Angeliki Kritikakou

Olivier Sentieys

Guillaume Hubert

Youri Helen

Jean-Francois Coulon

Patrice Deroux-Dauphin

FLODAM: Cross-Layer Reliability Analysis Flow for Complex Hardware Designs

Keywords: Soft Errors, Radiation, Cross-Layer Reliability Analysis, Single/Multiple-Bit Faults, Fabrication Technology

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Critical embedded applications, typically found in avionics, automotive, railway, defense, nuclear and medical domains, have to provide guarantees in terms of operating safety and reliability. However, due to the increased level of chip integration, the reduced transistor sizes and the lower supply voltages of modern technologies [START_REF] Ibe | Impact of Scaling on Neutron-Induced Soft Error in SRAMs From a 250 nm to a 22 nm Design Rule[END_REF], [START_REF] Dixit | The impact of new technology on soft error rates[END_REF], the hardware designs are becoming more and more sensitive to environmental sources [START_REF] Rehman | Reliable Software for Unreliable Hardware: A Cross Layer Perspective[END_REF], such as radiation. Radiation-induced faults can be caused by particles from the atmosphere and their impact on the reliability must be carefully assessed [START_REF] Baumann | Soft errors in advanced computer systems[END_REF]. Ionizing particles can create charges in semiconductor and their density depends on the ion species and their energy, characterized by the Linear Energy Transfer (LET) [START_REF] Hubert | Musca sep3 contributions to investigate the direct ionization proton upset in 65nm technology for space, atmospheric and ground applications[END_REF]. The transferred energy from particles can (i) corrupt the state of sequential logic, by flipping bits stored in a memory cell or a flip-flop, and (ii) impact the combinational logic, by creating current-voltage transients, known as Single-Event-Transient (SET). The SET is propagated in the forward cone of the impacted combinational cell and it can be eventually latched by sequential logic [START_REF] Seifert | Soft Error Susceptibilities of 22 nm Tri-Gate Devices[END_REF]. As a result, one or several bits are modified leading to Single-Event-Upset (SEU) or Multiple-Event-Upset (MEU). Such errors can jeopardize the system execution, since their appearance in hardware can lead to failures in the application.

Evaluating the behavior of a system under radiation-induced faults is an essential part of the system design process. Two main trends exist, i.e., the first category characterises the induced faults by analysing the radiation impact at the lower hardware design layers, i.e., Technology and Circuit (T&C) Ref.

Layer Fault model T&C Gate RTL Microarch. Application SEU MEU SET [START_REF] Hubert | Musca sep3 contributions to investigate the direct ionization proton upset in 65nm technology for space, atmospheric and ground applications[END_REF], [START_REF] Riera | A detailed methodology to compute soft error rates in advanced technologies[END_REF] ---- [START_REF] Mutlu | Characterization of the impact of soft errors on iterative methods[END_REF] ----- [START_REF] Wang | Examining ace analysis reliability estimates using fault-injection[END_REF], [START_REF] Kaliorakis | Differential fault injection on microarchitectural simulators[END_REF] ----- [START_REF] Wilkening | Calculating architectural vulnerability factors for spatial multi-bit transient faults[END_REF], [START_REF] George | Transient fault models and avf estimation revisited[END_REF] ---- [START_REF] Chang | Hamartia: A fast and accurate error injection framework[END_REF] --- [START_REF] Sartor | A fast and accurate hybrid fault injection platform for transient and permanent faults[END_REF] --FLODAM layers, while the second category is applied at higher hardware and application layers to characterize the system execution under transient faults (or soft errors). Table I summarizes the State-of-the-Art (SoA) vulnerability approaches, and compare them with regards to the layers and fault models they consider. Radiation analysis at technology and circuit layers can take into account the circuit layout, the fabrication technology, the radiation and operational environments. For instance, soft errors due to neutron strikes through a SPICE simulator are characterized in [START_REF] Riera | A detailed methodology to compute soft error rates in advanced technologies[END_REF] and Monte-Carlo simulations are used to compute neutron, proton, heavy ion and α emitter contamination in [START_REF] Hubert | Musca sep3 contributions to investigate the direct ionization proton upset in 65nm technology for space, atmospheric and ground applications[END_REF]. Although these approaches accurately characterize the impact of radiation on the physical circuit, they remain at low hardware design layers and do not analyse the propagation of such faults to the system execution.

For reliability analysis at higher layers, a trade-off exists between the accuracy and the time for analysing complex hardware designs. Fault injection at the application level is fast, but less accurate. It is agnostic of the hardware state, fault injection occurs only between instructions and in application variables [START_REF] Mutlu | Characterization of the impact of soft errors on iterative methods[END_REF]. To improve accuracy, the underlying hardware should be taken into account. Fault injection approaches at the hardware level typically use random fault models following uniform distributions [START_REF] George | Transient fault models and avf estimation revisited[END_REF]. The majority of existing approaches focus on single-bit faults in sequential logic, e.g., the fault model is based on a random single bit-flip in the microarchitectural state [START_REF] Wang | Examining ace analysis reliability estimates using fault-injection[END_REF] and a bit-flip on the value of one storage element [START_REF] Kaliorakis | Differential fault injection on microarchitectural simulators[END_REF]. Some approaches consider multiple-bit flips in sequential logic, e.g., multiple architectural vulnerability analysis is computed for faults affecting a number of contiguous bits in SRAMs [START_REF] Wilkening | Calculating architectural vulnerability factors for spatial multi-bit transient faults[END_REF], [START_REF] George | Transient fault models and avf estimation revisited[END_REF]. Last, approaches consider faults occurring both in the sequential and the combinational logic. For instance, random single and multiple faults, occurring to instructions that use arithmetic units, are analysed in [START_REF] Chang | Hamartia: A fast and accurate error injection framework[END_REF]. An Instruction-Set Simulator (ISS) executes the application and invokes a gate-level simulator to inject and propagate the fault. A hybrid fault injection framework combines Register Transfer Level (RTL) and gate-level simulation, injecting an SET of one clock cycle following a uniform probability [START_REF] Sartor | A fast and accurate hybrid fault injection platform for transient and permanent faults[END_REF]. Existing approaches characterise the impact of transient faults on the system execution, typically considering random faults and uniform distributions.

Although these two categories are complementary, few works consider their combination in order to accurately analyse the impact of the radiation from the environment on the hardware design and application workload under study. A recent cross-layer approach considers technology, circuit, hardware and application layers based on Bayesian models for single-bit faults in memory components [START_REF] Vallero | Syra: Early system reliability analysis for cross-layer soft errors resilience in memory arrays of microprocessor systems[END_REF]. However, multiple-bit faults and combinational logic faults cannot be neglected and should be included in the reliability analysis.

The FLODAM project 1 addresses this limitation by providing a novel cross-layer reliability analysis from the semiconductor layer up to the application layer. The proposed method and associated flow can quantify the risks of faults under a given context, taking into account the characteristics of the environmental radiation, the physical hardware design and the application. The main contributions of FLODAM cross-layer reliability flow are:

• Realistic modelling of radiation-induced faults at the circuit level, considering the environmental conditions, the types and LET of particles, the fabrication technology and the hardware design.

• Efficient gate-level analysis, based on an approach that combines single-cycle simulation with statistical analysis with high confidence level. • An open-source RISC-V processor is used as the case study for the FLODAM flow. The obtained results describe the fault models at the circuit level, the error patterns at the gate level and the vulnerability metrics at the microarchitecture level. FLODAM is a national collaborative research project between industry and academia, funded by the French Ministry of Defence (Direction générale de l'Armement -DGA), at its final phase. FLODAM provides to semiconductor manufacturers and embedded system designers a methodology and design automation tools to quantify, in an accurate and fast way, the reliability of hardware designs against transient faults, caused by single radiation particles of natural origin, as defined in the IEC 62396 standard.

The rest of this paper is organized as follows. Section II describes FLODAM cross-layer reliability analysis flow. Section III presents representative results obtained by applying FLODAM flow on the open-source RISC-V processor. Section IV presents the conclusions and the lessons learnt from this project.

II. FLODAM CROSS-LAYER RELIABILITY ANALYSIS

A. Overview

The overview of the cross-layer reliability analysis flow of FLODAM is depicted in Figure 1. The first step to obtain an accurate analysis is to use fault models that reflect the reality of the system's environment, e.g., actual occurring faults on a given hardware design during a flight from Paris to Los Angeles due to single energy particles. To achieve that, we model radiation-induced faults, taking into account both the specific environmental conditions and the characteristics of the specific hardware. To accomplish that, FLODAM cross-layer reliability analysis flow uses a technology and circuit layer simulation tool that characterises the impact of ionizing particles, under different scenarios, on the hardware design under study. The tool is based on a multi-physics approach that is able to analyse the impact of radiation on the physical design. The output is a set of databases that model the distribution of the transient faults at the circuit layer, depending on the cell type, size, inputs and radiation scenario. The transient faults are Single Event Upsets (SEU) and Multiple Event Upsets (MEU) for sequential cells and Single Event Transients (SET) for combinational cells. Such technology-and circuitlayer analysis is required once per fabrication technology and radiation scenario.

