Pierrick Tranouez
email: pierrick.tranouez@univ-rouen.fr

Antoine Dutot
email: antoine.dutot@univ-lehavre.fr

Different goals in multiscale simulations and how to reach them

Keywords: Multiscale, clustering, dynamic graphs, adaptation

In this paper we sum up our works on multiscale programs, mainly simulations. We first start with describing what multiscaling is about, how it helps perceiving signal from a background noise in a flow of data for example, for a direct perception by a user or for a further use by another program. We then give three examples of multiscale techniques we used in the past, maintaining a summary, using an environmental marker introducing an history in the data and finally using a knowledge on the behavior of the different scales to really handle them at the same time.

Introduction: What this paper is about, and what it's not

Although we delved into different applications and application domains, the computer science research goals of our team has remained centered on the same subject for years. It can be expressed in different ways that we feel are, if not exactly equivalent, at least closely connected. It can be defined as managing multiple scales in a simulation. It also consists in handling emergent structures in a simulation. It can often also be seen as dynamic heuristic clustering of dynamic data 3 . This paper is about this theme, about why we think it is of interest and what we've done so far in this direction. It is therefore akin to a state of the art kind of article, except more centered on what we did. We will allude to what others have done, but the focus of the article is presenting our techniques and what we're trying to do, like most articles do, and not present an objective description of the whole field, as the different applications examples could make think : we're sticking to the same computer science principles overall. We're taking one step back from our works to contemplate them all, and not the three steps which would be necessary to encompass the whole domain, as it would take us beyond the scope of this book.

2 Perception: filtering to make decisions I look at a fluid flow simulation but all I'm interested in is where does the turbulence happen, in a case where I couldn't know before the simulation [Tranouez 2005a]. I use a multi-participant communication system in a crisis management piece of software and I would like to know what are the main interests of each communicant based on what they are saying [Lesage 1999]. I use an Individual-Based Model (IBM) of different fish species but I'm interested in the evolution of the populations, not the individual fish [Prevost 2004]. I use a traffic simulation with thousands of cars and a detailed town but what I want to know is where the traffic jams are (coming soon).

In all those examples, I use a piece of software which produces huge amounts of data but I'm interested in phenomena of a different scale than the raw basic components. What we aim at is helping the user of the program to reach what he is interested in, be this user a human (Clarification of the representation) or another program (Automatic decision making). Although we're trying to stay general in this part, we focused on our past experience of what we actually managed to do, as described in "Some techniques to make these observations in a time scale comparable to the observed", this is not gratuitous philosophy.

Clarification of the representation

This first step of our work intends to extract the patterns on the carpet from its threads [Tranouez 1984]. Furthermore, we want it to be done in "real (program) time", meaning not a posteriori once the program is ended by examining its traces [Servat 1998], and sticking as close as possible to the under layer, the one pumping out dynamic basic data. We don't want the discovery of our structures to be of a greater time scale than a step of the program it works upon.

How to detect these structures? For each problem the structure must be analyzed, to understand what makes it stand out for the observer. This implies knowing the observer purpose, so as to characterize the structure. The answers are problem specific, nevertheless rules seem to appear.

In many situations, the structures are groups of more basic entities, which then leads to try to fathom what makes it a group, what is its inside, its outside, its frontier, and what makes them so.

Quite often in the situation we dealt with, the groups members share some common characteristics. The problem in that case belongs to a subgenre of clustering, where the data changes all the time and the clusters evolve with them, they are not computed from scratch at each change.

The other structures we managed to isolate are groups of strongly communicating entities in object-oriented programs like multiagent simulations. We then endeavored to manage these cliques.

In both cases, the detected structures are emphasized in the graphical representation of the program. This clarification lets the user of the simulation understand what happens in its midst. Because modeling, and therefore understanding, is clarifying and simplifying in a chosen direction a multi-sided problem or phenomenon, our change of representation participates to the understanding of the operator. It is therefore also a necessary part of automating the whole understanding, aiming for instance at computing an artificial decision making.

Automatic decision making

Just like the human user makes something of the emerging phenomena the course of the program made evident, other programs can use the detected organizations.

For example in the crisis management communication program, the detected favorite subject of interest of each of the communicant will be used as a filter for future incoming communications, favoring the ones on connected subjects. Other examples are developed below, but the point is once the structures are detected and clearly identified, the program can use models it may have of them to compute its future trajectory. It must be emphasized that at this point the structures can themselves combine into groups and structures of yet another scale, recursively. We're touching there an important component of complex system [Simon 1996]. We may hope the applications of this principle to be numerous, such as robotics, where perceiving structures in vast amounts of data relatively to a goal, and then being able to act upon these accordingly is a necessity.

We're now going to develop these notions in examples coming from our past works.

Some techniques to make these observations in a time scale comparable to the observed

The examples of handling dynamic organization we chose are taken from two main applications, one of a simulation of a fluid flow, the other of the simulation of a huge cluster of computed processes, distributed over a dynamic network of computing resources, such as computers. The methods titled "Maintaining a summary of a simulation" and "Reification: behavioral methods" are theories from the hydromechanics simulation, while "Traces of the past help understand the present" refers to the computing resources management simulation. We will first describe these two applications, so that an eventual misunderstanding of what they are does not hinder later the clarity of our real purpose, the analysis of multiscale handling methods.

In a part of a more general estuarine ecosystem simulation, we developed a simulation of a fluid flow. This flow uses a particle model [Leonard 1980], and is described in details in [Tranouez 2005a] or [Tranouez 2005b]. The basic idea is that each particle is a vorticity carrier, each interacting with all the others following Biot-Savart laws. As fluid flows tend to organize themselves in vortices, from all spatial scales from a tens of angstrom to the Atlantic Ocean, this is these vortices we tried to handle as the multiscale characteristic of our simulation. The two methods we used are described below.

The other application, described in depth in [Dutot 2005], is a step toward automatic distribution of computing over computing resources in difficult conditions, as:

• The resources we want to use can each appear and disappear, increase or decrease in number. • The computing distributed is composed of different object-oriented entities, each probably a thread or a process, like in a multiagent system for example (the system was originally imagined for the ecosystem simulation alluded to above, and the entities would have been fish, plants, fluid vortices etc., each acting, moving . . . ) Furthermore, we want the distribution to follow two guidelines:

• As much of the resources as possible must be used,

• Communications between the resources must be kept as low as possible, as it should be wished for example if the resources are computers and the communications therefore happen over a network, bandwidth limited if compared to the internal of a computer.

This the ultimate goal of this application, but the step we're interested in today consists in a simulation of our communicating processes, and of a program which, at the same time the simulated entities act and communicate, advises how they should be regrouped and to which computing resource they should be allocated, so as to satisfy the two guidelines above.

Maintaining a summary of a simulation

The first method we would like to describe here relates to the fluid flow simulation. The hydrodynamic model we use is based on an important number of interacting particles. Each of these influences all the others, which makes n 2 interactions, where n is the number of particles used. This makes a great number of computations. Luckily, the intensity of the influence is inversely proportional to the square of the distance separating two particles. We therefore use an approximation called Fast Multipoles Method (FMM), which consists in covering the simulation space with grids, of a density proportional to the density of particles (see Figure 2-a). The computation of the influence of its colleagues over a given particle is then done exactly for the ones close enough, and averaged on the grid for those further. All this is absolutely monoscale.

As the particles are vorticity carriers, it means that the more numerous they are in a region of space, the more agitated the fluid they represent is. We would therefore be interested in the structures built of close, dense particles, surrounded by sparser ones. A side effect of the grids of the FMM, is that they help us do just that. It is not that this clustering is much easier on the grids, it's above all that they are an order of magnitude less numerous, and organized in a tree, which makes the group detection much faster than if the algorithm was ran on the particles themselves. Furthermore, the step by step management of the grids is not only cheap (it changes the constant of the complexity of the particles movement method but not the order) but also needed for the FMM.

We therefore detect structures on

• Dynamic data (the particles characteristics)

• With little computing added to the simulation, which is what we aimed at.

The principle here is that through the grids we maintain a summary of the simulation, upon which we can then run static data algorithm, all this at a cheap computing price.

Traces of the past help understand the present

The second method relates to the detection of communication clusters inside a distributed application. The applications we are interested in are composed of a large number of object-oriented entities that execute in parallel, appear, evolve and, sometimes, disappear. Aside some very regular applications, often entities tend to communicate more with some than with others. For example in a simulation of an aquatic ecosystem, entities representing a species of fish may stay together, interacting with one another, but flee predators. Indeed organizations appear groups of entities form. Such simulations are a good example of applications we intend to handle, where the number of entities is often too large to compute a result in an acceptable time on one unique computer.

