Béatrice Mazoyer
email: bmazoyer@ina.fr

Julia Cagé
email: julia.cage@sciencespo.fr

Céline Hudelot
email: celine.hudelot@centralesupelec.fr

Marie-Luce Viaud
email: mlviaud@ina.fr

Real-time collection of reliable and representative tweets datasets related to news events

Keywords: Twitter, news, tweets retrieval

This paper is part of a wider work studying the co-influences of Twitter and the production of information by traditional media. A strong prerequisite of this study is to collect, with the limitations of the Twitter API, tweets linked to media events that are representative of the real Twitter activity. This paper describes two proposed approaches to handle this important task. The first one, inspired by information retrieval, puts the focus on query formulation. It consists on bridging the vocabulary gap between traditional news articles and tweets, by iteratively modifying the queries sent to the Twitter API depending on the tweets retrieved by previous queries. The second approach consists in streaming a representative sample of all emitted tweets and dynamically clustering them in events. We also discuss approaches to evaluate the collected datasets under the point of view of their representativity of the real activity on Twitter.

Introduction

How does information propagate online? Does it propagate differently on news websites and on social media? What is the role of social networks and in particular of Twitter in breaking news?

In an ideal world, to compare news production on social media and on mainstream media, one would need the universe during a given period of time (e.g. the year 2017) and a geographical location (e.g. France, the UK or the US) of documents published on the one hand on social media and on the other hand on mainstream media. Unfortunately, given the limitation of the Twitter API, it is not possible for the researcher to capture the universe of the documents (or tweets) published on Twitter. Does it mean that the research wont be able to answer the previously formulated questions? No. Because, the researcher can rather use a random sample of the documents, as long as this sample is representative. Why do we need representativity?

Assume that you want to answer the following question: what is the probability for a news story broken on social media to make it to the mainstream media? With the entire set of news stories broken on social media, it will be pretty simple to answer this question; but we dont have this data. Now assume that we get access to a subsample of the documents published on Twitter, but that this sample is not representative. For example, assume that this sample of tweets is such that the tweets characteristics (perhaps because the API provides the researcher with documents tweeted by users with more followers) are such that these documents have a higher probability to make it to the mainstream media. Then using this biased subsample will lead the researcher to overestimate the probability for a news story broken on social media to appear on mainstream media.

The same issue will arise if the researcher wants to tackle the follow-up question: what are the determinants of the success of a news story initially broken on social media? Imagine that the researcher is using a selected sample of tweets that is not representative. Imagine for example that this sample of tweets comes mainly from journalists working for a given media, e.g. Le Monde, and that, at the same time, within the set of tweets posted by Le Mondes journalists, only the successful ones are part of the sample, then the results of the empirical analysis will be biased in favor of Le Monde. In other words, when the researcher will study the impact of the company for which the journalist work (independent variable) on the probability for the news story broken on Twitter to make it to mainstream media (dependent variable), the coefficient obtained for Le Monde will ovestimate the real causal impact of the company.

It seems very difficult to correct for this bias. Hence the necessity to have a representative sample of tweets, i.e. a sample of tweets such that the tweets included in our sample do not differ from the tweets that are not included along all the dimensions that may have a direct impact on the dependent variable of interest.

In this paper, we aim at collecting, in real-time, tweets linked to French news that are representative of the total Twitter activity concerning media events. For now, we only work on tweets in French, which is an advantage in terms of volume. Indeed, since the Twitter API puts high restrictions on the volume of tweets returned, we get a largest proportion of emitted tweets by working on a language that is not very represented on Twitter (French tweets represent 1.8% of all emitted tweets). Based on the different ways of collecting tweets with the Twitter API, we explore two types of approaches to collect tweets:

1. Vocabulary-constrained collection based on keywords extracted from AFP dispatches. We use the stream of dispatches from the French news agency "Agence France Presse" (AFP). This news agency, like Reuters or Associated Press (AP), has a large network of journalists in many countries, that covers a wide range of news topics. Previous research has shown that this source covered 95% of all events covered by French news media in 2013 [START_REF] Cagé | The production of information in an online world: Is copy right?[END_REF]. 2. Random tweets collection followed by clustering. To get a continuous stream of random tweets, we studied Twitter Sample API 4 . Indeed, previous works show that the tweets from that API are "a representative sample of the true activity on Twitter" [START_REF] Morstatter | When is it biased?: assessing the representativeness of twitter's streaming API[END_REF]. However, the number of French tweets in this sample is very small (2400 tweets per hour on average). It is likely that small events do not generate enough French tweets to be represented in that sample. We thus describe, in Section 3.3, an approach to increase this sample of French tweets.

State of the art

Collecting representative sets of tweets means performing together the tweets collection and the evaluation of its representativity. We therefore present in this part: [START_REF] Cagé | The production of information in an online world: Is copy right?[END_REF] techniques to build sets of relevant tweets and (2) methods to evaluate tweets sets. A tool designed to collect representative tweets sets should comprise both aspects.

Query building strategies

To our knowledge, few approaches have been proposed in the literature to perform targeted tweet collection based on news articles. Indeed, the literature related to the task of linking tweets and news [START_REF] Kothari | Detecting Comments on News Articles in Microblogs[END_REF][START_REF] Zhao | Comparing Twitter and Traditional Media Using Topic Models[END_REF][START_REF] Guo | Linking Tweets to News: A Framework to Enrich Short Text Data in Social Media[END_REF][START_REF] Mele | Linking News across Multiple Streams for Timeliness Analysis[END_REF][START_REF] Hu | ET-LDA: Joint Topic Modeling for Aligning Events and their Twitter Feedback[END_REF][START_REF] Hua | Topical Analysis of Interactions Between News and Social Media[END_REF] mainly used existing datasets, mostly in English. These articles do not address the issue of real-time collection of representative tweets. Becker et al. [START_REF] Becker | Automatic Identification and Presentation of Twitter Content for Planned Events[END_REF] do not directly work on news articles, but they define query building strategies based on the title, description, time and location of a selected set of expected events to collect related tweets. Tanev et al. [START_REF] Tanev | Enhancing Event Descriptions through Twitter Mining[END_REF] extract 1-grams and 2-grams from the title and first sentence of news articles, and weight them depending on their IDF in a one million news articles collection. They build queries out of these n-grams and explore several queryexpansion strategies based on co-occurrences in their news articles corpus. Ning et al. [START_REF] Ning | Uncovering News-Twitter Reciprocity via Interaction Patterns[END_REF] build chains of articles concerning the same event and first collect tweets containing the articles' urls, then extract a list of top ten keywords from those tweets and collect tweets containing these keywords. This method is an attempt to bridge the vocabularies between tweets and news by using keywords from the tweets. Castillo et al. [START_REF] Castillo | Characterizing the life cycle of online news stories using social media reactions[END_REF], that share common objectives with our work, also use urls to retrieve tweets related to news articles. However, our observations show that most tweets containing a link to an article share only the title or first sentence of that article, without additional vocabulary. This approach is restrictive and may lead to miss a large part of Twitter activity concerning news. Our work aims at building queries that contain Twitter-specific terms in order to achieve higher recall than previously introduced methods, without losing precision.

Representativity of tweets collection

Few papers address the problem of the representativity of the collected tweets: we could only find three studies addressing this issue [START_REF] Morstatter | Is the Sample Good Enough? Comparing Data from Twitter's Streaming API with Twitter's Firehose[END_REF][START_REF] Morstatter | When is it biased?: assessing the representativeness of twitter's streaming API[END_REF][START_REF] Wang | Should we use the sample? analyzing datasets sampled from twitter's stream api[END_REF]. Morstatter et al. and Wang et al. [START_REF] Morstatter | Is the Sample Good Enough? Comparing Data from Twitter's Streaming API with Twitter's Firehose[END_REF][START_REF] Wang | Should we use the sample? analyzing datasets sampled from twitter's stream api[END_REF] propose methods to evaluate the Twitter Filter API 5 . This API returns a stream of tweets that match one or several predicates given as parameters. Both articles point out some biases in the collected datasets. In another article [START_REF] Morstatter | When is it biased?: assessing the representativeness of twitter's streaming API[END_REF], Morstatter et al. evaluate the Sample API and conclude to its representativity of the global Twitter activity. However this study does not consider methods to collect larger representative datasets in a specific language as we do for French in this paper.

Methodology

Fig. 1. Diagram of our keywords mining approach on three recent events

We present here two approaches for tweet collection:

-A tool designed to build queries in real time based on the stream of AFP dispatches and expand them using Twitter specific vocabulary. This approach is novel with respect to existing works [START_REF] Becker | Automatic Identification and Presentation of Twitter Content for Planned Events[END_REF][START_REF] Tanev | Enhancing Event Descriptions through Twitter Mining[END_REF][START_REF] Ning | Uncovering News-Twitter Reciprocity via Interaction Patterns[END_REF] since the step of query expansion using word embeddings, while developed in other contexts such as web retrieval [START_REF] Diaz | Query Expansion with Locally-Trained Word Embeddings[END_REF][START_REF] Roy | Using Word Embeddings for Automatic Query Expansion[END_REF], has, as far as we know, never been used in the context of tweets retrieval. -A method to get a significant and representative set of random tweets in French. Our work here aims at maximizing the volume of our random sample, while keeping its distribution characteristics the closest to the one of the random sample collected with Twitter Sample API. Events will then be detected by clustering following the same procedure as [START_REF] Liu | Reuters Tracer: A Large Scale System of Detecting & Verifying Real-Time News Events from Twitter[END_REF].

