Dynamic Specific Heat and Glass Transitions

Abstract

The dynamical properties such as fragility, and non-Debye behavior of glass-forming materials have been studied by the frequency-dependent dynamic specific heat. Kubo’s formula on the fluctuation-dissipation theorem defines dynamic specific heat using the correlation function of enthalpy fluctuations. Dynamic specific heat is important for analyzing and understanding various relaxation processes. The dielectric relaxation is caused only by polar atomic motions, while the enthalpy relaxation is caused by total degrees of freedom of atomic motions. This chapter introduces two experimental methods to measure dynamic specific heat: (1) temperature-modulated differential scanning calorimetry (MDSC) and (2) photoacoustic spectroscopy. The experimental results of the dynamical properties of glass transitions in oxide glasses with covalent bond network structures and hydrogen-bonded glass-forming materials are reviewed.

1. Introduction

When a simple liquid is cooled from a high temperature, it undergoes a liquid-solid phase transition and crystallizes below a melting temperature, T_m. In contrast, when a complex liquid is cooled down below T_m, it changes into a supercooled liquid, which is a nonequilibrium state. Upon further cooling, a liquid-glass transition occurs at a glass transition temperature, T_g and it changes into a nonequilibrium glassy state. Figure 1 shows the enthalpy of liquid phase (AB), supercooled liquid state (BD), glass state (DE), and crystal phase (CG) as a function of temperature. The enthalpy of a liquid phase and supercooled liquid state crosses to that of a crystal phase at the Kauzmann temperature [1], T_K (F), which is an ideal glass transition temperature determined by the static condition. Another ideal glass transition temperature is the Vogel-Fulcher temperature [2], T_VF, which is determined by the Vogel-Fulcher law on the main structure relaxation process. In most glasses T_K is close to T_VF.

Figure 1.

The enthalpy is plotted as a function of temperature. Above T_m only a liquid phase (AB) exists. Below T_m a simple liquid crystallizes and crystal phase (CG) appears. While, a complex liquid keeps a supercooled liquid state (BD) between T_m and T_g. Below T_g a glass state (DE) appears.

Upon cooling a supercooled liquid, the relaxation time of a main structural relaxation process increases and undergoes a liquid-glass transition into a glass state at, T_g, where the relaxation time is 100 s. Both supercooled liquid and glass belong to nonequilibrium states. Understanding the dynamical properties of glass-forming materials is very important to clarify the mechanism of a liquid-glass transition. Their structural relaxation processes have been extensively studied by dielectric and ultrasonic spectroscopies, which are responsible for dielectric and elastic relaxations, respectively [3, 4].

According to the linear-response theory, dynamic specific heat is defined by Kubo’s formula on the fluctuation-dissipation theorem using the correlation function of enthalpy fluctuations [5]. The dynamic-specific heat provides a general understanding on the relaxation phenomena. Because the enthalpy relaxation contains the total degrees of freedom of atomic motions including polar atomic motions. In contrast, dielectric relaxation contains only polar atomic motions, and other nonpolar atomic motions do not contribute to the dielectric relaxation [6]. Therefore, it is important to study the structural relaxation processes of glass-forming materials by the frequency-dependent dynamic specific heat [7].

2. Dynamic susceptibility

Various physical quantities show time-dependent responses when an external field is applied to a system to be studied. Such a time-dependent response in a nonequilibrium state is analyzed by a response function or a relaxation function in linear-response theory, assuming that an external field is too weak to induce a nonlinear response. In observing a time-dependent response of physical quantities in a frequency-domain, dynamical susceptibility shows a frequency dispersion. The frequency dispersions of complex dielectric constants and complex elastic constants have been measured to study dynamical processes in a nonequilibrium state.

2.1 Complex dielectric susceptibility

In nonequilibrium statistical physics, dynamic (or complex) dielectric susceptibility is defined by the fluctuation-dissipation theory on the fluctuation of electric flux density, D(t) [5]. In the classical limit, it holds that

$$\epsilon \left(\omega \right)={\epsilon }_0+\frac{\mathit{i\omega }}{k_{B}T}{\int }_0^{\infty }⟨\left[D\left(t\right)-\overline{D}\right]\left[D\left(0\right)-\overline{D}\right]⟩e^{\mathit{i\omega t}}\mathit{dt}.$$E1

Here, k_B is Boltzmann constant, D(t) is electric flux density as a function of time, $⟨⟩$ is an expected value, and $\overline{D}$is the average value of D(t).

