Amélie Désiré

Bruno Paillard
email: bruno.paillard@pierre-fabre.com.

Joël Bougaret

Michel Baron

Guy Couarraze

Keywords: Extrusion-spheronization, scale-up, response surface, screw extruder, industrial scale, robustness, flexibility

Extruder scale-up assessment in the process of extrusion-spheronization: comparison of radial and axial systems by a design of experiments approach

Amélie Désiré, Bruno Paillard, Joel Bougaret, Michel Baron,

Introduction

Scale-up, which predicts the transition from lab batch to pilot and then to industrial batches, is a fundamental step in the drug development. As a matter of fact, increasing batch size may lead to a change in equipment, manufacturing site or process, and it is essential to evaluate the consequences of these changes [START_REF] Hattami | Scale up during pharmaceutical development and industrial production[END_REF] . The objective, by scaling-up a lab process to industrial scale, is to check that the industrial process is feasible with or without process modifications (as minimum as possible) and reproducible, to demonstrate that the product obtained at industrial scale is identical to the one elaborated at lab scale and also to demonstrate the formulation and process robustness when the batch size increases. Critical parameters must be identified for each manufacturing scale in order to set the process limits.

Authors described different approaches for scalingup a process to another scale: (i) monitoring of one Extruder scale-up assessment in the process of extrusionspheronization: comparison of radial and axial systems by a design of experiments approach representative parameter; (ii) search of scale-up dimensionless numbers that completely characterize the process to express geometric and dynamic similarities between two scales [START_REF] Iyer | Scale-up of Extrusion and Spheronization[END_REF][START_REF] Leuenberger | New trends in the production of pharmaceutical granules: the classical batch concept and the problem of scale-up[END_REF] ; (iii) construction of experimental designs to establish models able to estimate the quality of the product when the process conditions are changed. The design of experiments approach is a powerful tool for optimisation and process scaling-up: it allows a rational study of experimental parameters influence on selected responses with a shorter experimentation time and an improved process comprehension [START_REF] Lewis | Pharmaceutical Experimental Design[END_REF][START_REF] Singh | Optimizing drug delivery systems using systematic "design of experiments[END_REF] . It also provides assurance of quality after a scale-up operation by a "quality by design" approach, where all critical sources of variability are identified and explained, and where product quality can be accurately and reliably predicted over the design space established.

Among the different methods to produce pellets, the process of extrusion-spheronization is of particular interest, and is widely described in the literature [START_REF] Vervaet | Extrusion-spheronisation. A literature review[END_REF][START_REF] Gandhi | Extrusion and spheronization in the development of oral controlled-release dosage forms[END_REF][START_REF] Trivedi | Pharmaceutical approaches to preparing pelletized dosage forms using the extrusion-spheronization process[END_REF][START_REF] Galland | Twin product/process approach for pellet preparation by extrusion/spheronisation. Part I: hydro-textural aspects[END_REF][START_REF] Galland | Hydro-textural characterisation of wet granular media shaped by extrusion/ sphéronisation[END_REF][START_REF] Sinha | Extrusion-spheronization: process variables and characterization[END_REF] . The scale-up of the extrusion-spheronization process involves the separate scale-up of each of the five process steps: dry mixing, granulation, extrusion, spheronization, and drying. This study focuses on the extrusion step scale-up. Among the different types of extruders on the market (screw-feed, gravity feed, ram extruders), this work is limited to screw-feed extruders, which can be classified in three categories according to the design of the screen: axial, dome, and radial [START_REF] Trivedi | Pharmaceutical approaches to preparing pelletized dosage forms using the extrusion-spheronization process[END_REF] . This study compares the scale-up of a lab mono-screw extruder and of an industrial twin-screw extruder, for radial and axial systems. Because extrusion is a continuous process, the pilot devices correspond to industrial equipments, the batch size depending on the process time.

Previous studies compared three single-screw extrusion systems (radial, dome, and axial) at lab scale to evaluate, with the help of designs of experiments, their suitability to provide good results in terms of product quality, when formulation and process parameters changed [START_REF] Désiré | A Comparison of three extrusion systems -Part I: The influence of water content and extrusion speed on pellet properties[END_REF][START_REF] Désiré | A Comparison of three extrusion systems -Part II: The influence of formulation and spheronization conditions on pellet properties[END_REF] . Few other authors studied the comparative influence of radial and axial single-screw extruders at lab scale using different formulations, on the extrusion process characteristics [START_REF] Doeuff | An approach of extrusion and spheronization, comparison of two extruders by means of experimental designs[END_REF][START_REF] Sonaglio | Effects of extrusion and formulation parameters on the production of paracetamolmicrocrystalline cellulose extrudates[END_REF] , and on the final product quality [START_REF] Doeuff | An approach of extrusion and spheronization, comparison of two extruders by means of experimental designs[END_REF][START_REF] Sonaglio | Approach to the development of high dose paracetamol spheres by extrusion/ spheronization[END_REF][START_REF] Rabisková | The influence of the extrusion die on pellet characteristics[END_REF] . Only few authors carried out studies at industrial scale on twin-screw extruders for extrusion-spheronization. Most of them used radial twin-screw extruders, either for a specific extrusion study [START_REF] Shah | Physicomechanical characterization of the extrusionspheronization process. I. Instrumentation of the extruder[END_REF][START_REF] Shah | Physicomechanical characterization of the extrusion-spheronization process. Part II: Rheological determinants for successful extrusion and spheronization[END_REF][START_REF] Martin | Mechanics of paste flow in radial screen extruders[END_REF] , or for a global extrusionspheronization study [START_REF] Häring | Spheronisation variables and theophylline pellet characteristics[END_REF][START_REF] Pérez | Influence of the drying technique on theophylline pellets prepared by extrusion-spheronization[END_REF][START_REF] Elbers | Effect of amount and composition of granulation liquid on mixing, extrusion, and spheronization[END_REF] . Others used dome twin-screw extruder [START_REF] Kanbe | Manufacture of fine spherical granules by an extrusion/spheronization method[END_REF] . Axial twin-screw extruder was mentioned as being used for extrusion-spheronization in one study where the axial extruder was actually a system for wet granulation [START_REF] Schmidt | Comparison between a twin-screw extruder and a rotary ring die press, I. Influence of formulation variables[END_REF] . One patent detailed quite extensively the development of twin-screw extruder technology from axial extruders, through radial screen extruders, to dome extruders [START_REF] Fujimoto | Screw-type extrusion granulating apparatus, especially for producing very fine granules[END_REF] . No author studied the screw extrusion scale-up in the process of extrusion-spheronization.

The aim of the study is to compare radial and axial screw extrusion systems for their capacity to scale-up, and to study the influence of formulation (drug substance solubility and concentration) and process variables (spheronization time) between the two scales for the two extrusion systems. The other steps are kept at lab scale in order to focus on the extrusion step scale-up. A response surface design of experiments is set up to underline the influence of the different variables and their interactions. The objective is to identify the type of extruder that suits the best for scaling, i.e. the one which gives the best results in terms of productivity and pellet characteristics ("product quality"), the one which shows less impact on the product after scale-up ("scalability"), the one which shows the least influence on these same properties when the formula used changes ("robustness"), and the one which allows the possibility to adjust or improve pellets properties with spheronization variables ("flexibility").

This study provides a global vision of process scale-up by using a surface response design of experiments, responding to a quality by design approach. This study analyzes the results by crossing them, introducing the notions of robustness, flexibility, and scalability.