Vocabulary-constrained collection

Our query building tool is composed of several modules designed to (1) extract relevant keywords from AFP dispatches and build queries with them, (2) expand those queries using Twitter vocabulary, (3) validate that these queries return tweets linked to the event. All steps are summarized in Figure 1.

AFP dispatches comprise a title, a timestamp and a body containing a few lines of text describing a news event. 900 dispatches are published every day in average. We define a news event very simply by a new AFP dispatch with a title containing less than k words in common with the titles of previous dispatches published in a time window T . We do not proceed to more complex dispatch clustering since our observations show that when AFP journalists publish several dispatches on a given event, they usually keep the same title and only update the main text with additional details. In practice we have set k to 3 and T to 24 hours. For each event, we aim at building a set of queries that will retrieve tweets related to that event.

Keywords extraction For a given dispatch, we extract named entities from the title and the body. We then build queries using combinations of those entities following a number of rules: two locations cannot form a query (to avoid queries like "Alep Syria"), a query must contain at least two entities and less than four entities. This method is quite similar from what is done by Becker et al. [START_REF] Becker | Automatic Identification and Presentation of Twitter Content for Planned Events[END_REF]. We also take the exact title of the dispatch as a query, if it contains no named entities. The tweets containing these queries are then collected.

Query expansion We combine several methods to build new queries from collected tweets:

-Urls extraction: like Ning et al. [START_REF] Ning | Uncovering News-Twitter Reciprocity via Interaction Patterns[END_REF], we extract urls from collected tweets and extend our query set with these urls. -Keywords detection: we build a TF-IDF matrix from all terms of the tweets collected in the past 7 days. All tweets related to the same event form a document. We pick terms with a TF-IDF score higher than a defined threshold (different for 1-grams, 2-grams and 3-grams) -Synonyms: We use the CBOW model [START_REF] Mikolov | Efficient Estimation of Word Representations in Vector Space[END_REF] with negative sampling [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF] to build vector representations of words in Twitter vocabulary and find potential "synonyms" (terms used in very similar contexts) to expand our first queries. We train the model on a sample of French tweets collected with the Sample API in the past 7 days to get a vector representation of Twitter words. We define as "synonyms" of a word the words that have a cosine similarity higher than a certain threshold with that word-vector. For instance, using 0.85 as a threshold, we get "fh" (abbreviation for "François Hollande") as synonym for "hollande". We then build new queries by replacing terms of previous queries with their synonyms if they have some.

Query validation

The previous steps allow us to combine a large number of words to build queries that might be related to the given event. However, these methods can also produce wrong queries, either not precise enough ("Elysee Macron" for example, since l'"Elysée" is the common name for the French president's office), or clearly pointing to something or someone else than the entities extracted from the original dispatch. For instance, the synonyms of "sarkozy" are "sarko" (which is correct), "juppe" and "fillon" (who are politicians from the same party as Nicolas Sarkozy, but not the same person). We therefore need a method to filter incorrect queries. We can make the following hypothesis: if a news event is discussed on Twitter, the tweets containing words related to that news should form a peak in a time window of a few hours before or after the time of the event. If the tweets collected through a certain query do not peak around the event's time, it is either because the news is not discussed on Twitter, or because the query is not linked to the event in question. Starting from that assumption we are working on an algorithm to detect such anomalies on the time series of tweets and thus remove irrelevant queries.

Iteration The keywords extraction and query validation steps are repeated to increase the number of tweets linked to the considered event, until the set of collected tweet does not provide any new query.

Evaluation of the tweets collection tool

To the best of our knowledge, there is no dataset corresponding to our task. We are currently designing a user interface to manually label tweets. This interface will allow two types of evaluations:

-Precision: given the text of an AFP dispatch and the list of tweets collected by our tool as presumably linked to the dispatch, the user will assess if each tweet is correctly attributed. -Recall: the user has to manually formulate queries to retrieve as many tweets as possible in relationship to a given AFP dispatch, and assess if the returned tweets are indeed linked to the dispatch. This method, however, is only a partial solution to the problem of evaluating the proportion of tweets not found by our tool, since the user will not necessarily find all possible terms to query relevant tweets. Another way of approximating the recall would be to compare the proportion of tweets in each collected set to the size of clusters identified with our second approach.

Random tweets collection

Twitter Sample API returns 1% of the world stream randomly selected [START_REF] Morstatter | When is it biased?: assessing the representativeness of twitter's streaming API[END_REF], among which 1.8% on average are in French. 1% of the French stream represents in average only 2400 tweets an hour. A previous article [START_REF] Liu | Reuters Tracer: A Large Scale System of Detecting & Verifying Real-Time News Events from Twitter[END_REF] has estimated the proportion of news-related documents in tweets corpora to less than 0.2%. If this ratio is the same for French tweets, it means that we would receive between 4 and 5 news-related tweets in French per hour by using the Sample API, a volume much too small for our goals. Using the Filter API on "neutral" terms with "French" as language parameter allows us to collect a larger volume of tweets in French, but Twitter does not communicate about the tweets distribution. Since we need to ensure that the stream of collected tweets is representative of the tweets emitted on Twitter at the same moment, we designed some methods to compare it with the stream from the Sample API.

Joseph et al. [START_REF] Joseph | Two 1% s Don't Make a Whole: Comparing Simultaneous Samples from Twitter's Streaming API[END_REF] compare several samples collected with the Filter API on the same keywords using different connection tokens 6 : they find that two samples taken at the same time with the same keywords as inputs are "nearly identical" (96% of the tweets are the same, and the sets of tweets that are not have a very similar structure in terms of number of hashtags, urls and mentions per tweet). It is thus not possible to use several connections with the same keywords to get a higher number of tweets. However, spreading different keywords over several API connections should return a higher number of tweets. If these tweets are representative of the real activity on Twitter, the distribution of words within it should be the same (proportionally) to the one of tweets from the Sample API. To ensure that it is the case, we worked on optimizing the collect parameters (number of connections, number of keywords, distribution of keywords over the different tokens) by performing the steps described below:

Finding the most frequent keywords on Twitter. We collected tweets from the Sample API during three months and selected French tweets (retweets excluded). After removing punctuation, capital letters and non-Latin characters, we built a list of the 200 most frequent words in the French Twitter vocabulary.

Clustering keywords depending on their co-occurrences. We built a matrix of co-occurrences of the first 50, 100 and 200 most frequent words in the sampled tweets, and used it to cluster keywords in n clusters (n ∈ [START_REF] Morstatter | When is it biased?: assessing the representativeness of twitter's streaming API[END_REF][START_REF] Zhao | Comparing Twitter and Traditional Media Using Topic Models[END_REF]). By doing so, we aimed at putting together terms that are frequently used together, to collect sets of tweets with the smallest possible intersection. In total, we had 9 ways of collecting tweets (50, 100 or 200 words, spread over 2, 3 or 4 clusters). To control that our clustering approach was the best, we also tried to randomly spread the same number of words (50, 100 or 200) over the same number of Twitter access tokens (2, 3 or 4). We ran each test during 24 hours and compared the returned tweets with those obtained using the Sample API during the same period of 24 hours.

Running tests simultaneously. In order to select the best collection method we had to run series of tests in parallel: it is not possible compare tests conducted on different periods of times since the differences between the results could be due to a different tweets distribution between periods. However, we only had 13 Twitter access tokens, and thus could not run all tests simultaneously. Since a collection method requires 2 to 4 tokens, we could only run 6 to 3 tests in parallel.

Evaluation of random tweets collection

We used Kullback-Leibler divergence to compare the words distributions of the set of collected tweets to the distribution from the Sample API. The best collection method should have a words distribution very similar to the distribution of words collected with the Sample API, and thus the Kullback-Leibler divergence between the two distributions should be close to zero. We also evaluated our collection method with regards to the volume of collected tweets: we used a simple ratio between the volume collected with each method and the volume collected using the Sample API during the same period.

Results

Fig. 2. Daily evolution of the KL divergence between each collection method and the Sample API Fig. 3. Daily evolution of the ratio between the volume of collected tweets with each collection method and the volume collected with the Sample API Note: we have no results on the 01/22, since the connection to the API was broken.

Tweets collection tool

As explained in Section 3.2, we lack a labeled test-set to perform a correct evaluation of our tweets retrieval method. Our work so far has been directed towards the implementation of a robust architecture to stream and analyze a high number of documents in near real-time. The development of our test interface will allow us to provide concrete results in the near future. We also plan to compare our strategy with the state of the art, presented in section 2.1.

Random tweets collection

Our results are displayed in Figures 2 and3. We get consistent results over time: the collection method using words clustered by co-occurrences performs better in terms of proximity to the Sample than other methods using random distribution of words. The volume of collected tweets greatly differs from one collection method to another (from 50 to 75 times higher than the Sample), but the trend also remains stable over time 7 . However, these results need to be reproduced over a longer period to be confirmed, and we need to conduct wider tests on all collection methods.