The second step is the gate-level analysis, based on statistical fault injection through a single-cycle simulator, in order to characterise with significant statistical confidence the propagation of radiation-induced faults (modelled at the circuit level during the first step). This simulation is able to analyse logical masking and latching window masking, since the hardware design frequency, the area, the delay, the type of cells and the netlist of the hardware design are taken into account. The output is a set of databases describing singlebit and multi-bit error patterns that avoided masking and finally latched in the hardware design registers. Such gatelevel analysis is applied once per hardware design. The third step analyses the impact of single-bit and multibit error patterns, occurring at microarchitecture layer, on the system execution, taking into account the application workload. This is achieved initially through fault injection using a fast CABA simulator executing the application code. To further improve the analysis time at this step, FLODAM develops a fault injection emulation tool running on an FPGA platform. The microarchitecture-level analysis is applied per application workload.

B. Technology and Circuit Analysis

The FLODAM flow leverages MUSCA-SEP3 [START_REF] Hubert | Musca sep3 contributions to investigate the direct ionization proton upset in 65nm technology for space, atmospheric and ground applications[END_REF] and AT-MORAD [START_REF] Cavoli | Study of atmospheric muon interactions in si nanoscale devices[END_REF] towards the development of the DiaQuant tool in order to analyse the physical impact of radiation to the hardware circuit, therefore considering the environment and the physical circuit of the hardware design. As an output, DiaQuant creates a set of databases of fault models per technology cell, along with their probability to occur depending on the cell type, size and inputs. Figure 2 The environmental models are based on generating environment databases according to the considered missions (e.g., aircraft trajectories, ground trajectories, orbits) and space weather conditions (e.g., solar activity, solar flare). The physical models correspond to the device description, including the circuit layout, fabrication technology, semiconductor active zones, passivation, metallization layers and package. The technology and circuit level analysis is based on Monte-Carlo simulation using a sequential model of the various physical mechanisms occurring when a particle hits a circuit, and potentially leading to a radiation-induced fault. The model describes (i) the particle transport in materials (modifications of the particle primary characteristics via the interaction with the structure, shielding, package and over layers), (ii) the electron-hole pairs and charge generation in semiconductor, (iii) the transport and collection of the charges when electron/holes reach the electrodes, (iv) the transient pulses induced on the different electrodes (drains, sources and taps), and (v) the final effects at the circuit layer. When the circuit is impacted by a particle i of energy E i , the first step is to identify the cells and transistors potentially impacted. Depending on i, E i , the size of the transistor and the positioning of the particle with respect to the Drain-Grid-Source topology, the most relevant induced current is identified. After this analysis, DiaQuant selects a current from an I(t) database, depending on the characteristics of the transistor to be impacted (size, type), the particle and its characteristics, to be injected at the circuit layer. Following the aforementioned approach, a set of databases is generated with fault models per cell from the technology library used for the considered design, and for the different particles encountered in atmospheric and space environments.

C. Gate-Level Analysis

As exhaustive fault injection is not possible for complex hardware designs, the aim of the gate-level analysis is to create statistically representative models at this layer for radiation-induced faults occurring at both combinational and sequential cells of the hardware design. To obtain such fault models with statistical confidence and within reasonable time, the gate-level analysis is performed through statistical fault injection per pipeline stage. This is achieved through singlecycle gate-level simulation using the fault models obtained by the technology and circuit analysis. The number of faults N to be injected is defined from the required confidence level in the statistical analysis, i.e., N = t 2 ×p×(1-p)

e 2
, where t is the critical value related to the statistical confidence interval, e the error margin, and p the percentage of the possible fault population individuals that are assumed to lead to errors [START_REF] Tuzov | Accurate Robustness Assessment of HDL Models Through Iterative Statistical Fault Injection[END_REF].

In order to be able to inject faults, the netlist of the hardware design is extended with an injection block inserted at the output of each cell. With these injection blocks, we can insert SEUs, MEUs and SETs anywhere, at any time and with any duration in the netlist. Note that, in order not to affect the timing characteristics of the hardware design, the propagation delays of all the injection blocks are set to zero. The inputs of the hardware design are randomly generated, e.g., for the execution stage of a processor the inputs are instructions randomly generated from the Instruction Set Architecture (ISA) using random operands. For each such input, a faultfree cycle is executed to obtain the golden reference, i.e., the fault-free output. Based on the technology and circuit layer analysis, the higher the area of a technology cell, the higher its probability to be affected by radiation. Therefore, the selection of the cell, where the fault is injected, it driven by the area of the cell. If the selected cell is of sequential type, the fault is injected directly to the pipeline register. If the selected cell is of combinational type, an SET is inserted into the netlist. The time offset, when the SET is inserted, is randomly chosen within a clock cycle, since radiation may affect the hardware at whatever time instance. The SET duration is given from the databases obtained during the technology and circuit layer analysis. Then, the SET is injected and the output is latched by register at the output of the design stage. If the result in the register is different than the golden reference, the injected fault has led to a single-bit or multiple-bit error. The number and the position of faulty bits are logged, in order to create a set of databases with error patterns.

D. Microarchitecture-Level Analysis

The microarchitecture-level analysis is based on fault injection that is able to evaluate the masking due to the microarchitecture and the application. The fault injection is driven by the error patterns, modelled during the gate-level analysis and describe the single-bit and multiple-bit faults to be injected at the hardware design registers. Prior to any fault injection, we execute the application under study, without faults, in order to obtain a set of golden references: (i) the application output, (ii) the system state (memory and registers), and (iii) the number of cycles required for the execution of the application. Then, the fault injection tool executes the application and injects faults to the hardware design registers, while the application runs. The cycle to inject the faults is chosen randomly between the first cycle and the total number of cycles needed for the fault-free execution, since radiation may impact the system any time. The location, where the faults are injected, is driven by the size of the combinational and sequential logic of the overall hardware design. The larger the area, the higher its probability to be selected. The characteristics of the fault (i.e., how many and which register bits are affected), to be injected in the register of the selected hardware pipeline stage, are provided by the gate-level error patterns. The more times a specific error has appeared, the higher is its probability to be injected, during the microarchitecture analysis. After the fault injection and upon application termination, the results are compared to the golden references, categorising the impact of faults as:

• Hang (H): The execution time has exceeded a threshold, and thus, it is assumed that it has entered an infinite loop. • Crash (C): The execution of the application has terminated unexpectedly and an exception has been thrown (out of bound memory access, misaligned PC, hardware trap, etc.) • Application Output Mismatch (AOM): The application output is different than the golden reference. The FLODAM microarchitecture-level analysis is initially implemented by a Cycle-Accurate and Bit-Accurate (CABA) simulator. To further improve the time of fault injection campaigns at this level, a fault injection hardware emulation architecture running on an FPGA has been developed to inject faults at the flip-flops of the hardware design following the patterns issued from gate-level analysis. In this way, fault injection campaigns are performed at the speed of the design on the target FPGA.

III. CASE STUDY: VULNERABILITY ANALYSIS OF A

RISC-V PROCESSOR An open-source 32-bit RISC-V processor [START_REF] Rokicki | What You Simulate Is What You Synthesize: Designing a Processor Core from C++ Specifications[END_REF] is used as a case study, consisting of a standard 5-stage pipeline, including a forwarding mechanism, a hardware multiplier in its execution stage, a Register File (RF) with 32 registers in the write-back stage, and an instruction and data levelone (L1) cache memory. The considered processor is fully Multi-cycle operators stall Fig. 3: RISC-V core with 5-stage pipeline [START_REF] Rokicki | What You Simulate Is What You Synthesize: Designing a Processor Core from C++ Specifications[END_REF] specified using C++ and has been synthesized to the gatelevel using Mentor Graphics Catapult High-Level Synthesis and Synopsys Design Compiler with a target frequency of 500 MHz. The target fabrication technology is 28nm FDSOI from ST-Microelectronics using a supply voltage of 1.0V. The sequential logic of the processor (including RF) corresponds to 45.85% of the total area and the combinational logic to 54.15%.

