X Balandraud
email: xavier.balandraud@ifma.fr

J.-B Le Cam

Keywords: Rubber elasticity, thermomechanical couplings, heat source, heat diffusion equation. 1 Introduction

The present paper deals with the specificities of the thermal response of rubber under cyclic mechanical loading at constant ambient temperature. This question is important, since the stabilized thermal response is used in fatigue life criteria, especially for the fast evaluation of fatigue life. For this purpose, entropic coupling in a thermo-hyperelastic framework is first used to predict the variation in the heat source produced or absorbed by the material during cyclic loading. The heat diffusion equation is then used to deduce temperature variations under adiabatic and non-adiabatic conditions. The influence of several parameters on the stabilized thermal response is studied: signal shape, frequency, minimum and maximum stretch levels, multiaxiality of the mechanical state. The results show that, in the steady-state regime, the mean value between the maximum and minimum temperature variations over a mechanical cycle is different from zero. This is due to the specific variation in the heat source, which depends on both the stretch rate and the stretch level. This result has numerous consequences, in particular for fatigue. Indeed, the stabilized mean value between the maximum and minimum temperature variations during fatigue tests does not reflect only fatigue damage, since the entropic coupling also leads to a value different from zero. This is a major difference with respect to materials exhibiting only isentropic coupling, such as metallic materials.

literature, the Payne (or Fletcher-Gent) effect [4; 5; 6; 7; 8], cavitation [9; 10; 11] and stress-induced crystallization [12; 13; 14; 15; 16]. Classically, rubber-like elasticity is described as an entropic effect [START_REF] Lion | Coupling effects in dynamic calorimetry: frequency-dependent relations for specific heat and thermomechanical responses -a one-dimensional approach based on thermodynamics with internal state variables[END_REF], which leads to heat production (absorption) during loading (unloading). The first experimental observations of this effect in the literature were those obtained by Gough1 [18] and Joule [START_REF] Joule | On some thermodynamic properties of solids[END_REF], and dealt with the increase in temperature of rubber during stretching. In spite of these first results, no further experimental study was fully dedicated to the thermomechanical characterization of elastomers, although this is a key point to study and to predict phenomena such as fatigue damage [20; 21; 22; 23], and to establish material constitutive equations. From a theoretical point of view, the work by Chadwich and Creasy [START_REF] Chadwick | Modified entropic elasticity of rubberlike materials[END_REF] is one of the very few studies to echo the experimental study of Joule. Until now, no study has allowed us to link this theoretical modeling with experimental results, even for homogeneous thermomechanical fields. It can be noted that new characterization methods in this field have recently been developed and applied to rubbers by coupling optical and infrared thermography measurements. These methods provide full kinematic and thermal fields without any contact with the rubber specimen under study [10; 11; 25; 26; 27; 28; 29; 31; 32] and their coupling allows us to calculate heterogeneous heat source fields (typically at the crack tip [START_REF] Pottier | Study on the use of motion compensation technique to determine heat sources. application to large deformations on cracked rubber specimens[END_REF]). In the future, the comparison between such experimental fields and adequate theoretical approaches should be of wide interest to the understanding of the thermomechanics of rubbers.

The present study deals with the predicted temperature variation in rubber under cyclic loading at constant ambient temperature. The heat sources produced or absorbed by the stretched material are obtained from the loading conditions by using the framework of continuum thermodynamics [33; 34] with isotropic thermo-hyperelastic behaviour. The heat sources considered here are only due to entropic coupling. The temperature variations are then deduced from the heat sources by solving the heat diffusion equation. For the sake of processing simplicity (in order to avoid full 3D calculations), applications are performed here with homogeneous deformation fields. This approach does not alter the conclusions of the study and enables us to focus on the specificities of the thermal response of rubber.

The paper is composed of four sections. The general thermomechanical framework and the heat diffusion equation are presented in Section 2. Section 3 provides some hypotheses to predict temperature variation during the deformation process. The numerical simulation setup is presented in Section 4, including the loading conditions of the cyclic tests to be simulated. Section 5 is devoted to the analysis of the results, for adiabatic and non-adiabatic conditions. Finally, before summarizing the results obtained, some remarks are made concerning the effect of the volumetric part of the heat source (effect of internal energy variation) on the thermal response.

Thermomechanical framework

Thermodynamics with internal variables

It is assumed that any thermomechanical system out of equilibrium can be considered as the sum of several homogeneous subsystems at equilibrium. In this context, constitutive equations are established by considering the deformation process as issuing from a quasi-static thermodynamic process. This process can be dissipative. The equilibrium state of each volume material element is defined by n state variables: the absolute temperature T , the deformation gradient tensor F and m (= n -2) internal tensorial variables ξ α . When the free-energy function Ψ is chosen as the Helmholtz free-energy function, it is written in terms of internal energy e(F, η, ξ α ) and entropy η(F, T, ξ α ) as follows:

Ψ (F, T, ξ α ) = e(F, η, ξ α ) -T η(F, T, ξ α ) (1)
The three quantities Ψ , e and η refer to a unit volume of the reference configuration.

The local version of the second principle of thermodynamics leads to the local form of the Clausius-Duhem inequality in the reference configuration:

P : Ḟ -ρ( ė -T η) - 1 T Q • Grad T 0 ( 2
)
where P is the first Piola-Kirchhoff stress tensor (or nominal stress tensor) and Q the Piola-Kirchhoff heat flux (or nominal heat flux). ρ is the density of the material in the reference configuration. Here, a point placed above a symbol denotes the temporal derivative of the considered quantity. The operators "•" and ":" are the scalar product (dot product) and the tensorial product (trace of the matrix product), respectively. Note that the external heat source due to radiation, commonly denoted R, has been removed from the local form of the Clausius-Duhem inequality by using the local form of the balance of energy. The term -1 T Q • Grad T (related to heat conduction) corresponds to the thermal dissipation D the . Based on physical observations, heat flows from the warmer to the colder region of a body. It is assumed that heat conduction follows Fourier's law :

Q = -κ 0 Grad T, (3)
where κ 0 is a positive semi-definite tensor characterizing the thermal conductivity of the material. As the absolute temperature T is considered as non-negative, it follows that:

D the = - 1 T Q • Grad T 0 (4)
The quantity P : Ḟ-ρ( ė-T η) corresponds to the intrinsic dissipation D int (also named mechanical dissipation). It is considered as non-negative. Thus,

D int = P : Ḟ -ρ( ė -T η) 0 ( 5
)
This quantity is related to any irreversible mechanical mechanism (for instance fatigue damage, internal friction, viscosity...). Note that any change in the material microstructure may also have a part of mechanical irreversibility. The quantity is also written [START_REF] Holzapfel | Non linear solid mechanics -a continuum approach for engineering[END_REF]:

D int = P : Ḟ -ρ ∂Ψ ∂F T,ξα : Ḟ -ρ ∂Ψ ∂T F,ξα Ṫ -ρ m β=1 ∂Ψ ∂ξ β F,T,ξα =ξ β : ξ β 0 (6)
In the present formulation, constitutive state equations derive from the Helmholtz free energy function:

η = - ∂Ψ ∂T F,ξα (7)
P rev = ρ ∂Ψ ∂F T,ξα (8)
A β = ρ ∂Ψ ∂ξ β F,T,ξα =ξ β (9)
where P rev and A β are the thermodynamic forces associated with variables F and ξ β , respectively. Note that in the case of a reversible transformation (irreversible part P irrev of the stress equal to zero), we have:

P = P rev = ρ ∂Ψ ∂F T,ξα (10)

Calorimetry

The specific heat capacity, denoted C F,ξα in the following, is the energy required to produce a unit increase in temperature of a unit reference volume of a body while maintaining the deformation and internal variables fixed. It is defined by:

C F,ξα = -T ∂ 2 Ψ ∂T ∂T F,ξα > 0 (11)
In the following, the specific heat capacity is considered as constant during the deformation process and is simply denoted C.

Heat diffusion equation

Using the two principles of thermodynamics, the local form of the balance of energy and convenient entropy form are respectively written in the material description:

ρ ė = P : Ḟ -Div Q + R (12) ρT η = -Div Q + D int + R ( 13
)
where R is the external heat source (radiation, for instance). From Eq. 7, we obtain:

ρT η = -ρT ˙ ∂Ψ ∂T = -ρT ∂ 2 Ψ ∂T ∂T Ṫ -ρT ∂ 2 Ψ ∂T ∂F : Ḟ -ρT m β=1 ∂ 2 Ψ ∂T ∂ξ β : ξ β (14)
As mentioned previously, the term -T ∂2 Ψ ∂T ∂T corresponds to the specific heat capacity C. According to Eqs. 8, 9 and 10, we have:

ρ ∂ 2 Ψ ∂T ∂F = ∂P ∂T , ρ ∂ 2 Ψ ∂T ∂ξβ =
∂Aβ ∂T , and consequently Eq. 14 can be re-written:

ρT η = ρC Ṫ -T ∂P ∂T : Ḟ -T m β=1 ∂A β ∂T : ξ β (15)
The local heat diffusion equation derives from Eqs. 12, 13 and 15. It is written using Eq. 3:

ρC Ṫ -Div(κ 0 Grad T ) = D int + T ∂P ∂T : Ḟ + T m β=1 ∂A β ∂T : ξ β + R (16)
Let us denote s h the overall heat source 2 induced by the deformation process:

s h = D int + T ∂P ∂T : Ḟ + T m β=1 ∂A β ∂T : ξ β (17)
The terms T ∂P ∂T : Ḟ correspond to the heat source due to entropic coupling. It is denoted s ec in the following. The term T ∂Aβ ∂T : ξ β corresponds to the other thermomechanical couplings (for instance related to phase change in the material) and is denoted s otc in the following. Thus the overall heat source is re-written:

s h = D int + s ec + s otc (18)
Finally, the heat diffusion equation is written:

ρC Ṫ -Div(κ 0 Grad T ) = s h + R (19)
3 Prediction of temperature variation during the deformation process Section 3.1 gives some assumptions required to simplify the heat diffusion equation in the case of homogeneous deformation fields. These assumptions seek to simplify the calculations and do not alter the conclusions of the study. Section 3.2 presents the thermo-hyperelasticity framework which enables us to define the heat sources s ec due to entropic coupling from the loading conditions.

