Maxime

H Tawidian

T Lecas

M Mikikian Gremi

Characterization of low frequency instabilities in a Krypton dusty plasma

Keywords: numbers: 52, 27, Lw, 52, 35, -g

published or not. The documents may come

Characterization of low frequency instabilities in a

Krypton dusty plasma

Introduction

In addition to the neutral species, dusty plasmas [START_REF] Shukla | Colloquium: Fundamentals of dust-plasma interactions[END_REF] are constituted of three charged components: ions, electrons and bigger particles called dust particles. The size of these solid particles can vary from a few nanometers to millimeters. In astrophysics, dusty plasmas can be observed for example in comets tails or in some planetary atmospheres [START_REF] Kovačević | A candidate analog for carbonaceous interstellar dust: Formation by reactive plasma polymerization[END_REF][START_REF] Sciamma-O'brien | Optical constants from 370 nm to 900 nm of titan tholins produced in a low pressure rf plasma discharge[END_REF]. In industry, the formation of dust particles was observed in plasma processing reactors in the late 80s [START_REF] Selwyn | In situ laser diagnostic studies of plasma generated particulate contamination[END_REF]. Their appearance was an unexpected problem in the field of microelectronics where cleanliness is a drastic requirement for the fabrication process of semiconductors. Several techniques [START_REF] Kurimoto | Fine particle removal by a negatively-charged fine particle collector in silane plasma[END_REF][START_REF] Iizuka | Control of fine particles by timeaveraged external forces in plasmas[END_REF][START_REF] Li | Removing dust particles from a large area discharge[END_REF][START_REF] Ticos | Removal of floating dust in glow discharge using plasma jet[END_REF][START_REF] Mikikian | Dusty plasmas: synthesis, structure and dynamics of a dust cloud in a plasma[END_REF] were then proposed to remove them away from the processed surfaces. In fusion reactors like ITER, the presence of dust particles remains a critical issue [START_REF] S I Krasheninnikov | Recent progress in understanding the behavior of dust in fusion devices[END_REF][START_REF] Arnas | Analyses of dust samples collected in the mast tokamak[END_REF]. Whereas these nano sized particles are harmful in many processes, nowadays they are also considered for several applications like solar cells [START_REF] Roca I Cabarrocas | Low temperature plasma deposition of silicon thin films: From amorphous to crystalline[END_REF] and memories. In laboratories, dust particles can be easily grown using several methods like reactive gases [START_REF] Berndt | Some aspects of reactive complex plasmas[END_REF][START_REF] Bleecker | Detailed modeling of hydrocarbon nanoparticle nucleation in acetylene discharges[END_REF][START_REF] Hundt | Real-time monitoring of nucleation-growth cycle of carbon nanoparticles in acetylene plasmas[END_REF][START_REF] Calafat | Carbon Nanoparticle/Hydrogenated amorphous carbon composite thin films formed in ECR plasma[END_REF][START_REF] Géraud-Grenier | Photoluminescence of hydrogen amorphous carbon nitrile particles obtained in a 13.56 MHz dusty plasma[END_REF][START_REF] Despax | Cyclic powder formation during pulsed injection of hexamethyldisiloxane in an axially asymmetric radiofrequency argon discharge[END_REF][START_REF] Massereau-Guilbaud | Electron temperature evolution in a low pressure dusty rf nitrogen-rich methane plasma[END_REF][START_REF] Schulze | Controlled particle generation in an inductively coupled plasma[END_REF][START_REF] Greiner | Imaging Mie ellipsometry: dynamics of nanodust clouds in an argon-acetylene plasma[END_REF] or material sputtering [START_REF] Praburam | Experimental observation of very low-frequency macroscopic modes in a dusty plasma[END_REF][START_REF] Samsonov | Instabilities in a dusty plasma with ion drag and ionization[END_REF][START_REF] Mikikian | Formation and behaviour of dust particle clouds in a radio-frequency discharge: results in the laboratory and under microgravity conditions[END_REF][START_REF] Arnas | Thermal balance of carbon nanoparticles in sputtering discharges[END_REF][START_REF] Michau | Field reversal and particle growth in dc discharge[END_REF]. Thanks to these methods, molecular precursors are created in the discharge and are at the origin of chemical reactions leading to the growth of dust particles. As soon as these particles reach a sufficient size, they acquire a negative charge by capturing plasma electrons. This loss of electrons can lead to modifications in the plasma characteristics [START_REF] Goertz | Effects of charge depletion in dusty plasmas[END_REF][START_REF] Hübner | Dust-induced modulation of the atomic emission in a dusty argon discharge[END_REF][START_REF] Mitic | Spectroscopic evaluation of the effect of the microparticles on radiofrequency argon plasma[END_REF] and when the dust particle density is huge, different types of low frequency (from a few to a few hundreds Hz) instabilities can be observed: dust particle growth instabilities [START_REF] Praburam | Experimental observation of very low-frequency macroscopic modes in a dusty plasma[END_REF][START_REF] Samsonov | Instabilities in a dusty plasma with ion drag and ionization[END_REF][START_REF] Mikikian | Low frequency instabilities during dust particle growth in a radio-frequency plasma[END_REF][START_REF] Mikikian | Plasma emission modifications and instabilities induced by the presence of growing dust particles[END_REF] including plasmoids [START_REF] Mikikian | Carousel instability in a capacitively-coupled RF dusty plasma[END_REF][START_REF] Tawidian | Plasma inhomogeneities near the electrodes of a capacitively-coupled radio-frequency discharge containing dust particles[END_REF][START_REF] Mikikian | Merging and splitting of plasma spheroids in a dusty plasma[END_REF][START_REF] Tawidian | Zoom into dusty plasma instabilities[END_REF], successive generation instabilities [START_REF] Despax | Cyclic powder formation during pulsed injection of hexamethyldisiloxane in an axially asymmetric radiofrequency argon discharge[END_REF][START_REF] Cavarroc | Instabilities during the growth of dust successive generations in silane-based plasmas[END_REF][START_REF] Couëdel | Self-excited void instability during dust particle growth in a dusty plasma[END_REF] and the heartbeat instability [START_REF] Mikikian | Mixed mode oscillations in complex plasma instabilities[END_REF][START_REF] Mikikian | Threshold phenomena in a throbbing complex plasma[END_REF][START_REF] Pustylnik | On the heterogeneous character of the heartbeat instability in complex (dusty) plasmas[END_REF][START_REF] Heidemann | Comprehensive experimental study of heartbeat oscillations observed under microgravity conditions in the PK-3 plus laboratory on board the international space station[END_REF]. Some plasma instabilities (MHz range) related to dust particle growth can also be observed in electron cyclotron resonance plasmas [START_REF] Drenik | Observation of plasma instabilities related to dust particle growth mechanisms in electron cyclotron resonance plasmas[END_REF].

