M I Den Hertog

C Cayron

P Gentile

F Dhalluin

F Oehler

T Baron

J L Rouviere

Hidden defects in silicon nanowires

Keywords:

Recent publications have reported the presence of hexagonal phases in Si nanowires. Most of these reports were based on 'odd' diffraction patterns and HRTEM images , -'odd' means that these images and diffraction patterns could not be obtained on perfect silicon crystals in the classical diamond cubic structure. We analyze the origin of these 'odd' patterns and images by studying the case of various Si nanowires grown using either Ni or Au as catalyst in combination with P or Al doping. Two models could explain the experimental results : (i) the presence of an hexagonal phase or (ii) the presence of defects that we call 'hidden' defects because they cannot be directly observed on most images. We show that in many cases one direction of observation is not sufficient to distinguish between the two models. Several directions of observations have to be used. Secondly, conventional TEM images i.e. bright field two beam and dark field images, are of great value in the identification of 'hidden' defects. In addition, slices of nanowires perperpendicular to the growth axis can be very useful. In the studied nanowires no hexagonal phase with long range order is found and the 'odd' images and diffraction patterns are mostly due to planar defects causing superposition of different crystal grains. Finally, we show that in Raman experiments the defect rich NWs can give rise to a Raman peak shifted to 504-511 cm -1 with respect to the Si bulk peak at 520 cm -1 , indicating that Raman cannot be used to identify an hexagonal phase.

I. INTRODUCTION

Silicon nanowires (SiNWs) are studied widely for use in the semiconductor industry as building blocks for electrical and electro-optical devices. In some semiconductor NW systems such as gallium arsenide 1,2 both the cubic and hexagonal phase can be observed, which gives rise to interesting physical properties and provides an additional parameter to design functional nanostructures. For SiNWs a controversy is present in literature. In silicon thin films and silicon under high pressure more than twelve forms of silicon have been reported 3,4 , among others the 2H wurtzite structure. Hexagonal structures were also found in silicon and germanium thin films deposited by laser ablation 5,6 and recrystallized by an electron beam 7 . The existence of silicon polytypes was deduced using x-ray diffraction 8 , High Resolution Transmission Electron Microscopy (HRTEM) 9 and Raman studies 10 . In the TEM studies the polytypes appear as a sequence of stacking faults and twin defects, locally ordered on a few nanometers, but no long range order on tens of nanometers was demonstrated. Moreover many articles claim the existence of the 2H (or 6H) hexagonal wurtzite phase in SiNWs [11][12][13][14] . The evidence is generally based on transmission electron microscopy (TEM) images in combination with Raman spectra, where a shift from the 520 cm -1 frequency, characteristic of diamond cubic silicon, to lower wavenumbers is observed 13 . Interestingly very similar TEM images and electron diffraction patterns (EDPs) obtained on thin silicon and gold films and SiNWs have been attributed to double diffraction between superposed twinned crystals [15][16][17][18][19] . Furthermore some of the 'odd' diffraction spots have been attributed to streaking effects caused by very thin defect sequences 19,20 , to which we refer as nanotwins. Such streaking effects can be understood intuitively using the Ewald sphere scheme. A very thin crystal bounded by defects will diffract as a very thin sample with a consequently very long reciprocal lattice rod, that easily intersects with the Ewald sphere and creates a streaking effect. For very thin fcc crystals it has been proposed that at the surface the ABCABC cubic stacking can be incomplete, therefore some of the crystal can be described by the 2H wurtzite phase, which can create extra diffraction spots 21 . This idea was also used for NWs 22 . Although this hypothesis can explain the experimentally observed EDP's/HRTEM images successfully, this does not exclude the presence of defects.

Interestingly it has also been shown that defects such as dislocations could itself induce the growth of nanostructures, as demonstrated for PbSe NWs 23 . In the Si NWs described in our study no dislocations were observed and therefore we do not expect similar effects.

It seems important to note that the insertion of polytypes with long range order in III-V and II-VI NW systems is remarkably different from most of the claimed observations of polytypes in SiNWs. In III-V and II-VI NWs the different phases are generally separated by grain boundaries perpendicular to the growth direction, what we call here horizontal 1,2,24 . However in SiNWs the polytypes are generally found in the middle of the NW and no horizontal boundary is present between the normal diamond cubic structure and the other crystal phase 13 .

In a previous work 19 we have shown by a comparison of experimental and simulated diffraction patterns that 'odd' diffraction patterns (where odd means that these diffraction patterns do not exist in the perfect diamond cubic crystal) obtained along different directions of observation on SiNWs can be interpreted as a diamond cubic crystal structure with the presence of micro-and nanotwins. This analysis showed that it is not generally possible to determine the crystal phase (hexagonal, cubic or cubic with defects) from one single direction of observation. Despite our warning 19 recent publications claim to show unambiguous evidence of the existence of an ordered 2H hexagonal phase and 9R phase in SiNWs by using correlated TEM with Raman spectroscopy 13 and electron diffraction 14 , but regarding only one direction of observation in TEM.

The aim of this paper is to complete the work of Ref. 19 (based mostly on EDPs) by describing additional methods to distinguish by TEM the two possible models which are (i) the existence of new phases or (ii) the presence of (twin)defects. The NWs are studied by TEM from several directions and also along the NW growth direction. Focused ion beam (FIB) milling is used to prepare slices of NWs to image defects in the NW section along the growth direction. Furthermore we show that conventional two beam TEM images can be of great value to distinguish between the presence of 'hidden ' defects or random stacking faults oriented along the electron beam direction. Raman spectra were obtained on Ni catalyzed NWs and are compared with literature. Finally, in the discussion part we comment on the TEM images presented in 13,14 and show that indeed such images can be explained by the presence of micro-and nanotwins in the classic diamond cubic structure.

II. EXPERIMENTAL DETAILS

Collaborating with several teams working on Si NW growth, we have observed many nanowires 19,[25][26][27][28][29][30] . In this work we focus on NWs giving unusual HRTEM images or EDPs. We have selected 2 types of nanowires for this paper : (i) Si NWs grown with Ni catalyst and silane as the gas source or (ii) doped Si NWs grown with Au catalyst. Among, these observed samples, we selected the nanowires that have unusual images or diffraction patterns. It turned out that this happened primarily for two particular growth directions : 110 and 112 . The NWs having these growth directions will be respectively called 110 NWs and 112 NWs. For one particular sample defects could also be observed in other growth direction such as 111 and 100 . However, in general 111 NWs, i.e. nanowires with a 111 growth direction, did not have odd images or diffraction patterns.

The NWs were grown by chemical vapor deposition via the vapor-liquid-solid process on a Si substrate in a low pressure chemical vapor deposition (LPCVD) reactor at a base pressure held constant at 20 mbar. Silane (SiH 4 ) was the source of Silicon. Hydrogen (H 2 ) was used as the carrier gas.

For the Ni catalyst, 2 nm of nickel was deposited in a vacuum of 10 -6 Pa on a cleaned (100) silicon wafer. The substrate was then loaded in the reactor and annealed between 500 and 900 o C under a H 2 flow for several minutes, in order to de-wet the Ni layer and form nanocatalysts. After cooling down the substrate to the growth temperature, SiH 4 was introduced by keeping the total pressure (SiH 4 and H 2 ) constant. Wires were grown at different silane partial pressures ranging from, 0.054 to 0.3 mbar and at temperatures ranging from 550 o C to 800 o C. Si NWs with different growth directions were obtained. The observed NWs grow primarily in the 110 direction.

Similar conditions were used for the NWs grown with the gold catalyst, except that the NWs were tentatively doped either with P using diphosphine 29 or Al using trimethylalumina 25 and grown on (111) silicon.

The samples for TEM were prepared in several ways. For the Ni-catalyst samples, the Si substrate with the NWs on top of it was cleaved and the NWs were observed from the side, still attached to the substrate as described in the supporting information of Ref. 29 . A slice of a Ni-catalyst SiNW was prepared by Focussed Ion Beam (FIB) milling with an FEI Strata 400 equipment. Details of the FIB preparation are described in the supporting information. The Au-catalyst samples were either prepared by the cleaving technique or the NWs were deposited on a holey carbon grid. HRTEM images and Electron Diffraction Patterns (EDP's) were realized on a JEOL 4000EX (400 kV), equipped with a GATAN 2Kx2K CCD camera. Scanning TEM (STEM) images were obtained on a FEI TITAN TEM. Raman experiments were realized on an InVia Renishaw spectrometer allowing measurements in the visible and the near Infra-Red spectral ranges, equipped with an air-cooled CCD detector, a microscope, and a 50 × (NA = 0.95) objective. Raman spectra were obtained using the 514,5 nm line of an Argon-Krypton ion laser resulting in a probe size diameter of around 1 µm.

