Maciej K Mańko
email: maciej.manko@ug.edu.pl

Catriona Munro
email: catriona.munro@college-de-france.fr

Lucas Leclère

Establishing bilateral symmetry in hydrozoan planula larvae, a review of siphonophore early development

Siphonophores are colonial hydrozoans, characterized by directional asymmetric colony organization and unparalleled zooid functional specialization. Recent genomic studies have offered an evolutionary perspective on how this morphological complexity arose, but characterization of symmetry breaking in siphonophores is still largely missing. The advent of culturing methods of siphonophores has opened the possibility to investigate in detail their larval development. Here, bringing together historical data on early development with new immunohistochemical data, we review the diversity of developmental trajectories that leads to the formation of bilaterally symmetric planula larva across the siphonophore phylogeny. We then discuss siphonophore development within the broader, cnidarian context to highlight open questions and set the framework for future work on siphonophores.

Introduction

Siphonophores are a clade of hydrozoans found from the ocean surface to the deep sea that are especially known for their highly integrated, functionally specialized colonial development [START_REF] Totton | A Synopsis of the Siphonophora[END_REF][START_REF] Mapstone | Global Diversity and Review of Siphonophorae (Cnidaria: Hydrozoa)[END_REF]. Siphonophores are widespread and abundant predators in pelagic ecosystems, playing critical trophic roles in the pelagic 'jelly web' [START_REF] Choy | Deep pelagic food web structure as revealed by in situ feeding observations[END_REF][START_REF] Haddock | The evolution of siphonophore tentilla for specialized prey capture in the open ocean[END_REF][START_REF] Hetherington | Integrating siphonophores into marine food-web ecology[END_REF][START_REF] Boosten | Loss of the benthic life stage in Medusozoa and colonization of the open ocean[END_REF]. Unlike other gelatinous zooplankton, siphonophores are typically extremely fragile and are difficult to collect intact using plankton nets, and must be collected by remotely operated vehicle, scuba diving, or from the surface in locations with upwellings where they can be found close to the coast [START_REF] Dunn | Molecular Phylogenetics of the Siphonophora (Cnidaria), with Implications for the Evolution of Functional Specialization[END_REF]. One such location is Villefranche-sur-Mer, where significant progress describing embryonic development in many siphonophore species was made by Claude and Danielle Carré (Carré andCarré 1991, 1993).

Siphonophores are colonial organisms, with each of the bodies within the colony (called zooids) connected to a central stem with a shared gastrovascular cavity. These zooids are considered to be homologous to free-living polyps and medusae, but are physically integrated such that they do not function individually but rather as a colonial 'super-organism' [START_REF] Mackie | Siphonophores, Bud Colonies, and Superorganism[END_REF][START_REF] Totton | A Synopsis of the Siphonophora[END_REF]. Each of these zooids is functionally specialized for particular tasks, including feeding, swimming, and reproducing (for a glossary of specialized siphonophore terms, including the names of each of these zooids and their function, see table 1). Siphonophores have traditionally been divided into three main groups based on colony morphology: Cystonectae, Physonectae, and Calycophorae [START_REF] Totton | A Synopsis of the Siphonophora[END_REF]. Cystonectae, a monophyletic group including the famous Portuguese man of war Physalia physalis, are sister to the clade Codonophora, which includes all other siphonophores (Figure 1A) [START_REF] Dunn | Molecular Phylogenetics of the Siphonophora (Cnidaria), with Implications for the Evolution of Functional Specialization[END_REF][START_REF] Munro | Improved phylogenetic resolution within Siphonophora (Cnidaria) with implications for trait evolution[END_REF]. The Codonophora comprises the clade Calycophorae nested within the paraphyletic 'Physonectae' (Figure 1A) [START_REF] Dunn | Molecular Phylogenetics of the Siphonophora (Cnidaria), with Implications for the Evolution of Functional Specialization[END_REF][START_REF] Munro | Improved phylogenetic resolution within Siphonophora (Cnidaria) with implications for trait evolution[END_REF].

In recent years, significant progress has been made in describing the body plan and development of the 'adult' polygastric colony (=bearing multiple feeding polyps, called gastrozooids) at molecular and morphological levels [START_REF] Dunn | Complex colony-level organization of the deep-sea siphonophore Bargmannia elongata (Cnidaria, Hydrozoa) is directionally asymmetric and arises by the subdivision of pro-buds[END_REF][START_REF] Haddock | A re-examination of siphonophore terminology and morphology, applied to the description of two new prayine species with remarkable biooptical properties[END_REF][START_REF] Dunn | The evolution of colony-level development in the Siphonophora (Cnidaria:Hydrozoa)[END_REF][START_REF] Siebert | Re-evaluation of characters in Apolemiidae (Siphonophora), with description of two new species from Monterey Bay, California[END_REF][START_REF] Siebert | Stem cells in Nanomia bijuga (Siphonophora), a colonial animal with localized growth zones[END_REF]. Within cystonects, zooids arise as single buds along the stem [START_REF] Dunn | The evolution of colony-level development in the Siphonophora (Cnidaria:Hydrozoa)[END_REF], although P. physalis has a more complex colony organization, resulting from the development of a body plan adapted to life at the ocean surface [START_REF] Totton | Studies on Physalia physalis (L.). Part 1. Natural history and morphology[END_REF][START_REF] Munro | Morphology and development of the Portuguese man of war, Physalia physalis[END_REF]. Within Codonophora, most zooids arise from one or two growth zones, starting with a pro-bud that subdivides to give rise to different zooid types [START_REF] Dunn | The evolution of colony-level development in the Siphonophora (Cnidaria:Hydrozoa)[END_REF]. As the stem elongates, these zooids continue to develop and are carried down along the stem; other zooids can arise later along the stem, for example, additional bracts and palpons grow within already differentiated cormidia [START_REF] Siebert | Re-evaluation of characters in Apolemiidae (Siphonophora), with description of two new species from Monterey Bay, California[END_REF]. As a result of probud subdivision, zooids are found in reiterating units, called cormidia, along the stem [START_REF] Dunn | The evolution of colony-level development in the Siphonophora (Cnidaria:Hydrozoa)[END_REF]. Siphonophore polygastric colonies are linear and are typically oriented vertically along an anterior-posterior axis, reflecting the typical direction of locomotion (Figure 1B,C,D, but note P. physalis A-P axis is oriented horizontally, see [START_REF] Munro | Morphology and development of the Portuguese man of war, Physalia physalis[END_REF]. The anterior pole of the polygastric colony corresponds to the aboral pole of the larva [START_REF] Haddock | A re-examination of siphonophore terminology and morphology, applied to the description of two new prayine species with remarkable biooptical properties[END_REF]. The secondary, dorso-ventral axis is always present, with the growth zone, pro-buds, and cormidia on the ventral side (Figure 1C,D). Additional left-right asymmetries have been documented across siphonophore taxa [START_REF] Dunn | Complex colony-level organization of the deep-sea siphonophore Bargmannia elongata (Cnidaria, Hydrozoa) is directionally asymmetric and arises by the subdivision of pro-buds[END_REF][START_REF] Dunn | The evolution of colony-level development in the Siphonophora (Cnidaria:Hydrozoa)[END_REF].

Siphonophores also reproduce sexually to generate new polygastric colonies. The gametes develop within specialized zooids, the gonophores, and are spawned in the water column. The environmental and molecular factors triggering gamete maturation and release are still to be identified in siphonophores, even though dark-light transitions are likely to play major roles in many species, as observed in other hydrozoans [START_REF] Artigas | A gonad-expressed opsin mediates light-induced spawning in the jellyfish Clytia[END_REF][START_REF] Takeda | Identification of jellyfish neuropeptides that act directly as oocyte maturation-inducing hormones[END_REF]. Depending on the species, siphonophores are either monoecious (egg or sperm bearing gonophores are both found on the same colony) or dioecious (colonies have either egg or sperm bearing gonophores but not both) [START_REF] Totton | A Synopsis of the Siphonophora[END_REF]. After fertilization, embryonic development gives rise to a planula larva that develops directly into the colonial form. As a result of this unique development, siphonophore planulae are unusual as compared with other hydrozoan larvae in that they exhibit bilateral symmetry. Here, we review current knowledge of colonial body plan establishment during embryonic and larval development in siphonophores, and highlight several key open questions.

Major transitions in the siphonophore life cycle

While early and late stages of development have been described in detail in several codonophoran species, no information exists on cystonect embryogenesis and very little on later developmental stages (Figure 1B,C,D), all based on fixed material [START_REF] Okada | Développement post-embryonnaire de la Physalie Pacifique[END_REF][START_REF] Okada | Les jeunes Physalies[END_REF][START_REF] Munro | Morphology and development of the Portuguese man of war, Physalia physalis[END_REF]). We will thus primarily focus in this review on codonophoran development. In all codonophoran species studied so far, a series of unipolar and equal cleavages, followed by gastrulation by primary delamination give rise to a typical cnidarian planula larva. Spatial organization of the planula endoderm leads to the emergence of the ventral endoderm, a future site of zooid budding. The developmental trajectories of Calycophorae and 'physonects' diverge as soon as budding starts, primarily due to the absence of a gas-filled float (the pneumatophore) in calycophorans. The budding gives rise to either calyconula (in Calycophorae) or siphonula (in 'physonects' and cystonects) larvae. The chronology and sequence of budding appear to be species-specific, though in the majority of Calycophorae it starts with the nectophore, while in siphonula larva the pneumatophore tends to arise first (Figure 1C,D). The early budding is often accompanied by shedding of larval zooids (larval zooids differ in morphology from the sametype zooid in the polygastric colony). Simultaneously, the embryonic body differentiates into the protozooid (primary gastrozooid or feeding body); the vitelline reserves are used so that the gastric cavity hollows, the endodermal gland cells differentiate at the oral pole of the protozooid, the aboral portion of the protozooid swells and differentiates into nematogenic tissue, and the tentacle buds between the protozooid and nectophore through ecto-endodermic evagination. The appearance of the subsequent zooids, primarily the additional cormidial units, each bearing a single gastrozooid, mark the transition into the polygastric colony. The time needed to reach this stage depends on the water temperature [START_REF] Carré | A complete life cycle of the calycophoran siphonophore Muggiaea kochi (Will) in the laboratory, under different temperature conditions: ecological implications[END_REF][START_REF] Sherlock | Effects of temperature on the development and survival of Nanomia bijuga (Hydrozoa, Siphonophora)[END_REF], though total siphonophore life-span is unknown.

Gametes and fertilization

Siphonophore eggs are relatively large (e.g., 180 µm in mediterranean Nanomia bijuga Carré 1969, 500 µm in Lensia conoidea Carré 1967) and transparent; they undergo meiotic maturation prior to spawning, and as in other cnidarians both polar bodies are extruded prior to fertilization [START_REF] Carré | Ordre des siphonophores[END_REF]. Fully grown eggs are polarised with the germinal vesicle displaced to one pole, and following the meiotic divisions an extracellular structure called the cupule that overlays the polar bodies -this is the region of the egg where fertilization will take place [START_REF] Carré | Sperm Chemotaxis in Siphonophores[END_REF]. The presence of cupule has been identified in only a few species across the siphonophore phylogeny [START_REF] Carré | Sperm Chemotaxis in Siphonophores[END_REF], with no information on cystonects. The cupule forms early during oocyte maturation, probably through exocytosis [START_REF] Carré | Sperm Chemotaxis in Siphonophores[END_REF]. It is highly chemoattractive to spermatozoa in a species-specific manner (it contains a peptide that modifies the curvature of the sperm trajectories, [START_REF] Cosson | Sperm chemotaxis in siphonophores. II. Calciumdependent asymmetrical movement of spermatozoa induced by the attractant[END_REF], and is shed quickly from the egg after fertilization [START_REF] Carré | Sperm Chemotaxis in Siphonophores[END_REF]. Siphonophore spermatozoa have a typical cnidarian morphology, with a conical nucleus, a few mitochondria and a single flagellum [START_REF] Carré | Sperm Chemotaxis in Siphonophores[END_REF]. Their centriolar apparatus includes a striated rod extending anteriorly, which has been implicated in an acrosomal reaction during gamete fusion [START_REF] Carré | Existence d'un complexe acrosomal chez les spermatozoïdes du cnidaire Muggiaea kochi (Siphonophore, Calycophore): différenciation et réaction acrosomale[END_REF].

