Jose Ordonez-Miranda

Radiative Thermostat Driven by the Combined Dynamics of Electrons, Phonons and Photons

Keywords: numbers: 65.60.+a, 65.80.-g, 65.90.+i

Based on the metal-insulator transition of two phase-change materials (PCMs) with asymmetric thermal conductivities and emissivities sensitive to temperature, a radiative thermostat capable of maintaining a constant temperature without consuming energy is proposed. This is done by deriving explicit expressions for the heat flux and temperature profiles of the thermostat surrounded by these PCMs and demonstrating that it can be maintained at a temperature nearly equal to the common transition temperature of both PCMs, as the environmental one changes more than 20 K. It it shown that the combined operation of two PCMs is much better than non-PCMs to develop energyfree termostats, specially those supporting poor thermal conduction or dominant heat radiation. The obtained results thus establish a fundamental theoretical method to exploit the nonlinear heat transport driven by electrons, phonons and photons, for saving energy on temperature preservation and thermal insulation.

I. INTRODUCTION

Heat is one of the main forms of wasted energy and its control is hence of critical importance to efficiently manage the energy resources of nature, which is under increasing stress due to the growing energy-hungry world population and global warming issues. The management of heat transport has become very challenging, as revealed by, for instance, the high percentage (68.5%) of energy lost in developed countries as waste heat that is mainly released into the environment [START_REF] Laboratory | Energy flow charts: Charting the complex relationships among energy, water, and carbon[END_REF]. Around 15% of the remaining energy is used for thermal insulation or preserving the temperature of industrial fields, buildings, homes, etc. [START_REF] Goldstein | [END_REF]. These representative figures show that the reduction of the energy consumption is a huge scientific, technological and societal challenge in the energy sector, that has been partially addressed through the nonlinear heat transport.

Strongly nonlinear heat conduction (radiation) shows up in materials with temperature dependent thermal conductivities (emissivities), whose tailoring allows modulating heat currents [3].

Over the past two decades, many research groups have been exploiting these thermally driven properties to control the heat transport via phonons, electrons, and photons [4][5][6][7][8][9][10][11][12]. Several conceptual devices, such as thermal diodes [13][14][15][16][17][18], thermal transistors [19][20][21][22][23], thermal memories [24] and thermal memristors [25,26] were proposed; and the working principles of some of them were experimentally demonstrated [27][28][29]. Furthermore, based on two phase change materials (PCMs) with asymmetric thermal conductivities, Shen et al. [30] developed a temperaturetrapping theory and proposed an energy-free thermostat able to self-maintain a desired constant temperature * jose.ordonez@cnrs.pprime.fr without consuming energy, regardless of the sizeable changes of its environmental temperature. More recently, Wang et al. [31] introduced thermoelectric effects in this latter theory and proposed a negative-energy thermostat capable of generating electricity with energyfree maintenance of a constant ambient temperature. These two latter works [30,31] were also applied to significantly improve the performance of thermal and thermoelectric cloaks, whose central regions are kept at pretty much constant temperatures, as the ambient ones vary in relatively large intervals. The operation of these two thermostats and cloaks is based on conductive heat currents and therefore their performances are limited by the inevitable presence of interface thermal resistances between the involved PCMs. This limitation could, however, be overcome by a radiative thermostat supporting photon heat currents between their contactless terminals, as is the case of nanophotonic thermostats based on multilayer structures and operating with solar radiation [32].

The purpose of this work is to demonstrate the proof of principle of a radiative thermostat, a cavity able to self-maintain a constant temperature despite being exposed to sizeable changes of its environmental temperature. This is achieved by using two PCMs as the cavity walls, whose asymmetric thermal conductivities and emissivities significantly change with temperature, within a narrow interval of temperatures. The proposed thermostat is thus a cavity that capitalizes on the non-linear heat transport driven by both conductive and radiative heat currents, and therefore its working principle is different than the conductive thermostat [30,31], passive radiative one [32], and other devices of temperature regulation [33][34][35] consisting of a layered system with a surface emitting radiation to the outer space through the atmosphere's transparency windows in the mid-infrared. The cavity could thus be applied as a constant temperature environment of objects.

II. THEORETICAL MODELING 1 ( ) T x 3 ( ) T x 2 ( ) Constant T x  1 1 ( ) k T 3 3 ( ) k T 2 k 1 ( ) T x 3 ( ) T x 2 ( ) Constant T x  1 1 ( ) k T 3 3 ( ) k T 2 k 1 ( ) T x 3 ( ) T x 2 ( ) Constant T x  1 1 ( ) k T 3 3 ( ) k T 2 k 1 ( ) T x 3 ( ) T x 2 ( ) Constant T x  1 1 ( ) k T 3 3 ( ) k T 2 k 1 l cv 1 2 l l + 1 2 3 l l l + + 0 x q q q T h T c
FIG. 1: Scheme of a cavity with a nearly constant temperature maintained by two PCMs. The solid line stands for the temperature profile generated by the temperature dependence of the asymmetric thermal conductivities (dashed lines) of the PCMs. The surfaces x = 0 and x = l 1 + l 2 + l 3 are set at the steady-state temperatures T h and T c (< T h ), respectively.

Let us consider two PCMs exchanging heat by radiation and conduction through a cavity, as a result of the temperature difference T h -T c > 0 of their external surfaces, as shown in Fig. 1. In order to minimize the impact of the enviromental temperature variations on the cavity temperature profile, i consider that the left-hand (right-hand) side PCM behaves as a thermal insulator at the relatively high (low) temperature T h (T c ), while it becomes a thermal conductor for lower (higher) temperatures. According to the Fourier's law of heat conduction, these conditions can be achieved when the thermal conductivity (k 1 ) of the left-hand side PCM increases as the temperature decreases, while the one (k 3 ) of the right-hand side PCM exhibits the opposite behavior, as depicted in Fig. 1 through the dashed lines. More explicitly, this law along with the Stefan-Boltzmann's one and the principle of energy conservation establish that the steady-state heat flux q propagating along the three layers is given by [START_REF] Modest | Radiative Heat Transfer[END_REF]

q = -k 1 (T 1 ) dT 1 dx = θ -ψ ρ 2 + σε θ 4 -ψ 4 = -k 3 (T 3 ) dT 3 dx , (1)
where T 1 and T 3 are the respective temperatures within the PCM1 (0 ≤ x ≤ l 1 ) and PCM3 (l

1 ≤ x ≤ l 1 + l 2 ), θ = T 1 (l l ), ψ = T 3 (l l + l 2 ), σ is the Stefan- Boltzmann constant, ε = ε 1 (θ) -1 + ε 3 (ψ) -1 -1 -1
is the effective emissivity and the thermal resistance ρ 2 = l 2 /k 2 + 2/h, with ε 1 (θ) and ε 3 (ψ) being the respective emissivities of the facing walls of PCM1 and PCM3, which exchange heat by radiation, conduction and convection through the intracavity gas of thermal conductivity k 2 and convective heat transfer coefficient h. For air, k 2 = 25 mWm -1 K -1 and h = 5 Wm -2 K -1 , within a wide range of temperatures [START_REF] Incropera | Fundamentals of Heat and Mass Transfer[END_REF]. The effective emissivity ε driving the intracavity radiation is defined by the Stefan-Boltzmann's law [START_REF] Modest | Radiative Heat Transfer[END_REF] and its values are independent of the optical properties of the intracavity gas, because this gas is assumed to be transparent to the infrared radiation exchanged by PCM1 and PCM3, as is the case of air for temperatures around the room temperature considered in this work [32]. Given that q is a constant, Eq. ( 2) establishes that -dT n /dx decreases as k n (T n ) (n = 1 and 3) increases and vice versa, as depicted in Fig. 1.

Considering that both PCMs have a metal-insulator transition at the temperature T 0 , the required temperature dependence of their thermal conductivities k n and mean emissivities ε n , for n = 1 and 3, can be described as follows

k n (T ) = k ln + k hn -k ln 1 + e -β(T -T0) , ( 2a
)
ε n (T ) = ε ln + ε hn -ε ln 1 + e -β(T -T0) , ( 2b
)
where k ln (ε ln ) and k hn (ε hn ) are the thermal conductivities (emissivities) at low (T T 0 ) and high (T T 0 ) temperatures, respectively; and β is the phasetransition slope of k n (T ) and ε n (T ) at T = T 0 . In order to fulfill the external thermal insulation condition depicted in Fig. 1, i consider that k l1 > k h1 and k l3 < k h3 , which along with Eq.(2a) determines the behaviors of k 1 (T ) and k 3 (T ) shown in Fig. 2(a). Note that the PCM1 at low temperature (T T 0 ) is in its metallic phase, which switches to an insulating one at high temperature (T T 0 ). Taking into account that metals usually exibit emissitities lower than insulators, i consider that ε l1 < ε h1 , as shown in Fig. 2(b). Therefore, the PCM1 emissivity, as the thermal conductivity k 1 , is bounded between its values for the metallic and insulating phases, such that the ratio ε h1 /ε l1 > 1 allows enhancing (reducing) the radiative heat flux with PCM1 in its insulating (metallic) phase, as established by Eq. [START_REF] Laboratory | Energy flow charts: Charting the complex relationships among energy, water, and carbon[END_REF]. For the emissitivy of PCM3, on the other hand, i take ε l3 > ε h3 (Fig. 2(b)), given that k 3 (T ) has the opposite behavior of k 1 (T ). In practice, the temperature dependence of k 1 (T ) and ε 1 (T ) can be obtained with polyethylene nanofibers [START_REF] Zhang | [END_REF] and Ge 2 Sb 2 Te 5 [39], while that of k 3 (T ) and ε 3 (T ) is exhibited by VO 2 [3,[40][41][42] and nitinol [43]. The phase transition of these four PCMs is driven by correlated atomic interactions and therefore the values of their thermal and optical properties at low and high temperatures can be tailored through dopings or by varying the deposition parameters, as was reported for the metal-insulation transition of VO 2 [44][45][46][47]. Furthermore, the temperature variations of k n (T ) and ε n (T ), for n = 1 and 3, can also be achieved by means of alternating layered structures involving strips of shape memory alloys, as detailled by Shen et al. [30] and Wang et al. [31]. This latter multistep approximation method is particularly suitable to make sure that the values of the transition parameters T 0 and β are the same for both materials surrounding the cavity, given that they will be composed by the same shape memory alloy. In general, the thermal and optical properties of these alloys and PCMs vary significantly due to their metal-insulator transition in a relatively narrow interval of temperatures. This phase transition is the reason why the operation of the radiative thermostat (Fig. 1) is driven by the collective dynamics of electrons, phonons, and photons.

T e m p e r a t u r e , T P C M t h e r m a l c o n d u c t i v i t y ,

k n T 0 k l 1 k l 3 k h 1 k h 3 ( k l 1 + k h 1 ) / 2
The temperature profiles T 1 (x) and T 3 (x) along with the heat flux q can be determined by integrating Eq. ( 1) for the thermal conductivity in Eq. (2a) and the boundary conditions

