Massé Anthony
email: anthony.masse@univ-nantes.fr

Arab Oumnia

Séchet Véronique

Jaouen Pascal

Pontié Maxime

Sabiri Nour-Eddine

Plantier Séverine

Anthony Massé

Performances of dead-end ultrafiltration of seawater: from the filtration and backwash efficiencies to the membrane fouling mechanisms

Keywords: Seawater, Ultrafiltration, Fouling membrane, Alginic acids, Humic acids

The present work investigates the fouling mechanisms of PVDF hollow fibre membrane (0.03 μm) during the dead end ultrafiltration at a fixed permeate flux (outside to inside configuration) of complex synthetic seawater composed by humic acids, alginic acids, inorganic particles and numerous salts at high concentrations. Short term ultrafiltration experiments at 100 L.h-1.m-2 show that the optimal specific filtered volume seems to be equal to 50 L.m-2. A residual fouling resistance equal to 2.1010 m-1 is added after each cycle of filtration during 8h of ultrafiltration at 100 L.h-1.m-2 and 50 L.m-2. Most of the fouling is reversible (80%). Organics are barely (15% of humic acids) retained by the membrane. Backwash efficiency drops during operation which induces less organics into backwash waters. Humic acids could preferentially accumulate on the membrane early in the ultrafiltration and alginic acids after the build-up of a fouling pre-layer. Colloids and particulates could accumulate inside a heterogeneous fouling layer and/or the concentrate compartment of the membrane module before being more largely recovered inside backwash waters.

Introduction

Ultra-and Microfiltration membranes are widely used for seawater treatment instead of or after granular filters located at aquaculture or desalination plants [START_REF] Maurel | Dessalement de l'eau de mer et des eaux saumâtres et autres procédés non conventionnels d'approvisionnement en eau douce[END_REF][START_REF] Massé | Dead-end and tangential ultrafiltration of natural salted water : influence of operating parameters on performance and energy consumption[END_REF][START_REF] Sabiri | Performance of a sand filter in removal of micro-algae from seawater in aquaculture production systems[END_REF][START_REF] Plantier | Performance of a sand filter in removal of algal bloom for SWRO pre-treatment[END_REF][START_REF] Guilbaud | Seawater pretreatment by dead-end micro and ultrafiltration in pressure-driven inside feed[END_REF][START_REF] Haure | Molluscan shellfish safety, chapter 3: Storage and Detoxification of Bivalve Molluscs as a Tool in a Marketing Strategy[END_REF]. Porous membrane units are often more compact than conventional sand filtration ones [START_REF] Van Hoof | Dead-end ultrafiltration as alternative pretreatment to reverse osmosis in seawater desalination: a case study[END_REF][START_REF] Xu | Pilot study of inside-out and outside in hollow fiber UF modules as direct pretreatment of seawater at low temperature for reverse osmosis[END_REF] and allow to limit the (bio)-fouling of downstream units such as Reverse Osmosis (RO) membranes, and therefore the volume of chemicals used [START_REF] Knops | Economic evaluation of a new ultrafiltration membrane for pretreatment of seawater reverse osmosis[END_REF]. Moreover, Ultra-and Microfiltration enable a constant and good quality of the produced water [START_REF] Xu | Pilot study of inside-out and outside in hollow fiber UF modules as direct pretreatment of seawater at low temperature for reverse osmosis[END_REF][START_REF] Zhang | Pilot testing of outside-in UF pretreatment prior to RO for high turbidity seawater desalination[END_REF][START_REF] Xu | Performance of a ceramic ultrafiltration membrane system in pretreatment to seawater desalination[END_REF]. Pretreatment by Ultrafiltration can represent 10-40% of the total energy consumption of RO desalination plants [START_REF] Knops | Economic evaluation of a new ultrafiltration membrane for pretreatment of seawater reverse osmosis[END_REF]. The total cost of a RO plant can be 2-7% lower in the case of an UF pretreatment comparatively to a sand filter one [START_REF] Knops | Economic evaluation of a new ultrafiltration membrane for pretreatment of seawater reverse osmosis[END_REF]. Nevertheless, the mitigation of UF membrane fouling remains a key issue for sustainable operation of desalination plants.

Two modes of dead-end membrane ultrafiltration are widely used for the pretreatment of seawater: filtration under pressure from the lumen of fibers to the exterior or in the opposite direction and filtration by suction with submerged membranes [START_REF] Knops | Economic evaluation of a new ultrafiltration membrane for pretreatment of seawater reverse osmosis[END_REF]. Polyvinylidene fluoride, polysulfone or polyethersulfone membranes are often used [START_REF] Pearce | Introduction to membranes: Manufacturers' comparison: Part 3, Filtration + Separation[END_REF][START_REF] Xu | Ultrafiltration as pretreatment of seawater desalination: critical flux, rejection and resistance analysis[END_REF]. Permeate and backwash fluxes range from 20 to 150 L.h -1 .m -2 and from 60 to 350 L.h -1 .m -2 respectively and are usually encountered for seawater pretreatment by porous membranes [START_REF] Knops | UF pretreatment for cold seawater RO[END_REF][START_REF] Voutchkov | Seawater pretreatment, Bangkok, Edition: Water treatment academy[END_REF][START_REF] Gasia-Bruch | Field experinence with a 20,000 m 3 /d integrated membrane seawater desalination plant in Cyprus. Presented at EuroMed 2010 -Desalination for Clean Water and Energy: Cooperation among Mediterranean Countries of Europe and MENA Region[END_REF]. Backwash and filtration cycles can last from a few seconds and up to several minutes or hours respectively [START_REF] Heijman | Heterogeneous fouling in dead-end ultrafiltration[END_REF][START_REF] Jezowska | Ultrafiltration as direct pre-treatment of seawater: a case study[END_REF]. Therefore, the present work deals with the fouling mechanisms of membranes during seawater ultrafiltration in dead-end mode (from the outside to the inside of the PVDF hollow fiber membrane). The mass balances were carried out by considering the backwash waters and permeates. Thus, it is possible to identify at any time of the filtration, the compounds that participate to the membrane fouling. Hence, the changes of fouling behavior could be monitored.

Characteristics of natural seawater vary according to season, weather, tide, location, etc.

Seawater is mostly composed of water (96%). The remaining 4% are dissolved and particulate organics or minerals such as polysaccharides, proteins, humics, phytoplankton, salts, etc. On average, the salinity ranges from 25 to 45 g.L -1 . The marine organic carbon is mainly (>92 %) present in dissolved form (DOC). The particulate organic carbon represents 2-8% of the total organic carbon (TOC); the living part of the particulate organic carbon (POC) is very small except during algal blooms. On average, the concentrations of particulate organic carbon and nitrogen in coastal waters or surface waters are between 0.01 and 1 mg L -1 ; much higher concentrations are frequently found in marine phytoplankton blooms. DOC concentration ranges from 0.5 to 20 mg.L -1 for coastal seawater can be found [START_REF] Benner | Abundance, size distribution, and stable carbon and nitrogen isotopic compositions of marine organic matter isolated by tangential-flow ultrafiltration[END_REF][START_REF] Mccarthy | Major Biochemical composition of dissolved high molecular weight organic matter in seawater[END_REF][START_REF] Kennish | Practical Handbook of marine Science[END_REF]. Colloidal and soluble organics can be released by phytoplankton and/or result from cell lysis, solubilization of particles, etc. Humic substances constitute a part of this matter (60 to 80% of DOC) [START_REF] Moonkhum | Review of seawater natural organic matter fouling and reverse osmosis transport modeling for seawater reverse osmosis desalination[END_REF], which are themselves divided into fulvic acids (80%) and humic acids (20%). Sugars represent around 10% of DOC meaning a few hundred micrograms per liter of seawater. They come mainly from phytoplankton [START_REF] Ittekkot | Dissolved free and combined carbohydrates during a phytoplankton bloom in the northern North Sea[END_REF] and include sugars such as glucose, fructose, galactose, mannose and xylose [START_REF] Mccarthy | Major Biochemical composition of dissolved high molecular weight organic matter in seawater[END_REF][START_REF] Biersmith | Carbohydrates in phytoplancton and freshly produced dissolved organic matter[END_REF]. Finally, the lipids are found in trace amounts. As a matter of fact, due to this high variability of natural seawater composition, it is difficult to repeat similar filtration experiments unless several plants operate in parallel. In fact, an infinite change in the process control or seawater composition can induce huge changes on fouling. As a consequence, membrane fouling behavior can not be attributed to a defined parameter. For the present study, the ultrafiltration experiments of the synthetic seawater are replicated. Based on the average composition of natural seawater, a synthetic and reproducible saline suspension is prepared. Special attention is paid to the reconstitution of seawater's organic matter and more particularly the polysaccharides and humic substances due to their significant fouling potential [START_REF] Li | Organic fouling and chemical cleaning of nanofiltration membranes: measurements and mechanisms[END_REF][START_REF] Ye | Fouling mechanisms of alginate solutions as model extracellular polymeric substances[END_REF][START_REF] Jermann | Interplay of different NOM fouling mechanism during ultrafiltration for drinking water production[END_REF][START_REF] Baghoth | An urban water cycle perspective of natural organic matter (NOM): NOM in drinking water, wastewater effluent, storm water, and seawater[END_REF][START_REF] Katsoufidou | An experimental study of UF membrane fouling by humic acids and sodium alginate solutions: the effect of backwashing on flux recovery[END_REF][START_REF] Katsoufidou | An experimental study of UF membrane fouling by humic acid on flux recovery[END_REF][START_REF] Katsoufidou | UF membranes fouling by mixtures of humic acids and sodium alginate: Fouling mechanism and reversibility[END_REF]. Moreover, the fouling potential of organic matter changes in the presence of salts due to the change of interactions between the compounds and the membrane [START_REF] Jermann | Interplay of different NOM fouling mechanism during ultrafiltration for drinking water production[END_REF][START_REF] Katsoufidou | An experimental study of UF membrane fouling by humic acids and sodium alginate solutions: the effect of backwashing on flux recovery[END_REF][START_REF] Katsoufidou | An experimental study of UF membrane fouling by humic acid on flux recovery[END_REF][START_REF] Katsoufidou | UF membranes fouling by mixtures of humic acids and sodium alginate: Fouling mechanism and reversibility[END_REF][START_REF] Gray | The significance of interactions between organic compounds on low pressure membrane fouling[END_REF][START_REF] Katsoufidou | Experimental study of microfiltration membrane fouling by sodium alginate and flux recovery by backwashing[END_REF][START_REF] Van Den Brink | Effect of free calcium concentration and ionic strength on alginate fouling in cross-flow membrane filtration[END_REF][START_REF] Zazouli | Fouling effects of humic and alginic acids in nanofiltration and influence of solution composition[END_REF].