A. Technology and Circuit Layer Results

Thanks to the DiaQuant tool, we can obtain fault models for 28nm FDSOI technology cells considering different radiation scenarios. In this use case, we considered several flights departing from Paris to New York, Los Angeles, Johannesburg and Sao Paulo, and vice versa. Each flight is described from a set of realistic flight plans from the Eurocontrol Demand Data Repository (DDR) databases. Each flight plan is characterised by the information regarding the timing (date, time) and the trajectory (latitude, longitude and altitude). Figure 4 represents the set of considered flights. We can observe the diversity of the flight plans for a given flight, e.g., the maximum latitude of a flight from Paris to Los Angeles is between 67°and 45°N

orth, which implies a potential variability in the radiation impacting the aeroplane during the flight.

For each flight plan, the natural radiation environment can be calculated, which describes the differential energy spectrum of neutron, proton and muon for each point of the flight plan trajectory. Analysis takes place considering different particle types, particle energies and technology cells. The transistors potentially impacted when a particle i of energy E i hits a technology cell and the most relevant induced currents are identified. A current database is created depending on the transistor size and type, the particle and its characteristics. Then, current injection at the circuit level takes place leading to a set databases with fault models characterised with distributions of SET widths in the cell. For instance, Figure 5 summarizes the distribution of the width of the SET pulses created when neutrons hit the 28nm FDSOI cells, normalized to the cell Fig. 4: Flight plans from and to Paris area and input sizes. In this radiation scenario, we considered an LET equal to 58M eV.cm 2 .µm -1 , a temperature 25°C and an incidence angle of 90°. Overall, 212, 000 injections took place on 91 logic gates of the fabrication technology and the analysis took XX time. We observe that the majority of SET widths are below 500ps (89.457%), while a minority is larger than 1ns (0.651%) and than 1.5ns (0.13%) for our case study.

B. Gate-Level Results

The gate-level analysis provides the error patterns, created due to injected faults at the circuit layer, that managed to avoid logical and timing window masking and latched at the output register. Considering a 99.8% confidence interval with 5% error margin for each input set values, we injected 10 3 for each of the 10 3 different inputs per pipeline stage (10 6 total fault injections per stage). The time required to perform gatelevel analysis for the five RISC-V pipeline stages is 1, 039s with Questa Advanced Simulator 10.7b running on a secondgeneration Intel Xeon CPU. Figure 6 depicts the gate-level analysis results for the RISC-V execution stage, considering an SET pulse width equal to 400ps and an operating frequency of 500MHz. Figure 6a shows the probability of each bit of the output register of the execution stage to be erroneous. Some bits in the register have higher error probabilities, such as the bits corresponding to the ALU output (bit 32 up to bit 63) and data resulting from logical operations on values from Control and Status Registers (CSRs) (bit 94 up to bit 125). Figure 6b shows how many bits were faulty due to a single SET. The higher number of observed erroneous bits for the execution stage is 52 bits, which is 41.3% of the pipeline register size. Experiments for different pulse widths have shown similar results. Regarding SET propagation, for pulse widths 100ps and below, the SETs are not often propagated within the latching window of the register, and with a pulse width below 50ps, no latching of SETs is observed.

C. Microarchitecture-Level Results

Although the error patterns managed to survive at the microarchitecture-level, they may not necessarily lead to a visible error during the system execution. Using a 99.8% confidence interval and a 1% error margin, we injected 23, 874 faults, using the error patterns obtained from the gate-level analysis. The microarchitecture layer output is the distribution of vulnerability metrics. Figure 7 compares the vulnerability results for four benchmarks (matrix multiplication, qsort, strsearch, blowfish), obtained by FLODAM flow and by typical approaches which inject SEUs in the microarchitecture with a uniform fault occurring probability for all registers. Overall, with the FLODAM flow, we observe that less faults are masked, compared to standard microarchitecture-level analysis.

The CABA simulator performance is 18.2 million instructions per second on average, using an 8 th generation Intel i7 CPU running at 2.40 GHz. The time required to perform the

IV. CONCLUSIONS AND LESSONS LEARNT

FLODAM proposes a cross-layer reliability analysis from the semiconductor layer up to the application layer, able to quantify, in an accurate way, the reliability of hardware designs against soft errors, caused by single radiation particles. The main highlights and insights of FLODAM results are:

• Particles with different characteristics have different impacts when hitting sequential and combinational cells of the design. As a result, the environmental conditions should not be neglected during the reliability analysis of the system in order to obtain accurate results. • Particles impacting combinational logic can lead to SET of different width pulses. Depending on the characteristics of the particles and the hardware design, SET can be latched in the registers and alter a significant number of bits, leading to a significant number of MEUs. • The MEUs derived from SETs can be significantly large in size and they not disturb only adjacent bits. Thus, radiationinduced faults should not be modeled only with SEU, but also with (significantly large) MEUs. • Performing analysis considering the impact of radiation to combinational logic leads to less masked faults at the application layer, compared to typical methods using randomly injected faults based on uniform distribution. • Combining single-cycle gate-level fault injection and microarchitecture-level fault injection, the reliability analysis is significantly reduced compared to full gate-level fault injection. Further time reduction is obtained by migrating the microarchitecture-level analysis from the CABA simulator to the FLODAM fault injection hardware emulator.

  1 https://flodam.fr

Fig. 1 :

 1 Fig. 1: FLODAM cross-layer reliability analysis flow.

Fig. 2 :

 2 Fig. 2: DiaQuant technology and circuit layer analysis tool.

Fig. 5 :

 5 Fig. 5: SET distribution normalized to cell area and input sizes

  (a) Error probability for each register bit (b) Size of error patterns.

Fig. 6 :

 6 Fig. 6: Gate-level results of the RISC-V execution stage.

Fig. 7 :

 7 Fig. 7: Vulnerability metrics.

TABLE I :

 I Comparison with representative SoA approaches.

  Table II depicts the relative area the pipeline stages.

	Pipeline stage Fetch Decode Execute Memory WriteBack
	Area	6.01% 11.02% 35.47%	5.10%	42.41%

TABLE II :

 II Relative area of RISC-V pipeline stages.

  TableIII. The FLODAM fault injection emulation tool, currently without any optimisation, can run at 50 MHz on FPGA Xilinx Zynq XCZ 7045, improving by a factor of 2.5× the time required for the fault injection campaigns. Note that, performing similar analysis using full gate-level methods would require a prohibited time.

	Bench.	matmux	qsort	blowfish strsearch	average
	Time 388min 40sec 327min 20sec 466min 427min 20sec 406min 32sec

TABLE III :

 III Time of microarchitecture-level analysis.


Asmaa Hadj Ahmed

Jean-Vincent Daurelle
email: jean-vincent.daurelle@univ-amu.fr

Vincent Fourmond
email: vincent.fourmond@imm.cnrs.fr.

Optimizing the mass transport of wall-tube electrodes for protein film electrochemistry

Keywords: hydrodynamic electrode, wall-tube, computational fluid dynamics, protein film electrochemistry

Protein Film Electrochemistry (PFE) is a technique in which an enzyme is directly wired to an electrode and its catalytic turnover rate is measured under the form of an electrical current. This technique has proved useful for the study of a number of enzymes, but requires fast transport of the enzymatic substrate towards the electrode. In a previous work (Fadel et al, Phys. Chem. Chem. Phys., 2019, 21, 12360), we have proposed a new design based on the wall-tube electrode that provides better transport than the rotating disc electrode, which is usually employed for PFE studies. In the present work, we use computational fluid dynamics to explore the effects of the various parameters of the cell, and propose a semi-empirical formula suitable to predict the mass-transport coefficient and the wall shear stress on the electrode. We use a 3D-printed cell to experimentally validate our predictions.

Introduction

Protein Film Electrochemistry (PFE) is an electrochemical technique that is used for studying the mechanism of redox enzymes. It consists in immobilizing a film of enzymes on an electrode in a configuration where the electron transfer is direct, and measuring the response in current to changes in conditions [1,2]. The current generated from the exchange of electrons between the enzymes and the electrode is proportional to the turnover rate of the enzyme, which makes it possible to learn about various aspects of the reactivity of the immobilized enzyme [3][4][5]. As the enzymatic substrate is consumed at the electrode by the catalytic reaction, PFE requires fast transport of the substrate towards the electrode. Sometimes, the enzymatic reaction is slow enough that passive diffusion suffices [6]; however, most enzymes require faster transport, which is usually provided by using a rotating disc electrode (RDE) as working electrode [7]. However, our group has recently reached a limitation of the RDE setup while working on CO dehydrogenases, the enzymes which catalyze the reversible oxidation of CO to CO 2 [8]: these enzymes are so fast that even it the highest practical rotation rates, mass-transport of CO is limiting [9,10], which greatly complicates the study of this enzyme.

