1. Introduction

A wide range of problems in analytical chemistry, pharmaceutics, biophysics, and medicine implies studies of the unstable systems behavior, such as organic compounds and biological objects exposed to any external factor, including the increased temperature, pressure, magnetization, humidity, etc. Such exposure changes the structure, chemical composition, and related electrodynamic response of an analyte. Quantitative characterization of these transformations, including the in situ determination of reactant concentrations during the solid-phase chemical reactions, cause a challenging problem in chemical physics [1–5].

Nowadays, to study the crystal forms and evaluate the solid-state chemical reactions, a variety of techniques are applied. Among them, we should mention the X-ray diffractometry and nuclear magnetic resonance spectroscopy [6–8], infrared and Raman spectroscopy [9–12], circular dichroism spectroscopy [13], thermogravimetric and differential thermal analysis [14,15], as well as differential scanning calorimetry [16]. Since each individual method measures its unique set of physical properties and possesses its own limitations, two or more of methods are usually combined to uncover the phase and crystalline state of an analyte. However, not all of them are capable of the in situ applications. For example, X-ray diffractometry is a standard method to study the crystalline structure [17,18], that requires either preparation of a single-crystal sample (it can be a daunting task), or quite complex analysis of polycrystalline substances [19]. Optical methods, such as infrared dielectric and Raman spectroscopy, characterize mainly the intra-molecular vibrations, chemical bonds, and atomic groups. Despite they pave the way for the in situ measurements [20,21], it is difficult for them to characterize the crystalline hydrates with different water contents, as well as to access the dehydration kinetics. The optical methods might also suffer from the electromagnetic-wave scattering on the analyte heterogeneities.

Most recently, terahertz (THz) / far-infrared spectroscopy was found to be an effective tool in a variety of applications [22–24]. THz dielectric response of an analyte contains information about the low-frequency inter-molecular vibrational modes [25,26], hydrogen-bond stretching, van der Waals interactions [27–29], and torsion vibrations [30] in chemical and biological compounds, biomolecules, and pharmaceutical materials. It also reflects vibrational properties, conformational state, and structural dynamics of drugs [28,31], biomolecules [32–36], molecular isomers [37–39], polymorphs [40], hydrates [27,41–44], ices [45,46], and cocrystals [28,47,48], as well as the analyte structural features [49].

Among all THz tools, THz pulsed spectroscopy (TPS), that was vigorously explored during the past few decades [22], attracts special attention. TPS yields information about the frequency-dependent amplitude and the phase of a THz pulse in a broad spectral range, as a result of a single rapid measurement. Thus, it enables the reconstruction of the frequency-dependent complex dielectric permittivity of an analyte directly from the measured data without any physical assumptions or application of the Kramers-Kronig transform. Therefore, TPS can be used to study the kinetics of physical processes and chemical reactions lasting from few minutes to hours and even days. TPS was already applied to study dissolution of organic crystals and temperature-induced transformations of their composition [41,50], kinetics of water vapor adsorption by porous media and biological tissues [51,52], hydration changes [53–55], water bonding in biological liquid and tissues [56–60], glycation of tissue and blood [61,62], polymerization an degradation processes [63–65], cocrystal formation [66,67], diffusion and capillary flows [68,69] and even phase transformations between the solid state modifications of an analyte [29,43,50,70–74]. Despite a widespread TPS use in monitoring of the solid-state chemical reactions, the applied experimental protocols, physical models, and data processing methods vary considerably. This affects the measured data reproducibility and even makes impossible a comparison between the data from different studies.