5 Related work

Detecting events in the Twitter stream

A large number of methods have been proposed for detecting "stories" [START_REF] Petrović | Streaming first story detection with application to twitter[END_REF] or "events" [START_REF] Atefeh | A Survey of Techniques for Event Detection in Twitter[END_REF][START_REF] Aggarwal | Event detection in social streams[END_REF][START_REF] Weng | Event Detection in Twitter[END_REF][START_REF] Liang | Dynamic Clustering of Streaming Short Documents[END_REF][START_REF] Zhang | TrioVe-cEvent: Embedding-Based Online Local Event Detection in Geo-Tagged Tweet Streams[END_REF] in a continuous stream of tweets. However these methods do not link the detected clusters to news articles. Sankaranarayanan et al. [START_REF] Sankaranarayanan | TwitterStand: news in tweets[END_REF] use handpicked "seeders" (Twitter accounts that are known to publish news) and train a naive Bayes classifier to filter news tweets from "junk" tweets. News tweets are then clustered based on TF-IDF. Liu et al. [START_REF] Liu | Reuters Tracer: A Large Scale System of Detecting & Verifying Real-Time News Events from Twitter[END_REF] also adopt a twosteps approach to detect news clusters in a continuous stream of tweets. First, they use a noise-filtering algorithm based both on a set of filtering rules and on information credibility features [START_REF] Castillo | Information credibility on twitter[END_REF]. Then, they perform a clustering step depending on a similarity metric containing named entities, verbs and common nouns. These methods do not treat the problem of linking the collected tweets to news articles, but they are interesting to perform a first selection of relevant tweets.

Topic Models

Many methods designed to link tweets and news are based on topic models. A topic model takes a collection of documents as input and returns a set of topics and their distribution within the collection (a document is considered as a mix of several topics). Latent Dirichlet Allocation (LDA) [START_REF] Blei | Latent dirichlet allocation[END_REF] is currently the most commonly used topic model. Zhao et al. [START_REF] Zhao | Comparing Twitter and Traditional Media Using Topic Models[END_REF] perform LDA on a collection of news articles and an LDA-like algorithm on a collection of tweets and match the two sets of topics based on the similarity of words distribution. Other works develop algorithms to jointly learn the topics of tweets and news datasets [START_REF] Hu | ET-LDA: Joint Topic Modeling for Aligning Events and their Twitter Feedback[END_REF], [START_REF] Hua | Topical Analysis of Interactions Between News and Social Media[END_REF]. Guo et al. [START_REF] Guo | Linking Tweets to News: A Framework to Enrich Short Text Data in Social Media[END_REF] design a more specific latent variable model including features like hashtags, named entities and temporal relations to link each tweet of their dataset to the closest news article. The previous works perform well on aligning tweets to related news articles, but they are based on static datasets and do not tackle the dynamic nature of news events.

In 2006 Blei and Lafferty proposed Dynamic Topic Model (DTM) [START_REF] Blei | Dynamic topic models[END_REF] a model able to discover the evolution of topics over time. Mele et al. adapted this model to news events [START_REF] Mele | Linking News across Multiple Streams for Timeliness Analysis[END_REF] and used clustering to link the documents covering the same event across several news streams (news websites, RSS feed, Twitter accounts of media outlets).

Overall, several works address the task of linking tweets and news but they rely either on static datasets or on stream of tweets from manually selected users. We could not find any approach questioning the representativity of the collected tweets. Moreover, most approaches treat tweets and news articles only as text documents and do not take their multimodal nature (urls, mentions of users, pictures, videos) into account.

In this paper, we present two types of methods to collect tweets related to news events. The first one is a query formulation approach, that consists in generating potential queries related to an event and using the time repartition of collected tweets to detect and reject wrong queries. The second one is an approach based on clustering randomly collected events. We detailed the tests used to ensure that the collected tweets have a close to random distribution and showed from our first results that neutral words clustered by co-occurrence tend to perform better.

In future works, we plan to develop a user interface that will enable the labeling of numerous tweets, and give us a way of assessing the performance of our search tool. We will also work on methods to dynamically link tweets to news events, with a focus on processing tweets not only as text documents, but rather as multimodal objects.

https://developer.twitter.com/en/docs/tweets/sample-realtime/api-reference/getstatuses-sample

https://developer.twitter.com/en/docs/tweets/filter-realtime/api-reference/poststatuses-filter

To use the Twitter API, a connection token is required. Twitter limits the access to its data by generating only one connection token per Twitter account. The total number of tweets that one can get with only one token is limited to 1% of the global tweets volume at a given moment.

The drop in KL-divergence and in volume of collected tweets on the 01/21 could be linked to the fact that it was a Sunday, but we need to reproduce the experiment over a longer period to confirm that this drop is not a coincidence.


Erwan Paineau
email: erwan-nicolas.paineau@u-psud.fr

Geoffrey Monet

Véronique Peyre

Claire Goldmann

Stéphan Rouzière

Pascale Launois

Colloidal Stability of Imogolite Nanotubes Dispersions: A Phase Diagram Study

In this article, we revisit the colloidal stability of clay imogolite nanotubes by studying the effect of electrostatic interactions on geo-inspired synthetic nanotubes in aqueous dispersions. The nanotubes in question are double-walled aluminogermanate imogolite nanotubes (Ge-DWINTs) with a well-defined diameter (4.3 nm) and with an aspect ratio around 4. Surface charge properties are assessed by electrophoretic measurements, revealing that the outer surfaces of Ge-DWINT are positively charged up to high pH values. Series of Ge-DWINT dispersions have been prepared by osmotic stress to control both ionic strength of the dispersion and the volume fraction in nanotubes. Optical observations coupled to Small and Wide-Angle X-ray Scattering (SAXS/WAXS) experiments allow us to unravel different nanotube organizations. At low ionic strength (IS < 10 -2 mol.L -1 ), Ge-DWINTs are fully dispersed in water while they form an arrested gel phase above a given concentration threshold, which shifts towards higher volume fraction with increasing ionic strength. The swelling law, derived from the evolution of the mean intertube distance as a function of the nanotube concentration, evidences a transition from isotropic swelling at low volume fractions to one-dimensional swelling at higher volume fractions. These results show that the colloidal stability of Ge-DWINT is driven by repulsive interactions for ionic strengths lower than 10 -2 mol.L -1 . By contrast, higher salt concentrations lead to attractive interactions that destabilize the colloid suspension, inducing nanotube coagulation into larger structures that settle over time or form opaque gels. Detailed simulations of the WAXS diagram reveal that aggregates are mainly formed by an isotropic distribution of small bundles (less than 4 nanotubes) in which the nanotubes organized themselves in parallel orientation. Altogether, these measurements allow us to give the first overview of the phase diagram of colloidal dispersions based on geo-inspired imogolite-like nanotubes.

