L Laiarinandrasana

G Boisot

C Fond

G Hochstetter

TOUGHNESS IMPROVEMENT OF POLYAMIDE 11 ASSESSED VIA QUASI-STATIC TENSILE TESTS ON NOTCHED ROUND BARS

Keywords: Rubber toughened polymers, Damage mechanics, Fracture mechanics, Finite element, Void growth, Cavitation, Polymer mechanics

Toughening a polymeric material by adding rubber particles is a common technique used for glassy polymers and measured by means of Charpy or Izod impact tests. These latter are determined under dynamic conditions, thus at high strain rate or equivalently at very low temperature for polymeric materials. This work is devoted to three grades of semi-crystalline PolyAmide 11 (PA11), used in a large number of engineering components. Service conditions are close to the glass transition temperature. Therefore, tests with quasi-static loading are expected to be more appropriate. The aim of the study is to find a methodology allowing the quantification of toughness improvement. To this end, an experimental database was constituted. Three grades of PA11 were involved, that enable to analyze effects of: i) aging, ii) addition of a plasticizer iii) addition of rubber particles. Tensile tests were then carried out on notched round bars. Trends of load versus notch opening displacement curves are investigated. Longitudinal cross-sections on specimens issued from interrupted tests and fracture surfaces were examined in order to study deformation and damage mechanisms. An attempt is made to link toughness improvement to the increase in global fracture energy.

INTRODUCTION

Toughness of polymeric materials is often characterized by the impact strength (from Charpy or Izod tests). They are dymamic tests at high strain rate level or equivalently at very low temperature below T g (glassy polymers). When the service temperature is near or above T g , impact strength may be inappropriate to determine the failure behaviour of the material. Indeed for fully ductile material, when the energy brought by the striker is less than the impact strength, the specimen will not break but only plastically deforms with a large deflection. It is essential to find a relevant parameter to predict the durability of structural polymers. Fracture mechanics approaches give an accurate definition of fracture toughness: the value at failure of the stress intensity factor for linear elastic fracture mechanics (LEFM: small scale yielding), or the value at crack initiation of the J-integral for elastic plastic fracture mechanics (EPFM: extended plastic zone). Comprehensive research has been carried out for years on metallic materials concerning the equivalence between impact strength and fracture toughness [START_REF] Folch | Application of coupled brittle-ductile model to study correlation between Charpy energy and fracture values[END_REF][START_REF] Rossol | Determination of the fracture toughness of a low alloy steel by the instrumented impact test[END_REF][START_REF] Tanguy | Local approach to fracture based prediction of the ∆T 56J and the ∆T KIC,100 shifts due to irradiation for an A508 pressure vessel steel[END_REF]. On polymers, some attempts were made in this way but instead of the J-integral, the essential work of fracture was used. Requested plane strain conditions are difficult to fulfil. This paper is devoted to three grades of Polyamide 11 (PA11). It follows the local approach to fracture methodology to assess the ductile failure of the materials. Indeed, for the PA11 of interest, mechanical tests were carried out on specific notched round tensile bars. Taking advantage of previous studies at EFC17 conference [START_REF] Boisot | Failure of PolyAmide 11 using a damage finite element model[END_REF], one specimen geometry was chosen in order to analyze the effect of aging together with an amount of plasticizer and an addition of rubber toughening. Examinations of the void growth mechanisms were performed in order to identify the local failure criteria, depending on each material. The global mechanical response in terms of net stress versus axial displacement is analyzed. The mechanical parameters are studied and those that are relevant to predict the toughness improvement are highlighted.

MATERIAL, SPECIMEN AND TESTING

The materials of interest consist of 3 grades of Polyamide 11 (PA11) provided by ARKEMA: i) semi-crystalline neat PA11 (index of cristallinity = 25%); ii) "aged P40" is a plasticized PA11, aged for 7 days in pressurized water (pH7) at 140°C; iii) "aged P20R" is a plasticized, toughened by rubber particles (~250nm in diameter) PA11 aged for 7 days in pressurized water (pH7) at 140°C. For each material, some mecha nical characteristics were determined with the help of Dynamic Mechanical Analysis (DMA) tests, performed at a strain range of 0.5% at 5Hz. Figure 1 shows the evolution of the storage moduli and the damping coefficient tan δ. Further information concerning the materials is provided in Table 1.

The aim of the work is to investigate, on the PA11 material, the effects of plasticizer addition followed by aging (P40) as well as the effects of rubber toughening (aged P20R) based on data obtained on aged P40 material. Following the study of Boisot [START_REF] Boisot | Failure of PolyAmide 11 using a damage finite element model[END_REF] concerning experimentation and damage modelling of PA11, where various kinds of specimens were a) b)

tan δ E' (MPa) tested (smooth, notched round bars and single edge notched bending), the present study focuses on the circumferentially notched round bars with a notch root radius of 4mm (referred to as NT4). Figure 2 shows a sketch of the NT4 specimen and indicates its characteristic dimensions. According to Bridgman [START_REF] Bridgman | The stress distribution at the neck of a tension specimen[END_REF], the NT4 specimen exhibits stress triaxiality ratio τ profile for which the maximum value is located at the centre of the minimum cross section. Equation [START_REF] Folch | Application of coupled brittle-ductile model to study correlation between Charpy energy and fracture values[END_REF] gives the expression of this initial value of τ. Note that τ also represents the part of mean stress (hydrostatic pressure) applied to the section.

Materials

NT4 specimens were tested in tension at 0°C with tw o crosshead speeds of 0.05mm/s and 3mm/s respectively. During the tests the load, displacement and reduction in diameter were recorded. Each loading condition was repeated twice or three times to check the reproducibility of the test results. Some tests were interrupted before fracture in order to allow scanning electron microscope (SEM) observations of longitudinal cross-sections. Furthermore, fracture surfaces were analyzed to investigate deformation and damage mechanisms.

EXPERIMENTAL RESULTS

Mechanical response

Figure 3 illustrates plots of the net stress (σ net ) defined as the load divided by the net section on the first y-axis and the reduction in diameter ((Φ 0 -Φ)/Φ 0 ) on the second y-axis, with respect to the axial displacement. The same scale was applied to the plots for comparison R b r 0

• Notch root radius : r 0 = 4mm

• Nominal radius : R = 7mm

• Minimum section radius b = 2mm

• Stress triaxiality ratio [START_REF] Bridgman | The stress distribution at the neck of a tension specimen[END_REF] maximum at the centre of the section:

1 1 2 3 4 4 5 6 + + = τ 0 r 2 b 1 ln 3 1 (1)
purposes. Only results at 0.05mm/s are presented here, the trends being the same at 3mm/s.

Figure 3a shows the effects of ageing by comparing the mechanical response of PA11 and aged P40. For aged P40, the peak stress is lower whereas the corresponding applied displacement is higher, compared with the PA11 curve. This is assumed to be the effect of the plasticizer added on aged P40. The displacement and reduction in diameter at failure of aged P40 (4mm and 20% respectively) are very small compared with those of PA11 (10mm and 46% respectively). Actually, aged P40 did not exhibit any extended necking zone. This drop of deformation at failure is attributed to embrittlement of aged P40 during ageing. The initial values of stiffness are similar; this confirms the Young's moduli given in Table 1. In figure 3b, aged P40 and aged P20R responses are similar at the beginning of the tests. The peak stress of aged P20R is reached sooner than that of aged P40. This may be interpreted as the net stress for which cavitation of rubber particles starts in the centre of the net section. This is then followed by extension of cavitation phenomenon towards the surface (notch root) but also by elongation of already formed voids. Displacement and reduction in diameter at failure are increased again (8mm and 45% for aged P20R). This is assumed to be a gain in ductility due to the addition of rubber particles.

Microscopic examinations

As mentioned above, some tests were arrested aiming at investigating deformation and damage mechanisms. Recall that PA11 showed voiding [START_REF] Boisot | Mechanisms and mechanical modelling of deformation, damage and fracture of neat and rubber-toughened PolyAmide 11[END_REF][START_REF] Boisot | The role of volume change in the ductile/brittle transition of PA11[END_REF] during deformation, due to the presence of pre-existing voids (initial void volume fraction = 1%). The tests were stopped just before the load dropping down, at the onset of the complete failure of the specimen. The specimen was then longitudinally cut in order to observe the microstructure in the centre of the specimen. Figure 4a clearly shows a macro-crack within the aged P40 material, whereas aged P20R (fig. 4b) exhibits a homogeneous distribution in voids. Note that voids come from cavitation of rubber particles. No coalescence of these voids was detected and the porosity seems to have not yet reached a critical value.

According to these observations, final rupture of the specimen is expected to occur in two different ways:

-For aged P40, shearing and/or tearing of the remaining ligament of material between the crack and the notch root; b) a)

σ net (MPa) σ net (MPa) (Φ 0 -Φ)/Φ 0 (Φ 0 -Φ)/Φ 0
-For aged P20R, inter-void fibril stretching induces their hardening. As soon as a critical defect initiates the break of one fibril, the crack propagates catastrophically. The materials of interest exhibit void growth during deformation. This evolution of porosity induces a volume change of the materials. The higher the stress triaxiality ratio, the higher the void growth and hence the higher the volume change and its effect on the mechanical response of the material. Indeed a volumetric deformation has to be accounted for in a model. To this end, constitutive relationships must include a link between the mean stress and the volume change (see for instance Drucker-Prager model), in addition to the classical von Mises deviatoric criterion. Boisot [START_REF] Boisot | Mechanisms and mechanical modelling of deformation, damage and fracture of neat and rubber-toughened PolyAmide 11[END_REF] suggested the use of mechanics of porous media assisted by finite element analysis [START_REF] Besson | Large scale object-oriented finite element code design[END_REF]. A modified porous model was used. It was able to successfully capture the evolution of porosity as well as the location of its maximum value. Furthermore, two local failure criteria were proposed depending on the stress triaxiality ratio: -a critical void volume fraction of about 50% for high triaxiality; -a critical plastic strain of about 1.1 for low triaxiality.

The abovementioned observations suggest that PA11 and aged P20R follow the critical plastic strain criterion, at the end of the neck extension. Indeed neck extension decreases the triaxiality down to 0.33 (value for an uniaxial tension specimen). Conversely, aged P40 obeys a critical porosity criterion that is likely to be reached just before the stress plateau in figure 3. Detailed study of the triaxiality, porosity and the failure criteria is proposed in [START_REF] Boisot | Mechanisms and mechanical modelling of deformation, damage and fracture of neat and rubber-toughened PolyAmide 11[END_REF].

Note that strain rate effects can also be accounted for with the mechanics of porous media model implemented in a FE code.