III. EXPERIMENTAL RESULTS

A. 110 Si nanowires with Ni catalyst

This section is dedicated to Si nanowires grown with Ni as catalyst. These nanowires have relatively large diameters (in the 100 -300 nm range) and grow generally in the 110 direction. Fig. 1 shows typical odd high resolution images and electron diffraction patterns that can be encountered in this sample. The TEM sample was prepared by the clieving technique 29 and NWs grow mostly in directions not perpendicular to the substrate surface. Therefore the determination of the growth direction of NWs is not so straightforward from a single TEM image, as only a projection of the NW is obtained in these images. For instance, in Fig. 1a, the growth direction seems to be [ 1 12 ], which would be true if the NW was perpendicular to the electron beam, but it was found to be of 110 type by acquiring several HRTEM images taken at different tilt settings and measuring on the CCD camera the angle of the edge of the nanowire. This analysis shows that in Fig. 1 the NW makes an angle of 30 • with the horizontal image plane. The high resolution image of Fig. 1a is composed of two regions. At the edge of the nanowire -region noted Ia in Fig. 1a -the image corresponds simply to a diamond cubic silicon structure observed nearly along a 110 direction. More inside the nanowire -region IIa -the crystal has an additional periodicity of three (1 11) planes. The contrast in this region is changing slightly due to thickness variations. Fig. 1b shows the diffraction pattern taken in the microscope. The additional spots due to the new period are clearly visible and will be discussed in the next paragraphs. Then, while keeping the vertical (1 11) planes excited, the sample was tilted by approximately 19 • until the diffraction pattern of Fig. 1d was obtained. Along this new direction, high resolution images were taken. One of them is displayed in Fig. 1c. Again this HRTEM image has several regions. At the edge of the nanowire -region Ib -only {111} inclined planes can be seen. More inside the nanowire -region IIb -clear spots are obtained. What is surprising is that both the EDP of Fig. 1d and the region IIb of the HRTEM image of Fig. 1c contain 3 families of planes with interplanar spacings close to 0.31 nm and rotated to each other by of 56 • and 62 • degrees (Fig. 1c-d). Such a direction of observation does not exist in diamond cubic silicon and one could try to build hexagonal structures that display such characteristics.

However, the real nature of the NW appears in conventional TEM (CTEM) images (Fig. 2). Fig. 2a-b are obtained by tilting the sample several degrees off the [110] orientation while keeping one family of {111} planes excited. Then an objective aperture is introduced in order to select either the excited {111} spot -this is a darkfield-or the transmitted beam -this is the bright field. In the dark field image, regions of different contrast and with triangular shapes can be observed. These regions correspond to different crystallographic grains. A given grain will be bright only if the selected {111} spot is in Bragg condition in this grain. On the contrary, a given grain, like the triangular one marked by a T in Fig. 2a will be dark if the selected {111} does not belong to the given grain. If the selected {111} spot belongs to different superposed grains, fringes, also known as alpha-fringes 31 can appear at the interface (like region F in Fig. 2a). Definitely Fig. 2a shows that the NW is not formed of a unique crystal but contains many different grains with different orientations. It should be noted that due to the presence of many grains the deduction of the growth direction can be more complicated than for single crystalline NWs.

To have more details on these defects, a slice of a [011] nanowire perpendicular to the growth direction was realized by FIB on another nanowire. A rather large wire was selected (diameter of 350 nm) in order to facilitate the manipulation of the nanowire after its separation from the substrate. The preparation of the sample is described in the supporting information. Fig. 3 shows several images of the slice. Again, simple conventional images ( Fig. 3a) show that the wire is not uniform but composed of several grains. Fortunately, the slice is quite thin and as evidenced on high resolution STEM images (Fig. 3b,c,d,f), all the observed grains are observed along a [011] direction. Five different types of orientations are found in Fig. 3 hereafter noted A, B, C, D andE. These grains can be related to each others by rotations around the [011] growth direction of the NW (see supporting information for details on transformation matrices) or equivalently by a succession of twining on successive {111} planes. Let us choose the orientation named B in Fig. 3a as our reference crystal. By applying a rotation of angle θ = atan(2 √ 2) ≃ 70.53 • (respectively -θ) to B, the crystal C (respectively A) is obtained. Grain C (respectively A) can be deduced from grain B by a twinning on a (11 1) plane (respectively ( 11 1) plane ) of grain B, which we note (11 1)B. Starting from grain C (respectively A), a rotation of angle θ (respectively -θ) will give grains D (respectively E). The NW is mainly composed of grains A and B, and twins on ( 11 1)B planes but some areas, like areas c,e,f in Fig. 3b nearly contain all the 5 types of grains. Most of the grain boundaries are Σ = 3 GBs located on {111} planes. However also {112} Σ = 3 are encountered (Fig. 3d) [START_REF] Thibault | Electronic Structure and Properties of Semiconductors, chapter 7, Grain Boundaries in Semiconductors[END_REF] and a series of defects on alternating {111} and {112} planes can lead to an average {100} plane (Fig. 3d) in grain B, wich is close to an average {110} plane in grain A. As can be seen on Fig. 3b, the FIB milling has increased the thickness of the amorphous layer at the surface of the NW (about 40 nm of amorphous layer can be seen, where it has a thickness of 2 nm on traditional images of NWs), but it cannot have introduced all the observed defects, because these defects were also present in Fig. 2 and such defects are never observed on FIB prepared specimens of bulk silicon. Such complicated twinning in NWs has also been observed in silver NW's [START_REF] Chen | [END_REF] , where a fivefold twin separates the wire in five equal parts. When the NW is studied from a direction perpendicular or nearly perpendicular to the growth axis, as it is the case in Fig. 1, different crystal grains will superpose in the image/EDP. For instance, let us use the orientations of Fig. 3 In fact, as the two grains are superposed along the electron beam, a beam diffracted by the first grain (let us suppose it is B) can be diffracted again by the second grain (let us suppose that it is grain A). As it is shown in Fig. 4c, many additionnal spots appear due this double diffraction effect. For instance, the spot 1 /3( 11 1)A in Fig. 4c can appear with the following mechanism : firstly the incident beam is diffracted by grain B in the (1 13)B direction, which is equal to the 1 /3( 71 7)A direction, and secondly this diffracted beam is again diffracted by grain A in the (20 2)A direction. The final direction of this double diffracted beam is then 1 /3( 71 7)A+(20 2)A= 1 /3( 11 1)A. In the direct space, the 1 /3( 11 1)A spot corresponds to a period of 3 times the {111} interplanar distance (3 d 111 ). Fig. 4d-e illustrates that this period appears naturally when the two grains A and B are superposed along the beam. We verified using multislice simulations that the experimentally observed contrast on the 3 d 111 period is well reproduced with the hypothesis of a superposition of two grains along the electron beam, see the supporting information. The contrast in region IIa in (Fig. 1a) would then simply come from the superposition of grain A and B, the region Ia at the edge of the wire containing only grain A. Having determined a possible structure coherent with Fig. 1a andb, let us try to understand Fig. 1c andd 1c, region Ib contains one {111} family of planes that exactly coincides with the (111)E family of planes of Fig. 5c. So region Ib corresponds to a grain E alone, while region IIb corresponds to a superposition of grains B and E along the electron beam. So while tilting the sample from the orientation of Fig. 1a to the orientation of Fig. 1b, the studied area of the nanowire was changed slightly without notice. In Fig. 1a this is an area with a superposition of grain A and B at the center with a grain A at the edge, while in Fig. 1b there is a superposition of grain B and E at the center with a grain E at the edge. This is not very surprising as (i) the nanowire contains many grains as can be seen in Fig. 2 and as TEM goniometers are not all eucentric and always produce a shift when tilting. All this analysis shows that a superposition of grains is at the origin of the experimental data and it is not necessary to introduce exotic crystal phases to explain them.

Raman spectra were obtained on these NWs. The majority of the studied wires showed a Raman spectrum similar to that of bulk diamond cubic silicon which has a peak at 520 cm -1 . However, as show in Fig. 6, some NWs analyzed in the present study showed a shift of the Raman peak to lower wavenumbers. In these cases the spectra obtained on the extremities and mid- dle of the NW were different (Fig. 6). At one end the vibrational mode was relatively broad and observed around 484 cm -1 , which could be attributed to amorphous silicon 34 . In the middle a peak at 504 cm -1 was observed, whereas the other wire extremity showed a peak close to the value of diamond cubic silicon (519 cm -1 ). In literature calculations can be found showing that the Raman spectrum of the silicon wurtzite structure contains a peak around 503-506 cm -110, 35,36 . In other experimental papers 37,38 Raman peaks at either 504-511 cm -1 or 517 cm -1 are observed. They are attributed to the presence of defects that create smaller grains inside the NW. Indeed, due to quantum confinement, the Raman frequency should be downshifted for decreasing grain sizes Since all TEM evidence presented in the foregoing unambiguously shows that no other crystal structure than diamond cubic silicon is present, the shifted Raman signal is attributed to the high twin density, as the defects can create locally 2H ordering on the range of a few atomic planes 39 and as this local ordering can be seen in some areas of Fig. 3c ande. Intuitively it seems that a varying defect density from the perfect wurtzite phase on one side and the diamond cubic phase on the other side can tune the Raman signal everywhere between 503 and 520 cm -1 . Also a decreasing crystal grain size will increasingly downshift the Raman resonance frequency 38 .