INTRODUCTION

The state of dispersions of clay nanoparticles is important in many industrial applications, from the production of nanocomposites to food packaging, [START_REF] Duncan | Applications of Nanotechnology in Food Packaging and Food Safety: Barrier Materials, Antimicrobials and Sensors[END_REF] and it is also involved in various environmental processes. [START_REF] Zhou | Clay Particles Destabilize Engineered Nanoparticles in Aqueous Environments[END_REF] When one refers to the colloidal properties of clays, it usually concerns properties related to swelling clay nanosheets. [START_REF] Swartzen-Allen | Surface and Colloid Chemistry of Clays[END_REF][START_REF] Lagaly | Colloid Clay Science[END_REF] Since the pioneering work of Langmuir, 5 a wide literature has been devoted to the colloidal stability of di-or trioctahedral swelling clay minerals such as smectites and hectorites. [START_REF] Norrish | The Swelling of Montmorillonite[END_REF][START_REF] Ramsay | Colloidal Properties of Synthetic Hectorite Clay Dispersions: I. Rheology[END_REF][START_REF] Mourchid | On Viscoelastic, Birefringent, and Swelling Properties of Laponite Clay Suspensions: Revisited Phase Diagram[END_REF][START_REF] Abend | Sol-gel Transitions of Sodium Montmorillonite Dispersions[END_REF][START_REF] Michot | Phase Diagrams of Wyoming Na-Montmorillonite Clay. Influence of Particle Anisotropy[END_REF][START_REF] Tombacz | Colloidal Behavior of Aqueous Montmorillonite Suspensions: The Specific Role of PH in the Presence of Indifferent Electrolytes[END_REF][START_REF] Michot | Liquid-crystalline Aqueous Clay Suspensions[END_REF][START_REF] Ruzicka | More on the Phase Diagram of Laponite[END_REF][START_REF] Paineau | Liquid-Crystalline Nematic Phase in Aqueous Suspensions of a Disk-Shaped Natural Beidellite Clay[END_REF] These charged clay nanosheets exhibit a wide variety of states (isotropic sols, liquid-crystal phases, repulsive and attractive arrested phases, flocs and aggregated sediments) depending not only on the size of the clay particles or the effect of charge location, but also on the ionic strength IS of the solution (𝐼𝑆 ∑ 𝑐 𝑧 , where the sum performed over all types 'i' of ions of the solution, with concentration ci and with charge zi, is simply equal to the concentration of monovalent ions here), the valence and hydration properties of ions in the suspension. [START_REF] Lagaly | Colloid Chemistry of Clay Minerals: The Coagulation of Montmorillonite Dispersions[END_REF][START_REF] Paineau | Aqueous Suspensions of Natural Swelling Clay Minerals. 1. Structure and Electrostatic Interactions[END_REF][START_REF] Ruzicka | Observation of Empty Liquids and Equilibrium Gels in a Colloidal Clay[END_REF][START_REF] Paineau | Liquid-Crystalline Properties of Aqueous Suspensions of Natural Clay Nanosheets[END_REF][START_REF] Michot | Coagulation of Na-Montmorillonite by Inorganic Cations at Neutral PH. A Combined Transmission X-Ray Microscopy, Small Angle and Wide Angle X-Ray Scattering Study[END_REF][START_REF] Rosenfeldt | In-Depth Insights into the Key Steps of Delamination of Charged 2D Nanomaterials[END_REF] Compared to this large body of literature, there is surprisingly few works dealing with the colloidal stability of one-dimensional clay particles such as sepiolite nanorods [START_REF] Woolston | Isotropic-nematic Phase Transition in Aqueous Sepiolite Suspensions[END_REF] or halloysite nanotubes. [START_REF] Lisuzzo | Colloidal Stability of Halloysite Clay Nanotubes[END_REF] It might be related to the challenges of stabilizing in aqueous media this type of anisometric objects that remain stable for only few hours. Different strategies have been therefore employed to extend colloidal stability, either by adsorption of anionic compounds or dispersion in hexametaphosphate. [START_REF] Lisuzzo | Colloidal Stability of Halloysite Clay Nanotubes[END_REF][START_REF] Woolston | Isotropic-Nematic Phase Transition in Aqueous Sepiolite Suspensions[END_REF][START_REF] Luo | Liquid Crystalline Phase Behavior and Sol-gel Transition in Aqueous Halloysite Nanotube Dispersions[END_REF] Imogolite-like nanotubes (INTs) offer an interesting alternative since these aluminosilicate and aluminogermanate nanotubes can form stable aqueous dispersions without the assistance of a stabilizing agent. [START_REF] Paineau | A Liquid-Crystalline Hexagonal Columnar Phase in Highly-Dilute Suspensions of Imogolite Nanotubes[END_REF] Belonging to clay minerals, these inorganic nanotubes consist of a curved octahedral [Al(OH)3] layer on which isolated [(Si,Ge)O3(OH)] tetrahedron units are connected upright to the octahedral vacancy via covalent bonding between three mutual oxygen atoms (Figure 1a). A nanotube wall is constructed from (OH)3Al2O3(Si,Ge)(OH) elementary units, where hydroxyl groups and atoms are labelled here from outside to inside the wall. [START_REF] Cradwick | a Hydrated Aluminum Silicate of Tubular Structure[END_REF][START_REF] Monet | Structural Resolution of Inorganic Nanotubes with Complex Stoichiometry[END_REF] Naturally occurring worldwide either in weathered volcanic soils or in Spodosols, [START_REF] Nanzyo | Non-Crystalline Inorganic Constituents of Soil[END_REF] INTs can also be synthesized easily by using low-temperature sol-gel methods, [START_REF] Farmer | Synthesis of Imogolite -Tubular Aluminum Silicate Polymer[END_REF] while controlling both their inner diameter (1.5-3 nm), shape (single-walled: SW; or double-walled: DW structures, Figure 1b), or surface properties. [START_REF] Monet | Structural Resolution of Inorganic Nanotubes with Complex Stoichiometry[END_REF][START_REF] Yucelen | Shaping Single-Walled Metal Oxide Nanotubes from Precursors of Controlled Curvature[END_REF][START_REF] Amara | Tunable-Diameter, Metal Oxide Nanotubes for Trapping of Organic Molecules[END_REF] Thanks to these unique behaviors, INTs are becoming appealing for a wide range of potential applications, [START_REF] Poncelet | Industrial Implications in the Uses of Tubular Clay Minerals[END_REF][START_REF] Paineau | Imogolite Nanotubes: A Flexible Nanoplatform with Multipurpose Applications[END_REF] ranging from nanocomposites [START_REF] Shikinaka | Design of Stimuli-Responsive Materials Consisting of the Rigid Cylindrical Inorganic Polymer "Imogolite[END_REF] to molecular filtration [START_REF] Liou | Investigating the Potential of Single-Walled Aluminosilicate Nanotubes in Water Desalination[END_REF] or for energy production [START_REF] Elliott | Chemically Selective Alternatives to Photoferroelectrics for Polarization-Enhanced Photocatalysis: The Untapped Potential of Hybrid Inorganic Nanotubes[END_REF] to cite a few. Currently, it remains a very challenging task to control the packing of nanotubes before or during the processing into films, membranes or complex hierarchical structures. [START_REF] Ma | Methyl Methacrylate) Grafted Imogolite Nanotubes Prepared through Surface-Initiated ARGET ATRP[END_REF] From an applied point of view, a deep understanding of the colloidal stability of these systems is therefore necessary since most applications start from dispersions of INTs.

The swelling and colloidal properties of these nanotubes were quickly recognized as early as their discovery as macroscopic gel-like films in weathered pumice beds. [START_REF] Yoshinaga | Imogolite in Some Ando Soils[END_REF] Starting from INTs suspensions, it is easy to obtain individually dispersed nanotubes in acidic and low ionic strength aqueous medium. [START_REF] Karube | Hysteresis of the Colloidal Stability of Imogolite[END_REF] However, INTs have a strong tendency to form instantaneously transparent gels or to bundle irreversibly in bulky flocs at alkaline pH or for high ionic strength. [START_REF] Karube | Hysteresis of the Colloidal Stability of Imogolite[END_REF][START_REF] Philipse | On the Density and Structure Formation in Gels and Clusters of Colloidal Rods and Fibers[END_REF][START_REF] Tsuchida | Effects of PH and Ionic Strength on Electrokinetic Properties of Imogolite[END_REF][START_REF] Paineau | Effect of Ionic Strength on the Bundling of Metal Oxide Imogolite Nanotubes[END_REF] For instance, it has long been recognized that addition of a concentrated ammonia solution (typically 30 wt%) leads to the formation of gels, [START_REF] Farmer | Synthesis of Imogolite -Tubular Aluminum Silicate Polymer[END_REF] a straightforward method to retrieve a concentrated suspension but with potentially aggregated nanotubes at this alkaline pH. More interestingly, dispersions of INTs also show well-ordered liquid-crystalline phases for relatively low volume fraction, [START_REF] Paineau | A Liquid-Crystalline Hexagonal Columnar Phase in Highly-Dilute Suspensions of Imogolite Nanotubes[END_REF][START_REF] Kajiwara | Lyotropic Mesophase of Imogolite,1. Effect of Polydispersity on Phase-Diagram[END_REF][START_REF] Amara | Single-Step Formation of Micron Long (OH)3Al2O3Ge(OH) Imogolite-like Nanotubes[END_REF] if the nanotubes have a large enough aspect ratio. All these features are related to the surface charge properties of INTs and the peculiar difference in bond valence leading to positively (negatively) charged outer (inner) surface. [START_REF] Gustafsson | The Surface Chemistry of Imogolite[END_REF][START_REF] Teobaldi | Hydroxyl Vacancies in Single-Walled Aluminosilicate and Aluminogermanate Nanotubes[END_REF][START_REF] Poli | Large-Scale Density Functional Theory Simulation of Inorganic Nanotubes: A Case Study on Imogolite Nanotubes[END_REF] If we refer to the classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, the colloidal stability of charged particles is a balance between repulsive electrostatic forces due to the diffuse double layer of counterions and attractive van der Waals forces. This delicate equilibrium is clearly dependent on the ionic strength of the Ge-DWINTs are fully dispersed in aqueous suspensions while they form a gel phase above a given volume fraction threshold. By contrast, higher salt concentrations induce aggregation phenomena with the progressive formation of small bundles in suspension. These results are completed by using Small-Angle X-ray Scattering (SAXS) experiments to assess correlations between nanotubes depending on the position in the phase diagram as well as to establish a general swelling law.

EXPERIMENTAL SECTION

Chemicals. Aluminum perchlorate nonahydrate (Al(ClO4)3.9H2O, ACS reagent, ≥ 98%), tetraethoxygermane (TEOG, ≥99.95%), anhydrous NaOH (ACS reagent, ≥ 97%), NaCl (ACS reagent, ≥ 99%), polyethylene glycol (PEG20000, M = 20000 g.mol -1 ) and ethanol solution (96%)

were purchased from Sigma Aldrich. KBr (IR grade) was purchased from Merck. All products were used as received.