T 1 (0) = T h and T 1 (l 1 ) = θ (T 3 (l 1 + l 2 ) = ψ and T 3 (l 1 + l 2 + l 3 ) = T c ).
Final results can be written as follows

q = k h1 -k l1 β 1 l 1 ln f 1 (T h ) f 1 (θ) = k h3 -k l3 β 3 l 3 ln f 3 (ψ) f 3 (T c ) = θ -ψ ρ 2 + σε θ 4 -ψ 4 , (3a)
f 1 (T 1 (x)) = [f 1 (T h )] 1-x/l1 [f 1 (θ)] x/l1 , (3b)
f 3 (T 3 (x)) = [f 3 (ψ)] 1-(x-l1-l2)/l3 [f 3 (T c )] (x-l1-l2)/l3 , (3c)
where

f n (T ) = [1 + Z n (T )] [Z n (T )] -λn , Z n (T ) = exp [-β n (T -T 0 )] and λ n = k hn /(k hn -k ln ), for n = 1 and 3. Taking into account that k l1 > k l1 (k l3 < k h1 ), the function f 1 (T ) (f 3 (T )
) decreases (increases) monotonically as the temperature T rises and therefore the heat flux q in Eq. (3a) only takes positive values, in agreement with the second law of thermodynamics, for T h > T c . Given that the heat flux q is a constant, the third equality in Eq. (3a) establishes a tradeoff between the conductive and radiative contributions, such that the latter one strengthens as the conductive thermal resistance ρ 2 increases. The temperatures T 1 (x) and T 3 (x) are determined by the respective numerical solutions of Eqs. (3b) and (3c), after solving the strongly non-linear system of Eqs. (3a), for the temperatures θ and ψ of the inner walls. In order to capitalize on the full phase transition of the PCMs and derive an approximate solution of Eqs. (3a), the external temperatures T h and T c are chosen in such a way that they fulfill the condition T h T 0 T c . In this way, the cavity is thermally insulated from the external environment and therefore the temperatures θ and ψ of the inner walls are expected to satisfy the condition β |θ -T 0 | 1 and β |ψ -T 0 | 1, as graphically shown below (Fig. 3(a)). Under these conditions, Eqs. (3a) become a system of two linear equations for θ and ψ, which yields

θ -ψ = 2R 2 R(r + 1) T h -T c + 2 ln(2) (r -1) β . (4)
where

R = R 1 + R 2 + R 3 , R n = l n / kn , kn = (k ln + k hn )/2, for n = 1 and 3, R 2 = ρ -1 2 + 4σ εT 3 0 -1 , ε = ε-1 1 + ε-1 3 -1 -1 , εn = (ε ln + ε hn )/2, and r = k l1 /k h1 .
For the sake of simplicity and clarity, Eq. ( 4) was derived by using the condition k l1 = k h3 and k h1 = k l3 , which establishes the mirror-like symmetry of k 1 (T ) and k 3 (T ) around the transition temperature T = T 0 . Equation ( 4) indicates that the temperature difference θ -ψ can be minimized by reducing the thermal resistance R 2 , enhancing the thermal conductivity contrast r > 1 and getting steep phase transitions (high β values). As these three parameters are determined by the thermal and optical properties of the considered PCMs, the values of θ -ψ are expected to be much smaller than the corresponding ones obtained with non-PCMs (θ -ψ = R 2 (T h -T c )/R for r = 1). Note also that the radiative contribution boosts the reduction of θ -ψ through a high emissivity ε and/or transition temperature T 0 , which only appears due to the intracavity radiative heat transport.

III. RESULTS AND DISCUSSION

The thermal performance of the radiative thermostat is now numerically analyzed by means of the temperature profiles for different values of the environmental temperatures and thermal resistance ρ 2 . To show how the cavity temperature could be maintened at room temperature, we choose T 0 = 300 K and change the temperature T h through values higher than T 0 to capitalize on the phase transition of noth PCMs. For the sake of comparison, the temperature T c = 290 K is kept constant, given that its variations through values T c T 0 do not change either the thermal conductivity or emissivity of the PCM3, and therefore they will not significantly affect the thermostat temperature profile.

Figures 3(a) and 3(b) show the inner walls' temperatures θ and ψ along with their difference θ -ψ > 0 as functions of T h , respectively, for three representative values of the thermal resistance ρ 2 . Note that both θ and ψ increases almost linearly with T h , when this temperature is smaller than the transition one (T h < T 0 ). This behavior is typically generated by non-PCMs and appears due to the fact that the environment temperature T h is too low to excite the phase transition of both PCMs (see Fig. 2). By contrast, for T h > T 0 , the 1). This latter one was obtained for non-PCMs with thermal conductivities k n = kn and emissivities ε n = εn .

= l 3 = 3 cm, T 0 = 300 K, β = 2 K -1 , k l1 = k h3 = 50 Wm -1 K -1 , k h1 = k l3 = 1 Wm -1 K -1 , ε l1 = ε h3 = 0.2, ε h1 = ε l3 = 0.
slopes of the inner walls' temperatures reduce drastically and their values tend to keep close to T 0 , especially for the two smaller values of ρ 2 . In this case, the temperature difference θ -ψ is smaller than 1 K and is well predicted by Eq. ( 4), as shown by the blue and red dashed lines in Fig. 3(b). This key fact indicates that the temperature of the central material can be maintained pretty much constant and equal to the transition temperature of both PCMs, without the injection of external energy, when the environment temperatures vary through values higher and lower than T 0 (T h > T 0 > T c ). In practice, the thermal resistance ρ 2 = 0.5 cm 2 KW -1 could be set with either a 2-cm thick thermostat of copper (l 2 = 2 cm and k 2 = 400 Wm -1 K -1 ) or a 1.25 µm-thick cavity of air (l 2 = 1.25 µm and k 2 = 25 mWm -1 K -1 ).

Thicker air cavities yield a higher thermal resistance ρ 2 , which increases the temperature difference of the inner walls to values higher than 1 K. However, the values of θ -ψ generated by the combined effect of two PCMs for different values of ρ 2 , are relatively smaller than the corresponding ones obtained with non-PCMs, as shown by the dotted lines in Fig. 3(b). Therefore, the temperature profile of the central material (solid or gas) becomes considerably less sensitive to the environment temperature gradients, when is placed in between two PCMs with asymmetric thermal and optical properties than when it is in the middle of two non-PCMs. The temperature and thermal conductivity distributions inside the three-layer system are shown in Fig. 4 as functions of position, for a 2 cm-thick central layer of copper. The dashed line represents the temperature profile obtained when the two PCMs are replaced by non-PCMs with thermal conductivities and emissivities independent of temperature. Note that near the outer surfaces x = 0 and 8 cm, the temperature exhibits a pretty much linear behavior due to the relatively constant and low values of the 3) and (2a) inside the three-layer system, as functions of position, respectively. The dashed line stands for the prediction of Eq. ( 1) for non-PCMs with thermal conductivities k n = kn and emissivities ε n = εn . Calculations were done for a Cu central layer with l 2 = 2 cm, ρ 2 = 0.5 cm 2 KW -1 , T h = 310 K and the data in caption of Fig. 3.

thermal conductivity of both PCMs. For other internal positions, on the other hand, these conductivities change drastically to keep the temperature nearly constant and equal to the transition temperature T 0 = 300 K, specially within the central layer. This fact is consistent with the one shown in Fig. 3 and is, of course, generated by the combined phase transitions of the two PCMs.