B. Doped Si nanowires with Au catalyst

In the case of gold catalyzed NWs (grown in the presence of TMA or phosphine) that grow mostly in the 112 direction, we observed two different cases. Either images similar to Fig. 1a where the 3 d 111 period is very regular, were observed (see supporting information Fig. 11). Or images like Fig. 9 were observed. The question is again to determine if the observed images are due to a superposition of grains or to new phases. Each case is studied in turn.

Doped Si nanowires grown with TMA

Gold catalyzed NWs grown in the presence of TMA were observed to grow in various growth directions being 112 , 111 or 100 and would often change growth direction along the NW. In the supporting information (Fig. 11a ) a detail of a 112 NW having a 3 d 111 period in its center is shown. In Fig. 7 a detail of a 111 NW is shown. Three regions of the NW separated by (111) planes can be distinguished. They are marked as I, II and III. Region I has the normal diamond cubic structure. Region II shows the 3 d 111 period everywhere. In region III the 3 d 111 period disappears in the middle of the image without the observation of a grain boundary and the left side of region III has again the normal diamond cubic structure. As for the 011 NW and as schematically shown in Fig. 4, it is clear that in this NW, the 3 d 111 period can be explained by a superposition of two crystal grains along the electron beam that are in a twin relation, which is supported by the smooth disappearance of the 3 d 111 period in region III of Fig. 7 that is attributed to an inclined defect that reaches the NW surface. This smooth disappearance would be more difficult to explain if a new phase was present in the center of the wire and occupied the whole thickness of the nanowire. The easiest way to distinguish between these two possibilities (superposition of grains or new phase) is to acquire a conventional image as shown in Fig. 8b obtained on a different NW. This image is obtained using the common {111} beam having tilted the sample 11 • off the ZA. One can notice bright fringes in the center of the wire which have no relation in number to the 3 d 111 periods observed in the HRTEM image of Fig. 8a obtained at the same region of the NW. In fact, in litterature, such fringes are known as alpha fringe contrast and come from the interferences between the two superposed grains [START_REF] Williams | Transmission Electron Microscopy[END_REF] . Alpha fringes are due to grain boundaries inclined to the electron beam that result in a varying thickness of both crystal grains. The interference of beams passing through both grains create fringes comparable to thickness fringes. The periodicity of alpha fringes depends on many parameters and is therefore difficult to analyze, but we can distiguish two different situations (i) the periodicity of the fringes is equal to the number of 3 d 111 periods indicating a regular repetition of a grain A and grain B with grain boundaries parallel to the electron beam extending through the entire NW volume where only one of two grains is in a difracting condition. Such a regular repetition can be viewed as an anomalous crystal phase other than the diamond cubic structure and will be discussed in the next section. (ii) the periodicity of the fringes is not equal to the number of 3 d 111 periods (as is the case in Fig. 8) and must therefore be due to alpha fringe contrast caused by the superposition of two crystal grains both in a diffracting condition of the observed 112 NWs had variations of contrast on {111} planes parallel to the growth direction, that can be seen in Fig. 9. On the HRTEM image of Fig. 9a the NW is observed along a [011] direction and has a diameter of 66 nm. Clearly at the left edge of the NW there are two grains, A and B, that are in a twin relationship and that occupy the whole thickness of the NW as the contrast of the HRTEM image is sharp. As the electron beam is not perfectly along the common [011] direction but a few degrees off, the A and B grains are not observed exactly in the same direction and grains in the A orientation are darker than the ones in the B orientations. In the center of the NW there are a series of dark lines on (1 11)B planes. The first three dark lines on the left are regularly separated by a distance equal to 3 d 111 . Then there is another group of two lines separated again by 3 d 111 . The question is whether these 3 d 111 regions (like the one outlined by a white double arrow) are due to a local ordering of stacking faults (see Fig. 9e) or whether they are due to a superposition of two grains along the electron beam (like in Fig. 4c). The answer is again given by conventional TEM images. Fig. 9b is a conventional TEM image obtained by selecting the ( 11 1) plane of crystal A that does not correspond to a {111} plane in B. If there were a superposition of two grains (as shown in Fig. 4f) the contrast on the 3 first periods (delimited by an arrow in Fig. 9a) would be uniform as this area would be composed of a constant amount of crystal A, however we clearly see several regions and the number of bright lines in Fig. 9b corresponds to the number and location of 3 d 111 periods in Fig. 9a (for example compare the regions oulined by the arrow) which is exactly the situation as described by (i) in the previous section. So definitely, contrary to Fig. 8, we have a succession of stacking faults crossing the whole thickness of the nanowire. The exact stacking order is determined by analyzing HRTEM images and by looking at the alignment of the white dots along the [ 21 1]B direction (Fig. 9d). When the tunnels of the [011] direction of observation are white (as in Fig. 9d), the cubic stacking abc of orientation B is kept as long as the tunnels are aligned along [ 21 1]B. When a shift in the dot alignment takes place, it is the inverse stacking cba of orientation A that occurs and locally there is a small portion of crystal A around the stacking fault (see Fig. 9d). For instance, following the line in Fig. 9d, we have the following stacking : abc-bca-cab-abc ie 3 staking faults regularly distributed every 3 d 111 period. On the right of these regularly distributed stacking faults, there is a series of random stacking faults.

IV. DISCUSSION

Having observed a selection of nanowires having odd images/diffraction patterns, we have found two different situations. In undoped 110 NWs grown with a Ni catalyst and Al doped NWs grown with Au catalyst, the odd images/EDPs are definitely due to a superposition of twins along the electron beam. So in this first case, the odd situations are due to (mostly {111}) planar defectsour 'hidden' defects that create superposition of twinned crystals. In P doped 112 NWs grown with Au catalyst, we observed a more or less regular distribution of SFs that cross the whole diameter of the nanowire. In this second case, there is no superposition of twins or defects along the electron beam. Of course, if the stacking faults would be regularly distributed there would be a creation of hexagonal phases as claimed in 11,13 , which would be 2H if the stacking faults are every 2 d 111 planes or 9R if the stacking faults are distributed every 3 d 111 planes. However, in all our observed samples, the stacking faults were never distributed regularly on large distances and therefore it seems misleading to describe the Si nanowires as having an hexagonal phase. Interestingly these vertical SFs on 112 NWs and inclined defects are frequently observed when NWs are tentatively doped. So it seems likely that dopants or impurities are at the origin of the observed defects and might segregate at these defects. These faulted nanowires could be a way to incorporate dopants or impurities and could have interesting properties if the defects were more controlled. In any case more experimental data will be necessary to prove that such stacking faults are really due to impurities.

So given the respective TEM and Raman characterizations performed on Ni and Au catalyzed silicon NWs it seems that no hexagonal phase with long range order is present in the studied NWs that grow in majority in the 112 or 110 direction. In all our observed cases the appearance of a regular period of 3 d 111 is due to a superposition of different crystal grains caused by inclined (hidden) defects. Some clues can indicate when this situation is likely to happen : (i) the anomalous crystal phase is only observed in the center of the NW, the thicker region, and not at the edge of the NW where the crystal is thin, which makes it very probable that this odd contrast is caused by superposition of different crystal grains (e.g. Fig. 1), (ii) twins are often found in the vicinity of the odd region and (iii) the odd contrast can disappear smoothly in the middle of the NW, without the presence of a visible grain boundary or defect (e.g. Fig. 7). In the studied NW systems 'hidden' defects were evidenced either by a FIB slice of the NW or more easily and generally by taking conventional TEM images of the NWs. 'hidden' twins are then revealed by inhomogeneous contrast in dark-field and two beam images (Fig. 2a-b) or alpha-fringe contrast (Fig. 2a and Fig. 8b). One should not use only HRTEM images associated with only one diffraction pattern -as it is often the case -to claim the existence of hexagonal phases. We have seen in Fig. 1 that two HRTEM images of the same NW obtained in two directions can also be misleading if the NW contains different regions with different structures.

Of course, recent improvements in TEM techniques should allow other ways to characterize the NWs. For instance, one could eventually use the STEM sectioning technique 40 to visualize the individual grains. By using a large convergent beam and a probe-Cs-corrector it was shown that when the probe is focused at the top surface of the sample, essentially the top grain is visualized. When the probe is focused at the bottom surface of the sample, a superposition of the top and bottom grain was obtained 41 . Another solution would be to realize a 3D-analysis of grains like in Ref. 42. However these experiments are not routine analysis yet. On the contrary qualitative CTEM images can be realized quickly and efficiently on any kind of TEM.

