Marie Bernardin

Frédérique Bessueille-Barbier

Agnès Le Masle

Charles-Philippe Lienemann

Sabine Heinisch
email: sabine.heinisch@univ-lyon1.fr

Theoretical and experimental considerations on solute dispersion

Keywords:

come

Suitable interface for coupling liquid chromatography to

inductively coupled plasma-mass spectrometry for the analysis of organic matrices.

Introduction System (SIS) and a possible flow-splitter prior to SIS. Flow splitting can be required in case of organic matrices to reduce the organic solvent amount entering plasma which may lead to plasma instabilities.

Although extra-column dispersion is usually well taken into account with conventional UV detection it has been little studied in the context of LC-ICP-MS and moreover never quantified. Our objective is to assess the loss in column plates and hence in both separation quality and sensitivity which may be generated by the coupling of LC and ICP-MS in the specific case of organic matrices. In this first study, this is done (1) from a theoretical approach; (2) from 55 experimental studies reported in LC-ICP-MS and (3) from our experimental results highlighting the critical impact of the flow splitter on extra-column dispersion depending on both flow-rate and split ratio. It turns out by evaluating the 55 reported studies by means of theoretical calculations, that the loss in plates due to extra-

Introduction

The need for determining elemental species concentrations rather than total element concentrations has grown within the past decades. According to Templeton et al. [START_REF] Templeton | Guidelines for terms related to chemical speciations and fractionation of elements : definitions, structural aspects, and methodological approaches[END_REF][START_REF] Michalke | Element speciation definitions, analytical methodology, and some examples[END_REF] speciation analysis consists in identifying and quantifying different chemical species of a particular element in a given sample. It has become an important field of research over the past few years in several areas including biochemistry, environmental chemistry, ecotoxicology, pharmaceuticals, petrochemicals, and nutrition science [START_REF] Grotti | Speciation analysis by small-bore HPLC coupled to ICP-MS[END_REF].

Nowadays hyphenated techniques such as Liquid Chromatography (LC) hyphenated with Inductively Coupled Plasma Mass Spectrometry (LC-ICP-MS) are widely used both to obtain elemental information and to discriminate species in a given matrix. LC techniques, including ion exchange chromatography (IEC) [START_REF] Ammann | Arsenic speciation by gradient anion exchange narrow bore ion chromatography and high resolution inductively coupled plasma mass spectrometry detection[END_REF], reversed-phase liquid chromatography (RPLC) [START_REF] Amayo | Identification and Quantification of Arsenolipids Using Reversed-Phase HPLC Coupled Simultaneously to High-Resolution ICPMS and High-Resolution Electrospray MS without Species-Specific Standards[END_REF], ion-pairing chromatography (IPC) [START_REF] Lohmayer | Ion-pair chromatography coupled to inductively coupled plasma-mass spectrometry (IPC-ICP-MS) as a method for thiomolybdate speciation in natural waters[END_REF], size exclusion chromatography (SEC) [START_REF] Boulyga | Application of SEC-ICP-MS for comparative analyses of metal-containing species in cancerous and healthy human thyroid samples[END_REF][START_REF] Vargas | Development of a chromatographic methodology for the separation and quantification of V, Ni and S compounds in petroleum products[END_REF] and hydrophilic interaction liquid chromatography (HILIC) [START_REF] Nygren | Hydrophilic interaction liquid chromatography (HILIC) coupled to inductively coupled plasma mass spectrometry (ICPMS) utilizing a mobile phase with a low-volatile organic modifier for the determination of cisplatin, and its monohydrolyzed metabolite[END_REF][START_REF] Xie | Retention of arsenic species on zwitterionic stationary phase in hydrophilic interaction chromatography[END_REF], have been used for speciation analysis [START_REF] Delafiori | Clinical applications of HPLC-ICP-MS element speciation: A review[END_REF]. Different hyphenated techniques can be combined together to achieve more exhaustive characterization of complex matrices. For example, LC-ESI-MS can be associated to LC-ICP-MS to obtain both structural information and elemental information [START_REF] Amayo | Identification and Quantification of Arsenolipids Using Reversed-Phase HPLC Coupled Simultaneously to High-Resolution ICPMS and High-Resolution Electrospray MS without Species-Specific Standards[END_REF][START_REF] Beuvier | Développement d'une méthode de séparation chromatographique couplée aux spectrométries de masse à source d'ionisation électrospray (ESI-MS) et à[END_REF][START_REF] Pereira | Arsenic containing medium and long chain fatty acids in marine fish oil identified as degradation products using reversed-phase HPLC-ICP-MS/ESI-MS[END_REF][START_REF] Frindt | Multidimensional monitoring of anaerobic/aerobic azo dye based wastewater treatments by hyphenated UPLC-ICP-MS/ESI-Q-TOF-MS techniques[END_REF].

Most speciation analyses are performed in aqueous matrices by using ion-exchange chromatography and deal with environmental samples [START_REF] Sutton | Liquid chromatography-inductively coupled plasma mass spectrometry[END_REF]. In this specific case, the mobile phase is not critical for the coupling of both techniques since the amount of organic solvent in the mobile phase is limited (small percentage of organic modifier sometimes added, usually without disturbing plasma stability). However, when using RPLC [START_REF] Bendahl | Sample introduction systems for reversed phase LC-ICP-MS of selenium using large amounts of methanol-comparison of systems based on membrane desolvation, a spray chamber and direct injection[END_REF][START_REF] Balcaen | Hyphenation of reverse-phase HPLC and ICP-MS for metabolite profiling-application to a novel antituberculosis compound as a case study[END_REF][START_REF] Wolf | A comparison between HPLC-dynamic reaction cell-ICP-MS and HPLC-sector field-ICP-MS for the detection of glutathione-trapped reactive drug metabolites using clozapine as a model compound[END_REF][START_REF] Ackley | A comparison of nebulizers for microbore LC-ICP-MS with mobile phases containing methanol[END_REF][START_REF] Döker | Rapid extraction and reverse phase-liquid chromatographic separation of mercury(II) and methylmercury in fish samples with inductively coupled plasma mass spectrometric detection applying oxygen addition into plasma[END_REF][START_REF] Trümpler | Detoxification of mercury species-an in vitro study with antidotes in human whole blood[END_REF][START_REF] Moreno | Simultaneous speciation and preconcentration of ultra trace concentrations of mercury and selenium species in environmental and biological samples by hollow fiber liquid phase microextraction prior to high performance liquid chromatography coupled to inductively coupled plasma mass spectrometry[END_REF][START_REF] Klencsár | Determination of the total drug-related chlorine and bromine contents in human blood plasma using high performance liquid chromatography-tandem ICP-mass spectrometry (HPLC-ICP-MS/MS)[END_REF][START_REF] Meermann | HPLC/ICP-MS in Combination with "Reverse" Online isotope dilution in drug metabolism studies[END_REF][START_REF] Jensen | Application of inductively coupled plasma mass spectrometry and high-performance liquid chromatography-with parallel electrospray mass spectrometry to the investigation of the disposition and metabolic fate of 2-, 3-and 4-iodobenzoic acids in the rat[END_REF][START_REF] Chen | Determination of trace mercury species by high performance liquid chromatography-inductively coupled plasma mass spectrometry after cloud point extraction[END_REF][START_REF] Gabel-Jensen | Separation and identification of the selenium-sulfur amino acid S-(methylseleno)cysteine in intestinal epithelial cell homogenates by LC-ICP-MS and LC-ESI-MS after incubation with methylseleninic acid[END_REF][START_REF] Gabel-Jensen | LC-ICP-MS and LC-ESI-(MS)n identification of Se-methylselenocysteine and selenomethionine as metabolites of methylseleninic acid in rat hepatocytes[END_REF] or HILIC [START_REF] Nygren | Hydrophilic interaction liquid chromatography (HILIC) coupled to inductively coupled plasma mass spectrometry (ICPMS) utilizing a mobile phase with a low-volatile organic modifier for the determination of cisplatin, and its monohydrolyzed metabolite[END_REF][START_REF] Xie | Retention of arsenic species on zwitterionic stationary phase in hydrophilic interaction chromatography[END_REF][START_REF] Xie | Separation of Organoarsenicals by Means of Zwitterionic Hydrophilic Interaction Chromatography (ZIC®-HILIC) and Parallel ICP-MS/ESI-MS Detection[END_REF][START_REF] Far | Detection and identification of hydrophilic selenium compounds in selenium-rich yeast by size exclusion-microbore normal-phase HPLC with the on-line ICP-MS and electrospray Q-TOF-MS detection[END_REF][START_REF] Falta | Quantification of cisplatin, carboplatin and oxaliplatin in spiked human plasma samples by ICP-SFMS and hydrophilic interaction liquid chromatography (HILIC) combined with ICP-MS detection[END_REF][START_REF] Grevenstuk | Identification of the tri-Al tricitrate complex in Plantago almogravensis by hydrophilic interaction LC with parallel ICP-MS and electrospray Orbitrap MS/MS detection[END_REF][START_REF] Rellán-Álvarez | Identification of a Tri-Iron(III), Tri-Citrate Complex in the Xylem Sap of Iron-Deficient Tomato Resupplied with Iron: New Insights into Plant Iron Long-Distance Transport[END_REF][START_REF] Vidmar | Speciation of cisplatin in environmental water samples by hydrophilic interaction liquid chromatography coupled to inductively coupled plasma mass spectrometry[END_REF], a large amount of organic solvent is introduced into the plasma. The problems involved by this introduction was thoroughly discussed by Leclercq et al [START_REF] Leclercq | Tutorial: Introduction of organic/hydro-organic matrices in inductively coupled plasma optical emission spectrometry and mass spectrometry: A tutorial review. Part I. Theoretical considerations[END_REF][START_REF] Leclercq | Tutorial: Introduction of organic/hydro-organic matrices in inductively coupled plasma optical emission spectrometry and mass spectrometry: A tutorial review. Part II. Practical considerations[END_REF] in a recent review.