The present study is mainly focused on the analysis of dust particle growth instabilities (DPGI). The so-called DPGI, detailed in Sec. 3, is investigated by recording the discharge current during dust particle growth obtained by material sputtering. This signal reveals the temporal evolution of the instability. A Fourier analysis shows the existence of different instability regimes and complex frequency evolutions (Sec. [START_REF] Selwyn | In situ laser diagnostic studies of plasma generated particulate contamination[END_REF]). An alternation of ordered and stochastic regimes is evidenced. The ordered regimes are characterized by a well-defined frequency value that evolves smoothly as a function of time. On the contrary, during the stochastic regime the frequency is changing very quickly and erratically around a mean value. The evolution of these regimes is studied as a function of the gas pressure (Sec. 5) and some instability characteristics (appearance time, duration, typical frequencies) are compared. It appears that the simple measurement during an experiment of the instability appearance time is a good indication of the entire instability evolution (Sec. 6). The experimental results are discussed in Sec. 7.

Experimental setup

The experiments are performed in the PKE-Nefedov [START_REF] Nefedov | PKE-Nefedov: plasma crystal experiments on the International Space Station[END_REF] reactor where the plasma is created by a capacitively coupled radio-frequency (13.56 MHz in push-pull mode) discharge with a typical power of 3 W. The two parallel electrodes (4.2 cm in diameter, separated by 3 cm) are surrounded by guard rings (figure 1). Dust particles are grown in the plasma by sputtering previously injected micrometer size dust particles made of polymer (melamine formaldehyde) and lying on the electrodes [START_REF] Mikikian | Formation and behaviour of dust particle clouds in a radio-frequency discharge: results in the laboratory and under microgravity conditions[END_REF]. After each experiment, the grown dust particles fall down on the bottom electrode and this deposition is also sputtered during the experiments. Krypton is used as the sputtering gas with a typical pressure between 1.4 and 2.4 mbar. The growth process is highly sensitive to gas purity, and this effect can be amplified by the fact that experiments are performed without continuous gas flow. Before each experiment, the gas line and the reactor are pumped down to a low base pressure of 10 -6 mbar in order to eliminate the molecular impurities from the previous experiment. This pumping method is favorable to form a high dust particle density and to obtain relatively reproducible results [START_REF] Samsonov | Instabilities in a dusty plasma with ion drag and ionization[END_REF][START_REF] Mikikian | Formation and behaviour of dust particle clouds in a radio-frequency discharge: results in the laboratory and under microgravity conditions[END_REF]. For each studied pressure, several experiments are performed in order to obtain statistical measurements and estimate error bars.

Dust particle growth instabilities

Once the plasma is switched on, the dust particle growth process starts. This growth is easily identified by measuring the temporal evolution of the amplitude of the discharge current fundamental harmonic (DC component, figure 2). Indeed, as dust particles are growing, they attach more and more free electrons. It results in the global decrease of the discharge current [START_REF] Mikikian | Dusty plasmas: synthesis, structure and dynamics of a dust cloud in a plasma[END_REF]. After a few tens of seconds, dust particles strongly disturb the plasma equilibrium and DPGI start. They are easily detected in figure 2 and correspond to an increase of the signal fluctuation around its mean value. During DPGI, the plasma is highly unstable and shows very complex structure and behaviour [START_REF] Mikikian | Merging and splitting of plasma spheroids in a dusty plasma[END_REF]. While dust particles were easily observed in the Ar case [START_REF] Mikikian | Low frequency instabilities during dust particle growth in a radio-frequency plasma[END_REF], they are hardly visible with Kr. Our visual observations (at the extinction of the plasma thanks to laser light scattering) tend to show that the use of Kr leads to the formation of a denser cloud of smaller dust particles. When the plasma is turned off, the dust cloud is more easily observed as the oscillating plasma glow is no more disturbing the observation. In addition, dust particles are very light and immersed in a neutral gas at rest (static pressure) and at a relatively high pressure. Thus, the dust cloud falling can take several seconds due to the small gravitational force and the non negligible friction with the neutral gas. It is then possible to observe the dust cloud more easily than during the plasma ON phase. No clear void is detected in the center of the dust cloud and DPGI appear during the growth of a single dust particle generation. In the presented experiments, as the void does not appear, no cyclic formation of dust particles is obtained [START_REF] Berndt | Some aspects of reactive complex plasmas[END_REF][START_REF] Despax | Cyclic powder formation during pulsed injection of hexamethyldisiloxane in an axially asymmetric radiofrequency argon discharge[END_REF][START_REF] Greiner | Imaging Mie ellipsometry: dynamics of nanodust clouds in an argon-acetylene plasma[END_REF][START_REF] Cavarroc | Successive generations of dust in complex plasmas: A cyclic phenomenon in the void region[END_REF][START_REF] Hong | Size dependence of optical properties and internal structure of plasma grown carbonaceous nanoparticles studied by in situ Rayleigh-Mie scattering ellipsometry[END_REF]. It is confirmed by figure 2 where no very low-frequency oscillations of the signal (period of a few minutes) is detected. Some sudden amplitude drops in the electrical signal are noticed when using Kr. These drops last for a few seconds and are evidenced just after the plasma ignition and during the first part of the instabilities as clearly observed in figure 2. Later in the growth process, the existence of these drops is no more evidenced. They are also not obtained without dust particle growth. For these reasons, we do not attribute these drops to any problem in the electrical wiring circuit. Their origin is currently under investigation and their connection with dust particle growth is considered.