Methods

Raw materials

Pellets were prepared from the binary mixture of a drug substance (DS) and microcrystalline cellulose (MCC). Two drug substances DS1 and DS2 supplied by Pierre Fabre Research Institute were tested, DS1 corresponding to an antidepressant, and DS2 corresponding to theophylline. The two drug substances have been chosen for their different solubility in water (1250 g/L for DS1 and 8 g/L for DS2). MCC (Avicel® PH101), supplied by FMC Biopolymer (Philadelphia, USA), is insoluble in water. Three ratios of DS/MCC were tested for both drug substances: 20:80, 36:64 and 52:48 (%w/w).

Purified water was used as liquid binder, and was adjusted for the six different formulations. The optimal water quantity required for each of the six formulations, i.e. the percentage of water which gives the best yield of good quality pellets (% w/w in relation to the amount of solid mass), was determined by preliminary experiments at lab scale.

Description of the experimental design

A response surface design of experiments was built with Design Expert® software, version 7.0.1.0 (Stat-Ease, Minneapolis, USA). The mathematical model targeted for each response studied was a quadratic model with first-order interactions. Five factors were studied: drug solubility in water [DS1-DS2] (g/L), drug concentration [20-36-52] (%), extrusion system [radial-axial], extruder scale [lab-pilot], and spheronization time [2-3-4] (min). To analyze the results, drug solubility, extruder system, and extruder scale were considered as qualitative factors, whereas drug concentration and spheronization time were continuous factors.

The design of experiments was built as a set of 8 designs, each corresponding to one combination of three qualitative factors. For each of these 8 designs, a response surface was built where DS concentration and spheronization time were tested, and three replicates of the central point were run (level 0, i.e. 36% of DS and 3 min of spheronization time). The same experiments were thus repeated for both DS solubility, on the two extrusion systems, at lab and industrial scales. The whole experimental design included a total of 64 (8 × 8) experiments. Figure 1 summarizes factors and responses selected for the global design of experiments. The responses associated are described in the section "Characterizations ".

Pellet preparation

Process conditions were set according to the flow chart shown in Figure 2. After dry blending of MCC and DS, wetting and granulation were performed in an Aoustin kneader (RPA Process, Nanterre, France). The water quantity was adjusted at lab scale according to the formula. The value of water quantity for which pellets could be formed was found to be dependent on the drug solubility and its concentration. The optimal water amount at lab scale was found to decrease with DS solubility and concentration increase [START_REF] Désiré | A Comparison of three extrusion systems -Part II: The influence of formulation and spheronization conditions on pellet properties[END_REF] . The water quantities used at industrial scale were those optimized at lab scale.

The wet mass obtained was extruded with a lab MG-55 or an industrial EXDCS-100 extruder, both from FujiPaudal (The FitzPatrick Company Europe, Sint-Niklaas, Belgium), using the radial or the axial extrusion system. The extrudates were then spheronized in a spheronizer 250 (Caleva, Sturminster Newton, Dorset, UK) and the pellets obtained were dried in a drying oven (Heraeus, Hanau, Germany), and calibrated (Retsch GmbH, Haan, Germany) to obtain a [1400-2000] µm fraction.

MG-55 presents a single screw (160 × 55 mm), whereas EXDCS-100 presents two contra-rotatives screws (31.6 × 9.8 cm). For the extrusion step, whatever the extrusion system and scale, a screen die of 2 mm in diameter and a screw speed of 40 rpm were used to allow comparisons. In the lab and industrial axial systems, a flat end-plate screen is placed in front of the one or two flat blades positioned to the end of the screw(s), so that the extrudates are ejected out of the screens at the end of the extruder. In the lab and industrial radial systems, two semi-cylindrical screens are placed along each side of the one or two counter rotating extruder head(s) positioned to the end of the screw(s), so that the extrudates are ejected perpendicularly to the motion of the screw(s), each head consisting of a truncated cone with three attached blades. For both scales, the extrusion screen presents a 1-and 2-mm screen thickness, respectively for radial and axial system.

Characterizations

The different responses, corresponding to the different characterizations, are reported in Figure 1.

Extrusion process

Extrusion rate and extrusion yield were studied in order to follow the productivity of each extruder system.

The extrusion rate R E (g.s -1 ) was calculated according to:

R M T E -1 E E (g.s ) = (g) (s) (1)
where M E corresponds to the mass of wet extrudates recovered at the end of the extrusion step, and T E to the time necessary to extrude this mass. The extrusion yield Y E (%) was calculated according to:

Y M M E E WM (%) = (g) (g) 100 × (2)
where M WM and M E correspond respectively to the mass of wet powder initially introduced in the extruder and to the mass of wet extrudates at the end of extrusion. An extrusion rate and an extrusion yield are expected as high as possible to improve the productivity of the process.

Pellet hole fraction

Size distribution Particle size distribution of the pellets was determined with a laser diffractometer (Mastersizer 2000, Malvern Instruments, Malvern, UK), equipped with a dry powder dispersing system (Scirocco 2000; Malvern Instruments, Malvern, UK ), using a dispersion pressure of 1 bar, and by a microscopic method (Morphologi G2), described below. Particle size distributions were characterized by their volume median diameter (d 0.5 , µm) and their pellet size dispersion (d g ).

For the design of experiments analysis, the volume median diameter D [4-3] (µm) calculated by laser diffraction was used: a sieving of 2000 µm opening was carried out before the measure, the pellet fractions over this size being then re-entered in the results. Pellet mean size must be inferior to the diameter of the extruder die diameter (2 mm) considering densification and water evaporation during the spheronization and the drying stages.

For the design of experiments analysis, the pellet size dispersion calculated by microscopy was used. The pellet size dispersion was calculated from d 0.5 , d 0.1 and d 0.9 , according to the following equation:

d d d d g 0.9 0 .1 0.5 = ( ) - (3)
where d 0.1 and d 0.9 values are the particle diameters corresponding to respectively 10% and 90% of the cumulative distribution. Pellet size dispersion is required to be as narrow as possible, which is expressed by a low d g value (less than 3), to facilitate a possible later coating or capsule-filling operation.

Morphology The morphological analysis of pellets was performed by means of a particle-image analyzer (Morphologi G2; Malvern Instruments, Malvern, UK). Analysis was carried out on around 300 pellets. Pellet elongation (E) was calculated according to the following formula:

E = 1 width length -       (4)
Because the pellets are possibly filled into capsules, they should present good flow characteristics and be as spherical as possible. For a perfect disk, the value of the elongation factor equals 0. It is desirable to obtain pellets with the least possible elongation.

Usable yield

The pellet whole fraction was sieved on 1400-2000-μm sieves (Retsch, Haan, Germany) for 2 min, at a frequency of 60 Hz with an amplitude of 1 mm. The 1400-2000-μm fraction of pellets was considered to be the usable fraction. The authors used this fraction for pellet characterizations to eliminate the effect of size on the pellet mechanical properties. The usable yield had to be as high as possible.

Pellet usable fraction

Roughness Roughness analysis of the pellet yield fraction was assessed by measuring solidity factor using a Morphologi G2 (Malvern Instruments, Malvern, UK). Analysis was carried out on around 300 pellets from the usable yield fraction. Solidity factor (S) was calculated according to the following formula:

S A = (A+B) (5)
in which, A is pellet area and A+B is the area enclosed by the convex hull (A+B). High solidity is desirable because it corresponds to low roughness; rough pellets may generate fines or have poor flow characteristics. Surface roughness of the pellets is also an important characteristic when considering possible coating or compression into tablets.