To distribute these applications it would be interesting to both have approximately the same number of entities on each computing resource to balance the load, but also to avoid as much as possible to use the network, that costs significantly more in terms of latency than the internals of a computer. Our goal is therefore to balance the load and minimize network communications. Sadly, these criteria are conflicting, and we must find a tradeoff.

Our method consists in the use of an ant metaphor. Applications we use are easily seen as a graph, which is a set of connected entities. We can map a -Each color corresponds to a detected aggregate b -Each color corresponds to a computing ressource Fig. 2. Detection of emergent structures in two applications with distinct methods entities to vertices of the graph, and communications between these entities to the edges of the graph. This graph will follow the evolution of the simulation. When an entity appear, a vertex will appear in the graph, when a communication will be established between two entities, an edge will appear between the two corresponding vertices. We will use such a graph to represent the application, and will try to find clusters of highly communicating entities in this graph by coloring it, assigning a color to each cluster. This will allow to identify clusters as a whole and use this information to assign not entities, but at another scale, clusters to computing resources (Figure 2-b).

For this, we use numerical ants that crawl the graph as well as their pheromones, olfactory messages they drop, to mark clusters of entities. We use several distinct colonies of ants, each of a distinct color, that drop colored pheromones. Each color corresponds to one of the computing resources at our disposal. Ants drop colored pheromones on edges of the graph when they cross them. We mark a vertex as being of the color of the dominant pheromone on each of its incident edges. The color indicates the computing resource where the entity should run.

To ensure our ants color groups of highly communicating entities of the same color to minimize communications, we use the collaboration between ants: ants are attracted by pheromones of their own color, and attracted by highly communicating edges. To ensure the load is balanced, that is to ensure that the whole graph is not colored only in one color if ten colors are available, we use competition, ants are repulsed by the pheromones of other colors.

Pheromones in nature being olfactory molecules, they tend to evaporate. Ants must maintain them so they do not disappear. Consequently, only the interesting areas, zones where ants are attracted, are covered by pheromones and maintained. When a zone becomes less interesting, ants leave it and pheromone disappear. When an area becomes of a great interest, ants colonize it by laying down pheromones that attract more ants, and the process self-amplifies.

We respect the metaphor here since it brings us the very interesting property of handling the dynamics. Indeed, our application continuously changes, the graph that represents it follows this, and we want our method to be able to discover new highly communicating clusters, while abandoning vertices that are no more part of a cluster. As ants continuously crawl through the graph, they maintain the pheromone color on the highly communicating clusters. If entities and communications of the simulation appear or disappear, ants can quickly adapt to the changes. Colored pheromones on parts where a cluster disappeared evaporate and ants colonize new clusters in a dynamic way. Indeed, the application never changes completely all the time; it modifies itself smoothly. Ants lay down "traces" of pheromones and do not recompute the color of each vertex at each time, they reuse the already dropped pheromone therefore continuously giving a distribution advice at a small computing price, and adapting to the reconfigurations of the underlying application.

Reification: behavioral methods

This last example of our multiscale handling methods was also developed on the fluid flow simulation (Figure 3). Once more, we want to detect structures in a dynamic flow of data, without getting rid of the dynamicity by doing a full computation on each step of the simulation. The idea here is doing the full computation only once in a good while, and only relatively to the unknown parts of our simulation. We begin with detecting vortices on the basic particles once. Vortices will be a rather elliptic set of close particles of the same rotation sense. We then introduce a multiagent system of the vortices (Figure 3-right). We have indeed a general knowledge of the way vortices behave. We know they move like a big particle in our Biot-Savard model, and we model its structural stability through social interactions with the surrounding basic particles, the other vortices and the obstacles, through which they can grow, shrink or die (be dissipated into particles). The details on this can be found in [Tranouez 2005a]. Later on, we occasionally make a full-blown vortex detection, but only on the remaining basic particles, as the already detected vortexes are managed by the multiagent system.

In this case, we possess knowledge on the structures we want to detect, and we use it to build actually the upper scale level of the simulation, which at the same time lightens ulterior structures detection. We are definitely in the category described in Automatic decision making.

Conclusion

Our research group works on complex systems and focuses on the computer representation of their hierarchical/holarchical characteristics [START_REF] Koestler | Janus. A Summing Up[END_REF]], [Simon 1996], [Kay 2000]. We tried to illustrate that describing a problem at different scales is a well-spread practice at least in the modeling and simulating community. We then presented some methods for handling the different scales, with maintaining a summary, using an environmental marker introducing a history in the data and finally using knowledge on the behavior of the different scales to handle them at the same time.

We now believe we start to have sound multiscale methods, and must focus on the realism of the applications, to compare the sacrifice in details we make when we model the upper levels rather than just heavily computing the lower ones. We save time and lose precision, but what is this trade-off worth precisely?

Fig. 1 .

 1 Fig. 1. Studies of water passing obstacles and falling by Leonardo Da Vinci, c. 1508-9. In Codex Leicester.

Fig. 3 .

 3 Fig. 3. Fluid flow around an obstacle. On the left, the initial state. On the right, a part of the flow, some steps later (the ellipses are vortices)

We will of course later on describe in more details what we mean by all this.


Denis E Corpet

Ecological factors influencing the transfer of plasmids in vitro and in vivo

Plasmid transfer in the gut depends on the concentrations of recipient and donor bacteria and on their fertility. The main determining factors for the former are flow rate, concentration of limiting substrate, growth rates, adhesion, contamination and segregation and of the latter plasmid content, growth phase, and ecological conditions. These factors have to be measured with experimental models, and included in mathematical models, to estimate their relative incidence.

Introduction

Numerous experiments have been conducted to determine factors influencing plasmid transfer. In this article attention will be focused on the ecology of Escherichia coli in the gut or in anaerobic continuous flow cultures. A review of the literature distinguishes two types of experiments. In some of them, the potential plasmid donor and recipient bacteria are dominant in the ecosystem while, in others, they are overcome by a predominant flora.

I. Potential mating bacteria dominating the ecosystem

Examples of this include batch cultures (broth or agar plate), continuous flow cultures (chemostat) in the absence of other bacteria, germ-free animals (mice, chicken, pigs) dosed with a mating mixture [START_REF] Sansonetti | Parameters controlling interbacterial plasmid spreading in a gnotoxenic chicken gut system, influence of plasmid and bacterial mutations[END_REF]Corpet, unpublished results), very young animals [START_REF] Smith | The transfer of antibiotic resistance between strains of enterobacteria in chicken, calves and pigs[END_REF], humans or animals treated with antibiotics [START_REF] Datta | R factors in E. coli in faeces after oral chemotherapy in general practice[END_REF][START_REF] Anderson | Viability of, and transfer of a plasmid from, E. coli K12 in the human intestine[END_REF], and humans or animals with an enteric infection (leading to the dominance of enterobacteria in the gut). In these ecological situations, the transfer of plasmids occurred rapidly and extensively.

II. Potential mating bacteria sub-dominant in the ecosystem

Examples of this include continuous flow cultures (with diluted broth, rapid flow rate or brief incubation time [START_REF] Lebek | R-selection of subbacteriostatic tetracycline concentrations[END_REF], or with associated complex ftora), gnotobiotic animals colonized by bacteria [START_REF] Freter | Experimental and mathematical models of E. coli plasmid transfer in vitro and in vivo[END_REF][START_REF] Andremont | Plasmid-mediated susceptibility to intestinal microbial antagonisms in E. coli[END_REF], and adult humans or animals [START_REF] Wiedemann | Die Uebertragung extrachromosomaler Resistenzfaktoren in der Darmflora und ihre Hemmung[END_REF]. In these ecological situations, the plasmid transfer is a rare event. This rarity may be due to specific transfer inhibitors synthesized by the anaerobes.

In the systems cited under (I), above, donor and recipient bacteria grow actively and reach high densities, while in ecosystems cited under (II), the concentration of potential mating bacteria is low, and their growth phase is almost static. This raises the question of the different factors involved in plasmid transfer, and their relative importance. The question was studied by [START_REF] Freter | Experimental and mathematical models of E. coli plasmid transfer in vitro and in vivo[END_REF], and their approach, both experimental and mathematical, has produced clear answers which are considered below.

Factors involved in plasmid transfer

The plasmid transfer, estimated by the number of new conjugants created/unit of time, is directly proportional to the number of recipient and donor bacteria/unit volume, because mating requires the presence of both cells. It depends also on the fertility of the strains, i.e. their success in donating or accepting a plasmid.

The number of recipient bacteria/ml [N] depends on:

(a) The flow rate constant [Ro] In the digestive tract, the flow rate depends on the animal species, diet and flora (e.g. germfree (axenic) rodents have slow intestinal transit), whereas in a chemostat it is fixed. The flow causes a wash-out of the bacteria, decreasing their number [-Ro x N], and it brings new nutrients to the remaining populations, allowing continuous growth.