Some key issues have to be considered for coupling LC to ICP-MS in case of organic matrices [START_REF] Grotti | Speciation analysis by small-bore HPLC coupled to ICP-MS[END_REF]: (i) metal contamination from the chromatographic system and/or the stationary phase and/or the mobile phase [START_REF] Grotti | Speciation analysis by small-bore HPLC coupled to ICP-MS[END_REF], (ii) plasma instabilities due to the solvent load, especially in case of organic mobile phases [START_REF] Grotti | Speciation analysis by small-bore HPLC coupled to ICP-MS[END_REF], (iii) signal fluctuations in gradient elution depending on plasma parameters [START_REF] Klencsár | Development and validation of a novel quantification approach for gradient elution reversed phase highperformance liquid chromatography coupled to tandem ICP-mass spectrometry (RP-HPLC-ICP-MS/MS) and its application to diclofenac and its related compounds[END_REF] and (iv) solute dispersion into the interface located between LC and ICP-MS. The interface is made of a sample introduction system (SIS) and a possible flow splitter prior to SIS which may be required, in case of organic matrices, to reduce the amount of solvent entering plasma and hence to decrease plasma instabilities [START_REF] Grotti | Speciation analysis by small-bore HPLC coupled to ICP-MS[END_REF]. The solute dispersion in the interface unit is a critical issue because that can result in additional solute band broadening and hence in significant loss in both sensitivity and separation quality.

Although extra-column dispersion is usually well taken into account with conventional UV detection, it has been little studied in the context of LC-ICP-MS. In the present study, we made therefore an attempt to assess the extent to which the interface contributes to solute band broadening. This was done by (i) estimating from published studies the likely loss in plates due to solute dispersion in the interface and (ii) showing the critical impact of the flow splitter on extra-column dispersion depending on both flow-rate and split ratio. To support the first approach, a synoptic table has been built (Table 2) which summarizes 55 studies carried out on organic matrices in LC-ICP-MS and gives, for each study, an estimation of the interface contribution to solute band broadening. A further second part of our study will be dedicated to the comparison of a large number of commercially available SIS regarding the extra-column dispersion.

Theoretical considerations

Solute dispersion can be assessed by the peak variance. The total solute dispersion (total variance, ² ) comes from both dispersion inside the column (column variance, ² ) and extra-column dispersion (extra-column variance, ² ).

Extra-column dispersion results from the injection process ( ²

), the different tubing ( ² ) and the detection ( ² ) [START_REF] Martin | Study of the pertinency of pressure in liquid chromatography[END_REF].

Because variances can be added if the corresponding dispersion process are independent of each other, the total peak variance can be written as

² = ² + ² (1)
Similarly, the extra-column variance is the sum of individual contributions according to

² = ² + ² + ² (2)
For Gaussian peaks, the total peak variance in volume units can be given by the measured peak width at half peak height ( . ) according to

= ² . . (3)
Where is the mobile phase flow-rate.

For very bad peak shapes, tThe second order central moment has to be used to provide a reliable variance valueThese latter, in volume units, is given by

," = ² # $ % & ' ($ ' ) # ($ ' ) (4)
Where * + is the mean residence time and ,$*' the intensity as a function of time.

Extra-column variance can be approximated by removing the column and replacing it by a zero dead volume union connector. In this case, extra-column variance is calculated from Eq. (3).

The column variance, expressed in length unit is related to both the column length and the column plate height, H , by

² = L H (5)
H varies with the mobile phase linear velocity, /, and its variation may be fitted by the van Deemter equation [START_REF] Van Deemter | Longitudinal diffusion and resistance to mass transfer as causes of nonideality in chromatography[END_REF] or by the Knox equation [START_REF] Knox | Kinetic Conditions for Optimum Speed and Resolution in Column Chromatography[END_REF] using reduced parameters h col (H /d p ) and ν (/.d p /D m , d p being the average particle diameter and D m, the molecular diffusion coefficient of the solute in the mobile phase). At the minimum of the curve, typical values for h col and ν are 3 and 5 respectively.

Considering the solute linear velocity at the time the solute is eluted from the column (//(1+0 ), 0 being the retention factor at elution), the column variance can be expressed in volume units according to ² = 1 2 $345 6 '²7 89: ;

< is the column dead volume, related to the column length, the internal diameter, = and the column porosity, ε t by

< = π>ε $= 4' ⁄ (7)
Under isocratic conditions, 0 depends on the retention volume and is given by

0 = A & A -1 (8)
Under gradient conditions, 0 depends on both the gradient conditions and the solute properties. However a rough estimation of 0 can be made when the linear solvent strength theory (LSST) can be applied and the solvent strength parameter, D is known [START_REF] Snyder | Gradient elution in high-performance liquid chromatography : I. Theoretical basis for reversed-phase systems[END_REF][START_REF] D'attoma | On-line comprehensive two-dimensional separations of charged compounds using reversed-phase high performance liquid chromatography and hydrophilic interaction chromatography. Part I: Orthogonality and practical peak capacity considerations[END_REF], using the following relation

0 = 3 .E F ∆H I J (9)
With * the column dead time, ∆K the gradient composition range, and * L the gradient time.

Typical values of S (S being the absolute value of the slope of of the linear relationship between the logarithm of the retention factor and the stronger solvent volume fraction) are 4 for small molecules, 20 for peptides and much higher for larger molecules [START_REF] D'attoma | On-line comprehensive two-dimensional separations of charged compounds using reversed-phase high performance liquid chromatography and hydrophilic interaction chromatography. Part I: Orthogonality and practical peak capacity considerations[END_REF].

Thus, From Eqs. ( 6) and ( 7), the column variance for a given peak in a given chromatogram can be estimated provided that the column geometry is known, and the retention factor can be determined.

Similarly to Eq. ( 6), the total variance can be written

² = 1 2 $345 6 '²7 I9IM: ; ( 10
)
Where H is the total plate height resulting from the two dispersion processes.

The ratio, N , between column and total variances corresponds to the ratio between the two plate heights and hence to the ratio between effective and column plate numbers according

P I9IM: = Q 6RR68IST6 Q 89: (11)
Where U VV " and U are effective and column plate numbers respectively.

The term, N represents the fraction of remaining plates for a given solute, in given chromatographic and instrumental conditions.

The peak height and the resolution between two peaks are inversely proportional to the peak standard deviation ( ). As a result, the fractions of remaining peak height and remaining resolution are given by N.

The mobile phase flow-rate is related to the reduced linear velocity by

= ν A ; W X Y (12)
From Eqs. ( 6) and ( 12), Table 1 gives an overview of optimum flow-rates and corresponding column variances (0 =3) depending on both column internal diameter and particle size.

Depending on SIS, the flow entering the plasma source is expected to be a critical parameter regarding the performance of ICP-MS. According to Table 1, the internal diameter has to be chosen in accordance with the required flow-rate. However as shown in Fig. 1, any decrease in column internal diameter leads to a severe decrease in column variance, and hence to a huge decrease in both plate number and sensitivity if the dispersion in SIS is significant. The percentage of remaining plates was calculated from Eq. ( 11) as a function of extra-column variance for different column internal diameters and two different particle sizes (5 and 1.7 µm). The calculations were performed for a 0 value of 3 and a column length providing 10000 plates (Fig. 1a. 15 cm with 5 µm and 5 cm with 1.7 µm) and 30000 plates (Fig. 1b 15 cm with 1.7 µm). Extra-column variance was considered in the range 0.1 to 1000 µL² that covers current LC-UV instruments except nanoLC instruments. With the aim of maintaining more than 80 % of the plates (β² > 0.8) and hence more than 90 % of the peak height (β > 0.9), it appears that the maximum allowable values for the extra-column variance are (1) 1000 and 100 µL² for conventional columns (4.6 mm i.d) packed with 5 µm and 1.7 µm particles respectively;

(2) 50 and 5 µL² for narrow bore columns (2.1 mm i.d) packed with 5 µm and 1.7 µm particles respectively; 3 and 0.3 µL² for micro bore columns (1 mm i.d) packed with 5 µm and 1.7 µm particles respectively. These values are slightly higher for 30000 plates with 1.7 µm particles (Fig. 1b). With capillary columns (300 µm i.d.), the extra-column variance should be much lower than 0.1 µL². This figure also highlights the range of extra-column variance covered by commercially available HPLC and UHPLC instruments with UV detection.

With the objective of reaching 10000 plates, it appears that HPLC-UV instruments are not suitable for conventional columns packed with 1.7 µm particles and that most UHPLC-UV instruments cannot be used with narrow bore columns packed with 1.7 µm particles.

Furthermore, even very efficient UHPLC-UV instrument (i.e. extra-column variance as low as 5 µL²) are not suitable for micro bore columns.

In this context, it is of prime importance to assess the additional extra-column variance brought by the interface between LC and ICP-MS. Considering the curve shapes in Fig. 1, it appears that, in any case, the remaining plates might decrease from 80 % to 70 %, 50 %, and 30 % by increasing the extra-column variance by a factor of 2, 5 and 10 respectively.

As previously discussed, the interest for LC-ICP-MS in case of organic matrices has significantly grown during the past few years. In our opinion, instrument performances are often not considered enough, methods being developed without prior evaluation of the interface contribution to solute dispersion. To assess the additional dispersion brought by the interface in LC-ICP-MS, 55 studies reported in the area of organic matrix speciation have been reviewed and discussed in term of extra-column dispersion in the next section. With a view to reducing the flow entering SIS, an alternative to the reduction of internal diameter may be the use of a flow-splitter prior to SIS. A T-union was indeed used for quantification with online isotopic dilution [START_REF] Jensen | Application of inductively coupled plasma mass spectrometry and high-performance liquid chromatography-with parallel electrospray mass spectrometry to the investigation of the disposition and metabolic fate of 2-, 3-and 4-iodobenzoic acids in the rat[END_REF][START_REF] Wind | Modified direct injection high efficiency nebulizer with minimized dead volume for the analysis of biological samples by microand nano-LC-ICP-MS[END_REF][START_REF] Stefánka | Down-scaling narrowbore LC-ICP-MS to capillary LC-ICP-MS: a comparative study of different introduction systems[END_REF][START_REF] Zoorob | Evaluation of the direct injection nebulizer in the coupling of high-performance liquid chromatography to inductively coupled plasma mass spectrometry[END_REF][START_REF] Giusti | Interfacing reversed-phase nanoHPLC with ICP-MS and on-line isotope dilution analysis for the accurate quantification of selenium-containing peptides in protein tryptic digests[END_REF][START_REF] Kannamkumarath | Determination of levothyroxine and its degradation products in pharmaceutical tablets by HPLC-UV-ICP-MS[END_REF][START_REF] Rottmann | Development of an on-line isotope dilution technique with HPLC/ICP-MS for the accurate determination of elemental species[END_REF]. However such unit may lead to significant additional extra-column band broadening. The contribution of flow splitting to extra-column dispersion is therefore studied in the last section.