Dynamic/complex dielectric susceptibility has been discussed by linear-response theory. When the response of electric flux density, D(t), is linearly proportional to the external field, E(t), dielectric flux density is given by,

$$D\left(t\right)={\epsilon }_{\infty }E\left(t\right)+\left({\epsilon }_0-{\epsilon }_{\infty }\right){\int }_{-\infty }^t\varphi \left(t-t^{\prime }\right)E\left(t^{\prime }\right)d{t}^{\prime }.$$E2

Here ${\epsilon }_{\infty }$, ${\epsilon }_0$, and $\varphi \left(t\right)$ are dielectric constant of high-frequency limit, static dielectric constant, and dielectric response function, respectively. Figure 2 shows the impulse response of D(t).

Figure 2.

(a) Application of impulse electric field with a peak height E(0) at t = 0. (b) Impulse response of D(t), where ϕ(t) is a response function and D(0) = ε₀E(0).

By Fourier transformation from time-domain to frequency-domain, it holds that

$$D\left(\omega \right)={\int }_0^{\infty }D\left(t\right)e^{\mathit{i\omega t}}\mathit{dt}=\epsilon \left(\omega \right)E\left(\omega \right).$$E3

Here, the complex dielectric constant, $\epsilon \left(\omega \right)$, is related to a dielectric response function, $\varphi \left(t\right)$ .

$$\epsilon \left(\omega \right)={\epsilon }_{\infty }+\left({\epsilon }_0-{\epsilon }_{\infty }\right){\int }_0^{\infty }\varphi \left(t\right)e^{\mathit{i\omega t}}\mathit{dt},$$E4

Dielectric relaxation function, $\mathrm{\Psi }\left(t\right),$ is defined by

$$\varphi \left(t\right)=-\frac{d}{\mathit{dt}}\mathrm{\Psi }\left(t\right).$$E5

By the substitution of Eq. (5) for Eq. (4), we obtain

$$\epsilon \left(\omega \right)={\epsilon }_0+\left({\epsilon }_0-{\epsilon }_{\infty }\right)\mathit{i\omega }{\int }_0^{\infty }\Psi \left(t\right)e^{\mathit{i\omega t}}\mathit{dt}$$E6

In the Debye model with a relaxation time τ [8], the relaxation function is given by

$$\mathrm{\Psi }\left(t\right)=\mathit{exp}\left(-\frac{t}{\tau }\right).$$E7

The complex dielectric constant is given by

$$\epsilon \left(\omega \right)={\epsilon }_{\infty }+\frac{\left\{{\epsilon }_0-{\epsilon }_{\infty }\right\}}{1+\mathit{i\omega \tau }}.$$E8

Real and imaginary parts of complex dielectric constants are given by

$${\epsilon }^{\prime }\left(\omega \right)={\epsilon }_{\infty }+\frac{{\epsilon }_0-{\epsilon }_{\infty }}{1+{\omega }^2{\tau }^2},{\epsilon }^{"}\left(\omega \right)=\frac{\left({\epsilon }_0-{\epsilon }_{\infty }\right)\mathit{\omega \tau }}{1+{\omega }^2{\tau }^2}.$$E9

Figure 3 shows real and imaginary parts of complex dielectric constant with ${\epsilon }_{\infty }$ = 20, ${\epsilon }_0$ = 50, and τ = 1.6 × 10⁻⁹ s.

Figure 3.

Real and imaginary parts of complex dielectric constant with ${\epsilon }_{\infty }$ = 20, ${\epsilon }_0$ = 50, and τ = 1.6 × 10⁻⁹ s.

As the frequency increases from a low frequency, the real part gradually decreases from the low-frequency limit, ${\epsilon }_0$, and in the vicinity of ω = 1/τ remarkable decreases. For further increase in frequency, it approaches the high-frequency limit, ${\epsilon }_{\infty }$. The imaginary part shows the maximum at ω = 1/τ and ε = (${\epsilon }_0$ − ${\epsilon }_{\infty }$)/2. In a liquid-glass transition, upon cooling from a high-temperature liquid phase, the relaxation time of structural relaxation becomes longer toward a liquid-glass transition temperature, T_g. In a dielectric measurement, the relaxation time is determined by the maximum frequency of the imaginary part of the dielectric constant by the equation ωτ = 1.

Figure 4 shows the temperature dependence of the observed imaginary parts of glass-forming propylene glycol with T_g = 173 K in the large frequency range from 1 mHz to 10 GHz [9]. The broadband dielectric spectra were measured by the combination of two kinds of measurement. The low-frequency range between 1 mHz and 10 MHz was measured by the Impedance/Gain-phase analyzer (Solartron SI 1260). The high-frequency range between 1 MHz and 10 GHz was measured by the time-domain reflectometry (TDR) system, which is the combination of HP54750A digital Oscilloscope and HP54754A Differential TDR module. Upon cooling from a high temperature, the peak frequency at 315 K in a GHz range remarkably decreases down to that in a mHz range. The structural relaxation time obeys the Vogel-Fulcher-Tammann law.