(b) The concentration of the limiting substrate [S]

The concentration of substrate that enters the system is fixed in vitro (composition of the broth) but depends partially on the composition of the food in vivo. The limiting substrate is mainly metabolized by the predominant flora (anaerobes), which compete with E. coli for nutrients. The resulting concentration of substrate [Sr] is available for E. coli. The flow rate brings fresh material into the system. increasing the substrate concentration [+Ro x (Sr -S)], whereas the growth of donor and recipient bacteria decreases the substrate concentration. The nature of the limiting substrate in vivo cannot be known, but the numerical value of its concentration can be estimated from the number of bacteria produced in the gut, in relation to the specific yield [Y] of the substrate (the ratio of grains of bacteria produced/grams of substrate used).

(c) Growth rate [Psi] The intrinsic characteristics of growth of the dominant strain are the maximum growth rate [Mu] (at optimal concentration of the substrate) and the saturation constant for the limiting substrate [Ks] (which is equal to the substrate concentration which leads to half the maximum growth rate). According to the classical Monod equation, the growth rate is a function of the limiting substrate concentration [Psi = Mu x S/(Ks+S)]. The growth rate may be lowered by natural inhibitors (bile salts, volatile fatty acids, etc.) or ingested antibiotics. A measurement of the absolute bacterial replication in vivo was proposed by [START_REF] Hooke | Quantitative determination of bacterial replication in vivo[END_REF], by inoculation of the ecosystem with a mixture of thermosensitive mutants and wild type strains, and the measure of their ratio. The growth rate allowing the increase in the number of bacteria [+Psi x N], correlatively decreases the substrate concentration [-Psi x N/Y].

(d) Adhesion to the wall Adhesion may be nonspecific; e.g. [START_REF] Cohen | Adhesion of a human fecal E. coli strain to mouse colonic mucus[END_REF] demonstrated adhesion of an E. coli strain of human origin to the mouse colonic mucosa; the adhesive structures were the lipopolysaccharide and a mucosal glycoprotein. Alternatively it may be specific for the strain (e.g. a plasmid governing the synthesis of adhesin). The characteristics of the wall also must play a role and will vary with the animal species and the immune status. Very low levels of antibiotics can inhibit the adhesion of bacteria, without any other detectable effect. Bound bacteria avoid being washed out of the gut.

The number of plasmid donor bacteria/ml [NR] depends on:

(a) Contamination The input of donor bacteria in the gut depends on the frequency of contamination and on the density of the inoculum (number of ingested plasmids). For example, [START_REF] Linton | The colonization of the human gut by antibiotic resistant E. coli from chickens[END_REF] described the contamination of the human gut by resistant E. coli from chicken carcasses. Petrocheilou, Richmond & Bennett (1977) clearly showed the spread of an R-plasmid from a wife to her husband and, according to [START_REF] Datta | R factors in E. coli in faeces after oral chemotherapy in general practice[END_REF], resistant E. cnli appearing after antimicrobial therapy were never the original predominant serotype with acquired resistance, neither was there overgrowth of the resistant serotype previously prescrit in small numbers; there was always colonization of the bowel with a new resistant serotype. The quantitative incidence of contamination still remains to be determined in normal life. According to numerous authors (e.g. [START_REF] Andremont | Plasmid-mediated susceptibility to intestinal microbial antagonisms in E. coli[END_REF] most R-plasmids have a detrimental effect on the in-vivo growth rates of their hosts bacteria. Some plasmids, however, increase the invivo fitness of the bacterium carrying them, for reasons which are not known. The growth rate of the donor bacteria depends on its maximum growth rate [MuR], on its saturation constant for the limiting substrate [KsR], and on [S]. Antibiotics will not decrease the growth rate of resistant bacteria.

(d) Adhesion of donor bacteria

Generally, the donor bacterium reaches the gut where plasmid-free bacteria are dominant. Adhesion sites are occupied by the first invader; later ones have few chances to find available sites, unless they have specific adhesion structures, namely type 1 pili, or attachment antigens coded by plasmids (e.g. K99). The production of adhesin is itself a function of growth rate [START_REF] Van Verseveld | Production of fimbrial adhesins K99 and F41 by enterotoxigenic E. coli as a function of growth rate domain[END_REF]. Duval-iflah, Raibaud & Rousseau (1981) demonstrated a phenotypic adaptive mechanism which occurred in E. coli associated with germ-free mice for al least 30 days, and allowed the adapted bacteria systematically to overcome any non-adapted strain.

(e) Segregation

The rate of segregation is low and is dependent on the plasmid. Its effect is proportional to the number of plasmid-bearing bacteria that are multiplying, because the loss of plasmid occurs when a bacterium divides into two new cells, one of which is plasmid-free. Segregation, which has no detectable effect in vitro, in the long-term determines equilibrium in vivo.

Specific parameters of the transfer itself

These factors are included in the concept of fertility which is formalized by the transfer rate constant [Gamma]. It is the intrinsic ability to transfer the plasmid in given conditions (if it was really an intrinsic constant, the conditions should not modify it). The number of new conjugants/unit of time is [+Gamma x N x NR].

(a) Plasmid

The nature of the plasmid is the most important determinant of the fertility. Some plasmid characteristics are well known, for example, the ability to conjugate (non-conjugative plasmids needing the presence of a conjugative helper plasmid to cotransfer) and the repressed status. In newly formed conjugants, the transfer functions are stimulated, and consequently [Gamma] increases a lot. Most of the in-vivo transfers are in fact due to derepressed new conjugants [START_REF] Sansonetti | Parameters controlling interbacterial plasmid spreading in a gnotoxenic chicken gut system, influence of plasmid and bacterial mutations[END_REF]. After a certain number of generations, the transfer functions are repressed again, except in permanently derepressed drd mutants.

(b) Donor and recipient bacteria

The ability to give and accept a plasmid are characteristics of the mating strains, and depends on their stat.us.

(i) Growth phase. The effect of the static phase of growth was studied in vitro by [START_REF] Levin | The kinetics of conjugative plasmid transmission: fit of a simple mass action model[END_REF] and by [START_REF] Freter | Experimental and mathematical models of E. coli plasmid transfer in vitro and in vivo[END_REF] in vivo, injection of morphine to mice strongly reduces their intestinal motility, and consequently Ro. S and Psi. They found that the transfer was severely impaired by static conditions. In particular, there was a strong relationship between the growth rate of the donor bacterium and its fertility. In the gut, the transient potential donor strains are generally prevented from multiplying by the antagonistic effect of the resident microflora. This explains the considerably lower fertility of a strain in the presence of a complex flora than in a test tube or in a sterile gut.

(ii) Wild type smooth strains are less efficient donors and recipients than rough strains. This fact explains the maintenance and spread of non-conjugative plasmids among populations of wildtype E. coli strains in vivo for a long time, because the low intrinsic efficiency for plasmid transfer ensures the presence of a large plasmid-free population which is potentially able to act as recipients to any plasmid.

(iii) Conjugative systems; the con mutation in the recipient bacterium strongly decreases the rate of transfer [START_REF] Sansonetti | Parameters controlling interbacterial plasmid spreading in a gnotoxenic chicken gut system, influence of plasmid and bacterial mutations[END_REF].

(iv) The presence of an incompatible plasmid in the potential recipient, because two plasmids of the same incompatibility group are mutually exclusive, or of an enhancer plasmid, e.g. plasmid pBR322 the presence of which in the potential recipient increases the efficacy of transfer [START_REF] Freter | Experimental and mathematical models of E. coli plasmid transfer in vitro and in vivo[END_REF].

(c) Ecological conditions Some environmental factors have a proven effect, but it is difficult to know whether they act on the mating strains (population size and growth phase) or on the transfer itself. These factors are the physico-chemical conditions (rH, pH, etc.) [START_REF] Wiedemann | Die Uebertragung extrachromosomaler Resistenzfaktoren in der Darmflora und ihre Hemmung[END_REF] and the temperature; for example, the thermosensitive H2 plasmid transfers outside the gut and is then ingested by calves [START_REF] Timoney | Experimental ecological studies on H2 plasmids in the intestine and faeces of the calf[END_REF]. Another factor is the mixing effect of peristaltic movements of the gut, which may separate mating cells. The localization of the bacteria (adhesion to the wall) and their topographical location along the gut may have an effect on mating, if potential donor and recipient cells cannot meet. Specific interfering substances, which would inhibit the transfer itself, have been invoked frequently but their presence in vivo is still hypothetical. On the other hand, some antibiotics, for example bambermycins [START_REF] Corpet | The effect of bambermycin, carbadox, chlortetracycline and olaquindox on antibiotic resistance in intestinal coliforms: a new animal model[END_REF], are well known to inhibit the replication or the transfer of plasmids.

