Ronaldo De Carvalho

Augusto A Thomas Quack

Christoph G Grevelding

Christoph Grunau
email: christoph.grunau@univ-perp.fr

Chromatin structure around long non-coding RNA (lncRNA) genes in Schistosoma mansoni gonads

Keywords: Schistosoma mansoni, posttranslational histone modifications, long non-coding RNA, chromatin, gonads

In this study, we employed a total of eight distinct modifications of histone proteins (H3K23ac, H3K27me3, H3K36me3, H3K4me3, H3K9ac, H3K9me3, H4K12ac, and H4K20me1) to discern the various chromatin colors encompassing lncRNA genes in both mature and immature gonads of the human parasite Schistosoma mansoni. Our investigation revealed that these chromatin colors exhibit a tendency to aggregate based on the similarities in their metagene shapes, leading to the formation of less than six distinct clusters. Moreover, these clusters can be further grouped according to their resemblances by shape, which are co-linear with specific regions of the genes and potentially associated with transcriptional stages.

Introduction

Schistosomiasis, a parasitic disease caused by members of the Schistosoma genus, represents a significant public health burden in tropical and subtropical regions, affecting 250 million people worldwide. The life cycle of these parasitic flatworms involves complex interactions with both intermediate and definitive hosts, resulting in chronic infections that can lead to severe morbidity and mortality if left untreated. Despite its substantial impact on global health, schistosomiasis has received relatively limited attention in comparison to other neglected tropical diseases, and the intricacies of its molecular mechanisms remain incompletely understood [START_REF] Mcmanus | Schistosomiasis[END_REF]. Among the platyhelminths schistosomes are exceptional in evolution because they are the only representatives, which have evolved separate sexes. Furthermore, schistosomes exhibit a nearly unique sexual biology because the maturation of the female gonad depends on a permanent pairing-contact with a male partner [START_REF] Grevelding | Schistosoma[END_REF][START_REF] Kunz | Schistosome male-female interaction: induction of germ-cell differentiation[END_REF][START_REF] Loverde | Schistosome female reproductive development[END_REF][START_REF] Popiel | Reproductive development of female Schistosoma mansoni (Digenea: Schistosomatidae) following bisexual pairing of worms and worm segments[END_REF]. Among others, pairing induces the final differentiation of the ovary, which in an unpaired females consists of stem cell-like, undifferentiated oogonia that do not enter meiosis. After pairing, however, mitoses of gonadal cells are induced that finally lead to the differentiation of oocytes entering meiosis [START_REF] Hollander | Schistosoma mansoni: male stimulation and DNA synthesis by the female[END_REF][START_REF] Fitzpatrick | Dioecious Schistosoma mansoni express divergent gene repertoires regulated by pairing[END_REF]. These and other changes are accompanied by a remarkable growth of the paired female, and they are mediated by complex signalling systems as well as many genes expressed in a pairing-dependent manner, including gonadal genes. Although in the male no morphological changes are observed following pairing, previous transcriptomics data of proteincoding genes of whole worms (and their gonads) revealed an influence of the pairing status also on gene expression in males, including their testes [START_REF] Lu | Schistosome sex matters: a deep view into gonad-specific and pairing-dependent transcriptomes reveals a complex gender interplay[END_REF][START_REF] Lu | Males, the Wrongly Neglected Partners of the Biologically Unprecedented Male-Female Interaction of Schistosomes[END_REF]. Recent advances in genomics and transcriptomics have shed light on the role of long non-coding RNAs (lncRNAs) in the regulation of gene expression and the modulation of biological processes across various organisms. LncRNAs, a class of non-protein-coding transcripts with a length exceeding 200 nucleotides, have emerged as crucial players in diverse cellular and physiological functions, including development, differentiation, and disease pathogenesis [START_REF] Ponting | Evolution and functions of long noncoding RNAs[END_REF]. Although the role of lncRNAs has been extensively investigated in various organisms, their involvement in schistosome biology remains largely unexplored territory. Understanding the regulatory mechanisms governing schistosome lncRNA gene expression would allow for elucidating key aspects of their biology, including host-parasite interactions, immune evasion, and developmental transitions. In this context, the exploration of lncRNAs in schistosomes presents a promising avenue for uncovering novel regulatory elements and mechanisms that may contribute to the parasites' adaptation and survival within their hosts. In Schistosoma mansoni, lncRNA have been shown to affect pairing status, survival, and reproduction in adult worms, making them potential therapeutic targets [START_REF] Silveira | Long noncoding RNAs are essential for Schistosoma mansoni pairingdependent adult worm homeostasis and fertility[END_REF]. Additionally, lncRNAs have been identified in different developmental stages of S. mansoni, suggesting their involvement in stage-specific gene expression and potential as biomarkers [START_REF] Rocha | Long non-coding RNAs as possible therapeutic targets in protozoa, and in Schistosoma and other helminths[END_REF]. Furthermore, the expression patterns of lncRNAs in S. mansoni have been characterized at the singlecell level, revealing tissue-specific expression and regulated expression programs [START_REF] Morales-Vicente | Single-cell RNAseq analyses show that long non-coding RNAs are conspicuously expressed in Schistosoma mansoni gamete and tegument progenitor cell populations[END_REF]. In Schistosoma japonicum infection, lncRNAs have been implicated in the formation of liver granulomas and the polarization of macrophages, providing a potential target for treatment [START_REF] Zhao | Knocking Down Gm16685 Decreases Liver Granuloma in Murine Schistosomiasis Japonica[END_REF]. Intergenic, antisense, and sense lncRNAs are differentially expressed after treatment 5-AzaC, an inhibitor of DNA/RNA methylases, in S. mansoni females. These differentially expressed lncRNAs belong to coexpression network modules related to male metabolism, and they are also differentially expressed in unpaired females compared with paired females and their ovaries, respectively [START_REF] Amaral | Long non-coding RNA levels can be modulated by 5-azacytidine in Schistosoma mansoni[END_REF]. Re-analysis of single-cell RNA-sequencing data from adult S. mansoni identified the expression patterns of lncRNAs in different cell types including specific lncRNAs in the gonads [START_REF] Rocha | Long non-coding RNAs as possible therapeutic targets in protozoa, and in Schistosoma and other helminths[END_REF]. Given the stage-and cell specific expression of lncRNA and its sensitivity to an epigenetic stressor [START_REF] Amaral | Long non-coding RNA levels can be modulated by 5-azacytidine in Schistosoma mansoni[END_REF], we hypothesized that lncRNA gene function could be influenced by chromatin structure around their genes. Chromatin, the complex of DNA and proteins that constitutes the structural framework of eukaryotic genomes, plays a pivotal role in controlling gene function and maintaining genome integrity. Over the past few decades, extensive research has revealed the remarkable diversity and complexity of chromatin organization, leading to the emergence of the concept of "chromatin colors" or "chromatin flavors." These metaphoric terms describe, in analogy to the additive primary colors that can produce the full spectrum of visible colors, the idea that individual compositions of the chromatin can be combined into different functional states ("colors") that it can adopt to orchestrate gene regulation and cellular identity [START_REF] Strahl | The language of covalent histone modifications[END_REF]. We used here 8 different posttranslational histone modifications (H3K23ac, H3K27me3, H3K36me3, H3K4me3, H3K9ac, H3K9me3, H4K12ac, and H4K20me1) to identify the chromatin colors surrounding lncRNA genes in gonads, ovaries and testes, isolated from paired and unpaired S. mansoni. We found that chromatin colors cluster by similarity of their metagene shapes into less than 6 clusters, which can be grouped into similar shapes that appear to be associated with parts of the genes, the pairing status of the adults from which the gonads originated, and potentially with transcription steps.