Papers claiming the observation of the hexagonal structure in Si NWs often use the shifted Raman frequency to 503-506 cm -1 as evidence in combination with TEM images, where generally only one direction of observation is regarded 13 . A recent article 43 reports odd HRTEM images on SiNWs in combination with a shifted Raman signal to 503 cm -1 , however only normal diamond cubic silicon is observed in X-ray diffraction (XRD) experiments (see ref 43 and the supporting information). This is a clear indication that Raman spectroscopy is not the appropriate technique to distinguish between the hexagonal phase and the diamond cubic phase with defects because a shifted Raman signal does not necessarily imply the presence of 2H or 9R phases 38,43 . From our point of view Raman spectra with only one associated HRTEM image cannot be a proof of the presence of hexagonal phases.

We do not refute the existence of a 2H structure in all the Si NWs. Such structures can exist but are in general localized to a very limited area and are constituted by a limited number of stacking faults, as shown in Ref. 38. Most publications claiming the existence of a 2H structure do not bring sufficient evidence and the results can be interpreted by a classical Si structure with nano and microtwins 19 . A very convincing example of 2H hexagonal silicon is the growth of a silicon shell on a GaP NW with the 2H hexagonal structure 44 , where it is shown that the crystal structure of the core GaP NW that has the 2H hexagonal structure can be successfully transferred to the Si shell. In this case none of the 'odd' features as described previously (i -iii) can be observed. The presence of massive 2H structures in pure Si NWs inserted perpendicular to the NW axis (as in the GaAs NWs 1,2 ) between diamond cubic silicon regions is more convincingly reported by Arbiol et al. 45 in the very special case of Si NWs elaborated at high temperature (>700 o C). However, the FFT of their Figure 4 45 shows odd features (the distance X1-X2 =X2-X3). Therefore conclusive evidence whether an hexagonal phase or a cubic phase with defects is present can only be obtained by using several directions of observation in TEM, especially using conventional TEM images in association with HRTEM/EDP images, or by correlating TEM and XRD measurements.

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V. CONCLUSION

In conclusion studying several cases of Si NWs presenting odd images or diffraction patterns, we do not observe massive hexagonal phases extending over tens of nm. In some situations of doped 112 NWs, there is a tendency to form more or less regular vertical stacking faults. But from our point of view it seems misleading to consider this local ordering as hexagonal Si phases. In most cases, odd diffraction patterns/ TEM images are due to a superposition of different crystal grains along the electron beam. Generally these grains are in a twin relationship and form what we call the 'hidden' defects that are at the origin of the odd contrasts. The easiest way to check whether there is a superposition or not of grains along the electron beam is to perform conventional TEM images. In the study of 110 NWs, a slice of the NW, perpendicular to the growth direction and realized by FIB, was very helpful to understand the different grains present in the NWs. Finally we show that defect rich NWs can give rise to shifted Raman peaks, which can be attributed to the small crystal grain size created by the high defect density 38 .

VI. ACKNOWLEDGEMENTS

The authors thank the European Community for funding under the FP6 Marie Curie Host Fellowships for Early Stage Research Training (EST)"CHEMTRONICS" contract No. MEST-CT-2005-020513. We would like to acknowledge V. Salvador (CEA-Grenoble -INAC) for the skilful FIB sample preparation. We thank Eddy Romain-Latu (CEA-Grenoble -DOPT -LETI), Denis Rouchon (INPG, LEPMI -LETI) and Michel Mermoux (LEPMI, CNRS-INPG-UJF) for help with the Raman measurements. We thank Heinz Schmid, Mikael T. Björk, Heike Riel, Siegfried Karg and Walter Riess (IBM Zurich) for one NW sample.

VII. SUPPORTING INFORMATION A. FIB sample preparation

For FIB sample preparation the NW was attached to the omniprobe tip using W. Then the NW was removed from the substrate by FIB milling at the NW base. The NW was approached to the copper TEM grid (Fig. 10a) and attached to the TEM grid by W deposition(Fig. 10b). The NW was gradually thinned to electron transparency by using progressively decreasing acceleration voltages to minimize the thickness of the amorphized surface regions on the TEM sample (Fig. 10c-e). After TEM observation of the NW slice as shown in Fig. 3 a gentle 20s Argon ion milling was attempted to remove some of the amorphous surface layers. Unfortunately this has broken the sample (Fig. 10f). The SEM image Fig. 10f shows that the thickness of the slice can be estimated around 130 nm. The FIB milling/ embedding has partly damaged some surface of the nanowires, but it cannot have introduced all the observed defects, because these defects were also present in Fig. 2.

B. Multislice simulation

In Fig. 11a-b two TEM images of a region with odd contrast, similar to the one in Fig. 1a and Fig. 7, are shown and compared to JEMS multislice simulations 46 at Scherzer defocus. In the simulation the grains A and B were simply superposed along the direction of observation with respective thicknesses t A and t B . The complete crystal thickness was around 50 nm, corresponding to the NW diameter. The electron plane wave first propagated in grain B then in the twinned grain A. The experimentally observed contrast and the 3 d 111 period are well reproduced in the simulations.

C. Transformation matrices

In this appendix, the transformation matrices that link the different grains A, B, C, D, and E as described in Fig. 3 are given. B is taken as the reference crystal. Let X be a vector whose coordinates in crystal B (respectively A) are given by the column vector X B (respectively X A ). The transformation matrix from grain B to grain A (noted B to A) links the two column vectors: When a NW containing multiple grains is tilted in the TEM the direction of observation changes differently for the various grains. This effect is illustrated schematically in Fig. 12 for grains A and B of Fig. 3 where stereograms of grain A and B are shown if the NW is observed along the growth direction (as in Fig. 3) and if the NW is observed from the side along {110} and {123} type directions respectively (as in Fig. 1). Such a schematic is helpful for better understanding of the observation of the different grains and correct indexing of diffraction patterns obtained on superposed grains.

FIG. 1 :

 1 FIG. 1: The same NW on two different ZA's. (a) HRTEM image of a [011] NW observed on the [110] direction. (b) Corresponding diffraction pattern, an extra period of ∼ 1 nm is present. (c) HRTEM image of the same NW tilted over 19 • -rotating around the (1 11) plane normal -with respect to the image/EDP in a and b. (d) The corresponding EDP, the angles are indicated and the distances are equal to the interplanar distance of a {111} plane (0.31 nm).

FIG. 2 :

 2 FIG. 2: (a) Darkfield image made using the 1g excitation. The NW was tilted off the [110] ZA by several degrees. (b) Bright field image at the same orientation.

  to explain Fig. 1b. The diffraction of Fig. 1b can be reproduced by a superposition of twinned grains A and B observed along the [110]B=[101]A direction whose projection on the (011) plane is marked by a dashed arrow in Fig. 3b. In fact the [110]B direction points out of the plane and makes an angle of 30 • with this dashed

FIG. 3 :

 3 FIG. 3: (a) Bright field (BF) STEM image at low magnification of a NW section, prepared perpendicular to the NW growth direction, observed along the [011] direction. Numerous defects can be observed. Damage due to FIB milling has formed a damaged layer of ∼ 40 nm around the NW core. (b) BF STEM at higher magnification. (c)BF STEM zoom of a complicated ensemble of defects. Four twin variants can be identified. (d) BF STEM zoom of a series of alternating twin defects on {111} and {112} planes. (e) (Filtered) HAADF STEM zoom showing 4 different twin variants. The twin grain boundaries are marked by black lines.

  , which are obtained by rotating the nanowire of Fig. 1a along the (1 11)B direction with an angle of plus or minus 19 • . Grain B observed in Fig. 1a-b in a [110]B direction will be thus observed in Fig. 1c-d along either a [231] or [32 1]. Let us suppose that this orientation is [32 1]. Similar results will be obtained with the other possibility. As shown in Fig. 5a grain B oriented along [32 1] produces an EDP with a systematic row of diffracted spots generated by the (1 11)B plane. If we found a second grain oriented on a 321 direction whose systematic row generated by its {111} planes makes an angle with the (1 11)B systematic row of 56 • , the EDP of Fig. 1d would be explained: the extraspots of Fig. 1d being generated by double-diffraction between the two systematic rows. Using the transformation matrices given in the supporting information we find that when grain B is observed along [32 1], the different grains A,C,E,D are observed respectively along directions [3 12]A, [ 13 2]C,[ 12 3]E and 1 /3[ 11 21]D. Grains A, C and E have the desired direction of observation. Their {111} diffracting planes are respectively the ( 11 1)A, (111)C and (111)E and the respective angles of these planes with the (1 11)B plane are equal to 0 • in grain A, 56.25 • in grain C and 180-56.25 • in grain E. So a superposition of grains B and C or B and E will reproduce exactly the odd diffraction of Fig. 1d. Let us suppose this is grain E. Coming back to Fig.