LC-ICP-MS Interface

Peak band broadening is rarely taken into consideration even when the sample matrix becomes complex. Most of the time, authors work with only a few standards instead of a complex mixture to optimize the separation techniques and even less often on a real sample. For instance, Raber et al. [START_REF] Raber | Quantitative speciation of arsenic compounds when using organic solvent gradients in HPLC-ICPMS[END_REF] focused on eleven arsenic species using an LC separation prior to ICP-MS detection but, as can be seen, the peaks were poorly resolved. A low measured plate number with respect to the theoretical column plate number, can be due to (i) solute dispersion in the chromatographic system (injection system, tubing, UVdetection), (ii) a loss in column plates (depending on the history of the column), or (iii) an additional dispersion generated by the interface between LC and ICP-MS. The interface consists in a Sample Introduction System (SIS) and a possible flow-splitter prior to SIS which can be required, in case of organic matrices, to reduce the amount of solvent entering plasma and hence to decrease plasma instabilities. SIS is usually divided into two devices, the nebulizer, and the spray chamber. The nebulizer converts the liquid from the separation technique into a heterogeneous aerosol made of different droplet sizes. The aerosol is then usually sorted out in a spray chamber. Larger droplets are carried to the waste while smaller ones are sent to the plasma source for atomization/excitation/ionization [START_REF] Leclercq | Tutorial: Introduction of organic/hydro-organic matrices in inductively coupled plasma optical emission spectrometry and mass spectrometry: A tutorial review. Part I. Theoretical considerations[END_REF]. Nebulizers, spray chambers, and flow-splitters represent critical devices for ensuring a high-performance coupling. Particular attention must be paid both to the quality of the aerosol produced through the nebulizer/spray chamber [START_REF] Todoli | Liquid Sample Introduction in ICP Spectrometry : A Practical Guide[END_REF] and to the contribution of the whole interface to peak band broadening [START_REF] Grotti | Speciation analysis by small-bore HPLC coupled to ICP-MS[END_REF][START_REF] Stefánka | Down-scaling narrowbore LC-ICP-MS to capillary LC-ICP-MS: a comparative study of different introduction systems[END_REF][START_REF] Zoorob | Evaluation of the direct injection nebulizer in the coupling of high-performance liquid chromatography to inductively coupled plasma mass spectrometry[END_REF][START_REF] Acon | A direct injection high efficiency nebulizer interface for microbore high-performance liquid chromatography-inductively coupled plasma mass spectrometry[END_REF][START_REF] Todoli | Fast determination of arsenosugars in algal extracts by narrow bore high-performance liquid chromatography-inductively coupled plasma mass spectrometrv[END_REF][START_REF] Lokits | Interfaces for capillary LC with ICPMS detection: A comparison of nebulizers/spray chamber configurations[END_REF], both features being able to significantly affect the sensitivity. A well-documented summary of SIS devices was reported by Leclerq and al. [START_REF] Leclercq | Tutorial: Introduction of organic/hydro-organic matrices in inductively coupled plasma optical emission spectrometry and mass spectrometry: A tutorial review. Part II. Practical considerations[END_REF].

Major advantages and drawbacks were discussed in terms of sensitivity and ease of use, but other analytical performances such as efficiency, resolution, sensitivity or extra-column solute dispersion, were not taken into account to compare the different devices. In the present work, this feature has been subjected to a broad exploratory study on 55 published studies (from 1995 to 2017) dealing with the speciation of organic matrices. The corresponding analytical conditions, including LC conditions, type of nebulizers, type of spray chambers and use or not of a flow-splitter, are listed in Table 2 and Table S1 of Supplementary Information, depending on whether enough data were available or not.

When data were available (Table 2), an evaluation of the total solute dispersion (total variance, ² ) was made from Eq. ( 3) by considering the most retained peak (symmetrical peaks only). The column variance, ² , was evaluated from Eq. ( 6). The extra-column variance, ² , was estimated from Eq. (1). To easily compare the percentage of remaining plates (β²), this value was systematically determined for a retention factor of 3, by column variance rescaling using Eq. ( 6). Corresponding β² values are given in Table 2. The reasons for low β² values can be strongly related to the instrument characteristics including the type of interface but also the level of matching between the column geometry and the LC instrument.

Fig. 2 illustrates the relative distribution of nebulizers and spray chambers used for speciation in LC-ICP-MS in case of organic solvent matrices regarding the dimension of the column used. In many studies, it was difficult to draw relevant conclusions because of a lack of information about SIS and/or the column particle diameter and/or even sometimes, the mobile phase flow-rate. However when the calculations were possible, the following comments can be provided depending on whether the column was conventional, narrowbore, micro-bore or capillary:

(i) conventional columns:

According to Fig. 2, three different nebulizers have been used: pneumatic micro-concentric nebulizer (50 %) operated with flow-rates in the range 40 to 1000 µL/min, pneumatic concentric nebulizer (10 %) operated with flow-rates higher than 1000 µL/min and hydraulic high-pressure nebulizer (10 %). With the first and second ones [START_REF] Balcaen | Hyphenation of reverse-phase HPLC and ICP-MS for metabolite profiling-application to a novel antituberculosis compound as a case study[END_REF][START_REF] Döker | Rapid extraction and reverse phase-liquid chromatographic separation of mercury(II) and methylmercury in fish samples with inductively coupled plasma mass spectrometric detection applying oxygen addition into plasma[END_REF][START_REF] Moreno | Simultaneous speciation and preconcentration of ultra trace concentrations of mercury and selenium species in environmental and biological samples by hollow fiber liquid phase microextraction prior to high performance liquid chromatography coupled to inductively coupled plasma mass spectrometry[END_REF][START_REF] Meermann | HPLC/ICP-MS in Combination with "Reverse" Online isotope dilution in drug metabolism studies[END_REF][START_REF] Jensen | Application of inductively coupled plasma mass spectrometry and high-performance liquid chromatography-with parallel electrospray mass spectrometry to the investigation of the disposition and metabolic fate of 2-, 3-and 4-iodobenzoic acids in the rat[END_REF], the liquid is introduced through a horizontal capillary and the gas conducted through an external tube around the liquid capillary. With the third one [START_REF] Manuel Marchante-Gayo | Comparison of different nebulisers and chromatographic techniques for the speciation of selenium in nutritional commercial supplements by hexapole collision an react cell ICP-MS[END_REF], the liquid is forced through a highly turbulent hydraulic nozzle. Similarly, three different spray chambers are used in association with the preceding nebulizers: cyclonic spray chambers (30 %), double pass spray chamber (20 %) and desolvation units (10 %). With cyclonic spray chambers, the aerosol is tangentially introduced and first impacts against the front walls generate thinner aerosol which is transported towards the injector [START_REF] Todoli | Liquid Sample Introduction in ICP Spectrometry : A Practical Guide[END_REF]. The double pass spray chambers is composed of two concentric tubes, the external one promoting droplet elimination through impacts against the wall and reducing aerosol fluctuation due to pump pulsing, the internal one eliminating coarse droplets and inducing a laminar flow aerosol [START_REF] Todoli | Liquid Sample Introduction in ICP Spectrometry : A Practical Guide[END_REF]. Finally, desolvation units are designed to reduce the solvent amount into the plasma with either a desolvation membrane to remove organic/hydro-organic matrices in association with a micro-concentric nebulizer [START_REF] Bendahl | Sample introduction systems for reversed phase LC-ICP-MS of selenium using large amounts of methanol-comparison of systems based on membrane desolvation, a spray chamber and direct injection[END_REF][START_REF] Manuel Marchante-Gayo | Comparison of different nebulisers and chromatographic techniques for the speciation of selenium in nutritional commercial supplements by hexapole collision an react cell ICP-MS[END_REF][START_REF] Møller | Quantification of pharmaceutical peptides in human plasma by LC-ICP-MS sulfur detection[END_REF] or an ultrasonic nebulizer [START_REF] Axelsson | The potential of inductively coupled plasma mass spectrometry detection for high-performance liquid chromatography combined with accurate mass measurement of organic pharmaceutical compounds[END_REF]. The main drawback of desolvation systems is the possible loss of analytes through the desolvation membrane. As reported, the sensitivity achieved with a desolvation micro-concentric nebulizer was dependent on the analyte structure [START_REF] Møller | Quantification of pharmaceutical peptides in human plasma by LC-ICP-MS sulfur detection[END_REF][START_REF] Jensen | Comparison of direct injection nebulizer and desolvating microconcentric nebulizer for analysis of chlorine-, bromine-and iodine-containing compounds by reversed phase HPLC with ICP-MS detection[END_REF].