To overcome this limitation, we have chosen to design a new electrochemical cell with improved transport properties. The mass transport should be as homogeneous as possible, to avoid heterogeneity of substrate concentration on the electrode, which would greatly complicate the studies as the enzymatic response is highly non-linear in function of the substrate concentration. This property is known as "uniform accessibility" [11]. Moreover, since the cell will be used by immobilizing enzymes on the surface of the working electrode, a particular interest should be given to the forces applied by the fluid at the surface (the shear stress) since it could be one of the main reasons for the desorption of the film of enzyme [12,13]. Jet electrodes, in which the flow impinges normally from an inlet toward a disk electrode [14], are good candidates for solving the transport limitation since they ensure high and controllable mass transport [15]. The jet electrodes can be divided into two sub-types depending on the size of the electrode with respect to the size of the inlet: the "wall jet" configuration [16], in which the diameter of the inlet is smaller then the one of the electrode, which is widely applied for electrochemical detection in liquid chromatography [17,18], and the "wall tube" configuration [19], in which the electrode is smaller than the inlet. Only the latter provides uniform accessibility [20,11].

After an initial screening of a number of geometries using computational fluid dynamics, we have recently proposed a design based on wall-tube electrodes [21], which provides a factorof-three improvement with respect to the classical RDE setup (assuming a rotation rate of 5000 rpm). However, in our previous work, we relied only on computational fluid dynamics simulations to predict the mass transport in given conditions. This contrasts with the ease of use of the RDE, for which a reliable analytical formula is available for predicting the mass transfer properties [22]. Hydrodynamics in impinging jets have been addressed by many studies in mass and in heat transfer [23,24]. Homann and coworkers developed a mathematical solution to describe the boundary layer flow in the stagnation region of a uniform flowing field on an infinite plate [25]. This solution was extended to the case of a uniform impinging jet, where it was proved that a boundary layer of a constant thickness is present in the stagnation region. Chia and coworkers demonstrated that the mass transfer rate at the impinged surface is rel- atively uniform in the case of wall-tube electrodes [26]. Semiempirical solutions of mass transport in a uniform impinging jet were developed in previous studies [27][28][29]11] but they were limited to certain configurations. In spite of the amount of studies about the mass transport in jet electrodes, there were only few that addressed shear stress.

In this work, we have developed semi-empirical formulas suitable to predict the mass-transport properties and the shear stress for wall-tube cells such as the one we have previously proposed [21]. We have used an approach based on computational fluid dynamics, using first a systematic sensitivity study to determine the most influencing parameters, and then focusing on them to propose an analytical formula that reproduces our data within 10%. We then demonstrate the validity of our formulas by verifying them experimentally using the reduction of ferricyanide in a 3D-printed cell.

Results

Figure 1 represents a schematic view of the experimental setup, similar to the one we have used in our previous work [21]. The fluid comes through the inlet in the center of the scheme (orange downward arrow), impinges on the working electrode (the gray area at the bottom) and is evacuated via symmetric outlets that are initially inclined with respect to the vertical axis (blue upward arrows). The reference and counter electrodes are near the working electrode (they are not represented in figure 1). The cell is designed in two parts, one with the inlet and outlets, and the other with the electrodes. For more detailed information, see ref. 21. Numerical simulations were performed using commercial finite volumes modelling package starccm+ from CD Adapco to study the flow and transport properties of the cells in silico. We used a physical model that is based on a laminar incompressible flow at steady state regime in a 2D axis symmetric domain with a mesh of a base size of 50 µm. The size of the mesh cells decrease when approaching the working electrode so we can properly model the strong gradient of concentration in this zone (diffusion layer). The boundary conditions that were used are as following: we imposed a given velocity and a fixed concentration in the inlet of the cell, a split value condition at the outlets, a non slip condition on the walls. At the working electrode, we assumed an infinitely fast catalytic reaction that consumes all the reaction's substrate, so its concentration is zero at the surface of the electrode. More details about the meshing and the boundary conditions can be found in supplementary section S1.

These simulations were used to compute the velocity profiles in order to determine the value of the wall shear-stress at the electrode, but, most importantly, to compute the flux of electroactive species towards the electrode, in order to determine the mass-transport coefficient m, defined by:

j = m × (c(∞) -c(0)) (1)
in which j is the flux of species at the electrode, c(∞) the concentration of species in the upstream flow and c(0) the concentration of species at the electrode (in our case, c(0) = 0). In contrast to our previous work [21], we simplified the 3D design into a 2D design with a rotational invariance, which provided both finer grained results, less discretization artifacts, and faster computation time. We show in supplementary section S2 (and figure S3) that the two models predict the same flux within less than 10% for values of r between 0 and at least the value of the inlet radius.

Sensitivity study

In a first part, we studied the influence of the design parameters of the setup on the properties of the cell. We chose to focus on six parameters (figure 1): P1, the inlet radius, P2, the distance between the center of the cell and the center of the outlets, P3, the diameter of the outlets, P4, the distance between the inlet and the electrode, P5, the angle of the outlets, and P6, the input flow rate. The electrode radius plays also a very important role in the properties of the setup, but as it can be varied at will by just changing the size of the region in which the flux is considered (see below), we haven't included it as a parameter.

We used a design of experiments approach to systematically study the influence of these six parameters on the masstransport properties of the cell. We used a full-factorial design with two levels [START_REF] Antony | Design of experiments for engineers and scientists[END_REF], which consists in defining a "low" and a "high" level for all the parameters and running simulations for all the possibilities (here, 2 6 = 64 combinations). For the levels, we used values that correspond to the limits of what we could experimentally build (see table 1).

We ran the 64 simulations, extracted the mass-transport coefficient (equation ( 1)) and applied the statistical treatment described in ref. [START_REF] Antony | Design of experiments for engineers and scientists[END_REF] The single effects of all parameters on the simulated mass transport coefficient from the sensitivity study for r elect = 0.1 mm. Average is the average of the mass-transport coefficient over all the 64 simulations. The list of all the effects including multi-parameter changes is found in supplementary table S1 (section S3).

for each parameter (and combination of parameters), which are the difference between the average values of all the "high level" simulations and the average values of the "low level" simulations. The greater the effect (in absolute value), the more impact the parameter has on the mass transport. Table 2 lists all the single-parameter effects. It shows that, out of the six parameters, only three have a significant influence: P1, P4 and P6, respectively the inlet radius, the inlet-to-electrode distance and the flow rate (the full list of effects, including the effect of jointly changing two or more parameters can be found in supplementary Table S1, section S3). The inlet radius has a strong negative effect, the flow rate a slightly smaller positive effect, and the inlet-electrode distance has about 10 times less effect as the inlet radius. The effects of the other parameters, are at least one order of magnitude smaller than that of P4. These results are in line with other works, which found that the inlet diameter, the flow rate and the distance between inlet and electrode were the most important parameters [16,29].

Derivation of semi-empirical formulas

In a second step, we proceeded to propose semi-empirical formulas to predict the values of the mass-transport coefficient and the wall shear-stress, restricting ourselves to the dependence on the most influencing parameters determined in the first step.

Theoretical background: the hydrodynamic constant. We consider a stationary incompressible flow. According to Homann [25], the axial (v) and the radial (u) velocities in the stagnation region can be written under the following forms:

u = arφ (η) (2a) v = -2 √ aνφ(η) (2b)
in which a is a constant, called the "hydrodynamic constant" (of dimension the reciprocal of time), η is the dimensionless axial coordinate, and φ(η) is a function. η is defined as:

η = z a/ν ( 3
)
where ν is the kinematic viscosity.

Frossling proposed a Taylor expansion for the function φ(η) [START_REF] Frossling | [END_REF]. For the purposes of this article, we have truncated the expansion to its first term [29]:

φ(η) = 0.656 × η 2 (4)
This expression is highly general since it has been applied to wall-jet cells [29] but it is also valid in the case of the rotating disc electrode. We first demonstrate that the knowledge of the value of a is sufficient to determine the mass-transport coefficient and the shear stress; we then focus on extracting the values of a from the simulations and deriving an empirical formula for the prediction of a. Note that the actual value of the coefficient in equation ( 4) has little importance here, since changing the coefficient is equivalent to changing the value of a; we have used 0.656 to be consistent with previous studies [START_REF] Frossling | [END_REF]. We now compute the concentration profile c of the electroactive species in the solution. The electroactive species is consumed at the electrode, with a rate supposed to be sufficiently fast that c = 0 at the electrode surface. Similarly to what has been done before [29], we assume that the concentration is independent of the radial coordinate, so that c is only a function of η. Under this hypothesis, and using the expression of the velocities in (2), the steady-state convection-diffusion equation for c can be expressed as:

d 2 c dη 2 + 2 Sc φ(η) dc dη = 0 (5)
where we have introduced the Schmidt number,

Sc = ν/D (6)
Double integration of (5) yields:

c(∞) -c(0) κ 0 = ∞ 0 exp(-2 Sc Φ(η))dη (7)
in which κ 0 is an integration constant (the value of the first derivative of c at 0), c(0) is the concentration at the electrode, c(∞) at a large distance from the electrode (in the bulk), and Φ is a primitive of φ defined by:

Φ(η) = η 0 φ η dη (8)
Equation ( 1) for the mass-transport coefficient m can be rewritten thus:

D dc dz = m × (c(∞) -c(0)) (9)
Integrating of ( 7) using the expression of φ(η) given in ( 4) and injecting the result into equation ( 9) yields:

m = β a 1/2 D 2/3 ν -1/6 ( 10
)
where β = 0.85002. Note that equation ( 10) is formally identical to the equation giving the mass transport coefficient for a rotating disc electrode, if one uses a = 0.533 × ω, in which ω is the angular velocity of the RDE. The wall shear stress τ w is defined as:

τ w = ρν × du dz z=0 (11)
Using the expressions of u and φ(η) from equations ( 2) and ( 4), we obtain:

τ w = 1.312 × ρν 1/2 a 3/2 r (12)
Equations ( 12) and (10) show that it is possible to deduce m and τ w from the value of the hydrodynamic constant a.