Synthesis of aluminogermanate double-walled nanotubes (Ge-DWINTs). Ge-DWINTs were synthesized by sol-gel process using the method proposed by Levard et al. [START_REF] Levard | Synthesis of Large Quantities of Single-Walled Aluminogermanate Nanotube[END_REF] Typically, the synthesis is performed in PTFE beakers by adding Ge(OEt)4 to an aluminum perchlorate solution with an [Al]/[Ge] ratio equal to 2 under vigorous stirring. In order to obtain only DW nanotube structures, the initial aluminum concentration is set at 0.25 mol.L -1 . [START_REF] Thill | Physico-Chemical Control over the Single-or Double-Wall Structure of Aluminogermanate Imogolite-like Nanotubes[END_REF] The solution is slowly hydrolyzed by addition of a NaOH solution (0.25 mol.L -1 ) until a [OH]/[Al] of 2 is reached and then, this mixture is placed into an oven at 95°C for 5 days. At the end of the ageing process, the suspension was recovered at room temperature and subsequently dialyzed (Spectra/Por, cut-off 10 kDa, Spectrum) against ultrapure water until the conductivity dropped below 5 µS.cm -1 . The initial suspension stock will be referenced as DW0 hereafter.

Nanotubes characterization. The characterization of synthesized nanotubes was performed from an aliquot of the DW0 suspension. Transmission Electron Microscopy (TEM) observations on individual nanotubes were performed on a JEOL1400 microscope operating at 80 kV to avoid any electron beam damage. The use of standard procedure by depositing a drop of the dialyzed suspension on the TEM grid always leads to the formation of thread-like structures forming entangled networks of INTs. [START_REF] Mukherjee | Phenomenology of the Growth of Single-Walled Aluminosilicate and Aluminogermanate Nanotubes of Precise Dimensions[END_REF] To overcome this drawback, observation of individual nanotubes was achieved by simply using a modification of the method proposed by Yang & Su, [START_REF] Huixian | Individual Dispersion of Synthetic Imogolite Nanotubes via Droplet Evaporation[END_REF] and Yucelen et al. [START_REF] Yucelen | A Generalized Kinetic Model for the Formation and Growth of Single-Walled Metal Oxide Nanotubes[END_REF] Typically, a dilute suspension of DW0 (~ 1 mg.L -1 ) was prepared in ethanol and a drop was deposited on a carbon-coated copper grid lying on an absorbent paper and dried in air. The morphological parameters, i.e. the mean length and diameter of DW0, were measured over 500 individual particles by using Fiji software. [START_REF] Schindelin | Fiji: An Open-Source Platform for Biological-Image Analysis[END_REF] The polydispersity index PDI was also determined as:

𝑃𝐷𝐼 〈𝑋 〉 〈𝑋〉 ⁄ 〈𝑋〉 ⁄ 1
with 〈X〉 being either the mean length L or diameter D of the nanotubes.

Fourier Transform Infrared Spectroscopy (FTIR) spectrum of DW0 was recorded using a Nicolet iS50 FTIR spectrometer equipped with a KBr beamsplitter and a DTGS/KBr detector. The transmission measurement was performed at 4 cm -1 resolution in the low IR region (1300-400 cm - 1 range). A pressed pellet was prepared by mixing about 1.5 mg of dry DW0 powder with 150 mg of KBr.

Measurement of the electrophoretic mobility µ was undertaken using a Zetasizer Nano ZS (Malvern) at a fixed wavelength of 632 nm in a folded capillary polycarbonate cell with gold electrodes. The potential applied between the electrodes was set around 150 V. The measurements were performed in triplicate at 25°C on a dilute DW0 aqueous suspension (~ 2 mg.L -1 ). Small amounts of a 0.1 mol.L -1 NaOH solution were added stepwise (pH ~ 0.5) under stirring and, after an equilibrium time of 1 min, the electrophoretic mobility was determined as a function of pH.

Samples preparation by osmotic stress. Osmotic stress (here, dialysis against a polymer solution) is a convenient way to obtain concentrated suspensions of colloidal charged particles, [START_REF] Parsegian | Osmotic Stress for the Direct Measurement of Intermolecular Forces[END_REF] and has been widely employed to explore the phase diagram of clay nanosheets. [START_REF] Michot | Phase Diagrams of Wyoming Na-Montmorillonite Clay. Influence of Particle Anisotropy[END_REF][START_REF] Michot | Liquid-crystalline Aqueous Clay Suspensions[END_REF][START_REF] Paineau | Liquid-Crystalline Nematic Phase in Aqueous Suspensions of a Disk-Shaped Natural Beidellite Clay[END_REF][START_REF] Paineau | Aqueous Suspensions of Natural Swelling Clay Minerals. 1. Structure and Electrostatic Interactions[END_REF] Maillet showed that concentrated suspensions of aluminogermanate single-walled imogolite nanotubes can also be prepared by osmotic compression using Dextran; however control of electrostatic interactions had not been undertaken. [START_REF] Maillet | Structure et Croissance de Nanotubes de Ge-Imogolite Simple et Double-Paroi[END_REF] To achieve this goal, batches of suspensions were prepared from the initial stock of DW0 suspension at a fixed ionic strength (IS = 10 -4 -1 mol.L -1 ). [START_REF] Paineau | Liquid-Crystalline Nematic Phase in Aqueous Suspensions of a Disk-Shaped Natural Beidellite Clay[END_REF] Typically, dialysis membranes (Spectra/Por, cut-off 10 kDa, Spectrum) were filled with stock suspension and placed in a 1 L reservoir of PEG20000 solution prepared by dilution with the appropriate amount of NaCl. Under such conditions, the ionic strength is fixed in the reservoir, which avoids problems related with the Donnan effect. [START_REF] Dubois | Osmotic Pressure and Salt Exclusion in Electrostatically Swollen Lamellar Phases[END_REF] The osmotic pressure was imposed during two weeks to obtain concentrated suspensions. The Ge-DWINTs suspensions were then recovered and the volume fraction  was determined as: In both cases, the intensity versus scattering vector modulus Q (Q = 4π/λ sin(θ), where λ is the incident wavelength and 2θ is the scattering angle) curves were deduced from the azimuthal angular integration [0, 2π] of the scattering patterns, previously corrected for water and glass scattering using homemade software.