Toughness

In fracture mechanics concepts, the toughness is either the critical value at failure of the stress intensity factor in LEFM, or the value at crack initiation for the J-integral for EPFM. A given material toughness can then be determined by using any pre-cracked specimen recommended by fracture mechanics theory. The area under load versus crack opening displacement curve allows the determination of the toughness, until failure for LEFM and until the peak load for EPFM respectively. For polymeric materials, it is difficult to perform these tests for several reasons such as:

-the plane strain condition is not always fulfilled; some researchers use the essential work of fracture theory, valid under plane stress condition; -viscoelastic/viscoplastic deformation of the materials limits the use of no time dependent elasticity/plasticity constitutive relationships; -the porosity (hence the volume change) and the high strain (finite deformation) are not correctly taken into account.

In the present study, the notch has a finite root radius (4mm) and therefore cannot be considered as sharp crack. Investigations on PA11 round bars with various notch root radii [START_REF] Boisot | Mechanisms and mechanical modelling of deformation, damage and fracture of neat and rubber-toughened PolyAmide 11[END_REF][START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part 1 yield criteria and flow rules for porous ductile media[END_REF] were already perfomed leading to many interesting conclusions. This paper introduces another parameter that, as a first approximation, can be linked to fracture toughness.

Fracture energy concepts

From figure 3, and similarly to LEFM and EPFM approaches, the area under the net stress versus applied displacement curve is representative of the total energy needed by the specimen to fracture. Additionally, although it includes both (plastic) deformation energy and cracking energy, it is similar to what is measured in impact tests such as Charpy or Izod tests. The main difference is that impact tests are done under bending stress and with dynamic conditions. Under quasi-static conditions and when test temperatures are close to T g , impact tests might be inappropriate. Indeed if the material is tough enough, the energy of the striker may not be sufficient to break the specimen. Only large plastic strain of the specimen occurs in this case. For the three materials, areas under net stress vs. displacement curves were measured. Improvement of fracture toughness is demonstrated with the increase in fracture energy from aged P40 (160kJ/m 2 ) to aged P20R (300kJ/m 2 ). This is mainly due to the increase in strain at failure, which is similar to that of PA11. Indeed, experimental data on aged P40 and aged P20R are quite similar in terms of peak stress and Young's modulus. Furthermore, PA11 and aged P20R curves essentially differ in the values of peak stress. Although both materials are supposed to be tough, aged P20R fracture energy is only half of that of PA11. The toughness improvement can be predicted by the length of the stress plateau (ductility) in a stress vs displacement curve as depicted in figure 3. At microscopic scale, as mentioned previously, the origin of this improvement of toughness is the homogeneous distribution of rubber particles that prevents from void coalescence during the necking process.

CONCLUSIONS

An attempt was made to characterise the ageing, plasticizing and rubber toughening effects on three grades of PA11. Once the key physico-chemical parameters of each material were presented, the specific NT4 specimen geometry was detailed. Tensile tests on this NT4 specimen were carried out. Experimental results consist of the net stress and the reduction in diameter vs. applied displacement curves. Some tests were arrested before failure of the specimens. These latter were cut longitudinally in order to perform SEM examinations of the deformation and damage (void growth) mechanisms. It was shown that aging leaded to a reduced strain at failure (embrittlement) due to the formation of a macro-crack within the material. Conversely, rubber toughened aged P20R exhibited enhanced ductility because of homogeneously distributed voids that do not coalesce. Failure of the specimen occurs after a large plateau where the necking zone extension is observed. The fracture energy parameter, corresponding to the area under the net stress vs. displacement curve was able to assess the toughness improvement thanks to the addition of rubber particles. It could be concluded that from material processing viewpoint, the toughness improvement is likely due to the good spatial distribution of rubber particles that prevents from void coalescence during the necking process.

Figure 1 :

 1 Figure 1: DMA results on all PA11 materials: a) storage moduli; b) evolution of tan δ with temperature

Figure 2 :

 2 Figure 2: Sketch and characteristic dimensions of a notched round bar

Figure 3 :

 3 Figure 3: Net stress and diametrical reduction versus applied displacement for all PA11 materials: a) PA11 and aged P40; b) aged P40 and aged P20R

Figure 4 :

 4 Figure 4: SEM observations of longitudinal cross-sections after tests arrested at the onset of failure: a) aged P40 presence of a macro-crack ; b) aged P20R: well dispersed voids, critical void volume fraction not reached

Figure 5 :

 5 Figure 5: SEM observations of fracture surfaces: a) aged P40, diffuse ductile surface; b)aged P20R, localized crack initiation site after extended necking DISCUSSIONS

  Table 2 summarizes these fracture energies (last column) together with other previously mentioned characteristics, in particular, the peak stress and the displacement at break that interfere in the fracture energy calculation.Ductile and tough materials are known to have high values of fracture energy. Regarding the values in Table2, it is shown that the fracture energy of aged P40 (160kJ/m 2 ) is 4 times less than that of PA11 (620kJ/m 2 ). Aging resulted in embrittlement of the material by reducing the strain at failure and the peak stress. Adding 3% plasticizer content in aged P40 did not compensate this embrittlement. Note that the T g of aged P40 is 25°C lower than that of PA11, which theoretically would have enhanced the ductility of aged P40 (test temperature 0°C closer to the T g of aged P40). But the shape of E' evolution clearly shows (figure1) that at 0°C the Young's moduli are similar (typically 1700MPa).

	Materials	T g	E	Peak stress	Displ. at break	∆Φ/Φ 0 at break	Fracture energy
	PA11	50°C 1700MPa	62MPa	10mm	46%	620±30kJ/m 2
	Aged P40	25°C 1700MPa	48MPa	4mm	20%	160±10 kJ/m 2
	Aged P20R 30°C 1500MPa	46MPa	8mm	45%	300±15kJ/m 2

Table 2 :

 2 Values of fracture energy of PA11, aged P40 and aged P20R calculated via the net stress vs displacement curves


Keywords: modélisation et contrôle de systèmes biologiques, réseaux métaboliques, réseaux génétiques

Les réactions métaboliques sont régulées par des enzymes mais, en même temps, les gènes codant pour ces enzymes peuvent être régulées par les produits des réseaux métaboliques. Ce mécanisme de régulation en boucle fermée permet l'adaptation d'un système métabolique aux changements environnementaux. Nous étudions un système composé d'une chaine métabolique couplée à un réseau génétique, où un des métabolites joue le rôle de régulateur global de l'expression enzymatique. Le métabolite régulateur peut être soit répresseur soit activateur de chaque gène. Sous l'hypothèse que l'échelle de temps du système métabolique est plus rapide que celle du réseau génétique, le système composé peut être simplié. Dans ce cas, nous développons des critères géométriques pour caractériser les équilibres du système et analyser la monoou bi-stabilité du système en termes des paramètres.

Introduction

Metabolism and gene expression are two fundamental levels of cellular regulation. Gene expression can impact metabolic levels through changes in enzyme concentration and, conversely, metabolic species can inuence gene transcription and hence modulate enzyme synthesis [START_REF] Kotte | Bacterial adaptation through distributed sensing of metabolic uxes[END_REF]. These two levels have specic functions and properties, and it remains a challenge to characterize the interconnections between them. The congurations of metabolic-genetic interactions can lead to a diverse range of dynamic behaviors, each one of which denes a specic metabolic phenotype. Our understanding of natural regulatory circuits is important not only for revealing the design principles that underlie observed metabolic dynamics, but also for our ability to design synthetic circuits that enable new phenotypes [START_REF] Alon | An introduction to systems biology: design principles of biological circuits[END_REF].

A specic phenotype depends, among others, on: the regulatory topology (i.e. which metabolites regulate which enzymes), logic (i.e. activation or repression), and the sensitivity of regulation (i.e. graded or switch-like regulation). In this paper we present a mathematical analysis of these properties by studying the interconnection between an unbranched metabolic network and a one-to-all genetic control circuit. The gene circuit implements a form of global control in the sense that one metabolic species modulates the activity of all enzymes.

The model integrates the classical kinetic equations to represent metabolite dynamics and piecewise ane (PA) systems to represent gene regulation (Section 2). To analyze the model we consider that metabolic reactions happen in a much faster time scale when compared to gene transcription or translation [START_REF] Alon | An introduction to systems biology: design principles of biological circuits[END_REF]. PA models [START_REF] Casey | Piecewise-linear models of genetic regulatory networks: equilibria and their stability[END_REF] provide a convenient way of encoding switch-like regulation with relatively few parameters (only expression rates and regulatory thresholds), whereas the time scale separation allows for an approximation of the metabolic dynamics by a system of algebraic equations. This approximation ultimately leads to a reduction of the model to a 2-dimensional PA system dened in conic domains (Section 3). Solutions of these type of systems can be obtained by dening the system as a dierential inclusion and using a construction due to [START_REF] Aleksei | Dierential equations with discontinuous righthandsides[END_REF]. The conic geometry introduces new features in the solutions at the switching domains, and we characterize the existence and stability of the possible sliding motion and singular equilibria (Section 4). This analysis is applied to study a particular conguration of the globally regulated metabolic chain (Section 5).

Global genetic control

We consider an unbranched metabolic network with one metabolite as a global regulator of enzyme expression. A schematic diagram of such class of networks is shown in Figure 1, where s i denotes the concentration of the i th metabolite and v i is the rate of the i th reaction (catalyzed by an enzyme with concentration e i ).

As a way of accounting for the mass exchange between the network and its environment, we assume that the metabolic substrate s 0 is constant. For the sake of generality, in this paper we deal with networks of n metabolites and n + 1 enzymes regulated by a single global regulator s -1 ( > 1). The rate of change of both metabolite and enzyme concentrations can be described by

ṡi = v i (s i-1 , e i ) -v i+1 (s i , e i+1 ), (1)
ėi = κ 0 i + κ 1 i σ i (s -1 , θ i ) -γ i e i . (2)
where κ 0 i , κ 1 i , θ i , γ i are positive parameters. The metabolic model (1) arises from the mass balance between the reactions that produce and consume s i , whereas the model for the enzyme concentrations (2) comes from the balance between protein synthesis and degradation (modeled as a linear process). The constant κ 0 i represents a basal expression level of protein e i , whereas κ 1 i and the functions σ i model the eect of the global regulator on the synthesis rates.