Simplified heat diffusion equation

Let us consider an instance of mechanical loading such that the deformation field is homogeneous in the specimen at any time. Due to the very small thermal diffusivity of elastomers (order of magnitude of 10 -7 m 2 .s -1 ), the temperature field is almost homogeneous in the specimen at any time. In fact, temperature gradients exist close to the specimen boundary. These are related to the heat exchanges with the surroundings. However, these small temperature gradients do not prevent the temperature at the surface from being nearly equal to the mean temperature in the specimen. Let us denote θ the temperature variation with respect to the equilibrium temperature T ref in the reference state, corresponding to the undeformed state (T ref is equal to the ambient temperature). The heat exchanges at the specimen boundary are linearized assuming that the temperature variations θ are small. After some calculations which are not detailed here, the 3D heat diffusion equation can be rewritten as follows [36; 37; 38; 39; 40; 41; 31; 32]:

ρC θ + θ τ = s h ( 20
)
where τ is a time constant characterizing the heat exchanges between the specimen and its surroundings. This formulation can be referred to as the "0D" formulation. For similar simplifications of the heat diffusion equation, the reader can also refer to Refs. [42; 43; 44; 45; 46; 47], which are devoted to wires or thin plates, leading respectively to 1D or 2D formulations.

It is worth noting that, experimentally, τ depends on the specimen material (through its thermal diffusivity), the environment (ambient air and material used for the grips of the testing machine) and the specimen geometry (through the surface-area-to-volume ratio). For instance, the value of τ depends on the air flow around the specimen: the higher the convection, the smaller the parameter τ . Let us note that changes in the specimen shape (due to large deformations) modify the surface-area-to-volume ratio. The area of the surface in contact with the grips of the testing machine and the material of the grips also have an influence. A value of τ should be measured for each testing configuration (machine used, environment, stretch level, etc). Experimentally, τ is usually identified using a natural return to room temperature. Indeed, in this case the temperature variation is governed by the heat diffusion equation, with no heat source produced by the material itself ( θ + θ/τ = 0). In the present numerical study, τ is considered as a constant. For the experimental characterization of τ , the reader can refer to Ref. [START_REF] Martinez | Mechanisms of deformation in crystallizable natural rubber. Part 2: quantitative calorimetric analysis[END_REF].

The value of τ characterizes the level of non-adiabaticity of the mechanical test. If τ = 0, the test is isothermal (the heat produced or absorbed by the material due to stretching is instantaneously exchanged with the outside). A perfectly adiabiatic evolution corresponds to τ = ∞. More generally, if τ is much larger than the test duration, the evolution can be considered as adiabatic over this duration.

Thermally and mechanically isotropic hyperelasticity

In the present study, no intrinsic dissipation is taken into account, meaning that D int = 0. Only the contribution of entropic coupling is considered in the heat source expression, i.e. s h = s ec . Such material behaviour is observed for unfilled rubbers, for instance in silicone rubbers that exhibit a perfect non-linear and reversible behaviour with neither Mullins effect nor hysteresis loop, meaning that all loading and unloading curves are superimposed [START_REF] Meunier | Experimental and numerical study of the mechanical behaviour of an unfilled silicone rubber[END_REF].

The prediction of the heat sources produced during the deformation process requires the choice of a free energy density Ψ . Here, the material is assumed to behave as a hyperelastic material that is mechanically incompressible. As high strain levels are not investigated here, the Neo-Hookean model is chosen [START_REF] Treloar | The elasticity of a network of long chain molecules (I and II)[END_REF]. The free energy function is then given by the following strain energy density:

ρΨ (F, T ) = W (F, T ) = 1 2 N k T (I 1 -3) ( 21
)
where N is the number of network chains per unit volume, k is Boltzmann's constant, T is the temperature in Kelvin and I 1 is the first invariant of the left Cauchy-Green deformation tensor FF T .

If the material is mechanically incompressible, the deformation gradient tensor for a biaxial loading is given by:

F = λ e 1 ⊗ e 1 + λ B e 2 ⊗ e 2 + λ -(B+1) e 3 ⊗ e 3 (22)
where:

e 1 , e 2 and e 3 are three orthonormal vectors of the 3D euclidian space, -λ is the stretch ratio in the e 1 direction, defined as the ratio of the current length to the initial length in the e 1 direction, -the operator ⊗ between two vectors is such that [a ⊗ b] ij = a i b j , -B is the coefficient of biaxiality [48; 49]. It is equal to -0.5, 0 and 1 for uniaxial tension, pure shear and equibiaxial tension, respectively.

The strain energy density is written:

W (λ, T ) = 1 2 N k T λ 2 + λ 2B + λ -2(B+1) -3 (23)
The heat source is then given by:

s h = s ec = T ∂P ∂T : Ḟ = T ∂ 2 W ∂T ∂λ dλ dt (24)
Using this expression, the heat diffusion equation is:

ρ C θ + θ τ = N kT λ + Bλ 2B-1 -(B + 1)λ -2B-3 dλ dt (25)
Remark 1 -The quantity N kT appearing in the heat source expression contains the value of the temperature in Kelvin. For temperature variations which do not exceed a few degrees, N kT remains in fact nearly equal to N kT ref , where T ref is the initial temperature (in other words, the temperature variations have no significant influence on the elastic properties of the material). For instance, a temperature variation of 1 degree around ambient temperature leads to a heat source variation of 100 × 1/293 = 0.34%.

For uniaxial tension, pure shear and equibiaxial tension, we have respectively:

ρ C θ + θ τ = N kT ref λ - 1 λ 2 dλ dt ; ( 26
) ρ C θ + θ τ = N kT ref λ - 1 λ 3 dλ dt ; (27)
and

ρ C θ + θ τ = 2N kT ref λ - 1 λ 5 dλ dt (28)
Remark 2 -The heat source due to entropic coupling depends on the stretch rate dλ/dt, but also on the stretch ratio through the term λ+Bλ 2B-1 -(B +1)λ -2B-3 . This is a major difference compared to materials exhibiting only an isentropic-type thermoelastic coupling (metallic materials in particular). In this latter case, the heat source depends only on the strain rate. This property has inspired the so-called thermoelastic stress analysis technique [START_REF] Dulieu-Barton | Development and application of thermoelastic stress analysis[END_REF], which enables stress distribution to be measured in metallic structural components from full temperature field measurements. The equations written in this section are used below to predict the thermal response of rubber for different loading conditions. Section 4 presents the numerical setup, including in particular the different tests to be simulated.

Numerical simulation setup

This section describes the different tests and provides some information about the numerical solution of the heat diffusion equation.

Material parameter -The material is assumed to behave as a neo-Hookean material. In this case, the only material parameter is the number of chain segments N per unit volume. As the temperature variations in the simulations below will not exceed a few degrees, the quantity N kT is considered as constant (see Remark 1 above). It is taken as equal to 2 MPa. This is the order of magnitude obtained from experimental tests. This choice has been already done in previous theoretical studies (see for instance Ref. [START_REF] Verron | A multiaxial criterion for crack nucleation in rubber[END_REF]).

Level of non-adiabaticity of the thermodynamic evolution -Two series of tests are simulated below: one under adiabatic condition and another under non-adiabatic conditions. The adiabatic case corresponds to a time constant τ which is infinite. In practice, the term θ/τ is removed from the heat diffusion equation. For the non-adiabatic case, the time constant τ is taken as equal to 60 s in the present study. This is the order of magnitude of τ measured for a 1 × 5 × 20 mm 3 natural rubber specimen (see for instance Refs. [START_REF] Martinez | Mechanisms of deformation in crystallizable natural rubber. Part 2: quantitative calorimetric analysis[END_REF] and [START_REF] Martinez | Filler effects on the thermomechanical response of stretched rubbers[END_REF]).

Loading conditions -Cyclic tests are considered here. Several parameters are studied: shape of the signal applied, frequency, strain ratio and multiaxiality of the mechanical state. Two types of signal are considered: triangular and sinusoidal. The former case enables us to exclude the influence of the stretch rate variation from the analysis (dλ/dt is constant during the loading stage and the unloading stage). The latter case is classically used for experimental cyclic tests.