Cleavages and gastrulation

Cleavages are unipolar and equal until the 32 cell stage [START_REF] Freeman | Experimental studies on embryogenesis in hydrozoans (Trachylina and Siphonophora) with direct development[END_REF][START_REF] Carré | Ordre des siphonophores[END_REF]. At the 32 cell stage, the blastomeres consist of an outer cytoplasmic region containing the nucleus and organelles, and an inner cytoplasmic region containing the vitelline reserves [START_REF] Carré | Ordre des siphonophores[END_REF]. At this point, divisions have been reported as superficial and partial, with furrows only penetrating the outer cytoplasmic region containing the nucleus [START_REF] Carré | Ordre des siphonophores[END_REF]. The periblastula stage thus consists of an outer flat layer of cells surrounding an internal anucleated mass of vitelline vesicles [START_REF] Carré | Ordre des siphonophores[END_REF]. Gastrulation occurs via primary delamination, where certain cells divide tangentially or obliquely relative to the surface of the embryo, resulting in a sheet of endodermal cells directly below the sheet of ectodermal cells [START_REF] Freeman | Experimental studies on embryogenesis in hydrozoans (Trachylina and Siphonophora) with direct development[END_REF][START_REF] Carré | Ordre des siphonophores[END_REF]. During the transition to the planula larva stage, these endodermal cells -that are located in the region that is by convention called the ventral side of the planula -continue to divide in this region, while other endodermal cells migrate inwards between the vesicles of vitellus [START_REF] Carré | Ordre des siphonophores[END_REF]. The precise site of gastrulation (if any) is not known, however [START_REF] Carré | Ordre des siphonophores[END_REF] imply that gastrulation occurs on the ventral side ("certain endodermal cells stay in the place where they were formed, multiply, forming a layer that lines the ventral face of the larva", translated from French), and our confocal data (Figure 2, Figure 3) also point towards an endodermal layer only on the ventral side. The site of origin of cells that migrated and are interspersed amongst the vitelline vesicles is unclear. [START_REF] Carré | Ordre des siphonophores[END_REF] point out that the internal mass of vitellus with endodermal cells interspersed were referred to by many authors, including themselves in earlier papers, as the "primary" endoderm, while the ventral "thickening" of cells were referred to as the "secondary" endoderm. Given the unclear order of appearance of these two populations of endodermal cells, or the fate of the cells interspersed among the vitelline vesicles, we will not be using these terms in this review, and instead will refer primarily to the 'ventral endoderm' when discussing this thickened ventral layer that plays a critical role in establishing the larval body plan.

Diversity of larval trajectories

The ciliated planula larva is bilaterally symmetrical, it is elongated with an oral and aboral pole, and the endoderm consists of the ventral layer, as well as scattered cells amongst the vitelline vesicles (Figure 2, Figure 3) [START_REF] Carré | Ordre des siphonophores[END_REF]. The planula stage is very short in duration (typically less than 24 hours [START_REF] Carré | Ordre des siphonophores[END_REF], after which the first zooid buds or structures, such as tentacles, begin to form. Larval development has been described in detail in a number of species with variation between species in terms of budding sequence and development (Figure 4).

The budding sequence has been described for several 'physonect' siphonula larvae, and the first buds vary considerably in both budding location and identity. In N. bijuga the pneumatophore arises first, via invagination at the aboral pole of the larva (Figure 2A), subsequently the first larval tentacle buds on the ventral side (Figure 2B) (Carré 1969a). In Halistemma rubrum, the pneumatophore bud forms first on the dorsal side close to the aboral pole, simultaneously, the first nectophore bud forms on the ventral side close to the aboral pole (Carré 1971, Figure 4A). In Cordagalma ordinatum [referred to as C. cordiformis] (Carré 1973, Figure 4C) and in Agalma elegans [START_REF] Fewkes | On the development of Agalma[END_REF], however, the bud of the larval bract forms first, ventrally close to the aboral pole, then subsequently the pneumatophore forms via invagination on the dorsal side close to the aboral pole. The third bud gives rise to the nectophore, much like in Physophora hydrostatica [START_REF] Haeckel | Zur Entwicklungsgeschichte der Siphonophoren[END_REF]. Lastly, in Forskalia edwardsii, three buds arise simultaneously on the ventral side that will give rise to the pneumatophore, larval bract, and larval tentacle (Figure 4B, Carré 1967b). The pneumatophore continues to elongate, meanwhile in the middle and oral regions of the larva, a protozooid begins to differentiate (Figure 2C,D). At the base of the differentiated protozooid, a secondary tentacle forms on the opposite side of the first tentacle (Figure 2E). The narrowing at the level of the tentacle bulb (Figure 2B-D) will later stretch to give rise to the colony stem.

Within Calycophorae, the first bud to form on the planula body is almost uniformly the nectophore (e.g. Muggiaea kochii [START_REF] Carré | A complete life cycle of the calycophoran siphonophore Muggiaea kochi (Will) in the laboratory, under different temperature conditions: ecological implications[END_REF], Sphaeronectes koellikeri [referred to as S. gracilis] Carré 1969b), with just two described exceptions. In Abylopsis tetragona, both the nectophore and bract bud appear at the same time (Figure 3C, D; Carré 1967a), while in Lilyopsis medusa a conspicuous outgrowth starts to project from the planula body (Figure 4F, [START_REF] Carré | Le développemet larvaire de Lilyopsis rosea (Chun, 1885) siphonophore calycophore, prayidae[END_REF]. The nectophore budding usually starts on the planula ventral side, although there are species in which nectophore bud appears on the aboral pole (e.g. Sphaeronectes irregularis, Figure 4G, Carré 1969b) or even close to the oral pole (e.g. L. medusa, Figure 4F, [START_REF] Carré | Le développemet larvaire de Lilyopsis rosea (Chun, 1885) siphonophore calycophore, prayidae[END_REF]. Irrespectively, nectophore development always progresses analogously to hydrozoan medusae, via ectoderm delamination and formation of the entocodon. Below the nectophore bud, on the ventral side of the larva, the primary tentacle develops (e.g. in L. conoidea, Figure 4D, Carré 1967b). After that, the oral region of the planula differentiates to form the protozooid (Figure 3F), which will later develop a mouth opening (Figure 3G). The most notable deviation from this otherwise conserved trajectory is the fate of planula body in members of the two calycophoran families: Hippopodidae and Prayidae [START_REF] Metschnikoff | Studien über die Entwickelung der Medusen und Siphonophoren[END_REF][START_REF] Carré | Le développemet larvaire de Lilyopsis rosea (Chun, 1885) siphonophore calycophore, prayidae[END_REF]. In their larvae, after entocodon formation, an extensive deposition of mesoglea starts between the endoderm and both the planula ectoderm and nectophore bud ectoderm, so that the entirety of the embryonic body becomes enclosed within the nectophore and forms the somatocyst (Figure 4F).

Post-larval development

Later on, both siphonula and calyconula transition into the post-larva, which is typically characterized by a presence of fully differentiated zooids (Carré 1969a), though there is no clear definition of this stage. As the post-larva continues to grow, its stem elongates and the additional zooids are budded from the growth zone(s). When two growth zones are present, the aboral one buds nectophores, while the other buds cormidia, consisting of all other zooid types. If only a single growth zone is present, it buds only cormidia.

Cormidia are budded on the same side as were the primary zooids of the planula, i.e., ventral, with the exception of S. koellikeri (Figure 4G), whose cormidia in the polygastric colony are budded from planula dorsal side. The nectophores, on the other hand, can be budded off either form the same side of the stem as the cormidia (e.g. in F. edwardsii , Carré 1967b) or the opposite (nectophores -dorsal, cormidia -ventral; A. elegans, Fewkes 1885).

As development progresses, some caducous larval zooids detach, while others are retained in the polygastric colony (see [START_REF] Pugh | A review of the family Sphaeronectidae (Class Hydrozoa, Order Siphonophora), with the description of three new species[END_REF]. In calycophorans, the larval nectophores, for example, are typically retained within Sphaeronectidae (Carré 1969b) and Prayidae [START_REF] Carré | Le développemet larvaire de Lilyopsis rosea (Chun, 1885) siphonophore calycophore, prayidae[END_REF], and replaced by one or more definitive ones in Diphyidae (Carré andCarré 1991, 1993), as evidenced by differences in the morphology of larval vs. definitive zooids (size, course of radial canals, ridge patterns on the exumbrella; [START_REF] Carré | Étude du développement d'Halistemma rubrum (Vogt, 1952) Siphonophore Physonecte Agalmidae[END_REF]. The larval bracts, on the other hand, are carried over during development in agalmatiid 'physonects' [START_REF] Fewkes | On the development of Agalma[END_REF][START_REF] Carré | Étude du développement de Cordagalma cordiformis Totton, 1932, Siphonophore Physonectae, Agalmidae[END_REF], and seem caducous in many other codonophorans (F. edwardsii Carré 1967b andA. tetragona Carré 1967a). The fate of larval zooids thus appears specific to particular phylogenetic groups.

Establishing the colony body plan Polarity and axis specification

Fully grown siphonophore oocytes, as noted above, are polarised with a germinal vesicle displaced to one pole. After meiosis is complete, the female pronucleus remains positioned at the animal pole, immediately below the extracellular cupule and the two polar bodies. [START_REF] Carré | Ordre des siphonophores[END_REF] noted that the female pronucleus itself is polarised, with digitations in the nuclear membrane towards the cell membrane along which most of the nuclear pores are located. Within the gonophore prior to oocyte release, the egg is oriented with the germinal vesicle pole facing the release opening [START_REF] Carré | Ordre des siphonophores[END_REF]. The planula larvae are bilaterally symmetrical, with an oral pole that will give rise to the future protozooid, and an aboral pole which is defined in some species by the presence of a pneumatophore or nectophore (Figure 5). They also have a dorsal-ventral axis, with the ventral side characterised by the presence of a thickened layer of endoderm. It is along this side where the first zooids begin to bud.