When calculations were possible, the calculated percentage of remaining plates (β²) was found to be lower than 35 % (i.e. 24 [START_REF] Balcaen | Hyphenation of reverse-phase HPLC and ICP-MS for metabolite profiling-application to a novel antituberculosis compound as a case study[END_REF], 16 % [START_REF] Meermann | HPLC/ICP-MS in Combination with "Reverse" Online isotope dilution in drug metabolism studies[END_REF], 31 % [20], 14 % [START_REF] Jensen | Application of inductively coupled plasma mass spectrometry and high-performance liquid chromatography-with parallel electrospray mass spectrometry to the investigation of the disposition and metabolic fate of 2-, 3-and 4-iodobenzoic acids in the rat[END_REF], 34 % [START_REF] Chen | Determination of ultra-trace amount methyl-, phenyland inorganic mercury in environmental and biological samples by liquid chromatography with inductively coupled plasma mass spectrometry after cloud point extraction preconcentration[END_REF]) except in one case (i.e. 86 % [START_REF] Chen | Determination of trace mercury species by high performance liquid chromatography-inductively coupled plasma mass spectrometry after cloud point extraction[END_REF]). Unfortunately, in this latter case, the information about the interface was not available and as a result, no relevant conclusion could be drawn. As seen in Fig. 2 and mentioned earlier, in many cases, the necessary information about nebulizer (30 %) and spray chamber (40 %) was not provided. It is important to note that, with conventional columns leading to high column variance values, the measured loss in plates is likely due either to the interface or to column ageing or to both, but probably not to the contribution of LC instrument to extra-column solute dispersion. As a result, it can be concluded that, in most cases, the separation achieved in the column was partially or completely lost in the interface. As examples, with a micro-concentric nebulizer in association with a Peltier Cooled Cyclonic spray chamber PC 3 (supplied by Elemental Scientific), only 24 % of the plates remained [START_REF] Balcaen | Hyphenation of reverse-phase HPLC and ICP-MS for metabolite profiling-application to a novel antituberculosis compound as a case study[END_REF] and 14 % in association with a double pass spray chamber [START_REF] Jensen | Application of inductively coupled plasma mass spectrometry and high-performance liquid chromatography-with parallel electrospray mass spectrometry to the investigation of the disposition and metabolic fate of 2-, 3-and 4-iodobenzoic acids in the rat[END_REF]. It can be noticed that only one study deals with the use of low cyclonic spray chamber among the 55 publications reported [START_REF] Todoli | Fast determination of arsenosugars in algal extracts by narrow bore high-performance liquid chromatography-inductively coupled plasma mass spectrometrv[END_REF]. Finally, in association with a desolvation unit, the very bad peak shapes did not allow to conclude about the impact of the interface [START_REF] Manuel Marchante-Gayo | Comparison of different nebulisers and chromatographic techniques for the speciation of selenium in nutritional commercial supplements by hexapole collision an react cell ICP-MS[END_REF].

(ii) Narrow bore columns

The distribution of the nebulizers was different from that for conventional columns with four different ones used: micro-concentric nebulizers (67 %), parallel path nebulizers (12 %) with the liquid interacting with the high-velocity gas stream coming tangentially into contact with it [START_REF] Todoli | Liquid Sample Introduction in ICP Spectrometry : A Practical Guide[END_REF], cross-flow nebulizers (4 %) with liquid and gas outlets perpendicularly mounted on a polymer body and concentric nebulizers (4 %). Most of the time, micro-concentric nebulizers [START_REF] Nygren | Hydrophilic interaction liquid chromatography (HILIC) coupled to inductively coupled plasma mass spectrometry (ICPMS) utilizing a mobile phase with a low-volatile organic modifier for the determination of cisplatin, and its monohydrolyzed metabolite[END_REF][START_REF] Ackley | A comparison of nebulizers for microbore LC-ICP-MS with mobile phases containing methanol[END_REF][START_REF] Trümpler | Detoxification of mercury species-an in vitro study with antidotes in human whole blood[END_REF][START_REF] Klencsár | Determination of the total drug-related chlorine and bromine contents in human blood plasma using high performance liquid chromatography-tandem ICP-mass spectrometry (HPLC-ICP-MS/MS)[END_REF][START_REF] Gabel-Jensen | Separation and identification of the selenium-sulfur amino acid S-(methylseleno)cysteine in intestinal epithelial cell homogenates by LC-ICP-MS and LC-ESI-MS after incubation with methylseleninic acid[END_REF][START_REF] Gabel-Jensen | LC-ICP-MS and LC-ESI-(MS)n identification of Se-methylselenocysteine and selenomethionine as metabolites of methylseleninic acid in rat hepatocytes[END_REF][START_REF] Xie | Separation of Organoarsenicals by Means of Zwitterionic Hydrophilic Interaction Chromatography (ZIC®-HILIC) and Parallel ICP-MS/ESI-MS Detection[END_REF][START_REF] Rellán-Álvarez | Identification of a Tri-Iron(III), Tri-Citrate Complex in the Xylem Sap of Iron-Deficient Tomato Resupplied with Iron: New Insights into Plant Iron Long-Distance Transport[END_REF][START_REF] Vidmar | Speciation of cisplatin in environmental water samples by hydrophilic interaction liquid chromatography coupled to inductively coupled plasma mass spectrometry[END_REF][START_REF] Kannamkumarath | Determination of levothyroxine and its degradation products in pharmaceutical tablets by HPLC-UV-ICP-MS[END_REF][START_REF] Todoli | Fast determination of arsenosugars in algal extracts by narrow bore high-performance liquid chromatography-inductively coupled plasma mass spectrometrv[END_REF][START_REF] Manuel Marchante-Gayo | Comparison of different nebulisers and chromatographic techniques for the speciation of selenium in nutritional commercial supplements by hexapole collision an react cell ICP-MS[END_REF][START_REF] Vidler | The determination of methylmercury in biological samples by HPLC coupled to ICP-MS detection[END_REF][START_REF] Birka | Sensitive quantification of gadolinium-based magnetic resonance imaging contrast agents in surface waters using hydrophilic interaction liquid chromatography and inductively coupled plasma sector field mass spectrometry[END_REF][START_REF] Telgmann | Speciation of Gd-based MRI contrast agents and potential products of transmetalation with iron ions or parenteral iron supplements[END_REF][START_REF] Hemström | Alternative organic solvents for HILIC separation of cisplatin species with on-line ICP-MS detection[END_REF][START_REF] Trümpler | Interaction of thimerosal with proteins-ethylmercury adduct formation of human serum albumin and β-lactoglobulin A[END_REF][START_REF] Hogeback | Investigating the adduct formation of organic mercury species with carbonic anhydrase and hemoglobin from human red blood cell hemolysate by means of LC/ESI-TOF-MS and LC/ICP-MS[END_REF] were associated to a cyclonic spray chamber (54 % of the spray chambers used). A specific micro-concentric nebulizer (i.e. microflow PFA-ST or PFA-LC nebulizer supplied by Elemental scientific) was used in several studies [START_REF] Vidler | The determination of methylmercury in biological samples by HPLC coupled to ICP-MS detection[END_REF][START_REF] Telgmann | Speciation of Gd-based MRI contrast agents and potential products of transmetalation with iron ions or parenteral iron supplements[END_REF][START_REF] Hogeback | Investigating the adduct formation of organic mercury species with carbonic anhydrase and hemoglobin from human red blood cell hemolysate by means of LC/ESI-TOF-MS and LC/ICP-MS[END_REF]. It is made of a copolymer, with an internal capillary more recessed than other micro-concentric nebulizers [START_REF] Todoli | Liquid Sample Introduction in ICP Spectrometry : A Practical Guide[END_REF]. However, very low percentages of remaining plates were estimated in these cases. The highest value was 70 % [START_REF] Todoli | Fast determination of arsenosugars in algal extracts by narrow bore high-performance liquid chromatography-inductively coupled plasma mass spectrometrv[END_REF] while all other ones were below 20 % and even some of them below 5 % [START_REF] Xie | Separation of Organoarsenicals by Means of Zwitterionic Hydrophilic Interaction Chromatography (ZIC®-HILIC) and Parallel ICP-MS/ESI-MS Detection[END_REF][START_REF] Birka | Sensitive quantification of gadolinium-based magnetic resonance imaging contrast agents in surface waters using hydrophilic interaction liquid chromatography and inductively coupled plasma sector field mass spectrometry[END_REF]. In the particular study of Vidler et al. [START_REF] Vidler | The determination of methylmercury in biological samples by HPLC coupled to ICP-MS detection[END_REF], the calculated retention factor was found to be lower than 1 (i.e. about 0.5) leading to 16 % of remaining plates only. In this study a micro concentric nebulizer was used, unfortunately, the spray chamber was not mentioned. However, considering a retention factor of 3, the calculated fraction of remaining plates is tremendously improved (i.e. 59 %), suggesting that, in this particular case, the loss in plates was likely due to LC system volumes rather to the interface.

Interestingly, the use of two different detectors, ICP-MS and ESI-TOF [START_REF] Rellán-Álvarez | Identification of a Tri-Iron(III), Tri-Citrate Complex in the Xylem Sap of Iron-Deficient Tomato Resupplied with Iron: New Insights into Plant Iron Long-Distance Transport[END_REF] allowed us to indirectly assess the extra-column dispersion induced by the interface between LC and ICP-MS, assuming no dispersion in both ESI-TOF and ICP-MS. From our calculations, the total variance was 1360 µL² with LC-ICP-MS against 490 µL² with LC-ESI-MS, thereby leading to an impressive peak variance of 870 µL² for the interface alone, far too high considering the expecting narrow-bore column variance (i.e. <200 µL² as shown in Table 1). Unfortunately, no information about the interface was available [START_REF] Rellán-Álvarez | Identification of a Tri-Iron(III), Tri-Citrate Complex in the Xylem Sap of Iron-Deficient Tomato Resupplied with Iron: New Insights into Plant Iron Long-Distance Transport[END_REF] which could have permitted to draw some interesting trends.