Conclusions

The factors interfering with plasmid transfer are too numerous to be measured simultaneously in vivo. Parameters must be isolated in experimental models, such as continuous flow cultures or gnotobiotic animals.

These factors are too numerous to be comprehended simultaneously, particularly in their quantitative relationships. One has to build a mathematical model to estimate the relative effect of each factor. This was done by [START_REF] Freter | Experimental and mathematical models of E. coli plasmid transfer in vitro and in vivo[END_REF], and the differential equations they propose can be used, with and without modifications, to test new hypotheses and to conduct more accurate experiments. The results of computer simulations based on measurements made in experimental models indicate that:

(i) the reduction of growth rate generally observed in plasmid-bearing bacteria explains the general dominance of antibiotic-susceptible bacteria in mammals' digestive tracts; and

(ii) the efficiency of in-vivo plasmid -transfer does not differ from that occurring in vitro. The observed differences (the very low rates of in-vivo plasmid transfer) are referable mainly to the population densities of E. coli in the gut of normally reared mammals.

( b )

 b Factors which are the same for donor and recipient bacteria The flow rate constant [Ro], the concentration of the limiting substrate [Sr and S], the presence of inhibitors and the structure of the wall are the saine for recipient and donor bacteria but have différent effects in the two populations, because of factors listed below. (c) Growth rate [PsiR].


Diego Milian

Heon Choi

Tilahun Tsehaye

Peter Fischer

Jens Tübke

Cristina Iojoiu

Denis C D Roux

Nadia El Kissi
email: nadia.elkissi@univ-grenoble-alpes.fr

Nadia El Kissi

Formulation and characterization of zinc slurries for zinc slurry -air redox flow battery: A rheological approach

This study presents rheometry as an optimization tool to increase discharge performance of a zinc slurry -air redox flow battery (RFB). Incorporation of electrode suspension slurry to metal-air batteries, in which dense metal solid microparticles are suspended in a conductive fluid with the aid of gelling agents, was investigated. Electrochemical performance of zinc anode slurries was characterized in a single redox flow battery cell in discharge mode.

Rheological behavior of each of the slurries is investigated and the key parameters underlying the cell performance were determined. Combined rheological and electrochemical effects were examined and an optimal PAA concentration of 0.40 wt. % was found for the zinc slurry system, in which slurry stability is not compromised and conductivity is satisfactory. Furthermore, at the reported gelling agent optimal concentration, a maximum peak power density of 24 mW/cm 2 is obtained from polarization curves, improving previous reported performances using the same cell configuration by a factor of 2.4. These results bring more understanding in the development of slurry-based RFBs with a rheometric approach.

Introduction

As current energy development aims towards technologies powered by renewable sources [START_REF] Breyer | Global overview on grid-parity[END_REF], necessity of efficient energy storage systems has driven research to explore new materials and configurations. Redox flow batteries (RFBs hereinafter) have been gaining attention as their versatile and flexible operation is suitable to potentiate intermittent renewable energy sources like wind and solar energy. Their ability to decouple energy storage and power generation [START_REF] Ye | Viscosity, conductivity, and electrochemical property of dicyanamide ionic liquids[END_REF], improves electrical grid efficiency and power quality by allowing better resource management for peak shaving and valley filling. Among different possible chemistries, zinc-air is one of the most interesting due to its high theoretical specific energy density of 1350 Wh/kg [START_REF] Bockelmann | Electrically rechargeable zinc-oxygen flow battery with high power density[END_REF]) and zinc availability in Earth's crust. Zinc -air batteries have been commercialized since early 20th century (Mainar et al. 2018a), mostly as primary batteries. In a typical alkaline zinc-air battery, the electrochemical reactions proceeding in the cell during discharge mode are the following:

ℎ : + 2 + 4 4 (1)
( = 0.40 # $% & ) () : *) + 4 -2 *)( ) , (2)
( = -1.25 # $% & ) *)( ) , → *) + + 2 00 1 2 3 ): 2*) + 2*) (3)
( = 1.65 #)
Nevertheless, the development of a rechargeable zinc -air battery has been hindered due to different issues, largely, zinc anode limitations, which have been studied in depth by A.

Mainar (Mainar et al. 2018b, a, c), and other authors [START_REF] Yang | Improved discharge capacity and suppressed surface passivation of zinc anode in dilute alkaline solution using surfactant additives[END_REF][START_REF] Yang | Effects of Carboxymethyl Cellulose on the Electrochemical Characteristics and Dendrite Growth of Zinc in Alkaline Solution[END_REF]Banik andAkolkar 2013, 2015;[START_REF] Liu | Suppressing the dendritic growth of zinc in an ionic liquid containing cationic and anionic zinc complexes for battery applications[END_REF]Schmid and Willert-Porada 2017;Schmid et al. 2017). In order to reduce zinc anode problems, metal-air batteries such as zinc -air, have been redesigned as flowing systems instead of static ones [START_REF] Han | Metal-Air Batteries: From Static to Flow System[END_REF].

In particular, zinc slurry -air flow battery is an interesting approach as zinc particles are dispersed in the electrolyte media with the aid of gelling agents, enhancing electrode surface and energy density [START_REF] Duduta | Semi-solid lithium rechargeable flow battery[END_REF][START_REF] Huang | Next-generation, high-energy-density redox flow batteries[END_REF]. Many of the advantages of having a flowable electrode comprise the reduction of dendrite formation [START_REF] Jiang | Towards a uniform distribution of zinc in the negative electrode for zinc bromine flow batteries[END_REF], and non-localized electrode shape change and passivation layers [START_REF] Yang | Improved discharge capacity and suppressed surface passivation of zinc anode in dilute alkaline solution using surfactant additives[END_REF]). The flow of the slurry and charge transfer in single cell is optimized by using flow fields (Choi et al. 2020a), promoting uniform distribution over geometric area.

However, zinc microparticles are largely dense (5 67 = 7.13 g/cm 3 ), and in order to increase active material utilization, microparticles need to be suspended in the electrolyte media so the flow will be homogeneous throughout the battery. Thus, the fluid that will be used must avoid the accumulation of particles at the bottom of the container at rest or in the transporting conduits during flow. In this way, it will be possible to obtain a homogeneous distribution of zinc particles throughout the process inside the battery. These particles can then be brought in contact with the electrode, where their discharge takes place. Hence, rheology plays an important role in RFBs as network properties are important for slurry formulation. Moreover, electrochemical performance is directly linked to flow behavior because mass transfer coefficients, ions diffusion and pump losses, among others, depend on electrolyte viscosity [START_REF] Smith | Studies on pressure losses and flow rate optimization in vanadium redox flow battery[END_REF][START_REF] Iyer | Assessing the impact of electrolyte conductivity and viscosity on the reactor cost and pressure drop of redox-active polymer flow batteries[END_REF][START_REF] Barton | Quantifying the impact of viscosity on mass-transfer coefficients in redox flow batteries[END_REF].

In this study, zinc slurry -air RFB is designed to work under highly alkaline conditions.

Alkaline electrolytes are recurrently used in zinc-air batteries due to their high ionic conductivity and solubility of zinc species [START_REF] Mainar | Alkaline aqueous electrolytes for secondary zinc-air batteries: an overview[END_REF], both of which are important in the redox reaction. Aqueous electrolytes are mixed with gelling agents and additives to prepare gel polymer electrolytes (GPEs). However, such a harsh alkaline environment (i.e., 10 M KOH used by Choi et al. for zinc slurry -air RFB (Choi et al. 2020a, b)) limits the availability of gelling agents that are able to suspend dense zinc particles, compromising slurry stability. Among available gelling agents, Carbopol 940® and attapulgite were chosen to prepare zinc slurries.

Carbopol 940®, also known as polyacrylic acid (PAA), is used frequently as a model fluid in experiments together with numerical simulation because of its stability and well-known properties at near-neutral pH, where yield stress behavior is enhanced (refer to [START_REF] Piau | Carbopol gels: Elastoviscoplastic and slippery glasses made of individual swollen sponges. Meso-and macroscopic properties, constitutive equations and scaling laws[END_REF] for extensive study in PAA elasto-visco-plastic behavior). PAA has been investigated as electrolyte additive for application in alkaline metal-air batteries like Al-air and Zn-air batteries. [START_REF] Zhang | All-solid-state Al-air batteries with polymer alkaline gel electrolyte[END_REF]) investigated PAA gel performance in an Al-air battery where its conductivity (460 mS/cm) was found to be close to that of aqueous electrolytes. Moreover, GPEs using PAA as gelling agent have been proved to show higher conductivity compared to GPEs incorporating other gelling agents like poly(vinyl alcohol) (PVA) and poly (4-vinylbenzenesulfonate-co-[3-(methacryloylamino)propyl]trimethylammonium chloride) (PAM) at 6 M KOH [START_REF] Tran | A study of alkaline gel polymer electrolytes for rechargeable zinc-air batteries[END_REF].