Within the polygastric colonies, directional asymmetries have also been described during probud subdivision. In Bargmannia elongata polygastric colonies, for example, gonozooids attach to the left, and there are two lateral left bracts and one right bract [START_REF] Dunn | Complex colony-level organization of the deep-sea siphonophore Bargmannia elongata (Cnidaria, Hydrozoa) is directionally asymmetric and arises by the subdivision of pro-buds[END_REF]. The directionality of this displacement is consistent across individuals of the same species, and arises from asymmetric budding during pro-bud subdivision [START_REF] Dunn | Complex colony-level organization of the deep-sea siphonophore Bargmannia elongata (Cnidaria, Hydrozoa) is directionally asymmetric and arises by the subdivision of pro-buds[END_REF]. Meanwhile P. physalis colonies are dimorphic and may be either left or right handed (this in turn determines sailing direction), with the attachment point of the tentacle of the tentacular palpon as the first indicator of this asymmetric budding pattern in fixed larval samples [START_REF] Okada | Les jeunes Physalies[END_REF][START_REF] Totton | Studies on Physalia physalis (L.). Part 1. Natural history and morphology[END_REF][START_REF] Munro | Morphology and development of the Portuguese man of war, Physalia physalis[END_REF].

Experimental data on siphonophore embryos provide some intriguing insights into aboral-oral and dorsal-ventral axis specification and patterning. In a series of experiments conducted on the 'physonect' Nanomia cara and calycophoran Muggiaea atlantica, [START_REF] Freeman | Experimental studies on embryogenesis in hydrozoans (Trachylina and Siphonophora) with direct development[END_REF] used chalk marks to demonstrate that the site of the first cleavage furrow marks the future oral region of the larva. In N. cara, oral-aboral specification is established prior to the 8-cell stage, while in M. atlantica, the aboral half is specified by the 8-cell stage but the oral half is capable of regulation [START_REF] Freeman | Experimental studies on embryogenesis in hydrozoans (Trachylina and Siphonophora) with direct development[END_REF]. To trace the origin of bilateral symmetry in M. atlantica he drew a chalk ring around the egg along the cleavage furrow. Only a few cases (6 out of 30) were suitable for analysis and in all of these planulae one side of the chalk ring coincided with the side of the ventral endoderm [START_REF] Freeman | Experimental studies on embryogenesis in hydrozoans (Trachylina and Siphonophora) with direct development[END_REF]. In cutting experiments separating dorsal and ventral halves in M. atlantica embryos, [START_REF] Freeman | Experimental studies on embryogenesis in hydrozoans (Trachylina and Siphonophora) with direct development[END_REF] found that in all cases the ventral halves gave normal larvae, while for the dorsal halves the outcome depended on the stage at which the cutting experiments occurred. At 6-7 h of development, dorsal/ventral sides cannot be distinguished, however when cut at this time, both halves develop into normal larvae. If cutting occurred during early stages (around 15-17 h) the ventral endoderm formed on the dorsal side opposite the cut site, meanwhile if it occurred at later stages (22-24 h and 30-36 h) the ventral endoderm took longer to form and budding was disrupted. Taken together, these data suggest that asymmetries in the egg that are set up before first cleavage direct the development of both the oral-aboral and dorsal-ventral axes. The first cleavage furrow has been shown to initiate at or close to the site of polar body formation in other hydrozoans, including Clytia gregaria [referred to as Phialidium gregarium] and Hydractinia echinata [START_REF] Freeman | Hydrozoan eggs can only be fertilized at the site of polar body formation[END_REF], implying that the region of the egg with the cupule corresponds to the site of the first cleavage furrow and therefore will give rise to the oral pole of the larva (Figure 5). No cellular feature defining either of the dorsal or ventral poles in the egg has yet to be identified. Interestingly, based on the descriptions of [START_REF] Carré | Ordre des siphonophores[END_REF], the initiation site of gastrulation in the embryo may correspond to the future ventral side.

Pneumatophore and nectophore budding

The transition to siphonula involves the budding of pneumatophore, always found at the aboral pole of the polygastric colony. The gas-filled float development starts from invagination of a few ectodermal cells (Figure 6B.1) at a species-specific position (see 'Diversity of larval trajectories' section above), which then push inside onto the endoderm (Figure 6B.2). After a few days, the roughly circular bud formed by the ectodermal cells, arranged around a primary cavity (Figure 6B.3) is completely enclosed within the planula body by the overlying ectoderm. At this stage, the ectodermal secretion of chitin within the cavity begins and cells located basally in the bud divide to give rise to the aeriform ectoderm (Figure 6B.4), which later proliferate to cover the entirety of the primary cavity (Figure 6B.5). Such internal reorganization of the bud leads to the accumulation of the chitin in a single layer covering the aeriform tissue, which starts to produce gas. Subsequently, the bud, which now defines the aboral/anterior pole (Figure 5), is being pushed out of the planula body outline, together with the associated layer of endoderm (Figure 6B.5). The pneumatophore bud is thus covered by the ectoderm and two-layered endoderm while itself is made up of two layers of ectoderm, separated by chitin (Figure 6B.6). In some species, the additional folding of the endodermal layer gives rise to a few longitudinal cavities (8 in N. bijuga Carré 1969a, 6 in H. rubrum Carré 1971) stretching from the tip to the base of pneumatophore which remain in contact with gastrovascular cavity.

The development of codonophoran swimming bells, the nectophores, parallels that of medusae in other hydrozoans. The nectophore budding usually starts on the ventral side of the planula, although it may begin elsewhere (Figure 4). The budding begins with the delamination of the ectoderm (Figure 6A.1), with the newly appearing cell mass pushing back on to the ventral endoderm (Figure 6A.2). As a result of the continued cell proliferation within that hemispherical cell mass it begins to separate from the overlying ectoderm (Figure 6A.3), up until formation of the typical medusa nodule, the entocodon (Figure 6A.4). The entocodon, doubled exteriorly by the ventral endoderm, grows considerably, acquiring more spherical shape and hollowing internally; it then starts to project externally from the planula body, with its long axis (ostium -tip of the nectophore) perpendicular to the planula oral-aboral axis (Figure 6A.5). The nectophore bud enlarges asymmetrically, with growth more pronounced at the aboral pole, leading to the repositioning of the nectophore, so that its long axis becomes parallel to the oral-aboral axis of planula body (Figure 6A.6). Simultaneously, the ventral endoderm, lying between the outer ectoderm and the entocodon ectoderm, begins to hollow, giving rise to the gastrovascular canals (radial and ring) and the somatocyst; additionally, the nectophore musculature starts to differentiate at roughly the same time (Figure 6A.6). Finally, an orifice opens at the site of contact between the entocodon and the planula ectoderm, marking the stage at which the nectophore becomes functional.

The pneumatophore is not traditionally considered to be a zooid, contrary to nectophores, as it arises once during embryonic development, like the protozooid, but unlike non-larval zooids which largely arise from the growth zones in the polygastric colony. However, it is made of different tissue types, has complex anatomy and performs a small number of specialized functions that are critical to the colony. The superficial resemblance of pneumatophore and nectophore early buds have fueled attempts at homologizing these two structures [START_REF] Metschnikoff | Studien über die Entwickelung der Medusen und Siphonophoren[END_REF][START_REF] Chun | Über den Bau und die morphologische Auffassung der Siphonophoren[END_REF], while others refute their claims [START_REF] Garstang | The morphology and relations of the Siphonophora[END_REF]. Histological analyses of siphonophore embryos have provided arguments for both sides of the debate (see e.g., [START_REF] Carré | Ordre des siphonophores[END_REF], although they predominantly point to a lack of homology between pneumatophore and nectophore [START_REF] Mackie | Siphonophore Biology[END_REF]. Pneumatophore budding site varies between species, is not restricted to the aboral pole of planula, and can even bud from the same position as nectophores (Figure 4). However, the pneumatophore aeriform ectoderm, postulated by [START_REF] Metschnikoff | Studien über die Entwickelung der Medusen und Siphonophoren[END_REF] as homologous to the medusoid manubrium, arises through proliferation of ectoderm (Figure 6B.5), and not through ecto-endodermal evagination as the manubrium does in other medusozoans (note that siphonophore nectophores do not have a manubrium, Carré 1967b). Although the pneumatophore primary cavity can be interpreted as homologous to nectophore subumbrella cavity (Figure 6A.4/A.5, Figure 6B.3/B.4), it is later replaced by the definitive cavity arising internally, within the chitin-delimited zone (Figure 6B.6, Carré 1967b). Lastly, nectophore radial canals arise through cavitation of the endoderm, while pneumatophore cavities are separated from the gastrovascular cavity through the endodermal folding (Carré 1969a).

Establishing growth zones

Budding in siphonophores is highly restricted spatially, with zooids arising from either one (Calycophorae) or two growth zones ('physonectae';[START_REF] Dunn | The evolution of colony-level development in the Siphonophora (Cnidaria:Hydrozoa)[END_REF]. The growth zones are sites of high cell proliferation, and interstitial stem cells (i-cells) are found in the ectoderm and endoderm of the horn (anterior structure in the growth zone where pro-buds arise) [START_REF] Siebert | Stem cells in Nanomia bijuga (Siphonophora), a colonial animal with localized growth zones[END_REF]. The developmental origin of either growth zone during larval development has not been described in detail, however, it seems likely that the primary growth zone originates from the planula ventral endoderm. The subsequent spatial separation into two growth zones in 'physonectae' occurs during the siphonula stage, when stem elongation moves them apart (Carré 1969a). In calycophorans, however, the calyconula body elongates only orally, with the embryonic body between protozooid and larval nectophore remaining competent to bud both the cormidia and definitive nectophores. Even when mature, the stem of certain calycophorans retains the ability to replace damaged or lost nectophore(s) [START_REF] Carré | Sur le genre Sulculeolaria Blainville, 1834 (Siphonophora, Calycophorae, Diphyidae)[END_REF], hence suggesting that they either also possess two, albeit unseparated, growth zones, or that their single growth zone retains its larval budding plasticity.

Larval zooids bud differently from zooids in the polygastric colony that arise from growth zones. During larval development, the body of the planula differentiates into the protozooid and other zooids are budded individually from the ventral endoderm. Meanwhile, once growth zones are established, budding in polygastric colony growth zones occur via pro-bud subdivision, i.e. a single bud gives rise to the whole cormidium [START_REF] Carré | Ordre des siphonophores[END_REF][START_REF] Dunn | The evolution of colony-level development in the Siphonophora (Cnidaria:Hydrozoa)[END_REF]. This transition occurs early in the larval development, in Forskalia edwardsii for example, the young siphonula still without any nectophores, starts to bud cormidia from the zone between protozooid and larval bracts (Carré 1967b). Little is known about the patterning and specification of either larval zooids or zooids arising by pro-bud subdivision, or indeed whether these two processes are similar.

Molecular perspectives on the early development of siphonophores

Siphonophores exhibit unparalleled zooid functional specialization and have highly organized polygastric colonies where zooids are arranged in an ontogenetic series from anterior to posterior in repeating species-specific units of zooids. It is no surprise therefore that siphonophore embryonic development is also unusual as compared with other cnidarian species. The work of Danielle and Claude Carré, along with Gary Freeman and others, have provided an excellent foundation for our understanding of early siphonophore development, however many open questions remain that also have implications for our broader understanding of cnidarian development and body plan specification.