RESULTS AND DISCUSSION

Characterization of initial stock (DW0) colloid sample. INTs are easily obtained by sol-gel methods. However, both the morphology and the surface properties of INTs strongly depend on the synthesis route used. [START_REF] Yucelen | Shaping Single-Walled Metal Oxide Nanotubes from Precursors of Controlled Curvature[END_REF][START_REF] Thill | Physico-Chemical Control over the Single-or Double-Wall Structure of Aluminogermanate Imogolite-like Nanotubes[END_REF][START_REF] Chemmi | Key Steps Influencing the Formation of Aluminosilicate Nanotubes by the Fluoride Route[END_REF] We first characterized the initial stock of dialyzed DW0 by using transmission electron microscopy (TEM) observations. A representative TEM micrograph is presented in Figure 2a, evidencing well-dispersed individual nanotubes. The nanotube length (NLD) and diameter distributions (NDD), deduced from the analysis of more than 500 nanotubes, are plotted as histograms in Figure 2b,c and the deduced morphological parameters are summarized in Table 1. The NLD follows a log-normal distribution with a mean length 〈L〉 = 18  1.6 nm but with a relatively high PDIL value (Table 1), as expected for INTs samples synthesized in similar conditions. [START_REF] Amara | Single-Step Formation of Micron Long (OH)3Al2O3Ge(OH) Imogolite-like Nanotubes[END_REF][START_REF] Yucelen | A Generalized Kinetic Model for the Formation and Growth of Single-Walled Metal Oxide Nanotubes[END_REF] By contrast, the NDD remains highly monodisperse (PDID = 7%) with a normally distributed diameter centered at 〈D〉 = 4.3  0.3 nm, in close agreement with previous results obtained from X-ray scattering experiments. [START_REF] Amara | Single-Step Formation of Micron Long (OH)3Al2O3Ge(OH) Imogolite-like Nanotubes[END_REF] Accordingly, the aspect ratio 〈L〉∕〈D〉 of these nanotubes remains extremely low (Table 1), showing that they represent an attractive model of nanotubes with well-defined morphology. Figure 3a presents the IR spectrum obtained on the 1300-400 cm -1 range, an area related to the INT backbone. The band at 915 cm -1 and the doublet at 825 and 805 cm -1 are specific to Ge-O stretching vibrations for well-formed tubular structures. [START_REF] Paineau | Effect of Ionic Strength on the Bundling of Metal Oxide Imogolite Nanotubes[END_REF][START_REF] Amara | Single-Step Formation of Micron Long (OH)3Al2O3Ge(OH) Imogolite-like Nanotubes[END_REF][START_REF] Wada | Effects of Substitution of Germanium for Silicon in Imogolite[END_REF] Below, we find a large absorption band, typical of the imogolite-like local structure. [START_REF] Paineau | Effect of Ionic Strength on the Bundling of Metal Oxide Imogolite Nanotubes[END_REF][START_REF] Bishop | Spectral and Hydration Properties of Allophane and Imogolite[END_REF] The bands located at 690 and 555 cm -1 are assigned to Al-O stretching vibrations arising from the INT structure, while the peaks at 465 and 420 cm -1 seem to be related to various -OH, Ge-O-Al and Ge-O bending modes. [START_REF] Paineau | Effect of Ionic Strength on the Bundling of Metal Oxide Imogolite Nanotubes[END_REF][START_REF] Bishop | Spectral and Hydration Properties of Allophane and Imogolite[END_REF] In addition, the band between 1200 and 1100 cm -1 corresponds to residual perchlorates anions adsorbed on the outside surfaces of DW0. Since the surface charge of INTs seems to be at the origin of the nanotube affinity to ion adsorption, [START_REF] Gustafsson | The Surface Chemistry of Imogolite[END_REF] this property was assessed by performing electrokinetic measurements on a dilute DW0 sample (see Method section) to reduce the probability of double-layer interaction. Figure 3b presents the evolution of the electrophoretic mobility as a function of the pH. The DW0 suspension displays a positive mobility µ in the whole range of pH studied. The value of µ is fairly constant around 4.5 10 -8 m².V -1 .s -1 for pH ≤ 6, indicating saturated protonation of the nanotube outer surface (-AlOH2 + groups) [START_REF] Gustafsson | The Surface Chemistry of Imogolite[END_REF][START_REF] Buining | Phase Behavior of Aqueous Dispersions of Colloidal Boehmite Rods[END_REF] for such pH values. Then it decreases almost linearly with pH similarly as observed previously for single-walled aluminosilicate [START_REF] Tsuchida | Effects of PH and Ionic Strength on Electrokinetic Properties of Imogolite[END_REF][START_REF] Karube | Electrophoretic Behavior of Imogolite under Alkaline Conditions[END_REF][START_REF] Su | The Electrophoretic Mobility of Imogolite and Allophane in the Presence of Inorganic Anions and Citrate[END_REF] and aluminogermanate nanotubes. [START_REF] Paineau | Effect of Ionic Strength on the Bundling of Metal Oxide Imogolite Nanotubes[END_REF] Interestingly, the use of the isoelectrical point (IEP), i.e. the pH at which the electrophoretic mobility is zero, has been proposed as an alternative method to evaluate the formation of INTs. [START_REF] Arancibia-Miranda | Use of Isoelectric Point and PH to Evaluate the Synthesis of a Nanotubular Aluminosilicate[END_REF] The increase of IEP to alkaline values seems to reflect a decrease of the contribution of both terminal ≡Al-OH and ≡Ge-OH groups on the INT surface charge. [START_REF] Arancibia-Miranda | Use of Isoelectric Point and PH to Evaluate the Synthesis of a Nanotubular Aluminosilicate[END_REF] . In the present work, the -Al-OH groups remain protonated over a wide range of pH with an IEP value around 12, more than two orders of magnitude higher than the IEP value reported for Ge-SWINTs. [START_REF] Paineau | Effect of Ionic Strength on the Bundling of Metal Oxide Imogolite Nanotubes[END_REF] Although the latter form much longer nanotubes (lower contribution of tube-ends on the electrophoretic mobility), [START_REF] Amara | Tunable-Diameter, Metal Oxide Nanotubes for Trapping of Organic Molecules[END_REF] this result thus suggests that, even if the DW0 sample has low aspect ratio (Table 1), it is made of well-formed INTs with a low ratio of edge defects.

Phase diagram.

Starting from the initial stock of DW0, concentrated Ge-DWINTs suspensions were prepared at different ionic strengths (IS = 10 -4 , 10 -3 , 10 -2 , 10 -1 and 1 mol.L -1 ) to scan the /IS phase diagram. These suspensions were transferred in glass vials for naked-eye optical observations, either in natural light (Figure 4) or between crossed polarizers (Figure S1). At high ionic strengths (IS ≥ 10 -1 mol.L -1 ), the predominance of attractive van der Waals interactions induces the aggregation of INTs. All samples form opaque aggregates that slowly settle down after few days, the proportion of sediment increasing with the volume fraction (Figure 4a,b). When the volume fraction reaches values higher than 1.5%, these aggregates no longer sediment but form instead an arrested (attractive gel) phase (Figure S2). In contrast, a clear change appears at lower ionic strength. Indeed, all samples display stable transparent colloidal dispersions over a wide range of volume fractions (Figure 4c-e). Whatever the IS value (10 -4 -10 -2 mol.L -1 ), the viscosity gradually increases with  and, above a certain volume fraction, the dispersions form instantaneously a gel phase (Figure 4d,e), [START_REF] Philipse | On the Density and Structure Formation in Gels and Clusters of Colloidal Rods and Fibers[END_REF][START_REF] Liz-Marzán | Synthesis of Platinum Nanoparticles in Aqueous Host Dispersions of Inorganic (Imogolite) Rods[END_REF] which remains completely transparent. These gel phases become birefringent for  > 1% (Figure S1), meaning that Ge-DWINTs are "frozen" with preferred orientations. From these macroscopic optical observations, we were able to establish the first phase diagram representing the effect of ionic strength as a function of the volume fraction in aqueous suspensions of Ge-DWINT (Figure 4f). As stated in the introduction, the colloidal stability of charged nanoparticles like Ge-DWINTs can be divided, in principle, into two domains controlled by the ionic strength of the medium. At low ionic strength, the Debye screening length 𝜅 of the diffuse double layer is large (e.g. 𝜅 0.304 √𝐼𝑆 ⁄ = 30.4 nm for IS = 10 -4 mol.L -1 ), inducing the stabilization of nanotubes by electrostatic repulsions. The liquid/gel transition also displays a positive slope, which hints at the repulsive nature of the interactions. [START_REF] Paineau | Liquid-Crystalline Nematic Phase in Aqueous Suspensions of a Disk-Shaped Natural Beidellite Clay[END_REF][START_REF] Michot | Sol/Gel and Isotropic/Nematic Transitions in Aqueous Suspensions of Natural Nontronite Clay. Influence of Particle Anisotropy. 1. Features of the I/N Transition[END_REF] At higher ionic strength, the Debye length is strongly reduced. Abend & Lagaly have clearly demonstrated that the interactions between clay nanosheets particles become attractive at a ionic strength of about 0.01 mol.L -1 (also referred as the critical coagulation concentration). [START_REF] Abend | Sol-gel Transitions of Sodium Montmorillonite Dispersions[END_REF] Similar behavior was also observed in colloidal dispersions of boehmite rods. [START_REF] Buining | Phase Behavior of Aqueous Dispersions of Colloidal Boehmite Rods[END_REF] We observe the same effect here since coagulation of Ge-DWINTs starts to occur above this critical concentration, where 𝜅 is of the same order of magnitude as the diameter of Ge-DWINTs. Finally, it can be pointed out that the general appearance of this phase diagram for 1D charge nanoparticles is in good agreement with those reported for 2D charged layered materials such as swelling clays, 9,10,12,14 layered double hydroxide, [START_REF] Zhang | Phase Behavior of Aqueous Suspensions of Mg2Al Layered Double Hydroxide: The Competition among Nematic Ordering, Sedimentation, and Gelation[END_REF] or takovites. [START_REF] Michot | Sol-Gel Transitions in Aqueous Suspensions of Synthetic Takovites. The Role of Hydration Properties and Anisotropy[END_REF] Nanotubes local organization. The transition from repulsive to attractive interactions should affect the local structural organization of Ge-DWINTs. We investigate this effect by first carrying out WAXS experiments on the most concentrated samples obtained at each IS value (Figure 5a). All WAXS diagrams exhibit the expected intensity modulations characteristic of a doublewalled nanotube in the 4-10 nm -1 Q-range, [START_REF] Paineau | A Liquid-Crystalline Hexagonal Columnar Phase in Highly-Dilute Suspensions of Imogolite Nanotubes[END_REF] which are related to the interferences between inner and outer tubes. [START_REF] Cambedouzou | X-Ray Diffraction as a Tool for the Determination of the Structure of Double-Walled Carbon Nanotube Batches[END_REF] Beyond this proof of the DW structure, the most striking feature is probably the stepwise flattening of the first oscillation centered at Q = 2.2 nm -1 along with a shape modification of the subsequent oscillations, in particular at Q = 5 nm -1 , for IS ≥ 10 -2 mol.L -1 (Figure 5a).