FIG. 4 :FIG. 5 :

 45 FIG. 4: (a) Electron diffraction of grain B along the [110]B direction. (b)Combined [110]B EDP and [101]A EDP, grain A and B are twin crystals, the twin plane being (11 1)A=( 11 1)B. (c) EDP when the electron beam crosses first grain B then grain A. Double diffraction creates additional spots, in particular the 1 /3{111} spot. (d) Atomic model of a 111 twin grain boundary between grain A and B. Grain B is on the left and grain A on the right of the grain boundary, as depicted on the scheme. (e) Projection of a superposition of grains A and B along the electron beam. The period of 3 times the {111} interplanar distance associated to the 1 /3{111} spatial frequency appears. (f) Schematic illustrating a possible configuration of grain A and B in the NW volume and how these grains superpose in the TEM image, based on Fig. 3d.

FIG. 6 :

 6 FIG.6: Example of three Raman spectra made on the same NW in the middle and on both ends of the NW.

2 .

 2 Doped 112 Si nanowires grown with phosphine

Fig. 9 FIG. 7 :

 97 Fig.9shows the case of gold catalyzed NWs grown in the presence of phosphine29 and growing in the 112 direction. These nanowires were grown on a 111 Si substrate. Most of the nanowires had a 111 growth direction and these 112 NWs were in fact in minority. Most

FIG. 9 :

 9 FIG. 9: (a) HRTEM image of a part of gold catalyzed 112 NW observed along the [011] direction. Atoms are black. Two crystal orientations, marked as A and B, in a twin relationship are present. The vertical twin plane is chosen as ( 11 1)B =(11 1)A. The length of the white double arrowed line is 10 d111 ∼ 3.14 nm (b) Dark field image of the same nanowire by tilting the sample away from the [011] direction in order to be on a two-beam condition with crystal A (beam excited g=( 11 1)A) and to have no excited beam on crystal B. (c) Bright field general view of the nanowire on the [011]. (d) HRTEM image zooming on the area in a outlined by the double white arrow. An atomic model of the projected structure is superimposed on the experimental images. The stacking sequence of ( 11 1)B planes is marked by the letter a-b-c and outlined by lines joining the center of the tunnels of the structure. The order a-b-c is characteristic of the orientation B, while the reverse order c-b-a, outlined by ellipses around the letter c-b-a, is characteristic of the orientation A. Stacking faults make the boundaries between grain A and B. Alternatively, one can consider that there are locally some 2H hexagonal phases b-c-b-c or c-a-c-a or a-b-a-b in the orientation B. (e)Schematic of a NW section on the [ 21 1] ZA, which is the growth direction. The projection of the stacking faults in the TEM image is illustrated.

FIG. 10 :

 10 FIG. 10: SEM images obtained during FIB preparation steps preparing a slice of NW that can be observed along the growth direction. (a) The NW attached with W to the W tip, the omniprobe, is approached to the TEM grid. (b) The NW is attached using W deposition to the TEM grid. (c) The bottom part of the NW is gradually removed by FIB milling using progressively decreasing acceleration voltage to minimize amorphization of the specimen surface. (d) The upper part of the NW is removed by FIB milling as in (c). (e) The sample is finally thinned to electron transparency. (f) After TEM observation a gentle 20s Argon ion milling was attempted to remove some of the amorphous surface layers. Unfortunately this has broken the sample.

FIG. 11 :

 11 FIG.11: (a-b) HRTEM images of a defect zone on the [110] ZA. The best match with multislice simulations of two superposed crystal grains related by a twin defect placed on a {110} plane is shown in the inset at Scherzer defocus. The thickness of both crystal grains, indicated by t1 and t2 is indicated. The characteristics of the JEOL 4000 EX were included in the simulations.

  X A = B to A * X B . It is a rotation matrix around the [011]A=[011]B axis with an angle of about -70.53 • (see Fig. 3). For instance [ 11 1]B (respectively [110]B) has coordinates [11 1]A in grain A (respectively [101]A)

FIG. 12 :

 12 FIG.12: Schematic illustrating the different directions of observation of grain A and B observed in Fig.3with stereograms. Most to the left grain A and B are observed along the growth direction as in Fig.3, in the center grain A and B are observed along a {110} type direction. On the right grain A and B are observed along a {123} type direction. The arrows in the center of the stereogram indicate how the NW is positioned in the image plane (as pointed out previously the NW is not lying flat in the image plane and the electron beam is not perpendicular to the NW). The arrows below the stereogram indicate how the projection of the electron beam enters the NW slice and a small copy of Fig.3b and eis repeated here for clarity. The schematics of Fig.4fand Fig.5dare based on how the different grains would superpos in the respective directions of observations if the electron beam would enter the NW slice as indicated.


Q B Nguyenf

A Mébarkif

F Mercierj

R Ami Saadaf

M Reimeringerj

Thé Domino Effect and Integrated Probabilistic Approaches for Risk Analysis

Keywords: domino effect, industrial explosions accident, mechanical impact, tanks, projectile, pénétration, perforation, risk, reliabiliry, probabilistic methods

The paper develops a probabilistic approach in order to deal with domino effects that may occur in industrial facilities: an explosion or accident may generate various sets of projectiles that may impact other existing facilities (tanks under high-pressure, etc) and may generate other sets of projectiles and so on. Three main parts are considered: -Source term: for thé first set of generated projectiles, probabilistic distributions are considered for thé number, masses, velocities, departure angles, geometrical form, dimensions, and constitutive materials properties. The authors hâve collected existing models from thé literature. -Target term: for thé set of impacted targets, probabilistic distributions are considered for thé number of impacting projectiles, velocities, incidence angles and energy at impact, constitutive materials properties, dimensions of thé impacted targets, and projectiles pénétration depths into thé targets. In this paper, new models for thé impact are proposed to calculate thé pénétration depth after impact: case of cylindrical rods impacting rectangular plates, both are métal made. The theoretical results are compared to thé expérimental data (4 data sets) collected from thé literature with thé following features: projectiles mass ranging from 0.1g up to 250 kg, projectiles velocity ranging from 10 m/s up to 2100 m/s, projectiles diameters ranging from 1.5 mm up to 90 mm, target strength ranging from 300 MPa up to 1400 MPa and incidence angles ranging from 0° up to 70°. -Domino effect term: évaluation of thé risks of second set of explosions that may take place in thé impacted components. Monte Carlo simulations are used in order to calculate thé différent probabilities: probability of impact, distribution of thé pénétration depth and probability of domino effect.

Introduction

Many types of equipment under pressure may exist in industrial installations: tanks containing gas or highly pressurized liquids, for instance. When reaching critical levels of overpressure, overheating or mechanical demand, a catastrophic séquence, may rise. Actually, thé tanks may suddenly explode in many fragments that should be considered as projectiles threatening thé other equipments or installations in their neighbourhood, [START_REF] Mercier | Risques liés aux impacts de fragments lors des accidents industriels[END_REF]. Thèse projectiles can impact other equipments, penetrating them partially or perforating them. Depending on thèse degrees of perforation, new accidents may take place in thé impacted objects, leading therefore to séries of accidents known as domino effect. Many studies deal with thé prévention or mitigation of thé domino effect conséquences, [START_REF] Gubinelli | A simplified model for the assessment of the impact probability of fragments[END_REF][START_REF] Hauptmanns | A procédure for analyzing thé flight of missiles from explosions of cylindrical vessels[END_REF][START_REF] Hauptmanns | A Monte-Carlo based procédure for treating thé flight of missiles from tank explosions[END_REF][START_REF] Holden | Fragment hazards form failures of pressurised liquefied gas vessels[END_REF].

The présent paper develops a global methodology in order to study thé domino effect, detailing in a probabilistic framework ail thé components of thé catastrophic séquence. The probabilistic distributions of thé whole involved parameters are derived from existing bibliography and data or are postulated otherwise: fragment number, initial velocity, initial angles, mass, geometrical form... The trajectory of thé projectiles and their impact with surrounding objects are also investigated. Mechanical models are developed in order to calculate thé pénétration depth and thé mechanical damage caused to thé impacted targets. The risk or probability of domino effect occurrence is calculated by Monte Carlo simulations, [START_REF] Bless | Penetration mechanics of yawed rods[END_REF][START_REF] Bukharev | Model of the penetration of a metal barrier by a rod projectile with an angle of attack[END_REF][START_REF] Lepareux | Expérimental and numerical studies of impacts on stainless steel plates subjected to rigid missiles at low velocity[END_REF][START_REF] Neilson | Empirical équations for thé perforation of mild steel plates[END_REF].

General framework

Thé overall domino effect séquences are described in Figure la. This domino effect may be detailed by describing each of thé elementary steps or cycles. Each cycle requires three elementary components (steps): a source term (explosion or génération of thé fragments), thé projectiles trajectory term (angles, velocities and displacements from thé source), and thé target term (impact and interaction between thé projectile and thé target), see Figure lb. Thus, each elementary branch of thé domino effect requires three detailed steps, see Figure 2: -The génération of thé preliminary accident that gives rise to thé fragments projection: thé factory site contains one or more tanks under pressure of gas, heated liquids or mechanical aggression. Under thé effect of thé overpressure due to thé gas or thé liquid, or thé mechanical aggression, thé tank may explode and generate fragments. This probabilistic term describes thé probability of génération of thé fragments, P gen . -In their trajectory, thé fragments may hit other equipment in their neighbourhood. This possibility of impact is defined by thé probability of impact, P imp .