Even though the non-linear behavior of temperature within the PCMs is totally different than the linear one (dashed line) obtained with non-PCMs, the temperature values inside the cavity are pretty much the same in both cases. This is reasonable due to the relatively high thermal conductivity of the copper layer and it is expected to change for a cavity with lower (higher) thermal conductivity (thermal resistance ρ 2 ), as shown in Fig. 5. While the cavity temperature tends to keep close to T 0 = 300 K, the corresponding one obtained with non-PCMs significantly deviates from this temperature by many degrees, in particular for ρ 2 = 50 cm 2 KW -1 .

It is thus clear that the combined operation of two PCMs is better than non-PCMs to develop energy-free thermostats capable of maintaining a nearly constant temperature as the ambient temperature gradients change.

IV. CONCLUSIONS

The proof of principle of a radiative thermostat has been demonstrated by exploiting the metal-insulator transition of two phase change materials with asymmetric thermal conductivities and emissivities. It has been shown that the thermostat is capable of maintaining a constant temperature equal to the transition one of the

APPENDIX

In practice, the asymmetric thermal conductivities of PCM1 and PCM3 shown in Fig. 1(a), can conveniently be implemented by means of layered structures made up of still and movable parts, as shown in Fig. 6(a). Following the experiments reported in the literature [30], the still part (bottom layer) is composed of five phosphor copper films (k p = 54 Wm -1 K -1 ) separated by five silicon grease (k s = 4 Wm -1 K -1 ) layers. On the other hand, the movable part (top layer) is made up of five bimetallic strips placed on top of the silicon grease layers. Each of these strips is composed of a phosphor copper film assambled with a shape memory alloy (SMA), such that their length and width are equal to the corresponding ones of the phosphor copper films. These latter films are slightly thicker than the silicon grease layers to level the bimetallic strips. The SMA is an alloy able to change its shape with temperature. In particular, a two-way SMA "remembers" two different shapes related to a phase transformation, one at low tempratures and another one at high temperatures, when it is heated up (or cooled down) through its transition temperature T t [48,49]. Based on this fact, a two-way SMA able to tilt up for temperatures T T t = 300 K and fully level for T T t , is chosen for builing the PCM1 and PCM3 of the proposed radiative thermostat. Taking into account that these PCMs are set at high and low temperatures (see Fig. 1), their bimetallic strips will be driven up and down with an opposite deformation, but a common transition temperature T 0 = T t = 300 K. According to Fig. 6(a), the "experimental" thermal conductivities k 1 and k 3 of the respective PCM1 and PCM3 can then be estimated by the well-known model of thermal resistances in series, as follows

2N k 1 = N k p + n k s + N -n k c , ( 5a
) 2N k 3 = N k p + n k c + N -n k s , (5b)
where n = 0, 1, ..., N = 5 is the number of bimetallic strips tilted up (leveled) of PCM1 (PCM3) and k c is the thermal conductivity of one of these strips.

Considering that a bimetallic strip is composed of a phosphor copper film attached to a SMA one of thermal conductivity k SM A , its effective thermal conductivity is determined by the model of thermal resistance in parallel: k c = (l p k p + l SM A k SM A )/(l p + l SM A ), with l p and l SM A being the heights through which the heat propagate inside the phosphor copper and SMA films, respectively. Taking into account that k SM A can also change with temperature [49], the condition l SM A l p is assumed, such that k c ≈ k p , which is nearly independent of temperature, for the range of temperatures considered in this work. Figure 6(b) shows the "experimental" predictions (solid lines) of Eqs. (5a) and (5b) in comparison with the respective "theoretical" ones (dashed lines) predicted by Eq. (2a) for the thermal conductivities of PCM1 and PCM3. One can see that the solid curves are similar to the corresponding dashed ones describing the asymmetric temperature variations of k 1 and k 3 between 295 and 305 K. In practice, these variations can be smoother by increasing the number 2N of layers, as established by Eqs. (5a) and (5b). The proposed methodology based on a SMA could thus be implemented to observe the operation of a radiative thermostat, in a similar way as was done with the conductive one [30]. ) Temperature, T (K)

k 1 k 3 (b)
FIG. 6: (a) Schemes of the PCM1 and PCM3 composed of layered systems based on bimetallic strips. These strips tilt up for low temperatures (T 300 K) and level for high ones (T 300 K), due to its composition with a SMA, as demonstrated in the literature [30]. The thermal conductivities of PCM1 and PCM3 thus depend on the shape of the bimetallic strips and their temperature dependences are shown in (b). The dashed lines stand for the predictions of Eq. (2b), while the solid ones are predicted by Eqs. (5a) and (5b), assuming the leveling of 0, 1, 2, 3, 4, and 5 bimetallic strips. Calculations in (b) were done considering that the shape change of a strip is generated by a temperature variation of 1 K followed by 0.5 K between the transformations of two consecutive strips.

0 e l 1 e l 3 e h 1 e h 3 ( e l 1 + e h 1 3 eFIG. 2 :

 1131132 FIG. 2: Temperature dependence of the (a) thermal conductivities and (b) mean emissivities of the two PCMs involved in Fig. 1. The black line in (b) stands for the effective emissivity ε defined in Eq.(1).