Thus, divalent ions of seawater could significantly increase the fouling by the organic matter [START_REF] Cornelissen | Influence of Calcium-NOM Complexes on Fouling of Nanofiltration Membranes in Drinking Water Production[END_REF]. Therefore, several types of ions are added to the synthetic seawater.

For the present work, the fouling mechanisms of an ultrafiltration membrane are studied for a (i) high salinity mixture with organic and inorganic multi components rather than only one or two components (ii) hollow fiber dead-end filtration with fluxes as well as duration of filtration and backwash cycles in the range of values to those found for the industrial plants (iii) long term ultrafiltrations respectively for a lab scale. Critical conditions of synthetic seawater ultrafiltration are determined thanks to short term experiments. These critical conditions are chosen for longer experiments of ultrafiltration (8h) in order to study the fouling behavior.

Material and methods

Synthetic seawater preparation

Natural seawater contains the same families of components all over the world. However, the contents of the compounds vary according to tides, seasons, temperatures, regions, weather, etc. (Table 1).

Insert table 1

Therefore, synthetic seawater, as much as possible reproducible, is prepared in order to carry out several experiments with similar conditions and define an average behavior of the UF (Table 1). Synthetic seawater should take into account both mineral and especially organic compounds (from particulate to dissolved compounds) since the latter pose big problems of fouling on reverse osmosis pre-treatment [START_REF] Moonkhum | Review of seawater natural organic matter fouling and reverse osmosis transport modeling for seawater reverse osmosis desalination[END_REF][START_REF] Kennedy | Colloidal organic matter fouling of UF membranes: roles of NOM composition and size[END_REF]. For each class of compounds, mineral and organic, one compound was selected as a model.

Mineral matters -Dissolved fraction

Most of the research on membrane fouling is performed with lightly salted solutions or with only one or two salts [START_REF] Jermann | Interplay of different NOM fouling mechanism during ultrafiltration for drinking water production[END_REF][START_REF] Katsoufidou | An experimental study of UF membrane fouling by humic acids and sodium alginate solutions: the effect of backwashing on flux recovery[END_REF][START_REF] Katsoufidou | UF membranes fouling by mixtures of humic acids and sodium alginate: Fouling mechanism and reversibility[END_REF][START_REF] Jermann | Influence of interactions between NOM and particules on UF fouling mechanisms[END_REF]. For the present study, Instant ocean TM salts are dissolved in osmosed water in order to obtain a concentration of 35 g salts .L -1 . These salts are regularly used for aquariums in order to obtain a reproducible ionic composition with several salts.

Mineral matters -Particulate fraction

Bentonite from Riedel de Haën has been selected as model for particulate mineral material. It is already present in natural water and has a size inferior to 200 µm. Kieselguhr, a silica-based mineral, is also added to the synthetic seawater. This diatom is largely present in natural seawater. Both bentonite and kieselguhr are maintained in an oven at 100°C before the preparation of synthetic seawater and 5mg.L -1 of each are used for seawater reconstitution.

Organic matters -Dissolved/colloidal fraction

Dissolved and colloidal organic matters (DOM) are mainly composed of humic substances and polysaccharides. They are strongly involved in the fouling of reverse osmosis and ultrafiltration membranes of seawater desalination plants [START_REF] Huang | Natural organic matter fouling of low-pressure, hollow-fiber membranes: effects of NOM source and hydrodynamic conditions[END_REF][START_REF] Kimura | Irreversible membrane fouling during ultrafiltration of surface water[END_REF]. Humic and alginic acids are selected as model compounds.

A concentrated solution of humic acid is prepared. One gram of humic acid (HA) (supplied by Acros Organics) is added to 500 mL of osmosed water. Then, 62.4 mL of NaOH solution (2N) are added before completing up to one liter with osmosed water and stirring for 24 h in the dark. One liter of this concentrated suspension is added to 1 L of Instant ocean TM salted water at 70g salts /L. Finally, the pH is adjusted to 7-8 by addition of HCl (2N) and the solution is filtered through GF/F Whatman filters (0.7 µm) in order to retain the complexes (particles) formed between humic acids and salts. The filtrate is recovered and represents the acid humic stock solution which is stored in the dark at 4°C during less than 24h before the preparation of synthetic seawater.

An alginic acid stock solution is also prepared. Thirty five grams of Instant ocean TM salts are dissolved in 500 mL of osmosed water. Then, 0.2 gram of alginic acid (supplied by Sigma Aldrich) is introduced before completing up to 1L with osmosed water. Finally, the suspension is filtered with a 0.7 µm GF/F Whatman filter and stored at 4 °C. It represents the alginic acid stock solution.

The organic carbon concentration of humic and alginic acid stock solutions is determined the day before each synthetic seawater preparation in order to determine volumes to be added in order to obtain the suitable concentrations (0.8mg C .L -1 for humic acids and 0.2 mg C .L -1 for alginic acids). Humic and then alginic acids are added after bentonite and kieselguhr during the synthetic seawater preparation.

The synthetic seawater was prepared 4 times with a composition similar to a natural one (Table 1). It is worth noting that TOC concentration of synthetic seawater is higher than organic carbon concentration coming from humic plus alginic acids. A part of this surplus of organic carbon can come from dusts, residuals (cleaning chemicals, organic matter from the last experiment), osmosed water which is used for the reconstitution of seawater, etc.

Experimental set-up:

The pilot plant (Fig. 1) is equipped with a volumetric membrane pump (Grundfos DMX 221) which allows the feeding of the external side of the hollow fiber membrane with seawater.

Pressure and temperature sensors (Wika -accuracy: 0.5-1% of span) are located before the module (0-2.5 bar) and in the permeate exit (0-4 bar). Permeate and backwash flowrates are measured thanks to an electromagnetic flowmeter (Techfluid-Flomid 0 FX). Pressure is applied on a balloon located downstream from the membrane module and filled by osmosed water, when backwash is planned. Specific filtered volumes (permeate flux multiplied by time of filtration) are comprised between 25 to 100 L.m -2 at 100 L.h -1 .m -2 and backwash fluxes are set to 340 L.h -1 .m -2 during 30 seconds. Each filtration sequence includes a filtration step followed by 5 seconds of relaxation then 30 seconds of backwash and a final step of 5 seconds of relaxation before beginning another sequence. As reminded in the introduction of this article, operating conditions (flux values, duration of cycles, membrane material, mode of filtration, etc.,) correspond to the ones classically encountered in the desalination plants. For the present study, 4 (for short term experiments) or 16 cycles of filtration (for experiments of 8h) are carried out, in dead-end mode from the outside to the inside of hollow fiber membrane. Surface of polyvinylidene difluoride (PVDF) membrane is equal to 0.28 m 2 and the mean pore diameter is equal to 0.03 µm.

Insert Fig 1

Sampling and analysis:

During all the duration of ultrafiltration, synthetic seawater of feed tank is continuously mixed. The composition of feed water remains constant during the ultrafiltration. Feed water is sampled before and during each trial in order to determine organic carbon, turbidity, UV 254nm . Permeates and water from backwashes are collected in vials previously washed and stored at 450°C during one night in order to be decontaminated.

During long term experiments, at the end of a filtration sequence (one cycle of filtration followed by one cycle of backwash), the whole permeate and backwash waters corresponding to this sequence are sampled and analyzed. It concerns the sequences numbered 5 to 16 for the permeate waters and 2, 3, 5, 6, 8, 9, 11, 12, 14, 15 for the backwash ones. Otherwise, the samples which correspond to the filtration cycles numbered 1 to 4 or numbered 4, 7, 10, 13, 16 for the permeate and backwash waters respectively, are taken each and during five minutes or seconds respectively. Turbidity is measured with HACH 2100AN Turbidimeter. Ionic chromatography is used to measure the ion concentration. It is composed of two units, D-120 and ICS-900, each associated with an ion exchange resin (stationary phase), one for the cations and the other for anions. Sulfuric acid (17 mM) and sodium carbonate (9 mM) solutions are used as eluent (mobile phase) for the anionic and cationic chromatography. Detection is carried out by conductivity and concentrations are determined using internal standards. Data are transferred to the computer with Chromeleon software Version 6.0. All samples are filtered to 0.45µm before measurement by ionic chromatography. Total organic carbon (TOC) is analyzed with Shimazu TOC-L. Dissolved Organic Carbon (DOC) is obtained after the filtration of sample at 0.7 µm. Humic acids (HA) are aromatic compounds which absorb at 254 nm. The absorbance measurements at 254 nm are performed using a UV/Visible spectrophotometer from Perkin Elmer. HA calibration curves are carried out with saline standard solutions (35g salt /L). The Silt Density Index is determined at 2.1 bars according to ASTM D-4189-95 standard. It is defined as follows (equation 1):

(1) With t 0 , the time required to filter the first 500 mL of the suspension (s), t F , the time (s) required to obtain 500 mL of permeate after a period of time equal to t T (min), usually 15 minutes. Millipore 0.45µm nitrocellulose membrane (Ref. HAWP 04700) is used for SDI measurements.