Determination of the hydrodynamic constant as a function of the parameters. With the aim of determining the dependence of the hydrodynamic constant a on the flow rate Q v , the inlet diameter r jet and h the distance nozzle-electrode, we ran a series of 6 × 6 × 3 = 108 simulations with all the possible combinations of the parameter values listed in table 3. For all these simulations, we have extracted the value of a by fitting either equation (2a) or (2b) to the values of the velocities in a neighbourhood of the electrode (see supplementary figures S4 and S5 in supplementary section S4 for more information). Figure 2 shows the 108 values of a, plotted as a function of the inlet radius r jet . The data show a clear decrease of a as r jet increases, along with a strong increase as Q v increases (the different values of Q v are distinguished by the shape of the points). The dependence on h is less marked, but it is nevertheless visible that a decrease in h leads to a slight increase in a.

To remove the largest part of the influence of the parameters on a, we have worked on a dimensionless hydrodynamic constant a * defined this way:

a * = a π r 3 jet Q v ( 13
)
We consider here that the hydrodynamic constant only depends on four dimensional parameters: r jet , Q v , h, and the kinematic viscosity ν, that have two independent dimensions in total (time and distance). The π theorem therefore states that it can be expressed as a function of two independent dimensionless parameters. We have chosen the following:

α 1 = r jet h α 2 = Q v νh (14)
We sought an approximation for a * in the form of a simple power law:

a * = σ × α 1 β 1 × α 2 β 2 (15)
in which σ, β 1 and β 2 are constants to determine. We used a simple systematic search to find the values of σ, β 1 and β 2 that yield the minimum root mean squared difference between the predicted and measured values of a * . We found:

σ = 3.04993 β 1 = 0.1685 β 2 = -0.0995 (16)
Figure 3 represents the values of the a * predicted from equation (15) with the coefficients in (16) of a * determined from the simulations. The data show that all the points from the simulations lie in the 10% region on either side of the y = x line, showing that the formula can predict hydrodynamic constants with a high accuracy.

Expressions for the mass-transport coefficient and the shear stress. Substituting (15) back into equations ( 10) and ( 12) yields the following expressions:

m = 0.83753D 2/3 ν -0.11692 Q v 0.45025 h -0.0345 r jet -1.41575 (17a)
τ w = 1.255ρν 0.64925 Q v 1.35075 h -0.1035 r jet -4.24725 r (17b)

Figures 4 and supplementary figure S6 show that the above expressions correctly predict the values of m and of τ w deduced from the simulations. While this is not surprising, considering that the values of m, τ w , and a are all deduced from the same velocity profiles, the quality of the predictions confirm the derivations above, and show that the semi-empirical formulas (17) reliably predict the results of our 2D simulations. The dependency on a of equations ( 10) and ( 12) imply that a 10 % error on a corresponds to a 5 % error on m and a 15 % error on τ w .

Homogeneity of the mass transport

For the cell to be used for electrochemical studies of enzymes, it is important that the flux is homogeneous, so that the concentration of the enzymatic substrate at the electrode is homogeneous [21]. In the simulations, we have used a very large electrode surface (r elec = 1.25 mm) over which we assumed c = 0. This made it possible to systematically probe the flux over a large surface and determine the maximum radius of the electrode which receives a homogeneous flux. We assume that electrode parts further away from the axis cannot influence the flux in electrode parts closer to the axis. This assumption is compatible with the direction of the flow, since it seems very unlikely that consumption of the electroactive species further downstream has any influence on the concentration of species upstream. We have verified that this is the case in supplementary section S5.

Figure 5 shows the flux of species at the electrode as a function of the distance from the axis for two different simulations. The yellow curve (for which r jet = 0.3 mm and h = 0.15 mm) shows the typical behaviour observed in most of the simulations. The flux hardly changes between r = 0, and a certain value, here r ≈ r jet = 0.3 mm, and it then decreases with r. On the other hand, the green curve (for r jet = 0.5 mm) is less common, but was found in a few simulations, especially at high values of r jet /h ratio. In this case, after a region in which the flux varies little, it first increases significantly around r = 0.4 mm < r jet , peaking about r = r jet at a value 30% higher than the value at r = 0 before decreasing slowly at higher values of r. For each of the simulations, we have determined the value r max for which the flux does not vary more than 10% between r = 0 and r = r max . The values are plotted in figure 6 as a function of r jet , and for the different values of h and Q v used in the simulations. Overall, the points follow a linear relationship as a function of r jet , as is attested by the fact that they mostly lie between the y = 0.6x and y = x lines. In general, increasing the flow rate increases r max : the circles (for which Q v = 1 mL/min) correspond to the lowest values of r max , while the triangles (Q v = 10 mL/min) correspond to the highest values of r max .

However, the individual dependence of r max on r jet can sometimes be more complex, as is attested by data for h = 0.15 mm and Q v = 1 mL/min in figure 6. For r jet ≤ 0.4 mm, the value of r max increases with r jet , but for larger values of r jet , the value of r max decreases; this behaviour is related to the occurrence of the "atypical" behaviour discussed above in which the flux increases significantly about r = r jet , before decreasing again (green curve in figure 5).

From the overall data in figure 6, we can conclude that the requirement that the variation in flux (or mass-transport coefficient) remains smaller than 10% is fulfilled when the radius of the electroactive area is smaller than 0.6 × r jet .

Experimental validation

To validate the relevance of the prediction of equation ( 10) in a real experimental setting, we built, using 3D-printing, a cell corresponding to r jet = 0.3 mm and h = 0.35 mm. We performed cyclic voltammetry of a K 3 Fe(CN) 6 solution for a number of flow rates. The results are plotted in figure 7. Figure 7A shows the raw voltammograms, which show plateaus at low potential that are typical of a transport-limited reduction process. The fluctuations visible are linked to the argon flux necessary to keep the system as anaerobic as possible. From the values of the plateau current, it is possible to deduce the masstransport coefficient. We have plotted it against the power 0.45 of the flow rate in figure 7B (circles), along with the prediction of equation (10). The agreement is excellent, fully validating the whole approach.

Discussion

In this article, we have built upon our previous work, in which we have described a wall-tube cell designed for the studies of very fast enzymes [21]. Although we demonstrated that the cell was already improving upon the mass-transport properties of the RDE, which is commonly used for studying enzymes [1,2], our previous work relied on CFD simulations for predicting the mass-transport properties of the cell, which is far too time-and resource-consuming for practical uses. Instead, for practical operation of the cell, and fine-tuning of its properties (geometric parameters), analytical formulas are required. The present work arises from attempts to i) validate the previously proposed formulas (or propose new ones) to accurately predict the mass-transport in our configuration; ii) verify the conditions under which the mass transport can be considered homogeneous and iii) propose formulas for predicting the shear stress, which has not been addressed in the literature before. The computation of the shear stress is important because it is likely that excessive shear stress could lead to washing of the enzymatic film [13], which has to be prevented as much as possible.

We used a two step approach to provide semi-empirical formula for predicting both the mass-transport coefficient and the shear stress values. First, with a coarse sensitivity study using a design of experiments approach applied to mass-transport values determined from simulations, we showed that only the flow rate, the jet radius and the distance between the inlet and the electrode have significant influence on the mass transport coefficient. Then, in a second approach, we use finer grained simulations with a greater number of parameter values to explore more systematically the parameter space. We used these data to propose semi-empirical formulas for both the mass-transport coefficient and the shear stress in the system. These formulas were able to reproduce simulated mass-transport coefficients and shear stress values with good accuracy, and were also very accurate to predict the experimental current obtained in a 3Dprinted cell (figure 7).