Attapulgite is a type of magnesium aluminium silicate clay with a three-dimensional needlelike morphology [START_REF] Shen | Preparation and rheology of polyamide-6/attapulgite nanocomposites and studies on their percolated structure[END_REF]) and chemical formula 3MgO-1.5Al2O3-8SiO2-9H2O. When dispersed at high shear rates, attapulgite is able to work as a gel-forming clay [START_REF] Haden | ATTAPULGITE: ITS PROPERTIES AND APPLICATIONS[END_REF]. The shape and size of its needles, as well as its gelling mechanism, result in a good resistance to high concentrations of electrolytes [START_REF] Abdo | Clay nanoparticles modified drilling fluids for drilling of deep hydrocarbon wells[END_REF]. Under these latter conditions, yield stress is enhanced [START_REF] Chang | The effect of pH, ionic strength, and temperature on the rheology and stability of aqueous clay suspensions[END_REF][START_REF] Qian | Enhancing thixotropy of fresh cement pastes with nanoclay in presence of polycarboxylate ether superplasticizer (PCE)[END_REF], making it a good candidate to prepare zinc slurries. Besides its rheology modifier properties, attapulgite also shows properties that enhance the electrochemical performance of zinc slurry-air batteries. Attapulgite has been previously investigated to formulate zinc slurries for automotive application and it was found that the addition of this clay reduces anode corrosion, even more when combined with ZnO (Sonneveld 1991).

This study aims to assess the effect of PAA gelling agent on slurry electrochemical and rheological properties in a custom-made zinc slurry -air RFB. Furthermore, we present here how rheometry can be used to assess performance of such devices as RFBs. The effect of PAA concentration on slurry electrochemical performance and stability is thoroughly investigated by electrochemical cell tests and rheological characterization. Lastly, an optimal concentration of PAA is discussed in terms of the trade-off between performance and stability. This study links material formulation in harsh alkaline environment with electrochemical experimentation and rheological characterization, highlighting the importance of material formulation for RFBs.

Materials and methods

Slurry preparation

In order to have reproducible results, a slurry preparation protocol is followed to produce each of the samples before either rheology or electrochemical characterization. Slurry composition was taken from a previous study using the same experimental set-up (Choi et al. 2020a), except for gelling agents presented in a previous study [START_REF] Milián | Rheological behavior of gel polymer electrolytes: yield stress and viscoelasticity[END_REF] Germany) and ZnO (Alfa Aesar, United States) was added to the solution. The sample was mixed using a high shear homogenizer Ultra Turrax T25 (Ika, Germany) at a speed of 11 000 rpm for 1 minute to disperse the clay particles, thickening the mixture and forming a gel.

GPEs composition together with their rheological characterization can be found in [START_REF] Milián | Rheological behavior of gel polymer electrolytes: yield stress and viscoelasticity[END_REF], where is reported that all GPEs were found to have a yield stress fluid behavior.

Further information of materials used to formulate GPEs can be found in supplementary information S1, together with a scheme illustrating the gel structure.

Zinc slurries

Zinc particles battery grade (GN 7-4/200Bi/200In, Grillo GmbH, Germany) were used as received and size characterization is shown in Figure 1.

Figure 1a shows a picture of zinc particles taken with a scanning electron microscopy (SEM, ZEISS GeminiSEM 500). It is seen that zinc particles show rounded shapes varying from Once GPEs are prepared, zinc slurries are prepared in the following way. Zinc particles are added into freshly prepared GPEs. Particles are first dispersed using a spatula to avoid lumps and then slurries are homogenized with Ultra Turrax at 11 000 rpm for 1 minute. This preparation ensures homogeneous distribution of zinc particles in the suspending fluid. After preparation, samples were ready for either electrochemical or rheological characterization.

The final slurry concentrations can be found in Table 1. PAA concentrations higher than 0.6 wt. % were not investigated because of their final high apparent viscosity, which in practice made it impossible to flow samples through the RFB cell. b)

Electrochemical characterization

Electrochemical tests were performed using a home-made single cell presented in a previous study (Choi et al. 2020a) to determine performances of zinc slurries in a zinc slurry -air RFB. Single cell is operated in discharge mode. A scheme of the working principle of the anode slurry is depicted in supplementary information S2. The experimental set-up used for this study contained a graphite bipolar plate, flow frame flow field, separator (Cellophane™ PØØ purchased from FUTAMURA, Hamburg, Germany), gaskets and current collectors, correctly fixed by the aid of two end plates. Zinc slurry is kept in a non-stirred container and is pumped to the single cell using a peristaltic pump.

Zinc slurry and synthetic air were pumped in the anode and cathode compartments with a flow rate of 160 mL/min and 100 mL/min, respectively. During cell testing, current-voltage characteristic curves were obtained by using a BaSyTec GSM Battery Test System (BaSyTec GmbH, Germany). When anolyte and catholyte are flowing through the battery, constant current is applied and steady state voltage is obtained.

Finally, slurries conductivity (σ) was obtained by electrochemical impedance spectroscopy (EIS) using a response analyzer (Hewlett Packard, United States) at 20 °C.

Zinc sedimentation in slurries

Zinc particle sedimentation in slurries is critical not only in RFB application but also in rheological characterization. If sedimentation takes place in the battery, slurries will become two-phase systems leading to clogging and/or lower battery performances due to less particles utilization. Correspondingly, if sedimentation takes place during rheological characterization, initial sample properties will change during test, as sedimentation is in competition with deformation. Therefore, slurries sedimentation time is a critical factor as it can interfere in sample characterization.

Sedimentation kinetics of recently prepared slurry samples were determined by tracking particle position over time. Slurries were placed in a rectangular cell of width 4.5 mm and depth 10 mm. Pictures of slurries at different times were recorded using a digital action camera (Xiaomi Yi Cam, China) and were analyzed using ImageJ software. ImageJ can process images in grayscale and motion can be tracked after defining a pixel/distance ratio.

Sedimentation velocity Vs was determined from the slope of the linear decrease at the beginning of the test (supplementary information S3). Furthermore, it can be determined the time at which sample has undergone sedimentation at different percentages defined by the ratio between particle position h and initial position h0 loss. We define in this study a critical sedimentation threshold of 5%, equivalent to a ratio h/h0 = 0.95. Beyond 5% of sedimentation, sample cannot be characterized as its properties would have already changed from its initial state.

Rheometry

Zinc slurries rheological behavior was studied using a stress-controlled rheometer AR-G2

(TA instruments, United States) and a vane-in-cup geometry. Vane geometry (4-blades, blade width = 24 mm, h = 45.5 mm), was specially designed to study slurries in this study ensuring a homogeneous shearing [START_REF] Marchal | Quantitative Analysis of Mixer-Type Rheometers[END_REF] while avoiding slippage [START_REF] Barnes | Rotating vane rheometry-a review[END_REF][START_REF] Hermoso | High pressure mixing rheology of drilling fluids[END_REF]. A scheme of the set-up can be found in a previous publication [START_REF] Milián | Rheological behavior of gel polymer electrolytes: yield stress and viscoelasticity[END_REF]. Transparent cup (D = 48 mm) is equipped with a grid (thickness = 2 mm), in order to eliminate slippage effects [START_REF] Owens | Improved rheometry of yield stress fluids using bespoke fractal 3D printed vanes[END_REF].

Vane tool was calibrated prior to performing samples rheological characterization. It is possible to assume that the vane geometry corresponds to a cylinder geometry because the flow profile generated by the vane in a Couette is similar to the one generated by concentric cylinders geometries [START_REF] Baravian | Vane rheometry with a large, finite gap[END_REF][START_REF] Ovarlez | Flows and heterogeneities with a vane tool: Magnetic resonance imaging measurements[END_REF]. Therefore, the rotational speed and torque obtained with the vane-in-cup is converted into shear rate and shear stress, respectively, using a Couette analogy approach (Aït-kadi et al. 2002):

89 = : ;9 < 9 (4) = = : > Γ (5)
Where < 9 is the rotational velocity, Γ is the measured torque and : ;9 and : > are the geometrical correction factors. Geometrical factors permit to translate rotational speed and torque obtained with vane-in-cup, into shear rate and shear stress, respectively. Complete calculation of geometrical factors is available in a previous study [START_REF] Milián | Rheological behavior of gel polymer electrolytes: yield stress and viscoelasticity[END_REF].

Finally, factors calculated with vane and cup dimensions are : ;9 = 2.30 rad -1 and : > = 13 450.7 Pa N -1 m -1 , corresponding to an analogous inner cylinder of diameter d = 26 mm.