Fouling resistance definition and calculation

The permeate flux (J p ) and the transmembrane pressures (TMP) are used for the calculation of total resistance and then fouling resistance (R foul ) knowing the membrane resistance (R m ) (equation 2):

(

The viscosity of salted permeate at 20°C (µ) is taken to 1.08×10 -3 Pa.s. For each filtration cycle, the fouling resistance (R foul. ) is defined as the sum of two terms, a reversible fouling (R rev ) which can be removed thanks to a backwash and a residual fouling (R res ) resistance which remains after backwash (Fig. 2)

Insert Fig 2

Reversible and residual fouling resistances can be calculated for the cycle (n) by using the equation numbered 3, 4, 5:

(3) (4) [START_REF] Guilbaud | Seawater pretreatment by dead-end micro and ultrafiltration in pressure-driven inside feed[END_REF] The indexes i and f refer to the initial and final fouling resistance during one cycle of filtration.

Results

Study of critical conditions on short term ultrafiltrations

Insert Fig 3

Few changes of normalized permeability are observed during the ultrafiltration of synthetic seawater to a specific filtered volume equal to 25 L.m -2 . However, the permeability drops from one cycle to another for 50, 75 and 100 L.m -2 which means membrane fouling during ultrafiltration. For specific volume equal to 100 L.m -2 , the backwashes do not allow recovering the initial permeability; the filtration operates above critical conditions. For 75 L.m -2 , the normalized permeability decreases drastically during each cycle of filtration meaning an increase of transmembrane pressure since the permeate flux is fixed to 100 L.h - 1 .m -2 . Hence, the maximum pressure before membrane module breakage could be rapidly exceeded beyond four cycles of ultrafiltration. At 50 L.m -2 , the ultrafiltration seems to operate close to the critical conditions since few residual fouling is observed. For this condition, synthetic seawater is compared to a natural one during four cycles of ultrafiltration (Fig. 3b).

Normalized permeability decreases and contrary to the synthetic seawater an important residual fouling remains after backwashes. It is worth noting that the present behavior of natural seawater ultrafiltration will change according to the day, the season, the weather, the tide, etc. because the composition of water will change. Therefore, it is difficult to define generic optimal operating conditions for ultrafiltration of natural seawater.

In conclusion, for the remainder of this study, the fouling mechanisms will be studied, during long term ultrafiltration (8 hours) of synthetic seawater when the specific filtered volume will be equal to 50 L.m -2 at a permeate flux of 100 L.h -1 .m -2 .

Filtration efficiency and fouling mechanisms during 8 h of ultrafiltration

Hydraulic performances

Changes of fouling resistance during ultrafiltration are shown in figure 4.

Insert Fig 4

The average fouling resistance increases slowly from the beginning to the end of operation where it reaches around 3.10 11 m -1 . So, the fouling velocity is approximately equal to 6.10 8 m - 1 .min -1 during the 8 h of the filtration period. The fouling resistance at the end of the first filtration cycle, meaning before any backwashes, is of the same order of magnitude as the one found by Castaing et al. [START_REF] Castaing | Immersed hollow fibres microfiltration (MF) for removing undesirable micro-algae and protecting semi-closed aquaculture basins[END_REF] after the same time of dead-end microfiltration (0.2 µm) of a seawater containing 1000 microalgae.L -1 , meaning a total organic carbon concentration close to this study (around 2 mg.L -1 ).

For each cycle of filtration, the fouling resistance divides into residual and reversible ones as defined in the material and methods paragraph (Fig. 5).

Insert Fig 5

As already remarked with experiments performed on 4 cycles of filtration (Fig. 3), few residual fouling accumulates and the cumulative residual fouling resistance increases quasi linearly during filtration. The backwashes do not allow removing all the foulants of the membrane surface which is why a residual fouling resistance equal to approximately 2.10 10 m - 1 is added at each cycle. Most of the fouling is reversible since, on average, only 20% of fouling remain after a backwash. No residual fouling is observed after the first cycle of filtration (Fig 5).

Quality of the produced water

The average size of bentonite and kieselguhr is equal to 17-21 µm [START_REF] Massé | Seawater ultrafiltration: role of particles on organic rejections and permeate fluxes[END_REF]. Hence, these particles are fully retained by the membrane and can't participate in the turbidity and SDI 15 of permeates contrary to organics and salts alone or recombined together (Fig. 6).

Insert Fig 6

The values of SDI 15 and turbidity of permeates vary between 2.0 and 3.5 %.min -1 (average equal to 2.6 %. min -1 ) and 0.1 to 0.4 NTU respectively; more than 6.6 %.min -1 and 3.1 NTU are found for the feeding. The SDI 15 of permeates are relatively high even if a spiral wound reverse osmosis module could probably be fed by this water without severe clogging.

The Total Organic Carbon of permeates is also analyzed (Fig 7).

Insert Fig 7

In average, the TOC concentration of permeates is equal to 2.40.8 mg C .L -1 while organic carbon concentration coming from humic acids is equal to 0.680.03 mg C .L -1 (Fig. 8).

Insert Fig 8

Hence, according to the standard deviations, the average permeate TOC concentration is more or less equal to the feeding one (1.8 ± 0.4 mg C .L -1 ) or even higher. In the latter case, the highest concentration into the permeates could be explained by the organic matter accumulation within the membrane module (in the retentate side). Indeed, backwashes are not efficient enough to remove all organics from the module as we could see later in this article (Fig. 10). Thus, the TOC concentration in the vicinity of the membrane surface would be higher than within the feed tank and assuming the rejection rate constant, more TOC could be passed through the membrane. Thus, the organic rejection rate is low. Similarly, Castaing et al. [START_REF] Castaing | Investigating submerged ultrafiltration (UF) and microfiltration (MF) membranes for seawater pre-treatment dedicated to total removal of undesirable micro-algae[END_REF] obtained low total organic carbon rejection, ranging from 66 to 74%, during the deadend filtration on 10 kDa, 300 kDa and 0.2 µm submerged membranes of microalgae rich seawater.

Only 15% of humic acids are retained by the membrane. This rejection rate is low compared to Katsoufidou et al. 's results [START_REF] Katsoufidou | UF membranes fouling by mixtures of humic acids and sodium alginate: Fouling mechanism and reversibility[END_REF], who obtained a rejection rate equal to 60% during the ultrafiltration of a mixture containing humic acids, alginates, NaCl (20 mM) and Ca 2+ (1 mM). Nevertheless, for our study, stock solutions of humic and alginic acids are pre-filtered with a 0.7 µm filter which induces the removal of the biggest colloids from the raw seawater.

For the first cycles of filtration and may be subsequently, the permeate contains less humic acids at the beginning of a cycle than afterwards (Fig. 8). The membrane module is never drained during operation therefore the water which is filtrated at the beginning of a cycle corresponds to that trapped in the module at the end of the backwash. Backwash water which leaves the plant at the end of the cycle "n" contains less or as much humic acids as the seawater in the feed tank (Fig. 10). Thus, the average permeate collected during the first 5 minutes of the "n+1" cycle contains less humic acids than the raw seawater especially since the permeate pipe still contains osmosed water of backwash; it dilutes the average permeate recovered at the beginning of each cycle.

Backwash efficiency

Analysis of an average sample, representative of each backwash water

The physico-chemical analyses of water coming from backwashes allow to determine the amount of total organic carbon withdrawn and that coming from humic acids (Fig. 9).

Insert Fig 9

The amount of TOC decreases drastically during the early backwash cycles and decelerates afterwards. The organic matter could be better removed from membrane surface and module in the early backwashes rather than after several cycles. At the beginning of the filtration, the organic matter would deposit directly on the membrane material versus heterogeneous fouling layer afterwards. The interactions between the organic matter and the membrane can be weaker than between organic matter ones which could explain the drop of organic matter inside backwash waters as ultrafiltration progresses.

The amount of recovered humic acids in the backwash waters seems to increase slightly as filtration progresses (Fig. 9). Backwash efficiency to remove humic acids could be enhanced during filtration. The amount of humic acids reaching the membrane and present in the permeates is relatively constant. Hence, the amount of humic acid that remains on the membrane surface after one backwash could decrease during ultrafiltration.

To sum up, the amount of TOC from backwash waters decreases while the organic matter coming from humic acids would increase slightly (Fig. 9). Hence, the alginic acid amount in the backwash waters could progressively decrease since organic matter is mainly brought by alginic and humic acids even if a part of the organics of feeding is not identified (Table 1). On the other hand, the amounts of alginic acids brought by the reconstituted seawater to the membrane are always constant and could slightly decrease in the permeates as the filtration progresses (Fig. 7 and8). Higher accumulation of alginic acids, rather than humic acids, could occur on the membrane and pipe surfaces and/or in the piping and membrane module. More and more alginic acids could accumulate during the filtration. This result is consistent with the results obtained by Jermann et al. [START_REF] Jermann | Influence of interactions between NOM and particules on UF fouling mechanisms[END_REF], Katsoufidou et al. [START_REF] Katsoufidou | UF membranes fouling by mixtures of humic acids and sodium alginate: Fouling mechanism and reversibility[END_REF] and Resosudarmo et al. [START_REF] Resosudarmo | Analysis of UF membrane fouling mechanisms caused by organic interactions in seawater[END_REF] who found an increase of alginate rejection during the filtration.

Analysis of fractionated sampling of water's backwashes

Waters coming from backwashes numbered 1, 4, 7, 10, 13, 16 are collected each and during five seconds in order to be analyzed (Fig. 10). Water volumes obtained during five seconds of backwash are added and placed on X-axis. For each volume, the amount of TOC and organic carbon from humic acids is determined; the cumulative values are indicated on Y-axis.

Insert Fig 10

At the end of several backwashes, the velocity of recovery of TOC or organic matter coming from humic acids decreases since a plateau seems to be taking shape (Fig. 10 a,b). However, even after 30 seconds of backwash, organics are still recovered. Hence, the duration and/or intensity of backwashes are not sufficient to remove all organic matter from membrane or retentate compartment of membrane module.

The turbidity of backwash waters is almost constant for the early cycles of backwash and increases for the last ones (Fig. 10c). The amount of colloids and particles (alginic, humic acids, salts, kieselguhr and bentonite) alone or recombined could increase in the backwash waters. It is likely that colloids and particles accumulate not only on the fouled surfaces but also into the bulk of dead-end volumes of the plant. Indeed, it should be noted that no drainage of the membrane module or plant occurs while the installation is running.