The impacted targets may suffer partial damage or complète pénétration of thé projectile. Depending on thé targets properties and their critical damage, an explosion may rise in this targeted élément. The probability of explosion after impact, P rap , defines thé risk of another branch occurrence in thé domino effect séquence. As a first step, thé présent paper focuses mainly on thé case of one elementary séquence in thé domino effect séquences. The target is supposed to be a planar métal plate. The projectile is supposed to be a métal rod, see Figure lb. The risk of domino effect occurrence may be expressed as:

where: P gen = probability of génération of projectiles (source term), Pi mp = probability of impact, P mp = probability of failure of thé impacted target.

In fact, thé général expression is: P f =P(E<0)= Jf e (e)de= {jf r (r).f s (s).dr.ds, andE = R-S (2.) E<0 R-S<0

With: R= random résistance (which values are r), S= random mechanical demand (which values are s), Pf= probability of failure, E= limit state fonction (E < 0: defines thé failure domain, E > 0: defines thé safety domain, E = 0: defines thé limit state surface).

The risk évaluation requires thé knowledge of thé probability density fonctions f r (.) and 4 Source terms

Source term features

An industrial accident may generate several fragments having various forms, sizes, initial velocity, and initial departure angles. It is therefore required to define correctly thé distributions and thé features of thé source term: fragments number, fragments forms and size, their masses, their initiais departure angles (horizontal and vertical angles), their initial velocities at departure, and their aerodynamic coefficients (lift and drag coefficients).

Fragment number

A tank explosion may generate one or many fragments according to thé critical pressure, to thé cracks propagation, to thé tank constitutive material and to thé connection between thé elementary mechanical components. According to scientific reports collected from INERIS [START_REF] Ineris | Effets mécaniques de projectiles impactant des sphères de gaz liquéfié[END_REF][START_REF] Ineris | Analyse du choc d'un projectile sur un réservoir calorifuge de GPL[END_REF][START_REF] Ineris | Méthode pour l'Identification et la Caractérisation des effets Dominos[END_REF][START_REF] Ineris | Les éclatements de réservoirs, Phénoménologie et modélisation des effets. Direction des Risques Accidentels[END_REF][START_REF] Ineris | Calculs des effets mécaniques d'un BLEVE de citerne ferroviaire[END_REF][START_REF] Ineris | Etude de la tenue mécanique d'un talus à une onde de souffle et un impact[END_REF], thé typical explosion (BLEVE) of a cylindrical tank produces a limited number of massive fragments: generally two or three, and very seldom more than four or five. fragments number [START_REF] Ineris | Calculs des effets mécaniques d'un BLEVE de citerne ferroviaire[END_REF] projectiles from thé cylindrical tank under gas or liquid pressure [START_REF] Holden | Fragment hazards form failures of pressurised liquefied gas vessels[END_REF] Holden [START_REF] Holden | Fragment hazards form failures of pressurised liquefied gas vessels[END_REF] quotes, for 31 BLEVEs having produced 76 fragments, an average number of 2.45 fragments, corresponding to 2.87 without fire production and 2.34 with fire production. In fact, thé accidents generate 1, 2, 3 or 4 fragments (except a case without fire). A test of BLEVE, with a carnage of 45 m 3 containing 5 tons of propane, produced 4 fragments from thé tank envelop and thé BLEVE tests on propane reserves of 400 litres produced 1 up to 3 fragments [START_REF] Ineris | Méthode pour l'Identification et la Caractérisation des effets Dominos[END_REF][START_REF] Ineris | Les éclatements de réservoirs, Phénoménologie et modélisation des effets. Direction des Risques Accidentels[END_REF][START_REF] Ineris | Calculs des effets mécaniques d'un BLEVE de citerne ferroviaire[END_REF][START_REF] Ineris | Etude de la tenue mécanique d'un talus à une onde de souffle et un impact[END_REF].

Moreover, according to thé analysis and thé experiments of Baum [START_REF] Baum | The velocity of end-cap and rocket missiles generated by failure of a gas pressurised vessel containing particulate material[END_REF], thé rupture starts with thé circumferential welding and terminâtes with thé split of bottom part of thé tank (without or with a part of thé ring). Therefore, it générâtes 2 up to 4 fragments: bottom of thé tank (with or without thé ring) and pièces of thé ring. The size of thé tank does not hâve an influence on thé fragments number. Larger thé tank, larger thé fragments, but thé size of thé tank does not hâve incidence on thé number of fragments [START_REF] Ineris | Calculs des effets mécaniques d'un BLEVE de citerne ferroviaire[END_REF].

For thé cylindrical tank, Hauptmanns [START_REF] Hauptmanns | A procédure for analyzing thé flight of missiles from explosions of cylindrical vessels[END_REF] finds out that thé number of fragments follows a log normal distribution.

Therefore, in thé présent study, thé fragment number is supposed to follow a log normal distribution, truncated within thé interval [1 up to 4].

Fragments shape

The form of thé fragments dépends mainly on both thé rupture type and thé cracks propagation in thé tank. However, thé cracks initiâtes in thé circumferential part and propagates until it cuts one of thé two end-caps; then, it unfolds thé ring [START_REF] Ineris | Effets mécaniques de projectiles impactant des sphères de gaz liquéfié[END_REF][START_REF] Ineris | Calculs des effets mécaniques d'un BLEVE de citerne ferroviaire[END_REF][START_REF] Ineris | Etude de la tenue mécanique d'un talus à une onde de souffle et un impact[END_REF].

It may also happen that thé end-cap does not split from thé ring, thé ring remains linked to thé end-cap as a rocket [START_REF] Ineris | Etude de la tenue mécanique d'un talus à une onde de souffle et un impact[END_REF]. The ring générâtes sometimes 1 or 2 pièces [START_REF] Ineris | Calculs des effets mécaniques d'un BLEVE de citerne ferroviaire[END_REF].

The fragment shape is also variable and unpredictable. Holden [START_REF] Holden | Fragment hazards form failures of pressurised liquefied gas vessels[END_REF] gives a mère classification of thé fragments types according to thé number of linked caps and number of circumferential cracks; thé types of fragments are similar to those collected from INERIS.

According to Gubinelli [START_REF] Gubinelli | A simplified model for the assessment of the impact probability of fragments[END_REF], some types of fragments of a BLEVE might be classified as: cylindrical, end-cap, end-cap attached to thé ring, and thé tube. Baum [START_REF] Baum | Failure of a horizontal pressure vessel containing a high temperature liquid: the velocity of end-cap and rocket missiles[END_REF] has studied also thé end-cap (in case of constrained tanks) and thé end-cap with thé cylindrical binding ring (in case of unrestrained tank).

In thé présent study, it is therefore assumed that thé fragments are one among thé end-cap, thé ring attached to thé end-cap, or thé flattened pièce of thé ring (thé plate).

Fragment mass

There is no suitable information available, in thé existing literature, about thé fragments masses distribution. Therefore, as a first step, ail thé fragments are supposed to hâve thé same mass. Their form is supposed to be a pièce of sphère [START_REF] Hauptmanns | A Monte-Carlo based procédure for treating thé flight of missiles from tank explosions[END_REF]. On thé other hand, for cylindrical tanks, Hauptmanns [START_REF] Hauptmanns | A procédure for analyzing thé flight of missiles from explosions of cylindrical vessels[END_REF] shows that thé fragments masses factor (ratio between thé fragment mass / total mass) follows a Beta distribution.

However, when thé shape and thé size of a fragment are known, thé fragment mass may easily be calculated.

In thé présent study, thé fragment masses are then derived from thé shape and thé size of each fragment, as fragment size is supposed to follow a log normal distribution.

Initial velocity

It is also required to define thé velocity of thé fragments resulting from an industrial accident (explosion, etc.). Aquaro [1] performed expérimental studies on a réduction scale with mass fragments thronging within thé interval [20 up to 200] kg and velocity ranging within thé interval [100 up to 30] m/s, respectively. The fragment velocity might therefore be derived according to thé fragment mass and thé pressure before combustion.

Actually, thé initial velocity of thé fragments requires thé knowledge of thé kinetic energy (Ek) [START_REF] Hauptmanns | A procédure for analyzing thé flight of missiles from explosions of cylindrical vessels[END_REF][START_REF] Hauptmanns | A Monte-Carlo based procédure for treating thé flight of missiles from tank explosions[END_REF]. For this purpose, Ek is deduced from thé total energy (E), on thé basis of thé methods suggested by Baker and Baum [START_REF] Baker | Pros and cons of TNT équivalence for industrial explosion accidents[END_REF][START_REF] Baum | Rocket missiles generated by failure of a high pressure liquid storage vessel[END_REF].