Oscillatory rheometry

Small deformation rheometry was used to characterize slurries viscoelastic properties representative of samples close to rest (no flow through the battery). Oscillatory rheometry consists in applying a sinusoidal deformation 8( ) = 8 sin(C ), with 8 as the oscillation amplitude and C as the angular frequency. In each test, storage and loss moduli (G' and G'' respectively), are recorded as function of deformation or frequency.

The linear viscoelastic region (LVR) of slurries, in which viscoelastic properties are independent of sample deformation, was determined by performing a strain amplitude sweep from 0.1 to 1000% at 1 Hz.

Viscoelastic behavior was obtained by applying a frequency sweep from 0.1 to 4 Hz at a deformation located in samples LVR. These deformations 8 were 0.1% for S1 and 1% for all the other slurries (S2-S5).

Large deformation rheometry

Slurry flow curves were obtained by large deformation tests. Large deformation tests are important to understand how slurries will behave while pumping in a RFB. Steady shear rheometry was performed by decreasing shear rate steps in which the stress response is measured. As explained in one of our previous studies [START_REF] Milián | Rheological behavior of gel polymer electrolytes: yield stress and viscoelasticity[END_REF], just decreasing shearing steps are considered in order to avoid heterogeneous shearing zones [START_REF] Ovarlez | On the existence of a simple yield stress fluid behavior[END_REF]. Shear rate was decreased from 100 to 0.01 s -1 .

Depending on PAA concentration, slurries undergo sedimentation at different rates.

Rheological characterization was set so the experiment time will be shorter than the time a

given sample shows 5 % sedimentation, as explained in subsection 2.3.

To quantify rheological parameters, shear stress vs shear rates curves were fitted to a Herschel-Bulkley model:

= = = D + :89 7 (6)
Where = is the shear stress, = D is the yield stress, K is the consistency index, 89 is the shear rate and n is the flow index. To compare all slurries under the same conditions, current density values at a constant voltage of 0.7 V are represented as function of PAA concentration in the inset of Figure 2a.

Results

Electrochemical discharge performance

This voltage is representative for zinc-air batteries as it has been shown that cycling stability and energy efficiency is optimal at 0.7 V [START_REF] Li | Advanced zinc-air batteries based on high-performance hybrid electrocatalysts[END_REF]. It is observed that current density increases from 15 to 32.5 mA/cm 2 as PAA concentration increases from 0.2 to 0.4 wt. %.

Contrarily, when PAA concentration is increased further than 0.4 wt. % up to 0.6 wt. %, current density decreases to a value of 23 mA/cm 2 . S3 out-stands among the different slurries by showing the highest current density at 0.7 V.

b)

In an equivalent representation, Figure 2b analyses power density as a function of current density for each of the slurries. As current density increases, power density increases up to a maximum point before decreasing and forming a parabola. The maximum point is known as peak power density, and it is depicted as function of PAA concentration in the inset of Figure 2b. It is observed that peak power density increases from 10 to 24 mW/cm 2 as PAA concentration increases from 0.2 to 0.4 wt. %. When concentration of PAA is increased in slurry formulation, peak power density decreases to a value of 16.5 mW/cm 2 for a PAA concentration of 0.6 wt. %. These results suggest that concentration of 0.4 wt. % corresponding to S3 is close to a limit between two behaviors, one that favors battery performance and other that produces the contrary effect. To determine the origin of this phenomena, conductivity and rheometric measurements were performed on slurries.

To broaden our understanding of slurries electrochemical properties, slurries ionic conductivity was measured by EIS. Figure 3 represents ionic conductivity as function of PAA concentration.

Figure 3. Slurries conductivity as function of PAA concentration at 20 °C. Conductivity of pure KOH 10 M at 20 °C is 0.4737 S/cm [START_REF] Gilliam | A review of specific conductivities of potassium hydroxide solutions for various concentrations and temperatures[END_REF].

It is observed that when PAA concentration increases, conductivity decreases following a quadratic behavior with a decay faster from 0.2 to 0.4 wt. % of PAA in slurries, and slower above 0.4 wt. %. Moreover, slurries conductivities is found to be lower compared to that of pure KOH 10 M, having a value of 0.4737 S/cm [START_REF] Gilliam | A review of specific conductivities of potassium hydroxide solutions for various concentrations and temperatures[END_REF]). In fact, when increasing PAA concentration, a viscosity increase is imminent, as it will be presented in a following section (subsection 3.3.2). Slurries may benefit from such an increase in viscosity to slow down sedimentation kinetics, but electrochemical activity can be reduced. Similar This link between viscosity and conductivity can be explained thanks to Stokes-Einstein equation ( 7). The diffusion coefficient E F of a given molecule (i.e., electrolyte) is inversely proportional to the viscosity G of its surrounding media:

E F = H I J KLF M (7)
Where N O is the Boltzmann's constant, T is the temperature and 1 P is the effective hydrodynamic radius. Therefore, microstructural changes in the electrolyte structure, resulting in a viscosity increase, results in a decreased ion mobility, which is strongly correlated to its conductivity, as reviewed by M-C. Pang et al. [START_REF] Pang | Mechanical behaviour of inorganic solidstate batteries: can we model the ionic mobility in the electrolyte with Nernst-Einstein's relation?[END_REF].

Consequently, GPEs display a trade-off between their conductivity and their mechanical ability to suspend zinc particles.

Results presented in this section illustrate that there is an optimal PAA concentration (in the range of studied concentrations), delivering a maximum cell performance and that slurry ionic conductivity is highly dependent on the gelling agent concentration, and hence the viscosity of the electrolyte. To investigate why slurry peak power density and voltage (Figure 2) are enhanced at 0.4 wt. % PAA, corresponding to S3 formulation, rheological characterization needs to be performed. Nevertheless, sedimentation tests are performed to study PAA effect on sedimentation kinetics, but also allowing to know for how long slurries are stable in order to be characterized.

Zinc sedimentation

Sedimentation tests were performed tracking zinc particle position in zinc slurries and results are shown in Figure 4. The ratio between the recorded height and the initial height as a function of time is represented in Figure 4 for each slurry. Results revealed that different sedimentation kinetics are found depending on PAA concentration used to prepare slurries.

S1 shows the fastest h/h0 ratio decrease at times t < 2000 s. Sedimentation velocity vs is calculated from the linear height h decrease and it is found to be vs1 = 1.8 µm/s. Then h stabilizes slowly until reaching a plateau at ~ 10 000 s (2.78 hours). For S2, h/h0 ratio decreases slowly compared to S1. The sedimentation velocity is found to be vs2 = 0.2 µm/s, which is 9 times slower compared to S1. Finally, S2 sedimentation finishes after 15 000 s (4.17 hours).

Contrastingly, samples S3, S4 and S5 remained homogeneous for a longer time and no significant sedimentation (h/h0 lower than 2 %) was observed in 55 h. Table 2 reports the times at which each slurry undergo sedimentation up to a given percentage of h/h0 loss. Moreover, it can be highlighted that linear velocity is in the order of 0.267 cm/s (flow rate of 160 mL/min through a section of 2 x 5 cm), whereas sedimentation velocity is in the order of µm/s. This indicates that when flow is happening, sedimentation cannot take place as linear velocity is 4 orders of magnitude greater.

These results are important since if sedimentation occurs while battery is in operation, it could happen that just the electrolyte will be recirculated and fewer zinc particles will be in touch with the bipolar plate, decreasing the capacity of the battery. Similarly, when battery is at rest it could happen that zinc particles can settle in the flow field or in the tubing of the battery, clogging the system and needing manual intervention.

Accordingly, the protocol to determine battery performance was adjusted so the time of the test is shorter than the sedimentation time at 5%. Similarly, rheological characterization presented in following section was performed so the time of each experiment was shorter than the time to reach 5 % of sedimentation, point at which the sample is not considered homogeneous anymore. When amplitude strain continues to increase, a crossover point is met where G' = G'' and indicates the transition point from elastic solid to viscous liquid behavior. As deformation continues to increase beyond crossover point, G'' becomes greater than G', indicating that sample initial network is being deformed indefinitely. In this non-linear region, moduli are strongly dependent of deformation and sample is under conditions similar to flow. This behavior is observed for all slurries in figure 5a, but as PAA concentration decreases, moduli values decrease as well, S1 being the slurry with lowest G' and G'' values.

Furthermore, figure 5b shows the results of the oscillation frequency sweep. It is observed that for all slurries G' is greater than G'' in the range of frequency tested from 0.1 to 4 Hz.

Moreover, modulus G' is parallel to G'' and they show a slight dependence to frequency as moduli values increase with frequency with an average slope of 0.14. Moreover, there is not a moduli crossover in the range of frequencies analyzed.

b)

The effect of PAA is also observed in Figure 5b. As PAA concentration increases from S1 to S5, moduli values increase accordingly, with S5 showing the highest moduli values and S1

the lowest moduli values, respectively.