The molecular players underlying axis specification and symmetry breaking in siphonophores, setting up the oral-aboral and dorsal-ventral axes during early development, as well as directional asymmetry of the cormidia, have yet to be investigated. Recent work on other hydrozoan species, notably Clytia hemisphaerica [START_REF] Momose | Two Oppositely Localised Frizzled RNAs as Axis Determinants in a Cnidarian Embryo[END_REF][START_REF] Momose | A maternally localised Wnt ligand required for axial patterning in the cnidarian Clytia hemisphaerica[END_REF], Dynamena pumila [START_REF] Vetrova | From apolar gastrula to polarized larva: Embryonic development of a marine hydroid, Dynamena pumila[END_REF], Hydractinia echinata [START_REF] Duffy | Wnt signaling promotes oral but suppresses aboral structures in Hydractinia metamorphosis and regeneration[END_REF][START_REF] Kraus | The embryonic development of the cnidarian Hydractinia echinata[END_REF], or on the anthozoan sea anemone Nematostella vectensis (e.g., [START_REF] Lebedeva | Cnidarian-bilaterian comparison reveals the ancestral regulatory logic of the β-catenin dependent axial patterning[END_REF] all showed that the Wnt-β-catenin signaling pathway plays a major and conserved role in setting up the oral-aboral axis during embryogenesis. Activation of this pathway at the animal pole of early cleaving Clytia embryos, through deposition of Wnt ligand and receptors on the animal and vegetal sides of the egg, leads to the activation of a set of transcription factors and signaling molecules setting up the oral pole [START_REF] Momose | Two Oppositely Localised Frizzled RNAs as Axis Determinants in a Cnidarian Embryo[END_REF][START_REF] Momose | A maternally localised Wnt ligand required for axial patterning in the cnidarian Clytia hemisphaerica[END_REF]. Considering that the animal pole also gives rise to the oral pole in siphonophores [START_REF] Freeman | Experimental studies on embryogenesis in hydrozoans (Trachylina and Siphonophora) with direct development[END_REF], it is likely that the Wnt-β-catenin signaling plays a similar role in specifying the primary body axis in siphonophore embryos as it does in other cnidarians. The primary axis of siphonophore planulae is also evident in their oriented swimming toward the aboral pole. This process is driven by beating of cilia covering the entire planula ectoderm (Carré 1967, Carré 1969). A Frizzled-dependent Planar Cell Polarity pathway was shown to control this process in Clytia embryos [START_REF] Momose | A conserved function for Strabismus in establishing planar cell polarity in the ciliated ectoderm during cnidarian larval development[END_REF], and is likely similarly involved in siphonophores.

The molecular factors establishing the secondary dorsal-ventral axis in siphonophore planulae are also to be uncovered, notably those responsible for the second symmetry breaking before gastrulation and those responsible for the accumulation of endodermal cells on the ventral side and activation of the first zooid budding, leading to the formation of a bilateral planula larva. Besides siphonophores, bilateral symmetry in cnidarian embryos is only evident in anthozoan larvae, in the folding of their inner cavity [START_REF] Genikhovich | On the evolution of bilaterality[END_REF]. The BMP pathway was shown to play a major role in the formation of this secondary axis in the sea anemone N. vectensis [START_REF] Genikhovich | On the evolution of bilaterality[END_REF]. Early budding of zooids on the ventral side of siphonophore planulae bears resemblance to the asymmetric budding of polyp or stolon observed in most colonial hydrozoans, suggesting that similar patterning mechanisms may be involved (e.g. [START_REF] Hobmayer | WNT signalling molecules act in axis formation in the diploblastic metazoan Hydra[END_REF][START_REF] Bagaeva | сWnt signaling modulation results in a change of the colony architecture in a hydrozoan[END_REF][START_REF] Sanders | Frizzled3 expression and colony development in hydractiniid hydrozoans[END_REF]. Polyp budding in Hydra was notably shown to involve a non-BMP member of the TGF-β family [START_REF] Watanabe | Nodal signalling determines biradial asymmetry in Hydra[END_REF]. Expression studies of Wnt, BMP and other TGF-β signaling components as well as conserved downstream targets (e.g. Brachyury, [START_REF] Schwaiger | An ancestral Wnt-Brachyury feedback loop in axial patterning and recruitment of mesoderm-determining target genes[END_REF]) in several siphonophore species would be particularly helpful to start uncovering mechanisms of symmetry breaking and axis specification in siphonophores. Considering the great variability in the order and position of the different zooids budded during early development it is entirely possible that different factors are involved in secondary axis formation in the different siphonophore clades.

A peculiarity of siphonophore colonies is that they display only one or two growth and budding zones in very precise locations, as compared to most other colonial hydrozoans in which growth budding zones are widely and less precisely distributed within the colony. It was hypothesized that the spatial restriction of i-cells only in mature zooids and growth zones could play a role in the reduced developmental plasticity of most parts of the colony [START_REF] Siebert | Stem cells in Nanomia bijuga (Siphonophora), a colonial animal with localized growth zones[END_REF]. It is currently unknown how these zones are defined during colony formation, which mechanisms precisely define their number and location and how interstitial stem cells are restricted to these zones. Detailed cellular and molecular descriptions of the development of the growth zone in several siphonophores, coupled with single cell sequencing approaches, would provide important insights. Involvement of the Notch and FGF signaling pathways in the formation and maintenance of growth zones in siphonophores should also be investigated as they play major role in the regulation of budding or tentacle morphogenesis in several hydrozoans [START_REF] Münder | Notch signalling defines critical boundary during budding in Hydra[END_REF][START_REF] Lange | Dynamic expression of a Hydra FGF at boundaries and termini[END_REF][START_REF] Gahan | Functional studies on the role of Notch signaling in Hydractinia development[END_REF][START_REF] Hou | Repetitive accumulation of interstitial cells generates the branched structure of Cladonema medusa tentacles[END_REF].

RNA sequencing approaches during development could also highlight genes involved in the specification and patterning of different zooids during budding, as well as the pneumatophore, and give insights into their homology within Siphonophora, and with the medusa and polyp zooids of other hydrozoans. Considering the difficulty of getting a large number of individuals of the different stages, future development of spatial transcriptomic approaches could be particularly interesting in this regard.

The next challenge is bringing siphonophores and their larvae into the lab. There is optimism for this, as recent advancements in culturing are opening up possibilities to maintain fragile siphonophore species more easily [START_REF] Patry | Diffusion tubes: a method for the mass culture of ctenophores and other pelagic marine invertebrates[END_REF]. With the increasing availability of genomic resources for siphonophore species, including zooid-specific transcriptomes of polygastric colonies [START_REF] Siebert | Differential Gene Expression in the Siphonophore Nanomia bijuga (Cnidaria) Assessed with Multiple Next-Generation Sequencing Workflows[END_REF][START_REF] Munro | Evolution of Gene Expression across Species and Specialized Zooids in Siphonophora[END_REF], and the recent advancement of morphological and molecular techniques applicable to hard-to-get wild planktonic organisms, the moment is ripe for a renewed understanding of siphonophore development integrating morphological, cellular and molecular levels.

Figures and tables

Figure 2 .

 2 Figure 2. Larval development of Nanomia bijuga starting from the planula larva. Nuclei were stained with Hoechst dye (red) and F-actin was stained with phalloidin (black). A-D single confocal plane. E, Maximum intensity projection of multiple confocal planes. Orientation shown in D, aboral (A, up) -oral (O, down) and dorsal (D, left) -ventral (V, right). Scale bar is 100 µm. Abbreviations: PB-pneumatophore bud, TB -tentacle bud, Pn -pneumatophore, T -tentacle, Pr -protozooid. Immunostaining methods included as a supplementary file.

Figure 3 .

 3 Figure 3. Larval development of Abylopsis tetragona starting from the planula larva. Nuclei were stained with Hoechst dye (red) and F-actin was stained with phalloidin (black). A-F single confocal plane (note, in F, bract is out of plane). G, Maximum intensity projection of multiple confocal planes. Orientation shown in G, aboral (A, up) -oral (O, down) and dorsal (D, left) -ventral (V, right). Scale bar is 100 µm. Abbreviations: Ect -Ectoderm, End -Endoderm, NB -nectophore bud, BrB -bract bud, N -nectophore, Br -Bract. Immunostaining methods included as a supplementary file.

Figure 4 .

 4 Figure 4. Developmental trajectories across codonophoran siphonophores; 'physonects': A. Halistemma rubrum, B. Forskalia edwardsii, C. Cordagalma ordinatum and Calycophorae: D. Lensia conoidea, E. Sphaeronectes irregularis, F. Lilyopsis medusa, G. Sphaeronectes koellikeri. In sections A-F, the top row contains a photograph of an actual young planula (left) and a schematic (right) with an approximation of the delimitation of the ventral endoderm, while the bottom row shows a photograph of a later stage of the development. Section G shows development from the planula (top) to the polygastric colony (bottom). Photographs were reproduced from Carré 1967b (B. and D.), 1969b (E. and G.), 1971 (A.), 1973 (C.); Carré and Carré 1969 (F.) with permission from the publishers of Cahiers de biologie marine and Contributions to Zoology. All pictures are oriented so that: top-aboral, bottom-oral, left-dorsal, right-ventral. Labels as in Figures 2-3, plus M-mesoglea, F-filament. Scale bar is 250 µm.

Figure 5 .

 5 Figure 5. Comparison of body axes in developing siphonophores (Nanomia bijuga and Muggiaea atlantica, mid and bottom row, respectively) and a leptothecate hydrozoan (Clytia hemisphaerica, top row). Blue shading marks the oral pole. Abbreviations: A -aboral, Ant. -anterior, O -oral, Post. -Posterior, D -dorsal, V -ventral. Not drawn to scale. Clytia drawings after Munro et al. 2023; Nanomia and Muggiaea larvae redrawn from Russell 1938; Carré 1969a, respectively.

Figure 6 .

 6 Figure 6. Comparison of budding process between Lensia conoidea nectophore (A.1-6) and Forskalia edwardsii pneumatophore Color-coding has been used to distinguish germ layers (ectoderm teal, endoderm beige), mesoglea (violet), primary ectodermal cavity (light blue) and nuclei (dark grey). Thick black solid line represents a layer of chitin, while the thick dark brown solid line corresponds to muscle fibers. Redrawn after Carré 1967b. Scale bar is 20 µm.

Table 1 .

 1 Glossary of siphonophore terms

	Term	Definition
	Bract	Gelatinous zooid, likely required for buoyancy, defense
	Calyconula	Larval stage in calycophorans (see 'Siphonula' below),
		variously used to describe the stage when the first zooids
		start to bud, or when the protozooid is functional
	Cormidium	Reiterating sequence of zooids found along the stem
	Gastrozooid	Feeding zooid, considered homologous to polyps, typically
		bears a tentacle to catch prey
	Gonozooid	Reproductive zooid, polyp, that bears gonophores
		(reproductive zooids, medusae). Contain eggs or sperm.
	Medusa	One of two main adult body forms in Cnidaria, typically
		planktonic and swimming e.g. "jellyfish" lifecycle stage in
		Aurelia aurita, Clytia hemisphaerica
	Nectophore	Swimming zooid, considered homologous to medusae
	Palpon	Digestive/defensive zooid, considered homologous to
		polyps
	Polygastric colony	'Adult' colony -poly (many) gastric (gastrozooids)
	Polyp	One of two main adult body forms in Cnidaria, typically
		sessile and benthic with tentacles surrounding a mouth
	Pneumatophore	Gas filled float, typically at the anterior of the polygastric
		colony.
	Protozooid	First gastrozooid (see above) that forms during larval
		development.
	Siphonula	Larval stage in 'physonects' and cystonects (see
		'Calyconula' above), variously used to describe the stage
		when the first zooids start to bud, or when the protozooid is
		functional.
	Somatocyst	Endodermal diverticulum functioning as a colony
		lipid/nutrient reservoir
	Stem	Central stem/stalk to which all zooids are attached.
	Zooids	Bodies/units that are functionally specialized to different
		tasks and are evolutionarily homologous to free living

Acknowledgements

We thank Evelyn Houliston for comments on the manuscript and productive discussions. This work has received financial support from EMBRC-France (grant no. OOV-2016-638 and OOV-EMBRC FR-AAP2017-972 to CM and grant no. OOV-EMBRC FR-AAP2018-2180 to MKM), and the National Science Centre in Poland (grant no. 2018/29/N/NZ8/01305 awarded to MKM in collaboration with LL).