From expérimental studies, Baum [START_REF] Baum | Rupture of a gas-pressurized cylindrical vessel: the velocity of a detached end-cap[END_REF] proposed thé adéquate formula for thé endcap velocity of vertical cylindrical tank being detached and being accelerated by thé exit of gas due to failure of thé circumferential welding, see Figure 6. They correspond to thé "zéro mass of end-cap" model, thé "broad mass of end-cap" model and thé "velocity limit of thé end-cap" model. From other tests, Baum [START_REF] Baum | Failure of a horizontal pressure vessel containing a high temperature liquid: the velocity of end-cap and rocket missiles[END_REF] investigated thé end-cap velocity according to thé kind of materials contained in thé tank and thé température of thé substances. Two types of fragment are analyzed: end-cap and thé end-cap linked to thé ring (thé rocket), [START_REF] Baum | The velocity of end-cap and rocket missiles generated by failure of a gas pressurised vessel containing particulate material[END_REF]. From thé results of thé tests performed by Baum [START_REF] Baum | Rocket missiles generated by failure of a high pressure liquid storage vessel[END_REF][START_REF] Baum | The velocity of end-cap and rocket missiles generated by failure of a gas pressurised vessel containing particulate material[END_REF][START_REF] Baum | Failure of a horizontal pressure vessel containing a high temperature liquid: the velocity of end-cap and rocket missiles[END_REF][START_REF] Baum | The velocity of large missiles resulting from axial rupture of gas pressurised cylindrical vessels[END_REF], thé initial velocity is well described by thé log normal distribution.

Initial departure angle

Vertical angle

From thé existing literature, there is no reliable information about thé horizontal angle. It is assumed therefore, as a first step, that thé initial vertical angle follows a uniform distribution within thé interval [-90°; 90°], [START_REF] Hauptmanns | A procédure for analyzing thé flight of missiles from explosions of cylindrical vessels[END_REF][START_REF] Hauptmanns | A Monte-Carlo based procédure for treating thé flight of missiles from tank explosions[END_REF].

Horizontal angle

From analysis of accidents that hâve occurred, Holden [START_REF] Holden | Fragment hazards form failures of pressurised liquefied gas vessels[END_REF] investigated thé distribution of thé fragments initial trajectory in thé tank principal axes System, see Figures 8a-b. According to thé statistical analysis for 15 accidentai BLEVE having produced 59 fragments [START_REF] Holden | Fragment hazards form failures of pressurised liquefied gas vessels[END_REF], approximately 60% of thé fragments are projected in an angular sector of 60° around thé axis of thé tank, and thé remainder is projected rather perpendicularly to thé tank major axis. Also, it has been found thé end-cap follow thé major axis of thé tank whereas thé pièces belonging to thé ring follow thé minor axis [START_REF] Ineris | Calculs des effets mécaniques d'un BLEVE de citerne ferroviaire[END_REF]. According to Holden [START_REF] Holden | Fragment hazards form failures of pressurised liquefied gas vessels[END_REF], a uniform distribution might be considered in thé case of a cylindrical tank; actually thé ratios of fragments horizontal departures angles, see Figure 8b: 20% of thé angles range within [30°-150°], 30% of thé angles range within [150°-210°], 20% of thé angles range within [210°-330°], and 30% of thé angles range within [330°-30], [START_REF] Hauptmanns | A procédure for analyzing thé flight of missiles from explosions of cylindrical vessels[END_REF].

In thé présent study, thé initial horizontal angle are supposed to follow thé uniform distribution detailed as follows: 20% of thé angles range within [30°-150°], 30% of thé angles range within [150°-210°], 20% of thé angles range within [210°-330°], and 30% of thé angles range within [330°-30]. The lift rises for dissymmetric faces of thé fragment: upper surface and lower surface, [START_REF] Ineris | Les éclatements de réservoirs, Phénoménologie et modélisation des effets. Direction des Risques Accidentels[END_REF]. Thus, thé lift should be considered for thé end-cap and thé end-cap attached to thé ring. This lift coefficient is expressed as: C L = 0.351 +0.6723.a (3.) With: a = thé inclination angle. CL is supposed to be equal to zéro for thé angles ranging outside thé interval [0°, 10°], [START_REF] Hauptmanns | A procédure for analyzing thé flight of missiles from explosions of cylindrical vessels[END_REF].

In thé présent paper, it is assumed, as a first step, that thé lift coefficient follows a log normal distribution.

Drag coefficient, C D

The drag phenomenon dépends on several factors, i.e. thé geometry, thé surface roughness and thé orientation in respect to thé velocity direction. An acceptable range values for CD, in thé case of tank end-caps, is [0.8 -1.1] and a range values [1.1 -1.8] for thé rings. A uniform distribution for thé drag coefficient, CD, might be considered within thé given intervais, [START_REF] Hauptmanns | A procédure for analyzing thé flight of missiles from explosions of cylindrical vessels[END_REF][START_REF] Hauptmanns | A Monte-Carlo based procédure for treating thé flight of missiles from tank explosions[END_REF].

In thé présent study, thé following hypothèses are adopted: -For end-cap fragments, thé lift coefficient follows a uniform distribution within thé interval [0.351 -0.468] and thé drag coefficient follows a uniform distribution within thé interval [0.8 -1.1]; For thé ring, thé effect of thé lift can be neglected. Thus thé drag coefficient follows a uniform distribution within thé interval [1.1 -1.8].

Thé generated fragments may impact potential targets on their trajectory, being therefore a mechanical threat for thèse target, as it may damage it seriously. The trajectory of thé projectiles has then to be exactly known. This kind of problem has already been studied in order to investigate thé possible thé collision between a fragment (thé projectile) and a tank (thé target) in its vicinity, Hauptmanns [START_REF] Hauptmanns | A procédure for analyzing thé flight of missiles from explosions of cylindrical vessels[END_REF][START_REF] Hauptmanns | A Monte-Carlo based procédure for treating thé flight of missiles from tank explosions[END_REF]. The movement of thé fragment has been analysed in bidimensional (2D) approach, see Figure 9. The simplified form of thé possible impact analysis supposes that thé fragment trajectory can be described by thé trajectory of a point (its barycentre). For this purpose, Gubinelli [START_REF] Gubinelli | A simplified model for the assessment of the impact probability of fragments[END_REF] présents also thé model known as "minimal distances". The simplified forms are thé following: rectangular form in xy plane and circular form on xz plane.

The présent study deals with thé possible impact and requires also thé detailed information in order to evaluate thé mechanical damage caused to thé targets by thé impact. A three dimensional analysis is therefore considered. However, for simplification purposes at thé présent step, thé rotation effect of thé fragment during thé movement is neglected. The équations of motion for thé centre of thé fragment are as follows, see Figure 10: The detailed équations are described in thé Appendix that présents thé analytical solutions of thé nonlinear differential équations System. From thèse équations and thé coordinates of thé potential targets, thé impact occurs when exists a possible intersection between thé fragment and thé target equipment. If there is a possible impact, thé solutions provide thé coordinates of thé intersection, thé velocities and angles at impact. for thé impacting projectile: its velocity of arrivai v p , its mass m p , its shape, its size and its constitutive material properties, for thé impacted target (metallic plate in thé présent study): its thickness and its constitutive material properties. The impact may cause a partial damage to thé target. It may also cause a total pénétration or a critical damage that may cause explosion of thé target if concerned, see Figure 11. The fragment may penetrate inside thé target and créâtes a crater, reducing therefore thé residual resisting thickness of thé target ate thé impacted zone. Three scénarios might be considered, see Figures 1112Partial pénétration in thé critical domain: The fragment pénétrâtes partially but thé residual thickness of thé plate at thé zone of impact remains larger than critical thickness of thé plate, e cr (Curve 1, see Figure 12), Partial pénétration beyond thé critical domain: The fragment pénétrâtes partially thé target but thé residual thickness ranges within thé critical interval [O, e cr ] (Curve 2, Figure 12). Therefore, due to thé effect of thé gas or liquid internai pressure, thé target may not resist this high pressure: cracks may appear with possible explosion of this damaged target. Complète perforation: The kinetic energy of thé fragment is so large that thé projectile may penetrate completely thé target, and cause explosion of thé target (Curve 3, Figure 12). The last two cases also represent threaten therefore thé integrity of thé impacted target after thé impact. For this purpose, mechanical models should be run in order to calculate thé pénétration depth and compare thé residual thickness to thé threshold thickness that may avoid thé explosion of thé impacted target.