Tables 1 and2 summarize the different uniaxial tests to be carried out, for adiabatic and nonadiabatic conditions respectively. The first two columns give the minimum and maximum stretch ratios applied, λ min and λ max respectively. The strain ratio R ǫ is defined as the ratio between the minimum strain and the maximum strain. f is the frequency of the cyclic mechanical loading. The main objective of the present study is to investigate the stabilized thermal response, which is obtained after a certain period of time. For adiabatic conditions, the stabilization of the temperature oscillation is obtained from the first mechanical cycle. For non-adiabatic conditions, the stabilized oscillation of the temperature is reached after a duration which can be approximated to 3τ .

-Tests #1 and #2 correspond to uniaxial tension (UT) loading, processed under adiabatic conditions.

These tests aim at highlighting the effect of the signal shape (triangular or sinusoidal) on the heat source and temperature variations under adiabatic conditions, for which Eq. 20 becomes ρC θ = s h . As a consequence, the loading frequency does not influence to the temperature variation level (It is arbitrarily fixed to 0.5 Hz for both simulations). -Tests #3 to #8 enable us to study the effect of the loading parameters under non-adiabatic conditions. Apart from test #5, the loading frequency is equal to 0.5 Hz and the test duration is chosen equal to 300s, which corresponds to 5τ . This enables us to observe both the transient and stabilized regimes. Test #5 is a special case, considered to study the effect of the loading frequency. Several frequencies in the range [0.001 Hz; 0.2 Hz] are tested. This range enables us to study the influence of the heat exchanges with the outside during one mechanical cycle. Indeed, the longer the period 1/f of the mechanical cycle, the stronger the effect of the heat exchanges with the outside environment during the cycle. The test duration is adapted to the tested frequency in order to observe several cycles. In the table, UT, EQT and PS correspond to uniaxial tension, equibiaxial tension and pure shear, respectively.

Data to be extracted from the temperature variations -Under non-adiabatic conditions, the thermal response evolves from one cycle to another before stabilizing. In order to follow the global thermal response, we define by θ the mean value between the maximum and minimum temperature variations over a mechanical cycle by:

θ = θ max + θ min 2 (29)
where θ min and θ max are the minimum and maximum temperature variations over a mechanical cycle. Moreover, the mean temperature variation over one cycle, defined as the temporal integration of θ over the cycle, is also calculated for each cycle.

Numerical resolution -The resolution of Eq. 26, Eq. 27 and Eq. 28 is based on an implicit Euler scheme. Particular attention is paid to time discretization, since accurate results require a very refined temporal discretization. The number of time steps per mechanical cycle is critical. After some preliminary tests, this number is fixed to 10 4 . Such a refined temporal discretization is particularly required for triangular loading, which exhibits a non-continuous variation in the heat source, induced by the non-continuous variation of dλ/dt.

Unit used for heat sources -The heat source s h is expressed in W.m -3 . However, it can be useful to divide this quantity by ρC:

s = s h ρC (30)
In the following parts of the paper, the term "heat source" is still used for the quantity s. This later is expressed in • C.s -1 . Using this unit enables us to directly read the temperature rate in the case of adiabatic evolution. Obviously, in case of non-adiabatic conditions, the temperature rate is different from the heat source s because of the heat exchanges with the outside. From Eq. 20, the heat diffusion equation becomes:

θ + θ τ = s ( 31
)
The heat is the temporal integration of the heat sources. When divided by ρC, the heat is expressed in • C.

Results and analysis

Tests under adiabatic conditions

Tests under adiabatic conditions are first presented in order to give preliminary information about the heat source and temperature variations in a simple case. Indeed, according to Eq. 31, we have θ = s. The temperature rate is equal to the heat source.

Figure 1 presents the evolution of the heat source s and of the temperature variation θ over two mechanical cycles for test #1 (uniaxial triangular signal). As expected for an entropic material, a certain amount of heat is produced during loading [18; 19; 53]. This quantity can be calculated by integrating the heat source over the loading duration. During unloading, the heat source is negative. The absolute value of the heat absorbed is equal to the heat produced during loading. As a consequence, the adiabatic nature of the evolution (no heat exchange with the outside) leads to a temperature variation equal to zero at the end of each mechanical cycle. As already mentioned in Section 3.2 (see Remark 2 above), the heat source value does not depend only on the stretch rate. It depends also on the value of stretch ratio λ (see Eq. 24), so the heat source evolves during loading and unloading. For materials exhibiting only isentropic-type thermoelastic coupling (metallic materials in particular), the heat source is constant for triangular-shaped loadings, with only an opposite sign between loading and unloading. As the temperature variation oscillates here between 0 and a maximum value θ max , the mean value between the maximum and minimum variations over a mechanical cycle is obviously different from zero: θ = 0.

Figure 2 shows the results obtained for test #2 (uniaxial sinusoidal signal). The shape of the heat source curve is not exactly sinusoidal. This is clearly visible at low strains. A nearly horizontal plateau can be seen around zero deformation. An important remark can be made here. This type of response could be wrongly attributed to the effect of the variation in internal energy at low strains (isentropic coupling), but let us recall that isentropic coupling is not considered in the modeling here. Besides, isentropic coupling would have led to opposite signs for the heat sources (negative during loading and positive during unloading). See also Section 5.4. As a result, the temperature variation at low strain levels is affected by a particularity of the variation in heat sources: we can see a "flat valley" around zero deformation. This thermal response was observed experimentally in Ref. [START_REF] Pottier | Study on the use of motion compensation technique to determine heat sources. application to large deformations on cracked rubber specimens[END_REF].

Tests under non-adiabatic conditions

In the present section, non-adiabatic conditions are considered to investigate the stabilized thermal regime.

Let us recall that the intrinsic dissipation is equal to zero in all our simulations (D int = 0). In such a situation, it is worth noting that, for purely isentropic materials (metals in particular), heat exchanges with the outside environment of the specimen lead to a value of θ equal to zero in the steady-state regime; the temperature variation θ oscillates symmetrically around zero after the transient regime.

Let us now analyze the stabilized temperature variation for hyperelastic materials.

Effect of non-adiabaticity itself

Figure 3 presents the evolution of the temperature variation for a triangular signal under non-adiabatic conditions (test #3). Two main results can be drawn from this figure :   -Quantity θ, which corresponds to the mean value between the maximum and minimum variations over a mechanical cycle, decreases over the duration of the test. This can be simply explained as follows. During the first loading stage, the temperature increases because the heat source is positive. However, a certain quantity of heat is given to the outside because the material is warmer than the outside (this heat corresponds to the temporal integration of θ/τ ). Thus, the temperature at the end of the first loading stage is lower than the value that would be obtained for adiabatic test conditions. During unloading, the heat source is negative, leading to a temperature decrease. This phenomenon is amplified by the fact that the material continues to give heat to the outside as long as it is warmer than the outside (as long as θ/τ is positive). As a consequence, the temperature variation at the end of the first mechanical cycle is lower than zero. This phenomenon continues throughout the cycles. -It can be observed that the mean value θ between the maximum and minimum variations over a mechanical cycle reaches a stabilized value which is different from zero: see Figure 4 which shows two stabilized cycles (at the end of test #3). Here, θ is equal to 0.4043 • C. This effect is due to the particular variation in the heat source over time. This result can be observed whenever the chosen time constant τ is different from zero. This is a significant result, typical of rubber-like materials.

It is worth noting that this phenomenon is not due to intrinsic dissipation (D int equal to zero in the simulations).

Remark 3 -Unlike θ, the mean temperature variation over one cycle is equal to zero in the steadystate regime. Indeed, according to Eq. 31, we have θ = τ sθ . First, the temporal integration of the heat source s over a mechanical cycle is equal to zero, i.e. the heat produced is equal to that absorbed. Second, in the steady-state regime, the temporal integration of θ is also equal to zero. As a consequence, the mean temperature variation over one cycle is also equal to zero in the steady-state regime.

Effect of signal shape

This section will highlight the fact that the value of θ in the steady-state regime depends on the signal shape. Test #4 uses the same parameters as test #3, but the signal shape is now sinusoidal. Figures 5 and6 show the evolution of the temperature variation θ over the test and for two stabilized cycles (at the end of the test), respectively. As observed previously for a triangular signal shape, the value of θ is different from zero in the steady-state regime. The stabilized value of θ obtained here (0.3317 • C) is different from that obtained for the triangular signal (0.4043 • C). As a general result, the stabilized value of θ depends on the signal shape.

