Chloe Girard

David Zwicker

Raphael Mercier

The regulation of meiotic crossover distribution: a coarse solution to a century-old mystery?

Meiotic crossovers, which are exchanges of genetic material between homologous

chromosomes, are more evenly and distantly spaced along chromosomes than expected by chance. This is because the occurrence of one crossover reduces the likelihood of nearby crossover events -a conserved and intriguing phenomenon called crossover interference. Although crossover interference was first described over a century ago, the mechanism allowing coordination of the fate of potential crossover sites half a chromosome away remains elusive. In this review, we discuss the recently published evidence supporting a new model for crossover patterning, coined the coarsening model, and point out the missing pieces that are still needed to complete this fascinating puzzle.

[Introduction]

Meiotic crossovers (COs) between homologous chromosomes lead to the rearrangement of parental alleles, generating and maintaining genetic diversity. They also provide a physical link between homologs, which is required, in many species, for the correct segregation of chromosomes during the first meiotic division. The number and distribution of COs along chromosomes are neither uniform nor random and show highly conserved properties.

First, in most species, each pair of chromosomes requires at least one CO to guarantee their correct segregation at metaphase I, the so-called obligate crossover [START_REF] Jones | Meiotic Crossing-over: Obligation and Interference[END_REF]. Second, despite a large number of DNA double-strand breaks (DSBs), which are the potential precursors of COs, only a few COs are typically formed per chromosome pair, irrespective of the size of chromosomes across species, e.g., Plasmodium chromosome 11 (2 Mb), canola chromosome 6 (28 Mb) and pig chromosome 6 (157 Mb) [START_REF] Choi | The reference genetic linkage map for the multinational Brassica rapa genome sequencing project[END_REF][START_REF] Wang | The genome of the mesopolyploid crop species Brassica rapa[END_REF][START_REF] Tortereau | A high density recombination map of the pig reveals a correlation between sex-specific recombination and GC content[END_REF][START_REF] Miles | Indels, structural variation, and recombination drive genomic diversity in Plasmodium falciparum[END_REF][START_REF] Fernandes | Unleashing meiotic crossovers in hybrid plants[END_REF] all exhibit around three COs per meiosis. Third, COs are more distantly distributed along chromosomes than expected by chance. This is the result of a phenomenon called crossover interference, by which the formation of a CO at one locus interferes with and prevents the formation of another CO in its vicinity on the same chromosome. But how do potential crossover sites half a chromosome away communicate with each other to coordinate their fate and establish the final distribution? More than a century after its first detection in Drosophila [START_REF] Sturtevant | The linear arrangement of six sex-linked factors in Drosophila, as shown by their mode of association[END_REF][START_REF] Sturtevant | The behavior of the chromosomes as studied through linkage[END_REF][START_REF] Muller | The Mechanism of Crossing-Over. II. IV. The Manner of Occurrence of Crossing-Over[END_REF], and its subsequent observation in a large range of species, the nature of the CO interference signal and its spreading mechanism are still the subject of lively debates [START_REF] Berchowitz | Genetic interference: don't stand so close to me[END_REF][START_REF] Zickler | A few of our favorite things: Pairing, the bouquet, crossover interference and evolution of meiosis[END_REF][START_REF] Von Diezmann | Let's get physical -mechanisms of crossover interference[END_REF][START_REF] Lloyd | Crossover patterning in plants[END_REF].

Crossover formation

Meiotic COs arise from programmed DNA double-strand breaks and their repair by the homologous recombination machinery. Between 2-to 200-fold more DSBs than COs are formed (depending on the species, [START_REF] Serrentino | The spatial regulation of meiotic recombination hotspots: Are all DSB hotspots crossover hotspots?[END_REF]): only a (small) subset will be selected to become crossovers, while the vast majority will be repaired as non-crossovers, i.e., a non-reciprocal copy of DNA sequence which does not result in chromatid exchange.

In most species, two co-existing molecular pathways are responsible for crossover formation. Class I COs, which account for most events, are promoted by a group of conserved factors collectively referred to as the ZMM proteins based on Saccharomyces cerevisiae nomenclature (Zip1/2/3/4, Msh4/5, and Mer3). These COs can be distinguished cytologically as they are the sites of accumulation of specific procrossover proteins (e.g., MLH1/3 in plants and mammals, COSA-1 in C. elegans, Figure 1 [START_REF] Anderson | Distribution of Crossing Over on Mouse Synaptonemal Complexes Using Immunofluorescent Localization of MLH1 Protein[END_REF][START_REF] Chelysheva | An easy protocol for studying chromatin and recombination protein dynamics during Arabidopsis thaliana meiosis: immunodetection of cohesins, histones and MLH1[END_REF][START_REF] Yokoo | COSA-1 reveals robust homeostasis and separable licensing and reinforcement steps governing meiotic crossovers[END_REF]). Class II COs do not rely on ZMMs for their formation but on a set of endonucleases, prominently Mus81 and its partner Mms4 [START_REF] Hollingsworth | The Mus81 solution to resolution: generating meiotic crossovers without Holliday junctions[END_REF]. Class I, ZMMdependent crossovers are sensitive to interference in all species studied thus far: they tend to be distantly spaced along chromosomes [START_REF] Anderson | Distribution of Crossing Over on Mouse Synaptonemal Complexes Using Immunofluorescent Localization of MLH1 Protein[END_REF][START_REF] Lhuissier | The Mismatch Repair Protein MLH1 Marks a Subset of Strongly Interfering Crossovers in Tomato[END_REF][START_REF] Kan | Comparative analysis of meiotic progression in female mice bearing mutations in genes of the DNA mismatch repair pathway[END_REF][START_REF] Wang | Per-Nucleus Crossover Covariation and Implications for Evolution[END_REF][START_REF] Morgan | Diffusion-mediated HEI10 coarsening can explain meiotic crossover positioning in Arabidopsis[END_REF]. On the other hand, class II COs are not (or are much less) sensitive to interference [START_REF] Hollingsworth | The Mus81 solution to resolution: generating meiotic crossovers without Holliday junctions[END_REF][START_REF] Anderson | Combined fluorescent and electron microscopic imaging unveils the specific properties of two classes of meiotic crossovers[END_REF]. They represent a minority of events in most eukaryotes. Class II COs are often considered as a backup repair mechanism for the recombination intermediates that fail to mature into class I COs or non-COs. Importantly, when the ratio between class I and class II COs is modified (e.g., in mutants that abolish class I COs, or increase class II COs), this leads to an apparent reduction of CO interference, while the process of interference itself is not modified [START_REF] Stahl | Defining and Detecting Crossover-Interference Mutants in Yeast[END_REF][START_REF] Séguéla-Arnaud | Multiple mechanisms limit meiotic crossovers: TOP3α and two BLM homologs antagonize crossovers in parallel to FANCM[END_REF]. This needs to be kept in mind when attributing roles in CO interference based on mutant phenotypes. In the remainder of this review, we will consider exclusively class I, ZMM-dependent, interfering crossovers. Meiotic recombination progresses within the highly ordered environment of chromosome organization. At the onset of meiosis, chromatin is organized as an array of loops, the bases of which are tethered by proteins to form an axis [START_REF] Zickler | Meiotic chromosomes: integrating structure and function[END_REF]. The doublestrand break machinery is assembled on this axis, and following the initiation of recombination, homologs will pair in a loose alignment all along their length [START_REF] Zickler | Meiotic chromosomes: integrating structure and function[END_REF][START_REF] Hunter | Meiotic recombination: The essence of heredity[END_REF].

The two axes are then "zipped up" to form the tripartite synaptonemal complex (SC): a transverse filament (Zip1 in S. cerevisiae) polymerizes progressively between the axes along the length of the homologs in a process called synapsis [START_REF] Zickler | Meiotic chromosomes: integrating structure and function[END_REF]. It is widely accepted that the mechanistic metric for interference is µm of axis/SC, rather than Mb of DNA, meaning that the interference signal propagates along the length of chromosomes axes and/or along the SC [START_REF] Zickler | A few of our favorite things: Pairing, the bouquet, crossover interference and evolution of meiosis[END_REF][START_REF] Hunter | Meiotic recombination: The essence of heredity[END_REF]. However, whether the tripartite SC or only the chromosome axis is required for the propagation of interference is a matter of ongoing debate ( [START_REF] Von Diezmann | Let's get physical -mechanisms of crossover interference[END_REF] and see below).

ZMM pro-crossover proteins act in concert to promote the maturation of recombination intermediates into COs. Among them, the RING-family E3 ligase Zip3/Zhp-3/HEI10/RNF212/Vilya homologs exhibit a conserved behavior (hereafter called HEI10 for simplicity), as illustrated in Figure 1 [START_REF] Chelysheva | An easy protocol for studying chromatin and recombination protein dynamics during Arabidopsis thaliana meiosis: immunodetection of cohesins, histones and MLH1[END_REF][START_REF] Bhalla | ZHP-3 acts at crossovers to couple meiotic recombination with synaptonemal complex disassembly and bivalent formation in C. elegans[END_REF][START_REF] Reynolds | RNF212 is a dosage-sensitive regulator of crossing-over during mammalian meiosis[END_REF][START_REF] De Muyt | E3 ligase Hei10: A multifaceted structure-based signaling molecule with roles within and beyond meiosis[END_REF][START_REF] Qiao | Antagonistic roles of ubiquitin ligase HEI10 and SUMO ligase RNF212 regulate meiotic recombination[END_REF][START_REF] Lake | Vilya, a component of the recombination nodule, is required for meiotic double-strand break formation in Drosophila[END_REF][START_REF] Lake | Narya, a RING finger domain-containing protein, is required for meiotic DNA double-strand break formation and crossover maturation in Drosophila melanogaster[END_REF]. Numerous small foci form on chromosomes as they synapse, colocalizing with the tripartite SC (Figure 1A). Super-resolution microscopy shows that HEI10 foci decorate the very middle of the SC and do not fill the axis or the entire space between the axes [START_REF] Reynolds | RNF212 is a dosage-sensitive regulator of crossing-over during mammalian meiosis[END_REF][START_REF] Capilla-Pérez | The synaptonemal complex imposes crossover interference and heterochiasmy in Arabidopsis[END_REF]. As meiosis progresses, less numerous but larger foci form (Figure 1B). This tendency for bigger foci culminates in the formation of a few intense foci corresponding to CO sites (Figure 1C). In the mouse, two RING-family E3 ligases -RNF212 and HEI10 -are essential for CO formation, and, intriguingly, RNF212 displays the cytological behavior shown by HEI10 homologs in other clades [START_REF] Reynolds | RNF212 is a dosage-sensitive regulator of crossing-over during mammalian meiosis[END_REF]. One may interpret this gradual accumulation at specific sites merely as a downstream manifestation of a CO designation decision that occurred earlier in meiotic prophase (e.g., before synapsis). In this view, HEI10 accumulation would only be the read-out of the CO designation process. Alternatively, the recently reported coarsening model proposes that this accumulation directly reflects the CO designation process itself and sees HEI10 accumulation as the driver of CO designation: only when HEI10 accumulation reaches a threshold does an embedded recombination intermediate become a CO ([22] see also below).