To mitigate this challenge, in this paper, a novel approach to quantitatively characterize the solid-phase chemical reactions is proposed, which relies on the temperature-dependent TPS measurements. It defines the physically-reasonable relation between the dielectric response of an analyte and the concentration of a reagent at each point of the two-dimensional (2D) space formed by the analyte temperature and the reaction time. For this aim, the Lorentz oscillator, the sum rule, and the Arrhenius theory are used. Practical utility of the developed approach is then demonstrated experimentally, by studying kinetics of the thermostimulated molecular decomposition of polycrystalline $\alpha$-Lactose Monohydrate ($\alpha$-LM) [43,74]. Lactose has a monoclinic crystal symmetry, with a space group (P$2_1$) and the maximal crystallite size of $0.1$ mm. As shown in Fig. 1, it has two anomers – $\alpha$-LM (4-0-$\beta$-D-galactopyranosil-$\alpha$-D-glucopyranose monohydrate) and anhydrous form $\beta$-L (4-O-$\beta$-D-galactopyranosyl-$\beta$-D-glucopyranose) [75]. Unlike $\beta$-L, $\alpha$-LM contains intracrystalline water, as a structural element of the molecule. Heating of $\alpha$-LM leads to its thermal decomposition and formation of new phases, accompanying with irreversible changes in its dielectric response [74]. Analysis of the $\alpha$-LM THz dielectric response evolution in time in the $300$–$425$ K temperature range revealed that the reaction rate increases in the range of $V_\mathrm{1} = 9 \times 10^{-4}$–$10^{-2}$ min$^{-1}$ with an increase of the analyte temperature in the $T = 313$–$393$ K range. The Arrhenius activation energy is $E_\mathrm{a} \simeq 45.4$ kJ/mol.

 

Fig. 1. Schematic of the crystal lattice basis for the lactose anomers $\alpha$-LM and $\beta$-L.

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2. Approach to characterize a solid-state chemical reaction

As shown in Fig. 2, to study kinetics of a solid-state chemical reaction, evolution of the frequency-dependent THz dielectric response of an analyte should be analyzed in a 2D space, formed by the analyte temperature $T$ in [K] and the reaction time $t$ in [min]. The proposed approach implies analysis of the irreversibly changed with a chemical reaction THz response of an analyte, that is represented in form of the dynamic conductivity. Evolution of the analyte response is considered in the two heating modes.

 

Fig. 2. Schematic of the resonant line evolution in the analyte conductivity spectrum $\sigma \left ( \nu, T, t \right )$ with increasing analyte temperature $T$ and reaction time $t$. Here, ”slow” and ”fast” heating modes are highlighted.

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2.1 Estimation of the complex dielectric permittivity

In both ”slow” and ”fast” heating modes, analyte measurements are performed at different points of the $T$–$t$ space. Based on the TPS data, the complex dielectric permittivity of an analyte $\widetilde {\varepsilon } \left ( \nu, T, t \right )$ is estimated

At each $\left ( T, t \right )$-point, the collected THz spectra are then described by the classical Lorentz oscillator model

2.2 Estimation of the reagent concentration

Based on TPS measurements in a ”slow heating” mode, a $j^\mathrm{th}$ resonant absorption line is selected for further analysis (Fig. 2), so that it changes irreversibly with a chemical transform and, thus, attributed to changes in a reagent content. Analysis of the conductivity contour of this resonance provides an estimate for the reagent concentration $C \left ( T, t \right )$ at each analyzed $\left ( T, t \right )$-point. For this aim, the frequency-domain integral of the $j^\mathrm{th}$ conductivity peak is calculated analytically using the sum rule and the Lorentz kernel parameters as follows

Estimation of concentration $C$ exactly via the conductivity integration is essential, since such integral is linearly proportional to the number of dipoles (charges) $K_j$, underlying the considered spectral resonance with their effective charge $q_j$ and mass $m_j$. Indeed, the sum rule is well-known in condensed matter physics and electrodymanics, and it is directly related to the charge conservation law [76]. The result of such integration does not depend on possible changes in the shape of the analyzed conductivity peak (for example, with increasing temperature of an analyte). Other commonly used parameters, like peak values of the absorption coefficient $\alpha$ or dielectric loss $\varepsilon ''$ [41,50], are not so linearly related to the number of charges $K_j$ and reagent content $C$. Therefore, they can somewhat reduce accuracy of the chemical reaction characterization.