Pycnometric density Pycnometric density of pellets, D pycno (g.cm [START_REF] Leuenberger | New trends in the production of pharmaceutical granules: the classical batch concept and the problem of scale-up[END_REF] ), was determined using a helium pycnometer (Accupyc 1330, Micromeritics Instrument, Norcross, USA) Samples were degassed under 6.5 Pa vacuum (VacPrep 061, Micromeritics Instrument) for 2 days at about 25°C. Measurements were performed using a 10 cm 3 cell, and repeated until the value stabilized. The mean pycnometric density was calculated from the final three stabilized data points.

Friability Friability F(%) was measured on approximately 20 g of pellets from the usable yield fraction, to which were added 40 g of 6-mm glass beads. After 30 min of blending in a 200-mL flask in a shaker-mixer (Turbula, GlenMills, Clifton, NJ, USA) at 42 rpm, the mass retained on a 1400-μm sieve was weighed, and the friability F(%) was calculated according to the following equation:

F M M M (%) = [( i f) / i] 100 - × ( 6
)
where Mi is the mass of granules before the test (i.e. 20 g) and Mf is the mass of granules retained by the sieve after the test. The test was performed in triplicate. The friability test showed the pellet surface resistance to abrasion, which should be as high as possible to avoid abrasion during further processing.

Diametral crushing force

The resistance to crushing R (N) was tested on 20 pellets of the usable yield fraction with a durometer (Computest, Kraemer Elektronik, Darmstadt, Germany). The diametral crushing force measured indicates the mechanical robustness of the pellets. It should be as high as possible to avoid pellet breakage during further processing.

Design of experiments interpretation

The mathematical model generated for each response Y was a quadratic model with first-order interactions, built according to the following equation: (7) in which X i and X j represent the levels of the factors; a 0 is the intercept representing the mean of the measured response data; and a i and a j , a ii and a jj , and a ij correspond to the coefficients of first-order terms, the coefficients of second-order quadratic terms, and the coefficient of second-order interaction terms, respectively. The coefficient corresponding to a factor or an interaction shows its importance on the studied response. The symbol ε represents pure error. To simplify the design of experiments interpretation, the coefficients of second-order quadratic terms were not presented in this study. The coefficient values were expressed in coded units in order to compare their relative effect to that of the others. Analysis of variance (ANOVA) was performed to determine the significance of the model. A term that had a probability value lower than 0.05 was considered as significant. A probability value greater than 0.10 was regarded as not significant.

The ideal industrial extrusion system is the one which gives the best results in terms of productivity and pellet characteristics ("quality"), the one which shows less impact on the product after scale-up ("scalability"), the one which shows the least influence on these same properties when the formula used changes ("robustness"), and the one which allows the possibility to adjust or improve pellet properties with spheronization variables ("flexibility").

Results and discussion

Experiment feasibility

Pictures of extrusion step on the two extrusion systems at lab and industrial scales are presented in Figure 3. For axial extrusion, all experiments were feasible at industrial Y a a X a X a X X a X a X = + + + + + + 0 i i j j i j i j i i i

. Extrusion of the wet mass in the MG-55 lab system: (i) radial, (ii) axial, and in the EXDCS-100 industrial system: (iii) radial, (iv) axial.

scale, using water quantities optimized at lab scale. No water readjustment was thus necessary after scale-up. For radial extrusion, using water quantities optimized at lab scale, experiments were not all feasible at industrial scale.

As a matter of fact, a dewatering phenomenon appeared for some batches at industrial scale for radial system, corresponding to the wet masses containing more than 40% of water. During the extrusion of wet mass, after a more or less short time depending on the batches, the following events appeared chronologically: (i) extrusion rate decreased, (ii) extrudates formed out were very wet, (iii) water flowed along the screen (drainage), (iv) extrusion stopped. In these cases, the extrudates recovered after the beginning of the phenomenon were not usable for the next spheronization steps.

This phenomenon can be explained by the low quantities of wet mass used in our work. The extrusion of a high quantity of product, as normally expected on an industrial extruder, should avoid the drainage phenomenon: as a matter of fact, in this case the wet mass should stay for less time in the device as pushed by the incoming product, thus decreasing the possibility of water migration.

Design of experiments interpretation

The effects of factors and main interactions between the factors, deduced from the analysis of the experimental design, are summarized with bar graphs on Figure 4. The red bars correspond to the factors that significantly influence the response Y x .

No pellets could be formed for the batches which have shown a drainage phenomenon (4 batches): these batches were not included in the design of experiments. Nevertheless, this lack of data showed no impact on the design of experiments interpretation. The drainage phenomenon should nevertheless be kept in mind for the discussion.

Global analysis

Global analysis allowed identifying the significant factors and interactions on the different responses. It thus allowed studying the global influence of formulation and spheronization variables, and the global impact of extrusion system and scale-up, and their interactions, on productivity and pellet quality. It thus showed a global vision of the experimental design.

Formulation variables showed to have a significant impact on all the responses, excepted on pellet circularity. The influence of drug solubility and concentration on product characteristics is linked to the corresponding water quantity used, as described in a previous work [START_REF] Désiré | A Comparison of three extrusion systems -Part II: The influence of formulation and spheronization conditions on pellet properties[END_REF] . Spheronization time increase showed a favourable effect on pellet morphology and mechanical properties (pellet dispersion, elongation, roughness and friability decrease) and increased pellet diameter.

The system of extrusion showed a significant impact on all the responses, except on the pellet pycnometric density. The extrusion scale showed a significant impact on all characteristics, except on pellet circularity. Moreover, in addition to the numerous interactions with formulation and spheronization variables, the design of experiments underlined interactions between extrusion system and scale for a majority of responses: this therefore requires a more detailed analysis.

The objective of the study is not to describe the influence of formulation and spheronization variables on the different responses, but to use their variations to better compare the different extruder systems at different scales. The goal is to "stress" the manufacturing process, i.e. to vary the various parameters identified as critical to a certain limit of change and examine the results they provide on the different systems and scales.

The ideal pilot extrusion system is the one which gives the best results in terms of productivity and pellet characteristics ("quality"), the one which shows less impact on the product after scale-up ("scalability"), the one which shows the least influence on these same properties when the formula used changes ("robustness"), and the one which allows the possibility to adjust or improve pellet properties with the spheronization variables change ("flexibility"). For this purpose, the influence of the different variables was studied by analyzing (and crossing) the results according to the scale or/and the extrusion system, by detailing the global design of experiments into several designs.

Analysis by scale ("quality" study)

Analysis by scale (lab vs industrial scale) allowed identifying the extrusion system giving the best results in terms of productivity and pellet characteristics at industrial scale, compared to lab scale. The two different scales were analysed separately as two distinct designs of experiments (Table 1) in order to study the influence of the extrusion system (E) according to the scale. Analysis by scale also allowed studying the influence of formulation and spheronization variables according to the scale.

For both lab and industrial scales, the extrusion system showed a significant influence on all responses, except on pellet pycnometric density at lab scale, and on pellet elongation, roughness and yield at industrial scale. Industrial scale thus showed a better robustness in terms of pellet morphology when the extrusion system changes, compared to lab scale.

For both lab and industrial scales, axial system showed the best characteristics in terms of productivity (except for extrusion rate at lab scale) and pellet quality. Axial system also produced pellets with higher diameter compared to radial system. These conclusions showed that in addition to the better robustness at industrial scale when extrusion system changes, the axial system identified at lab scale as the best one in terms of product quality is refunded as the best one at industrial scale.