What is interference, and how to measure it?

Crossover interference is a term that describes the tendency of crossovers to form farther away from one another than would be expected by chance. Literally, one CO interferes with the presence of another CO nearby on the same chromosome. This prevents the occurrence of closely spaced pairs of crossovers and generates a more even spacing of COs than would be expected if they were distributed independently from one another.

These two ways of describing the same phenomenon, (i) the larger and more even distances between neighboring COs than expected and (ii) the lack of close double COs, are reflected in two classical methods to detect and measure CO interference at the chromosome level (Figure 2). The first one examines the distribution of intercrossover distances (Figure 2B, [START_REF] Mcpeek | Modeling interference in genetic recombination[END_REF][START_REF] De Boer | Two levels of interference in mouse meiotic recombination[END_REF]): in the presence of interference, the distribution of inter-crossover distances is shifted towards larger values (purple) than if CO events were randomly distributed along the chromosome (grey). The distribution of the distances can be conveniently fitted by a gamma distribution [START_REF] Broman | Characterization of Human Crossover Interference[END_REF], whose shape parameter n gives a measurement of the strength of interference. One limitation of this approach is that the distribution can be affected by phenomena other than interference, complicating its interpretation [START_REF] Zickler | A few of our favorite things: Pairing, the bouquet, crossover interference and evolution of meiosis[END_REF]38]. The second way to measure interference is to calculate the Coefficient of Coincidence (CoC; Figure 2C), which effectively measures the lack of double COs compared to the expected number if COs would occur independently from each other. It compares the observed frequency of simultaneous COs in two intervals (e.g., between intervals I1 and I2, the numerator in Figure 2C), to the expected number if COs were independent (product of CO frequencies of each interval, the denominator in Figure 2C). This classic way of measuring CO interference locally [START_REF] Muller | The Mechanism of Crossing-Over. II. IV. The Manner of Occurrence of Crossing-Over[END_REF] can be extended to an entire chromosome by calculating CoC for every pair of intervals possible. A CoC curve is obtained by plotting the CoC values against the distance ! between the two intervals (Figure 2C). In the presence of interference, very few double crossovers are observed for close-by intervals, and the CoC is close to 0 on the left of the curve (blue in Figure 2C). For more distant intervals (longer !), interference vanishes, and the observed frequency of double COs approaches the expected value (CoC = 1). One possible measurement of interference strength is the length for which the value of the CoC reaches 0.5 [38]. This latter method to examine interference, which is perhaps less intuitive than the former, was shown to be more robust to other alterations of the recombination process [START_REF] Zickler | A few of our favorite things: Pairing, the bouquet, crossover interference and evolution of meiosis[END_REF]38].

These measures of interference need large datasets that can be of various kinds. One can use cytological markers of CO sites (e.g., Mlh1, COSA-1, recombination nodules) and determine their distribution along meiotic chromosomes. The metric used to describe CO distribution is, in this case, the length of axis/SC (in µm) between two adjacent events, which is likely the most relevant mechanistically [START_REF] Anderson | Combined fluorescent and electron microscopic imaging unveils the specific properties of two classes of meiotic crossovers[END_REF][START_REF] De Muyt | E3 ligase Hei10: A multifaceted structure-based signaling molecule with roles within and beyond meiosis[END_REF]38]. COs can also be detected genetically using DNA polymorphisms between the two parents, an approach that has become more powerful with increased sequencing capability [START_REF] Mancera | High-resolution mapping of meiotic crossovers and non-crossovers in yeast[END_REF][START_REF] Wijnker | The genomic landscape of meiotic crossovers and gene conversions in Arabidopsis thaliana[END_REF][START_REF] Rowan | Rapid and Inexpensive Whole-Genome Genotyping-by-Sequencing for Crossover Localization and Fine-Scale Genetic Mapping[END_REF][START_REF] Rowan | An ultra high-density Arabidopsis thaliana crossover map that refines the influences of structural variation and epigenetic features[END_REF][43]. In this case, the distance between two adjacent COs is measured in the number of DNA base pairs. While this measurement can be very powerful, allowing highprecision mapping of CO sites and analysis of large populations, it has some limitations. First, class I and class II COs are indistinguishable by this approach, which could alter the measure of interference (see above). Second, the conversion from the DNA space in Mb to the SC space (in µm), which is relevant for mechanistic inference, may be delicate as the compaction of DNA varies along chromosomes (Table 1). Third, when sequencing gametes, only half of the events are detected, which modifies inter-CO distances [START_REF] Veller | Limitations of gamete sequencing for crossover analysis[END_REF] but not CoC curves [START_REF] Wang | Crossover maturation inefficiency and aneuploidy in human female meiosis[END_REF].

Intriguingly, interference acts at varying distances along chromosomes depending on the species (Table 1). In S. cerevisiae for instance, interference acts at a distance of roughly 0.5 µm (only 10% of the longest chromosome, [38]) while in mouse it covers most of a full chromosome [START_REF] Anderson | Distribution of Crossing Over on Mouse Synaptonemal Complexes Using Immunofluorescent Localization of MLH1 Protein[END_REF][START_REF] De Boer | Two levels of interference in mouse meiotic recombination[END_REF][START_REF] Petkov | Crossover interference underlies sex differences in recombination rates[END_REF][START_REF] Wang | Crossover patterns under meiotic chromosome program[END_REF]. In C. elegans this distance exceeds the length of all wild-type chromosomes ensuring that one and only one CO per chromosome pair is formed [START_REF] Hillers | Chromosome-Wide Control of Meiotic Crossing over in C. elegans[END_REF][START_REF] Libuda | Meiotic chromosome structures constrain and respond to designation of crossover sites[END_REF]. This means that the mechanism of interference, if universal, must be able to act at both small (0.5 µm of SC) and large ranges (20 µm).

A complete picture of CO patterning includes quantitative measurements of interference, CO counts and CO distribution. These different aspects of CO pattering are intertwined. Notably, CO interference influences CO counts, but also distribution.

For example, if interference extends to a large proportion of the chromosome length, chromosome with exactly two COs would tend to have a CO at each end, while a CO in the middle would tend to be alone [START_REF] Libuda | Meiotic chromosome structures constrain and respond to designation of crossover sites[END_REF][START_REF] Wang | Crossover Interference, Crossover Maturation, and Human Aneuploidy[END_REF].

The beam-film model of crossover interference It has been widely thought that CO designation must emit a signal that propagates outward and suppresses CO formation over a certain distance on the chromosome. This led to the formulation of a compelling model in which redistribution of mechanical stress away from designated sites would prevent other COs from forming in the vicinity, a model which has come to be known as the beam-film model [38,[START_REF] Kleckner | A mechanical basis for chromosome function[END_REF]. In physical systems, any local increase or decrease in mechanical stress at one position tends to redistribute outward from that point. In the beam-film model, the chromosomes are under mechanical stress, and at their surfaces lie an array of precursors (the doublestrand breaks). Eventually, one of these breaks reaches a threshold and undergoes a stress-promoted molecular change: CO designation. This process results in a local relaxation of stress that immediately redistributes outward from the designated event, preventing other precursors from reaching the required stress threshold to initiate CO designation in the vicinity of the first event, establishing interference. The most attractive feature of this model is that the medium for communication between COs is built into the meiotic chromosome structure: the meshwork of DNA/protein interactions at the axis should be capable of accumulating and transducing mechanical stress [START_REF] Sun | Condensin controls mitotic chromosome stiffness and stability without forming a structurally contiguous scaffold[END_REF].

Supporting this model, topoisomerase II (Topo2), an enzyme able to relax over-and underwound DNA molecules [START_REF] Baxter | A model for chromosome condensation based on the interplay between condensin and topoisomerase II[END_REF] has been shown to be involved in modulating CO interference [38].

The coarsening model of crossover interference

An alternative model, coined the coarsening model, reverses the perspective and posits that interference does not operate through the transmission of a suppressing signal but by accumulation of a pro-crossover factor at future CO sites at the expense of other neighboring ones, therefore establishing crossover interference. It was originally described, conceptualized, and supported using cytological data in Arabidopsis [START_REF] Morgan | Diffusion-mediated HEI10 coarsening can explain meiotic crossover positioning in Arabidopsis[END_REF]. A very similar model was independently developed by another team using data from C. elegans [START_REF] Zhang | Crossover patterning through kinase-regulated condensation and coarsening of recombination nodules[END_REF]. This model crystallizes a corpus of previous data and ideas accumulated in the field [START_REF] Yokoo | COSA-1 reveals robust homeostasis and separable licensing and reinforcement steps governing meiotic crossovers[END_REF][START_REF] Reynolds | RNF212 is a dosage-sensitive regulator of crossing-over during mammalian meiosis[END_REF][START_REF] Rog | The synaptonemal complex has liquid crystalline properties and spatially regulates meiotic recombination factors[END_REF].