However, the WAXS diagrams do not exhibit any narrow Bragg peaks, suggesting that the local structural organization, if any, is rather restricted to bundles with limited size. [START_REF] Paineau | Effect of Ionic Strength on the Bundling of Metal Oxide Imogolite Nanotubes[END_REF] X-ray scattering diagrams can be used to reveal accurate information on both morphology and spatial organization of imogolite-like nanotubes. [START_REF] Paineau | A Liquid-Crystalline Hexagonal Columnar Phase in Highly-Dilute Suspensions of Imogolite Nanotubes[END_REF][START_REF] Monet | Structural Resolution of Inorganic Nanotubes with Complex Stoichiometry[END_REF][START_REF] Paineau | Effect of Ionic Strength on the Bundling of Metal Oxide Imogolite Nanotubes[END_REF] For small scattering wave-vectors (Q < 10 nm -1 ), WAXS diagrams can be fitted within the homogenous approximation. In the present case, Ge-DWINT is modelled by two hollow cylinders with equal wall thickness twall, a distance tgap in between and electronic density ρINT (0.91 e -.Å -3 ) as schematized in Figure 5b. The Ge-DWINT is defined by its external radius Re = Ri + 2twall + tgap with Ri the inner radius of the inner tube and tgap the gap thickness between the inner and outer tubes. For the sake of clarity, we consider nanotubes with infinite length in the calculation of WAXS diagrams. For interested readers, a complete description of the X-ray formalism used to fit the WAXS data of nanotubes can be found elsewhere. [START_REF] Rols | Diffraction by Finite-Size Crystalline Bundles of Single Wall Nanotubes[END_REF][START_REF] Amara | Hexagonalization of Aluminogermanate Imogolite Nanotubes Organized into Closed-Packed Bundles[END_REF] Briefly, the scattered intensity from Ge-DWINTs of infinite length, with random orientations, is simply the product of the squared form factor P(Q) of the nanotube by the structure factor S(Q) of the bundle, renormalized to the wave-vector:

𝐼 𝑄 ∝ 1 𝑄 𝑃 𝑄 𝑆 𝑄 3
where P(Q) and S(Q) are defined as:

𝑃 𝑄 𝜌 𝑓 𝑄 𝜌 𝑓 𝑄 𝐹 𝐹 𝐹 𝐹 4𝑎 𝑆 𝑄 𝑝 𝐽 𝑄𝑅 4𝑏
with fINT and fw the normalized X-ray scattering factor of an imogolite unit and a water molecule, respectively; ρw (0.334 e -.Å -3 ) is the electronic density of water inside and outside the nanotubes, FR is the Fourrier transform of the projection of a full cylinder of radius R along its axis.The term 𝑝 yields the proportion of each type of bundle formed by Nb nanotubes in the sample while 𝑅 is the modulus of the vector joining the centers of tubes j and k in a bundle, in the plane perpendicular to its axis (Figure 5c). We chose as a first approximation a hexagonal packing configuration with a lattice parameter a, which is ultimately the structure adopted by nanotubes in large bundles. [START_REF] Paineau | A Liquid-Crystalline Hexagonal Columnar Phase in Highly-Dilute Suspensions of Imogolite Nanotubes[END_REF][START_REF] Paineau | Effect of Ionic Strength on the Bundling of Metal Oxide Imogolite Nanotubes[END_REF] Experimental WAXS diagrams obtained at IS = 10 -4 mol.L -1 are well reproduced considering dispersions of individual nanotubes (Nb = 1, p = 1) with Ri = 0.77 nm, twall = 0.58 nm and tgap = 0.2 nm (blue dotted curve in Figure 5a), the calculated 2Re = 4.26 nm being in close agreement to the 〈D〉 value determined from TEM measurements (Table 1). It must also be emphasized that the transition from an isotropic liquid (IL,  = 0.26%) to a birefringent gel phase (BG,  = 0.99%) does not modify the WAXS diagram (Figure S3), implying in both cases the presence of individual nanotubes only. Similar behaviors are also observed for samples prepared at IS = 10 -3 mol.L -1 (Figure 5a), which confirm the predominance of repulsive interactions between Ge-DWINTs in this range of IS values, in agreement with optical observations (Figure 4). For IS ≥ 10 -2 mol.L -1 , the WAXS curves no longer match obviously the scattering diagram of individual nanotubes. A structure factor needs to be taken into consideration in eq. ( 3) to account for experimental data.

We first explored different scenarios of bundling, varying both the number of tubes per bundle and the lattice parameter a (Figure S4). The slight deformation of the oscillations coupled with the lack of Bragg peaks indicate the presence of small bundles, containing only a few tubes. However, the presence of only one population of bundles (Nb fixed) can be ruled out since the simulated WAXS diagrams do not allow to reproduce the deformation of the scattered intensity. Instead, a qualitative fit is obtained considering a mixture of bundles (Nb < 4) with a mean lattice parameter a = 4.5 nm , corresponding to a 2 Å gap between the nanotubes (Figure 5a). Such a value is reasonable if one considers intertube interactions and the possible presence of anions bounded on the outer surface of the nanotubes. It can be pointed out that errors on the proportion of each type of bundle are relatively significant due to the relative limited size of the bundles as evidenced for Ge-SWINTs, [START_REF] Paineau | Effect of Ionic Strength on the Bundling of Metal Oxide Imogolite Nanotubes[END_REF] and similar WAXS diagrams may be obtained by slightly altering 𝑝 values (Figure S4d). (Figure 6). We have seen that Ge-DWINTs are no longer organized individually but are stacked locally in small bundles forming micrometer-sized aggregates of Ge-DWINTs, as evidence from optical observations (Figure 4a,b). Assuming a similar behavior as the one reported for 2D swelling clays, [START_REF] Michot | Coagulation of Na-Montmorillonite by Inorganic Cations at Neutral PH. A Combined Transmission X-Ray Microscopy, Small Angle and Wide Angle X-Ray Scattering Study[END_REF] the appearance of this Q -2 slope can be reasonably interpreted as resulting from the formation of these aggregates, due to the onset of dominating attractive interactions. By using eq. 5, the evolution of the average interparticle distance as a function of the volume fraction, the so-called swelling (or dilution) law, can be obtained. Assuming a local hexagonal configuration of the nanotubes as the packing limit that nanotubes can reach, [START_REF] Paineau | A Liquid-Crystalline Hexagonal Columnar Phase in Highly-Dilute Suspensions of Imogolite Nanotubes[END_REF][START_REF] Paineau | Effect of Ionic Strength on the Bundling of Metal Oxide Imogolite Nanotubes[END_REF] the relation between d and volume fraction in the case of 1D particles is of the form: [START_REF] Maier | Liquidlike Order of Charged Rodlike Particle Solutions[END_REF][START_REF] Belamie | Structure and Chirality of the Nematic Phase in α-Chitin Suspensions[END_REF] 𝑑 𝜋 √3 8 ⁄ / 〈𝐷〉𝜙 . 6

with 〈𝐷〉 being the mean diameter of the nanotube. Despite the limited range of the volume fraction we investigated, dispersions of Ge-DWINTs seem to follow such swelling law related to a twodimensional swelling for the most concentrated samples (Figure 7). Similar evolution has already been observed for aqueous dispersions of aluminosilicate and aluminogermanate single-walled imogolite nanotubes. [START_REF] Maillet | Structure et Croissance de Nanotubes de Ge-Imogolite Simple et Double-Paroi[END_REF][START_REF] Levitz | Intermittent Brownian Dynamics over a Rigid Strand: Heavily Tailed Relocation Statistics in a Simple Geometry[END_REF] It is also worth noting that the interparticle distances deviate from linearity due to the predominance of isotropic swelling occurring at low volume fractions ( -0.5 ≥ 20). As shown from eq. 6, the slope of the swelling law should be used to derive the average diameter 〈𝐷〉 of Ge-DWINTs. In our case, the obtained value is close to 4 nm, in agreement with the diameter of individual nanotubes derived either from TEM measurements (Table 1) or WAXS simulations (Figure 5a). It should be noted that the so-called percolation threshold 𝜙 of homogeneous gels of one-dimensional objects writes: [START_REF] Philipse | On the Density and Structure Formation in Gels and Clusters of Colloidal Rods and Fibers[END_REF] 𝜙 𝐿 𝐷 0.7 7

Here, the minimum value of 𝜙 is found to be around 0.5% for nanotubes with aspect ratio of 4.2, leading to a value 𝜙 0.02 significantly lower than 0.7. The gelation of our colloidal

Ge-DWINTs at low ionic strength is related to a complex situation due to the interplay between electrostatic interactions, translational and rotational Brownian motion and orientation phenomena rather than a long-range connectivity between the nanotubes. For IS ≥ 10 -2 mol.L -1 , X-ray scattering and optical results reveal the existence of small bundles arranged in structures much larger than the size of individual nanotubes. From a fundamental point of view, it would certainly be relevant to assess the correlation length in these aggregated structures, a tricky investigation that goes beyond the purpose of this study.