Conflict of Interest Statement

The authors have no conflicts of interest to declare.


J Stafiej

D Di Caprio

J P Badiali

Theoretical analysis of the competition between coulombic and specific interactions at charged interfaces

We present a model of an inhomogenous electrolyte solution based on the Landau-Ginzburg approach. It contains coulombic interactions, ideal entropy contribution and non local specific interactions. With this model we can study the specific interaction of the electrolyte with the electrode surface. Numerical calculations of the ionic density, potential and charge density profiles have been performed assuming reasonable values of the phenomenological parameters. We also use numerical methods to obtain differential capacitance as a function of charge density and the magnitude of electrode/electrolyte specific interactions.

It is a new aspect of this approach that ions can be excluded from the immediate vicinity of the electrode surface.

Introduction

Electrified interfaces investigated in electrochemistry represent an example of complex systems. They have several components and a lot of phenomena take place e. g. solvation of ions, adsorption at the electrode surface. The existence of strong electric fields induces non linear phenomena when the charge of the interface is changed. In such systems we have to deal with several length scales. The coulombic interaction introduces the Debye length which depends on ionic concentration. The short range interactions between various components introduce the sizes of the species which are concentration independent. The interaction between the electrode and the solution side of the interface leads to another characteristic length. Taking into account the existence of these several length scales we introduce a simple model in order to investigate the competition between electrostatic and non electrostatic phenomena at a charged interface.

Our approach is based on a simple Landau-Ginzburg hamiltonian that we have already used to investigate several aspects concerning both the interfacial and bulk properties of ionic systems [1]- [4]. In these works we study a theoretical system in which we assume that the solvent has no relevant contribution. The basic ingredients are the distributions of cations ρ + (⃗ r) and anions ρ -(⃗ r) across the interface. In the spirit of a field theoretical approach we treat ρ + (⃗ r) and ρ -(⃗ r) as two fields [START_REF] Parisi | Statistical Field Theory[END_REF]. The partition function of the interface is calculated via a functional integral in which all possible forms of ρ + (⃗ r) and ρ -(⃗ r) give a contribution weighted by the Boltzmann factor

exp{-βH[ρ + (⃗ r), ρ -(⃗ r)]} where the hamiltonian H[ρ + (⃗ r), ρ -(⃗ r)] is in fact a free energy functional of ρ + (⃗ r) and ρ -(⃗ r).
If H is restricted to the coulombic interaction and the ideal configurational entropy then, as we have shown [1], the mean field description leads to the non linear Gouy Chapman theory (NLGCT). By adding to the hamiltonian quadratic terms of the form a ij ρ i (⃗ r)ρ j (⃗ r)d⃗ r, where i, j = ±, we introduce some specificity of the system via the coefficients a ij . Quite surprisingly, such a simple model leads to qualitatively new behaviours compared to the NLGCT [2]. Near the point of zero charge and at low concentrations our results are identical to those of the NLGCT but at a higher concentration there can be a maximum or a minimum on the capacitance vs charge density the electrode we find two finite limits instead of the unlimited increase predicted by the NLGCT. If a ++ ̸ = a --the capacitance vs charge density curves are asymmetric with respect to zero charge density and otherwise they are symmetric.

The same kind of approach has been used to investigate the structure of bulk ionic solutions [4]. In this case an exact calculation has been performed. We have shown that the fluctuations around the mean field value of the ionic density lead to the exact Debye-Hückel limiting law. In addition, we have investigated the role of the symmetry between ions in the critical behaviour at low concentrations. In particular the Stillinger-Lovett conditions have been investigated near the critical point.

From our point of view the main interest in this kind of approach is to introduce physical ingredients with clear physical meaning. As we have seen already with very simple ingredients we can predict a large number of different behaviours.

For the charged interfaces only the electrostatic interaction between the electrode and the solution side has been considered. In this paper we go further and introduce a short range potential between the electrode and the solution. We assume that the effect of this potential is localized at the electrode surface. The magnitude of this potential can be positive or negative i. e. it can either favour adsorption or desorption. This paper is organized as follows. In Section 1 we present the ingredients of the hamiltonian. In Section 2 the mean field theory (MFT) is considered. We give also the main points of the method of numerical solution of the MFT equations. In the last Section we present and discuss some results.

The hamiltonian

We define the system as a 1-1 mixture of ions carrying elementary charges ±e of opposite signs at a given temperature T = kβ -1 in a dielectric continuum with the dielectric constant ε equal to that of a pure solvent. The overall bulk ionic density ρ b corresponds to twice the salt concentration. The electrolyte solution is limited to the halfspace, z > 0, by an impenetrable wall representing the electrode surface. We want to describe the gross features of the competition between the coulombic interaction with the other non coulombic interactions which may be present in this system. Therefore we select the hamiltonian H as simple as possible. It consists of the bulk part and the surface part:

H[ρ + (⃗ r), ρ -(⃗ r)] = H bulk + H surf (1)
The bulk part is further split into several contributions:

H bulk [ρ + (⃗ r), ρ -(⃗ r)] = H coul + H entropic + H quad + H nloc (2)
described in detail in ref. [4] and the surface part of the hamiltonian is introduced in ref. [3]. Let us just briefly review the ingredients of the hamiltonian H.

The coulombic interaction is given by:

H coul [ρ + (⃗ r), ρ -(⃗ r)] = 1 2 K 2 D 4πβρ b q(⃗ r)q( ⃗ r ′ ) |⃗ r -⃗ r ′ | d⃗ rd⃗ r ′ (3)
where

q(⃗ r) = ρ + (⃗ r) -ρ -(⃗ r)
is the charge density expressed in the elementary charge units as the difference of the densities of univalent cations and anions, respectively.

K 2 D = e 2 βρ b /ε is the square of the inverse Debye length. Let us note that 1 e δH coul δq(⃗ r) = e 4πε q( ⃗ r ′ ) |⃗ r -⃗ r ′ | d ⃗ r ′ = V (⃗ r) (4)
is just the electrostatic potential produced by the charge distribution eq( ⃗ r ′ ) at the point ⃗ r. The effect of the solvent on the Coulombic interaction is taken into account by the dielectric constant ε.

The configurational entropy gives the following contribution to the hamiltonian [1]:

H entropic [ρ + (⃗ r), ρ -(⃗ r)] = 1 β {ρ + (⃗ r)[-ln ρ + (⃗ r) ρ + 1] + ρ -(⃗ r)[-ln ρ -(⃗ r) ρ + 1]}d⃗ r (5)
where the meaning of the parameter ρ has been discussed in ref. [1].

The local quadratic contribution is given by:

H quad [ρ + (⃗ r), ρ -(⃗ r)] = 1 2βρ b {a ++ [ρ + (⃗ r)] 2 + 2a +-ρ + (⃗ r)ρ -(⃗ r) + a --[ρ -(⃗ r)] 2 }d⃗ r (6)
The quantities a ++ , a +-and a --already mentioned in the introduction are dimensionless parameters describing the correlations between ions. They contain the specific properties of the system. They are a part of the potential of mean force between ions and thus they take into account implicitely some aspects of the short range ion-solvent interaction. The effect of the local quadratic term on the differential capacitance has been already discussed to some extent in our previous work [2]. In this term we focus on the interaction of the fields at a single point in the space.

The nonlocal term establishes a link between a point in the space and its neighbourhood. We write this term as: that there is already a nonlocal interaction associated with the charge distribution taken into account via the coulombic interaction. In principle, there should be no difficulty in considering the general case when b ij 's are not necessarily the same but our simplification is greatly advantageous also for the numerical reasons. Note that b has a dimension of a length.

H nloc [ρ + (⃗ r), ρ -(⃗ r)] = 1 2βρ b {b 2 ++ [∇ρ + (⃗ r)] 2 + 2b 2 +-∇ρ + (⃗ r)∇ρ -(⃗ r) + b 2 --[∇ρ -(⃗ r)] 2 }d⃗ r (7)
In addition to the above ingredients which enter H bulk and describe the bulk solution we introduce a specific surface hamiltonian for which we assume the following form:

H surf [ρ + (⃗ r), ρ -(⃗ r)] = 1 β hδ(z)[ρ + (⃗ r) + ρ -(⃗ r)]d⃗ r (8)
Thus in addition to behaving like a charged impenetrable wall with charge density σ we assume that the wall has a given affinity to adsorb (h < 0) or desorb (h > 0) ions with a strength h. We also assume that this effect is localized near the wall and that its range is sufficiently short compared with other lengths that we can represent it by a Dirac δ function. This function is commonly used in the liquid state theory to represent short range potentials. When we use it we have in mind a narrow potential profile localized at the electrode surface and producing the effect equivalent to a step potential profile of magnitude 1 kT over the distance h or the potential step profile h kT over the distance 1 in a given unit system. As above, for simplicity we assume that the wall behaves the same way for anions and cations. There is also no theoretical difficulty in considering a wall behaving in a different way for each kind of ions.

To summarize, we consider an interface described by the hamilonian (1) in which there are several length scales involved: i) the Debye length K -1 D , ii) b which controls the order of magnitude of the gradient of the total density and iii) h which, as will be seen below, eq. ( 12) determines the gradient of the total density profile in contact with the wall.

With the above hamiltonian one might attempt an exact calculation. However, as a first approximation, we stay on the level of the MFT.

MFT calculations

In MFT we assume that only the fields which minimize H are important. This requirement gives us the MFT equations:

δβH[ρ + (⃗ r), ρ -(⃗ r)] δρ + (⃗ r) ρ+ (⃗ r),ρ -(⃗ r) = βµ + δβH[ρ + (⃗ r), ρ -(⃗ r)] δρ -(⃗ r) ρ-(⃗ r),ρ -(⃗ r) = βµ - (9)
which yield the MFT profiles ρ+ (⃗ r), ρ-(⃗ r) and

F M F T = H[ρ + (⃗ r), ρ-(⃗ r)] is identified
with the free energy of the system. In our case these equations read for z > 0:

+βeV (z) + log 2ρ + (z) ρ b + a ++ ρ + (z) ρ b + a +- ρ -(z) ρ b - b 2 ρ b d 2 dz 2 [ρ + (z) + ρ -(z)] = 1 2 (a ++ + a +-) -βeV (z) + log 2ρ -(z) ρ b + a -- ρ -(z) ρ b + a +- ρ + (z) ρ b - b 2 ρ b d 2 dz 2 [ρ + (z) + ρ -(z)] = 1 2 (a --+ a +-) (10)
where because of the symmetry of the system we consider the dependence only on one coordinate, z. We have eliminated the Lagrange multipliers µ + , µ -taking care of the constant number of molecules in the system using the conditions for the bulk. For z = 0, where we place the wall, we obtain the following boundary conditions:

- dV dz z=0 = ⃗ E(z = 0) = σ ε (11) d dz [ρ + (z) + ρ -(z)] z=0 = ρ b h b 2 (12)
equations which can be transformed into a system of differential equations which is in general much easier to handle.