Results

500bp is the optimal bin size for ChromstaR analysis

ChromstaR [START_REF] Taudt | Atlas of Schistosoma mansoni long non-coding RNAs and their expression correlation to protein-coding genes[END_REF]) is a R-package for the analysis of ChIP-Seq data based on Hidden-Markov-Model (HMM). Regions of the genome that are enriched in specific histone modifications ("peaks") are identified based on transition probabilities. ChromstaR allows to combine individual histone modifications into combinational states ("chromatin colours"). It requires to establish empirically a bin size prior to the full analysis. For this we used comparison with other peakcalling methods and visual inspection under IGV for the relatively sharp H3K4me3 peaks. Best concordance of peak detection between MACS2, Peakranger, and ChromstaR was obtained at a bin size of 500 bp with input BAM as reference. Visual inspection of peakcalling using MACS bedgraph, and Peakranger, and ChromstaR wig files, and BED files from ChromstaR revealed excellent concordance of peak representation and peak detection by ChromstaR with these parameters. Metagene profiles were similar for the tree methods and showed high similarity between replicates. The modified lines in the univariate fits of ChromstaR were flat, an empirical indicator for good model fitting. In conclusion, further analyses were done with binsize 500 bp and input BAM as references.

There are sex-specific differences in chromatin profiles around lncRNA genes in gonads of S. mansoni

Based on a previously established organs isolation method [START_REF] Hahnel | Whole-organ isolation approach as a basis for tissue-specific analyses in Schistosoma mansoni[END_REF], we used pairing-experienced females (bisex females, bF) and males (bM) as well as used pairing-unexperienced females (sF, single sex female) and males (sM) for isolation of their gonads, ovaries from bF (bO) and sF (sO) and testes of bM (bT) and sM (sT). Next, we performed ChIP-Seq with the abovementioned 8 antibodies. We aligned ChIP-seq reads to the genome and generated metagene profiles i.e. log(observed/expected) of color enrichment spanning 2 kb upstream of the transcription start site (TSS) to 2 kb downstream of the transcription end site (TES). We noticed that some profiles had similar shapes and used unsupervised clustering to sort similar profiles into clusters. We allowed up to 6 clusters but considered only clusters with at least 4 colors. Cluster 0 contains the "flatliners" i.e., no enrichment around lncRNA genes. Number of profiles in clusters are listed in table 1. By visual inspection of cumulated log(obs/exp) metagene profiles, we realized that these profiles can be grouped by their shapes (figure 1). These groups were different in both sexes and varied also between mature and immature gonads. We tentatively assigned the different cluster groups to parts of genes. It should be notes that clusters were independently numbered e.g. cluster 1 in bT has not the same composition as cluster 1 in sT. For the composition of the clusters, contact the authors.

Table 2: Assignment of clusters to part of genes (see also figure 1) Abbreviations as in the main text.

Part of gene sT bT sO bO

TSS Cluster 3 Cluster 1 Cluster 3 Gene body

Cluster 6, 2 Cluster 3, 2 Cluster 1 Cluster 2

TES and downstream

Cluster 1, 5 Cluster 5

Absent in gene body

Cluster 2

Cluster 1 Fig. 1a. Metagene profiles of testes of single-sex, pairing-unexperienced males (sT), and bisex, pairing-experienced males (bT), combined into clusters by shape and associated with gene parts. X-axis 2 kb upstream of TSS and 2 kb downstream of TES. Y-axis log(obs/exp) totaled over all profiles in the cluster. Histograms show the absolute number of occurrences of a histone modification in the chromatin colors. Squares surround groups of profiles with similar shape.

Discussion

Our results reveal a complex landscape of chromatin modifications surrounding long non-coding RNA (lncRNA) genes in the gonads of S. mansoni, with a theoretical capacity to generate 256 distinct "chromatin colors" through combinations of eight histone modifications.

Remarkably, in the gonads, we observed nearly the full spectrum of these combinations around lncRNA genes, yet these many modifications cluster into fewer than six categories based on the similarities of their metagene profiles. This clustering seems to be associated with distinct steps of gene transcription and represents a significantly reduced complexity compared to what might be expected by chance. Such findings suggest a functional role for histone modifications in as a regulatory mechanism that selectively employs certain histone modifications to orchestrate lncRNA transcription. The discovery that the composition of chromatin colors varies between male and female gonads additionally indicates a sex-dependent difference. Furthermore, the pairing status of the worms obviously adds another layer of complexity to the regulation of lncRNA transcription. This sexual dimorphism and pairing-status specificity in chromatin modification patterns suggest that the regulation of lncRNA transcription is finely tuned to the physiological and developmental context of the parasite, especially with respect to the pairing status. The differential chromatin landscapes between males and females and between different pairing statuses imply that there is a strong epigenetic component in form of histone modifications that might play key roles in mediating sexspecific and pairing-dependent gene expression programs.

Our results suggest that the precise control of lncRNA transcription via specific patterns of histone modifications may represent a critical aspect of these regulatory functions. It is conceivable that aberrations in these chromatin modification patterns could lead to dysregulated lncRNA expression and contribute to developmental malfunction. Indeed, a recent functional study in S. mansoni identified lncRNA involvement in regulation cell proliferation in adults and gonads as well as reproduction including egg development [START_REF] Silveira | Long noncoding RNAs are essential for Schistosoma mansoni pairingdependent adult worm homeostasis and fertility[END_REF] In conclusion, our study provides additional compelling evidence for the functional role of histone modifications in the control of lncRNA transcription in S. mansoni gonads, revealing the possibility of a regulatory system that is sensitive to the parasites´ developmental and physiological context. These findings open new avenues for understanding the mechanisms of gene regulation at the chromatin level and their implications for development, and physiology of the parasite, and probably also for the control of schistosomiasis, a NTD for which new treatment concepts are urgently needed [START_REF] Bergquist | Controlling schistosomiasis with praziquantel: How much longer without a viable alternative[END_REF]Silveira et al., 2022)

Materials and Methods

Life cycle maintenance of S. mansoni and ethics statement

To maintain the life cycle of the Liberian strain of S. mansoni, [START_REF] Grevelding | The female-specific W1 sequence of the Puerto Rican strain of Schistosoma mansoni occurs in both genders of a Liberian strain[END_REF]

Organ extraction and N-ChIP

To generate worm populations of one sex, we performed monomiracidial infections of snails, in which clonal cercarial populations (single-sex cercariae) develop, and which we subsequently used for final host infections to obtain pairing-unexperienced females or males. Mixed-sex worm populations were obtained by polymiracidial snail infections. Gonad isolation was performed by a detergent-and enzyme-based isolation procedure [START_REF] Hahnel | Whole-organ isolation approach as a basis for tissue-specific analyses in Schistosoma mansoni[END_REF] using about 50 pairingexperienced females and males, and 100 pairing-unexperienced females, which are much smaller than paired females, as are their ovaries. For vitality check, 5-10 organs of each sample were stained 10 µl Trypan Blue (0.4%; Sigma) and checked by phase-contrast microscopy (Olympus IX 81 microscope). N-ChIP was done as described before [START_REF] Cosseau | Native chromatin immunoprecipitation (N-ChIP) and ChIP-Seq of Schistosoma mansoni: Critical experimental parameters[END_REF]. Antibodies were purchased from commercial suppliers.

Alignment and ChromstaR optimization

Uniquely aligned reads for N-ChIP with antiH3K4me3 on two biological replicates of pairing experienced ovaries (bF) and pairing unexperienced ovaries (sF) were downsampled to 2.5 Mio reads per sample and the resulting BAM files were used as input for peak calling in MACS2 (Effective genome size 350 Mb, building a model with lower mfold bound 5, upper mfold bound 50, band width for picking regions to compute fragment size 300, Peak detection based on qvalue, Minimum FDR (q-value) cutoff for peak detection 0.05), Peakranger 1.17 (Peak detection mode at low resolution, read extension length 300 bp), both with input BAM DU Sm-SsOv-INPUT. ChromstaR was evoked with default parameters and 250, 500, 750 and 1 000 bp bin size, with or without input. Metagene profiles were generated with DeepTools for MACS and Peakranger results, and with plotEnrichment function of ChromstaR using 5 275 plus strand protein coding genes based on the Smv7 annotation (ftp://ftp.sanger.ac.uk/pub/project/pathogens/Schistosoma/mansoni/v7/an notation/Sm_v7.1.gff). Result files were visually inspected on IGV. We used version 7 of the genome assembly (ftp://ftp.sanger.ac.uk/pub/project/pathogens/Schistosoma/mansoni/v7/se quence/Smansoni_v7.fa).

Metagene analysis around lncRNA genes and clustering by color

Coding sequences of lncRNA were obtained as BED file from [START_REF] Taudt | Atlas of Schistosoma mansoni long non-coding RNAs and their expression correlation to protein-coding genes[END_REF]. Metagene profiles were generated with plotEnrichment function of ChromstaR using 8,328 lncRNA coding genes on the plus strand. Metagene profiles over all combinations were exported from ChromstaR as individual TSV files. Hierarchical clustering was performed using the hclust function of R (figure 2).

Fig. 1b .

 1b Fig. 1b. Metagene profiles of ovaries of single sex, pairing-unexperienced females (sO), and bisex, pairing-experienced females (bO). Axis as for figure 1a.

Fig. 2 .

 2 Fig. 2. Schematic representation of the bioinformatics workflow used to perform clustering of metagene profiles.

Table 1 :

 1 Numbers of testes from single-sex males (sT; extracted from worms representing clonal, unisexual worm populations), bisex testes (bT; extracted from worms representing clonal, bisexual worm populations), single-sex ovaries (sO; extracted form ss females) and bisex ovaries (bO; extracted from bs females). In bold given are clusters that were used for further analyses.

	cluster	sT	bT	sO	bO
	0 (flatline)	19	58	45	37
	1	57	160	163	140
	2	162	13	41	68
	3	6	18	2	5
	4	2	1	1	1
	5	6	5	1	1
	6	4	1	1	2

  , we used Biomphalaria glabrata as intermediate host and hamster as final hosts in accordance with the European Convention for the Protection of Vertebrate Animals Used for Experimental and Other Scientific Purposes (ETS No 123; revised Appendix A). Animal experiments were approved by the Regional Council Giessen (V54-19 c 20/15 c GI 18/10).

Acknowledgments

This study is set within the framework of the "Laboratoires d'Excellences (LABEX)" TULIP (ANR-10-LABX-41). With the support of the Wellcome Trust (107475/Z/15/Z) and the LabEx CeMEB, an ANR "Investissements d'avenir" program (ANR-10-LABX-04-01).

End Matter

Author Contributions and Notes C.G.G and C.G designed research, R.A and T.Q performed research; and all authors wrote the paper. The authors declare no conflict of interest. This article contains supporting information that can be requested from the authors.


Giglio Eric

Charge relaxation rates in insulating straight capillaries

Keywords: numbers: 34, 80, Dp, 34, 80, Pa

The charge relaxation of accumulated charge patches in insulating straight capillaries is investigated theoretically. The model assumes that charges accumulate only at the inner and outer insulator-vacuum interface of the capillary but not in the bulk. We give an analytical solution to the coupled equations that describe the surface charge dynamics at both interfaces. We provide a tool to calculate easily the characteristic relaxation times in a straight capillary of any dimension, possibly surrounded by a conducting cylinder. The latter allows for different scenarios found in experimental setups, and is applicable to both nano and macro-capillaries. We propose an original experimental technique to monitor the charge relaxation in a straight glass capillary and show how to use the presented model to extract the bulk and surface conductivity of the insulator from the measured data. In the supplementary material, we provide a script in ®Mathematica that allows the reader to compute comfortably the decay rates for all straight insulating capillaries the reader is interested in.

I. INTRODUCTION

When low energetic ions hit an insulating surface, charge carriers are injected at the impact points and trapped by defects. As a result, charge accumulates in the dielectric, generating an electric field that, if sufficiently strong, deviates the following beam particles, preventing them to hit the insulator surface. This basic property led to the discovery that slow ions can pass through insulator capillaries (having an aspect ratio, length over diameter, larger than 50) without any change, neither in the ion charge state nor in the ion kinetic energy, even when the geometrical conditions do not allow it. The phenomenon is called charged particle guiding by insulating capillaries and it has become an intensively studied field since its discovery by the pioniering work by N. Stoltherfoht for polyethylene terephthalate (PET) nano-capillaries [1] and T. Ikeda for glass macro-capillaries [2], more than 10 years ago.

On the experimental front, charge relaxation in insulating capillaries was recently investigated by several authors. The charge decay of a previously charged conical glass capillary was monitored by recording the deflection of a by-passing ion beam on a PSD [3]. The measured decay time of 2000 min was used to adjust the surface conductivity in simulations. Dubois et al. monitored the decay of charge patches in a glass tube [4]. After charging a glass tube by a tilted 1 keV Ar + beam, the charge patch was allowed to discharge for a given time. After that time, the beam was re-injected, and the initial transmitted fraction was given as a function of the discharge time. In Nagy et al., a micrometer sized proton beam of 1 MeV, tilted by 1 • , was center injected into a Teflon capillary with macroscopic dimensions [5]. With increasingly accumulated charge, the patch got sufficiently strong to deflect the beam out of the capillary on a screen downstream. A dynamical equilibrium between the charges being deposited and flowing away in form of leakage current was found. The leakage current was estimated to 10 % of the injected current.

While nowadays the guiding of ions through glasscapillaries due to charge patches is qualitatively understood, the complex nature of the electric conduction in such insulators makes quantitative theoretical predictions still a challenging task. Indeed, for a given ion beam, the guiding is entirely controlled by the dynamics of the charge patches in the dielectric, which in turn depend on the electrical properties of insulator, on the possible presence of electrodes that affect the electric field inside the capillary and on the charge transfer at the interfaces.

In the past, numerous theoretical approaches succeeded to capture the guiding properties of insulating capillaries, by using in their simulations appropriate charge relaxation times (also called decay times) for the exponential decay of the accumulated charges [6,7]. Typically, by varying the decay times, observed trends in the guiding process of ion beam through capillaries could be reproduced numerically [8][9][10][11][12][13]. The success of those models showed that the charge dynamics in insulators is already well described by a simple exponential decay of the accumulated charge distribution. It seems thus that having an accurate relationship between the charge relaxation rate of an accumulated charge distribution and the geometrical and electrical properties of the capillary, would be beneficial for (i) making reliable theoretical predictions, by knowing the electrical properties of the capillary, or (ii) deducing the electrical properties of the capillary by measuring the decay rates. Being able to refine the computing of the relaxation rates would allow for deeper understanding of the influence of the bulk and surface conductivity and dimensions of the capillary on the charge dynamics in capillaries.

An initial guess of the decay rate was usually obtained by considering the bulk conductivity, diffusion and dielec-tric constant of the insulating capillary. For example, in macro-capillaries, the order of magnitude of the depletion rate was estimated using the expression 2κ b /[ε 0 (1 + ε r )], with κ b being the bulk conductivity and ε r the dielectric constant of the insulator [8,12,13]. The screening factor 2/(1 + ε r ) corresponds to the screening of a charge lying at the insulator-vacuum interface of an infinite dielectric plane. Alternatively, the more common bulk relaxation rate κ b /(ε 0 ε r ) could be used. In nano-capillaries, the relaxation rate was estimated by Schiessl et al. using the surface diffusion constant [11].

Usually the surface conductivity κ s of the insulatorvacuum interface was not directly used for estimating the decay rates, supposedly because the surface conductivity of an insulating surface is mainly due to adsorbed impurities and is thus not a well-known quantity and because the relation between the surface conductivity and the relaxation rates is not easy to establish. That does not mean that surface currents were not considered in the simulations. For nano-capillaries, the surface currents were evaluated by means of a drift model, where the accumulated charge carriers were field-driven along the surface, with a velocity proportional to their surface mobility [14,15], and surface currents were even shown to be the dominant relaxation channel. For macroscopic glass capillaries, the surface currents were taken proportional to the insulator-vacuum surface conductivity [3]. However, for the estimation of the total charge relaxation rate, the influence of the surface conductivity κ s was systematically ignored.

The present work models the charge dynamics in straight insulating capillaries and extracts from the equations the relaxation rates that characterize the charge dynamics. It gives the explicit dependency of the relaxation rates on the geometric dimensions and macroscopic electrical properties of the capillary. Much attention is payed to the electric field that drives the charges in the insulator, so that reliable relaxation rates can be deduced from the model. Here, the electric field is evaluated by considering the appropriated boundary conditions that describe the capillary and its environment. In other words, the electric field includes the exact contribution of induced polarization charges (image charges) that appear at the interfaces separating two dielectric media and that screen the accumulated charges in the dielectric. The contribution of the surface conductivity (at the insulator-vacuum interface) to the total charge relaxation rate is clearly established and can be estimated quantitatively with the model.

In a preliminary work, Giglio et al. evaluated the charge relaxation rates of a surface charge distribution at the inner surface of a straight glass capillary with the outer surface grounded [16]. The charge dynamics equation at the inner surface was solved by projecting the charge distribution onto a multipole expansion. The results showed that each multipole moment has its own relaxation rate, with the latter being the weighted sum of the bulk and surface conductivity. For each moment, the weights are different and depend only on the dimensions of the capillary. Those findings where then used to analyze the experimental results given in [4].

In this manuscript we will go a step further and treat the more general case where the outer surface of the capillary is no longer electrically grounded but simply the dielectric-vacuum interface where charges can accumulate and be injected. We add to the model a third conducting cylindrical interface that surrounds the capillary. Depending on the radius of the conducting cylindrical interface, the model can describe various setup scenarios found in the literature. The presented model can be applied to straight macroscopic glass capillaries as well as to straight nano-capillaries in membranes and thin insulating foils.

The manuscript is structured as follow. In section II, we give the the coupled differential equations that describe the charge dynamics in the capillary. We give an analytic solution of the differential equations that satisfies exactly the boundary conditions, which in turn describe the capillary and its environment. In section III, we present a Gedankenexperiment where the analytic solution is used to predict an observable as a function of the dimensions and electrical properties of a straight capillary. We show how to use the theoretical prediction to deduce the bulk and surface conductivity of the capillary from the measured observable. In the supplementary material, the author provides a script for ®Mathematica [18] that calculates the charge relaxation rates for a given capillary. The script can also evaluate the decay of the electric field in the capillary, if the initial charge distribution is given.

II. THEORETICAL MODEL

Consider a straight insulating glass capillary of length H, with R 1 and R 2 being the inner and outer radius of the capillary tube, as depicted in Fig. 1. The electric properties of the capillary are given by the relative permittivity ε r > 1, the bulk conductivity κ b and surface conductivity κ s , which we consider here all constant. The surface conductivity of an insulating surface is usually due to adsorbed impurities and expressed in Siemens (S). In the atmosphere for example, the surface conductivity of glass samples depends strongly on the humidity in the atmosphere, meaning that the adsorbed water ions are the main contributors to the anomalous large surface conductivity for glasses in the atmosphere. In near vacuum, the humidity is clearly low and the surface conductivity decreases by several orders of magnitude with respect to the humid atmosphere. It was found by Gruber et al. [19] that the surface conductivity in Borosilicate glass follows an Arrhenius type law, with a typical value of κ s = 10 -16 S at room temperature.

We suppose that the capillary is surrounded by an electrically conducting (metal) cylinder of radius R 3 and length H, sharing the same symmetry axis than the cap-FIG. 1: Cut of the capillary along the xOz plane containing the symmetry axis. Black fat lines stand for grounded surfaces, namely the S3 interface, the entrance and the outlet of the capillary. Grey rectangles represent the dielectric bulk. Blue lines stand for vacuum-insulator interfaces of the inner (S1) and outer (S2) capillary surface, carrying the surface charge densities σ (1,2) . illary. Let us label S 1 the vacuum-insulator interface of the inner capillary surface and S 2 the insulator-vacuum interface of the outer capillary surface. S 3 stands for the vacuum-conductor interface of the cylindrical surface of radius R 3 . The potential V 3 of the S 3 interface may be imposed or floating. With respect to the previous model presented in [16], interface S 3 is an important addition, making the present model quite general. In particular, it allows us to impose a Dirichlet boundary condition to the electric potential, defining it uniquely. The latter suggests that in experiments, a conducting cylinder should surround the capillary if a reliable comparison between experiments and theory is desired. We study here merely the case, found in many experimental setups, where the interface S 3 is electrically grounded, V 3 = 0 [3,13,[19][START_REF] Giglio | Instrum[END_REF][21][22][23][24][25] We further suppose that the capillary is mounted behind a grounded collimator plate with an inlet hole for the ion beam of radius R col < R 1 , avoiding the charging-up of the entrance.

In alkali silicate glasses like borosilicate, the mobility of holes, electrons and anions is usually small compared to the mobility of alkali ions, so that the charge transport in ionic glasses at room temperature is entirely due to mobile alkali ions [26][27][28][29][30]. We further assume that the rate τ -1 h at which the injected holes migrate into the bulk is negligible compared to the charge relaxation rate due to the bulk conductivity,

τ -1 h τ -1 b = κ b ε 0 ε r . (1)
The latter suggests that, as soon as a hole migrates from the surface into the bulk, the hole is "quickly" relaxed by the migration of alkali ions (away from the hole) so that no charge accumulates in the bulk. As a result, the charge accumulates only at the interfaces and may be represented by the surface charge densities σ (1) and σ (2) at the inner and outer interfaces respectively. The electric field that drives the alkali ions in the bulk is thus divergence-free, ∇ • E = 0. Using cylindrical coordinates (r, θ, z), the dynamics of the surface charge densities σ (1) ≡ σ(R 1 , θ, z, t) and σ (2) ≡ σ(R 2 , θ, z, t) are given by two surface continuity equations, which are coupled by the electric field E that depends on the surface charges σ (1,2) ,

∂σ (1) ∂t = -κ b E r - 1 R 1 ∂ ∂θ [κ s E θ ] - ∂ ∂z [κ s E z ] + γ (1) , (2)
∂σ (2) ∂t = κ b E r - 1 R 2 ∂ ∂θ [κ s E θ ] - ∂ ∂z [κ s E z ] + γ (2) , (3)
where ( 2) is evaluated at r = R 1 and (3) at r = R 2 .

The first right hand terms in equations (2-3) are proportional to the bulk conductivity κ b and stand for the mobile charge carriers that are field driven from the inner to the outer surface (or vice-verso) by the radial component E r of the electric field. The second and third right hand terms account for surface currents along the u z and u θ directions and are proportional to the surface conductivity κ s . The source terms γ (1,2) (θ, z, t) represent the deposited charge per unit time and area at the interfaces S 1 and S 2 , respectively. If the capillary is well screened from stray electrons [21], no charges are injected at the outer surface and γ (2) = 0. Such fully screened glass capillaries were used for example in [3] in order to measure the discharge rates of pre-charged capillaries and to put into evidence the self-organized focusing by conical glass capillaries. The source term γ (1) stands for the injected holes at the inner interface by impacting beam ions and is thus proportional to the difference between the injected current I in and the transmitted current I out (t), γ (1) (θ, z, t) ∝ (I in -I out (t)). Note that γ (1) can also include secondary electrons generated at the impact point and absorbed elsewhere at the inner surface. The analytic solution of coupled continuity equations (2-3) is obtained using a multipole expansion technique, already used in [16].

A. Multipole expansion

We assume that the beam axis and capillary axis lie in the xOy plane so that the accumulated charge in the straight capillary has xOy plane symmetry. The interfaces S 1 and S 2 are in contact with the grounded collimator at z = 0 and grounded exit at z = H. Hence, no charges accumulate at the grounded entrance and exit so that the free surface densities are zero at z = 0 and z = H. We may then expand the surface densities σ (1,2) on the following basis which accounts for the xOy plane symmetry and zero density at the capillary entrance,

σ (1) (θ, z, t) = m,n σ (1) mn (t) cos(mθ) sin(k n z) , (4) σ (2) (θ, z, t) = m,n σ (2) mn (t) cos(mθ) sin(k n z) . (5)
where m ≥ 0 stands for the angular moment of the distribution. Choosing the wave number k n of index n ≥ 1 to be defined as

k n = nπ H , (6)
ensures that the surface densities are zero at the exit of the capillary. We further assume that the grounded entrance and exit hinder charges to be injected at z = 0 and z = H, so that the injection cross-sections γ (1) and γ (2) are expanded similarly,

γ (1) (θ, z, t) = m,n γ (1) mn (t) cos(mθ) sin(k n z) , γ (2) (θ, z, t) = m,n γ (2) mn (t) cos(mθ) sin(k n z) .
The electric field being divergence-free in the bulk and outside the bulk, the electric potential V satisfies the Laplace equation everywhere, ∇ 2 V = 0. We distinguish 3 regions in space, bound by the three interfaces S 1 , S 2 and S 3 . We further assume that the potential is zero (grounded) at the entrance z = 0 and exit z = H of the capillary. Within the above assumptions, a general solution of the Laplace equation in cylindrical coordinates for all three regions of space is eventually given by,

V (1) (r, θ, z, t) = m,n A mn (t)I m (k n r) cos(mθ) sin(k n z), 0 ≤ r ≤ R 1 (7) V (2) (r, θ, z, t) = m,n [B mn (t)I m (k n r) + C mn (t)K m (k n r)] × cos(mθ) sin(k n z) , R 1 ≤ r ≤ R 2 (8) V (3) (r, θ, z, t) = m,n [D mn (t)I m (k n r) + E mn (t)K m (k n r)] × cos(mθ) sin(k n z) , R 2 ≤ r ≤ R 3 (9)
where I m () and K m () are modified Bessel functions of order m ≥ 0. Note that a zero potential condition at the exit of the capillary is not strictly required here, but it simplifies the mathematical expressions without loosing much of the general scope. We also note that for r < R col , the potential V (1) is obviously not imposed at the entrance and V (1) is thus not rigorously zero for z = 0. However, for r < R col the potential V (1) at the entrance can be assumed sufficiently close to zero so that the V (1) (z = 0) = 0 conditions can nevertheless be imposed without introducing a significant bias in the final results The five terms A mn (t), B mn (t) . . . E mn (t) found in (7)(8)(9) are defined by the boundary and kinematic conditions given below.

B. Boundary Conditions

In order to define unambiguously the potentials in the three domains, we need to add 5 boundary conditions. The electric potential is continues at the three interfaces. Consequently, the potentials V (1,2,3) defined in the three domains must satisfy the boundary conditions, (S 1 ) :

V (1) (R 1 , θ, z, t) = V (2) (R 1 , θ, z, t) (10) (S 2 ) : V (2) (R 2 , θ, z, t) = V (3) (R 2 , θ, z, t) (11)
(S 3 ) :

V (3) (R 3 , θ, z, t) = V 3 = 0 , (12)
with V 3 = 0 standing for the grounded potential of the conducting interface S 3 . At the interface separating two dielectric media, the electric field normal to the interface is discontinues. In the presence of free surface charges one has, (S 1 ) : -ε r ∂V (2) ∂r r=R1

+ ∂V (1)

∂r r=R1 = σ (1) (t) ε 0 (13) (S 2 ) : - ∂V (3) ∂r r=R2 + ε r ∂V (2) ∂r r=R2 = σ (2) (t) ε 0 . ( 14
)
From the above boundary conditions, we deduce that the terms A mn (t), B mn (t) . . . E mn (t) found in (7-9) can be written as a linear combination of both timedependent surface charge densities moments σ (1,2) mn (t) in the form, A mn (t) = a (1) mn σ (1) mn (t) + a (2) mn σ (2) mn (t) (15) . . . E mn (t) = e (1) mn σ (1) mn (t) + e (2) mn σ (2)

mn (t) (16)
The constant coefficients a

(i) mn . . . e (i)
mn , i = 1, 2 are known quantities that depend on the parameter R 1 , R 2 , H, ε r of the capillary as well as on R 3 . This means that the influence of the dimensions of the capillary, its dielectric constant and the dimensions of the surrounding metal cylinder are hidden in those constant coefficients. Details about how to evaluate the coefficients a

(i) mn . . . e (i)
mn , i = 1, 2 are given in the Appendix A. Eventually, the expression of the electric potentials V (2) (r, θ, z, t), which will be needed in the next section, reads

V (2) = mn b (1) mn I m (k n r) + c (1) mn K m (k n r) σ (1) mn (t) + b (2) mn I m (k n r) + c (2) mn K m (k n r) σ (2) mn (t) × cos(mθ) sin(k n z) , (17)
where the dependence on time is explicitly given by the time-evolution of the surface charge densities at the inner and outer capillary interfaces. Note that, as V (2) satisfies the required boundary conditions (10)(11)(12)(13)(14), it automatically includes the contributions from the induced image charges at the interfaces S 1 , S 2 , and S 3 .

C. General Solution of the time evolution of the surface charges

The expressions of the electric field that appear in the equations (2-3) can all be deduced from the potential V (2) (Eq. 17), which is defined for R 1 ≤ r ≤ R 2 , so that we may re-write the equations (2-3) in the form, ∂σ (1) ∂t = κ b ∂V (2) ∂r r=R1

-

κ s m 2 R 2 1 + k 2 n V (2) r=R1 + γ (1) ∂σ (2) ∂t = -κ b ∂V (2) ∂r r=R2 -κ s m 2 R 2 2 + k 2 n V (2) r=R2 + γ (2) (18)
Injecting the multipole expansions of σ (1,2) (4) , γ (1,2) (5) and V (2) (17) into ( 18) and projecting the latter onto the basis elements cos(mθ) sin(k n z) yields for each set of indices (m, n) a linear matrix equation of the form,

  σ(1) mn σ(2) mn   = -   f (11) mn f (12) mn f (21) mn f (22) mn     σ (1) mn σ (2) mn 
 +   γ (1) mn γ (2) mn   (19)
where all the dependencies on the geometry and electric properties of the insulating capillary now hidden in the time-independent coefficients

f (1j) mn = -κ b I m (k n R 1 ) + κ s m 2 R 2 1 + k 2 n I m (k n R 1 ) b (j) mn + -κ b K m (k n R 1 ) + κ s m 2 R 2 1 + k 2 n K m (k n R 1 ) c (j) mn (20)
f (2j) mn = κ b I m (k n R 2 ) + κ s m 2 R 2 2 + k 2 n I m (k n R 2 ) b (j) mn + κ b K m (k n R 2 ) + κ s m 2 R 2 2 + k 2 n K m (k n R 2 ) c (j) mn ( 21
)
where the prime symbol stands for the derivative with respect to the radial component and j = 1, 2. Remarkably, there is no coupling between the modes in (19), neither for the angular mode m nor for the axial mode n. As a result, (19) can be solved independently for each mode (m, n). This however is the case only for straight capillary tubes. In the case of tapered capillaries, at least the axial modes n would be coupled to other axial modes n .

For convenience, a ®Mathematica script that evaluates the matrix elements f

(i,j)
mn is given in appendix B.

D. Homogeneous solution

In this work we consider only the homogeneous solution of (19), that is, the solution in the absence of source terms, γ

(1,2) mn (t) = 0. Such a solution describes the charge relaxation of a previously accumulated charge distribution in the case where no beam is further injected into the capillary. The homogeneous solution of ( 19) is easily found if the matrix

F mn = {f (ij) mn } is non-defective (diagonalizable) so that 1/τ (i) mn δ ij = T -1 mn F mn T mn (22)
where the characteristic relaxation rates 1/τ

mn and 1/τ

(2)
mn are the two eigenvalues of the matrix F mn and where T mn is the change of basis matrix. The relaxation rates 1/τ (1) mn are thus functions of the dimensions (R 1 , R 2 , H) and electric properties (ε r , κ s , κ b ) of the capillary as well as the radius R 3 of the grounded cylinder surrounding the capillary. Note that because the relaxation rates 1/τ (1,2) mn are obtained by diagonalizing the F mn matrices, they are in general non-linear functions of the bulk and surface conductivity of the capillary. In the particular case where the outer surface S 2 is grounded, the elements f

mn = f (11)
mn , which is a linear functions of κ s and κ b . The latter finding was already discussed in [16].

A solution of the homogeneous case is eventually given by   σ

where the σ (i) mn (0) are the initial (previously accumulated) surface charge distributions at the interfaces S i . Introducing the auxiliary projector P mn defined by

P mn = T mn   1 0 0 0   T -1 mn , (24)
and noting σ mn (t) = σ

(1)

mn (t) σ (2) mn (t)
, one can express the solution (23) in the following compact form σ mn (t) = e -t/τ (1) mn P mn + e -t/τ (2) mn (I -P mn ) • σ mn (0) , (25) where I is the identity matrix. Equation (25) yields that, in the absence of source terms, each moment σ mn (t) relaxes exponentially in time with two time constants τ (1) mn and τ

(2) mn . Eventually injecting (25) into the multipole expansion ( 4) and (5) gives the explicit time evolution of the surface charge densities σ (1,2) (θ, z, t) that satisfy the surface charge equations ( 2) and (3) without source terms. Similarly, injecting (25) into (17) yields the electric potential that drives the charge carriers in the insulator bulk. Finally, injecting (25) into (A14) yields the electric potential inside the capillary that guides the ion beam.

E. Stationary source terms

Consider the case where the source term γ mn (t) = γ (1)

mn (t) γ (2) mn (t)
is time independent, so that ∂ t γ mn (t) = 0 for all m, n. Such a situation may correspond to a stationary beam that injects charge at constant rate at the inner surface of a (tilted) capillary. We further assume that the capillary is initially electrically neutral, σ mn (0) = 0. The time evolution of the surface charge distributions is then given by σ mn (t) = 1 -e -t/τ (1) mn τ (1) mn P mn γ mn +

1 -e -t/τ (2) mn τ (2) mn (I -P mn ) • γ mn . (

Relationship ( 26) is handy if one wants to estimate the charge distribution of a patch in the capillary after a given charging time t. The relationship holds in the case of a constant or slowly (adiabatically) evolving source term.

III. DEFLECTION OF THE BEAM BY A CHARGE PATCH

The above model is quite general and useful to determine the charge relaxation rates in many different scenarios, and is not limited to macroscopic glass-capillaries. For instance the model can also be applied to nanocapillaries in insulating foils. The inner radius R 1 , length H, dielectric constant ε r and bulk conductivity κ b of the nano-capillaries are usually given. Caution must be paid to the choice of R 2 and R 3 for the model to represent reliably the nano-capillaries in question as they are not so clearly defined as in glass macro-capillaries. We give here two examples. For nano-capillaries in PET foils, like the one used in [1], we suggest to set the radius R 2 to half of the inter-capillary distance, which in [1] was about R 2 2.5 µm. As no grounded S 3 interface surrounds the nano-capillaries in PET foils, R 3 should be set to an arbitrary large value so that R 3 R 2 . The nano-capillaries in n-doped silicon membrane found in [24], have an insulating SiO 2 thickness of 100 nm. The radius R 2 may then be set to R 2 = R 1 +100 nm. Because the n-doped silicon is a semi-conducting media, we can Glass bulk is represented by glue-gray rectangles and is surrounded by grounded layer (black full lines). The yellow parts stand for insulator rings keeping the glass tube in place. Upper panel: a tilted beam injects a charge patch at the inner surface. Lower panel: transmitted beam is deflected by the charge patch and its position ∆x(t) is recorded on the PSD located downstream.

assume that the potential of the silicon media in between the nano-capillaries is grounded so that R 3 = R 2 . The remaining unknown in both examples is then the surface conductivity. A detailed study of the charge relaxation in nano-capillaries and the analyses of the experimental data with this model will be presented elsewhere. In the present work, we will merely focus on straight glass capillaries and show how to use the model to estimate the surface and bulk conductivity of a given insulating glass capillary that was previously charged by an ion beam.

A. Modeling a Gedankenexperiment

In the previous section, we saw how to calculate the relaxation rates, knowing the bulk and surface conductivity for a given capillary. In the following Gedankenexperiment, we will show how to estimate the bulk and surface conductivity of the capillary unambiguously, by monitoring the charge relaxation rate of a previously injected charge patch.

We consider here a straight glass capillary, with the dimensions H = 30 mm , R 1 = 0.43 mm, R 2 = 0.75 mm. The capillary is made out of Borosilicate glass, characterized by a dielectric constant ε r = 4.6. Glass tubes like the one supposed in this Gedankenexperiment are commercially available, for example, by Warner Instruments. We consider two different configurations. In the first configuration, labeled (a), the conducting interface S 3 has a much larger inner radius than the outer surface of the capillary, R 2 R 3 = 5 mm. In the second configuration, labeled (b), the outer surface is grounded, R 3 = R 2 . Both configurations are easily set up in a dedicated experiment like the one presented in [3]. The idea is to use the different charge relaxation rates that yield case (a) and (b) in order to estimate the surface and bulk conductivity of the capillary.

The experimental methodology is the following. In a first step, the axis of the capillary is tilted by 1.5 • with respect to the beam axis. A charge patch, centered at θ = 0,

z c = R 1 / tan(1.5 • )
16 mm, is injected at the inner surface by a uniform ion beam collimated to a diameter of b = 0.1 mm. The intensity I in of the injected current must be adjusted so as to produce a single patch which is strong enough to deflect a part of the injected beam through the capillary outlet. The charge patch has a characteristic length of d = b/ tan(1.5 )

3.8 mm in the z-direction, see upper panel of Fig. 2. As soon as the charge patch is sufficiently strong to deflect a part of the beam thought the capillary outlet, the injected intensity is reduced to several fA. Reducing the beam intensity is important in order to avoid that the charge patch is neutralized by secondary electrons generated by beam ions hitting the border of the collimator hole at the entrance. At the same time, the capillary axis is aligned with the beam axis (zero tilt angle). The previously accumulated charge patch deflects now the injected beam ions of charge q and momentum p z u z by an angle α, which can be monitored on a position sensitive detector (PSD) downstream, see lower panel of Fig. 2. In a second step, the time evolution of the deflection angle α(t) of the transmitted beam is recorded. The methodology is repeated for setup (a) and (b).

We will now model the time evolution of the measured deflection angle of the Gedankenexperiment and give an analytic expression of the deflection angle α(t) using the findings of section II. The x-component of the electric field evaluated at the symmetry axis is given by

E x (0, 0, z, t) = - ∂V (1) (r, θ = 0, z, t) ∂r r=0 = - n=1 nπ 2H   a (1) 1n σ (1)
1n (t) + a (2) 1n σ
(2)

1n (t) amn• σmn(t)    × sin(k n z) (27)
with the constant coefficients a

1,n and a

1,n being defined in the Appendix A. Note that only dipole angular moments, m = 1, give a non-zero contribution to the electric field evaluated at the symmetry axis. Injecting (25) into (27) allows expressing the time-evolution of E x (0, 0, z, t),

E x (0, 0, z, t) = - n=1 nπ 2H     e -t/τ (1) 1n a 1n • P 1n • σ 1n (0) g (1) n +e -t/τ (2) 1n a 1n • (I -P 1n ) • σ 1n (0) g (2) n     × sin(k n z) . (28)
The scalars g

n and g

n give the amplitudes of the two exponential decay functions for each n and depend explicitly on the initial charge distributions σ 1n (0) at the inner and outer surface of the capillary.

In the next step, the capillary axis is aligned with the beam axis (zero tilt angle). The injected intensity is reduced to several fA, in order to avoid that the charge patch is neutralized by secondary electrons generated by beam ions hitting the border of the collimator hole at the entrance. The charge patch deflects now the beam ions of charge q and momentum p z u z by an angle α, which can be monitored on a position sensitive detector (PSD) downstream, see lower panel of Fig. 2. The time evolution of the deflection angle α(t) is given by tan

(α(t)) = p x (t) p z = q 2E k H 0 E x (0, 0, z, t)dz = 1 2V s n=1
,odd e -t/τ (1) 1n g (1) n + e -t/τ (2) 1n g (

where E k = qV s is the kinetic energy of the ion beam. The constant amplitudes g

n and g

n can be evaluated if the initial surface charge moments σ 1,n (0) are known.

B. Numerical application

Borosilicate glass has typically a bulk conductivity of κ b = 10 -13 S/m at room temperature, which yields a bulk relaxation time of,

τ b = ε r ε r κ b 400 s . (30)
We will later compare this value to the decay times obtained for α(t) in configurations (a) and (b). We further suppose a surface conductivity of κ s = 10 -16 S, close to the value measured by [19] at room temperature. Both κ b and κ s may be regarded as initial guesses.

In order to evaluate α(t) numerically, using (29), we need to provide a spatial profile of the accumulated charge distribution at the interfaces after the initial charging step. To avoid unnecessary complexity, we assume that a single patch has been injected in the capillary. Note that the time evolution of the deflection angle α(t) in not very sensitive to the exact profile of the

R3 R2 (a) R3 = R2 (b) n τ (1) 1,n τ (2) 1,n g (1) n g (2) n τ (1) n τ (2) 1,n g (1) n g (2) n (s)
(s) (V) (V) (s) (s) (V) (V
n .

injected charge patch as the profile controls mainly the initial amplitude of the deflection angle. For the sake simplicity, we assume that the charge patch at the interface S 1 has a Gaussian profile of the form

σ (1) (θ, z, t = 0) = Q bπd exp - (z -z c ) 2 d 2 exp - (R 1 θ) 2 b 2 ,
(31) where Q = 50 pC is the total charge of the patch. The value of Q will be justified later. The lengths d and b are related to the beam size and tilt angle and were already defined in section III A. The Gaussian profile is already a good approximation of the charge distribution generated by the thin ion beam at the inner surface of the capillary, as we could check with our simulations. At the interface S 2 , the charge distribution is assumed zero, σ 2 (θ, z, 0) = 0.

In a first step, we take the multipole expansion (4) of the surface charge density σ (1) (θ, z, t = 0). We retain only the dipole angular moments σ (1) 1,n (0), which enter Eq. 28 . The Gaussian charge patch has a typical length along the Oz axis of d = 3.8 mm and is thus already well approached by an expansion limited to the first n ≤ N = (H/d) + 1 9 modes. In a second step, we evaluate for both cases (a) and (b), the characteristic times 1/τ (1,2) 1,n , for all n ≤ N using, for example, the Mathematica script found in the supplementary material and described in Appendix B. In a third step, we calculate for both setups the amplitudes g

(1)
n and g

(2)
n defined in (28). An option in the provided Mathematica script allows computing all the required amplitudes.

In table I we give, for the first n ≤ 9 the rates 1/τ

(1,2) 1,n
and amplitudes the g

(1,2) n that contribute to the deflection angle in (29). First we note that for case (a), g

(2) n is systematically one order of magnitude smaller than g n to the deflection angle α. Second, we note that for both configurations (a) and (b), the rates 1/τ (1) 1,n vary by less than 10% over the range n ∈ [1,9]. We may thus approach τ g (1) n G1

(32)

= 1 2V s G 1 e -t/τ (1) 1,1 (33)
The model eventually predicts that the deflection amplitude α(t) of the transmitted beam decays exponentially in time with a decay time equal to the relaxation time τ

1,1 obtained by diagonalizing the matrix F 1,1 , see (22) The constant G 1 has the dimension of an electric potential and controls the amplitude of the deflection angle α.

In the last line of table I, we give G 1 for both configurations (a) and (b). Assuming for the extraction potential V s of the ion source a value of V s = 1.5 kV, we get for the case (a) the time evolution of the deflection angle, which in the small angle approximation reads (in radians),

α(t) 1 60 exp (-t/94) R 3 R 2 . ( 34
)
In setup (b) where the outer surface of the capillary is grounded, the deflection amplitude is reduced by more than a factor 2 and evolves with a time constant of 133 s, about three times larger than for case (a),

α(t) 1 150 exp (-t/204) . R 3 = R 2 . (35)
Comparing the decay times of setup (a), 94 s and (b), 204 s to the significantly larger bulk relaxation time τ b = 400 s, highlights the non-negligible influence of the surface conductivity on the decay rates, in the case where κ s = 10 -16 S and κ b = 10 -13 S/m. We note that it is important not to overcharge the patch. In configuration (a), a total charge patch of Q = 50 pC yields a deflection amplitude of 1/60 rad, which is sufficiently low so that the deflected beam will not hit the opposite capillary wall before exiting the capillary, while the deflection is sufficiently large to be measured on a PSD some 300 mm downstream. In the previous section, we showed that the model can predict the decay rate of the deviation angle α(t), knowing the bulk and surface conductivities of a glass tube. In Fig. 3, we give the dependency of 1/τ

1,1 as a function of the surface conductivity κs for both configurations (a) and (b). Full and empty circles are obtained using a bulk conductivity κ b = 10 -13 S/m; full and empty squares were calculated using κ b = 5 × 10 -13 S/m. be deduced from the curves with filled circles and filled squares. The linear relationship was already pointed out in [16]. For setup (a) however, the relation is nonlinear as is shown by the curve with the empty squares close to the origin. This is not surprising as the rates are the zeros of the characteristic polynomial of order two of the matrix F mn , see (22).