Therefore, this viscoelastic behavior observed for zinc slurries suggest that slurries show properties characteristic of gel-like materials. This is very important as zinc particles form part of a gelled network in a zinc slurry -air RFB.

Zinc slurries flow curves

To understand slurries behavior representative of flow, flow curves are obtained by shearing tests. In Figure 6, steady state shear stress as function of shear rate is observed for all slurries. It is worth to mention again that flow curves were obtained by down-ramping shear rate. Moreover, rod climbing (also known as Weissenberg effect (Weissenberg 1947)), was observed at shear rates > 10 s -1 (video, supplementary information), where no steady state was found, as observed in our previous study with gel polymer electrolytes [START_REF] Milián | Rheological behavior of gel polymer electrolytes: yield stress and viscoelasticity[END_REF]. Therefore, results presented in this study are obtained for a range of shear rates comprised between 0.01 and 10 s -1 .

Figure 6. Shear stress as function of shear rate for zinc slurries at 20 °C. Solid lines represent Herschel-Bulkley fitting (Equation 6).

S1 was analyzed at shear rates from 1 to 10 s -1 and it is observed that shear stress increases slightly with shear rate. Lower shear rates lead to sedimentation of the initial sample before reaching steady state, even when shearing time was shorter than the sedimentation time (420 s for S1 to reach 5 % of sedimentation, Table 2). An explanation to the observed early-settling process may lay on the protocol used to obtain the flow curves.

As shear rate is decreased, slurry viscosity decreases from the beginning of the test. In the fluidized slurry state, yield stress is absent and gelled network cannot avoid zinc particles to settle towards the bottom of the cup. This settling process has also been observed by other researchers dealing with suspensions and irreversible particle migration [START_REF] Ovarlez | Three-dimensional jamming and flows of soft glassy materials[END_REF][START_REF] Ovarlez | Flows and heterogeneities with a vane tool: Magnetic resonance imaging measurements[END_REF][START_REF] Owens | Improved rheometry of yield stress fluids using bespoke fractal 3D printed vanes[END_REF]).

Next, S2 shows higher shear stress values as PAA concentration is increased, and sedimentation was not observed during the test. As shear rate increases from 0.01 to 10 s -1 , shear stress continuously increases. However, a different behavior is observed for S3, which was prepared with 0.4 wt. % of PAA. S3 shows that as shear rate is increased, shear stress remains first constant at shear rates < 1 s -1 and then shear stress increases as shear rate increases. S4 and S5 show similar behavior compared to S3: a shear stress plateau up to a shear stress of 0.1 s -1 , and then shear stress increases as shear rate is increased.

Overall, slurries analyzed in here show a behavior corresponding to yield stress fluids, except for S1. This rheological behavior is sought because yield stress will help to maintain zinc particles in suspension. In addition, shear-thinning behavior is also observed, decreasing viscosity as shear rate is increased. It is important to remember that viscosity decrease eases pumping of slurry throughout a RFB, favoring global energetic efficiency.

Shear stress as function of shear rate curves were fitted to Herschel-Bulkley model (Equation 6) and fitting results are reported in Figure 7. In the case of S1, as there were not enough experimental points, no fitting was done. However, for S1 only, an estimated shear stress value was taken. This was the value of the shear stress at the lowest shear rate obtained on Figure 6. In Figure 7 it is observed that yield stress increases by a factor of 4 from 11.6 to 45.3 Pa as PAA concentration is increased from 0.2 to 0.4 wt. %. When PAA concentration is further increased up to 0.6 wt. %, the increase in yield stress is weaker compared to the other slurries. Yield stress for S4 and S5 are 48.2 and 54.9 Pa, respectively.

Herschel-Bulkley parameters can be found in Table 3. It is observed that shear thinning behavior is found by the model as flow index n < 1. Furthermore, consistency index K was found to increase with increasing PAA concentration.

Finally, it is observed that concentration of 0.4 wt. % is close to an optimal concentration from which a yield stress behavior is very well defined in comparison with the slurries prepared with concentrations lower to 0.4 wt. %. Discussion comparing both electrochemical and rheological results is found in next section.

Discussion

As seen across electrochemical and rheological characterization, PAA concentration plays an important role in the formulation of zinc slurries for RFB.

From electrochemical characterization, polarization curves (Figure 2), it is observed that slurry performance is enhanced when PAA concentration increases from S1 to S3, the latter reaching a concentration of 0.4 wt. % of PAA. At this optimal concentration, the single cell exhibits the higher electrochemical performance i.e. a current density of 32.5 mA/cm 2 at 0.7

V and a peak power density of 24 mW/cm 2 . Contrastingly, when PAA concentration is further increased up to 0.6 wt. % (corresponding to S5), single cell performance decreases, as observed in the insets of Figure 2.

These results show not only that there is an optimal PAA concentration where electrochemical performance during discharge is maximized, but also that S3 formulation improves the performance of a zinc slurry -air RFB compared to a previous study. Choi et al. (Choi et al. 2020a), reported a current density value of 15 mA/cm 2 at 0.7 V and 10 mW/cm 2 using the same set-up as the one used in this study (flow frame geometry for anode side), but a different zinc slurry formulation without attapulgite. From this comparison, electrochemical performance was improved by a factor of ~ 2.4 in terms of peak power density, by using S3 as zinc anode slurry.

In order to understand how formulation can affect the performance of a zinc slurry -air RFB, rheological characterization at small and large deformation was performed. Oscillatory rheology results represented viscoelastic properties of slurries at rest. It was found from frequency sweep (Figure 5b), that all slurries behaved as elastic-solids. Indeed, G' was found to be greater than G'' and independent of frequency, in the range of frequencies tested. Therefore, this viscoelastic behavior suggests that there is a percolated network consisting mainly in the gel suspending fluid where zinc particles are dispersed. Gel-like

behavior is important at rest because there is a mechanical resistance coming from the network that will prevent particles from settling. Nevertheless, it is crucial to assess if network strength is adequate to prevent zinc particles from settling when no flow is imposed.

Certainly, yield stress suggests that there is a structure in the slurries that prevents zinc particles from settling. Nonetheless, the gelled network used in slurry has to possess a minimum yield stress to maintain zinc particles in suspension. This minimum yield stress will depend on zinc particles characteristics and it is calculated as follows.

4.1. Minimum required yield stress = D Slurries are prepared by using dense zinc particles prone to settling in aqueous medium.

Knowing zinc particles size distribution (Figure 1b), it is possible to determine the gel strength necessary to maintain a single particle in suspension. This is done by the equilibrium of forces acting on a zinc particle (supplementary information S4).

A single zinc particle (D50 = 76.4 µm, 5 =7.13 g/cm 3 ), is under the action of gravity force Fg, buoyancy force FB and gel strength Fy. Fg drags the particles towards the bottom of a container following gravity action. FB is the fluid force opposing to particle weight. Finally, Fy is the force corresponding to the gelled network and it will representative of its suspending ability. The equilibrium of a single particle in suspension can be represented by following equation:

Q D + Q O -Q = 0 (8)
Furthermore, FB and Fg can be expressed in terms of 5 density, g gravity constant and V volume:

Q O = 5 RST • V • # W X S (9) Q = Y • V = 5 W X S • V • # W X S (10)
By substituting equations ( 9) and ( 10) into (8), we obtain:

Q D = # W X S • V • Z5 W X S -5 S [ (11)
It is assumed that zinc particles are spherical, so the volume is equal to # W X S =

, \

]1 W X S \ . Knowing that stress can be expressed as = = Q/(, where A is the contact particle area (( = ]1 W X S ), it is possible to calculate the stress needed to maintain single particle in suspension. This stress will be analogous to yield stress:

= D = , \ 1 W X S • V • Z5 W X S -5 S [ (12)
By substituting in equation ( 12) the values of radius (r = 38.2 µm, Figure 1), density corresponding to zinc particles (5 = 7.13 g/cm 3 ) and the density of the suspending fluid (5 _`a = 1.40 g/cm 3 , obtained from 2000 DMA density meter), a yield stress value = D67 = 2.85 c is obtained. In order to maintain a single zinc particle in suspension, GPE yield stress value should be larger than = D67 .

In the next subsection, we analyze shortly rheological behavior of GPEs, notably their suspending ability through their yield stress.

GPEs as suspending fluids

Rheology of GPEs was analyzed in a previous work [START_REF] Milián | Rheological behavior of gel polymer electrolytes: yield stress and viscoelasticity[END_REF]) and yield stresses = D

were determined for each of the suspending fluids. Shear stress as function of shear rate curves were fitted to Herschel-Bulkley model (equation 6) for each GPE. Yield stress values and fitting parameters are displayed in Table 4 for each GPE.