Table 4 summarizes some of the formulas available in the literature for the prediction of the mass-transport coefficient in wall-tube or wall-jet geometries, along with the smallest errors, the largest errors and the root-mean-squared deviation between the predicted value of mass transport and that measured on the 108 simulations of the second step of this work. For most of 4: Standard deviation and maximum deviation of the difference between the predicted and the simulated mass-transport coefficient for the case of the 108 simulations. g(Sc) is an asymptotic series whose development is: g(Sc) = 1 -0.084593Sc -1/3 -0.0016368Sc -2/3 -0.0057398Sc -1 + 0.0014288Sc -3/4 + ... the formulas, the root-mean-squared deviation is in the order of 10% or significantly more, while the root-mean-squared deviation with respect to equation ( 17) is less than 3%. Only the prediction of Chin and Tsang comes reasonably close to our proposition, with less than 5% error in average, and a maximum error not much above that of (17). We could not find formulas that predict the wall shear stress in wall-jet and wall-tube electrodes in the literature, perhaps because the main objective of the published works is to predict the value of the current that one can obtain from a given experimental setup, but also probably because it is experimentally more challenging to measure the wall shear-stress, and it is hence harder to verify the given formulas experimentally. Rather than using an analytical formula to predict the shear stress, Esteban and coworkers use a combination of equations (10) and (12) to predict the shear stress from an experimentally determined mass-transport coefficient [12], in the context of corrosion studies. Here, we provide an analytical formula for predicting the shear stress. Although we could not validate our predictions for the shear stress experimentally, considering that both the mass-transport coefficient and the shear stress are derived from the same equations, based on the determination of the hydrodynamic constant a, we are confident that a good accuracy in the prediction of experimental mass-transport coefficients is an indication of the quality of the prediction of the shear stress too. It should be noted that, provided the equations ( 10) and ( 12) are valid (which is the case for both the wall-tube electrode and the RDE), the shear stress is related to the mass-transport coefficient through the following formula:

τ w = 2.13 × ρ ν r D 2 × m 3 ( 18
)
so that the shear stress scales as the cube of the mass transport coefficient. Therefore, the only way to greatly increase the mass-transport coefficient without a significant increase in the shear stress is to decrease the electrode size, since the shear stress is proportional to the distance from the rotational symmetry axis. An important requirement to conduct protein film electrochemistry studies is the need for uniform accessibility: the masstransport should be homogeneous across the whole surface of the electrode, so that the concentration of substrate as perceived by all the molecules of enzymes is homogeneous. The rule-ofthumb found in the literature is that uniform accessibility is obtained in a wall-jet system when the electrode is smaller than the inlet [12]; this configuration is known as wall-tube. However, our data (figure 5) suggests that there can be significant heterogeneity even when the wall-tube condition is satisfied, and that for a more homogeneous mass-transport coefficient, a condition of r electrode ≤ 0.6 × r jet is necessary. Our data also show that, under certain conditions, the center of the electrode does not correspond to the maximum of the mass transport coefficient, but that the transport increases slightly up to the size of the inlet, before decreasing (figure 5, green curve). MacPherson and coworkers experimentally observed a similar behaviour on a microjet electrode, which is similar to the setup we propose here, save for the smaller dimensions. Using electrodes significantly smaller than the inlet size, they could use a micropositioning device to map the mass transport coefficient as a function of the position of the electrode, and they also observed a slight increase when moving away from the center of the inlet [32].

Other works have emphasized the presence of recirculation patterns in wall-jet or wall-tube setups, in particular in the socalled "wall-jet" region, in which the radial velocity starts to decrease. These were predicted in some simulations [33], also in the case of the RDE [34], and observed experimentally in the case of the microjet electrode [35]. In the case of our simulations, we could not find traces of recirculation patterns in all the simulations (see for instance supplementary figure S10). Presumably, this is due to the fact that the horizontal flow proceeds in a restricted space of height h comparable or even smaller than the inlet diameter, which leaves little space for recirculation patterns to occur.

Equation (17) shows that there cannot be an "optimal" setup, since the mass-transport coefficient can be made arbitrarily large by decreasing the size of the jet. A practical setup must therefore be a compromise between the mass-transport capacities obtained at a given flow rate and the practicality of building a cell with a small jet diameter, and an even smaller electrode diameter. The possibility to make very thin platinum electrodes were exploited by MacPherson and coworkers to build a micro- jet setup with an electrode size of 25 µm and a nozzle diameter in the 80 to 120 µm range, with the highest mass-transport reported to date with a wall jet system [36]. However, this sys-tem requires a horizontal micropositioning device to ensure the electrode is centered with respect to the inlet. With the application to PFE in mind, larger electrodes will have to be used, because the electrode of choice for PFE studies is the pyrolytic graphite edge electrode [2,1,7], which cannot easily be machined to sizes below 0.4 mm. Together with the practical implications of using 3D-printed parts for the inlet, this sets a reasonable size of the inlet diameter around 0.6 mm, which is what we have used in the experimental validation of figure 7. However, the formulas and guidelines proposed in this article are likely valid for even smaller ranges, which means they could be used for much smaller electrodes.

The motivation of this work is to provide faster mass-transport than the setup usually employed for PFE studies, the RDE, as this setup, even used at very high rates, wasn't enough to prevent depletion for studying fast enzymes like the CO dehydrogenase [9]. For both the RDE and the wall-tube, the mass-transport is given by equation (10), with only the value of a changing from one setup to the other. Therefore, the factor of increase of the transport over the RDE is given by:

m WT m RDE = a WT a RDE 1/2 = 1.35 ν 0.04975 Q v 0.45025
h 0.0345 r jet 1.41575 ω 1/2 (19) in which the quantities indexed with RDE correspond to the RDE while those indexed with WT refer to the wall-tube electrode. Assuming a reasonable maximum value of 5 krpm for the rotation rate of the RDE, using the values of r jet = 0.5 mm and h = 0.35 mm and Q v = 37 mL/min as described in our previous work [21] yields the factor of 3 described previously. Using r jet = 0.3 mm, the flow rate required for a factor-of-three improvement drops to Q v = 10 mL/min. As was noted above (equation ( 18)), the wall-tube design in itself does not provide an advantage over the RDE with respect to the shear stress, since both setups share the same relationship between masstransport coefficient and shear-stress. However, while for the RDE, the only possibility for increasing mass-transport is to increase the rotation rate, but practical rates for rotating disc electrodes seldom go above 5 krpm. On the contrary, to increase the transport in the case of the wall tube electrode, one can either decrease the jet size or increase the flow rate, which provides more flexibility to achieve high mass transport coefficients.

Figure 1 :

 1 Figure 1: Schematic representation of the geometry of the cell, indicating the 6 parameters varied in the systematic approach. The detailed meaning and values of the parameters are given in table 1

Figure 2 :

 2 Figure 2: Values of the hydrodynamic constant a determined from the simulations for the 108 combinations of parameters from table 3, as a function of the radius of the inlet, r jet . The different symbol shapes correspond to different values of Q v : 1 mL/min for circles, 5 mL/min for squares and 10 mL/min for triangles. The different values of h are color-coded, as indicated in the graph itself. The other parameters were: distance inlet-outlet (P2 = 1 mm), outlet thickness (P3 = 0.36 mm), inlet-outlet angle (P5 = 26 • ). The value of the electrode radius has no influence on the hydrodynamic parameters.

Figure 3 :

 3 Figure 3 represents the values of the a * predicted from equation (15) with the coefficients in (16) as a function of the values

  coef (μm/s)

Figure 4 :Figure 5 :

 45 Figure 4: Comparison between simulated mass transport coefficient and the predicted one from equation (17) for the 108 simulations.

Figure 6 :

 6 Figure6: r max , the maximal value of the electrode radius for which the values of the flux for 0 ≤ r ≤ r max do not vary more than 10% as a function of the inlet radius, for different values of h and Q v . The values of h can be deduced from the color of the symbol (see the legend inside the graph), while the shape of the symbols correspond to different values of Q v : Q v = 1 mL/min for the circles, Q v = 5 mL/min for the squares and Q v = 10 mL/min for the triangles. The dot and the dash lines correspond respectively to y = x and y = 0.6x.