As mentioned above, the HEI10-Zhp3/4-Zip3-RNF212-Vilya ZMM proteins exhibit very specific dynamics during prophase, evolving from many small foci to a few large aggregates that localize at CO sites on chromosomes [START_REF] Bhalla | ZHP-3 acts at crossovers to couple meiotic recombination with synaptonemal complex disassembly and bivalent formation in C. elegans[END_REF][START_REF] Reynolds | RNF212 is a dosage-sensitive regulator of crossing-over during mammalian meiosis[END_REF][START_REF] De Muyt | E3 ligase Hei10: A multifaceted structure-based signaling molecule with roles within and beyond meiosis[END_REF][START_REF] Lake | Vilya, a component of the recombination nodule, is required for meiotic double-strand break formation in Drosophila[END_REF][START_REF] Lake | Narya, a RING finger domain-containing protein, is required for meiotic DNA double-strand break formation and crossover maturation in Drosophila melanogaster[END_REF][START_REF] Capilla-Pérez | The synaptonemal complex imposes crossover interference and heterochiasmy in Arabidopsis[END_REF][START_REF] Agarwal | Zip3 provides a link between recombination enzymes and synaptonemal complex proteins[END_REF]]. The coarsening model proposes that this progressive accumulation is driven by onedimensional diffusion of HEI10 molecules along the SC. This initiates a "coarsening" process when bigger foci tend to capture more material than smaller foci, so larger aggregates grow at the expense of nearby smaller ones (Figure 3). As large aggregates siphon off nearby HEI10 molecules, they tend to form at a distance from one another, spontaneously creating interference. The HEI10 large foci then attract pro-CO factors that can implement the formation of a CO (resolution) at each site, such as MLH1 in plants and mammals or COSA-1 in C. elegans (Figure 1 and Figure 3A).

Recombination intermediates devoid of HEI10/MLH1 foci (Figure 3) would be matured into non-COs by anti-CO factors such as Sgs1(atRECQ4)-Top3-Rmi [START_REF] Séguéla-Arnaud | Multiple mechanisms limit meiotic crossovers: TOP3α and two BLM homologs antagonize crossovers in parallel to FANCM[END_REF][START_REF] Kaur | Top3-Rmi1 DNA Single-Strand Decatenase Is Integral to the Formation and Resolution of Meiotic Recombination Intermediates[END_REF], and infrequently as class II COs. Note that this process provides an immediate explanation for the obligate crossover as, even if allowed to proceed to completion, it leads to the formation of a single large aggregate, and thus a minimum of one CO (see supplemental videos). If interrupted before completion, it leads to a limited number of aggregates that are distributed at a distance from each other. One major change of paradigm from previous models of crossover interference is that HEI10 accumulation does not only reflect the selection of the CO sites -as a readout of an upstream decision -but is the actual driving force determining CO positions.

For this model to work, two conditions must be satisfied. First, HEI10 molecules, that initially load onto the SC at multiple positions, must diffuse along the SC, but not (or at a much lower rate) between separate SCs/chromosomes (first equation in Figure 3B).

The liquid-like properties of the SC may contribute to these HEI10 dynamics [START_REF] Zhang | Crossover patterning through kinase-regulated condensation and coarsening of recombination nodules[END_REF][START_REF] Rog | The synaptonemal complex has liquid crystalline properties and spatially regulates meiotic recombination factors[END_REF][START_REF] Stauffer | Diffusion through a liquid crystalline compartment regulates meiotic recombination[END_REF]. ZHP-3/4 has been shown experimentally in C. elegans to remain dynamic even after accumulation in foci [START_REF] Zhang | Crossover patterning through kinase-regulated condensation and coarsening of recombination nodules[END_REF]. The second prerequisite is that larger HEI10 aggregates should retain more HEI10 molecules than smaller aggregates (second equation in Figure 3B). The important parameters of the model that determine the eventual number of crossovers are: (i) the length ! of the SC, (ii) the initial amount of HEI10 loaded on the SC, (iii) the diffusivity " of HEI10, (iv) the rate Λ of HEI10 exchange, (v) the duration $ of coarsening (duration of pachytene), and (vi) the minimum size of HEI10 focus (threshold) that changes the fate of an underlying recombination intermediate into a CO designated site. The equations defined in Figure 3B allow us to make predictions that can be tested in different species. This model satisfactorily accounts for a number of observations. (i) The less HEI10 foci per chromosome, the brighter each is [START_REF] Morgan | Diffusion-mediated HEI10 coarsening can explain meiotic crossover positioning in Arabidopsis[END_REF]: foci on chromosomes with only one HEI10 late focus are brighter than foci on chromosomes with two or more. The coarsening model readily predicts this distinct behavior as the growth of each HEI10 focus is fueled by HEI10 proteins coming from the shrinking/disappearance of neighboring foci on the same chromosome. (ii) Intermediate Hei10 foci, which mark both sites that will and will not become crossovers, are interfering in Sordaria [START_REF] De Muyt | E3 ligase Hei10: A multifaceted structure-based signaling molecule with roles within and beyond meiosis[END_REF], which is expected if foci are growing at the expense of neighboring ones. (iii) CO count is sensitive to HEI10/RNF212 dosage in plants, mice, pig, sheep, cattle, deer and humans [START_REF] Reynolds | RNF212 is a dosage-sensitive regulator of crossing-over during mammalian meiosis[END_REF][START_REF] Kong | Sequence Variants in the RNF212 Gene Associate with Genome-Wide Recombination Rate[END_REF][START_REF] Kong | Common and low-frequency variants associated with genome-wide recombination rate[END_REF][START_REF] Chowdhury | Genetic Analysis of Variation in Human Meiotic Recombination[END_REF][START_REF] Fledel-Alon | Variation in Human Recombination Rates and Its Genetic Determinants[END_REF][START_REF] Sandor | Genetic Variants in REC8, RNF212, and PRDM9 Influence Male Recombination in Cattle[END_REF][START_REF] Johnston | Conserved Genetic Architecture Underlying Individual Recombination Rate Variation in a Wild Population of Soay Sheep ( Ovis aries )[END_REF][START_REF] Kadri | Coding and noncoding variants in HFM1, MLH3, MSH4, MSH5, RNF212, and RNF212B affect recombination rate in cattle[END_REF][START_REF] Petit | Variation in Recombination Rate and Its Genetic Determinism in Sheep Populations[END_REF][START_REF] Ziolkowski | Natural variation and dosage of the HEI10 meiotic E3 ligase control Arabidopsis crossover recombination[END_REF][START_REF] Johnston | A Genomic Region Containing REC8 and RNF212B Is Associated with Individual Recombination Rate Variation in a Wild Population of Red Deer (Cervus elaphus)[END_REF][START_REF] Johnsson | Genetic variation in recombination rate in the pig[END_REF][START_REF] Durand | Joint control of meiotic crossover patterning by the synaptonemal complex and HEI10 dosage[END_REF]. This could be understood in the context of the coarsening model by assuming that HEI10 dosage determines the amount of HEI10 molecules on the SC. Everything else being equal, HEI10 dosage then directly determines the CO count [START_REF] Morgan | Diffusion-mediated HEI10 coarsening can explain meiotic crossover positioning in Arabidopsis[END_REF]. (iv) A strong correlation between SC length and CO number is observed. This is true within a meiocyte and between meiocytes, notably when comparing male versus female meiosis [START_REF] Wang | Per-Nucleus Crossover Covariation and Implications for Evolution[END_REF][START_REF] Kleckner | Coordinate variation in meiotic pachytene SC length and total crossover/chiasma frequency under conditions of constant DNA length[END_REF]. Assuming a fixed amount of HEI10 loaded per µm of SC, the SC length linearly determines the total amount of HEI10 introduced into the system and, consequently, CO number [START_REF] Durand | Joint control of meiotic crossover patterning by the synaptonemal complex and HEI10 dosage[END_REF]. The SC length also influences the coarsening itself because it affects the time it takes HEI10 to diffuse along. However, this effect is likely minor compared to the effect of the total amount loaded initially. (v) The HEI10 dosage and SC length have a combined effect; increasing the HEI10 dosage increases CO proportionally to the SC length [START_REF] Durand | Joint control of meiotic crossover patterning by the synaptonemal complex and HEI10 dosage[END_REF]. (vi) CO interference among class I COs is abolished in the absence of the transverse filament of the SC in Arabidopsis [START_REF] Capilla-Pérez | The synaptonemal complex imposes crossover interference and heterochiasmy in Arabidopsis[END_REF][START_REF] France | ZYP1 is required for obligate crossover formation and crossover interference in Arabidopsis[END_REF]73] and rice [START_REF] Wang | The Central Element Protein ZEP1 of the Synaptonemal Complex Regulates the Number of Crossovers during Meiosis in Rice[END_REF][START_REF] Wang | Increasing the Genetic Recombination Frequency by Partial Loss of Function of the Synaptonemal Complex in Rice[END_REF]. This is interpreted in the context of the coarsening model as follows: in the absence of the SC, diffusion of HEI10 molecules occurs within the whole nucleoplasm and these can form foci on recombination intermediates to promote CO formation throughout the nucleus. Diffusion being no longer constrained by the SC, the process is now blind to chromosomes, abolishing CO interference, the obligate CO, and the male-female CO difference that is imposed by different SC lengths [START_REF] Capilla-Pérez | The synaptonemal complex imposes crossover interference and heterochiasmy in Arabidopsis[END_REF][START_REF] Durand | Joint control of meiotic crossover patterning by the synaptonemal complex and HEI10 dosage[END_REF]73].