Table 4. Herschel-Bulkley parameters obtained from fitting with equation ( 6) for GPEs [START_REF] Milián | Rheological behavior of gel polymer electrolytes: yield stress and viscoelasticity[END_REF]. Reprinted with permission of Springer Nature. Therefore, when minimum yield stress to suspend zinc particles is met by increasing PAA concentration up to 0.4 wt. % corresponding to S3, electrochemical performance is enhanced (Figure 2). Nevertheless, when PAA concentration is further increased, electrochemical performance is not improved, as the addition of gelling agent decreases ionic conductivity (Figure 3). As discussed by K. Smith et al. [START_REF] Smith | Studies on pressure losses and flow rate optimization in vanadium redox flow battery[END_REF], an optimal yield stress value must exist, as mechanical energy dissipation and ion mobility depends on the viscosity of the system. Similarly, Z. Zhang et al. [START_REF] Zhang | All-solid-state Al-air batteries with polymer alkaline gel electrolyte[END_REF]) not only demonstrated that an optimal conductivity of PAA-based alkaline gels was attained at a specific PAA concentration, but also further increase in gelling concentration led to lower conductivities.

Sample

= D [Pa] K [Pa s n ] n [ -] R 2 GPE1 1.
Finally, zinc slurries key rheological and electrochemical properties evaluated in this study are summarized in Figure 8. possess a yield stress large enough to avoid zinc particles sedimentation. Hence, it is then close to this PAA critical concentration where there should be an optimum concentration in which power peak density can be maximized, meaning that there is not PAA in excess hindering conductivity, and network properties are able to suspend zinc particles homogeneously. Consequently, once this optimum concentration is exceeded (PAA > 0.4 wt. %), peak power density decreases even if more PAA in the formulation leads to more stable slurries in terms of sedimentation, but less appropriate for RFB application in terms of low conductivity and viscosity increase.

Conclusions

As seen in electrochemical results, as PAA concentration increases to an optimal concentration of 0.40 wt. % of PAA, corresponding to S3, a maximum peak power density is obtained during the discharge of a zinc slurry -air RFB. Moreover, it was found by rheometric means that the optimal concentration of gelling agent is needed to have at least a yield stress = D 67 > 2.85 Pa, mandatory to maintain zinc microparticles (D50 = 76.4 µm) in suspension. This result was also reflected on sedimentation tests, where S1 and S2 settle as gelled network does not have the ability to maintain zinc particles homogeneously suspended. Further increase in PAA concentration leads to lower single cell performances as slurry conductivity decreases.

Hence, rheology is found to be a useful tool to characterize slurries and understand key rheological parameters affecting discharge performance of a zinc slurry -air RFB. Yield stress shows to be not only important as stability enhancer but also as a parameter favoring the performance of a network formed by a gel and zinc particles. Furthermore, we present here a method in which just by knowing the size and density of the particles to be suspended, it is possible to calculate the yield stress, and therefore the concentration of gelling agent, needed to obtain the best electrochemical performance of the slurry system.

This study is an insight on the electrochemical performance of a zinc microparticles slurrybased RFB observed from a rheological point of view. In such way, metal-air RFB can continue to be developed as a high energy density and economic alternative to ensure a transition to more sustainable energy sources.

Figure 1 .

 1 Figure 1. Zinc particles battery grade size and shape characterization. a) SEM picture of zinc particles. b) Zinc particles size distribution obtained by laser diffraction distribution (Mastersizer 2000).

Figure 2 .

 2 Figure 2. Zinc slurries polarization curves. a) Voltage as function of current density. Inset indicates current density at 0.7 V as function of PAA concentration. b) Power density as function of current density. Inset displays peak power density as function of PAA concentration.

Figure 2

 2 Figure 2 reveals the effect of PAA concentration used in slurry formulation on cell electrochemical performance. Current-voltage characteristic curve (Figure 2a), records the cell voltage at an imposed current. It is observed that cell voltage drops at different rates for different PAA concentrations as current density is increased. It is remarked that S3 reaches the higher current density values among all tested slurries.

  conductivity decrease was observed by Sekhon[START_REF] Sekhon | Conductivity behaviour of polymer gel electrolytes: Role of polymer[END_REF] for lithium salts by adding different polymers and Yuan et. al(Yuan et al. 2018) for ionic liquids.

Figure 4 .

 4 Figure 4. Slurries sedimentation kinetics at 20 °C for each studied slurry.

Figure 5 .

 5 Figure 5. Slurry oscillatory rheometry. a) Log-log plot of G' and G'' as function of strain amplitude at 1 Hz, 20 °C. Solid symbols represent storage modulus G' and open symbols represent loss modulus G''. b) Log-log plot of G' and G'' as function of frequency at a deformation located in slurries LVR, at 20 °C. Solid symbols represent storage modulus G' and open symbols represent loss modulus G''.

Figure 7 .

 7 Figure 7. Yield stress as function of PAA concentration for slurries. As there are few points for S1, a yield stress was not considered but just the shear stress measured at the lowest shear rate in Figure 6 was considered. Solid lines serve just as guide for the eyes.

Figure 8 .

 8 Figure 8. Yield stress, peak power density and ionic conductivity as function of PAA concentration.

Figure 8

 8 Figure 8 displays the combined effects of yield stress and conductivity on the performance of the single RFB cell. As PAA concentration increases in slurries, peak power density increases steadily together with yield stress up to a concentration of 0.4 wt. %. At this PAA concentration, maximum peak power density is reached and suspending fluid (GPE3),

Table 1 .

 1 Zinc slurries composition.

	Sample	KOH 10 M wt. %	PAA wt. %	Attapulgite wt. %	ZnO wt.%	Zn wt. %
	S1		0.20
	S2		0.30
	S3	61*	0.40	0.9	4.0	33.8
	S4		0.50
	S5		0.60
	*KOH wt. % concentration was adjusted to each slurry to have a sum of components equal to 100 %.

Table 2 .

 2 Characteristic sedimentation times for S1 and S2 at 20 °C. S3, S4 and S5 do notshow a sedimentation loss equal or higher than 5 % in 200 000 s (~ 55 h).

	Sedimentation	S1	S2	S3 S4 S5
	5 %	420 s 3240 s -	-	-
	7 %	660 s 8760 s -	-	-
	10 %	1200 s	-	-	-	-
	14 %	9600 s	-	-	-	-
	It is observed that slurry S1 shows a settling of 5 % at a time of 420 s, whereas S2 reaches
	this threshold at 3240 s, almost 8 times longer. Finally, S3, S4 and S5 remain stable (less
	than 2 % of sedimentation) in the timespan analyzed. Finally, sample height stabilizes at a
	h/h0 loss of 14 % for S1 and 7 % for S2, respectively.
	Finally, sedimentation tests show that S1 and S2 settle within an hour of being prepared,
	contrary to S3, S4 and S5, slurries that remained stable for at least the length of the
	experiment (~ 55 h).

Table 3 .

 3 Herschel-Bulkley fitting parameters for zinc slurries obtained from equation 6. GPEs yield stress.

	Sample	PAA [wt. %]	= D [Pa]	K [Pa s n ]	n [ -]	R 2
	S1*	0.20	11.6	-	-	-
	S2	0.30	22.3	8.10	0.35	0.99
	S3	0.40	45.3	2.67	0.80	0.98
	S4	0.50	48.2	13.00	0.57	0.99
	S5	0.60	54.9	25.21	0.64	0.99

  Each of the GPEs correspond to the slurries with the same number (i.e. GPE5 is the suspending fluid of S5). As expected, it is observed that when PAA increases, yield stress value increases as well.

	As previously described, there is a condition to comply to maintain zinc particles in
	suspension, being that gelled network yield stress ought to be greater than minimum yield
	stress to suspend zinc particles (= D > = D 67 = 2.85 c ). It is observed in Table 4, that only
	three of the GPEs comply with this condition, being GPE3, GPE4 and GPE5. In other words,
	slurries prepared with at least 0.4 wt. % of PAA, corresponding to S3 (Table 1), will remain
	homogeneous for longer durations with no significant sedimentation observed in 55 h.
	Consequently, it is with no surprise that electrochemical and rheological behaviors are
	different when analyzing samples prone to sedimentation. Moreover, results obtained during
	sedimentation experiments (Figure 4), confirm that S1 and S2 exhibited sedimentation
	profiles, whereas S3, S4 and S5, prepared with GPEs with = D > 2.85 c are stable for the
	whole extent of the experiment.
		28	1.12	0.60
	GPE2	2.51	5.09	0.63
	GPE3	4.44	9.62	0.62	0.99
	GPE4	7.75	20.27	0.61
	GPE5	12.15	38.58	0.59

Acknowledgements

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement number 765289.

Conflicts of interest

The authors declare no conflict of interest.

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