Evidence of different instability regimes

In figure 3, the AC part of the discharge current is recorded from the plasma ignition till its extinction at a pressure of 1.8 mbar. Measuring the AC component gives a much higher resolution than the DC component (figure 2) for analyzing the DPGI frequency evolution. The plasma ignition is detected as a sharp peak 3 s after the recording start (due to the time scale used in figure 3, it is superimposed on the yaxis). The instability begins several tens of seconds after the plasma ignition and lasts about 2000 s. The DPGI frequency evolution is evidenced by calculating the corresponding Fourier spectrogram (figure 4). In order to emphasize small ordered domains, the spectrogram intensity has been normalized separately inside each 100 s range [START_REF] Mikikian | Low frequency instabilities during dust particle growth in a radio-frequency plasma[END_REF] (from 0 to 100 s the intensity has been normalized to its maximum value inside this time domain and so on). From figures 3 and 4, it clearly appears that the instability is characterized by a well defined succession of phases. These phases can be either ordered with a smoothly evolving frequency, or stochastic with fast and erratic frequency variations. In figure 4, the main stochastic phase clearly appears between 400 and 1300 s. These phases are also well identified in figure 3 where they are characterized by different instability amplitudes. In figure 4, when several frequencies are detected simultaneously, the colour code allows to identify the dominant frequency. Indeed, the electrical measurements show that the instability is constituted of well defined patterns with a varying number of peaks. Depending on the respective peak amplitudes, the dominant frequency corresponds either to the main frequency pattern (constituted of several peaks) or to the peak separation within the pattern [START_REF] Tawidian | Zoom into dusty plasma instabilities[END_REF].

At the beginning of the instability, a fast increase of the frequency is observed until 400 s (figure 4). During this period, the instability alternates between ordered and more stochastic phases. Then, a long stochastic phase starts at 400 s up to 1300 s. The frequency fluctuates around 170 Hz quickly and erratically but stays within a rather small value range. During this period, the envelope amplitude of the electrical signal stays roughly at the same height (figure 3) and the most highlighted frequency (figure 4) corresponds to the frequency between consecutive peaks. During this phase no specific pattern can be determined due to the stochastic evolution. Then, the frequency decreases to about 55 Hz at around 1300 s. This change appears as an increase of the signal amplitude in figure 3, and to the appearance of a clear two peak pattern, the dominant frequency being the main pattern frequency (and not the frequency corresponding to the two peak separation). Finally, the instability ends with an ordered phase characterized by a high frequency (≃ 200 Hz in figure 4) and a small amplitude (figure 3).

In this section, we have briefly identified different phases and showed a complex evolution of the main frequency. In order to better understand the instability behaviour, its modification as a function of the pressure is now investigated.

Pressure dependence of DPGI

The evolution of the RF current amplitude (DC component) as a function of time is presented for different pressures from 1.4 to 2.4 mbar in figure 5. At the plasma ignition, it is observed that the higher the pressure, the higher the rise of the signal. Then, the current decrease due to dust particle growth is slightly different as a function of the pressure. For instance, at 1.4 and 1.6 mbar, the decrease is slow and the amplitude variation is small. At 1.8 mbar, the current decrease becomes faster and a clear gap can be observed between the curves at 1.6 and 1.8 mbar. At 2 and 2.4 mbar, the current decrease continues to accelerate. The gap observed between 2 mbar and 2.4 mbar could be the place of a missing measurement at 2.2 mbar (not available due to experimental problems during the measurement campaign). These measurements show that at higher pressures (above 1.6 mbar) the growth kinetics is faster and the dust density is higher (the bigger is the amount of formed dust particles, the bigger is the loss of electrons). These hypotheses are consistent with the visual observation of the dust cloud falling at the extinction of the plasma (the dust cloud is hardly visible by laser light scattering during the plasma phase due to the small dust particle size).

In the same range of pressures (from 1.4 to 2.4 mbar), the instability characteristics are determined. The evolution of the appearance time and total duration of DPGI are presented in figure 6. Error bars are estimated from a statistics based on several experiments performed in the same conditions. A long pumping period (at least one half-day) between each experiment guarantees that the amount of impurities (strongly affecting the dust particle growth process) is low and rather similar for each experiment. At 1.4 mbar, the instability appears around 220 s after the plasma ignition, while at 2.4 mbar it occurs more quickly (35 s). Increasing the pressure results in a continuous decrease of the appearance time that can be easily fitted by a decreasing exponential function. It means that the necessary conditions to trigger DPGI are obtained faster at high pressure. Anew, it is consistent with the observation that higher pressures result in the growth of a higher dust density (and thus smaller dust sizes) as shown in figure 5.

These conditions seem to be more favorable to trigger DPGI than situations with a lower density of bigger dust particles obtained at low pressures [START_REF] Mikikian | Low frequency instabilities during dust particle growth in a radio-frequency plasma[END_REF].

Concerning the instability duration, it increases with the pressure from 1200 s at 1.4 mbar up to 2400 s at 2.4 mbar. The duration trend can also be fitted by an exponential function in the form ab.exp(-c.P ). At high pressures, the amount of dust particles is large enough to sustain the instability for a long duration. From the behaviours observed in figure 6, it can be deduced that the higher the pressure, the shorter the appearance time and the longer the duration of the instability. Thus, a high dust particle density is required to induce well developed instabilities. During experiments, this observation can provide an easy indication of the amount of grown dust particles. Dust particle growth kinetics can also be estimated from the evolution of the DPGI frequency. For this purpose, the starting (at the instability beginning) and final (just before the instability end) frequencies are measured as a function of the pressure (figure 7). At 1.4 mbar, the starting frequency is around 35 Hz and it increases with the pressure up to a value of about 58 Hz at 2 mbar (an increase of 1.6 times). The final frequency is around 75 Hz at 1.4 mbar, and it increases with the pressure up to 350 Hz at 2 mbar (an increase of 4.6 times). The ratio between the final and starting frequencies is also highly dependent on the pressure, from a factor 2 at 1.4 mbar, 3 at 1.6 mbar, 4.5 at 1.8 mbar and 6 at 2 mbar. At 2.4 mbar, the signal is mostly dominated by a stochastic behaviour, and thus starting and final frequencies cannot be clearly defined. A relatively small difference in the starting frequency can thus result in rather different final frequencies confirming the pressure dependence of dust particle growth kinetics. In order to better investigate the frequency evolution during dust particle growth, the four main phases of DPGI (figure 4) are analyzed as a function of the gas pressure. As shown in section 4, these successive phases are:

-From the beginning up to the main stochastic regime (till 400 s). The evolution of the frequency during this period is discussed in [START_REF] Tawidian | Zoom into dusty plasma instabilities[END_REF].