(vi) Disturbing the integrity of the SC, through diminishing the amount of SC protein or removal of SC proteins, allows for more crossovers to form per chromosome [START_REF] Libuda | Meiotic chromosome structures constrain and respond to designation of crossover sites[END_REF][START_REF] Hurlock | Identification of novel synaptonemal complex components in C. Elegans[END_REF][START_REF] Köhler | Dynamic molecular architecture of the synaptonemal complex[END_REF]. This could be due to the disruption of ZHP-3 diffusion along chromosomes that would prevent the ultimate siphoning of all proteins into a single focus.

Open questions

The coarsening model provides an intuitive basis for crossover interference, but many questions need to be addressed before it can be regarded as a convincing and comprehensive model:

-How is HEI10 diffusion constrained to the SC? The SC must have an affinity for HEI10, so as not to lose these molecules to the nucleoplasm. Multiple initial HEI10 foci are situated in the central part of the SC [START_REF] Reynolds | RNF212 is a dosage-sensitive regulator of crossing-over during mammalian meiosis[END_REF][START_REF] Capilla-Pérez | The synaptonemal complex imposes crossover interference and heterochiasmy in Arabidopsis[END_REF], suggesting a direct or indirect affinity for the N-terminus of the transverse element protein or proteins of the central element [START_REF] Ur | Architecture and Dynamics of Meiotic Chromosomes[END_REF]. Interestingly, HEI10 and ZHP-4 RING domains (in S. macrospora and C. elegans, respectively) are required for HEI10/ZHP-3 loading on chromosomes [START_REF] De Muyt | E3 ligase Hei10: A multifaceted structure-based signaling molecule with roles within and beyond meiosis[END_REF][START_REF] Nguyen | C. elegans ZHP-4 is required at multiple distinct steps in the formation of crossovers and their transition to segregation competent chiasmata[END_REF], suggesting an important role for post-translational modifications in the loading of HEI10 on the SC. It should also be noted that Hei10 diffusion and coarsening has not yet been observed in real time in vivo, due to the inherent difficulty to track individual small recombination foci in a living organism for hours at a time. Leveraging the recent advances of gentle super-resolution microscopy will provide important information about HEI10 behavior during crossover formation.

-What drives coarsening? HEI10 forms foci that grow with time, suggesting that HEI10 has some effective, as yet undescribed self-association properties. Up to now, the effective coarsening of HEI10 (i.e., some foci growing while other shrink) in real time has not been observed, due to technical challenges. To trigger coarsening, larger HEI10 foci should have a stronger affinity for HEI10, thus outcompeting smaller ones. This could be explained by phase separation or, perhaps more likely, by a catalytic activity provided by HEI10 itself and/or associated proteins promoting post-translational modifications of HEI10 and/or associated proteins [START_REF] Zhang | Crossover patterning through kinase-regulated condensation and coarsening of recombination nodules[END_REF][START_REF] Rog | The synaptonemal complex has liquid crystalline properties and spatially regulates meiotic recombination factors[END_REF]. Along those lines, CDK2 phosphorylation activity in C. elegans is required for the aggregation of ZHP-3 [START_REF] Haversat | Robust designation of meiotic crossover sites by CDK-2 through phosphorylation of the MutSγ complex[END_REF].

-What triggers the maturation of a given HEI10 focus into a CO-designated site (i.e., MLH1/COSA-1-positive)? One possibility is the size of the focus, but the mechanism responsible remains unknown.

-What stops the coarsening process? If unstopped, the coarsening process would lead to a single focus and a single CO per chromosome (supplemental movies). However, 2-3 COs are typically formed per chromosome in many species. Thus, in these species, the coarsening process must be stopped before completion, presumably by triggering desynapsis and progression to the next step of meiotic prophase (diplotene). The current mathematical implementations of the model presume a fixed time, after which the coarsening is stopped. An attractive alternative possibility is the existence of a checkpoint that would trigger desynapsis, interrupting HEI10 diffusion and, therefore its coarsening, when satisfied. The checkpoint may depend on the maturation of the first HEI10 foci into CO-designated sites.

-What is the relationship between HEI10 foci and DNA recombination intermediates? For each focus to make a CO it must embed a DSB repair intermediate compatible with CO formation (a double Holliday junction). We could hypothesize that the multiple initial HEI10 foci form at DSB repair sites. This is not the most plausible model, however, as HEI10 foci outnumber the estimated numbers of DSB sites in some species [START_REF] Capilla-Pérez | The synaptonemal complex imposes crossover interference and heterochiasmy in Arabidopsis[END_REF]. Moreover, ZHP-3 and RNF212 loading onto synapsed chromosomes is independent of DSB formation [START_REF] Reynolds | RNF212 is a dosage-sensitive regulator of crossing-over during mammalian meiosis[END_REF][START_REF] Jantsch | Targeted gene knockout reveals a role in meiotic recombination for ZHP-3, a Zip3-related protein in Caenorhabditis elegans[END_REF] and they also load onto DNA-free poly-complexes [START_REF] Rog | The synaptonemal complex has liquid crystalline properties and spatially regulates meiotic recombination factors[END_REF][START_REF] Köhler | Dynamic molecular architecture of the synaptonemal complex[END_REF]. Instead, the initial HEI10 loading could depend solely on the tripartite SC, and recombination intermediates would locally favor the subsequent coarsening process. RNF212 localizes to DSB repair sites in the absence of the SC, suggesting an affinity of this family of E3 ligases for recombination intermediates [START_REF] Reynolds | RNF212 is a dosage-sensitive regulator of crossing-over during mammalian meiosis[END_REF]. ZMM proteins, such as Msh4/5 or Zip2-Spo16, can bind recombination intermediates and could in turn attract HEI10 [START_REF] Pyatnitskaya | Crossing and zipping: molecular duties of the ZMM proteins in meiosis[END_REF][START_REF] Pyatnitskaya | The Zip4 protein directly couples meiotic crossover formation to synaptonemal complex assembly[END_REF].

-What are the targets of HEI10 E3 ligase activity? And what are the roles of these targets in coarsening and recombination? Cytologically, SUMOylation of the chromosome axes is partially dependent on HEI10 and RNF212, in Sordaria and mouse respectively [START_REF] De Muyt | E3 ligase Hei10: A multifaceted structure-based signaling molecule with roles within and beyond meiosis[END_REF][START_REF] Rao | A SUMO-ubiquitin relay recruits proteasomes to chromosome axes to regulate meiotic recombination[END_REF].

-Could the coarsening model coexist with other mechanisms of interference? One attractive possibility is that some mechanisms could act at relative short distances, while the coarsening mechanism would superimpose interference at longer ones.

Both DSB interference [START_REF] Garcia | Tel1ATM-mediated interference suppresses clustered meiotic double-strand-break formation[END_REF] and stress-mediated interference [38,[START_REF] Kleckner | A mechanical basis for chromosome function[END_REF] could impose a first layer of interference, preselecting recombination intermediates, among which a minority will be further selected through the coarsening process to become COs.

In species with a large excess of DSBs and in which interference acts at long distances (i.e., half a chromosome, Table 1), the contribution of DSB interference is probably minor. In species with a low CO/DSB ratio [START_REF] Serrentino | The spatial regulation of meiotic recombination hotspots: Are all DSB hotspots crossover hotspots?[END_REF] and/or short interference range, this contribution may be more important.

Answering these questions, which will either challenge or support the coarsening model, will require the combination of genetics, advanced microscopy, biochemistry, and modeling, which promise exciting lines of research. [Perspectives]

1.

Meiotic crossovers shuffle genetic information between generations in eukaryotes. Because of crossover interference, they tend to form away from each other along chromosomes through an elusive mechanism.

2.

An emerging model proposes that the coarsening of a conserved procrossover factor drives crossover distribution and interference.

3.

While the model accounts for numerous observations, many aspects need to be further explored.

Figure 1 .

 1 Figure 1. Cytological behavior of Hei10 homologs in Arabidopsis thaliana, Caenorhabditis elegans and Sordaria macrospora. A. HEI10 first appears along chromosomes as multiple small foci (STED in Arabidopsis, in between axes marked by REC8, see inset) and/or a continuous signal (confocal microscopy in C. elegans and wide field microscopy in Sordaria with axes marked by Spo76/Pds5-TdTomato). B. As meiosis progresses, bigger HEI10 foci form while others diminish in size. C. Toward the end of meiotic prophase, a limited number of large HEI10 foci remain that mark CO sites, and which colocalize with MLH1 in Arabidopsis and COSA-1 in C. elegans. Scale bars 2 µm. Credits: S. Durand for A. thaliana, S. Köhler for C. elegans, C. Girard for S. macrospora.

Figure 2 .

 2 Figure 2. Measuring crossover interference.There are two standard ways of measuring interference from the positions of CO events along a chromosome. Both methods can be applied on cytological (A) or genetic data. B. In the first method, the distribution of distances between events (purple d in A) is plotted. In the presence of interference, the distribution is shifted to higher values (purple) compared to the expected in the absence of interference (grey). The parameter n of the best fit gamma distribution is used as a measurement of interference. C. In the second method, chromosomes are divided into intervals and the Coefficient of Coincidence (CoC) is calculated for all possible pairs of intervals. The CoC values are plotted against the distance between the two intervals (blue L in A). In the presence of interference, the CoC curve is close to 0 for short distances (blue dotted line), while in the absence of interference, the curve is flat at 1 (grey dotted line).

Figure 3 .

 3 Figure 3. The coarsening model. A. Schematic of crossover formation within the SC as envisioned by the coarsening model B. A mathematical description of the coarsening model. The HEI10 concentration %(', )) is defined along the entire SC of length !. The first equation describes its time evolution by diffusion (first term) and exchange with + HEI10 foci at positions ' ! (second term). The second equation describes the evolution of the foci sizes -! , which grow by taking up HEI10 from the SC when the equilibrium concentration .-! "# is smaller than the concentration %(' ! ) on the SC. After initializing the system with many small foci, coarsening ensues, since larger foci exhibit a smaller equilibrium concentration, so fewer, larger foci remain after the finite simulation time $. C. Graphical output of a simulation showing the distribution and size of HEI10 foci for one chromosome after 1 min, 10 min, 1 hour, 5 hours and 10 hours of the coarsening process. See also supplemental video 1 to observe the complete process on five chromosomes of different sizes, and additional simulation replicates in supplemental movies 2 & 3. D. Kymograph corresponding to the simulation in C. Shades of colors represent HEI10 density at each position. From the many foci initiated at time 0, the coarsening process yields a few large foci while the others vanish.