The regulatory function σ i (s -1 , θ i ) represents the lumped eect of gene expression control by a transcription factor, together with its interaction with the regulator s -1 . To account for the typical switch-like nature of transcriptional regulation, σ i (x, θ) is assumed to be a step function; depending on whether gene expression is activated or repressed by s -1 , we assign σ i = σ + or σ i = σ -= 1 -σ + , respectively, with

σ + (x, θ) = 0, x < θ 1, x > θ . (3)
This class of regulatory functions is widely used used in the analysis of genetic networks [START_REF] Ropers | Qualitative simulation of the carbon starvation response in Escherichia coli[END_REF][START_REF] Chaves | Methods of robustness analysis for boolean models of gene control networks[END_REF] and was rst suggested in [START_REF] Glass | The logical analysis of continuous, nonlinear biochemical control networks[END_REF]. In the sequel we will not presuppose a specic form of the enzyme kinetics; instead to keep the analysis as general as possible, we make the following assumption on the enzyme kinetics.

Assumption 1. The metabolic reaction rates are linear in the enzyme concentrations and non-decreasing functions of the metabolite concentrations, so that the rate functions are written as

v i (s i-1 , e i ) = g i (s i-1 )e i , (4)
where g i is the enzyme turnover rate (i.e. the reaction rate per unit of enzyme concentration) and saties

∂g i (s i-1 ) ∂s i-1 ≥ 0. (5)
The monotonicity condition in (5) accounts for a broad class of saturable enzyme kinetics that includes, in particular, Michaelis-Menten and Hill kinetics [START_REF] Cornish-Bowden | Fundamentals of Enzyme Kinetics[END_REF]. In the rest of the paper we aim at characterizing the equilibria of the feedback system in (1) [START_REF] Casey | Piecewise-linear models of genetic regulatory networks: equilibria and their stability[END_REF]. Metabolic dynamics operate in a much shorter time scale than their genetic counterpart [START_REF] Alon | An introduction to systems biology: design principles of biological circuits[END_REF]. This property allows the approximation of the nonlinear dynamics in [START_REF] Alon | An introduction to systems biology: design principles of biological circuits[END_REF] by an algebraic relationship between the enzymes and metabolite concentrations. If the metabolites are assumed to be in quasi steady state (QSS) with respect to the enzyme concentrations, then we set ṡi (t) = 0 for all t ≥ 0 to obtain

g i+1 (s i (t)) = g i (s i-1 (t)) e i (t) e i+1 (t) , (6)
Equation ( 6) holds for every i = 1, 2, . . . , n and hence it is equivalent to

g i (s i-1 (t)) = g 1 (s 0 ) e 1 (t) e i (t) . ( 7
)
The trajectory of the global regulator can be then computed by solving the equation

g (s -1 (t)) = g 1 (s 0 ) e 1 (t) e (t) . (8)
If the function g i does not saturate, a nonnegative solution of ( 7) exists at all times. However, for saturable functions one must guarantee that this is true by stating appropriate assumptions on the parameters (for instance, of the form

κ 0 i /γ i ≥ (g 1 (s 0 )/ max(g i ))(κ 0 1 + κ 1 1 )/γ 1 )
. For space reasons we will assume that solutions of (7) do exist and omit the details. A key aspect of this approximation is that the solution of (1)(2) depends only on two proteins. The dynamics of the complete feedback system can thus be fully characterized by analyzing the 2-dimensional phase plane of the dierential equations

ė1 = κ 0 1 + κ 1 1 σ 1 (s -1 , θ 1 ) -γ 1 e 1 , ė = κ 0 + κ 1 σ (s -1 , θ ) -γ e , (9)
subject to s -1 satisfying (8).

Equivalent piecewise ane system in conic domains

The algebraic equation in ( 8) can be interpreted as a mapping from R ≥0 to R 2 ≥0 , whereby each value of the regulator s -1 maps into a half-line in the (e 1 , e ) plane. Moreover, as a consequence of the monotonicity of g , the partition of R ≥0 induced by the thresholds can be mapped into a partition of R 2 ≥0 : if

s -1 < θ i then g (s -1 ) < g (θ i ), (10)
which combined with (8) yields

e > β i e 1 , (11)

RR n°7468

with

β i = g 1 (s 0 )/g (θ i ).
The relation in (11) denes a cone in the x = (e 1 , e ) plane

D i = x ∈ R 2 ≥0 : x 2 > β i x 1 , (12)
and we dene its complementary cone as Di = R 2 ≥0 \ (D i ∪ S i ) with S i the half-line

S i = x ∈ R 2 ≥0 : x 2 = β i x 1 . (13)
The half-line S i is a subset of the (e 1 , e ) plane where the regulator reaches the switching threshold θ i . The dynamics of the reduced system in ( 9) depend on the value of s -1 with respect to the thresholds θ 1 and θ . Assume, without loss of generality, that θ 1 < θ (the problem can be treated analogously in the case θ 1 > θ , and the case θ 1 = θ is treated as in Section 4). With the previous denitions we can establish the following relations

s -1 < θ 1 ⇐⇒ x ∈ R 1 , θ 1 < s -1 < θ ⇐⇒ x ∈ R 1 , s -1 > θ ⇐⇒ x ∈ R , (14)
where

R 1 = D 1 , R 1 = D1 ∩ D and R = D .
In the sequel we refer to S i as a switching domain, whereas the cones R j are called regular domains (see [START_REF] Casey | Piecewise-linear models of genetic regulatory networks: equilibria and their stability[END_REF] for detailed denitions). The system in ( 9) is equivalent to a piecewise ane (PA) system [START_REF] Habets | A control problem for ane dynamical systems on a full-dimensional polytope[END_REF] in conic domains

ẋ = h(x) -Γx, (15)
where

h(x) =      h 1 x ∈ R 1 h 1 x ∈ R 1 h x ∈ R , Γ = γ 1 0 0 γ . ( 16
)
The vectors h 1 , h 1 and h are constant and their values depend on whether s -1 activates or represses the expression of enzymes e 1 and e . For example, in the case of repression (i.e. σ 1 = σ -and σ = σ -) we have

h 1 = κ 0 1 + κ 1 1 κ 0 + κ 1 , h 1 = κ 0 1 κ 0 + κ 1 , h = κ 0 1 κ 0 . ( 17
)
These vectors determine the location of the focal points of the PA system, dened as

φ 1 = Γ -1 h 1 , φ 1 = Γ -1 h 1 , φ = Γ -1 h . ( 18
)
The conic partition of the (e 1 , e ) phase plane and the focal points are shown in Figure 2. For any x(t 0 ) in a regular domain, e.g. x(t 0 ) ∈ R 1 , the right-hand side of (20) is well dened and its solution satises a standard ane dierential equation, that is

x(t) = φ 1 + e Γ(t0-t) x(t 0 ) -φ 1 , t ≥ t 0 (19)
so that x(t) monotonically approaches φ 1 , possibly reaching the switching domain S 1 , where the vector eld of (20) is not dened, and thus a specialized analysis is required. As we shall see in the next section, the location of the focal points plays a major role in the dynamics of (15).

INRIA e 1 φ 1 κ 0 1 γ1 e S 1 S φ 1 φ R 1 κ 0 +κ 1 γ R 1 κ 0 γ R κ 0 1 +κ 1 1 γ1
Figure 2: Conic partition of the state space and focal points when both e 1 and e are repressed by metabolite s -1 .

Piecewise ane systems in cones

To understand the dynamics of the PA system in (15), in this section we rst study the solutions of a 2-dimensional PA system dened only in two cones (as opposed to three cones, cf. Figure 2). Consider the half-line S = x ∈ R 2 ≥0 : x 2 = βx 1 with β > 0, and the PA system

ẋ = f (x) x ∈ D f g(x) x ∈ D g , (20)
where

D f = x ∈ R 2 ≥0 : x 2 > βx 1 and D g = R 2 ≥0 \ (S ∪ D f ).
We denote the generating vector of S as η = 1 β T ; the vector elds

f (x) = f 1 (x) f 2 (x) T and g(x) = f 1 (x) f 2 (x)
T are ane and given by

f (x) = h f -Γx, g(x) = h g -Γx, (21)
where h f and h g are entrywise positive vectors, and

Γ = diag {γ 1 , γ 2 } > 0 .
The focal points of f (x) and g(x) are given by φ f = Γ -1 h f and φ g = Γ -1 h g , respectively, and are assumed to satisfy φ f , φ g / ∈ S. Solutions of dierential equations with discontinuous vector elds are typically characterized with a construction due to Filippov [START_REF] Aleksei | Dierential equations with discontinuous righthandsides[END_REF]. This method proceeds by extending (20) to a dierential inclusion

ẋ ∈ H(x), ∀x ∈ S, (22)
where H(x) is a set-valued function dened as the closed convex hull of f (x) and g(x), i.e.

H(x) = z ∈ R 2 : z = αf (x) + (1 -α)g(x), 0 ≤ α ≤ 1 .
The solutions of (22) are understood in the following sense (see also [START_REF] Gouzé | A class of piecewise linear dierential equations arising in biological models[END_REF][START_REF] Casey | Piecewise-linear models of genetic regulatory networks: equilibria and their stability[END_REF] for more details).

Denition 4.1 or a given ρ 0 , a solution of ( 22) in [0, T ] is an absolutely con-

tinuous function ρ : [0, T ] → R 2 ≥0 such that ρ(0) = ρ 0 and ρ(t) ∈ H(ρ(t)) for almost all t ∈ [0, T ].

RR n°7468

Solutions in the switching domain

Depending on the directions of the vector elds f (x) and g(x), Filippov's construction may not allow for uniqueness of solutions in the switching domains [START_REF] Aleksei | Dierential equations with discontinuous righthandsides[END_REF]. When uniqueness can be guaranteed, then solutions of (22) can: (a) cross to a regular domain or, (b) slide along the switching surface S. Roughly speaking, case (a) occurs when f (x) and g(x) point in similar directions in a vicinity of S, so that the vectors in H(x) point toward a regular domain irrespective of α. In case (b) both vector elds point towards the switching domain, so that one can nd a unique value of α such that H(x) points in the direction of S (in this case we say that the solution exhibits stable sliding motion in S). Uniqueness of solutions is lost when both vector elds point away from the switching domain, in which case solutions starting at S cannot be uniquely dened and any small perturbation will drive x away from S (referred to as an unstable sliding motion).