- 1 2 = 5 cm 2 KW - 1 2FIG. 3 :

 113 FIG. 3: (a) Inner walls temperatures and (b) their difference as functions of the temperature T h . Calculations were done withT c = 290 K, l 1 = l 3 = 3 cm, T 0 = 300 K, β = 2 K -1 , k l1 = k h3 = 50 Wm -1 K -1 , k h1 = k l3 = 1 Wm -1 K -1 , ε l1 = ε h3 = 0.2, ε h1 = ε l3 = 0.8 and three values of ρ 2 . The solid, dashed, and dotted lines in (b) stand for the respective predictions of Eqs. (3a), (4), and (1). This latter one was obtained for non-PCMs with thermal conductivities k n = kn and emissivities ε n = εn .

  FIG. 3: (a) Inner walls temperatures and (b) their difference as functions of the temperature T h . Calculations were done withT c = 290 K, l 1 = l 3 = 3 cm, T 0 = 300 K, β = 2 K -1 , k l1 = k h3 = 50 Wm -1 K -1 , k h1 = k l3 = 1 Wm -1 K -1 , ε l1 = ε h3 = 0.2, ε h1 = ε l3 = 0.8 and three values of ρ 2 . The solid, dashed, and dotted lines in (b) stand for the respective predictions of Eqs. (3a), (4), and (1). This latter one was obtained for non-PCMs with thermal conductivities k n = kn and emissivities ε n = εn .

1 K - 1 ) k 1 k 3 TP C M 1 P C M 3 T h e r m o s t a t FIG. 4 :

 11313t4 FIG.4: Temperature and thermal conductivity distributions obtained with Eqs. (3) and (2a) inside the three-layer system, as functions of position, respectively. The dashed line stands for the prediction of Eq. (1) for non-PCMs with thermal conductivities k n = kn and emissivities ε n = εn . Calculations were done for a Cu central layer with l 2 = 2 cm, ρ 2 = 0.5 cm 2 KW -1 , T h = 310 K and the data in caption of Fig.3.

- 1 r 2 = 5 c m 2 K W - 1 r 2 = 5 0 c m 2 K W - 1 PFIG. 5 :

 12212215 FIG.5: Temperature profiles within the layered system shown in Fig.1, for three values of ρ 2 . The dashed lines stand for the corresponding predictions of Eq. (1) for non-PCMs with thermal conductivities k n = kn and emissivities ε n = εn . Calculations were done with l 2 = 2 cm, T h = 310 K and the data in caption of Fig.3.


Michel A Isidro-Ojeda

Jose Ordonez-Miranda

Luis C Malacarne

Juan J Alvarado-Gil

Thermal and thermoelastic response of materials with thick-disk geometry excited by a ring-shaped laser beam

Keywords: numbers: Valid appear here Ring-shaped, Photothermal effects, Thermal lensing

I. INTRODUCTION

Laser technology introduced new ideas for the design of experimental setups and photothermal spectroscopy is not an exception. This spectroscopic technique is based on the transformation of light into heat at the sample. And it has been used for materials characterization. So, the knowledge of the laser beams is crucial for the correct interpretation of the heat transfer through the materials. When a laser beam impinges a material, different effects occur. Gaussian laser beams have been widely used and characterized, also to being used in numerous experiments, [START_REF] Klein | Describing beam-aberration effects induced by laser-light transmitting components: a short account of raytheon's contribution[END_REF][START_REF] Winkler | Heating by optical absorption and the performance of interferometric gravitational-wave detectors[END_REF][START_REF] Sparks | Optical distortion by heated windows in high-power laser systems[END_REF][START_REF] Mansuripur | Diffraction modeling of optical path for magneto-optical disk systems[END_REF][START_REF] Klein | Optical distortion coefficients of high-power laser windows[END_REF][START_REF] Peng | Influence of thermal deformations of the output windows of high-power laser systems on beam characteristics[END_REF][START_REF] Klein | Analytical stress modeling of high-energy laser windows: application to fusion-cast calcium fluoride windows[END_REF] which has been used in various photothermal techniques for the characterization of materials. Research on other forms of distribution, such as the top-hat laser beam, has also been studied. [START_REF] Astrath | Top-hat cwlaser-induced time-resolved mode-mismatched thermal lens spectroscopy for quantitative analysis of low-absorption materials[END_REF][START_REF] Zhao | Compensation of strong thermal lensing in high-opticalpower cavities[END_REF] There is an excellent work describing the temperature profile for a ring-profile laser beam. [START_REF] Kim | Influence of a ring-shaped pump beam on temperature distribution and thermal lensing in end-pumped solid state lasers[END_REF] Many techniques are based on laser beam technologies in which the laser beam profile is very important for the correct description of the phenomena, some examples are thermal lens spectroscopy [START_REF] Shen | Three-dimensional model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin-film samples[END_REF][START_REF] Luterotti | Ultrasensitive determination of trans-β-carotene in rat and beef livers by means of high-performance liquid chromatography coupled with thermal lens detection[END_REF][START_REF] Franko | Analytical thermal lens instrumentation[END_REF][START_REF] Franko | Determination of trans-β-carotene and other carotenoids in blood plasma using high-performance liquid chromatography and thermal lens detection[END_REF][START_REF] Astrath | Timeresolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20 k[END_REF] and thermal mirror spectroscopy. [START_REF] Astrath | Time-resolved thermal mirror for nanoscale surface displacement detection in low absorbing solids[END_REF][START_REF] Malacarne | Nanoscale surface displacement detection in high absorbing solids by time-resolved thermal mirror[END_REF][START_REF] Zanuto | Thermal mirror spectrometry: An experimental investigation of optical glasses[END_REF][START_REF] Bianchi | Resonant excited state absorption and relaxation mechanisms in tb 3+-doped calcium aluminosilicate glasses: an investigation by thermal mirror spectroscopy[END_REF] These have been used successfully in characterizing many materials. To a) michel.isidro15@gmail.com improve the design of an experimental setup used in optical experiments using many lenses, mirrors and different optical components; it is necessary to know what happens to the instrumentation used, especially in experiments that include high-power laser beams; for example, when samples with very low optical absorption (water, alcohols, glasses, etc.) are explored. Optical path difference (OPD) is the basis of these methods. Analytical solutions to the complex thermoelastic problem have been approached using simplifications, these are known as thin-disk [START_REF] Zhu | Analytical model of optical path difference in an end-pumped yb: Yag thin-disk laser with nonuniform pumping light[END_REF] or long-rod 5 approximations. A complete analysis for material with thick-disk geometry that is exposed to a Gaussian laser beam has been shown in a previous work. [START_REF] Isidro-Ojeda | Laser induced wavefront distortion in thick-disk material: An analytical description[END_REF] The present study considers a ring-shaped laser beam, it should be mentioned that this type of laser beam can provide constant power at the center of a sample at steady-state. The finite element method (FEM) has been used to predict how a system behaves in different situations. [START_REF] Sazegari | Structural and optical behavior due to thermal effects in end-pumped yb: Yag disk lasers[END_REF][START_REF] Milani | Calculating optical path difference in end-pumped yb: Yag thin disk lasers[END_REF][START_REF] Zhu | Analytical model of thermal effect and optical path difference in endpumped yb: Yag thin disk laser[END_REF][START_REF] Aminpour | Optical path difference of slanted edge diode-pumped yb: Yag/yag thin-disk laser[END_REF][START_REF] Anufriev | Measurement of the phonon mean free path spectrum in silicon membranes at different temperatures using arrays of nanoslits[END_REF] For this reason, the FEM is an excellent complement to the theoretical analysis developed in this work. Using the equations obtained and the FEM, different numerical analyzes were performed and a good agreement was calculated between methods. Furthermore, our equations for a ring-shaped laser beam were compared with a Gaussian profile to observe the differences between temperature, displacement, stresses and OPD profiles.