MFT Differential Equations

It is convenient to introduce dimensionless reduced variables:

ẑ = K D z (13) g j (ẑ) = 2ρ j (z) ρ b where j = +, - (14)
v(ẑ) = 2βeV (z) (15) b = bK D (16) Ê(ẑ) = σ(ẑ) = - dv dẑ (17) ĥ(ẑ) = 2K D h (18)
They differ from their usual counterparts by conversion constants being combinations of the Debye inverse length

K D = e(ρ b β/ε) 1 2
, temperature T, concentration ρ b and the universal constants. We use K -1 D as a unit of length. From now on we will use reduced variables exclusively and we shall refer to g + (ẑ), g -(ẑ), v(ẑ), σ(ẑ), ĥ(ẑ) as ionic, electric potential, electric field and wall affinity (h field) profiles omitting the ˆsign. Let us note that the boundary conditions set equivalence between the external wall parameter σ on the wall and the electric field at the position of the wall and also between the wall ionophobicity (or philicity) and the gradient of the overall ionic density at the position of the wall. Both fields jointly describe the effect of the wall which gradually diminishes away from the wall to vanish far in the bulk. By virtue of the Poisson equation and using the reduced variables we can transform the system of the two integro-differential equations into the following system of four non-linear first-order differential equations:

2 g + (z) + a ++ -a +- dg + (z) dz - 2 g -(z) + a ---a +- dg -(z) dz = 2σ(z) (19) dg + (z) dz + dg -(z) dz = h(z) b 2 (20) dσ(z) dz = g + (z) -g -(z) (21) dh(z) dz = 2 log g + (z)g -(z) + (a ++ + a +-)g + (z) + (a --+ a +-)g -(z) -(a ++ + 2a +-+ a --) (22)
which have to be solved taking into account the boundary conditions. Two such conditions are imposed on the wall:

σ(z = 0) = σ 0 (23) h(z = 0) = h 0 (24)
We also have infinitely far from the wall:

g + (z → ∞) = 1 and g -(z → ∞) = 1 (25)
and then it follows that:

σ(z → ∞) = 0 (26) h(z → ∞) = 0 (27)

Numerical methods

For a system of non linear differential equation we have no analytical solution in general.

We resort to a numerical procedure and it is most straightforward to use a solver for the initial value problem. We cannot use it for the surface characterized by a given σ 0 and h 0 since for arbitrarily selected values of g + 0 = g + (z) and g - 0 = g -(z) for z = 0 we do not satisfy the conditions of the bulk phase at infinity. However far from the wall we can obtain the asymptotic solution of our problem by the linearization around the bulk values: g + (z) ≡ 1, g -(z) ≡ 1, σ(z) ≡ 0 and h(z) ≡ 0. The linearized problem can be solved as discussed in our previous paper [2] for sufficiently small but otherwise arbitrary values of σ a and h a at an arbitrarily selected dividing point z a > 0 as if there were a wall just slightly affecting the bulk uniform profiles. The matching condition sets the initial values of the ionic profiles at the dividing point equal to those calculated from the linearized equations. Then we launch the numerical integration backwards. Of course, we cannot tell the values of σ 0 and h 0 before actually making the integration. However, we are sure to get the physical ionic profiles with the correct asymptotic behaviour corresponding to the wall characterized by the calculated values of σ 0 and h 0 . The above described approach is in fact essentially the same as that discussed in ref. [2] for the case with no square gradient i. e. b = 0. In the present case, however, we have a two component group of transformations:

σ 0 = σ 0 (σ a , h a , z a ) ( 28
)
h 0 = h 0 (σ a , h a , z a ) (29)
and then in contrast to the former case we can observe non monotonous ionic density profiles [3].

The quantity of a particular experimental interest is the differential capacitance.

Assuming that the wall at z = 0 does not change its ionophobicity (ionophilicity) when charged we can calculate the differential capacitance as:

C = dσ 0 dv 0 h 0 (30)
For the numerical calculation we use a simple shooting method [START_REF] Press | Numerical Recipes in FORTRAN. The Art of Scientific Computing[END_REF] by which we apply a solver for an initial value problem to solve the boundary value problem.

3 Results

Choice of the parameters

To obtain meaningful results we have to specify the values of the phenomenological parameters related to the microscopic background of the studied phenomena. To mimic an aqueous electrolyte solution at room temperature we assume T = 298.15K and ε = 78. Since we concentrate on the effect of the competition between electrostatic and non electrostatic phenomena, we set the values of the local coupling constants a ij = 0. The value of b in this case gives the correlation length for the non coulombic interaction. We set its value to a typical value of a molecular diameter in ionic solutions which is of the order of 0.3nm. This corresponds to b = 1 in a 1M aqueous electrolyte solution for which K -1 D = 0.3nm is also of the order of the molecular diameter. In a 1M solution the deviations in the ionic solution properties from the Debye-Hückel limiting law become already important which makes our selection reasonable. We consider the values of h of the order of 1 in reduced units which corresponds to the step potential profile 1kT high and 0.15nm wide or 1.5kT high and 1nm wide.

Conclusions

Already with a very simple hamiltonian we can describe various behaviours of the system which are qualitatively different from the Gouy-Chapman theory. We can have exclusion of both ions from the vicinity of the electrode surface. For a strongly charged electrode we have a competition between exclusion and electrostatic attraction of the counterion leading to the maximum of the counterion profile. We have also seen that the type of the electrode surface, given in our case by h as a measure of the electrode's ionophilicity or ionophobicity, influences the capacitance curve to a large extent if we adopt values reasonable from the point of view of a molecular description. We can hope that if we overcome the simplifications of the present treatment we will arrive at a more extensive description of the electrified interfaces.

Figure Captions

---------------------- the ionophobicity parameter, h 0 = 6.2 corresponding to a 0.9 nm x kT potential step. (1) of a weakly charged wall corresponding to Fig. 1 (2) of a strongly charged wall corresponding to Fig. 2 Fig. 4 Charge density profiles for the two cases:

(1) of a weakly charged wall corresponding to Fig. 1 (2) of a strongly charged wall corresponding to Fig. 2 Fig. 5 Differential capacitance curves calculated according to the Gouy-Chapman theory (GC) and according to the formula (30) for b=1 and with values of the ionophobicity parameter h indicated for each curve.

  To have the simplest hamiltonian possible in what follows we take the same value b for all b ij 's, b ++ = b --= b +-= b. Hence this non local term relates only to the total density of ions -the sum of the densities for the cation and anion. Note

Fig. 1

 1 Fig.1 Ionic profiles g + (z) (dotted line) and g -(z) (solid line) in a 1M solution at a charged wall. The characteristic length for specific non local interactions in the solution is b = 0.3nm. The values of the parameters characterizing the wall placed at z = 0 are: charge density σ 0 = 0.012Cm -2 , the ionophobicity parameter, h 0 = 2.7 corresponding to a 0.4 nm x kT potential step.

Fig. 2

 2 Fig.2 Ionic profiles g + (z) (dotted line) and g -(z) (solid line) for a solution as in Fig.1 but at a different wall. The values of the parameters characterizing the wall placed at z = 0 are: charged density σ 0 = 0.16Cm -2 ,

Fig. 3

 3 Fig.3 Potential profiles for the two cases:

Acknowledgement

This paper results from the cooperation of our laboratories in the framework of the agreement between C.N.R.S in France and the Polish Academy of Sciences

Ionic profiles

In Figures 1 and2 we present two examples of profiles. They correspond to an electrode surface which is repulsive for both kind of ions. We choose the value of h such that both profiles vanish at the surface, g + (z = 0) = 0 and g -(z = 0) = 0. Of course, for a higher value of charge density we need a higher value of h to make the counterion profile vanish. We can see that this behaviour is totally different from that of NLGCT already for reasonable values of h at the wall. For instance, if we interpret h as a height of the potential step profile extending 0.1nm from the electrode it is 4kT and 9kT for the cases presented in Figures 1 and2 respectively. If the short range interaction dominates the coulombic interaction (Figure 1) the profiles increase monotonically from zero to their bulk values far from the wall. If, in contrast, the surface is strongly charged (Figure 2) we find a maximum in the distribution of counterions while the distribution of coions remains monotonic though exhibits an inflection point. The coions are excluded from the vicinity of the wall. Although the profiles in Figures 1 and2 seem to have a simple spatial dependence a more detailed mathematical analysis shows that this is not true [8]. Near the wall the profiles exhibit a non-analytic behaviour.

In Figure 3 we present the profiles of the electric potential V(z) corresponding to the cases studied in Figures 1 and2. These profiles are quantitatively different but otherwise both monotonic and there is no special feature from which to expect the qualitative difference in counterion profiles observed in the two cases. The potential profiles are also qualitatively similar to that of NLGCT. If we take, however, the second derivative of the potential profiles we obtain the charge density profiles which are non monotonous in contrast to NLGCT (Figure 4).

Differential capacitance

We expect that the differential capacitance dependence on charge density is more sensitive to the details of the present model than the corresponding dependence of the potential drop. The capacitance, as given by formula (30) may depend also on h i. e.

the type of the surface in contact with solution. This is illustrated on Figure 5 where we present the capacitance curves calculated for b = 1 and several values of h. For h = 0 the capacitance at the pzc coincides with the Gouy-Chapman capacitance. In this case the capacitance is determined by the response of the system to small values of σ keeping h = 0. Then the profiles deviate only slightly from the uniform bulk ones and are described by the linearized Gouy-Chapman theory. Away from the pzc and h = 0 the effect of the gradient term couples with the non linear entropy term giving rise to a behaviour qualitatively different from that of the NLGCT. For non vanishing h it is interesting to discuss the results from the geometrical point of view.

By integration of the Poisson equation we obtain the expression of the potential drop across the interface:

We can also consider that this integral introduces the first moment < z > or the center of gravity of the charge distribution. From this last expression we can interpret the potential across the interface as equivalent to that of two charged planes separated by the distance < z >. The integral capacitance is just ε/ < z > while the differential capacitance is given by: 1

At the pzc, 1/C =< z > pzc /ε. If h < 0 the wall tends to adsorb both ions and we may expect than < z > pzc is smaller than the corresponding value in the NLGCT where we have < z > pzc = K -1 D . Thus at the pzc for h < 0 it is natural to find C(h < 0) > C GC = εK D . In the case h > 0 the wall tends to push ions away towards the bulk. Then < z > pzc > K -1 D and, as expected, C(h > 0) < C GC . When we move away from the pzc the results are much more sophisticated. The differential capacitance is no longer given by < z > but also by σd < z > /dσ. The case h = 0 is no longer identical to the Gouy-Chapman theory since with (12) we fix as zero the derivative of the total density at the wall. This makes the center of gravity of the charge move away from the electrode. Then we may expect a decrease of the capacitance compared to NLGCT. In the domain close to pzc capacitance is not only smaller than that of the NLGCT but in contrast to the NLGCT decreases with charge forming a local maximum at the pzc surrounded by two minima. Then it starts to increase in the region where the contribution from σ d<z> dσ becomes important. A similar behaviour is observed for h = -1.


D Di Caprio
email: dicaprio@ccr.jussieu.fr

J Stafiej

J P Badiali

Field theory for ionic systems. From fluctuations and structure at a hard wall to thermodynamics

Keywords: Statistical field theory, double layer, interfacial structure, surface tension, capacitance

We use a field theoretical approach to study ionic systems. In this paper we illustrate this formalism in a grand canonical ensemble for the long range Coulomb potential. For the inhomogeneous system near a hard neutral plane wall we go beyond the well known Debye -Hückel electrolyte results. The results are compared with well established sum rules for ionic systems and thermodynamic relations are verified.