-The main central stochastic regime (from 400 s up to around 1300 s).

-The frequency drop phase.

-The last ordered phase (around 2000 s).

In figure 8, the first part of the instability is observed for four values of the pressure (1.4, 1.6, 1.8, and 2 mbar). For a better comparison, the same time and frequency scales are used for each pressure. Some conclusions drawn from figures 6 and 7 (on the appearance time and starting frequencies) can also be found again in this figure. At 1.4 mbar, several ordered phases are observed at the beginning of the instability before the long stochastic phase. The frequency increases slowly with a speed of about 0.07 Hz/s. At 1.6 mbar, the first part of the instability becomes shorter and an alternation of ordered and stochastic behaviour is observed before the beginning of the long stochastic phase at around 500 s. During this first part, the frequency increases with a speed of about 0.25 Hz/s. At 1.8 mbar, the rise of the frequency is faster with a speed of 0.5 Hz/s, and the first part of the instability becomes mostly stochastic. At 2 mbar, the first part of the instability is brief and the frequency increases at a speed of 0.9 Hz/s. These evolutions show that at high pressure, the frequency increases faster than at low pressure and is related to the formation of a high dust density.

The first phases of the instability are followed by the main stochastic regime, a relatively robust phase appearing during all our experiments. In figure 9, the long stochastic phase frequency is shown as a function of the pressure. The frequency mean value is around 55 Hz at 1.4 mbar, and increases up to 240 Hz at 2 mbar. At 2.4 mbar, this frequency can strongly vary from one experiment to another. It is mainly due to the strong stochastic behaviour of DPGI at pressures above 2 mbar as already mentioned for figure 7. Indeed, in our experimental conditions these pressures lead to very strong stochastic instabilities that can evolve quite differently. Nevertheless, as we performed many experiments for each pressure, we found that the average frequency at 2.4 mbar fit rather well with the global trend as function of the pressure. Thus, we can conclude that as for the starting and final frequencies (figure 7), the central stochastic frequency also increases with the pressure.

After this long stochastic phase, the frequency decreases to a smaller value (figure 10 (a) 1.4, (b) 1.6, (c) 1.8 and (d) 2 mbar). This is mainly due to the fact that for this phase, the frequency corresponds to the pattern frequency and no more to the consecutive peak frequency. This phase is constituted of two parts, a well-ordered one and a more stochastic one. As expected, this phase is more ordered at low pressure. At 1.4 mbar, it begins around 1180 s (figure 10 (a)) where a frequency at 32 Hz and its harmonics are observed for a short time until 1220 s, then the frequency at 65 Hz becomes dominant. At 1.6 mbar, the phase is clearly evidenced around 1250 till 1600 s (figure 10 (b)) before turning into a more stochastic behaviour. At 1.8 mbar (figure 10 (c)), it is globally more stochastic, meaning that a pattern exists but it is disturbed by the appearance of random peaks. Finally at pressures above 2 mbar, the stochastic behaviour is completely dominant and at 2.4 mbar this phase is the final one before the end of the instability: the last ordered phase (figure 4) is not observed.

At pressures lower than 2.4 mbar, another phase occurs before the instability end. This final phase has a clear ordered behaviour as shown in figure 11. At high pressure, the phase is shorter than at low pressure, and the shape of the frequency evolution is different. At 1.4 mbar, the frequency increases slowly with a speed of about 0.09 Hz/s then slightly decreases at the end. At 1.6 mbar, the frequency increases with a speed of 0.10 Hz/s and the last decrease is less marked. At 1.8 mbar, the speed is around 0.13 Hz/s and the last decrease is not observed. At 2 mbar, this phase is divided into two parts, with a rising speed of about 0.63 Hz/s. The frequency evolution becomes faster as a function of the pressure as we already observed at the beginning of the instability (figure 8). In figure 11, it clearly appears that the duration of this ordered phase decreases with the pressure. The splitting of this phase at 2 mbar is a first indication of the weakness of the ordered behaviours at high pressure. It is a sign of the future disappearance of this phase at 2.4 mbar. The progressive disappearance of the ordered phases at high pressure is clearly evidenced in figure 12. In this figure, the relative contributions of the ordered and stochastic phases have been measured. The durations of all the stochastic phases occurring during a DPGI sequence have been summed up. The same procedure has been used for the ordered regimes. At 1.4 mbar, the two types of phases have roughly the same duration, whereas the stochastic phases become dominant as the pressure is increased. At 2.4 mbar, the ordered phases are almost absent, and DPGI are mainly stochastic during their whole duration. It means that when the dust particle density is huge (and thus the dust particle size is small), the instability cannot enter in an ordered regime. The disturbance induced by the dust particles is too strong to allow the instability to have an ordered oscillating behaviour.

Interdependency between instability parameters

In the previous section, the evolution of the instability characteristics has been studied as a function of the pressure. In the present section, relations between these different characteristics is emphasized. During an experiment, the first parameter that can be easily measured is the instability appearance time. The knowledge of this single parameter provides information on the instability evolution. In figure 13, the instability duration is plotted as a function of the appearance time. It clearly shows the result of figure 6 that the longer the appearance time, the shorter the instability duration.

When the instability appears 35 s after the plasma ignition, it can last for 2400 s.

When it starts 220 s after the plasma ignition, the total duration of the instability is only 1200 s. Figure 13 shows that there is roughly an inversely linear dependence between the instability duration and appearance time. The duration of DPGI can thus be roughly predicted as soon as they appear. Thanks to the appearance time, the different frequencies of the instability (starting, during the long stochastic phase, and final) can also be determined as shown in figure 14 (the evolution of the starting frequency is zoomed in the upper right insert). The frequency trends have been fitted by decreasing exponential functions. Figure 14 clearly shows the relation between the frequencies discussed previously in figure 7. A quick start of the instability (60 s after plasma ignition) shows a strong temporal increase of the mean frequency (a factor of 6 between the starting and the final frequencies). For a longer appearance time (220 s after plasma ignition), the factor between the starting and the final frequencies becomes only 2. It means that the earlier is the appearance of the instability, the bigger is the global increase of the frequency.

These dependences with the appearance time show that DPGI is a relatively robust phenomenon and that small fluctuations in the starting conditions do not change its long term evolution. This conclusion has to be slightly moderated for the high pressures. It can be useful for the experimenters, as the knowledge of the appearance time is enough to know rather accurately the instability evolution.