To compare the results obtained, other models were used that have been used to describe the steady-state temperature profile. These models for a Gaussian 26 and ringshaped 27 laser beam show a very good agreement with the results of this work for both the analytical model and the FEM. This gives the confidence to continue with the solution of the thermoelastic equation. Furthermore, these results obtained for a ring-shaped laser beam could be used for the determination of thermal and optical properties as has been done using different laser beams profiles. The goal of this work is to provide analytical expressions to describe the temperature profile produced by a ringshaped laser beam in a material with thick-disk geometry and the consequent deformation. And to obtain a more general expressions for laser beams at steady-state.

II. THEORETICAL MODEL

A. Temperature distribution Fig. 1 shows that both the sample and laser beams can be described using a cylindrical coordinate system. This geometric configuration is very convenient considering that the sample has a thick-disk shape and that the laser beam profile is ring-shaped. So, the heat diffusion equation is given by

∂∆T (r, z, t) ∂t -D∇ 2 ∆T (r, z, t) = Q(r, z), (1)
where ∆T (r, z, t) is the temperature difference according to the radius r, thickness z and time t. D = k/ (ρc) is the thermal diffusivity where k, ρ and c are the thermal conductivity, density and specific heat of the sample, respectively. Q(r, z) is the t-independent heat source. Q(r, z) = Q 0 f (r)e -Aez , where 8,27,28

Q 0 = 1 2 Q1 ω 2 e -ω 2 i
, for a ring-shaped laser beam

Q1 ω 2 G
, for a Gaussian laser beam ,

where

Q 1 = 2AP 0 (1 -R)/(πcρ).
A is the optical absorption coefficient at the wavelength of the laser beam. P 0 , ω e and ω i are the power, the external and the internal radius of the laser beam, respectively. ω G is the radius of the Gaussian laser beam where its intensity decreases to 1/e 2 . R is the material's surface reflectivity. For optical systems with low optical absorption, the approximation e -Aez → 1 can be used. Laser beam profile f (r) is as follows

f (r) = θ(r -ω i )θ(ω e -r)
, for a ring-shaped beam e -2r 2 /ω 2 G , for a Gaussian beam ,

(3) where θ is the Heaviside function. Hence, Eq. ( 1) is rewritten z-independent as Eq. ( 4) is subject to the boudary condition ∆T (a, t) = 0 and the initial condition ∆T (r, 0) = 0. According to the method of separation of variables, the temperature distribution can be described by an infinite sum of zero-order Bessel functions J 0 , as follows

∂∆T (r, t) ∂t -D 1 r ∂ ∂r r ∂ ∂r ∆T (r, t) = Q(r). (4)
∆T (r, t) = ∞ n=1 A n (t)J 0 α n r a , (5)
where α n are the zeros of J 0 . To find A n (t), the laser beam must be represented similarly to ∆T (r, t), in order to do that, we write down

f (r) = ∞ n=1 B n J 0 α n r a , (6)
where B n is found taking into account the orthogonality conditions. Therefore, we get the expressions for B n as follows

B n,R = 1 aJ 2 1 (α n ) 2 α n ω e J 1 α n ω e a -ω i J 1 α n ω i a (7a)
and

B n,G = 1 aJ 2 1 (α n ) ω 2 G 2a e -1 2 ( αn ω G 2a ) 2 , (7b)
for a ring-shaped and Gaussian laser beam, respectively. By replacing one of Eqs. (7) in Eq. ( 6), the laser beam profile is obtained in terms of a sum of first-order Bessel functions J 1 . It can be seen in Eq. (7a) that when ω e = ω i the value of B n,R is equal to zero, this means that there is no heat source as expected. When ω i = 0 and taking into account that ω e,G ≤ a, then B n,R = 2B n,G . This last result suggests that Q(r) for a Gaussian profile is a consequence of ring-shaped Q(r) at steady-state and when l 0 a. Substituting Eq. ( 5) and Eq. ( 6) in the heat diffusion equation given by Eq. ( 4), we have the next differential equation

d dt A n (t) + Dα 2 n a 2 A n (t) -B n Q 0 = 0, ( 8
)
whose solution is given by

A n (t) =B n Q 0 a 2 Dα 2 n 1 -e -t τ α 2 n , (9)
where τ = a 2 /D is the thermal diffusion time and B n is given by Eqs. (7). Eq. ( 9) is very important because it describes the shape and behavior of the laser beam. Where the second term (in parentheses) contains the time dependency, where it can be seen that when t → ∞ this term is equal to 1, this way the steady-state is obtained. This last approach is very important in practice because laser beams need stabilization before using them in experiments. Substituting Eq. ( 9) in Eq. ( 5), with B n given by Eq. (7a), we have an equation that describes the temperature profile when a ring-shaped laser beam excites a sample. This equation matches very well the set of equations (3) found by Kim et al. [START_REF] Kim | Influence of a ring-shaped pump beam on temperature distribution and thermal lensing in end-pumped solid state lasers[END_REF]

B. Thermoelastic equation

Heating produces a deformation in the sample. For this reason, it is important to understand the thermoelastic behavior that said heat produces. It is called thermoelasticity because the deformation of the structure does not exceed the elastic limit and when the heat source disappears the deformation also does. 29 Thermoelasticity theory makes possible to calculate the stresses in a body produced by a temperature field. This theory includes heat conduction, strains and stresses due to a heat source. 30 As has been shown in a previous work, [START_REF] Isidro-Ojeda | Laser induced wavefront distortion in thick-disk material: An analytical description[END_REF] the displacement vector (u i ) and the stress components (σ ij ) are obtained from

u i (r, z, t) =∂ i Ψ (r, z, t) + 1 1 -2ν 2 (1 -ν) δ zi ∇ 2 -∂ zi ψ (r, z, t) (10)
and

σ ij (r, z, t) =2µ ∂ ij -δ ij ∇ 2 Ψ (r, z, t) + 2µ 1 -2ν ∂ z νδ ij ∇ 2 -∂ ij + 1 -ν) ∇ 2 (δ iz ∂ j + δ jz ∂ i ) ψ (r, z, t) , (11)
where ∇ 2 is the Laplacian operator, δ ij is the the Kronecker delta function.

µ = Y / [2 (1 + ν)]
where ν is the Poisson ratio and Y is the Young's modulus of the sample. Using the thermoelastic displacement potential Ψ and Love function ψ, the thermoelastic equation can be expressed by

∇ 2 Ψ(r, t) = m∆T (r, t) (12)
and

∇ 2 ∇ 2 ψ(r, z, t) = 0. (13)
Eqs. ( 12) and ( 13) are known as the Poisson and Love equations, respectively. m = α T (1 + ν) / (1ν), where α T is the linear thermal expansion coefficient. Using Eq. ( 5), solutions of ( 12) and ( 13) take the following form

Ψ(r, t) = - ∞ n=1 ma 2 α 2 n A n (t)J 0 α n r a (14)
and

ψ (r, z, t) = ∞ n=1 J 0 αnr a (α n /a) 2 B 1 + B 2 α n a z e αnz/a + C 1 + C 2 α n a z e -αnz/a , (15)
respectively. The solutions for the constants B 1 , B 2 , C 1 and C 2 can be consulted in the reference, [START_REF] Isidro-Ojeda | Laser induced wavefront distortion in thick-disk material: An analytical description[END_REF] when σ zz = σ rz = 0 at z = 0 and z = l 0 . To account for edge deformation for a thick-disk material, it is necessary to use 30

u z → u z + 2N ν (z -l 0 /2) Y , (16)
σ rr → σ rr -N and σ φφ → σ φφ -N. (17)
Then, the displacement on the faces of the sample can be expressed as

u z (r, 0, t) = -u z (r, l 0 , t) = ∞ n=1 A n (t)α T α n -2α n h(α n )l 0 (ν + 1) J 0 α n r a + l 0 νJ 1 (α n ) ν -1 (4h(α n )ν -1) (18)
with

h(α n β) ≡ cosh (α n β) -1 α n β [sinh (α n β) + α n β] , (19)
where β = l 0 /a. Fig. 2 shows the behavior of Eq. ( 19) for the first thirty-five α n values that produce J 0 (α n ) = 0. It can be seen that when β → 0 it is obtained that h(α n β) = 1/4, this means that the thin-disk approximation is reached. Now, using the general equations for the OPD shown in a previous work [START_REF] Isidro-Ojeda | Laser induced wavefront distortion in thick-disk material: An analytical description[END_REF] and equations found here, we have a complete set of equations for the correct description of the OPD when a laser beam heats a material with thickdisk configuration.

III. RESULTS AND DISCUSSION

Fig. 3 shows the temperature distribution produced by a ring-shaped laser beam and one with a Gaussian profile. Data used for the calculus can be found in a previous work [START_REF] Isidro-Ojeda | Laser induced wavefront distortion in thick-disk material: An analytical description[END_REF] for a BK7 sample. It can be seen a very good agreement between Eq. ( 5) with A n (t) given by Eq. ( 9) and B n given by Eq. (7a), found in this work and represented by continuous lines, and Eqs. (3) reported by Kim et al. [START_REF] Kim | Influence of a ring-shaped pump beam on temperature distribution and thermal lensing in end-pumped solid state lasers[END_REF] represented by diamonds. It is shown how the temperature profile changes as a function of ω i . The value of ω e is kept constant. When ω i becomes zero, it gives a top-hat profile. In the same figure, the result of a Gaussian laser beam (Eq. ( 5) with Eq. ( 9) and Eq. (7b)), found in a previous work is shown represented by the dashed line. [START_REF] Isidro-Ojeda | Laser induced wavefront distortion in thick-disk material: An analytical description[END_REF] The latter is compared with the equation found by Gordon et al., [START_REF] Gordon | Long-transient effects in lasers with inserted liquid samples[END_REF] represented by triangles, also obtaining a very good agreement. It is important to mention that these curves were obtained at steady-state, this means that t → ∞. The FEM was used to validate the theoretical model proposed in this work. Circles and squares are for ring and Gaussian FEM simulations.

Fig. 4 shows the temperature dependence as a function of r and ω i for the x-axis and y-axis, respectively. The outer radius for the ring-shaped laser beam was set at ω e = 1 mm. It can be seen that for r = ω e the value of ∆T decreases independently of the value given for the internal radius ω i . Furthermore, the behavior of the decrease of ∆T is the same after that value of r = ω e . It is important to say that when the ω i = ω e there is no representation because there is no laser beam.

In Fig. 5 is shown the temperature distribution in a sample for (a) a ring-shaped and (b) a Gaussian laser beam. These were generated using the models found in this and a previous work. [START_REF] Isidro-Ojeda | Laser induced wavefront distortion in thick-disk material: An analytical description[END_REF] These graphs allow observing how the sample is affected by the incidence of a ringshaped and a Gaussian laser beam. Furthermore, in the models, these calculations can be carried out as a function of time, then the behavior of the sample since the laser beam is applied on the sample can be recorded until it reaches the steady-state. Fig. 6 shows the displacement calculations using Eq. (18). The FEM was used as a validation method, and we can see that there is a good agreement between both methods. In addition, a comparison with the Gaussian model is shown. For the ring model, ω e = 1 mm is maintained, while the value of ω i takes the values 0.8, 0.7, 0.6 and 0.5 mm. It can be seen that when the internal radius goes to zero, the maximum value of u z increases until reaching a behavior similar to that of a Gaussian laser beam.

Figs. 7 and8 show the radial and angular stress along the z-axis, respectively. It can be seen that the absolute maximum value is presented in the middle of the sample, this means at z = l 0 /2. As ω i goes to zero, the absolute maximum value increases. It should be noted that when ω i → 0, the data for the ring-shaped laser beam takes the form of σ rr and σ φφ for a Gaussian laser beam.

The stress along the z-axis for σ zz is presented in Fig. 9. It can be seen how σ zz goes from positive to negative values at the center of the sample as the internal radius is decreasing. The data shown correspond to ω i = 0.8, 0.7, 0.6 and 0.5 mm keeping the value of ω e = 1 mm as was done previously. Fig. 10 shows total OPD in a sample with l 0 = a = 2 mm. It can be observed how at the center of the graph it is shaped like a top-hat when a ring-shaped laser beam is used to heat. The fall of the curve that represents the ∆S appears in the internal radius of the laser beam, that is, in the value of ω i . Remember that these data were calculated using the steady-state.

To appreciate how the value of ∆S increases with time, different calculations were performed, which are shown in Fig. 11. These calculations were made at the center of the sample, this means when r = 0. As can be seen in the figure, ∆S does not rise instantaneously for the ringshaped laser beam, observe continuous lines, this makes sense because at the center of this type of laser beam, there is no heat source. This does not happen for a Gaussian laser beam as can be seen in the same figure, where it can be seen that the increase of ∆S is instantaneous, observe dashed lines. Furthermore, it can be seen that at t = 5 seconds, the value of the total OPD is almost constant, for this reason, a steady-state approximation can be used successfully for the data calculations when studying low absorption samples with thick-disk geometry by using a ring-shaped laser beam.

IV. CONCLUSIONS

We have solved the heat diffusion and thermoelastic equation for a laser beam with a ring profile. We have shown some differences concerning the profile of the Gaussian laser beam. In both cases, we have used the FEM as a validation method, and we have obtained very good results. Expressions that have been found are more general for the description of a laser beam, going from a ring-shaped to a Gaussian-shape at steady-state. We have calculated the percentage differences between expressions found and FEM for the temperature, displacement and optical path difference at the center(edge) of the sample; these are approximately 0.14 % (0 %), 0.11 % (30.50 %) and 0.09 % (4.86 %), respectively. The ad- vantage of a ring-shaped laser beam over the Gaussian laser beam is that the former shows a homogeneous circular temperature distribution at steady-state, which is known as a top-hat distribution. These calculations can be used in the design of optical instrumentation and the characterization of low absorption samples with thickdisk geometry when a laser beam with a ring profile is used.

FIG. 1 .

 1 FIG.1. Geometry of the laser beam profile and sample. This is important for the coordinates' system used in the solution of the heat and thermoelastic equations. (a) Ring-shaped laser beam: a, l0 are the radius and thickness of the sample, respectively. ωi and ωe are the internal and external radius of the laser beam, respectively. (b) Gaussian laser beam: a, l0 are the radius and thickness of the sample, respectively. ωG is the radius of the Gaussian laser beam where its relative intensity is 1/e 2 .

FIG. 2 .

 2 FIG. 2.Behavior of h(αnβ) for J0(αn) = 0, n = 1, ..., 35. It can be seen that when h(β → 0) = 1/4, this means the thin-disk approximation (l a).

FIG. 3 .

 3 FIG.3. Temperature profile created by a ring-shaped laser beam varying the internal radius ωi with a constant external radius ωe. ωG is the radius for a Gaussian laser beam. All these data were created at steady-state, this means t → ∞.

FIG. 4 .

 4 FIG. 4. Temperature difference, ∆T , as a function of r (xaxis) and ωi (y-axis). The value of ωe is set at 1 mm. Also, a = l0 = 2 mm.

FIG. 5 .FIG. 6 .

 56 FIG. 5. Temperature distribution generated with the (a) ringshaped and (b) Gaussian model.

FIG. 7 .FIG. 8 .

 78 FIG.7. Radial stress for a material with thick-disk geometry at r = 0 by using a ring-shaped laser beam. The sample has the dimensions l0 = a = 2 mm. It is shown a comparison with the FEM and the model that describes a Gaussian laser beam.

FIG. 9 .FIG. 10 .

 910 FIG.9. Stresses along the z-axis are shown when a ringshaped laser beam at r = 0 is used. It can be seen how the behavior of the graph goes from positive values to negative values when ωi is decreasing. The absolute maximum value is obtained when ωi = 0, recovering the data for a Gaussian laser beam with ωG = 1 mm as can be seen.

  FIG. 11. OPD time-resolved for the first 5 seconds at the center of material with thick-disk geometry.

S. Timoshenko and J. Goodier, "Theory of elasticity, by s. timoshenko and j.n. goodier, ... 2nd edition," (1951).

 30 W. Nowacki, "Thermoelasticity, by w. nowacki," International Series of Monographs on Aeronautics and Astronautics. Division I: Solid and Structural Mechanics, V.3 (1962).