Effect of loading frequency

All the previous tests were performed at 0.5 Hz. Test #5 is devoted to the effect of the loading frequency on the stabilized value of θ. Test #5 uses the same parameters as test #4 apart from the frequency, which varies between 0.001 and 0.2 Hz. For f = 0.001 Hz, the period of the mechanical signal is equal to 1000 s: this value is much greater than 3τ , so the temperature variation evolution is influenced by the heat exchanges with the outside environment during the cycle. On the contrary, for f = 0.2 Hz, the period is equal to 5 s: this value is much lower than 3τ , so the heat exchanges are very low during one cycle. The stabilized value of θ versus the frequency is given in Figure 7. Two curves are plotted in this figure. They correspond to two different values of N kT ref : 1 and 2 MPa. Several comments can be made:

the ratio between the two curves is the same as the ratio between the elastic parameters; -by decreasing the frequency, the stabilized value of θ tends towards zero. This was expected, as the test tends to be isothermal; -by increasing the frequency from zero, the stabilized value of θ first increases and then reaches an asymptotic value. Note that for N kT ref = 2 MPa, this value is equal to that obtained previously for test #4 (0.3317 • C at 0.5 Hz); -the maximum value of θ is reached for high loading frequencies (such that the signal period is inferior to about 3τ ), for which the heat exchanges have insufficient time to occur over one cycle.

These results are of importance for the prediction of fatigue life. Indeed, most approaches based on heat build-up measurements to evaluate quickly the fatigue life of hyperelastic materials use cyclic tests between 0.5 and 10 Hz. Our results show that the corresponding value of θ is maximum in this range of frequencies and can be wrongly associated with heat build-up due to fatigue.

Effect of strain ratio R ǫ

Test #6 is similar to test #4 (in particular, both have the same stretch ratio amplitude), but the strain ratio R ǫ is equal to 1 2 instead of 0. Results are presented in Figures 8 and9. For isentropic materials such as metals, the temperature amplitude θ max -θ min would be the same in the stabilized regime (the loading frequency of 0.5 Hz is high enough to be nearly in adiabatic conditions over one mechanical cycle) and θ would be equal to zero. Comparing test #4 (Fig. 6) and test #6 (Fig. 9) shows that the temperature amplitudes θ max -θ min are different. Moreover, the (non-null) stabilized value of θ depends on the considered zone of strain-stress relationship. For the present simulations, θ equals 0.2819 • C for λ oscillating between 1 and 3, and 0.3317 • C for λ oscillating between 2 and 4.

Remark on the effect of the multiaxiality of the mechanical state

Tests #4, #7 and #8 investigate the effect of three loading conditions, uniaxial tension (UT), equibiaxial tension (EQT) and pure shear (PS) respectively, classically used to characterize the mechanical response of rubber-like materials. Figure 10 groups the three stabilized thermal responses. As expected from the form of Eqs. 26 to 28, UT and PS exhibit very close heat source values, while the heat sources for EQT are approximately two times greater. For the present numerical simulations, the following results are found in the steady-state regime: θ = 0.3317 • C for UT, 0.3287 • C for PS and 1.2508 • C for EQT.

Remark on the effect of the internal energy variation

As previously reported in Refs. [START_REF] Joule | On some thermodynamic properties of solids[END_REF] and [START_REF] Anthony | Equations of state for naturals and synthetic rubber like materials: unaccelerated natural soft rubber[END_REF], vulcanized rubber is characterized by slight cooling during stretching in the very low strain range (typically for λ < 1.1), followed by strong heating when continuing the stretch increase. This phenomenon is referred to as the Gough-Joule effect in the literature and leads to a thermoelastic inversion. Indeed, a competition takes place between the effect of the internal energy variation (isentropic coupling) and the effect of entropy variation (entropic coupling). At low strain levels, the heat source is mainly composed of the contribution of the internal energy variation (negative heat source for a positive strain rate). At higher strain levels, the heat source is mainly composed of the contribution of the entropy variation (positive heat source for a positive strain rate). However, both contributions exist whatever the strain level. For instance, it can be seen from Eqs. 26-28 that the heat source due to entropic coupling is different from zero for small strains, too, i.e. as soon as the strain is different from zero. As a consequence, even for cyclic tests at small strains, the value of θ is different from zero in the steady-state regime.

The contribution of the internal energy variation is rarely taken into account in the literature, although it significantly affects the thermal response of rubber at low strains. Ref. [START_REF] Chadwick | Modified entropic elasticity of rubberlike materials[END_REF] proposed a modified entropic theory to account for this phenomenon using a volumetric part of the strain energy density that contains an energetic contribution to the heat source produced by the material. Figure 11 presents the evolution of the stretch ratio, the heat source and the temperature variation with test conditions similar to test #4 but including the effect of the internal energy variation. The heat source variation is in correct agreement with the experimental results obtained at low strains by Pottier et al. [START_REF] Pottier | Study on the use of motion compensation technique to determine heat sources. application to large deformations on cracked rubber specimens[END_REF]. The stabilized value of θ is different from zero: it is here equal to 0.3261 • C. This value is slightly lower than that obtained for test #4 (0.3317 • C). Isentropic coupling does not strongly change the conclusions.

Conclusion

This paper has investigated the thermal stabilization of rubber-like material under cyclic loading conditions. Results have shown that the mean value between the maximum and the minimum temperature variations differs from zero in the steady-state regime. This is due to the particular distribution of the heat source over time, and more especially to the fact that the heat source depends on both the strain and the strain rate. This phenomenon is observed while the temporal integration of the heat source is null over each cycle and no intrinsic dissipation occurs. This effect clearly distinguishes rubber from most other materials, whose thermoelastic response is driven by isentropic coupling. Moreover, the stabilized mean value between the maximum and minimum temperature variations depends on the strain ratio, the frequency and the multiaxiality of the mechanical state. The special thermosensitivity of rubber engenders many consequences, in particular for fatigue life prediction. Indeed, the stabilized mean value between the maximum and minimum temperature variations during fatigue tests does not reflect only self-heating due to fatigue, since entropic coupling leads also to a value that is not equal to zero. Finally, let us recall that small strain levels were investigated in the present study, leading to small thermal effects of the entropic coupling (but important for fatigue prediction). The effect would be much higher for large loading amplitudes, which besides could have an influence on thermo-oxidation of rubber.

Fig. 1

 1 Fig.1Test #1 (adiabatic conditions, triangular signal, λmin = 1, λmax = 3, Rǫ = 0, f = 0.5 Hz). Top: stretch ratio λ vs. time. Middle: heat source s vs. time. Bottom: temperature variation θ vs. time.

Fig. 2

 2 Fig.2Test #2 (adiabatic conditions, sinusoidal signal, λmin = 1, λmax = 3, Rǫ = 0, f = 0.5 Hz). Top: stretch ratio λ vs. time. Middle: heat source s vs. time. Bottom: temperature variation θ vs. time.

Fig. 3

 3 Fig. 3 Test #3 (non-adiabatic conditions, triangular signal, λmin = 1, λmax = 3, Rǫ = 0, f = 0.5 Hz). Temperature variation θ vs. time.

Fig. 4 Fig. 5

 45 Fig. 4 Last two cycles of test #3.

Fig. 6

 6 Fig. 6 Last two cycles of test #4.

Fig. 7 Fig. 8 Fig. 9

 789 Fig. 7 Test #5: stabilized value of θ versus loading frequency.

Fig. 10

 10 Fig.10Results obtained under uniaxial tension (UT), pure shear (PS) and equib-iaxial tension (EQT) for the last two cycles for non-adiabatic conditions. The other parameters are those used for test #4. Top: stretch ratio λ vs. time. Bottom: temperature variation θ vs. time.

Table 1

 1 Uniaxial tensile (UT) tests under adiabatic conditions (τ = ∞) Test number λmin [-] λmax [-] Rǫ [-] f (Hz)

"If one end of a slip of Caoutchouc be fastened to a rod of metal or wood, and a weight be fixed to the other extremity, in order to keep it in a vertical position; the thong will be found to become shorter with heat and longer with cold",Gough, 1805

The term 'heat source' is used in this paper to mean the heat power density which is produced or absorbed by the material. Note that the terms 'heat source' and 'heat' must be distinguished. The heat (in J/m

) is the temporal integration of the heat source (in W/m 3 ).


Y Candéla

J.-B Jager

R Pratibha Nalini

G L Lin

M Terrier

V Calvo

F Gourbilleau

J Hare

Photoluminescence of silicon nano-clusters functionalized microtoroids

Keywords:

We report on the first realization of on-chip toroid microcavities embedding size-controlled silicon nanoclusters (Si-nc), produced by reactive magnetron co-sputtering. We demonstrate functionalized whispering gallery microresonators with low volume and high quality factor. Emission properties are optimized by tailoring the size and the shape of the toroids and the Si-nc distribution. Coalescence of Si-nc during the melting of the toroid is minimized by the design of the microdisk preforms. Photoluminescence in the whispering gallery modes of 13.5 µm diameter microtoroids, is observed in both near-field and far-field.