Discussion and Conclusions

In this paper, we characterized instabilities triggered by the presence of a high density of grown dust particles in a Kr plasma. These instabilities are analyzed using electrical measurements: the amplitude of the discharge current is recorded from the ignition till the extinction of the plasma. The Fourier spectrogram brings to light the temporal evolution of the frequencies during the instability. In particular, a succession of different ordered and stochastic phases are identified as in the Ar case [START_REF] Mikikian | Low frequency instabilities during dust particle growth in a radio-frequency plasma[END_REF]. The differences between the two gases mainly concerns the frequency values and evolution within the phases. Frequencies are higher in Kr certainly due to a higher dust particle density and a smaller dust particle size. A systematic comparison between the two gases has to be performed with experiments completely dedicated to this defined study. In this paper we focused our analysis on the measurement of the instability characteristics as a function of the Kr pressure. It appears that the higher the pressure, the shorter the appearance time (figure 6), the longer the duration (figure 6) and the higher the frequencies (figures 7 and 9). These results are consistent with the observation performed in figure 5, that at high pressure the decrease of the current amplitude is sharper. It means that the growth kinetics is faster [START_REF] Mikikian | Formation and behaviour of dust particle clouds in a radio-frequency discharge: results in the laboratory and under microgravity conditions[END_REF][START_REF] Boufendi | Dusty plasma for nanotechnology[END_REF] and the dust density is certainly higher (usually connected with a smaller dust particle size [START_REF] Mikikian | Low frequency instabilities during dust particle growth in a radio-frequency plasma[END_REF]). Indeed, our experimental observations tend to show that a fast growth kinetics and a high dust density lead to a rapid appearance of DPGI. As the disturbance is strong and the discharge is saturated by dust particles, the instability can thus last longer. For the frequencies, a direct correlation is more difficult to define. Indeed, a simple approximation consists in considering that the typical DPGI frequencies are related to the dust plasma frequency [START_REF] Shukla | Colloquium: Fundamentals of dust-plasma interactions[END_REF][START_REF] Mikikian | Dusty plasmas: synthesis, structure and dynamics of a dust cloud in a plasma[END_REF][START_REF] Cavarroc | Self-excited instability occurring during the nanoparticle formation in an Ar -SiH 4 low pressure radio frequency plasma[END_REF]

ω pd = Q 2 d n d ǫ 0 m d ∝ n d r d
(where n d is the dust density, Q d the dust charge, m d the dust mass and r d the dust radius). As n d is higher and r d is smaller at high pressure, it is logical to obtain a global increase of the typical DPGI frequencies with the pressure (figures 7 and 9). Nevertheless, this simple consideration cannot clearly explain the frequency temporal evolution. Whereas, the frequency increase at the beginning (figure 8) could be related to an increase of n d due to the sputtering process, the stochastic phase and the last ordered phase cannot be easily related to ω pd . In the long stochastic phase, the mean frequency is relatively constant. It is not consistent with dust particles. An hypothesis can be that we have a slowing down of the dust particle growth during this regime, and then, rather constant values of n d and r d . The last ordered phase is characterized by a frequency increase. As n d is not expected to increase at this stage of the dust particle growth process (no indication of a new dust particle generation is observed) the relation with ω pd is rather difficult. This behaviour shows that more complex phenomena are driving DPGI and further investigations need to be conducted to analyze the evolution of dust and plasma properties. For example, the use of in-situ diagnostics like ellipsometry [START_REF] Greiner | Imaging Mie ellipsometry: dynamics of nanodust clouds in an argon-acetylene plasma[END_REF][START_REF] Hong | Size dependence of optical properties and internal structure of plasma grown carbonaceous nanoparticles studied by in situ Rayleigh-Mie scattering ellipsometry[END_REF] will give access to dust particle size and will allow a better correlation with DPGI evolution. Finally, we emphasized that the single measurement of the appearance time provides an estimation of the other instability properties thanks to linear and exponential dependences. It can be an easy and useful indication for the experimenter as soon as the instability begins. In the future, the evolution of the instability amplitude will be also further investigated. Indeed, as shown in figure 3, the amplitude variations could be connected to dust particle growth dynamics.

Figure 1 .

 1 Figure 1. Scheme of the PKE-Nefedov reactor.

Figure 2 .

 2 Figure 2. Amplitude of the rf current (DC component) as a function of time.

Figure 3 .Figure 4 .

 34 Figure 3. Alternative part of the discharge current amplitude evidencing DPGI.

Figure 5 .Figure 6 .

 56 Figure 5. Amplitude of the rf current (DC component) as a function of time for different pressures.

Figure 7 .

 7 Figure 7. (a) Starting and (b) final frequencies of the instability as a function of the pressure.

FrequencyFigure 8 .

 8 Figure 8. Fourier spectrogram of the first phases for different pressures, (a) 1.4 mbar, (b) 1.6 mbar, (c) 1.8 mbar and (d) 2 mbar.

Figure 9 .

 9 Figure 9. Central stochastic phase frequency as a function of the pressure.

Figure 10 .

 10 Figure 10. Fourier spectrogram of the frequency drop phase for different pressures, (a) 1.4 mbar, (b) 1.6 mbar, (c) 1.8 and (d) 2 mbar.

Figure 11 .

 11 Figure 11. Spectrogram of the last ordered phase before the end of the instabilities, (a) at 1.4 mbar, (b) 1.6 mbar, (c) 1.8 mbar and (d) 2 mbar.

Figure 12 .

 12 Figure 12. Contribution of the ordered (⊓ ⊔) and stochastic (×) regimes as a function of the pressure.

Figure 13 .

 13 Figure 13. Duration of the entire instability as a function of the appearance time.

Figure 14 .

 14 Figure 14. Starting (⊓ ⊔), stochastic (×), and final (•) frequencies of the instability as a function of the appearance time.