Analysis by extrusion system ("scalability" study)

Analysis by extrusion system (radial vs axial) allowed identifying the easiest scaled-up system, i.e. the one which showed less impact on product characteristics after the scale-up. The two different extruder systems were analysed separately as two distinct designs of experiments (Table 2) in order to study the significant impact of scale-up according to the extruder. Analysis by extrusion system also allowed studying the influence of formulation and spheronization variables according to the extrusion system.

For both extrusion systems, the scale-up had a favourable impact on extrusion rate, but a negative effect on extrusion yield (due to the humid mass quantities used at industrial scale similar to the extruder dead zone). For both extrusion systems, the scale-up led to a pellet size increase and hardness decrease, considered as unfavourable effects, because it needs to readjust water quantity at industrial scale to produce pellets of similar size and hardness to those obtained at lab scale. The scale-up nevertheless led to an improved dispersion for both systems.

For radial system, scale-up showed no impact on pellet elongation, yield and density, and a favourable effect on pellet roughness and friability. For axial system, scale-up showed no influence on pellet roughness and friability, a favourable impact on pellet density, but a negative effect on pellet circularity and yield. Considering these results, it appears that radial system showed a little lower impact after scale-up than axial system. Moreover, these results do not take into account that several experiments could not be reproduced at industrial scale, especially with formulations containing high water quantities, causing drainage problems. Considering that, it can be concluded that axial system is the easiest scaled-up system in our work.

Analysis by scale and extrusion system ("robustness" and "flexibility" study)

Analysis by extrusion system and scale allowed to identify the kind of extruder which gave the best results in terms of process robustness and flexibility. The different extruders and scales were analysed separately as four distinct designs of experiments (Table 3) to study, for each combination, the significant effects of formulation and spheronization factors. The best system is the one which shows the less influence on the product properties when the formula used changes ("robustness"), and the one which allows the possibility to adjust or improve pellet properties with spheronization variables ("flexibility"). Whatever the extrusion system, the industrial scale led globally to a very slightly improved system robustness, and lower flexibility, compared to lab scale.

At lab scale, the results are not statistically all analyzable (notably elongation and usable yield for axial Table 1. Detailed experimental design analysis by scale. Influence of the radial to axial system change (i.e. influence of axial system by comparison with radial system): no impact; positive impact; negative impact.

Responses Lab scale Industrial scale

A B C D A B C D Y1 ↗ NS ↗ ↘ NS NS ↗ ↗ Y2 ↗ NS NS ↗ ↘ NS ↗ ↗ Y3 NS ↗ ↗ ↗ ↗ NS NS ↗ Y4 ↗ ↘ ↘ ↘ ↘ ↘ NS ↘ Y5 NS ↘ NS ↘ NS ↘ ↗ NS Y6 NS NS ↗ ↗ NS NS ↗ NS Y7 NS ↗ ↗ ↗ NS ↗ ↗ NS Y8 ↘ ↘ ↘ NS NS NS ↘ ↗ Y9 NS NS ↘ ↘ ↘ NS ↘ ↘ Y10 ↘ NS ↘ ↗ ↘ NS ↘ ↗ NS =
no significant effect of the factor on the response. ↗ = the value of the response significantly increases with the factor (i.e. when the factor increases) ↘ = the value of the response significantly decreases with the factor (i.e. when the factor decreases). A = DS concentration, B = Spheronization time, C = DS solubility, D = Extrusion system. Y values are defined in Figure 1.

Table 2. Detailed experimental design analysis by extrusion system. Influence of the lab to industrial scale-up: no impact; positive impact; negative impact.

Responses

Radial system

Axial system

A B C E A B C E Y1 ↘ NS ↗ ↗ NS NS NS ↗ Y2 ↘ NS ↗ ↘ NS NS ↗ ↘ Y3 ↘ ↗ ↗ ↗ ↗ ↗ ↗ ↗ Y4 NS ↘ ↘ ↘ ↗ ↘ ↗ ↘ Y5 NS ↘ NS NS ↗ ↘ ↗ ↗ Y6 ↗ NS ↗ NS ↘ NS NS ↘ Y7 NS ↗ ↗ ↗ NS NS ↗ NS Y8 NS NS ↘ NS NS NS ↘ ↗ Y9 ↘ NS ↘ ↘ NS NS NS NS Y10 ↘ NS NS ↘ ↘ NS ↘ ↘ NS =
no significant effect of the factor on the response. ↗ = the value of the response significantly increases with the factor (i.e. when the factor increases). ↘ = the value of the response significantly decreases with the factor (i.e. when the factor decreases). A = DS concentration, B = Spheronization time, C = DS solubility, E = Extrusion scale. Y values are defined in Figure 1.

system), but a previous study showed that radial system presented the best robustness and flexibility [START_REF] Désiré | A Comparison of three extrusion systems -Part II: The influence of formulation and spheronization conditions on pellet properties[END_REF] . Conversely, at industrial scale, axial system presented the best robustness (with more particularly an absence of impact of drug solubility on pellet size) and a slightly better flexibility (even if this one is limited at industrial scale).

conclusion

Compared to lab scale, the conclusions observed at industrial scale are the same in terms of product quality (axial system giving better results), and different in terms of robustness and flexibility (axial system being better at industrial scale, conversely to radial system at lab scale), which confirms the importance to test the two systems at industrial scale before investing in one industrial equipment.

Considering the results, axial system appears as the system giving the best results, and the easiest scaled-up one, without any process or formulation readjustment, by comparison with lab scale. The design of experiments approach thus allowed studying critical parameters and their interactions on the different responses, and identifying the more efficient extrusion system, through different design analyse.

Experimental designs allow building design spaces where selected responses are into specifications whatever the formulation and/or process parameters. It firstly evaluates potential risks in scale-up operation: if the design space is applicable to multiple operational scales, it is described in terms of relevant scale-independent parameters. It also evaluates the ability of the process to tolerate variability of materials and changes in the process and equipment without negative impact on quality. It thus enters in quality risk management, in a quality by design context. ↘ NS NS = no significant effect of the factor on the response. ↗ = the value of the response significantly increases with the factor (i.e. when the factor increases). ↘ = the value of the response significantly decreases with the factor (i.e. when the factor decreases). A = DS concentration, B = Spheronization time, C = DS solubility. Y values are defined in Figure 1.

Figure 1 .

 1 Figure 1. Experimental design: factors and responses.

Figure 2 .

 2 Figure 2. Pellet elaboration flow chart.

Figure 4 .

 4 Figure 4. The effects of factors and main interactions on various responses Y x . A = DS concentration, B = Spheronization time, C = DS solubility, D = Extrusion system, E = Extrusion scale, AD = Interaction between DS concentration and extrusion system, AE = Interaction between DS concentration and extrusion scale, BD = Interaction between spheronization time and extrusion system, BE = Interaction between spheronization time and extrusion scale, CD = Interaction between DS solubility and extrusion system, CE = Interaction between DS solubility and extrusion scale, DE = Interaction between extrusion system and extrusion scale. Y values are defined in Figure 1.

Table 3 .

 3 Detailed experimental design analysis by scale and extrusion system. Study of robustness (A and/or C = NS) and flexibility (B = ↗ or ↘).