Introduction

Due to its indirect band-gap, Silicon was for a long time considered as a poor light emitter. The discovery in the 90's of an efficient emission from porous Si produced by electrochemistry [START_REF] Canham | Silicon quantum wire array fabrication by electrochemical and chemical dissolution of wafers[END_REF] or nanostructured Si obtained by other methods [START_REF] Brus | Luminescence of silicon materials: Chains, sheets, nanocrystals, nanowires, microcrystals, and porous silicon[END_REF] led to lively interest [START_REF] Kovalev | Breakdown of the kconservation rule in Si nanocrystals[END_REF][START_REF] Wolkin | Electronic states and luminescence in porous silicon quantum dots: the role of oxygen[END_REF]. It was early recognized that nanometer scale confinement results in a weakening of the indirect bandgap properties. Active researches yield to evidence of optical gain in Silicon nanocrystals [START_REF] Pavesi | Optical gain in silicon nanocrystals[END_REF]. This increased the activity [START_REF] Dovrat | Radiative versus nonradiative decay processes in silicon nanocrystals probed by time-resolved photoluminescence spectroscopy[END_REF][START_REF] Sychugov | Narrow luminescence linewidth of a silicon quantum dot[END_REF] in silicon photonics because an on-chip a Si laser would be a major achievement towards CMOS compatible photonics, but direct laser action has not yet been achieved.

In this quest, a critical point is the choice of high-performance cavities which can be combined with this active material. In this perspective, the Whispering Gallery Modes (WGM) of fused silica microcavities, exhibiting ultra high-Q up to 10 10 and small mode volumes V down to 100 λ 3 constitute an appealing option. The high Q/V figures of merit achieved in these cav- ities have allowed to observe low-threshold lasers [START_REF] Sandoghdar | Very low threshold whispering-gallery mode microsphere laser[END_REF], nonlinearities [START_REF] Kippenberg | Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity[END_REF] and quantum effects [START_REF]Optical processes in microcavities[END_REF]. The on-chip microtoroids introduced in [START_REF] Armani | Ultra-high-Q toroid microcavity on a chip[END_REF] are peculiarly suited in the perspective of thin-film based all-silicon devices.

Silicon nanoclusters (Si-nc) are usually obtained from sub-stoichiometric silica, generally called "silicon-rich silicon oxide" (SRSO). Thermal annealing then enables phase segregation of the excess silicon and Si-nc nucleation. SRSO itself is obtained by different methods including laser pyrolysis [START_REF] Wolkin | Electronic states and luminescence in porous silicon quantum dots: the role of oxygen[END_REF], chemical vapor deposition [START_REF] Pavesi | Optical gain in silicon nanocrystals[END_REF], electron beam evaporation [START_REF] Verbert | Efficient coupling of Er-doped silicon-rich oxide to microdisk whispering gallery modes[END_REF][START_REF] Verbert | Room temperature emission from er-doped silicon-rich oxide microtorus[END_REF] and ionic implantation [START_REF] Kalkman | Fabrication and characterization of erbium-doped toroidal microcavity lasers[END_REF].

An interesting variant is the reactive co-magnetron sputtering technique [START_REF] Gourbilleau | Si-rich/SiO nanostructured multilayers by reactive magnetron sputtering[END_REF][START_REF] Zacharias | Size-controlled highly luminescent silicon nanocrystals: A SiO/SiO 2 superlattice approach[END_REF]. This process not only produces SRSO, but also enables the production of multilayers (ML) consisting in an alternation of SiO 2 and SRSO films, and allows to accurately define the SRSO sublayer thickness. As the growth of Si-nc during the annealing step is constrained by the SRSO sublayer, this method allows to take control on their size distribution and on the resulting emission bandwidth. This unique property makes this technique very appealing for optical applications where size monodispersity could be a decisive advantage.

To the best of our knowledge, two papers have been devoted to Si-nc ML functionalized microcavities : planar microcavities in [START_REF] Belarouci | Microcavity enhanced spontaneous emission from silicon nanocrystals[END_REF] and WGM microdisks in [START_REF] Zhang | Visible range whispering-gallery mode in microdisk array based on size-controlled Si nanocrystals[END_REF]. In this paper, we report for the first time on the combination of very-high quality factor WGM microtoroids with size controlled Si-nc produced by magnetron sputtering. We have successfully fabricated ML SRSO microdisks, and converted them to Si-nc doped microtoroids by laser fusion. We have then demonstrated an efficient coupling of the Si-nc emission with low order WGM of the microcavities. The obtained photoluminescence (PL) signal and its limiting factors will be presented and analyzed.

Sample fabrication

The production of microtoroids relies on several steps, the first of which consist in the fabrication of a silica microdisk on a silicon pedestal. The processing steps used for this purpose are described elsewhere [START_REF] Armani | Ultra-high-Q toroid microcavity on a chip[END_REF]. In brief, starting from an oxidized silicon substrate, lithography followed by wet etching in HF leads to silica disks on silicon, and then dry etching of the silicon creates the pedestals, isolating the disks from the silicon surface.

SRSO microdisks

The active material consists of successive layers of pure and sub-stoichiometric silica deposited by reactive magnetron co-sputtering [START_REF] Gourbilleau | Si-rich/SiO nanostructured multilayers by reactive magnetron sputtering[END_REF]. This technique uses two targets of Si and SiO 2 in an Ar -H 2 plasma. The injection of H 2 in the vacuum chamber allows the reduction of the SiO 2 target to favor a higher Si incorporation. During deposition the substrate temperature is set to 500°C and the chamber pressure is set to 0. In both cases the pump is filtered out by using notch or Long pass filter respectively. The far-field (scattered light) is also collected using the thinned multimode fiber MMF F2 and analyzed with the same spectrometer.

The SRSO sublayer thickness determines the mean diameter of the Si-nc, hence the central wavelength of the emission, ranging from visible to near infrared for thickness varying from 1 nm to 4 nm. For the purpose of this paper, we focus on SRSO sublayer thickness of 3 nm, leading to a central emission wavelength in the MLs structure of about 775 nm.

The SRSO MLs are deposited on top of a pure silica microdisk. This approach, introduced in [START_REF] Verbert | Efficient coupling of Er-doped silicon-rich oxide to microdisk whispering gallery modes[END_REF], avoids the specific difficulty of SRSO wet chemical etching. The supporting silica microdisks have a diameter varying from 20 µm to 50 µm and a thickness ranging from 0.5 to 1.5 µm (see Fig. 1-c).

The samples are subsequently annealed at 1050°C for 1 h under pure N 2 atmosphere, in order to nucleate the Si-nc. Fig. 1-(a) shows a typical image of the material after annealing. In Fig. 1-(b) is plotted the emission spectrum of a microdisk embedding Si-nc, optically pumped at 532 nm.

Functionalized microtoroids

The deposited microdisks are then fused close to the focus of a CO 2 laser Gaussian beam, with a waist of 110 µm. We used a circular polarization in order to keep a cylindrical symmetry. The silicon pedestal acting as an efficient heat sink, only the disk border reaches the fusion temperature, and surface tension shapes the molten material into a toroid.

As the melting temperature of silicon is lower than that of SiO 2 , the embedded Si-nc can be damaged by the laser fusion. In order to minimize this effect, the microdisks are molten with a laser power as small as possible. This melting threshold was determined by fusing identical microdisks with different powers. It was found about 10 kW/cm 2 , thought dependent on the microdisk geometry.

The microtoroid shape was characterized with a scanning electronic microscope (SEM) and an atomic force microscope. The observed surface roughness is under detection limit, with an amplitude below 1 nm, for 1 µm 2 scan.

Experiment

The experimental setup is shown in Fig. 2. As in [START_REF] Lin | Excitation mapping of whispering gallery modes in silica microcavities[END_REF], the microtoroid is hold on a three axis PZT-actuated translation stage and is coupled to a tapered fiber. Three different lasers are alternatively or simultaneously used. The first one (L1) is a tunable extended-cavity laser diode (ECLD) at λ 1 = 775 nm, which is used to characterize the cold cavity spectrum next to the maximum of emission. The two others, used as pump lasers, are a frequency-doubled Nd 3+ :YAG providing 25 mW at λ 2 = 532 nm (L2) on one hand and a single mode tunable ECLD at λ 3 = 405 nm with a power of 13 mW (L3) on the other hand. The L2/L3 can be combined with L1 using a dichroic mirror and coupled to the same taper for evanescent coupling to the microtoroid.

With a coupling gap of ∼ 0.5 µm the intrinsic microcavity losses dominate the coupling losses. When scanning L1 frequency one detects the resonances in the vicinity of 775 nm as absorption dips on the transmitted power, monitored on photodiode PD (as plotted in Fig. 4). This provides the microtoroid quality factor Q e at emission wavelength. In the same way, the tunable laser L3 can be used to characterize the WGM spectrum in the absorption range, and estimate the quality factor Q p .

The light emitted into the WGM by the Si-nc is detected in two ways. On the one hand, it is evanescently coupled out by the taper and led to the output fiber F1 (near-field signal). On the other hand, it is scattered in the toroid and collected through a thinned multimode fiber F2 (far-field signal) like in [START_REF] Orucevic | Transmittance and near-field characterization of sub-wavelength tapered optical fibers[END_REF][START_REF] Redding | Comparison of raised-microdisk whisperinggallery-mode characterization techniques[END_REF][START_REF] Srinivasan | Photoluminescence measurements of quantum-dotcontaining semiconductor microdisk resonators using optical fiber taper waveguides[END_REF]. At the output of both fibers the pump is removed by using a long pass filter LP (OG590). The two signals are then analyzed by an imaging spectrograph with a f = 300 mm focal length.