Acknowledgments

The PKE-Nefedov chamber has been made available by the Max-Planck-Institute for Extraterrestrial Physics, Germany, under the funding of DLR/BMBF under Grant No. 50WM9852. This work was supported by the French National Research Agency (ANR), project INDIGO n • ANR-11-JS09-010-01


Ali Jaffal
email: ali.jaffal@malix.univ-paris1.fr

Bénédicte Le

Towards an Automatic Extraction of Smartphone Users' Contextual Behaviors

Keywords: user behavior extraction, automatic interpretation, context analysis, smartphone traces analysis, Formal Concept Analysis, Galois lattices

This paper presents a new method for automatically extracting smartphone users' contextual behaviors from the digital traces collected during their interactions with their devices. Our goal is in particular to understand the impact of users' context (e.g., location, time, environment, etc.) on the applications they run on their smartphones. We propose a methodology to analyze digital traces and to automatically identify the significant information that characterizes users' behaviors. In earlier work, we have used Formal Concept Analysis and Galois lattices to extract relevant knowledge from heterogeneous and complex contextual data; however, the interpretation of the obtained Galois lattices was performed manually. In this article, we aim at automating this interpretation process, through the provision of original metrics. Therefore our methodology returns relevant information without requiring any expertise in data analysis. We illustrate our contribution on real data collected from volunteer users.

I. INTRODUCTION

Pervasive Information Systems extend traditional Information Systems by taking into account users' environment and mobility. This mobility has become possible through the expansion of lightweight devices such as smartphones [START_REF] Kourouthanassis | A Design Theory for Pervasive Information Systems[END_REF]. Understanding the influence of context and environment on the way people use their mobile devices represents a challenge [START_REF] Fernandes | Model-Driven Development for Pervasive Information Systems[END_REF]. The ArtDeco project [START_REF] Anastasi | Methodologies and Technologies for Networked Enterprises[END_REF] is dedicated to the extraction of knowledge -and its representation-from heterogeneous information collected from applications and sensors in companies. In another context, the U-CAT project [START_REF] López-Cózar | COST278 Final Workshop and ITRW on Applied Spoken Language Interaction in Distributed Environments[END_REF] aims at developing a system for educational activities that rely on various mobile devices.

In earlier work [START_REF] Jaffal | Unified and Conceptual Context Analysis in Ubiquitous Environments[END_REF], we have used Formal Concept Analysis (FCA) and Galois latticesintroduced in Section II -to study the impact of smartphone users' context and environment on the applications they execute on their mobile devices. We have obtained promising results, however we have performed the interpretation of the obtained Galois lattices manually. Our goal is now to automate this interpretation task, so that our approach can be useful for people who are not experts in Formal Concept Analysis, or in data analysis in general. An overview of our methodology for an automated extraction of smartphone users' contextual behaviors from digital traces is represented on Figure 1. The various steps of this process are explained in Section II. In this paper, we illustrate our methodology and the underlying analysis techniques on a sample of real data, collected from a group of master students of our university. Those students agreed to fill in questionnaires where they reported, during one week, the applications they have used on their smartphones, as well as the associated contexts (e.g., geographical location, time, type of connection network, etc.).

Note that although we illustrate our theoretical contribution in the area of pervasive information systems, it can be applied to other application domains (see Section IV).

II. BACKGROUND AND RELATED WORK

Formal Concept Analysis (FCA) is at the intersection between mathematics and data analysis [START_REF] Priss | Formal Concept Analysis in Information Science[END_REF]. The input of a FCA algorithm is a set of data elementscalled objectscharacterized by (Boolean) features -called attributes. The binary relation between objects and attributes is represented in a table called formal context. In our case study, the objects are the applications executed on the smartphone and the attributes corresponds to user context. Each piece of information related to context is called context element in the following. Figure 2 is an example formal context, which has been derived from the questionnaire of one of our students. In this formal context, the applications executed by this student are listed in the rows of the table and the various contextual elements associated to the applications appear in the columns. Note that the list of applications and context elements may vary as they are specified by users themselves. This specific student has used the following applications: Gmail, SMS, telephone, VDM (web site where people tell their unlucky daily stories), Flappy bird (game), and Youtube. In this case, we may consider that these applications belong to two broad categories: communication and leisure. The various context elements identified by this user are related to her location (university, home and public transportation), but also to the type of network connection (3G) and the moment the day (morning, afternoon, evening). For example, the second row in the formal context of Figure 2 shows that this specific student has used the SMS application at the university, in the public transportation, from a 3G network and in the afternoon. From each formal context given as an input, FCA groups objects into clusters -called formal concepts-according to their common attributes [START_REF] Wille | Formal Concept Analysis as Mathematical Theory of Concepts and Concept Hierarchies[END_REF]. In the case study presented here, applications are thus clustered according to the context elements they have in common. One strength of FCA over most other clustering approaches is that the resulting clusters are overlapping (i.e., objects and attributes may appear simultaneously in several clusters). Moreover, the semantics of each cluster (formal concept) is explicit: the existence of each formal concept is justified by the common attributes that characterize the clustered objects.

The result of the FCA process is a Galois lattice that represents the partial order relation between formal concepts. Figure 3 represents the Galois lattice generated from the formal context of Figure 2. The manual interpretation of this lattice leads to useful information about the smartphone usage of the associated student and about the impact of contextual elements on the applications she has executed on her mobile device. Each concept (cluster) of the lattice groups applications according to the context elements they have in common. For example, the concept number 9 in the lattice represented on Figure 3 contains 3 applications: Flappy Bird, Gmail and SMS, that have all been used from a 3G network, in the public transportation and at the university. The concepts that are close to the top of the lattice are the most generic concepts, i.e., the concepts that contain many applications but have few context elements in common. Those applications therefore have a low conceptual usage similarity (we provide a formal definition of this notion in Section III).

On the other hand, the concepts that are close to the bottom of the lattice are specific, i.e., they contain few applications that share many context elements, and they are thus very similar, in terms of conceptual usage similarity. Let us give a few examples of possible interpretations of this lattice: the concept number 14 -the upper bound of the lattice-contains the 3G context element that is common to all applications: this context element can therefore be considered as "universal" for this specific user. Similarly, the Gmail application is almost universal as it appears in a concept (number 1) that is very close to the lower bound of the lattice.

In addition to the interpretation of the bounds of the lattice, we may identify some applications that frequently appear simultaneously in the formal concepts of the lattice, i.e., that are conceptually similar: this is the case for Gmail and SMS, that appear together in five concepts (one third of the total number of concepts in the lattice). This observation may seem obvious, as both applications are dedicated to communication. However, we also learn that these applications are usually associated to afternoon, public transportation and university. Similarly, some applications belonging to the leisure category appear in the same concepts, and are used in conjunction with other context elements: for example, VDM and Youtube appear together in concept number 7, associated to home and morning.