Figure 4.

The temperature and frequency dependences of the imaginary part of dielectric susceptibility of glass-forming propylene glycol.

In the Debye relaxation model, the equation of semi-circle holds for ${\epsilon }^{\prime }\left(\omega \right)$and ${\epsilon }^{″}\left(\omega \right).$

$${\left\{{\epsilon }^{\prime }\left(\omega \right)-\frac{{\epsilon }_0+{\epsilon }_{\infty }}{2}\right\}}^2+{\left\{{\epsilon }^{″}\left(\omega \right)\right\}}^2={\left(\frac{{\epsilon }_0-{\epsilon }_{\infty }}{2}\right)}^2$$E10

The plot of ${\epsilon }^{\prime }\left(\omega \right)$ versus ${\epsilon }^{″}\left(\omega \right)$ is called a Cole-Cole plot as shown in Figure 5.

Figure 5.

Cole-Cole plot of a complex dielectric constant.

In the non-Deby case, the relaxation time has a distribution, and a complex dielectric constant is given by the summation of relaxation processes with a distribution function of relaxation time, g(τ):

$$\epsilon \left(\omega \right)={\epsilon }_{\infty }+\left({\epsilon }_0-{\epsilon }_{\infty }\right){\int }_0^{\infty }\frac{g\left(\tau \right)}{1+\mathit{i\omega \tau }}\mathit{d\tau }.$$E11

If the complex dielectric constant over a very large frequency is available, the distribution function can be determined by inverse-integral transformation. However, for the analysis of experimental results with a limited frequency range, the following empirical formulae with few fitting parameters have been used.

1. Davidson-Cole equation [10]

   $$\epsilon \left(\omega \right)={\epsilon }_{\infty }+\frac{{\epsilon }_0-{\epsilon }_{\infty }}{{\left(1-\mathit{i\omega \tau }\right)}^{\gamma }},0<\gamma <1$$E12

2. Cole-Cole equation [11]

   $$\epsilon \left(\omega \right)={\epsilon }_{\infty }+\frac{{\epsilon }_0-{\epsilon }_{\infty }}{1-{\left(\mathit{i\omega \tau }\right)}^{\alpha }},0<\alpha <1$$E13

3. Haviliak-Negami equation [12]

   $$\epsilon \left(\omega \right)={\epsilon }_{\infty }+\frac{{\epsilon }_0-{\epsilon }_{\infty }}{{\left\{1-{\left(\mathit{i\omega \tau }\right)}^{\alpha }\right\}}^{\lambda }},0<\alpha ,\lambda <1$$E14

In the time-domain, Kohlrausch-Wiliams-Watts (KWW) function has been used in non-Debye relaxation [13, 14]:

$$\mathrm{\Psi }\left(t\right)=\mathit{exp}\left\{-{\left(\frac{t}{\tau }\right)}^{\beta }\right\},0\le \beta \le 1$$E15

The numerical study on the relation of parameters between time-domain KWW function and the frequency-domain Haviliak-Negami equation was reported [15].

$$\mathit{ln}\left(\frac{{\tau }_{\mathit{HN}}}{{\tau }_{\mathit{KWW}}}\right)\cong 2.6{\left(1-\beta \right)}^{0.5}exp\left(-3\beta \right),\mathit{\alpha \gamma }\cong {\beta }^{1.23}.$$E16

Here, ${\tau }_{\mathit{HN}}$and ${\tau }_{\mathit{KWW}}$are the relaxation times of Haviliak-Negami equation and KWW function, respectively.

2.2 Dynamic specific heat

In the equilibrium statistical mechanics, a specific heat at a constant pressure C_p is defined by

$$C_p=\frac{\partial ⟨H⟩}{\partial T},$$E17

where H is enthalpy. In this definition, C_p is time-independent and includes the contribution of the total degrees of freedom.

According to the nonequilibrium statistical mechanics, dynamic (or complex) specific heat is C_p(ω) is defined by the fluctuation-dissipation theorem [16, 17] using the correlation function of enthalpy fluctuations. In the classical limit, it holds that

$$C_p\left(\omega \right)=C_p^0+\frac{\mathit{i\omega V}}{k_B{T}^2}{\int }_0^{\infty }⟨\left[h\left(t\right)-\overline{h}\right]\left[h\left(0\right)-\overline{h}\right]⟩e^{\mathit{i\omega t}}\mathit{dt},$$E18

where, h(t) is the time-dependent enthalpy at constant pressure, $\overline{h}$ is the averaged h(t), and V is the volume.