(ii) Micro bore and capillary columns

When the column internal diameter is further decreased, the trend is towards the use of total consumption nebulizers (35 % and 18 % for microbore and capillary columns respectively) at the expense of micro-concentric ones (67 %, 47 % and 18 % for narrow-bore, microbore and capillary columns respectively) (Fig. 2). Total consumption nebulizers do not require spray chambers or desolvation systems and can work with low flow-rates (typically a few tens of µL/min). The primary aerosol is entirely sent to the plasma. Two different total consumption nebulizers were used with organic/hydro-organic matrices, namely Direct Injection Nebulizer (DIN, supplied by CETAC Technologies) and Direct Injection High-Efficiency Nebulizer (DIHEN, supplied by Meinhard). DIN has been used since 1992 [START_REF] Shum | Elemental speciation by liquid chromatographyinductively coupled plasma mass spectrometry with direct injection nebulization[END_REF] and was often adapted before use [START_REF] Bendahl | Sample introduction systems for reversed phase LC-ICP-MS of selenium using large amounts of methanol-comparison of systems based on membrane desolvation, a spray chamber and direct injection[END_REF][START_REF] Shum | Elemental speciation by liquid chromatographyinductively coupled plasma mass spectrometry with direct injection nebulization[END_REF]. Despite its low volume, highly recommended for hyphenated techniques, its main drawback is its small internal capillary (i.e. 60 µm) which can be easily clogged [START_REF] Leclercq | Tutorial: Introduction of organic/hydro-organic matrices in inductively coupled plasma optical emission spectrometry and mass spectrometry: A tutorial review. Part II. Practical considerations[END_REF][START_REF] Todoli | Liquid Sample Introduction in ICP Spectrometry : A Practical Guide[END_REF]. Moreover, finding the correct distance between the tip and the plasma is not easy considering the capillary thinness. For the above reasons, this nebulizer is no more commercialized [START_REF] Caumette | Sensitivity improvement in ICP MS analysis of fuels and light petroleum matrices using a microflow nebulizer and heated spray chamber sample introduction[END_REF]. DIHEN is a pneumatic nebulizer in glass or quartz, working with liquid flow-rates between 1 and 100 µL/min. It is characterized by a larger dead volume than DIN. It is therefore always used with an inserted capillary to reduce its dead volume [START_REF] Leclercq | Tutorial: Introduction of organic/hydro-organic matrices in inductively coupled plasma optical emission spectrometry and mass spectrometry: A tutorial review. Part II. Practical considerations[END_REF][START_REF] Stefánka | Down-scaling narrowbore LC-ICP-MS to capillary LC-ICP-MS: a comparative study of different introduction systems[END_REF][START_REF] Zoorob | Evaluation of the direct injection nebulizer in the coupling of high-performance liquid chromatography to inductively coupled plasma mass spectrometry[END_REF][START_REF] Acon | A direct injection high efficiency nebulizer interface for microbore high-performance liquid chromatography-inductively coupled plasma mass spectrometry[END_REF][START_REF] Wind | Modified direct injection high efficiency nebulizer with minimized dead volume for the analysis of biological samples by micro-and nano-LC-ICP-MS[END_REF][START_REF] Brennan | Nano-HPLC-inductively coupled plasma mass spectrometry for arsenic speciation[END_REF][START_REF] Bendahl | Interfacing capillary electrophoresis with inductively coupled plasma mass spectrometry by direct injection nebulization for selenium speciation[END_REF]. The optimum flow-rate for total consumption nebulizers is a few µL/min, which is also the range of flow-rates adapted to capillary LC (i.e. < 10 µL/min as seen in Table 1). Nevertheless in all studied cases (see Table 2), the estimated percentage of remaining plates was lower than 5 %.

Finally, two different SIS (a total consumption nebulizer and a micro-concentric nebulizer combined with a cyclonic spray chamber) were compared for a given capillary column while keeping the same separation conditions [START_REF] Wind | Modified direct injection high efficiency nebulizer with minimized dead volume for the analysis of biological samples by micro-and nano-LC-ICP-MS[END_REF]. In both cases, the estimated percentage of remaining plates was found to be lower than 1 %, highlighting once again the importance of extra-column band broadening, especially in case of thin columns and the necessity of trying to ensure the lowest possible contribution of the interface.

In summary, it has been pointed out that it is essential that the interface is optimized in relation to the separation conditions and especially to the column geometry. In most cases among 26 published chromatograms, the estimated percentage of remaining plates (calculated with a retention factor of 3) were found to be lower than 50 %, 20 %, 5 % and 1 % for conventional, narrow-bore, micro-bore and capillary columns respectively. These low percentage values clearly show that the interfacs are mostly inapropriate. Moreover, it is important to point out that the extra column dispersion is never considered. Further investigations were carried out in our Lab to compare commercially available SIS with respect to extra-column dispersion. These results will be extensively discussed in Part II.

Adapting the flow-rate before entering SIS can be a good option when using a total consumption nebulizer designed to work at low flow-rates to limit the amount of organic solvent entering the plasma. A zero-dead volume T-union along with suitable tubing can be used to adjust the flow-rate just prior entering SIS [START_REF] Beuvier | Développement d'une méthode de séparation chromatographique couplée aux spectrométries de masse à source d'ionisation électrospray (ESI-MS) et à[END_REF][START_REF] Jensen | Application of inductively coupled plasma mass spectrometry and high-performance liquid chromatography-with parallel electrospray mass spectrometry to the investigation of the disposition and metabolic fate of 2-, 3-and 4-iodobenzoic acids in the rat[END_REF][START_REF] Stefánka | Down-scaling narrowbore LC-ICP-MS to capillary LC-ICP-MS: a comparative study of different introduction systems[END_REF][START_REF] Zoorob | Evaluation of the direct injection nebulizer in the coupling of high-performance liquid chromatography to inductively coupled plasma mass spectrometry[END_REF][START_REF] Wind | Modified direct injection high efficiency nebulizer with minimized dead volume for the analysis of biological samples by micro-and nano-LC-ICP-MS[END_REF]. Nonetheless, such device is also expected to give rise to additional extra-column dispersion. Zoorob et al. [START_REF] Zoorob | Evaluation of the direct injection nebulizer in the coupling of high-performance liquid chromatography to inductively coupled plasma mass spectrometry[END_REF] indeed showed that a split ratio of 1:20 generated more dispersion in the interface than DIN alone. However, most authors, using a flow splitter did not consider the additional dispersion generate by such devices [START_REF] Meermann | HPLC/ICP-MS in Combination with "Reverse" Online isotope dilution in drug metabolism studies[END_REF][START_REF] Jensen | Application of inductively coupled plasma mass spectrometry and high-performance liquid chromatography-with parallel electrospray mass spectrometry to the investigation of the disposition and metabolic fate of 2-, 3-and 4-iodobenzoic acids in the rat[END_REF][START_REF] Stefánka | Down-scaling narrowbore LC-ICP-MS to capillary LC-ICP-MS: a comparative study of different introduction systems[END_REF][START_REF] Wind | Modified direct injection high efficiency nebulizer with minimized dead volume for the analysis of biological samples by micro-and nano-LC-ICP-MS[END_REF]. However, it was evaluated under fast separation conditions by injecting, without column, 20 µL of a potassium iodide solution [START_REF] Kannamkumarath | Determination of levothyroxine and its degradation products in pharmaceutical tablets by HPLC-UV-ICP-MS[END_REF]. Flow splitting was used to reduce the acetonitrile concentration and hence to improve ICP-MS performance. The authors visually compared peak shapes with and without flow splitting and deduced that the T-connection did not have a significant impact on the dispersion. However, the comparison was qualitative and not quantitative. To the best of our knowledge, no quantitative study on additional variance brought by flow splitting depending on both flow-rate and split ratio has been reported. This issue is addressed in the next section.

Determination of peak variance resulting from flow splitting

The aim of the present study was to quantify the additional peak variance due to the presence of a flow-splitter. As discussed above, total consumption spray chambers can be used with low flow-rates (up to a few tens of µL/min) to reduce the solvent load into the plasma. However, such flow-rates are much too low when using narrow bore columns (i.e.

2.1 mm i.d.) and sub 2 µm particles as usually done in UHPLC. A flow-splitter can be used to solve this problem. It consists in a zero-dead volume T-union located between the column outlet and the detector and intended to divide the main flow into two different flows, the highest being directed to the waste and the lowest to the detector, thereby significantly reducing the flow entering the plasma source. The difficulty may stem from the volumes involved by additional tubing and from the split itself as both may lead to significant additional band broadening, especially when a low split ratio has to be considered. This issue is addressed below by assessing the extra-column variance induced by such device depending on flow-rate, internal diameter of tubing and split ratio.

. Material and reagents

Methanol was HPLC grade from Sigma Aldrich (Steinheim, Germany). Water was obtained from Elga water purification system (Veolia water STI, Le Pless Robinson, France).

Methylparaben was used as test compound and supplied by Sigma-Aldrich (Steinheim, Germany).

Apparatus

The instrument used was an Acquity UPLC I-Class 2DLC liquid chromatography system. It includes two high-pressure binary solvent manager with a maximum delivery of 2 mL/min, an autosampler with a 5 µL injection sample loop, a column oven with a maximum temperature of 90°C and two different detectors (TUV and PDA) with identical flow-cells, leading to identical signal intensity. They can be used one after the other with or without flow-splitter between both. The wavelength was set at 254 nm with a sampling rate of 40 Hz.

The instrument control was performed by Mass Lynx software. The maximum backpressure allowed in the first detector cell was 1000 psi.

Procedure and methods

The mobile phase was a mixture of 50/50 Water/MeOH (v/v). Methylparaben was diluted in the same mixture with a concentration of 75 ppm. This study was carried out without column. All peak variances were calculated from the second order moment of the peak, Eq. ( 4), using in-house calculation tool.The injected volume was 1 µL. A P-727 (0.57 µL)

T-union was used for flow splitting (from Upchurch, Cluzeau, Sainte-Foy-La-Grande, France).

This T-union was chosen according to a previous study on different commercially available Tunions which concluded that this one was the best adapted to low dispersion [START_REF]Développement de méthodes bidimensionnelles en ligne LCxLC-MS pour l'analyse de composés chargés[END_REF]. The setup used to measure the peak variance due to flow splitting is shown in Fig. 3. Tube #1 was located between UV-detector #1 and the T-union, Tube #2, between the T-union and UV Detector #2 (replacing ICP-MS detection for this study) and Tube #3 between the T-union and the waste. Four different settings were considered. The corresponding tubing geometries are given in Table 3. The theoretical split ratio, z th , is calculated from the Poiseuille law, considering the dimensions of Tube #2 and #3, Eq. ( 13):

Z [ = + \ + \ 4+ with a = ; S S b (13)
L i and d i being the length and the internal diameter of Tube #"i" respectively.

The measured split ratio, z meas , was obtained from the measured flow-rate, F2, in Tube #2 (z meas =F 2 /F 1 ). In the absence of flow splitting (Setting #A), the split ratio was 1. The difference between calculated and measured values in Table 3 may be imputed to internal diameter irregularity.