					Lab scale					Industrial scale
			Radial			Axial			Radial			Axial
	Responses	A	C	B	A	C	B	A	C	B	A	C	B
	Y1	↗	↗	NS	NS	↗	NS	↘	↗	NS	NS	NS	NS
	Y2	↗	NS	NS	↗	↗	NS	↘	↗	NS	NS	↗	NS
	Y3	↘	↗	↗	↗	↗	↗	↘	↗	NS	↗	NS	NS
	Y4	NS	↘	↘	↗	NS	↘	↘	NS	↘	NS	↗	↘
	Y5	NS	↘	↘	NS	NS	↘	NS	NS	NS	↗	↗	↘
	Y6	↗	↗	↗	NS	NS	NS	↗	↗	NS	↘	NS	NS
	Y7	NS	↗	↗	NS	↗	NS	NS	↗	NS	NS	↗	NS
	Y8	↘	↘	NS	↘	↘	NS	NS	↘	NS	NS	↘	NS
	Y9	NS	↘	NS	NS	NS	NS	↘	↘	NS	NS	NS	NS
	Y10	↘	NS	NS	↘	↘	NS	↘	↗	NS	NS

Declaration of interest

The authors report no conflicts of interest.


Farmers delivering in local short food supply chains: mobility practices, territories, and proximities. A case study in Hauts-de-France (France)

Extended Abstract -ERSA 2019 -Special Session S21. Circularities and proximities: stakeholders, scales, resources. Céline Raimbert, IFSTTAR/AME/SPLOTT Gwenaëlle Raton, IFSTTAR/AME/SPLOTT Farmers delivering in local short food supply chains: mobility practices, territories, and proximities. A case study in Hauts-de-France (France).

With the rise of local short food supply chains (SFSC), which outlets multiply and diversify, more and more producers are becoming physical actors of food supply chains. By reducing the number of intermediaries, the producers are in charge of multiple commercial and logistical tasks. They involve developing more or less close relationships with a potential or established clientele (canvassing, communicating, ensuring follow-up in demand areas), but also, very often, delivering. As soon as the issue is no longer about observing the catchment area of a store but the mobility practices of actors from the productive space towards consumption basins, it modifies the focus of the geographer. This implies questioning producers' representations of spaces and territorial resources and the processes at work in their preferences to certain delivered spaces. What does the mobility of producers in SFSC say about their links built/under construction with the various spaces adjacent to farms (urban or multi-polarized areas, peri-urban or rural areas)? From the point of view of these mobility practices, what are the emerging territories? What are their size, shape and orientation?

This communication aims at analyzing the mobility practices of producers of SFSC from their farm to their delivered outlets and the territories they build then. Their perimeter might be extensive or concentrated, continuous or fragmented. We postulate that they are the result of specific mobility strategies and representations of spaces and territorial resources. By relying on the case study of six producers involved in SFSC and located in the Grand Hainaut region (Nord, France), we will try to qualify these territories, in light of two types of dynamics: (1) the influence of urban areas, as consumption basins (depending on their size and their attractiveness); (2) the role of production basins (production concentration, homogeneity or heterogeneity of productive profile). In the first case, we mobilize population density indicators (number of clients that can be reached in a minimum journey), but also the socio-economic attributes of the inhabitants. In the second case, issues of competition and collaboration are at stake: the number of producers located in the territory and their productive profile. Besides, we mobilize the concepts of chosen or suffered geographical proximities [START_REF]Retour sur la notion de proximité géographique[END_REF] to address the question of the representations of the different spaces and the territorial resources they contain (Gumuchian & Pecqueur, 2007). Thus, we assume that, depending on producers' perceptions -positive or negative -of the territorial resources from both consumption and production basins, geographical proximities are one of the determinants of mobility strategies of producers in SFSC.

Our communication is composed of three main points:

1) A description of our case studies, i.e. the six investigated producers and their territory: the Grand Hainaut. It is composed of numerous multi-polarized spaces, which allows a fertile discussion about urban tropism and preferences to deliver different urban areas.

2) A presentation of our methods, which consist in:

-Defining precisely the mobility territory of each producer: localization of farm and delivery points, calculation of different types of distance between farm and delivery points (ellipsoid and standard deviational ellipse) and characteristics of the delivery journey (frequency, organization of delivery tour or not). For this first task, we map the mobility territory of each producer on this model:

Map1. The territory mobility of producer E. Conception by author 2.

-Determine the type of areas that are frequented: types of environment where the farm and the delivery points are located including type of urban areas, type of agricultural production profile, socioeconomic indicators of both spaces;

-Lighting up the distances covered by the notion of geographical proximity: thanks to this concept, we can address another dimension of the distance. It is used for relativizing real distances by the mobility capital of the concerned individuals and by their own perception of distance and its costs and externalities (Pecqueur & Zimmermann, 2004;[START_REF] Rallet | Proximité et localisation[END_REF][START_REF] Talbot | Proximité et institutions : nouveaux éclairages. Présentation du numéro[END_REF].

3) In the results, we compare the mobility territory of the six investigated producers and try to highlight some recurrent mobility strategies such as:

-Concentration strategies: In this case, mobility territories are tight and the distances are relatively short. Considered as sufficient and adequate, local territorial resources are used and valued by the producers. In our case studies, these strategies prevail in producers whose farms are located in or near dense urban areas and for which the access costs of these abundant resources are low. Nevertheless, they can take place in less favored contexts. Then, the higher access cost to territorial resources is compensated by organizational strategies (multiplication and/or diversification of outlets, systematization of delivery tour, compartmentalization and extension of mobility territory). In any case, the geographical proximity is perceived positively and can be considered chosen.

-Extroversion strategies: When they adopt this kind of strategies, the producers voluntarily extricate themselves from an existing area of influence (urban concentration and/or production concentration) and may travel long distances to reach their preferred territories. Among the investigated producers, those concerned by this type of strategy are located in marginal areas (implying, for instance, strong rural features or weak socio-economic dynamics) which imply suffered geographical proximity.

-Diversification strategies: In most of the cases, they are combined with other strategies. For example, we observe producers who value local territorial resources and show particular attachment to the place of origin (farm) (concentration strategy), but also express a desire for territorial diversification towards other environments, implying further distances from farm and spaces with different types of urban areas, socio-economic characteristics and/or production profiles. These strategies tend to extend, and sometimes, to fragment the mobility territory.

Finally, the delivery imperative of SFSC French producers has meant an increase of opportunities available and a high demand in some urban context. The limitation of the number of intermediaries invite producers to (re)question the potential of the territory: to whom should I sale my products?

Where is the target consumers located? Who are my competitors? Where are they concentrated? In addition, the significant time spent on deliveries [START_REF] Raton | Organisations Logistiques Intelligentes des Circuits courts en vallée de Seine, rapport final OLICO-Seine[END_REF] encourages attention to the distances travelled and the location of outlets. From then on, the location of the farm is central, such as the attachment of local environment, sometimes favored or on the contrary avoided.


M Nazar

A Ibrahim

Z Khatir

N Degrenne

Z Al-Masry

Wire-bond Contact Degradation Modelling for Remaining Useful Lifetime Prognosis of IGBT Power Modules

Reliability of power electronic equipment can be enhanced with the ability to predict its health state to avoid damage and permit making necessary maintenance. In this paper, an analytical physics of failure model is built that describes the contact resistance between wire-bonds and pad-metallization in an IGBT semiconductor device. On one hand, this model is based on the theory of the electrical contact resistance and the other hand on the spread of the current from the contact to the aluminum film. When contact degradations occur, the model can relate the crack propagation in the wire-bond interconnection to the collectoremitter voltage (𝑉 𝐶𝐸 ) and may allow identifying the health of the device. If the voltage is measured during actual operation, the length of the contact radius can be estimated and crack propagation rate can be calculated and used to predict the remaining useful life. The approach is tested with experimental accelerated aging tests.