In these experiments a challenging difficulty lies in the strong infrared (IR) PL induced in the fibers by the short wavelength lasers used to pump the Si-nc. In many cases this spurious IR PL hides the interesting signal in both F1 and F2. This problem originates in the composition of the fiber used in the experiment. Our tapers are usually produced with IR single mode fibers (SMF) which have pure silica cladding and doped core: the deleterious signal comes from the doping elements. In contrast, the SMF adapted to short wavelength have a silica core and doped cladding, which makes them unsuitable for taper fabrication. When using blue laser L3, this spurious PL is significantly stronger than with L2. Therefore we abandoned resonant intracavity pumping with L3 and decided to use the non-tunable green laser L2.

The usual tapers, made from IR-fiber SMF 125-A-42, are single-mode at Si-nc emission wavelength. However, at the pump wavelength λ 2 , they are multimode. The corresponding low effective index could excite poorly confined higher order WGM. Nevertheless, this effect is minimized by injecting L2 into the taper through a SMF. Finally, the rather high power of this laser allows to efficiently pump the Si-nc located in the fundamental mode.

In the last stage of the experiment, the taper fabrication was improved in a decisive way to eliminate this fiber core PL limitation. This consisted in fabricating an "hybrid" taper made from two different fibers. A small length ∼ 15 mm of pure silica cladding SMF was spliced, on both sides, with pure silica core fibers (3M FS-SN-3224) suitable for visible light operation. This combination allows to minimize the fiber PL while preserving the ability to produce a good quality taper, with a transmittance T ≥ 90% at λ 1 = 775 nm.

Results

We present successively the results obtained with two samples A and B of different thickness and diameters. Their geometrical parameters are summarized in table 1. Sample A was produced in a first stage of the experiment, and the analysis of its characteristics led to engineer sample B, which features the main results of this work. For both samples, after annealing, the emission wavelength band, spanning from 730 nm to 830 nm, corresponds to a distribution of Si-nc diameters centered at 3 nm with a dispersion estimated to 0.3 nm.

We first used pure silica microdisks presenting a moderate thickness and a large distance between the pedestal and the edge (sample A in Table 1). The resulting microtoroid exhibit fundamental TE modes with quality factors Q e = 2 × 10 7 and Q p = 10 6 . For all other observed modes (10 per FSR), Q e is reduced to about 4 × 10 6 . In spite of these good quality factors, the Si-nc emission was to weak to be isolated from the spurious fiber emission in F1.

An explanation of this small emission eficiency is provided by a structural analysis. After the optical characterization, the toroid was cut using a focused ion beam (FIB) in order to study the silicon distribution after melting. SEM images of the section are shown in Fig. 3-(a) and (b). These pictures evidence the usual rolled-up shape of the toroid originating from the large pedestal-edge distance [START_REF] Kalkman | Fabrication and characterization of erbium-doped toroidal microcavity lasers[END_REF]. Moreover, at high resolution, one observes a spiral shaped distribution of structures with a typical diameter of about 100 nm, which result from Si-nc coalescence, as confirmed by an energy dispersive X-ray spectrometry measurement, to be published elsewhere. To illustrate this feature we show in the inset (e) of Fig. 3 the cross section of a toroid very similar to that of sample A, but with a much higher contrast. One actually observes the mid-size nanostructures created by coalescence of Si-nc, favored by advection during the melting. One also distinguishes the pure silica bottom layer of the microdisk, and above it a silicon doped layer.

If some nanometer sized Si-nc survived the melting process, their spatial distribution must basically be the same as for the larger visible ones. Due to the roll-up shape, the depth of the active region matches the 1.5 µm thickness of the supporting microdisk, while the radial extension of the fundamental mode is about 2 λ 1 below the surface. This results in a nearly vanishing coupling between the emitting Si-nc and high-Q WGM, and explains the dim signal.

Achieving a better coupling, requires a thinner silica substrate and a better control of the toroid shape. For this purpose, the fabrication process is modified with 3 times thinner supporting silica microdisks and a 3 times smaller distance pedestal-edge, as described for sample B in Table 1. We keep the same pedestal diameter, and the same total disk thickness by increasing by a factor 4 the number of SRSO layers, with a reduced spacing to improve the Si-nc concentration. In what follows we focus on this improved sample.

Fig. 3-(c) displays SEM images of the cross section of this toroid, demonstrating that is not rolled up. Actually, decreasing the amount of fused material inhibits the large displacement associated to the usual roll-up shape, and results in a 3 times smaller minor diameter. On the zoom presented in Fig. 3-(d) there is no indication of coalescence : in contrast to (b) and (e) the observable nanostructures have a much smaller apparent size. We observe no sign of advection, as the preserved Si-nc are almost homogeneously spread in the whole section.

Putting in contact the hybrid taper with the microtoroid allows to maximize the output coupling. The resulting signals, simultaneously collected by using the two fibers F1 and F2, are analyzed with SP using a 300 grooves/mm grating (resolution ≈ 1 nm). In Fig. 4, we have plotted the corresponding spectra. Fig. 4-(a) is a picture of the experiment clarifying the two output ports F1 and F2, and Fig. 4-(c) is the absorption dip observed when scanning L1 in undercoupled conditions, exhibiting a reduced albeit still large Q e in sample B. Fig. 4-(b) is the "near-field" signal detected at the hybrid taper output. The emission into the WGM with its characteristic quasi-periodicity (FSR ≈ 9 nm) is clearly observed. This spectrum is superimposed on a large background, which results from the PL created in the central part of the taper and in the auxiliary multimode fibre F3 leading to SP. Only the PL emitted in the WGM is coupled back to the taper, therefore the broadband component of Si-nc emission does not contribute to this background. Fig. 4-(d) shows the "far-field" signal coming from the MMF F2, in the same conditions. As a fraction of the scattered pump is collected by F2, it generates a large spurious PL which is the dominant contribution of the background. Thanks to the scattering losses of the microtoroid, the Si-nc emission structured by the WGM is still present, but with a reduced contrast. A comparison between (b) and (d) demonstrates that the taper coupling has a better extraction efficiency than the omnidirectional scattering: in spite of a larger background, it enables a 20 times better detection efficiency of the WGM.

When working in non-contact conditions we recover the high quality factor of the WGM. The F1 signal spectrum is recorded in SP, with a 1,200 grooves/mm grating enabling a higher resolution. The pump-induced fibre PL background was first recorded with a large gap (g ∼ 10 µm), in order to digitally remove it from the detected signal. In Fig. 5 is plotted the resulting spectrum obtained by this procedure. One recognizes the 9.0 nm periodicity of the WGM spectrum and 4 modes in each FSR. The observed linewidth of about 0.5 nm is here instrumental-limited. The two strongest peaks in a given FSR are separated by about 4.5 nm, compatible with the TE-TM shift expected for this small microtoroid.

Conclusion

We have for the first time produced on-chip silica toroidal microcavities embedding Si-nc of controlled size, and have investigated their photoluminescence. This combination of the best material with the highest Q/V silicon based microcavities was achieved by the reactive mag-netron sputtering of SRSO on silica microdisks, followed by laser reflow and surface tension reshaping. These active cavities feature a Q-factor larger by one order of magnitude than the one obtained for microdisks in [START_REF] Kekatpure | Fundamental photophysics and optical loss processes in Si-nanocrystaldoped microdisk resonators[END_REF][START_REF] Ghulinyan | Whispering-gallery modes and light emission from a silicon-nanocrystal-based single microdisk resonator[END_REF], and is mostly limited by diffraction and not by scattering.

The optical characterization of the functionalized cavities has been realized in both the nearfield (evanescently coupled taper) and the far-field (scattering collected by a fiber tip). It shows that rather large toroids, due to their "rolled-up" shape, maximize the quality factor, but gives a geometric mismatch between the emitters distribution and the WGM field, resulting in a very weak emission. In contrast, significantly smaller toroids, presenting an optimized symmetric shape, at the expanse of a significantly reduced Q-factor, feature a good coupling of the Si-nc to the WGM and give rise to a well resolved WGM emission.

The technical limitation coming from the spurious photoluminescence of the fibers was overcome by the realization of an hybrid taper. A further improvement in the taper fabrication technique is required to enable a better coupling of the pump in a resonant WGM. Such a configuration providing a much higher sensitivity would allow a better material optimization and the investigation of the fundamental loss mechanisms in Si-nc.

Acknowledgment

This work has been supported by the Region le-de-France in the framework of C'Nano IdF, the nanoscience competence center of Paris Region. Y. C. acknowledges support granted by the French Ministry of Higher Education and Research.

Fig. 1 .

 1 Fig. 1. (a) Transmission electron microscope image of a typical SRSO ML sample obtained with reactive co-magnetron sputtering technique, followed by thermal annealing at 1050°C for 1h. The white lines underline the boundaries of the MLs. In the inset, a zoom on a Si-nc shows the crystalline planes (b) Emission spectra of silicon nano-clusters functionalized microdisks pumped at 532 nm (c) Schematic of a functionalized silica microdisk on a silicon pedestal.

Fig. 2 .

 2 Fig.2. Three Laser L1, L2 and L3 combined on a dichroic beam-splitter DM, are injected in the tapered fiber to characterize or to pump the microtoroid. The taper (F1) output signal is split to either measure the total transmitted power or analyze the PL with spectrometer SP. In both cases the pump is filtered out by using notch or Long pass filter respectively. The far-field (scattered light) is also collected using the thinned multimode fiber MMF F2 and analyzed with the same spectrometer.

Fig. 3 .

 3 Fig. 3. SEM images of microtoroids cut with FIB. (a) Sample A general view, (b) zoom on the roll of sample A. Inset (e) : toroid similar to sample A, selected for the visibility of the coalesced Si clusters. (c) Sample B general view, (d) zoom on the roll of sample B toroid. On (d) and (e) the dashed line shows the boundary between silica and deposited material.

Fig. 4 .

 4 Fig. 4. (a) Top view of the experiment. (c) Absorption dip for a fundamental mode observed when scanning L1. (b) Spectrum of the near field signal at output of the hybrid taper. (d) Spectrum of the far field signal in the same conditions. frequency.

Fig. 5 .

 5 Fig. 5. Same as Fig. 4-(b) but acquired at higher resolution and after subtraction of the fiber PL background.

Table 1 .

 1 Geometrical parameters of the microdisks, multilayers, and resulting microtoroids used in the experiment. T d , D d and D p denote the disk thickness, diameter and pedestal diameter, respectively. N, T SRSO and T SiO2 denote the number of bilayers, SRSO thickness and Silica thickness, respectively. D and d are the outer and minor diameters of the toroid.

		Silica disk (µm)		SRSO		Toroid (µm)
	Parameters	T d D d	D p	N	T SRSO	T SiO2	D	d
	Sample A	1.5 50	7	70 3 nm 15 nm	33	8
	Sample B	0.5 20	7	150 3 nm 7 nm 13.5	3


Andreas Niekler

Gregor Wiedemann
email: gregor.wiedemann@uni-leipzig.de

Gerhard Heyer

Leipzig Corpus Miner -A Text Mining Infrastructure for Qualitative Data Analysis

HAL is

Introduction

For several years humanists, social scientists and media analysts working with text as primary data have been opening up to large scale text analysis procedures. This development, sometimes referred to by the term "Digital Humanities", is fueled by increased availability of digital text and algorithms for identification of (semantic) structures in unstructured data. Nonetheless, the complexity of handling 'Text Mining' (TM) procedures as well as problems in managing of 'big data' prevents those approaches from being used by a wider audience lacking a computer science background.

To facilitate the handling of large document collections and make use of algorithmic Text Mining procedures for content analysis (CA) we built the "Leipzig Corpus Miner" (LCM). With CA we refer to a broad set of methods and corresponding methodologies for analyzing textual data common in various scientific disciplines. These may include classic (quantitative) content analysis [START_REF] Krippendorff | Content analysis: An introduction to its methodology[END_REF] as well as rather qualitative approaches like discourse analysis [START_REF] Laclau | Hegemony and socialist strategy[END_REF], grounded theory [START_REF] Glaser | Grounded theory: Strategien qualitativer Forschung[END_REF] or qualitative content analysis [START_REF] Mayring | Qualitative Inhaltsanalyse: Grundlagen und Techniken[END_REF]. In this respect, techniques integrated into the LCM do not replicate analysis procedures of these methods exactly. Rather they offer a set of tools which enable analysts to support certain steps of an applied method and to extend the size of collections under investigation to a degree which could not be handled manually.

In contrast to most computer-assisted CA studies which employ only a single or very few TM procedures [START_REF] Wiedemann | Opening up to big data: Computer-assisted analysis of textual data in social sciences[END_REF] LCM allows for application of multiple procedures which may be integrated systematically into complex analysis workflows. Thus, results of single processes are not restricted to be interpreted in an isolated manner. Beyond that, they may be used as input data for further mining processes. For example, a list of key terms automatically extracted from a subset of a reference corpus may be utilized to retrieve documents of interest in a second target corpus. Semantic topics automatically retrieved from Topic Models [START_REF] Blei | Probabilistic topic models: Surveying a suite of algorithms that offer a solution to managing large document archives[END_REF] may be applied to disambiguate homonymous term usage and thus, help defining concepts under investigation for individual CA studies.

The systematic integration of TM procedures in a coherent user-friendly application enables content analysts to employ complex algorithms while simultaneously retaining control over major parameters or configurations of analysis process chains. Utilizing those procedures on large document collections may improve quality of qualitative data analysis especially in terms of reliability and reproducibility. It thereby gives analysts without deeper knowledge of foundations of natural language processing (NLP) the ability to develop best practices for computer-assisted large scale text analysis. Because NLP experts usually lack background knowledge about requirements and methodologies in the humanities and social sciences such best practices could not be developed by either discipline alone. Providing a common interface on text analysis is the major advantage of using a framework like the LCM.

Architecture

The LCM is a combination of different technologies which provide a qualitative data analysis environment accessible by an interface targeted towards domain experts unfamiliar with NLP. Analysts are put in a position to work on their data with more methodical rather than technical understanding of the algorithms. Applied technologies behind the user interface need to support analysts in tasks such as data storage, retrieval, processing and presentation. A schematic overview of the architecture and workflow is given in figure 1.

Storage and processing of large amounts of text data are key tasks within the proposed environment. To process data we use UIMA, an architecture to identify structures in unstructured textual data [START_REF] Ferrucci | Uima: An architectural approach to unstructured information processing in the corporate research environment[END_REF]. Within this architecture data readers, data writers and data processing classes can be chained together in order to define standardized workflows for datasets. In UIMA data processing is done by so called "annotators" which (usually) add additional information to the analyzed documents. UIMA employs "stand off" markup for annotations which in contrast to inline markup does not alter the original text. Therefore, annotations can be stored in a separate data structure [START_REF] Burghardt | Stand off-Annotation für Textdokumente: Vom Konzept zur Implementierung (zur Standardisierung?)[END_REF]. Annotations could include sentence information, token separations or classification labels. But annotators can also be used to aggregate single text documents for further processing such as topic modeling. We defined UIMA processes for the following tasks:

-Initial processing of XML sources: To initially process raw sets of text data we defined a process which identifies areas like titles and paragraphs, separates sentences, applies tokenization and POS-Tagging, detects entities and multi word units (MWU) and performs entity resolution. OpenNLP1 , the Stanford NER tool [START_REF] Finkel | Incorporating non-local information into information extraction systems by gibbs sampling[END_REF], a Wikipedia based multi word unit detection and the JRC-Names resource [START_REF] Steinberger | JRC-NAMES: A freely available, highly multilingual named entity resource[END_REF] for entity resolution are utilized within this tool chain. We trained custom models for sentence detection and tokenization with resources provided by the Leipzig Corpora Collection (LCC) [START_REF] Biemann | The leipzig corpora collection -monolingual corpora of standard size[END_REF]. -Classification, Corpus Linguistics (CL) and Machine Learning (ML):

The UIMA based ClearTK [START_REF] Ogren | ClearTK: A UIMA toolkit for statistical natural language processing[END_REF] framework is applied to provide convenient access to classification and machine learning libraries to support certain analysis steps. We also implemented inference algorithms for different Topic Models and corpus linguistic analyses based on ClearTK.

Annotations made by processing chains, e.g. tokens, sentences, entities or classification labels, can vary throughout different analysis approaches. We therefore decided to employ a schema free data storage which gives us the flexibility to delete or add annotation structures to stored text documents. For this, we use MongoDB2 -a NoSQL database implementation which stores datasets as JavaScript Object Notation (JSON) structures. It supports the distribution of very large datasets to different shards and machines. This is a very important feature as we want to open the system for new large corpora. Every imported corpus can be accessed separately within the architecture. Alongside with data storage the user-friendly accessibility of corpora by different indexes is an additional requirement. As we want to provide indexes accessible by a convenient query language based on metadata fields and full-text of documents we use the Solr3 enterprise search server. For this a set of metadata fields and full-text contents for indexing each corpus can be defined. This information is provided to Solr which uses annotations made by UIMA directly to populate its indexes. As well as MongoDB we can distribute the Solr indexes to a cloud and thus are able to index and store new large corpora. Indexes are used for full-text search, faceted search and search on meta data fields.

Access to the datasets and indexes is facilitated by a web-based front end (fig. 2). This web application, based on the YAML CSS framework 4 , accesses stored corpora and datasets through Java Servlets5 hosted by an Application Server. The front end to back end communication is managed by AJAX calls via the jQuery6 javascript framework. Presentations of results, graphs and charts are implemented in D3 [START_REF] Bostock | D3: Data-driven documents[END_REF], a visualization library to create, manipulate and animate SVG Objects.

One central objective of the LCM is to enable analysts to perform Text Mining tasks without explicit guidance by NLP experts. For that reason we implemented a middleware (UIMAWS), a webservice to start, stop and manage UIMA processes for certain tasks. The analysis tasks are described in detail in section 3 of this paper. The UIMAWS is deployed to a dedicated server and is able to manage multiple instances of UIMA processes. The webservice communicates information on the progress of running processes. Results of completed tasks will be written to the database and visualized by the front end. The UIMA processes always operate on a finite set of documents which are referenced by their ID within MongoDB. Those identifications can be retrieved by querying the Solr index, a custom information retrieval algorithm or manually compiled lists of documents. Starting and managing of UIMA processes can be done in the front end as shown in figure 3.

Analysis capabilities

The LCM integrates several procedures for retrieving, annotating and mining textual data. Flexibility in combining these tools lends support to various analysis interests ranging from quantitative corpus linguistics to qualitative reconstructivist methodologies. We briefly introduce the inbuilt technologies and give an example of a workflow in a concrete research environment in section 4.

-Information retrieval: Assuming the availability of a large document collection, e.g. complete volumes of a daily newspaper over several decades, a common need is to identify documents of interest for certain research questions. The LCM provides two ways of IR for this task. First, a full text index allows for key term search over the entire collection (see section 2). But, for CA relevancy often cannot be tied to a handful of key terms. Thus, we provide a second retrieval method based on contextualized dictionaries which can be retrieved from a reference corpus of paradigmatic documents covering a certain research interest. A list of several hundred key terms, called 'dictionary', is extracted automatically from this reference corpus and co-occurrence patterns of these terms are measured. Relevancy in our target collection is then defined through occurrence of dictionary terms in their desired co-occurrence contexts from the reference collection. This method allows not only for retrieving coherent document sets on a single subjectmatter (e.g. 'Iraq war'), but rather for distinguishing modes of speech over various topics (e.g. military cadence in politics or sports). Document collections retrieved by either IR system can be stored for further procedures. -Lexicometrics: The LCM has implemented computation and visualization of basic corpus linguistic measures on stored collections. It allows for fre-quency analysis, co-occurrence analysis (figure 4) and automatic extraction of key terms. As documents contain a time-stamp in their metadata, visualizations can aggregate these measurements over days, weeks, months or years. Researchers may choose if they want to have absolute or relative frequencies displayed. For co-occurrence analysis LCM allows selecting different significance measures to compute term relations (raw counts, Tanimoto, Dice, Mutual-Information, Log-Likelihood) [START_REF] Büchler | Medusa: Performante Textstatistiken auf großen Textmengen: Kookkurrenzanalyse in Theorie und Anwendung[END_REF]. -Topic models: For analysis of topical structures in large text collections Topic Models have been shown to be useful in recent studies [START_REF] Niekler | ASV Monitor: Creating comparability of machine learning methods for content analysis[END_REF]. Topic Models are statistical models which infer probability distributions over latent variables, assumed to represent topics, in text collections as well as in single documents. So far the LCM has implemented models of Latent Dirichlet Allocation (LDA) [START_REF] Blei | Latent dirichlet allocation[END_REF], Hierarchical Dirichlet Process (HDP), Pitman-Yor Process (HPY) [START_REF] Teh | Hierarchical Bayesian nonparametric models with applications[END_REF] and Online LDA [START_REF] Hoffman | Online learning for latent dirichlet allocation[END_REF], a very fast inference algorithm for the LDA model [START_REF] Blei | Latent dirichlet allocation[END_REF]. Results from Topic Models can be seen as an unsupervised clustering which gives analysts the opportunity to identify and describe thematic structures on a macro level as well as to select subsets of documents from a collection for further refinement of the analysis or for a "close reading" process. Distributions of topics can be visualized over time (figure 5). -Classification: Supervised learning from annotated text to assist coding of documents or parts of documents promises to be one major innovation to CA applications [START_REF] Scharkow | Automatische Inhaltsanalyse und maschinelles Lernen[END_REF]. The LCM allows for manual annotation of complete documents or snippets of documents with category labels (figure 6). The analyst may initially develop a hierarchical category system and / or refine it during the process of annotation. Annotated text parts are used as training examples for automatic classification processes which output category labels for unseen analysis units (e.g. sentences, paragraphs or documents). For that the LCM integrated several classification approaches like SVM, Naive Bayes and Semantically Smoothed Naive Bayes [START_REF] Zhou | Semantic smoothing for bayesian text classification with small training data[END_REF]. Feature engineering can be performed by the analyst up to a certain degree by configuring the classification process (e.g. restriction to certain POS-types for training). An iterated process of manual labeling and evaluation of (best) automatically retrieved labels may replicate "Hybrid Active Learning" [START_REF] Lughofer | Hybrid active learning (hal) for reducing the annotation efforts of operators in classification systems[END_REF].

Example use case

The LCM was initially built to facilitate a CA study on political theory. Within the German "ePol-project"7 political scientists conduct large scale text analyses with the help of NLP researchers. The project aims to identify changes in discursive patterns of policy justifications in public media. By identifying certain patterns and measuring their quantities the project strives to verify or reject central hypotheses about a phenomenon called "Post-democracy". One central question is: Has there been an "economization" of justifications in some or all policy fields during the last decades? [START_REF] Lemke | Die Ökonomisierung des Politischen: Entdifferenzierungen in kollektiven Entscheidungsprozessen[END_REF]. To answer this question a corpus consisting of 3.5 million newspaper articles from 1946 to 2012 is investigated. The LCM was built with the requirement to support analyses of and inquiries into these rather abstract questions. Figure 7 shows a simplified workflow of the analysis process which is realized with the help of the aforementioned procedures.

1. A subset of relevant documents for the analysis is retrieved from the complete corpus of 3.5 million documents. To identify "neoliberal" themes and modes of speech a reference corpus consisting of 36 works from neoliberal theorists was compiled. A dictionary of 500 automatically extracted key terms and their sentence co-occurrences were computed. With our IR system we select 10,000 (potentially) relevant documents from the initial corpus. 2. A topic model computation over this subset identifies thematic clusters which help to describe its content from a macro perspective. The topic model result allows to distinguish of policy fields mentioned within the articles. It also enables analysts to remove documents from the selected subset which contain topics irrelevant to the specific purpose of this analysis (e.g. articles mainly concerned with foreign policy are not considered relevant for this study). 3. Best ranked documents from step 1 of each topic (step 2) are investigated manually by the political scientists to identify argumentative structures in different policy fields. A hierarchical category system is derived from common, distinctive structures. Relevant sections of the documents were then annotated with category labels. 4. An automatic classification procedure is invoked on the unlabeled data to identify more text parts containing argumentative structures. The NLP group supports the analysts by identifying discriminating linguistic features for this task. 5. Text snippets identified in the previous step (supposedly containing arguments of interest) are presented to the analysts ranked by certainty of label assignments. Analysts can verify or reject the results manually. In this active learning paradigm we calculate internal precision / recall measures while the analysts are evaluating the process qualitatively. If those ongoing evaluations show satisfactory results, the process of creating training data is concluded.

6. The classification procedure is run on the entire collection under investigation. Results can be described qualitatively and quantitatively (e.g. proportions of categories over time slices).

Future Work

The LCM supports manual Content Analysis (CA) via basic corpus linguistic procedures as well as supervised state-of-the-art Text Mining techniques. In a next step we will incorporate a user rights management system which allows for access restrictions to different corpora in the LCM. In addition, work-flows will be further enhanced to combine intermediate results of different procedures. For example, a centralized management of (semi)automatically generated dictionaries could be useful to exploit controlled lists of key terms throughout different steps of the entire process chain (e.g. computing co-occurrences just for key terms; using occurrence of terms in dictionaries as additional feature for classification tasks). To support more inductive research approaches, we plan to integrate unsupervised data-driven analysis procedures into the LCM. These could help analysts especially between steps 2 and 3 of our proposed process chain to identify categories of interest in unknown data. This includes the identification of stable as opposed to volatile semantic concepts in certain topics over time [START_REF] Heyer | Change of topics over time and tracking topics by their change of meaning[END_REF].

In classification and annotation processes we enable analysts to define categories and training data for those categories. Within this process many evaluation measures, frequently used by CA scholars, could be applied. In future developments we will incorporate more of these measures regarding cooperative annotation and categorization tasks. This will be a further step towards integration with common social science research methods away from a narrow computer linguistic oriented work-flow. Its our hope that the LCM may contribute to improve the comparability, reliability and validity of social science research standards.

Fig. 1 .

 1 Fig. 1. Architecture of the LCM.

Fig. 2 .

 2 Fig. 2. Browser-based user interface of the LCM. Fig. 3. The GUI enables users to manage various Text Mining tasks on sub collections.

Fig. 4 .

 4 Fig. 4. The LCM provides various data visualizations on sub collections, e.g. cooccurrence graphs...

Fig. 5 .

 5 Fig. 5. ... or topic model distributions over time.

Fig. 6 .

 6 Fig. 6. The UI provides assistance for manual annotation of document parts which subsequently may be used as training examples for classification.

8

 8

Fig. 7 .

 7 Fig. 7. Schema of a Content Analysis workflow realized within the LCM.

http://opennlp.apache.org

http://www.mongodb.org

http://lucene.apache.org/solr

http://www.yaml.de

http://www.oracle.com/technetwork/java/index-jsp-135475.html

http://jquery.com

http://www.epol-project.de

The corpus consists of complete volumes of DIE ZEIT, TAZ, Süddeutsche Zeitung and a representative sample of Frankfurter Allgemeine Zeitung.

Data Mining. pp. 289-300 (2008)