Next we identify the above scenarios for the PA system dened in (20). To that end we dene the sets

Ω - f = {x ∈ D f ∪ S : f 2 (x) -βf 1 (x) ≤ 0} , Ω + f = {x ∈ D f ∪ S : f 2 (x) -βf 1 (x) > 0} , Ω - g = {x ∈ D g ∪ S : g 2 (x) -βg 1 (x) ≤ 0} , Ω + g = {x ∈ D g ∪ S : g 2 (x) -βg 1 (x) > 0} . (23)
Lemma 4.1 [Crossings between regular domains] The solutions of (20) cross from D g to D f in the set

L gf = Ω + f ∩ Ω + g ⊆ S, (24)
and cross from D f to D g in the set

L f g = Ω - f ∩ Ω - g ⊆ S, (25)
Proof: We only prove the rst case (i.e. that if the solution reaches L gf dened in (24), then it crosses from D g to D f ); the converse case follows analogously. From Filippov's construction, the vector eld in S has the form

ẋ = αf 1 (x) + (1 -α)g 1 (x) αf 2 (x) + (1 -α)g 2 (x) , (26)
with 0 ≤ α ≤ 1. By the denition of Ω + f and Ω + g , for any point x ∈ L gf we have that

αf 2 (x) + (1 -α)g 2 (x) > αβf 1 (x) + (1 -α)βg 1 (x), (27)
> β (αf 1 (x) + (1 -α)g 1 (x)) ,
which implies that the vector eld points to D f for any 0 ≤ α ≤ 1. In other words, the set H(x) in ( 22) is fully contained in the regular domain D f and hence the trajectory crosses the switching domain. 2

Lemma 4.2 [Stable sliding motion]

The solutions of (20) exhibit stable sliding motion in the set

L s = Ω - f ∩ Ω + g ⊆ S. (28)

INRIA

Geometric representation

The results of last section dene a partition of the switching domain into regions for crossings and sliding motions that has a convenient geometric representation. An alternative denition of the sets in (23) can be constructed as follows: dene the normal vector of S as

η ⊥ = -β 1 T , then x ∈ Ω - f if and only if f (x), η ⊥ ≤ 0, which after substitution of f (x) = h f -Γx becomes Γ(x -φ f ), η ⊥ ≥ 0. ( 36
)
Analogous denitions can be constructed for the other sets in (23). The boundary between Ω - f and Ω + f is the half-line

C f = x ∈ R 2 ≥0 : Γ(x -φ f ), η ⊥ = 0 , (37)
whereas the boundary between Ω - g and Ω + g is

C g = x ∈ R 2 ≥0 : Γ(x -φ g ), η ⊥ = 0 . ( 38
)
The focal points satisfy φ f ∈ C f and φ g ∈ C g , respectively. Moreover, C f and C g are parallel and thus the sets L s and L s correspond to the intersection between S and the band generated by C f and C g . Figure 3 illustrates this idea; a necessary condition for sliding motion (stable or unstable) is that the band between C f and C g intersects S in R 2 ≥0 . Moreover, inspection of Figure 3 suggests that the partition precludes the existence of stable and unstable sliding motion in the same switching domain (i.e. at least one of the sets L s and L s is empty). Next we provide a geometric condition for characterizing these scenarios. Lemma 4.4 Let

ϑ 1 = ∠ Γ φ f -φ g , η ⊥ . If ϑ 1 ∈ -π, - π 2 ∪ π 2 , π , (39)
unstable sliding motion cannot exist in S. Conversely, if

ϑ 1 ∈ - π 2 , π 2 , ( 40
)
stable sliding motion cannot exist in S.

Proof: C f and C g intercept the vertical axis of R 2 3), which leads to the angle conditions in (39)(40). 2

≥0 at p f = γ -1 2 Γφ f , η ⊥ and p g = γ -1 2 Γφ g , η ⊥ , respectively. Whether L s = ∅ or L s = ∅ depends on sgn p f -p g = sgn Γ φ f -φ g , η ⊥ (see Figure

Equilibria

The focal points are locally stable equilibria of (20) provided that they belong to their respective cone, i.e. φ f ∈ D f or φ g ∈ D g (stability follows from γ 1 , γ 2 > 0). However, as suggested by Figure 3, it is possible that the trajectories reach an equilibrium that lies in the switching domain. This kind of equilibrium is sometimes termed singular equilibrium [START_REF] Casey | Piecewise-linear models of genetic regulatory networks: equilibria and their stability[END_REF] and must be understood in Filippov's sense (that is, at a singular equilibrium the convex hull H(x) in ( 22) contains the origin). Depending on the location of the focal points, a singular equilibrium may exist in L s ; we characterize this statement more precisely in the next lemma.

INRIA x1 φ f Ω - f Ω + g Cg S Ω + f φ g C f Ω - g Ls x 2 Ls x 2 S x1 C f φ g φ f Ω + f Ω - f Cg Ω + g Ω - g
Lemma 4.5 [Singular equilibrium] Assume that L s = ∅; let L φ be the line containing φ f and φ g and ϑ 2 = ∠ φ f -φ g , η ⊥ . The point

φ s = L s ∩ L φ , (41)
is a singular equilibrium of (20). Moreover, if

ϑ 2 ∈ -π, - π 2 ∪ π 2 , π , (42)
then φ s is locally stable, and if

ϑ 2 ∈ - π 2 , π 2 , (43)
then φ s is unstable.

Proof: The proof follows by looking at the form of the vector eld along S when solutions are dened with Filippov's method. When x ∈ L s the solution satises

ẋ = αf (x) + (1 -α)g(x), (44)
with α = α(x) given in (34). Substituting α(x) in (44) we get

ẋ = A f g (x)η (g 2 (x) -βg 1 (x)) -(f 2 (x) -βf 1 (x)) , (45)
where A f g (x) is given by

A f g (x) = γ 1 γ 2 x T P φ f -φ g + φ f T P φ g , (46)
with P = 0 1 -1 0 so that x T P x = 0 for all x ∈ R 2 . A point φ s ∈ L s is a singular equilibrium of (20) if it satises A f g (φ s ) = 0. The equation A f g (x) = 0 is satised by both focal points, i.e. A f g φ f = A f g (φ g ) = 0, and so the curve

L φ = x ∈ R 2 ≥0 : A f g (x) = 0 , (47) RR n°7468
is the line containing both focal points. We thus conclude that any singular equilibrium must be located at φ s = L φ ∩ L s . The stability of φ s follows by examining the direction of the vector eld in (45). We know that

(g 2 (x) -βg 1 (x)) -(f 2 (x) -βf 1 (x)) > 0, (48)
for all x ∈ L s , and hence the direction of the right-hand side of (45) depends only on the sign of A f g (x) along L s . The function A f g (x) evaluated along L s (i.e. when x = x 1 • η) denes a line

A f g (x)| x∈Ls = γ 1 γ 2 φ f -φ g , η ⊥ x 1 + φ f T P φ g , (49)
with slope

∂ ∂x 1 A f g (x)| x∈Ls = γ 1 γ 2 φ f -φ g , η ⊥ . ( 50
)
Note that the line in ( 49) is transversal to the line L φ and they intersect at φ s (because A f g (φ s ) = 0 and φ s ∈ L s ). Therefore A f g (x) changes sign at x = φ s , so the local stability of φ s depends on the sign of the slope in (50); namely

φ s is stable if φ f -φ g , η ⊥ < 0 unstable if φ f -φ g , η ⊥ > 0 , (51)
which are equivalent to the angle conditions in (42)(43) (note that φ fφ g , η ⊥ = 0 since φ f , φ g / ∈ S by assumption). 2 The result in Lemma 4.5 provides a geometric condition to check the existence of singular equilibrium points for the PA system (20). These equilibria can be locally stable or unstable, a property that can be graphically checked with a simple angle condition; the stable and unstable cases are illustrated in Figure 4(a) and Figure 4(b), respectively. The PA system can also lack a singular equilibrium, in which case L s ∩ L φ = ∅ and the solutions may exhibit sliding motion in L s but eventually escape to one of the regular domains; this scenario is shown in Figure 4(c).

(a)

φ f φ g φs η ⊥ Ls Cg ϑ2 S C f (b) φ g C f S Ls φs φ f Cg η ⊥ ϑ2 (c) φ g Cg φ f C f S Ls
Figure 4: Singular equilibrium for the PA system in (20).

INRIA 5 Equilibria of genetic control circuit

In this section we return to the original problem of determining the metabolite and enzyme equilibria of the system in (1)(2), after its reduction to a PA system dened in three conic domains, see (15). If θ 1 = θ , then the region R 1 is empty, and the equilibria can be obtained directly with the analysis in Section 4. In the case θ 1 = θ , the idea is to use Lemmas 4.4 and 4.5 by splitting the three cone case into a pair of two cone problems: one for the switching domain S 1 with the focal points φ 1 and φ 1 , and another one for S with the focal points φ 1 and φ . As discussed in Section 3.2, the location of the focal points depends on the regulatory logic in [START_REF] Kotte | Bacterial adaptation through distributed sensing of metabolic uxes[END_REF], which is comprised in the functions σ 1 and σ . If θ 1 = θ and regardless of the particular regulatory logic, one pair of focal points will share the same vertical coordinate and another pair will share the horizontal coordinate. We can then use our results to show that the system can have one or two stable equilibria. Due to space constraints we only focus in the case of Figure 2, with θ 1 < θ . Lemma 5.1 [Equilibrium of genetic circuit] Consider the PA system in (15) with σ 1 = σ -and σ = σ -, θ 1 < θ , and the focal points located as in Figure 2. Then, (15) has only one equilibrium point φ 1 s , which is singular, locally stable, and located at

φ 1 s = L φ ∩ S 1 , (52)
where L φ is the line containing φ 1 and φ 1 .

Proof: In Figure 2 we see that the focal points lie outside their corresponding regular domains, so they cannot be equilibria of (15). The only option then is to look for singular equilibria in S 1 or S . Denote the normal vectors to S 1 and S as η ⊥ 1 and η ⊥ , respectively. We proceed by cases: φ f = φ 1 and φ g = φ are aligned vertically, and hence

- π 2 ≤ ∠ Γ φ 1 -φ , η ⊥ ≤ 0, (53)
for all γ 1 , γ 2 > 0; from Lemma 4.4, (53) implies that S cannot contain any stable sliding motion.