We can also focus on context elements and identify some conceptually similar context elements, such as transportation, afternoon and 3G, which are associated to a high proportion of common applications: Gmail, telephone and SMS. Such conclusions may be very valuable for a mobile phone service provider, who can thus better understand customers' preferences and habits. However two limitations need to be overcome: first, this interpretation should be automated in order to select the most significant conclusions. Moreover, it is important to know which observations made at the individual scale can be generalized. Therefore, the results obtained from the analysis of individual lattices should be compared and summarized in order to distinguish frequent behaviors from more typical ones.

The goal of the work presented in this paper is to design and implement metrics in order to perform an automatic interpretation of individual lattices (to understand individual users' behaviors), as well as comparisons of several lattices (to compare the behaviors of several users). Other methods are dedicated to lattice reduction, in order to make their visualization easier. Our work is orthogonal to the initiatives, as we focus on a systematic and relevant interpretation of the lattice, whatever its size.

In the following Section, we propose several metrics designed to automate the interpretation and comparison of Galois lattices. Several metrics such as stability and support [START_REF] Jay | Analysis of social communities with iceberg and stability-based concept lattices[END_REF], [START_REF] Klimushkin | Approaches to the selection of relevant concepts in the case of noisy data[END_REF], [START_REF] Kuznetsov | Reducing the Representation Complexity of Lattice-Based Taxonomies[END_REF] have been proposed to analyze Galois lattices, but their goal is to reduce them by keeping only the most relevant concepts [START_REF] Ventos | Alpha galois lattices: An overview[END_REF]. However, even reduced lattices may remain too large to be interpreted manually. Moreover, such analyses still require expertise in Formal Concept Analysis. Our contribution is thus complementary to these approaches, as it aims at providing intuitive interpretation of lattices, for non-expert users. The authors of [START_REF] Grand | Empreintes conceptuelles et spatiales pour la caractérisation des réseaux sociaux[END_REF] have proposed some « conceptual » metrics that they have applied to the characterization of social network members. However, those metrics focus on the characterization of objects and attributes without analyzing the impact they may have on one another.

III. METRICS FOR THE AUTOMATIC INTERPRETATION OF GALOIS LATTICES

In order to go beyond existing metrics for the interpretation of Galois lattices, we propose three metrics that we detail in the following.

A. Conceptual weight

The first metric, called conceptual weight, represents the frequency of a given object (resp. attribute) in the lattice: it corresponds to the proportion of concepts in the lattice that contain this object (resp. attribute).

Given an application Ai:

Similarly, given a context element Ej:

Figure 4 illustrates the computation of this conceptual weight, for the Gmail application from the lattice of Figure 3. This application appears in 8 concepts of the lattice, out of 15 concepts in total: its conceptual weight is therefore equal to 0.53, as reported in Figure 5, with the weights of all other applications. This « conceptual » weight is based on the clustering performed by Formal Concept Analysis, and returns different results from the frequency that can be computed from the formal context given as an input to FCA. For example, the frequency of the Gmail application in the formal context of Figure 2 is equal to 0.71 (as this application is associated to 5 context elements out of 7 in total). The values of conceptual weight help understand the relative importance of applications with regard to context elements, for a specific user (i.e., lattice). This metric identifies the applications that frequently appear in the lattice, and that are therefore associated to common context elements.

If we consider the conceptual weight of morning and afternoon context elements, illustrated in Figure 6, we may notice that although their frequency in the input formal context of Figure 2 is identical (=0.5), their conceptual weights are different: indeed, morning is associated to a greater number of application clusters than afternoon. One limitation of the conceptual weight is that it focuses on a single application (or context element). In the following Section, we propose the conceptual usage similarity metric that compares two applications or context elements.

B. Conceptual usage similarity

We define the conceptual usage similarity between two context elements Ei and Ej as follows:

Similarly, we define the conceptual usage similarity between two applications Ai and Aj as follows:

This conceptual usage similarity metric allows comparing two specific context elements, or two specific applications, which was not possible with the conceptual weight described in the previous Section. A high value of the conceptual usage similarity metric between two context elements Ei and Ej, i.e., close to 100%, indicates that most applications associated with the context element Ei are also associated to the context element Ej. For example, public transportation and 3G network have a conceptual usage similarity equal to 73%, while public transportation and afternoon have a 45% conceptual usage similarity.

If we consider the temporal context elements, we notice that morning and afternoon are more strongly related to the 3G network than evening is, which is understandable as other types of network access are available in the evening (probably from home). Another observation is that morning is more similar to evening than it is from afternoon, in terms of application usage.

Figure 8 displays the values of conceptual usage similarity between 3G and evening for all students who answered our questionnaires. These values have been computed from 305 lattices. This diagram shows that the usage similarity between both context elements is rather low in most lattices, despite a few exceptions, which confirms the observation that was made at an individual scale (in Figure 7). Other observations are possible if we compare the overall values of conceptual usage similarity between other context elements. For example, Figure 9 and Figure 10 show that university is in average more similar to afternoon than to morning. After studying context elements, we now focus on applications. Figure 11 represents the conceptual usage similarity between all applications of the lattice of Figure 3. Some applications appear expectably similar (e.g., Gmail and SMS, with a similarity above 63%, or Youtube and VDM with almost 50%), but it is important to understand that the conceptual usage similarity reflects where applications are used in similar contexts; therefore two very different types of applications may be conceptually similar in terms of usage: this is the case for Flappy bird and telephone, which have a high conceptual usage similarity, although they are dedicated to leisure and communication respectively. We can also notice that, in terms of conceptual usage, Flappy bird is more similar to telephone than to Youtube for this user, which could seem surprising. In addition, Figure 13 shows that the conceptual usage similarity between Gmail and telephone is globally significantly lower than the similarity between Gmail and SMS. Gmail and telephone applications for all students Figure 14 provides another example of conclusion, related to two different types of applications: telephone and Flappy bird game. For many users, both applications have nothing in common (similarity equal to zero), but for the others the similarity is surprisingly high; this means that they use both applications in the same conditions, despite their very different natures. The conceptual usage similarity metric presented in this Section considers applications and context elements separately. In the following, we propose a metric that reflects the mutual impact of applications and the associated context elements.