Then determination of the absolute conversion factor between the concentration $C$ (either molar or by volume / mass) and the contour integral values $\Delta \varepsilon _j \nu _j^2$ is performed, considering the relation between the specific absorption line and the selected reagent. Without heating the analyte, no chemical reaction is observed; therefore, the integral of the conductivity contour corresponds to the initial $100$% content of a reagent ($C_\mathrm{init} = 1.0$). Upon the fast heating and at static temperatures $T_j$, the chemical reaction proceeds, the reagent concentration $C$ and the integral values $\Delta \varepsilon _j \nu _j^2$ decrease monotonically with time. Finally, with saturation of the chemical reaction, the analyzed resonance disappears from the THz spectrum, which is caused by the absence of a reagent in an analyte ($C_\mathrm{fin} = 0.0$). From these conditions, the reagent concentration $C \left ( T, t \right )$ is calculated unequivocally at each analyzed $\left ( T, t \right )$-point. Despite the convenient analytical integration of the Lorentz conductivity contour (Eq. (4)), the latter often features more complex behaviour with no straightforward analytical model and therefore requires numerical integration.

Finally, from the ”slow heating” mode measurements, the temperature range is selected for the ”fast heating” experiments, when the chemical reaction is finished in a reasonable time, while the considerable temperature-induces broadening of the analyzed spectra resonance and the fast high-temperature analyte decomposition are simultaneously prevented.

2.3 Estimation of the chemical reaction parameters

The ”fast heating” measurement yields the chemical reaction parameters (Fig. 2). In this mode, evolution in time of the $j^\mathrm{th}$ conductivity contour is studied at different static temperatures $T_j$, after a rapid sample heating; and the reagent concentration $C \left ( T, t \right )$ is calculated from Eqs. (1)–(4).

The observed transient processes $C \left ( T, t \right )$ can be fitted by various kinetic equations to estimate parameters of a chemical reaction [2,77]. In this study, a simple exponential model is considered

Knowing the temperature-dependent reaction rate $V_{1} \left ( T \right )$, the activation energy $E_\mathrm{a}$ in [J/Kmol] is determined using the Arrhenius formula

3. Experimental study of the $\alpha$-LM thermal decomposition

Now we proceed to application of the developed approach to study of $\alpha$-LM thermostimulated molecular decomposition, aimed at highlighting its practical utility.

3.1 In-house TPS system

In our experiments, an in-house TPS system is used, that operates in the frequency range of $\nu = 0.3$–$3.7$ THz, and that was detailed Ref. [78]. The low-frequency limit of the spectral operation range ($0.3$ THz) is attributed to the diffraction losses as a result of THz beam scattering on a finite sample aperture. In turn, the high frequency limit ($3.7$ THz) appeared due to both the drop of TPS signal at such high frequencies and the increasing THz-wave absorption by an analyte with frequency.

Our setup uses a TOptica FemtoFErb $780$ fiber laser to pump and probe a pair of LT-GaAs photoconductive antennas (Fraunhofer ITWM, Germany), that serve as an emitter and a detector of THz pulses. The THz beam path is vacuumized using the oil-free Scroll (Iwata ISP250) and turbomolecular (Phieffer vacuum) pumps, with the resultant pressure of $10^{-2}$ mbar, as measured by a Tyracont VD$85$ piezo/Pirani sensors. The vacuum pumping prevents an impact of water vapor along the THz beam path on the TPS data, and, thus, extends the TPS spectral operation range. Moreover, it also prevents possible reactions between the $\alpha$-LM (or products of its thermal decomposition) and atmospheric oxygen.

A judiciously-designed heat unit is mounted inside the THz beam path to accommodate a measured sample, and heat it to the temperatures ranging form $T = 300$ to $425$ K with the temperature maintenance error down to $0.5$ K. The temperature is controlled by a k-type thermocouple, that is mounted directly on the sample holder. The described high-temperature unit is similar to that from Ref. [74]. At the same time, for the low-temperature measurements an in-house cryostat is applied, which is described in Ref. [78].

The THz dielectric response of a sample is estimated at different $\left ( T, t \right )$ points (Fig. 2). At each point, both reference (i.e., corresponding to an empty THz beam path) and sample (i.e., measured with a sample inside the THz beam path) waveforms are collected and processed pair-by-pair for reducing an impact of time-domain drift of the TPS system parameters. Particularly, in the ”fast heat” experiments, the waveforms $E \left ( t \right )$ are collected with the time step of $10$ min during exposure at a given temperature $T_j$. In Fig. 3, examples of the reference and sample spectra $\left | E \left ( \nu \right ) \right |$ and related TPS waveforms $E \left ( t \right )$ are shown, that are measured before the sample heating.

 

Fig. 3. Example of the reference and sample TPS signals in (a) frequency-domain $\left | E \left ( \nu \right ) \right |$ and (b) time-domain $E \left ( t \right )$. At low ($<0.3$ THz) and high ($>3.7$ THz) frequencies, red-colored areas define the spectral ranges, in which the TPS sensitivity drops and which are excluded from our analysis.

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The sample complex dielectric permittivity $\varepsilon \left ( \nu, T, t \right )$ is estimated from the TPS waveforms by minimizing a discrepancy between the experimental data and the analytical model. The latter is based on a plane-wave approximation and describes all peculiarities of the THz-wave – sample interaction, including the Fresnel reflections at interfaces, wave attenuation and phase delay in a bulk media, as well as interference effects. Detailed description of the dielectric permittivity reconstruction procedure can be found in Ref. [79].

3.2 Sample preparation

Flat tablets are made of $\alpha$-LM powder (SuperTab $30$GR DFE pharma, Germany) using a hydraulic press (LabTools, Russia) and an evacuated mold (Specac, UK). The compressing force of $20$–$30$ kN is applied for $\simeq 40$ min to each individual sample. This pressure interval is somewhat optimal for preparation of the $\alpha$-LM tablets, since it minimizes the sample porosity, while also preserves it from cracking. The sample diameter is $13$ mm and exceeds the TPS beam spot, while the resultant tablet thickness is in the range of $0.8$–$1.5$ mm.

Each $\alpha$-LM tablet is used in a single experiment with its unique trajectory in the $T$–$t$-space (Fig. 2). After cooling, each sample is weighed using an AnD HR-$250$AZG electronic balance with the accuracy of $0.1$ mg prior to the high-temperature measurements. The density of pressed $\alpha$-LM is then estimated, relying on the measured dimensions and mass of a samples. Mean density is found to be $\simeq 1.4$ g/cm$^3$, which is $\simeq 90$% level of the $1.543$ g/cm$^3$ X-ray density of the $\alpha$-LM single crystal [75].

3.3 Results of the ”slow heating” measurements

In Fig. 4, results of the ”slow heating” experiment are shown, where (a) illustrates an evolution of the $\alpha$-LM conductivity $\sigma \left ( \nu \right )$ during its heating up to $425$ K, while (b) shows an equal data set for the subsequent analyte cooling down to $300$ K, which overall took $\simeq 150$ min. From the differences between the analyte conductivity spectra before and after heating (compare the green-colored $\sigma$-curves at $350$ K in (a) and (b)), it follows that the analyzed solid-state chemical reaction is irreversible in nature. It leads to formation of new stable phases, with their active absorption bands at the frequencies of $\nu \geq 0.6$ THz or the wavenumbers of $k \geq 20$cm$^{-1}$. Thus, it can be concluded that heating leads to the $\alpha$-LM thermolysis. In the presented conductivity spectra $\sigma \left ( \nu \right )$, peaks L$_1$–L$_2$ deserve particular attention, since they either disappear (L$_1$ and L$_4$) or appear (L$_2$, L$_3$, and L$_5$) with the $\alpha$-LM thermolysis. All these peaks can be generally analyzed using the developed approach, but we selected the lowest-frequency resonance L$_1$ at $\nu \simeq 0.53$ THz ($k \simeq 17.6$ cm$^{-1}$), because it is unambiguously associated with $\alpha$-LM, isolated from others bands, and, thus, less distorted by their tails.

 

Fig. 4. Results of the ”slow heating” experiment. (a) Evolution of the $\alpha$-LM conductivity $\sigma \left ( \nu \right )$ during its heating from $350$ to $425$ K. (b) Equal data for the subsequent analyte cooling. In (a) and (b), by markers $L_1$–$L_5$, the conductivity peaks that either disappears or appears with heating are pointed out by the red or blue colored areas, respectively, among which the low-frequency peak $L_1$ is selected for further analysis. (c) Temperature-dependent Lorentz model parameters $\Delta \epsilon _{L1} \left ( T \right )$ and $\gamma _{L1} \left ( T \right )$ for the peak $L_1$, where the grade-shaded temperature range is chosen to study the chemical reaction kinetics.

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After representing the observed peaks by the multiple Lorentz model (Eq. (2)), evolution of the L$_1$-peak parameters is analyzed to reveal optimal temperatures for studying the chemical reaction kinetics. In Fig. 4(c), a dielectric contribution $\Delta \varepsilon _1 \left ( T \right )$ and a damping constant $\gamma _1 \left ( T \right )$ of the L$_1$ peak are shown as functions of the analyte temperature $T$. In the low-temperature range of $\simeq 10$–$100$ K, $\Delta \varepsilon _1$ increases with $T$ and $\gamma _1$ remains constant, indicating stability of $\alpha$-LM at low temperatures, while the low-temperature anomaly may be somehow attributed to structural instability of the $\alpha$-LM crystal lattice. In $\simeq 100$–$310$ K range, $\Delta \varepsilon _1$ appears to be constant and $\gamma _1$ increases almost linearly, that can be attributed to the common temperature-induced resonance broadening. In $\simeq 310$–$400$ K range, a pronounced drop of $\Delta \varepsilon _1$ is evident, indicating changes in the analyte chemical composition. At higher temperatures $>400$ K, these changes are notable as well, but analysis of the conductivity peak parameters appears to be less reliable due to a considerable resonance broadening. Thus, to study kinetics of the $\alpha$-LM thermolysis in the follow-up ”fast heat” mode, the $\simeq 310$–$400$ K range is chosen.

3.4 Results of the ”fast heating” measurements

Then, $\alpha$–LM samples are measured in the ”fast heating” mode for the distinct desired temperatures within the $313$–$393$ K range. In Fig. 5(a), the $\alpha$–LM conductivity $\sigma \left ( \nu \right )$ evolution in time is shown at the constant analyte temperature of $T = 373$ K. Such a time-dependent character is inherent to all analyzed temperatures in the selected range, while only the transient process parameters differ. In Fig. 5(b), area of the L$_1$ conductivity peak is calculated for the considered temperatures, as a function of the reaction time $t$ with the $10$ min step. In Fig. 5(c), related time-temperature regimes are shown, including the heating, exposure, and cooling stages. Thus computed values of integral represent changes in the $\alpha$-LM concentration $C$, as defined by Eq. (4). From Fig. 5(b), we notice that the observed $\alpha$-LM concentration $C$ changes exponentially in time $t$, which is usual for such first-order chemical reactions as thermolysis [41].

 

Fig. 5. Results of the ”fast heating” experiment. (a) Evolution of the $\alpha$-LM conductivity spectrum $\sigma \left ( \nu \right )$ at the constant temperature of $373$ K. (b) Evolution of the $L_1$ peak integral (Eq. (4)) at the different constant temperatures in the range of $T = 313$–$393$ K. (c) Regimes of the analyte heating, exposure at the desired constant temperatures, and subsequent cooling. (d) Changes in the weight of the $\alpha$-LM samples due to their dehydration as a result of the thermal exposure and further rehydration as a results of the $24$-h-long exposure to an ambient humid atmosphere.

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In Fig. 5(d), relative changes in the sample weight are shown, that are measured, first, after the sample exposure and cooling and, second, after a sample handling at an ambient humid atmosphere for $24$-hours. No significant decrease in the sample mass is notable after the ”fast heating” experiment, while the observed loss of the sample mass is attributed to evaporation of the intercrystalline weakly bonded water. In turn, an increase in the sample mass after its exposure to a humid ambient atmosphere is due to adsorption of atmospheric water by a sample [51]. These weigh measurements revealed stability of the sample mass and dimensions during the considered chemical reaction, which is important for its characterization via the developed technique.

In Figs. 6(a), the observed transient processes $C \left ( t \right )$ are represented in log-scale and fitted by the kinetic equation (Eq. (5)). The resultant reaction rate increases in the range of $V_\mathrm{1} = 9 \times 10^{-4}$–$10^{-2}$ min$^{-1}$ while the temperature rises from $T = 313$ to $393$ K. Moreover, in Fig. 6(b), estimates of the initial $\alpha$-LM content $C_\mathrm{init}$ in an analyte (i.e. at the point after the analyte heating and before its measurements at static $T$) are plotted in form of both molar fraction and molar mass. These estimates are obtained by comparing the L$_1$ conductivity contour integrals before and after the fast heating, while for the intact sample integration results in $0.29$ $\Omega ^{-1}$cm$^{-2}$. Notice that such approach allows monitoring of $\alpha$-LM content in any analyte via calculation of the L$_1$ conductivity peak. In Fig. 6(c), the Arrhenius plot is shown, which represents the reaction rate $V_\mathrm{1}$ in log-scale, as a function of the inverse temperature $T^{-1}$. The observed dependence $V_\mathrm{1} \left ( T^{-1} \right )$ is then fitted by Eq. (6), with the resultant Arrhenius activation energy of $E_\mathrm{a} = 45.4$ kJ/mol or $10.8$ kcal/mol. Quite good agreement between the experimental points and the analytical model is observed, justifying that the $\alpha$-LM thermolysis is governed by the Arrhenius theory.

 

Fig. 6. Estimates of the chemical reaction kinetic parameters. (a) Fitting of the time-dependent $\alpha$-LM concentrations $C \left ( t \right )$, by the exponential kinetic equation (Eq. (5)) at the different analyte temperatures in the range of $T =313$–$393$ K. (b) Fraction and molar mass of $\alpha$-LM in analyte at the beginning of its exposure to different temperatures $T$. (c) Arrhenius plot for the $\alpha$-LM decomposition, where the experimental reaction rates $V_1 = \left ( T \right )$ are fitted by Eq. (6) with the resultant activation energy $E_\mathrm{a} = 45.4$ kJ/mol.

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Considering the classification of molecular bonds from Refs. [80], we conclude that this activation energy corresponds to strong hydrogen bond and can be associated with intramolecular water C$_{12}$H$_{22}$O$_{11}$:H$_{2}$O. This somewhat justifies correctness of our considerations.

4. Discussions

In general, spectroscopy of chemical reactions in solids can be a more complex problem. Indeed, in many cases, along with the considered transformations of the conductivity spectrum in time, that are mainly caused by changes in the reagent content and symmetry of vibrations, structural transformations of an analyte, changes in its density and weight might occur and, thus, distort (in a different manner) the measured dielectric response of an analyte in a broad frequency range. Dielectric response of multicomponent samples, including porous ones, depends on both the volume fractions of reagents and the depolarizing factors, and can be described by common models of the effective medium theory [22]. For example, effective dielectric permittivity of an analyte is governed by the Lichtenecker relation [81]

Both the sample density and depolarization can be assessed for the considered study of the $\alpha$-LM thermolisys. The measured mean density of samples was $\simeq 1.4$ g/ cm$^{-3}$ or $90$% of the X-ray density. For such a high density, expected errors in the estimates of the dielectric permittivity and dynamic conductivity should not exceed $10$%. An impact of depolarization and morphology are included in the Lichtenecker coefficient $\alpha$. It was reported In Ref. [82], that for $\alpha$-LM one can expect this coefficient to be close to zero – namely, $\alpha \simeq 0.05$. This indicates isotropic distribution of crystallites and negligible effect of depolarization on the measured dielectric response function.

Due to strong polarity of water molecules H$_2$O, adsorbed moisture also affects the dielectric response of $\alpha$-LM. Changes in the THz response caused by trapped water were studied in Ref. [51], involving the stable SiO$_2$-based nanoporous glass matrix. The dielectric response fucntion of materials with initially low dielectric constant can increase considerably due to a water uptake, while an impact of the absorbed water is usually unstable in time and depends on the environmental conditions. At the same time, we can somewhat suppress an impact of this factor on the measured data, because trapped water molecules contribute to the resultant dielectric permittivity of an analyte in form of a very broadband relaxation, the bandwidth of which are several orders higher than that of quite local resonance features observed in the dynamic conductivity spectra of $\alpha$-LM. These broad bands can be described by the overdamped Lorentz oscillators within a dielectric permittivity model (Eq. (2)) and, then, excluded from our analysis.

When analyzing the temperature evolution of the conductivity peak, that is isolated from others and unambiguously associated with a reagent content, we have demonstrated practical utility of the developed approach in an ideal case. At the same time, description of the complex dielectric permittivity spectra by the multiple Lorentz oscillator model (Eq. (2)) allows us to study (in the same way) more complex dielectric spectra with overlapped resonance contours and (to some extend) relaxation features. Efficiency of the developed method can be further improved by its combined use with analytical and numerical analysis based on the density functional theory, molecular dynamics simulations, and factor-group analysis. Finally, we notice that, despite dehydration of $\alpha$-LM samples was systematically studied in this paper, the developed TPS-based method has a general character and can be applied to characterize other hydrates [27,41–44,83,84]. For this aim, the temperature regimes of the ”fast” and ”slow” experiments, the spectral operation range, and the analyzed dynamic conductivity peak should be selected properly based on the experimental data or literature.

5. Conclusions

In this work, we determined the reaction rates and Arrhenius activation energy of thermoactivated solid state reaction and estimated molar fraction of tested phase by analyzing the bands parameters in the conductivity spectra in the THz frequency range. The study of the temperature evolution of the conductivity spectra was performed using two types of heating modes – namely, the continuous heat and periodic spectra acquisition at stabilized temperatures. The simulation of the initial data was carried out using both the Fresnel equations and classical oscillator model, which ensured minimal distortion in the shape of the determined line contour due to interference in plane-parallel sample. It was shown that a thermoactivated solid-state reaction in $\alpha$-LM is thermolysis, which leads to the formation of new stable phases in a multicomponent sample. We found that the changes in the $L_1$ band parameters recorded by TPS are observed at exposition temperatures of no less than $333$ K ($60^\circ$C). The values of $\alpha$-LM concentration $C_0$ in the beginning of exposition were determined. The temperature dependence of the reaction rate is obtained and the Arrhenius activation energy is calculated to be $E_\mathrm{a} ~ 45.4$ kJ/mol. The obtained value of $E_a$ is characteristic of a strong hydrogen bond, which indicates a relationship between $L_1$ resonance and rotational dynamics of intracrystalline water. The use of calibration spectra of intact samples expands the analytical capabilities of spectroscopic methods for studying of solid-state reactions by assessing the molar fraction of reagents. The application of the proposed approach to the analysis of optical conductivity spectra allows determination of both the kinetic parameters of the solid state reaction and control the molar fractions of reagents in heterophase samples, which is promising for the problem of pharmaceutics and biophotonics.