φ f = φ 1 and φ g = φ 1 are aligned horizontally, and thus we have

-π ≤ ∠ Γ φ 1 -φ 1 , η ⊥ 1 ≤ - π 2 , (54)
for all γ 1 , γ 2 > 0; from Lemma 4.4 (54) implies that S 1 can contain a region for stable sliding motion; the existence, location and stability of a singular equilibrium in S 1 follow directly from Lemma 4.5 and inspection of Figure 2. 2

The equilibrium enzyme concentrations for the feedback system in (1)(2) can be directly obtained with Lemma 5.1. Assuming that θ 1 < θ and given RR n°7468 that φ 1 s ∈ S 1 is a singular equilibrium, we have s -1 = θ 1 and hence (see Figure 2)

ē1 = 1 β 1 ē , ē = κ 0 + κ 1 γ . (55)
The remaining equilibrium enzymes can be computed from (2) as

ēi = κ 0 i + κ 1 i σ i (θ 1 , θ i ) γ i , i = 1, (56)
where the threshold are assumed to satisfy θ i = θ 1 for i = 1. The equilibrium metabolites can be computed by solving the algebraic equation (cf. ( 7))

g i (s i-1 ) = g 1 (s 0 ) ē1 ēi , i > 1, i = . (57)
As an illustrative example, in Figure 5 we plot the (e 1 , e ) phase plane and the metabolite trajectories for the system in (1)(2), together with its QSS approximation in [START_REF] Kotte | Bacterial adaptation through distributed sensing of metabolic uxes[END_REF]. The simulations in Figure 5 were obtained for a system of length n = 2 with σ

1 = σ -, σ = σ -, substrate s 0 = 1, Michaelis-Menten kinetics (g i = k cat i s i-1 /(K m i + s i-1 ), with k cat i = 2 • 10 2 and K m i = 5, i = 1, 2 , 3)
, regulatory parameters θ i = {0.5, 1, 1.5}, κ 0 i = {0.08, 0.15, 0.3}, κ 1 i = 10κ 0 i , and degradation rates γ i = {0.25, 0.9, 0.8}.

Discussion & outlook

The analysis carried out in this paper shows how genetic regulation of enzyme activity can generate dierent metabolic equilibria. This mechanism can control metabolic adaptations to environmental changes and is based on the interaction between metabolic species and transcription factors which modulate enzyme biosynthesis.

Key elements in our analysis are the use of a PA model for gene regulation and exploiting the time scale separation between metabolic and genetic dynamics. Applying a quasi steady state approximation ultimately permits to study the n-species gene circuit by a 2-dimensional PA system dened in conic domains. The algebraic equations in [START_REF] Gouzé | A class of piecewise linear dierential equations arising in biological models[END_REF] will be a good approximation of the metabolite trajectories provided that the time constants of the enzyme kinetics are much smaller than protein half-lifes, which in turn are inversely proportional to their degradation rates γ i . In addition, enzyme saturation may invalidate the INRIA QSS approximation when a positive solution of (7) fails to exist. This issue has been briey mentioned, and will be addressed in an upcoming paper.

The analysis of the reduced PA system with Filippov's method [START_REF] Aleksei | Dierential equations with discontinuous righthandsides[END_REF] allows the derivation of geometric conditions on the protein expression and degradation rates to identify the behavior of solutions at the switching domain. We have obtained two angle conditions to check the existence of stable or unstable sliding motion (Lemma 4.4), and the existence of singular equilibria at the switching domains (Lemma 4.5). By combining these two results we can characterize the equilibria for a given combination of gene regulatory parameters. Because of length constraints we have limited the analysis to the setup of Lemma 5.1 (see Figure 2), but the procedure can be applied to any combination of activation or repression feedback loops.

3 Figure 1 :

 31 Figure 1: Global gene regulation for an unbranched network.

INRIA 3 scale separation 3 . 1

 331 Time Quasi steady state approximation

Figure 3 :

 3 Figure 3: Switching domain with stable (left) and unstable (right) sliding motion.

2 Figure 5 :

 25 Figure 5: Numerical example; eect of QSS approximation.

RR n°7468

http://www.inria.fr ISSN 0249-6399

Proof: We rst prove by contradiction that for all x(t 0 ) ∈ L s the vector elds are such that x(t) cannot leave the switching domain in an interval (t 0 , t 0 + ∆ ] . Dene the absolutely continuous function

Suppose that there exists ∆ > 0 such that z(t) > 0 for t ∈ ( t 0 , t 0 + ∆ ]. If x(t 0 ) ∈ S we have that z(t 0 ) = 0, so by continuity it must be that ż > 0 for t ∈ ( t 0 , t 0 + ∆ 0 ] and some 0 < ∆ 0 ≤ ∆. In addition, from the denition of the PA system in (20), if z(t) > 0 then ẋ = f (x) for t ∈ ( t 0 , t 0 + ∆ 0 ] and so

However, the right-hand side of (30) is continuous in t, and when x(t 0 ) ∈ L s it follows that ż ≤ 0 for t ∈ ( t 0 , t 0 + ∆ 0 ], which is a contradiction. The converse argument can be used to show that z(t) < 0 for t ∈ ( t 0 , t 0 + ∆ ] leads to a contradiction. We thus conclude that z(t) = 0 for t ∈ [t 0 , t 0 + ∆], and so

The proof follows by checking that the vector elds for x ∈ L s are compatible with Filippov's construction, see [START_REF] Gouzé | A class of piecewise linear dierential equations arising in biological models[END_REF]. If there is sliding motion in L s , then there exists ∆ > 0 such that

Since x(t) must be a solution in Filippov's sense for t ∈ [t 0 , t 0 + ∆], then there must exist 0 ≤ α ≤ 1 such that

Combining ( 31) and (32) we get

Solving for α in (33) gives

For x ∈ L s it holds that (f 2 -βf 1 ) ≤ 0 and (g 2 -βg 1 ) > 0, therefore α(x) is unique for all x ∈ L s and satises 0 ≤ α(x) ≤ 1. 2

Lemma 4.3 [Unstable sliding motion] The solutions of (20) cannot be uniquely dened in the set

Proof: Consider the function z(t) dened in (29). As opposed to the proof of Lemma 4.2, in this case it can be shown that for x(t 0 ) ∈ L s both z(t) > 0 and z(t) < 0 for t ∈ ( t 0 , t 0 + ∆ ] are possible solutions. Note that another possible solution can be dened by picking α as in (34) so that x(t) slides along S. 2

RR n°7468


Maryline Chetto
email: maryline.chetto@univ-nantes.fr

Hui Zhang
email: hui.zhang@univ-nantes.fr

Performance Evaluation of Real-Time Scheduling Heuristics for Energy Harvesting Systems

Energy constrained systems can increase their usable lifetimes by extracting energy from their environment. This is known as energy harvesting. This paper investigates scheduling issues in uni-processor real time embedded systems using regenerative energy. Task scheduling should account for the properties of the regenerative energy source which fluctuates, capacity of the energy storage as well as deadlines of the time critical tasks that characterize most of real time embedded systems. In this context, designing efficient scheduling strategies is significantly more complex compared to conventional real-time scheduling. In this paper we compare several scheduling heuristics with the optimal algorithm known as LSA (Lazy Scheduling Algorithm). We report results of an experiment study in terms of percentage of deadlines satisfied.

Introduction

Real-time embedded systems play a considerable role in our society, and they cover a spectrum from the very simple to the very complex. Examples of current systems include the control of domestic appliances like washing machines and televisions, the control of automobile engines, telecommunication systems, military command and control systems, industrial process control, flight control systems, and space shuttle and aircraft avionics... For example, a system that monitors temperature in a nuclear power plant would require that the readings be reported to a base station within enough time for a proper response to be made to a rapid increase in the temperature.

Harvesting energy in surrounding environment to power embedded systems for the lifespan appears nowadays to be the alternative to conventional batteries. Harvesting systems constructed to date extract power efficiently from the source. However, they do not use it adequately under real-time running conditions. As a result, they need a much larger harvester (e.g. solar panel) than necessary to yield the same level of power as a more efficient one, or they rely on a larger, more expensive, higher capacity battery than needed in order to sustain extended operation.

In this new context, the main problem to solve comes from the instantaneous power level that tends to vary over a wide. The autonomous nature of operation makes it imperative that the system learns its own energy environment and adapts its power consumption accordingly. Goal of this adaptation is to maximize the utility of the application in a long-term perspective. The resulting mode of operation is sometimes called energy neutral operation.

Then, the crucial issue is to find scheduling mechanisms that can adapt the performance to the available energy profile. Up to now, when designing a real-time embedded system, the first concern has been usually time, leaving energy efficiency as a hopeful consequence of empiric decisions. Now, the primary concern is that power from solar panels or other free sources that cannot be stored (or stored with limited capacity) should be fully consumed greedily, or else this energy will be wasted.

In a real-time environment where tasks have to meet deadlines and execute periodically, energy harvesting and task scheduling are strongly dependent since they have to jointly handle timing constraints and variability of available energy.

In that paper, we propose five scheduling heuristics, easy to implement with limited overhead. Our objective is to compare their performance to an optimal scheduler, namely LSA with high overhead. The main question is to appreciate precisely the relative performance of all these schedulers thanks to simulation. All of them are based on the famous Earliest Deadline First rule. We report results of an experiment study in terms of percentage of satisfied deadlines and wasted energy. The quantitative analysis is achieved without considering computation overheads.

The remainder of this paper is organized as follows. In section 2, we describe the main issues in energy harvesting and briefly describe projects that involve energy harvesting. Section 3 gives background materials around real-time scheduling and precisely describes the optimal uni-processor scheduler, LSA, under energy harvesting assumptions. In section 4, we present five scheduling heuristics and we propose to compare them to LSA. Section 5 reports results of a simulation study and enables us to bring to light that some heuristics may have similar performance to LSA without incurring high overhead. We conclude in section 6 with a brief discussion on ongoing research.

Energy harvesting 1 Motivations

With the multitude of mobile devices that are used in all areas of social and commercial life it is increasingly important to design systems in an energy-efficient manner. These autonomous systems (including sensoractuator networks) are being envisioned to carry out complex task sets under real-time requirements without human intervention. However, they require power in order to operate, and if power outages occur, critical data may be lost. The true autonomy of such systems depends on their reliable and guaranteed operation for extended times without maintenance.

Most prior wireless monitoring systems in last decades have relied on continuous power supplied by batteries such as lithium-ion cells. Their disadvantage is that they become depleted, must be periodically replaced or recharged and consequently place hard restrictions on products' usability, lifetime, and cost of ownership. Moreover, while processing power roughly doubles ever two years, battery technology advances at a much more sluggish pace (battery capacity has doubled every 10 years). In addition to the very slow growth in their energy capacity, traditional batteries have a limit to the total practical energy density they can provide.

Even if it is possible to increase their energy density by tenfold within a few years, we must still consider practical safety concerns. First, given improper use, batteries with extremely high energy densities can become dangerous, explosive devices. Second, in many embedded applications, battery replacement is impractical or has high labour costs associated with maintenance.

Besides, batteries suffer from self-discharge, temperature, and other environmental effects that work to bound their usable lifetime, even in the case of zero use. Consequently for long-term, economical deployment, embedded systems must gather energy from the environment around it, a technique known as energy harvesting or energy scavenging.

It concerns as well the high technology sectors as the general public products in which wireless embedded systems are used in a variety of applications, such as environmental applications (forest fire and flood detection, monitoring of drinking water and level air pollution), military applications (battlefield surveillance, reconnaissance of enemy forces), health applications (tele-monitoring of human physiological data, tracking and monitoring of doctors and patients), home applications (intrusion detection), or commercial applications (monitoring of product quality, climate control in large buildings). While some of these applications are marginal today, they will become commonplace one day. Devices with maintenance-free life of hundreds of years can now be envisaged if we provide them with efficient strategies for harvesting, storing and managing environmental energy. The current perspectives of this market are thus very big and promising.