The specific heat of nonequilibrium materials is generally not a constant on time. The specific heat of nonequilibrium states has been discussed using a response function ϕ(t). The frequency-domain enthalpy is defined by the change of enthalpy h(t) ≡ δH = H(t) − H_eq, where H_eq is the value at an equilibrium state, is given by

$$h\left(\omega \right)={\int }_0^{\infty }h\left(t\right)e^{\mathit{i\omega t}}\mathit{dt}=C_p\left(\omega \right)T\left(\omega \right)$$E19

The dynamic-specific heat $C_p\left(\omega \right)$ is related to the enthalpy response function $\varphi \left(t\right)$ .

$$C_p\left(\omega \right)=C_p^{\infty }+\left(C_p^0-C_p^{\infty }\right){\int }_0^{\infty }\varphi \left(t\right)e^{\mathit{i\omega t}}\mathit{dt},$$E20

where $C_p^{\infty }$ is the value that equilibrates instantaneously, and $C_p^0$ is the static specific heat in an equilibrium state.

The specific-heat spectroscopy was developed by Birge [6], using an alternating current calorimetry technique, is not generally in use in laboratories and requires expert skill to detect weak third harmonic signals. In fact, this method measures the product of specific heat and thermal conductivity, i.e., the thermal effusivity [18]. Photoacoustic spectroscopy was also applied to study the dynamic specific heat of glass-forming materials [7]. Recently, temperature-modulated DSC (MDSC) is commercially available, and measurement can be performed in routine DSC scan [19]. In this chapter, at first, MDSC and the application to liquid-glass transitions of oxide network glasses are described. Secondly, photoacoustic spectroscopy is described, and the application to liquid-glass transitions of alcohols is described.

3. Temperature-modulated differential scanning calorimetry

In the early 1990s, a new type of differential scanning calorimetry (DSC), modulated temperature DSC (MDSC), was developed by Reading et al. [20, 21]. It has been shown that MDSC is a powerful tool to investigate the dynamical properties of glass transitions [22, 23]. MDSC is the extended equipment of a standard DSC. A small sinusoidal temperature perturbation is superimposed on the linear temperature ramping of the standard DSC, as shown in Figure 6. The MDSC was analyzed by Schawe based on the linear-response theory [24, 25].

Figure 6.

The temperature-modulated DSC (MDSC) applies a small temperature fluctuation, generally sinusoidal temperature oscillation, to linear temperature ramping of conventional differential scanning calorimetry.

The calorimeter response is given by

$$\frac{\mathit{dQ}}{\mathit{dt}}=C_p\frac{\mathit{dT}}{\mathit{dt}}+f\left(t,T\right).$$E21

Here, dQ/dt, dT/dt, $C_p$, and $f\left(t,T\right)$ are heat flow, heating rate, specific heat, and kinetic component, respectively. This heating rate-dependent component is called reversing, and the other independent one is nonreversing. MDSC can simultaneously measure these two components and the total DSC heat flow.

In DSC, the differential heat flow (HF) was determined by the temperature difference between the sample and the reference. The resultant HF is analyzed by the deconvolution of the sample’s response into the underlying heating rate [26]. The raw resultant HF is averaged over the period of more than one modulation. After the averaged HF was subtracted from the resultant HF, the modulated contribution to the resultant HF is calculated by the discrete Fourier transformation (DFT) to obtain the amplitude of modulated contribution A_HF, and the phase lag (ϕ) between the modulated contribution and modulation in the heating rate. The phase angle is corrected according to the calculation reported in Ref. [27]. The absolute value of C_p^* (ω) is calculated by the amplitudes of modulated contribution A_HF and the heating rate A_q as shown in Eq. (22). The real part C_p′ and imaginary part C_p″ are calculated by the phase lag(ϕ) from C_p^*:

$$\left|C_p^{\ast }\right|=\frac{A_{\mathit{HF}}}{M{A}_q},$$E22

where M is the mass of the sample.

Modulated temperature differential scanning calorimetry (MDSC) measurements were performed by using TA instruments DSC2920 to obtain a thermal relaxation process in the vicinity of a glass transition temperature [26]. Figure 7 shows the absolute value of C_p* and phase lag(ϕ) observed in the vicinity of a glass transition temperature of glucose. It has been shown that MDSC can provide the frequency-dependence of the specific heat in the order of 0.01–0.1 Hz. The information obtained by MDSC determines the frequency-dependent specific heat C_p*(ω). Then, $C_p^{\prime }\left(\omega \right)$ and $C_p^{″}\left(\omega \right)$ are calculated using a phase lag ϕ,

Figure 7.

The curves of the absolute value of complex specific heat |C_p| and phase lag of glucose obtained by MDSC [26].

$$C_p^{\ast }\left(\omega \right)=\left|C_p^{\ast }\left(\omega \right)\right|e^{-\mathtt{\varphi }}=C_p^{\prime }\left(\omega \right)-i{C}_p^{"}\left(\omega \right)$$E23

$$C_p^{\prime }\left(\omega \right)=\left|C_p^{\ast }\left(\omega \right)\right|\mathit{cos}\mathtt{\varphi },C_p^{"}\left(\omega \right)=\left|C_p^{\ast }\left(\omega \right)\right|\mathit{sin}\mathtt{\varphi },$$E24

where $C_p^{\prime }\left(\omega \right)$ relates to an in-phase component of the modulation and $C_p^{"}\left(\omega \right)$ relates to an out-of-phase component. The temperature dependence of real $C_p^{\prime }$ and imaginary $C_p^{"}$ parts of glass-forming glucose is shown in Figure 8.

Figure 8.

The real and imaginary parts of dynamic specific heat in the vicinity of a glass transition temperature of glucose.

The dynamic heat capacity of sodium borate glasses was measured by DSC 2920 and DSC T-zero Q200 (TA Instruments, Tokyo, Japan) [28]. A glass sample of about 10.0 mg was put into an aluminum pan. The sample pan was heated over T_g with a linear heating rate of 0.5 or 1.0°C/min. The temperature modulation conditions are amplitude ±1 °C and the period between 20 and 200 s. As a purge gas, dry nitrogen gas flowed in a sample chamber with a flow rate of 20 ml/min. The frequency dependences of the real and imaginary parts of the dynamic heat capacity of xNa₂O-(1 − x)B₂O₃ glass (x = 0.41) in the vicinity of glass transition temperature are shown in Figure 9 [28].

Figure 9.

Frequency dependences of the real and imaginary parts of the dynamic specific heat of xNa₂O-(1 − x)B₂O₃ glass (x = 0.41) observed by T-zero DSCQ200.

In Figure 9, the peak temperature of C_p”(ω), T_g(ω), increases as the temperature modulation frequency increases. The α-relaxation time τ_α is related to the modulation period, and it was determined by

$${\tau }_{\alpha }=\frac{1}{\omega }=\frac{P}{2\pi },$$E25

where P is the temperature modulation period. Figure 10 shows the Angell plot [29] on the observed values of τ_α, where the horizontal axis is normalized by static T_g, which was determined as the temperature when τ_α becomes 100 s. By the Angell plot, fragility index m is determined by

Figure 10.

Angell plot of xNa₂O-(1 − x)B₂O₃ glasses with x = 0.11 and 0.41 Mol%. The horizontal axis is normalized by T_g. The straight lines are the fits by a least square method.

$$m={\left[\frac{d{\mathit{\text{logτ}}}_{\alpha }}{d\left(\frac{T_g}{T}\right)}\right]}_{T=T_g}.$$E26

The range of m changes from strong glass-forming liquids with m ≈ 16 to fragile glass-forming liquids with m ≈ 200 [30]. From the Angell plots of x = 11 and 41 mol% in Figure 10, the fragility index m was calculated by Eq. (26), corresponding to the steepness of the linear curve. Such a determination of fragility index using MDSC was also reported in lithium borate glasses [31], glucose, fructose [26], ethylene glycol-glucose aqueous solutions [32], and poly (propylene glycol)s [33].

The Na₂O content dependence of the fragility index m of xNa₂O-(1 − x)B₂O₃ glasses is shown in Figure 11 with the reference values summarized by Chryssikos et al. [34]. The values observed by the MDSC are in good agreement with the experimental result of viscosity. The agreement indicates the reliability of the MDSC in studying the fragility of glass-forming materials. It is found that the steepness increases as the Na₂O content increases. This result indicates the change of the fragility from “strong” to “fragile” as the Na₂O content increases below x = 0.45. According to the Vilgis model, the degree of fragility relates to the distribution of the coordination number of a key element. In the sodium borate glasses, the composition dependence of m and the variation of the coordination number of boron atoms have similarities. It indicates that the origin of the fragility can be related to the distribution of the coordination number of boron. A similar result was also reported in lithium borate glass using MDSC [31].

Figure 11.

Composition dependence of the fragility index m of xNa₂O-(1 − x)B₂O₃ glasses. The values observed by the MDSC are plotted by solid circles. The reference values summarized by Chryssikos et al. [34] are plotted by open circles.

For the study on the non-Debye nature of structural relaxation, the Cole-Col plot of xLi₂O-(1 − x)B₂O₃ glass (x = 0.40) is shown in Figure 12. The imaginary part, C_p″ is plotted against the real part C_p′. Experimental conditions: heating rate 1°C/min, amplitude of modulation 1.0°C, period of modulation 100 s. This plot shows the temperature averaged non-Debye nature in the vicinity of a dynamic glass transition temperature. Circles denote the experimental results. The solid line denotes the fitted values by the HN equation. The values of parameters are α = 0.94 and γ = 0.67, and both parameters are less than 1.0, and non-Debye nature is observed [17].

Figure 12.

Cole-Cole plot of dynamic specific heat of 40 Li₂O·60 B₂O₃. Circles denote the experimental results. The solid line denotes the fitted values by Haviliak-Negami equation (α = 0.94 and γ = 0.67).

The origin of non-Debye nature relates to the coexistence of the different types of intermediate structural units with the three- and four-coordinated boron atoms. In alkali borate glasses, the addition of Li₂O leads to the formation of tetrahedral B∅₄, where ∅ is bridging oxygen. Therefore, the heterogeneity of the intermediate structure occurs under a low composition range of Li₂O. The distribution of intermediate structures causes the distribution of relaxation times and the non-Debye nature. Such a non-Debye nature was also observed by the Cole-Cole plot of complex heat capacity in sodium borate glass [28], fructose, and glucose [26].

4. Photoacoustic method

In 1880, the photophonic phenomena were discovered by Alexander Graham Bell. When a solid in an enclosed cell is illuminated by a rapidly modulated beam of sunlight shines, it is possible to hear an audible sound by a hearing tube attached to the cell [35]. Tyndall and Rontgen reported that an audible sound could also be emitted when a gas in an enclosed cell is illuminated with intermitted Drummond’s limelight [36, 37]. Bell subsequently experimented with the production of sound by radiant energy with a variety of solids, liquids, and gases [38]. After 100 years had passed, the interest in photoacoustic effects and related experimental techniques increased. The first conference on photoacoustic effect took place in February 1981 in Bad Honnef in Germany [39]. Spectroscopy and detection of minute concentrations, monitoring chemical reactions, energy conversion processes, and calorimetric applications to phase transitions were extensively discussed. By using a finely focused laser beam, photoacoustic imaging (PAI) is also possible, and a liquid-solid interface was observed in the vicinity of a melting temperature [40]. Recently, PAI is an emerging noninvasive imaging modality. PAI provides better resolution than pure ultrasonic imaging and deeper penetration than pure optical imaging [41].

Rosencwaig and Gersho reported a quantitative derivation for the acoustic signal of solid and semisolid matter in a photoacoustic cell in terms of the optical, thermal, and geometric parameters of the system [42]. A one-dimensional model of the heat flow in the cell resulting from the absorbed light energy was calculated.

A standard cylindrical photoacoustic cell is shown in Figure 13 by the cross-sectional view. A sample is located between backing material and gas. Here, the thermal diffusion length of a sample, μ_s, the thermal diffusion length of gas, μ_g, the thermal conductivity (cal/cm s °C), k_i, the density (g/cm³), ρ_i, the specific heat (cal/g°C), C_pi, the thermal diffusivity i (cm²/s) α_i = k_i/ρ_iC_i, the thermal diffusion coefficient (cm), a_i = (ω/2α_i)^1/2; the thermal diffusion length (cm) μ_i = 1/a_i, where subscript i denotes s, g, and b for the solid, gas, and backing material, respectively. The modulated frequency of the incident light to a cell is ω in radians per second.

Figure 13.

One-dimensional model of photoacoustic effect in gas-microphone method on the optically thick sample (L_s > μ_β > μ_s), where μ_s and μ_g are thermal diffusion lengths of a sample and gas, respectively. μ_β = 1/β is the optical absorption length.

The thermal diffusion equation in the solid, taking into account the distributed heat source, can be written by

$$\frac{{\partial }^2\varphi }{\partial x^2}=\frac{1}{{\alpha }_s}\frac{\partial \varphi }{\partial t}-\mathit{\text{Aexp}}\left(\mathit{\beta x}\right)\left[1+exp\left(\mathit{j\omega t}\right)\right],\text{for}-L_s\le x\le 0.$$E27

Here, $A=\beta I_0\eta /2k_s$, ϕ and η are temperature and efficiency, respectively.

The full expression for the incremental pressure,

$$\mathrm{\delta }P\left(t\right)=Qexp\left\{j\left(\mathit{\omega t}-\pi /4\right)\right\},\text{where}Q=\gamma P_0\theta /\sqrt{2}{l}_g{a}_g{T}_0.$$E28

The full expression for δP(t) is complicated for interpretation. However, it is understandable by considering typical conditions. For optically opaque (μ_β = 1/β ≪ 1) and thermally thick (μ_s ≪ l; μ_β < μ_s) solids, it holds that

$$Q≌\frac{\left(1-j\right){\mu }_s}{2a_g}\left(\frac{{\mu }_s}{k_s}\right)Y,\text{where}Y=\frac{\gamma P_0{I}_0}{2\sqrt{2}{l}_g{T}_0}.$$E29

This case is shown in Figure 14(a) [42, 43].

Figure 14.

(a) Optically thin and thermally thick (μ_β < μ_s < L_s), (b) optically very thin and thermal thick (μ_β ≪ μ_s < L_s). The modulated light is absorbed in an opaque film, and thermal waves are generated at interfaces with adjacent gas and a sample.

The photoacoustic signals of glass-forming liquids were measured in the condition of Figure 14(b) [7, 44]. The modulated light is absorbed in an opaque film, and thermal waves are generated at interfaces with adjacent gas (0 < x < L_g) and a sample (−L_s < x < 0). During a cycle of modulation, thermal waves propagate μ_s and μ_g to minus and plus directions along the x-axis, respectively.

According to their theory on the case of an optically very thin and thermal thick sample, the photoacoustic signal is related to squared complex thermal effusivity, $C_p\left(\nu \right)\kappa \left(\nu \right)$,which is the product of complex specific heat and complex thermal conductivity, as below. Linear-response theory defines dynamic thermal conductivity by the fluctuation of heat flux density [5]:

$$\left|C_p\left(\nu \right)\kappa \left(\nu \right)\right|=A_{\mathit{pt}}{\left(\nu \right)}^{-2}.$$E30

$${\mathit{tan}}^{-1}\left\{\frac{{\left(C_p\left(\nu \right)\kappa \left(\nu \right)\right)}^{"}}{{\left(C_p\left(\nu \right)\kappa \left(\nu \right)\right)}^{\prime }}\right\}=-2{\varphi }_{\mathit{pt}}\left(\nu \right).$$E31

Here, $A_{\mathit{pt}}\left(\nu \right)$ and ${\varphi }_{\mathit{pt}}\left(\nu \right)$ are the frequency-dependent photoacoustic amplitude and the phase lag, respectively. The $C_p\left(\nu \right)$ and $\kappa \left(\nu \right)$ are dynamic specific heat and dynamic thermal conductivity, respectively. From Eqs. (30) and (31), the frequency-dependence of the product of a dynamic specific heat and a dynamic thermal conductivity can be determined by the frequency-dependence of observed $A_{\mathit{pt}}\left(\nu \right)$ and ${\varphi }_{\mathit{pt}}\left(\nu \right)$. Figure 15 shows the temperature dependence of A_pt and −2 ${\varphi }_{\mathit{pt}}$ of propylene glycol at 80 Hz [44]. Upon cooling from a high temperature, the relaxation frequency (the inverse of the relaxation time) decreases and approaches the modulation frequency of an incident light. When the relaxation frequency becomes the same as the modulation frequency, the maximum of −2 ${\varphi }_{\mathit{pt}}$ appears and A_pt decreases.

Figure 15.

Temperature dependence of amplitude and phase of photoacoustic signals in propylene glycol [44].

Cole-Cole plot of dynamic $C_p$κ of propylene glycol is shown in Figure 16 [44]. These values of the Davidson-Cole exponent of photoacoustic method were compared with the values determined by the dielectric and the ultrasonic measurements, as shown in Table 1. It is found that the Davidson-Cole exponent of photoacoustic method in the two liquids is smaller than those of the dielectric and ultrasonic measurements [45, 46]. The dielectric relaxation is related to only the fluctuation of polarization. In contrast, a photoacoustic response is related to thermal fluctuations, which contain all degrees of freedom including polarization and strain. Therefore, the distribution of relaxation time of photoacoustic method can be broader than those of the dielectric and ultrasonic measurements. Therefore, it is reasonable that Davidson-Cole exponents of photoacoustic method can be smaller than those of the dielectric and elastic measurements. This fact is seen in Table 1. There are two and three hydroxyl groups in a molecule for propylene glycol and glycerol, respectively. Since the dominant interaction among molecules is the intermolecular hydrogen bonding, the nearest interaction of molecules of glycerol is more complex than that of propylene glycol, and the distribution of relaxation time of glycerol can be broader than that of propylene glycol. Therefore, the exponent of glycerol is smaller than that of propylene glycol, which is smaller than those of n- and i-propanol [47].

Figure 16.

Cole-Cole plot of dynamic $C_p$κ of propylene glycol. The solid line shows calculated values by the Davidson-Cole equation at ${\beta }_{\mathit{CD}}$=0.52, where ($C_p$κ)′ and ($C_p$κ)” are real and imaginary parts of dynamic $C_p$κ [44].

Method	Response	Glycerol	Propylene glycol
Dielectric [45]	Polarization	0.58 ± 0.03	0.66 ± 0.01
Ultrasonic [46]	Strain	0.42 ± 0.05
Photoacoustic [44]	Enthalpy	0.35 ± 0.03	0.52 ± 0.05

Table 1.

Comparison of Davidson-Cole exponents of dielectric, ultrasonic, and photoacoustic measurements in glass-forming glycerol and propylene glycol.

Upon cooling a liquid from a high temperature, several kinds of relaxation processes, an α-relaxation, slow and fast β-processes and the low-energy excitation, boson peaks were observed in the large frequency range between the μHz and THz ranges as shown in Figure 17 [9, 48, 49]. The relaxation time of the main α-structural relaxation process in a fragile liquid diverges at T_VF, and it obeys the following Vogel-Tammann-Fulcher law reflecting the cooperative motion in a cage [2].

Figure 17.

Various relaxation processes and boson peaks of propylene glycol. The solid line shows the Vogel-Tammann-Fulcher (VTF) law observed by various measurements.

$$\tau ={\tau }_0\mathit{exp}\left(\frac{B}{T-T_{\mathit{VF}}}\right).$$E32

Here ${\tau }_0$and B are constants. However, in a strong liquid, such as a silica glass, T_VF is far below T_g, and the relaxation time approximately obeys the Arrhenius law. Therefore, no remarkable change in relaxation time is observed in the vicinity of T_g.

Propylene glycol undergoes a liquid-glass transition at about T_g = 168 K. It is one of the typical glass-forming materials with the strong glass-forming tendency [48, 49]. It belongs to an intermediate liquid with a fragility index of m = 48 in the strong-fragile classification [33]. The temperature dependences of α relaxation determined by dielectric and photoacoustic measurements together with the boson peak and fast relaxations determined by the Raman and Brillouin scattering measurements are shown in Figure 7. The boson peak frequency is nearly constant. The fast relaxation determined from the polarized Brillouin scattering spectra is also temperature-independent. In the supercooled liquid, this process is faster than other relaxation processes determined by dielectric and photoacoustic measurements [9, 44]. This process may be attributed to the fast β-process predicted by the mode coupling theory on glass transitions, which is faster than the α-relaxation process [50]. Another relaxation process was determined from the Rayleigh wing in a depolarized Brillouin scattering spectrum, and it was attributed to the α-relaxation at a GHz range [49]. The γ-relaxation was also determined from the width of a longitudinal acoustic (LA) peak, assuming the coupling between the relaxation process and LA mode [51]. Below a GHz range, the relaxation time of the α-relaxation process was observed by dielectric spectroscopy, which detects polar motions [9]. While dynamic heat capacity detects both polar and nonpolar motions. The mean relaxation time of the α-relaxation process obeys the Vogel-Tammann-Fulcher law. For the detailed study on the relaxation processes and vibrational properties in a liquid-glass transition, a complete picture of relaxation map is required in the very large frequency range, in which various kinds of relaxation processes and low-energy excitation are included.

5. Conclusions

In a linear-response theory, the dynamic specific heat is defined by Kubo’s formula on the fluctuation-dissipation theorem using the correlation function of enthalpy fluctuations. It is important that the dynamic specific heat can provide us with a more general interpretation of various relaxation and dynamical processes. In fact, the origin of the enthalpy relaxation is attributed to the total degrees of freedom of atomic motions. While, the dielectric relaxation is related only to polar atomic motions, and ultrasonic spectroscopy is related only to strain. This chapter introduces two experimental methods to measure physical properties related to dynamic specific heat, (1) temperature-modulated differential calorimetry (MDSC) and (2) photoacoustic spectroscopy. Then, the dynamical properties of glass transitions in oxide network glasses and hydrogen-bonded alcohols are reviewed on dynamic glass transition temperature, fragility index, and the distribution of relaxation time.

In contrast, in broadband dielectric spectroscopy, which covers from micro- to terahertz frequency range, the frequency range of observable dynamic heat capacity is limited in the low-frequency range. MDSC detects dynamic specific heat, while specific-heat spectroscopy and photoacoustic methods are based on thermal effusion [18] and detect dynamic thermal effusion, which relates to the product of dynamic specific heat and dynamic thermal conductivity. Therefore, further development of experimental methods is necessary to extend the frequency range of measurements of dynamic specific heat or to detect dynamic specific heat and dynamic thermal conductivity independently.