Introduction

The reliability of power electronic devices is an important issue for most power electronic systems. Electric components such as IGBTs are subject to high environmental and functional stresses (ambient temperature, vibrations…). All these factors have a strong impact on device reliability and lifetime.

Prediction of the Remaining Useful Lifetime (RUL) with a high confidence level is needed to assure reliability. It is important economically to have predictive maintenance for such devices than having preventive maintenance performed at fixed time intervals. One way to have a prediction approach is to have a model that describes the degradation in the power device. Two main kinds of models are presented in the literature for prognostic: the damage accumulation models and the condition based prediction models. The damage accumulation models can estimate the lifetime of components operating under given usage conditions. The effect of past usage is explicitly modeled by summing up the life consumption due to each stress undergone by the component in operation [START_REF] Manson | thermal stress law-cycle fatigue[END_REF][START_REF] Yang | Physics-offailure lifetime prediction models for wire bond interconnects in power electronic modules[END_REF]. Research is ongoing to improve these types of models, for example, N. Dornic [START_REF] Dornic | Stress-Based Model for Lifetime Estimation of Bond Wire Contacts Using Power Cycling Tests and Finite-Element Modeling[END_REF] gives the dependence of the 𝑁 𝑓 (number of cycles until failure) with the stress duration 𝑡 𝑜𝑛 and junction temperature swing amplitudes 𝛥𝑇 𝑗 . On the other hand, condition-based prediction models are based on the measurement and extrapolation of a precursor to failure. It assesses the behavior of individual units not only on their usage history but also on expected future load profiles. In other words, it utilizes a degradation model that predicts the future state based on input that describes the current system state and the expected load levels on the system. These component-specific models are generally adaptable and can be developed based on the physics of the system like [START_REF] Degrenne | Diagnostics and Prognostics of Wire-Bonded Power Semi-Conductor Modules subject to DC Power Cycling with Physically-Inspired Models and Particle Filter[END_REF][START_REF] Hu | Cost-Effective Prognostics of IGBT Bond Wires with Consideration of Temperature Swing[END_REF], or can be learned from run to failure data using data-driven methods like [START_REF] Sreenuch | Probabilistic Monte-Carlo Method for Modelling and Prediction of Electronics Component Life[END_REF][START_REF] Alghassi | Prognostics and Health Management of Power Electronics[END_REF]. Sreenuch in [START_REF] Sreenuch | Probabilistic Monte-Carlo Method for Modelling and Prediction of Electronics Component Life[END_REF] used a probabilistic Monte-Carlo method for modeling and online prediction of the remaining useful life (RUL) in an electronic component. His study is based on the 𝑉 𝐶𝐸 data collected from the electronic component, also Zhen Liu in [START_REF] Liu | Remaining useful life estimation of insulated gate biploar transistors (IGBTS) based on a novel volterra K-nearest neighbor optimally pruned extreme learning machine (VKOPP) model using degradation data[END_REF] used the Machine learning method to predict RUL for IGBT semiconductors using degradation data.

The remainder of the paper is organized as follows. Section 2 defines different types of degradations that occur in IGBT devices due to aging. Section 3 presents the analytical model used for RUL prediction. It describes the evolution of the dominant degradation that takes place in the studied module. It is based on the theory of contact resistance and on the spreading resistance in a thin metallization film. In section 4, an online prognostics model methodology is presented to predict the RUL. The steps for building the model are illustrated. This model is tested on experimental results with different stress conditions.

Aging behaviors and Degradations

During the aging of power electronic devices, different types of degradations can take place as shown in Fig. 1.

Fig.1: Possible degradation location in IGBT power module

The degradation types are: wire-bond degradation that usually leads to heel cracks, wire-bond lift-off due to crack propagation in the wire-bond interconnection with the metallization, top metallization degradation due to grains recrystallization, die-attach degradation that leads to cracks at the die bottom connection, cracks in the metallization ceramic interface, cracks in the ceramic itself, and substrate attach that leads to solder cracks.

In the IGBT module, wire-bond lift-off is a dominant degradation (see Fig. 2a[9]). For this reason, the focus in this paper is related to the crack propagation leading to the lift-off. (Fig. 2b) presents crack propagation in the wire-bond interconnection with the metallization.. A way to observe the 𝑉 𝐶𝐸 evolution with degradation due to thermal fatigue is using accelerated aging tests like power cycling. Fig. 3 shows experimental data for 4 IGBT modules of the same type aged under the same stress conditions. As data from the field are not easy to obtain, these types of accelerated results are used hereafter for model testing and validation. The only degradation observed in these aging tests is the lift-off due to crack propagation. Hence, it is assumed that the voltage variation in the wire-bond contact is the only responsible of the voltage variation observed between the collector and the emitter terminals.

In Fig. 3, the experimental accelerated conditions used are from power pulses with 3s heating time and 6s cooling time with power value that leads to 110°C temperature variation, from 35°C to 145°C. This figure gives the 𝑉 𝐶𝐸 evolution during power cycling of the 4 devices under tests (DUTs). The lifetime of the devices can be divided into two main zones. In zone 1, the voltage increase is due to the crack propagation in the wire-bond interconnection with the metallization. At the end of this zone, more than 80% of the lifetime is passed. Zone 2 presents the increase in voltage due to the lift-off process. The focus of this paper is related to the degradation in zone 1.

Contact Resistance Physical Model

In this section, an analytical model is built that estimates the increase of the crack length at the wire-bond contact from the increase in the 𝑉 𝐶𝐸 . This model is then used to predict the RUL, assuming a mean speed linear crack propagation. In comparison to literature, a study done by Degrenne and Mollov [START_REF] Degrenne | Diagnostics and Prognostics of Wire-Bonded Power Semi-Conductor Modules subject to DC Power Cycling with Physically-Inspired Models and Particle Filter[END_REF] presents a physical model that relates the crack propagation to on-state voltage. Using a specific algorithm, this model is used to predict the RUL with an accuracy of less than ±10%. This model assumed that crack propagation is linear. However, Hu in [START_REF] Hu | Cost-Effective Prognostics of IGBT Bond Wires with Consideration of Temperature Swing[END_REF] built a physical model that relates the crack propagation to the voltage using the Finite Element Method and taking into consideration nonlinear crack propagation. This model took into account also the effects of temperature variation and metallization degradation on the on-state voltage increase.

The model in this paper combines two different models. The first one describes the constriction resistance at the bond contact due to the crack from the wire-bond side. The second one gives the spreading resistance in the metallization due to the current lines flowing from the die, collected by the metallization pad and going through the bonding wire. The spreading of the current lines is also affected by the crack length. The model is simplified in order to consider a disc as a surface contact between a cylindrical bond-wire and the metallization as depicted in Fig. 4. Based on Timsit's relation [START_REF] Application | I I 1 1 I I[END_REF], the constriction resistance in the wire-bond side is given by: where 𝜌 (resistivity of the material), 𝑎 (radius of the contact area), and 𝑟 is the redius of the wire-bond. This constriction resistance is due to the current constricted to pass through the contact area (see Fig. 4). At first, when there is no crack and the contact is full, even if current lines are perturbed due to geometry, this is not strictly speaking a constriction zone.

Constriction Resistance in the wire-bond part

𝑅 𝐶 = 𝜌 4𝑎 [1 -1.
According to Eq.1, 𝑅 𝐶 is dependent on the radius of the area still in contact. Since crack propagates during the aging, the contact area decreases with the radius 𝑎 that leads to an increase in Constriction resistance 𝑅 𝐶 .