History

A number of projects have used energy harvesting technologies to deliver sustainable power for autonomous sensors. Photovoltaic energy harvesting is by far the most prevalent form of technology used in current projects in part due to the plentiful supply of light in many deployment settings, and the low cost of photovoltaic modules. Nodes conventionally store electrical energy in super-capacitors or batteries to achieve operation. Heliomote has a solar panel and two AA type Ni MH batteries [START_REF] Raghunathan | Design considerations for solar energy harvesting wireless embedded systems[END_REF]. The solar panel is directly connected to its battery through a diode. Even though ample power may be available on the solar panel, a wireless sensor node can still draw current from the battery. Prometheus has a super-capacitor as a primary buffer, a Li-Polymer battery, and a solar panel [START_REF] Jiang | Perpetual environmentally powered sensor networks[END_REF]. The solar panel first charges the supercapacitor, from which the system draws current when enough power is available on the solar panel. The system draws current from the battery only when the charge level of the primary buffer is less than a certain threshold, and it seldom draws power from the battery.

Heliomote and Prometheus have permitted to show that systems may operate perpetually through scavenging solar energy. However, the common drawback of these first prototypes is that they do not target at real-time and quality of service requirements that characterize most of embedded applications.

Several prototype systems incorporating vibration energy harvesting have been developed too. For example, the S5NAP uses a commercially-available electromagnetic vibration energy harvester to power an accelerometer based condition monitoring system. In this system, energy harvested from vibrations is buffered in super-capacitors to permit nodes to draw large bursts of power during radio transmissions and sensing operations [START_REF] George | Evaluation of a vibration-powered, wireless temperature sensor for health monitoring[END_REF].

Another project named ShiMmer uses piezoelectric transducers to evaluate a portion of a structure (i.e. a bridge) to determine if damage exists. It relies on a wireless platform that combines active sensing and localized processing with energy harvesting to provide long-lived structural health monitoring. One of the objectives of ShiMmer project is to create a robust and flexible software controller that can manage both the energy and the task execution [START_REF] Dondi | SHiMmer: a Wireless Harvesting Embedded System for Active Ultrasonic Structural Health Monitoring[END_REF].

Scheduling with energy constraints

Background materials

Most of embedded applications require periodic activities that have to be cyclically executed at fixed rates and within special deadlines. Typically, each periodic instance is assigned a relative deadline equal to the task period and is treated as a hard job. Thus, a periodic task is executed only if all its instances are guaranteed to complete within their deadlines. Schedulability analysis of periodic task sets can easily be performed both under fixed and dynamic priority assignments. In particular, a lot of work has been done for the Rate Monotonic (RM) and the Earliest Deadline First (EDF) algorithms [START_REF] Liu | Scheduling Algorithms for Multiprogramming in Hard Real-time Environment[END_REF]. Schedulability analysis has also been extended for the case in which tasks use shared resources or run in the presence of aperiodic activities, under fixed priority scheduling and in dynamic priority systems as well [START_REF] Chetto | Some Results of the Earliest Deadline Scheduling Algorithm[END_REF] [START_REF] Liu | Real-Time Systems[END_REF].

While EDF (dynamic priority depending on urgency) and RM (fixed priority depending on period) can support sophisticated task set characteristics such as deadlines, precedence constraints, shared resources, jitter, etc., they are all open loop scheduling algorithms. Open loop refers to the fact that once schedules are created they are not "adjusted" based on continuous feedback. Systems with open-loop schedulers are usually designed based on worst-case parameters. Such an approach can result in a highly underutilized system based on extremely pessimistic estimation of workload (or energy). While open-loop scheduling algorithms can perform well when the workload and the processing performance are accurately modelled, they perform poorly in unpredictable dynamic systems including regenerative energy dependent ones.

Only in the past decade, researchers started to address power and scheduling issues with the objective of either minimizing power usage under timing constraints or maximizing the system performance under the energy constraints. Nevertheless, they did not consider the rechargeability of the batteries. For example, EDF and RM scheduling have been extended to variable-voltage processors. The idea is to save power by slowing down the processor just enough to meet the deadlines [START_REF] Quan | Designing Embedded Processors, A Low Power Perspective[END_REF]. But solely applying these techniques has limitations in energy harvesting systems because they minimize CPU power, rather than they dynamically manage power according to the profiles of both available energy and processor workload.

The performance of a practical energy harvesting real-time system is measured by the deadline miss rate and heavily depends upon the stored energy and the energy harvested from the environment. Unfortunately, the scavenging power is time-varying and thus very unstable. Therefore, the accurate modelling for energy source plays a key role in designing a good policy to schedule the tasks and reduce the deadline miss rate.

An optimal scheduling algorithm

The first work that really makes adaptive power management for energy harvesting systems with real time constraints has been published in [START_REF] Moser | Real-time scheduling with regenerative energy[END_REF]. There, C. Moser et al. propose a real-time scheduling algorithm, called Lazy Scheduling Algorithm (LSA) that uses task postponement. Algorithm LSA is energyclairvoyant, i.e., the generated energy in the future is known. Taking into account available time as well as processable energy, an optimal task ordering can be determined based on the prediction of the available energy in the future.

This work deals with a mono-processor architecture that draws the energy from storage and uses it to process tasks (periodic or non periodic) with arrival time, deadline, and worst case execution time. The worst case execution time represents the maximum energy demand of the task. The arrival time of the task is not known beforehand. The deadline as well as the worst case execution time of the task is unknown before it is released. However, as long as the task is released, all these parameters are determined. They assume that tasks are preemptable and execute according to the earliest deadline first policy.

At any time, the energy source module harvests the energy from its ambient environment and then converts it into electrical energy. The electrical energy can be stored in the energy storage (battery), whose capacity is precisely known. The stored energy is assumed to be known at the system level at any time and is no more than the storage capacity. It is assumed that the energy storage is ideal and the battery can be recharged up to its capacity. Likewise, it can also be completely discharged to as less as zero. If the stored energy reaches the capacity, the incoming harvested energy overflows the storage and is discarded.

According to LSA, the processor executes all tasks at full power when the battery is full time, and the system starts executing a task if the task is ready and has the earliest deadline among all ready tasks and the system is able to keep on running at the maximum power until the deadline of the task.

Contrary to greedy scheduling algorithms, LSA hesitates to power tasks until it is necessary to respect timing constraints. In that sense, tasks are executed neither as soon as possible nor as late as possible. In this paper, the authors also discuss an admittance test that decides, whether a set of real-time tasks can be scheduled without violating deadlines. Another crucial question which has been solved is how to dimension the capacity of the battery that ensures continuous operation. The simulation study demonstrates that achievable capacity savings between 20% and 45% are obtained comparing the classical Earliest Deadline First algorithm. However, all theses measurements ignore on line computational costs.

While optimal in the case of a single speed processor, LSA algorithm has the following drawbacks:

• The consumption power of the task is assumed to be characterized by some value. This implies that for every task, its total energy consumption is directly connected to its execution time through the constant power of the processing device. However, in practice, the total energy which can be consumed by a task has no correlation with the worst case execution time.

• Renewable energy sources must be accurately modelled, otherwise the performance of LSA will be degraded.

• Scheduler LSA requires a lot of mathematical computations to be performed on-line. So, in practice, we have to consider its computational overhead, i.e. the cost of its operation both in terms of time and energy consumption.

Description of scheduling heuristics

To evaluate the effectiveness of the LSA algorithm on energy saving and performance improvement, we developed a discrete-event simulation and compared LSA to several scheduling heuristics, all using the simple and easy to implement earliest deadline rule:

• Heuristic 1: EDt. Before starting the execution of the highest priority task which is ready, a test is performed to compare the energy level of the battery to the total energy required by the task for its execution. If the energy available in battery is sufficient, the task is authorized to execute. Otherwise, the processor is put into sleep mode until the battery contains enough energy to run it. According to this scheduler, we never start execution of a task if there is no sufficient energy to execute it totally.

• Heuristic 2: EDi. All the tasks are processed as soon as possible according to EDF until the battery is empty. Then, the processor is put into sleep mode until the next release date i.e. the next instant corresponding to the arrival of another ready task. During that period, the battery will replenish and necessarily, the processor will be active at that instant for executing the highest priority task.

• Heuristic 3: EDd. As previously, all tasks execute as soon as possible according to EDF until there is no more energy available in the battery. Then, all ready tasks are discarded and the processor is put into sleep mode until the next release time.

• Heuristic 4: EDu. Compared to EDi, EDu is similar but lets the processor in sleep mode just during one time unit whenever the battery is empty.

• Heuristic 5: EDc. Compared to EDd, EDc is similar but just deletes the current active task instead of discarding all the tasks waiting for execution. The processor is put into sleep mode until the arrival of a new task even if the list of ready tasks may not be empty.

Edi, EDd, EDu and EDc execute tasks as soon as possible i.e. as long as the battery contains at least one unit of energy. These are typical greedy scheduling strategies which mainly differ each other in the way of managing energy lack situations. In contrast, EDt will test dynamically energy availability before running tasks and does not greedily consumes energy. Let us note that for every heuristic, as soon as a deadline is missed, the corresponding task is aborted for immediately stopping the overload situation and limiting wasted energy.

Experiments

Description of the simulator

In order to evaluate and compare the performance and the effectiveness of the scheduling heuristics to the optimal algorithm, we developed a discrete-event simulation in C/C++. The simulator has been designed specifically for any periodic task set under energy constraints. By using it, we can report details of the schedule produced for any task set with given energy storage capacity and energy source profile.

In our study, we consider the scheduling problem of periodic tasks. We assume that deadlines are less than or equal to periods and greater than or equal to computation times. So we use a task generator to produce 30 periodic task sets, each consisting of 6 tasks with a least common multiple of the periods (often called hyper-period) equal to 300 time units.

The rechargeable power is variable with time in practice. But in this paper, we report results when the source power is constant along time, equal to 6. In the first experiment, the battery capacity is 10. In other terms, 10/6 units of time are required to replenish the energy storage from the environmental power source.

Simulations have been processed over 5 hyperperiods with a processor which is characterized by its maximum consumption power, here equal to 8 (watts). In any case, we assume that the energy storage is fully charged at the beginning of the simulation.

Experiment 1: Varying the processing load

Figure 1 presents the ratios of satisfied deadlines for the heuristics and the optimal algorithm. It clearly shows that the performances of all algorithms degrade when the processor utilization factor increases. Figure 1 naturally shows that optimal LSA outperforms all other policies. And LSA has the softest degradation, and EDd has the strongest one. Velocities of degradations for the other heuristics are intermediate.

For 60% processor utilization, LSA permits to satisfy 90% task deadlines while EDd, the worst one satisfies about 40% task deadlines. And for 100% processor utilization, LSA succeeds in satisfying about 60% task deadlines while the performance of EDd drops to less than 10%. From a general view, higher is the processor utilization, higher is the gain of LSA compared to heuristics, given a power source profile and a battery capacity. For every processor load, EDd provide the worst performance. Nevertherless, performance improvement of LSA over EDi and EDu is always less than 20% and less than 10% for very highly loaded systems.

Experiment 2: Varying the battery capacity

We choose three different values for U, 0.2, 0.5 and 0.8 respectively representing low, medium and high system load. And for each load and a given source power, we make vary the capacity of the battery. The objective of such study amounts to determine optimal dimensioning of the battery for each scheduler and a given tolerance expressed in terms of missed deadlines.

Figure 2 (a) is relative to low processing load (U=0.2) and consequently to low energy requirement. When the battery capacity is less than 12, difference of performance between LSA and EDi is very large and as the battery capacity increases, the gaps are getting smaller. For a battery capacity equal to 3, 100% deadlines are satisfied under LSA, Edi and EDu while less than 10% respectively 30% under EDd respectively EDt and EDc. When the battery capacity is larger than 18, all schedulers perform quasi identically. When the battery capacity is greater than 27, all schedulers achieve exactly 100% satisfied deadlines. This simulation result clearly indicates that for low processing loads and low battery capacities, LSA, EDu and Edi outperform very significantly the other heuristics in terms of deadline meeting.

Figure 2 (b) shows that the performances of all schedulers are degraded when the processing load is 0.5. When the size of the battery is less than 30, no strategy can achieve 100% satisfied deadlines including LSA and EDi. To guarantee that LSA achieves 100% satisfied deadlines, the battery capacity should be larger than 40. Consequently, we see that if we make the load 2.5 times, we have to make the battery capacity more than 10 times to guarantee the same level of performance. As in previous curve, it clearly appears that behaviour of EDt, EDc and EDd are very sensitive to the size of the energy storage. To ensure 40% satisfied deadlines, they require a battery capacity at least equal to 5 while they require a battery capacity equal to 30 to guarantee 90% satisfied deadlines. In the same capacity range, LSA will respectively guarantee between 90% and 100% deadlines. ). Comparing to the previous curves, a higher battery capacity is required to provide the same performance level. No scheduler is able to provide 100% satisfied deadlines even if the battery capacity is more than 80. The system is highly time constraint and there is no flexibility for constructing the schedule which may be unfeasible even for high battery capacities. Except for very low battery capacities, all policies yield similar performance with a difference limited to 10%. There is consequently no significant motivation to increase the battery size for improving performance.

Conclusion

Careful energy management is the key to providing the best possible performance in real-time harvesting systems. In this paper we have presented scheduling heuristics in order to compare them to the optimal algorithm known as LSA. We have implemented the policies and reported results showing that the optimal policy outperforms the heuristics that we examined. Results were in terms of percentage of satisfied deadlines which is commonly used to measure the performance of real time systems. However, the experiment reveals that, under all processing loads, the gain is not significant compared to heuristic EDi that executes tasks greedily according to Earliest Deadline First until the energy storage unit be empty and then lets the processor idle until the next release time.

Moreover, practical implementation of LSA requires exact prediction of environmental energy in order to compute dynamically the exact start time of every task. Approximation on the above quantities will make LSA a sub-optimal scheduler, actually providing worst performance relative to the proposed heuristics while leading to higher computational overheads. This interesting issue needs more attention that will be in our immediate research plan. We are measuring the impact of approximating energy availability on the effective performance of LSA and the actual gain of LSA if still existing.

Figure 1 :

 1 Figure 1: Percentage of satisfied deadlines, making vary the processing load for a given battery capacity

8 Figure 2 :

 82 Figure2(c) reports the results for high processing loads (U=0.8). Comparing to the previous curves, a higher battery capacity is required to provide the same performance level. No scheduler is able to provide 100% satisfied deadlines even if the battery capacity is more than 80. The system is highly time constraint and there is no flexibility for constructing the schedule which may be unfeasible even for high battery capacities. Except for very low battery capacities, all policies yield similar performance with a difference limited to 10%. There is consequently no significant


C S Cojocaru

M Larijani

D S Misra

M K Singh

P Veis

F Le Normand

A new polarised hot filament chemical vapor deposition process for homogeneous diamond nucleation on Si(100)

Keywords: Diamond, HFCVD, Nucleation, Bias-enhanced HFCVD, Silicon

A new hot filament chemical vapor deposition with direct current plasma assistance (DC HFCVD) chamber has been designed for an intense nucleation and subsequent growth of diamond films on Si(100).Growth process as well as the If(V) characteristics of the DC discharge are reported. Gas phase constituents activation was obtained by a stable glow discharge between two grid electrodes coupled with two sets of parallel hot filaments settled in-between and polarised at the corresponding plasma potential. The sample is negatively biased with a small 10-15 V extraction potential with respect to the cathode grid.Such design allows to create a high density of both ions and radicals that are extracted and focussed onto the surface of the sample.The current density onto the sample can be finely tuned independently of the primary plasma.A homogeneous plasma fully covering the sample surface is visualized. Consequently, a high-density nucleation (>10 10 cm -2 ) occurs.

Introduction

Since the first successful report of diamond growth by the chemical vapor deposition (CVD) technique in the early 1980s w 1-3 x , the growth of heteroepitaxial or fibre textured films w 4,5 x is one of the main challenges to manage the outstanding properties of diamond.Tackling such a problem requires, however, to get a high nucleation density as well as an orientation of these nuclei on the heterosubstrate.Local epitaxial orientation or fibre textured diamond films were successfully grown on silicon substrates since approximately 10 years by introducing a bias nucleation step in the microwaveassisted chemical vapor deposition (MWCVD) process w 6-8 x .Hence, the bombardment of positively charged carbon species onto the surface was claimed to enhance diamond nucleation, although the true mechanism was far from being fully understood.By contrast, a similar bias-enhanced step introduced in the hot filament chem-ical vapor deposition (HFCVD) process reveals many problems that are presently neither fully understood nor experimentally well-resolved w 9-26 x .Instead of the MWCVD process, the HFCVD process is, however, a potential alternative to grow heteroepitaxial diamond films as it can be more easily scaled up and it is by far less expensive.In fact both the ignition and the sustainance of a stable glow discharge are facilitated in the presence of hot filaments that act as an effective source of electron emission, thus enhancing the ionisation through electron collisions with the gaseous molecules or radicals activated by the hot filaments.It has been proposed that electron emission can also occur from the intense secondary emission of diamond-coated substrate holder w 8,11 x .However, electron emission can also occur directly from the heated tungsten filaments in the HFCVD process, when the filaments were not used as anodes w 13 x .Moreover, the role of hydrocarbon ions as active species in the diamond nucleation has been addressed: either selective etching of the non-diamond species, formation of subplanted diamond clusters or enhanced surface diffusion were proposed as possible explanations of diamond nucleation w 15 x .In addition, drive was made up of an insulator ceramic MACORM, which was partially hollowed out in the center and filled with a molybdenum foil (200-mm thickness) to allow electrical contact on the rear of the substrate.The ceramic substrate holder was thoroughly cleaned before each run using SiC abrasives in order to remove the carbonaceous deposits that might initiate side electrical discharges, and subsequently ultrasonically rinsed 5 min into acetone and ethanol.The sample w Si(1 0 0) sample P-doped at 1-3 V cm; size 9x7x0.5 mm 3 , covered with a layer of SiO 2 (thickness 8 nm) x was strongly the ionic bombardment of the surface can be highly detrimental to the quality of diamond and the rugosity of the film w 12,16 x .Finally, due to the different nature of the substrate holder and the complex nature of the sample surface during the nucleation, the electric field may be spatially variable and time-dependent, leading to inhomogeneity and irreproducibility of the nucleation process w 27-29 x .In this paper, a new CVD set-up has been designed to investigate the nucleation of diamond films.Details of this CVD process as well as first electrical measurements are reported.

Experimental part

Sample preparation

The samples were prepared in a bakeable ultra high vacuum stainless steel chamber that had been previously clamped against the molybdenum foil with isolated tungsten springs.A chromel -alumel thermocouple, handled from outside, contacted the Mo foil and was used for temperature measurement and thermal regulation during growth step and acted as electric contact for sample negative biasing during the etching and nucleation steps.The two electrodes, the filaments and the sample, were biased according to the electrical scheme sketched in Fig. 2. The variable resistances R g , R p , R e as well as the fixed resistances R 1 and R 2 were calculated to stabilize both the DC discharge between the electrodes and to allow withdrawing a controlled current density of positively charged ionic species towards the substrate. The temperature was controlled and regulated by an independent infrared heater set on the rear side of the sample.The following procedure was followed to switch on the discharge during the nucleation step.First the gas mixture was admitted to the desired composition and the pressure was adjusted to 15 mbar.The filaments were switched onto the desired power as the discharge was easily ignited and stabilized by prior properly heating and biasing the filaments.The rheostat R g was used to set the desired polarisation voltage V p after the glow discharge ignition between the anode and the cathode.Further, with the R p rheostat the V A -F potential of the filaments was set so as the electron emission was compensated by the positive ions impinging the filaments (I f 0-1.5 mA).Finally, the extraction voltage described in detail w 30-32 x .The base vacuum was V e and the intensity current on the sample I e were 10 y9 mbar.The sample was first introduced into the adjusted at the desired values by adjusting the rheostats preparation chamber through a motion drive.Therefore, R e and R g , respectively. I e and V e values were kept cleaning procedures of the walls of the chamber as well constant at 1 mA and within 10-30 V, respectively. as filaments carburation could be performed in the Furthermore, the potentials V p and V e needed to be absence of the sample.The CVD deposition experiments were carried out with the same orthogonal sets of two parallel tungsten filaments (7 mm apart; diameter of continuously readjusted throughout the process with rheostats R p and R g .

Orders of magnitude of the plasma polarisation V p 0.15 mm) and separated by 12 mm. Two electrodes both and the extraction potential V e were 170-200 V and consisting in a molybdenum grid (open surface 68%) were fixed on the same moveable ensemble including the filaments (Fig. 1).The anode in the form of a square grid (size 20x20 mm 2 ) was settled at 6 mm above the top couple of parallel filaments.The cathode grid (size 15x15 mm 2 ) was positioned in-between the top and the bottom couples of filaments at equal distance (6 mm).The substrate holder mounted on the motion 10-30 V, respectively, and the discharge current I p was 40-100 mA, depending on the experimental conditions, but also on the cleanness of the chamber (substrate holders, walls, filaments).The extraction current was fixed in the range 1-5 mA corresponding to a current density within 1.5 -7.5 mA cm y2 . The substrate temperature during the nucleation step T N was monitored with an optical monochromatic pyrometer and it was checked 10 K.The temperature of the filaments T F was measured with an optical bichromatic pyrometer.The following sequences were carried out in a given deposition experiment:

Step Etching Nucleation Growth An etching procedure with a H 2 plasma for 2 min at as described above. -At the end of the nucleation step, the polarisation and extraction potentials V p and V e were switched off, the CVD parameters were readjusted to the desired growth conditions (pressure P, temperature T, methane concentration C). -The growth was finally stopped by first switching off the filament, then by stopping the carbon feed and finally by dropping the temperature (15 K min -1 ) under hydrogen.

The experimental parameters involved in the subsequent etching, nucleation and growth steps are included in Table 1.

The variable parameters are indicated between brackets with the extension range in the nucleation process.

a d FS is the distance between the bottom couple of filaments and the substrate (Fig. 1).

b V p is not an operating parameter.It depends on the quality of the diamond coating on the electrodes, the cleanliness of the walls, the substrate and filaments holders, and it might vary in the course of the process.

Preliminary to the deposition, some important operations were achieved to ensure a stable ignition of the discharge:

-If just implemented, the new filaments were carburized. 2).An intense blue light discharge focussed above the sample can be evidenced.The emission of the filaments on the top of the images is partly masked on the center by one pillar (Fig. 1).Conditions are V p 380 V; I p 12.5 mA; V e 10 V, I e 6 mA; T 973 K; 2.5% CH 4 in H 2 ; P 15 mbar; T f 2173 K.The power of the filaments and the heating lamp were dropped just before the flash. -In addition to the bake out, the chamber was in situ cleaned in order to remove the carbonaceous deposits.This was carried out by a H 2 plasma between the filaments and the walls for 3 min at 30 mbar.

Sample characterization

The films were characterized by scanning electron microscopy (SEM) performed in a JEOL JSM 6700F set-up working at 10 and 3 kV, respectively.The resolution was estimated to 0.005 mm in the best case.

Results and discussion

A view of the discharge is shown in Fig. 2, where most of the intense radiative emission by tungsten filaments was removed just before recording the images. This image is the sum of optically filtered images of mm 2 Si(1 0 0) sample.The density at the center of the sample can be estimated to approximately 10 10 cm y2 . The deviation of the density is not more than 20% along the sample.The primary plasma ignited and sustained by the hot filaments is very stable.Moreover, the presence of hot filaments allows to ignitiate the plasma at voltages lower than 200 V.Further, the characteristics I e -f (V e ) as well as the influence of the plasma primary potential I e -f (I p ) were studied.These studies were performed on a Si(1 0 0) sample precovered by ECR plasma processing with a SiO 2 layer of 8-nm thickness. This surface prevents the diamond nucleation and thus avoids strong modifications of the extraction intensity I e due to the intense secondary electron emission from diamond w 34,35 x .

Fig. 4 displays the characteristics I e -f (V e ) on SiO 2 y Si(1 0 0) at two fixed values of the primary plasma intensity I p .Within the range of extraction potential the emission line in the visible of different gas phase used (0-40 V), the extraction intensity I e increases species H a , H b , CH, CH q and C 2 H 2 (Table 2).Each singly filtered image displays identical discharge (not shown).Three distinct luminescent areas can be evidenced.One of them concerns the emission around the hot filaments on top of the images. Two different glow discharges with the characteristic purple blue color are evidenced, one above the cathode and the most intense one spreading out above the sample.The discharge focussed above the sample displays intense optical emission, due to the high density of excited species generated above a small surface, which homogeneously covers the sample surface entirely.The H b emission was more extended towards the cathode and the carbon species (CH and C 2 H 2 ) emissions are more concentrated over the sample.These observations sustain recent numerical unidimensional simulations for electron, ions and neutral densities performed by Hash et al. w 33 x for a DC HFCVD reactor.Therefore, I e can be used in the following as the true extraction current of ions impinging the surface of the sample.The high nucleation density and the homogeneity of the diamond deposit is checked after 15 min of nucleation followed by 60 min of growth (Fig. 3) at three locations along the diagonal of a 9x7 linearly with V e .However, the intensity as well as the slope markedly increase at higher I p .In Fig. 5 the extraction current I e is again linearly dependent on the primary plasma intensity I p at a fixed extraction voltage V e .It must be noted that without any extraction voltage (V e 0), the extraction intensity is not zero, the current density being more than 50% of the current density at V e 20 V. Conversely, the current density clearly goes to zero when I p decreases.

Many configurations have been introduced in the literature in order to achieve a bias step to the HFCVD process.Among these, the set-up described by Zhou et al. w 25 x displays some similitudes but also clear differences with the one described in this work.As in this work, a double bias was applied between an anode grid, filaments and the substrate was used as a cathode. Linear behaviours were reported both for the current extracted on the sample with the substrate bias voltage, and for the current between the anode and the filaments grid current.Moreover, large nucleation density was observed.However, the substrate voltage approximately 150 V, relative to the filaments, although decreased when compared with other set-up, remained large and it was expected that the energy of the ionised carbon precursors is high and led to a rapid degradation of the diamond nuclei.Reversely, in the set-up described in this work, by introducing a set of electrodes distinct from the sample a primary plasma is created with characteristics which are independent from the surface of the substrate.The introduction of a small negative extraction voltage between the cathode of the primary plasma and the substrate allows to draw part of these ionized carbon species and to impinge the surface of the sample with a lower kinetic energy, as expected from the many collisions occurring in the discharge ulations on a DC HFCVD reactor w 33 x .Further 1-h growth by a classical diamond HFCVD process yields diamond films that are characterized by SEM.Large density and high homogeneity over an area approxiabove the sample.This behaviour has been confirmed mately 9.7 mm 2 have been found. It is expected that for different gas compositions and for different activation or geometrical parameters w 35 x .Moreover, the filaments are maintained in our case at the potential of the plasma and therefore the emission of electrons by filaments is controlled.If not the case, the plasma becomes rapidly unstable.Now the linear behaviour of the current density on the sample with both the extraction voltage and the primary plasma intensity can be explained as follows.In the former case the extraction discharge above the sample acts as a resistive column through the many collisions occurring in this region.In the latter case, many ionic species accelerated in the primary plasma towards the open grid cathode cross it and are collected by the discharge established between the cathode and the sample, which is slightly negatively biased relative to this cathode.This is supported by the detection of a significant current density on the sample, even when no extraction voltage has been applied.

Of course in this rather complex system the properties of the ionic carbon species (mean and distribution of kinetic energy, density, nature) will depend both on the electrical parameters of the two discharge generated and on the geometric configuration, especially the distance between the cathode and the substrate w 33,35 x , in addition to the parameters of the HFCVD process.It is especially expected to get with this set-up a narrow and controlled distribution of the kinetic energy of the carbon ions precursor of the diamond nucleation, and therefore to achieve a control of this important step.Further works are done in this direction.

Conclusion

A new DC HFCVD chamber has been designed for an intense nucleation and subsequent growth of diamond films on Si(1 0 0).In a DC discharge and hot filamentsassisted nucleation step, the activation of the gas phase was obtained through a stable primary plasma created between polarised electrodes approximately 180 V with a couple of two parallel filaments settled in-between and set at the plasma potential.In addition, a second and small extraction discharge of 10 -15 V is introduced between the cathode and the sample, negatively charged. Such design allows to create a high density of both ions and radicals that are extracted and focussed onto the surface of the sample.The characteristics of the current density onto the sample can be finely tuned as a function of the primary plasma intensity or the extraction voltage drop.By qualitative optical emission analysis, preferential localization of the carbon-activated species is observed above the substrate confirming numerical sim-such a process will lead to a more accurate control of the nucleation step with a high density and a high reproducibility.

Fig. 1 .

 1 Fig.1. Scheme of the configuration electrodes-filaments-substrate.

Fig. 2 .

 2 Fig.2. Electrical scheme of the polarisation process in relation to a filtered image of the discharge at the distance cathode-substrate d fs 10 mm. The image of the discharge is the sum of the filtered image of H b , CH and C 2 H 2 emission lines at 486.2 nm, 431 and 518 nm, with widths 7.3; 7.4 and 8 nm and transmissions 54; 51 and 51%, respectively (Table2).An intense blue light discharge focussed above the sample can be evidenced.The emission of the filaments on the top of the images is partly masked on the center by one pillar (Fig.1).Conditions are V p 380 V; I p 12.5 mA; V e 10 V, I e 6 mA; T 973 K; 2.5% CH 4 in H 2 ; P 15 mbar; T f 2173 K.The power of the filaments and the heating lamp were dropped just before the flash.

Fig. 3 .

 3 Fig.3. SEM images of a diamond deposition after 15 min nucleation and subsequent 60 min growth along the diagonal of the Si(1 0 0) sample at (a) 1 mm; (b) 6 mm and (c) 11 mm from one corner.

Fig. 4 .

 4 Fig.4. Characteristic I e sf(V e ) of the extraction discharge for (square) I p s5 mA; (circle) I p s10 mA.Conditions are: CH 4 s2%; substrate SiO 2 ySi(1 0 0).

Fig. 5 .

 5 Fig.5. Plot I e sf(I p ). V e s10 V, (empty square) CH 4 s2%; (full square) CH 4 s4%; substrate SiO 2 ySi(1 0 0).

Table 1

 1 Parameters of the etching, nucleation and growth steps

	T (K)	973	973	1123
	CH 4 (%)	0	w2-4x	1
	D (sccm)	100	100	200
	p (mbar)	15	15	30

Table 2

 2 Observed optical emission lines and optical filters characteristics

	Fragments	Emission line	Optical filters (COHERENT ® )
		(nm)	Position (nm)	Width (nm)	Transmission (%)
	H a	656.2	656.3	11.58	57
	H b	486.2	488	7.28	54
	CH	431.1	430	7.38	51
	CH q	417.1
	CH q	422.2
	C 2 H 2	516-520	514.5	8.00	51

Acknowledgments

Fruitful discussions with C.Fuchs (PHASE, Strasbourg, France) are greatly appreciated.J.Faerber (IPCMSyGSI, Strasbourg, France) for SEM observations and P.Legagneux (Thales R&D, Orsay, France) for purchasing the SiO 2 ySi(1 0 0) samples are acknowledged.