The peak variance due to flow splitting (σ² split ) was assessed by subtracting the sum of the peak variance measured with Detector #1 (σ² D1 ) and that due to Tube #1 (σ² tubing 1 ) from the peak variance measured with Detector #2 (σ² D2 ), Eq. ( 14).

σ² split = σ² D2 -σ² D1 -σ² Tub #1 [START_REF] Frindt | Multidimensional monitoring of anaerobic/aerobic azo dye based wastewater treatments by hyphenated UPLC-ICP-MS/ESI-Q-TOF-MS techniques[END_REF] σ² tubing 1 was measured in the absence of flow splitting with Detector #1 and 2 in series (σ² Tube #1 = σ² D2 -σ² D1 ). It should be noted that σ² D2 corresponds to the total extra-column variance (σ² ext in Eq. ( 2)).

Results and discussion

Throughout this study, five different settings were studied (Table 3), which differentiated themselves from their measured split ratio and/or their tubing geometry. For the five settings, the dimensions of Tube #1 were the same and hence the variation of σ² Tube #1 with the total flow-rate (F 1 Similarly, the variation of σ² split with F 1 and hence with F 2 was studied in the presence of the T-union (Fig. 5). Two different situations were considered to assess to what extent the peak variance may vary: a decrease in the internal diameter of Tube #2 while keeping the split ratio nearly constant (Setting B and C (Fig. 5a)) and a strong decrease in the split ratio while keeping the internal diameter of Tube #2 constant (Setting C, D and E (Fig. 5b)).

The impact of the internal diameter of Tube #2 was investigated with two different internal diameters, 65 µm (setting B) and 25 µm (setting C). As shown in Fig. 5a, the curves, representing the variation of the peak variance with F 2, are very similar in both cases with only 2 µL² difference between both curves, indicating that the internal diameter of Tube #2 has little or no impact on the dispersion generated by the split. On the other hand, the impact of the split ratio was found to be very significant as highlighted in Fig. 5b. Considering the same resulting F 2 value (e.g. 8 µL/min), the measured peak variance increased from 6 to 34 µL² with a split ratio decreasing by a factor of about 2 (1:4.8 to 1:9.4), and up to 200 µL² with a split ratio decreasing by a factor of 10. The impact of the flow splitter on the dispersion can also be visually assessed by comparing the peaks obtained with the different settings. The x-axis is expressed in volume units to have a better comparison of peak broadening. For a given mobile phase flow-rate of 0.1 mL/min, Fig. 6a shows the degradation of the peak shape when the split ratio decreased from 1:1 (no flow splitting) to 1:50 (only 2 µL/min sent to UV-Detector 2). In this latter case, the variance due to the split reached 77µL². When considering a low F 2 value of 10 µL/min (Fig. 6b), the degradation is even more important with 208 µL² obtained when the split ratio is decreased down to 1:50. In addition to peak broadening, Fig. 6 clearly points out the problem of sensitivity that can be encountered if the interface and in particular the split device are not carefully chosen.

According to Fig. 1a, a total extra-column variance of only 100 µL² is sufficient to lose more than 70 % of the column plates in UHPLC conditions. Consequently, these results clearly underline the importance of finding the best trade-off between the split ratio which must be high enough, the flow entering ICP-MS which must be low enough and the column flow-rate which must be adapted to UHPLC conditions (typically in the range 400 to 1000 µL/min). In any case, the additional extra-column dispersion brought by flow splitting should be carefully evaluated. Micro-concentric nebulizers can be considered as a good option for the coupling of UHPLC with ICP-MS since they avoid the need for low split ratios. Such nebulizers usually work with flow-rates of around hundreds µL/min as those required with narrow bore columns. On the other hand, the use of a flow splitter in association with a total consumption spray chamber operated at around 10 µL/min, seems inappropriate for UHPLC purposes.

Conclusion and future trends

Within the last number of decades, analytical techniques have evolved from the simple determination of the total amount of a metal element to the determination of its chemical species. In speciation studies, ICP-MS is now routinely coupled with liquid chromatography (LC-ICP-MS).

Most applications are performed in an aqueous solvent and are therefore much easier to implement than those needing the introduction of a large amount of organic solvent into the plasma source. Unfortunately, very few analysts pay enough attention to the chromatographic aspects when coupling LC and ICP-MS. During method development, special attention should be focused on extra-column dispersion to minimize it, avoiding excessive dead volume. In view of this article, it clearly appears that efforts need to be made in this specific area before analyzing complex matrix. Indeed, the benefit of method development performed to enhance the analytical performance (resolution and sensitivity) could be lost if solute dispersion in the interface, located between separation and detection devices, is not minimized. A special attention must be paid to a better description of Sample Introduction System since it seems to be the critical part of the coupling and its characteristics are not always specified by the authors. For speciation analysis, it is higlhy recommended to thoroughly characterize the interface by evaluating extra column dispersion induced by SIS and ensuring that less than 20 % plates are loss. This will be extensively discussed in Part II.

Flow splitting can be used with total consumption spray chambers to reduce the flow-rate prior to the sample introduction system but it must be sized properly. As shown in this study, a low split ratio can significantly increase solute dispersion and hence solute dilution. Microconcentric nebulizers are expected to be more appropriate with possible flow-rates of around hundreds µL/min, thus without the need of flow splitting. Such nebulizers will be compared in the second part of this study which will be devoted to the comparison of commercially available Sample Introduction Systems in term of extra-column dispersion.

A last interesting point, in our view, is the impact of extra-column peak broadening on the whole analytical performance when using more complex hyphenated techniques such as 2D-LC-ICP-MS. In the future, if online comprehensive two-dimensional separations are more extensively developed, a particular attention should be paid to the set-up so that every additional dead volume could be assessed in terms of additional dispersion. 6) and [START_REF] Boulyga | Application of SEC-ICP-MS for comparative analyses of metal-containing species in cancerous and healthy human thyroid samples[END_REF] with k e =3; h=3 and ε t =0.7. Detector 2, and F 3 is the flow-rate sent to the waste. Tubing geometry is given in Table 3.

Figure captions

Figure 4: Variation of the variance due to the tube 1 (σ² Tube 1 ) as a function of F 1 . See Table 3 for tubing geometry and

Figure 1 :

 1 Figure 1: Percentage of remaining plates (β²) as a function of extra-column variance for different column geometries providing (a) 10000 and (b) 30000 plates. Calculations from Eqs. (6) and[START_REF] Boulyga | Application of SEC-ICP-MS for comparative analyses of metal-containing species in cancerous and healthy human thyroid samples[END_REF] with k e =3; h=3 and ε t =0.7.

Figure 2 :Figure 3 :

 23 Figure 2: Relative distribution of nebulizers and spray chambers depending on the column inner diameter. Based on 55 reported studies dealing with LC-ICP-MS for organic matrices (from 1995 to 2017)

Fig. 3

 3 for instrument set-up. Solute: methylparaben (75 ppm in 50/50 Water/MeOH v/v). 1 µL injected. No column. Mobile phase: 50/50 water/MeOH v/v. 254 nm.

Figure 5 :

 5 Figure 5: Variation of the variance due to the split (σ² split ) with F 2 . (a) effect of the tube i.d.

Figure 6 :

 6 Figure 6 : Overlay of the peaks obtained in Detector 2 with settings A, B, C, D and E. (a) Same F 1 (100 µL/min) resulting in different F 2 (100, 20.8, 20.8, 10.6, 2 µL/min respectively) and (b) different F 1 (10, 50, 50, 100, 400 µL/min respectively) resulting in same F 2 (10 µL/min). Measured extra-column variance (σ²ext ) is indicated at peak apex. Same other conditions as in Fig.4.

  ). F 1 was varied in the range 50-700 µL/min, 20-150 µL/min, 50-100 µL/min and 100-400 µL/min with settings B, C, D and E respectively. The experimental variation of σ² Tube 1 with F 1 is given in Fig.4. It could be fitted with an exponential function. As highlighted in Fig.4, the variance increases linearly with the flow-rate up to a value of nearly 200 µL/min and is almost unchanged beyond (i.e. around 8 µL²). This is in good agreement with reported studies, showing that peak variance values deviate from linearity at high flowrates with non ideal tubing (i.e. short, coiled and/or rugged tubing)[73].

Table 1 .

 1 Optimum flow-rates (F opt ), column plate number (N col ), column variance (σ² col ) depending on column internal diameter (d i ) and particle diameter (d p ). Calculations performed with = 3; = 10 -9 m²/s; = 150 mm; ν = 5; ℎ = 3; εt = 0.7, using Eqs. (5) to[START_REF] Beuvier | Développement d'une méthode de séparation chromatographique couplée aux spectrométries de masse à source d'ionisation électrospray (ESI-MS) et à[END_REF].

			= 5 µm		= 1.7 µm
			= 10000	= 30000
		(mm)	(µL/min)	σ² col (µL²)	(µL/min)	σ ² col (µL²)
	Conventional	4.6	698	4867	2052	1655
	Narrow bore	2.1	145	211	428	72
	Micro bore	1	33	11	97	4
	Capillary	0.3	3	0.088	9	0.03
	Nano bore	0.075	0.2	0.00034	0.5	0.00012

Table 2. Reported LC-ICP-MS studies on organic matrices. Evaluation of the percentage of remaining plates (β²).

Calculations with a retention factor of 3, from Eqs. [START_REF] Templeton | Guidelines for terms related to chemical speciations and fractionation of elements : definitions, structural aspects, and methodological approaches[END_REF] to [START_REF] Beuvier | Développement d'une méthode de séparation chromatographique couplée aux spectrométries de masse à source d'ionisation électrospray (ESI-MS) et à[END_REF], with S = 4 (small molecules) and S = 50 (large molecules).


B Flipon

L Milhem
email: luc.milhem@insa-rouen.fr

C Keller

R Quey

F Barbe

MODELLING OF POLYCRYSTALS USING WELL-CONTROLLED VORONOI-TYPE TESSELLATIONS AND ITS APPLICATIONS TO MICROMECHANICAL ANALYSES

Keywords: Polycrystals, Voronoi tessellation, Laguerre tessellation, Johnson-Mehl tessellation, Micromechanics

This paper provides a review of our developments on the use of Voronoi-type tessellations for the analysis of the deformation of polycrystalline structures. This work was initiated 20 years ago by a fruitful collaboration with Dominique Jeulin and is still guided by his works. We introduce different types of tessellations and explain how they can be used by presenting several micromechanics cases of study, from the most simple to the most complex configurations.