Metallisation's Resistance Model

The resistance of the metallization is due to the lateral current flowing vertically from the die (see Fig. 5). If the bondwire contact radius is 𝑎 and the radius of the considered metallization area is 𝑏, then the equivalent resistance of metallization, "seen" between 𝑟 = 𝑎 and 𝑟 = 𝑏 is given by eq.2, see appendix.

𝑅 𝑚𝑒𝑡. = 𝜌 2𝜋𝑡 ( 𝑏 2 𝑏 2 -𝑎 2 𝑙𝑛 𝑏 𝑎 - 1 2 ) ( 2
)
Where t is the metallization thickness and 𝜌 its resistivity. The effect of crack propagation on the metallization resistance is very important to be modeled. It is shown later that the metallization resistance variation is high in comparison to the constriction resistance.

This resistance in Eq.2 covers the resistance of the metallization form 𝑎 to 𝑏. This relation doesn't include the resistance increase in the metallization due to resistivity increase and/or structure change (recrystallization and grain grooving). Nevertheless, the resistance of metallization increases with aging due to power cycling as mentioned in [START_REF] Zhao | A study on the effect of microstructure evolution of the aluminum metallization layer on its electrical performance during power cycling[END_REF]. The literature reports that the metallization resistance can increase by a factor 2 to more than 10 depending on stress conditions [START_REF] Zhao | A study on the effect of microstructure evolution of the aluminum metallization layer on its electrical performance during power cycling[END_REF][START_REF] Arab | Experimental investigations of trench field stop IGBT under repetitive short-circuits operations[END_REF][START_REF] Pommier | Effect of die metallization layer ageing in the case of power semiconductor devices[END_REF]. Based on that, a factor "w" is multiplied to the resistance of Eq.2 to form Eq.3. No additional measurements were done to fix "w", however, it is assumed to be constant and equal to 2 based on the study of [START_REF] Zhao | A study on the effect of microstructure evolution of the aluminum metallization layer on its electrical performance during power cycling[END_REF] that carries almost the same conditions of power cycling tests used in this study.

𝑅 = 𝑤𝜌 2𝜋𝑡 ( 𝑏 2 𝑏 2 -𝑎 2 𝑙𝑛 ( 𝑏 𝑎 ) - 1 2 ) (3)
It is necessary to be mentioned that the wire-bond contact shape in IGBT is ellipsoidal. However, Holms mentioned in [START_REF] Holm | Electric Contacts[END_REF] that for ellipse contact shape the difference can be neglected if the multiplication of the ellipse's major and minor semi-axis is equal to 𝑎 2 (which is our situation). Moreover, the dimensions of the metallization are not circular; in this study "b" is calculated experimentally by fitting the whole experimental curve to the model. Besides, a parameter sensitivity study for "b" is done and it shows that changing b in an acceptable range between 1.3 mm and 2 mm has a very small effect in changing the predicted curves.

Proposed model

By combining two models described in 3.1 and 3.2, Eq.4 is built that describes the effect of crack propagation (contact area decrease) on the resistance between wire-bond and metallization.

Fig. 6 shows effect of the radius change of the bond contact on both resistances due to crack propagation. It is observed that with a decrease in the radius, the two resistances increase. It can be observed that the metallization resistance remains very high compared to that of constriction except when the contact is reduced to a very small radius. sensitivity of resistance to radius change increases with radius decrease, and it increases in high slope at the last few micrometers.

The values of parameters of Eq.1 & 3 used in this study based on the structure are 𝑏 = 2 𝑚𝑚, radius initial value 𝑎 0 = 332 𝜇𝑚, 𝑡 = 4𝜇𝑚, and 𝜌 = 3.8 * 10 -8 Ω𝑚 This model is used in the prognostics methodology to predict the remaining voltage variation at each measurement. Besides, as mentioned before, the variation of voltage calculated from Eq.3 equals the variation of collector-emitter voltage (𝛥𝑉 𝐶𝐸 ). The approach is more detailed in the next section.

Prognostics Approach

The prognostics approach for RUL estimation is based on Eq.4. At the current time, the 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸 value is measured (mth measurement), (see Fig. 7). This is used to calculate the current contact area radius 𝑎 𝑚 using Eqs.5-8 and compare it to the initial radius 𝑎 0 . If a difference is detected, this means that a crack has occurred or progressed. The initial contact area radius 𝑎 0 , as well as the successive contact radius 𝑎 𝑖 (i=1 to m) are used to compute the rate of crack propagation (𝐾 𝑚 ), between the mth and the (m-1)th measurements, using Eq.9. For prognostic purpose, a mean value 𝐾 ̅ of the successive 𝐾 𝑚 is used to predict the evolution of the contact area radius with the assumption of linear crack propagation using Eq.10.

The detailed process is as follows, at first, relative variation in 𝑉 𝐶𝐸 is calculated, applying Ohm's law:

𝛥𝑉 𝐶𝐸 𝑉 𝐶𝐸,𝑖𝑛𝑖 = 𝛥𝑅(𝑎) 15𝑊 * 𝐼 𝑉 𝐶𝐸,𝑖𝑛𝑖 (5)
where 𝑉 𝐶𝐸,𝑖𝑛𝑖 is the initial collector-emitter voltage and 𝐼 is the direct current used. 𝛥𝑅(𝑎) 15𝑊 is the resistance variation of 15 parallel wire-bonds connected to the metallization pad. Eleven of them are assumed degrading and four are not. This degradation distribution assumption is due to a study done that is related to the structural wire bonding on the device used, which leads the current to take less resistive paths and make degradation effect on 11 wire-bonds rather being on all of them. Which could give the possibility to assume degradation in 11 wire-bonds. Also it is assumed that the rate of degradation for the 11 damaged wire-bonds is the same.

The resistances 𝑅(𝑎 0 ) and 𝑅(𝑎 𝑚 ) are calculated from Δ𝑉 𝐶𝐸 using Eq.5-8, the parameter 𝑎 𝑚 is calculated after inverting Eq.4 and then used to compute 𝐾 𝑚 using Eq.9

𝐾 𝑚 = 𝑎 𝑚 -𝑎 𝑚-1 𝑛 𝑚 -𝑛 𝑚-1 (9)
Using the average value (𝐾 ̅ ) of the 𝐾 𝑚 values (𝐾 1 . . . , 𝐾 𝑚 ), in Eq.10 gives the predicted values of radius contact 𝑎 𝑛 until the crack propagation completes.

𝑎 𝑛 = 𝐾 ̅ 𝑛 + 𝑎 0 [START_REF] Application | I I 1 1 I I[END_REF] where 𝐾 ̅ is the average crack propagation rate. "n" represents the number of cycles for the prediction process at each planned measurement m.

By introducing the predicted values of 𝑎 𝑛 in Eqs.5-8, the values of 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 are calculated.

Which means that at each new measurement m, a prediction curve that predicts 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 is formed. This prediction curve stops when the contact area radius "𝑎 𝑛 " reaches zero, that means lift-off.