(k D -(-l)\k L )x + x = 0 ((-l) n .k D +k L )y 2 +y+g = 0, ( 4
For this purpose, various expérimental data for this kind of impacts are collected from thé existing literature [START_REF] Bless | Penetration mechanics of yawed rods[END_REF][START_REF] Bukharev | Model of the penetration of a metal barrier by a rod projectile with an angle of attack[END_REF][START_REF] Lepareux | Expérimental and numerical studies of impacts on stainless steel plates subjected to rigid missiles at low velocity[END_REF][START_REF] Mebarki | Modèle d'atténuation sismique : prédiction probabiliste des pics d'accélération[END_REF][START_REF] Mebarki | Mechanical behavior of metallic plates under metallic rods impact and industrial risks[END_REF][START_REF] Mebarki | A probabilistic model for thé vulnerability of métal plates under thé impact of cylindrical projectiles[END_REF][START_REF] Neilson | Empirical équations for thé perforation of mild steel plates[END_REF]. For instance, a mechanical model is developed in order to calculate thé pénétration depth for given ranges of thé projectiles velocities: for instance, Bless [START_REF] Bless | Penetration mechanics of yawed rods[END_REF], corresponds to a velocity of almost ~2.15km/s. Many other models hâve also been developed and compared to thé expérimental data, Neilson [START_REF] Neilson | Empirical équations for thé perforation of mild steel plates[END_REF], BRL and SRI models, [START_REF] Bukharev | Model of the penetration of a metal barrier by a rod projectile with an angle of attack[END_REF][START_REF] Lepareux | Expérimental and numerical studies of impacts on stainless steel plates subjected to rigid missiles at low velocity[END_REF]. The authors hâve recently issued a mechanical model that évaluâtes thé pénétration depth, h p , in thé case of metallic targets and metallic rods as projectiles, Mébarki [START_REF] Mebarki | Modèle d'atténuation sismique : prédiction probabiliste des pics d'accélération[END_REF][START_REF] Mebarki | Mechanical behavior of metallic plates under metallic rods impact and industrial risks[END_REF][START_REF] Mebarki | A probabilistic model for thé vulnerability of métal plates under thé impact of cylindrical projectiles[END_REF]. The pénétration depth dépends on thé incidence angle a of thé projectile at thé impact, see Figure 11 : -pénétration depth for thé case a ^ 0 -dp .cosa+J(dp .cosa) +4.tana| -1

V

V*u-S u 2. tana (6.) pénétration depth for thé case a = O With thé kinetic energy, E c =

Resistance

Thé realization of thé random variable résistance r = e t -e cr comprises a random variable, and a constant, e cr . In first approach, It makes it possible to suppose that thé random variable, and, follows thé normal law (u e , u e ). Thus thé random variable r also follows thé normal law (\i e -e cr , a e ).

Application and numerical simulation

In order to evaluate thé risk of domino effect, when dealing with industrial risks, Monte Carlo simulations hâve been performed with thé following hypothèses and probabilistic distributions:

-thé fragment number follows a log normal distribution, -thé type of thé fragments: end-cap of tank, end-caps of tank attached to thé ring and thé flattened ring are considered, -thé fragment mass is derived from thé knowledge of thé shape and thé size of each fragment, while it is supposed that thé size follows is a lognormal random variable, -dimensions of thé fragment follow a Gaussian distribution N(ndp, cfdp), -thé departure velocity of thé fragments follow a Gaussian distribution N(u. u , Furthermore, thé risk évaluation relies on thé following models and sets of hypothèses:

-Impact analysis: thé possible impact is analysed for thé case of various shapes of thé projectiles or targets (ellipsoids, cylinders, rectangles), Wang [START_REF] Wang | An algebraic condition for thé séparation of two ellipsoids[END_REF]. The data expected at this stage are: possibility of impact, velocity and angles at impact. Probability of impact might then be evaluated by Monte Carlo simulations.

-The mechanical damage at impact by using thé simplifiée! analytical model, [START_REF] Mebarki | Modèle d'atténuation sismique : prédiction probabiliste des pics d'accélération[END_REF][START_REF] Mebarki | Mechanical behavior of metallic plates under metallic rods impact and industrial risks[END_REF][START_REF] Mebarki | A probabilistic model for thé vulnerability of métal plates under thé impact of cylindrical projectiles[END_REF], that calculâtes thé pénétration depth of thé fragment and thé probability of critical damage of thé tank after impact. The Monte Carlo simulations are performed with thé following set of data and thé hypothèses presented above, see Table 1

Conclusions

Thé présent paper deals with thé domino effect in industrial Systems. The risk represents thé probability of occurrence of a domino effect as a conséquence of an explosion that générâtes projectiles. Thèse projectiles may impact other industrial components (such as tanks) at thé neighbourhood. The features of thé projectiles as well as thé features of thé impacted targets are described within a probabilistic framework. Three main steps are required: probabilistic analysis of thé source event (probability of explosion and génération of fragments, P gen ), probabilistic analysis of thé projectiles movement and their possible impact on targets (probability of impact, Pimp) and probabilistic analysis of thé target damage (probability of target rupture that may cause a new séquence of projectiles génération leading to a domino effect, Prup)-

The number of fragments, their size, their mass, their velocity at departure, and their departure angles are considered as random variables. The velocities and incidence angles at impact of potential targets are also considered as random variables. The mechanical features of thé impacted targets are described as random variables. The mechanical models developed for this purpose evaluate thé residual depth of thé targets and thé pénétration depth of thé projectiles after impact. Monte Carlo simulations are run and thé risk of domino effect is evaluated. The domino effect is thus be analyzed within a probabilistic framework. For thé simulations run as an example, thé probability of failure effect is in thé order of 4x10" 3 .

Appendix Analytical solutions of motion équations System

Figure 1 :Figure 2 :

 12 Figure 2: Domino effect séquences

Figure 3 :

 3 Figure3: Explosion générâtes several fragments[START_REF] Mebarki | Mechanical behavior of metallic plates under metallic rods impact and industrial risks[END_REF]

Figure 4 :

 4 Figure 4: Number of accidents (BLEVE) Figure 5: Number of fragments flying as vs.fragments number[START_REF] Ineris | Calculs des effets mécaniques d'un BLEVE de citerne ferroviaire[END_REF] projectiles from thé cylindrical tank under gas or liquid pressure[START_REF] Holden | Fragment hazards form failures of pressurised liquefied gas vessels[END_REF]

Figure 6 :

 6 Figure 6: End-cap of thé vertical tank Figure 7: (a) Horizontal constrained detached[START_REF] Baum | Rupture of a gas-pressurized cylindrical vessel: the velocity of a detached end-cap[END_REF] tank; (b) Unconstrained tank[START_REF] Baum | Failure of a horizontal pressure vessel containing a high temperature liquid: the velocity of end-cap and rocket missiles[END_REF]

Figure 8 : 4 . 7 Drag coefficient and lift coefficient 4 . 7 . 1

 847471 Figure8: Horizontal departure angles of thé projected fragments,[START_REF] Holden | Fragment hazards form failures of pressurised liquefied gas vessels[END_REF]

Figure 9 :

 9 Figure 9: Trajectory of thé fragment

  .) (k D -(-l)\k L )z + z = 0 where: n = 1 at descending part; n = 2 at ascending part. with: k D 4 Pair ' CD ' AD ;k L= { P ^-CL -AL ;(k D -k J = k m ;(k D +k L ) = (5.)

Figure 10 :

 10 Figure 10: Forces applied to thé fragment

Fragment

Figure 11 :

 11 Figure 11 : Impact of a projectile (fragment) on a target (a plate)

Figure 12 :

 12 Figure 12: Relationship between thé projectile velocity and thé pénétration depth 1 -Partial pénétration in thé critical domain, 2 -Partial pénétration beyond thé critical domain, 3 -Complète perforation

For

  thé assembled part of thé trajectory o Formulas x, z • k m = 0 : x = u x .t->t =-• k m ^ O, u x = O : x = O, no movements on axis X p .g.t + ln2 + -.l For thé descending part of thé trajectory o Formulas x, z • k p = 0: x = uPx .(t-tP)+xï x-x t k p ^ O, up x = O : x = Xp, pas de mouvements sur l'axe x kp^O, u Px >0: "_" , ln(upx-k p .(t-t P ) + l) u Px .k p .(t-t p )+l) k m < 0 : y = y p + jj-An[(cos(jgk^.{t -1, k m >0: Y = y P + M t P )+ In2 -I Jn With u x , u y , u z ,, initial velocities at thé departure point O and Xp, yp, zp, tp, coordinates and time at thé maximum point of thé fragment barycentre trajectory.

Table 1 -

 1 . Data and Monte Carlo simulations results

	Fragments	Target
	Rod form	Tank cylindrical horizontal form
	Radius	«0.3x0.2x0.3 (m)	Radius	2x1x2 (m)
	Length	«0.52 (m)	Length	7(m)
	Thickness	0.07 (m)	Thickness	0.07 (m)
	Initial velocity	«100-150 (m/s)	Centre coordinates	10x3x10(m)
			Resuit
	Probability of génération	100% (assumed that an accident has
			occurred)
	Probability of impact	0,6% (for 30000 simulations)
	Probability of ruin		71% (for 182 simulations of impact)
	Risk		0,43% (for 182 simulations of impacts)