Discussion and summary

It should be noted that the mechanisms of fouling which are assumed and described below, are established for a feed with organic carbon mainly brought by the humic acids containing several salts in high concentrations, in the presence of particles (bentonite and kieselguhr) and for specific conditions of dead-end filtration (PVDF membrane, 0.03 µm of pore diameter, permeate flux equal to 100 L.h -1 .m -2 , SFV equal to 50 L.m -2 , etc.).

The operating conditions used for this study do not induce a lot of fouling and notably residual fouling which accounts on average for 20% of total fouling resistance (Fig. 5). The fouling reversibility could be improved since deionized water has been used instead of permeate water for the backwashes [START_REF] Resosudarmo | Analysis of UF membrane fouling mechanisms caused by organic interactions in seawater[END_REF][START_REF] Li | Seawater ultrafiltration fouling control: Backwashing with demineralized water/SWRO permeate[END_REF]. Moreover, no sharp change of fouling intensity is noted during the filtration since fouling resistance changes quasi linearly with time (Fig. 4 and5). Therefore, the fouling layer seems not to be compressed at the transmembrane pressures which are reached in the present study contrary to Sioutopoulos and Karabelas [START_REF] Sioutopoulos | The effect of permeation flux on the specific resistance of polysaccharide fouling layers developing during dead-end ultrafiltration[END_REF] who observed a compression from a certain value of pressure beyond an initial phase of membrane coverage by alginate. Moreover, the fouling of membrane does not strongly affect the transmission rate of the humic acids since the permeate concentrations stay stable (Fig. 8).

Concerning the TOC and alginic acids, the transmission rate (calculated from the concentrations in the feed tank) could be constant or be slightly decreased.

Most of the salts passes through the membrane since it has been verified that no change of conductivity between the feed and permeates was observed (results not shown). Nevertheless, a highly salted water is used for the present study (NaCl ~ 450 mM, Ca 2+ ~ 10 mM and Mg 2+ ~ 50 mM) and a small amount of salts (not visible by measurements of conductivity) and more particularly divalent ones (Ca 2+ and Mg 2+ ) could participate in the fouling mechanisms of humic and alginic acids [START_REF] Li | Organic fouling and chemical cleaning of nanofiltration membranes: measurements and mechanisms[END_REF][START_REF] Jermann | Interplay of different NOM fouling mechanism during ultrafiltration for drinking water production[END_REF][START_REF] Katsoufidou | UF membranes fouling by mixtures of humic acids and sodium alginate: Fouling mechanism and reversibility[END_REF]. Thus, Resosudarmo et al. [START_REF] Resosudarmo | Analysis of UF membrane fouling mechanisms caused by organic interactions in seawater[END_REF] observed an increase of fouling phenomenon with a highly salted water (NaCl ~ 376 mM and Ca 2 + ~ 8 mM and Mg 2+ ~ 39 mM). Some "cationic bridges" could be formed between carboxylic groups of organic acids. Moreover, salts could mitigate the electrostatic repulsion between the membrane and compounds involved in the membrane fouling [START_REF] Li | Organic fouling and chemical cleaning of nanofiltration membranes: measurements and mechanisms[END_REF].

More and more humic acids seem to be recovered during backwashes, suggesting as previously observed [START_REF] Jermann | Interplay of different NOM fouling mechanism during ultrafiltration for drinking water production[END_REF][START_REF] Katsoufidou | UF membranes fouling by mixtures of humic acids and sodium alginate: Fouling mechanism and reversibility[END_REF][START_REF] Resosudarmo | Analysis of UF membrane fouling mechanisms caused by organic interactions in seawater[END_REF], their higher propensity to adsorb on virgin rather than already fouled membrane. Humic acids could have been irreversibly adsorbed on membrane material during the filtration of a mixture containing alginates, humic acids and Ca 2+ inducing smaller membrane pore size followed by a rejection of alginates [START_REF] Jermann | Influence of interactions between NOM and particules on UF fouling mechanisms[END_REF]. Thus, Humic acids could form a pre-layer favorable to the fouling of other compounds notably alginic acids.

Conversely, alginic acids seem preferentially removed by backwashes at the beginning of ultrafiltration and therefore would contribute to the initial reversible and final residual membrane fouling. Interactions between alginic acids and heterogeneous fouling layer/gel would be stronger than between alginic acids and the membrane. Thus, alginic acids could form a gel on the membrane surface which could become denser in the presence of calcium [START_REF] Resosudarmo | Analysis of UF membrane fouling mechanisms caused by organic interactions in seawater[END_REF].

Results would seem to show that all particles and colloids are not removed from the membrane and/or retentate compartment during the early cycles of backwashes (Fig. 10c).

Colloids and particles could participate in the fouling layer build-up. According to Jermann et al. [START_REF] Jermann | Influence of interactions between NOM and particules on UF fouling mechanisms[END_REF], a cake composed by kaolinite alone is almost totally reversible contrary to a kaolinite -humic acids cake or a kaolinite -humic acids -alginate cake. Humic acids would adsorb on to kaolinite which would in turn accumulate on the membrane. It should be noted that the structure of the fouling layer will be different according to the inorganic particle concentration within the raw seawater. At low concentrations, fouling layer will be composed of particles remoted from each other. At higher concentrations, alginates and humic acids will accumulate inside the porous layer and will form a more compact gel than particles alone [START_REF] Jermann | Influence of interactions between NOM and particules on UF fouling mechanisms[END_REF].

Nevertheless, the amount of inorganic particles brought to the membrane per cycle is close: 100 mg for Jermann's study versus 140 mg for this one. However, the membrane surfaces are different, 0.0042 m² versus 0.28 m² respectively.

Conclusion:

This work investigated the membrane fouling mechanisms during the dead-end ultrafiltration of synthetic seawater at 100 L.h -1 .m -2 . Duration of filtration cycles equal to 30 minutes followed by backwash at 340 L.h -1 .m -2 during 30 seconds allow a sustainable ultrafiltration during 8 hours. A graphical abstract gathers the main results obtained following the ultrafiltration of the synthetic seawater (Fig. 11). The fouling is mainly reversible. The deposit/gel and adsorbates do not induce a high increase of transmembrane pressure meaning high fouling. The deposit/gel would remain fairly permeable since almost all the humic acids and more generally total organic carbon passes through the membrane throughout operation.

Less and less organics are withdrawn by backwash waters during the operation even if more and more humic acids are recovered (Fig. 11). Mechanisms of fouling seem to change during the ultrafiltration according to the state of the membrane, pristine or already fouled: during the early stages of filtration, humic acids could preferentially participate in the membrane fouling compared with alginic acids whereas their magnitude would be reversed after several sequences of filtration/backwash. Colloids and particulates of seawater could be retained in the heterogeneous fouling layer and/or accumulate in the concentrate compartment of the membrane module before being recovered inside backwash waters. In order to confirm several assumptions enounced during the present work, future researches on fouling layer autopsies during ultrafiltration should be undertaken. These autopsies would allow better understanding of the fouling mechanisms. Graphical abstract of fouling mechanisms during ultrafiltration of the synthetic seawater (TOC: total organic carbon, OC-HA organic carbon from humic acids, OC-Alg: organic carbon from alginic acids).

R i (n) R foul. Time R i (n+1) R f (n+1) R f (n) R foul. (n) R rev (n) R res (n) Cycle (n) Cycle (n+1)

  Concerning short term experiments, 4 cycles of filtration are successively done, each one intersected by backwash at 250 L.h -1 .m -2 during 30 seconds. Different specific filtered volumes per cycle of filtration (SFV) are tested: 25, 50, 75 and 100 L.m -2 meaning a filtration cycle duration of 0.25, 0.5, 0.75, 1h at the permeate flux of 100 L.h -1 .m -2 . This value of permeate flux is in the range of those classically encountered for ultrafiltration of seawater on a large scale. These experiments aim to choose, for longer term experiments, the SFV below which few residual fouling will be observed in order to have a sustainable operation. Changes of normalized permeability (Lp (20 °C) / Lp 0(20 °C) ) during 4 cycles of filtration are shown in figure 3. Results for ultrafiltration of synthetic and natural seawater sampled at Le Croisic (Atlantic Ocean -France) are gathered.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Experimental set-up

Fig. 3 :Fig. 4 :

 34 Fig. 3: Normalized permeability at 20°C during filtration of synthetic () and natural seawater (o) at 100 L.h - 1 .m -2 for specific filtered volumes equal to 25 L.m -2 (a), 50 L.m -2 (b), 75 L.m -2 (c), 100 L.m -2 (d)

Fig. 5 :Fig. 6 :

 56 Fig. 5: Changes of cumulative residual fouling resistances as well as residual/total fouling resistance ratio during the ultrafiltration of synthetic seawater

Fig. 7 :

 7 Fig. 7 : Total organic carbon (TOC) concentration of permeates (average results of four experiments). P i designates the sample of permeate recovered during the i th filtration cycle. For the first four cycles of filtration, the permeates are recovered during and all five minutes of filtration.

Fig. 8 :

 8 Fig. 8 : Organic carbon concentration of permeates brought by humic acids (average results of two experiments). P i designates the sample of permeate recovered during the i th filtration cycle. For the first four cycles of filtration, the permeates are recovered during and all five minutes of filtration.

Fig. 9 Fig. 10 :

 910 Fig. 9 Amount of Total organic carbon (TOC) and organic carbon from humic acids (OC-HA) in the backwash waters (BW)

Fig. 11 .

 11 Fig. 11.

Table 1 :

 1 Characteristics of the prepared seawater and natural seawaters sampled at different worldwide locations[START_REF] Heijman | Heterogeneous fouling in dead-end ultrafiltration[END_REF][START_REF] Jezowska | Ultrafiltration as direct pre-treatment of seawater: a case study[END_REF][START_REF] Benner | Abundance, size distribution, and stable carbon and nitrogen isotopic compositions of marine organic matter isolated by tangential-flow ultrafiltration[END_REF][START_REF] Mccarthy | Major Biochemical composition of dissolved high molecular weight organic matter in seawater[END_REF][START_REF] Kennish | Practical Handbook of marine Science[END_REF][START_REF] Moonkhum | Review of seawater natural organic matter fouling and reverse osmosis transport modeling for seawater reverse osmosis desalination[END_REF][START_REF] Zazouli | Fouling effects of humic and alginic acids in nanofiltration and influence of solution composition[END_REF][START_REF] Cornelissen | Influence of Calcium-NOM Complexes on Fouling of Nanofiltration Membranes in Drinking Water Production[END_REF][START_REF] Copin-Montégut | Chimie de l'eau de mer[END_REF][START_REF] Gaid | Le dessalement des eaux par osmose inverse: l'expérience de Véolia Water[END_REF][START_REF] Leparc | Water quality and performance evaluation at seawater reverse osmosis plants through the use of advanced analytical tools[END_REF][START_REF] Petry | The El Coloso (Chile) reverse osmosis plant[END_REF]

				Prepared seawater (average on 4 synthetic seawaters)	Natural seawater
	Turbidity (NTU)	3.1 ± 0.5	0.1 -20
	pH			7-8	7.4 -8.5
	Suspended solids (mg.L -1 )	8.7 ± 1,7	-
	Humic acids (mg C .L -1 )	0.8	-
	Alginic acids (mg C .L -1 )	0.2	-
	SDI 15 (%/min)	> 6.6	> 6.6
	TOC (mg.L -1 )	1.8 ± 0,4	0.8 -4.0
	Conductivity (20°C) (mS.cm -1 )	42 ± 3	49 -62
	[Salts] (mg.L -1 )	34 684 ± 3
	Na +			10 074 ± 13	10 945 -15 000
	K +			332 ± 1	383 -690
	Mg 2+	1 274 ± 56	1 334 -1 660
	Ca 2+	414 ± 6	405 -780
	Cl -			20 173 ± 38	19 080 -26 500
	Br -			73 ± 1	45 -67
	NO 3	-	158 ± 1	0.2 -25
	SO 4	2-	2 185 ± 15	2 400 -2 965

(TOC: total organic carbon, OC-HA organic carbon from humic acids, OC-Alg: organic carbon from alginic acids)


J L Autran

D Munteanu

Diffusion-collection modeling of radiation-induced charge in semiconductor devices New insights into diffusion-collection modeling of radiation-induced charge in semiconductor devices

Charge diffusion from an ion track and its collection by a biased contact in a semiconductor domain is modeled and analyzed within the framework of the so-called diffusion-collection approach. We successively examine the case of charge diffusion from a point source and from a linear distribution, introducing and discussing the concept of collection velocity at the point where the collection current is evaluated. Analytical formulations of the collected charge, collection current and collection velocity are developed. Implications for the calculation of the soft error rate (SER) in complementary metal-oxide-semiconductor (CMOS) circuits exposed to ionizing particles are derived. Finally, our model provides new insights into the correct definition of the charge collection velocity in collection-diffusion models.

I. INTRODUCTION

The passage of an ionizing particle through a semiconductor device results in energy transfer and the creation of electron-hole pairs along the particle track [START_REF] Leroy | Principes of Radiation Interaction Matter and Detection[END_REF][START_REF] Autran | Soft errors: from particles to physics[END_REF] . Carriers in excess are then transported by ambipolar diffusion throughout the volume of the semiconductor until they recombine or are collected and extracted, generally under the influence of an electric field developed by a biased contact or a reversebiased p-n junction [START_REF] Murat | Electron and ion tracks in silicon: Spatial and temporal evolution[END_REF] . Modeling these different physical processes is essential to predict the transient electrical response of complementary metal-oxide-semiconductor (CMOS) devices and circuits subjected to ionizing particles and to evaluate their soft error rate (SER), i.e., the probability of an ionizing particle to cause a transient error in the circuit that affects its operation, without causing permanent damage [START_REF] Autran | Soft errors: from particles to physics[END_REF][START_REF] Munteanu | Modeling and simulation of single-event effects in digital devices and ics[END_REF] . Over the last forty years, several modelling and numerical simulation studies have been carried out in the field of radiationinduced charge diffusion and collection in CMOS devices and circuits. Among these works are several pioneering contributions by Kirkpatrick 5 , Messenger 6 , Edmonds [START_REF] Edmonds | Charged collected by diffusion from an ion track under mixed boundary conditions[END_REF] or Palau et al. [START_REF] Palau | Device simulation study of the seu sensitivity of srams to internal ion tracks generated by nuclear reactions[END_REF] , followed in the last two decades by further contributions in this field [START_REF] Palau | Monte carlo exploration of neutron-induced seu-sensitive volumes in srams[END_REF][START_REF] Lambert | Neutron-induced seu in bulk srams in terrestrial environment: Simulations and experiments[END_REF][START_REF] Merelle | Criterion for seu occurrence in sram deduced from circuit and device simulations in case of neutron-induced ser[END_REF][START_REF] Wrobel | A semi-empirical approach for heavy ion seu cross section calculations[END_REF][START_REF] Correas | Innovative simulations of heavy ion cross sections in 130 nm cmos sram[END_REF][START_REF] Correas | Simulation tool for the prediction of heavy ion cross section of innovative 130-nm srams[END_REF][START_REF] Artola | Collected charge anal-ysis for a new transient model by tcad simulation in 90 nm technology[END_REF][START_REF] Uznanski | Single event upset and multiple cell upset modeling in commercial bulk 65nm cmos srams and flip-flops[END_REF][START_REF] Artola | Seu prediction from set modeling using multi-node collection in bulk transistors and srams down to the 65 nm technology node[END_REF][START_REF] Martinie | Underground experiment and modeling of alpha emitters induced soft-error rate in cmos 65 nm sram[END_REF][START_REF] Autran | Physics-based analytical formulation of the soft error rate in cmos circuits[END_REF] , which have benefited from increasingly efficient computing resources and simulation tools. Our analysis of these works has uncovered challenges in computing carrier collection current and assessing the circuit soft error rate. Starting from the fundamental equations of diffusioncollection, we therefore aimed to revisit established results that have been questioned as to their ability to explain the physical mechanisms involved in charge collection and how this process is controlled. More precisely, in this work, we model and analyze the diffusion of charge from an ion track and its collection by a contact in silicon. First, we reassess the case of charge diffusion from a point source and examine the methods for modelling the collected current and charge by a small contact at a particular distance from the source, while considering two diverse formalisms for the current: a pure diffusion current or a conduction current that incorporates a collection velocity. For the two approaches, we derive ana-lytical expressions for the time dependence of both collected current and charge. In the second part, we revisit the charge diffusion-collection from a linear distribution (emulating an ion track with a constant linear energy deposition). We notice that the previous solutions derived for the point source result in different representations of the collected charge depending on the chosen formalism for the current (pure diffusion current or conduction current). Implications for the calculation of the SER of CMOS circuits exposed to ionizing particles are derived and discussed in the last part of this paper as well as the important issue of the collection velocity evaluation in diffusion-collection models. The series of analytical developments described below provide new insights into collection-diffusion models and the physical mechanisms involved, particularly in charge collection and the control of such a collection process. In addition, these models should enable a more accurate modeling of the SER of CMOS circuits, incorporating the essential physics of diffusion and collection of radiation-induced charge within the semiconductor.

II. CHARGE DIFFUSION AND COLLECTION FROM A POINT SOURCE

We first consider a point charge Q = qn 0 deposited at t = 0 in point M located in the bulk of a semiconductor domain. Despite this configuration implies an infinite homogeneous halfspace, bounded by a planar surface, as represented in Fig. 1, in the following we neglect the reflective boundary condition on the silicon surface and consider the 3D simulation domain as an infinite medium. This point charge corresponds to a number n 0 of carriers-in-excess, for example a number of electrons issued from a punctual energy transfer resulting in the creation of the same number of electron-hole pairs at t = 0 in M (the transport of holes will not be detailed in the following). Q is treated as a Dirac delta function of r, i.e., Q(r) = qn 0 (r). In the pure diffusion approach, the transport of the excess carrier density n e (m -3 ) is governed by a 3D spherical diffusion law in the semiconductor domain [START_REF] Kirkpatrick | Modeling diffusion and collection of charge from ionizing radiation in silicon devices[END_REF][START_REF] Messenger | Collection of charge on junction nodes from ion tracks[END_REF][START_REF] Edmonds | Charged collected by diffusion from an ion track under mixed boundary conditions[END_REF] : ∂ n e (r,t) ∂t -n e (r,t)

τ = D∇ 2 n e (r,t) (1)
where τ is the carrier lifetime and D is the ambipolar diffusion coefficient given by:

D = 2D n D p /(D n + D p ) (2)
with D n and D p the diffusion coefficients for electrons and holes respectively. The excess carrier density at time t and distance r originating from this point charge n 0 and solution of Eq. ( 1) can be written under the form [START_REF] Messenger | Collection of charge on junction nodes from ion tracks[END_REF] :

n e (r,t) = n 0 (4πDt) 3 2 exp - r 2 4Dt - t τ (3)
Neglecting carrier recombination in the following (τ → +∞), which represents a strong simplification of the model but an imperative to maintain its analytical character, Eq. ( 3) can be used to calculate the charge collected in point P (see Fig. 1) by a small contact of surface A S developing no electric field in the semiconductor. To do this, we have to evaluate the current density in point P and then integrate it on the contact surface and in time, from 0 to infinity. Two different formalisms can be used at this level to evaluate this quantity: the theory of diffusion or the general electrokinetic formulation of the conduction.

A. Pure diffusion current

The expression of the diffusion current density is:

⃗ J di f f (r,t) = qD ⃗ ∇(n e ) (4)
where the diffusion coefficient in this equation is the ambipolar diffusion because it corresponds to the considered carrier transport mechanism. Considering a small electrode centered in P (see Fig. 1) and the case where the diffusing carrier flux is perpendicular to the collecting surface, the diffusion current evaluated at the contact is:

I di f f (r,t) = A S ⃗ J di f f (r,t) • ⃗ dS ≈ qA S D ∂ n e (r,t) ∂ r (5)
where A S is the surface of the electrode (assumed to be sufficiently "small" with respect to all other geometrical dimensions to avoid numerical integration in the current expression).

The time integration of Eq. ( 5) leads to the expression of the charge that diffuses to the electrode surface at P (which is then collected), originating from qn 0 at M. From Eqs. (3) and ( 5), we obtain:

I di f f (r,t) = qA S n 0 (4πD) 3 2 × r 2t 5 2 × exp - r 2 4Dt
= qA S n e (r,t) × r 2t ( 6)

q di f f col (r,t) = t 0 I di f f (r,t ′ )dt ′ = qA S n 0 2(π) 3 2 r 2 × Γ 3 2 , r 2 4Dt (7)
where Γ(a, x) is the upper incomplete gamma function. For (t → +∞), the gamma function term in Eq. ( 7) reduces to the quantity √ π/2 and the maximum collected charge has the following expression:

q di f f col (r) = qn 0 × A S 4πr 2 (8)
Another interesting quantity is the ratio q di f f col (r)/qn 0 that corresponds to a collection efficiency for the collecting contact:

η di f f S = q di f f col (r) qn 0 = A S 4πr 2 (9)
Arranged under this form, the quantity Ω = A S /4πr 2 corresponds to the ratio of the solid angle of the collecting contact centered in point P and observed from point M to the solid angle for the entire space (see Fig. 1), as considered in ref. [START_REF] Wrobel | A semi-empirical approach for heavy ion seu cross section calculations[END_REF] . This collection efficiency thus appears to be proportional to the collector surface and inversely proportional to the square of the distance between the deposited charge and the collector.

At this level, we establish two interesting results for the times at which n e (r,t) and I di f f (r,t) admit a maximum. From the time derivative of Eqs. ( 3) and (6), it is easy to show that:

∂ n e (r,t) ∂t ∂ I di f f (r,t) ∂t

t=t n max = 0 ⇔ t n max = r 2 6D (10)
t=t i max = 0 ⇔ t i max = r 2 10D ( 11
)
This result is a priori counter-intuitive: the charge and the diffusion current do not pass through a maximum at the same time, contrary to what is assumed in ref. [START_REF] Palau | Device simulation study of the seu sensitivity of srams to internal ion tracks generated by nuclear reactions[END_REF] but the current reaches its maximum slightly before the charge. The maximum values for the charge and the current are the following:

n e,max (r) = n e (r,t n max ) = n 0 2 3 πer 2 -3 2 ( 12
)
I di f f ,max (r) = I di f f (r,t i max ) = 5 e 5 2 × qA S n 0 D (2π) 3 2 r 4 (13)
Fig. 2 shows the time evolution of the carrier density and the diffusion current evaluated using Eqs. ( 3) and ( 6) for a distance r = 1.5 µm from a point charge source Q = qn 0 .

The curves highlight the time shift between the two transients: the current transient peak precedes the carrier density peak, with a peak position extracted at 0.163 ns against 0.272 ns for the density curve. This example also numerically illustrates that the time position and magnitude of the maximum of the two peaks verify the analytical expressions of Eqs. (10) to (13) that give: t i max = 0.1631 ns, t n max = 0.2718 ns, I di f f ,max = 0.6231 µA, n e,max = 3.308 × 10 16 cm -3 .

B. Conduction current

Instead of choosing the diffusion current formalism, one can consider the general electrokinetic equation for the current density carried by electrons of density n given by:

⃗ J = qn⃗ v ( 14
)
where v is the electron diffusion velocity. Using Eq. ( 3) for the carrier density n e at point P (these carriers being transported by diffusion from source M to point P) and integrating Eq. ( 14) over the surface A S gives the collected current 8 :

I col (r,t) = qA S v col n e (r,t) (15)
where the carrier diffusion velocity is denoted as v col for "collection velocity" because it corresponds to the velocity at which the carriers are collected by crossing the collecting surface of the contact, in the absence of an electric field and transported only by the diffusion process.

The equality of the current densities expressed from Eqs. ( 4) and ( 14) at the point of collection leads to a general expression of this collection velocity under the form:

qD ⃗ ∇(n e ) = qn e ⃗ v col ⇒⃗ v col = D ⃗ ∇(n e ) n e (16)
The direct comparison of Eqs. ( 6) and ( 15), that must correspond to the same current, forces the term v col to take the following expression:

(6) = (15) ⇒ v col = r 2t (17)
This quantity has the physical dimension of a velocity that is position and time dependent. The expression of v col (r,t) given by Eq. ( 17) ensures that the conduction current equals the diffusion current at the position where the two quantities are evaluated. Of course, it is evident that using the conduction current given by Eq. ( 15) with the above collection velocity leads to the same collected charge as that given by Eqs. ( 7) and ( 8). We will come back in detail to this concept of collection velocity in section 5.

C. Case of lateral diffusion

The case of lateral diffusion should be examined since it is different from the previous case (see Fig. 1) where the carrier flux is perpendicular to the collecting electrode. We first examine the case of the ideal planar collecting contact, as represented in Fig. 3a and previously considered. The solid angle of the contact viewed from point M is, in this case, equal to:

Ω = A S × cos(α) 4πr 2 = A S ℓ 4πr 3 (18)
Considering Eq. ( 18), Eqs. ( 7) and ( 8) must then be rewritten under the following form:

q di f f col (r, ℓ,t) = qA S n 0 ℓ 2(π) 3 2 r 3 × Γ 3 2 , r 2 4Dt ( 19
)
FIG. 3. Schematic illustrations of collecting contacts: a) an ideal planar contact of surface A S and b) a "real" contact in CMOS circuits consisting in a reversely-biased junction with a layout surface A S and a lateral surface A L due to the depth extension (W SCR ) of the space charge region (SCR).

q di f f col (r, ℓ) = qn 0 × A S ℓ 4πr 3 ( 20
)
These expressions predict a collected charge equal to zero for ℓ = 0. In other words, as the point source moves closer to the surface, the collected charge reduces until it reaches zero for a source situated on the semiconductor's surface. However, this scenario does not correspond to reality, as collector contacts (also known as sensitive circuit nodes) in CMOS technologies are never planar contacts with no physical thickness, but rather ohmic contacts (N + N or P + P junctions with low resistance and non-rectifying properties) or reverse-biased junctions (N + P or P + N). Such devices have a physical extension in depth which corresponds to the space charge region of the junction. Therefore, a "real" collector contact, as schematically represented in Fig. 3b, is characterized by two surfaces:

• an horizontal surface (A S ) which represents the contact area at the "layout" level and which also corresponds to the lower face of the space charge region in the semiconductor.

• a vertical surface (A L ) which corresponds to the side surface of the space charge region in the semiconductor.

The diffusion current is thus obtained by integrating the current density on the whole collector surface inside the semiconductor (i.e., on the lateral + lower surfaces):

I di f f (r, l,t) = A S +A L ⃗ J di f f (r,t) • ⃗ dS ≈ qD ∂ n e (r, ℓ,t) ∂ r × A C (M) (21)
where A C (M) is the effective collection surface viewed from point M, i.e., the collector surface in direct view from the point source. The approximation made in Eq. ( 21) requires that the dimensions of the collector are small compared to the source-to-collector distance. Eqs. ( 7), ( 8) and all derived equations still apply in the case of lateral diffusion with A S FIG. 4. Schematic illustration of the case of a particle track considered as an ensemble of point charges qδ n 0 uniformly distributed along a straight segment of length L.

formally replaced by the quantity A C (M). From these equations, the collected charge is never equal to zero when M is located at the semiconductor surface (ℓ = 0). Because A C (M) is difficult to evaluate for a realistic junction geometry from analytical calculations, in the following we make the assumption that A C (M) ≈ A S whatever the point M in the semiconductor domain (sufficiently far enough from the contact), which is an acceptable approximation in case of "box" geometries of the collector.

III. CHARGE DIFFUSION AND COLLECTION FROM A LINE SOURCE

From the previous results in the effective collection surface approximation, the total collected charge at the collecting contact resulting from the contribution of a particle track can be evaluated from Eq. ( 8), considering a track as an ensemble of point charges uniformly distributed along a straight segment. We consider the simplified case, shown in Fig. 4, where a ionizing particle perpendicularly arrives to the semiconductor surface. The particle is assumed to lose the energy ∆E along the segment [IF] of length L. Considering an energy of electron-hole pair creation equal to E e,h for the target semiconductor material, the particle creates a total of N = ∆E/E e,h pairs.

For simplicity, we suppose a constant linear energy transfer (LET) value: this charge is thus uniformly deposited along the ion track. In this case, each elementary segment of length dℓ carries an elementary charge δ n 0 supposed punctual:

δ n 0 = N L × dℓ = ∆E E e,h L × dℓ (22)
where ℓ and dℓ are defined in Fig. 4. The integration of Eq. ( 8) along the particle track using Eq. ( 22) gives the total collected charge at the collecting contact:

Q di f f col = qA S N 4πL L 0 dℓ l 2 0 + ℓ 2 (23)
We obtain:

Q di f f col = Q di f f 0 × arctan L l 0 (24)
with

Q di f f 0 = qA S N 4πLl 0 (25)
Finally, and similarly to Eq. ( 9), we can define a collection efficiency for the collection from a line source. From Eqs. ( 24) and (25), we obtain:

η di f f S = Q di f f col qN = A S 4πLl 0 × arctan L l 0 ( 26
)
Eq. ( 26) reduces to Eq. ( 9) with r = l 0 when L/l 0 → 0.

IV. IMPLICATIONS FOR THE EVALUATION OF THE SOFT ERROR RATE OF CMOS CIRCUITS

Eq. ( 24) with the prefactor term defined in Eq. ( 25) shows that the collected charge at the level of a small collecting contact is a function of l 0 , the distance between the track and the contact, as defined in Fig. 4. Solving this equation allows us to define a critical distance l 0 = l crit from the contact below which any ionizing particle (characterized by a certain N/L value) will produce a collected charge of at least a given amount, called the critical charge Q crit . The critical charge is a key factor for measuring the circuit's vulnerability to radiation. It specifies the amount of charge necessary to trigger a transition from a logical low to a logical high or vice versa at a sensitive node of the circuit. The concepts of critical distance and critical charge are illustrated in Fig. 5. The product of the disk surface of radius l crit by the particle flux directly gives the soft error rate (SER), i.e. the rate at which errors occur at the level of the sensitive node (defined by the collecting surface A S in P) of an electronic system or device:

SER = πl 2 crit F ( 27
)
where F is the particle flux (m -2 s -1 ). This SER is typically expressed in FITs (Failure-In-Time), which is a measure of the number of failures that can be expected in one billion device hours of operation 2 . For memory devices, the SER is given in FIT/Mbit, i.e. in FIT for 1024 2 memory cells. We detail in the following the derivation of l crit and SER from the previous development used to evaluate the collected charge. As explained above, to determine l crit we have to solve Eqs. ( 24) and (25) for Q di f f col = Q crit and l 0 = l crit as follows: Unfortunately, l crit cannot be analytically determined from Eq. (28) because this equation is transcendental. Indeed, the term l crit appears two times in the equation, both in the prefactor and in the arctan argument, and only a numerical solving can be envisaged to determine l crit . To remain analytical, we propose in the following to replace in Eq. (28) the arctan function by the well-known approximation:

Q crit = qA S N 4πLl crit × arctan L l crit (28)
arctan(x) ≈ π 2 × x 1 + |x| (29)
Fig. 6 shows the comparison of the arctan function and its approximation, evidencing a very good agreement over all the real axis. Introducing Eq. (29) in Eq. ( 24) in place of the arctan term and rearranging the equation, we obtain:

Q di f f col = qA S N 4πl 0 L × π 2 × L/l 0 1 + L/l 0 = qA S N 8L 2 × (L/l 0 ) 2 1 + L/l 0 (30) Let x = L/l crit , Q di f f col = Q crit
can be rewritten under the form of a polynomial of degree two in x:

x 2 -Bx -B = 0 (31)
with:

B = 8L 2 Q crit qA S N (32)
The positive root of Eq. (31) directly gives l crit :

l crit = 2L B + √ B 2 + 4B (33)
The soft error rate is then expressed by:

SER = πl 2 crit F = 4πL 2 F (B + √ B 2 + 4B) 2 (34)
This SER versus Q crit will be compared to other SER expressions and values in the next section.

V. LINK WITH OTHER DIFFUSION-COLLECTION FORMALISMS BASED ON THE COLLECTION VELOCITY A. Time-independent collection velocity

In many studies carried out under the diffusion-collection framework, the expression of the collected current includes a collection velocity that remains constant over time [START_REF] Palau | Device simulation study of the seu sensitivity of srams to internal ion tracks generated by nuclear reactions[END_REF][START_REF] Palau | Monte carlo exploration of neutron-induced seu-sensitive volumes in srams[END_REF][START_REF] Lambert | Neutron-induced seu in bulk srams in terrestrial environment: Simulations and experiments[END_REF][START_REF] Merelle | Criterion for seu occurrence in sram deduced from circuit and device simulations in case of neutron-induced ser[END_REF][START_REF] Wrobel | A semi-empirical approach for heavy ion seu cross section calculations[END_REF][START_REF] Correas | Innovative simulations of heavy ion cross sections in 130 nm cmos sram[END_REF][START_REF] Correas | Simulation tool for the prediction of heavy ion cross section of innovative 130-nm srams[END_REF][START_REF] Artola | Collected charge anal-ysis for a new transient model by tcad simulation in 90 nm technology[END_REF][START_REF] Uznanski | Single event upset and multiple cell upset modeling in commercial bulk 65nm cmos srams and flip-flops[END_REF][START_REF] Martinie | Underground experiment and modeling of alpha emitters induced soft-error rate in cmos 65 nm sram[END_REF][START_REF] Autran | Physics-based analytical formulation of the soft error rate in cmos circuits[END_REF] . The consequence of this approximation in the present developments is to remove this term from the time integration of the current, that leads to a different expression for the collected charge, called q col (r), diffusing from a point source [START_REF] Autran | Physics-based analytical formulation of the soft error rate in cmos circuits[END_REF] :

q col (r) = qn 0 × A S v col 4πDr ( 35
)
The direct comparison of this expression with Eq. ( 8), that must correspond to the same quantity, forces the term v col to be equal in this case to:

(8) = (35) ⇒ v col = D r (36)
In such an approximation, v col appears to be only positiondependent. Because this expression is obtained by comparing two expressions of the collected charge, which are the results of the current integration over time, Eq. (36) must not be used in Eq. ( 15) to express the conduction current, since it would no longer be in balance with the diffusion current. In other words, this approximation of v col only makes sense at the level of the expression of the collected charge, not at the level of the current.

Eq. ( 36) can be used to calculate an average value of the collection velocity for a particle track defined in Fig. 4 (see Appendix). With such a constant value < v col > and similarly to what has been done in section 3, the integration of Eq. (35) along the particle track gives a different expression of the total collected charge Q col at the collecting contact:

Q col = qA S N < v col > 4πDL L 0 dℓ l 2 0 + ℓ 2 (37)
We obtain:

Q col = Q 0 × arsinh L l 0 ( 38
)
with

Q 0 = qA S N < v col > 4πDL ( 39
)
where arsinh(x) is the inverse hyperbolic sine, defined over the whole real line and given by:

arsinh(x) = ln x + x 2 + 1 ( 40
)
Finally and similarly to Eq. ( 26), we can define a collection efficiency:

η S = Q col qN = A S < v col > 4πDL × arsinh L l 0 (41)
Eq. ( 38) with prefactor defined by Eq. ( 39) represents a priori a severe approximation of Eqs. ( 24) and (25) because the collection velocity v col (which is position-dependent, as established in Eq. ( 36)) is now becoming a constant. This approximation is numerically evaluated in Figs. 7a, 7b and7c. This figure compares the collection efficiency computed from Eq. (41) considering the different average collection velocities defined in the Appendix with the collection efficiency defined by Eq. ( 26) exactly derived from the diffusion current. Results of Figs. 7a, 7b and 7c show that the agreement between the two efficiencies is reasonable for the different < v col > values. The smaller the range of the particle and the greater the distance between the particle strike and the contact, the better the agreement. Results are more satisfactory with < v col > given by Eq. (A1) or Eq. (A5) than with Eq. (A2). The reason is that Eqs. (A1) and (A5) give more weight to the contribution of the particle track which is closest to the collector, in contrast to a uniform average over the entire track.

As a result, average velocities given by Eqs. (A1) or (A5) lead to a slight overestimation of the collection efficiency for l 0 < 1 µm whereas Eq. (A2) clearly underestimates the collection efficiency for L > 2 µm and for l 0 < 2 µm. These numerical results suggest that if an average velocity < v col,4 > is taken as the average of the velocities given by Eqs. (A1) or (A2), the observed underestimation and overestimation should be compensated, and the agreement of the results could then be better. We thus introduce:

< v col,4 >= 1 4   D l 0 + D l 2 0 + L 2   + D 2L arsinh L l 0 (42) FIG. 7.
Comparison of the collection efficiency given by Eq. ( 26) exactly derived from the diffusion current with collection efficiencies computed from Eq. (41) considering the average collection velocities given by: a) Eq. (A1); b) Eq. (A5); c) Eq. (A2); and d) Eq. ( 42).

This result is fully confirmed in Fig. 7d with a very nice agreement observed between the two collection efficiencies for all values of l 0 and L ranging from 0.1 to 40 µm. Fig. 7d shows that the deliberate choice to remove the collection velocity from the integral term in Eq. (37) therefore appears to be numerically justified as long as a correct average velocity of the carriers diffusing from the track is used.

In practice, the error made by considering Eqs. ( 38)-(39) instead of Eqs. ( 24)-(25) should be limited by the different < v col > expressions if the track length L is limited to a few microns or, for longer ranges, if the track-to-contact distance l 0 is typically greater than a few microns.

As in section 4, to obtain the SER one has to solve Eq. ( 38) with Q col = Q crit and l 0 = l crit , i.e.:

Q crit = Q 0 × arsinh L l crit (43)
The main advantage of considering a constant collection velocity is that Eq. ( 43) can be solved analytically in two interesting cases: i) < v col > is reduced to a single numerical value or ii) < v col >=< v col,2 > given in the appendix (Eq. (A2)). In the first case, the SER is identical to the value reported in [START_REF] Autran | Physics-based analytical formulation of the soft error rate in cmos circuits[END_REF] . In the second case, due to the presence of the term arcsinh(L/l 0 ) in the expression of < v col >, this term can be factorized and the resulting SER is equal to: 34) and (44), from the numerical solving of Eq. ( 28), of Eq. ( 43) considering collection velocity given by Eq. ( 42) and from an exponential approximation (Eq. ( 19) in ref [START_REF] Autran | Physics-based analytical formulation of the soft error rate in cmos circuits[END_REF] ) close to the empirical expression proposed by Hazucha and Svensson [START_REF] Hazucha | Impact of cmos technology scaling on the atmospheric neutron soft error rate[END_REF] . Simulation parameters are: A S = 0.25 µm 2 , L = 2 µm, N = 1.38 × 10 6 , D = 61 cm 2 s -1 , F = 10 -3 cm -2 h -1 and v col = 3.5 × 10 5 cm s -1 for Eq. ( 19) in ref. [START_REF] Autran | Physics-based analytical formulation of the soft error rate in cmos circuits[END_REF] In other cases, i.e. for the numerical solving of Eq. ( 28) or Eq. ( 43) with the most accurate < v col > given by Eq. (42), a simple numerical solving of the equation Q col (l crit ) = Q crit can be performed, using the dichotomy or Newton-Raphson methods for example [START_REF] Press | Numerical recipes in c: The art of scientific computing, second edition[END_REF] .