Acknowledgements: We thank Marcel Ernst and Wayne Crismani for their helpful comments and Neysan Donnelly for proofreading the manuscript. We are grateful to Stéphanie Durand, Simone Köhler, Benjamin Pigeard and Karine Budin for the cytology images in Figure 1, and to Andrew Lloyd his help in designing Figure 3. Funding: D.Z. acknowledges funding from the European Union (ERC, EmulSim, 101044662). CG acknowledges funding from the French Agence National de la Recherche (ANR-20-CE20-0007). RM thanks the Max Planck society for core funding. Bibliography

Table 1: Distances at which interference acts in different species. Distances are rounded approximations from experimental data hereafter. For S. cerevisiae: we approximated the distance at which interference has an effect from the distance at which the CoC curve of Zip3 foci reaches 1 [38] and at the peak of the distribution of inter-CO distances detected genetically [START_REF] Mancera | High-resolution mapping of meiotic crossovers and non-crossovers in yeast[END_REF]. For S. macrospora, we used the distance at which the CoC curve of HEI10 foci reaches 1 [START_REF] De Muyt | E3 ligase Hei10: A multifaceted structure-based signaling molecule with roles within and beyond meiosis[END_REF]. For C. elegans, we used the peak of the distribution of inter-CO distances (marked by COSA-1) on the mnT12 fusion chromosome [START_REF] Libuda | Meiotic chromosome structures constrain and respond to designation of crossover sites[END_REF]. For M. musculus, we used the distance at which the CoC curve of Mlh1 foci [START_REF] Anderson | Distribution of Crossing Over on Mouse Synaptonemal Complexes Using Immunofluorescent Localization of MLH1 Protein[END_REF][START_REF] Wang | Crossover patterns under meiotic chromosome program[END_REF] and of genetic COs [START_REF] Petkov | Crossover interference underlies sex differences in recombination rates[END_REF] reaches 1. For A. thaliana, we used the peak of the distribution of inter-CO distances (marked by HEI10 [START_REF] Morgan | Diffusion-mediated HEI10 coarsening can explain meiotic crossover positioning in Arabidopsis[END_REF]) and the distance at which the CoC curve of genetic COs reaches 1 [START_REF] Durand | Joint control of meiotic crossover patterning by the synaptonemal complex and HEI10 dosage[END_REF]. For S. lycopersicum, we used the distance at which CO pairs (MLH1 foci) are as frequent as expected [START_REF] Anderson | Combined fluorescent and electron microscopic imaging unveils the specific properties of two classes of meiotic crossovers[END_REF].


domain: chim spi

text:
Jakob Langenbach

Camille Bakkali-Hassani

François Tournilhac

Sophie Norvez
email: sophie.norvez@espci.fr

FAST RETURN AND FULL ELASTIC RECOVERY IN SELF-HEALING ELASTOMERS FOR SOFT ROBOTIC ACTUATION

Keywords: Self-Healing Elastomers, Epoxidized Natural Rubber, Fibre Strain Sensor, Stiffness Mismatch, Soft Robotics

Soft robotics is an emergent branch of robotics, which involves incorporation of elastomeric elements. Addition of soft materials into robotic devices allows better flexibility and adaptability to the environment than when robots are made of hard and rigid components. However, the soft materials are more prone to damage than their rigid counterparts. Incorporation of self-healing materials into soft robots may permit to fix damages. Equipped with a sensor, the robot could also be able to detect the damage to initiate the repairing process autonomously. In this work, fibre strain sensors were integrated into self-healing elastomers. It is shown that the stiffness of the soft matrix and the sensor fibre should be close enough to avoid a premature rupture of the fibre due to shear stress at the matrix-fibre interface. Elastomeric matrices based on epoxidized natural rubber (ENR) capable of healing after damage were produced. For damage and deformation detection, the elastomer was filled with conductive carbon black to produce piezo-resistive strain sensors. Laser-cut sensor fibres were implemented into a compression sensor. The resulting actuator exhibited fast elastic return and recovery of mechanical and electrical properties after cut and self-healing.

INTRODUCTION

Soft robotics involves incorporation of elastomeric elements into robotic devices. To protect the soft materials from irreversible damage, incorporation of self-healing (SH) materials is desirable. In addition, the robot should be able to detect the damage to repairing itself autonomously [START_REF] Terryn | A review on self-healing polymers for soft robotics[END_REF]. In this work, fibre strain sensors were integrated into self-healing elastomers. It is shown that the stiffness of the soft matrix and the sensor fibre should be close enough to avoid a premature rupture of the fibre due to shear stress at the matrix-fibre interface. Moreover, mechanical actuation should be fast enough to permit large development of automatic processes. A drawback of common self-healing rubbery materials is the slowness of the material answer at room temperature, when healing is strongly associated to dissipation processes around the glass transition temperature of the elastomer. Natural rubber materials are promising candidates to overcome this problem, as their glass transition temperature lays far below room temperature. Elastomeric matrices based on epoxidized natural rubber (ENR) capable of healing after damage were produced here, and filled with conductive carbon black to produce piezo-resistive strain sensors for damage and deformation detection. Laser-cut sensor fibres were implemented into compression sensor exhibiting fast elastic return.

METHODS, RESULTS AND DISCUSSION

Stiffness matching in epoxy-vitrimers containing strain sensor fibre

Strain sensors consisting of an elastomeric material containing conductive fillers were introduced between autonomously self-healing H-bonding epoxy-acid matrices by sandwich welding at moderate temperature (Figure 1). The sensorized matrices were tested under cyclic tensile experiments (0-20% strain), measuring simultaneously the tensile stress and the electrical resistance of the sensor fibre. The experiments were conducted at 50°C, far enough from the Tg around 12-16°C, as the speed of elastic return is highly influenced by the distance from Tg. Stiffer samples containing 10 mol% epoxy excess (called C_10epoxy) were also synthesized for comparison with the stoichiometric sample called C_0epoxy. Figure 2 displays 20 cycles of tensile stress and relative resistance, which is the ratio of the change in resistance (R-R0) divided by the resistance at rest (R0):

Relative Resistance 0 0 R R R  
At room temperature (not shown), all samples show buckling during cyclic tensile loading due to the vicinity of glass transition, but the sensor fibres withstand the deformation without break.

Around room temperature, stiffness of matrix and fibre are very close. At 50°C, the matrix gets softer whereas the stiffness of the strain sensor remains constant. During cyclic testing at 50°C, the sensor fibre breaks in the matrix after three tensile cycles, as shown by the dashed line of the relative resistance increasing abruptly in Figure 2A. Conversely, the sensor fibre in the stiffer sample C_10epoxy (Figure 2B) shows a linear response with only a small drift along all 20 tensile cycles, with no rupture of the sensor fibre inside the matrix. The conductive filler reinforces strongly the strain sensor which is relatively stiff for an elastomer, which results in a stiffness mismatch between the elastic matrix and the integrated sensor fibre. The mismatch induces shear stresses at the fibre-matrix interface which may lead to the rupture of the sensor fibre inside the matrix. The sensor rupture appears in the shoulder region of the dog-bone samples, as shown by image taken in unpolarized light (Figure 3A,3B). Observations under polarized light show areas of strong shear stress especially in varying crosssection regions, leading to fibre rupture (Figure 3C,3D). Model strength calculations within areas of strong changing cross-section show that small stiffness changes can significantly decrease the shear stress to prevent a fibre rupture [START_REF] Langenbach | Adhesion and stiffness matching in epoxy-vitrimers/strain sensor fiber laminates[END_REF]. showing areas of high stress concentration due to extensive deformation.

Fast return and self-healing in crosslinked ENR

A better match between fibre and matrix, and then an enhanced adhesion, should be obtained using same materials for both components. Besides, fast elastic response is expected from elastomer having low Tg, as typically natural rubber derivatives. Indeed, in the vicinity of glass transition, viscous dissipations are important and samples recover their dimensions quite slowly [START_REF] Sordo | Design of selfhealing supramolecular rubbers with a tunable number of chemical cross-links[END_REF]. Such dissipation effects are apparent in the shape of the tan  curve (DMA). Figure 4 compares broad tan  curves of various H-bonding epoxy-acid matrices exhibiting self-healing behaviour at room temperature [START_REF] Langenbach | Adhesion and stiffness matching in epoxy-vitrimers/strain sensor fiber laminates[END_REF] and curves of rubbery matrices based on epoxidized natural rubber, showing low Tg and strong elastomeric properties at room temperature. We previously showed that epoxidized natural rubber (ENR) could be efficiently cross linked using dicarboxylic acids DA, reacting with the epoxy functions of ENR (Figure 5) [START_REF] Imbernon | Semi-interpenetrating networks in blends of epoxidized natural rubbers[END_REF]. By varying the nature of the main chain of DA, reprocessable materials were obtained, for instance using dithiodibutyric acid (DTDB) as a crosslinker (Figure 6A) [START_REF] Imbernon | Chemically crosslinked yet reprocessable epoxidized natural rubber via thermo-activated disulfide rearrangements[END_REF]. In this work, we demonstrate that a self-healing ENR10 crosslinked with H-bonding DA derivatives exhibits fast elastic response, using dynamic tensile tests done in a Bose machine at a frequency of 5 Hz with a strain amplitude of 6-30% (Figure 6B). The drift of the signal might be related to the presence of defects (dangling chains, sol molecules), introduced together with the H-bonding moieties which promote SH [START_REF] Langenbach | Adhesion and stiffness matching in epoxy-vitrimers/strain sensor fiber laminates[END_REF].