For instance, for m=1, 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 is measured at 8 kilocycles and is equal to 0.16% as shown in Fig. 8. This allows to calculate the area contact radius 𝑎 1 . Then, by having 𝑎 0 and 𝑎 1 , the first crack speed rate 𝐾 1 is calculated and assumed as constant for predicting the remaining radiuses until "a" reaches zero The predicted values of radiuses are reinserted in Eqs.5-8 to calculate the predicted evolution of the curve 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 . The first prediction can be observed in Fig. 8. By repeating the previous process for several measurements and plotting the predicted 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 , Fig. 9 could be observed. As it can be observed in Fig. 9, the prediction was done to estimate the remaining 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 curve at each step of measurement. The conventional failure criteria assumption on the collector-emitter voltage indicator is when the 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 reaches a 5% increase. However, it is important to note that this threshold is quickly achieved in the last part of aging, which is wire-bond lift-off (zone 2 Fig. 3). In Fig. 9, the experimental data represented by the dashed black curve is used to validate the predictions. The first and third predictions (at 8 kcycles and 16 kcycles) are less acceptable than the rest of the curves. This is due to the primary construction of the degradation propagation rate. However, this rate is then tuned after each new measurement due to computing the mean of crack propagation rates, and as observed for the remaining six predictions, the estimated 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 is close to the experimental observation and can be relied on. Fig. 11 and Fig. 12 present the predicted RUL and Real RUL change with the number of cycles for the two different test conditions applied in this study. They start where the first prediction curve took place, except for Fig. 12 the curve starts from 32 k.cycles due to the errors observed in the experimental data at the first 30 k.cycles. The values of predicted RULs could be also verified in Fig. 9 and Fig. 11 where the RUL in this study presents the wire-bond's lift-off.

As shown in Fig. 12, the predicted RUL from 13 to 18 kilocycles is overestimating the real RUL. The prediction curves are not very reliable. However, after 18 kcycles the RUL is being underestimated and near to the real RUL curve. In both figures, the predicted RUL underestimates the failure which could mean that no additional costs the company will pay to failure consequences.

One way to assess the model robustness is by applying it to different stress conditions. For that, the physical model (including its parameters) is also applied to data aged in different conditions.

The test conditions used in Fig. 13 are for an aging test under 𝛥𝑇 = 90°𝐶 and 𝑡 𝑜𝑛 = 3𝑠𝑒𝑐. As expected, these tests last much more than tests under 𝛥𝑇 = 110°𝐶. The lifetime for tests at 𝛥𝑇 = 90°𝐶 reaches around 100 kcycles whereas the lifetime of the tests at 𝛥𝑇 = 110°𝐶 reaches around 35 kcycles. As observed in Fig. 13 the first and second curves are far from the experimental curve. This is due to high variations in data measurements as observed at first 30 kcycles. Then, the prediction curves start to trend upward as the experimental curve.

Conclusion

In this study, a physics-based analytical model is built to relate crack propagation to 𝑉 𝐶𝐸 . Since 𝑉 𝐶𝐸 is monitored in realtime operation, then the crack length at each measurement can be estimated. Then a crack rate of propagation is computed and used to predict the crack lengths until the crack is completed and lift-off is achieved, assuming a linear crack propagation. These predicted crack lengths give the predicted collector-emitter voltage variation using the model built. The rate of crack propagation is averaged at each new measurement and used for the next prediction. The predicted 𝑉 𝐶𝐸 identifies the end of zone 1, which forms 80% of the total lifetime of the IGBT semiconductor device. The importance of this work lies under the online prediction ability of the important part of the lifetime in the IGBT; it could be improved to predict the whole life.

Fig. 2 :

 2 Fig.2: a) wire-bond lift-off[START_REF] Ciappa | Selected failure mechanisms of modern power modules[END_REF], b) crack propagation in the wire-bond interconnection with the metallization[START_REF] Degrenne | Diagnostics and Prognostics of Wire-Bonded Power Semi-Conductor Modules subject to DC Power Cycling with Physically-Inspired Models and Particle Filter[END_REF]

Fig. 3 :

 3 Fig.3: Collector-Emitter voltage variation, and its zones due to types of degradation

Fig. 4 :

 4 Fig.4: Model geometry with the constriction and the spreading areas.

Fig. 5 :

 5 Fig.5: Current flow in the metallization

Fig. 6 :

 6 Fig.6: Effect of contact area radius "a" change in Eq. 1 & 9 on the resistance

Fig. 7 : 7 )

 77 Fig.7: Vce measurements instants

Fig. 8 :

 8 Fig.8: Voltage variation percentage of the first prediction at 8 k.cycles compared to Experimental data

Fig. 9 :

 9 Fig.9: Prediction of the relative increase in 𝑉 𝐶𝐸 for IGBT device under the conditions 𝛥𝑇 = 110°𝐶 & 𝑡𝑜𝑛 = 3𝑠𝑒𝑐 Fig.9 can show the predicted curves for 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 at several numbers of cycles. The experimental results in Fig.9 refer to a device aging under 𝛥𝑇 = 110°𝐶 and 𝑡 𝑜𝑛 = 3 𝑠𝑒𝑐. The experimental data used are filtered with an average moving filter on 3 consecutive points.As it can be observed in Fig.9, the prediction was done to estimate the remaining 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 curve at each step of measurement. The conventional failure criteria assumption on the collector-emitter voltage indicator is when the 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 reaches a 5% increase. However, it is important to note that this threshold is quickly achieved in the last part of aging, which is wire-bond lift-off (zone 2 Fig.3). In Fig.9, the experimental data represented by the dashed black curve is used to validate the

Fig. 10 :

 10 Fig.10: Contact area radius measured (𝑎 𝑚 ) decrease with the number of cycles for the aging tests done under 𝛥𝑇 = 110°𝐶 𝑎𝑛𝑑 𝑡 𝑜𝑛 = 3𝑠𝑒𝑐 At each measurement done a new contact area radius is computed. Fig.10 presents the radius decrease variation with cycles. It is shown that the variation of radius 𝑎 with cycles is almost linear. This verifies the assumption of linearity used in crack propagation.

Fig. 11 :Fig. 12 :

 1112 Fig.11: Predict RUL change when applied on aging tests done under the test condition 𝛥𝑇 = 110°𝐶 and 𝑡 𝑜𝑛 = 3𝑠𝑒𝑐

Fig. 13 :

 13 Fig.13: Prediction of the relative increase in 𝑉 𝐶𝐸 for IGBT device under the conditions of 𝛥𝑇 = 90°𝐶 & 𝑡𝑜𝑛 = 3𝑠𝑒𝑐

Appendix

The metallization resistance is formulated considering Fig. 14.

Considering 𝐽 0 as the vertical current density in the die, 𝐼 1 the total current from die under the contact (𝐼 1 = 𝐽 0 𝜋𝑎 2 ) and 𝐼 2 the total current outside the contact (𝐼 2 = 𝐽 0 𝜋(𝑏 2 -𝑎 2 )), 𝑎 is the radius of the contact area, and 𝑏 is the outer radius of the metallization. The whole current is 𝐼 = 𝐼 1 + 𝐼 2 . One can observe that the current in the metallization 𝑖(𝑟) is negative in the radial oriented axis but the current variation 𝑑𝑖/𝑑𝑟 at radius r is positive and given by:

Eq.11 is integrated from 𝑟 to 𝑏 knowing that 𝑖(𝑏) = 0. The following equation is formed:

Where 𝑖(𝑟) is the lateral current in the metallization, the negative sign indicates that it is flowing in the reverse direction of the radial oriented axis. Let 𝑗(𝑟) be the lateral current density in the metallization at 𝑟:

where 𝜌 is the electrical resistivity of the material, and 𝑡 is the thickness of the metallization. By using Eqs.12 & 13 the Eq.14 is obtained:

Thus, after integration from 𝑎 to 𝑏 :

The equivalent resistance of metallization, seen between 𝑟 = 𝑎 and 𝑟 = 𝑏, can be written: