Heat conduction in ultrafast thin-film nanocalorimetry

Thermochimica Acta 640 (2016) 42–51

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Heat conduction in ultrafast thin-film nanocalorimetry Alexander A. Minakov a , Christoph Schick b,∗ a b

A.M. Prokhorov General Physics Institute, Vavilov st. 38, 119991 Moscow, Russia University of Rostock, Institute of Physics, Albert-Einstein-Str. 23-24, 18051 Rostock, Germany

a r t i c l e

i n f o

Article history: Received 30 May 2016 Received in revised form 19 July 2016 Accepted 29 July 2016 Available online 1 August 2016 Keywords: Thin-film nanocalorimetry Thermal contact resistance Non-Fourier heat conduction Temperature-modulated calorimetry Ultra-Fast calorimetry

a b s t r a c t Ultrafast nanocalorimetry on the basis of thin-film gauges was developed and utilized for studying phase-transition kinetics in thin-film samples. Ultrafast measurements at increasing rates of temperature change require a comprehensive analysis of the heat conduction processes. In this paper the analysis of the fast thermal response in thin-film samples at the boundary conditions corresponding to ultrafast calorimetric gauges is performed. The effects of interfacial thermal contact resistance and non-Fourier heat conduction are calculated analytically. The limits of validity of the classical diffusive heat conduction theory at high-frequency temperature-modulation measurements are studied. The alternatives of Fourier and non-Fourier heat conduction are considered. The conditions optimal for detection of the effect of non-Fourier heat conduction are defined. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Thin-film ultrafast calorimetry was intensively developed in the last two decades [1–16]. It provides opportunities to generate nonequilibrium states and to study phase-transition kinetics at sub-millisecond timescales. The capabilities of thin-film calorimetry have been greatly enhanced with introduction of the low-stress amorphous silicon nitride membrane technology [9]. The scheme of such a thin-film calorimeter gauge from Xensor Integration, Nl, which can be utilized for ultrafast thin-film nanocalorimetry [8,10,11] is shown in Fig. 1. The technical details for the gauges are available on the website of Xensor Integration [12]. Calorimeters based on such thin-film (silicon-nitride membrane) gauges were recently constructed [6,7] and they were also utilized for investigation of thermodynamic processes in thin-film samples at high-frequency temperature-modulation [13–15]. Small thermal inertia and high resolution ca. nJ/K of thin-film calorimeters are possible due to the very small addenda heat capacity (ca. 10 nJ/K) of the silicon-nitride membrane gauges. The controlled ultrafast cooling and heating is possible up to 107 K/s [8,16] with the gauges similar to that available from Xensor Integration. Temperature-modulated calorimetry at frequencies up to 106 Hz was also realized [15]. However, the potential capability of thinfilm calorimetric gauges is still not attained. As a realistic goal it

∗ Corresponding author. E-mail address: [email protected] (C. Schick). http://dx.doi.org/10.1016/j.tca.2016.07.023 0040-6031/© 2016 Elsevier B.V. All rights reserved.

is considered to achieve at least 107 Hz for thin-film samples in Helium gas as the cooling medium. Increasing of the rate of the temperature change in calorimetric experiments is essential for research of irreversible processes in glass forming and nanoscale materials. However, a comprehensive analysis of heat conduction processes is required for correct measurements. In fact, the interfacial thermal contact conductance is a very significant parameter for ultrafast thermal measurements. Besides, the limits of validity of the classical diffusive Fourier heat conduction theory at ultrafast measurements should be determined. The possibility of the effect of nonequilibrium (non-Fourier) heat conduction at ultrafast thermal experiments should be taken into account. Actually, the effect of non-Fourier heat conduction can be essential, when the sample parameters are sharply varying in space and time. Relaxation effects in heat conduction were considered primarily by Maxwell [17], Cattaneo [18], and Vernotte [19]. Hyperbolic heat conduction equations (see below) were proposed for fast thermal processes as an alternative to the classical parabolic Fourier’s law. Recent trends in nanoscale devices and manufacturing processes promotes rapidly growing interest in non-Fourier heat conduction. Non-Fourier heat-conduction effects were found in dynamic thermal response of Graphene [20], single-walled carbon nanotube [21], and metal-dielectric core-shell nanoparticles [22]. The short-time heat conduction phenomena are of interest for the thermal design of micro- or nanosystems. Non-Fourier heat-transfer equations were utilized for designing of metal-oxidesemiconductor field-effect transistors [23]. Several studies were performed to solve non-equilibrium thermophysical problems aris-

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Nomenclature Latin Symbols ˆ Bˆ A, Complex-valued amplitudes, K c Specific heat capacity, J kg−1 K−1 D Thermal diffusivity, m2 s−1 d Sample thickness, m Frequency, Hz f h Thermal contact conductance, W m−2 K−1 kˆ Complex wave number, m−1 lq Thermal-diffusion length, m lph Phonon mean-free-path, m lT Thermal-wave length, m N Number of phonon collisions n Integer number Volumetric heat power, W m−3 P q Heat flux vector, W m−2 q Heat flux modulus, W m−2 r Space variable vector, m Time variable, s t Temperature, K T T Interfacial temperature step, K DTˆh Normalized differential amplitude, K m s−1−1 V Speed of sound, m s−1 Space variable, m x Greek Symbols ˆ ˇ Complex-valued coefficient, dim. less  Coefficient, W−1 m2 K  Damping coefficient, m−1  Wave number, m−1  Thermal conductivity, W K−1 m−1 n Coefficient, s−1 Eigenvalue, m−1 n Density, kg m−3

Time constant, s ˚ Normalized heat power, K s−1 ␸ Phase shift, rad. ω Angular frequency, rad. s−1 Subscripts A Amplitude Dual-phase-lagging D H Hyperbolic h Heater n Number of fourier component Membrane m P Parabolic S Sample T Temperature

ing in materials processing [24,25]. Non-Fourier heat conduction theory was applied for explanation of physical properties of nonhomogeneous [26], functionally graded [27,28], porous [29,30], and composite materials [31,32] designed for specific functions. Numerous studies were performed to solve non-Fourier heat conduction equations at different conditions [33–37]. An analytical solution of a transient thermal response in a slab sample under a periodic surface heating was obtained [33]. The thermal resonance in a slab sample at periodic heating was investigated; the conditions of under- and overdamped thermal oscillations were established [34]. Standing wave behavior at non-Fourier heat conduction was analytically studied in a porous material [35]. Stationary temperature oscillations in a slab sample in contact with a semi-infinite

Fig. 1. Photograph of the gauge XEN-39394 utilized as a calorimetric measuring cell. The rectangular 2 × 3 mm2 silicon frame supporting the submicron siliconnitride membrane is bonded on a standard chip holder (a). Zoomed photograph of the central part of the membrane with 8 × 10 ␮m2 region heated by two parallel heater stripes and the hot junction of the thermocouple located in between the heater stripes (b). Schematic cross-sectional view of the gauge with the sample is shown in (c).

layer at periodic heating were investigated analytically for a broad frequency range [36]. The transient thermal response in thin-film samples was analyzed analytically in reference [37]. Comprehensive reviews of non-Fourier heat-conduction problems are available in the literature [38–42]. The concepts of local entropy-production and local temperature were reconsidered. Thus the extended irreversible thermodynamics was developed as a background for the non-Fourier heat-conduction theory [43–46]. It is worth mentioning that the numerical simulation for a onedimensional chain of particles demonstrates a crossover from diffusive parabolic long-wave heat conduction to hyperbolic heat conduction in the short-wave range [47]. Such behavior requires application of the parabolic Fourier equation and demonstrates the necessity of a hyperbolic heat conduction theory. However, this study reveals that the characteristic relaxation time in an oscillatory hyperbolic regime depends on the wavelength of the temperature perturbations. Furthermore, a ballistic-diffusive equation is required to describe heat-conduction in very thin samples, when the phonon mean-free-path is comparable with the sample thickness [48]. Without doubt the classical diffusive Fourier heatconduction theory is adequate for most applications; nevertheless, it may be insufficient for ultrafast processes in nanoscale systems. The aim of this paper is to study heat conduction processes at ultrafast calorimetric experiments and to answer the question: what are the limits of validity of the classical diffusive heat conduction at high-frequency temperature-modulation measurements? What is expected in ultrafast experiments at considerable non-Fourier heat conduction? The goal is to compare the alternatives of Fourier and non-Fourier heat conduction in thin-film samples at high-frequency temperature modulation. The analysis is performed for boundary conditions corresponding to thin-film nanocalorimetry; the interfacial thermal contact resistances are taken into account. The origins of the factors, which suppress the effect of non-Fourier heat conduction at temperature-modulation experiments, are analyzed.

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2. Non-Fourier heat conduction In the case of extremely fast thermal disturbances in a material the heat flux can lag behind the fast changes of the temperature gradient, as well as the thermal gradients can lag behind the fast changes of the heat flux. Consider a uniform and isotropic substance with the following thermal parameters: density , specific heat capacity c, thermal conductivity , and thermal diffusivity D = / c. We focus on dielectric materials; therefore, no free electron component contributes to the heat flux. Non-Fourier heat conduction with delays of the heat flux vector q and the temperature gradient can be expressed as follows [49] q(r, t + q ) = − · ∇ T (r, t + T ),

(2.1)

where q and T are the characteristic time constants describing the lags of the heat flux and the temperature gradient. Then in a firstorder approximation for small q and T regarding the measuring time scale (say at ω q << 1 and ω T << 1, where ω = 2 f and f is the frequency of the temperature modulation) it follows from Eq. (2.1) that q + q ∂q/∂t = − · ∇ T − T  · ∇ (∂T/∂t).

(2.2)

Combining Eq. (2.2) with the energy balance equation c · ∂T/∂t = −div(q) + P(r, t),

(2.3)

where P(r, t) is the power of a volumetric heat source, the hyperbolic heat conduction equation can be obtained



2

c · ∂T/∂t + q ∂ T/∂t 2







= div (∇ T ) + T div ∇ ∂T/∂t

 



+ P + q ∂P/∂t .

(2.4) In this paper we focus on small temperature perturbations in thin-film samples. The temperature distribution in a thin-film sample can be described by a function T (x, t) of one space variable x with Ox axis normal to the surface of the film. The thermal parameters of the sample can be considered as independent of the temperature at small temperature perturbations. Then the dual-phase-lagging hyperbolic equation, proposed by Tzou [50], follows from Eq. (2.4)



2





2



∂T/∂t + q ∂ T/∂t 2 = D · 1 + T ∂/∂t ∂ T/∂x2 + 1 + q ∂/∂t ˚(x, t),

(2.5) where ˚(x, t) = P(x, t)/ c. Suppose the delay of the temperature gradient is negligible with respect to the heat flux delay, then the Cattaneo − Vernotte [18,19] hyperbolic equation follows from Eq. (2.5) at T = 0 2

2





∂T/∂t + q ∂ T/∂t 2 = D · ∂ T/∂x2 + 1 + q ∂/∂t ˚(x, t).

(2.6)

The classical parabolic Fourier equation follows from Eq. (2.5) at

q = 0 and T = 0, when the lag of the heat flux behind the temperature gradient and the lag of the thermal gradient behind the heat flux can be neglected. 2

∂T/∂t = D · ∂ T/∂x2 + ˚(x, t).

(2.7)

Eqs. (2.2), (2.5), and (2.6) can be derived not only for small ω q and ω T . In fact, the characteristic time constants q and T can be interpreted as parameters describing the relaxation processes of the heat-flux and the temperature gradient. Then the hyperbolic heat conduction equations follow from convolution relationships between the heat-flux and the history of the temperature gradient without time-scale limitations, see [51,52]. Furthermore, the hyperbolic heat conduction equations follow from the Boltzmann kinetic equation at relaxation approximation [51].

In this paper we compare the alternatives described by standard parabolic Eq. (2.7), as well as hyperbolic Eqs. (2.5) and (2.6) for experimental conditions corresponding to thin-film calorimetry. We focus on heat conduction in polymer samples. Heat conduction in thin-film polymer samples can be considered without a ballistic contribution at least down to 100 nm thickness, since the phonon mean-free-path lph , being in the order of 10 nm, is quite short in polymer materials [53]. The phonon distribution function, as well as the temperature gradient and the heat flux, relaxes at a time-scale in the order of lph /V or even slower after a fast thermal disturbance. Suppose N is the number of phonon collisions required for relaxation. Thus the violation of the classical Fourier behavior can be observed at the time-scale in the order of N · lph /V . Then the characteristic time constants q and T can be in the order of 1 ns or even more in polymer samples at N ca. 102 , the phonon mean-freepath lph in the order of 10 nm, and the sound speed V ca. 103 m/s [53,54]. Besides, the heat exchange between translational and relatively slow configurational degrees of freedom can cause a slowing down of the relaxation processes in polymers. The relaxation of the temperature gradient and the heat flux after fast changes can be even much slower in nonhomogeneous, functionally graded, and composite materials, because of relatively slow heat exchange between subsystems in such materials [26–32,51]. Actually, experimental studies of such ultrafast thermal processes are required. The aim of this paper is to provide an appropriate method for the correct analysis of experimental data, which can be obtained at high-frequency temperature-modulation experiments for both possibilities of Fourier and non-Fourier heat conduction, respectively. The thermal parameters of a typical polymer and the amorphous silicon-nitride membrane as utilized for the model calculations are collected in Table 1 [9,53–60]. 3. Stationary temperature oscillations in thin-film samples Temperature-modulated calorimetry provides an appropriate and highly sensitive method for the investigation of irreversible processes in nonequilibrium systems. Subsequently we analyze the alternatives of Fourier and non-Fourier heat conduction in slab samples at the boundary conditions typical for thin-film temperature-modulated calorimetry. Let Ox axis be directed along the normal to the sample/heater interface, as shown in Fig. 1 Consider the experiment, when a stationary heat flux q(t) = Re (q0 · exp(iωt)) is supplied at x = 0 from a thin-film heater of negligible thickness to one face of a slab sample of thickness d. The opposite side of the sample at x = d remains at adiabatic condition, and the power of the volumetric heat source P(x, t) equals zero. Initially we consider the case without supporting membrane and assume the ideal thermal contact at the sample/heater interface. Thus, we assume infinite thermal contact conductance hS at the sample/heater interface, i.e. zero thermal contact resis−1 tance hS . Therefore, we suppose the interfacial temperature step TS (t) = Th (0, t) − TS (0, t) at the sample/heater interface equals zero, where Th and TS are the temperatures in the heater and the sample respectively. Generally the interfacial temperature step TS (t) is proportional to the interfacial thermal contact resistance and the heat flux flowing across the interface TS (t) = hS

−1

· q(t).

(3.1)

Next consider the temperature distribution T (x, t) inside the sample (the subscript S is omitted in this section). The Fourier or non-Fourier heat conduction is described by homogeneous parabolic or hyperbolic equations at P(x, t) = 0. 2

∂T/∂t − D · ∂ T/∂x2 = 0,

(3.2)

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Table 1 Thermal parameters utilized for model calculations (at room temperature and normal pressure). Material

Density, in g/cm3

Specific heat capacity, in J/

Amorphous silicon-nitride Typical polymer

3.4 1

0.7 2

2

2

(3.3)

2

Specific heat capacity, in J/m3

2

with a second kind boundary condition following from Eq. (2.2):



(3.10)

2 kˆ H = iω/D − q ω2 /D,



2 kˆ D

where T = 0 for the hyperbolic Eq. (3.3) and q = 0, T = 0 for the parabolic Eq. (3.2). The adiabatic boundary condition ∂T/∂x = 0 (at x = d) is the same for parabolic and hyperbolic equations. Denote T1 (x, t) and T2 (x, t) the steady-state solutions of the linear boundary value problem for homogeneous parabolic or hyperbolic equations at q(t) = q0 ·cos(ωt) and q(t) = q0 ·sin(ωt), respectively. Then T1 (x, t) + i · T2 (x, t) is a complex steady-state solution for the same linear boundary value problem with the heat flux qˆ (t) = q0 · exp(iωt) at x = 0. Furthermore, if Tˆ (x, t) is a complex steady-state solution of the linear value for a  boundary   problem  homogeneous equation, then Re Tˆ (x, t) and Im Tˆ (x, t) are also the solutions of that problem at q(t) = q0 ·cos(ωt) and q(t) = q0 ·sin(ωt) respectively. Then the steady-state solutions of the linear boundary value problems for homogeneous parabolic Eq. (3.2) and hyperbolic Eqs. (3.3), (3.4) can be expressed in the complex form Tˆ (x, t) = TˆA (x) · exp(iωt), where TˆA (x) = TA (x) · exp(iϕ), TA is a real-valued amplitude of the temperature modulation, and ϕ is a phase shift of the temperature oscillations with respect to the heat flux acting at the boundary x = 0. Actually the parameters TA and ϕ are measured in the experiment. Then from the boundary condition Eq. (3.5) we get for the steady-state solution: ˆ ∂TˆA (x)/∂x = −q0 · ˇ(ω)/, at x = 0,

(3.6)

ˆ where ˇ(ω) = (1 + iω q )/(1 + iω T ). The steady-state solution satisfying Eqs. (3.2)–(3.4) equals



ˆ + Bˆ · exp(−kx) ˆ ˆ · exp(kx) Tˆ (x, t) = exp(iωt) · A .

(3.7)

ˆ and Bˆ can be obtained from the boundary conThe coefficients A ditions at x = 0 and x = d. Then the amplitude of the temperature modulation inside the sample is equal to





1 + exp 2kˆ · (x − d) ˆ 1 − exp(−2kd)

,

(3.8)

ˆ ˆ where ˇ(ω) = 1 for parabolic Eq. (3.2), and k(ω) can be obtained from the dispersion relations, see Eqs. (3.10)–(3.12). Similarly the steady-state heat flux inside the sample equals qˆ A (x) · exp(iωt), where qˆ A (0) = q0 . Then from Eqs. (3.8) and (2.2) we get the amplitude qˆ A (x)



ˆ · qˆ A (x) = q0 · exp(−kx)





1 − exp 2kˆ · (x − d) ˆ 1 − exp(−2kd)

(3.11)

=



iω · 1 + T q ω2 − ( q − T ) · ω2



D · 1 + ( T ω)2

(3.5)



1.25·10−6 1.5·10−7

and kˆ = kˆ D for dual-phase-lagging Eq. (3.4)

1 + T · ∂/∂t · ∂T (x, t)/∂x = − 1 + q · ∂/∂t · q(t)/, at x = 0,

ˆ q0 · ˇ(ω) ˆ · TˆA (x) = · exp(−kx)  · kˆ

3 0.3

kˆ = kˆ H for hyperbolic Eq. (3.3)





Thermal diffusivity, in m2 /s

Indeed qˆ A (0) = q0 and qˆ A (d) = 0 in agreement with the boundary conditions. Next the substitution of Eq. (3.7) in Eqs. (3.2)–(3.4) provides kˆ = kˆ P for parabolic Eq. (3.2)

(3.4)



Km Thermal conductivity, in W/

kˆ P2 = iω/D,

∂ T/∂t 2 + (1/ q ) · ∂T/∂t − (D/ q ) · ∂ T/∂x2 − (D · T / q ) · ∂/∂t(∂ T/∂x2 ) = 0,



K

2.4·106 2·106

∂ T/∂t 2 + (1/ q ) · ∂T/∂t − (D/ q ) · ∂ T/∂x2 = 0,

2

gK

.

(3.9)



.

(3.12)

ˆ Denote k(ω) = (ω) + i(ω), then a superposition of thermal waves moving in opposite directions can be obtained from Eq. (3.7) ˆ · exp (i(ωt + x)) · exp(x) + Bˆ · exp (i(ωt − x)) · exp(−x), Tˆ (x, t) = A

(3.13)





ˆ where (ω) = Re k(ω) is a damping coefficient and (ω) =   ˆ Im k(ω) is a wave number. The thermal waves oscillate at the   ˆ distance of the thermal-wave length lT (ω) = 2 /Im k(ω) . In the case of Fourier heat conduction the damping coefficient P (ω) is equal to the wave number P (ω) = ω/2D, see Eq. (3.10). However, H (ω) > H (ω) as follows from Eq. (3.11). 2 H = 2H + q · ω2 /D.

(3.14)

Thus the thermal waves are underdamped at q > 0. On the contrary, the thermal waves are overdamped D (ω) > D (ω) at T > q , as follows from Eq. (3.12): 2 2D = D +



( T − q ) · ω2



1 + ( T ω)2 · D

.

(3.15)

Noteworthy, the damping of the thermal waves increases essentially at T > 0, even if T < q ; see Fig. 2 plotted according to Eq. (3.12).   Consider the frequency dependences of TA (0) and arg TˆA (0) measured at x = 0 for different d, q , and T . Suppose the amplitude of the heat flux q0 equals 2·105 W/m2 . Such heat flux is generated by 100 × 100 ␮m2 thin-film heater at the amplitude of the heating power 2 mW. Then the amplitude TA is in the order of 1 K at d = 200 nm, f = 0.1 MHz, and ·c = 2·106 J/Km3 , see Eq. (3.16) following from Eq. (3.8), when kd < < 1 at f < < D/ d2 TˆA =

q0 . iω cd

(3.16)

Generally Eq. (3.16) is utilized for heat capacity measurements. Thus at low frequencies the phase lag of the temperature oscilla  tions with respect to the heat flux is equal to arg TˆA = − /2 and the normalized amplitude f·d·TA , being equal to q0 /(2␲ c), is independent on the frequency f and the thickness d; see Fig. 3 plotted according to Eq. (3.8). The case of zero time lags T = 0 and q = 0 corresponds to the classical diffusive Fourier heat conduction. The normalized amplitudes f·d·TA start to differ from the constant value at moderate frequencies, when kd ≈ 1 at d·( ·f/D)1/2 ≈ 1, namely at f ≈ 0.1 MHz and f ≈ 2.5 MHz for d = 500 nm and 100 nm respectively.

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ˆ ) · d vs. normalized frequencyf · q at Fig. 2. Normalized complex wave number k(f ˆ and Re(kd) ˆ − lines marked by circles and triangudS = 500 nm and q = 100 ns: Im(kd) lar at T = 0 and 20 ns − solid and open symbols respectively; as well as for classical ˆ = Re(k) ˆ − half-filled squares. Fourier heat conduction at q = 0, T = 0, and Im(k)

Fig. 3. Normalized amplitude f·d·TA (0) vs. f at d = 500 nm, T = 0, and different q = 0, 50 ns, 100 ns, 200 ns, 500 ns – lines marked by solid squares, circles, triangular-up, triangular-down, diamonds respectively, as well as d = 100 nm, T = 0, and q = 0, 5 ns, 10 ns, 20 ns, 50 ns – open squares, circles, triangular-up, triangular-down, diamonds.

Besides, the oscillations of TA vs. f are observed at sufficiently large frequencies for appropriate q and d, see Fig. 3. Underdamped thermal oscillations can occur in a slab sample at sufficiently large q , see Eq. (A6). In fact, the oscillations occur, when



4D · q is larger than the charthe thermal-diffusion length lq = acteristic length d/␲ of the boundary value problem for a sample of thickness d. Thus, the condition for the thickness of the sample is as follows: d < lq .

(3.17)

A thin-film sample can serve as a resonator for underdamped thermal waves. The resonance condition is d = lT (ω) · n/2, n = 1, 2, 3, .  . ., where the length of the thermal wave lT (ω) equals  ˆ . Then the resonance frequencies fn (d, q , T ) can be 2 /Im k(ω) calculated from Eq. (3.18)





Im kˆ D (fn ) = n/d,

(3.18)

where kˆ D can be obtained from Eqs. (3.12) or (3.11) at T = 0. Thus the first resonance occurs at f1 = 0.53 MHz for d = 500 nm and

q = 500 ns, as well as 8.51 MHz for d = 100 nm and q = 50 ns (see Fig. 3). The resonance behavior of the amplitude TA vs. frequency f is well-defined at sufficiently large q and small d. The oscillations of TA vs. f are visible at q ≥ 5 ns in the sample 100 nm thick, see

Fig. 4. Amplitude TA (0) vs. d at f = 5 MHz for T = 0 and different q = 0, 25 ns, 50 ns, 100 ns − lines marked by solid squares, circles, triangular-up, triangular-down respectively, as well as q = 0 and T = 0, 25 ns, 50 ns, 100 ns − open squares, circles, triangular-up, triangular-down are shown in the insert.

Fig. 3. However, the oscillations are not observed at q < 100 ns in the sample 500 nm thick. The optimal resonator thickness lT /2 can be found for a definite f,

T , and q . For instance, the optimal resonator thickness equals 120 nm, 160 nm, and 200 nm at q = 100 ns, 50 ns, and 25 ns respectively, when f = 5 MHz and T = 0; see Fig. 4 plotted according to Eq. (3.8). Indeed, no resonance occurs for overdamped waves at q = 0 and different T .   Actually the measurements of TA and arg TˆA (0) provide a straight-forward method to detect the effect of non-Fourier heat conduction in thin-film samples; see Figs. 5 and 6 plotted according to Eq. (3.8) for d = 300 nm and 100 nm respectively. Resonance frequencies f1 = 1.90 MHz, f2 = 4.02 MHz, f3 = 6.06 MHz, see Fig. 5, can be obtained according to Eq. (3.18) for q = 100 ns, T = 0, and d = 300 nm, as well as f1 = 8.51 MHz, f2 = 17.24 MHz, f3 = 25.92 MHz, for q = 50 ns, T = 0, and d = 100 nm, see Fig. 6. However no resonance occurs for overdamped thermal waves at q = 0 and different T , see Figs. 5 and 6. The difference between the classical diffusive Fourier and nonFourier heat conduction is most pronounced in thin samples at large

q and appropriately high frequencies ω in the order of 1/ q . The effect of the thermal wave resonance can be observed in sufficiently thin samples at d < 2 D · q for T = 0, see Eq. (3.17). Thus, at small q ca. 10 ns the resonance of the thermal waves in polymer samples can be observed at d as thin as ca. 100 nm and at frequencies in the order of 10 MHz, see Figs. 3 and 6. In contrast, the thermal wave resonance is not observed at q < 100 ns in 500 nm thick samples, as shown in Fig. 3. Besides, the resonance is damped essentially at T > 0. Next we consider the effect of non-Fourier heat conduction at T = 0 for various d and q at boundary conditions corresponding to thin-film chip calorimetry. 4. Stationary temperature oscillations in thin-film samples inside a chip calorimeter The here considered thin-film chip calorimeter consists of an amorphous silicon-nitride membrane ca. 1 ␮m thick with a thinfilm heater and a temperature sensor located at the central part of the membrane, for details see [10,11,16]. The available gauges from Xensor Integration can be utilized for high-frequency temperaturemodulated calorimetry [13–15]. In this paper we focus on general aspects of thin-film calorimetry, without consideration of different kinds of gauges and particular arrangement of the temperature sensors, neglecting the specific details of the temperature measurements, which will be considered somewhere else.

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47



Fig. 5. Frequency dependence of the amplitude TA (0) – (a) and arg TˆA (0) – (b) at d = 300 nm for underdamped thermal waves at T = 0 and q = 0, 25 ns, 50 ns, 100 ns – lines marked by solid squares, circles, triangular-up, and triangular-down respectively; as well as for overdamped thermal waves at q = 0 and T = 0, 25 ns, 50 ns, 100 ns − open squares, circles, triangular-up, and triangular-down are shown in the inserts.





Fig. 6. Frequency dependence of the amplitude TA (0) – (a) and arg TˆA (0) – (b) at d = 100 nm for underdamped thermal waves at T = 0 and q = 0, 10 ns, 20 ns, 50 ns – lines marked by solid squares, circles, triangular-up, and triangular-down respectively; as well as for overdamped thermal waves at q = 0 and T = 0, 10 ns, 20 ns, 50 ns −open squares, circles, triangular-up, and triangular-down are shown in the inserts.

Fig. 7. Schematic cross-sectional view of the central part of the membrane with the heater and the sample; as well as the schematic temperature distribution vs. distance from the heater in the membrane and the sample; note the interfacial temperature steps Tm and TS at the membrane/heater and the sample/heater interfaces.

Consider a model, when a thin-film sample is positioned at the central heated part of the membrane. Let Ox axis is directed along the normal to the membrane. The heater is a thin film of negligible thickness located at x = 0 in between the membrane and the sample, see Fig. 7. Denote Tˆh is the amplitude of the temperaturemodulation in the heater. In this paper we focus on the behavior of the amplitude of the temperature-modulation in the heater; we assume the amplitude Tˆh can be measured in the experiment (in

fact, the amplitude close to Tˆh can be measured in the gauges available from Xensor Integration, see Fig. 1). Suppose a stationary heat flux q(t) = Re (q0 · exp(iωt)) is generated by the heater. The heat flux is distributed between the sample and the membrane. Thus, q0 = qˆ m (0) + qˆ S (0), where qˆ m (0) and qˆ S (0) are the complex amplitudes of the heat flux supplied at the boundary x = 0 into the membrane and the sample respectively. Indeed the sum qˆ m (0)  + qˆ S (0)is equal tothe real  valued amplitude q0 . The arguments arg qˆ m (0) and arg qˆ S (0) correspond to the phase shifts between the heat fluxes in the membrane and the sample at x = 0 with respect to the heat flux q(t). Denote thickness di , density i , specific heat capacity ci , thermal conductivity i , and thermal diffusivity Di = i / i ci , where index i equals m and S for membrane and sample respectively. Assume the heat conduction is described by the parabolic Eq. (3.2) in the membrane and the hyperbolic non-Fourier Eq. (3.3) in the sample. The amplitudes in the membrane Tˆm (x) and the sample TˆS (x)can be obtained from the boundary condition similar Eq. (3.6) at x = 0, and adiabatic boundary conditions: ∂Tˆm /∂x = 0 at x = − dm , as well as ∂TˆS /∂x = 0 at x = dS . Then similar Eq. (3.8)



qˆ m (0) · exp(kˆ m · x) · Tˆm (x) = m · kˆ m TˆS (x) =





1 + exp 2kˆ m · (−x − dm )

ˆ qˆ S (0) · ˇ(ω) · exp(−kˆ S · x) · ˆ S · kS

, (4.1)

1 − exp(−2kˆ m · dm )







1 + exp 2kˆ S · (x − dS ) 1 − exp(−2kˆ S · dS )

,(4.2)

where kˆ m and kˆ S are equal to 2 km = iω/Dm ,

(4.3)

48

A.A. Minakov, C. Schick / Thermochimica Acta 640 (2016) 42–51

ˆ kˆ S2 = iω · ˇ(ω)/D S.

(4.4)

The interfacial temperature step Tm at the membrane/heater interface can be described by an equation similar to Eq. (3.1). Thus, the interfacial temperature steps Tm and TS are proportional −1 −1 to the interfacial thermal contact resistances hm and hS at the membrane/heater and the sample/heater interfaces, respectively, see Eq. (3.1) and Fig. 7. Then the amplitudes of the temperature oscillations TˆS (0) and Tˆm (0) at the corresponding interfaces equal Tˆh − qˆ S (0)/hS and Tˆh − qˆ m (0)/hm , respectively. Therefore, as follows from Eqs. (4.1) and (4.2) at x = 0, the amplitude of the temperature oscillations in the heater equals





Tˆh = qˆ m (0) · 1/hm + ˆ m , as well as



(4.5)



ˆ , Tˆh = qˆ S (0) · 1/hS + ˆ S · ˇ

(4.6)

where (at i = m or i = S) ˆ i equals ˆ i =



1

·

kˆ i · i



1 + exp(−2kˆ i · di ) . 1 − exp(−2kˆ i · di )

(4.7)

Consequently from Eqs. (4.5), (4.6) and the energy balance q0 = qˆ m (0) + qˆ S (0) it follows that



Tˆh = q0 ·

 

ˆ 1/hm + ˆ m · 1/hS + ˆ S · ˇ



ˆ 1/hm + ˆ m + 1/hS + ˆ S · ˇ

 qˆ S (0) = q0 ·



1/hm + ˆ m



,

(4.8)



ˆ 1/hm + ˆ m + 1/hS + ˆ S · ˇ

.

(4.9)

Then according to Eq. (4.2)

TˆS (x) =

ˆ q0 · ˇ S · kˆ S

    1/hm + ˆ m · exp(−kˆ S · x) + exp kˆ S · (x − 2dS ) ·    . ˆ · 1 − exp(−2kˆ S · dS ) 1/hm + ˆ m + 1/hS + ˆ S · ˇ

(4.10)

Fig. 8. Normalized amplitude TA ·f·dm vs. f in the heater at q = 0 – lines marked by squares, as well as at q = 50 ns in the heater, sample/heater, and membrane/heater interfaces – circles, triangular-up, and triangular-down respectively for dS = 100 nm = 1 ␮m; hm and hS are equal to 109 W/m2 K. Corresponding dependences anddm  arg TˆA vs. f are shown in the insert.

Next we assume dm = 1 ␮m, hm = 108 W/m2 K, and the sample parameters are in the range hS = 105 –108 W/m2 K, dS = 100–500 nm, and q = 0–100 ns. The thermal parameters of the polymer sample and the membrane are collected in Table 1. The heat flux q0 = 2·105 W/m2 is the same as in the previous section. Primarily consider the amplitude Th at low frequencies. Then ˆ i approximately equals ˆ i ≈

1 i · di · kˆ i2

(4.11)

as follows from Eq. (4.7) at di ki << 1. Next the interfacial temperature steps qm /hm and qS /hS are insignificant regarding −1 the amplitude Th at low frequencies, when hi << i , see Eq. (4.8). Then the interfacial thermal contact resistance 1/hS can be neglected at f <<

Indeed, similar equations hold for qˆ m (0) and Tˆm (x). Next, consider the steady-state thermal oscillations in thin-film samples at different dm , dS , hm , hS , q , and T = 0. Then the thermal contact conductance is a very important parameter in fast calorimetry measurements. This parameter can be changed in a wide range depending on numerous factors. Generally, a “dry” thermal contact is not very perfect due to the roughness of the contacting surfaces. The thermal contact conductance between polished surfaces depends on the applied pressure and can be up to 104 –105 W/m2 K [61]. The thermal contact conductance between solid and liquid surfaces is more stable and considerable. Thus a thermal contact conductance ca. 107 W/m2 K was observed at the interface of mercury droplets and a silicone substrate [62]. The thermal contact conductance between water and solids for hydrophilic and hydrophobic interfaces was measured in the order of 108 W/m2 K [63]. The thermal contact conductance between carbon nanotubes can be as high as 109 –1010 W/m2 K [64,65]. Epitaxial thermal contact conductance between closely related materials can be in the order of 109 W/m2 K [66]. The thermal contact conductance in epitaxial structures between highly dissimilar materials can be varied in the range 107 –109 W/m2 K [67]. Then the thermal contact conductance of the epitaxial contact at the membrane/heater interface probably can be in the order of 108 W/m2 K. The thermal contact at the sample/heater interface usually is nice and stable due to strong adhesive forces acting between a polymer sample and substrate [6,11].

,

hS . 2 S cS dS

(4.12)

Therefore from Eq. (4.8) we get at f → 0 Tˆh =

q0 . iω( m cm dm + S cS dS )

(4.13)

Consider the case of thin sample of 100 nm thick, when m cm dm >> S cS dS . In that case, we normalize the temperaturemodulation amplitudes by multiplying on the product f·dm . Then, the frequency dependencies of TS (0)·f·dm , Tm (0)·f·dm , and Th ·f·dm can be obtained from Eqs. (4.8) and (4.10). First, consider the case of very strong thermal contacts hm = 109 W/m2 K and hS = 109 W/m2 K. Then the interfacial temperature step qS /h is insignificant in a wide frequency range, namely at f < < 109 Hz for dS = 100 nm, see Eq. (4.12). Thus the amplitudes Th , TS (0), and Tm (0) shown in Fig. 8 are practically the same. The effect of non-Fourier   heat conduction on the frequency dependences of Th and arg Tˆh is well-pronounced (see Fig. 8) at q = 50 ns and dS = 100 nm in a gauge with membrane of 1 ␮m thick, when the thermal contact conductance hm and hS are as large as 109 W/m2 K. Next, consider the effect of the thermal contact resistance on the amplitudes Th , TS (0), and Tm (0). The amplitudes Th , TS (0), and Tm (0) are significantly different at thermal contact parameters hm = 108 W/m2 K and hS = 107 W/m2 K, see Fig. 9. As a result the effect of non-Fourier   heat conduction on the frequency dependencies Th and arg Tˆh becomes almost invisible, even if the effect is   noticeable in TS (0) and arg TˆS (0) . Thus the visibility of the effect

A.A. Minakov, C. Schick / Thermochimica Acta 640 (2016) 42–51

49

Fig. 9. Frequency dependence of normalized amplitude f·dm ·TA in the heater, sample/heater, and membrane/heater interfaces – lines marked by squares,   circles, and triangular respectively at q = 0 – (a) and 50 ns – (b) for the sample 100 nm thick at hm = 108 W/m2 K and hS = 107 W/m2 K. Corresponding dependences arg TˆA vs. f are shown in the insert.

of non-Fourier heat conduction in dependency Th vs. frequency becomes appreciably suppressed at reasonable thermal contact parameters hm and hS , see Fig. 9. However, the visibility of that effect can be increased by measurements in differential mode. Consider a system of two coupled identical gauges: one gauge is loaded with a sample and the other is free. Denote Tˆh0 is the amplitude measured in the heater of the unloaded gauge. The amplitude Tˆh0 can be obtained from Eq. (4.8) at dS → 0, as well as from Eq. (4.5) at qˆ m (0) = q0





Tˆh0 = q0 · 1/hm + ˆ m .

(4.14)

A difference between the amplitudes Tˆh0 and Tˆh is measured in differential mode. As follows from Eqs. (4.8) and (4.14) the differential amplitude equals



Tˆh0 − Tˆh = q0 ·



1/hm + ˆ m

2

ˆ 1/hm + ˆ m + 1/hS + ˆ S · ˇ

.

(4.15)

Then the differential amplitude at f → 0 equals Tˆh0 − Tˆh ≈ q0 ·

S cS dS / m cm dm . iω( m cm dm + S cS dS )

(4.16)

Fig. 10. Normalized differential amplitude DTˆh vs. f at q = 0 and 50 ns − lines marked by squares and circles respectively for dS = 100 nm and 250 nm – solid and open symbols, as well as 500 nm at q = 0 and 100 ns – half-filled squares and circles at   hS = 107 W/m2 K and hm = 108 W/m2 K. Corresponding dependences of arg DTˆh vs. f are shown in the insert.

Thus the differential amplitude is proportional to the heat capacity of the sample at low frequencies [13,14]. Actually, the effect of non-Fourier heat conduction is more visible in differential mode, see Figs. 10 and 11. Consider the differential amplitude normalized by factor f·dm 2 /dS





2 DTˆh = Tˆh0 − Tˆh · f · dm /dS .

(4.17)

Frequency dependencies of normalized differential amplitude DTˆh at different thickness dS and thermal contact conductance hS are shown in Figs. 10 and 11. Thus the oscillations of the differential amplitude DTˆh vs. frequency due to the effect of non-Fourier heat conduction are well-defined at thermal contact conductance hS = 107 W/m2 K and hm = 108 W/m2 K in thin samples ca. 100 nm at q = 50 ns. On the other hand the oscillations are almost invisible in thick samples ca. 500 nm even at q = 100 ns, see Fig. 10. This result is in agreement with Eq. (3.17). Consider the influence of the thermal contact resistance on the visibility of the oscillations of DTˆh vs. f. Let the condition Eq. (3.17) ˆ ≈ be satisfied. Then for high frequencies ˆ S ˇ DS q /S , as follows from Eq. (4.7) at dS kS > 1 and ω q > 1. In fact, the oscillations are ˆ > 0.1/hS , see Eq. (4.15). Therefore, we visible approximately at ˆ S ˇ get the requirement

Fig. 11. Normalized differential amplitude DTˆh vs. frequency for dS = 100 nm at q = 0 and 50 ns – lines marked by squares and circles at hS = 3·107 W/m2 K, 106 W/m2 K, 3·105 W/m2 K, and 105 –W/m2 K – solid, open, half-filled-right, and half-filled-left   symbols respectively. Corresponding frequency dependences of arg DTˆh are shown in the insert.

hS > S / lq .

(4.18)

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A.A. Minakov, C. Schick / Thermochimica Acta 640 (2016) 42–51

Table 2 Requisite parameters for measurements of underdamped thermal oscillations in polymer samples. Characteristic time constant q ,in ns

Maximum sample thickness, lq ,in nm

Minimum thermal contact conductance S / lq ,in W/Km2

0.001 1 50

2.4 80 500

108 4·106 5·105

Thus the underdamped thermal oscillations can be measured at the parameters collected in Table 2. Thus the visibility of oscillations of DTˆh vs. frequency is strongly suppressed at decreasing thermal contact conductance. The effect of non-Fourier heat conduction is invisible at hS ca. 105 W/m2 K for

q = 50 ns, see Fig. 11, which is in agreement with Eq. (4.18). 5. Conclusions The effects of interfacial thermal contact resistance and non-Fourier heat conduction in thin-film samples are calculated analytically for the boundary conditions corresponding to thin-film chip calorimetry. Actually, high frequency temperaturemodulated calorimetry is an appropriate method for investigation of irreversible thermal processes in nonequilibrium systems. The experimental study of the non-Fourier heat conduction can be performed in sufficiently thin samples smaller than lq , where the



characteristic thermal diffusion length lq = 4D · q . However, the visibility of the effect of non-Fourier heat conduction becomes appreciably suppressed due to the limited thermal contact conductance hS at the sample/heater interface, even if the requirement dS < lq is satisfied. The visibility can be enhanced essentially at measurements in differential mode. Then, the oscillations of the differential amplitude DTˆh vs. frequency due to the effect of nonFourier heat conduction can be observed at hS > S / lq . Suppose the sample thickness is in the order of lq , then the thermal contact conductance should be at least in the order of S /dS . Thus the thermal contact conductance should be larger than 3·106 W/m2 K for polymer samples ca. 100 nm thick. The effect is practically impossible to observe in polymer samples at q in the order of 1 ps, because of the requirement of very large hS ca. 108 W/m2 K and very small dS ca. 1 nm, see Table 2. Next, the effect of non-Fourier heat conduction is insignificant at low frequencies, when f < D/ d2 and ω q << 1. For instance, non-Fourier heat conduction can be neglected at frequencies below 0.1 MHz for polymer samples ca. 1 ␮m and q < 1 ␮s, as well as at f < 1 MHz for samples ca. 100 nm and q < 100 ns, see Fig. 3. Moreover, the effect can be considerably smeared due to the averaging over a possible distribution of the time-lags q and T . Regarding the interfacial thermal contact resistance, hS is insignificant at f << hS /2 S cS dS , see Eq. (4.12). Nevertheless, the interfacial thermal contact resistance can be considerable in wide frequency range, see Fig. 11. Acknowledgement The financial support of the German Science Foundation (DFG SCHI 331/14 is gratefully acknowledged. Appendix. Condition of underdamped thermal oscillations in a slab sample Consider a slab sample of thickness d at adiabatic boundary conditions. Assume the temperature distribution T(x,t) in the sample is described by nonhomogeneous hyperbolic equation 2

2

∂T/∂t + q ∂ T/∂t 2 = D · ∂ T/∂x2 + F(x, t),

(A1)

with arbitrary function F(x,t). Thus the second boundary value problem is analyzed over the domain 0 ≤ x ≤ d, with homogeneous

boundary conditions ∂T/∂x = 0 at x = 0 and x = d. The problem, when a heat flux q(t) is supplied to one face of the sample, can be converted to the problem with adiabatic boundary conditions by the following substitution [68]: T (x, t) = T˜ (x, t) + (x2 /2d − x) · q(t)/.

(A2)

Substitution Eq. (A2) merely modify F(x,t) in Eq. (A1). Subsequently the boundary value problem with homogeneous boundary conditions is analyzed. The hyperbolic equation Eq. (A1) can be transformed to a standard form by substitution T(x,t) = exp(–t/2 q )·U(x,t) [68], which does not change the homogeneous boundary conditions. Then Eq. (A1) can be transformed to 2

2

∂ U/∂t 2 − U/4 q 2 = (D/ q ) · ∂ U/∂x2 + exp(t/2 q ) · F(x, t)/ q .(A3) The boundary value problem with homogeneous boundary conditions can be solved by separation of variables. The solution of Eq. (A3) can be presented as a series expansion over orthogonal Eigenfunctions Xn (x) = cos( n ·x) satisfying the boundary value problem at corresponding Eigenvalues  n = ␲n/d for n = 1, 2, 3,. . .. Then the oscillator equations are obtained for Fourier components of Eq. (A3): 2

∂ Un /∂t 2 + n 2 · Un = exp(t/2 q ) · Fn / q , where n

2

1 = 2

q

 D · q ·

 n 2 d

1 − 4

(A4)

 .

(A5)

The underdamped thermal oscillations occur at n 2 > 0, providing

q > d2 /4 2 D.

(A6)

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