We also calculate the differential capacitance in a slab using the linear response theory. The capacitance is calculated from the charge -charge correlations for the neutral system using the fluctuation -dissipation theorem.

Introduction

In a recent series of papers a considerable effort has been devoted to describe physicochemical systems with Coulombic interactions using alternative approaches to the classic statistical mechanics or density functional theories. These approaches are based on a field theoretical formalism [1,2]. The motivations for this interest are varied. In the case of systems with Coulomb potential there is a conviction that a field theoretical formalism is better suited to describe the effect of the long range Coulomb interaction. The other aspect is that in these approaches the fluctuations are treated in a different way. Some recent works [3][4][5][6] have shown that these fluctuations can be very important as they can lead to peculiar effects for instance the existence of charge induced attraction between likely charged objects.

The field theory presented here is simpler than the approaches [2] based on the Sine Gordon transformation which introduces an imaginary fluctuating field. As discussed by Fisher [7] the fluctuating imaginary field has no tangible physical meaning and it is difficult to develop intuitions about approximations based on the expansion of the Hamiltonian with respect to this field. In contrast we use fields which have a physical meaning and in the macroscopic limit they represent directly the ionic distributions coupled with the usual Coulomb potential.

In this paper we show that the field theoretical framework with a Hamiltonian based on ionic density fields provides a viable description of the system. Frusawa and Hayakawa [8] give a justification for such a treatment.

There is a set of basic relations characteristic for the field theoretical formalism which we show to acquire a new meaning when applied to the description of classical fluids. These relations appear to be similar to the well known relations in liquid state theory, like for instance the YBG equations [START_REF] Hansen | Theory of simple Liquids[END_REF] although as we will see the formulation is different.

Second, we show that the theory emphasizes different aspects in the physics of the system compared to other approaches such as the usual statistical me-chanics or density functional theories [START_REF] Evans | [END_REF]. The resulting approximations and interpretations can lead to new insights into the physics of the system. In the simple case of the neutral hard plane wall we show that the behavior of the charge fluctuations and their effect are non trivial. This is illustrated on the inhomogeneous charge -charge correlation functions at the level of the quadratic approximation. In the linear response theory we calculate the differential capacitance at the point of zero charge knowing the structure of the neutral system. We also go beyond the Debye approximation. Then as an effect of the fluctuations we have a desorption profile.

Another aim of this paper is to show that our approach is consistent with certain sum rules, thermodynamics and limiting laws. For instance the density profile is in agreement with the contact theorem [11] and the Gibbs adsorption isotherm. Differential capacitance at the point of zero charge is equal to the Gouy -Chapman value for small charges.

We use the grand canonical ensemble throughout the paper. In section 2 we present the field theoretical framework for ionic systems. In section 3 we apply the theory to the case of a slab geometry. We calculate the pressure as well as the inhomogeneous charge -charge correlation function. In the next section, we discuss on a few examples the effect of these fluctuations. Going beyond the Debye approximation, we calculate a desorption profile directly related to the charge fluctuations. The surface tension is obtained using the Gibbs adsorption isotherm. From the interfacial structure of the neutral system we calculate the differential capacitance. Finally we summarize the results in the conclusion.

2 Field theory

The Field theoretical framework

The theory is based on a Hamiltonian which is a functional of the density fields ρ i=± (r) for anions and cations. Integrating over all possible density distributions we obtain the partition function of the system

Ξ = N i Dρ i (r) e -βH[ρ i (r)]+βµ i ρ i (r)dr i (1)
where N is a normalization constant. Our formalism corresponds to the grand canonical ensemble, where µ i are the chemical potentials of the ions and for symmetric ions we assume µ + = µ -= µ. An equivalent representation is in terms of charge density and overall ionic density : q(r) = ρ + (r) -ρ -(r) and s(r) = ρ + (r) + ρ -(r). The Hamiltonian is a functional of the fields :

βH[q(r), s(r)] = βH e [q(r), s(r)] + βH c [q(r)] (2)
where the Coulomb contribution is given by

βH c [q(r)] = 1 4πε q(r)q(r ′ ) r drdr ′ (3)
where r = |r-r ′ | and β = 1/(k B T ) is the reciprocal temperature. H e [ρ + (r), ρ -(r)] is the ideal entropy contribution functional used in [8,12]

βH e [ρ + (r), ρ -(r)] = i=± ρ i (r) ln ρ i (r) ρ ref -1 dr (4)
which can be expressed alternatively in terms of q and s fields and ρ ref is an unspecified reference density. The grand potential is defined as

Ω ≡ -k B T ln Ξ
and we assume it corresponds to the thermodynamic definition γA-pV where p is the pressure, γ the surface tension A the area and V the volume of the system.

A new toolbox

Within the field theoretical framework we can find some general and useful equations. The first example is a relation similar to the equation of motion [START_REF] Feynman | Quantum Mechanics and Path Integrals[END_REF] obtained by requirement that Ξ be invariant under a shift of the dummy field integration variables : ρ i (r). The equation reads

β < δH δρ i (r) > -βµ i = 0 (5) < ln ρ i (r) > + j βV ij (r, r ′ ) < ρ j (r ′ ) > dr ′ -βµ i = 0 ( 6
)
where V ij (r, r ′ ) is the potential coupling the fields ρ i (r) to ρ j (r ′ ), and the average of any quantity A is defined

< A >= i Dρ i (r) A e -βH[ρ i (r)]+βµ ρ i (r)dr i i Dρ i (r) e -βH[ρ i (r)]+βµ ρ i (r)dr i (7)
Equations ( 5) or ( 6) are exact relations different from the saddle point equations of the mean field theories [START_REF] Justin | Quantum field theory and critical phenomena[END_REF]. They are also different from the density theory formalism [START_REF] Evans | [END_REF], as H does not represent the free energy of the system. The Hamiltonian enters into the above equation through its average.

Deriving this equation with respect to r we obtain what can be seen as the equivalent of the YBG [START_REF] Hansen | Theory of simple Liquids[END_REF] equation for the field theory.

∇ r < ρ i (r) > < ρ i (r) > = β∇ r V ext i (r) + β ∇ r V ij (r, r ′ ) < ρ j (r ′ ) > dr ′ (8)
where we introduce V ext i (r) as external fields. The equation is exact and distinct from the YBG equation. It is formally a closed relation in which only the one body distribution is involved and not the beginning of a hierarchy between correlation functions.

In an earlier paper [START_REF] Di Caprio | [END_REF] we have shown the analogy between the Ward Takahashi (WT) identities [START_REF] Justin | Quantum field theory and critical phenomena[END_REF] which result from the invariance of the system under a symmetry of the fields and relations found in liquid state theory. In the case of the translational invariance we have shown that relations obtained in [16,17] and sometimes known as Baxter relations for homonogeneous fluids have the equivalent WT type identities for the field theory.

The field theoretical framework can therefore be constructed around a few fundamental equations similar to well know relations in usual statistical mechanics although their detailed expression is different.

The model

The system we want to describe is an ionic solution confined by two hard plane walls of a surface A placed at z = L 1 and z = L 2 at a separation L = L 2 -L 1 from one another as shown in Fig. 1. The thermodynamic limit A → ∞ is considered from the beginning while for the slab thickness L we take it finite 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000

111111111
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0000000000000000000 1111111111111111111 00000000000000000000000000000000000000 11111111111111111111111111111111111111 0 1 0 1 ε ε ε L z L L 2 1
Fig. 1. Schematic representation of the model system.

first and then consider the limit L → ∞. The dielectric constant is uniform throughout the whole space and it corresponds to the dielectric constant of the pure solvent ε.

The Hamiltonian

For the open system the chemical potentials of ions are fixed and the ionic densities fluctuate around their averages in an a priori unconstrained fashion.

Hereafter, the Hamiltonian eq.( 2) is expanded around the saddle point average values q = 0 and s = ρ, which will be specified in the following. The ideal entropy becomes

βH e = ρ ln ρ 2ρ ref -1 V + δs(r) ln ρ 2ρ ref dr + 1 2ρ
δs 2 (r) + q 2 (r) dr -1 6ρ 2 δs 3 (r) + 3δs(r)q 2 (r) dr

+ 1 12ρ 3
δs 4 (r) + 6δs 2 (r)q 2 (r) + q 4 (r) dr + ...

where δs = s-ρ. The Coulomb Hamiltonian is not modified in this expansion.

The term related to the chemical potential is

s(r)dr = ρV + δs(r)dr (10)
We reorder the terms in the Hamitonian according to powers of δs. The term independent of δs is

βH 0 = ρ ln ρ ρ ref -1 V -βµρV (11)
The term linear in the fields is

βH 1 = L 1 <z 1 <L 2 ln ρ ρ ref -βµ δs(r 1 )dr 1 (12)
Now we choose ρ such that the linear term βH 1 vanishes. This provides a relation between µ and ρ βµ = ln ρ ρ ref [START_REF] Feynman | Quantum Mechanics and Path Integrals[END_REF] replacing µ in eq.( 11), we then have βH 0 = -ρV . The Hamiltonian is now

βH = -ρV + βH c [q(r)] + 1 2ρ δs 2 (r) + q 2 (r) dr - 1 6ρ 2 δs 3 (r) + 3δs(r)q 2 (r) dr + 1 12ρ 3
δs 4 (r) + 6δs 2 (r)q 2 (r) + q 4 (r) dr + ...

We denote by H 2 the quadratic part of the Hamiltonian and treat it as the reference system. Subsequent terms will be treated perturbatively βδH = -1 6ρ 2 δs 3 (r) + 3δs(r)q 2 (r) dr

+ 1 12ρ 3
δs 4 (r) + 6δs 2 (r)q 2 (r) + q 4 (r) dr

In the following we consider the quadratic Hamiltonian in which there is no coupling between the two fields q(r) and s(r).

The quadratic Hamiltonian

Diagonal representation of the quadratic Hamiltonian

The fields are expanded on the basis which is written [18,19] q(r) = K,ν>0

e iKR q ν ϕ ν (z), s(r) = K,ν>0
e iKR a ν ϕ ν (z) (16) where q ν and a ν are the parameters defining the fields and

ϕ ν (z) = e -iν(L 2 -z) ∓ e iν(L 1 -z) 2L 1 ∓ sin(νL) νL 1/2 (17)
these functions are eigenfunctions of the electric potential operator and the + and -refer to even and odd functions.

V (r) = e 4πε dr ′ q(r ′ ) |r -r ′ | (18) = e ε K e iKR ν>0 λ ν qν (K)ϕ ν (z) (19)
where

λ ν = 1/(K 2 + ν 2 ) with the eigenvalues ν ± verifying tan( ν + L 2 ) = - ν + K and tan( ν -L 2 ) = K ν - (20)
In this basis the quadratic Hamiltonian H 2 reads

βH 2 = A 2ρ K,ν(K) a ν a * ν + q ν q * ν (1 + K 2 D [ρ]λ ν ) (21)
where K D [ρ] = β ρe 2 /ε is the inverse Debye length with a ionic density ρ.

Calculation of the pressure

From the grand potential eq.( 5) we can calculate the pressure βpV = ρV + ln N DqDs e -βH 2 [q,s]

As we focus on electrostatic effects we calculate the excess pressure relative to the uncharged system.