Figure 7 : 6 (

 76 Figure 7: Experimental validation of the transport properties of one of the promising electrochemical cells (r jet = 0.3 mm; h = 0.35 mm): (A) Cyclic voltammetry for the reduction of Fe(CN) 3 - 6 at a graphite electrode of diameter 100 µm at ν = 100 mV.s -1 . The experiments were conducted at volume flow rate that ranged from 1 mL/min to 25 mL/min at room temperature, under degassed system with argon. (B) the limiting current (in absolute value) as a function of the flow rate. (C) Comparison of mass transport coefficient (m = I/nFA) extracted from the cyclic voltammetry experiments of the reduction of Fe(CN) 3 - 6 (green circles) with the predicted mass transport coefficient from equation (17) (black line), Q v = 1 mL/min. Conditions: [Fe(CN) 3- 6 ] = 1 mM, electrolyte: 0.5 M NaCl, room temperature. We determined the working electrode's surface to be 8.2 × 10 -9 m 2 .

Table 1 :

 1 . The results of this treatment are "effects" Values of the low and high levels of the parameters varied in the sensitivity studies. The definition of the parameters is in figure1.

		Inlet radius Distance inlet-outlet Outlet thickness distance nozzle-electrode inlet-outlet angle Flow rate
	Parameter	(P1)	(P2)	(P3)	(P4)	(P5)	(P6)
		mm	mm	mm	mm	•	mL/min
	Low level	0.1	0.05	0.1	0.1	10	1
	High level	0.5	1	1	0.5	80	5
		Parameters	Effects (mm.s -1 )
		Inlet radius (P1)	-1.26953
	Distance inlet-outlet (P2)	0.00194
	Outlet thickness (P3)	0.00509
	Distance nozzle-electrode (P4)	-0.11021
	Inlet-outlet angle (P5)	-0.00073
		Flow rate (P6)	0.58171
		Average	0.73959

Table 2 :

 2

Table 3 :

 3 Values of the different parameters used for the determination of the hydrodynamic constant. All the 6 × 6 × 3 = 108 combinations were used.

	Inlet radius Inlet-electrode distance Inlet flow rate
	r jet	h	Q v
	(mm)	(mm)	(mL/min)
	0.25	0.15	1
	0.30	0.20	5
	0.35	0.25	10
	0.40	0.30
	0.45	0.35
	0.5	0.40

Acknowledgements

The authors acknowledge support from CNRS, Aix-Marseille Université, Agence Nationale de la Recherche (ANR-15-CE05-0020, ANR-17-CE11-002), and the Excellence Initiative of Aix-Marseille University -A*MIDEX, a French "Investissements d'Avenir" programme (ANR-11-IDEX-0001-02), and the ANR-DFG project SHIELDS (PL 746/2-1). They also wish to acknowledge expert help from Alain Grossi and Sady Noel in preparing the 3D-printed cell and the electrodes, and Gaëlle Coustillier and Philippe Delaporte from the laboratory "Lasers, Plasmas et Procédés Photoniques" in Marseille for their preparation of graphite samples of very small size, and finally Jérôme Vicente for insightful discussions and comments. AHA and VF are part of FrenchBIC (http://frenchbic.cnrs.fr).

Conflict of interest statement

The authors have no conflict of interest to declare.


Ch Laurent

E Dooryhée

C Dufour

F Gourbilleau

M Levalois

G Nouet

E Pauji1jer

Modification by high energy ion irradiation of iron-alumina nano-composites

This work reports on the current observations by transmission electron microscopy (TEM) of nano-composites irradiated by Ge V heavy ions. The composite targets consist of nanometric Fe particles dispersed in a ceramic (AI 2 O 3 ) powder. Large metal particles ( ~ 30 nm diameter) are Jocated over the alumina grain surface whereas smaller particles ( ~ 10 nm) are directly epitaxied in the alumina grains. The TEM micrographs show lhat the surface distribution of the large particles and the orientation of the small Fe particles in the Al 2 O 3 micrometer grains are changed by the irradiation. At high fluences, the grains of alumina are destroyed.

Introduction

The imaging by transmission electron microscopy (TEM) has provided a major insight into the atomic details of the structural changes produced in a solid by swift heavy ion irradiation (SHI) [1][2][3][4][5][6]. Over the past years, the TEM studies have rather focused on the SHI effects in bulle solids which are not sensitive to purely ionizing radiations (photons or electrons). For example, TEM has revealed the formation of latent tracks in high :r;, supercon ductors such as YBa 2 Cu 3 O 7 [7]. the amorphization of NiZr 2 [8] or the phase change in Ti [9]. The description of the effects along the ion path (point defects, clusters of deîects, amorphization, strain) was iargely improved in Y 3 Fe 5 O 12 , BaFe 12 O 19 and other ferrimagnetic oxides [6,10).

The present work reports on the first TEM observations of SHI effects in an alumina based nana-composite. The nove! target investigated in the present study consists of nanometric iron particles dispersed in and over the mi crometer-sized grains of powdered n-Al 2 O 3 • Both materi als (Fe and Al 1 O 3 ) are highly resistant to low dcnsities of electronic excitations, and the effects of swift heavy ion 1 Ion irradiation performed at the GANIL accelerator, Caen, France.

• Corresponding author. Fax ( + 33) 31 45 47 [12). For dE/dx ;;i, 4 kcV / Â , a strong enhancement of the damage is observed in Fe, due to electronic excitations [12]. We are now investigating the response to SHI of nanometric iron dispersed in an insulat ing aiumina matrix. Interest in the response of the Fe-Al 2 O 3 nano-com posite to SHI is further stimulated by recent observations in Na-MgO [START_REF] Beranger | presented at 9th !nt. Conf. on Ion Bearn Modification of Materials[END_REF]: sodium dispersoïds ( ~ 10 nm diameter) embedded in a MgO single crystal vanish away under SHI. The decrease of the concentration of the alkali nano-pre cipitates in the MgO matrix is imparted to a SHI induced atomic mixing at the Na/MgO i.. riterface. In the prescnt article, the TEM micrographs depict the evolution of the Fe-Al 2 O 3 nana-structure under SHI. The distribution of the Fe nano-particles (location, orientation, sizc and con centration) and the stability of the composite are observed after SHI. The large densities of electronic excitations produced by the incident Ge V heavy ions are responsible for the observed features. These unique modifications in tum could affect the related properties of Fe-AI 2 O 3 such as the mechanical strengthening and the super-paramagne tism.

Experimental

Sample preparation and characterization

The 20 wt.% Fe-Al 2 O 3 nano-composite powder was prepared according to the method proposed by Devaux et al. [START_REF] Devaux | [END_REF]. The mixed complex oxalate (NH 4 )iAI 1 . 8 Fe 0 _ 2 - (C 2 0 4 ) 3 ] was prepared by co-precipitation of aluminium and iron oxalates. The mixed oxalate was decomposed in air at 673 K and next calcinated in air at 1373 K, yielding an intimate mixture of two mixed oxide phases as shown by X-ray diffraction (XRD). The major phase is an Al-rich solid solution of the a-Al 2 O 3 (corundum) type, whereas the minor phase is an Fe-rich solid solution of the a-Fe 2 O 3 (hematite) type. The latter phase forms because the solubil ity of hematite in a-Al 2 O 3 is limited to 10 wt.% [15]. The oxide powder was heated up at 1573 K in pure, dry hydrogen for 1 h, giving rise to the 20 wt.% Fe-Al 2 O 3 nano-composite powder. lt is studied by XRD and Mëssbauer spectroscopy.

The microstructure of the specimens was investigated using transmission electron microscopy (TEM) and related techniques such as electron micro-diffraction (ED) and energy-dispersive X-ray analysis (EDS).

Irradiation conditions

The Fe-Al 2 O 3 powder was laid and dried out over a 400 mesh copper grid after ultrasonic dispersion in ethanol. Each grid was observed by TEM as irradiated. The grids covered with powder were exposed to 1 GeV 181 Ta ions (incident energy E = 5.1 MeV /amu) at 80 K. The stop ping power of Ta is dominated by inelastic interactions.

100nm

The electronic energy Joss dE/d x is 5.4 and 3.6 keV / Â in Fe and Al 2 O 3 respectively. It exceeds the energy loss by elastic collisions by more than a factor 600. The fluences are 1.5 X 10 12 (#!), 3 X 10 12 (#2) and 10 13 cm -2 (specimen #3). The particle flux was limited to ~ 2 X 10 8 cm -2 s -1 for preventing any excessive heat ing.