Implementation in smart devices

Experiments were conducted to prospect an application of cross linked ENR10 containing conductive particles as a pressure sensor. Multi-lined shape was laser cut from a cross linked sheet and connected to wires. The sensor, put on the top of a human hand, reveals fast answer under cyclic pressure of a finger (Figure 7). As for applications in robotic devices, assemblies of sensorized self-healing pneumatic finger are currently under investigation to perform fast cyclic movements. The actuator is composed of a self-healing material based on ENR. Strain sensor fibres integrating carbon black particles for conductivity are glued to an ENR finger-like sample to build a piezo-resistive sensor fibre actuator. As fibre and finger consist of the same self-healing ENR material, both adhesion and fast elastic response are insured.

CONCLUSIONS AND OUTLOOK

When integrating fibre strain sensors into soft elastomers, stiffness of sensor fibre and matrix should be close enough to avoid a rupture of the fibre due to shear stress at the matrix-fibre interface. Moreover, when based on epoxidized natural rubber, fiber and matrix capable of healing after damage were produced, exhibiting fast elastic return and recovery of mechanical and electrical properties after cut and self-healing. Assemblies of sensorized self-healing pneumatic finger are currently under implementation for applications in soft robotic devices.

Figure 1 :

 1 Figure 1: Fibre strain sensors were integrated into self-healing elastomers to measure sample deformation (after [2]).

Figure 2 :

 2 Figure 2: Cyclic tests at 50°C (20 cycles at 0-20% strain) at a deformation rate of 50%.min -1 . Solid lines: stress values; dashed lines: relative resistance for A) stoichiometric sample C_0epoxy; B) stiffer sample C_10epoxy.

Figure 3 :

 3 Figure 3: A) Image of stretched sample showing spots of fracture of the sensor fibre; B) Zoom-in on the point of rupture in shoulder region observed with a binocular magnifier; C, D) A sample bearing with 1 N weight observed in polarized light between crossed polarizers oriented at 45° or parallel to the stretching direction, showing areas of high stress concentration due to extensive deformation.

Figure 4 :

 4 Figure 4: Dissipation factor tan δ measured in film-tension geometry (DMA) for A) various H-bonding epoxyacid matrices [2]; B) ENR matrices containing 50, 25, 10% epoxy content.

Figure 5 :

 5 Figure 5: Crosslinking of ENR using dicarboxylic acids DA (after [4]). Main chain of DA may vary (ex: DTDB contains reversible disulphide bonds).

Figure 6 :

 6 Figure 6: A) Recycling experiments with SH ENR: Tensile test before and after the recycling experiment (black: ENR25-DTDB; grey: ENR25-DA; dashed: before curing; line: after curing) (after [5]); B) Fast return within SH ENR10: 100 tensile cycles at 5 Hz frequency after 1 min of self-healing at room temperature.

Figure 7 :

 7 Figure 7: Multi-lined compression strain sensor laser-cut from ENR sheet embedding carbon black particles used as a pressure sensor.

ACKNOWLEDGEMENTS

This work was funded by the European Union's Horizon 2020 FET Open Project Self-Healing Soft Robotics [grant agreement no. 828818]. The authors thank ESPCI Paris-PSL, CNRS, for the supportive financial environment. Frank Clemens (EMPA Zurich) is warmly thanked for fruitful discussions.


Jean-François Olivieri

ab James T Hynes

Damien Laage

Water dynamics and sum-frequency generation spectra at electrode/aqueous electrolyte interfaces

The dynamics of water at interfaces between an electrode and an electrolyte is essential for the transport of redox species and for the kinetics of charge transfer reactions next to the electrode. However, while the effects of electrode potential and ion concentration on the electric double layer structure have been extensively studied, a comparable understanding of dynamical aspects is missing. Interfacial water dynamics presents challenges since it is expected to result from the complex combination of water-water, water-electrode and water-ion interactions. Here we perform molecular dynamics simulations of aqueous NaCl solutions at the interface with graphene electrodes and examine the impact of both ion concentration and electrode potential on interfacial water reorientational dynamics. We show that for all salt concentrations water dynamics exhibits a strongly asymmetric behavior: it slows down at increasingly positively charged electrodes but it accelerates at increasingly negatively charged electrodes. At negative potentials water dynamics is determined mostly by the electrode potential value, but in contrast at positive potentials it is governed both by ion-water and electrode-water interactions. We show how these strikingly different behaviors are determined by the interfacial hydrogen-bond network structure and by the ions' surface affinity. Finally, we indicate how the structural rearrangements impacting water dynamics can be probed via vibrational sum-frequency generation spectroscopy.

Introduction

Interfaces between an aqueous electrolyte and an electrode play a key role in a broad range of fundamental chemical processes including, e.g., energy conversion via photocatalytic water splitting, energy storage in batteries, and carbon dioxide reduction. Accordingly, a major effort, both experimental and theoretical, is being devoted to gaining a better comprehension of these interfaces 1,2 . Despite important advances, key gaps nonetheless remain in our molecular-level understanding of such interfaces' structure and dynamics.

Here we focus on two key questions. The first pertains to the structure of the electrical double layer formed at the electrode and how it changes with the applied electrode potential. Textbook descriptions of electrode/electrolyte interfaces traditionally rely on mean-field continuum pictures due to Hemholtz, Gouy, Chapman and Stern 3 . While these have been valuable guides for electrochemistry, it is well known that they miss important aspects; these include the discrete nature of water molecules and ions, ion correlations, specific molecular interactions and nonlocal effects 1 . Vibrational sum-frequency generation (SFG) speca PASTEUR, Department of Chemistry, École normale supérieure, PSL University, Sorbonne Université, CNRS, 75005 Paris, France. Email: damien.laage@ens.psl.eu b Department of Chemistry, University of Colorado, Boulder, Colorado 80309, United States troscopy offers an incisive experimental approach to selectively probe the interface and the orientational arrangement of water molecules there 1,[START_REF] Shen | [END_REF]5 . However, interpreting these spectra can be ambiguous: the respective contributions from the first so-called Stern layer and the diffuse layer are difficult to disentangle 1,[6][7][8] .

The second question deals with the electrode's impact on the electrolyte dynamics. While most studies so far have focused on structural aspects, the dynamics of interfacial water molecules has been much less examined, despite its essential role for the transport of redox species to and away from these interfaces and for the kinetics of interfacial charge transfer reactions.

The structure of the electrical double layer at charged surfaces has been extensively studied with both SFG spectroscopy experiments and molecular simulations using charged silica surfaces (see, e.g. refs. 5,[9][10][11][12][13][14] ). These studies have investigated, e.g., how the double layer structure varies with surface charge and the nature of the salt. However, a complexity arises here, since changing the silica surface charge is achieved by shifting the pH, which also implies changing the interface's chemical properties.

An alternate, and particularly attractive, prototypical electrified/aqueous solution interface is provided by graphene electrodes. In addition to its promising technological applications, graphene combines several important advantages: i) due to its semi-metallic nature, an external electric potential can be applied to control its surface charges; ii) buried water-electrode interfaces are challenging for SFG experiments, while the IR-transparent graphene sheet allows more incoming IR beam transmission than does a metal, iii) it provides a well-defined, atomically flat, interface. The feasibility of this graphene-based approach was recently demonstrated in pioneering SFG experiments on interfaces between electrified graphene and neat water [15][16][17][18][19][20] . In addition, in molecular simulations by some of us 21,22 , characterization of the dynamics of water molecules in contact with an electrified graphene interface showed that interfacial water dynamics is markedly different at positive and negative electrodes. While water reorientation dynamics is increasingly slowed down as the magnitude of an applied positive electrode potential increases, it changes in a non-monotonic fashion with the potential at negatively charged electrodes.

However, these just-described experimental and simulation results were obtained in the neat water situation; the key question of the impact of a salt on the interfacial properties at different electrode potentials thus remains unaddressed.

The present first effort to deal with this lacuna focuses on the paradigm interface between electrified graphene and NaCl aqueous solutions. We use molecular dynamics simulations to determine how the structure and dynamics of the interfacial water layer is affected by the electrode potential and by the salt. Further, we indicate how the combined impacts of the electrode and the ions on the interfacial structure are reflected in experimentally-accessible SFG spectra.

In the remainder of this contribution, we first describe our simulation methodology and determine the respective salt and electrode impacts on the interfacial water reorientation dynamics. We then characterize the structure of the electrical double layer and the orientation of the interfacial water molecules as salt is added and as the electrode potential is increased. Next, we calculate the SFG spectra and determine the interfacial and diffuse layer contributions. Finally, we offer some concluding remarks.

Methodology

Our simulation methodology extends the procedure described in our prior studies 21,22 of neat water/electrified graphene interfaces to include Na + Cl -ions at concentrations ranging from 0.1 to 5 mol/L. We performed classical molecular dynamics simulations of a slab of water molecules confined between two parallel single graphene plates separated by a 60 Å distance. The rigid graphene sheets are constructed from an ideal hexagonal lattice with a 1.42 Å distance between neighboring carbon atoms and each 240-atom plate is periodically replicated in the two directions parallel to the graphene plane. The resulting simulation box dimensions are 24.58×25.54×60.00 Å 3 . At each salt concentration, the number of water molecules is adjusted by an insertion/deletion procedure to match the experimental aqueous solution density 23,24 . The numbers of water molecules and of Na + Cl - ion pairs are respectively 1152/0 in neat water, 1152/2 at 0.1 mol/L, 1144/10 at 0.5 mol/L, 1133/21 at 1 mol/L, 1110/41 at 2 mol/L, 1083/62 at 3 mol/L, 1057/83 at 4 mol/L and 1033/104 at 5 mol/L. Water is described by the SPC/E potential 25 which has been shown 26 to provide an excellent description of water dynamics at ambient temperature. Ions are modeled by a scaled-charge ECCR force field which was shown 27 to provide a good implicit description of polarizability effects, and especially of their impact on water dynamics. Graphene carbon Lennard-Jones parameters are taken from ref. 28 where they were optimized to reproduce DFT-based molecular dynamics simulation results. The carbon-ion dispersive Lennard-Jones terms are scaled by the same 0.7 factor as suggested in ref. 28 for the carbon-water oxygen interaction. Long-range electrostatic interactions are described via a two-dimensional Ewald summation method with the Yeh-Berkowitz slab correction 29 . Opposite electrostatic potentials are applied to the two graphene plates. In order to maintain the electrostatic potential at each electrode atom fixed, the graphene carbon atom charges are allowed to fluctuate. We employ the constant potential 30,31 method which has already been successfully used for a broad range of electrode/electrolyte interfaces. Graphene carbon charges are described by Gaussian distributions whose amplitudes are determined at every step, using the constant potential implementation of ref. 32 . The Gaussian inverse width is fixed to be 2.55 Å -1 , as suggested in ref. 33 for carbon atoms.

Molecular dynamics simulations were run with LAMMPS 34 for a series of electrode potentials ranging from 0 to ±2.5 V and for a series of salt concentrations from 0 to 5 mol/L. While similar electrode potentials have been used in experiments on neat water/electrified graphene interfaces 15 , large voltages may cause electrode/electrolyte electron transfers in the presence of ions and they should only be considered to highlight the trends in our simulations. After an initial 1 ns-long equilibration, trajectories were propagated for 1 ns in the canonical ensemble at 298.15 K using a Nose-Hoover thermostat. Configurations were collected every 50 ps along this thermostatted trajectory. Our analyses are based on the independent microcanonical trajectories that were then propagated for 500 ps from these configurations, using a 10-fs output interval. All error bars reported in the following are 95% confidence intervals.

Phase-resolved vibrational SFG spectra were calculated using the empirical map developed in ref. 35 to connect the water OH bond dipole and polarizability tensor to the instantaneous electric field experienced by the water H atom along the OH direction. Our calculated spectra correspond to these that would be measured using a dilute HOD in a liquid D 2 O solution, in order to decouple the local OH mode from other OD stretch modes in the liquid. The switching function suggested in ref. 36 and dependent on the distance between the oxygen atom and the middle of the water slab is applied to determine the SFG spectra at each electrode separately.

Interfacial water dynamics

Water reorientation dynamics

We first characterize the dynamics of water molecules at the electrode interface and how it is impacted by the salt and the electrode potential. We focus on water OH group reorientation dynamics and calculate the reorientation time-correlation function relevant for ultrafast infrared anisotropy and NMR measurements

C OH 2 (t) = P 2 [u OH (0) • u OH (t)] (1)
where u OH (t) is the unit vector along the water OH group direction, P 2 is the second-order Legendre polynomial and the ... ensemble average is performed over all water OH groups within the first layer at t=0 (defined to include all water molecules whose oxygen atom is within 5 Å from the graphene plane 22 ). We then determine the reorientation time τ OH 2 as the time integral of C OH 2 (t). In prior simulation studies, we have already separately considered the impact of ions on water dynamics in the bulk (see, e.g., refs. 27,37,38 ) and the impact of an electrode on neat water dynamics (see, e.g., refs. 21,22 ). We now aim at determining how water dynamics is affected by the simultaneous combination of ions and of a charged electrode interface.

Impact of electrode potential

We consider how ions affect the way interfacial water dynamics changes with the electrode potential. Figure 1 contrasts the interfacial water reorientation times without ions and with 5 mol/L NaCl for a series of electrode potentials. The most striking result visible in Fig. 1 is that although this salt does not substantially affect interfacial water dynamics at negatively charged electrodes, it does have a large impact at positively charged electrodes.

Here it is useful to recall the neat water dynamical behavior that had been revealed in our recent studies 21,22 . In neat water, positive electrode potentials induce a large slowdown in water dynamics, as shown in Fig. 1. In contrast, increasingly negative potentials cause an acceleration of water dynamics, followed by a retardation for very negative potentials. The electrode's impact on water reorientation dynamics was determined to be shortranged 21 and Fig. 1 shows this also holds here: τ OH 2 in the middle of the water slab remains bulk-like and is almost insensitive to the electrode potential. for water OH groups initially in the first interfacial layer without salt (blue circles) and in the presence of 5 mol/L NaCl (red squares) for a series of electrode potentials. Grey symbols show the reorientation times of water OH groups in the middle of the slab, without salt (circles) and with 5 mol/L NaCl (squares).

In the presence of 5 mol/L NaCl, we first discuss the 0 V situation. Surprisingly, the introduction of salt has no significant effect on water dynamics in the middle of the slab. In contrast, in bulk solutions NaCl is known to cause a moderate slowdown of water dynamics 27,38 . For example, simulations using the same water and ion force field had shown 27 that in bulk 5 mol/L NaCl aqueous solution, τ OH 2 is slowed by 1.2 with respect to the neat water situation. In contrast, at the interface the salt increases the water reorientation time by a 1.4 factor with respect to the neat water situation.

We now turn to the impact of electrode potentials. For increasingly negative electrode potentials, the presence of salt does not change the behavior of interfacial water dynamics. We observe essentially the same acceleration for moderately negative potentials and slowdown for very negative potentials as in the neat water case: the τ OH 2 values are only slightly modified and the acceleration relative to the 0 V case is only somewhat enhanced by the salt. In contrast, for increasingly positive electrode potentials, the presence of salt substantially enhances the slowdown of interfacial water reorientation dynamics. For the largest electrode potentials considered here, τ OH 2 is almost twice as slow in the presence of salt than in its absence.

Impact of salt concentration

Next, we examine the impact of increasing salt concentration on interfacial dynamics at fixed electrode potentials, here chosen to be ± 0 V and ± 2.5 V. At the negative electrode, increasing the salt concentration causes a very moderate slowdown of interfacial water dynamics, and this retardation almost vanishes when the electrode potential absolute value increases (Fig. 2a). In contrast, at the positive electrode, adding more ions induces a pronounced slowdown, which markedly increases with increasing electrode potential (Fig. 2b).

a b

Fig. 2 Water reorientation time τ OH 2 for water OH groups initially in the first interfacial layer for increasing NaCl concentration at 0 V (blue) and at ±2.5 V (red) a) at the negatively charged electrode and b) at the positively charged electrode. Grey symbols show the values in the middle of the slab, without salt (circles) and with 5 mol/L NaCl (squares).

Structure of interfacial layer

Our prior studies on a neat water slab between graphene electrodes had revealed the importance of interfacial water orientation and hydrogen-bond (H-bond) exchanges in determining water reorientation dynamics 21,22 . In order to provide a molecular interpretation to the above observations, we now examine the orientational arrangement of interfacial water molecules and the structure of the electric double layer.

Orientation of interfacial water molecules

To characterize the orientational arrangement of water molecules at the graphene electrode, we calculate the probability distribution of the angle θ between each water OH group and the vector normal to the interface oriented towards the nearest graphene plate and of the distance ∆z between the water H atom and the nearest graphene electrode (Fig. 3a). The reference situation of neat water next to the 0 V electrode reveals three distinct peaks in the water orientational distribution within the first layer (Fig. 3b). The first one corresponds to a small fraction of OH groups pointing towards the electrode (θ < 40 • , ∆z 2Å). When the electrode is held at 0 V, carbon atoms are approximately apolar and these water OH groups are analogous to dangling OH groups at the air/water interface. The second peak arises from the largest fraction of interfacial OH groups which are tangent to the electrode surface (θ 90 • , ∆z 3.5Å) and which are engaged in a H-bond with another interfacial water molecule. The third population corresponds to OH groups pointing away from the electrode and towards the liquid (θ > 140 • , ∆z 5Å), donating a H-bond to a second-layer water molecule.

In the neat water case, our prior studies 21,22 have shown that the reorientation of interfacial water molecules due to surface charges is more complex than a simple dipole flip 13,15,39 . At the positive electrode the orientational arrangement of interfacial water molecules changes only marginally with the electrode potential (Fig. 3e). In contrast, at the negative electrode a substantial rearrangement occurs (Fig. 3c), which is responsible for the observed acceleration in interfacial reorientation dynamics. Due to the increasingly favorable interaction between water OH group and the graphene electrode when the potential is more negative and the graphene carbon atoms are more negatively charged, the fraction of interfacial water OH groups pointing towards the electrode grows. This is manifest in the change in probability distribution with respect to the 0 V neat water situation shown in Fig. 3c. This structural rearrangement impacts the water dynamics because τ OH 2 reorientation times depend on the OH group initial orientation. For tangent OH groups, the reorientation dynamics is moderately retarded with respect to that in the bulk due to the reduced number of potential H-bond acceptors to which these OH groups can jump and form a new H-bond. In contrast, because OH groups pointing towards the moderately negative electrode are engaged in an interaction which is weaker than that of an H-bond with another water molecule, their reorientation dynamics is facilitated with respect to that in the bulk, and therefore faster (for very negative electrode potentials, the interaction becomes stronger than a water-water H-bond and the reorientation dynamics is slower than in the bulk). The growing fraction of OH groups pointing towards the interface and whose reorientation is fast thus causes the observed acceleration in average interfacial dynamics at moderately negative electrodes.

We therefore examine how the interfacial orientational arrangement is impacted by the addition of salt. Figures 3d and3f present the changes in probability distribution between neat water at the 0 V interface and 5 mol/L NaCl at the -2.5 V and +2.5 V electrodes respectively. They show that at both electrodes the interfacial structural rearrangement in the presence of salt is qualitatively similar to that in the absence of salt (Fig. 3c ande). The negative electrode potential (Fig. 3d) causes an enrichment in the population pointing towards the electrode and a depletion of the H-bonded populations either tangent to the interface or pointing towards the slab, and these population transfers are slightly enhanced by the addition of salt. The positive electrode potential induces a depletion in the population pointing towards the interface. The impact of added salt on interfacial water dynamics at the positive electrode thus does not appear to be caused by a substantial change in the equilibrium between water OH groups pointing towards the interface and towards an H-bond acceptor. However, we note that the most important effect caused by the ions is the strong depletion in water OH groups pointing away from the positively charged electrode (Fig. 3f), and we will investigate its cause and implications in the following.