In this paragraph we want to show how to deduce the bulk and surface conductivity from the decay measurements. As we have two unknowns, (κ b , κ s ), we need two independent measurements of 1/τ (1) 1,1 . The latter requirement is fulfilled by configurations (a) and (b) as can be seen by the different slopes they yield in Fig. 3. Monitoring the decay of the deflection angle α(t) of the transmitted beam on a PSD and then fitting its time-evolution using expression (33) allows to deduce the experimental relaxation rate τ In the particular case where the measured decay rates of both configurations are the same, the model predicts that the charge relaxation rate is dominated by the bulk conductivity and the surface conductivity may be neglected in the model (2,3). On the contrary, if the measured rates (a) and (b) differ sensibly, then the surface conductivity has a non-negligible influence on the charge dynamics. In the latter case, using the provided Mathematica script and a Newton type search technique, a pair of (κ b , κ s ) that yields the measured 1/τ (1) 1,1 for case (a) and (b) should eventually be found. If no such pair of bulk and surface conductivities can be found, that would involve that the model does not describe realistically the the charge dynamics in the capillary. The experiment should thus provide a stringent test for the model.

IV. CONCLUSION

We proposed a simple model that describes the charge dynamics in straight insulating capillaries. The model is based on the assumption that the charges accumulate only at the inner and outer interface of the capillary. The current density through the bulk is proportional to the bulk conductivity. The surface currents along the interface are proportional to the surface conductivity. Within theses assumptions, the model is solved using a multipole expansion of the electric potential that satisfies the given boundary conditions. The model predicts the time evolution of the accumulated surface charge distribution in the presence of electrons and holes injected at the inner and outer surface. The model is applicable to a large number of systems, including nano-capillaries. It yields the charge relaxation rates that characterize the dynamics of charge patches as a function of the geometric dimensions and electric properties of the capillary. In the case of time independent source terms, it also allows to calculate the surface charge distributions that accumulate at the interfaces obtained after a given charging time.

We present a Gedankenexperiment that measures the relaxation rates of the dipole part of the accumulated charge at the capillary interfaces. We showed how to use the model in order to extract from the measurements the surface and bulk conductivity of the capillary. Upcoming experimental measurements, based on the proposed Gedankenexperiment should clarify if the surface conductivity can be omitted in the simulations and if the surface charge approximation is applicable in glass capillaries. Preliminary experimental work has already started.

For the readers interested in developing numerical codes for the charge transport through insulating capillaries, the model proposes a CPU efficient technique for the evaluation of the electric field, knowing the charge distribution at the capillary interfaces. We provide a script in ®Mathematica that calculates the quantities presented in the manuscript and which may be easily modified to calculate those quantities for various setups the reader imagines. Finally, the present model may be easily extended to account also for tapered capillaries. Work in this direction is underway. mn , i = 1, 2 which are needed in order to express the potentials V (1) and V (2) as functions of the surface charge densities σ (1) and σ (2) . In the following, we drop the indexes m and n in the mind the reader that the matrix elements depend on the indexes m, n, even if not explicitly stated. All the dependency on the dimensions (R 1 , R 2 , R 3 , H) of the straight capillary and surrounding electrode as well as the dielectric constant ε r of the insulator is captured by the matrix elements. The latter are thus time-independent and can be evaluated once for all for each mode (m, n). We will need in this work the explicit dependence of the electric potentials V (2) (t) on both time-dependent surface charge densities σ (1,2) mn (t). From the structure of (A10), we deduce that the terms A mn (t), B mn (t) and C mn (t) can be written as a linear combination of both surface charge densities σ A mn (t) = a (1) mn σ (1) mn (t) + a (2) mn σ (2) mn (t) (A11)

B mn (t) = b (1) mn σ (1) mn (t) + b (2) mn σ (2) mn (t) (A12)

C mn (t) = c (1) mn σ (1) mn (t) + c (2) mn σ (2) mn (t) (A13)

where the constant coefficients a

mn and c (i) mn , i = 1, 2 are known quantities obtained by inverting the matrix (A10) for each set of indexes (m, n). Eventually the potential V (1) (r, θ, z, t) and V (2) (r, θ, z, t) are expressed using explicitly the surface charge densities, V (1) = mn I m (k n r) a (1) mn σ (1) mn (t) + a (2) mn σ (2) mn (t)

× cos(mθ) sin(k n z) (A14) and 1) mn I m (k n r) + c (1) mn K m (k n r) σ (1) mn (t) + b (2) mn I m (k n r) + c (2) mn K m (k n r) σ (2) mn (t) × cos(mθ) sin(k n z) (A15)

V (2) = mn b (
The potential V (3) (r, θ, z, t) can be obtained similarly.

  mn are zero. The matrix F mn has then only one non-zero eigenvalue, namely τ(1)

FIG. 2 :

 2 FIG.2: Scheme of the experimental technique to monitor the relaxation of a previously injected charge patch in the case (a) where the outer surface of the capillary is not grounded. Glass bulk is represented by glue-gray rectangles and is surrounded by grounded layer (black full lines). The yellow parts stand for insulator rings keeping the glass tube in place. Upper panel: a tilted beam injects a charge patch at the inner surface. Lower panel: transmitted beam is deflected by the charge patch and its position ∆x(t) is recorded on the PSD located downstream.

  zero. For both cases, we neglect thus the contributions of g

  ∀n ≤ N , so that the expression of the deflection angle (29) simplifies, for both cases (a) and (b)

C.

  Deducing κ b and κs from the decay measurements of setups (a) and(b)

5 FIG. 3 :

 53 FIG. 3: Decay rate 1/τ

  of the dipole angular moment of the charge distribution. The rates measured in case (a) and (b) should permit us to uniquely estimate the surface conductivity κ s and bulk conductivity κ b of the glass tube.

  Appendix A: Expression of the potential as a function of the surface charge densitiesThis appendix details the evaluation of the time independent coefficient a

( 1 )

 1 mn (t) and σ

( 2 )

 2 mn (t), namely,

TABLE I :

 I Characteristic times τ1,n and amplitudes gn for m = 1 and n < 10. Left part of the table gives the values for setup (a) and right part of the table for setup (b). For n ≥ 10, the amplitudes gn are negligible and thus omitted. Last gives the total amplitude G1 = n g

					)
	1	94 295 83.5 4.5 204 ≥ 10 4 35.4 0
	3	94 293 -53.9 -3.0 204	" -23.4 "
	5	94 287 22.1 1.4 201	"	11.9 "
	7	93 280 -5.6 -0.4 197	"	-3.9 "
	9	92 270	0.6 0. 192	"	0.3 "
	G1(V )		48		19.7

Acknowledgments

This work was supported by the french research agency CNRS via the project PICS N • 245358 Hongrie 2018.

Bessel functions in order to simplify the notations,

From the boundary conditions (10)(11)(12)(13)(14) we deduce the relations

and

The above system of equations (A5 -A9) has 5 unknowns A mn , B mn , . . . , E mn and 5 independent linear equations, so that the unknowns are uniquely determined. It is convenient to put the above system of equations into a matrix form, Appendix B: Script for evaluating numerically the characteristic times τ

(i) mn

We provide a script for ®Mathematica 5 or above that calculates the characteristic times τ

mn and τ

mn for all m ∈ [0, M max ] and n ∈ [1, N max ]. The script named "rates.nb" takes as input the dimensions R 1 , R 2 , H, dielectric constant ε r and conductivities κ b and κ s of the straight capillary as well as the radius R 3 of the conducting interface S 3 . If there is no conducting S 3 interface, R 3 can be set to an arbitrary large value, typically R 3 H. Lengths are given in mm, surface conductivity in Siemens (S) and bulk conductivity is S/m. Finally, the routine needs the highest angular M max and axial N max mode for which it should perform the calculations. It outputs the matrix elements f mn , i = 1, 2, necessary to express the potential V (1) and V (2) in the form (A14) and (A15) respectively. The script is available as a separate file in the supplementary material.