Introduction

The mechanisms governing the elasto-plastic deformation, damage and fracture of metallic alloys are intimately related to their polycrystalline microstructures. This has motivated since the 90s the development of full-field modelling approaches to predict their behaviors. In the simulations, polycrystals are usually represented as Voronoi tessellations generated from a uniform random distribution of seed points, so-called "Poisson-Voronoi tessellations" [START_REF] Barbe | Intergranular and intragranular behavior of polycrystalline aggregates. Part 1: F.E. model[END_REF]. However, Poisson-Voronoi tessellations are, at best, only partly representative of experimental microstructures. This is especially the case for microstructures optimized to enhance their mechanical properties, which may contain columnar or elongated grains (arising from solidification or forming operations, respectively), grain-size gradients, wide or even bimodal grain size distributions, etc. In this paper, we present the series of developments on the modelling and micromechanics of complex experimental microstructures that we have carried out since our years spent with Dominique Jeulin, and that have been implemented in the specialized free (open-source) Neper software package [START_REF] Quey | Large-scale 3-D random polycrystals for the finite element method: Generation, meshing and remeshing[END_REF][START_REF] Quey | Optimal polyhedral description of 3D polycrystals: method and application to statistical and synchotron X-ray diffraction data[END_REF]. First, we show how particular choices in the spatial distribution of the seed points of a Voronoi tessellation can lead to well-controlled size-gradient structures of mixed equiaxed/columnar structures. Then, we introduce Laguerre tessellations (a generalization of the Voronoi algorithm to weighted tessellations) and demonstrate how they can be used to represent experimental microstructures described by multiple metrics, using optimization. Finally, we present Johnson-Mehl tessellations (a generalization of Voronoi tessellations to time-dependent nucleation and growth rates) and show how they can be used in the context of diffusive transformations of polycrystals. These developments are illustrated by specific cases of study on the micromechanics of polycrystals, which have been further addressed with finite element analyses. These developments not only lead to high-end tools for the modelling of existing materials but are also meant to provide a complementary, "numerical" strategy in material engineering approaches related to new manufacturing capabilities.

Voronoi tessellations 2.1 Definition

Let us define D as a domain of space and G i as a set of points distributed in D, defined by their coordinates. A Voronoi cell C i is associated to each seed point (G i ) as follows:

C i = {P (x) ∈ D | d(P, G i ) < d(P, G j ) ∀j = i} , (1)
where d(A, B) corresponds to the Euclidean distance between A and B. Figure 1, which is taken from the seminal work of Dominique Jeulin and applied in Ref. [START_REF] Barbe | Intergranular and intragranular behavior of polycrystalline aggregates. Part 1: F.E. model[END_REF], illustrates the construction of a Voronoi tessellation in 2D. Typically, the seed points are distributed randomly according to a uniform distribution, which leads to so-called "Poisson-Voronoi tessellations". Poisson-Voronoi tessellations are composed of convex cells of random sizes and shapes, and so are good first-order approximations of real polycrystals and are relatively easy to mesh into finite elements [START_REF] Quey | Large-scale 3-D random polycrystals for the finite element method: Generation, meshing and remeshing[END_REF]. This is why they have become a standard in polycrystal computations of elastoplastic deformation [START_REF] Barbe | Intergranular and intragranular behavior of polycrystalline aggregates. Part 2: results[END_REF][START_REF] Zeghadi | Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure -Part 2: crystal plasticity[END_REF], fatigue [START_REF] Guerchais | Micromechanical study of the loading path effect in high cycle fatigue[END_REF], intergranular fracture [START_REF] Benedetti | A three-dimensional cohesive-frictional grain-boundary micromechanical model for intergranular degradation and failure in polycrystalline materials[END_REF], cleavage fracture [START_REF] N'guyen | Micromechanical local approach to brittle failure in bainite high resolution polycrystals: A short presentation[END_REF], etc. A review of such applications to the full-field modelling of polycrystals is provided in [START_REF] Benedetti | Modelling Polycrystalline Materials: An Overview of Three-Dimensional Grain-Scale Mechanical Models[END_REF], and an extensive list of the works carried out with Neper (most of which using Voronoi tessellations) can be found on the Neper website [START_REF] Quey | Neper: polycrystal generation and meshing (version 3.1)[END_REF] (125 papers as of March 2018).

Application to size-gradient structures

Introducting a grain size gradient in a polycrystalline material, with smaller grains near the surface, can improve its resistance to fatigue [START_REF] Shao | Improvement of low-cycle fatigue resistance in TWIP steel by regulating the grain size and distribution[END_REF]. Such a microstructure can be represented by a Voronoi tessellation relatively simply: as the size of a cell is directly related to the distance between its seed and the surrounding seeds, a gradient of volume density of the seeds (in the domain) results in a gradient of size of the corresponding cells. In practice, the seeds can first be distributed on a regular array, where the distance between seeds is determined from the expected density at that location of the domain, and then slightly shifted about their nominal positions to retrieve random morphologies. The advantage of this method is twofold, as it controls the global size gradient as well as the minimum and maximum cell sizes. However, these properties are quite sensitive to the parameters of the seed distribution, so that they must be defined with great care. This is illustrated on Figure 2, where the complex interdependencies between the setting of the nominal regular array of seeds and the prescribed range of the perturbation distance lead to 2D microstructures with grain shapes and size gradient that are not representative of real materials. To better understand (and monitor) these interdependencies, a procedure based on an objective and reproducible construction of the inputs has been developed to explore the types of microstructures that can be obtained from a unidirectional gradient in a 2D domain: all the inputs have been defined in terms of a limited number of parameterized equations and the parameters space has been fully explored automatically. This enabled us to identify suitable values for the spacings between seeds and the perturbations distances that could then be used for the 3D tessellations. Figure 3 presents two examples of such microstructures, with low and high gradients, respectively.

Application to mixed equiaxed/columnar structures

Columnar grains can be obtained by directional solidification and are desirable in some applications, e.g. in turbine blades, to avoid the existence of grain boundaries perpendicular to the radial direction, which is subjected to high centrifugal forces. During elaboration, columnar grains usually appear near the outer surface of die-cast parts, as a result of very fast solidification and high temperature gradients. In this case, columnar grains generally coexist with equiaxed grains, which are located at the center, as illustrated on Fig. 4a. Equiaxed grains have random (crystallographic) orientations whereas columnar grains have orientations characterized by a crystallographic direction (typically 1 0 0 for cubic metals) parallel to the solidification direction, giving rise to an intense fibre texture. The deformation of such microstructures can only be predicted by full-field modelling, as they show grain morphologies and crystallographic orientation distributions distinctly different of those of "conventional" microstructures. This type of structures can be represented by Voronoi tessellations, by distributing the seeds as illustrated on Fig. 4. As for their deformation, we showed in Ref. [START_REF] Flipon | Influence of bimodal microstructure morphologies on local aand macroscopic mechanical responses[END_REF] that the crystallographic texture of the columnar grains drives their own plastic deformation but also the stress distribution of the equiaxed grains (Fig. 5).

3 Laguerre tessellations

Definition

Similarly to Voronoi tessellations, Laguerre tessellations are generated from seed points (G i ) of given coordinates, but these seed points are attributed weights, w i . A laguerre cell is associated to each seed point as follows:

C i = P (x) ∈ D | d(P, G i ) 2 -w i < d(P, G j ) 2 -w j ∀j = i (2)
The higher the weight of a seed, the larger its associated cell. A Voronoi tessellation is obtained if all weights are equal. Figure 6 illustrates the construction of a Laguerre tessellation in 2D. Laguerre tessellations are more general than Voronoi tessellations, as they can reproduce any tessellation composed of convex cells [START_REF] Lautensack | Random Laguerre tessellations[END_REF]. However, not only the seed positions but also the seed weights need to be properly controlled, which is not trivial. Still, it can be clearly seen from Fig. 6 that Laguerre tessellations can include wider grain size distributions than Voronoi tessellations. Most of the applications of Laguerre tessellations concern the modelling of microstructures of polycrystals or foams [START_REF] Morfa | Virtual modeling of polycrystalline structures of materials using particle packing algorithms and Laguerre cells[END_REF][START_REF] Lautensack | 3D image analysis of open foams using random tessellations[END_REF], while their applications to mechanical or physical analyses are much more rare.

Application to bimodal grain size distributions

Grain size refinement is known to improve the yield stress of metallic alloys but also to degrade their ductility, which can be detrimental for material forming. Several approaches have been developed to evade this strengthductility dilemna. One of the most promising approach consists in architecturing the microstructure in phases of different properties, which has lead, in particular, to the concept of bimodal polycrystals, for which ultrafine grains coexist with coarse grains. Such polycrystals have been elaborated by means of powder metallurgy and spark plasma sintering and, as shown in Fig. 7, exhibit properties significantly different of those of "conventional" materials [START_REF] Flipon | Elaboration of austenitic stainless steel with bimodal grain size distribution and investigation of their mechanical behavior[END_REF][START_REF] Flipon | Influence of bimodal microstructure morphologies on local aand macroscopic mechanical responses[END_REF]. These results then rise questions on the optimization of such bimodal materials: what should be the volume fractions of the two grain populations? and and how should these populations be spatially distributed? Full-field modelling provides efficient means to address these questions. Preliminary analyses have dealt with the representativeness of the considered volume element as regards its effective properties. It has been shown that a single coarse grain embedded in a matrix of ultrafine grains (Fig . 8a-b) is not convenient, and that the grain size distribution alone is not sufficient to describe the microstructure: the spatial arrangement of the coarse grains can play a significant role, as shown on Fig. 8c and9.