Ion density profiles

To further characterize the structure of the aqueous solution at the electrode interface, we calculate the density profiles of the Na + and Cl -ions along the axis between the two graphene plates, shown in Fig. 4 for NaCl aqueous solutions of increasing concentrations between electrodes held at ±2.5 V. It reveals a striking difference between the Cl -anions and Na + cations: while the Cl -anions strongly adsorb at the positively charged electrode, the Na + cations display a much weaker affinity for the negative electrode. In addition, these results obtained from molecular dy-namics simulations differ in several important ways from the predictions of mean-field theories. First, the electrical double layer terminology is misleading, since even next to the Cl -anions that strongly adsorb at the electrode no compensating layer of Na + cations forms in the second layer. The second aspect relates to the proposed Stern layer: we see that this description is much more adequate for Cl -anions than for Na + cations. In addition, even for Cl -, this layer should not be described as compact: for the solution of 5 mol/L average concentration, the local Cl - number density at the positive electrode implies that the effective local concentration is approximately 8 mol/L, with anion-anion nearest neighbor distances of 14 Å, much larger than the contact distance. Finally, the diffuse layer is shown to strongly differ from the exponential distribution expected from mean-field models, which again stresses the importance of accounting for specific molecular interactions. for Na + at the -2.5 V electrode.

Interpretation of ions' impact on interfacial water dynamics

The dramatically different behaviors of Na + and Cl -ions thus provide the molecular explanation for the contrasted impacts of ions on interfacial water dynamics (Fig. 1). Adding NaCl does not markedly change water reorientation dynamics at negatively charged electrodes because of the low propensity of ions to adsorb at this interface: water dynamics is thus governed by the water-electrode interactions and accelerated at moderately negative electrode potentials. In contrast, at the positively charged interface, the effective local concentration in Cl -anions is much larger than in the bulk solution, and water dynamics is controlled both by water-electrode and by water-ion interactions. We had shown in prior works 40,41 that water reorientation dynamics proceeds via two pathways: large-angular jumps caused by H-bond exchanges and diffusive tumbling of the local frame between successive H-bond exchanges. In a recent simulation study 27 , we showed that the slowdown caused by NaCl mostly arises from the retardation of the frame component, due to the salt-induced viscosity increase. The local increase in anion concentration at the positive electrode (Fig. 4) thus causes the slowdown of water reorientation seen in Fig. 1. In addition, the presence of Cl - anions in the interfacial layer explains the partial orientational rearrangement of interfacial water molecules shown in Fig. 3f: the depletion in water OH groups pointing towards the slab upon adding salt is due to the reorientation of interfacial molecules in order to donate H-bonds to the Cl -ions. Finally, the preferential adsorption of ions at the interface leads to a decrease in the local salt concentration in the middle of the slab, and explains the smaller effect on water reorientation dynamics in the middle of the slab than in bulk solutions with the same average concentration.

Vibrational sum-frequency generation spectra

We now examine whether the structural rearrangements of water at electrodes in the presence of salt shown here via simulations can be probed experimentally via SFG spectroscopy.

We note that the first SFG spectra of water at electrified graphene interfaces 15 employed a support layer which was recently suggested 20 to significantly affect the measured signal. Alternative setups with a free-standing graphene sheet 17,18 thus offer a promising route to exclusively probe the effect of the electrified graphene on the interfacial water molecules, including in the presence of ions 18 . Our simulated system resembles these free-standing setups, with the notable difference of the obviously shorter distance between the oppositely charged electrodes.

Spectra

We calculate the resonant susceptibility accessible in SFG measurements with heterodyne detection and in the ssp polarization scheme

χ (2) xxz,yyz (ω) = iω k B T ∞ 0 dt e iωt α xx,yy (t)µ z (0) ( 2
)
where α is the water polarizability tensor and µ is the dipole moment, both determined at every step of the simulation from the local electric field using the empirical map developed in ref. 35 .

When salt is added to the solution, the most dramatic impact on SFG spectra at both electrodes is the spectacular disappearance of the H-bonded band centered on 3400 cm -1 (Fig. 5). This band is known to arise from the static electric field created by the electrodes throughout the entire slab, which breaks the local symmetry and induces an SFG response. The progressive screening of this static field by the ions at the interface is thus expected to cause the strong decrease of this band's amplitude. However, closer analysis of the spectra reveals that the salt effect is more complex than a simple screening, since a change in the sign of the SFG susceptibility is discernable for example at the positive electrode for the larger salt concentrations. We therefore successively examine the SFG signals caused by the static field in the diffuse layer and by the interfacial layer.

Signal caused by static field

The long-range, so called χ [START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF] , signal caused by the static field created by charged interfaces has already been amply dis-a b Fig. 5 Calculated phase-resolved SFG spectra of water a) at the -0.5 V electrode and b) at the +0.5 V electrode, both for increasing NaCl concentrations.

cussed 5,6,8,16,[42][43][44][45] and we have shown its importance for neat water at electrified graphene interfaces in prior work 22 . In electrolytes, this signal arises from the diffuse layer and must be subtracted from the collected total SFG signal in order to gain specific information about the interfacial layer, which is key, e.g., for electrochemical reactions. This diffuse layer signal is proportional to the static electric field in the diffuse layer, and determining this field is critical to subtract the diffuse layer's signal. We calculated the electrostatic potential in the cell by solving the Poisson equation following our prior work 22 and Fig. 6a shows that beyond a few molecular layers from the electrode, the slope of the potential, i.e., the field, is constant. However, Fig. 6a further reveals the strong salt-dependent screening of the electrode field by the first few molecular layers. As shown in Fig. 6b, the field in the middle of the cell is much smaller than the external field imposed by the electrodes. A supplementary difficulty is added by the salt's impact which changes with the external field. These results thus show that the static field cannot be obtained from mean-field descriptions and that subtracting the diffuse layer χ [START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF] signal from the experimental spectrum is far from straightforward.

Signal due to interfacial layers

We finally turn to the SFG signal produced by the first molecular layers at the electrode interface. The results in Fig. 7 show that SFG spectra reveal important structural rearrangements caused by the ions, and that these effects are not restricted to the first layer but extend at least to the third molecular layer. We now show that these changes in the SFG signal can be interpreted via the structural analysis of Figs. 34.

At the negatively charged electrode, ions are shown to have a limited impact on the SFG spectra, in line with the small orientational rearrangement of interfacial water molecules in Fig. 3d. The largest change in the SFG signal comes from the second and third layers, due to the reorientation of water molecules caused by the presence of Na + cations in the second layer (Fig. 4).

Changes in the SFG contributions at the positive electrode are more dramatic. Ion-induced spectral changes in the first layer are explained by the orientational rearrangement shown in Fig. 3f: water OH groups that used to point away from the interface (leading to a negative χ (2) ) in the absence of salt reorient to donate a H-bond to Cl -ions present in the first layer, leading to a red- shifted positive contribution that grows with ion concentration and a decrease of the negative-going band. In the second and third layers, the χ (2) signal is observed to change sign at some frequencies. This is again ascribed to the reorientation of water OH groups that increasingly point towards the Cl -ions within the first layer when the salt concentration increases (see Fig. 3f).

Concluding remarks

Our study reveals that the impact of an electrified interface on the interfacial structure and dynamics of an aqueous electrolyte is strongly asymmetric with respect to the electrode potential. This arises from the molecular structure of water; the molecules' orientations are governed by H-bond formation more than by dipolefield interactions, and by the different surface propensities of anions and cations. For NaCl aqueous solutions, our results show that ions have a much larger structural and dynamical impact at the positive electrode due to the Cl -strong surface affinity. We show that the structural rearrangements of the water H-bond network and of the electrical double layer that cause the observed slowdown in water dynamics have an impact on the SFG spectra. However, determination of the SFG signal arising from the interfacial layers is not straightforward. It requires subtraction of the contribution from the diffuse layer: this scales with the static field in the middle of the cell, and due to interfacial screening this field is strongly reduced from the external field. Further studies will examine in detail how the nature of the salt affects the properties of electrode/electrolyte interfaces.

Fig. 1

 1 Fig. 1 Water reorientation time τ OH 2

Fig. 3 a

 3 Fig.3 a) Coordinates used to describe the orientation of interfacial water OH groups: angle θ between a water OH and the surface normal vector and distance ∆z between the water H atom and the nearest graphene plane. Normalized probability distributions b) for a neat water slab at 0 V, c) change in probability for neat water when charging the electrode from 0 V to -2.5 V, d) change in probability when going from neat water at 0 V to 5 mol/L NaCl at the -2.5 V electrode, e) change in probability for neat water when charging the electrode from 0 V to +2.5 V, f) change in probability when going from neat water at 0 V to 5 mol/L NaCl at the +2.5 V electrode.

Fig. 4

 4 Fig.4 Ion density profiles for NaCl aqueous solutions along the distance to the nearest graphene plate, a) for Cl -at the +2.5 V electrode, b) for Cl -at the -2.5 V electrode, c) for Na + at the +2.5 V electrode and d) for Na + at the -2.5 V electrode.

Fig. 6 a

 6 Fig. 6 a) Mean electrical potential profile across the cell between electrodes held at ±0.5 V for solutions of increasing NaCl concentration; b) electric field in the middle of the cell for increasing external field imposed by the electrodes in neat water (blue) and with 1 mol/L (red) and 5 mol/L (green) NaCl (dashes show the external field).

| 1-8

1-8 J o u r n a l N a me , [ y e a r ] , [ v o l . ] ,

J o u r n a l N a me , [ y e a r ] , [ v o l . ] ,

Acknowledgements

This work was supported in part by NSF grant CHE-1112564 (J.T.H.).

Notes and references

Conflicts of interest

There are no conflicts to declare.