C. Mutual impact

In a pervasive environment, context elements have an influence on application usage, and, reciprocally, applications are executed in specific contexts; we consider here that the applications users choose to execute on their smartphones reflect their behaviors. We first define the absolute mutual impact between an application Ai and a context element Ej as the proportion of concepts that contain both Ai and Ej.

We also define the relative mutual impact between an application Ai and a context element Ej:

The absolute version of mutual impact is useful to compare values of mutual impact in lattices which sizes may be very different. When considering a single lattice, relative mutual impact is more relevant.

Figure 15 represents the relative mutual impact between the applications and context elements of the lattice of Figure 3. This Figure may be interpreted as follows: we may first notice that the values of mutual impact vary significantly according to context elements and applications. For example, evening is only associated to the VDM application, and afternoon only to Gmail, SMS and telephone. On the other hand, home is not associated to Gmail or SMS. If we consider the university location, we find out that its maximum relative impact is obtained with Gmail (almost 30%), and that this impact is null with VDM or Youtube. The observations above relate to a specific user. Figure 16 shows that the value of 30% for the individual student of Figure 3 is higher than the average impact between Gmail and university. The behavior of this student with regard to the relative usage of Gmail at the university is therefore not frequently observed at the scale of the global population. The individual observation may thus not be generalized.

Additional conclusions may be drawn if we compare the overall values of relative mutual impact for other combinations of applications and context elements. Figure 17 shows that although Gmail and evening had no mutual impact for the individual user of Figure 2, this is not the case for all users; some users even have rather high values of mutual impact between Gmail and evening. Note that the results of the conceptual mutual impact can be presented in a dual way, as shown on Figure 19. Although the content is equivalent as both diagrams have been built from the same input data, the "client" of our analysis (e.g., a mobile service provider) may choose between both representations, which provide different observation angles. For example, we may see on Figure 19 that the values of mutual impact for Youtube -when they are not null-are rather homogeneous for all context elements (home, 3G network and morning). The same observation can be made for the SMS application. Fig. 19. Relative mutual impact computed from the lattice of Figure 3 IV. CONCLUSION AND PERSPECTIVES In this paper, we have proposed original metrics derived from Formal Concept Analysis, in order to automate the interpretation of Galois lattices. This interpretation methodology can be applied to any application domain: the only input required is one -or several-Galois lattices.

We have applied our proposal to the area of pervasive information systems. In our case study, each Galois latticeassociated to a specific smartphone user-clusters the applications that have been executed on the device according to the context in which they have been used. We have applied our conceptual metrics to real data collected from a sample of students. We have shown examples of automatic interpretations made possible by our metrics, which help understand the behavior of smartphone users, in terms of contextual usage of applications.

The results of smartphone users' contextual behavior can be valuable for mobile service providers. Indeed, from a service provider point of view, such conclusions can help recommend applications under specific contexts; new applications may also be developed for particular contextual situations, taking into account the mutual impact between applications and context elements, as well as usage similarity among applications and among context elements.

Each Galois lattice generated from the input data represents the behavior of a single smartphone user; we have also shown how our metrics, designed for the analysis of individual lattices, can also be used to compare several lattices -and therefore several users. This comparison may be used to classify users according to their contextual behavior, and can also help detect general trends. For the moment, the most significant combinations of applications and/or context elements (out of all possible pairs) have been chosen manually. We will also automate this pair selection process.

We have two main perspectives for the future, which will be addressed in parallel. On the one hand, we will design and implement additional metrics in order to go further in the interpretation of Galois lattices. We have focused so far on the analysis of concepts, objects and attributes; we will design metrics in order to exploit the links within the lattices, i.e., the generalization and specialization relationships between formal concepts. Indeed, the semantics of these relations can be very rich and should be taken into account during the interpretation process. We will also validate our results by collecting the feedback of users on the proposed interpretation.

On the other hand, we will apply our metrics to Galois lattices generated from other types of data. In the short term, we will apply them to e-learning data, where objects are pupils and attributes are related to evaluation criteria, such as scores, time required to answer a question, number of attempts before succeeding, etc. In the longer term, we will consider their application to the area of social networks, in order to propose an on original technique for social network analysis, in addition to traditional ones.

Fig. 1 .

 1 Fig. 1. Overview of the proposed contextual behavior extraction process

Fig. 2 .

 2 Fig. 2. Example formal context extracted from one specfic questionnaire

Fig. 3 .

 3 Fig. 3. Example Galois lattice extracted from the Formal context of Fig. 2

Fig. 4 .

 4 Fig. 4. Example of conceptual weight computation for Gmail application

Fig. 5 .

 5 Fig. 5. Conceptual weights of the applications from the lattice of Figure 3

Fig. 6 .

 6 Fig. 6. Conceptual weights of the context elements from the lattice of Fig. 3

Figure 7

 7 Figure 7 represents the conceptual usage similarity between all context elements of the lattice of Figure 3. We have chosen

Fig. 7 .

 7 Fig. 7. Conceptual usage similarity between all context elements of the lattice of Figure 3

Fig. 8 .

 8 Fig. 8. Comparison of the values of the conceptual usage similarity between 3G and transportation context elements for all students

Fig. 9 .Fig. 10 .

 910 Fig. 9. Comparison of the values of the conceptual usage similarity between university and afternoon context elements for all students

Fig. 11 .

 11 Fig. 11. Conceptual usage similarity between all applications of Figure 3

Figure 12

 12 Figure 12 represents the values of conceptual usage similarity between the Gmail and SMS applications over the 305 lattices. This Figure confirms the (expected) overall high value of conceptual usage similarity between both applications.

Fig. 12 .

 12 Fig. 12. Comparison of the values of the conceptual usage similarity betweenGmail and SMS applications for all students

Fig. 13 .

 13 Fig. 13. Comparison of the values of the conceptual usage similarity betweenGmail and telephone applications for all students

Fig. 14 .

 14 Fig. 14. Comparison of the values of the conceptual usage similarity between Flappy bird and telephone applications for all students

Fig. 15 .

 15 Fig. 15. Relative mutual impact computed from the lattice of Figure 3It is possible to compare the mutual impact between university and Gmail among all students. This value has been

Fig. 16 .

 16 Fig. 16. Relative mutual impact between Gmail and university for all students

Fig. 17 .

 17 Fig. 17. Relative mutual impact between Gmail and evening for all students On the other hand, Figure 18 confirms the extremely low impact between Youtube and university at the global scalewhich is reassuring for the teachers!