstract
	5 35	1,4,5 , G. Morin 4 , P. Le
	6 36	Pape 4 , P. Lefebvre 4 , M. Le Coz 1 , C. Cazala 1 Stable Pb isotope ratios were measured and compared to U distributions in three soil cores
	37 7 38	located in a wetland highly impacted by water discharge of a former U-mine. Pb isotope ratios showed notable alignments in binary mixing plots demonstrating the dissemination of
	8 39	1 Institut de Radioprotection et de Sûreté Nucléaire -PSE/ENV -SEDRE/LELI, Fontenay-aux-radioactive-enriched material from the U-mine. Thanks to theses alignments and to the
	Roses, 92262, France. 2 Institut de Physique du Globe de Paris, Sorbonne Paris Cité, Université Paris-Diderot, UMR CNRS 7154, 1 rue Jussieu, 75238 Paris Cedex, France. 3 Institut de Radioprotection et de Sûreté Nucléaire -PSE/ENV -SEDRE/USDR, Fontenay-aux-Roses, 92262, France. 5, France. measurement of 204 49 9 10 11 12 13 15 16
	50 17 51 18 52	5 Centre for Microbiology and Environmental Systems Science, University of Vienna, Althanstraße 14, UZA II, 1090 Vienna, Austria
	19 53
	54 20 55	*corresponding author: alkiviadis.gourgiotis@irsn.fr
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4 Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie (IMPMC), UMR 7590 14 CNRS-Sorbonne Université IRD-MNHN, case 115, 4 place Jussieu, 75252 Paris Cedex