β∆pV = (ρ -ρ b )V + ln DqDs e -βH 2 [q,s] DqDs e -βH 2 [q,s]| e=0 ( 22
)
where ∆p is the excess pressure of the refered to the pressure of the corresponding ideal gas. The density of the ideal gas is taken as ρ b the average density of the interacting gas. This density is different from ρ which corresponds only to the density of the saddle point approximation. H 2 [q, s]| e=0 is the Hamiltonian of the ideal gas, where the Coulomb interaction has been switched off. We then have

β∆pV = (ρ -ρ b )V + 1 2 K,ν(K)>0 ln K 2 + ν 2 K 2 + ν 2 + K 2 D [ρ] (23)
The sum in the above equation can be explicitely calculated [18]. In the limit L → ∞ it becomes.

ν ln K 2 + ν 2 K 2 + ν 2 + K 2 D = -2( K 2 + K 2 D -K)L -4 ln   K + K 2 + K 2 D 2K   (24)
The first term corresponds to the bulk contribution and we have

β∆p = ρ -ρ b + K 3 D [ρ] 12π (25)
The ionic density is different from that of the saddle point:

ρ b V ≡ < N > = ∂ ln Θ ∂ ρ dρ dβµ = ρ ∂ ln Θ ∂ ρ (26)
Where we have used the definition of ρ given in eq.( 13) With the expression of ln Θ derived from eq.( 25), we have

ρ b = ρ + K 3 D [ρ] 8π (27)
Finally the excess pressure is

∆p = - k B T K 3 D [ρ] 24π ( 28
)
At the lowest order in the expansion we can replace ρ by ρ b in K D . The result we obtain for the pressure is the same as in the canonical ensemble [18]. We emphasize that the density ρ b is not an external parameter of the system but has to be calculated perturbatively at each level of approximation as illustrated by eq.( 27).

Charge -charge correlation functions

The interest of the field theoretical approach is to stress the importance of fluctuactions and of the resulting structure as given by the correlation function.

We calculate the charge -charge correlations in the vicinity of a plane hard wall. The quadratic Hamiltonian is the same as the one used in [18,19] and the results are the same except that the density is replaced by ρ. The bulk correlation function does not depend on the distance to the walls:

< q(r)q(r ′ ) > b = ρ δ(r -r ′ ) - K 2 D [ρ] 4π e -K D [ρ]r r ( 29
)
The surface contribution is

< q(r)q(r ′ ) > s = i=1,2 ρe -K D [ρ]r * i 2π K 2 D [ρ] 2r * i + K D [ρ] r * 2 i - K 2 D [ρ]z 2 i -1 r * 3 i -3 K D [ρ]z 2 i r * 4 i -3 z2 i r * 5 i + ρ 2π K 2 dKJ 0 (Kρ)e -|z i | √ K 2 +K 2 D [ρ]
(30) [START_REF] Spanier | An Atlas of Functions[END_REF]. The first contribution is screened over the Debye length and r * i is the distance between the point r and the image r ′ * of the second point symmetric with respect to the wall at L 1 and L 2 respectively. It is similar to image contribution with screening however the dielectric constant is uniform across the system and there is no explicit image potential. The charge -charge correlations verify the asymptotic behaviour predicted in [21] and decay as 1/R 3 parallel to the wall without screening as a consequence of the local dipole moments created by the deformation of the screening cloud in the vicinity of the wall. The coefficient ρ can be replaced by ρ b in eq.( 29) when we take into account the standard one loop correction due to the q 4 coupling term in δH [START_REF] Justin | Quantum field theory and critical phenomena[END_REF]. As for the other ρ's which appear in K D [ρ] they can be systematically replaced by ρ b at the lowest order in the expansion. Integrating over r ′ , we can check that the inhomogeneous correlation function verifies the electroneutrality condition.

where z1 = z + z ′ -2L 1 , z2 = z + z ′ -2L 2 , ρ = |R -R ′ | and J 0 is a Bessel function
4 From fluctuations to thermodynamics.

In this section we will see how the above calculated charge fluctuations give rise to a nontrivial desorption profile and to the electrostatic contribution to the surface tension for the neutral system. We will also calculate the differential capacitance of the system in the linear response theory.

Desorption density profile.

The expansion of the ideal entropy beyond the quadratic term [18] : δH = -1 2ρ 2 δs(r)q 2 (r)dr couples charge and density fields, this term is treated as a perturbation. The average of any quantity with the full Hamiltonian is then written

⟨A⟩ = ⟨A⟩ 0 -⟨A βδH⟩ 0 + ... (31)
where the averages ⟨...⟩ and ⟨...⟩ 0 refer to the full Hamiltonian and to the quadratic Hamiltonian respectively. With the new Hamiltonian the density profile at the wall is modified. As we are interested in the profile at the wall we subtract the corresponding bulk contribution. Eq. (31) for the ionic density profile reads

⟨δs⟩ = q(r) q(r ′ )δs(r ′ ) 2 dr ′ 2ρ 2 b 0 - q(r) q(r ′ )δs(r ′ ) 2 dr ′ 2ρ 2 b 0,bulk (32)
where we use the fact that the profile ⟨δs⟩ 0 for the quadratic Hamiltonian is zero. In the quadratic Hamiltonian the fields q and s are independent. The expression for the profile can be obtained in the form

⟨δs(r)⟩ = - K 3 D 8π I[(z -L 1 )K D ] (33)
where

I(x) = ∞ 1 dt e -2xt /(t + √ t 2 -1)
2 is a fast decaying function. In the vicinity of one wall at z = L 1 the correction to the total density profile given by eq.( 33) is plotted in figure 2. We have I(0) = 1/3 which yields the contact value of the corrected profile ⟨δs(z = L 1 )⟩ = -K 3 D 24π consistent with the contact theorem [11] and the Debye-Hückel correction for the ionic gas pressure eq.( 28). Vice versa this shows that the quadratic Hamiltonian, where the profile at the wall is constant, does not fulfill the mechanical equilibrium expressed by the contact theorem.

Surface excess free energy

The excess free energy due to the electrostatics has been calculated in the bulk [1] and the contribution due to the surface has been calculated in two ways in [18]. Let us recollect the results. The bulk contribution to the free energy is

βF b = -V K 3 D 12π . ( 34
)
And the surface contribution to the free energy can be obtained directly from the partition function

βF s | L→∞ = AK 2 D 4π 2 ln 2 -1 4 . ( 35
)
An alternative way to calculate the free energy is using the charge -charge correlation functions and a charging process. We define

β Fλ ≡ dr 1 dr 2 βH c (r 1 , r 2 ; λ)g(r 1 , r 2 ; λ) (36)
where βH c (r 1 , r 2 ; λ) = λ 2 βe 2 8πε|r 1 -r 2 | and where we consider a charging process by which g(r 1 , r 2 ; λ) is the charge -charge correlation function for a system with elementary electric charge λe instead of e. The electrostatic contribution to the free energy for a system of elementary electric charges e is

1 0 dλ β Fλ = β F| e -β F| e=0 . (37)
The contribution from the bulk part of the correlation function eq. ( 29) yields

β Fb = -V K 3 D 12π + A K 2 D 16π , (38)
The contribution from the surface correlation given in eq. ( 30) is

β Fs | L→∞ = A K 2 D 8π (ln 2 -1). ( 39
)
Summing the two contributions, we find that we have indeed F b + F s | L→∞ = Fb + Fs | L→∞ . This shows the consistency of the two ways of calculating the free energy.

Finally the surface tension can also be calculated from the Gibbs adsorption isotherm

dγ = 2Γ salt dµ (40)
where Γ salt is the adsorption of the salt per unit surface area. The adsorption can be calculated from the density profile eq.( 33). Integrating the isotherm from zero density to the final density of the system we have that the surface tension is

βγ = K 2 D (2 ln 2 -1) 32π ( 41
)
This result is consistent with the results above.

δH as the perturbation in eq.( 31). The potential difference between the two walls can then be written in the following way :

V (L 2 ) -V (L 1 ) = V ext (r) + e 4πε q(r ′ ) |r -r ′ | dr ′ 1 -βe q(r ′′ )V ext (r ′′ )dr ′′ 0,z=L 2 -... 0,z=L 1 = δV ext + e 4πε q(r ′ ) |r -r ′ | dr ′ 0,z=L 2 -... 0,z=L 1 - βe 2 4πε q(r ′ ) |r -r ′ | dr ′ q(r ′′ )V ext (r ′′ )dr ′′ 0,z=L 2 -... 0,z=L 1 -βe V ext (L 2 ) q(r ′′ )V ext (r ′′ )dr ′′ 0,z=L 2 -... 0,z=L 1 (45)
where for each quantity its difference for z = L 2 and z = L 1 is taken. The first term is δV ext = V ext (z = L 2 ) -V ext (z = L 1 ) = -σeL/ε the potential difference due to the electric charges on the wall. The second term is the average of the Coulomb potential created by the charges in the ionic solution but taken for neutral walls, in this case electric potential is constant throughout the system and this difference is zero [12,22]. The last term is proportional to σ 2 and can be neglected in the linear response regime. Finally we have [23] V

(L 2 ) -V (L 1 ) = δV ext + σeL ε - 2σe εK D 1 -e -K D L = - 2σe εK D 1 -e -K D L (46)
Deriving eq.( 46) with respect to σ

C -1 = Ṽ (L 2 ) -Ṽ (L 1 ) -σe = 2 εK D
1 -e -K D L (47) which in the large L limit corresponds to the differential capacitance for the Gouy -Chapman theory.

Conclusion

In this paper on a specific example, an ionic system in a slab, we show that the field theoretical approach can be used for a coherent description. The specific features of systems with Coulomb potential are reproduced. The bulk charge correlation function obeys the required sum rules [22] and the inhomogeneous charge correlation function verifies the electroneutrality and the asymptotic behaviour parallel to the wall predicted by B. Jancovici [21].

The theory is based on fields which have a clear physical meaning. Our approach can be situated between the density functional theory and the usual statistical mechanics. As for the density functional theory the choice of ionic densities leads to a description closer to the macroscopic level and better understood. On the other hand with ionic densities as stochastic fields our treatment is close to the usual statistical mechanics and the assumptions at the level of the Hamiltonian are simpler than those at the level of the density functional.

With a simple extension of the Debye -Hückel theory we predict a nontrivial desorption profile at the interface of the ionic solution with a neutral hard wall.

In contrast to the Debye -Hückel linear approximation, this result restores the consistency of the approach with the contact theorem and Gibbs adsorption thermodynamics.

Different tools and ways of thinking specific to the field theoretical approach yield new points of view. For instance the desorption profile is described as a result of the coupling between the charge and density fields due to the ideal entropy. This is a different point of view from the standard liquid state theory where the ideal entropy is usually written as an explicit and independent contribution to the free energy.

Fig. 2 .

 2 Fig. 2. Plot of the function I related to the total density profile correction across the interface.

Differential capacitance.

000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000

Fig. 3. Schematic representation of the model system with wall with surface charge σ.

In this section the Coulomb Hamiltonian includes an external contribution accounting for the fact that the walls can be charged βH c [q(r)] = βe 2 4πε q(r)q(r ′ ) r drdr ′ + βe q(r)V ext (r)dr,

V ext (r) is the potential created by the surface charge densities σ and -σ on the walls. This contribution is

where the zero for the electric potential is set in the middle of the system. To calculate the capacitance of the system we consider the linear response theory for a small surface charge density σ. The Hamiltonian can be rewritten H = H σ=0 + H σ with H σ = e q(r)V ext (r)dr (44)

All averages are then performed with the Hamiltonian H σ=0 and H σ replaces