Application to specified grain size and sphericity distributions

When several grain properties must be controlled, it becomes extremely difficult to determine the seeds positions and weights of a Laguerre tessellation a priori, as was done previously. We then employ an optimization approach [START_REF] Quey | Optimal polyhedral description of 3D polycrystals: method and application to statistical and synchotron X-ray diffraction data[END_REF], for which the variables are the seed positions and weights and the objective function (O, to minimize) quantifies the difference between the target and current cell properties,

O = 1 n n s=1 A s 2 , ( 3
)
where each of the A s concern a particular property. As mentioned before, the properties can describe the morphology and arrangement of the grains, i.e. their sizes, shapes, positions, etc. Experimentally, grain properties such as their sizes or shapes are generally described by their distributions. To define A, it is then needed to compare the set of (discrete) cell values to a (continuous) target distribution. This is a well-known problem in experimental physics (or medecine), for which different metrics (or "tests") have been proposed from 1930-1970. We employ a test of the form

A s = +∞ -∞ [X s (x) -F s (x)] 2 F s (x) [1 -F s (x)] dx with F s = F • s X s = X • s (4)
where F represents the target cumulative distribution function, X represents the current cumulative distribution function, and s is a Gaussian smoothing kernel (used to facilitate optimization convergence). A s decreases as X s converges to F s and is equal to 0 when X s = F s . Figure 10 provides tessellations generated from different size and sphericity distributions. As microstructures with different properties can be generated automatically, it becomes easy to analyse microstructure-property relationships. Recent applications concern the simulation of thermal transport in nanocrystalline MoS 2 sheets, which is important in energy storage and microelectronics applications [START_REF] Sledzinska | Thermal conductivity of MoS2 polycrystalline nanomembranes[END_REF][START_REF] Sledzinska | Record Low Thermal Conductivity of Polycrystalline MoS2 films: Tuning the Thermal Conductivity by Grain Orientation[END_REF]. The aim is to reduce the the thermal diffusion of the material by counterbalancing the very fast diffusion in the crystalline structure by the high density of grain boundaries (of slower diffusion) of a nanocrystalline structure. The macroscopic thermal conductivity therefore depends on the average grain size, but it may also depend on the grain size distribution. This is illustrated on Fig. 11 in the case of nanocrystalline graphene. Note how wider grain size distributions result in higher conductivities due to the presence of preferential diffusion paths.

4 Johnson-Mehl tessellations

Definition

The Johnson-Mehl tessellation is obtained by combining nucleation (through a sequential intensity Ṅ (t)) and growth (with growth rate α(t)) [START_REF] Jeulin | Random tessellations and boolean random functions[END_REF]. Considering the sequence of seed points G i (x i , t i ), a cell C i is associated to each seed point (G i ) as follows:

C i = P (x) ∈ D | d(P, G i (x i , t i )) -α(t i ) t i < d(P, G j (x j , t j )) -α(t j ) t j ∀j = i , (5)
The nucleation ( Ṅ (t)) and growth (α(t)) laws can be arbitrary, so that Johnson-Mehl tessellations are more general than Voronoi tessellations and can include non-convex cells. Figure 12 (v f = 1) provides an example of a Johnson-Mehl tessellation. A Voronoi tessellation is obtained if all seed points nucleate at the same time and growth is isotropic. When a Johnson-Mehl tessellation is constructed in the context of phase transformation, the nucleation ( Ṅ (t)) and growth (α(t)) laws can be based on physical principles. Moreover, the transformation kinetics can then be described by the theory of Kolmogorov (1938), Johnson-Mehl (1939) and Avrami (1939), which is well-known for providing a simple and general law to describe the kinetics of formation of a product phase during a process of nucleation and growth. It takes the form v f (t) = 1 -exp(kt n ) where v f is the product phase volume fraction and (k, n) are parameters. This form generalizes the particular configuration of an isotropic growth at a constant velocity α which had been considered by Johnson and Mehl: with this condition and in two particular cases of nucleation, the variation dv f within a time interval dt can be integrated and the following evolution laws are obtained: • for a simultaneous nucleation with a density N 0 : v f (t) = 1 -exp -4π 3 α 3 N 0 t 3 ;

• for a constant nucleation rate Ṅ :

v f (t) = 1 -exp -π 3 α 3 Ṅ t 4 .
In the following, we will only consider the case of a constant growth rate.

Application to solid-solid phase transformations

The formation of Johnson-Mehl tessellations has been implemented in Neper and applied in Ref. [START_REF] Barbe | A numerical modelling of 3D polycrystal-to-polycrystal diffusive phase transformations involving crystal plasticity[END_REF] to describe the microstructure evolution of a polycrystal during a diffusive transformation. The general objective of the modelling is to account for the local elastoplastic interactions between grains of either the parent or the product phase to predict the residual stresses and the effective plasticity that are induced by a diffusive solid-solid phase transformation in a steel. The parent microstructure is a Voronoi tessellation which represents the austenitic (or ferritic) microstructure, and from which favourable nucleation sites of the product phase can be defined: grain corners, edges and surfaces, in decreasing order of likelihood. In Fig. 12, the nucleation sites are restricted to grain corners, and a constant growth velocity is considered. It can be seen how the resulting microstructure differs from a Voronoi tessellation, as grains are non convex. Fig. 13 illustrates the elastoplastic interactions that arise between grains during phase transformation, without any external loading (each grain has a crystal plasticity constitutive behavior corresponding to its phase). It appears that residual stresses and microplasticity are generated locally, which eventually influences the effective properties of the parent phase.

Voronoi;

v f = 0 v f = 0.06 v f = 0.45 v f = 1; Voronoi Voronoi; v f = 0 v f = 0.06 v f = 0.45 v f = 1; Johnson-Mehl

Conclusion

In this paper, we provided an overview of the work we carried out as a continuation of our years spent with Dominique Jeulin. Dominique's playground, morphological mathematics, as well as his own contributions, have of course been very important. We have developed several new strategies for representing random polycrystals by geometrical models and used the models in various applications. This approach is based on two main motivations:

• to reproduce existing materials with information on the microstructure which is as close to reality as possible;

• to generate polycrystals in a process of material engineering, i.e. to guide the development of architectured materials and their elaboration by new processing technologies.

In both cases, the grain size distribution is an important property, but we have shown that it cannot be considered alone to accurately describe polycrystalline microstructures. Indeed, it is also necessary to control (at least) the grain shape [START_REF] Quey | Optimal polyhedral description of 3D polycrystals: method and application to statistical and synchotron X-ray diffraction data[END_REF]. Laguerre tessellations offer full control on these properties, and we have presented an optimization approach that exploits them to their full potential. Jonhson-Mehl tessellations offer a more general formulation and are attractive in the context of phase transformations. >From a general point of view, on top of statistics, 2-point statistics are of course needed to characterize and control properties such as the percolation of large grains: in a polycrystal containing two distinctly different grain size populations, then, as shown in [START_REF] Flipon | Influence of bimodal microstructure morphologies on local aand macroscopic mechanical responses[END_REF], the spatial distribution of the coarse grains (whether they are isolated or clustured for example) can have a significant effect on the effective behaviour. Once again, Dominique's contributions to the control of phase distributions in a morphological model [START_REF] Kanit | Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry[END_REF][START_REF] Jean | Large scale computations of effective elastic properties of rubber with carbon black fillers[END_REF] or to the estimation of tortuosity in 3D [START_REF] Peyraga | Estimation of tortuosity and reconstruction of geodesic paths in 3D[END_REF] will be our sources of knowledge and knowhow.

Figure 1 :

 1 Figure 1: Construction of a Voronoi tessellation on a periodic grid [1]. (a) Distance function of a single seed point, (b) distance function of a set of seed points and (c) final result after cell construction and labelling.

Figure 2 :Figure 3 :

 23 Figure 2: Examples of some remarkable 2D size-gradient tessellations extracted from the exploration of the parameters space used to define the inputs of the tessellation.

Figure 4 :Figure 5 :Figure 6 :

 456 Figure 4: Mixed equiaxed/columnar structure: (a) Orientation map (inverse pole figure) of the section of a die-cast cylinder including both equiaxed grains and columnar grains. (b) Distribution of seed points, where the black points generate equiaxed cells and the red points generate columnar cells, seen as top and perspective views, (c) resulting Voronoi tessellation and (d) orientation distributions of the (left) columnar and (right) equiaxed grains, shown as 1 0 0 pole figures.

Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7: EBSD maps of bimodal 316L steel polycrystals elaborated by spark plasma sintering: (a) 75% 15µm-coarse-grains, 25% 0.5µm-fine-grains, (b) 25% 15µm-coarse-grains, 75% 0.5µm-fine-grains, (c) tensile responses of unimodal and bimodal 316L steel specimens elaborated by spark plasma sintering. After [6].

σFigure 10 :

 10 Figure 10: Tessellations generated from grain size and sphericity distributions. The normalized grain size distribution is a normal distribution of average 1 and standard deviation σ d . The sphericity distribution is a normal distribution of average s and standard deviation 0.03. (a) 2D tessellations of different values of σ d and s (100 cells). (a) 3D tessellations corresponding to a grain-growth polycrystal: σ d = 0.35, s = 0.855 (1000 cells).

Figure 11 :

 11 Figure 11: Thermal transport in nanocrystalline graphene sheets with the same average grain size (10 µm) but different grain size distributions.

Figure 12 :

 12 Figure 12: Two polycrystal-to-polycrystal microstructure evolutions corresponding to diffusive transformations at a constant growth velocity starting from the same Voronoi polycrystal. (Top) The nucleation is simultaneous with a given density N 0 and Voronoi polyhedra are obtained. (Bottom) Nucleation takes place at a constant rate Ṅ through time and non convex polyhedra are obtained. Each example is a particular case of a Johnson-Mehl tessellation. After [3].

Figure 13 :

 13 Figure 13: Stress-free transformation of a 85-grain austenite polycrystal to a 177-grain ferrite polycrystal. Volume fraction of ferrite is z = 0.67. (a) Grains of ferrite (parent austenitic grains are not represented). (b) Von Mises equivalent stress field on the contour of ferrite grains. (c) Equivalent total strain field on the contour of ferrite grains. A quarter of the polycrystal has been removed for visualisation sake. After [3].

  International Workshop on the Physics and Mechanics of Random Structures: from Morphology to Material Properties (June 17-22, 2018)