Results

Non-irradiated powder

XRD analysis of the nano-composite powder reveals the presence of metallic iron and a-Al 2 O 3 only. The reduction treatment was made at a temperature high enough to avoid the formation of a FeAl 2 O 4 spinel phase [16]. Analysis of the Mëssbauer spectra shows the absence of ferrous and ferric ions, thus confirming that the reduction is total at 1573 K [ 17]. The intra-granular Fe particles are formed by reduction of the substitutional Fe 3+ ions in the Al-rich solid solution. The ferromagnetic surface Fe parti des derive from the reduction of the Fe-rich zones and from the coalescence of smaller particles by diffusion at the surface and in the open porosity of the matrix [START_REF] Devaux | [END_REF]17]. TEM observations (Fig. 1) show the microstructure of the composite powder. The dimensions of the matrix grains are of the order of a few hundreds of nanometers. The EDS analyses confirm the global iron content (20 wt.%). Most of the Fe particles are smaller than 10 nm in size and uniformly located inside the alumina grains. Sorne larger ( � 30 nm) metal particles are located at the surface and in the open porosity of the oxide matrix. A previous study [18) has shown that both the alumina grains and the Fe particles are monocrystalline and that the intra-granular Fe particles are epitaxied in the alumina lattice according to the following general epitaxial relationship:

(111 )[ïÏ2k Il (OOOI)[JOÏO)a-AI 2 0 3

Latent tracks in irradiated specimens

TEM observations of specimen # 1 reveals the presence of tracks of cylindrical morphology, parallel to each other in an alumina grain (Fig. 2a). The diameter of a cylinder is approximately 2 nm. Such tracks are also observed in the other specimens, but a strong carbon contamination occur ring under the electron beam impairs the quality of the TEM micrographs. Crack-like, cylindrical defects much shorter than those described above are also observed in specimen # 1 (Fig. 2b). lt is noteworthy that there is an area where a much lower concentration of Fe particles is observed. The edge and some features of the alumina grain in this area look "rounded". lt should be noted that "doublet" tracks similar to those observed in Cr-Al 2 O 3 nano-composite powder irra diated with 3.7 MeV /amu 154 Sm ions [START_REF] Dooryhee | presented at 9th Int. Conf. on Ion Bearn Modification of Mate rials[END_REF] were not detected in the present specimens.

Evolution of the nana-structure in irradiated speci mens

A comparison of the ED patterns before (Fig. 3a) and after irradiation (Figs. 3b-3d) shows that the intra-granular Fe particles do no longer share the same orientation after b) irradiation and that the magnitude of ti 'ïe disorientation effect increases with increasing fluence. Indeed, one can see that some extra (110) Fe diffraction spots appear on the ED pattern of specimen # 1 (Fig. 3b). This is explained by the rotation of some Fe particles away from th. eir epitaxial position. The diffuse (110) Fe diffraction ring is detected on the ED pattern of specimen #2 (Fig. 3c), showing more and more disorientation. For specimen #3 The destruction of some composite grains is observed in specimens #2 and #3 (Fig. 4), but not in specimen # 1. lt can be seen that the destruction may either be partial or total, but is not systematically related with the fluence. Thus it is not yet possible to determine whether there is a threshold fluence at which grain destruction starts or whether this is a progressive effect. EDS analyses reveal that both Al and Fe are present in the fragments; most of which are as small as 10 nm in size (Fig. 4). ln the case of partial destruction, needle-shaped fragments are observed in the vicinity of the virgin grain (Fig. 4a).

Oe • • 0 • "• • 'o.. • OÏlO •--
EDS analyses were performed on specimen #3 during the TEM observations. The Fe content in the grains shown in

Discussion

Disorientation of the Fe nana-inclusions in A/ 2 O 3

Stress relaxation in the alumina lattice could account for the disorientation of the Fe particles. Indeed, the most frequently observed epitaxial relationships [START_REF] Devaux | [END_REF] correspond to a fairly high lattice mismatch (15-20%). lt must be remembered that the intra-granular Fe particles do not grow in a free manner. Their orientation is rather dictated by the matrix, which is crystallized in the stable cx-Al 2 0 3 form prior to the nucleation of the metallic iron. Thus, it may be considered that stress relaxation induced by SHI allows the disorientation of the Fe inclusions into a more energetically favourable position.

Role of the elastic collisions

The contribution of elastic nuclear collisions into the process of atomic mixing or bond breaking can be readily estimated. The number of target atoms moved out by elastic knock-on is overestimated using the TRIM code [START_REF] Biersack | [END_REF]. For a fluence <f>t = !0 12 Ta/cm 2 , the computed simulation yields 3 X w-5 vacancy per target atom (dpa) in Al 2 0 3 and 7 X w-5 dpa in Fe. An analytical treatment using the Lindhard potential [23] and the formulae of Norgett et al. [24] gives very similar values. Given a nano-particle of 10 nm diameter, no more than 6 iron atoms can be displaced away for !0 12 Ta/cm 2 • Nuclear collisions are by far insufficient for the creation of a mixed interface, and cannot break a sufficient number of bonds at the interface. The nuclear collisions are not responsible either for the destruction of the surface particles.

The calculated total cross section for atomic displace ment by direct elastic collisions is ~ 3 X w-17 cm 2 in pure cx-Al 2 0 3 • Canut et al. [25] measured the fraction of disorder produced by 0.9 Ge V 20 8 Pb ions (dE/dx = 4.1 keV / Â) in monocrystalline cx-Al 2 0 3 , by means of chan neling Rutherford backscattering spectrometry. The mea sured total cross section for effective atomic displacement is ~ 10-13 cm 2 (at 80 K). We infer from the above discussion that electronic excitations largely prevail in the process of intra-granular Fe particle rotation and in the destruction of the surface Fe particles.

lnelastic energy loss in Fe nano-particles

A crude estimate of the energy absorbed by the surface metal particle ( ~ 30 nm diameter) through electronic exci tations yields ~ 0.7 eV /atom per hit. We calculate that 80% of the incident energy is retained within the isolated particle (50% if diameter ~ 10 nm). This accounts for the kinetic energy lost out of the particle by escaping ejected electrons. The calculation consists of intersecting the track radial profile of the energy carried away by the electrons [START_REF] Waligorski | Nue![END_REF] and the finite cylinder-like volume of the nano-par ticle [START_REF] Spohr | Ion tracks and microtechnology: principles and applications[END_REF]. For a fluence of 10 12 cm -2 , the large particles receive 7 impacts on average Oess than I for intra-granular particles). The value ~ 0.7 eV /atom/projectile can be tentatively compared with the thermal energy absorbed by Fe at 80 K for complete melting: 0.8 eV /atom. Due to the compressive stress in the alumina matrix, the fusion energy of the intra-granular Fe particles is expected to be higher. Based on this thermodynamical difference, could the ther mal spike mode! [START_REF] Toulemonde | [END_REF] explain that the large, free particles (on surface) disappear whereas the small, embedded parti cles (intra-granular) are still observed after SHI?

S. Conclusion

We have examined the structural modifications induced by Ge V Ta ions in a nana-composite target (Fe-Al 2 0 3 ) at 80 K. The irradiated material is composed of a powder of micrometric alumina grains, with nano-particles of Fe (20 wt.%) dispersed inside and outside the ceramic grains. The epitaxial orientation of the intra-granular Fe inclusions is progressively lost with increasing fluence. The larger Fe particles located at the surface and in the open porosity of the oxide grains disappear from their original sites. Track like features are observed and fragmentation of the oxide grains occurs. These significant modifications mark the efficient conversion of track electronic excitations into atomic effects in materials (Fe and Al 2 0 3 ) a priori not much sensitive to low rates of electronic excitations.

Fig. 1 .

 1 Fig. 1. TEM micrograph of the non-irradiated 20 wt.% Fe-Al 2 O3 powder. The µm-size alumina mono-crystalline grain contains a uniform distribution of nano-crystalline, epitaxied Fe precipitates ( ~ 10 nm size). Larger Fe particles ( ~ 30 nm) are observed over the surface of the alumina grain.

Fig. 2 .

 2 Fig. 2. TEM micrographs showing (a) tracks (diameter == 2 nm) and (b) cracks in specimen # 1 (fluence = 1.5 X 10 12 Ta/ cm 2 ).

Fig. 3 .Fig. 4 .

 34 Fig. 3. ED patterns of the non-irradiated 20 wt.% Fe-Al 2 0 3 powder (a), specimen #1 at 1.5 X 10 12 Ta/cm 2 (b), #2 at 3 X 10 12 Ta/cm 2 (c) and #3 al 10 13 Ta/cm 2 (d). a) superimposed diffraction patterns of the two lattices, Fe and Al 2 0 3 , sharing the same orientation. b) ( 110) Fe diffraction extra spots appear. c) Fe nano-particles are more and more disoriented, contributing to the ( 110) Fe Scherrer ring. d) (200) and (211) Fe diffraction occurs, for a single orientation of the substrate alumina grain.

Fig. 1 is found to be approximately 7 .

 7 5 wt.%. These values are measured on very small zones inside these grains and closely match the intra-granular iron content found by other methods [20]. lt seems that the Fe particles located at the surface and in the open porosity of the oxide matrix are removed from their original location due to SHI.

47 54, e-mail: dooïhyee@ganac4.in2p3.fr.