CONCLUSIONS

In summary, we unravel the colloidal stability of double-walled aluminogermanate imogolite nanotubes. By carefully controlling both ionic strength of the medium and the volume fraction in nanotubes, we were able to propose a complete phase diagram from the combination of optical observations and SAXS/WAXS experiments. Although the overall topology of the phase diagram is similar to that of clay nanosheets, the position of the different phase boundary are different, which may be related to the difference in electrostatic double-layer interaction energy between clay minerals with differing geometries and aspect ratio. [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF] In the repulsive regime, i.e. at IS < 10 - 2 mol.L -1 , Ge-DWINTs are dispersed individually in aqueous suspensions and form stable, isotropic sol. These transparent suspensions turn out as a gel phase over a critical volume fraction threshold, whose position shifts to higher concentration when the ionic strength increases. In all cases, Ge-DWINTs behave like "swelling" clay nanosheets, the average interparticle distance following the inverse square root of the nanotube volume fraction. Above IS ≥ 10 -2 mol.L -1 , the Debye screening length is not sufficient to prevent nanotubes from coagulation. As a result, Ge-DWINTs are stacked together in bundles with a hexagonal close-packing configuration, forming aggregates, which collapse under gravity or form opaque (attractive) arrested phases depending on the nanotube concentration. Altogether, these results provide a basis for the colloidal stability of imogolite-like nanotubes, which could open new pathways for designing hydrogels, [START_REF] Shikinaka | Design of Stimuli-Responsive Materials Consisting of the Rigid Cylindrical Inorganic Polymer "Imogolite[END_REF] nanocomposites, [START_REF] Mauroy | Elaboration of Cellulose Nanocrystal/Ge-Imogolite Nanotube Multilayered Thin Films[END_REF] or membranes, [START_REF] Li | Effects of Aluminogermanate Imogolite Nanotube Orientation on Mass Transport across Polyamide Nanocomposite Membranes[END_REF] based on these geo-inspired INTs. Beyond applications, imogolite nanotubes are singular since their surface is highly curved, which in turn complicate the analytical calculation of electrostatic double-layer interaction energies. Here, we demonstrated that Ge-DWINTs can be considered as a model charged nanotube due to their well-defined diameter, high colloidal stability while controlling their aspect ratio by soft chemistry. [START_REF] Paineau | A Liquid-Crystalline Hexagonal Columnar Phase in Highly-Dilute Suspensions of Imogolite Nanotubes[END_REF][START_REF] Amara | Single-Step Formation of Micron Long (OH)3Al2O3Ge(OH) Imogolite-like Nanotubes[END_REF] Therefore, it

Figure 1 .

 1 Figure 1. (a) Local arrangement of AlO6 octahedra (in blue) and Ge(Si)O4 tetrahedra (in green) in imogolite-like nanotubes; O atoms are shown in red and H ones in white. (b) Schematic view of the atomic structure of a double-walled germanium-based imogolite nanotube (Ge-DWINT). Al, blue; Ge, green; O, red; H, white.

  medium. Determining the "phase diagram" as a function of the nanotube volume fraction and the suspension ionic strength is therefore essential to understand and predict the colloidal stability of INTs dispersions in aqueous medium. This study aims to propose the first complete phase diagram of INTs dispersions. Samples are synthetic double-walled Ge-INTs (Ge-DWINTs) as schematically represented in Figure 1b. Surface charge properties are assessed thanks to electrophoretic measurements, revealing that Ge-DWINTs outer surfaces are positively charged up to high pH values. Different suspensions have been prepared at fixed IS by osmotic stress to scan the volume fraction/IS phase diagram. Optical observations and Wide Angle X-ray Scattering (WAXS) experiments allow us to unravel different domains in this phase diagram for nanotube organization. At low ionic strength (IS ≤ 10 -2 mol.L -1 ),

𝜙 𝐶 𝜌 2 with

 2 C being the solid concentration of Ge-DWINTs determined by weight loss upon drying, taking into account the weight of residual NaCl, especially at high ionic strength.vol is the density of a Ge-INT (3.64 g.cm -3 ). For each IS value, series of samples have been prepared by dilution with NaCl solution at constant ionic strength to scan the /IS phase diagram. X-ray scattering measurements. Selected samples of Ge-DWINTs suspensions were held in borosilicate capillary tubes (WJM-Glas/Müller GmbH, DE) of 1 mm diameter and stored vertically after flame-sealing. Wide-Angle X-ray Scattering (WAXS) experiments were carried out on a Cu rotating anode (RU H3R, Rigaku Corporation, Japan) at a wavelength of CuK = 0.1542 nm delivered by a multilayer W/Si mirror (Osmic). Two-dimensional (2D) WAXS patterns were collected on a MAR345 detector (marXperts GmbH, Germany) with 150 µm pixel size, placed at a sample-to-detector distance of 300 mm. The experimental resolution can be approximated by a Gaussian with Full-Width at Half Maximum (FWHM) ~0.13 nm -1 . Small-Angle X-ray Scattering (SAXS) experiments were carried out at the TRUSAXS ID02 beamline of the synchrotron ESRF (Grenoble, France) at a wavelength of  = 0.095 nm. 57 2D SAXS patterns were acquired on a FReLoN (Fast-Readout Low-Noise) camera with 52.5 µm pixel size and placed in a vacuum detection tunnel at a sample to detector distance of 2 m.

Figure 2 .

 2 Figure 2. (a) Typical TEM micrograph of individually nanotubes obtained from a dilute suspension of DW0. Histograms of the (b) nanotube length distribution NLD (red curve: lognormal fitting) and (c) nanotube diameter distribution NDD (red curve: Gaussian fitting).

Figure 3 .

 3 Figure 3. (a) Infrared spectrum obtained on a powder of DW0. (b) Evolution of the electrophoretic mobility as a function of the pH on a dilute suspension of DW0.

Figure 4 .

 4 Figure 4. (a-e) Optical observations of aqueous suspensions of Ge-DWINT prepared at different ionic strengths, as a function of the volume fraction . (a) IS = 1 mol.L -1 ; (b) IS = 10 -1 mol.L -1 ; (c) IS = 10 -2 mol.L -1 ; (d) IS = 10 -3 mol.L -1 ; (e) IS = 10 -4 mol.L -1 . (f) Corresponding phase diagram as a function of Ge-DWINT volume fraction  and ionic strength. The dotted lines between the different regimes are only eye guide. IL (open circles): isotropic liquid; IG (black triangles):

Figure 5 .

 5 Figure 5. (a) Evolution of WAXS diagrams of Ge-DWINTs suspensions upon increasing the ionic strength: IS = 10 -4 mol.L -1 ( = 0.99%); IS = 10 -3 mol.L -1 ( = 1.49%); IS = 10 -2 mol.L -1 ( = 0.92%); IS = 10 -1 mol.L -1 ( = 1.74%); IS = 1 mol.L -1 ( = 2.80%). Dotted lines correspond to

  Nevertheless, our results clearly show that the formation of bundles in aqueous suspensions is related to a reduction of the electrostatic interactions by a screening effect of the nanotube surface charge via the condensation of anions, most probably on the outer surface of Ge-DWINTs.[START_REF] Gustafsson | The Surface Chemistry of Imogolite[END_REF] Structure of the suspensions and swelling law. To get a more general overview of the colloidal properties of Ge-DWINTs, their structural organization at larger distances was assessed from SAXS experiments in the different domains of the phase diagram. Starting from the lowest ionic strength and up to 10 -2 mol.L -1 , SAXS curves present similar decay as the calculated diagram for dispersions of individual nanotubes 𝐷𝑊 (Nb = 1, p = 1) (Figure6). The decay of the scattered intensity I with scattering vector modulus Q follows a Q -1 dependence, related to the onedimensional particle form factor P(Q) of the scattered INTs described in eqs. (3) and (4a). It means therefore that these samples are formed of individual nanotubes, regardless the colloidal state of the system (IL, IG or BG) or the volume fraction. In contrast, the slope of the SAXS curves systematically evolves to a -2 exponent when the ionic strength rises above IS ≥ 10 -1 mol.L -1

Figure 6 .

 6 Figure 6. Variation of the SAXS diagrams of Ge-DWINTs suspensions upon increasing the ionic strength: IS = 10 -4 mol.L -1 (blue); IS = 10 -3 mol.L -1 (red); IS = 10 -2 mol.L -1 (green); IS = 10 -1 mol.L - 1 (yellow); IS = 1 mol.L -1 (grey). Dotted lines correspond to WAXS curves obtained for the most concentrated samples at each IS value. Dashed line corresponds to calculated WAXS diagram using eq. 3 for individual Ge-DWINTs (𝐷𝑊 , Nb = 1, p = 1). All curves have been translated for the sake of clarity.

Figure 7 .

 7 Figure 7. Evolution of the interparticle distance obtained from the SAXS curves as a function of  -0.5 and for different ionic strength. The two black curves correspond to linear fit using eq. 6 (see text for details).

Table 1 .

 1 Morphological parameters deduced from the analysis of the TEM micrographs.

	〈𝑳〉 (nm)	PDIL (%)	〈𝑫〉 (nm)	PDID (%)	Aspect ratio
	18.0  1.6	49	4.3  0.3	7	4.2

would certainly be relevant to use them for investigating in depth the energy of interaction depending onthe separation distance, ionic strength and valence of the counterions, a pending issue for charged nanocylinders.

ASSOCIATED CONTENT Supporting Information. The following files are available free of charge.

Additional optical observations of aqueous suspensions between crossed polarizers and WAXS diagrams simulations (PDF) AUTHOR INFORMATION

Corresponding Author

*E-mail: erwan-nicolas.paineau@u-psud.fr. Phone: +33 (0) 1 69 15 60 51.

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENT

The present work has benefited from the Electronic Microscopy facility of Imagerie-Gif, (http://www.i2bc.paris-saclay.fr), member of IBiSA (http://www.ibisa.net), supported by "France-BioImaging" (ANR-10-INBS-04-01), and the Labex "Saclay Plant Science" (ANR-11-IDEX-0003-02). E.P. would like to thank Dr. P. Davidson for access to ID02 experiments (ESRF, proposal SC4867) and Dr. Lewis Sharpnack for his help during SAXS experiments.