SER

= πL 2 F × sinh Q crit Q 0 -2 (44)
Fig. 8 compares the SER versus Q crit curves for these different equations and resolution methods, considering the analytical SER expressions given by Eqs. ( 34) and ( 44), the numerical solving of Eq. ( 28) and of Eq. ( 43) with the collection velocity given by Eq. ( 42) and finally the exponential expression, close to the empirical expression proposed by Hazucha and Svensson [START_REF] Hazucha | Impact of cmos technology scaling on the atmospheric neutron soft error rate[END_REF] , proposed in ref. [START_REF] Autran | Physics-based analytical formulation of the soft error rate in cmos circuits[END_REF] (Eq. (19) in the reference). The simulation parameters are reported in the caption of Fig. 8. The different expressions obtained in the present work exhibit a non-exponential dependence of the SER on the critical charge. In particular, there is a large increase in the SER at low Q crit values. However, as shown in Fig. 8, over a significant interval of Q crit values (here between 1 and 2.2 fC), all SER(Q crit ) curves exhibit a quasi-exponential behavior. More generally, for relatively limited intervals of Q crit , these distributions can be approximated by an exponential law. This is generally the case for experimental data reported in the literature, where the critical charge ranges are limited for technological reasons. This could explain why such non-exponential variations are not observed and why the exponential dependence of the SER is systematically taken into account in its modeling. Although non-exponential, the expressions of the SER proposed in the framework of this study, in particular Eqs. ( 27)-(28) and their approximation (Eq.( 34)), remain fully compatible with the observed variations of the SER over limited intervals of variation of the critical charge.

B. About the nature and interpretation of collection velocity

In addition to the previous point, it seems that there has long been confusion in the literature about the nature of the collection velocity introduced in diffusion-collection models.

For example, in ref. [START_REF] Palau | Device simulation study of the seu sensitivity of srams to internal ion tracks generated by nuclear reactions[END_REF] but also in refs. [START_REF] Palau | Monte carlo exploration of neutron-induced seu-sensitive volumes in srams[END_REF][START_REF] Lambert | Neutron-induced seu in bulk srams in terrestrial environment: Simulations and experiments[END_REF][START_REF] Merelle | Criterion for seu occurrence in sram deduced from circuit and device simulations in case of neutron-induced ser[END_REF][START_REF] Wrobel | A semi-empirical approach for heavy ion seu cross section calculations[END_REF][START_REF] Correas | Innovative simulations of heavy ion cross sections in 130 nm cmos sram[END_REF][START_REF] Correas | Simulation tool for the prediction of heavy ion cross section of innovative 130-nm srams[END_REF][START_REF] Artola | Collected charge anal-ysis for a new transient model by tcad simulation in 90 nm technology[END_REF][START_REF] Uznanski | Single event upset and multiple cell upset modeling in commercial bulk 65nm cmos srams and flip-flops[END_REF][START_REF] Artola | Seu prediction from set modeling using multi-node collection in bulk transistors and srams down to the 65 nm technology node[END_REF][START_REF] Martinie | Underground experiment and modeling of alpha emitters induced soft-error rate in cmos 65 nm sram[END_REF][START_REF] Autran | Physics-based analytical formulation of the soft error rate in cmos circuits[END_REF] (including certain of our own past work), the authors consider that the collection velocity corresponds to "the average velocity of the carriers in the space charge region of the drain junction" 8 . As discussed below, this definition is not correct for the velocity used in Eq. ( 14) but it has had almost no impact on the published results because, until now, this velocity has not been calculated but rather extracted as a fitting parameter from numerical TCAD (Technology Computer-Aided Design) simulations. These simulations solve the Poisson and continuity equations in a self-consistent manner, usually within the framework of a drift-diffusion transport model. In this sense, they properly take into account the diffusion of carriers during their transport in the semiconductor, from the deposition to the collection points. As a result, the extracted velocity value is correct and the resulting calculated current corresponds to a pure diffusion current. Fig. 9 schematically represents the case of charge diffusion from a point source and its collection by a reverse-biased N + P junction, representative of a sensitive drain in a CMOS circuit. We focus on the particular situation at point P, located at the border of the space charge region created by the junction. On one hand, the carrier flux coming from M gives in P, on the bulk semiconductor side, a current density of pure diffusion expressed by Eq. ( 4) or by Eq. ( 14) with Eq. ( 42). On the other hand, when these carriers cross the boundary of the SCR, they are driven by the electric field F of the junction and give rise to a drift current density. This transformation of the nature of the current should not hide the fact that the continuity of the current densities must be respected: the diffusion current density at the boundary between the neutral region of the substrate (i.e. the diffusion domain) and the extremity of the junction SCR is equal to the carrier drift current density inside the SCR (i.e. the drift domain). The consequence for the charge collection mechanism of a reverse-biased junction is important: carriers arriving at the boundary of the SCR and being literally sucked in by the electric field of the junction are those that contribute to the collected charge. If carrier recombination is neglected, the intensity of the electric field inside the SCR therefore does not matter: for the carriers arriving at the edge of the SCR, their subsequent transport into the SCR does not change the amount of collected carriers. This quantity depends only on the diffusion current density evaluated at the limit of the SCR and integrated on the surface separating it from the neutral zone of the semiconductor. The reverse bias (V R ) of the junction has only one main effect in charge collection: it controls the depth of the SCR and thus modulates the distance between the carrier source and the SCR (distance between points M and P in Fig. 9). This modulation has only a limited effect on the collection of the charges deposited in the vicinity of the collector contact, i.e., for short l 0 distances. Taking the example of an N + P junction with respective doping levels of 3 × 10 20 cm -3 and 10 18 cm -3 (typical values for a deca-nanometer CMOS bulk technology) the extension of the SCR goes from 45 nm to 65 nm as the voltage V R varies from 0.5 V to 2 V. Modulating the depth of the SCR by no more than 20 nm will have an extremely limited effect on the charge collection efficiency of the junction.

VI. CONCLUSION

Despite a consequent amount of modeling and simulation works in the field of diffusion-collection of radiation-induced charges in semiconductor devices, some previous works have admitted questionable points for the understanding of the physical mechanisms involved, in particular in the collection of charges and how such a collection process is controlled. This work has precisely revisited, from the basic equations of diffusion-collection, some results considered as established in the literature. Thus, we have reexamined the case of charge diffusion from a point source and from a linear distribution, and determined analytical formulations for the collected charge from two different formalisms: the theory of diffusion and the general electrokinetic formulation of the conduction. We have shown that for a charge diffusing from a point source and being collected by a remote contact, the collected charge and current do not pass through a maximum at the same time, contrary to what it is widely assumed in the literature: the diffusion current reaches its maximum slightly before the charge. In the case of the conduction current formalism, we have reexamined the concept of collection velocity at the point where the collection current is evaluated. The implications for the calculation of the soft error rate in CMOS circuits exposed to ionizing particles have been explored. Finally, we have shown that in the diffusion-collection approach, the key quantity to evaluate is clearly the diffusion current density at the limit of the SCR of the collector contact. This quantity does not depend on the electric field inside the SCR but only on the distance from the carrier source (initial charge deposition) and on the parameters D and v col which characterize the carrier dynamics in the neutral zone of the semiconductor. The collection velocity included in the expression of the collected current and charge does not depend on the internal parameters of the collector contact; it is a parameter that does not go beyond the exclusive framework of the carrier diffusion mechanism.

FIG. 1 .

 1 FIG. 1. Schematic illustration of the diffusion from a point charge and the collection by a small contact of surface A S .

FIG. 2 .

 2 FIG. 2. Time evolution of the carrier density and the diffusion current evaluated from Eqs. (3) and (6), respectively, for a distance r = 1.5 µm from a point charge source Q = qn 0 . Simulation parameters are: n 0 = 1.52 × 10 6 electrons, D = 13.78 cm 2 s -1 and A S = 0.0323 µm 2 .

FIG. 5 .FIG. 6 .

 56 FIG.5. Illustration of the concepts of critical distance l crit and critical charge Q crit . Any particle (characterized by the ratio N/L) that impacts the disk centered in P and of radius l crit will result in a collected charge above Q crit , then inducing a soft error.

FIG. 8 .

 8 FIG.8. SER versus Q crit computed from analytical Eqs. (34) and (44), from the numerical solving of Eq. (28), of Eq. (43) considering collection velocity given by Eq. (42) and from an exponential approximation (Eq. (19) in ref[START_REF] Autran | Physics-based analytical formulation of the soft error rate in cmos circuits[END_REF] ) close to the empirical expression proposed by Hazucha and Svensson[START_REF] Hazucha | Impact of cmos technology scaling on the atmospheric neutron soft error rate[END_REF] . Simulation parameters are: A S = 0.25 µm 2 , L = 2 µm, N = 1.38 × 10 6 , D = 61 cm 2 s -1 , F = 10 -3 cm -2 h -1 and v col = 3.5 × 10 5 cm s -1 for Eq. (19) in ref.[START_REF] Autran | Physics-based analytical formulation of the soft error rate in cmos circuits[END_REF]

FIG. 9 .

 9 FIG. 9. Schematic illustration of the diffusion from a point charge and the collection by a reversely-biased N + P junction.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

AUTHOR'S CONTRIBUTIONS

All authors contributed equally to this work.

Appendix A: Appendix: Determination of the average collection velocity for a particle track From Eq. (35), different average values for an extended source, i.e. a particle track of length L located at the distance l 0 from the collecting contact (see Fig. 4), can be easily derived.

A first way is to roughly calculate the average value from the values evaluated at the two extremities of the track segment, i.e.:

A second way to evaluate v col is to calculate its average value this time along the whole track segment, i.e.:

Finally, a third way to evaluate v col is to weigh each segment dℓ by the quantity f (r) = 1/r 2 to take into account the weighed contribution of the segment to the total collected charge:

with the normalization constant K given by:

that finally gives: